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 |
|---|---|---|---|---|---|---|---|---|---|---|
num_itv_mul_boundl b1 b2 (x1 x2 : R) :
(BLeft 0%:Z <= b1 -> BLeft 0%:Z <= b2 ->
num_itv_bound R b1 <= BLeft x1 ->
num_itv_bound R b2 <= BLeft x2 ->
num_itv_bound R (mul_boundl b1 b2) <= BLeft (x1 * x2))%O. | Proof.
move: b1 b2 => [[] b1 | []//] [[] b2 | []//] /=; rewrite 4!bnd_simp.
- set bl := match b1 with 0%Z => _ | _ => _ end.
have -> : bl = BLeft (b1 * b2).
rewrite {}/bl; move: b1 b2 => [[|p1]|p1] [[|p2]|p2]; congr BLeft.
by rewrite mulr0.
by rewrite bnd_simp intrM -2!(ler0z R); apply: ler_pM.
- case: b1 =... | Lemma | num_itv_mul_boundl | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"BLeft",
"apply",
"bnd_simp",
"intrM",
"le_lt_trans",
"le_trans",
"ler0z",
"ler_pM",
"ler_pM2r",
"ler_wpM2l",
"ltW",
"lt_le_trans",
"ltr_pM",
"ltr_pM2l",
"ltr_pM2r",
"mul_boundl",
"mulr0",
"mulr_ge0",
"num_itv_bound"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_itv_mul_boundr b1 b2 (x1 x2 : R) :
(0 <= x1 -> 0 <= x2 ->
BRight x1 <= num_itv_bound R b1 ->
BRight x2 <= num_itv_bound R b2 ->
BRight (x1 * x2) <= num_itv_bound R (mul_boundr b1 b2))%O. | Proof.
case: b1 b2 => [b1b b1 | []] [b2b b2 | []] //= x1p x2p; last first.
- case: b2b b2 => -[[|//] | //] _ x20.
+ have:= @ltxx _ (itv_bound R) (BLeft 0%:~R).
by rewrite (lt_le_trans _ x20).
+ have -> : x2 = 0 by apply/le_anti/andP.
by rewrite mulr0.
- case: b1b b1 => -[[|//] |//] x10 _.
+ have:= @ltxx _... | Lemma | num_itv_mul_boundr | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"BLeft",
"BRight",
"apply",
"bnd_simp",
"intrM",
"itv_bound",
"last",
"le_anti",
"le_lt_trans",
"le_trans",
"ler_pM",
"ler_wpM2l",
"ler_wpM2r",
"lerz0",
"lt_le_trans",
"ltr_pM",
"ltr_pM2l",
"ltr_pM2r",
"ltrz0",
"ltxx",
"mul",
"mul0r",
"mul_boundr",
"mulr0",
"mulr_ge0_... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
BRight_le_mul_boundr b1 b2 (x1 x2 : R) :
(0 <= x1 -> x2 \in Num.real -> BRight 0%Z <= b2 ->
BRight x1 <= num_itv_bound R b1 ->
BRight x2 <= num_itv_bound R b2 ->
BRight (x1 * x2) <= num_itv_bound R (mul_boundr b1 b2))%O. | Proof.
move=> x1ge0 x2r b2ge0 lex1b1 lex2b2.
have /orP[x2ge0 | x2le0] := x2r; first exact: num_itv_mul_boundr.
have lem0 : (BRight (x1 * x2) <= BRight 0%R)%O.
by rewrite bnd_simp mulr_ge0_le0 // ltW.
apply: le_trans lem0 _.
rewrite -(mulr0z 1) BRight_le_num_itv_bound.
apply: mul_boundr_gt0 => //.
by rewrite -(@BRight... | Lemma | BRight_le_mul_boundr | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"BRight",
"BRight_le_num_itv_bound",
"apply",
"bnd_simp",
"le_trans",
"ltW",
"mul_boundr",
"mul_boundr_gt0",
"mulr0z",
"mulr_ge0_le0",
"num_itv_bound",
"num_itv_mul_boundr",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_num_itv_bound (x y : itv_bound int) :
(num_itv_bound R x >=< num_itv_bound R y)%O. | Proof.
by case: x y => [[] x | []] [[] y | []]//; apply/orP;
rewrite !bnd_simp ?ler_int ?ltr_int;
case: leP => xy; apply/orP => //; rewrite ltW ?orbT.
Qed. | Lemma | comparable_num_itv_bound | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"apply",
"bnd_simp",
"int",
"itv_bound",
"leP",
"ler_int",
"ltW",
"ltr_int",
"num_itv_bound"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_itv_bound_min (x y : itv_bound int) :
num_itv_bound R (Order.min x y)
= Order.min (num_itv_bound R x) (num_itv_bound R y). | Proof.
have [lexy | ltyx] := leP x y; [by rewrite !minEle le_num_itv_bound lexy|].
rewrite minElt -if_neg -comparable_leNgt ?le_num_itv_bound ?ltW//.
exact: comparable_num_itv_bound.
Qed. | Lemma | num_itv_bound_min | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"comparable_leNgt",
"comparable_num_itv_bound",
"int",
"itv_bound",
"leP",
"le_num_itv_bound",
"ltW",
"min",
"minEle",
"minElt",
"num_itv_bound"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_itv_bound_max (x y : itv_bound int) :
num_itv_bound R (Order.max x y)
= Order.max (num_itv_bound R x) (num_itv_bound R y). | Proof.
have [lexy | ltyx] := leP x y; [by rewrite !maxEle le_num_itv_bound lexy|].
rewrite maxElt -if_neg -comparable_leNgt ?le_num_itv_bound ?ltW//.
exact: comparable_num_itv_bound.
