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 |
|---|---|---|---|---|---|---|---|---|---|---|
zprimitive0 : zprimitive 0 = 0. | Proof.
by apply/polyP=> i; rewrite coef0 coef_map_id0 ?div0z // zcontents0 divz0.
Qed. | Lemma | zprimitive0 | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"apply",
"coef0",
"coef_map_id0",
"div0z",
"divz0",
"polyP",
"zcontents0",
"zprimitive"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zprimitive_eq0 p : (zprimitive p == 0) = (p == 0). | Proof.
apply/idP/idP=> /eqP p0; first by rewrite [p]zpolyEprim p0 scaler0.
by rewrite p0 zprimitive0.
Qed. | Lemma | zprimitive_eq0 | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"apply",
"p0",
"scaler0",
"zpolyEprim",
"zprimitive",
"zprimitive0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
size_zprimitive p : size (zprimitive p) = size p. | Proof.
have [-> | ] := eqVneq p 0; first by rewrite zprimitive0.
by rewrite {1 3}[p]zpolyEprim scale_poly_eq0 => /norP[/size_scale-> _].
Qed. | Lemma | size_zprimitive | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"eqVneq",
"scale_poly_eq0",
"size",
"size_scale",
"zpolyEprim",
"zprimitive",
"zprimitive0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgz_lead_primitive p : sgz (lead_coef (zprimitive p)) = (p != 0). | Proof.
have [-> | nz_p] := eqVneq; first by rewrite zprimitive0 lead_coef0.
apply: (@mulfI _ (sgz (zcontents p))); first by rewrite sgz_eq0 zcontents_eq0.
by rewrite -sgzM mulr1 -lead_coefZ -zpolyEprim sgz_contents.
Qed. | Lemma | sgz_lead_primitive | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"apply",
"eqVneq",
"lead_coef",
"lead_coef0",
"lead_coefZ",
"mulfI",
"mulr1",
"nz_p",
"sgz",
"sgzM",
"sgz_contents",
"sgz_eq0",
"zcontents",
"zcontents_eq0",
"zpolyEprim",
"zprimitive",
"zprimitive0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zcontents_primitive p : zcontents (zprimitive p) = (p != 0). | Proof.
have [-> | nz_p] := eqVneq; first by rewrite zprimitive0 zcontents0.
apply: (@mulfI _ (zcontents p)); first by rewrite zcontents_eq0.
by rewrite mulr1 -zcontentsZ -zpolyEprim.
Qed. | Lemma | zcontents_primitive | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"apply",
"eqVneq",
"mulfI",
"mulr1",
"nz_p",
"zcontents",
"zcontents0",
"zcontentsZ",
"zcontents_eq0",
"zpolyEprim",
"zprimitive",
"zprimitive0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zprimitive_id p : zprimitive (zprimitive p) = zprimitive p. | Proof.
have [-> | nz_p] := eqVneq p 0; first by rewrite !zprimitive0.
by rewrite {2}[zprimitive p]zpolyEprim zcontents_primitive nz_p scale1r.
Qed. | Lemma | zprimitive_id | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"eqVneq",
"nz_p",
"scale1r",
"zcontents_primitive",
"zpolyEprim",
"zprimitive",
"zprimitive0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zprimitive_monic p : p \in monic -> zprimitive p = p. | Proof. by move=> mon_p; rewrite {2}[p]zpolyEprim zcontents_monic ?scale1r. Qed. | Lemma | zprimitive_monic | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"monic",
"scale1r",
"zcontents_monic",
"zpolyEprim",
"zprimitive"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zprimitiveZ a p : a != 0 -> zprimitive (a *: p) = zprimitive p. | Proof.
have [-> | nz_p nz_a] := eqVneq p 0; first by rewrite scaler0.
apply: (@mulfI _ (a * zcontents p)%:P).
by rewrite polyC_eq0 mulf_neq0 ?zcontents_eq0.
by rewrite -{1}zcontentsZ !mul_polyC -zpolyEprim -scalerA -zpolyEprim.
