name
stringlengths 12
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stringclasses 1
value | informal_prefix
stringlengths 10
481
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stringlengths 50
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stringlengths 22
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stringclasses 1
value |
|---|---|---|---|---|---|---|
exercise_1_13b | test | /-- Suppose that $f$ is holomorphic in an open set $\Omega$. Prove that if $\text{Im}(f)$ is constant, then $f$ is constant.-/
| theorem exercise_1_13b {f : β β β} (Ξ© : Set β) (a b : Ξ©) (h : IsOpen Ξ©)
(hf : DifferentiableOn β f Ξ©) (hc : β (c : β), β z β Ξ©, (f z).im = c) :
f a = f b := | f : β β β
Ξ© : Set β
a b : βΞ©
h : IsOpen Ξ©
hf : DifferentiableOn β f Ξ©
hc : β c, β z β Ξ©, (f z).im = c
β’ f βa = f βb | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
| |
exercise_1_19a | test | /-- Prove that the power series $\sum nz^n$ does not converge on any point of the unit circle.-/
| theorem exercise_1_19a (z : β) (hz : abs z = 1) (s : β β β)
(h : s = (Ξ» n => β i in (range n), i * z ^ i)) :
Β¬ β y, Tendsto s atTop (π y) := | z : β
hz : Complex.abs z = 1
s : β β β
h : s = fun n => β i β range n, βi * z ^ i
β’ Β¬β y, Tendsto s atTop (π y) | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
| |
exercise_1_19c | test | /-- Prove that the power series $\sum zn/n$ converges at every point of the unit circle except $z = 1$.-/
| theorem exercise_1_19c (z : β) (hz : abs z = 1) (hz2 : z β 1) (s : β β β)
(h : s = (Ξ» n => β i in (range n), i * z / i)) :
β z, Tendsto s atTop (π z) := | z : β
hz : Complex.abs z = 1
hz2 : z β 1
s : β β β
h : s = fun n => β i β range n, βi * z / βi
β’ β z, Tendsto s atTop (π z) | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
| |
exercise_2_2 | test | /-- Show that $\int_{0}^{\infty} \frac{\sin x}{x} d x=\frac{\pi}{2}$.-/
| theorem exercise_2_2 :
Tendsto (Ξ» y => β« x in (0 : β)..y, Real.sin x / x) atTop (π (Real.pi / 2)) := | β’ Tendsto (fun y => β« (x : β) in 0 ..y, x.sin / x) atTop (π (Real.pi / 2)) | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
| |
exercise_2_13 | test | /-- Suppose $f$ is an analytic function defined everywhere in $\mathbb{C}$ and such that for each $z_0 \in \mathbb{C}$ at least one coefficient in the expansion $f(z) = \sum_{n=0}^\infty c_n(z - z_0)^n$ is equal to 0. Prove that $f$ is a polynomial.-/
| theorem exercise_2_13 {f : β β β}
(hf : β zβ : β, β (s : Set β) (c : β β β), IsOpen s β§ zβ β s β§
β z β s, Tendsto (Ξ» n => β i in range n, (c i) * (z - zβ)^i) atTop (π (f zβ))
β§ β i, c i = 0) :
β (c : β β β) (n : β), f = Ξ» z => β i in range n, (c i) * z ^ n := | f : β β β
hf :
β (zβ : β),
β s c,
IsOpen s β§ zβ β s β§ β z β s, Tendsto (fun n => β i β range n, c i * (z - zβ) ^ i) atTop (π (f zβ)) β§ β i, c i = 0
β’ β c n, f = fun z => β i β range n, c i * z ^ n | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
| |
exercise_3_4 | test | /-- Show that $ \int_{-\infty}^{\infty} \frac{x \sin x}{x^2 + a^2} dx = \pi e^{-a}$ for $a > 0$.-/
| theorem exercise_3_4 (a : β) (ha : 0 < a) :
Tendsto (Ξ» y => β« x in -y..y, x * Real.sin x / (x ^ 2 + a ^ 2))
atTop (π (Real.pi * (Real.exp (-a)))) := | a : β
ha : 0 < a
β’ Tendsto (fun y => β« (x : β) in -y..y, x * x.sin / (x ^ 2 + a ^ 2)) atTop (π (Real.pi * (-a).exp)) | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
| |
exercise_3_14 | test | /-- Prove that all entire functions that are also injective take the form $f(z) = az + b$, $a, b \in \mathbb{C}$ and $a \neq 0$.-/
| theorem exercise_3_14 {f : β β β} (hf : Differentiable β f)
(hf_inj : Function.Injective f) :
β (a b : β), f = (Ξ» z => a * z + b) β§ a β 0 := | f : β β β
hf : Differentiable β f
hf_inj : Injective f
β’ β a b, (f = fun z => a * z + b) β§ a β 0 | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
| |
exercise_5_1 | test | /-- Prove that if $f$ is holomorphic in the unit disc, bounded and not identically zero, and $z_{1}, z_{2}, \ldots, z_{n}, \ldots$ are its zeros $\left(\left|z_{k}\right|<1\right)$, then $\sum_{n}\left(1-\left|z_{n}\right|\right)<\infty$.-/
| theorem exercise_5_1 (f : β β β) (hf : DifferentiableOn β f (ball 0 1))
(hb : Bornology.IsBounded (Set.range f)) (h0 : f β 0) (zeros : β β β) (hz : β n, f (zeros n) = 0)
(hzz : Set.range zeros = {z | f z = 0 β§ z β (ball (0 : β) 1)}) :
β (z : β), Tendsto (Ξ» n => (β i in range n, (1 - zeros i))) atTop (π z) := | f : β β β
hf : DifferentiableOn β f (ball 0 1)
hb : Bornology.IsBounded (Set.range f)
h0 : f β 0
zeros : β β β
hz : β (n : β), f (zeros n) = 0
hzz : Set.range zeros = {z | f z = 0 β§ z β ball 0 1}
β’ β z, Tendsto (fun n => β i β range n, (1 - zeros i)) atTop (π z) | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
| |
exercise_1_1b | test | /-- If $r$ is rational $(r \neq 0)$ and $x$ is irrational, prove that $rx$ is irrational.-/
| theorem exercise_1_1b
(x : β)
(y : β)
(h : y β 0)
: ( Irrational x ) -> Irrational ( x * y ) := | x : β
y : β
h : y β 0
β’ Irrational x β Irrational (x * βy) | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_1_4 | test | /-- Let $E$ be a nonempty subset of an ordered set; suppose $\alpha$ is a lower bound of $E$ and $\beta$ is an upper bound of $E$. Prove that $\alpha \leq \beta$.-/
| theorem exercise_1_4
(Ξ± : Type*) [PartialOrder Ξ±]
(s : Set Ξ±)
(x y : Ξ±)
(hβ : Set.Nonempty s)
(hβ : x β lowerBounds s)
(hβ : y β upperBounds s)
: x β€ y := | Ξ± : Type u_1
instβ : PartialOrder Ξ±
s : Set Ξ±
x y : Ξ±
hβ : s.Nonempty
hβ : x β lowerBounds s
hβ : y β upperBounds s
β’ x β€ y | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_1_8 | test | /-- Prove that no order can be defined in the complex field that turns it into an ordered field.-/
| theorem exercise_1_8 : Β¬ β (r : β β β β Prop), IsLinearOrder β r := | β’ Β¬β r, IsLinearOrder β r | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_1_12 | test | /-- If $z_1, \ldots, z_n$ are complex, prove that $|z_1 + z_2 + \ldots + z_n| \leq |z_1| + |z_2| + \cdots + |z_n|$.-/
| theorem exercise_1_12 (n : β) (f : β β β) :
abs (β i in range n, f i) β€ β i in range n, abs (f i) := | n : β
f : β β β
β’ Complex.abs (β i β range n, f i) β€ β i β range n, Complex.abs (f i) | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_1_14 | test | /-- If $z$ is a complex number such that $|z|=1$, that is, such that $z \bar{z}=1$, compute $|1+z|^{2}+|1-z|^{2}$.-/
