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 |
|---|---|---|---|---|---|---|---|---|---|---|
subitvPl b1l b2l br :
b2l <= b1l -> {subset Interval b1l br <= Interval b2l br}. | Proof. by move=> ?; apply: subitvP; rewrite subitvE lexx andbT. Qed. | Lemma | subitvPl | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"apply",
"lexx",
"subitvE",
"subitvP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subitvPr bl b1r b2r :
b1r <= b2r -> {subset Interval bl b1r <= Interval bl b2r}. | Proof. by move=> ?; apply: subitvP; rewrite subitvE lexx. Qed. | Lemma | subitvPr | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"apply",
"lexx",
"subitvE",
"subitvP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_xx x cl cr y :
(y \in Interval (BSide cl x) (BSide cr x)) = cl && ~~ cr && (y == x). | Proof. by case: cl cr => [] []; rewrite [LHS]lteif_anti // eq_sym. Qed. | Lemma | itv_xx | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"eq_sym",
"lteif_anti"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
boundl_in_itv c x b :
(x \in Interval (BSide c x) b) = c && (BRight x <= b). | Proof. by rewrite itv_boundlr bound_lexx. Qed. | Lemma | boundl_in_itv | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BRight",
"bound_lexx",
"itv_boundlr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
boundr_in_itv c x b :
(x \in Interval b (BSide c x)) = ~~ c && (b <= BLeft x). | Proof. by rewrite itv_boundlr bound_lexx implybF andbC. Qed. | Lemma | boundr_in_itv | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BLeft",
"bound_lexx",
"itv_boundlr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bound_in_itv | := (boundl_in_itv, boundr_in_itv). | Definition | bound_in_itv | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"boundl_in_itv",
"boundr_in_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_in_itv bl br x : x \in Interval bl br -> bl < br. | Proof. by case/andP; apply/le_lt_trans. Qed. | Lemma | lt_in_itv | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"apply",
"le_lt_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lteif_in_itv cl cr yl yr x :
x \in Interval (BSide cl yl) (BSide cr yr) -> yl < yr ?<= if cl && ~~ cr. | Proof. exact: lt_in_itv. Qed. | Lemma | lteif_in_itv | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"lt_in_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_ge b1 b2 : ~~ (b1 < b2) -> Interval b1 b2 =i pred0. | Proof. by move=> ltb12 y; apply/contraNF: ltb12; apply/lt_in_itv. Qed. | Lemma | itv_ge | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"apply",
"lt_in_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_decompose i x : Prop | :=
let: Interval l u := i in
(match l return Prop with
| BSide b lb => lb < x ?<= if b
| BInfty b => b
end *
match u return Prop with
| BSide b ub => x < ub ?<= if ~~ b
| BInfty b => ~~ b
end)%type. | Definition | itv_decompose | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_dec : forall x i, reflect (itv_decompose i x) (x \in i). | Proof. by move=> ? [[? ?|[]][? ?|[]]]; apply: (iffP andP); case. Qed. | Lemma | itv_dec | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"apply",
"itv_decompose"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_rewrite i x : Type | :=
let: Interval l u := i in
(match l with
| BLeft a => (a <= x) * ((x < a) = false)
| BRight a => (a <= x) * (a < x) * ((x <= a) = false) * ((x < a) = false)
| -oo => forall x : T, x == x
| +oo => forall b : bool, unkeyed b = false
end *
match u with
| BRight b => (x ... | Definition | itv_rewrite | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BLeft",
"BRight",
"type"
] | we compute a set of rewrite rules associated to an interval | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
itvP x i : x \in i -> itv_rewrite i x. | Proof.
case: i => [[[]a|[]][[]b|[]]] /andP [] ha hb; rewrite /= ?bound_in_itv;
do ![split | apply/negbTE; rewrite (le_gtF, lt_geF)];
by [|apply: ltW | move: (lteif_trans ha hb) => //=; exact: ltW].
Qed. | Lemma | itvP | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"apply",
"bound_in_itv",
"itv_rewrite",
"le_gtF",
"ltW",
"lt_geF",
"lteif_trans",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_splitU1 b x : b <= BLeft x ->
Interval b (BRight x) =i [predU1 x & Interval b (BLeft x)]. | Proof.
move=> bx z; rewrite !inE/= !subitvE ?bnd_simp//= lt_neqAle.
by case: (eqVneq z x) => [->|]//=; rewrite lexx bx.
