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
ltrD2l x : {mono +%R x : y z / y < z}.	Proof. by move=> y z; rewrite !lt_neqAle lerD2l (inj_eq (addrI _)). Qed.	Lemma	ltrD2l	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "addrI", "inj_eq", "lerD2l", "lt_neqAle" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lerD2	:= (lerD2l, lerD2r).	Definition	lerD2	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "lerD2l", "lerD2r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltrD2	:= (ltrD2l, ltrD2r).	Definition	ltrD2	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "ltrD2l", "ltrD2r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lterD2	:= (lerD2, ltrD2).	Definition	lterD2	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "lerD2", "ltrD2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subr_ge0 x y : (0 <= y - x) = (x <= y).	Proof. by rewrite -(@lerD2r x) addrNK add0r. Qed.	Lemma	subr_ge0	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "add0r", "addrNK", "lerD2r" ]	Comparison and negation / opposite.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
oppr_ge0 x : (0 <= - x) = (x <= 0).	Proof. by rewrite -sub0r subr_ge0. Qed.	Lemma	oppr_ge0	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "sub0r", "subr_ge0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subr_gt0 x y : (0 < y - x) = (x < y).	Proof. by rewrite !lt_def subr_eq0 subr_ge0. Qed.	Lemma	subr_gt0	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "lt_def", "subr_eq0", "subr_ge0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subr_le0 x y : (y - x <= 0) = (y <= x).	Proof. by rewrite -[LHS]subr_ge0 opprB add0r subr_ge0. Qed.	Lemma	subr_le0	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "add0r", "opprB", "subr_ge0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subr_lt0 x y : (y - x < 0) = (y < x).	Proof. by rewrite -[LHS]subr_gt0 opprB add0r subr_gt0. Qed.	Lemma	subr_lt0	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "add0r", "opprB", "subr_gt0" ]	FIXME: rewrite pattern	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lerN2 : {mono -%R : x y /~ x <= y :> R}.	Proof. by move=> x y /=; rewrite -subr_ge0 opprK addrC subr_ge0. Qed.	Lemma	lerN2	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "addrC", "opprK", "subr_ge0" ]	FIXME: rewrite pattern	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltrN2 : {mono -%R : x y /~ x < y :> R}.	Proof. by move=> x y /=; rewrite leW_nmono. Qed.	Lemma	ltrN2	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "leW_nmono" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lterN2	:= (lerN2, ltrN2).	Definition	lterN2	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "lerN2", "ltrN2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lerNr x y : (x <= - y) = (y <= - x).	Proof. by rewrite (monoRL opprK lerN2). Qed.	Lemma	lerNr	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "lerN2", "opprK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltrNr x y : (x < - y) = (y < - x).	Proof. by rewrite (monoRL opprK (leW_nmono _)). Qed.	Lemma	ltrNr	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "leW_nmono", "opprK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lterNr	:= (lerNr, ltrNr).	Definition	lterNr	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "lerNr", "ltrNr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lerNl x y : (- x <= y) = (- y <= x).	Proof. by rewrite (monoLR opprK lerN2). Qed.	Lemma	lerNl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "lerN2", "opprK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltrNl x y : (- x < y) = (- y < x).	Proof. by rewrite (monoLR opprK (leW_nmono _)). Qed.	Lemma	ltrNl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "leW_nmono", "opprK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lterNl	:= (lerNl, ltrNl).	Definition	lterNl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "lerNl", "ltrNl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subr_lte0	:= (subr_le0, subr_lt0).	Definition	subr_lte0	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "subr_le0", "subr_lt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subr_gte0	:= (subr_ge0, subr_gt0).	Definition	subr_gte0	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "subr_ge0", "subr_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subr_cp0	:= (subr_lte0, subr_gte0).	Definition	subr_cp0	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "subr_gte0", "subr_lte0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lerD x y z t : x <= y -> z <= t -> x + z <= y + t.	Proof. by move=> lxy lzt; rewrite (@le_trans _ _ (y + z)) ?lterD2. Qed.	Lemma	lerD	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "le_trans", "lterD2" ]	Addition, subtraction and transitivity	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ler_ltD x y z t : x <= y -> z < t -> x + z < y + t.	Proof. by move=> lxy lzt; rewrite (@le_lt_trans _ _ (y + z)) ?lterD2. Qed.	Lemma	ler_ltD	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "le_lt_trans", "lterD2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltr_leD x y z t : x < y -> z <= t -> x + z < y + t.	Proof. by move=> lxy lzt; rewrite (@lt_le_trans _ _ (y + z)) ?lterD2. Qed.	Lemma	ltr_leD	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "lt_le_trans", "lterD2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltrD x y z t : x < y -> z < t -> x + z < y + t.	Proof. by move=> lxy lzt; rewrite ltr_leD // ltW. Qed.	Lemma	ltrD	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "ltW", "ltr_leD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lerB x y z t : x <= y -> t <= z -> x - z <= y - t.	Proof. by move=> lxy ltz; rewrite lerD // lterN2. Qed.	Lemma	lerB	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "lerD", "lterN2", "ltz" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ler_ltB x y z t : x <= y -> t < z -> x - z < y - t.	