| theorem exercise_1_14
(z : β) (h : abs z = 1)
: (abs (1 + z)) ^ 2 + (abs (1 - z)) ^ 2 = 4 := | z : β
h : Complex.abs z = 1
β’ Complex.abs (1 + z) ^ 2 + Complex.abs (1 - z) ^ 2 = 4 | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_1_17 | test | /-- Prove that $|\mathbf{x}+\mathbf{y}|^{2}+|\mathbf{x}-\mathbf{y}|^{2}=2|\mathbf{x}|^{2}+2|\mathbf{y}|^{2}$ if $\mathbf{x} \in R^{k}$ and $\mathbf{y} \in R^{k}$.-/
| theorem exercise_1_17
(n : β)
(x y : EuclideanSpace β (Fin n)) -- R^n
: βx + yβ^2 + βx - yβ^2 = 2*βxβ^2 + 2*βyβ^2 := | n : β
x y : EuclideanSpace β (Fin n)
β’ βx + yβ ^ 2 + βx - yβ ^ 2 = 2 * βxβ ^ 2 + 2 * βyβ ^ 2 | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_1_18b | test | /-- If $k = 1$ and $\mathbf{x} \in R^{k}$, prove that there does not exist $\mathbf{y} \in R^{k}$ such that $\mathbf{y} \neq 0$ but $\mathbf{x} \cdot \mathbf{y}=0$-/
| theorem exercise_1_18b
: Β¬ β (x : β), β (y : β), y β 0 β§ x * y = 0 := | β’ Β¬β (x : β), β y, y β 0 β§ x * y = 0 | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_2_19a | test | /-- If $A$ and $B$ are disjoint closed sets in some metric space $X$, prove that they are separated.-/
| theorem exercise_2_19a {X : Type*} [MetricSpace X]
(A B : Set X) (hA : IsClosed A) (hB : IsClosed B) (hAB : Disjoint A B) :
SeparatedNhds A B := | X : Type u_1
instβ : MetricSpace X
A B : Set X
hA : IsClosed A
hB : IsClosed B
hAB : Disjoint A B
β’ SeparatedNhds A B | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_2_25 | test | /-- Prove that every compact metric space $K$ has a countable base.-/
| theorem exercise_2_25 {K : Type*} [MetricSpace K] [CompactSpace K] :
β (B : Set (Set K)), Set.Countable B β§ IsTopologicalBasis B := | K : Type u_1
instβΒΉ : MetricSpace K
instβ : CompactSpace K
β’ β B, B.Countable β§ IsTopologicalBasis B | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_2_27b | test | /-- Suppose $E\subset\mathbb{R}^k$ is uncountable, and let $P$ be the set of condensation points of $E$. Prove that at most countably many points of $E$ are not in $P$.-/
| theorem exercise_2_27b (k : β) (E P : Set (EuclideanSpace β (Fin k)))
(hE : E.Nonempty β§ Β¬ Set.Countable E)
(hP : P = {x | β U β π x, (P β© E).Nonempty β§ Β¬ Set.Countable (P β© E)}) :
Set.Countable (E \ P) := | k : β
E P : Set (EuclideanSpace β (Fin k))
hE : E.Nonempty β§ Β¬E.Countable
hP : P = {x | β U β π x, (P β© E).Nonempty β§ Β¬(P β© E).Countable}
β’ (E \ P).Countable | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_2_29 | test | /-- Prove that every open set in $\mathbb{R}$ is the union of an at most countable collection of disjoint segments.-/
| theorem exercise_2_29 (U : Set β) (hU : IsOpen U) :
β (f : β β Set β), (β n, β a b : β, f n = {x | a < x β§ x < b}) β§ (β n, f n β U) β§
(β n m, n β m β f n β© f m = β
) β§
U = β n, f n := | U : Set β
hU : IsOpen U
β’ β f,
(β (n : β), β a b, f n = {x | a < x β§ x < b}) β§
(β (n : β), f n β U) β§ (β (n m : β), n β m β f n β© f m = β
) β§ U = β n, f n | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_3_2a | test | /-- Prove that $\lim_{n \rightarrow \infty}\sqrt{n^2 + n} -n = 1/2$.-/
| theorem exercise_3_2a
: Tendsto (Ξ» (n : β) => (sqrt (n^2 + n) - n)) atTop (π (1/2)) := | β’ Tendsto (fun n => β(n ^ 2 + n) - n) atTop (π (1 / 2)) | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_3_5 | test | /-- For any two real sequences $\left\{a_{n}\right\},\left\{b_{n}\right\}$, prove that $\limsup _{n \rightarrow \infty}\left(a_{n}+b_{n}\right) \leq \limsup _{n \rightarrow \infty} a_{n}+\limsup _{n \rightarrow \infty} b_{n},$ provided the sum on the right is not of the form $\infty-\infty$.-/
| theorem exercise_3_5
(a b : β β β)
(h : limsup a + limsup b β 0) :
limsup (Ξ» n => a n + b n) β€ limsup a + limsup b := | a b : β β β
h : limsup a + limsup b β 0
β’ (limsup fun n => a n + b n) β€ limsup a + limsup b | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_3_7 | test | /-- Prove that the convergence of $\Sigma a_{n}$ implies the convergence of $\sum \frac{\sqrt{a_{n}}}{n}$ if $a_n\geq 0$.-/
| theorem exercise_3_7
(a : β β β)
(h : β y, (Tendsto (Ξ» n => (β i in (range n), a i)) atTop (π y))) :
β y, Tendsto (Ξ» n => (β i in (range n), sqrt (a i) / n)) atTop (π y) := | a : β β β
h : β y, Tendsto (fun n => β i β range n, a i) atTop (π y)
β’ β y, Tendsto (fun n => β i β range n, β(a i) / βn) atTop (π y) | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_3_13 | test | /-- Prove that the Cauchy product of two absolutely convergent series converges absolutely.-/
| theorem exercise_3_13
(a b : β β β)
(ha : β y, (Tendsto (Ξ» n => (β i in (range n), |a i|)) atTop (π y)))
(hb : β y, (Tendsto (Ξ» n => (β i in (range n), |b i|)) atTop (π y))) :
β y, (Tendsto (Ξ» n => (β i in (range n),
Ξ» i => (β j in range (i + 1), a j * b (i - j)))) atTop (π y)) := | a b : β β β
ha : β y, Tendsto (fun n => β i β range n, |a i|) atTop (π y)
hb : β y, Tendsto (fun n => β i β range n, |b i|) atTop (π y)
β’ β y, Tendsto (fun n => β i β range n, fun i => β j β range (i + 1), a j * b (i - j)) atTop (π y) | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_3_21 | test | /-- If $\left\{E_{n}\right\}$ is a sequence of closed nonempty and bounded sets in a complete metric space $X$, if $E_{n} \supset E_{n+1}$, and if $\lim _{n \rightarrow \infty} \operatorname{diam} E_{n}=0,$ then $\bigcap_{1}^{\infty} E_{n}$ consists of exactly one point.-/
| theorem exercise_3_21
{X : Type*} [MetricSpace X] [CompleteSpace X]
(E : β β Set X)
(hE : β n, E n β E (n + 1))
(hE' : Tendsto (Ξ» n => Metric.diam (E n)) atTop (π 0)) :
β a, Set.iInter E = {a} := | X : Type u_1
instβΒΉ : MetricSpace X
instβ : CompleteSpace X
E : β β Set X
hE : β (n : β), E n β E (n + 1)
hE' : Tendsto (fun n => Metric.diam (E n)) atTop (π 0)
β’ β a, Set.iInter E = {a} | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_4_1a | test | /-- Suppose $f$ is a real function defined on $\mathbb{R}$ which satisfies $\lim_{h \rightarrow 0} f(x + h) - f(x - h) = 0$ for every $x \in \mathbb{R}$. Show that $f$ does not need to be continuous.-/
| theorem exercise_4_1a