Qed. | Lemma | zprimitiveZ | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"apply",
"eqVneq",
"mul_polyC",
"mulfI",
"mulf_neq0",
"nz_p",
"polyC_eq0",
"scaler0",
"scalerA",
"zcontents",
"zcontentsZ",
"zcontents_eq0",
"zpolyEprim",
"zprimitive"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zprimitive_min p a q :
p != 0 -> p = a *: q ->
{b | sgz b = sgz (lead_coef q) & q = b *: zprimitive p}. | Proof.
move=> nz_p Dp; have /dvdzP/sig_eqW[b Db]: (a %| zcontents p)%Z.
by rewrite dvdz_contents; apply/polyOver_dvdzP; exists q.
suffices ->: q = b *: zprimitive p.
by rewrite lead_coefZ sgzM sgz_lead_primitive nz_p mulr1; exists b.
apply: (@mulfI _ a%:P).
by apply: contraNneq nz_p; rewrite Dp -mul_polyC => ->; ... | Lemma | zprimitive_min | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"apply",
"contraNneq",
"dvdzP",
"dvdz_contents",
"lead_coef",
"lead_coefZ",
"mul0r",
"mul_polyC",
"mulfI",
"mulr1",
"mulrC",
"nz_p",
"polyOver_dvdzP",
"scalerA",
"sgz",
"sgzM",
"sgz_lead_primitive",
"sig_eqW",
"zcontents",
"zpolyEprim",
"zprimitive"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zprimitive_irr p a q :
p != 0 -> zprimitive p = a *: q -> a = sgz (lead_coef q). | Proof.
move=> nz_p Dp; have: p = (a * zcontents p) *: q.
by rewrite mulrC -scalerA -Dp -zpolyEprim.
case/zprimitive_min=> // b <- /eqP.
rewrite Dp -{1}[q]scale1r scalerA -subr_eq0 -scalerBl scale_poly_eq0 subr_eq0.
have{Dp} /negPf->: q != 0.
by apply: contraNneq nz_p; rewrite -zprimitive_eq0 Dp => ->; rewrite scale... | Lemma | zprimitive_irr | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"apply",
"contraNneq",
"lead_coef",
"mulr0",
"mulrC",
"nz_p",
"scale1r",
"scale_poly_eq0",
"scaler0",
"scalerA",
"scalerBl",
"sgz",
"subr_eq0",
"zcontents",
"zpolyEprim",
"zprimitive",
"zprimitive_eq0",
"zprimitive_min"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zcontentsM p q : zcontents (p * q) = zcontents p * zcontents q. | Proof.
have [-> | nz_p] := eqVneq p 0; first by rewrite !(mul0r, zcontents0).
have [-> | nz_q] := eqVneq q 0; first by rewrite !(mulr0, zcontents0).
rewrite -[zcontents q]mulr1 {1}[p]zpolyEprim {1}[q]zpolyEprim.
rewrite -scalerAl -scalerAr !zcontentsZ; congr (_ * (_ * _)).
rewrite [zcontents _]intEsg sgz_contents lead_... | Lemma | zcontentsM | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"absz_gt0",
"apply",
"coef0",
"coef_map",
"contraTneq",
"dvdn_leq",
"dvdzE",
"dvdz_contents",
"eqVneq",
"eqnP",
"eqn_leq",
"int",
"intEsg",
"intEsign",
"intr",
"lead_coefM",
"leqNgt",
"map_poly",
"map_polyZ",
"mul0r",
"mul1r",
"mulf_eq0",
"mulf_neq0",
"mulr0",
"mulr1"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
zprimitiveM p q : zprimitive (p * q) = zprimitive p * zprimitive q. | Proof.
have [pq_0|] := eqVneq (p * q) 0.
rewrite pq_0; move/eqP: pq_0; rewrite mulf_eq0.
by case/pred2P=> ->; rewrite !zprimitive0 (mul0r, mulr0).
rewrite -zcontents_eq0 -polyC_eq0 => /mulfI; apply; rewrite !mul_polyC.
by rewrite -zpolyEprim zcontentsM -scalerA scalerAr scalerAl -!zpolyEprim.