Proof. by move=> lxy lzt; rewrite ler_ltD // lterN2. Qed.	Lemma	ler_ltB	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "ler_ltD", "lterN2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltr_leB x y z t : x < y -> t <= z -> x - z < y - t.	Proof. by move=> lxy lzt; rewrite ltr_leD // lterN2. Qed.	Lemma	ltr_leB	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "lterN2", "ltr_leD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltrB x y z t : x < y -> t < z -> x - z < y - t.	Proof. by move=> lxy lzt; rewrite ltrD // lterN2. Qed.	Lemma	ltrB	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "lterN2", "ltrD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lerBlDr x y z : (x - y <= z) = (x <= z + y).	Proof. by rewrite (monoLR (addrK _) (lerD2r _)). Qed.	Lemma	lerBlDr	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "addrK", "lerD2r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltrBlDr x y z : (x - y < z) = (x < z + y).	Proof. by rewrite (monoLR (addrK _) (ltrD2r _)). Qed.	Lemma	ltrBlDr	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "addrK", "ltrD2r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lerBrDr x y z : (x <= y - z) = (x + z <= y).	Proof. by rewrite (monoLR (addrNK _) (lerD2r _)). Qed.	Lemma	lerBrDr	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "addrNK", "lerD2r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltrBrDr x y z : (x < y - z) = (x + z < y).	Proof. by rewrite (monoLR (addrNK _) (ltrD2r _)). Qed.	Lemma	ltrBrDr	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "addrNK", "ltrD2r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lerBDr	:= (lerBlDr, lerBrDr).	Definition	lerBDr	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "lerBlDr", "lerBrDr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltrBDr	:= (ltrBlDr, ltrBrDr).	Definition	ltrBDr	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "ltrBlDr", "ltrBrDr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lterBDr	:= (lerBDr, ltrBDr).	Definition	lterBDr	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "lerBDr", "ltrBDr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lerBlDl x y z : (x - y <= z) = (x <= y + z).	Proof. by rewrite lterBDr addrC. Qed.	Lemma	lerBlDl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "addrC", "lterBDr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltrBlDl x y z : (x - y < z) = (x < y + z).	Proof. by rewrite lterBDr addrC. Qed.	Lemma	ltrBlDl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "addrC", "lterBDr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lerBrDl x y z : (x <= y - z) = (z + x <= y).	Proof. by rewrite lerBrDr addrC. Qed.	Lemma	lerBrDl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "addrC", "lerBrDr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltrBrDl x y z : (x < y - z) = (z + x < y).	Proof. by rewrite lterBDr addrC. Qed.	Lemma	ltrBrDl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "addrC", "lterBDr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lerBDl	:= (lerBlDl, lerBrDl).	Definition	lerBDl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "lerBlDl", "lerBrDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltrBDl	:= (ltrBlDl, ltrBrDl).	Definition	ltrBDl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "ltrBlDl", "ltrBrDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lterBDl	:= (lerBDl, ltrBDl).	Definition	lterBDl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "lerBDl", "ltrBDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lerDl x y : (x <= x + y) = (0 <= y).	Proof. by rewrite -{1}[x]addr0 lterD2. Qed.	Lemma	lerDl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "addr0", "lterD2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltrDl x y : (x < x + y) = (0 < y).	Proof. by rewrite -{1}[x]addr0 lterD2. Qed.	Lemma	ltrDl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "addr0", "lterD2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lerDr x y : (x <= y + x) = (0 <= y).	Proof. by rewrite -{1}[x]add0r lterD2. Qed.	Lemma	lerDr	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "add0r", "lterD2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltrDr x y : (x < y + x) = (0 < y).	Proof. by rewrite -{1}[x]add0r lterD2. Qed.	Lemma	ltrDr	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "add0r", "lterD2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gerDl x y : (x + y <= x) = (y <= 0).	Proof. by rewrite -{2}[x]addr0 lterD2. Qed.	Lemma	gerDl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "addr0", "lterD2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gerBl x y : (x - y <= x) = (0 <= y).	Proof. by rewrite lerBlDl lerDr. Qed.	Lemma	gerBl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "lerBlDl", "lerDr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gtrDl x y : (x + y < x) = (y < 0).	Proof. by rewrite -{2}[x]addr0 lterD2. Qed.	Lemma	gtrDl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "addr0", "lterD2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gtrBl x y : (x - y < x) = (0 < y).	Proof. by rewrite ltrBlDl ltrDr. Qed.	Lemma	gtrBl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "ltrBlDl", "ltrDr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gerDr x y : (y + x <= x) = (y <= 0).	Proof. by rewrite -{2}[x]add0r lterD2. Qed.	Lemma	gerDr	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "add0r", "lterD2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gtrDr x y : (y + x < x) = (y < 0).	Proof. by rewrite -{2}[x]add0r lterD2. Qed.	Lemma	gtrDr	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "add0r", "lterD2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cprD	:= (lerDl, lerDr, gerDl, gerDl, ltrDl, ltrDr, gtrDl, gtrDl).	Definition	cprD	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "gerDl", "gtrDl", "lerDl", "lerDr", "ltrDl", "ltrDr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addr_ge0 x y : 0 <= x -> 0 <= y -> 0 <= x + y.	Proof. move=> x_ge0 y_ge0; have := lerD2r y 0 x. by rewrite add0r x_ge0 => /(le_trans y_ge0). Qed.	Lemma	addr_ge0	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "add0r", "le_trans", "lerD2r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addr_gt0 x y : 0 < x -> 0 < y -> 0 < x + y.	Proof. move=> x_gt0 y_gt0; have := ltrD2r y 0 x. by rewrite add0r x_gt0 => /(lt_trans y_gt0). Qed.	Lemma	addr_gt0	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "add0r", "lt_trans", "ltrD2r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
