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
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exercise_1_1_20 | test | /-- For $x$ an element in $G$ show that $x$ and $x^{-1}$ have the same order.-/
| theorem exercise_1_1_20 {G : Type*} [Group G] {x : G} :
orderOf x = orderOf x⁻¹ := | G : Type u_1
inst✝ : Group G
x : G
⊢ orderOf x = orderOf x⁻¹ | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_1_1_22b | test | /-- Deduce that $|a b|=|b a|$ for all $a, b \in G$.-/
| theorem exercise_1_1_22b {G: Type*} [Group G] (a b : G) :
orderOf (a * b) = orderOf (b * a) := | G : Type u_1
inst✝ : Group G
a b : G
⊢ orderOf (a * b) = orderOf (b * a) | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_1_1_29 | test | /-- Prove that $A \times B$ is an abelian group if and only if both $A$ and $B$ are abelian.-/
| theorem exercise_1_1_29 {A B : Type*} [Group A] [Group B] :
∀ x y : A × B, x*y = y*x ↔ (∀ x y : A, x*y = y*x) ∧
(∀ x y : B, x*y = y*x) := | A : Type u_1
B : Type u_2
inst✝¹ : Group A
inst✝ : Group B
⊢ ∀ (x y : A × B), x * y = y * x ↔ (∀ (x y : A), x * y = y * x) ∧ ∀ (x y : B), x * y = y * x | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_1_3_8 | test | /-- Prove that if $\Omega=\{1,2,3, \ldots\}$ then $S_{\Omega}$ is an infinite group-/
| theorem exercise_1_3_8 : Infinite (Equiv.Perm ℕ) := | ⊢ Infinite (Equiv.Perm ℕ) | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_1_6_11 | test | /-- Let $A$ and $B$ be groups. Prove that $A \times B \cong B \times A$.-/
| noncomputable def exercise_1_6_11 {A B : Type*} [Group A] [Group B] :
A × B ≃* B × A := | A : Type u_1
B : Type u_2
inst✝¹ : Group A
inst✝ : Group B
⊢ A × B ≃* B × A | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_1_6_23 | test | /-- Let $G$ be a finite group which possesses an automorphism $\sigma$ such that $\sigma(g)=g$ if and only if $g=1$. If $\sigma^{2}$ is the identity map from $G$ to $G$, prove that $G$ is abelian.-/
| theorem exercise_1_6_23 {G : Type*}
[Group G] (σ : MulAut G) (hs : ∀ g : G, σ g = 1 → g = 1)
(hs2 : ∀ g : G, σ (σ g) = g) :
∀ x y : G, x*y = y*x := | G : Type u_1
inst✝ : Group G
σ : MulAut G
hs : ∀ (g : G), σ g = 1 → g = 1
hs2 : ∀ (g : G), σ (σ g) = g
⊢ ∀ (x y : G), x * y = y * x | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_2_1_13 | test | /-- Let $H$ be a subgroup of the additive group of rational numbers with the property that $1 / x \in H$ for every nonzero element $x$ of $H$. Prove that $H=0$ or $\mathbb{Q}$.-/
| theorem exercise_2_1_13 (H : AddSubgroup ℚ) {x : ℚ}
(hH : x ∈ H → (1 / x) ∈ H):
H = ⊥ ∨ H = ⊤ := | H : AddSubgroup ℚ
x : ℚ
hH : x ∈ H → 1 / x ∈ H
⊢ H = ⊥ ∨ H = ⊤ | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_2_4_16a | test | /-- A subgroup $M$ of a group $G$ is called a maximal subgroup if $M \neq G$ and the only subgroups of $G$ which contain $M$ are $M$ and $G$. Prove that if $H$ is a proper subgroup of the finite group $G$ then there is a maximal subgroup of $G$ containing $H$.-/
| theorem exercise_2_4_16a {G : Type*} [Group G] {H : Subgroup G}
(hH : H ≠ ⊤) :
∃ M : Subgroup G, M ≠ ⊤ ∧
∀ K : Subgroup G, M ≤ K → K = M ∨ K = ⊤ ∧
H ≤ M := | G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : H ≠ ⊤
⊢ ∃ M, M ≠ ⊤ ∧ ∀ (K : Subgroup G), M ≤ K → K = M ∨ K = ⊤ ∧ H ≤ M | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_2_4_16c | test | /-- Show that if $G=\langle x\rangle$ is a cyclic group of order $n \geq 1$ then a subgroup $H$ is maximal if and only $H=\left\langle x^{p}\right\rangle$ for some prime $p$ dividing $n$.-/
| theorem exercise_2_4_16c {n : ℕ} (H : AddSubgroup (ZMod n)) :
∃ p : (ZMod n), Prime p ∧ H = AddSubgroup.closure {p} ↔
(H ≠ ⊤ ∧ ∀ K : AddSubgroup (ZMod n), H ≤ K → K = H ∨ K = ⊤) := | n : ℕ
H : AddSubgroup (ZMod n)
⊢ ∃ p, Prime p ∧ H = AddSubgroup.closure {p} ↔ H ≠ ⊤ ∧ ∀ (K : AddSubgroup (ZMod n)), H ≤ K → K = H ∨ K = ⊤ | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_3_1_22a | test | /-- Prove that if $H$ and $K$ are normal subgroups of a group $G$ then their intersection $H \cap K$ is also a normal subgroup of $G$.-/
| theorem exercise_3_1_22a (G : Type*) [Group G] (H K : Subgroup G)
[Normal H] [Normal K] :
Normal (H ⊓ K) := | G : Type u_1
inst✝² : Group G
H K : Subgroup G
inst✝¹ : H.Normal
inst✝ : K.Normal
