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 |
|---|---|---|---|---|---|---|---|---|---|---|
real_ceil_floor x : x \is real_num ->
ceil x = floor x + (x \isn't a int_num). | Proof.
case Ix: (x \is a int_num) => Rx /=.
by apply/eqP; rewrite addr0 ceilNfloor eqr_oppLR floorN.
apply/ceil_def; rewrite addrK; move: (real_floor_itv Rx).
by rewrite le_eqVlt -intrEfloor Ix /= => /andP[-> /ltW].
Qed. | Lemma | real_ceil_floor | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"addr0",
"addrK",
"apply",
"ceil",
"ceilNfloor",
"ceil_def",
"eqr_oppLR",
"floor",
"floorN",
"int_num",
"intrEfloor",
"le_eqVlt",
"ltW",
"real_floor_itv",
"real_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncn_floor x : truncn x = if 0 <= x then `|floor x|%N else 0%N. | Proof.
move: (floorP x); rewrite truncEfloor realE.
have [/le_floor|_]/= := boolP (0 <= x); first by rewrite floor0; case: floor.
by case: ifP => [/le_floor|_ /eqP->//]; rewrite floor0; case: floor => [[]|].
Qed. | Lemma | truncn_floor | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor",
"floor0",
"floorP",
"le_floor",
"realE",
"truncEfloor",
"truncn"
] | Relating Cnat and oldCnat. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
truncnP x :
if 0 <= x then (truncn x)%:R <= x < (truncn x).+1%:R else truncn x == 0%N. | Proof.
rewrite truncn_floor.
case: (boolP (0 <= x)) => //= /[dup] /le_floor + /ger0_real/real_floor_itv.
by rewrite floor0; case: (floor x) => // n _; rewrite absz_nat addrC -intS.
Qed. | Lemma | truncnP | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"absz_nat",
"addrC",
"floor",
"floor0",
"ger0_real",
"intS",
"le_floor",
"real_floor_itv",
"truncn",
"truncn_floor"
] | trunc and nat_num | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
truncn_itv x : 0 <= x -> (truncn x)%:R <= x < (truncn x).+1%:R. | Proof. by move=> x_ge0; move: (truncnP x); rewrite x_ge0. Qed. | Lemma | truncn_itv | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"truncn",
"truncnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncn_le x : ((truncn x)%:R <= x) = (0 <= x). | Proof. by case: ifP (truncnP x) => [+ /andP[] | + /eqP->//]. Qed. | Lemma | truncn_le | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"truncn",
"truncnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_truncnS_gt x : x \is real_num -> x < (truncn x).+1%:R. | Proof. by move/real_ge0P => [/truncn_itv/andP[]|/lt_le_trans->]. Qed. | Lemma | real_truncnS_gt | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"lt_le_trans",
"real_ge0P",
"real_num",
"truncn",
"truncn_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncn_def x n : n%:R <= x < n.+1%:R -> truncn x = n. | Proof.
case/andP=> lemx ltxm1; apply/eqP; rewrite eqn_leq -ltnS -[(n <= _)%N]ltnS.
have/truncn_itv/andP[lefx ltxf1]: 0 <= x by apply: le_trans lemx; apply: ler0n.
by rewrite -!(ltr_nat R) 2?(@le_lt_trans _ _ x).
Qed. | Lemma | truncn_def | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"eqn_leq",
"le_lt_trans",
"le_trans",
"ler0n",
"ltnS",
"ltr_nat",
"truncn",
"truncn_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncn_ge_nat x n : 0 <= x -> (n <= truncn x)%N = (n%:R <= x). | Proof.
move=> /truncn_itv /andP[letx ltxt1]; apply/idP/idP => lenx.
by apply: le_trans letx; rewrite ler_nat.
by rewrite -ltnS -(ltr_nat R); apply: le_lt_trans ltxt1.
Qed. | Lemma | truncn_ge_nat | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"le_lt_trans",
"le_trans",
"ler_nat",
"ltnS",
"ltr_nat",
"truncn",
"truncn_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncn_gt_nat x n : (n < truncn x)%N = (n.+1%:R <= x). | Proof.
case: ifP (truncnP x) => [x0 _ | x0 /eqP->]; first by rewrite truncn_ge_nat.
by rewrite ltn0; apply/esym/(contraFF _ x0)/le_trans.