posrE x : (x \is Num.pos) = (0 < x).	Proof. by []. Qed.	Lemma	posrE	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "pos" ]	Predicate definitions.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nnegrE x : (x \is Num.nneg) = (0 <= x).	Proof. by []. Qed.	Lemma	nnegrE	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "nneg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
realE x : (x \is Num.real) = (0 <= x) \|\| (x <= 0).	Proof. by []. Qed.	Lemma	realE	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "real" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
negrE x : (x \is Num.neg) = (x < 0).	Proof. by []. Qed.	Lemma	negrE	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "neg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nposrE x : (x \is Num.npos) = (x <= 0).	Proof. by []. Qed.	Lemma	nposrE	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "npos" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le0r x : (0 <= x) = (x == 0) \|\| (0 < x).	Proof. by rewrite le_eqVlt eq_sym. Qed.	Lemma	le0r	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "eq_sym", "le_eqVlt" ]	General properties of <= and <	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt0r x : (0 < x) = (x != 0) && (0 <= x).	Proof. exact: lt_def. Qed.	Lemma	lt0r	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "lt_def" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt0r_neq0 (x : R) : 0 < x -> x != 0.	Proof. by move=> /gt_eqF ->. Qed.	Lemma	lt0r_neq0	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "gt_eqF" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltr0_neq0 (x : R) : x < 0 -> x != 0.	Proof. by move=> /lt_eqF ->. Qed.	Lemma	ltr0_neq0	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "lt_eqF" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
oppr_gt0 x : (0 < - x) = (x < 0).	Proof. by rewrite ltrNr oppr0. Qed.	Lemma	oppr_gt0	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "ltrNr", "oppr0" ]	Comparison and opposite.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
oppr_gte0	:= (oppr_ge0, oppr_gt0).	Definition	oppr_gte0	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "oppr_ge0", "oppr_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
oppr_le0 x : (- x <= 0) = (0 <= x).	Proof. by rewrite lerNl oppr0. Qed.	Lemma	oppr_le0	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "lerNl", "oppr0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
oppr_lt0 x : (- x < 0) = (0 < x).	Proof. by rewrite ltrNl oppr0. Qed.	Lemma	oppr_lt0	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "ltrNl", "oppr0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gtrN x : 0 < x -> - x < x.	Proof. by move=> n0; rewrite -subr_lt0 -opprD oppr_lt0 addr_gt0. Qed.	Lemma	gtrN	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "addr_gt0", "opprD", "oppr_lt0", "subr_lt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
oppr_lte0	:= (oppr_le0, oppr_lt0).	Definition	oppr_lte0	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "oppr_le0", "oppr_lt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
oppr_cp0	:= (oppr_gte0, oppr_lte0).	Definition	oppr_cp0	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "oppr_gte0", "oppr_lte0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lterNE	:= (oppr_cp0, lterN2).	Definition	lterNE	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "lterN2", "oppr_cp0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ge0_cp x : 0 <= x -> (- x <= 0) * (- x <= x).	Proof. by move=> hx; rewrite oppr_cp0 hx (@le_trans _ _ 0) ?oppr_cp0. Qed.	Lemma	ge0_cp	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "le_trans", "oppr_cp0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gerN x : 0 <= x -> - x <= x.	Proof. by move=> x0; rewrite ge0_cp. Qed.	Lemma	gerN	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "ge0_cp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gt0_cp x : 0 < x -> (0 <= x) * (- x <= 0) * (- x <= x) * (- x < 0) * (- x < x).	Proof. move=> hx; move: (ltW hx) => hx'; rewrite !ge0_cp hx' //. by rewrite oppr_cp0 hx // (@lt_trans _ _ 0) ?oppr_cp0. Qed.	Lemma	gt0_cp	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "ge0_cp", "ltW", "lt_trans", "oppr_cp0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le0_cp x : x <= 0 -> (0 <= - x) * (x <= - x).	Proof. by move=> hx; rewrite oppr_cp0 hx (@le_trans _ _ 0) ?oppr_cp0. Qed.	Lemma	le0_cp	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "le_trans", "oppr_cp0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt0_cp x : x < 0 -> (x <= 0) * (0 <= - x) * (x <= - x) * (0 < - x) * (x < - x).	Proof. move=> hx; move: (ltW hx) => hx'; rewrite !le0_cp // hx'. by rewrite oppr_cp0 hx // (@lt_trans _ _ 0) ?oppr_cp0. Qed.	Lemma	lt0_cp	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "le0_cp", "ltW", "lt_trans", "oppr_cp0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ler_wpDl y x z : 0 <= x -> y <= z -> y <= x + z.	Proof. by move=> *; rewrite -[y]add0r lerD. Qed.	Lemma	ler_wpDl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "add0r", "lerD" ]	Addition with left member known to be positive/negative	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltr_wpDl y x z : 0 <= x -> y < z -> y < x + z.	