⊢ (H ⊓ K).Normal | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_3_2_8 | test | /-- Prove that if $H$ and $K$ are finite subgroups of $G$ whose orders are relatively prime then $H \cap K=1$.-/
| theorem exercise_3_2_8 {G : Type*} [Group G] (H K : Subgroup G)
[Fintype H] [Fintype K]
(hHK : Nat.Coprime (card H) (card K)) :
H ⊓ K = ⊥ := | G : Type u_1
inst✝² : Group G
H K : Subgroup G
inst✝¹ : Fintype ↥H
inst✝ : Fintype ↥K
hHK : (card ↥H).Coprime (card ↥K)
⊢ H ⊓ K = ⊥ | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_3_2_16 | test | /-- Use Lagrange's Theorem in the multiplicative group $(\mathbb{Z} / p \mathbb{Z})^{\times}$to prove Fermat's Little Theorem: if $p$ is a prime then $a^{p} \equiv a(\bmod p)$ for all $a \in \mathbb{Z}$.-/
| theorem exercise_3_2_16 (p : ℕ) (hp : Nat.Prime p) (a : ℕ) :
Nat.Coprime a p → a ^ p ≡ a [ZMOD p] := | p : ℕ
hp : p.Prime
a : ℕ
⊢ a.Coprime p → ↑a ^ p ≡ ↑a [ZMOD ↑p] | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_3_3_3 | test | /-- Prove that if $H$ is a normal subgroup of $G$ of prime index $p$ then for all $K \leq G$ either $K \leq H$, or $G=H K$ and $|K: K \cap H|=p$.-/
| theorem exercise_3_3_3 {p : Nat.Primes} {G : Type*} [Group G]
{H : Subgroup G} [hH : H.Normal] (hH1 : H.index = p) :
∀ K : Subgroup G, K ≤ H ∨ H ⊔ K = ⊤ ∨ (K ⊓ H).relindex K = p := | p : Nat.Primes
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : H.Normal
hH1 : H.index = ↑p
⊢ ∀ (K : Subgroup G), K ≤ H ∨ H ⊔ K = ⊤ ∨ (K ⊓ H).relindex K = ↑p | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_3_4_4 | test | /-- Use Cauchy's Theorem and induction to show that a finite abelian group has a subgroup of order $n$ for each positive divisor $n$ of its order.-/
| theorem exercise_3_4_4 {G : Type*} [CommGroup G] [Fintype G] {n : ℕ}
(hn : n ∣ (card G)) :
∃ (H : Subgroup G) (H_fin : Fintype H), @card H H_fin = n := | G : Type u_1
inst✝¹ : CommGroup G
inst✝ : Fintype G
n : ℕ
hn : n ∣ card G
⊢ ∃ H H_fin, card ↥H = n | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_3_4_5b | test | /-- Prove that quotient groups of a solvable group are solvable.-/
| theorem exercise_3_4_5b {G : Type*} [Group G] [IsSolvable G]
(H : Subgroup G) [Normal H] :
IsSolvable (G ⧸ H) := | G : Type u_1
inst✝² : Group G
inst✝¹ : IsSolvable G
H : Subgroup G
inst✝ : H.Normal
⊢ IsSolvable (G ⧸ H) | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_4_2_8 | test | /-- Prove that if $H$ has finite index $n$ then there is a normal subgroup $K$ of $G$ with $K \leq H$ and $|G: K| \leq n!$.-/
| theorem exercise_4_2_8 {G : Type*} [Group G] {H : Subgroup G}
{n : ℕ} (hn : n > 0) (hH : H.index = n) :
∃ K ≤ H, K.Normal ∧ K.index ≤ n.factorial := | G : Type u_1
inst✝ : Group G
H : Subgroup G
n : ℕ
hn : n > 0
hH : H.index = n
⊢ ∃ K ≤ H, K.Normal ∧ K.index ≤ n.factorial | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_4_2_9a | test | /-- Prove that if $p$ is a prime and $G$ is a group of order $p^{\alpha}$ for some $\alpha \in \mathbb{Z}^{+}$, then every subgroup of index $p$ is normal in $G$.-/
| theorem exercise_4_2_9a {G : Type*} [Fintype G] [Group G] {p α : ℕ}
(hp : p.Prime) (ha : α > 0) (hG : card G = p ^ α) :
∀ H : Subgroup G, H.index = p → H.Normal := | G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
p α : ℕ
hp : p.Prime
ha : α > 0
hG : card G = p ^ α
⊢ ∀ (H : Subgroup G), H.index = p → H.Normal | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_4_4_2 | test | /-- Prove that if $G$ is an abelian group of order $p q$, where $p$ and $q$ are distinct primes, then $G$ is cyclic.-/
| theorem exercise_4_4_2 {G : Type*} [Fintype G] [Group G]
{p q : Nat.Primes} (hpq : p ≠ q) (hG : card G = p*q) :
IsCyclic G := | G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
p q : Nat.Primes
hpq : p ≠ q
hG : card G = ↑p * ↑q
⊢ IsCyclic G | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_4_4_6b | test | /-- Prove that there exists a normal subgroup that is not characteristic.-/
| theorem exercise_4_4_6b :
∃ (G : Type*) (hG : Group G) (H : @Subgroup G hG), @Characteristic G hG H ∧ ¬ @Normal G hG H := | ⊢ ∃ G hG H, H.Characteristic ∧ ¬H.Normal | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_4_4_8a | test | /-- Let $G$ be a group with subgroups $H$ and $K$ with $H \leq K$. Prove that if $H$ is characteristic in $K$ and $K$ is normal in $G$ then $H$ is normal in $G$.-/
| theorem exercise_4_4_8a {G : Type*} [Group G] (H K : Subgroup G)