Qed. | Lemma | num_itv_bound_max | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"comparable_leNgt",
"comparable_num_itv_bound",
"int",
"itv_bound",
"leP",
"le_num_itv_bound",
"ltW",
"max",
"maxEle",
"maxElt",
"num_itv_bound"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_spec_mul (xi yi : Itv.t) (x : num_def R xi) (y : num_def R yi)
(r := Itv.real2 mul xi yi) :
num_spec r (x%:num * y%:num). | Proof.
rewrite {}/r; case: xi yi x y => [//| [xl xu]] [//| [yl yu]].
case=> [x /=/and3P[xr /= xlx xxu]] [y /=/and3P[yr /= yly yyu]].
rewrite -/(sign (Interval xl xu)) -/(sign (Interval yl yu)).
have ns000 : @Itv.num_sem R `[0, 0] 0 by apply/and3P.
have xyr : x * y \in Num.real by exact: realM.
case: (signP xlx xxu xr) ... | Lemma | num_spec_mul | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"BRight_le_mul_boundr",
"apply",
"comparable_ge_min",
"comparable_leP",
"comparable_le_max",
"comparable_num_itv_bound",
"ltW",
"mul",
"mul0r",
"mulNr",
"mul_boundrC",
"mulr0",
"mulrC",
"mulrN",
"mulrNN",
"num",
"num_def",
"num_itv_bound_max",
"num_itv_bound_min",
"num_itv_mul_... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_inum (xi yi : Itv.t) (x : num_def R xi) (y : num_def R yi) | :=
Itv.mk (num_spec_mul x y). | Canonical | mul_inum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"mk",
"num_def",
"num_spec_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_spec_min (xi yi : Itv.t) (x : num_def R xi) (y : num_def R yi)
(r := Itv.real2 min xi yi) :
num_spec r (Order.min x%:num y%:num). | Proof.
apply: Itv.spec_real2 (Itv.P x) (Itv.P y).
case: x y => [x /= _] [y /= _] => {xi yi r} -[lx ux] [ly uy]/=.
move=> /andP[xr /=/andP[lxx xux]] /andP[yr /=/andP[lyy yuy]].
apply/and3P; split; rewrite ?min_real//= num_itv_bound_min real_BSide_min//.
- apply: (comparable_le_min2 (comparable_num_itv_bound _ _)) => //.... | Lemma | num_spec_min | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"apply",
"comparable_le_min2",
"comparable_num_itv_bound",
"min",
"min_real",
"num",
"num_def",
"num_itv_bound_min",
"num_spec",
"real2",
"real_BSide_min",
"real_comparable",
"spec_real2",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_spec_max (xi yi : Itv.t) (x : num_def R xi) (y : num_def R yi)
(r := Itv.real2 max xi yi) :
num_spec r (Order.max x%:num y%:num). | Proof.
apply: Itv.spec_real2 (Itv.P x) (Itv.P y).
case: x y => [x /= _] [y /= _] => {xi yi r} -[lx ux] [ly uy]/=.
move=> /andP[xr /=/andP[lxx xux]] /andP[yr /=/andP[lyy yuy]].
apply/and3P; split; rewrite ?max_real//= num_itv_bound_max real_BSide_max//.
- apply: (comparable_le_max2 (comparable_num_itv_bound _ _)) => //.... | Lemma | num_spec_max | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"apply",
"comparable_le_max2",
"comparable_num_itv_bound",
"max",
"max_real",
"num",
"num_def",
"num_itv_bound_max",
"num_spec",
"real2",
"real_BSide_max",
"real_comparable",
"spec_real2",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
min_max_typ d | := MinMaxTyp {
min_max_sort : porderType d;
#[canonical=no]
min_max_sem : interval int -> min_max_sort -> bool;
#[canonical=no]
min_max_minP : forall (xi yi : Itv.t) (x : Itv.def min_max_sem xi)
(y : Itv.def min_max_sem yi),
let: r := Itv.real2 min xi yi in
Itv.spec min_max_sem r (Order.min x%:num... | Record | min_max_typ | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"def",
"int",
"interval",
"max",
"min",
"num",
"real2",
"spec"
] | We can't directly put an instance on Order.min for R : numDomainType
since we may want instances for other porderType
(typically \bar R or even nat). So we resort on this additional
canonical structure. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
min_typ_inum d (t : min_max_typ d) (xi yi : Itv.t)
(x : Itv.def (@min_max_sem d t) xi) (y : Itv.def (@min_max_sem d t) yi)
(r := Itv.real2 min xi yi) | :=
Itv.mk (min_max_minP x y). | Canonical | min_typ_inum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"def",
"min",
"min_max_typ",
"mk",
"real2"
] | The default instances on porderType, for min... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
max_typ_inum d (t : min_max_typ d) (xi yi : Itv.t)
(x : Itv.def (@min_max_sem d t) xi) (y : Itv.def (@min_max_sem d t) yi)
(r := Itv.real2 min xi yi) | :=
Itv.mk (min_max_maxP x y). | Canonical | max_typ_inum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"def",
"min",
"min_max_typ",
"mk",
"real2"
] | ...and for max | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
num_min_max_typ | := MinMaxTyp num_spec_min num_spec_max. | Canonical | num_min_max_typ | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"num_spec_max",
"num_spec_min"
] | Instance of the above structure for numDomainType | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
nat_num_spec (i : Itv.t) (n : nat) : nat_spec i n = num_spec i (n%:R : R). | Proof.
case: i => [//| [l u]]; rewrite /= /Itv.num_sem realn/=; congr (_ && _).