Qed. | Lemma | itv_splitU1 | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BLeft",
"BRight",
"bnd_simp",
"eqVneq",
"inE",
"lexx",
"lt_neqAle",
"predU1",
"subitvE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_split1U b x : BRight x <= b ->
Interval (BLeft x) b =i [predU1 x & Interval (BRight x) b]. | Proof.
move=> bx z; rewrite !inE/= !subitvE ?bnd_simp//= lt_neqAle.
by case: (eqVneq z x) => [->|]//=; rewrite lexx bx.
Qed. | Lemma | itv_split1U | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BLeft",
"BRight",
"bnd_simp",
"eqVneq",
"inE",
"lexx",
"lt_neqAle",
"predU1",
"subitvE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bound_meet bl br : itv_bound T | :=
match bl, br with
| -oo, _ | _, -oo => -oo
| +oo, b | b, +oo => b
| BSide xb x, BSide yb y =>
BSide (((x <= y) && xb) || ((y <= x) && yb)) (x `&` y)
end. | Definition | bound_meet | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"itv_bound"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bound_join bl br : itv_bound T | :=
match bl, br with
| -oo, b | b, -oo => b
| +oo, _ | _, +oo => +oo
| BSide xb x, BSide yb y =>
BSide ((~~ (x <= y) || yb) && (~~ (y <= x) || xb)) (x `|` y)
end. | Definition | bound_join | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"itv_bound"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bound_meetC : commutative bound_meet. | Proof.
case=> [? ?|[]][? ?|[]] //=; rewrite meetC; congr BSide.
by case: lcomparableP; rewrite ?orbF // orbC.
Qed. | Lemma | bound_meetC | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"bound_meet",
"lcomparableP",
"meetC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bound_joinC : commutative bound_join. | Proof.
case=> [? ?|[]][? ?|[]] //=; rewrite joinC; congr BSide.
by case: lcomparableP; rewrite ?andbT // andbC.
Qed. | Lemma | bound_joinC | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"bound_join",
"joinC",
"lcomparableP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bound_meetA : associative bound_meet. | Proof.
case=> [? x|[]][? y|[]][? z|[]] //=; rewrite !lexI meetA; congr BSide.
by case: (lcomparableP x y) => [|||->]; case: (lcomparableP y z) => [|||->];
case: (lcomparableP x z) => [|||//<-]; case: (lcomparableP x y);
rewrite //= ?andbF ?orbF ?lexx ?orbA //; case: (lcomparableP y z).
Qed. | Lemma | bound_meetA | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"bound_meet",
"lcomparableP",
"lexI",
"lexx",
"meetA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bound_joinA : associative bound_join. | Proof.
case=> [? x|[]][? y|[]][? z|[]] //=; rewrite !leUx joinA; congr BSide.
by case: (lcomparableP x y) => [|||->]; case: (lcomparableP y z) => [|||->];
case: (lcomparableP x z) => [|||//<-]; case: (lcomparableP x y);
rewrite //= ?orbT ?andbT ?lexx ?andbA //; case: (lcomparableP y z).
Qed. | Lemma | bound_joinA | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"bound_join",
"joinA",
"lcomparableP",
"leUx",
"lexx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bound_meetKU b2 b1 : bound_join b1 (bound_meet b1 b2) = b1. | Proof.
case: b1 b2 => [? ?|[]][? ?|[]] //=;
rewrite ?meetKU ?joinxx ?leIl ?lexI ?lexx ?andbb //=; congr BSide.
by case: lcomparableP; rewrite ?orbF /= ?andbb ?orbK.
Qed. | Lemma | bound_meetKU | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"bound_join",
"bound_meet",
"joinxx",
"lcomparableP",
"leIl",
"lexI",
"lexx",
"meetKU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bound_joinKI b2 b1 : bound_meet b1 (bound_join b1 b2) = b1. | Proof.
case: b1 b2 => [? ?|[]][? ?|[]] //=;
rewrite ?joinKI ?meetxx ?leUl ?leUx ?lexx ?orbb //=; congr BSide.
by case: lcomparableP; rewrite ?orbF ?orbb ?andKb.