(hHK : H ≤ K) [hHK1 : (H.subgroupOf K).Normal] [hK : K.Normal] :
H.Normal := | G : Type u_1
inst✝ : Group G
H K : Subgroup G
hHK : H ≤ K
hHK1 : (H.subgroupOf K).Normal
hK : K.Normal
⊢ H.Normal | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_4_5_13 | test | /-- Prove that a group of order 56 has a normal Sylow $p$-subgroup for some prime $p$ dividing its order.-/
| theorem exercise_4_5_13 {G : Type*} [Group G] [Fintype G]
(hG : card G = 56) :
∃ (p : ℕ) (P : Sylow p G), P.Normal := | G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
hG : card G = 56
⊢ ∃ p P, (↑P).Normal | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_4_5_15 | test | /-- Prove that a group of order 351 has a normal Sylow $p$-subgroup for some prime $p$ dividing its order.-/
| theorem exercise_4_5_15 {G : Type*} [Group G] [Fintype G]
(hG : card G = 351) :
∃ (p : ℕ) (P : Sylow p G), P.Normal := | G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
hG : card G = 351
⊢ ∃ p P, (↑P).Normal | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_4_5_17 | test | /-- Prove that if $|G|=105$ then $G$ has a normal Sylow 5 -subgroup and a normal Sylow 7-subgroup.-/
| theorem exercise_4_5_17 {G : Type*} [Fintype G] [Group G]
(hG : card G = 105) :
Nonempty (Sylow 5 G) ∧ Nonempty (Sylow 7 G) := | G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
hG : card G = 105
⊢ Nonempty (Sylow 5 G) ∧ Nonempty (Sylow 7 G) | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_4_5_19 | test | /-- Prove that if $|G|=6545$ then $G$ is not simple.-/
| theorem exercise_4_5_19 {G : Type*} [Fintype G] [Group G]
(hG : card G = 6545) : ¬ IsSimpleGroup G := | G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
hG : card G = 6545
⊢ ¬IsSimpleGroup G | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_4_5_21 | test | /-- Prove that if $|G|=2907$ then $G$ is not simple.-/
| theorem exercise_4_5_21 {G : Type*} [Fintype G] [Group G]
(hG : card G = 2907) : ¬ IsSimpleGroup G := | G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
hG : card G = 2907
⊢ ¬IsSimpleGroup G | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_4_5_23 | test | /-- Prove that if $|G|=462$ then $G$ is not simple.-/
| theorem exercise_4_5_23 {G : Type*} [Fintype G] [Group G]
(hG : card G = 462) : ¬ IsSimpleGroup G := | G : Type u_1
inst✝¹ : Fintype G
inst✝ : Group G
hG : card G = 462
⊢ ¬IsSimpleGroup G | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_4_5_33 | test | /-- Let $P$ be a normal Sylow $p$-subgroup of $G$ and let $H$ be any subgroup of $G$. Prove that $P \cap H$ is the unique Sylow $p$-subgroup of $H$.-/
| theorem exercise_4_5_33 {G : Type*} [Group G] [Fintype G] {p : ℕ}
(P : Sylow p G) [hP : P.Normal] (H : Subgroup G) [Fintype H] :
∀ R : Sylow p H, R.toSubgroup = (H ⊓ P.toSubgroup).subgroupOf H ∧
Nonempty (Sylow p H) := | G : Type u_1
inst✝² : Group G
inst✝¹ : Fintype G
p : ℕ
P : Sylow p G
hP : (↑P).Normal
H : Subgroup G
inst✝ : Fintype ↥H
⊢ ∀ (R : Sylow p ↥H), ↑R = (H ⊓ ↑P).subgroupOf H ∧ Nonempty (Sylow p ↥H) | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_7_1_2 | test | /-- Prove that if $u$ is a unit in $R$ then so is $-u$.-/
| theorem exercise_7_1_2 {R : Type*} [Ring R] {u : R}
(hu : IsUnit u) : IsUnit (-u) := | R : Type u_1
inst✝ : Ring R
u : R
hu : IsUnit u
⊢ IsUnit (-u) | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_7_1_12 | test | /-- Prove that any subring of a field which contains the identity is an integral domain.-/
| theorem exercise_7_1_12 {F : Type*} [Field F] {K : Subring F}
(hK : (1 : F) ∈ K) : IsDomain K := | F : Type u_1
inst✝ : Field F
K : Subring F
hK : 1 ∈ K
⊢ IsDomain ↥K | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_7_2_2 | test | /-- Let $p(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$ be an element of the polynomial ring $R[x]$. Prove that $p(x)$ is a zero divisor in $R[x]$ if and only if there is a nonzero $b \in R$ such that $b p(x)=0$.-/
| theorem exercise_7_2_2 {R : Type*} [Ring R] (p : Polynomial R) :
p ∣ 0 ↔ ∃ b : R, b ≠ 0 ∧ b • p = 0 := | R : Type u_1
inst✝ : Ring R
p : R[X]
⊢ p ∣ 0 ↔ ∃ b, b ≠ 0 ∧ b • p = 0 | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_7_3_16 | test | /-- Let $\varphi: R \rightarrow S$ be a surjective homomorphism of rings. Prove that the image of the center of $R$ is contained in the center of $S$.-/
| theorem exercise_7_3_16 {R S : Type*} [Ring R] [Ring S]