Proof. by move=> *; rewrite -[y]add0r ler_ltD. Qed.	Lemma	ltr_wpDl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "add0r", "ler_ltD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltr_pwDl y x z : 0 < x -> y <= z -> y < x + z.	Proof. by move=> *; rewrite -[y]add0r ltr_leD. Qed.	Lemma	ltr_pwDl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "add0r", "ltr_leD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltr_pDl y x z : 0 < x -> y < z -> y < x + z.	Proof. by move=> *; rewrite -[y]add0r ltrD. Qed.	Lemma	ltr_pDl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "add0r", "ltrD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ler_wnDl y x z : x <= 0 -> y <= z -> x + y <= z.	Proof. by move=> *; rewrite -[z]add0r lerD. Qed.	Lemma	ler_wnDl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "add0r", "lerD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltr_wnDl y x z : x <= 0 -> y < z -> x + y < z.	Proof. by move=> *; rewrite -[z]add0r ler_ltD. Qed.	Lemma	ltr_wnDl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "add0r", "ler_ltD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltr_nwDl y x z : x < 0 -> y <= z -> x + y < z.	Proof. by move=> *; rewrite -[z]add0r ltr_leD. Qed.	Lemma	ltr_nwDl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "add0r", "ltr_leD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltr_nDl y x z : x < 0 -> y < z -> x + y < z.	Proof. by move=> *; rewrite -[z]add0r ltrD. Qed.	Lemma	ltr_nDl	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "add0r", "ltrD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ler_wpDr y x z : 0 <= x -> y <= z -> y <= z + x.	Proof. by move=> *; rewrite addrC ler_wpDl. Qed.	Lemma	ler_wpDr	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "addrC", "ler_wpDl" ]	Addition with right member we know positive/negative	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltr_wpDr y x z : 0 <= x -> y < z -> y < z + x.	Proof. by move=> *; rewrite addrC ltr_wpDl. Qed.	Lemma	ltr_wpDr	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "addrC", "ltr_wpDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltr_pwDr y x z : 0 < x -> y <= z -> y < z + x.	Proof. by move=> *; rewrite addrC ltr_pwDl. Qed.	Lemma	ltr_pwDr	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "addrC", "ltr_pwDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltr_pDr y x z : 0 < x -> y < z -> y < z + x.	Proof. by move=> *; rewrite addrC ltr_pDl. Qed.	Lemma	ltr_pDr	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "addrC", "ltr_pDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ler_wnDr y x z : x <= 0 -> y <= z -> y + x <= z.	Proof. by move=> *; rewrite addrC ler_wnDl. Qed.	Lemma	ler_wnDr	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "addrC", "ler_wnDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltr_wnDr y x z : x <= 0 -> y < z -> y + x < z.	Proof. by move=> *; rewrite addrC ltr_wnDl. Qed.	Lemma	ltr_wnDr	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "addrC", "ltr_wnDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltr_nwDr y x z : x < 0 -> y <= z -> y + x < z.	Proof. by move=> *; rewrite addrC ltr_nwDl. Qed.	Lemma	ltr_nwDr	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "addrC", "ltr_nwDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltr_nDr y x z : x < 0 -> y < z -> y + x < z.	Proof. by move=> *; rewrite addrC ltr_nDl. Qed.	Lemma	ltr_nDr	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "addrC", "ltr_nDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
paddr_eq0 (x y : R) : 0 <= x -> 0 <= y -> (x + y == 0) = (x == 0) && (y == 0).	Proof. rewrite le0r; case/orP=> [/eqP->\|hx]; first by rewrite add0r eqxx. by rewrite (gt_eqF hx) /= => hy; rewrite gt_eqF // ltr_pwDl. Qed.	Lemma	paddr_eq0	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "add0r", "eqxx", "gt_eqF", "le0r", "ltr_pwDl" ]	x and y have the same sign and their sum is null	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
naddr_eq0 (x y : R) : x <= 0 -> y <= 0 -> (x + y == 0) = (x == 0) && (y == 0).	Proof. by move=> lex0 ley0; rewrite -oppr_eq0 opprD paddr_eq0 ?oppr_cp0 // !oppr_eq0. Qed.	Lemma	naddr_eq0	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "lex0", "opprD", "oppr_cp0", "oppr_eq0", "paddr_eq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addr_ss_eq0 (x y : R) : (0 <= x) && (0 <= y) \|\| (x <= 0) && (y <= 0) -> (x + y == 0) = (x == 0) && (y == 0).	Proof. by case/orP=> /andP []; [apply: paddr_eq0 \| apply: naddr_eq0]. Qed.	Lemma	addr_ss_eq0	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "apply", "naddr_eq0", "paddr_eq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sumr_ge0 I (r : seq I) (P : pred I) (F : I -> R) : (forall i, P i -> (0 <= F i)) -> 0 <= \sum_(i <- r \| P i) (F i).	Proof. exact: (big_ind _ _ (@ler_wpDl 0)). Qed.	Lemma	sumr_ge0	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "big_ind", "ler_wpDl", "seq" ]	big sum and ler	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sumr_le0 I (r : seq I) (P : pred I) (F : I -> R) : (forall i, P i -> F i <= 0) -> \sum_(i <- r \| P i) F i <= 0.	Proof. by move=> F0; elim/big_ind : _ => // i x Pi; exact/ler_wnDl. Qed.	Lemma	sumr_le0	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "F0", "big_ind", "ler_wnDl", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ler_sum I (r : seq I) (P : pred I) (F G : I -> R) : (forall i, P i -> F i <= G i) -> \sum_(i <- r \| P i) F i <= \sum_(i <- r \| P i) G i.	Proof. exact: (big_ind2 _ (lexx _) lerD). Qed.	Lemma	ler_sum	algebra.numeric_hierarchy	algebra/numeric_hierarchy/orderedzmod.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]	[ "big_ind2", "lerD", "lexx", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltrD2l x : {mono +%R x : y z / y < z}.

