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 |
|---|---|---|---|---|---|---|---|---|---|---|
rootCMr n x z : 0 <= x -> n.-root (z * x) = n.-root z * n.-root x. | Proof. by move=> x_ge0; rewrite mulrC rootCMl // mulrC. Qed. | Lemma | rootCMr | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"mulrC",
"root",
"rootCMl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
imaginaryCE : 'i = sqrtC (-1). | Proof.
have : sqrtC (-1) ^+ 2 - 'i ^+ 2 == 0 by rewrite sqrCi rootCK // subrr.
rewrite subr_sqr mulf_eq0 subr_eq0 addr_eq0; have [//|_/= /eqP sCN1E] := eqP.
by have := @Im_rootC_ge0 2 (-1) isT; rewrite sCN1E raddfN /= Im_i ler0N1.
Qed. | Lemma | imaginaryCE | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"Im_i",
"Im_rootC_ge0",
"addr_eq0",
"ler0N1",
"mulf_eq0",
"raddfN",
"rootCK",
"sqrCi",
"sqrtC",
"subr_eq0",
"subr_sqr",
"subrr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leif_rootC_AGM (I : finType) (A : {pred I}) (n := #|A|) E :
{in A, forall i, 0 <= E i} ->
n.-root (\prod_(i in A) E i) <= (\sum_(i in A) E i) / n%:R
?= iff [forall i in A, forall j in A, E i == E j]. | Proof.
move=> Ege0; have [n0 | n_gt0] := posnP n.
rewrite n0 root0C invr0 mulr0; apply/leif_refl/forall_inP=> i.
by rewrite (card0_eq n0).
rewrite -(mono_in_leif (ler_pXn2r n_gt0)) ?rootCK //=.
- by rewrite qualifE /= rootC_ge0 // prodr_ge0.
- by rewrite rpred_div ?rpred_nat ?rpred_sum.
exact: leif_AGM.
Qed. | Lemma | leif_rootC_AGM | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"card0_eq",
"forall_inP",
"invr0",
"leif_AGM",
"leif_refl",
"ler_pXn2r",
"mono_in_leif",
"mulr0",
"n_gt0",
"posnP",
"prodr_ge0",
"root",
"root0C",
"rootCK",
"rootC_ge0",
"rpred_div",
"rpred_nat",
"rpred_sum"
] | The proper form of the Arithmetic - Geometric Mean inequality. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
sqrtC0 : sqrtC 0 = 0. | Proof. exact: rootC0. Qed. | Lemma | sqrtC0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"rootC0",
"sqrtC"
] | Square root. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
sqrtC1 : sqrtC 1 = 1. | Proof. exact: rootC1. Qed. | Lemma | sqrtC1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"rootC1",
"sqrtC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtCK x : sqrtC x ^+ 2 = x. | Proof. exact: rootCK. Qed. | Lemma | sqrtCK | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"rootCK",
"sqrtC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrCK x : 0 <= x -> sqrtC (x ^+ 2) = x. | Proof. exact: exprCK. Qed. | Lemma | sqrCK | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"exprCK",
"sqrtC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtC_ge0 x : (0 <= sqrtC x) = (0 <= x). | Proof. exact: rootC_ge0. Qed. | Lemma | sqrtC_ge0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"rootC_ge0",
"sqrtC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtC_eq0 x : (sqrtC x == 0) = (x == 0). | Proof. exact: rootC_eq0. Qed. | Lemma | sqrtC_eq0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"rootC_eq0",
"sqrtC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtC_gt0 x : (sqrtC x > 0) = (x > 0). | Proof. exact: rootC_gt0. Qed. | Lemma | sqrtC_gt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"rootC_gt0",
"sqrtC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtC_lt0 x : (sqrtC x < 0) = false. | Proof. exact: rootC_lt0. Qed. | Lemma | sqrtC_lt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"rootC_lt0",
"sqrtC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtC_le0 x : (sqrtC x <= 0) = (x == 0). | Proof. exact: rootC_le0. Qed. | Lemma | sqrtC_le0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"rootC_le0",
"sqrtC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ler_sqrtC : {in Num.nneg &, {mono sqrtC : x y / x <= y}}. | Proof. exact: ler_rootC. Qed. | Lemma | ler_sqrtC | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ler_rootC",
"nneg",
"sqrtC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ltr_sqrtC : {in Num.nneg &, {mono sqrtC : x y / x < y}}. | Proof. exact: ltr_rootC. Qed. | Lemma | ltr_sqrtC | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ltr_rootC",
"nneg",
"sqrtC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqr_sqrtC : {mono sqrtC : x y / x == y}. | Proof. exact: eqr_rootC. Qed. | Lemma | eqr_sqrtC | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"eqr_rootC",
"sqrtC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtC_inj : injective sqrtC. | Proof. exact: rootC_inj. Qed. | Lemma | sqrtC_inj | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"rootC_inj",
"sqrtC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtCM : {in Num.nneg &, {morph sqrtC : x y / x * y}}. | Proof. by move=> x y _; apply: rootCMr. Qed. | Lemma | sqrtCM | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"nneg",
"rootCMr",
"sqrtC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrtC_real x : 0 <= x -> sqrtC x \in Num.real. | Proof. by rewrite -sqrtC_ge0; apply: ger0_real. Qed. | Lemma | sqrtC_real | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"ger0_real",
"real",
"sqrtC",
"sqrtC_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqrCK_P x : reflect (sqrtC (x ^+ 2) = x) ((0 <= 'Im x) && ~~ (x < 0)). | Proof.