Qed. | Lemma | truncn_gt_nat | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"le_trans",
"ltn0",
"truncn",
"truncnP",
"truncn_ge_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncn_lt_nat x n : 0 <= x -> (truncn x < n)%N = (x < n%:R). | Proof. by move=> ?; rewrite real_ltNge ?ger0_real// ltnNge truncn_ge_nat. Qed. | Lemma | truncn_lt_nat | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ger0_real",
"ltnNge",
"real_ltNge",
"truncn",
"truncn_ge_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_truncn_le_nat x n :
x \is real_num -> (truncn x <= n)%N = (x < n.+1%:R). | Proof. by move=> ?; rewrite real_ltNge// leqNgt truncn_gt_nat. Qed. | Lemma | real_truncn_le_nat | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"leqNgt",
"real_ltNge",
"real_num",
"truncn",
"truncn_gt_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncn_eq x n : 0 <= x -> (truncn x == n) = (n%:R <= x < n.+1%:R). | Proof.
by move=> xr; apply/eqP/idP => [<-|]; [exact: truncn_itv|exact: truncn_def].
Qed. | Lemma | truncn_eq | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"truncn",
"truncn_def",
"truncn_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_truncn : {homo truncn : x y / x <= y >-> (x <= y)%N}. | Proof.
move=> x y lexy; move: (truncnP x) (truncnP y).
case: ifP => [x0 /andP[letx _] | x0 /eqP->//].
case: ifP => [y0 /andP[_] | y0 /eqP->]; [|by rewrite (le_trans x0 lexy) in y0].
by move=> /(le_lt_trans lexy) /(le_lt_trans letx); rewrite ltr_nat ltnS.
Qed. | Lemma | le_truncn | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"le_lt_trans",
"le_trans",
"ltnS",
"ltr_nat",
"truncn",
"truncnP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natrK : cancel (GRing.natmul 1) truncn. | Proof. by move=> m; apply: truncn_def; rewrite ler_nat ltr_nat ltnS leqnn. Qed. | Lemma | natrK | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"leqnn",
"ler_nat",
"ltnS",
"ltr_nat",
"natmul",
"truncn",
"truncn_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natrEtruncn x : (x \is a nat_num) = ((truncn x)%:R == x). | Proof.
by apply/natrP/eqP => [[n ->]|<-]; [rewrite natrK | exists (truncn x)].
Qed. | Lemma | natrEtruncn | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"nat_num",
"natrK",
"natrP",
"truncn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
archi_boundP x : 0 <= x -> x < (bound x)%:R. | Proof.
move=> x_ge0; case/truncn_itv/andP: (normr_ge0 x) => _.
exact/le_lt_trans/real_ler_norm/ger0_real.
Qed. | Lemma | archi_boundP | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"bound",
"ger0_real",
"le_lt_trans",
"normr_ge0",
"real_ler_norm",
"truncn_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncnK : {in nat_num, cancel truncn (GRing.natmul 1)}. | Proof. by move=> x; rewrite natrEtruncn => /eqP. Qed. | Lemma | truncnK | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"nat_num",
"natmul",
"natrEtruncn",
"truncn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncn0 : truncn 0 = 0%N. | Proof. exact: natrK 0%N. Qed. | Lemma | truncn0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"natrK",
"truncn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncn1 : truncn 1 = 1%N. | Proof. exact: natrK 1%N. Qed. | Lemma | truncn1 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"natrK",
"truncn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncnD :
{in nat_num & nneg_num, {morph truncn : x y / x + y >-> (x + y)%N}}. | Proof.
move=> _ y /natrP[n ->] y_ge0; apply: truncn_def.
by rewrite -addnS !natrD !natrK lerD2l ltrD2l truncn_itv.
Qed. | Lemma | truncnD | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"addnS",
"apply",
"lerD2l",
"ltrD2l",
"nat_num",
"natrD",
"natrK",
"natrP",
"nneg_num",
"truncn",
"truncn_def",
"truncn_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncnM : {in nat_num &, {morph truncn : x y / x * y >-> (x * y)%N}}. | Proof. by move=> _ _ /natrP[n1 ->] /natrP[n2 ->]; rewrite -natrM !natrK. Qed. | Lemma | truncnM | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"nat_num",
"natrK",
"natrM",
"natrP",
"truncn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncnX n : {in nat_num, {morph truncn : x / x ^+ n >-> (x ^ n)%N}}. | Proof. by move=> _ /natrP[n1 ->]; rewrite -natrX !natrK. Qed. | Lemma | truncnX | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"nat_num",
"natrK",
"natrP",
"natrX",
"truncn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncn_gt0 x : (0 < truncn x)%N = (1 <= x). | Proof.
case: ifP (truncnP x) => [x0 | x0 /eqP<-]; first by rewrite truncn_ge_nat.
by rewrite ltnn; apply/esym/(contraFF _ x0)/le_trans.
