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Coq-MathComp

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deg_le2_poly_ge0 : (forall x, 0 <= p.[x]) -> delta <= 0.
Proof. have [age0|alt0] := leP 0 a; first exact: deg_le2_poly_delta_ge0. move=> pge0; move: degp; rewrite leq_eqVlt => /orP[/eqP|] degp'; last first. by move: alt0; rewrite /a nth_default ?ltxx. have [//|dge0] := leP delta 0. pose r1 := (- b - sqrt delta) / (2 * a). pose r2 := (- b + sqrt delta) / (2 * a). pose x0 :=...
Lemma
deg_le2_poly_ge0
algebra.numeric_hierarchy
algebra/numeric_hierarchy/numfield.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "div", "path", "fintype", "bigop", "ssrAC", "finset", "fingroup", "nmodule", "order", "interval", "rings_modules_and_algebras", "divalg", "decfield", "poly", "ordere...
[ "age0", "apply", "deg2_poly_factor", "deg_le2_poly_delta_ge0", "degp", "delta", "hornerE", "last", "leNgt", "leP", "leq_eqVlt", "ltW", "lt_max", "ltr01", "ltrDl", "ltxx", "max", "mulrA", "mulr_gt0", "nmulr_rge0", "nth_default", "r1", "r2", "sqrt", "subr_gt0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
deg_le2_poly_le0 : (forall x, p.[x] <= 0) -> delta <= 0.
Proof. move=> ple0; rewrite /delta -sqrrN -[c]opprK mulrN -mulNr -[-(4 * a)]mulrN. by rewrite -!coefN deg_le2_poly_ge0 ?size_polyN// => x; rewrite hornerN oppr_ge0. Qed.
Lemma
deg_le2_poly_le0
algebra.numeric_hierarchy
algebra/numeric_hierarchy/numfield.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "div", "path", "fintype", "bigop", "ssrAC", "finset", "fingroup", "nmodule", "order", "interval", "rings_modules_and_algebras", "divalg", "decfield", "poly", "ordere...
[ "coefN", "deg_le2_poly_ge0", "delta", "hornerN", "mulNr", "mulrN", "opprK", "oppr_ge0", "size_polyN", "sqrrN" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ring_display : Order.disp_t.
Proof. exact. Qed.
Fact
ring_display
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" ]
[ "disp_t" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le x y
:= nonneg (y - x).
Definition
le
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" ]
[ "nonneg" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_refl : reflexive le.
Proof. by move=> x; rewrite /le subrr nonneg0. Qed.
Fact
le_refl
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", "subrr" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_anti : antisymmetric le.
Proof. by move=> x y /andP[xy yx]; apply/subr0_eq/nonneg_definite; rewrite ?opprB. Qed.
Fact
le_anti
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", "le", "opprB", "subr0_eq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_trans : transitive le.
Proof. by move=> x y z => /nonnegD /[apply]; rewrite addrCA [_ - x]addrAC subrr add0r. Qed.
Fact
le_trans
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", "addrAC", "addrCA", "apply", "le", "subrr" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ler_wD2l : forall x, {homo +%R x : y z / le y z}.
Proof. by move=> x y z; rewrite /le [x + _]addrC addrKA. Qed.
Fact
ler_wD2l
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", "addrKA", "le" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ler
:= (@Order.le ring_display _) (only parsing).
Notation
ler
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", "ring_display" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"@ 'ler' R"
:= (@Order.le ring_display R) (at level 10, R at level 8, only parsing) : function_scope.
Notation
@ 'ler' R
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", "ring_display" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltr
:= (@Order.lt ring_display _) (only parsing).
Notation
ltr
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", "ring_display" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"@ 'ltr' R"
:= (@Order.lt ring_display R) (at level 10, R at level 8, only parsing) : function_scope.
Notation
@ 'ltr' R
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", "ring_display" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ger
:= (@Order.ge ring_display _) (only parsing).
Notation
ger
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" ]
[ "ge", "ring_display" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"@ 'ger' R"
:= (@Order.ge ring_display R) (at level 10, R at level 8, only parsing) : function_scope.
Notation
@ 'ger' R
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" ]
[ "ge", "ring_display" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gtr
:= (@Order.gt ring_display _) (only parsing).
Notation
gtr
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", "ring_display" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"@ 'gtr' R"
:= (@Order.gt ring_display R) (at level 10, R at level 8, only parsing) : function_scope.
Notation
@ 'gtr' R
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", "ring_display" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lerif
:= (@Order.leif ring_display _) (only parsing).
Notation
lerif
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" ]
[ "leif", "ring_display" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"@ 'lerif' R"
:= (@Order.leif ring_display R) (at level 10, R at level 8, only parsing) : function_scope.
Notation
@ 'lerif' R
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" ]
[ "leif", "ring_display" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lterif
:= (@Order.lteif ring_display _) (only parsing).