apply: (iffP andP) => [[leI0x not_gt0x] | <-]; last first.
by rewrite sqrtC_lt0 Im_rootC_ge0.
have /eqP := sqrtCK (x ^+ 2); rewrite eqf_sqr => /pred2P[] // defNx.
apply: sqrCK; rewrite -real_leNgt ?rpred0 // in not_gt0x;
apply/Creal_ImP/le_anti;
by rewrite leI0x -oppr_ge0 -raddfN -defNx Im_rootC_ge0.
Qed. | Lemma | sqrCK_P | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"Creal_ImP",
"Im",
"Im_rootC_ge0",
"apply",
"eqf_sqr",
"last",
"le_anti",
"oppr_ge0",
"pred2P",
"raddfN",
"real_leNgt",
"rpred0",
"sqrCK",
"sqrtC",
"sqrtCK",
"sqrtC_lt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normC_def x : `|x| = sqrtC (x * x^* ). | Proof. by rewrite -normCK sqrCK. Qed. | Lemma | normC_def | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"normCK",
"sqrCK",
"sqrtC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
norm_conjC x : `|x^*| = `|x|. | Proof. by rewrite !normC_def conjCK mulrC. Qed. | Lemma | norm_conjC | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"conjCK",
"mulrC",
"normC_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normC_rect :
{in real &, forall x y, `|x + 'i * y| = sqrtC (x ^+ 2 + y ^+ 2)}. | Proof. by move=> x y Rx Ry; rewrite /= normC_def -normCK normC2_rect. Qed. | Lemma | normC_rect | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"normC2_rect",
"normCK",
"normC_def",
"real",
"sqrtC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normC_Re_Im z : `|z| = sqrtC ('Re z ^+ 2 + 'Im z ^+ 2). | Proof. by rewrite normC_def -normCK normC2_Re_Im. Qed. | Lemma | normC_Re_Im | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"Im",
"Re",
"normC2_Re_Im",
"normCK",
"normC_def",
"sqrtC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normCDeq x y :
`|x + y| = `|x| + `|y| ->
{t : C | `|t| == 1 & (x, y) = (`|x| * t, `|y| * t)}. | Proof.
move=> lin_xy; apply: sig2_eqW; pose u z := if z == 0 then 1 else z / `|z|.
have uE z: (`|u z| = 1) * (`|z| * u z = z).
rewrite /u; have [->|nz_z] := eqVneq; first by rewrite normr0 normr1 mul0r.
by rewrite normf_div normr_id mulrCA divff ?mulr1 ?normr_eq0.
have [->|nz_x] := eqVneq x 0; first by exists (u y)... | Lemma | normCDeq | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"AC",
"addIr",
"addNr",
"addrA",
"addrI",
"apply",
"divfK",
"divff",
"eqVneq",
"expf_eq0",
"exprMn",
"exprMn_n",
"invC_norm",
"invfM",
"mul",
"mul0r",
"mulIf",
"mulNrn",
"mulr1",
"mulrA",
"mulrAC",
"mulrC",
"mulrCA",
"mulrDl",
"mulrDr",
"mulrN",
"mulr_natr",
"no... | Norm sum (in)equalities. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
normC_sum_eq (I : finType) (P : pred I) (F : I -> C) :
`|\sum_(i | P i) F i| = \sum_(i | P i) `|F i| ->
{t : C | `|t| == 1 & forall i, P i -> F i = `|F i| * t}. | Proof.
have [i /andP[Pi nzFi] | F0] := pickP [pred i | P i & F i != 0]; last first.
exists 1 => [|i Pi]; first by rewrite normr1.
by case/nandP: (F0 i) => [/negP[]// | /negbNE/eqP->]; rewrite normr0 mul0r.
rewrite !(bigD1 i Pi) /= => norm_sumF; pose Q j := P j && (j != i).