Qed. | Lemma | truncn_gt0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"le_trans",
"ltnn",
"truncn",
"truncnP",
"truncn_ge_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncn0Pn x : reflect (truncn x = 0%N) (~~ (1 <= x)). | Proof. by rewrite -truncn_gt0 -eqn0Ngt; apply: eqP. Qed. | Lemma | truncn0Pn | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"eqn0Ngt",
"truncn",
"truncn_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sum_truncnK I r (P : pred I) F : (forall i, P i -> F i \is a nat_num) ->
(\sum_(i <- r | P i) truncn (F i))%:R = \sum_(i <- r | P i) F i. | Proof. by rewrite natr_sum => natr; apply: eq_bigr => i /natr /truncnK. Qed. | Lemma | sum_truncnK | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"eq_bigr",
"nat_num",
"natr_sum",
"truncn",
"truncnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prod_truncnK I r (P : pred I) F : (forall i, P i -> F i \is a nat_num) ->
(\prod_(i <- r | P i) truncn (F i))%:R = \prod_(i <- r | P i) F i. | Proof. by rewrite natr_prod => natr; apply: eq_bigr => i /natr /truncnK. Qed. | Lemma | prod_truncnK | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"eq_bigr",
"nat_num",
"natr_prod",
"truncn",
"truncnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natr_sum_eq1 (I : finType) (P : pred I) (F : I -> R) :
(forall i, P i -> F i \is a nat_num) -> \sum_(i | P i) F i = 1 ->
{i : I | [/\ P i, F i = 1 & forall j, j != i -> P j -> F j = 0]}. | Proof.
move=> natF /eqP; rewrite -sum_truncnK// -[1]/1%:R eqr_nat => /sum_nat_eq1 exi.
have [i /and3P[Pi /eqP f1 /forallP a]] : {i : I | [&& P i, truncn (F i) == 1
& [forall j : I, ((j != i) ==> P j ==> (truncn (F j) == 0))]]}.
apply/sigW; have [i [Pi /eqP f1 a]] := exi; exists i; apply/and3P; split=> //.
by ap... | Lemma | natr_sum_eq1 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"eqr_nat",
"f1",
"forallP",
"nat_num",
"sigW",
"split",
"sum_nat_eq1",
"sum_truncnK",
"truncn",
"truncnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natr_mul_eq1 x y :
x \is a nat_num -> y \is a nat_num -> (x * y == 1) = (x == 1) && (y == 1). | Proof. by do 2!move/truncnK <-; rewrite -natrM !pnatr_eq1 muln_eq1. Qed. | Lemma | natr_mul_eq1 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"muln_eq1",
"nat_num",
"natrM",
"pnatr_eq1",
"truncnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natr_prod_eq1 (I : finType) (P : pred I) (F : I -> R) :
(forall i, P i -> F i \is a nat_num) -> \prod_(i | P i) F i = 1 ->
forall i, P i -> F i = 1. | Proof.
move=> natF /eqP; rewrite -prod_truncnK// -[1]/1%:R eqr_nat prod_nat_seq_eq1.
move/allP => a i Pi; apply/eqP; rewrite -[F i]truncnK ?natF// eqr_nat.
by apply: implyP Pi; apply: a; apply: mem_index_enum.