: β (f : β β β), (β (x : β), Tendsto (Ξ» y => f (x + y) - f (x - y)) (π 0) (π 0)) β§ Β¬ Continuous f := | β’ β f, (β (x : β), Tendsto (fun y => f (x + y) - f (x - y)) (π 0) (π 0)) β§ Β¬Continuous f | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_4_3 | test | /-- Let $f$ be a continuous real function on a metric space $X$. Let $Z(f)$ (the zero set of $f$ ) be the set of all $p \in X$ at which $f(p)=0$. Prove that $Z(f)$ is closed.-/
| theorem exercise_4_3
{Ξ± : Type} [MetricSpace Ξ±]
(f : Ξ± β β) (h : Continuous f) (z : Set Ξ±) (g : z = fβ»ΒΉ' {0})
: IsClosed z := | Ξ± : Type
instβ : MetricSpace Ξ±
f : Ξ± β β
h : Continuous f
z : Set Ξ±
g : z = f β»ΒΉ' {0}
β’ IsClosed z | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_4_4b | test | /-- Let $f$ and $g$ be continuous mappings of a metric space $X$ into a metric space $Y$, and let $E$ be a dense subset of $X$. Prove that if $g(p) = f(p)$ for all $p \in P$ then $g(p) = f(p)$ for all $p \in X$.-/
| theorem exercise_4_4b
{Ξ± : Type} [MetricSpace Ξ±]
{Ξ² : Type} [MetricSpace Ξ²]
(f g : Ξ± β Ξ²)
(s : Set Ξ±)
(hβ : Continuous f)
(hβ : Continuous g)
(hβ : Dense s)
(hβ : β x β s, f x = g x)
: f = g := | Ξ± : Type
instβΒΉ : MetricSpace Ξ±
Ξ² : Type
instβ : MetricSpace Ξ²
f g : Ξ± β Ξ²
s : Set Ξ±
hβ : Continuous f
hβ : Continuous g
hβ : Dense s
hβ : β x β s, f x = g x
β’ f = g | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_4_5b | test | /-- Show that there exist a set $E \subset \mathbb{R}$ and a real continuous function $f$ defined on $E$, such that there does not exist a continuous real function $g$ on $\mathbb{R}$ such that $g(x)=f(x)$ for all $x \in E$.-/
| theorem exercise_4_5b
: β (E : Set β) (f : β β β), (ContinuousOn f E) β§
(Β¬ β (g : β β β), Continuous g β§ β x β E, f x = g x) := | β’ β E f, ContinuousOn f E β§ Β¬β g, Continuous g β§ β x β E, f x = g x | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_4_8a | test | /-- Let $f$ be a real uniformly continuous function on the bounded set $E$ in $R^{1}$. Prove that $f$ is bounded on $E$.-/
| theorem exercise_4_8a
(E : Set β) (f : β β β) (hf : UniformContinuousOn f E)
(hE : Bornology.IsBounded E) : Bornology.IsBounded (Set.image f E) := | E : Set β
f : β β β
hf : UniformContinuousOn f E
hE : Bornology.IsBounded E
β’ Bornology.IsBounded (f '' E) | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_4_11a | test | /-- Suppose $f$ is a uniformly continuous mapping of a metric space $X$ into a metric space $Y$ and prove that $\left\{f\left(x_{n}\right)\right\}$ is a Cauchy sequence in $Y$ for every Cauchy sequence $\{x_n\}$ in $X$.-/
| theorem exercise_4_11a
{X : Type*} [MetricSpace X]
{Y : Type*} [MetricSpace Y]
(f : X β Y) (hf : UniformContinuous f)
(x : β β X) (hx : CauchySeq x) :
CauchySeq (Ξ» n => f (x n)) := | X : Type u_1
instβΒΉ : MetricSpace X
Y : Type u_2
instβ : MetricSpace Y
f : X β Y
hf : UniformContinuous f
x : β β X
hx : CauchySeq x
β’ CauchySeq fun n => f (x n) | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_4_15 | test | /-- Prove that every continuous open mapping of $R^{1}$ into $R^{1}$ is monotonic.-/
| theorem exercise_4_15 {f : β β β}
(hf : Continuous f) (hof : IsOpenMap f) :
Monotone f := | f : β β β
hf : Continuous f
hof : IsOpenMap f
β’ Monotone f | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_4_21a | test | /-- Suppose $K$ and $F$ are disjoint sets in a metric space $X, K$ is compact, $F$ is closed. Prove that there exists $\delta>0$ such that $d(p, q)>\delta$ if $p \in K, q \in F$.-/
| theorem exercise_4_21a {X : Type*} [MetricSpace X]
(K F : Set X) (hK : IsCompact K) (hF : IsClosed F) (hKF : Disjoint K F) :
β (Ξ΄ : β), Ξ΄ > 0 β§ β (p q : X), p β K β q β F β dist p q β₯ Ξ΄ := | X : Type u_1
instβ : MetricSpace X
K F : Set X
hK : IsCompact K
hF : IsClosed F
hKF : Disjoint K F
β’ β Ξ΄ > 0, β (p q : X), p β K β q β F β dist p q β₯ Ξ΄ | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_5_1 | test | /-- Let $f$ be defined for all real $x$, and suppose that $|f(x)-f(y)| \leq (x-y)^{2}$ for all real $x$ and $y$. Prove that $f$ is constant.-/
| theorem exercise_5_1
{f : β β β} (hf : β x y : β, |(f x - f y)| β€ (x - y) ^ 2) :
β c, f = Ξ» x => c := | f : β β β
hf : β (x y : β), |f x - f y| β€ (x - y) ^ 2
β’ β c, f = fun x => c | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_5_3 | test | /-- Suppose $g$ is a real function on $R^{1}$, with bounded derivative (say $\left|g^{\prime}\right| \leq M$ ). Fix $\varepsilon>0$, and define $f(x)=x+\varepsilon g(x)$. Prove that $f$ is one-to-one if $\varepsilon$ is small enough.-/
| theorem exercise_5_3 {g : β β β} (hg : Continuous g)
(hg' : β M : β, β x : β, |deriv g x| β€ M) :
β N, β Ξ΅ > 0, Ξ΅ < N β Function.Injective (Ξ» x : β => x + Ξ΅ * g x) := | g : β β β
hg : Continuous g
hg' : β M, β (x : β), |deriv g x| β€ M
β’ β N, β Ξ΅ > 0, Ξ΅ < N β Function.Injective fun x => x + Ξ΅ * g x | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_5_5 | test | /-- Suppose $f$ is defined and differentiable for every $x>0$, and $f^{\prime}(x) \rightarrow 0$ as $x \rightarrow+\infty$. Put $g(x)=f(x+1)-f(x)$. Prove that $g(x) \rightarrow 0$ as $x \rightarrow+\infty$.-/
| theorem exercise_5_5
{f : β β β}
(hfd : Differentiable β f)
(hf : Tendsto (deriv f) atTop (π 0)) :
Tendsto (Ξ» x => f (x + 1) - f x) atTop atTop := | f : β β β
hfd : Differentiable β f
hf : Tendsto (deriv f) atTop (π 0)
β’ Tendsto (fun x => f (x + 1) - f x) atTop atTop | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_5_7 | test | /-- Suppose $f^{\prime}(x), g^{\prime}(x)$ exist, $g^{\prime}(x) \neq 0$, and $f(x)=g(x)=0$. Prove that $\lim _{t \rightarrow x} \frac{f(t)}{g(t)}=\frac{f^{\prime}(x)}{g^{\prime}(x)}.$-/
| theorem exercise_5_7
{f g : β β β} {x : β}
(hf' : DifferentiableAt β f 0)
(hg' : DifferentiableAt β g 0)
(hg'_ne_0 : deriv g 0 β 0)
(f0 : f 0 = 0) (g0 : g 0 = 0) :
Tendsto (Ξ» x => f x / g x) (π x) (π (deriv f x / deriv g x)) := | f g : β β β
x : β
hf' : DifferentiableAt β f 0
hg' : DifferentiableAt β g 0
hg'_ne_0 : deriv g 0 β 0
f0 : f 0 = 0
g0 : g 0 = 0
β’ Tendsto (fun x => f x / g x) (π x) (π (deriv f x / deriv g x)) | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_5_17 | test | /-- Suppose $f$ is a real, three times differentiable function on $[-1,1]$, such that $f(-1)=0, \quad f(0)=0, \quad f(1)=1, \quad f^{\prime}(0)=0 .$ Prove that $f^{(3)}(x) \geq 3$ for some $x \in(-1,1)$.-/