rewrite -normr_eq0 in nzFi; set c := F i /... | Lemma | normC_sum_eq | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"F0",
"addrA",
"apply",
"bigD1",
"divfK",
"divff",
"eq_le",
"last",
"le_trans",
"lerD2l",
"lerD2r",
"ler_normD",
"ler_norm_sum",
"mul0r",
"mulKf",
"mulrC",
"normCDeq",
"normfV",
"normr0",
"normr1",
"normrM",
"normr_eq0",
"normr_id",
"pickP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normC_sum_eq1 (I : finType) (P : pred I) (F : I -> C) :
`|\sum_(i | P i) F i| = (\sum_(i | P i) `|F i|) ->
(forall i, P i -> `|F i| = 1) ->
{t : C | `|t| == 1 & forall i, P i -> F i = t}. | Proof.
case/normC_sum_eq=> t t1 defF normF.
by exists t => // i Pi; rewrite defF // normF // mul1r.
Qed. | Lemma | normC_sum_eq1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"mul1r",
"normC_sum_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normC_sum_upper (I : finType) (P : pred I) (F G : I -> C) :
(forall i, P i -> `|F i| <= G i) ->
\sum_(i | P i) F i = \sum_(i | P i) G i ->
forall i, P i -> F i = G i. | Proof.
set sumF := \sum_(i | _) _; set sumG := \sum_(i | _) _ => leFG eq_sumFG.
have posG i: P i -> 0 <= G i by move/leFG; apply: le_trans.
have norm_sumG: `|sumG| = sumG by rewrite ger0_norm ?sumr_ge0.
have norm_sumF: `|sumF| = \sum_(i | P i) `|F i|.
apply/eqP; rewrite eq_le ler_norm_sum eq_sumFG norm_sumG -subr_ge0... | Lemma | normC_sum_upper | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"add0r",
"apply",
"eqVneq",
"eq_bigr",
"eq_le",
"ger0_norm",
"le_trans",
"ler_norm_sum",
"mulfI",
"mulr1",
"mulr_suml",
"normC_sum_eq",
"normr_eq0",
"normr_ge0",
"psumr_eq0P",
"subrK",
"subr_ge0",
"subrr",
"sumrB",
"sumr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normCBeq x y :
`|x - y| = `|x| - `|y| -> {t | `|t| == 1 & (x, y) = (`|x| * t, `|y| * t)}. | Proof.
set z := x - y; rewrite -(subrK y x) -/z => /(canLR (subrK _))/esym-Dx.
have [t t_1 [Dz Dy]] := normCDeq Dx.
by exists t; rewrite // Dx mulrDl -Dz -Dy.
Qed. | Lemma | normCBeq | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"Dx",
"mulrDl",
"normCDeq",
"subrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"n .-root" | := (@nthroot _ n). | Notation | n .-root | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"nthroot"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r1 | := (- b - sqrtC delta) / (2 * a). | Let | r1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"delta",
"sqrtC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r2 | := (- b + sqrtC delta) / (2 * a). | Let | r2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"delta",
"sqrtC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_factor : p = a *: ('X - r1%:P) * ('X - r2%:P). | Proof. by apply: deg2_poly_factor; rewrite ?pnatr_eq0// sqrtCK. Qed. | Lemma | deg2_poly_factor | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"pnatr_eq0",
"r1",
"r2",
"sqrtCK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_root1 : root p r1. | Proof. by apply: deg2_poly_root1; rewrite ?pnatr_eq0// sqrtCK. Qed. | Lemma | deg2_poly_root1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"pnatr_eq0",
"r1",
"root",
"sqrtCK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_root2 : root p r2. | Proof. by apply: deg2_poly_root2; rewrite ?pnatr_eq0// sqrtCK. Qed. | Lemma | deg2_poly_root2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"pnatr_eq0",
"r2",
"root",
"sqrtCK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r1 | := (- b - sqrtC delta) / 2. | Let | r1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"delta",
"sqrtC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r2 | := (- b + sqrtC delta) / 2. | Let | r2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"delta",
"sqrtC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_factor : p = ('X - r1%:P) * ('X - r2%:P). | Proof. by apply: deg2_poly_factor; rewrite ?pnatr_eq0// sqrtCK. Qed. | Lemma | deg2_poly_factor | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"pnatr_eq0",
"r1",
"r2",
"sqrtCK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
age0 : 0 <= a. | Hypothesis | age0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
agt0 : 0 < a. | Proof. by rewrite lt_def aneq0. Qed. | Let | agt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"aneq0",
"lt_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
a4gt0 : 0 < 4 * a. | Proof. by rewrite mulr_gt0 ?ltr0n. Qed. | Let | a4gt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ltr0n",
"mulr_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_min x : p.[- b / (2 * a)] <= p.[x]. | Proof.
rewrite [p]deg2_poly_canonical ?pnatr_eq0// -/a -/b -/c /delta !hornerE/=.
by rewrite ler_pM2l// lerD2r mulNr addNr expr0n sqr_ge0.