Qed. | Lemma | zprimitiveM | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"apply",
"eqVneq",
"mul0r",
"mul_polyC",
"mulfI",
"mulf_eq0",
"mulr0",
"polyC_eq0",
"pred2P",
"scalerA",
"scalerAl",
"scalerAr",
"zcontentsM",
"zcontents_eq0",
"zpolyEprim",
"zprimitive",
"zprimitive0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dvdpP_int p q : p %| q -> {r | q = zprimitive p * r}. | Proof.
case/Pdiv.Idomain.dvdpP/sig2_eqW=> [[c r] /= nz_c Dpr].
exists (zcontents q *: zprimitive r); rewrite -scalerAr.
by rewrite -zprimitiveM mulrC -Dpr zprimitiveZ // -zpolyEprim.
Qed. | Lemma | dvdpP_int | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"dvdpP",
"mulrC",
"scalerAr",
"sig2_eqW",
"zcontents",
"zpolyEprim",
"zprimitive",
"zprimitiveM",
"zprimitiveZ"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
int_Smith_normal_form m n (M : 'M[int]_(m, n)) :
{L : 'M[int]_m & L \in unitmx &
{R : 'M[int]_n & R \in unitmx &
{d : seq int | sorted dvdz d &
M = L *m (\matrix_(i, j) (d`_i *+ (i == j :> nat))) *m R}}}. | Proof.
move: {2}_.+1 (ltnSn (m + n)) => mn.
elim: mn => // mn IHmn in m n M *; rewrite ltnS => le_mn.
have [[i j] nzMij | no_ij] := pickP (fun k => M k.1 k.2 != 0); last first.
do 2![exists 1%:M; first exact: unitmx1]; exists nil => //=.
apply/matrixP=> i j; apply/eqP; rewrite mulmx1 mul1mx mxE nth_nil mul0rn.
ex... | Lemma | int_Smith_normal_form | algebra | algebra/intdiv.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"bigop",
"prime",
"nmodule",
"order",
"perm",
"rings_modules_and_algebras",
"divalg",
"poly",
"polydiv",
"zmodp",
"matrix",
"ord... | [
"Bezoutz",
"Da",
"a1",
"absz_gt0",
"add0r",
"addNKr",
"addnC",
"addnS",
"addr0",
"allP",
"apply",
"big_ord1",
"big_ord_recl",
"block_mx",
"block_mxEh",
"cofactor",
"col'",
"col_mxEu",
"contraNneq",
"dM",
"det1",
"det_lblock",
"det_scalar1",
"det_ublock",
"divz",
"di... | Integral spans. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
itv_bound (T : Type) : Type | := BSide of bool & T | BInfty of bool. | Variant | itv_bound | 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 | |
BLeft | := (BSide true). | Notation | BLeft | 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 | |
BRight | := (BSide false). | Notation | BRight | 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 | |
"'-oo'" | := (BInfty _ true) : order_scope. | Notation | '-oo' | 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 | |
"'+oo'" | := (BInfty _ false) : order_scope. | Notation | '+oo' | 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 | |
interval (T : Type) | := Interval of itv_bound T & itv_bound T. | Variant | interval | 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 | |
pair_of_interval T (I : interval T) : itv_bound T * itv_bound T | :=
let: Interval b1 b2 := I in (b1, b2). | Coercion | pair_of_interval | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"interval",
"itv_bound"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"`[ a , b ]" | := (Interval (BLeft a) (BRight b)) : order_scope. | Notation | `[ a , b ] | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BLeft",
"BRight"
] | We provide the 9 following notations to help writing formal intervals | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"`] a , b ]" | := (Interval (BRight a) (BRight b)) : order_scope. | Notation | `] a , b ] | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BRight"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"`[ a , b [" | := (Interval (BLeft a) (BLeft b)) : order_scope. | Notation | `[ a , b [ | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BLeft"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"`] a , b [" | := (Interval (BRight a) (BLeft b)) : order_scope. | Notation | `] a , b [ | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BLeft",
"BRight"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"`] '-oo' , b ]" | := (Interval -oo (BRight b)) : order_scope. | Notation | `] '-oo' , b ] | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BRight"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"`] '-oo' , b [" | := (Interval -oo (BLeft b)) : order_scope. | Notation | `] '-oo' , b [ | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BLeft"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"`[ a , '+oo' [" | := (Interval (BLeft a) +oo) : order_scope. | Notation | `[ a , '+oo' [ | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BLeft"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"`] a , '+oo' [" | := (Interval (BRight a) +oo) : order_scope. | Notation | `] a , '+oo' [ | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BRight"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"`] -oo , '+oo' [" | := (Interval -oo +oo) : order_scope. | Notation | `] -oo , '+oo' [ | 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 | |