| theorem exercise_5_17
{f : β β β}
(hf' : DifferentiableOn β f (Set.Icc (-1) 1))
(hf'' : DifferentiableOn β (deriv f) (Set.Icc 1 1))
(hf''' : DifferentiableOn β (deriv (deriv f)) (Set.Icc 1 1))
(hf0 : f (-1) = 0)
(hf1 : f 0 = 0)
(hf2 : f 1 = 1)
(hf3 : deriv f 0 = 0) :
β x, x β Set.Ioo (-1 : β) 1 β§ deriv (deriv (deriv f)) x β₯ 3 := | f : β β β
hf' : DifferentiableOn β f (Set.Icc (-1) 1)
hf'' : DifferentiableOn β (deriv f) (Set.Icc 1 1)
hf''' : DifferentiableOn β (deriv (deriv f)) (Set.Icc 1 1)
hf0 : f (-1) = 0
hf1 : f 0 = 0
hf2 : f 1 = 1
hf3 : deriv f 0 = 0
β’ β x β Set.Ioo (-1) 1, deriv (deriv (deriv f)) x β₯ 3 | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| |
exercise_2_1_18 | test | /-- If $G$ is a finite group of even order, show that there must be an element $a \neq e$ such that $a=a^{-1}$.-/
| theorem exercise_2_1_18 {G : Type*} [Group G]
[Fintype G] (hG2 : Even (card G)) :
β (a : G), a β 1 β§ a = aβ»ΒΉ := | G : Type u_1
instβΒΉ : Group G
instβ : Fintype G
hG2 : Even (card G)
β’ β a, a β 1 β§ a = aβ»ΒΉ | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_2_1_26 | test | /-- If $G$ is a finite group, prove that, given $a \in G$, there is a positive integer $n$, depending on $a$, such that $a^n = e$.-/
| theorem exercise_2_1_26 {G : Type*} [Group G]
[Fintype G] (a : G) : β (n : β), a ^ n = 1 := | G : Type u_1
instβΒΉ : Group G
instβ : Fintype G
a : G
β’ β n, a ^ n = 1 | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_2_2_3 | test | /-- If $G$ is a group in which $(a b)^{i}=a^{i} b^{i}$ for three consecutive integers $i$, prove that $G$ is abelian.-/
| def exercise_2_2_3 {G : Type*} [Group G]
{P : β β Prop} {hP : P = Ξ» i => β a b : G, (a*b)^i = a^i * b^i}
(hP1 : β n : β, P n β§ P (n+1) β§ P (n+2)) : CommGroup G := | G : Type u_1
instβ : Group G
P : β β Prop
hP : P = fun i => β (a b : G), (a * b) ^ i = a ^ i * b ^ i
hP1 : β n, P n β§ P (n + 1) β§ P (n + 2)
β’ CommGroup G | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_2_2_6c | test | /-- Let $G$ be a group in which $(a b)^{n}=a^{n} b^{n}$ for some fixed integer $n>1$ for all $a, b \in G$. For all $a, b \in G$, prove that $\left(a b a^{-1} b^{-1}\right)^{n(n-1)}=e$.-/
| theorem exercise_2_2_6c {G : Type*} [Group G] {n : β} (hn : n > 1)
(h : β (a b : G), (a * b) ^ n = a ^ n * b ^ n) :
β (a b : G), (a * b * aβ»ΒΉ * bβ»ΒΉ) ^ (n * (n - 1)) = 1 := | G : Type u_1
instβ : Group G
n : β
hn : n > 1
h : β (a b : G), (a * b) ^ n = a ^ n * b ^ n
β’ β (a b : G), (a * b * aβ»ΒΉ * bβ»ΒΉ) ^ (n * (n - 1)) = 1 | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_2_3_16 | test | /-- If a group $G$ has no proper subgroups, prove that $G$ is cyclic of order $p$, where $p$ is a prime number.-/
| theorem exercise_2_3_16 {G : Type*} [Group G]
(hG : β H : Subgroup G, H = β€ β¨ H = β₯) :
IsCyclic G β§ β (p : β) (Fin : Fintype G), Nat.Prime p β§ @card G Fin = p := | G : Type u_1
instβ : Group G
hG : β (H : Subgroup G), H = β€ β¨ H = β₯
β’ IsCyclic G β§ β p Fin, p.Prime β§ card G = p | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_2_5_23 | test | /-- Let $G$ be a group such that all subgroups of $G$ are normal in $G$. If $a, b \in G$, prove that $ba = a^jb$ for some $j$.-/
| theorem exercise_2_5_23 {G : Type*} [Group G]
(hG : β (H : Subgroup G), H.Normal) (a b : G) :
β (j : β€) , b*a = a^j * b := | G : Type u_1
instβ : Group G
hG : β (H : Subgroup G), H.Normal
a b : G
β’ β j, b * a = a ^ j * b | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_2_5_31 | test | /-- Suppose that $G$ is an abelian group of order $p^nm$ where $p \nmid m$ is a prime. If $H$ is a subgroup of $G$ of order $p^n$, prove that $H$ is a characteristic subgroup of $G$.-/
| theorem exercise_2_5_31 {G : Type*} [CommGroup G] [Fintype G]
{p m n : β} (hp : Nat.Prime p) (hp1 : Β¬ p β£ m) (hG : card G = p^n*m)
{H : Subgroup G} [Fintype H] (hH : card H = p^n) :
Subgroup.Characteristic H := | G : Type u_1
instβΒ² : CommGroup G
instβΒΉ : Fintype G
p m n : β
hp : p.Prime
hp1 : Β¬p β£ m
hG : card G = p ^ n * m
H : Subgroup G
instβ : Fintype β₯H
hH : card β₯H = p ^ n
β’ H.Characteristic | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_2_5_43 | test | /-- Prove that a group of order 9 must be abelian.-/
| def exercise_2_5_43 (G : Type*) [Group G] [Fintype G]
(hG : card G = 9) :
CommGroup G := | G : Type u_1
instβΒΉ : Group G
instβ : Fintype G
hG : card G = 9
β’ CommGroup G | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_2_5_52 | test | /-- Let $G$ be a finite group and $\varphi$ an automorphism of $G$ such that $\varphi(x) = x^{-1}$ for more than three-fourths of the elements of $G$. Prove that $\varphi(y) = y^{-1}$ for all $y \in G$, and so $G$ is abelian.-/
| theorem exercise_2_5_52 {G : Type*} [Group G] [Fintype G]
(Ο : G β* G) {I : Finset G} (hI : β x β I, Ο x = xβ»ΒΉ)
(hI1 : (0.75 : β) * card G β€ card I) :
β x : G, Ο x = xβ»ΒΉ β§ β x y : G, x*y = y*x := | G : Type u_1
instβΒΉ : Group G
instβ : Fintype G
Ο : G β* G
I : Finset G
hI : β x β I, Ο x = xβ»ΒΉ
hI1 : 0.75 * β(card G) β€ β(card { x // x β I })
β’ β (x : G), Ο x = xβ»ΒΉ β§ β (x y : G), x * y = y * x | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_2_7_7 | test | /-- If $\varphi$ is a homomorphism of $G$ onto $G'$ and $N \triangleleft G$, show that $\varphi(N) \triangleleft G'$.-/
| theorem exercise_2_7_7 {G : Type*} [Group G] {G' : Type*} [Group G']
(Ο : G β* G') (N : Subgroup G) [N.Normal] :
(Subgroup.map Ο N).Normal := | G : Type u_1
instβΒ² : Group G
G' : Type u_2
instβΒΉ : Group G'
Ο : G β* G'
N : Subgroup G
instβ : N.Normal
β’ (Subgroup.map Ο N).Normal | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_2_8_15 | test | /-- Prove that if $p > q$ are two primes such that $q \mid p - 1$, then any two nonabelian groups of order $pq$ are isomorphic.-/
| def exercise_2_8_15 {G H: Type*} [Fintype G] [Group G] [Fintype H]
[Group H] {p q : β} (hp : Nat.Prime p) (hq : Nat.Prime q)
(h : p > q) (h1 : q β£ p - 1) (hG : card G = p*q) (hH : card G = p*q) :
G β* H := | G : Type u_1
H : Type u_2
instβΒ³ : Fintype G
instβΒ² : Group G
instβΒΉ : Fintype H
instβ : Group H
p q : β
hp : p.Prime
hq : q.Prime
h : p > q
h1 : q β£ p - 1
hG hH : card G = p * q