Proof. by move=> y z; rewrite !lt_neqAle lerD2l (inj_eq (addrI _)). Qed.

Lemma

ltrD2l

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "addrI", "inj_eq", "lerD2l", "lt_neqAle" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lerD2

:= (lerD2l, lerD2r).

Definition

lerD2

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "lerD2l", "lerD2r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltrD2

:= (ltrD2l, ltrD2r).

Definition

ltrD2

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "ltrD2l", "ltrD2r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lterD2

:= (lerD2, ltrD2).

Definition

lterD2

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "lerD2", "ltrD2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subr_ge0 x y : (0 <= y - x) = (x <= y).

Proof. by rewrite -(@lerD2r x) addrNK add0r. Qed.

Lemma

subr_ge0

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "add0r", "addrNK", "lerD2r" ]

Comparison and negation / opposite.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

oppr_ge0 x : (0 <= - x) = (x <= 0).

Proof. by rewrite -sub0r subr_ge0. Qed.

Lemma

oppr_ge0

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "sub0r", "subr_ge0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subr_gt0 x y : (0 < y - x) = (x < y).

Proof. by rewrite !lt_def subr_eq0 subr_ge0. Qed.

Lemma

subr_gt0

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "lt_def", "subr_eq0", "subr_ge0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subr_le0 x y : (y - x <= 0) = (y <= x).

Proof. by rewrite -[LHS]subr_ge0 opprB add0r subr_ge0. Qed.

Lemma

subr_le0

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "add0r", "opprB", "subr_ge0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subr_lt0 x y : (y - x < 0) = (y < x).

Proof. by rewrite -[LHS]subr_gt0 opprB add0r subr_gt0. Qed.

Lemma

subr_lt0

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "add0r", "opprB", "subr_gt0" ]

FIXME: rewrite pattern

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lerN2 : {mono -%R : x y /~ x <= y :> R}.

Proof. by move=> x y /=; rewrite -subr_ge0 opprK addrC subr_ge0. Qed.

Lemma

lerN2

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "addrC", "opprK", "subr_ge0" ]

FIXME: rewrite pattern

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltrN2 : {mono -%R : x y /~ x < y :> R}.

Proof. by move=> x y /=; rewrite leW_nmono. Qed.

Lemma

ltrN2

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "leW_nmono" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lterN2

:= (lerN2, ltrN2).

Definition

lterN2

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "lerN2", "ltrN2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lerNr x y : (x <= - y) = (y <= - x).

Proof. by rewrite (monoRL opprK lerN2). Qed.

Lemma

lerNr

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "lerN2", "opprK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltrNr x y : (x < - y) = (y < - x).

Proof. by rewrite (monoRL opprK (leW_nmono _)). Qed.

Lemma

ltrNr

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "leW_nmono", "opprK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lterNr

:= (lerNr, ltrNr).

Definition

lterNr

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "lerNr", "ltrNr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lerNl x y : (- x <= y) = (- y <= x).

Proof. by rewrite (monoLR opprK lerN2). Qed.

Lemma

lerNl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "lerN2", "opprK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltrNl x y : (- x < y) = (- y < x).

Proof. by rewrite (monoLR opprK (leW_nmono _)). Qed.

Lemma

ltrNl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "leW_nmono", "opprK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lterNl

:= (lerNl, ltrNl).

Definition

lterNl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "lerNl", "ltrNl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subr_lte0

:= (subr_le0, subr_lt0).

Definition

subr_lte0

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "subr_le0", "subr_lt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subr_gte0

:= (subr_ge0, subr_gt0).

Definition

subr_gte0

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "subr_ge0", "subr_gt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subr_cp0

:= (subr_lte0, subr_gte0).

Definition

subr_cp0

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "subr_gte0", "subr_lte0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lerD x y z t : x <= y -> z <= t -> x + z <= y + t.

Proof. by move=> lxy lzt; rewrite (@le_trans _ _ (y + z)) ?lterD2. Qed.

Lemma

lerD

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "le_trans", "lterD2" ]

Addition, subtraction and transitivity

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ler_ltD x y z t : x <= y -> z < t -> x + z < y + t.

Proof. by move=> lxy lzt; rewrite (@le_lt_trans _ _ (y + z)) ?lterD2. Qed.

Lemma

ler_ltD

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "le_lt_trans", "lterD2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltr_leD x y z t : x < y -> z <= t -> x + z < y + t.

Proof. by move=> lxy lzt; rewrite (@lt_le_trans _ _ (y + z)) ?lterD2. Qed.

Lemma

ltr_leD

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "lt_le_trans", "lterD2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltrD x y z t : x < y -> z < t -> x + z < y + t.

Proof. by move=> lxy lzt; rewrite ltr_leD // ltW. Qed.

Lemma

ltrD

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "ltW", "ltr_leD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lerB x y z t : x <= y -> t <= z -> x - z <= y - t.

Proof. by move=> lxy ltz; rewrite lerD // lterN2. Qed.

Lemma

lerB

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "lerD", "lterN2", "ltz" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ler_ltB x y z t : x <= y -> t < z -> x - z < y - t.

Proof. by move=> lxy lzt; rewrite ler_ltD // lterN2. Qed.

Lemma

ler_ltB

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "ler_ltD", "lterN2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltr_leB x y z t : x < y -> t <= z -> x - z < y - t.

Proof. by move=> lxy lzt; rewrite ltr_leD // lterN2. Qed.