Qed. | Lemma | bound_joinKI | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"andKb",
"bound_join",
"bound_meet",
"joinKI",
"lcomparableP",
"leUl",
"leUx",
"lexx",
"meetxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bound_leEmeet b1 b2 : (b1 <= b2) = (bound_meet b1 b2 == b1). | Proof.
case: b1 b2 => [[]t[][]|[][][]] //=; rewrite ?eqxx// => t';
rewrite [LHS]/<=%O /eq_op ?andbT ?andbF ?orbF/= /eq_op/= /eq_op/=;
case: lcomparableP => //=; rewrite ?eqxx//=; [| | |].
- by move/lt_eqF.
- move=> ic; apply: esym; apply: contraNF ic.
by move=> /eqP/meet_idPl; apply: le_comparable.
- by move/lt_e... | Lemma | bound_leEmeet | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"apply",
"bound_meet",
"eqxx",
"lcomparableP",
"le_comparable",
"lt_eqF",
"meet_idPl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bound_le0x b : -oo <= b. | Proof. by []. Qed. | Lemma | bound_le0x | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bound_lex1 b : b <= +oo. | Proof. by case: b => [|[]]. Qed. | Lemma | bound_lex1 | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_meet i1 i2 : interval T | :=
let: Interval b1l b1r := i1 in
let: Interval b2l b2r := i2 in Interval (b1l `|` b2l) (b1r `&` b2r). | Definition | itv_meet | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"interval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_join i1 i2 : interval T | :=
let: Interval b1l b1r := i1 in
let: Interval b2l b2r := i2 in Interval (b1l `&` b2l) (b1r `|` b2r). | Definition | itv_join | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"interval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_meetC : commutative itv_meet. | Proof. by case=> [? ?][? ?] /=; rewrite meetC joinC. Qed. | Lemma | itv_meetC | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"itv_meet",
"joinC",
"meetC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_joinC : commutative itv_join. | Proof. by case=> [? ?][? ?] /=; rewrite meetC joinC. Qed. | Lemma | itv_joinC | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"itv_join",
"joinC",
"meetC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_meetA : associative itv_meet. | Proof. by case=> [? ?][? ?][? ?] /=; rewrite meetA joinA. Qed. | Lemma | itv_meetA | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"itv_meet",
"joinA",
"meetA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_joinA : associative itv_join. | Proof. by case=> [? ?][? ?][? ?] /=; rewrite meetA joinA. Qed. | Lemma | itv_joinA | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"itv_join",
"joinA",
"meetA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_meetKU i2 i1 : itv_join i1 (itv_meet i1 i2) = i1. | Proof. by case: i1 i2 => [? ?][? ?] /=; rewrite meetKU joinKI. Qed. | Lemma | itv_meetKU | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"itv_join",
"itv_meet",
"joinKI",
"meetKU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_joinKI i2 i1 : itv_meet i1 (itv_join i1 i2) = i1. | Proof. by case: i1 i2 => [? ?][? ?] /=; rewrite meetKU joinKI. Qed. | Lemma | itv_joinKI | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"itv_join",
"itv_meet",
"joinKI",
"meetKU"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_leEmeet i1 i2 : (i1 <= i2) = (itv_meet i1 i2 == i1). | Proof.
by case: i1 i2 => [? ?] [? ?]; rewrite /eq_op/=/eq_op/= eq_meetl eq_joinl.
Qed. | Lemma | itv_leEmeet | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"eq_joinl",
"eq_meetl",
"itv_meet"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_le0x i : Interval +oo -oo <= i. | Proof. by case: i => [[|[]]]. Qed. | Lemma | itv_le0x | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_lex1 i : i <= `]-oo, +oo[. | Proof. by case: i => [?[|[]]]. Qed. | Lemma | itv_lex1 | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_itvI x i1 i2 : (x \in i1 `&` i2) = (x \in i1) && (x \in i2). | Proof. exact: lexI. Qed. | Lemma | in_itvI | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"lexI"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
BSide_min s (x y : T) :
BSide s (Order.min x y) = Order.min (BSide s x) (BSide s y). | Proof. exact: comparable_BSide_min. Qed. | Lemma | BSide_min | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"comparable_BSide_min",
"min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
BSide_max s (x y : T) :
BSide s (Order.max x y) = Order.max (BSide s x) (BSide s y). | Proof. exact: comparable_BSide_max. Qed. | Lemma | BSide_max | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"comparable_BSide_max",
"max"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_bound_total : total (<=%O : rel (itv_bound T)). | Proof. by move=> [[]?|[]][[]?|[]]; rewrite /<=%O //=; case: ltgtP. Qed. | Lemma | itv_bound_total | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"itv_bound",
"ltgtP",
"rel",
"total"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_meetUl : @left_distributive (interval T) _ Order.meet Order.join. | Proof.