- by case: l => [[] l |//]; rewrite !bnd_simp ?pmulrn ?ler_int ?ltr_int.
- by case: u => [[] u |//]; rewrite !bnd_simp ?pmulrn ?ler_int ?ltr_int.
Qed. | Lemma | nat_num_spec | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"bnd_simp",
"ler_int",
"ltr_int",
"nat",
"nat_spec",
"num_sem",
"num_spec",
"pmulrn",
"realn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natmul_itv (i1 i2 : Itv.t) : Itv.t | :=
match i1, i2 with
| Itv.Top, _ => Itv.Top
| _, Itv.Top => Itv.Real `]-oo, +oo[
| Itv.Real i1, Itv.Real i2 => Itv.Real (mul i1 i2)
end. | Definition | natmul_itv | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_spec_natmul (xi ni : Itv.t) (x : num_def R xi) (n : nat_def ni)
(r := natmul_itv xi ni) :
num_spec r (x%:num *+ n%:num). | Proof.
rewrite {}/r; case: xi x ni n => [//| xi] x [| ni] n.
by apply/and3P; case: n%:num => [|?]; rewrite ?mulr0n ?realrMn.
have Pn : num_spec (Itv.Real ni) (n%:num%:R : R).
by case: n => /= n; rewrite [Itv.nat_sem ni n](nat_num_spec (Itv.Real ni)).
rewrite -mulr_natr -[n%:num%:R]/((Itv.Def Pn)%:num).
by rewrite (... | Lemma | num_spec_natmul | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"apply",
"mulr0n",
"mulr_natr",
"nat_def",
"nat_num_spec",
"nat_sem",
"natmul_itv",
"num",
"num_def",
"num_spec",
"num_spec_mul",
"realrMn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natmul_inum (xi ni : Itv.t) (x : num_def R xi) (n : nat_def ni) | :=
Itv.mk (num_spec_natmul x n). | Canonical | natmul_inum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"mk",
"nat_def",
"num_def",
"num_spec_natmul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_spec_int (i : Itv.t) (n : int) :
num_spec i n = num_spec i (n%:~R : R). | Proof.
case: i => [//| [l u]]; rewrite /= /Itv.num_sem num_real realz/=.
congr (andb _ _).
- by case: l => [[] l |//]; rewrite !bnd_simp intz ?ler_int ?ltr_int.
- by case: u => [[] u |//]; rewrite !bnd_simp intz ?ler_int ?ltr_int.
Qed. | Lemma | num_spec_int | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"bnd_simp",
"int",
"intz",
"ler_int",
"ltr_int",
"num_real",
"num_sem",
"num_spec",
"realz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_spec_intmul (xi ii : Itv.t) (x : num_def R xi) (i : num_def int ii)
(r := natmul_itv xi ii) :
num_spec r (x%:num *~ i%:num). | Proof.
rewrite {}/r; case: xi x ii i => [//| xi] x [| ii] i.
by apply/and3P; case: i%:inum => [[|n] | n]; rewrite ?mulr0z ?realN ?realrMn.
have Pi : num_spec (Itv.Real ii) (i%:num%:~R : R).
by case: i => /= i; rewrite [Itv.num_sem ii i](num_spec_int (Itv.Real ii)).
rewrite -mulrzr -[i%:num%:~R]/((Itv.Def Pi)%:num).... | Lemma | num_spec_intmul | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"apply",
"int",
"mulr0z",
"mulrzr",
"natmul_itv",
"num",
"num_def",
"num_sem",
"num_spec",
"num_spec_int",
"num_spec_mul",
"realN",
"realrMn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intmul_inum (xi ni : Itv.t) (x : num_def R xi) (n : num_def int ni) | :=
Itv.mk (num_spec_intmul x n). | Canonical | intmul_inum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"int",
"mk",
"num_def",
"num_spec_intmul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_itv_bound_keep_pos (op : R -> R) (x : R) b :
{homo op : x / 0 <= x} -> {homo op : x / 0 < x} ->
(num_itv_bound R b <= BLeft x)%O ->
(num_itv_bound R (keep_pos_bound b) <= BLeft (op x))%O. | Proof.
case: b => [[] [] [| b] // | []//] hle hlt; rewrite !bnd_simp.
- exact: hle.
- by move=> blex; apply: le_lt_trans (hlt _ _) => //; apply: lt_le_trans blex.
- exact: hlt.
- by move=> bltx; apply: le_lt_trans (hlt _ _) => //; apply: le_lt_trans bltx.
Qed. | Lemma | num_itv_bound_keep_pos | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"BLeft",
"apply",
"bnd_simp",
"keep_pos_bound",
"le_lt_trans",
"lt_le_trans",
"num_itv_bound"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_itv_bound_keep_neg (op : R -> R) (x : R) b :
{homo op : x / x <= 0} -> {homo op : x / x < 0} ->
(BRight x <= num_itv_bound R b)%O ->
(BRight (op x) <= num_itv_bound R (keep_neg_bound b))%O. | Proof.
case: b => [[] [[|//] | b] | []//] hge hgt; rewrite !bnd_simp.
