fact
stringlengths 8
1.54k
| type
stringclasses 19
values | library
stringclasses 8
values | imports
listlengths 1
10
| filename
stringclasses 98
values | symbolic_name
stringlengths 1
42
| docstring
stringclasses 1
value |
|---|---|---|---|---|---|---|
predC_itvra : [predC Interval a +oo] =i Interval -oo a.
Proof. by move=> y; rewrite inE/= -predC_itvl negbK. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] |
algebra/interval.v
|
predC_itvr
| |
predC_itvi : [predC i] =i [predU Interval -oo i.1 & Interval i.2 +oo].
Proof.
case: i => [a a']; move=> x; rewrite inE/= itv_splitI negb_and.
by symmetry; rewrite inE/= -predC_itvl -predC_itvr.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] |
algebra/interval.v
|
predC_itv
| |
real_BSide_minb x y : x \in Num.real -> y \in Num.real ->
BSide b (Order.min x y) = Order.min (BSide b x) (BSide b y).
Proof. by move=> xr yr; apply/comparable_BSide_min/real_comparable. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] |
algebra/interval.v
|
real_BSide_min
| |
real_BSide_maxb x y : x \in Num.real -> y \in Num.real ->
BSide b (Order.max x y) = Order.max (BSide b x) (BSide b y).
Proof. by move=> xr yr; apply/comparable_BSide_max/real_comparable. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] |
algebra/interval.v
|
real_BSide_max
| |
mem0_itvcc_xNxx : (0 \in `[- x, x]) = (0 <= x).
Proof. by rewrite itv_boundlr [in LHS]/<=%O /= oppr_le0 andbb. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] |
algebra/interval.v
|
mem0_itvcc_xNx
| |
mem0_itvoo_xNxx : 0 \in `]- x, x[ = (0 < x).
Proof. by rewrite itv_boundlr [in LHS]/<=%O /= oppr_lt0 andbb. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] |
algebra/interval.v
|
mem0_itvoo_xNx
| |
oppr_itvba bb (xa xb x : R) :
(- x \in Interval (BSide ba xa) (BSide bb xb)) =
(x \in Interval (BSide (~~ bb) (- xb)) (BSide (~~ ba) (- xa))).
Proof.
by rewrite !itv_boundlr /<=%O /= !implybF negbK andbC lteifNl lteifNr.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] |
algebra/interval.v
|
oppr_itv
| |
oppr_itvoo(a b x : R) : (- x \in `]a, b[) = (x \in `]- b, - a[).
Proof. exact: oppr_itv. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] |
algebra/interval.v
|
oppr_itvoo
| |
oppr_itvco(a b x : R) : (- x \in `[a, b[) = (x \in `]- b, - a]).
Proof. exact: oppr_itv. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] |
algebra/interval.v
|
oppr_itvco
| |
oppr_itvoc(a b x : R) : (- x \in `]a, b]) = (x \in `[- b, - a[).
Proof. exact: oppr_itv. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] |
algebra/interval.v
|
oppr_itvoc
| |
oppr_itvcc(a b x : R) : (- x \in `[a, b]) = (x \in `[- b, - a]).
Proof. exact: oppr_itv. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] |
algebra/interval.v
|
oppr_itvcc
| |
miditv(R : numDomainType) (i : interval R) : R :=
match i with
| Interval (BSide _ a) (BSide _ b) => (a + b) / 2%:R
| Interval -oo%O (BSide _ b) => b - 1
| Interval (BSide _ a) +oo%O => a + 1
| Interval -oo%O +oo%O => 0
| _ => 0
end.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] |
algebra/interval.v
|
miditv
| |
mid_in_itv: forall ba bb (xa xb : R), xa < xb ?<= if ba && ~~ bb ->
mid xa xb \in Interval (BSide ba xa) (BSide bb xb).
Proof.
by move=> [] [] xa xb /= ?; apply/itv_dec; rewrite /= ?midf_lte // ?ltW.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] |
algebra/interval.v
|
mid_in_itv
| |
mid_in_itvoo: forall (xa xb : R), xa < xb -> mid xa xb \in `]xa, xb[.
Proof. by move=> xa xb ?; apply: mid_in_itv. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] |
algebra/interval.v
|
mid_in_itvoo
| |
mid_in_itvcc: forall (xa xb : R), xa <= xb -> mid xa xb \in `[xa, xb].