{φ : R →+* S} (hf : Function.Surjective φ) :
φ '' (center R) ⊂ center S := | R : Type u_1
S : Type u_2
inst✝¹ : Ring R
inst✝ : Ring S
φ : R →+* S
hf : Function.Surjective ⇑φ
⊢ ⇑φ '' Set.center R ⊂ Set.center S | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_7_4_27 | test | /-- Let $R$ be a commutative ring with $1 \neq 0$. Prove that if $a$ is a nilpotent element of $R$ then $1-a b$ is a unit for all $b \in R$.-/
| theorem exercise_7_4_27 {R : Type*} [CommRing R] (hR : (0 : R) ≠ 1)
{a : R} (ha : IsNilpotent a) (b : R) :
IsUnit (1-a*b) := | R : Type u_1
inst✝ : CommRing R
hR : 0 ≠ 1
a : R
ha : IsNilpotent a
b : R
⊢ IsUnit (1 - a * b) | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_8_2_4 | test | /-- Let $R$ be an integral domain. Prove that if the following two conditions hold then $R$ is a Principal Ideal Domain: (i) any two nonzero elements $a$ and $b$ in $R$ have a greatest common divisor which can be written in the form $r a+s b$ for some $r, s \in R$, and (ii) if $a_{1}, a_{2}, a_{3}, \ldots$ are nonzero elements of $R$ such that $a_{i+1} \mid a_{i}$ for all $i$, then there is a positive integer $N$ such that $a_{n}$ is a unit times $a_{N}$ for all $n \geq N$.-/
| theorem exercise_8_2_4 {R : Type*} [Ring R][NoZeroDivisors R]
[CancelCommMonoidWithZero R] [GCDMonoid R]
(h1 : ∀ a b : R, a ≠ 0 → b ≠ 0 → ∃ r s : R, gcd a b = r*a + s*b)
(h2 : ∀ a : ℕ → R, (∀ i j : ℕ, i < j → a i ∣ a j) →
∃ N : ℕ, ∀ n ≥ N, ∃ u : R, IsUnit u ∧ a n = u * a N) :
IsPrincipalIdealRing R := | R : Type u_1
inst✝³ : Ring R
inst✝² : NoZeroDivisors R
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : GCDMonoid R
h1 : ∀ (a b : R), a ≠ 0 → b ≠ 0 → ∃ r s, gcd a b = r * a + s * b
h2 : ∀ (a : ℕ → R), (∀ (i j : ℕ), i < j → a i ∣ a j) → ∃ N, ∀ n ≥ N, ∃ u, IsUnit u ∧ a n = u * a N
⊢ IsPrincipalIdealRing R | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_8_3_5a | test | /-- Let $R=\mathbb{Z}[\sqrt{-n}]$ where $n$ is a squarefree integer greater than 3. Prove that $2, \sqrt{-n}$ and $1+\sqrt{-n}$ are irreducibles in $R$.-/
| theorem exercise_8_3_5a {n : ℤ} (hn0 : n > 3) (hn1 : Squarefree n) :
Irreducible (2 : Zsqrtd $ -n) ∧
Irreducible (⟨0, 1⟩ : Zsqrtd $ -n) ∧
Irreducible (1 + ⟨0, 1⟩ : Zsqrtd $ -n) := | n : ℤ
hn0 : n > 3
hn1 : Squarefree n
⊢ Irreducible 2 ∧ Irreducible { re := 0, im := 1 } ∧ Irreducible (1 + { re := 0, im := 1 }) | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_8_3_6b | test | /-- Let $q \in \mathbb{Z}$ be a prime with $q \equiv 3 \bmod 4$. Prove that the quotient ring $\mathbb{Z}[i] /(q)$ is a field with $q^{2}$ elements.-/
| theorem exercise_8_3_6b {q : ℕ} (hq0 : q.Prime)
(hq1 : q ≡ 3 [ZMOD 4]) {R : Type} [Ring R]
(hR : R = (GaussianInt ⧸ span ({↑q} : Set GaussianInt))) :
IsField R ∧ ∃ finR : Fintype R, @card R finR = q^2 := | q : ℕ
hq0 : q.Prime
hq1 : ↑q ≡ 3 [ZMOD 4]
R : Type
inst✝ : Ring R
hR : R = (GaussianInt ⧸ span {↑q})
⊢ IsField R ∧ ∃ finR, card R = q ^ 2 | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_9_1_10 | test | /-- Prove that the ring $\mathbb{Z}\left[x_{1}, x_{2}, x_{3}, \ldots\right] /\left(x_{1} x_{2}, x_{3} x_{4}, x_{5} x_{6}, \ldots\right)$ contains infinitely many minimal prime ideals.-/
| theorem exercise_9_1_10 {f : ℕ → MvPolynomial ℕ ℤ}
(hf : f = λ i => MvPolynomial.X i * MvPolynomial.X (i+1)):
Infinite (minimalPrimes (MvPolynomial ℕ ℤ ⧸ span (range f))) := | f : ℕ → MvPolynomial ℕ ℤ
hf : f = fun i => MvPolynomial.X i * MvPolynomial.X (i + 1)
⊢ Infinite ↑(minimalPrimes (MvPolynomial ℕ ℤ ⧸ span (range f))) | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_9_4_2a | test | /-- Prove that $x^4-4x^3+6$ is irreducible in $\mathbb{Z}[x]$.-/
| theorem exercise_9_4_2a : Irreducible (X^4 - 4*X^3 + 6 : Polynomial ℤ) := | ⊢ Irreducible (X ^ 4 - 4 * X ^ 3 + 6) | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_9_4_2c | test | /-- Prove that $x^4+4x^3+6x^2+2x+1$ is irreducible in $\mathbb{Z}[x]$.-/
| theorem exercise_9_4_2c : Irreducible
(X^4 + 4*X^3 + 6*X^2 + 2*X + 1 : Polynomial ℤ) := | ⊢ Irreducible (X ^ 4 + 4 * X ^ 3 + 6 * X ^ 2 + 2 * X + 1) | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_9_4_9 | test | /-- Prove that the polynomial $x^{2}-\sqrt{2}$ is irreducible over $\mathbb{Z}[\sqrt{2}]$. You may assume that $\mathbb{Z}[\sqrt{2}]$ is a U.F.D.-/
| theorem exercise_9_4_9 :