- exact: hgt.
- by move=> xltb; apply: hgt; apply: lt_le_trans xltb _; rewrite lerz0.
- exact: hge.
- by move=> xleb; apply: hgt; apply: le_lt_trans xleb _; rewrite ltrz0.
Qed. | Lemma | num_itv_bound_keep_neg | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"BRight",
"apply",
"bnd_simp",
"keep_neg_bound",
"le_lt_trans",
"lerz0",
"lt_le_trans",
"ltrz0",
"num_itv_bound"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_spec_inv (i : Itv.t) (x : num_def R i) (r := Itv.real1 inv i) :
num_spec r (x%:num^-1). | Proof.
apply: Itv.spec_real1 (Itv.P x).
case: x => x /= _ [l u] /and3P[xr /= lx xu].
rewrite /Itv.num_sem/= realV xr/=; apply/andP; split.
- apply: num_itv_bound_keep_pos lx.
+ by move=> ?; rewrite invr_ge0.
+ by move=> ?; rewrite invr_gt0.
- apply: num_itv_bound_keep_neg xu.
+ by move=> ?; rewrite invr_le0.
+ ... | Lemma | num_spec_inv | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"apply",
"inv",
"invr_ge0",
"invr_gt0",
"invr_le0",
"invr_lt0",
"num",
"num_def",
"num_itv_bound_keep_neg",
"num_itv_bound_keep_pos",
"num_sem",
"num_spec",
"real1",
"realV",
"spec_real1",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inv_inum (i : Itv.t) (x : num_def R i) | := Itv.mk (num_spec_inv x). | Canonical | inv_inum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"mk",
"num_def",
"num_spec_inv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_itv_bound_exprn_le1 (x : R) n l u :
(num_itv_bound R l <= BLeft x)%O ->
(BRight x <= num_itv_bound R u)%O ->
(BRight (x ^+ n) <= num_itv_bound R (exprn_le1_bound l u))%O. | Proof.
case: u => [bu [[//|[|//]] |//] | []//].
rewrite /exprn_le1_bound; case: (leP _ l) => [lge1 /= |//] lx xu.
rewrite bnd_simp; case: n => [| n]; rewrite ?expr0//.
have xN1 : -1 <= x.
case: l lge1 lx => [[] l | []//]; rewrite !bnd_simp -(@ler_int R).
- exact: le_trans.
- by move=> + /ltW; apply: le_trans.
hav... | Lemma | num_itv_bound_exprn_le1 | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"BLeft",
"BRight",
"apply",
"bnd_simp",
"expr0",
"expr1",
"exprNn",
"exprS",
"expr_le1",
"exprn_ge0",
"exprn_le1_bound",
"leP",
"le_trans",
"ler1_real",
"lerNl",
"ler_int",
"ler_piMl",
"ltW",
"mulN1r",
"num_itv_bound",
"opprK",
"oppr_ge0",
"real",
"real_ge0P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_spec_exprn (i : Itv.t) (x : num_def R i) n (r := Itv.real1 exprn i) :
num_spec r (x%:num ^+ n). | Proof.
apply: (@Itv.spec_real1 _ _ (fun x => x^+n) _ _ _ _ (Itv.P x)).
case: x => x /= _ [l u] /and3P[xr /= lx xu].
rewrite /Itv.num_sem realX//=; apply/andP; split.
- apply: (@num_itv_bound_keep_pos (fun x => x^+n)) lx.
+ exact: exprn_ge0.
+ exact: exprn_gt0.
- exact: num_itv_bound_exprn_le1 lx xu.
Qed. | Lemma | num_spec_exprn | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"apply",
"exprn",
"exprn_ge0",
"exprn_gt0",
"num",
"num_def",
"num_itv_bound_exprn_le1",
"num_itv_bound_keep_pos",
"num_sem",
"num_spec",
"real1",
"realX",
"spec_real1",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprn_inum (i : Itv.t) (x : num_def R i) n | :=
Itv.mk (num_spec_exprn x n). | Canonical | exprn_inum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"mk",
"num_def",
"num_spec_exprn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_spec_exprz (xi ki : Itv.t) (x : num_def R xi) (k : num_def int ki)
(r := Itv.real2 exprz xi ki) :
num_spec r (x%:num ^ k%:num). | Proof.
rewrite {}/r; case: ki k => [|[lk uk]] k; first by case: xi x.
case: xi x => [//|xi x]; rewrite /Itv.real2.
have P : Itv.num_sem
(let 'Interval l _ := xi in Interval (keep_pos_bound l) +oo)
(x%:num ^ k%:num).
case: xi x => lx ux x; apply/and3P; split=> [||//].