Qed. | Lemma | natr_prod_eq1 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"allP",
"apply",
"eqr_nat",
"mem_index_enum",
"nat_num",
"prod_nat_seq_eq1",
"prod_truncnK",
"truncnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
raddfZ_nat a u : a \is a nat_num -> f (a *: u) = a *: f u. | Proof. by move=> /natrP[n ->]; apply: raddfZnat. Qed. | Lemma | raddfZ_nat | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"nat_num",
"natrP",
"raddfZnat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rpredZ_nat (S : addrClosed V) :
{in nat_num & S, forall z u, z *: u \in S}. | Proof. by move=> _ u /natrP[n ->]; apply: rpredZnat. Qed. | Lemma | rpredZ_nat | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"addrClosed",
"apply",
"nat_num",
"natrP",
"rpredZnat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
raddfZ_int a u : a \is a int_num -> f (a *: u) = a *: f u. | Proof. by move=> /intrP[m ->]; rewrite !scaler_int raddfMz. Qed. | Lemma | raddfZ_int | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"int_num",
"intrP",
"raddfMz",
"scaler_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rpredZ_int (S : zmodClosed V) :
{in int_num & S, forall z u, z *: u \in S}. | Proof. by move=> _ u /intrP[m ->] ?; rewrite scaler_int rpredMz. Qed. | Lemma | rpredZ_int | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"int_num",
"intrP",
"rpredMz",
"scaler_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aut_natr nu : {in nat_num, nu =1 id}. | Proof. by move=> _ /natrP[n ->]; apply: rmorph_nat. Qed. | Lemma | aut_natr | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"id",
"nat_num",
"natrP",
"rmorph_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aut_intr nu : {in int_num, nu =1 id}. | Proof. by move=> _ /intrP[m ->]; apply: rmorph_int. Qed. | Lemma | aut_intr | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"id",
"int_num",
"intrP",
"rmorph_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trunc_itv | := truncn_itv (only parsing). | Notation | trunc_itv | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"truncn_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trunc_def | := truncn_def (only parsing). | Notation | trunc_def | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"truncn_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncK | := truncnK (only parsing). | Notation | truncK | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"truncnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trunc0 | := truncn0 (only parsing). | Notation | trunc0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"truncn0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trunc1 | := truncn1 (only parsing). | Notation | trunc1 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"truncn1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncD | := truncnD (only parsing). | Notation | truncD | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"truncnD"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncM | := truncnM (only parsing). | Notation | truncM | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"truncnM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncX | := truncnX (only parsing). | Notation | truncX | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"truncnX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trunc_gt0 | := truncn_gt0 (only parsing). | Notation | trunc_gt0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"truncn_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trunc0Pn | := truncn0Pn (only parsing). | Notation | trunc0Pn | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"truncn0Pn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sum_truncK | := sum_truncnK (only parsing). | Notation | sum_truncK | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"sum_truncnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
prod_truncK | := prod_truncnK (only parsing). | Notation | prod_truncK | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"prod_truncnK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trunc_floor | := truncn_floor (only parsing). | Notation | trunc_floor | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"truncn_floor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ge_floor | := real_floor_le (only parsing). | Notation | real_ge_floor | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"real_floor_le"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_lt_succ_floor | := real_floorD1_gt (only parsing). | Notation | real_lt_succ_floor | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"real_floorD1_gt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_gt_pred_ceil | := real_floorD1_gt (only parsing). | Notation | real_gt_pred_ceil | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"real_floorD1_gt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_le_ceil | := real_ceil_ge (only parsing). | Notation | real_le_ceil | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"real_ceil_ge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceil_le | := le_ceil (only parsing). | Notation | ceil_le | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"le_ceil"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natrE | := natrEtruncn (only parsing). | Notation | natrE | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"natrEtruncn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
le_ceil_tmp | := le_ceil (only parsing). | Notation | le_ceil_tmp | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"le_ceil"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_floor_ge_int_tmp | := real_floor_ge_int (only parsing). | Notation | real_floor_ge_int_tmp | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"real_floor_ge_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_ceil_le_int_tmp | := real_ceil_le_int (only parsing). | Notation | real_ceil_le_int_tmp | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"real_ceil_le_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
upper_nthrootP x i : (bound x <= i)%N -> x < 2%:R ^+ i. | Proof.
case/truncn_itv/andP: (normr_ge0 x) => _ /ltr_normlW xlt le_b_i.
by rewrite (lt_le_trans xlt) // -natrX ler_nat (ltn_trans le_b_i) // ltn_expl.