Irreducible (X^2 - C Zsqrtd.sqrtd : Polynomial (Zsqrtd 2)) := | ⊢ Irreducible (X ^ 2 - C Zsqrtd.sqrtd) | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_11_1_13 | test | /-- Prove that as vector spaces over $\mathbb{Q}, \mathbb{R}^n \cong \mathbb{R}$, for all $n \in \mathbb{Z}^{+}$.-/
| def exercise_11_1_13 {ι : Type*} [Fintype ι] :
(ι → ℝ) ≃ₗ[ℚ] ℝ := | ι : Type u_1
inst✝ : Fintype ι
⊢ (ι → ℝ) ≃ₗ[ℚ] ℝ | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| |
exercise_13_3b | test | /-- Show that the collection $$\mathcal{T}_\infty = \{U | X - U \text{ is infinite or empty or all of X}\}$$ does not need to be a topology on the set $X$.-/
| theorem exercise_13_3b : ¬ ∀ X : Type, ∀s : Set (Set X),
(∀ t : Set X, t ∈ s → (Set.Infinite tᶜ ∨ t = ∅ ∨ t = ⊤)) →
(Set.Infinite (⋃₀ s)ᶜ ∨ (⋃₀ s) = ∅ ∨ (⋃₀ s) = ⊤) := | ⊢ ¬∀ (X : Type) (s : Set (Set X)), (∀ t ∈ s, tᶜ.Infinite ∨ t = ∅ ∨ t = ⊤) → (⋃₀ s)ᶜ.Infinite ∨ ⋃₀ s = ∅ ∨ ⋃₀ s = ⊤ | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| |
exercise_13_4a2 | test | /-- If $\mathcal{T}_\alpha$ is a family of topologies on $X$, show that $\bigcup \mathcal{T}_\alpha$ does not need to be a topology on $X$.-/
| theorem exercise_13_4a2 :
∃ (X I : Type*) (T : I → Set (Set X)),
(∀ i, is_topology X (T i)) ∧ ¬ is_topology X (⋂ i : I, T i) := | ⊢ ∃ X I T, (∀ (i : I), is_topology X (T i)) ∧ ¬is_topology X (⋂ i, T i) | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
def is_topology (X : Type*) (T : Set (Set X)) :=
univ ∈ T ∧
(∀ s t, s ∈ T → t ∈ T → s ∩ t ∈ T) ∧
(∀s, (∀t ∈ s, t ∈ T) → sUnion s ∈ T)
| |
exercise_13_4b2 | test | /-- Let $\mathcal{T}_\alpha$ be a family of topologies on $X$. Show that there is a unique largest topology on $X$ contained in all the collections $\mathcal{T}_\alpha$.-/
| theorem exercise_13_4b2 (X I : Type*) (T : I → Set (Set X)) (h : ∀ i, is_topology X (T i)) :
∃! T', is_topology X T' ∧ (∀ i, T' ⊆ T i) ∧
∀ T'', is_topology X T'' → (∀ i, T'' ⊆ T i) → T' ⊆ T'' := | X : Type u_1
I : Type u_2
T : I → Set (Set X)
h : ∀ (i : I), is_topology X (T i)
⊢ ∃! T',
is_topology X T' ∧
(∀ (i : I), T' ⊆ T i) ∧ ∀ (T'' : Set (Set X)), is_topology X T'' → (∀ (i : I), T'' ⊆ T i) → T' ⊆ T'' | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
def is_topology (X : Type*) (T : Set (Set X)) :=
univ ∈ T ∧
(∀ s t, s ∈ T → t ∈ T → s ∩ t ∈ T) ∧
(∀s, (∀t ∈ s, t ∈ T) → sUnion s ∈ T)
| |
exercise_13_5b | test | /-- Show that if $\mathcal{A}$ is a subbasis 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_5b {X : Type*}
[t : TopologicalSpace X] (A : Set (Set X)) (hA : t = generateFrom A) :
generateFrom A = generateFrom (sInter {T | is_topology X T ∧ A ⊆ T}) := | X : Type u_1
t : TopologicalSpace X
A : Set (Set X)
hA : t = generateFrom A
⊢ generateFrom A = generateFrom (⋂₀ {T | is_topology X T ∧ A ⊆ T}) | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
def is_topology (X : Type*) (T : Set (Set X)) :=
univ ∈ T ∧
(∀ s t, s ∈ T → t ∈ T → s ∩ t ∈ T) ∧
(∀s, (∀t ∈ s, t ∈ T) → sUnion s ∈ T)
| |
exercise_13_8a | test | /-- Show that the collection $\{(a,b) \mid a < b, a \text{ and } b \text{ rational}\}$ is a basis that generates the standard topology on $\mathbb{R}$.-/
| theorem exercise_13_8a :
IsTopologicalBasis {S : Set ℝ | ∃ a b : ℚ, a < b ∧ S = Ioo ↑a ↑b} := | ⊢ IsTopologicalBasis {S | ∃ a b, a < b ∧ S = Ioo ↑a ↑b} | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| |
exercise_16_1 | test | /-- Show that if $Y$ is a subspace of $X$, and $A$ is a subset of $Y$, then the topology $A$ inherits as a subspace of $Y$ is the same as the topology it inherits as a subspace of $X$.-/
| theorem exercise_16_1 {X : Type*} [TopologicalSpace X]
(Y : Set X)
(A : Set Y) :
∀ U : Set A, IsOpen U ↔ IsOpen (Subtype.val '' U) := | X : Type u_1
inst✝ : TopologicalSpace X
Y : Set X
A : Set ↑Y
⊢ ∀ (U : Set ↑A), IsOpen U ↔ IsOpen (Subtype.val '' U) | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| |
exercise_16_6 | test | /-- Show that the countable collection \[\{(a, b) \times (c, d) \mid a < b \text{ and } c < d, \text{ and } a, b, c, d \text{ are rational}\}\] is a basis for $\mathbb{R}^2$.-/
| theorem exercise_16_6
(S : Set (Set (ℝ × ℝ)))
(hS : ∀ s, s ∈ S → ∃ a b c d, (rational a ∧ rational b ∧ rational c ∧ rational d
∧ s = {x | ∃ x₁ x₂, x = (x₁, x₂) ∧ a < x₁ ∧ x₁ < b ∧ c < x₂ ∧ x₂ < d})) :
IsTopologicalBasis S := | S : Set (Set (ℝ × ℝ))
hS :