β’ G β* H | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_2_10_1 | test | /-- Let $A$ be a normal subgroup of a group $G$, and suppose that $b \in G$ is an element of prime order $p$, and that $b \not\in A$. Show that $A \cap (b) = (e)$.-/
| theorem exercise_2_10_1 {G : Type*} [Group G] (A : Subgroup G)
[A.Normal] {b : G} (hp : Nat.Prime (orderOf b)) :
A β (Subgroup.closure {b}) = β₯ := | G : Type u_1
instβΒΉ : Group G
A : Subgroup G
instβ : A.Normal
b : G
hp : (orderOf b).Prime
β’ A β Subgroup.closure {b} = β₯ | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_2_11_7 | test | /-- If $P \triangleleft G$, $P$ a $p$-Sylow subgroup of $G$, prove that $\varphi(P) = P$ for every automorphism $\varphi$ of $G$.-/
| theorem exercise_2_11_7 {G : Type*} [Group G] {p : β} (hp : Nat.Prime p)
{P : Sylow p G} (hP : P.Normal) :
Subgroup.Characteristic (P : Subgroup G) := | G : Type u_1
instβ : Group G
p : β
hp : p.Prime
P : Sylow p G
hP : (βP).Normal
β’ (βP).Characteristic | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_3_2_21 | test | /-- If $\sigma, \tau$ are two permutations that disturb no common element and $\sigma \tau = e$, prove that $\sigma = \tau = e$.-/
| theorem exercise_3_2_21 {Ξ± : Type*} [Fintype Ξ±] {Ο Ο: Equiv.Perm Ξ±}
(h1 : β a : Ξ±, Ο a = a β Ο a β a) (h2 : Ο β Ο = id) :
Ο = 1 β§ Ο = 1 := | Ξ± : Type u_1
instβ : Fintype Ξ±
Ο Ο : Equiv.Perm Ξ±
h1 : β (a : Ξ±), Ο a = a β Ο a β a
h2 : βΟ β βΟ = id
β’ Ο = 1 β§ Ο = 1 | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_4_1_34 | test | /-- Let $T$ be the group of $2\times 2$ matrices $A$ with entries in the field $\mathbb{Z}_2$ such that $\det A$ is not equal to 0. Prove that $T$ is isomorphic to $S_3$, the symmetric group of degree 3.-/
| def exercise_4_1_34 : Equiv.Perm (Fin 3) β* Matrix.GeneralLinearGroup (Fin 2) (ZMod 2) := | β’ Equiv.Perm (Fin 3) β* GL (Fin 2) (ZMod 2) | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_4_2_6 | test | /-- If $a^2 = 0$ in $R$, show that $ax + xa$ commutes with $a$.-/
| theorem exercise_4_2_6 {R : Type*} [Ring R] (a x : R)
(h : a ^ 2 = 0) : a * (a * x + x * a) = (x + x * a) * a := | R : Type u_1
instβ : Ring R
a x : R
h : a ^ 2 = 0
β’ a * (a * x + x * a) = (x + x * a) * a | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_4_3_1 | test | /-- If $R$ is a commutative ring and $a \in R$, let $L(a) = \{x \in R \mid xa = 0\}$. Prove that $L(a)$ is an ideal of $R$.-/
| theorem exercise_4_3_1 {R : Type*} [CommRing R] (a : R) :
β I : Ideal R, {x : R | x*a=0} = I := | R : Type u_1
instβ : CommRing R
a : R
β’ β I, {x | x * a = 0} = βI | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_4_4_9 | test | /-- Show that $(p - 1)/2$ of the numbers $1, 2, \ldots, p - 1$ are quadratic residues and $(p - 1)/2$ are quadratic nonresidues $\mod p$.-/
| theorem exercise_4_4_9 (p : β) (hp : Nat.Prime p) :
(β S : Finset (ZMod p), S.card = (p-1)/2 β§ β x : ZMod p, x^2 = p) β§
(β S : Finset (ZMod p), S.card = (p-1)/2 β§ Β¬ β x : ZMod p, x^2 = p) := | p : β
hp : p.Prime
β’ (β S, S.card = (p - 1) / 2 β§ β x, x ^ 2 = βp) β§ β S, S.card = (p - 1) / 2 β§ Β¬β x, x ^ 2 = βp | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_4_5_23 | test | /-- Let $F = \mathbb{Z}_7$ and let $p(x) = x^3 - 2$ and $q(x) = x^3 + 2$ be in $F[x]$. Show that $p(x)$ and $q(x)$ are irreducible in $F[x]$ and that the fields $F[x]/(p(x))$ and $F[x]/(q(x))$ are isomorphic.-/
| theorem exercise_4_5_23 {p q: Polynomial (ZMod 7)}
(hp : p = X^3 - 2) (hq : q = X^3 + 2) :
Irreducible p β§ Irreducible q β§
(Nonempty $ Polynomial (ZMod 7) β§Έ span ({p} : Set $ Polynomial $ ZMod 7) β+*
Polynomial (ZMod 7) β§Έ span ({q} : Set $ Polynomial $ ZMod 7)) := | p q : (ZMod 7)[X]
hp : p = X ^ 3 - 2
hq : q = X ^ 3 + 2
β’ Irreducible p β§ Irreducible q β§ Nonempty ((ZMod 7)[X] β§Έ span {p} β+* (ZMod 7)[X] β§Έ span {q}) | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_4_6_2 | test | /-- Prove that $f(x) = x^3 + 3x + 2$ is irreducible in $Q[x]$.-/
| theorem exercise_4_6_2 : Irreducible (X^3 + 3*X + 2 : Polynomial β) := | β’ Irreducible (X ^ 3 + 3 * X + 2) | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_5_1_8 | test | /-- If $F$ is a field of characteristic $p \neq 0$, show that $(a + b)^m = a^m + b^m$, where $m = p^n$, for all $a, b \in F$ and any positive integer $n$.-/
| theorem exercise_5_1_8 {p m n: β} {F : Type*} [Field F]
(hp : Nat.Prime p) (hF : CharP F p) (a b : F) (hm : m = p ^ n) :
(a + b) ^ m = a^m + b^m := | p m n : β
F : Type u_1
instβ : Field F
hp : p.Prime
hF : CharP F p
a b : F
hm : m = p ^ n
β’ (a + b) ^ m = a ^ m + b ^ m | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_5_3_7 | test | /-- If $a \in K$ is such that $a^2$ is algebraic over the subfield $F$ of $K$, show that a is algebraic over $F$.-/
| theorem exercise_5_3_7 {K : Type*} [Field K] {F : Subfield K}
{a : K} (ha : IsAlgebraic F (a ^ 2)) : IsAlgebraic F a := | K : Type u_1
instβ : Field K
F : Subfield K
a : K
ha : IsAlgebraic (β₯F) (a ^ 2)
β’ IsAlgebraic (β₯F) a | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_5_4_3 | test | /-- If $a \in C$ is such that $p(a) = 0$, where $p(x) = x^5 + \sqrt{2}x^3 + \sqrt{5}x^2 + \sqrt{7}x + \sqrt{11}$, show that $a$ is algebraic over $\mathbb{Q}$ of degree at most 80.-/
| theorem exercise_5_4_3 {a : β} {p : β β β}
(hp : p = Ξ» (x : β) => x^5 + sqrt 2 * x^3 + sqrt 5 * x^2 + sqrt 7 * x + 11)
(ha : p a = 0) :
β p : Polynomial β , p.degree < 80 β§ a β p.roots β§
β n : p.support, β a b : β€, p.coeff n = a / b := | a : β
p : β β β
hp : p = fun x => x ^ 5 + ββ2 * x ^ 3 + ββ5 * x ^ 2 + ββ7 * x + 11
ha : p a = 0
β’ β p, p.degree < 80 β§ a β p.roots β§ β (n : { x // x β p.support }), β a b, p.coeff βn = βa / βb | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_5_6_14 | test | /-- If $F$ is of characteristic $p \neq 0$, show that all the roots of $x^m - x$, where $m = p^n$, are distinct.-/
| theorem exercise_5_6_14 {p m n: β} (hp : Nat.Prime p) {F : Type*}
[Field F] [CharP F p] (hm : m = p ^ n) :
card (rootSet (X ^ m - X : Polynomial F) F) = m := | p m n : β
hp : p.Prime
F : Type u_1
instβΒΉ : Field F
instβ : CharP F p
hm : m = p ^ n
β’ card β((X ^ m - X).rootSet F) = m | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| |
exercise_2_26 | test | /-- Prove that a set $U \subset M$ is open if and only if none of its points are limits of its complement.-/
| theorem exercise_2_26 {M : Type*} [TopologicalSpace M]