have xr : x%:num \is Num.real by case:... | Lemma | num_spec_exprz | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"apply",
"bnd_simp",
"exprn",
"exprz",
"exprz_ge0",
"exprz_gt0",
"int",
"keep_pos_bound",
"last",
"natz",
"num",
"num_def",
"num_itv_bound_keep_pos",
"num_sem",
"num_spec",
"num_spec_exprn",
"pmulrn",
"real",
"real2",
"realV",
"realX",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprz_inum (xi ki : Itv.t) (x : num_def R xi) (k : num_def int ki) | :=
Itv.mk (num_spec_exprz x k). | Canonical | exprz_inum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"int",
"mk",
"num_def",
"num_spec_exprz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_spec_norm {V : normedZmodType R} (x : V) :
num_spec (Itv.Real `[0, +oo[) `|x|. | Proof. by apply/and3P; split; rewrite //= ?normr_real ?bnd_simp ?normr_ge0. Qed. | Lemma | num_spec_norm | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"apply",
"bnd_simp",
"normr_ge0",
"normr_real",
"num_spec",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
norm_inum {V : normedZmodType R} (x : V) | := Itv.mk (num_spec_norm x). | Canonical | norm_inum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"mk",
"num_spec_norm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrt_itv (i : Itv.t) : Itv.t | :=
match i with
| Itv.Top => Itv.Real `[0%Z, +oo[
| Itv.Real (Interval l u) =>
match l with
| BSide b 0%Z => Itv.Real (Interval (BSide b 0%Z) +oo)
| BSide b (Posz (S _)) => Itv.Real `]0%Z, +oo[
| _ => Itv.Real `[0, +oo[
end
end. | Definition | sqrt_itv | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"Posz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_spec_sqrt (i : Itv.t) (x : num_def R i) (r := sqrt_itv i) :
num_spec r (Num.sqrt x%:num). | Proof.
have: Itv.num_sem `[0%Z, +oo[ (Num.sqrt x%:num).
by apply/and3P; rewrite /= num_real !bnd_simp sqrtr_ge0.
rewrite {}/r; case: i x => [//| [[bl [l |//] |//] u]] [x /= +] _.
case: bl l => -[| l] /and3P[xr /= bx _]; apply/and3P; split=> //=;
move: bx; rewrite !bnd_simp ?sqrtr_ge0// sqrtr_gt0;
[exact: lt_le_tr... | Lemma | num_spec_sqrt | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"apply",
"bnd_simp",
"le_lt_trans",
"lt_le_trans",
"num",
"num_def",
"num_real",
"num_sem",
"num_spec",
"split",
"sqrt",
"sqrt_itv",
"sqrtr_ge0",
"sqrtr_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrt_inum (i : Itv.t) (x : num_def R i) | := Itv.mk (num_spec_sqrt x). | Canonical | sqrt_inum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"mk",
"num_def",
"num_spec_sqrt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtC_itv (i : Itv.t) : Itv.t | :=
match i with
| Itv.Top => Itv.Top
| Itv.Real (Interval l u) =>
match l with
| BSide b (Posz _) => Itv.Real (Interval (BSide b 0%Z) +oo)
| _ => Itv.Top
end
end. | Definition | sqrtC_itv | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"Posz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_spec_sqrtC (i : Itv.t) (x : num_def R i) (r := sqrtC_itv i) :
num_spec r (sqrtC x%:num). | Proof.
rewrite {}/r; case: i x => [//| [l u] [x /=/and3P[xr /= lx xu]]].
case: l lx => [bl [l |//] |[]//] lx; apply/and3P; split=> //=.
by apply: sqrtC_real; case: bl lx => /[!bnd_simp] [|/ltW]; apply: le_trans.
case: bl lx => /[!bnd_simp] lx.
- by rewrite sqrtC_ge0; apply: le_trans lx.
- by rewrite sqrtC_gt0; apply:... | Lemma | num_spec_sqrtC | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"apply",
"bnd_simp",
"le_lt_trans",
"le_trans",
"ltW",
"num",
"num_def",
"num_spec",
"split",
"sqrtC",
"sqrtC_ge0",
"sqrtC_gt0",
"sqrtC_itv",
"sqrtC_real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtC_inum (i : Itv.t) (x : num_def R i) | := Itv.mk (num_spec_sqrtC x). | Canonical | sqrtC_inum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"mk",
"num_def",
"num_spec_sqrtC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_spec_zero : nat_spec (Itv.Real `[0, 0]%Z) 0. | Proof. by []. Qed. | Lemma | nat_spec_zero | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"nat_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zeron_inum | := Itv.mk nat_spec_zero. | Canonical | zeron_inum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"mk",
"nat_spec_zero"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_spec_add (xi yi : Itv.t) (x : nat_def xi) (y : nat_def yi)
(r := Itv.real2 add xi yi) :
nat_spec r (x%:num + y%:num). | Proof.
have Px : num_spec xi (x%:num%:R : int).
by case: x => /= x; rewrite (@nat_num_spec int).
have Py : num_spec yi (y%:num%:R : int).
by case: y => /= y; rewrite (@nat_num_spec int).
rewrite (@nat_num_spec int) natrD.
rewrite -[x%:num%:R]/((Itv.Def Px)%:num) -[y%:num%:R]/((Itv.Def Py)%:num).
exact: num_spec_add... | Lemma | nat_spec_add | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"Px",
"add",
"int",
"nat_def",
"nat_num_spec",
"nat_spec",
"natrD",
"num",
"num_spec",
"num_spec_add",
"real2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addn_inum (xi yi : Itv.t) (x : nat_def xi) (y : nat_def yi) | :=
Itv.mk (nat_spec_add x y). | Canonical | addn_inum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"mk",
"nat_def",
"nat_spec_add"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_spec_succ (i : Itv.t) (n : nat_def i)
(r := Itv.real2 add i (Itv.Real `[1, 1]%Z)) :
nat_spec r (S n%:num). | Proof.
pose i1 := Itv.Real `[1, 1]%Z; have P1 : nat_spec i1 1 by [].
by rewrite -addn1 -[1%N]/((Itv.Def P1)%:num); apply: nat_spec_add.