Lemma

ltr_leB

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "lterN2", "ltr_leD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltrB x y z t : x < y -> t < z -> x - z < y - t.

Proof. by move=> lxy lzt; rewrite ltrD // lterN2. Qed.

Lemma

ltrB

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "lterN2", "ltrD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lerBlDr x y z : (x - y <= z) = (x <= z + y).

Proof. by rewrite (monoLR (addrK _) (lerD2r _)). Qed.

Lemma

lerBlDr

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "addrK", "lerD2r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltrBlDr x y z : (x - y < z) = (x < z + y).

Proof. by rewrite (monoLR (addrK _) (ltrD2r _)). Qed.

Lemma

ltrBlDr

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "addrK", "ltrD2r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lerBrDr x y z : (x <= y - z) = (x + z <= y).

Proof. by rewrite (monoLR (addrNK _) (lerD2r _)). Qed.

Lemma

lerBrDr

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "addrNK", "lerD2r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltrBrDr x y z : (x < y - z) = (x + z < y).

Proof. by rewrite (monoLR (addrNK _) (ltrD2r _)). Qed.

Lemma

ltrBrDr

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "addrNK", "ltrD2r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lerBDr

:= (lerBlDr, lerBrDr).

Definition

lerBDr

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "lerBlDr", "lerBrDr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltrBDr

:= (ltrBlDr, ltrBrDr).

Definition

ltrBDr

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "ltrBlDr", "ltrBrDr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lterBDr

:= (lerBDr, ltrBDr).

Definition

lterBDr

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "lerBDr", "ltrBDr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lerBlDl x y z : (x - y <= z) = (x <= y + z).

Proof. by rewrite lterBDr addrC. Qed.

Lemma

lerBlDl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "addrC", "lterBDr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltrBlDl x y z : (x - y < z) = (x < y + z).

Proof. by rewrite lterBDr addrC. Qed.

Lemma

ltrBlDl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "addrC", "lterBDr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lerBrDl x y z : (x <= y - z) = (z + x <= y).

Proof. by rewrite lerBrDr addrC. Qed.

Lemma

lerBrDl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "addrC", "lerBrDr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltrBrDl x y z : (x < y - z) = (z + x < y).

Proof. by rewrite lterBDr addrC. Qed.

Lemma

ltrBrDl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "addrC", "lterBDr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lerBDl

:= (lerBlDl, lerBrDl).

Definition

lerBDl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "lerBlDl", "lerBrDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltrBDl

:= (ltrBlDl, ltrBrDl).

Definition

ltrBDl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "ltrBlDl", "ltrBrDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lterBDl

:= (lerBDl, ltrBDl).

Definition

lterBDl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "lerBDl", "ltrBDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lerDl x y : (x <= x + y) = (0 <= y).

Proof. by rewrite -{1}[x]addr0 lterD2. Qed.

Lemma

lerDl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "addr0", "lterD2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltrDl x y : (x < x + y) = (0 < y).

Proof. by rewrite -{1}[x]addr0 lterD2. Qed.

Lemma

ltrDl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "addr0", "lterD2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lerDr x y : (x <= y + x) = (0 <= y).

Proof. by rewrite -{1}[x]add0r lterD2. Qed.

Lemma

lerDr

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "add0r", "lterD2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltrDr x y : (x < y + x) = (0 < y).

Proof. by rewrite -{1}[x]add0r lterD2. Qed.

Lemma

ltrDr

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "add0r", "lterD2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gerDl x y : (x + y <= x) = (y <= 0).

Proof. by rewrite -{2}[x]addr0 lterD2. Qed.

Lemma

gerDl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "addr0", "lterD2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gerBl x y : (x - y <= x) = (0 <= y).

Proof. by rewrite lerBlDl lerDr. Qed.

Lemma

gerBl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "lerBlDl", "lerDr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gtrDl x y : (x + y < x) = (y < 0).

Proof. by rewrite -{2}[x]addr0 lterD2. Qed.

Lemma

gtrDl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "addr0", "lterD2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gtrBl x y : (x - y < x) = (0 < y).

Proof. by rewrite ltrBlDl ltrDr. Qed.

Lemma

gtrBl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "ltrBlDl", "ltrDr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gerDr x y : (y + x <= x) = (y <= 0).

Proof. by rewrite -{2}[x]add0r lterD2. Qed.

Lemma

gerDr

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "add0r", "lterD2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gtrDr x y : (y + x < x) = (y < 0).

Proof. by rewrite -{2}[x]add0r lterD2. Qed.

Lemma

gtrDr

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "add0r", "lterD2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cprD

:= (lerDl, lerDr, gerDl, gerDl, ltrDl, ltrDr, gtrDl, gtrDl).

Definition

cprD

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "gerDl", "gtrDl", "lerDl", "lerDr", "ltrDl", "ltrDr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addr_ge0 x y : 0 <= x -> 0 <= y -> 0 <= x + y.

Proof. move=> x_ge0 y_ge0; have := lerD2r y 0 x. by rewrite add0r x_ge0 => /(le_trans y_ge0). Qed.

Lemma

addr_ge0

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "add0r", "le_trans", "lerD2r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addr_gt0 x y : 0 < x -> 0 < y -> 0 < x + y.

Proof. move=> x_gt0 y_gt0; have := ltrD2r y 0 x. by rewrite add0r x_gt0 => /(lt_trans y_gt0). Qed.

Lemma

addr_gt0

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "add0r", "lt_trans", "ltrD2r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

posrE x : (x \is Num.pos) = (0 < x).