(U : Set M) : IsOpen U β β x β U, Β¬ ClusterPt x (π UαΆ) := | M : Type u_1
instβ : TopologicalSpace M
U : Set M
β’ IsOpen U β β x β U, Β¬ClusterPt x (π UαΆ) | import Mathlib
open Filter Real Function
open scoped Topology
| |
exercise_2_32a | test | /-- Show that every subset of $\mathbb{N}$ is clopen.-/
| theorem exercise_2_32a (A : Set β) : IsClopen A := | A : Set β
β’ IsClopen A | import Mathlib
open Filter Real Function
open scoped Topology
| |
exercise_2_46 | test | /-- Assume that $A, B$ are compact, disjoint, nonempty subsets of $M$. Prove that there are $a_0 \in A$ and $b_0 \in B$ such that for all $a \in A$ and $b \in B$ we have $d(a_0, b_0) \leq d(a, b)$.-/
| theorem exercise_2_46 {M : Type*} [MetricSpace M]
{A B : Set M} (hA : IsCompact A) (hB : IsCompact B)
(hAB : Disjoint A B) (hAβ : A β β
) (hBβ : B β β
) :
β aβ bβ, aβ β A β§ bβ β B β§ β (a : M) (b : M),
a β A β b β B β dist aβ bβ β€ dist a b := | M : Type u_1
instβ : MetricSpace M
A B : Set M
hA : IsCompact A
hB : IsCompact B
hAB : Disjoint A B
hAβ : A β β
hBβ : B β β
β’ β aβ bβ, aβ β A β§ bβ β B β§ β (a b : M), a β A β b β B β dist aβ bβ β€ dist a b | import Mathlib
open Filter Real Function
open scoped Topology
| |
exercise_2_92 | test | /-- Give a direct proof that the nested decreasing intersection of nonempty covering compact sets is nonempty.-/
| theorem exercise_2_92 {Ξ± : Type*} [TopologicalSpace Ξ±]
{s : β β Set Ξ±}
(hs : β i, IsCompact (s i))
(hs : β i, (s i).Nonempty)
(hs : β i, (s i) β (s (i + 1))) :
(β i, s i).Nonempty := | Ξ± : Type u_1
instβ : TopologicalSpace Ξ±
s : β β Set Ξ±
hsβΒΉ : β (i : β), IsCompact (s i)
hsβ : β (i : β), (s i).Nonempty
hs : β (i : β), s i β s (i + 1)
β’ (β i, s i).Nonempty | import Mathlib
open Filter Real Function
open scoped Topology
| |
exercise_3_1 | test | /-- Assume that $f \colon \mathbb{R} \rightarrow \mathbb{R}$ satisfies $|f(t)-f(x)| \leq|t-x|^{2}$ for all $t, x$. Prove that $f$ is constant.-/
| theorem exercise_3_1 {f : β β β}
(hf : β x y, |f x - f y| β€ |x - y| ^ 2) :
β c, f = Ξ» x => c := | f : β β β
hf : β (x y : β), |f x - f y| β€ |x - y| ^ 2
β’ β c, f = fun x => c | import Mathlib
open Filter Real Function
open scoped Topology
| |
exercise_3_63a | test | /-- Prove that $\sum 1/k(\log(k))^p$ converges when $p > 1$.-/
| theorem exercise_3_63a (p : β) (f : β β β) (hp : p > 1)
(h : f = Ξ» (k : β) => (1 : β) / (k * (log k) ^ p)) :
β l, Tendsto f atTop (π l) := | p : β
f : β β β
hp : p > 1
h : f = fun k => 1 / (βk * (βk).log ^ p)
β’ β l, Tendsto f atTop (π l) | import Mathlib
open Filter Real Function
open scoped Topology
| |
exercise_4_15a | test | /-- A continuous, strictly increasing function $\mu \colon (0, \infty) \rightarrow (0, \infty)$ is a modulus of continuity if $\mu(s) \rightarrow 0$ as $s \rightarrow 0$. A function $f \colon [a, b] \rightarrow \mathbb{R}$ has modulus of continuity $\mu$ if $|f(s) - f(t)| \leq \mu(|s - t|)$ for all $s, t \in [a, b]$. Prove that a function is uniformly continuous if and only if it has a modulus of continuity.-/
| theorem exercise_4_15a {Ξ± : Type*}
(a b : β) (F : Set (β β β)) :
(β x : β, β Ξ΅ > 0, β U β (π x),
(β y z : U, β f : β β β, f β F β (dist (f y) (f z) < Ξ΅)))
β
β (ΞΌ : β β β), β (x : β), (0 : β) β€ ΞΌ x β§ Tendsto ΞΌ (π 0) (π 0) β§
(β (s t : β) (f : β β β), f β F β |(f s) - (f t)| β€ ΞΌ (|s - t|)) := | Ξ± : Type u_1
a b : β
F : Set (β β β)
β’ (β (x Ξ΅ : β), Ξ΅ > 0 β β U β π x, β (y z : βU), β f β F, dist (f βy) (f βz) < Ξ΅) β
β ΞΌ, β (x : β), 0 β€ ΞΌ x β§ Tendsto ΞΌ (π 0) (π 0) β§ β (s t : β), β f β F, |f s - f t| β€ ΞΌ |s - t| | import Mathlib
open Filter Real Function
open scoped Topology
| |
exercise_2_3_2 | test | /-- Prove that the products $a b$ and $b a$ are conjugate elements in a group.-/
| theorem exercise_2_3_2 {G : Type*} [Group G] (a b : G) :
β g : G, b* a = g * a * b * gβ»ΒΉ := | G : Type u_1
instβ : Group G
a b : G
β’ β g, b * a = g * a * b * gβ»ΒΉ | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| |
exercise_2_8_6 | test | /-- Prove that the center of the product of two groups is the product of their centers.-/
| noncomputable def exercise_2_8_6 {G H : Type*} [Group G] [Group H] :
center (G Γ H) β* (center G) Γ (center H) := | G : Type u_1
H : Type u_2
instβΒΉ : Group G
instβ : Group H
β’ β₯(center (G Γ H)) β* β₯(center G) Γ β₯(center H) | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
--center of (G Γ H) equivalent, preserves multiplication with (center G) Γ (center H)
| |
exercise_3_2_7 | test | /-- Prove that every homomorphism of fields is injective.-/
| theorem exercise_3_2_7 {F : Type*} [Field F] {G : Type*} [Field G]
(Ο : F β+* G) : Injective Ο := | F : Type u_1
instβΒΉ : Field F
G : Type u_2
instβ : Field G
Ο : F β+* G
β’ Injective βΟ | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
open RingHom
| |
exercise_3_7_2 | test | /-- Let $V$ be a vector space over an infinite field $F$. Prove that $V$ is not the union of finitely many proper subspaces.-/
| theorem exercise_3_7_2 {K V : Type*} [Field K] [AddCommGroup V]
[Module K V] {ΞΉ : Type*} [Fintype ΞΉ] (Ξ³ : ΞΉ β Submodule K V)
(h : β i : ΞΉ, Ξ³ i β β€) :
(β (i : ΞΉ), (Ξ³ i : Set V)) β β€ := | K : Type u_1
V : Type u_2
instβΒ³ : Field K
instβΒ² : AddCommGroup V
instβΒΉ : Module K V
ΞΉ : Type u_3
instβ : Fintype ΞΉ
Ξ³ : ΞΉ β Submodule K V
h : β (i : ΞΉ), Ξ³ i β β€
β’ β i, β(Ξ³ i) β β€ | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| |
exercise_6_4_2 | test | /-- Prove that no group of order $p q$, where $p$ and $q$ are prime, is simple.-/
| theorem exercise_6_4_2 {G : Type*} [Group G] [Fintype G] {p q : β}
(hp : Prime p) (hq : Prime q) (hG : card G = p*q) :
IsSimpleGroup G β false := | G : Type u_1
instβΒΉ : Group G
instβ : Fintype G
p q : β
hp : Prime p
hq : Prime q
hG : card G = p * q
β’ IsSimpleGroup G β false = true | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| |
exercise_6_4_12 | test | /-- Prove that no group of order 224 is simple.-/
| theorem exercise_6_4_12 {G : Type*} [Group G] [Fintype G]
(hG : card G = 224) :
IsSimpleGroup G β false := | G : Type u_1
instβΒΉ : Group G
instβ : Fintype G
hG : card G = 224
β’ IsSimpleGroup G β false = true | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| |