Qed. | Lemma | deg2_poly_min | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"addNr",
"deg2_poly_canonical",
"delta",
"expr0n",
"hornerE",
"lerD2r",
"ler_pM2l",
"mulNr",
"pnatr_eq0",
"sqr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_minE : p.[- b / (2 * a)] = - delta / (4 * a). | Proof.
rewrite [p]deg2_poly_canonical ?pnatr_eq0// -/a -/b -/c -/delta !hornerE/=.
by rewrite mulNr addNr expr0n add0r -mulNr mulrC -[LHS]mulrA invfM divfK.
Qed. | Lemma | deg2_poly_minE | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"add0r",
"addNr",
"deg2_poly_canonical",
"delta",
"divfK",
"expr0n",
"hornerE",
"invfM",
"mulNr",
"mulrA",
"mulrC",
"pnatr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_gt0 : reflect (forall x, 0 < p.[x]) (delta < 0). | Proof.
apply/(iffP idP) => [dlt0 x | /(_ (- b / (2 * a)))]; last first.
by rewrite deg2_poly_minE ltr_pdivlMr// mul0r oppr_gt0.
apply: lt_le_trans (deg2_poly_min _).
by rewrite deg2_poly_minE ltr_pdivlMr// mul0r oppr_gt0.
Qed. | Lemma | deg2_poly_gt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"deg2_poly_min",
"deg2_poly_minE",
"delta",
"last",
"lt_le_trans",
"ltr_pdivlMr",
"mul0r",
"oppr_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_ge0 : reflect (forall x, 0 <= p.[x]) (delta <= 0). | Proof.
apply/(iffP idP) => [dlt0 x | /(_ (- b / (2 * a)))]; last first.
by rewrite deg2_poly_minE ler_pdivlMr// mul0r oppr_ge0.
apply: le_trans (deg2_poly_min _).
by rewrite deg2_poly_minE ler_pdivlMr// mul0r oppr_ge0.
Qed. | Lemma | deg2_poly_ge0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"deg2_poly_min",
"deg2_poly_minE",
"delta",
"last",
"le_trans",
"ler_pdivlMr",
"mul0r",
"oppr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ale0 : a <= 0. | Hypothesis | ale0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
degpN : size (- p) = 3. | Proof. by rewrite size_polyN. Qed. | Let | degpN | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"size",
"size_polyN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
b2a : - (- p)`_1 / (2 * (- p)`_2) = - b / (2 * a). | Proof. by rewrite !coefN mulrN divrNN. Qed. | Let | b2a | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"coefN",
"divrNN",
"mulrN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deltaN : (- p)`_1 ^+ 2 - 4 * (- p)`_2 * (- p)`_0 = delta. | Proof. by rewrite !coefN sqrrN -mulrN opprK mulrN mulNr. Qed. | Let | deltaN | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"coefN",
"delta",
"mulNr",
"mulrN",
"opprK",
"sqrrN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_max x : p.[x] <= p.[- b / (2 * a)]. | Proof. by rewrite -lerN2 -!hornerN -b2a deg2_poly_min// coefN oppr_ge0. Qed. | Lemma | deg2_poly_max | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"b2a",
"coefN",
"deg2_poly_min",
"hornerN",
"lerN2",
"oppr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_maxE : p.[- b / (2 * a)] = - delta / (4 * a). | Proof.
apply/eqP; rewrite [eqbRHS]mulNr -eqr_oppLR -hornerN -b2a.
by rewrite deg2_poly_minE// deltaN coefN mulrN divrNN.
Qed. | Lemma | deg2_poly_maxE | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"b2a",
"coefN",
"deg2_poly_minE",
"delta",
"deltaN",
"divrNN",
"eqbRHS",
"eqr_oppLR",
"hornerN",
"mulNr",
"mulrN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_lt0 : reflect (forall x, p.[x] < 0) (delta < 0). | Proof.
rewrite -deltaN; apply/(iffP (deg2_poly_gt0 _ _)); rewrite ?coefN ?oppr_ge0//.
- by move=> gt0 x; rewrite -oppr_gt0 -hornerN gt0.
- by move=> lt0 x; rewrite hornerN oppr_gt0 lt0.
Qed. | Lemma | deg2_poly_lt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"coefN",
"deg2_poly_gt0",
"delta",
"deltaN",
"gt0",
"hornerN",
"lt0",
"oppr_ge0",
"oppr_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_le0 : reflect (forall x, p.[x] <= 0) (delta <= 0). | Proof.
rewrite -deltaN; apply/(iffP (deg2_poly_ge0 _ _)); rewrite ?coefN ?oppr_ge0//.
- by move=> ge0 x; rewrite -oppr_ge0 -hornerN ge0.
- by move=> le0 x; rewrite hornerN oppr_ge0 le0.
