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exercise_1_1_25 | valid | /-- Prove that if $x^{2}=1$ for all $x \in G$ then $G$ is abelian.-/
| theorem exercise_1_1_25 {G : Type*} [Group G]
(h : β x : G, x ^ 2 = 1) : β a b : G, a*b = b*a := | G : Type u_1
instβ : Group G
h : β (x : G), x ^ 2 = 1
β’ β (a b : G), a * b = b * a | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_1_1_34 | valid | /-- If $x$ is an element of infinite order in $G$, prove that the elements $x^{n}, n \in \mathbb{Z}$ are all distinct.-/
| theorem exercise_1_1_34 {G : Type*} [Group G] {x : G}
(hx_inf : orderOf x = 0) (n m : β€) :
x ^ n β x ^ m := | G : Type u_1
instβ : Group G
x : G
hx_inf : orderOf x = 0
n m : β€
β’ x ^ n β x ^ m | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_1_6_4 | valid | /-- Prove that the multiplicative groups $\mathbb{R}-\{0\}$ and $\mathbb{C}-\{0\}$ are not isomorphic.-/
| theorem exercise_1_6_4 :
IsEmpty (Multiplicative β β* Multiplicative β) := | β’ IsEmpty (Multiplicative β β* Multiplicative β) | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_1_6_17 | valid | /-- Let $G$ be any group. Prove that the map from $G$ to itself defined by $g \mapsto g^{-1}$ is a homomorphism if and only if $G$ is abelian.-/
| theorem exercise_1_6_17 {G : Type*} [Group G] (f : G β G)
(hf : f = Ξ» g => gβ»ΒΉ) :
β x y : G, f x * f y = f (x*y) β β x y : G, x*y = y*x := | G : Type u_1
instβ : Group G
f : G β G
hf : f = fun g => gβ»ΒΉ
β’ β (x y : G), f x * f y = f (x * y) β β (x y : G), x * y = y * x | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_2_1_5 | valid | /-- Prove that $G$ cannot have a subgroup $H$ with $|H|=n-1$, where $n=|G|>2$.-/
| theorem exercise_2_1_5 {G : Type*} [Group G] [Fintype G]
(hG : card G > 2) (H : Subgroup G) [Fintype H] :
card H β card G - 1 := | G : Type u_1
instβΒ² : Group G
instβΒΉ : Fintype G
hG : card G > 2
H : Subgroup G
instβ : Fintype β₯H
β’ card β₯H β card G - 1 | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_2_4_4 | valid | /-- Prove that if $H$ is a subgroup of $G$ then $H$ is generated by the set $H-\{1\}$.-/
| theorem exercise_2_4_4 {G : Type*} [Group G] (H : Subgroup G) :
closure ((H : Set G) \ {1}) = β€ := | G : Type u_1
instβ : Group G
H : Subgroup G
β’ Subgroup.closure (βH \ {1}) = β€ | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_2_4_16b | valid | /-- Show that the subgroup of all rotations in a dihedral group is a maximal subgroup.-/
| theorem exercise_2_4_16b {n : β} {hn : n β 0}
{R : Subgroup (DihedralGroup n)}
(hR : R = Subgroup.closure {DihedralGroup.r 1}) :
R β β€ β§
β K : Subgroup (DihedralGroup n), R β€ K β K = R β¨ K = β€ := | n : β
hn : n β 0
R : Subgroup (DihedralGroup n)
hR : R = Subgroup.closure {DihedralGroup.r 1}
β’ R β β€ β§ β (K : Subgroup (DihedralGroup n)), R β€ K β K = R β¨ K = β€ | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_3_1_3a | valid | /-- Let $A$ be an abelian group and let $B$ be a subgroup of $A$. Prove that $A / B$ is abelian.-/
| theorem exercise_3_1_3a {A : Type*} [CommGroup A] (B : Subgroup A) :
β a b : A β§Έ B, a*b = b*a := | A : Type u_1
instβ : CommGroup A
B : Subgroup A
β’ β (a b : A β§Έ B), a * b = b * a | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_3_1_22b | valid | /-- Prove that the intersection of an arbitrary nonempty collection of normal subgroups of a group is a normal subgroup (do not assume the collection is countable).-/
| theorem exercise_3_1_22b {G : Type*} [Group G] (I : Type*)
(H : I β Subgroup G) (hH : β i : I, Normal (H i)) :
Normal (β¨
(i : I), H i) := | G : Type u_1
instβ : Group G
I : Type u_2
H : I β Subgroup G
hH : β (i : I), (H i).Normal
β’ (β¨
i, H i).Normal | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_3_2_11 | valid | /-- Let $H \leq K \leq G$. Prove that $|G: H|=|G: K| \cdot|K: H|$ (do not assume $G$ is finite).-/
| theorem exercise_3_2_11 {G : Type*} [Group G] {H K : Subgroup G}
(hHK : H β€ K) :
H.index = K.index * H.relindex K := | G : Type u_1
instβ : Group G
H K : Subgroup G
hHK : H β€ K
β’ H.index = K.index * H.relindex K | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_3_2_21a | valid | /-- Prove that $\mathbb{Q}$ has no proper subgroups of finite index.-/
| theorem exercise_3_2_21a (H : AddSubgroup β) (hH : H β β€) : H.index = 0 := | H : AddSubgroup β
hH : H β β€
β’ H.index = 0 | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_3_4_1 | valid | /-- Prove that if $G$ is an abelian simple group then $G \cong Z_{p}$ for some prime $p$ (do not assume $G$ is a finite group).-/
| theorem exercise_3_4_1 (G : Type*) [CommGroup G] [IsSimpleGroup G] :
IsCyclic G β§ β G_fin : Fintype G, Nat.Prime (@card G G_fin) := | G : Type u_1
instβΒΉ : CommGroup G
instβ : IsSimpleGroup G
β’ IsCyclic G β§ β G_fin, (card G).Prime | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_3_4_5a | valid | /-- Prove that subgroups of a solvable group are solvable.-/
| theorem exercise_3_4_5a {G : Type*} [Group G]
(H : Subgroup G) [IsSolvable G] : IsSolvable H := | G : Type u_1
instβΒΉ : Group G
H : Subgroup G
instβ : IsSolvable G
β’ IsSolvable β₯H | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_3_4_11 | valid | /-- Prove that if $H$ is a nontrivial normal subgroup of the solvable group $G$ then there is a nontrivial subgroup $A$ of $H$ with $A \unlhd G$ and $A$ abelian.-/