∀ s ∈ S,
∃ a b c d,
rational a ∧
rational b ∧ rational c ∧ rational d ∧ s = {x | ∃ x₁ x₂, x = (x₁, x₂) ∧ a < x₁ ∧ x₁ < b ∧ c < x₂ ∧ x₂ < d}
⊢ IsTopologicalBasis S | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
def rational (x : ℝ) := x ∈ range ((↑) : ℚ → ℝ)
| |
exercise_18_8a | test | /-- Let $Y$ be an ordered set in the order topology. Let $f, g: X \rightarrow Y$ be continuous. Show that the set $\{x \mid f(x) \leq g(x)\}$ is closed in $X$.-/
| theorem exercise_18_8a {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
[LinearOrder Y] [OrderTopology Y] {f g : X → Y}
(hf : Continuous f) (hg : Continuous g) :
IsClosed {x | f x ≤ g x} := | X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : LinearOrder Y
inst✝ : OrderTopology Y
f g : X → Y
hf : Continuous f
hg : Continuous g
⊢ IsClosed {x | f x ≤ g x} | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| |
exercise_18_13 | test | /-- Let $A \subset X$; let $f: A \rightarrow Y$ be continuous; let $Y$ be Hausdorff. Show that if $f$ may be extended to a continuous function $g: \bar{A} \rightarrow Y$, then $g$ is uniquely determined by $f$.-/
| theorem exercise_18_13
{X : Type*} [TopologicalSpace X] {Y : Type*} [TopologicalSpace Y]
[T2Space Y] {A : Set X} {f : A → Y} (hf : Continuous f)
(g : closure A → Y)
(g_con : Continuous g) :
∀ (g' : closure A → Y), Continuous g' → (∀ (x : closure A), g x = g' x) := | X : Type u_1
inst✝² : TopologicalSpace X
Y : Type u_2
inst✝¹ : TopologicalSpace Y
inst✝ : T2Space Y
A : Set X
f : ↑A → Y
hf : Continuous f
g : ↑(closure A) → Y
g_con : Continuous g
⊢ ∀ (g' : ↑(closure A) → Y), Continuous g' → ∀ (x : ↑(closure A)), g x = g' x | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| |
exercise_20_2 | test | /-- Show that $\mathbb{R} \times \mathbb{R}$ in the dictionary order topology is metrizable.-/
| theorem exercise_20_2
[TopologicalSpace (ℝ ×ₗ ℝ)] [OrderTopology (ℝ ×ₗ ℝ)]
: MetrizableSpace (ℝ ×ₗ ℝ) := | inst✝¹ : TopologicalSpace (Lex (ℝ × ℝ))
inst✝ : OrderTopology (Lex (ℝ × ℝ))
⊢ MetrizableSpace (Lex (ℝ × ℝ)) | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| |
exercise_21_6b | test | /-- Define $f_{n}:[0,1] \rightarrow \mathbb{R}$ by the equation $f_{n}(x)=x^{n}$. Show that the sequence $\left(f_{n}\right)$ does not converge uniformly.-/
| theorem exercise_21_6b
(f : ℕ → I → ℝ )
(h : ∀ x n, f n x = x ^ n) :
¬ ∃ f₀, TendstoUniformly f f₀ atTop := | f : ℕ → ↑I → ℝ
h : ∀ (x : ↑I) (n : ℕ), f n x = ↑x ^ n
⊢ ¬∃ f₀, TendstoUniformly f f₀ atTop | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
abbrev I : Set ℝ := Icc 0 1
| |
exercise_22_2a | test | /-- Let $p: X \rightarrow Y$ be a continuous map. Show that if there is a continuous map $f: Y \rightarrow X$ such that $p \circ f$ equals the identity map of $Y$, then $p$ is a quotient map.-/
| theorem exercise_22_2a {X Y : Type*} [TopologicalSpace X]
[TopologicalSpace Y] (p : X → Y) (h : Continuous p) :
QuotientMap p ↔ ∃ (f : Y → X), Continuous f ∧ p ∘ f = id := | X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
p : X → Y
h : Continuous p
⊢ QuotientMap p ↔ ∃ f, Continuous f ∧ p ∘ f = id | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| |
exercise_22_5 | test | /-- Let $p \colon X \rightarrow Y$ be an open map. Show that if $A$ is open in $X$, then the map $q \colon A \rightarrow p(A)$ obtained by restricting $p$ is an open map.-/
| theorem exercise_22_5 {X Y : Type*} [TopologicalSpace X]
[TopologicalSpace Y] (p : X → Y) (hp : IsOpenMap p)
(A : Set X) (hA : IsOpen A) : IsOpenMap (p ∘ Subtype.val : A → Y) := | X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
p : X → Y
hp : IsOpenMap p
A : Set X
hA : IsOpen A
⊢ IsOpenMap (p ∘ Subtype.val) | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| |
exercise_23_3 | test | /-- Let $\left\{A_{\alpha}\right\}$ be a collection of connected subspaces of $X$; let $A$ be a connected subset of $X$. Show that if $A \cap A_{\alpha} \neq \varnothing$ for all $\alpha$, then $A \cup\left(\bigcup A_{\alpha}\right)$ is connected.-/
| theorem exercise_23_3 {X : Type*} [TopologicalSpace X]
[TopologicalSpace X] {A : ℕ → Set X}
(hAn : ∀ n, IsConnected (A n))
(A₀ : Set X)
(hA : IsConnected A₀)
(h : ∀ n, A₀ ∩ A n ≠ ∅) :
IsConnected (A₀ ∪ (⋃ n, A n)) := | X : Type u_1
inst✝¹ inst✝ : TopologicalSpace X
A : ℕ → Set X
hAn : ∀ (n : ℕ), IsConnected (A n)
A₀ : Set X
hA : IsConnected A₀