Qed. | Lemma | deg2_poly_le0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"coefN",
"deg2_poly_ge0",
"delta",
"deltaN",
"ge0",
"hornerN",
"le0",
"oppr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sqa2 : 4 * a ^+ 2 = (2 * a) ^+ 2. | Proof. by rewrite exprMn -natrX. Qed. | Let | sqa2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"exprMn",
"natrX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nz2 : 2 != 0 :> F. | Proof. by rewrite pnatr_eq0. Qed. | Let | nz2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"pnatr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r1 | := (- b - sqrt delta) / (2 * a). | Let | r1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"delta",
"sqrt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r2 | := (- b + sqrt delta) / (2 * a). | Let | r2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"delta",
"sqrt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_factor : 0 <= delta -> p = a *: ('X - r1%:P) * ('X - r2%:P). | Proof. by move=> dge0; apply: deg2_poly_factor; rewrite ?sqr_sqrtr. Qed. | Lemma | deg2_poly_factor | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"delta",
"r1",
"r2",
"sqr_sqrtr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_root1 : 0 <= delta -> root p r1. | Proof. by move=> dge0; apply: deg2_poly_root1; rewrite ?sqr_sqrtr. Qed. | Lemma | deg2_poly_root1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"delta",
"r1",
"root",
"sqr_sqrtr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_root2 : 0 <= delta -> root p r2. | Proof. by move=> dge0; apply: deg2_poly_root2; rewrite ?sqr_sqrtr. Qed. | Lemma | deg2_poly_root2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"delta",
"r2",
"root",
"sqr_sqrtr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_noroot : reflect (forall x, ~~ root p x) (delta < 0). | Proof.
apply/(iffP idP) => [dlt0 x | /(_ r1)].
case: ltgtP aneq0 => [agt0 _|alt0 _|//]; rewrite rootE; last first.
exact/lt0r_neq0/(deg2_poly_gt0 degp (ltW alt0)).
rewrite -oppr_eq0 -hornerN.
apply/lt0r_neq0/deg2_poly_gt0; rewrite ?size_polyN ?coefN ?oppr_ge0 ?ltW//.
by rewrite sqrrN -mulrA mulrNN mulrA.
by... | Lemma | deg2_poly_noroot | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"agt0",
"aneq0",
"apply",
"coefN",
"deg2_poly_gt0",
"deg2_poly_root1",
"degp",
"delta",
"hornerN",
"last",
"lt0r_neq0",
"ltNge",
"ltW",
"ltgtP",
"mulrA",
"mulrNN",
"oppr_eq0",
"oppr_ge0",
"r1",
"root",
"rootE",
"size_polyN",
"sqrrN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
a2gt0 : 0 < 2 * a. | Proof. by rewrite mulr_gt0 ?ltr0n. Qed. | Let | a2gt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ltr0n",
"mulr_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aa4gt0 : 0 < 4 * a * a. | Proof. by rewrite mulr_gt0 ?ltr0n. Qed. | Let | aa4gt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ltr0n",
"mulr_gt0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xb4 x : (x + b / (2 * a)) ^+ 2 * (4 * a * a) = (x * (2 * a) + b) ^+ 2. | Proof.
have -> : 4 * a * a = (2 * a) ^+ 2 by rewrite expr2 mulrACA -natrM mulrA.
by rewrite -exprMn mulrDl mulfVK ?mulf_neq0 ?pnatr_eq0.
Qed. | Let | xb4 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"expr2",
"exprMn",
"mulfVK",
"mulf_neq0",
"mulrA",
"mulrACA",
"mulrDl",
"natrM",
"pnatr_eq0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_gt0l x : x < r1 -> 0 < p.[x]. | Proof.
move=> xltr1; have [? | dge0] := ltP delta 0; first exact: deg2_poly_gt0.
have {}xltr1 : sqrt delta < - (x * (2 * a) + b).
by rewrite ltrNr -ltrBrDr addrC -ltr_pdivlMr.
rewrite [p]deg2_poly_canonical// -/a -/b -/c -/delta !hornerE/=.
rewrite mulr_gt0// subr_gt0 ltr_pdivrMr// xb4 -sqrrN.
rewrite -ltr_sqrt ?sqrt... | Lemma | deg2_poly_gt0l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"addrC",
"deg2_poly_canonical",
"deg2_poly_gt0",
"delta",
"exprn_gt0",
"hornerE",
"le_lt_trans",
"ler_norm",
"ltP",
"lt_le_trans",
"ltrBrDr",
"ltrNr",
"ltr_pdivlMr",
"ltr_pdivrMr",
"ltr_sqrt",
"mulr_gt0",
"r1",
"sqrrN",
"sqrt",
"sqrtr_ge0",
"sqrtr_sqr",
"subr_gt0",
"xb4"
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_gt0r x : r2 < x -> 0 < p.[x]. | Proof.