Qed. | Lemma | upper_nthrootP | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"bound",
"ler_nat",
"lt_le_trans",
"ltn_expl",
"ltn_trans",
"ltr_normlW",
"natrX",
"normr_ge0",
"truncn_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncnS_gt x : x < (truncn x).+1%:R. | Proof. exact: real_truncnS_gt. Qed. | Lemma | truncnS_gt | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"real_truncnS_gt",
"truncn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncn_le_nat x n : (truncn x <= n)%N = (x < n.+1%:R). | Proof. exact: real_truncn_le_nat. Qed. | Lemma | truncn_le_nat | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"real_truncn_le_nat",
"truncn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floor_itv x : (floor x)%:~R <= x < (floor x + 1)%:~R. | Proof. exact: real_floor_itv. Qed. | Lemma | floor_itv | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor",
"real_floor_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floor_le x : (floor x)%:~R <= x. | Proof. exact: real_floor_le. Qed. | Lemma | floor_le | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor",
"real_floor_le"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floorD1_gt x : x < (floor x + 1)%:~R. | Proof. exact: real_floorD1_gt. Qed. | Lemma | floorD1_gt | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor",
"real_floorD1_gt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floor_ge_int x n : (n <= floor x) = (n%:~R <= x). | Proof. exact: real_floor_ge_int. Qed. | Lemma | floor_ge_int | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor",
"real_floor_ge_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floor_lt_int x n : (floor x < n) = (x < n%:~R). | Proof. exact: real_floor_lt_int. Qed. | Lemma | floor_lt_int | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor",
"real_floor_lt_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floor_eq x n : (floor x == n) = (n%:~R <= x < (n + 1)%:~R). | Proof. exact: real_floor_eq. Qed. | Lemma | floor_eq | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor",
"real_floor_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floorDzr : {in @int_num R, {morph floor : x y / x + y}}. | Proof. by move=> x xz y; apply/real_floorDzr/num_real. Qed. | Lemma | floorDzr | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"floor",
"int_num",
"num_real",
"real_floorDzr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floorDrz x y : y \is a int_num -> floor (x + y) = floor x + floor y. | Proof. by move=> yz; apply/real_floorDrz/yz/num_real. Qed. | Lemma | floorDrz | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"floor",
"int_num",
"num_real",
"real_floorDrz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floor_ge0 x : (0 <= floor x) = (0 <= x). | Proof. exact: real_floor_ge0. Qed. | Lemma | floor_ge0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor",
"real_floor_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floor_le0 x : (floor x <= 0) = (x < 1). | Proof. exact: real_floor_le0. Qed. | Lemma | floor_le0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor",
"real_floor_le0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceil_itv x : (ceil x - 1)%:~R < x <= (ceil x)%:~R. | Proof. exact: real_ceil_itv. Qed. | Lemma | ceil_itv | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"real_ceil_itv"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceilB1_lt x : (ceil x - 1)%:~R < x. | Proof. exact: real_ceilB1_lt. Qed. | Lemma | ceilB1_lt | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"real_ceilB1_lt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceil_ge x : x <= (ceil x)%:~R. | Proof. exact: real_ceil_ge. Qed. | Lemma | ceil_ge | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"real_ceil_ge"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceil_le_int x n : (ceil x <= n) = (x <= n%:~R). | Proof. exact: real_ceil_le_int. Qed. | Lemma | ceil_le_int | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"real_ceil_le_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceil_gt_int x n : (n < ceil x) = (n%:~R < x). | Proof. exact: real_ceil_gt_int. Qed. | Lemma | ceil_gt_int | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"real_ceil_gt_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceil_eq x n : (ceil x == n) = ((n - 1)%:~R < x <= n%:~R). | Proof. exact: real_ceil_eq. Qed. | Lemma | ceil_eq | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"real_ceil_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceilDzr : {in @int_num R, {morph ceil : x y / x + y}}. | Proof. by move=> x xz y; apply/real_ceilDzr/num_real. Qed. | Lemma | ceilDzr | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"ceil",
"int_num",
"num_real",
"real_ceilDzr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceilDrz x y : y \is a int_num -> ceil (x + y) = ceil x + ceil y. | Proof. by move=> yz; apply/real_ceilDrz/yz/num_real. Qed. | Lemma | ceilDrz | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"ceil",
"int_num",
"num_real",
"real_ceilDrz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceil_ge0 x : (0 <= ceil x) = (-1 < x). | Proof. exact: real_ceil_ge0. Qed. | Lemma | ceil_ge0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"real_ceil_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceil_le0 x : (ceil x <= 0) = (x <= 0). | Proof. exact: real_ceil_le0. Qed. | Lemma | ceil_le0 | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"real_ceil_le0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceil_floor x : ceil x = floor x + (x \isn't a int_num). | Proof. exact: real_ceil_floor. Qed. | Lemma | ceil_floor | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil",