| theorem exercise_3_4_11 {G : Type*} [Group G] [IsSolvable G]
{H : Subgroup G} (hH : H β β₯) [H.Normal] :
β A β€ H, A.Normal β§ β a b : A, a*b = b*a := | G : Type u_1
instβΒ² : Group G
instβΒΉ : IsSolvable G
H : Subgroup G
hH : H β β₯
instβ : H.Normal
β’ β A β€ H, A.Normal β§ β (a b : β₯A), a * b = b * a | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_4_3_26 | valid | /-- Let $G$ be a transitive permutation group on the finite set $A$ with $|A|>1$. Show that there is some $\sigma \in G$ such that $\sigma(a) \neq a$ for all $a \in A$.-/
| theorem exercise_4_3_26 {Ξ± : Type*} [Fintype Ξ±] (ha : card Ξ± > 1)
(h_tran : β a b: Ξ±, β Ο : Equiv.Perm Ξ±, Ο a = b) :
β Ο : Equiv.Perm Ξ±, β a : Ξ±, Ο a β a := | Ξ± : Type u_1
instβ : Fintype Ξ±
ha : card Ξ± > 1
h_tran : β (a b : Ξ±), β Ο, Ο a = b
β’ β Ο, β (a : Ξ±), Ο a β a | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_4_2_14 | valid | /-- Let $G$ be a finite group of composite order $n$ with the property that $G$ has a subgroup of order $k$ for each positive integer $k$ dividing $n$. Prove that $G$ is not simple.-/
| theorem exercise_4_2_14 {G : Type*} [Fintype G] [Group G]
(hG : Β¬ (card G).Prime) (hG1 : β k : β, k β£ card G β
β (H : Subgroup G) (fH : Fintype H), @card H fH = k) :
Β¬ IsSimpleGroup G := | G : Type u_1
instβΒΉ : Fintype G
instβ : Group G
hG : Β¬(card G).Prime
hG1 : β (k : β), k β£ card G β β H fH, card β₯H = k
β’ Β¬IsSimpleGroup G | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_4_4_6a | valid | /-- Prove that characteristic subgroups are normal.-/
| theorem exercise_4_4_6a {G : Type*} [Group G] (H : Subgroup G)
[Characteristic H] : Normal H := | G : Type u_1
instβΒΉ : Group G
H : Subgroup G
instβ : H.Characteristic
β’ H.Normal | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_4_4_7 | valid | /-- If $H$ is the unique subgroup of a given order in a group $G$ prove $H$ is characteristic in $G$.-/
| theorem exercise_4_4_7 {G : Type*} [Group G] {H : Subgroup G} [Fintype H]
(hH : β (K : Subgroup G) (fK : Fintype K), card H = @card K fK β H = K) :
H.Characteristic := | G : Type u_1
instβΒΉ : Group G
H : Subgroup G
instβ : Fintype β₯H
hH : β (K : Subgroup G) (fK : Fintype β₯K), card β₯H = card β₯K β H = K
β’ H.Characteristic | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_4_5_1a | valid | /-- Prove that if $P \in \operatorname{Syl}_{p}(G)$ and $H$ is a subgroup of $G$ containing $P$ then $P \in \operatorname{Syl}_{p}(H)$.-/
| theorem exercise_4_5_1a {p : β} {G : Type*} [Group G]
{P : Subgroup G} (hP : IsPGroup p P) (H : Subgroup G)
(hH : P β€ H) : IsPGroup p H := | p : β
G : Type u_1
instβ : Group G
P : Subgroup G
hP : IsPGroup p β₯P
H : Subgroup G
hH : P β€ H
β’ IsPGroup p β₯H | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_4_5_14 | valid | /-- Prove that a group of order 312 has a normal Sylow $p$-subgroup for some prime $p$ dividing its order.-/
| theorem exercise_4_5_14 {G : Type*} [Group G] [Fintype G]
(hG : card G = 312) :
β (p : β) (P : Sylow p G), P.Normal := | G : Type u_1
instβΒΉ : Group G
instβ : Fintype G
hG : card G = 312
β’ β p P, (βP).Normal | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_4_5_16 | valid | /-- Let $|G|=p q r$, where $p, q$ and $r$ are primes with $p<q<r$. Prove that $G$ has a normal Sylow subgroup for either $p, q$ or $r$.-/
| theorem exercise_4_5_16 {p q r : β} {G : Type*} [Group G]
[Fintype G] (hpqr : p < q β§ q < r)
(hpqr1 : p.Prime β§ q.Prime β§ r.Prime)(hG : card G = p*q*r) :
Nonempty (Sylow p G) β¨ Nonempty (Sylow q G) β¨ Nonempty (Sylow r G) := | p q r : β
G : Type u_1
instβΒΉ : Group G
instβ : Fintype G
hpqr : p < q β§ q < r
hpqr1 : p.Prime β§ q.Prime β§ r.Prime
hG : card G = p * q * r
β’ Nonempty (Sylow p G) β¨ Nonempty (Sylow q G) β¨ Nonempty (Sylow r G) | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_4_5_18 | valid | /-- Prove that a group of order 200 has a normal Sylow 5-subgroup.-/
| theorem exercise_4_5_18 {G : Type*} [Fintype G] [Group G]
(hG : card G = 200) :
β N : Sylow 5 G, N.Normal := | G : Type u_1
instβΒΉ : Fintype G
instβ : Group G
hG : card G = 200
β’ β N, (βN).Normal | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_4_5_20 | valid | /-- Prove that if $|G|=1365$ then $G$ is not simple.-/
| theorem exercise_4_5_20 {G : Type*} [Fintype G] [Group G]
(hG : card G = 1365) : Β¬ IsSimpleGroup G := | G : Type u_1
instβΒΉ : Fintype G
instβ : Group G
hG : card G = 1365
β’ Β¬IsSimpleGroup G | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_4_5_22 | valid | /-- Prove that if $|G|=132$ then $G$ is not simple.-/
| theorem exercise_4_5_22 {G : Type*} [Fintype G] [Group G]
(hG : card G = 132) : Β¬ IsSimpleGroup G := | G : Type u_1
instβΒΉ : Fintype G
instβ : Group G
hG : card G = 132
β’ Β¬IsSimpleGroup G | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_4_5_28 | valid | /-- Let $G$ be a group of order 105. Prove that if a Sylow 3-subgroup of $G$ is normal then $G$ is abelian.-/
| def exercise_4_5_28 {G : Type*} [Group G] [Fintype G]
(hG : card G = 105) (P : Sylow 3 G) [hP : P.Normal] :
CommGroup G := | G : Type u_1
instβΒΉ : Group G
instβ : Fintype G
hG : card G = 105
P : Sylow 3 G
hP : (βP).Normal