"`[ a , b ]" | := (Interval (BLeft a) (BRight b)) : ring_scope. | Notation | `[ a , b ] | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BLeft",
"BRight"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"`] a , b ]" | := (Interval (BRight a) (BRight b)) : ring_scope. | Notation | `] a , b ] | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BRight"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"`[ a , b [" | := (Interval (BLeft a) (BLeft b)) : ring_scope. | Notation | `[ a , b [ | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BLeft"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"`] a , b [" | := (Interval (BRight a) (BLeft b)) : ring_scope. | Notation | `] a , b [ | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BLeft",
"BRight"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"`] '-oo' , b ]" | := (Interval -oo (BRight b)) : ring_scope. | Notation | `] '-oo' , b ] | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BRight"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"`] '-oo' , b [" | := (Interval -oo (BLeft b)) : ring_scope. | Notation | `] '-oo' , b [ | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BLeft"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"`[ a , '+oo' [" | := (Interval (BLeft a) +oo) : ring_scope. | Notation | `[ a , '+oo' [ | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BLeft"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"`] a , '+oo' [" | := (Interval (BRight a) +oo) : ring_scope. | Notation | `] a , '+oo' [ | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BRight"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"`] -oo , '+oo' [" | := (Interval -oo +oo) : ring_scope. | Notation | `] -oo , '+oo' [ | 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_bound_display (disp : Order.disp_t) : Order.disp_t. | Proof. exact. Qed. | Fact | itv_bound_display | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"disp_t"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
interval_display (disp : Order.disp_t) : Order.disp_t. | Proof. exact. Qed. | Fact | interval_display | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"disp_t"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_bound_can :
cancel (fun b : itv_bound T =>
match b with BSide b x => (b, Some x) | BInfty b => (b, None) end)
(fun b =>
match b with (b, Some x) => BSide b x | (b, None) => BInfty _ b end). | Proof. by case. Qed. | Lemma | itv_bound_can | 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 | |
interval_can :
@cancel _ (interval T)
(fun '(Interval b1 b2) => (b1, b2)) (fun '(b1, b2) => Interval b1 b2). | Proof. by case. Qed. | Lemma | interval_can | 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 | |
le_bound b1 b2 | :=
match b1, b2 with
| -oo, _ | _, +oo => true
| BSide b1 x1, BSide b2 x2 => x1 < x2 ?<= if b2 ==> b1
| _, _ => false
end. | Definition | le_bound | 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 | |
lt_bound b1 b2 | :=
match b1, b2 with
| -oo, +oo | -oo, BSide _ _ | BSide _ _, +oo => true
| BSide b1 x1, BSide b2 x2 => x1 < x2 ?<= if b1 && ~~ b2
| _, _ => false
end. | Definition | lt_bound | 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 | |
lt_bound_def b1 b2 : lt_bound b1 b2 = (b2 != b1) && le_bound b1 b2. | Proof. by case: b1 b2 => [[]?|[]][[]?|[]] //=; rewrite lt_def. Qed. | Lemma | lt_bound_def | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"le_bound",
"lt_bound",
"lt_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_bound_refl : reflexive le_bound. | Proof. by move=> [[]?|[]] /=. Qed. | Lemma | le_bound_refl | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"le_bound"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_bound_anti : antisymmetric le_bound. | Proof. by case=> [[]?|[]] [[]?|[]] //=; case: comparableP => // ->. Qed. | Lemma | le_bound_anti | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"comparableP",
"le_bound"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_bound_trans : transitive le_bound. | Proof.
by case=> [[]?|[]] [[]?|[]] [[]?|[]] lexy leyz //;
apply: (lteif_imply _ (lteif_trans lexy leyz)).