by move=> [? ?][? ?][? ?]; rewrite /Order.meet /Order.join /= -meetUl -joinIl.
Qed. | Lemma | itv_meetUl | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"interval",
"join",
"joinIl",
"meet",
"meetUl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_splitU c a b : a <= c <= b ->
forall y, (y \in Interval a b) = (y \in Interval a c) || (y \in Interval c b). | Proof.
case/andP => leac lecb y.
rewrite !itv_boundlr !(ltNge (BLeft y) _ : (BRight y <= _) = _).
case: (leP a) (leP b) (leP c) => leay [] leby [] lecy //=.
- by case: leP lecy (le_trans lecb leby).
- by case: leP leay (le_trans leac lecy).
Qed. | Lemma | itv_splitU | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BLeft",
"BRight",
"itv_boundlr",
"leP",
"le_trans",
"ltNge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_splitUeq x a b : x \in Interval a b ->
forall y, (y \in Interval a b) =
[|| y \in Interval a (BLeft x), y == x | y \in Interval (BRight x) b]. | Proof.
case/andP => ax xb y; rewrite (@itv_splitU (BLeft x)) ?ax ?ltW //.
by congr orb; rewrite (@itv_splitU (BRight x)) ?bound_lexx // itv_xx.
Qed. | Lemma | itv_splitUeq | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BLeft",
"BRight",
"bound_lexx",
"itv_splitU",
"itv_xx",
"ltW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_total_meet3E i1 i2 i3 :
i1 `&` i2 `&` i3 \in [:: i1 `&` i2; i1 `&` i3; i2 `&` i3]. | Proof.
case: i1 i2 i3 => [b1l b1r] [b2l b2r] [b3l b3r]; rewrite !inE /eq_op /=.
case: (leP b1l b2l); case: (leP b1l b3l); case: (leP b2l b3l);
case: (leP b1r b2r); case: (leP b1r b3r); case: (leP b2r b3r);
rewrite ?eqxx ?orbT //= => b23r b13r b12r b23l b13l b12l.
- by case: leP b13r (le_trans b12r b23r).
- by case:... | Lemma | itv_total_meet3E | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"eqxx",
"inE",
"leP",
"le_trans",
"lt_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_total_join3E i1 i2 i3 :
i1 `|` i2 `|` i3 \in [:: i1 `|` i2; i1 `|` i3; i2 `|` i3]. | Proof.
case: i1 i2 i3 => [b1l b1r] [b2l b2r] [b3l b3r]; rewrite !inE /eq_op /=.
case: (leP b1l b2l); case: (leP b1l b3l); case: (leP b2l b3l);
case: (leP b1r b2r); case: (leP b1r b3r); case: (leP b2r b3r);
rewrite ?eqxx ?orbT //= => b23r b13r b12r b23l b13l b12l.
- by case: leP b13r (le_trans b12r b23r).
- by case:... | Lemma | itv_total_join3E | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"eqxx",
"inE",
"leP",
"le_trans",
"lt_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
predC_itvl a : [predC Interval -oo a] =i Interval a +oo. | Proof.
case: a => [b x|[]//] y.
by rewrite !inE !subitvE/= bnd_simp andbT !lteBSide/= lteifNE negbK.