β’ CommGroup G | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_5_4_2 | valid | /-- Prove that a subgroup $H$ of $G$ is normal if and only if $[G, H] \leq H$.-/
| theorem exercise_5_4_2 {G : Type*} [Group G] (H : Subgroup G) :
H.Normal β β
(β€ : Subgroup G), Hβ β€ H := | G : Type u_1
instβ : Group G
H : Subgroup G
β’ H.Normal β β
β€, Hβ β€ H | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_7_1_11 | valid | /-- Prove that if $R$ is an integral domain and $x^{2}=1$ for some $x \in R$ then $x=\pm 1$.-/
| theorem exercise_7_1_11 {R : Type*} [Ring R] [IsDomain R]
{x : R} (hx : x^2 = 1) : x = 1 β¨ x = -1 := | R : Type u_1
instβΒΉ : Ring R
instβ : IsDomain R
x : R
hx : x ^ 2 = 1
β’ x = 1 β¨ x = -1 | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_7_1_15 | valid | /-- A ring $R$ is called a Boolean ring if $a^{2}=a$ for all $a \in R$. Prove that every Boolean ring is commutative.-/
| def exercise_7_1_15 {R : Type*} [Ring R] (hR : β a : R, a^2 = a) :
CommRing R := | R : Type u_1
instβ : Ring R
hR : β (a : R), a ^ 2 = a
β’ CommRing R | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_7_2_12 | valid | /-- Let $G=\left\{g_{1}, \ldots, g_{n}\right\}$ be a finite group. Prove that the element $N=g_{1}+g_{2}+\ldots+g_{n}$ is in the center of the group ring $R G$.-/
| theorem exercise_7_2_12 {R G : Type*} [Ring R] [Group G] [Fintype G] :
β g : G, MonoidAlgebra.of R G g β center (MonoidAlgebra R G) := | R : Type u_1
G : Type u_2
instβΒ² : Ring R
instβΒΉ : Group G
instβ : Fintype G
β’ β g : G, (MonoidAlgebra.of R G) g β Set.center (MonoidAlgebra R G) | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_7_3_37 | valid | /-- An ideal $N$ is called nilpotent if $N^{n}$ is the zero ideal for some $n \geq 1$. Prove that the ideal $p \mathbb{Z} / p^{m} \mathbb{Z}$ is a nilpotent ideal in the ring $\mathbb{Z} / p^{m} \mathbb{Z}$.-/
| theorem exercise_7_3_37 {p m : β} (hp : p.Prime) :
IsNilpotent (span ({βp} : Set $ ZMod $ p^m) : Ideal $ ZMod $ p^m) := | p m : β
hp : p.Prime
β’ IsNilpotent (span {βp}) | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_8_1_12 | valid | /-- Let $N$ be a positive integer. Let $M$ be an integer relatively prime to $N$ and let $d$ be an integer relatively prime to $\varphi(N)$, where $\varphi$ denotes Euler's $\varphi$-function. Prove that if $M_{1} \equiv M^{d} \pmod N$ then $M \equiv M_{1}^{d^{\prime}} \pmod N$ where $d^{\prime}$ is the inverse of $d \bmod \varphi(N)$: $d d^{\prime} \equiv 1 \pmod {\varphi(N)}$.-/
| theorem exercise_8_1_12 {N : β} (hN : N > 0) {M M': β€} {d : β}
(hMN : M.gcd N = 1) (hMd : d.gcd N.totient = 1)
(hM' : M' β‘ M^d [ZMOD N]) :
β d' : β, d' * d β‘ 1 [ZMOD N.totient] β§
M β‘ M'^d' [ZMOD N] := | N : β
hN : N > 0
M M' : β€
d : β
hMN : M.gcd βN = 1
hMd : d.gcd N.totient = 1
hM' : M' β‘ M ^ d [ZMOD βN]
β’ β d', βd' * βd β‘ 1 [ZMOD βN.totient] β§ M β‘ M' ^ d' [ZMOD βN] | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_8_3_4 | valid | /-- Prove that if an integer is the sum of two rational squares, then it is the sum of two integer squares.-/
| theorem exercise_8_3_4 {R : Type*} {n : β€} {r s : β}
(h : r^2 + s^2 = n) :
β a b : β€, a^2 + b^2 = n := | R : Type u_1
n : β€
r s : β
h : r ^ 2 + s ^ 2 = βn
β’ β a b, a ^ 2 + b ^ 2 = n | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_8_3_6a | valid | /-- Prove that the quotient ring $\mathbb{Z}[i] /(1+i)$ is a field of order 2.-/
| theorem exercise_8_3_6a {R : Type} [Ring R]
(hR : R = (GaussianInt β§Έ span ({β¨0, 1β©} : Set GaussianInt))) :
IsField R β§ β finR : Fintype R, @card R finR = 2 := | R : Type
instβ : Ring R
hR : R = (GaussianInt β§Έ span {{ re := 0, im := 1 }})
β’ IsField R β§ β finR, card R = 2 | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_9_1_6 | valid | /-- Prove that $(x, y)$ is not a principal ideal in $\mathbb{Q}[x, y]$.-/
| theorem exercise_9_1_6 : Β¬ Submodule.IsPrincipal
(span ({MvPolynomial.X 0, MvPolynomial.X 1} : Set (MvPolynomial (Fin 2) β))) := | β’ Β¬Submodule.IsPrincipal (span {MvPolynomial.X 0, MvPolynomial.X 1}) | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_9_3_2 | valid | /-- Prove that if $f(x)$ and $g(x)$ are polynomials with rational coefficients whose product $f(x) g(x)$ has integer coefficients, then the product of any coefficient of $g(x)$ with any coefficient of $f(x)$ is an integer.-/
| theorem exercise_9_3_2 {f g : Polynomial β} (i j : β)
(hfg : β n : β, β a : β€, (f*g).coeff = a) :
β a : β€, f.coeff i * g.coeff j = a := | f g : β[X]
i j : β
hfg : β β β a, (f * g).coeff = βa
β’ β a, f.coeff i * g.coeff j = βa | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_9_4_2b | valid | /-- Prove that $x^6+30x^5-15x^3 + 6x-120$ is irreducible in $\mathbb{Z}[x]$.-/
| theorem exercise_9_4_2b : Irreducible
(X^6 + 30*X^5 - 15*X^3 + 6*X - 120 : Polynomial β€) := | β’ Irreducible (X ^ 6 + 30 * X ^ 5 - 15 * X ^ 3 + 6 * X - 120) | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_9_4_2d | valid | /-- Prove that $\frac{(x+2)^p-2^p}{x}$, where $p$ is an odd prime, is irreducible in $\mathbb{Z}[x]$.-/
| theorem exercise_9_4_2d {p : β} (hp : p.Prime β§ p > 2)
{f : Polynomial β€} (hf : f = (X + 2)^p):
Irreducible (β n in (f.support \ {0}), (f.coeff n : Polynomial β€) * X ^ (n-1) :
Polynomial β€) := | p : β
hp : p.Prime β§ p > 2
f : β€[X]
hf : f = (X + 2) ^ p