exercise_10_1_13 | test | /-- An element $x$ of a ring $R$ is called nilpotent if some power of $x$ is zero. Prove that if $x$ is nilpotent, then $1+x$ is a unit in $R$.-/
| theorem exercise_10_1_13 {R : Type*} [Ring R] {x : R}
(hx : IsNilpotent x) : IsUnit (1 + x) := | R : Type u_1
instβ : Ring R
x : R
hx : IsNilpotent x
β’ IsUnit (1 + x) | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| |
exercise_10_6_7 | test | /-- Prove that every nonzero ideal in the ring of Gauss integers contains a nonzero integer.-/
| theorem exercise_10_6_7 {I : Ideal GaussianInt}
(hI : I β β₯) : β (z : I), z β 0 β§ (z : GaussianInt).im = 0 := | I : Ideal GaussianInt
hI : I β β₯
β’ β z, z β 0 β§ (βz).im = 0 | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| |
exercise_10_4_7a | test | /-- Let $I, J$ be ideals of a ring $R$ such that $I+J=R$. Prove that $I J=I \cap J$.-/
| theorem exercise_10_4_7a {R : Type*} [CommRing R] [NoZeroDivisors R]
(I J : Ideal R) (hIJ : I + J = β€) : I * J = I β J := | R : Type u_1
instβΒΉ : CommRing R
instβ : NoZeroDivisors R
I J : Ideal R
hIJ : I + J = β€
β’ I * J = I β J | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| |
exercise_11_2_13 | test | /-- If $a, b$ are integers and if $a$ divides $b$ in the ring of Gauss integers, then $a$ divides $b$ in $\mathbb{Z}$.-/
| theorem exercise_11_2_13 (a b : β€) :
(ofInt a : GaussianInt) β£ ofInt b β a β£ b := | a b : β€
β’ ofInt a β£ ofInt b β a β£ b | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| |
exercise_11_4_6a | test | /-- Prove that $x^2+x+1$ is irreducible in the field $\mathbb{F}_2$.-/
| theorem exercise_11_4_6a {F : Type*} [Field F] [Fintype F] (hF : card F = 7) :
Irreducible (X ^ 2 + 1 : Polynomial F) := | F : Type u_1
instβΒΉ : Field F
instβ : Fintype F
hF : card F = 7
β’ Irreducible (X ^ 2 + 1) | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| |
exercise_11_4_6c | test | /-- Prove that $x^3 - 9$ is irreducible in $\mathbb{F}_{31}$.-/
| theorem exercise_11_4_6c : Irreducible (X^3 - 9 : Polynomial (ZMod 31)) := | β’ Irreducible (X ^ 3 - 9) | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| |
exercise_11_13_3 | test | /-- Prove that there are infinitely many primes congruent to $-1$ (modulo $4$).-/
| theorem exercise_11_13_3 (N : β):
β p β₯ N, Nat.Prime p β§ p + 1 β‘ 0 [MOD 4] := | N : β
β’ β p β₯ N, p.Prime β§ p + 1 β‘ 0 [MOD 4] | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| |
exercise_13_6_10 | test | /-- Let $K$ be a finite field. Prove that the product of the nonzero elements of $K$ is $-1$.-/
| theorem exercise_13_6_10 {K : Type*} [Field K] [Fintype KΛ£] :
(β x : KΛ£, x) = -1 := | K : Type u_1
instβΒΉ : Field K
instβ : Fintype KΛ£
β’ β x : KΛ£, x = -1 | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| |
exercise_1_2 | test | /-- Show that $\frac{-1 + \sqrt{3}i}{2}$ is a cube root of 1 (meaning that its cube equals 1).-/
| theorem exercise_1_2 :
(β¨-1/2, Real.sqrt 3 / 2β© : β) ^ 3 = -1 := | β’ { re := -1 / 2, im := β3 / 2 } ^ 3 = -1 | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
open scoped BigOperators
| |
exercise_1_4 | test | /-- Prove that if $a \in \mathbf{F}$, $v \in V$, and $av = 0$, then $a = 0$ or $v = 0$.-/
| theorem exercise_1_4 {F V : Type*} [AddCommGroup V] [Field F]
[Module F V] (v : V) (a : F): a β’ v = 0 β a = 0 β¨ v = 0 := | F : Type u_1
V : Type u_2
instβΒ² : AddCommGroup V
instβΒΉ : Field F
instβ : Module F V
v : V
a : F
β’ a β’ v = 0 β a = 0 β¨ v = 0 | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| |
exercise_1_7 | test | /-- Give an example of a nonempty subset $U$ of $\mathbf{R}^2$ such that $U$ is closed under scalar multiplication, but $U$ is not a subspace of $\mathbf{R}^2$.-/
| theorem exercise_1_7 : β U : Set (β Γ β),
(U β β
) β§
(β (c : β) (u : β Γ β), u β U β c β’ u β U) β§
(β U' : Submodule β (β Γ β), U β βU') := | β’ β U, U β β
β§ (β (c : β), β u β U, c β’ u β U) β§ β (U' : Submodule β (β Γ β)), U β βU' | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| |
exercise_1_9 | test | /-- Prove that the union of two subspaces of $V$ is a subspace of $V$ if and only if one of the subspaces is contained in the other.-/
| theorem exercise_1_9 {F V : Type*} [AddCommGroup V] [Field F]
[Module F V] (U W : Submodule F V):
β U' : Submodule F V, (U'.carrier = βU β© βW β (U β€ W β¨ W β€ U)) := | F : Type u_1
V : Type u_2
instβΒ² : AddCommGroup V
instβΒΉ : Field F
instβ : Module F V
U W : Submodule F V
β’ β U', U'.carrier = βU β© βW β U β€ W β¨ W β€ U | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| |
exercise_3_8 | test | /-- Suppose that $V$ is finite dimensional and that $T \in \mathcal{L}(V, W)$. Prove that there exists a subspace $U$ of $V$ such that $U \cap \operatorname{null} T=\{0\}$ and range $T=\{T u: u \in U\}$.-/
| theorem exercise_3_8 {F V W : Type*} [AddCommGroup V]
[AddCommGroup W] [Field F] [Module F V] [Module F W]
(L : V ββ[F] W) :
β U : Submodule F V, U β (ker L) = β₯ β§
(range L = range (domRestrict L U)) := | F : Type u_1
V : Type u_2
W : Type u_3
instββ΄ : AddCommGroup V
instβΒ³ : AddCommGroup W
instβΒ² : Field F
instβΒΉ : Module F V
instβ : Module F W
L : V ββ[F] W
β’ β U, U β ker L = β₯ β§ range L = range (L.domRestrict U) | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| |
exercise_5_1 | test | /-- Suppose $T \in \mathcal{L}(V)$. Prove that if $U_{1}, \ldots, U_{m}$ are subspaces of $V$ invariant under $T$, then $U_{1}+\cdots+U_{m}$ is invariant under $T$.-/
| theorem exercise_5_1 {F V : Type*} [AddCommGroup V] [Field F]
[Module F V] {L : V ββ[F] V} {n : β} (U : Fin n β Submodule F V)
(hU : β i : Fin n, Submodule.map L (U i) = U i) :
Submodule.map L (β i : Fin n, U i : Submodule F V) =
(β i : Fin n, U i : Submodule F V) := | F : Type u_1
V : Type u_2
instβΒ² : AddCommGroup V
instβΒΉ : Field F
instβ : Module F V
L : V ββ[F] V
n : β
U : Fin n β Submodule F V
hU : β (i : Fin n), Submodule.map L (U i) = U i
β’ Submodule.map L (β i : Fin n, U i) = β i : Fin n, U i | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| |
exercise_5_11 | test | /-- Suppose $S, T \in \mathcal{L}(V)$. Prove that $S T$ and $T S$ have the same eigenvalues.-/
| theorem exercise_5_11 {F V : Type*} [AddCommGroup V] [Field F]
[Module F V] (S T : End F V) :
(S * T).Eigenvalues = (T * S).Eigenvalues := | F : Type u_1
V : Type u_2
instβΒ² : AddCommGroup V