Notation
lterif
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" ]
[ "lteif", "ring_display" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"@ 'lteif' R"
:= (@Order.lteif ring_display R) (at level 10, R at level 8, only parsing) : function_scope.
Notation
@ 'lteif' R
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" ]
[ "lteif", "ring_display" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparabler
:= (@Order.comparable ring_display _) (only parsing).
Notation
comparabler
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" ]
[ "comparable", "ring_display" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"@ 'comparabler' R"
:= (@Order.comparable ring_display R) (at level 10, R at level 8, only parsing) : function_scope.
Notation
@ 'comparabler' R
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" ]
[ "comparable", "ring_display" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
maxr
:= (@Order.max ring_display _).
Notation
maxr
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" ]
[ "max", "ring_display" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"@ 'maxr' R"
:= (@Order.max ring_display R) (at level 10, R at level 8, only parsing) : function_scope.
Notation
@ 'maxr' R
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" ]
[ "max", "ring_display" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
minr
:= (@Order.min ring_display _).
Notation
minr
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" ]
[ "min", "ring_display" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"@ 'minr' R"
:= (@Order.min ring_display R) (at level 10, R at level 8, only parsing) : function_scope.
Notation
@ 'minr' R
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" ]
[ "min", "ring_display" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pos_num_pred
:= fun x : R => ltr 0 x.
Definition
pos_num_pred
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" ]
[ "ltr" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pos_num : qualifier 0 R
:= [qualify x | pos_num_pred x].
Definition
pos_num
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_num_pred" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
neg_num_pred
:= fun x : R => ltr x 0.
Definition
neg_num_pred
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" ]
[ "ltr" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
neg_num : qualifier 0 R
:= [qualify x : R | neg_num_pred x].
Definition
neg_num
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_num_pred" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nneg_num_pred
:= fun x : R => ler 0 x.
Definition
nneg_num_pred
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" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nneg_num : qualifier 0 R
:= [qualify x : R | nneg_num_pred x].
Definition
nneg_num
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_num_pred" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
npos_num_pred
:= fun x : R => ler x 0.
Definition
npos_num_pred
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" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
npos_num : qualifier 0 R
:= [qualify x : R | npos_num_pred x].
Definition
npos_num
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_num_pred" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
real_num_pred
:= fun x : R => (ler 0 x) || (ler x 0).
Definition
real_num_pred
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" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
real_num : qualifier 0 R
:= [qualify x : R | real_num_pred x].
Definition
real_num
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_num_pred" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Rpos_pred
:= pos_num_pred (only parsing).
Notation
Rpos_pred
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_num_pred" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Rpos
:= pos_num (only parsing).
Notation
Rpos
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_num" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Rneg_pred
:= neg_num_pred (only parsing).
Notation
Rneg_pred
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_num_pred" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Rneg
:= neg_num (only parsing).
Notation
Rneg
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_num" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Rnneg_pred
:= nneg_num_pred (only parsing).
Notation
Rnneg_pred
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_num_pred" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Rnneg
:= nneg_num (only parsing).
Notation
Rnneg
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_num" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Rnpos_pred
:= npos_num_pred (only parsing).
Notation
Rnpos_pred
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_num_pred" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Rnpos
:= npos_num (only parsing).
Notation
Rnpos
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_num" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Rreal_pred
:= real_num_pred (only parsing).
Notation
Rreal_pred
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_num_pred" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Rreal
:= real_num (only parsing).
Notation
Rreal
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_num" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
le
:= ler (only parsing).
Notation
le
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" ]
Shorter qualified names, when Num.Def is not imported.
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lt
:= ltr (only parsing).
Notation
lt
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" ]
[ "ltr" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ge
:= ger (only parsing).
Notation
ge
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" ]
[ "ger" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
gt
:= gtr (only parsing).
Notation
gt
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" ]
[ "gtr" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
leif
:= lerif (only parsing).
Notation
leif
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" ]
[ "lerif" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lteif
:= lterif (only parsing).
Notation
lteif
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" ]
[ "lterif" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comparable
:= comparabler (only parsing).
Notation
comparable
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" ]
[ "comparabler" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
max
:= maxr.
Notation
max
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" ]
[ "maxr" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
min
:= minr.
Notation
min
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" ]
[ "minr" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
pos
:= pos_num.
Notation
pos
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_num" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
neg
:= neg_num.
Notation
neg
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_num" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nneg
:= nneg_num.
Notation
nneg
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_num" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
npos
:= npos_num.
Notation
npos
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_num" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
real
:= real_num.
Notation
real
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_num" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<=%R"
:= le : function_scope.
Notation
<=%R
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" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
">=%R"
:= ge : function_scope.
Notation
>=%R
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" ]
[ "ge" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<%R"
:= lt : function_scope.
Notation
<%R
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" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
">%R"
:= gt : function_scope.