Qed. | Lemma | predC_itvl | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"bnd_simp",
"inE",
"lteBSide",
"lteifNE",
"subitvE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
predC_itvr a : [predC Interval a +oo] =i Interval -oo a. | Proof. by move=> y; rewrite inE/= -predC_itvl negbK. Qed. | Lemma | predC_itvr | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"inE",
"predC_itvl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
predC_itv i : [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 | predC_itv | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"inE",
"itv_splitI",
"predC_itvl",
"predC_itvr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_itv_bound S 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 | map_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... | [
"itv_bound"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_itv_bound_comp S 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 | map_itv_bound_comp | 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... | [
"itv_bound",
"map_itv_bound"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_itv S 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 | map_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... | [
"interval",
"map_itv_bound"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
map_itv_comp S 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 | map_itv_comp | 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... | [
"interval",
"map_itv",
"map_itv_bound_comp"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
opp_bound b | :=
match b with
| BSide b x => BSide (~~ b) (intZmod.oppz x)
| BInfty b => BInfty _ (~~ b)
end. | Definition | opp_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... | [
"oppz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
opp_bound_ge0 b : (BLeft 0%R <= opp_bound b)%O = (b <= BRight 0%R)%O. | Proof. by case: b => [[] b | []//]; rewrite /= !bnd_simp oppr_ge0. Qed. | Lemma | opp_bound_ge0 | 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",
"bnd_simp",
"opp_bound",
"oppr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
opp_bound_gt0 b : (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 | opp_bound_gt0 | 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",
"bnd_simp",
"opp_bound",
"oppr_ge0",
"oppr_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
opp i | :=
let: Interval l u := i in Interval (opp_bound u) (opp_bound l). | Definition | opp | 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... | [
"opp_bound"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add_boundl b1 b2 | :=
match b1, b2 with
| BSide b1 x1, BSide b2 x2 => BSide (b1 && b2) (intZmod.addz x1 x2)
| _, _ => BInfty _ true
end. | Definition | add_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... | [
"addz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add_boundr b1 b2 | :=
match b1, b2 with
| BSide b1 x1, BSide b2 x2 => BSide (b1 || b2) (intZmod.addz x1 x2)
| _, _ => BInfty _ false
end. | Definition | add_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... | [
"addz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add i1 i2 | :=
let: Interval l1 u1 := i1 in let: Interval l2 u2 := i2 in
Interval (add_boundl l1 l2) (add_boundr u1 u2). | Definition | 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... | [
"add_boundl",
"add_boundr",
"l1",
"l2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
signb | := EqZero | NonNeg | NonPos. | Variant | signb | 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... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sign_boundl b | :=
let: b0 := BLeft 0%Z in
if b == b0 then EqZero else if (b <= b0)%O then NonPos else NonNeg. | Definition | sign_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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sign_boundr b | :=
let: b0 := BRight 0%Z in
if b == b0 then EqZero else if (b <= b0)%O then NonPos else NonNeg. | Definition | sign_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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
signi | := Known of signb | Unknown | Empty. | Variant | signi | 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... | [
"signb"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sign i : 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 | sign | 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... | [
"sign_boundl",
"sign_boundr",
"signi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_boundl b1 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 | 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",
"mulz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_boundr b1 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 | 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",
"mulz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_boundrC b1 b2 : mul_boundr b1 b2 = mul_boundr b2 b1. | Proof.
by move: b1 b2 => [[] [[|?]|?] | []] [[] [[|?]|?] | []] //=; rewrite mulnC.
Qed. | Lemma | mul_boundrC | 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_boundr",
"mulnC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_boundr_gt0 b1 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 | mul_boundr_gt0 | 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",
"mul_boundr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul i1 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... | Definition | 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... | [
"l1",
"l2",
"max",
"min",
"mul_boundl",
"mul_boundr",
"opp",
"opp_bound",
"sign"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
min i j | :=
let: Interval li ui := i in let: Interval lj uj := j in
Interval (Order.min li lj) (Order.min ui uj). | Definition | 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... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
max i j | :=
let: Interval li ui := i in let: Interval lj uj := j in
Interval (Order.max li lj) (Order.max ui uj). | Definition | 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... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
keep_nonneg_bound b | :=
match b with
| BSide _ (Posz _) => BLeft 0%Z
| BSide _ (Negz _) => -oo%O
| BInfty _ => -oo%O
end. | Definition | keep_nonneg_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... | [
"BLeft",
"Posz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