Qed. | Lemma | nat_spec_succ | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"P1",
"add",
"addn1",
"apply",
"nat_def",
"nat_spec",
"nat_spec_add",
"num",
"real2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
succn_inum (i : Itv.t) (n : nat_def i) | := Itv.mk (nat_spec_succ n). | Canonical | succn_inum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"mk",
"nat_def",
"nat_spec_succ"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_spec_double (i : Itv.t) (n : nat_def i) (r := Itv.real2 add i i) :
nat_spec r (n%:num.*2). | Proof. by rewrite -addnn nat_spec_add. Qed. | Lemma | nat_spec_double | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"add",
"addnn",
"nat_def",
"nat_spec",
"nat_spec_add",
"num",
"real2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
double_inum (i : Itv.t) (x : nat_def i) | := Itv.mk (nat_spec_double x). | Canonical | double_inum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"mk",
"nat_def",
"nat_spec_double"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_spec_mul (xi yi : Itv.t) (x : nat_def xi) (y : nat_def yi)
(r := Itv.real2 mul xi yi) :
nat_spec r (x%:num * y%:num). | Proof.
have Px : num_spec xi (x%:num%:R : int).
by case: x => /= x; rewrite (@nat_num_spec int).
have Py : num_spec yi (y%:num%:R : int).
by case: y => /= y; rewrite (@nat_num_spec int).
rewrite (@nat_num_spec int) natrM.
rewrite -[x%:num%:R]/((Itv.Def Px)%:num) -[y%:num%:R]/((Itv.Def Py)%:num).
exact: num_spec_mul... | Lemma | nat_spec_mul | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"Px",
"int",
"mul",
"nat_def",
"nat_num_spec",
"nat_spec",
"natrM",
"num",
"num_spec",
"num_spec_mul",
"real2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
muln_inum (xi yi : Itv.t) (x : nat_def xi) (y : nat_def yi) | :=
Itv.mk (nat_spec_mul x y). | Canonical | muln_inum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"mk",
"nat_def",
"nat_spec_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_spec_exp (i : Itv.t) (x : nat_def i) n (r := Itv.real1 exprn i) :
nat_spec r (x%:num ^ n). | Proof.
have Px : num_spec i (x%:num%:R : int).
by case: x => /= x; rewrite (@nat_num_spec int).
rewrite (@nat_num_spec int) natrX -[x%:num%:R]/((Itv.Def Px)%:num).
exact: num_spec_exprn.
Qed. | Lemma | nat_spec_exp | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"Px",
"exprn",
"int",
"nat_def",
"nat_num_spec",
"nat_spec",
"natrX",
"num",
"num_spec",
"num_spec_exprn",
"real1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expn_inum (i : Itv.t) (x : nat_def i) n | := Itv.mk (nat_spec_exp x n). | Canonical | expn_inum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"mk",
"nat_def",
"nat_spec_exp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_spec_min (xi yi : Itv.t) (x : nat_def xi) (y : nat_def yi)
(r := Itv.real2 min xi yi) :
nat_spec r (minn x%:num y%:num). | Proof.
have Px : num_spec xi (x%:num%:R : int).
by case: x => /= x; rewrite (@nat_num_spec int).
have Py : num_spec yi (y%:num%:R : int).
by case: y => /= y; rewrite (@nat_num_spec int).
rewrite (@nat_num_spec int) -minEnat natr_min.
rewrite -[x%:num%:R]/((Itv.Def Px)%:num) -[y%:num%:R]/((Itv.Def Py)%:num).
exact: ... | Lemma | nat_spec_min | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"Px",
"int",
"min",
"minEnat",
"minn",
"nat_def",
"nat_num_spec",
"nat_spec",
"natr_min",
"num",
"num_spec",
"num_spec_min",
"real2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
minn_inum (xi yi : Itv.t) (x : nat_def xi) (y : nat_def yi) | :=
Itv.mk (nat_spec_min x y). | Canonical | minn_inum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"mk",
"nat_def",
"nat_spec_min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_spec_max (xi yi : Itv.t) (x : nat_def xi) (y : nat_def yi)
(r := Itv.real2 max xi yi) :
nat_spec r (maxn x%:num y%:num). | Proof.
have Px : num_spec xi (x%:num%:R : int).
by case: x => /= x; rewrite (@nat_num_spec int).
have Py : num_spec yi (y%:num%:R : int).
by case: y => /= y; rewrite (@nat_num_spec int).
rewrite (@nat_num_spec int) -maxEnat natr_max.
rewrite -[x%:num%:R]/((Itv.Def Px)%:num) -[y%:num%:R]/((Itv.Def Py)%:num).