β’ Irreducible (β n β f.support \ {0}, β(f.coeff n) * X ^ (n - 1)) | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_9_4_11 | valid | /-- Prove that $x^2+y^2-1$ is irreducible in $\mathbb{Q}[x,y]$.-/
| theorem exercise_9_4_11 :
Irreducible ((MvPolynomial.X 0)^2 + (MvPolynomial.X 1)^2 - 1 : MvPolynomial (Fin 2) β) := | β’ Irreducible (MvPolynomial.X 0 ^ 2 + MvPolynomial.X 1 ^ 2 - 1) | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
|
exercise_13_1 | valid | /-- Let $X$ be a topological space; let $A$ be a subset of $X$. Suppose that for each $x \in A$ there is an open set $U$ containing $x$ such that $U \subset A$. Show that $A$ is open in $X$.-/
| theorem exercise_13_1 (X : Type*) [TopologicalSpace X] (A : Set X)
(h1 : β x β A, β U : Set X, x β U β§ IsOpen U β§ U β A) :
IsOpen A := | X : Type u_1
instβ : TopologicalSpace X
A : Set X
h1 : β x β A, β U, x β U β§ IsOpen U β§ U β A
β’ IsOpen A | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
|
exercise_13_4a1 | valid | /-- If $\mathcal{T}_\alpha$ is a family of topologies on $X$, show that $\bigcap \mathcal{T}_\alpha$ is a topology on $X$.-/
| theorem exercise_13_4a1 (X I : Type*) (T : I β Set (Set X)) (h : β i, is_topology X (T i)) :
is_topology X (β i : I, T i) := | X : Type u_1
I : Type u_2
T : I β Set (Set X)
h : β (i : I), is_topology X (T i)
β’ is_topology X (β i, T i) | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
def is_topology (X : Type*) (T : Set (Set X)) :=
univ β T β§
(β s t, s β T β t β T β s β© t β T) β§
(βs, (βt β s, t β T) β sUnion s β T)
|
exercise_13_4b1 | valid | /-- Let $\mathcal{T}_\alpha$ be a family of topologies on $X$. Show that there is a unique smallest topology on $X$ containing all the collections $\mathcal{T}_\alpha$.-/
| theorem exercise_13_4b1 (X I : Type*) (T : I β Set (Set X)) (h : β i, is_topology X (T i)) :
β! T', is_topology X T' β§ (β i, T i β T') β§
β T'', is_topology X T'' β (β i, T i β T'') β T'' β T' := | X : Type u_1
I : Type u_2
T : I β Set (Set X)
h : β (i : I), is_topology X (T i)
β’ β! T',
is_topology X T' β§
(β (i : I), T i β T') β§ β (T'' : Set (Set X)), is_topology X T'' β (β (i : I), T i β T'') β T'' β T' | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
def is_topology (X : Type*) (T : Set (Set X)) :=
univ β T β§
(β s t, s β T β t β T β s β© t β T) β§
(βs, (βt β s, t β T) β sUnion s β T)
|
exercise_13_5a | valid | /-- Show that if $\mathcal{A}$ is a basis for a topology on $X$, then the topology generated by $\mathcal{A}$ equals the intersection of all topologies on $X$ that contain $\mathcal{A}$.-/
| theorem exercise_13_5a {X : Type*}
[TopologicalSpace X] (A : Set (Set X)) (hA : IsTopologicalBasis A) :
generateFrom A = generateFrom (sInter {T | is_topology X T β§ A β T}) := | X : Type u_1
instβ : TopologicalSpace X
A : Set (Set X)
hA : IsTopologicalBasis A
β’ generateFrom A = generateFrom (ββ {T | is_topology X T β§ A β T}) | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
def is_topology (X : Type*) (T : Set (Set X)) :=
univ β T β§
(β s t, s β T β t β T β s β© t β T) β§
(βs, (βt β s, t β T) β sUnion s β T)
|
exercise_13_6 | valid | /-- Show that the lower limit topology $\mathbb{R}_l$ and $K$-topology $\mathbb{R}_K$ are not comparable.-/
| theorem exercise_13_6 :
Β¬ (β U, Rl.IsOpen U β K_topology.IsOpen U) β§ Β¬ (β U, K_topology.IsOpen U β Rl.IsOpen U) := | β’ (Β¬β (U : Set β), TopologicalSpace.IsOpen U β TopologicalSpace.IsOpen U) β§
Β¬β (U : Set β), TopologicalSpace.IsOpen U β TopologicalSpace.IsOpen U | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
def lower_limit_topology (X : Type) [Preorder X] :=
generateFrom {S : Set X | β a b, a < b β§ S = Ico a b}
def Rl := lower_limit_topology β
def K : Set β := {r | β n : β, r = 1 / n}
def K_topology := generateFrom
({S : Set β | β a b, a < b β§ S = Ioo a b} βͺ {S : Set β | β a b, a < b β§ S = Ioo a b \ K})
|
exercise_13_8b | valid | /-- Show that the collection $\{(a,b) \mid a < b, a \text{ and } b \text{ rational}\}$ is a basis that generates a topology different from the lower limit topology on $\mathbb{R}$.-/
| theorem exercise_13_8b :
(generateFrom {S : Set β | β a b : β, a < b β§ S = Ico βa βb}).IsOpen β
(lower_limit_topology β).IsOpen := | β’ TopologicalSpace.IsOpen β TopologicalSpace.IsOpen | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
def lower_limit_topology (X : Type) [Preorder X] :=
generateFrom {S : Set X | β a b, a < b β§ S = Ico a b}
|
exercise_16_4 | valid | /-- A map $f: X \rightarrow Y$ is said to be an open map if for every open set $U$ of $X$, the set $f(U)$ is open in $Y$. Show that $\pi_{1}: X \times Y \rightarrow X$ and $\pi_{2}: X \times Y \rightarrow Y$ are open maps.-/
| theorem exercise_16_4 {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
(Οβ : X Γ Y β X)
(Οβ : X Γ Y β Y)
(hβ : Οβ = Prod.fst)
(hβ : Οβ = Prod.snd) :
IsOpenMap Οβ β§ IsOpenMap Οβ := | X : Type u_1
Y : Type u_2
instβΒΉ : TopologicalSpace X
instβ : TopologicalSpace Y
Οβ : X Γ Y β X
Οβ : X Γ Y β Y
hβ : Οβ = Prod.fst
hβ : Οβ = Prod.snd
β’ IsOpenMap Οβ β§ IsOpenMap Οβ | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
|
exercise_17_4 | valid | /-- Show that if $U$ is open in $X$ and $A$ is closed in $X$, then $U-A$ is open in $X$, and $A-U$ is closed in $X$.-/
| theorem exercise_17_4 {X : Type*} [TopologicalSpace X]