keep_pos_bound b | :=
match b with
| BSide b 0%Z => BSide b 0%Z
| BSide _ (Posz (S _)) => BRight 0%Z
| BSide _ (Negz _) => -oo
| BInfty _ => -oo
end. | Definition | keep_pos_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... | [
"BRight",
"Posz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
keep_nonpos_bound b | :=
match b with
| BSide _ (Negz _) | BSide _ (Posz 0) => BRight 0%Z
| BSide _ (Posz (S _)) => +oo%O
| BInfty _ => +oo%O
end. | Definition | keep_nonpos_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... | [
"BRight",
"Posz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
keep_neg_bound b | :=
match b with
| BSide b 0%Z => BSide b 0%Z
| BSide _ (Negz _) => BLeft 0%Z
| BSide _ (Posz _) => +oo
| BInfty _ => +oo
end. | Definition | keep_neg_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... | [
"BLeft",
"Posz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inv i | :=
let: Interval l u := i in
Interval (keep_pos_bound l) (keep_neg_bound u). | Definition | 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... | [
"keep_neg_bound",
"keep_pos_bound"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprn_le1_bound b1 b2 | :=
if b2 isn't BSide _ 1%Z then +oo
else if (BLeft (-1)%Z <= b1)%O then BRight 1%Z else +oo. | Definition | exprn_le1_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... | [
"BLeft",
"BRight"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprn i | :=
let: Interval l u := i in
Interval (keep_pos_bound l) (exprn_le1_bound l u). | Definition | 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... | [
"exprn_le1_bound",
"keep_pos_bound"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exprz i1 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. | Definition | 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... | [
"Posz",
"exprn",
"keep_pos_bound",
"l2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
keep_sign i | :=
let: Interval l u := i in
Interval (keep_nonneg_bound l) (keep_nonpos_bound u). | Definition | keep_sign | 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... | [
"keep_nonneg_bound",
"keep_nonpos_bound"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
keep_nonpos i | :=
let 'Interval l u := i in
Interval -oo%O (keep_nonpos_bound u). | Definition | keep_nonpos | 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... | [
"keep_nonpos_bound"
] | used in ereal.v | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
keep_nonneg i | :=
let 'Interval l u := i in
Interval (keep_nonneg_bound l) +oo%O. | Definition | keep_nonneg | 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... | [
"keep_nonneg_bound"
] | used in ereal.v | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
t | := Top | Real of interval int. | Variant | t | 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",
"interval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub (x y : t) | :=
match x, y with
| _, Top => true
| Top, Real _ => false
| Real xi, Real yi => subitv xi yi
end. | Definition | sub | 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... | [
"subitv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
spec (i : t) (x : T) | := if i is Real i then sem i x else true. | Definition | 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... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
def (i : t) | := Def {
r : T;
#[canonical=no]
P : spec i r
}. | Record | def | 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... | [
"spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
typ i | := Typ {
sort : Type;
#[canonical=no]
sort_sem : interval int -> sort -> bool;
#[canonical=no]
allP : forall x : sort, spec sort_sem i x
}. | Record | 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... | [
"allP",
"int",
"interval",
"sort",
"spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mk {T f} i x P : @def T f i | := @Def T f i x P. | Definition | mk | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
from {T f i} {x : @def T f i} (phx : phantom T (r x)) | := x. | Definition | from | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fromP {T f i} {x : @def T f i} (phx : phantom T (r x)) | := P x. | Definition | fromP | 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"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
num_sem (R : numDomainType) (i : interval int) (x : R) : bool | :=
(x \in Num.real) && (x \in map_itv intr i). | Definition | num_sem | 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",
"interval",
"intr",
"map_itv",
"real"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nat_sem (i : interval int) (x : nat) : bool | := Posz x \in i. | Definition | nat_sem | 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",
"int",
"interval",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
posnum (R : numDomainType) & phant R | :=
def (@num_sem R) (Real `]0, +oo[). | Definition | posnum | 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",
"num_sem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nonneg (R : numDomainType) & phant R | :=
def (@num_sem R) (Real `[0, +oo[). | Definition | nonneg | 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",
"num_sem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
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 | real1 | 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",
"interval"
] | a few lifting helper functions | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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 | real2 | 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",
"interval"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
spec_real1 T 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 | spec_real1 | 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",
"interval",
"real1",
"spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
spec_real2 T 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 | spec_real2 | 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",
"interval",
"real2",
"spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"{ 'itv' R & i }" | := (def (@num_sem R) (Itv.Real i%Z)) : type_scope. | Notation | { 'itv' R & i } | 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",
"num_sem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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