"floor",
"int_num",
"real_ceil_floor"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ge_floor | := floor_le (only parsing). | Notation | ge_floor | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor_le"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
lt_succ_floor | := floorD1_gt (only parsing). | Notation | lt_succ_floor | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floorD1_gt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gt_pred_ceil | := ceilB1_lt (only parsing). | Notation | gt_pred_ceil | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceilB1_lt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floor_le_tmp | := floor_le (only parsing). | Notation | floor_le_tmp | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor_le"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floor_ge_int_tmp | := floor_ge_int (only parsing). | Notation | floor_ge_int_tmp | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"floor_ge_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ceil_le_int_tmp | := ceil_le_int (only parsing). | Notation | ceil_le_int_tmp | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"ceil_le_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natr_aut x : (nu x \is a nat_num) = (x \is a nat_num). | Proof. by apply/idP/idP=> /[dup] ? /(aut_natr nu) => [/fmorph_inj <-| ->]. Qed. | Lemma | natr_aut | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"apply",
"aut_natr",
"fmorph_inj",
"nat_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intr_aut x : (nu x \is a int_num) = (x \is a int_num). | Proof. by rewrite !intrE -rmorphN !natr_aut. Qed. | Lemma | intr_aut | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"int_num",
"intrE",
"natr_aut",
"rmorphN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj_natr x : x \is a nat_num -> x^* = x. | Proof. by move/Rreal_nat/CrealP. Qed. | Lemma | conj_natr | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"CrealP",
"Rreal_nat",
"nat_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conj_intr x : x \is a int_num -> x^* = x. | Proof. by move/Rreal_int/CrealP. Qed. | Lemma | conj_intr | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"CrealP",
"Rreal_int",
"int_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Znat_def (n : int) : (n \is a nat_num) = (0 <= n). | Proof. by []. Qed. | Lemma | Znat_def | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"int",
"nat_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ZnatP (m : int) : reflect (exists n : nat, m = n) (m \is a nat_num). | Proof. by case: m => m; constructor; [exists m | case]. Qed. | Lemma | ZnatP | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"int",
"nat",
"nat_num"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
NumDomain_isArchimedean R | := (NumDomain_hasTruncn R) (only parsing). | Notation | NumDomain_isArchimedean | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Build T U | := (NumDomain_hasTruncn.Build T U) (only parsing). | Notation | Build | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncn_itv x : 0 <= x -> (truncn x)%:R <= x < (truncn x).+1%:R. | Proof. by move=> x_ge0; move: (truncn_subproof x); rewrite x_ge0. Qed. | Fact | truncn_itv | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"truncn",
"truncn_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floor (x : R) : int | :=
if 0 <= x then Posz (truncn x)
else if x < 0 then - Posz (truncn (- x) + ~~ int_num x) else 0. | Definition | floor | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"Posz",
"int",
"int_num",
"truncn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
floor_subproof x :
if x \is real_num then (floor x)%:~R <= x < (floor x + 1)%:~R else floor x == 0. | Proof.
rewrite /floor intrE !natrE negb_or realE.
case: (comparableP x 0) (@truncn_itv x) => //=;
try by rewrite -PoszD addn1 -pmulrn => _ ->.
move=> x_lt0 _; move: (truncn_subproof x); rewrite lt_geF // => /eqP ->.
rewrite gt_eqF //=; move: x_lt0.
rewrite [_ + 1]addrC -opprB !intrN lerNl ltrNr andbC -oppr_gt0.
move:... | Fact | floor_subproof | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"Posz",
"PoszD",
"addn0",
"addn1",
"addrA",
"addrC",
"comparableP",
"eqVneq",
"floor",
"gt_eqF",
"gtrBl",
"intrB",
"intrE",
"intrN",
"last",
"le_eqVlt",
"lerNl",
"lexx",
"ltW",
"lt_geF",
"ltrNr",
"mulr1z",
"natrE",
"opprB",
"oppr_gt0",
"pmulrn",
"realE",
"real_n... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
truncnE x : truncn x = if floor x is Posz n then n else 0. | Proof.
rewrite /floor.
case: (comparableP x 0) (truncn_subproof x) => [+ /eqP ->| |_ /eqP ->|] //=.
by case: (_ + _)%N.
Qed. | Fact | truncnE | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"Posz",
"comparableP",
"floor",
"truncn",
"truncn_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
intrP x : reflect (exists n, x = n%:~R) (int_num x). | Proof.
rewrite intrE !natrE; apply: (iffP idP) => [|[n ->]]; last first.
by case: n => n; rewrite ?NegzE ?opprK natrK eqxx // orbT.
rewrite -eqr_oppLR !pmulrn -intrN.
by move=> /orP[] /eqP<-; [exists (truncn x) | exists (- Posz (truncn (- x)))].
Qed. | Fact | intrP | algebra | algebra/archimedean.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"ssrint",
"Order.TTheory",
... | [
"NegzE",
"Posz",
"apply",
"eqr_oppLR",
"eqxx",
"int_num",
"intrE",
"intrN",
"last",
"natrE",
"natrK",
"opprK",
"pmulrn",
"truncn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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