(U A : Set X) (hU : IsOpen U) (hA : IsClosed A) :
IsOpen (U \ A) β§ IsClosed (A \ U) := | X : Type u_1
instβ : TopologicalSpace X
U A : Set X
hU : IsOpen U
hA : IsClosed A
β’ IsOpen (U \ A) β§ IsClosed (A \ U) | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
|
exercise_18_8b | valid | /-- Let $Y$ be an ordered set in the order topology. Let $f, g: X \rightarrow Y$ be continuous. Let $h: X \rightarrow Y$ be the function $h(x)=\min \{f(x), g(x)\}.$ Show that $h$ is continuous.-/
| theorem exercise_18_8b {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
[LinearOrder Y] [OrderTopology Y] {f g : X β Y}
(hf : Continuous f) (hg : Continuous g) :
Continuous (Ξ» x => min (f x) (g x)) := | X : Type u_1
Y : Type u_2
instβΒ³ : TopologicalSpace X
instβΒ² : TopologicalSpace Y
instβΒΉ : LinearOrder Y
instβ : OrderTopology Y
f g : X β Y
hf : Continuous f
hg : Continuous g
β’ Continuous fun x => min (f x) (g x) | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
|
exercise_19_6a | valid | /-- Let $\mathbf{x}_1, \mathbf{x}_2, \ldots$ be a sequence of the points of the product space $\prod X_\alpha$. Show that this sequence converges to the point $\mathbf{x}$ if and only if the sequence $\pi_\alpha(\mathbf{x}_i)$ converges to $\pi_\alpha(\mathbf{x})$ for each $\alpha$.-/
| theorem exercise_19_6a
{n : β}
{f : Fin n β Type*} {x : β β Ξ a, f a}
(y : Ξ i, f i)
[Ξ a, TopologicalSpace (f a)] :
Tendsto x atTop (π y) β β i, Tendsto (Ξ» j => (x j) i) atTop (π (y i)) := | n : β
f : Fin n β Type u_1
x : β β (a : Fin n) β f a
y : (i : Fin n) β f i
instβ : (a : Fin n) β TopologicalSpace (f a)
β’ Tendsto x atTop (π y) β β (i : Fin n), Tendsto (fun j => x j i) atTop (π (y i)) | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
|
exercise_21_6a | valid | /-- Define $f_{n}:[0,1] \rightarrow \mathbb{R}$ by the equation $f_{n}(x)=x^{n}$. Show that the sequence $\left(f_{n}(x)\right)$ converges for each $x \in[0,1]$.-/
| theorem exercise_21_6a
(f : β β I β β )
(h : β x n, f n x = x ^ n) :
β x, β y, Tendsto (Ξ» n => f n x) atTop (π y) := | f : β β βI β β
h : β (x : βI) (n : β), f n x = βx ^ n
β’ β (x : βI), β y, Tendsto (fun n => f n x) atTop (π y) | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
abbrev I : Set β := Icc 0 1
|
exercise_21_8 | valid | /-- Let $X$ be a topological space and let $Y$ be a metric space. Let $f_{n}: X \rightarrow Y$ be a sequence of continuous functions. Let $x_{n}$ be a sequence of points of $X$ converging to $x$. Show that if the sequence $\left(f_{n}\right)$ converges uniformly to $f$, then $\left(f_{n}\left(x_{n}\right)\right)$ converges to $f(x)$.-/
| theorem exercise_21_8
{X : Type*} [TopologicalSpace X] {Y : Type*} [MetricSpace Y]
{f : β β X β Y} {x : β β X}
(hf : β n, Continuous (f n))
(xβ : X)
(hx : Tendsto x atTop (π xβ))
(fβ : X β Y)
(hh : TendstoUniformly f fβ atTop) :
Tendsto (Ξ» n => f n (x n)) atTop (π (fβ xβ)) := | X : Type u_1
instβΒΉ : TopologicalSpace X
Y : Type u_2
instβ : MetricSpace Y
f : β β X β Y
x : β β X
hf : β (n : β), Continuous (f n)
xβ : X
hx : Tendsto x atTop (π xβ)
fβ : X β Y
hh : TendstoUniformly f fβ atTop
β’ Tendsto (fun n => f n (x n)) atTop (π (fβ xβ)) | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
|
exercise_22_2b | valid | /-- If $A \subset X$, a retraction of $X$ onto $A$ is a continuous map $r: X \rightarrow A$ such that $r(a)=a$ for each $a \in A$. Show that a retraction is a quotient map.-/
| theorem exercise_22_2b {X : Type*} [TopologicalSpace X]
{A : Set X} (r : X β A) (hr : Continuous r) (h : β x : A, r x = x) :
QuotientMap r := | X : Type u_1
instβ : TopologicalSpace X
A : Set X
r : X β βA
hr : Continuous r
h : β (x : βA), r βx = x
β’ QuotientMap r | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
|
exercise_23_2 | valid | /-- Let $\left\{A_{n}\right\}$ be a sequence of connected subspaces of $X$, such that $A_{n} \cap A_{n+1} \neq \varnothing$ for all $n$. Show that $\bigcup A_{n}$ is connected.-/
| theorem exercise_23_2 {X : Type*}
[TopologicalSpace X] {A : β β Set X} (hA : β n, IsConnected (A n))
(hAn : β n, A n β© A (n + 1) β β
) :
IsConnected (β n, A n) := | X : Type u_1
instβ : TopologicalSpace X
A : β β Set X
hA : β (n : β), IsConnected (A n)
hAn : β (n : β), A n β© A (n + 1) β β
β’ IsConnected (β n, A n) | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
|
exercise_23_4 | valid | /-- Show that if $X$ is an infinite set, it is connected in the finite complement topology.-/
| theorem exercise_23_4 {X : Type*} [TopologicalSpace X] [CofiniteTopology X]
(s : Set X) : Infinite s β IsConnected s := | X : Type u_1
instβΒΉ : TopologicalSpace X
instβ : CofiniteTopology X
s : Set X
β’ Infinite βs β IsConnected s | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
set_option checkBinderAnnotations false
|
exercise_23_9 | valid | /-- Let $A$ be a proper subset of $X$, and let $B$ be a proper subset of $Y$. If $X$ and $Y$ are connected, show that $(X \times Y)-(A \times B)$ is connected.-/
| theorem exercise_23_9 {X Y : Type*}
[TopologicalSpace X] [TopologicalSpace Y]
(Aβ Aβ : Set X)
(Bβ Bβ : Set Y)
(hA : Aβ β Aβ)
(hB : Bβ β Bβ)
(hA : IsConnected Aβ)
(hB : IsConnected Bβ) :
IsConnected ({x | β a b, x = (a, b) β§ a β Aβ β§ b β Bβ} \
{x | β a b, x = (a, b) β§ a β Aβ β§ b β Bβ}) := | X : Type u_1