move=> xgtr2; have [? | dge0] := ltP delta 0; first exact: deg2_poly_gt0.
have {}xgtr2 : sqrt delta < x * (2 * a) + b.
by rewrite -ltrBlDr addrC -ltr_pdivrMr.
rewrite [p]deg2_poly_canonical// -/a -/b -/c -/delta !hornerE/=.
rewrite mulr_gt0// subr_gt0 ltr_pdivrMr// xb4.
rewrite -ltr_sqrt ?sqrtr_sqr ?(lt_le_tra... | Lemma | deg2_poly_gt0r | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"addrC",
"deg2_poly_canonical",
"deg2_poly_gt0",
"delta",
"exprn_gt0",
"hornerE",
"le_lt_trans",
"ler_norm",
"ltP",
"lt_le_trans",
"ltrBlDr",
"ltr_pdivrMr",
"ltr_sqrt",
"mulr_gt0",
"r2",
"sqrt",
"sqrtr_ge0",
"sqrtr_sqr",
"subr_gt0",
"xb4"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_lt0m x : r1 < x < r2 -> p.[x] < 0. | Proof.
move=> /andP[r1ltx xltr2].
have [dle0 | dgt0] := leP delta 0.
by move: (lt_trans r1ltx xltr2); rewrite /r1 /r2 ler0_sqrtr// oppr0 ltxx.
rewrite [p]deg2_poly_canonical// !hornerE/= -/a -/b -/c -/delta.
rewrite pmulr_rlt0// subr_lt0 ltr_pdivlMr// xb4 -ltr_sqrt// sqrtr_sqr ltr_norml.
by rewrite -ltrBlDr addrC -lt... | Lemma | deg2_poly_lt0m | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"addrC",
"deg2_poly_canonical",
"delta",
"hornerE",
"leP",
"ler0_sqrtr",
"lt_trans",
"ltrBlDr",
"ltrBrDr",
"ltr_norml",
"ltr_pdivlMr",
"ltr_pdivrMr",
"ltr_sqrt",
"ltxx",
"oppr0",
"pmulr_rlt0",
"r1",
"r2",
"sqrtr_sqr",
"subr_lt0",
"xb4"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_ge0l x : x <= r1 -> 0 <= p.[x]. | Proof.
rewrite le_eqVlt => /orP[/eqP->|xltr1]; last exact/ltW/deg2_poly_gt0l.
have [dge0|dlt0] := leP 0 delta; last by apply: deg2_poly_ge0 => //; apply: ltW.
by rewrite le_eqVlt (rootP (deg2_poly_root1 dge0)) eqxx.
Qed. | Lemma | deg2_poly_ge0l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"deg2_poly_ge0",
"deg2_poly_gt0l",
"deg2_poly_root1",
"delta",
"eqxx",
"last",
"leP",
"le_eqVlt",
"ltW",
"r1",
"rootP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_ge0r x : r2 <= x -> 0 <= p.[x]. | Proof.
rewrite le_eqVlt => /orP[/eqP<-|xgtr2]; last exact/ltW/deg2_poly_gt0r.
have [dge0|dlt0] := leP 0 delta; last by apply: deg2_poly_ge0 => //; apply: ltW.
by rewrite le_eqVlt (rootP (deg2_poly_root2 dge0)) eqxx.
Qed. | Lemma | deg2_poly_ge0r | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"deg2_poly_ge0",
"deg2_poly_gt0r",
"deg2_poly_root2",
"delta",
"eqxx",
"last",
"leP",
"le_eqVlt",
"ltW",
"r2",
"rootP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_le0m x : 0 <= delta -> r1 <= x <= r2 -> p.[x] <= 0. | Proof.
move=> dge0; rewrite le_eqVlt andb_orl => /orP[/andP[/eqP<- _]|].
by rewrite le_eqVlt (rootP (deg2_poly_root1 dge0)) eqxx.
rewrite le_eqVlt andb_orr => /orP[/andP[_ /eqP->]|].
by rewrite le_eqVlt (rootP (deg2_poly_root2 dge0)) eqxx.
by move=> ?; apply/ltW/deg2_poly_lt0m.