Proof. by move=> xa xb ?; apply: mid_in_itv. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] |
algebra/interval.v
|
mid_in_itvcc
| |
mem_miditvi : (i.1 < i.2)%O -> miditv i \in i.
Proof.
move: i => [[ba a|[]] [bb b|[]]] //= ab; first exact: mid_in_itv.
by rewrite !in_itv -lteifBlDl subrr lteif01.
by rewrite !in_itv lteifBlDr -lteifBlDl subrr lteif01.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] |
algebra/interval.v
|
mem_miditv
| |
miditv_le_lefti b : (i.1 < i.2)%O -> (BSide b (miditv i) <= i.2)%O.
Proof.
case: i => [x y] lti; have := mem_miditv lti; rewrite inE => /andP[_ ].
by apply: le_trans; rewrite !bnd_simp.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] |
algebra/interval.v
|
miditv_le_left
| |
miditv_ge_righti b : (i.1 < i.2)%O -> (i.1 <= BSide b (miditv i))%O.
Proof.
case: i => [x y] lti; have := mem_miditv lti; rewrite inE => /andP[+ _].
by move=> /le_trans; apply; rewrite !bnd_simp.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] |
algebra/interval.v
|
miditv_ge_right
| |
in_segmentDgt0Prx y z :
reflect (forall e, e > 0 -> y \in `[x - e, z + e]) (y \in `[x, z]).
Proof.
apply/(iffP idP)=> [xyz e /[dup] e_gt0 /ltW e_ge0 | xyz_e].
by rewrite in_itv /= lerBDr !ler_wpDr// (itvP xyz).
by rewrite in_itv /= ; apply/andP; split; apply/ler_addgt0Pr => ? /xyz_e;
rewrite in_itv /= lerBDr => /andP [].
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] |
algebra/interval.v
|
in_segmentDgt0Pr
| |
in_segmentDgt0Plx y z :
reflect (forall e, e > 0 -> y \in `[- e + x, e + z]) (y \in `[x, z]).
Proof.
apply/(equivP (in_segmentDgt0Pr x y z)).
by split=> zxy e /zxy; rewrite [z + _]addrC [_ + x]addrC.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice",
"From mathcomp Require Import div fintype bigop order ssralg finset fingroup",
"From mathcomp Require Import ssrnum"
] |
algebra/interval.v
|
in_segmentDgt0Pl
| |
map_itv_boundS T (f : S -> T) (b : itv_bound S) : itv_bound T :=
match b with
| BSide b x => BSide b (f x)
| BInfty b => BInfty _ b
end.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
map_itv_bound
| |
map_itv_bound_compS T U (f : T -> S) (g : U -> T) (b : itv_bound U) :
map_itv_bound (f \o g) b = map_itv_bound f (map_itv_bound g b).
Proof. by case: b. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
map_itv_bound_comp
| |
map_itvS T (f : S -> T) (i : interval S) : interval T :=
let 'Interval l u := i in Interval (map_itv_bound f l) (map_itv_bound f u).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
map_itv
| |
map_itv_compS T U (f : T -> S) (g : U -> T) (i : interval U) :
map_itv (f \o g) i = map_itv f (map_itv g i).
Proof. by case: i => l u /=; rewrite -!map_itv_bound_comp. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
map_itv_comp
| |
opp_boundb :=
match b with
| BSide b x => BSide (~~ b) (intZmod.oppz x)
| BInfty b => BInfty _ (~~ b)
end.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
opp_bound
| |
opp_bound_ge0b : (BLeft 0%R <= opp_bound b)%O = (b <= BRight 0%R)%O.
Proof. by case: b => [[] b | []//]; rewrite /= !bnd_simp oppr_ge0. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
opp_bound_ge0
| |
opp_bound_gt0b : (BRight 0%R <= opp_bound b)%O = (b <= BLeft 0%R)%O.
Proof.
by case: b => [[] b | []//]; rewrite /= !bnd_simp ?oppr_ge0 ?oppr_gt0.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
opp_bound_gt0
| |
oppi :=
let: Interval l u := i in Interval (opp_bound u) (opp_bound l).