Notation
>%R
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" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<?=%R"
:= leif : function_scope.
Notation
<?=%R
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" ]
[ "leif" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<?<=%R"
:= lteif : function_scope.
Notation
<?<=%R
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" ]
[ "lteif" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
">=<%R"
:= comparable : function_scope.
Notation
>=<%R
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" ]
[ "comparable" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"><%R"
:= (fun x y => ~~ (comparable x y)) : function_scope.
Notation
><%R
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" ]
[ "comparable" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<= y"
:= (ge y) : ring_scope.
Notation
<= y
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" ]
[ "ge" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<= y :> T"
:= (<= (y : T)) (only parsing) : ring_scope.
Notation
<= y :> T
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" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
">= y"
:= (le y) : ring_scope.
Notation
>= y
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" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
">= y :> T"
:= (>= (y : T)) (only parsing) : ring_scope.
Notation
>= y :> T
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" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"< y"
:= (gt y) : ring_scope.
Notation
< y
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" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"< y :> T"
:= (< (y : T)) (only parsing) : ring_scope.
Notation
< y :> T
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" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"> y"
:= (lt y) : ring_scope.
Notation
> y
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" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"> y :> T"
:= (> (y : T)) (only parsing) : ring_scope.
Notation
> y :> T
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" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x <= y :> T"
:= ((x : T) <= (y : T)) (only parsing) : ring_scope.
Notation
x <= y :> T
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" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x >= y"
:= (y <= x) (only parsing) : ring_scope.
Notation
x >= y
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" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x >= y :> T"
:= ((x : T) >= (y : T)) (only parsing) : ring_scope.
Notation
x >= y :> T
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" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x < y"
:= (lt x y) : ring_scope.
Notation
x < y
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" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x < y :> T"
:= ((x : T) < (y : T)) (only parsing) : ring_scope.
Notation
x < y :> T
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" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x > y"
:= (y < x) (only parsing) : ring_scope.
Notation
x > y
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" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x > y :> T"
:= ((x : T) > (y : T)) (only parsing) : ring_scope.
Notation
x > y :> T
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" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x <= y <= z"
:= ((x <= y) && (y <= z)) : ring_scope.
Notation
x <= y <= z
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" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x < y <= z"
:= ((x < y) && (y <= z)) : ring_scope.
Notation
x < y <= z
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" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x <= y < z"
:= ((x <= y) && (y < z)) : ring_scope.
Notation
x <= y < z
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" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x < y < z"
:= ((x < y) && (y < z)) : ring_scope.
Notation
x < y < z
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" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x <= y ?= 'iff' C"
:= (lerif x y C) : ring_scope.
Notation
x <= y ?= 'iff' C
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" ]
[ "lerif" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x <= y ?= 'iff' C :> R"
:= ((x : R) <= (y : R) ?= iff C) (only parsing) : ring_scope.
Notation
x <= y ?= 'iff' C :> R
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" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x < y ?<= 'if' C"
:= (lterif x y C) : ring_scope.
Notation
x < y ?<= 'if' C
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" ]
[ "lterif" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x < y ?<= 'if' C :> R"
:= ((x : R) < (y : R) ?<= if C) (only parsing) : ring_scope.
Notation
x < y ?<= 'if' C :> R
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" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
">=< y"
:= [pred x | comparable x y] : ring_scope.
Notation
>=< y
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" ]
[ "comparable" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
">=< y :> T"
:= (>=< (y : T)) (only parsing) : ring_scope.
Notation
>=< y :> T
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" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x >=< y"
:= (comparable x y) : ring_scope.
Notation
x >=< y
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" ]
[ "comparable" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
">< y"
:= [pred x | ~~ comparable x y] : ring_scope.
Notation
>< y
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" ]
[ "comparable" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
">< y :> T"
:= (>< (y : T)) (only parsing) : ring_scope.
Notation
>< y :> T
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" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x >< y"
:= (~~ (comparable x y)) : ring_scope.
Notation
x >< y
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" ]
[ "comparable" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lerD2l x : {mono +%R x : y z / y <= z}.
Proof. exact: homo_mono (addKr _) (ler_wD2l _) (ler_wD2l _). Qed.
Lemma
lerD2l
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" ]
[ "addKr", "ler_wD2l" ]
Monotony of addition
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
lerD2r x : {mono +%R^~ x : y z / y <= z}.
Proof. by move=> y z; rewrite ![_ + x]addrC lerD2l. Qed.
Lemma
lerD2r
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", "lerD2l" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltrD2r x : {mono +%R^~ x : y z / y < z}.
Proof. by move=> y z; rewrite !lt_neqAle lerD2r (inj_eq (addIr _)). Qed.
Lemma
ltrD2r
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" ]
[ "addIr", "inj_eq", "lerD2r", "lt_neqAle" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d