Proof. by []. Qed.

Lemma

posrE

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "pos" ]

Predicate definitions.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nnegrE x : (x \is Num.nneg) = (0 <= x).

Proof. by []. Qed.

Lemma

nnegrE

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "nneg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

realE x : (x \is Num.real) = (0 <= x) || (x <= 0).

Proof. by []. Qed.

Lemma

realE

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "real" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

negrE x : (x \is Num.neg) = (x < 0).

Proof. by []. Qed.

Lemma

negrE

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "neg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nposrE x : (x \is Num.npos) = (x <= 0).

Proof. by []. Qed.

Lemma

nposrE

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "npos" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le0r x : (0 <= x) = (x == 0) || (0 < x).

Proof. by rewrite le_eqVlt eq_sym. Qed.

Lemma

le0r

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "eq_sym", "le_eqVlt" ]

General properties of <= and <

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lt0r x : (0 < x) = (x != 0) && (0 <= x).

Proof. exact: lt_def. Qed.

Lemma

lt0r

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "lt_def" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lt0r_neq0 (x : R) : 0 < x -> x != 0.

Proof. by move=> /gt_eqF ->. Qed.

Lemma

lt0r_neq0

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "gt_eqF" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltr0_neq0 (x : R) : x < 0 -> x != 0.

Proof. by move=> /lt_eqF ->. Qed.

Lemma

ltr0_neq0

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "lt_eqF" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

oppr_gt0 x : (0 < - x) = (x < 0).

Proof. by rewrite ltrNr oppr0. Qed.

Lemma

oppr_gt0

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "ltrNr", "oppr0" ]

Comparison and opposite.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

oppr_gte0

:= (oppr_ge0, oppr_gt0).

Definition

oppr_gte0

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "oppr_ge0", "oppr_gt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

oppr_le0 x : (- x <= 0) = (0 <= x).

Proof. by rewrite lerNl oppr0. Qed.

Lemma

oppr_le0

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "lerNl", "oppr0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

oppr_lt0 x : (- x < 0) = (0 < x).

Proof. by rewrite ltrNl oppr0. Qed.

Lemma

oppr_lt0

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "ltrNl", "oppr0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gtrN x : 0 < x -> - x < x.

Proof. by move=> n0; rewrite -subr_lt0 -opprD oppr_lt0 addr_gt0. Qed.

Lemma

gtrN

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "addr_gt0", "opprD", "oppr_lt0", "subr_lt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

oppr_lte0

:= (oppr_le0, oppr_lt0).

Definition

oppr_lte0

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "oppr_le0", "oppr_lt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

oppr_cp0

:= (oppr_gte0, oppr_lte0).

Definition

oppr_cp0

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "oppr_gte0", "oppr_lte0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lterNE

:= (oppr_cp0, lterN2).

Definition

lterNE

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "lterN2", "oppr_cp0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ge0_cp x : 0 <= x -> (- x <= 0) * (- x <= x).

Proof. by move=> hx; rewrite oppr_cp0 hx (@le_trans _ _ 0) ?oppr_cp0. Qed.

Lemma

ge0_cp

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "le_trans", "oppr_cp0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gerN x : 0 <= x -> - x <= x.

Proof. by move=> x0; rewrite ge0_cp. Qed.

Lemma

gerN

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "ge0_cp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gt0_cp x : 0 < x -> (0 <= x) * (- x <= 0) * (- x <= x) * (- x < 0) * (- x < x).

Proof. move=> hx; move: (ltW hx) => hx'; rewrite !ge0_cp hx' //. by rewrite oppr_cp0 hx // (@lt_trans _ _ 0) ?oppr_cp0. Qed.

Lemma

gt0_cp

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "ge0_cp", "ltW", "lt_trans", "oppr_cp0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le0_cp x : x <= 0 -> (0 <= - x) * (x <= - x).

Proof. by move=> hx; rewrite oppr_cp0 hx (@le_trans _ _ 0) ?oppr_cp0. Qed.

Lemma

le0_cp

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "le_trans", "oppr_cp0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lt0_cp x : x < 0 -> (x <= 0) * (0 <= - x) * (x <= - x) * (0 < - x) * (x < - x).

Proof. move=> hx; move: (ltW hx) => hx'; rewrite !le0_cp // hx'. by rewrite oppr_cp0 hx // (@lt_trans _ _ 0) ?oppr_cp0. Qed.

Lemma

lt0_cp

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "le0_cp", "ltW", "lt_trans", "oppr_cp0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ler_wpDl y x z : 0 <= x -> y <= z -> y <= x + z.

Proof. by move=> *; rewrite -[y]add0r lerD. Qed.

Lemma

ler_wpDl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "add0r", "lerD" ]

Addition with left member known to be positive/negative

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltr_wpDl y x z : 0 <= x -> y < z -> y < x + z.

Proof. by move=> *; rewrite -[y]add0r ler_ltD. Qed.

Lemma

ltr_wpDl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "add0r", "ler_ltD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltr_pwDl y x z : 0 < x -> y <= z -> y < x + z.

Proof. by move=> *; rewrite -[y]add0r ltr_leD. Qed.

Lemma

ltr_pwDl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "add0r", "ltr_leD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltr_pDl y x z : 0 < x -> y < z -> y < x + z.

Proof. by move=> *; rewrite -[y]add0r ltrD. Qed.

Lemma

ltr_pDl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "add0r", "ltrD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ler_wnDl y x z : x <= 0 -> y <= z -> x + y <= z.