Arguments opp /.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
opp
| |
add_boundlb1 b2 :=
match b1, b2 with
| BSide b1 x1, BSide b2 x2 => BSide (b1 && b2) (intZmod.addz x1 x2)
| _, _ => BInfty _ true
end.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
add_boundl
| |
add_boundrb1 b2 :=
match b1, b2 with
| BSide b1 x1, BSide b2 x2 => BSide (b1 || b2) (intZmod.addz x1 x2)
| _, _ => BInfty _ false
end.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
add_boundr
| |
addi1 i2 :=
let: Interval l1 u1 := i1 in let: Interval l2 u2 := i2 in
Interval (add_boundl l1 l2) (add_boundr u1 u2).
Arguments add /.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
add
| |
signb:= EqZero | NonNeg | NonPos.
|
Variant
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
signb
| |
sign_boundlb :=
let: b0 := BLeft 0%Z in
if b == b0 then EqZero else if (b <= b0)%O then NonPos else NonNeg.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
sign_boundl
| |
sign_boundrb :=
let: b0 := BRight 0%Z in
if b == b0 then EqZero else if (b <= b0)%O then NonPos else NonNeg.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
sign_boundr
| |
signi:= Known of signb | Unknown | Empty.
|
Variant
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
signi
| |
signi : signi :=
let: Interval l u := i in
match sign_boundl l, sign_boundr u with
| EqZero, NonPos
| NonNeg, EqZero
| NonNeg, NonPos => Empty
| EqZero, EqZero => Known EqZero
| NonPos, EqZero
| NonPos, NonPos => Known NonPos
| EqZero, NonNeg
| NonNeg, NonNeg => Known NonNeg
| NonPos, NonNeg => Unknown
end.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
sign
| |
mul_boundlb1 b2 :=
match b1, b2 with
| BInfty _, _
| _, BInfty _
| BLeft 0%Z, _
| _, BLeft 0%Z => BLeft 0%Z
| BSide b1 x1, BSide b2 x2 => BSide (b1 && b2) (intRing.mulz x1 x2)
end.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
mul_boundl
| |
mul_boundrb1 b2 :=
match b1, b2 with
| BLeft 0%Z, _
| _, BLeft 0%Z => BLeft 0%Z
| BRight 0%Z, _
| _, BRight 0%Z => BRight 0%Z
| BSide b1 x1, BSide b2 x2 => BSide (b1 || b2) (intRing.mulz x1 x2)
| _, BInfty _
| BInfty _, _ => +oo%O
end.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
mul_boundr
| |
mul_boundrCb1 b2 : mul_boundr b1 b2 = mul_boundr b2 b1.
Proof.
by move: b1 b2 => [[] [[|?]|?] | []] [[] [[|?]|?] | []] //=; rewrite mulnC.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
mul_boundrC
| |
mul_boundr_gt0b1 b2 :
(BRight 0%Z <= b1 -> BRight 0%Z <= b2 -> BRight 0%Z <= mul_boundr b1 b2)%O.
Proof.
case: b1 b2 => [b1b b1 | []] [b2b b2 | []]//=.
- by case: b1b b2b => -[]; case: b1 b2 => [[|b1] | b1] [[|b2] | b2].
- by case: b1b b1 => -[[] |].
- by case: b2b b2 => -[[] |].
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
mul_boundr_gt0
| |
muli1 i2 :=
let: Interval l1 u1 := i1 in let: Interval l2 u2 := i2 in
let: opp := opp_bound in
let: mull := mul_boundl in let: mulr := mul_boundr in
match sign i1, sign i2 with
| Empty, _ | _, Empty => `[1, 0]
| Known EqZero, _ | _, Known EqZero => `[0, 0]
| Known NonNeg, Known NonNeg =>
Interval (mull l1 l2) (mulr u1 u2)
| Known NonPos, Known NonPos =>
Interval (mull (opp u1) (opp u2)) (mulr (opp l1) (opp l2))
| Known NonNeg, Known NonPos =>
Interval (opp (mulr u1 (opp l2))) (opp (mull l1 (opp u2)))
| Known NonPos, Known NonNeg =>
Interval (opp (mulr (opp l1) u2)) (opp (mull (opp u1) l2))
| Known NonNeg, Unknown =>
Interval (opp (mulr u1 (opp l2))) (mulr u1 u2)
| Known NonPos, Unknown =>
Interval (opp (mulr (opp l1) u2)) (mulr (opp l1) (opp l2))
| Unknown, Known NonNeg =>
Interval (opp (mulr (opp l1) u2)) (mulr u1 u2)
| Unknown, Known NonPos =>
Interval (opp (mulr u1 (opp l2))) (mulr (opp l1) (opp l2))
| Unknown, Unknown =>
Interval
(Order.min (opp (mulr (opp l1) u2)) (opp (mulr u1 (opp l2))))
(Order.max (mulr (opp l1) (opp l2)) (mulr u1 u2))
end.