h : ∀ (n : ℕ), A₀ ∩ A n ≠ ∅
⊢ IsConnected (A₀ ∪ ⋃ n, A n) | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| |
exercise_23_6 | test | /-- Let $A \subset X$. Show that if $C$ is a connected subspace of $X$ that intersects both $A$ and $X-A$, then $C$ intersects $\operatorname{Bd} A$.-/
| theorem exercise_23_6 {X : Type*}
[TopologicalSpace X] {A C : Set X} (hc : IsConnected C)
(hCA : C ∩ A ≠ ∅) (hCXA : C ∩ Aᶜ ≠ ∅) :
C ∩ (frontier A) ≠ ∅ := | X : Type u_1
inst✝ : TopologicalSpace X
A C : Set X
hc : IsConnected C
hCA : C ∩ A ≠ ∅
hCXA : C ∩ Aᶜ ≠ ∅
⊢ C ∩ frontier A ≠ ∅ | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| |
exercise_23_11 | test | /-- Let $p: X \rightarrow Y$ be a quotient map. Show that if each set $p^{-1}(\{y\})$ is connected, and if $Y$ is connected, then $X$ is connected.-/
| theorem exercise_23_11 {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
(p : X → Y) (hq : QuotientMap p)
(hY : ConnectedSpace Y) (hX : ∀ y : Y, IsConnected (p ⁻¹' {y})) :
ConnectedSpace X := | X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
p : X → Y
hq : QuotientMap p
hY : ConnectedSpace Y
hX : ∀ (y : Y), IsConnected (p ⁻¹' {y})
⊢ ConnectedSpace X | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| |
exercise_24_3a | test | /-- Let $f \colon X \rightarrow X$ be continuous. Show that if $X = [0, 1]$, there is a point $x$ such that $f(x) = x$. (The point $x$ is called a fixed point of $f$.)-/
| theorem exercise_24_3a [TopologicalSpace I] [CompactSpace I]
(f : I → I) (hf : Continuous f) :
∃ (x : I), f x = x := | I : Type u_1
inst✝¹ : TopologicalSpace I
inst✝ : CompactSpace I
f : I → I
hf : Continuous f
⊢ ∃ x, f x = x | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| |
exercise_25_9 | test | /-- Let $G$ be a topological group; let $C$ be the component of $G$ containing the identity element $e$. Show that $C$ is a normal subgroup of $G$.-/
| theorem exercise_25_9 {G : Type*} [TopologicalSpace G] [Group G]
[TopologicalGroup G] (C : Set G) (h : C = connectedComponent 1) :
IsNormalSubgroup C := | G : Type u_1
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : TopologicalGroup G
C : Set G
h : C = connectedComponent 1
⊢ IsNormalSubgroup C | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| |
exercise_26_12 | test | /-- Let $p: X \rightarrow Y$ be a closed continuous surjective map such that $p^{-1}(\{y\})$ is compact, for each $y \in Y$. (Such a map is called a perfect map.) Show that if $Y$ is compact, then $X$ is compact.-/
| theorem exercise_26_12 {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
(p : X → Y) (h : Function.Surjective p) (hc : Continuous p) (hp : ∀ y, IsCompact (p ⁻¹' {y}))
(hY : CompactSpace Y) : CompactSpace X := | X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
p : X → Y
h : Function.Surjective p
hc : Continuous p
hp : ∀ (y : Y), IsCompact (p ⁻¹' {y})
hY : CompactSpace Y
⊢ CompactSpace X | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| |
exercise_28_4 | test | /-- A space $X$ is said to be countably compact if every countable open covering of $X$ contains a finite subcollection that covers $X$. Show that for a $T_1$ space $X$, countable compactness is equivalent to limit point compactness.-/
| theorem exercise_28_4 {X : Type*}
[TopologicalSpace X] (hT1 : T1Space X) :
countably_compact X ↔ limit_point_compact X := | X : Type u_1
inst✝ : TopologicalSpace X
hT1 : T1Space X
⊢ countably_compact X ↔ limit_point_compact X | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
def countably_compact (X : Type*) [TopologicalSpace X] :=
∀ U : ℕ → Set X,
(∀ i, IsOpen (U i)) ∧ ((univ : Set X) ⊆ ⋃ i, U i) →
(∃ t : Finset ℕ, (univ : Set X) ⊆ ⋃ i ∈ t, U i)
def limit_point_compact (X : Type*) [TopologicalSpace X] :=
∀ U : Set X, Infinite U → ∃ x ∈ U, ClusterPt x (𝓟 U)
| |
exercise_28_6 | test | /-- Let $(X, d)$ be a metric space. If $f: X \rightarrow X$ satisfies the condition $d(f(x), f(y))=d(x, y)$ for all $x, y \in X$, then $f$ is called an isometry of $X$. Show that if $f$ is an isometry and $X$ is compact, then $f$ is bijective and hence a homeomorphism.-/
| theorem exercise_28_6 {X : Type*} [MetricSpace X]
[CompactSpace X] {f : X → X} (hf : Isometry f) :
Function.Bijective f := | X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompactSpace X
f : X → X
hf : Isometry f