exact: ... | Lemma | nat_spec_max | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"Px",
"int",
"max",
"maxEnat",
"maxn",
"nat_def",
"nat_num_spec",
"nat_spec",
"natr_max",
"num",
"num_spec",
"num_spec_max",
"real2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
maxn_inum (xi yi : Itv.t) (x : nat_def xi) (y : nat_def yi) | :=
Itv.mk (nat_spec_max x y). | Canonical | maxn_inum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"mk",
"nat_def",
"nat_spec_max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_min_max_typ | := MinMaxTyp nat_spec_min nat_spec_max. | Canonical | nat_min_max_typ | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"nat_spec_max",
"nat_spec_min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_spec_factorial (n : nat) : nat_spec (Itv.Real `[1%Z, +oo[) n`!. | Proof. by apply/andP; rewrite bnd_simp lez_nat fact_gt0. Qed. | Lemma | nat_spec_factorial | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"apply",
"bnd_simp",
"fact_gt0",
"lez_nat",
"nat",
"nat_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
factorial_inum n | := Itv.mk (nat_spec_factorial n). | Canonical | factorial_inum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"mk",
"nat_spec_factorial"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_spec_Posz n : num_spec (Itv.Real `[0, +oo[) (Posz n). | Proof. by apply/and3P; rewrite /= num_real !bnd_simp. Qed. | Lemma | num_spec_Posz | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"Posz",
"apply",
"bnd_simp",
"num_real",
"num_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Posz_inum n | := Itv.mk (num_spec_Posz n). | Canonical | Posz_inum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"mk",
"num_spec_Posz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_spec_Negz n : num_spec (Itv.Real `]-oo, (-1)]) (Negz n). | Proof. by apply/and3P; rewrite /= num_real !bnd_simp. Qed. | Lemma | num_spec_Negz | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"apply",
"bnd_simp",
"num_real",
"num_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Negz_inum n | := Itv.mk (num_spec_Negz n). | Canonical | Negz_inum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"mk",
"num_spec_Negz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nR | := (num_def R i). | Notation | nR | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
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"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"num_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num | := (@num R (@Itv.num_sem R) i). | Notation | num | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
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"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"num_sem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_eq : {mono num : x y / x == y}. | Proof. by []. Qed. | Lemma | num_eq | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
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"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_le : {mono num : x y / (x <= y)%O}. | Proof. by []. Qed. | Lemma | num_le | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_lt : {mono num : x y / (x < y)%O}. | Proof. by []. Qed. | Lemma | num_lt | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
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"ssralg",
"orderedzmod",
"numdomain",
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"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_min : {morph num : x y / Order.min x y}. | Proof. by move=> x y; rewrite !minEle num_le -fun_if. Qed. | Lemma | num_min | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
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"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"min",
"minEle",
"num",
"num_le"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_max : {morph num : x y / Order.max x y}. | Proof. by move=> x y; rewrite !maxEle num_le -fun_if. Qed. | Lemma | num_max | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
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"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"max",
"maxEle",
"num",
"num_le"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_abs_eq0 (a : R) : (`|a|%:nng == 0%:nng) = (a == 0). | Proof. by rewrite -normr_eq0. Qed. | Lemma | num_abs_eq0 | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"normr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_le_max a x y :
(a <= Num.max x%:num y%:num) = (a <= x%:num) || (a <= y%:num). | Proof. by rewrite -comparable_le_max// real_comparable. Qed. | Lemma | num_le_max | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"comparable_le_max",
"max",
"num",
"real_comparable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_ge_max a x y :
(Num.max x%:num y%:num <= a) = (x%:num <= a) && (y%:num <= a). | Proof. by rewrite -comparable_ge_max// real_comparable. Qed. | Lemma | num_ge_max | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"comparable_ge_max",
"max",
"num",
"real_comparable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_le_min a x y :
(a <= Num.min x%:num y%:num) = (a <= x%:num) && (a <= y%:num). | Proof. by rewrite -comparable_le_min// real_comparable. Qed. | Lemma | num_le_min | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
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"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"comparable_le_min",
"min",
"num",
"real_comparable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_ge_min a x y :
(Num.min x%:num y%:num <= a) = (x%:num <= a) || (y%:num <= a). | Proof. by rewrite -comparable_ge_min// real_comparable. Qed. | Lemma | num_ge_min | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
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"ssralg",
"orderedzmod",
"numdomain",
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"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"comparable_ge_min",
"min",
"num",
"real_comparable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_lt_max a x y :
(a < Num.max x%:num y%:num) = (a < x%:num) || (a < y%:num). | Proof. by rewrite -comparable_lt_max// real_comparable. Qed. | Lemma | num_lt_max | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
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"ssralg",
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"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"comparable_lt_max",
"max",
"num",
"real_comparable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_gt_max a x y :
(Num.max x%:num y%:num < a) = (x%:num < a) && (y%:num < a). | Proof. by rewrite -comparable_gt_max// real_comparable. Qed. | Lemma | num_gt_max | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
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"Order.TTheory",
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"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"comparable_gt_max",
"max",
"num",
"real_comparable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_lt_min a x y :
(a < Num.min x%:num y%:num) = (a < x%:num) && (a < y%:num). | Proof. by rewrite -comparable_lt_min// real_comparable. Qed. | Lemma | num_lt_min | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"comparable_lt_min",
"min",
"num",
"real_comparable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_gt_min a x y :
(Num.min x%:num y%:num < a) = (x%:num < a) || (y%:num < a). | Proof. by rewrite -comparable_gt_min// real_comparable. Qed. | Lemma | num_gt_min | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
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"ssrint",