Arguments mul /.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
mul
| |
mini j :=
let: Interval li ui := i in let: Interval lj uj := j in
Interval (Order.min li lj) (Order.min ui uj).
Arguments min /.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
min
| |
maxi j :=
let: Interval li ui := i in let: Interval lj uj := j in
Interval (Order.max li lj) (Order.max ui uj).
Arguments max /.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
max
| |
keep_nonneg_boundb :=
match b with
| BSide _ (Posz _) => BLeft 0%Z
| BSide _ (Negz _) => -oo%O
| BInfty _ => -oo%O
end.
Arguments keep_nonneg_bound /.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
keep_nonneg_bound
| |
keep_pos_boundb :=
match b with
| BSide b 0%Z => BSide b 0%Z
| BSide _ (Posz (S _)) => BRight 0%Z
| BSide _ (Negz _) => -oo
| BInfty _ => -oo
end.
Arguments keep_pos_bound /.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
keep_pos_bound
| |
keep_nonpos_boundb :=
match b with
| BSide _ (Negz _) | BSide _ (Posz 0) => BRight 0%Z
| BSide _ (Posz (S _)) => +oo%O
| BInfty _ => +oo%O
end.
Arguments keep_nonpos_bound /.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
keep_nonpos_bound
| |
keep_neg_boundb :=
match b with
| BSide b 0%Z => BSide b 0%Z
| BSide _ (Negz _) => BLeft 0%Z
| BSide _ (Posz _) => +oo
| BInfty _ => +oo
end.
Arguments keep_neg_bound /.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
keep_neg_bound
| |
invi :=
let: Interval l u := i in
Interval (keep_pos_bound l) (keep_neg_bound u).
Arguments inv /.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
inv
| |
exprn_le1_boundb1 b2 :=
if b2 isn't BSide _ 1%Z then +oo
else if (BLeft (-1)%Z <= b1)%O then BRight 1%Z else +oo.
Arguments exprn_le1_bound /.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
exprn_le1_bound
| |
exprni :=
let: Interval l u := i in
Interval (keep_pos_bound l) (exprn_le1_bound l u).
Arguments exprn /.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
exprn
| |
exprzi1 i2 :=
let: Interval l2 _ := i2 in
if l2 is BSide _ (Posz _) then exprn i1 else
let: Interval l u := i1 in
Interval (keep_pos_bound l) +oo.
Arguments exprz /.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
exprz
| |
keep_signi :=
let: Interval l u := i in
Interval (keep_nonneg_bound l) (keep_nonpos_bound u).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
keep_sign
| |
keep_nonposi :=
let 'Interval l u := i in
Interval -oo%O (keep_nonpos_bound u).
Arguments keep_nonpos /.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
keep_nonpos
| |
keep_nonnegi :=
let 'Interval l u := i in
Interval (keep_nonneg_bound l) +oo%O.
Arguments keep_nonneg /.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
keep_nonneg
| |
t:= Top | Real of interval int.
|
Variant
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
t
| |
sub(x y : t) :=
match x, y with
| _, Top => true
| Top, Real _ => false
| Real xi, Real yi => subitv xi yi
end.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
sub
| |
spec(i : t) (x : T) := if i is Real i then sem i x else true.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
spec
| |
def(i : t) := Def {
r : T;
#[canonical=no]
P : spec i r
}.
|
Record
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
def
| |
typi := Typ {
sort : Type;
#[canonical=no]
sort_sem : interval int -> sort -> bool;
#[canonical=no]
allP : forall x : sort, spec sort_sem i x
}.
|
Record
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
typ
| |
mk{T f} i x P : @def T f i := @Def T f i x P.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
mk
| |
from{T f i} {x : @def T f i} (phx : phantom T (r x)) := x.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
from
| |
fromP{T f i} {x : @def T f i} (phx : phantom T (r x)) := P x.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
fromP
| |
num_sem(R : numDomainType) (i : interval int) (x : R) : bool :=
(x \in Num.real) && (x \in map_itv intr i).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
num_sem
| |
nat_sem(i : interval int) (x : nat) : bool := Posz x \in i.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
nat_sem
| |
posnum(R : numDomainType) of phant R :=
def (@num_sem R) (Real `]0, +oo[).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
posnum
| |
nonneg(R : numDomainType) of phant R :=
def (@num_sem R) (Real `[0, +oo[).