Qed. | Lemma | le_bound_trans | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"apply",
"le_bound",
"lteif_imply",
"lteif_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bound_lexx c1 c2 x : (BSide c1 x <= BSide c2 x) = (c2 ==> c1). | Proof. by rewrite /<=%O /= lteifxx. Qed. | Lemma | bound_lexx | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"c1",
"c2",
"lteifxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bound_ltxx c1 c2 x : (BSide c1 x < BSide c2 x) = (c1 && ~~ c2). | Proof. by rewrite /<%O /= lteifxx. Qed. | Lemma | bound_ltxx | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"c1",
"c2",
"lteifxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ge_pinfty b : (+oo <= b) = (b == +oo). | Proof. by case: b => [|] []. Qed. | Lemma | ge_pinfty | 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 | |
le_ninfty b : (b <= -oo) = (b == -oo). | Proof. by case: b => // - []. Qed. | Lemma | le_ninfty | 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 | |
gt_pinfty b : (+oo < b) = false. | Proof. by []. Qed. | Lemma | gt_pinfty | 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 | |
lt_ninfty b : (b < -oo) = false. | Proof. by case: b => // -[]. Qed. | Lemma | lt_ninfty | 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 | |
ltBSide x y (b b' : bool) :
(BSide b x < BSide b' y) = (x < y ?<= if b && ~~ b'). | Proof. by []. Qed. | Lemma | ltBSide | 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 | |
leBSide x y (b b' : bool) :
(BSide b x <= BSide b' y) = (x < y ?<= if b' ==> b). | Proof. by []. Qed. | Lemma | leBSide | 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 | |
lteBSide | := (ltBSide, leBSide). | Definition | lteBSide | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"leBSide",
"ltBSide"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltBRight_leBLeft b x : (b < BRight x) = (b <= BLeft x). | Proof. by move: b => [[] b|[]]. Qed. | Lemma | ltBRight_leBLeft | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BLeft",
"BRight"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leBRight_ltBLeft b x : (BRight x <= b) = (BLeft x < b). | Proof. by move: b => [[] b|[]]. Qed. | Lemma | leBRight_ltBLeft | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BLeft",
"BRight"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
BLeft_ltE x y (b : bool) : (BSide b x < BLeft y) = (x < y). | Proof. by case: b. Qed. | Let | BLeft_ltE | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BLeft"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
BRight_leE x y (b : bool) : (BSide b x <= BRight y) = (x <= y). | Proof. by case: b. Qed. | Let | BRight_leE | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BRight"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
BRight_BLeft_leE x y : (BRight x <= BLeft y) = (x < y). | Proof. by []. Qed. | Let | BRight_BLeft_leE | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BLeft",
"BRight"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
BLeft_BRight_ltE x y : (BLeft x < BRight y) = (x <= y). | Proof. by []. Qed. | Let | BLeft_BRight_ltE | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BLeft",
"BRight"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
BRight_BSide_ltE x y (b : bool) : (BRight x < BSide b y) = (x < y). | Proof. by case: b. Qed. | Let | BRight_BSide_ltE | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BRight"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
BLeft_BSide_leE x y (b : bool) : (BLeft x <= BSide b y) = (x <= y). | Proof. by case: b. Qed. | Let | BLeft_BSide_leE | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BLeft"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
BSide_ltE x y (b : bool) : (BSide b x < BSide b y) = (x < y). | Proof. by case: b. Qed. | Let | BSide_ltE | 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 | |