Y : Type u_2
instβΒΉ : TopologicalSpace X
instβ : TopologicalSpace Y
Aβ Aβ : Set X
Bβ Bβ : Set Y
hAβ : Aβ β Aβ
hBβ : Bβ β Bβ
hA : IsConnected Aβ
hB : IsConnected Bβ
β’ IsConnected ({x | β a b, x = (a, b) β§ a β Aβ β§ b β Bβ} \ {x | β a b, x = (a, b) β§ a β Aβ β§ b β Bβ}) | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
|
exercise_24_2 | valid | /-- Let $f: S^{1} \rightarrow \mathbb{R}$ be a continuous map. Show there exists a point $x$ of $S^{1}$ such that $f(x)=f(-x)$.-/
| theorem exercise_24_2 {f : (Metric.sphere 0 1 : Set β) β β}
(hf : Continuous f) : β x, f x = f (-x) := | f : β(Metric.sphere 0 1) β β
hf : Continuous f
β’ β x, f x = f (-x) | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
|
exercise_25_4 | valid | /-- Let $X$ be locally path connected. Show that every connected open set in $X$ is path connected.-/
| theorem exercise_25_4 {X : Type*} [TopologicalSpace X]
[LocPathConnectedSpace X] (U : Set X) (hU : IsOpen U)
(hcU : IsConnected U) : IsPathConnected U := | X : Type u_1
instβΒΉ : TopologicalSpace X
instβ : LocPathConnectedSpace X
U : Set X
hU : IsOpen U
hcU : IsConnected U
β’ IsPathConnected U | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
|
exercise_26_11 | valid | /-- Let $X$ be a compact Hausdorff space. Let $\mathcal{A}$ be a collection of closed connected subsets of $X$ that is simply ordered by proper inclusion. Then $Y=\bigcap_{A \in \mathcal{A}} A$ is connected.-/
| theorem exercise_26_11
{X : Type*} [TopologicalSpace X] [CompactSpace X] [T2Space X]
(A : Set (Set X)) (hA : β (a b : Set X), a β A β b β A β a β b β¨ b β a)
(hA' : β a β A, IsClosed a) (hA'' : β a β A, IsConnected a) :
IsConnected (ββ A) := | X : Type u_1
instβΒ² : TopologicalSpace X
instβΒΉ : CompactSpace X
instβ : T2Space X
A : Set (Set X)
hA : β (a b : Set X), a β A β b β A β a β b β¨ b β a
hA' : β a β A, IsClosed a
hA'' : β a β A, IsConnected a
β’ IsConnected (ββ A) | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
|
exercise_27_4 | valid | /-- Show that a connected metric space having more than one point is uncountable.-/
| theorem exercise_27_4
{X : Type*} [MetricSpace X] [ConnectedSpace X] (hX : β x y : X, x β y) :
Β¬ Countable (univ : Set X) := | X : Type u_1
instβΒΉ : MetricSpace X
instβ : ConnectedSpace X
hX : β x y, x β y
β’ Β¬Countable βuniv | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
|
exercise_28_5 | valid | /-- Show that X is countably compact if and only if every nested sequence $C_1 \supset C_2 \supset \cdots$ of closed nonempty sets of X has a nonempty intersection.-/
| theorem exercise_28_5
(X : Type*) [TopologicalSpace X] :
countably_compact X β β (C : β β Set X), (β n, IsClosed (C n)) β§
(β n, C n β β
) β§ (β n, C n β C (n + 1)) β β x, β n, x β C n := | X : Type u_1
instβ : TopologicalSpace X
β’ countably_compact X β
β (C : β β Set X),
((β (n : β), IsClosed (C n)) β§ (β (n : β), C n β β
) β§ β (n : β), C n β C (n + 1)) β β x, β (n : β), x β C n | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
def countably_compact (X : Type*) [TopologicalSpace X] :=
β U : β β Set X,
(β i, IsOpen (U i)) β§ ((univ : Set X) β β i, U i) β
(β t : Finset β, (univ : Set X) β β i β t, U i)
|
exercise_29_1 | valid | /-- Show that the rationals $\mathbb{Q}$ are not locally compact.-/
| theorem exercise_29_1 : Β¬ LocallyCompactSpace β := | β’ Β¬LocallyCompactSpace β | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
|
exercise_29_10 | valid | /-- Show that if $X$ is a Hausdorff space that is locally compact at the point $x$, then for each neighborhood $U$ of $x$, there is a neighborhood $V$ of $x$ such that $\bar{V}$ is compact and $\bar{V} \subset U$.-/
| theorem exercise_29_10 {X : Type*}
[TopologicalSpace X] [T2Space X] (x : X)
(hx : β U : Set X, x β U β§ IsOpen U β§ (β K : Set X, U β K β§ IsCompact K))
(U : Set X) (hU : IsOpen U) (hxU : x β U) :
β (V : Set X), IsOpen V β§ x β V β§ IsCompact (closure V) β§ closure V β U := | X : Type u_1
instβΒΉ : TopologicalSpace X
instβ : T2Space X
x : X
hx : β U, x β U β§ IsOpen U β§ β K, U β K β§ IsCompact K
U : Set X
hU : IsOpen U
hxU : x β U
β’ β V, IsOpen V β§ x β V β§ IsCompact (closure V) β§ closure V β U | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
|
exercise_30_13 | valid | /-- Show that if $X$ has a countable dense subset, every collection of disjoint open sets in $X$ is countable.-/
| theorem exercise_30_13 {X : Type*} [TopologicalSpace X]
(h : β (s : Set X), Countable s β§ Dense s) (U : Set (Set X))
(hU : β (x y : Set X), x β U β y β U β x β y β x β© y = β
) :
Countable U := | X : Type u_1
instβ : TopologicalSpace X
h : β s, Countable βs β§ Dense s
U : Set (Set X)
hU : β (x y : Set X), x β U β y β U β x β y β x β© y = β
β’ Countable βU | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
|
exercise_31_2 | valid | /-- Show that if $X$ is normal, every pair of disjoint closed sets have neighborhoods whose closures are disjoint.-/
| theorem exercise_31_2 {X : Type*}
[TopologicalSpace X] [NormalSpace X] {A B : Set X}
(hA : IsClosed A) (hB : IsClosed B) (hAB : Disjoint A B) :
β (U V : Set X), IsOpen U β§ IsOpen V β§ A β U β§ B β V β§ closure U β© closure V = β
:= | X : Type u_1
instβΒΉ : TopologicalSpace X
instβ : NormalSpace X
A B : Set X
hA : IsClosed A
hB : IsClosed B
hAB : Disjoint A B