Qed. | Lemma | deg2_poly_le0m | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"deg2_poly_lt0m",
"deg2_poly_root1",
"deg2_poly_root2",
"delta",
"eqxx",
"le_eqVlt",
"ltW",
"r1",
"r2",
"rootP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r1 | := (- b + sqrt delta) / (2 * a). | Let | r1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"delta",
"sqrt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r2 | := (- b - sqrt delta) / (2 * a). | Let | r2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"delta",
"sqrt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aNge0 : 0 <= (- p)`_2. | Proof. by rewrite coefN oppr_ge0. Qed. | Let | aNge0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"coefN",
"oppr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r1N : (- (- p)`_1 - sqrt delta) / (2 * (- p)`_2) = r1. | Proof. by rewrite !coefN -opprD mulrN divrNN. Qed. | Let | r1N | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"coefN",
"delta",
"divrNN",
"mulrN",
"opprD",
"r1",
"sqrt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r2N : (- (- p)`_1 + sqrt delta) / (2 * (- p)`_2) = r2. | Proof. by rewrite !coefN mulrN divrN -mulNr opprK opprD. Qed. | Let | r2N | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"coefN",
"delta",
"divrN",
"mulNr",
"mulrN",
"opprD",
"opprK",
"r2",
"sqrt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_lt0l x : x < r1 -> p.[x] < 0. | Proof. by move=> ?; rewrite -oppr_gt0 -hornerN deg2_poly_gt0l// deltaN r1N. Qed. | Lemma | deg2_poly_lt0l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"deg2_poly_gt0l",
"deltaN",
"hornerN",
"oppr_gt0",
"r1",
"r1N"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_lt0r x : r2 < x -> p.[x] < 0. | Proof. by move=> ?; rewrite -oppr_gt0 -hornerN deg2_poly_gt0r// deltaN r2N. Qed. | Lemma | deg2_poly_lt0r | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"deg2_poly_gt0r",
"deltaN",
"hornerN",
"oppr_gt0",
"r2",
"r2N"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_gt0m x : r1 < x < r2 -> 0 < p.[x]. | Proof.
by move=> ?; rewrite -oppr_lt0 -hornerN deg2_poly_lt0m// deltaN r1N r2N.
Qed. | Lemma | deg2_poly_gt0m | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"deg2_poly_lt0m",
"deltaN",
"hornerN",
"oppr_lt0",
"r1",
"r1N",
"r2",
"r2N"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_le0l x : x <= r1 -> p.[x] <= 0. | Proof. by move=> ?; rewrite -oppr_ge0 -hornerN deg2_poly_ge0l// deltaN r1N. Qed. | Lemma | deg2_poly_le0l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"deg2_poly_ge0l",
"deltaN",
"hornerN",
"oppr_ge0",
"r1",
"r1N"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_le0r x : r2 <= x -> p.[x] <= 0. | Proof. by move=> ?; rewrite -oppr_ge0 -hornerN deg2_poly_ge0r// deltaN r2N. Qed. | Lemma | deg2_poly_le0r | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"deg2_poly_ge0r",
"deltaN",
"hornerN",
"oppr_ge0",
"r2",
"r2N"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_ge0m x : 0 <= delta -> r1 <= x <= r2 -> 0 <= p.[x]. | Proof.
by move=> ? ?; rewrite -oppr_le0 -hornerN deg2_poly_le0m ?deltaN// r1N r2N.
Qed. | Lemma | deg2_poly_ge0m | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"deg2_poly_le0m",
"delta",
"deltaN",
"hornerN",
"oppr_le0",
"r1",
"r1N",
"r2",
"r2N"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
a2 : 2 * a = 2. | Proof. by rewrite a1 mulr1. Qed. | Let | a2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"a1",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
a4 : 4 * a = 4. | Proof. by rewrite a1 mulr1. Qed. | Let | a4 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"a1",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_min x : p.[- b / 2] <= p.[x]. | Proof. by rewrite -a2 deg2_poly_min -/a ?a1 ?ler01. Qed. | Lemma | deg2_poly_min | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"a1",
"a2",
"ler01"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deltam : delta = b ^+ 2 - 4 * a * c. | Proof. by rewrite a1 mulr1. Qed. | Let | deltam | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"a1",
"delta",
"mulr1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_minE : p.[- b / 2] = - delta / 4. | Proof. by rewrite -a2 -a4 deltam deg2_poly_minE. Qed. | Lemma | deg2_poly_minE | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"a2",
"a4",
"delta",
"deltam"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_gt0 : reflect (forall x, 0 < p.[x]) (delta < 0). | Proof. by rewrite deltam; apply: deg2_poly_gt0; rewrite // -/a a1 ler01. Qed. | Lemma | deg2_poly_gt0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"a1",
"apply",
"delta",
"deltam",
"ler01"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_ge0 : reflect (forall x, 0 <= p.[x]) (delta <= 0). | Proof. by rewrite deltam; apply: deg2_poly_ge0; rewrite // -/a a1 ler01. Qed. | Lemma | deg2_poly_ge0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"a1",
"apply",
"delta",
"deltam",
"ler01"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r1 | := (- b - sqrt delta) / 2. | Let | r1 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"delta",
"sqrt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
r2 | := (- b + sqrt delta) / 2. | Let | r2 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"delta",
"sqrt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_factor : 0 <= delta -> p = ('X - r1%:P) * ('X - r2%:P). | Proof. by move=> dge0; apply: deg2_poly_factor; rewrite ?sqr_sqrtr. Qed. | Lemma | deg2_poly_factor | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"delta",
"r1",
"r2",
"sqr_sqrtr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_noroot : reflect (forall x, ~~ root p x) (delta < 0). | Proof. by rewrite deltam; apply: deg2_poly_noroot. Qed. | Lemma | deg2_poly_noroot | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"apply",
"delta",
"deltam",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_gt0l x : x < r1 -> 0 < p.[x]. | Proof.
by move=> ?; apply: deg2_poly_gt0l; rewrite // -/a ?a1 ?ler01 ?mulr1.