BSide_leE x y (b : bool) : (BSide b x <= BSide b y) = (x <= y). | Proof. by case: b. Qed. | Let | BSide_leE | 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 | |
BInfty_leE a : a <= BInfty T false. | Proof. by case: a => [[] a|[]]. Qed. | Let | BInfty_leE | 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 | |
BInfty_geE a : BInfty T true <= a. | Proof. by case: a => [[] a|[]]. Qed. | Let | BInfty_geE | 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 | |
BInfty_le_eqE a : (BInfty T false <= a) = (a == BInfty T false). | Proof. by case: a => [[] a|[]]. Qed. | Let | BInfty_le_eqE | 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 | |
BInfty_ge_eqE a : (a <= BInfty T true) = (a == BInfty T true). | Proof. by case: a => [[] a|[]]. Qed. | Let | BInfty_ge_eqE | 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 | |
BInfty_ltE a : (a < BInfty T false) = (a != BInfty T false). | Proof. by case: a => [[] a|[]]. Qed. | Let | BInfty_ltE | 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 | |
BInfty_gtE a : (BInfty T true < a) = (a != BInfty T true). | Proof. by case: a => [[] a|[]]. Qed. | Let | BInfty_gtE | 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 | |
BInfty_ltF a : (BInfty T false < a) = false. | Proof. by case: a => [[] a|[]]. Qed. | Let | BInfty_ltF | 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 | |
BInfty_gtF a : (a < BInfty T true) = false. | Proof. by case: a => [[] a|[]]. Qed. | Let | BInfty_gtF | 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 | |
BInfty_BInfty_ltE : BInfty T true < BInfty T false. | Proof. by []. Qed. | Let | BInfty_BInfty_ltE | 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 | |
bnd_simp | := (BLeft_ltE, BRight_leE,
BRight_BLeft_leE, BLeft_BRight_ltE,
BRight_BSide_ltE, BLeft_BSide_leE, BSide_ltE, BSide_leE,
BInfty_leE, BInfty_geE, BInfty_BInfty_ltE,
BInfty_le_eqE, BInfty_ge_eqE, BInfty_ltE, BInfty_gtE, BInfty_ltF, BInfty_gtF,
@lexx _ T, @ltxx _ T, @eqxx T). | Definition | bnd_simp | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BInfty_BInfty_ltE",
"BInfty_geE",
"BInfty_ge_eqE",
"BInfty_gtE",
"BInfty_gtF",
"BInfty_leE",
"BInfty_le_eqE",
"BInfty_ltE",
"BInfty_ltF",
"BLeft_BRight_ltE",
"BLeft_BSide_leE",
"BLeft_ltE",
"BRight_BLeft_leE",
"BRight_BSide_ltE",
"BRight_leE",
"BSide_leE",
"BSide_ltE",
"eqxx",
"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_BSide_min s (x y : T) : (x >=< y)%O ->
BSide s (Order.min x y) = Order.min (BSide s x) (BSide s y). | Proof. by rewrite !minEle bnd_simp => /comparable_leP[]. Qed. | Lemma | comparable_BSide_min | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"bnd_simp",
"comparable_leP",
"min",
"minEle"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable_BSide_max s (x y : T) : (x >=< y)%O ->
BSide s (Order.max x y) = Order.max (BSide s x) (BSide s y). | Proof. by rewrite !maxEle bnd_simp => /comparable_leP[]. Qed. | Lemma | comparable_BSide_max | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"bnd_simp",
"comparable_leP",
"max",
"maxEle"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subitv i1 i2 | :=
let: Interval b1l b1r := i1 in
let: Interval b2l b2r := i2 in (b2l <= b1l) && (b1r <= b2r). | Definition | subitv | 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 | |
subitv_refl : reflexive subitv. | Proof. by case=> /= ? ?; rewrite !lexx. Qed. | Lemma | subitv_refl | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"lexx",
"subitv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subitv_anti : antisymmetric subitv. | Proof.
by case=> [? ?][? ?]; rewrite andbACA => /andP[] /le_anti -> /le_anti ->.
Qed. | Lemma | subitv_anti | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"le_anti",
"subitv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subitv_trans : transitive subitv. | Proof.
case=> [yl yr][xl xr][zl zr] /andP [Hl Hr] /andP [Hl' Hr'] /=.
by rewrite (le_trans Hl' Hl) (le_trans Hr Hr').