β’ β U V, IsOpen U β§ IsOpen V β§ A β U β§ B β V β§ closure U β© closure V = β
| import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
|
exercise_32_1 | valid | /-- Show that a closed subspace of a normal space is normal.-/
| theorem exercise_32_1 {X : Type*} [TopologicalSpace X]
(hX : NormalSpace X) (A : Set X) (hA : IsClosed A) :
NormalSpace {x // x β A} := | X : Type u_1
instβ : TopologicalSpace X
hX : NormalSpace X
A : Set X
hA : IsClosed A
β’ NormalSpace { x // x β A } | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
|
exercise_32_2b | valid | /-- Show that if $\prod X_\alpha$ is regular, then so is $X_\alpha$. Assume that each $X_\alpha$ is nonempty.-/
| theorem exercise_32_2b
{ΞΉ : Type*} {X : ΞΉ β Type*} [β i, TopologicalSpace (X i)]
(h : β i, Nonempty (X i)) (h2 : RegularSpace (Ξ i, X i)) :
β i, RegularSpace (X i) := | ΞΉ : Type u_1
X : ΞΉ β Type u_2
instβ : (i : ΞΉ) β TopologicalSpace (X i)
h : β (i : ΞΉ), Nonempty (X i)
h2 : RegularSpace ((i : ΞΉ) β X i)
β’ β (i : ΞΉ), RegularSpace (X i) | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
|
exercise_32_3 | valid | /-- Show that every locally compact Hausdorff space is regular.-/
| theorem exercise_32_3 {X : Type*} [TopologicalSpace X]
(hX : LocallyCompactSpace X) (hX' : T2Space X) :
RegularSpace X := | X : Type u_1
instβ : TopologicalSpace X
hX : LocallyCompactSpace X
hX' : T2Space X
β’ RegularSpace X | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
|
exercise_33_8 | valid | /-- Let $X$ be completely regular, let $A$ and $B$ be disjoint closed subsets of $X$. Show that if $A$ is compact, there is a continuous function $f \colon X \rightarrow [0, 1]$ such that $f(A) = \{0\}$ and $f(B) = \{1\}$.-/
| theorem exercise_33_8
(X : Type*) [TopologicalSpace X] [RegularSpace X]
(h : β x A, IsClosed A β§ Β¬ x β A β
β (f : X β I), Continuous f β§ f x = (1 : I) β§ f '' A = {0})
(A B : Set X) (hA : IsClosed A) (hB : IsClosed B)
(hAB : Disjoint A B)
(hAc : IsCompact A) :
β (f : X β I), Continuous f β§ f '' A = {0} β§ f '' B = {1} := | X : Type u_1
instβΒΉ : TopologicalSpace X
instβ : RegularSpace X
h : β (x : X) (A : Set X), IsClosed A β§ x β A β β f, Continuous f β§ f x = 1 β§ f '' A = {0}
A B : Set X
hA : IsClosed A
hB : IsClosed B
hAB : Disjoint A B
hAc : IsCompact A
β’ β f, Continuous f β§ f '' A = {0} β§ f '' B = {1} | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
abbrev I : Set β := Icc 0 1
|
exercise_38_6 | valid | /-- Let $X$ be completely regular. Show that $X$ is connected if and only if the Stone-Δech compactification of $X$ is connected.-/
| theorem exercise_38_6 {X : Type*}
(X : Type*) [TopologicalSpace X] [RegularSpace X]
(h : β x A, IsClosed A β§ Β¬ x β A β
β (f : X β I), Continuous f β§ f x = (1 : I) β§ f '' A = {0}) :
IsConnected (univ : Set X) β IsConnected (univ : Set (StoneCech X)) := | Xβ : Type u_1
X : Type u_2
instβΒΉ : TopologicalSpace X
instβ : RegularSpace X
h : β (x : X) (A : Set X), IsClosed A β§ x β A β β f, Continuous f β§ f x = 1 β§ f '' A = {0}
β’ IsConnected univ β IsConnected univ | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
abbrev I : Set β := Icc 0 1
|
exercise_1_27 | valid | /-- For all odd $n$ show that $8 \mid n^{2}-1$.-/
| theorem exercise_1_27 {n : β} (hn : Odd n) : 8 β£ (n^2 - 1) := | n : β
hn : Odd n
β’ 8 β£ n ^ 2 - 1 | import Mathlib
open Real
open scoped BigOperators
|
exercise_1_31 | valid | /-- Show that 2 is divisible by $(1+i)^{2}$ in $\mathbb{Z}[i]$.-/
| theorem exercise_1_31 : (β¨1, 1β© : GaussianInt) ^ 2 β£ 2 := | β’ { re := 1, im := 1 } ^ 2 β£ 2 | import Mathlib
open Real
open scoped BigOperators
|
exercise_2_21 | valid | /-- Define $\wedge(n)=\log p$ if $n$ is a power of $p$ and zero otherwise. Prove that $\sum_{A \mid n} \mu(n / d) \log d$ $=\wedge(n)$.-/
| theorem exercise_2_21 {l : β β β}
(hl : β p n : β, p.Prime β l (p^n) = log p )
(hl1 : β m : β, Β¬ IsPrimePow m β l m = 0) :
l = Ξ» n => β d : Nat.divisors n, ArithmeticFunction.moebius (n/d) * log d := | l : β β β
hl : β (p n : β), p.Prime β l (p ^ n) = (βp).log
hl1 : β (m : β), Β¬IsPrimePow m β l m = 0
β’ l = fun n => β d : { x // x β n.divisors }, β(ArithmeticFunction.moebius (n / βd)) * (ββd).log | import Mathlib
open Real
open scoped BigOperators
|
exercise_3_1 | valid | /-- Show that there are infinitely many primes congruent to $-1$ modulo 6 .-/
| theorem exercise_3_1 : Infinite {p : Nat.Primes // p β‘ -1 [ZMOD 6]} := | β’ Infinite { p // ββp β‘ -1 [ZMOD 6] } | import Mathlib
open Real
open scoped BigOperators
|
exercise_3_5 | valid | /-- Show that the equation $7 x^{3}+2=y^{3}$ has no solution in integers.-/
| theorem exercise_3_5 : Β¬ β x y : β€, 7*x^3 + 2 = y^3 := | β’ Β¬β x y, 7 * x ^ 3 + 2 = y ^ 3 | import Mathlib
open Real
open scoped BigOperators
|
exercise_3_14 | valid | /-- Let $p$ and $q$ be distinct odd primes such that $p-1$ divides $q-1$. If $(n, p q)=1$, show that $n^{q-1} \equiv 1(p q)$.-/
| theorem exercise_3_14 {p q n : β} (hp0 : p.Prime β§ p > 2)
(hq0 : q.Prime β§ q > 2) (hpq0 : p β q) (hpq1 : p - 1 β£ q - 1)
(hn : n.gcd (p*q) = 1) :
n^(q-1) β‘ 1 [MOD p*q] := | p q n : β
hp0 : p.Prime β§ p > 2
hq0 : q.Prime β§ q > 2
hpq0 : p β q
hpq1 : p - 1 β£ q - 1
hn : n.gcd (p * q) = 1
β’ n ^ (q - 1) β‘ 1 [MOD p * q] | import Mathlib
open Real
open scoped BigOperators
|