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
nonneg
| |
real1(op1 : interval int -> interval int) (x : Itv.t) : Itv.t :=
match x with Itv.Top => Itv.Top | Itv.Real x => Itv.Real (op1 x) end.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
real1
| |
real2(op2 : interval int -> interval int -> interval int)
(x y : Itv.t) : Itv.t :=
match x, y with
| Itv.Top, _ | _, Itv.Top => Itv.Top
| Itv.Real x, Itv.Real y => Itv.Real (op2 x y)
end.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
real2
| |
spec_real1T f (op1 : T -> T) (op1i : interval int -> interval int) :
forall (x : T), (forall xi, f xi x = true -> f (op1i xi) (op1 x) = true) ->
forall xi, spec f xi x -> spec f (real1 op1i xi) (op1 x).
Proof. by move=> x + [//| xi]; apply. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
spec_real1
| |
spec_real2T f (op2 : T -> T -> T)
(op2i : interval int -> interval int -> interval int) (x y : T) :
(forall xi yi, f xi x = true -> f yi y = true ->
f (op2i xi yi) (op2 x y) = true) ->
forall xi yi, spec f xi x -> spec f yi y ->
spec f (real2 op2i xi yi) (op2 x y).
Proof. by move=> + [//| xi] [//| yi]; apply. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
spec_real2
| |
num:= r.
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
num
| |
Definition_ := [isSub for @Itv.r T f i].
HB.instance Definition _ : Order.POrder d itv := [POrder of itv by <:].
|
HB.instance
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
Definition
| |
itv_le_total_subproof: total (<=%O : rel nR).
Proof.
move=> x y; apply: real_comparable.
- by case: x => [x /=/andP[]].
- by case: y => [y /=/andP[]].
Qed.
HB.instance Definition _ := Order.POrder_isTotal.Build ring_display nR
itv_le_total_subproof.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
itv_le_total_subproof
| |
top_typ_specT f (x : T) : Itv.spec f Itv.Top x.
Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
top_typ_spec
| |
top_typT f := Itv.Typ (@top_typ_spec T f).
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
top_typ
| |
real_domain_typ_spec(R : realDomainType) (x : R) :
num_spec (Itv.Real `]-oo, +oo[) x.
Proof. by rewrite /Itv.num_sem/= num_real. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
real_domain_typ_spec
| |
real_domain_typ(R : realDomainType) :=
Itv.Typ (@real_domain_typ_spec R).
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
real_domain_typ
| |
real_field_typ_spec(R : realFieldType) (x : R) :
num_spec (Itv.Real `]-oo, +oo[) x.
Proof. exact: real_domain_typ_spec. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
real_field_typ_spec
| |
real_field_typ(R : realFieldType) :=
Itv.Typ (@real_field_typ_spec R).
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
real_field_typ
| |
nat_typ_spec(x : nat) : nat_spec (Itv.Real `[0, +oo[) x.
Proof. by []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
nat_typ_spec
| |
nat_typ:= Itv.Typ nat_typ_spec.
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
nat_typ
| |
typ_inum_spec(i : Itv.t) (xt : Itv.typ i) (x : Itv.sort xt) :
Itv.spec (@Itv.sort_sem _ xt) i x.
Proof. by move: xt x => []. Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
typ_inum_spec
| |
typ_inum(i : Itv.t) (xt : Itv.typ i) (x : Itv.sort xt) :=
Itv.mk (typ_inum_spec x).
|
Canonical
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
typ_inum
| |
unify{T} f (x y : T) := Unify : f x y = true.
#[export] Hint Mode unify + + + + : typeclass_instances.
|
Class
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
unify
| |
unify'{T} f (x y : T) := Unify' : f x y = true.
#[export] Instance unify'P {T} f (x y : T) : unify' f x y -> unify f x y := id.
#[export]
Hint Extern 0 (unify' _ _ _) => vm_compute; reflexivity : typeclass_instances.
|
Class
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
unify'
| |
unify_itvix iy := (unify Itv.sub ix iy).