Qed. | Lemma | subitv_trans | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"le_trans",
"subitv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pred_of_itv i : pred T | := [pred x | `[x, x] <= i]. | Definition | pred_of_itv | 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 | |
itvPredType | := PredType pred_of_itv. | Canonical | itvPredType | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"pred_of_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subitvE b1l b1r b2l b2r :
(Interval b1l b1r <= Interval b2l b2r) = (b2l <= b1l) && (b1r <= b2r). | Proof. by []. Qed. | Lemma | subitvE | 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_itv x i :
(x \in i) =
let: Interval l u := i in
match l with
| BSide b lb => lb < x ?<= if b
| BInfty b => b
end &&
match u with
| BSide b ub => x < ub ?<= if ~~ b
| BInfty b => ~~ b
end. | Proof. by case: i => [[? ?|[]][|[]]]. Qed. | Lemma | in_itv | 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_boundlr bl br x :
(x \in Interval bl br) = (bl <= BLeft x) && (BRight x <= br). | Proof. by []. Qed. | Lemma | itv_boundlr | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"BLeft",
"BRight"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itv_splitI bl br x :
(x \in Interval bl br) = (x \in Interval bl +oo) && (x \in Interval -oo br). | Proof. by rewrite !itv_boundlr andbT. Qed. | Lemma | itv_splitI | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"itv_boundlr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subitvP i1 i2 : i1 <= i2 -> {subset i1 <= i2}. | Proof. by move=> ? ? /le_trans; exact. Qed. | Lemma | subitvP | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"le_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subset_itv (x y z u : itv_bound T) : x <= y -> z <= u ->
{subset Interval y z <= Interval x u}. | Proof. by move=> xy zu; apply: subitvP; rewrite subitvE xy zu. Qed. | Lemma | subset_itv | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"apply",
"itv_bound",
"subitvE",
"subitvP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subset_itv_bound (r s u v : bool) x y : r <= u -> v <= s ->
{subset Interval (BSide r x) (BSide s y) <= Interval (BSide u x) (BSide v y)}. | Proof.
by move: r s u v=> [] [] [] []// *; apply: subset_itv; rewrite bnd_simp.
Qed. | Lemma | subset_itv_bound | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"apply",
"bnd_simp",
"subset_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subset_itv_oo_cc x y : {subset `]x, y[ <= `[x, y]}. | Proof. by apply: subset_itv; rewrite bnd_simp. Qed. | Lemma | subset_itv_oo_cc | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"apply",
"bnd_simp",
"subset_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subset_itv_oo_oc x y : {subset `]x, y[ <= `]x, y]}. | Proof. by apply: subset_itv; rewrite bnd_simp. Qed. | Lemma | subset_itv_oo_oc | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"apply",
"bnd_simp",
"subset_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subset_itv_oo_co x y : {subset `]x, y[ <= `[x, y[}. | Proof. by apply: subset_itv; rewrite bnd_simp. Qed. | Lemma | subset_itv_oo_co | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"apply",
"bnd_simp",
"subset_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subset_itv_oc_cc x y : {subset `]x, y] <= `[x, y]}. | Proof. by apply: subset_itv; rewrite bnd_simp. Qed. | Lemma | subset_itv_oc_cc | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"apply",
"bnd_simp",
"subset_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subset_itv_co_cc x y : {subset `[x, y[ <= `[x, y]}. | Proof. by apply: subset_itv; rewrite bnd_simp. Qed. | Lemma | subset_itv_co_cc | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"apply",
"bnd_simp",
"subset_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itvxx x : `[x, x] =i pred1 x. | Proof. by move=> y; rewrite in_itv/= -eq_le eq_sym. Qed. | Lemma | itvxx | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"eq_le",
"eq_sym",
"in_itv",
"pred1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
itvxxP y x : reflect (y = x) (y \in `[x, x]). | Proof. by rewrite itvxx; apply/eqP. Qed. | Lemma | itvxxP | algebra | algebra/interval.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"seq",
"choice",
"fintype",
"nmodule",
"order",
"Order.TTheory",
"IntervalCan.Exports"
] | [
"apply",
"itvxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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