"Order.TTheory",
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"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"comparable_gt_min",
"min",
"num",
"real_comparable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nR | := (num_def R (Itv.Real `[0%Z, +oo[)). | Notation | nR | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"num_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num | := (@num R (@Itv.num_sem R) (Itv.Real `[0%Z, +oo[)). | Notation | num | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"num_sem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_abs_le a x : 0 <= a -> (`|a|%:nng <= x) = (a <= x%:num). | Proof. by move=> a0; rewrite -num_le//= ger0_norm. Qed. | Lemma | num_abs_le | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
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"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"ger0_norm",
"num",
"num_le"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_abs_lt a x : 0 <= a -> (`|a|%:nng < x) = (a < x%:num). | Proof. by move=> a0; rewrite -num_lt/= ger0_norm. Qed. | Lemma | num_abs_lt | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
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"ssralg",
"orderedzmod",
"numdomain",
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"Order.TTheory",
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"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"ger0_norm",
"num",
"num_lt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itvnum_subdef : num_spec (Itv.Real (Interval l u)) x. | Proof. by apply/and3P. Qed. | Lemma | itvnum_subdef | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
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"ssralg",
"orderedzmod",
"numdomain",
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"ssrint",
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"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"apply",
"num_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ItvNum : num_def R (Itv.Real (Interval l u)) | := Itv.mk itvnum_subdef. | Definition | ItvNum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
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"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"itvnum_subdef",
"mk",
"num_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itvreal_subdef : num_spec (Itv.Real (Interval l u)) x. | Proof. by apply/and3P; split; first exact: num_real. Qed. | Lemma | itvreal_subdef | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
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"choice",
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"ssralg",
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"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"apply",
"num_real",
"num_spec",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ItvReal : num_def R (Itv.Real (Interval l u)) | :=
Itv.mk itvreal_subdef. | Definition | ItvReal | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"itvreal_subdef",
"mk",
"num_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv01_subdef : num_spec (Itv.Real `[0%Z, 1%Z]) x. | Proof. by apply/and3P; split; rewrite ?bnd_simp// ger0_real. Qed. | Lemma | itv01_subdef | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
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"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"apply",
"bnd_simp",
"ger0_real",
"num_spec",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Itv01 : num_def R (Itv.Real `[0%Z, 1%Z]) | := Itv.mk itv01_subdef. | Definition | Itv01 | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"itv01_subdef",
"mk",
"num_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
posnum_subdef : num_spec (Itv.Real `]0, +oo[) x. | Proof. by apply/and3P; rewrite /= gtr0_real. Qed. | Lemma | posnum_subdef | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
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"ssralg",
"orderedzmod",
"numdomain",
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"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"apply",
"gtr0_real",
"num_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PosNum : {posnum R} | := Itv.mk posnum_subdef. | Definition | PosNum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
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"canonicals",
"Ty... | [
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"posnum",
"posnum_subdef"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nngnum_subdef : num_spec (Itv.Real `[0, +oo[) x. | Proof. by apply/and3P; rewrite /= ger0_real. Qed. | Lemma | nngnum_subdef | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"apply",
"ger0_real",
"num_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
NngNum : {nonneg R} | := Itv.mk nngnum_subdef. | Definition | NngNum | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"mk",
"nngnum_subdef",
"nonneg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
posnum_spec (R : numDomainType) (x : R) :
R -> bool -> bool -> bool -> Type | :=
| IsPosnum (p : {posnum R}) : posnum_spec x (p%:num) false true true. | Variant | posnum_spec | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"num",
"posnum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
posnumP (R : numDomainType) (x : R) : 0 < x ->
posnum_spec x x (x == 0) (0 <= x) (0 < x). | Proof.
move=> x_gt0; case: real_ltgt0P (x_gt0) => []; rewrite ?gtr0_real // => _ _.
by rewrite -[x]/(PosNum x_gt0)%:num; constructor.
Qed. | Lemma | posnumP | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"PosNum",
"gtr0_real",
"num",
"posnum_spec",
"real_ltgt0P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nonneg_spec (R : numDomainType) (x : R) : R -> bool -> Type | :=
| IsNonneg (p : {nonneg R}) : nonneg_spec x (p%:num) true. | Variant | nonneg_spec | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"nonneg",
"num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nonnegP (R : numDomainType) (x : R) : 0 <= x -> nonneg_spec x x (0 <= x). | Proof. by move=> xge0; rewrite xge0 -[x]/(NngNum xge0)%:num; constructor. Qed. | Lemma | nonnegP | algebra | algebra/interval_inference.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"ssrnat",
"eqtype",
"choice",
"order",
"interval",
"ssralg",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
"Order.Syntax",
"GRing.Theory",
"Num.Theory",
"Itv.Exports",
"canonicals",
"Ty... | [
"NngNum",
"nonneg_spec",
"num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
simp | := (Monoid.Theory.simpm, oppr0). | Notation | simp | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"oppr0",
"simpm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
matrix : predArgType | := Matrix of {ffun 'I_m * 'I_n -> R}. | Variant | matrix | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | mostly inferred automatically | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
mx_val A | := let: Matrix g := A in g. | Definition | mx_val | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fun_of_matrix A (i : 'I_m) (j : 'I_n) | := mx_val A (i, j). | Definition | fun_of_matrix | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"mx_val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fun_of_matrix : matrix >-> Funclass. | Coercion | fun_of_matrix | algebra | algebra/matrix.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"nmodule",
"fingroup",
"perm",
"order",
"rings_modules_and_algebras",
"divalg",
"countalg",
"finalg",
"zmodp",
"GRing.Theory",
... | [
"matrix"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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