#[export] Instance top_wider_anything i : unify_itv i Itv.Top.
Proof. by case: i. Qed.
#[export] Instance real_wider_anyreal i :
unify_itv (Itv.Real i) (Itv.Real `]-oo, +oo[).
Proof. by case: i => [l u]; apply/andP; rewrite !bnd_simp. Qed.
|
Notation
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
unify_itv
| |
le_num_itv_bound(x y : itv_bound int) :
(num_itv_bound R x <= num_itv_bound R y)%O = (x <= y)%O.
Proof.
by case: x y => [[] x | x] [[] y | y]//=; rewrite !bnd_simp ?ler_int ?ltr_int.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
le_num_itv_bound
| |
num_itv_bound_le_BLeft(x : itv_bound int) (y : int) :
(num_itv_bound R x <= BLeft (y%:~R : R))%O = (x <= BLeft y)%O.
Proof.
rewrite -[BLeft y%:~R]/(map_itv_bound intr (BLeft y)).
by rewrite le_num_itv_bound.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
num_itv_bound_le_BLeft
| |
BRight_le_num_itv_bound(x : int) (y : itv_bound int) :
(BRight (x%:~R : R) <= num_itv_bound R y)%O = (BRight x <= y)%O.
Proof.
rewrite -[BRight x%:~R]/(map_itv_bound intr (BRight x)).
by rewrite le_num_itv_bound.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
BRight_le_num_itv_bound
| |
num_spec_sub(x y : Itv.t) : Itv.sub x y ->
forall z : R, num_spec x z -> num_spec y z.
Proof.
case: x y => [| x] [| y] //= x_sub_y z /andP[rz]; rewrite /Itv.num_sem rz/=.
move: x y x_sub_y => [lx ux] [ly uy] /andP[lel leu] /=.
move=> /andP[lxz zux]; apply/andP; split.
- by apply: le_trans lxz; rewrite le_num_itv_bound.
- by apply: le_trans zux _; rewrite le_num_itv_bound.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
num_spec_sub
| |
empty_itv:= Itv.Real `[1, 0]%Z.
|
Definition
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
empty_itv
| |
bottomx : ~ unify_itv i empty_itv.
Proof.
case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=.
by rewrite in_itv/= => /andP[] /le_trans /[apply]; rewrite ler10.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
bottom
| |
gt0x : unify_itv i (Itv.Real `]0%Z, +oo[) -> 0 < x%:num :> R.
Proof.
case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_].
by rewrite /= in_itv/= andbT.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
gt0
| |
le0Fx : unify_itv i (Itv.Real `]0%Z, +oo[) -> x%:num <= 0 :> R = false.
Proof.
case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=.
by rewrite in_itv/= andbT => /lt_geF.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
le0F
| |
lt0x : unify_itv i (Itv.Real `]-oo, 0%Z[) -> x%:num < 0 :> R.
Proof.
by case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=; rewrite in_itv.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
lt0
| |
ge0Fx : unify_itv i (Itv.Real `]-oo, 0%Z[) -> 0 <= x%:num :> R = false.
Proof.
case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=.
by rewrite in_itv/= => /lt_geF.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
ge0F
| |
ge0x : unify_itv i (Itv.Real `[0%Z, +oo[) -> 0 <= x%:num :> R.
Proof.
case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=.
by rewrite in_itv/= andbT.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
ge0
| |
lt0Fx : unify_itv i (Itv.Real `[0%Z, +oo[) -> x%:num < 0 :> R = false.
Proof.
case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=.
by rewrite in_itv/= andbT => /le_gtF.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
lt0F
| |
le0x : unify_itv i (Itv.Real `]-oo, 0%Z]) -> x%:num <= 0 :> R.
Proof.
by case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=; rewrite in_itv.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
le0
| |
gt0Fx : unify_itv i (Itv.Real `]-oo, 0%Z]) -> 0 < x%:num :> R = false.
Proof.
case: x => x /= /[swap] /num_spec_sub /[apply] /andP[_] /=.
by rewrite in_itv/= => /le_gtF.
Qed.
|
Lemma
|
algebra
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype choice",
"From mathcomp Require Import order ssralg ssrnum ssrint interval"
] |
algebra/interval_inference.v
|
gt0F
|
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