Proof. by move=> *; rewrite -[z]add0r lerD. Qed.

Lemma

ler_wnDl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "add0r", "lerD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltr_wnDl y x z : x <= 0 -> y < z -> x + y < z.

Proof. by move=> *; rewrite -[z]add0r ler_ltD. Qed.

Lemma

ltr_wnDl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "add0r", "ler_ltD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltr_nwDl y x z : x < 0 -> y <= z -> x + y < z.

Proof. by move=> *; rewrite -[z]add0r ltr_leD. Qed.

Lemma

ltr_nwDl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "add0r", "ltr_leD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltr_nDl y x z : x < 0 -> y < z -> x + y < z.

Proof. by move=> *; rewrite -[z]add0r ltrD. Qed.

Lemma

ltr_nDl

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "add0r", "ltrD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ler_wpDr y x z : 0 <= x -> y <= z -> y <= z + x.

Proof. by move=> *; rewrite addrC ler_wpDl. Qed.

Lemma

ler_wpDr

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "addrC", "ler_wpDl" ]

Addition with right member we know positive/negative

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltr_wpDr y x z : 0 <= x -> y < z -> y < z + x.

Proof. by move=> *; rewrite addrC ltr_wpDl. Qed.

Lemma

ltr_wpDr

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "addrC", "ltr_wpDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltr_pwDr y x z : 0 < x -> y <= z -> y < z + x.

Proof. by move=> *; rewrite addrC ltr_pwDl. Qed.

Lemma

ltr_pwDr

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "addrC", "ltr_pwDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltr_pDr y x z : 0 < x -> y < z -> y < z + x.

Proof. by move=> *; rewrite addrC ltr_pDl. Qed.

Lemma

ltr_pDr

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "addrC", "ltr_pDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ler_wnDr y x z : x <= 0 -> y <= z -> y + x <= z.

Proof. by move=> *; rewrite addrC ler_wnDl. Qed.

Lemma

ler_wnDr

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "addrC", "ler_wnDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltr_wnDr y x z : x <= 0 -> y < z -> y + x < z.

Proof. by move=> *; rewrite addrC ltr_wnDl. Qed.

Lemma

ltr_wnDr

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "addrC", "ltr_wnDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltr_nwDr y x z : x < 0 -> y <= z -> y + x < z.

Proof. by move=> *; rewrite addrC ltr_nwDl. Qed.

Lemma

ltr_nwDr

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "addrC", "ltr_nwDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltr_nDr y x z : x < 0 -> y < z -> y + x < z.

Proof. by move=> *; rewrite addrC ltr_nDl. Qed.

Lemma

ltr_nDr

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "addrC", "ltr_nDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

paddr_eq0 (x y : R) : 0 <= x -> 0 <= y -> (x + y == 0) = (x == 0) && (y == 0).

Proof. rewrite le0r; case/orP=> [/eqP->|hx]; first by rewrite add0r eqxx. by rewrite (gt_eqF hx) /= => hy; rewrite gt_eqF // ltr_pwDl. Qed.

Lemma

paddr_eq0

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "add0r", "eqxx", "gt_eqF", "le0r", "ltr_pwDl" ]

x and y have the same sign and their sum is null

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

naddr_eq0 (x y : R) : x <= 0 -> y <= 0 -> (x + y == 0) = (x == 0) && (y == 0).

Proof. by move=> lex0 ley0; rewrite -oppr_eq0 opprD paddr_eq0 ?oppr_cp0 // !oppr_eq0. Qed.

Lemma

naddr_eq0

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "lex0", "opprD", "oppr_cp0", "oppr_eq0", "paddr_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addr_ss_eq0 (x y : R) : (0 <= x) && (0 <= y) || (x <= 0) && (y <= 0) -> (x + y == 0) = (x == 0) && (y == 0).

Proof. by case/orP=> /andP []; [apply: paddr_eq0 | apply: naddr_eq0]. Qed.

Lemma

addr_ss_eq0

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "apply", "naddr_eq0", "paddr_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sumr_ge0 I (r : seq I) (P : pred I) (F : I -> R) : (forall i, P i -> (0 <= F i)) -> 0 <= \sum_(i <- r | P i) (F i).

Proof. exact: (big_ind _ _ (@ler_wpDl 0)). Qed.

Lemma

sumr_ge0

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "big_ind", "ler_wpDl", "seq" ]

big sum and ler

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sumr_le0 I (r : seq I) (P : pred I) (F : I -> R) : (forall i, P i -> F i <= 0) -> \sum_(i <- r | P i) F i <= 0.

Proof. by move=> F0; elim/big_ind : _ => // i x Pi; exact/ler_wnDl. Qed.

Lemma

sumr_le0

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "F0", "big_ind", "ler_wnDl", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ler_sum I (r : seq I) (P : pred I) (F G : I -> R) : (forall i, P i -> F i <= G i) -> \sum_(i <- r | P i) F i <= \sum_(i <- r | P i) G i.

Proof. exact: (big_ind2 _ (lexx _) lerD). Qed.

Lemma

ler_sum

algebra.numeric_hierarchy

algebra/numeric_hierarchy/orderedzmod.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "bigop", "nmodule", "order", "rings_modules_and_algebras", "Order.TTheory", "GRing.Theory", "Order.PreOCoercions", "Num.Syntax", "Num.Exports" ]

[ "big_ind2", "lerD", "lexx", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d