instβΒΉ : Field F
instβ : Module F V
S T : End F V
β’ (S * T).Eigenvalues = (T * S).Eigenvalues | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| |
exercise_5_13 | test | /-- Suppose $T \in \mathcal{L}(V)$ is such that every subspace of $V$ with dimension $\operatorname{dim} V-1$ is invariant under $T$. Prove that $T$ is a scalar multiple of the identity operator.-/
| theorem exercise_5_13 {F V : Type*} [AddCommGroup V] [Field F]
[Module F V] [FiniteDimensional F V] {T : End F V}
(hS : β U : Submodule F V, finrank F U = finrank F V - 1 β
Submodule.map T U = U) : β c : F, T = c β’ LinearMap.id := | F : Type u_1
V : Type u_2
instβΒ³ : AddCommGroup V
instβΒ² : Field F
instβΒΉ : Module F V
instβ : FiniteDimensional F V
T : End F V
hS : β (U : Submodule F V), finrank F β₯U = finrank F V - 1 β Submodule.map T U = U
β’ β c, T = c β’ LinearMap.id | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| |
exercise_5_24 | test | /-- Suppose $V$ is a real vector space and $T \in \mathcal{L}(V)$ has no eigenvalues. Prove that every subspace of $V$ invariant under $T$ has even dimension.-/
| theorem exercise_5_24 {V : Type*} [AddCommGroup V]
[Module β V] [FiniteDimensional β V] {T : End β V}
(hT : β c : β, eigenspace T c = β₯) {U : Submodule β V}
(hU : Submodule.map T U = U) : Even (finrank U) := | V : Type u_1
instβΒ² : AddCommGroup V
instβΒΉ : Module β V
instβ : FiniteDimensional β V
T : End β V
hT : β (c : β), T.eigenspace c = β₯
U : Submodule β V
hU : Submodule.map T U = U
β’ Even (finrank β₯U) | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| |
exercise_6_3 | test | /-- Prove that $\left(\sum_{j=1}^{n} a_{j} b_{j}\right)^{2} \leq\left(\sum_{j=1}^{n} j a_{j}{ }^{2}\right)\left(\sum_{j=1}^{n} \frac{b_{j}{ }^{2}}{j}\right)$ for all real numbers $a_{1}, \ldots, a_{n}$ and $b_{1}, \ldots, b_{n}$.-/
| theorem exercise_6_3 {n : β} (a b : Fin n β β) :
(β i, a i * b i) ^ 2 β€ (β i : Fin n, i * a i ^ 2) * (β i, b i ^ 2 / i) := | n : β
a b : Fin n β β
β’ (β i : Fin n, a i * b i) ^ 2 β€ (β i : Fin n, ββi * a i ^ 2) * β i : Fin n, b i ^ 2 / ββi | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| |
exercise_6_13 | test | /-- Suppose $\left(e_{1}, \ldots, e_{m}\right)$ is an or thonormal list of vectors in $V$. Let $v \in V$. Prove that $\|v\|^{2}=\left|\left\langle v, e_{1}\right\rangle\right|^{2}+\cdots+\left|\left\langle v, e_{m}\right\rangle\right|^{2}$ if and only if $v \in \operatorname{span}\left(e_{1}, \ldots, e_{m}\right)$.-/
| theorem exercise_6_13 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace β V] {n : β}
{e : Fin n β V} (he : Orthonormal β e) (v : V) :
βvβ^2 = β i : Fin n, ββͺv, e iβ«_ββ^2 β v β Submodule.span β (e '' Set.univ) := | V : Type u_1
instβΒΉ : NormedAddCommGroup V
instβ : InnerProductSpace β V
n : β
e : Fin n β V
he : Orthonormal β e
v : V
β’ βvβ ^ 2 = β i : Fin n, ββͺv, e iβ«_ββ ^ 2 β v β Submodule.span β (e '' Set.univ) | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| |
exercise_7_5 | test | /-- Show that if $\operatorname{dim} V \geq 2$, then the set of normal operators on $V$ is not a subspace of $\mathcal{L}(V)$.-/
| theorem exercise_7_5 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace β V]
[FiniteDimensional β V] (hV : finrank V β₯ 2) :
β U : Submodule β (End β V), U.carrier β
{T | T * adjoint T = adjoint T * T} := | V : Type u_1
instβΒ² : NormedAddCommGroup V
instβΒΉ : InnerProductSpace β V
instβ : FiniteDimensional β V
hV : finrank V β₯ 2
β’ β (U : Submodule β (End β V)), U.carrier β {T | T * adjoint T = adjoint T * T} | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| |
exercise_7_9 | test | /-- Prove that a normal operator on a complex inner-product space is self-adjoint if and only if all its eigenvalues are real.-/
| theorem exercise_7_9 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace β V]
[FiniteDimensional β V] (T : End β V)
(hT : T * adjoint T = adjoint T * T) :
IsSelfAdjoint T β β e : T.Eigenvalues, (e : β).im = 0 := | V : Type u_1
instβΒ² : NormedAddCommGroup V
instβΒΉ : InnerProductSpace β V
instβ : FiniteDimensional β V
T : End β V
hT : T * adjoint T = adjoint T * T
β’ IsSelfAdjoint T β β (e : T.Eigenvalues), (βT e).im = 0 | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| |
exercise_7_11 | test | /-- Suppose $V$ is a complex inner-product space. Prove that every normal operator on $V$ has a square root. (An operator $S \in \mathcal{L}(V)$ is called a square root of $T \in \mathcal{L}(V)$ if $S^{2}=T$.)-/
| theorem exercise_7_11 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace β V]
[FiniteDimensional β V] {T : End β V} (hT : T*adjoint T = adjoint T*T) :
β (S : End β V), S ^ 2 = T := | V : Type u_1
instβΒ² : NormedAddCommGroup V
instβΒΉ : InnerProductSpace β V
instβ : FiniteDimensional β V
T : End β V
hT : T * adjoint T = adjoint T * T
β’ β S, S ^ 2 = T | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| |
exercise_1_1_2a | test | /-- Prove the the operation $\star$ on $\mathbb{Z}$ defined by $a\star b=a-b$ is not commutative.-/
| theorem exercise_1_1_2a : β a b : β€, a - b β b - a := | β’ β a b, a - b β b - a | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_1_1_4 | test | /-- Prove that the multiplication of residue class $\mathbb{Z}/n\mathbb{Z}$ is associative.-/
| theorem exercise_1_1_4 (n : β) :
β (a b c : β), (a * b) * c β‘ a * (b * c) [ZMOD n] := | n : β
β’ β (a b c : β), βa * βb * βc β‘ βa * (βb * βc) [ZMOD βn] | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_1_1_15 | test | /-- Prove that $(a_1a_2\dots a_n)^{-1} = a_n^{-1}a_{n-1}^{-1}\dots a_1^{-1}$ for all $a_1, a_2, \dots, a_n\in G$.-/
| theorem exercise_1_1_15 {G : Type*} [Group G] (as : List G) :
as.prodβ»ΒΉ = (as.reverse.map (Ξ» x => xβ»ΒΉ)).prod := | G : Type u_1
instβ : Group G
as : List G
β’ as.prodβ»ΒΉ = (List.map (fun x => xβ»ΒΉ) as.reverse).prod | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_1_1_17 | test | /-- Let $x$ be an element of $G$. Prove that if $|x|=n$ for some positive integer $n$ then $x^{-1}=x^{n-1}$.-/
| theorem exercise_1_1_17 {G : Type*} [Group G] {x : G} {n : β}
(hxn: orderOf x = n) :
xβ»ΒΉ = x ^ (n - 1 : β€) := | G : Type u_1
instβ : Group G
x : G
n : β
hxn : orderOf x = n
β’ xβ»ΒΉ = x ^ (βn - 1) | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|