Qed. | Lemma | deg2_poly_gt0l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"a1",
"apply",
"ler01",
"mulr1",
"r1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_gt0r x : r2 < x -> 0 < p.[x]. | Proof.
by move=> ?; apply: deg2_poly_gt0r; rewrite // -/a ?a1 ?ler01 ?mulr1.
Qed. | Lemma | deg2_poly_gt0r | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"a1",
"apply",
"ler01",
"mulr1",
"r2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_lt0m x : r1 < x < r2 -> p.[x] < 0. | Proof.
by move=> ?; apply: deg2_poly_lt0m; rewrite // -/a ?a1 ?ler01 ?mulr1.
Qed. | Lemma | deg2_poly_lt0m | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"a1",
"apply",
"ler01",
"mulr1",
"r1",
"r2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_ge0l x : x <= r1 -> 0 <= p.[x]. | Proof.
by move=> ?; apply: deg2_poly_ge0l; rewrite // -/a ?a1 ?ler01 ?mulr1.
Qed. | Lemma | deg2_poly_ge0l | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"a1",
"apply",
"ler01",
"mulr1",
"r1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_ge0r x : r2 <= x -> 0 <= p.[x]. | Proof.
by move=> ?; apply: deg2_poly_ge0r; rewrite // -/a ?a1 ?ler01 ?mulr1.
Qed. | Lemma | deg2_poly_ge0r | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"a1",
"apply",
"ler01",
"mulr1",
"r2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg2_poly_le0m x : 0 <= delta -> r1 <= x <= r2 -> p.[x] <= 0. | move=> dge0 xm.
by apply: deg2_poly_le0m; rewrite -/a -/b -/c ?a1 ?mulr1 -/delta ?ler01.
Qed. | Lemma | deg2_poly_le0m | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"a1",
"apply",
"delta",
"ler01",
"mulr1",
"r1",
"r2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
degp : (size p <= 3)%N. | Hypothesis | degp | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"size"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
deg_le2_poly_delta_ge0 : 0 <= a -> (forall x, 0 <= p.[x]) -> delta <= 0. | Proof.
move=> age0 pge0; move: degp; rewrite leq_eqVlt => /orP[/eqP|] degp'.
exact/(Real.deg2_poly_ge0 degp' age0).
have a0 : a = 0 by rewrite /a nth_default.
rewrite /delta a0 mulr0 mul0r subr0 exprn_even_le0//=.
have [//|/eqP nzb] := eqP; move: (pge0 ((- 1 - c) / b)).
have -> : p = b *: 'X + c%:P.
apply/polyP => ... | Lemma | deg_le2_poly_delta_ge0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"age0",
"apply",
"coefE",
"deg2_poly_ge0",
"degp",
"delta",
"exprn_even_le0",
"hornerE",
"leq_eqVlt",
"leq_trans",
"ler0N1",
"ltnS",
"mul0r",
"mul1r",
"mulfV",
"mulr0",
"mulrAC",
"nth_default",
"polyP",
"simpm",
"subr0",
"subrK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deg_le2_poly_delta_le0 : a <= 0 -> (forall x, p.[x] <= 0) -> delta <= 0. | Proof.
move=> ale0 ple0; rewrite /delta -sqrrN -[c]opprK mulrN -mulNr -[-(4 * a)]mulrN.
rewrite -!coefN deg_le2_poly_delta_ge0 ?size_polyN ?coefN ?oppr_ge0// => x.
by rewrite hornerN oppr_ge0.
Qed. | Lemma | deg_le2_poly_delta_le0 | algebra.numeric_hierarchy | algebra/numeric_hierarchy/numfield.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"div",
"path",
"fintype",
"bigop",
"ssrAC",
"finset",
"fingroup",
"nmodule",
"order",
"interval",
"rings_modules_and_algebras",
"divalg",
"decfield",
"poly",
"ordere... | [
"ale0",
"coefN",
"deg_le2_poly_delta_ge0",
"delta",
"hornerN",
"mulNr",
"mulrN",
"opprK",
"oppr_ge0",
"size_polyN",
"sqrrN"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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