exercise_4_5 | valid | /-- Consider a prime $p$ of the form $4 t+3$. Show that $a$ is a primitive root modulo $p$ iff $-a$ has order $(p-1) / 2$.-/
| theorem exercise_4_5 {p t : β} (hp0 : p.Prime) (hp1 : p = 4*t + 3)
(a : ZMod p) :
IsPrimitiveRoot a p β ((-a) ^ ((p-1)/2) = 1 β§ β (k : β), k < (p-1)/2 β (-a)^k β 1) := | p t : β
hp0 : p.Prime
hp1 : p = 4 * t + 3
a : ZMod p
β’ IsPrimitiveRoot a p β (-a) ^ ((p - 1) / 2) = 1 β§ β k < (p - 1) / 2, (-a) ^ k β 1 | import Mathlib
open Real
open scoped BigOperators
|
exercise_4_8 | valid | /-- Let $p$ be an odd prime. Show that $a$ is a primitive root modulo $p$ iff $a^{(p-1) / q} \not \equiv 1(p)$ for all prime divisors $q$ of $p-1$.-/
| theorem exercise_4_8 {p a : β} (hp : Odd p) :
IsPrimitiveRoot a p β (β q : β, q β£ (p-1) β q.Prime β Β¬ a^(p-1) β‘ 1 [MOD p]) := | p a : β
hp : Odd p
β’ IsPrimitiveRoot a p β β (q : β), q β£ p - 1 β q.Prime β Β¬a ^ (p - 1) β‘ 1 [MOD p] | import Mathlib
open Real
open scoped BigOperators
|
exercise_5_13 | valid | /-- Show that any prime divisor of $x^{4}-x^{2}+1$ is congruent to 1 modulo 12 .-/
| theorem exercise_5_13 {p x: β€} (hp : Prime p)
(hpx : p β£ (x^4 - x^2 + 1)) : p β‘ 1 [ZMOD 12] := | p x : β€
hp : Prime p
hpx : p β£ x ^ 4 - x ^ 2 + 1
β’ p β‘ 1 [ZMOD 12] | import Mathlib
open Real
open scoped BigOperators
|
exercise_5_37 | valid | /-- Show that if $a$ is negative then $p \equiv q(4 a) together with p\not | a$ imply $(a / p)=(a / q)$.-/
| theorem exercise_5_37 {p q : β} [Fact (p.Prime)] [Fact (q.Prime)] {a : β€}
(ha : a < 0) (h0 : p β‘ q [ZMOD 4*a]) (h1 : Β¬ ((p : β€) β£ a)) :
legendreSym p a = legendreSym q a := | p q : β
instβΒΉ : Fact p.Prime
instβ : Fact q.Prime
a : β€
ha : a < 0
h0 : βp β‘ βq [ZMOD 4 * a]
h1 : Β¬βp β£ a
β’ legendreSym p a = legendreSym q a | import Mathlib
open Real
open scoped BigOperators
|
exercise_18_4 | valid | /-- Show that 1729 is the smallest positive integer expressible as the sum of two different integral cubes in two ways.-/
| theorem exercise_18_4 {n : β} (hn : β x y z w : β€,
x^3 + y^3 = n β§ z^3 + w^3 = n β§ x β z β§ x β w β§ y β z β§ y β w) :
n β₯ 1729 := | n : β
hn : β x y z w, x ^ 3 + y ^ 3 = βn β§ z ^ 3 + w ^ 3 = βn β§ x β z β§ x β w β§ y β z β§ y β w
β’ n β₯ 1729 | import Mathlib
open Real
open scoped BigOperators
|
exercise_2020_b5 | valid | /-- For $j \in\{1,2,3,4\}$, let $z_{j}$ be a complex number with $\left|z_{j}\right|=1$ and $z_{j} \neq 1$. Prove that $3-z_{1}-z_{2}-z_{3}-z_{4}+z_{1} z_{2} z_{3} z_{4} \neq 0 .$-/
| theorem exercise_2020_b5 (z : Fin 4 β β) (hz0 : β n, βz nβ < 1)
(hz1 : β n : Fin 4, z n β 1) :
3 - z 0 - z 1 - z 2 - z 3 + (z 0) * (z 1) * (z 2) * (z 3) β 0 := | z : Fin 4 β β
hz0 : β (n : Fin 4), βz nβ < 1
hz1 : β (n : Fin 4), z n β 1
β’ 3 - z 0 - z 1 - z 2 - z 3 + z 0 * z 1 * z 2 * z 3 β 0 | import Mathlib
open scoped BigOperators
open scoped BigOperators
|
exercise_2018_b2 | valid | /-- Let $n$ be a positive integer, and let $f_{n}(z)=n+(n-1) z+$ $(n-2) z^{2}+\cdots+z^{n-1}$. Prove that $f_{n}$ has no roots in the closed unit disk $\{z \in \mathbb{C}:|z| \leq 1\}$.-/
| theorem exercise_2018_b2 (n : β) (hn : n > 0) (f : β β β β β)
(hf : β n : β, f n = Ξ» (z : β) => (β i : Fin n, (n-i)* z^(i : β))) :
Β¬ (β z : β, βzβ β€ 1 β§ f n z = 0) := | n : β
hn : n > 0
f : β β β β β
hf : β (n : β), f n = fun z => β i : Fin n, (βn - ββi) * z ^ βi
β’ Β¬β z, βzβ β€ 1 β§ f n z = 0 | import Mathlib
open scoped BigOperators
|
exercise_2017_b3 | valid | /-- Suppose that $f(x)=\sum_{i=0}^{\infty} c_{i} x^{i}$ is a power series for which each coefficient $c_{i}$ is 0 or 1 . Show that if $f(2 / 3)=3 / 2$, then $f(1 / 2)$ must be irrational.-/
| theorem exercise_2017_b3 (f : β β β) (c : β β β)
(hf : f = Ξ» x => (β' (i : β), (c i) * x^i))
(hc : β n, c n = 0 β¨ c n = 1)
(hf1 : f (2/3) = 3/2) :
Irrational (f (1/2)) := | f : β β β
c : β β β
hf : f = fun x => β' (i : β), c i * x ^ i
hc : β (n : β), c n = 0 β¨ c n = 1
hf1 : f (2 / 3) = 3 / 2
β’ Irrational (f (1 / 2)) | import Mathlib
open scoped BigOperators
|
exercise_2010_a4 | valid | /-- Prove that for each positive integer $n$, the number $10^{10^{10^n}}+10^{10^n}+10^n-1$ is not prime.-/
| theorem exercise_2010_a4 (n : β) :
Β¬ Nat.Prime (10^10^10^n + 10^10^n + 10^n - 1) := | n : β
β’ Β¬(10 ^ 10 ^ 10 ^ n + 10 ^ 10 ^ n + 10 ^ n - 1).Prime | import Mathlib
open scoped BigOperators
|
exercise_2000_a2 | valid | /-- Prove that there exist infinitely many integers $n$ such that $n, n+1, n+2$ are each the sum of the squares of two integers.-/
| theorem exercise_2000_a2 :
β N : β, β n : β, n > N β§ β i : Fin 6 β β, n = (i 0)^2 + (i 1)^2 β§
n + 1 = (i 2)^2 + (i 3)^2 β§ n + 2 = (i 4)^2 + (i 5)^2 := | β’ β (N : β), β n > N, β i, n = i 0 ^ 2 + i 1 ^ 2 β§ n + 1 = i 2 ^ 2 + i 3 ^ 2 β§ n + 2 = i 4 ^ 2 + i 5 ^ 2 | import Mathlib
open scoped BigOperators
|
exercise_1998_a3 | valid | /-- Let $f$ be a real function on the real line with continuous third derivative. Prove that there exists a point $a$ such that-/
| theorem exercise_1998_a3 (f : β β β) (hf : ContDiff β 3 f) :
β a : β, (f a) * (deriv f a) * (iteratedDeriv 2 f a) * (iteratedDeriv 3 f a) β₯ 0 := | f : β β β
hf : ContDiff β 3 f
β’ β a, f a * deriv f a * iteratedDeriv 2 f a * iteratedDeriv 3 f a β₯ 0 | import Mathlib
open scoped BigOperators
|
Subsets and Splits