⊢ Function.Bijective f | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| |
exercise_29_4 | test | /-- Show that $[0, 1]^\omega$ is not locally compact in the uniform topology.-/
| theorem exercise_29_4 [TopologicalSpace (ℕ → I)] :
¬ LocallyCompactSpace (ℕ → I) := | inst✝ : TopologicalSpace (ℕ → ↑I)
⊢ ¬LocallyCompactSpace (ℕ → ↑I) | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
abbrev I : Set ℝ := Icc 0 1
| |
exercise_30_10 | test | /-- Show that if $X$ is a countable product of spaces having countable dense subsets, then $X$ has a countable dense subset.-/
| theorem exercise_30_10
{X : ℕ → Type*} [∀ i, TopologicalSpace (X i)]
(h : ∀ i, ∃ (s : Set (X i)), Countable s ∧ Dense s) :
∃ (s : Set (Π i, X i)), Countable s ∧ Dense s := | X : ℕ → Type u_1
inst✝ : (i : ℕ) → TopologicalSpace (X i)
h : ∀ (i : ℕ), ∃ s, Countable ↑s ∧ Dense s
⊢ ∃ s, Countable ↑s ∧ Dense s | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| |
exercise_31_1 | test | /-- Show that if $X$ is regular, every pair of points of $X$ have neighborhoods whose closures are disjoint.-/
| theorem exercise_31_1 {X : Type*} [TopologicalSpace X]
(hX : RegularSpace X) (x y : X) :
∃ (U V : Set X), IsOpen U ∧ IsOpen V ∧ x ∈ U ∧ y ∈ V ∧ closure U ∩ closure V = ∅ := | X : Type u_1
inst✝ : TopologicalSpace X
hX : RegularSpace X
x y : X
⊢ ∃ U V, IsOpen U ∧ IsOpen V ∧ x ∈ U ∧ y ∈ V ∧ closure U ∩ closure V = ∅ | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| |
exercise_31_3 | test | /-- Show that every order topology is regular.-/
| theorem exercise_31_3 {α : Type*} [PartialOrder α]
[TopologicalSpace α] (h : OrderTopology α) : RegularSpace α := | α : Type u_1
inst✝¹ : PartialOrder α
inst✝ : TopologicalSpace α
h : OrderTopology α
⊢ RegularSpace α | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| |
exercise_32_2a | test | /-- Show that if $\prod X_\alpha$ is Hausdorff, then so is $X_\alpha$. Assume that each $X_\alpha$ is nonempty.-/
| theorem exercise_32_2a
{ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)]
(h : ∀ i, Nonempty (X i)) (h2 : T2Space (Π i, X i)) :
∀ i, T2Space (X i) := | ι : Type u_1
X : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (X i)
h : ∀ (i : ι), Nonempty (X i)
h2 : T2Space ((i : ι) → X i)
⊢ ∀ (i : ι), T2Space (X i) | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| |
exercise_32_2c | test | /-- Show that if $\prod X_\alpha$ is normal, then so is $X_\alpha$. Assume that each $X_\alpha$ is nonempty.-/
| theorem exercise_32_2c
{ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)]
(h : ∀ i, Nonempty (X i)) (h2 : NormalSpace (Π i, X i)) :
∀ i, NormalSpace (X i) := | ι : Type u_1
X : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (X i)
h : ∀ (i : ι), Nonempty (X i)
h2 : NormalSpace ((i : ι) → X i)
⊢ ∀ (i : ι), NormalSpace (X i) | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| |
exercise_33_7 | test | /-- Show that every locally compact Hausdorff space is completely regular.-/
| theorem exercise_33_7 {X : Type*} [TopologicalSpace X]
(hX : LocallyCompactSpace X) (hX' : T2Space X) :
∀ x A, IsClosed A ∧ ¬ x ∈ A →
∃ (f : X → I), Continuous f ∧ f x = 1 ∧ f '' A = {0} := | X : Type u_1
inst✝ : TopologicalSpace X
hX : LocallyCompactSpace X
hX' : T2Space X
⊢ ∀ (x : X) (A : Set X), IsClosed A ∧ x ∉ A → ∃ f, Continuous f ∧ f x = 1 ∧ f '' A = {0} | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
abbrev I : Set ℝ := Icc 0 1
| |
exercise_34_9 | test | /-- Let $X$ be a compact Hausdorff space that is the union of the closed subspaces $X_1$ and $X_2$. If $X_1$ and $X_2$ are metrizable, show that $X$ is metrizable.-/
| theorem exercise_34_9
(X : Type*) [TopologicalSpace X] [CompactSpace X]
(X1 X2 : Set X) (hX1 : IsClosed X1) (hX2 : IsClosed X2)
(hX : X1 ∪ X2 = univ) (hX1m : MetrizableSpace X1)
(hX2m : MetrizableSpace X2) : MetrizableSpace X := | X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
X1 X2 : Set X
hX1 : IsClosed X1
hX2 : IsClosed X2
hX : X1 ∪ X2 = univ
hX1m : MetrizableSpace ↑X1
hX2m : MetrizableSpace ↑X2
⊢ MetrizableSpace X | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| |
exercise_43_2 | test | /-- Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces; let $Y$ be complete. Let $A \subset X$. Show that if $f \colon A \rightarrow Y$ is uniformly continuous, then $f$ can be uniquely extended to a continuous function $g \colon \bar{A} \rightarrow Y$, and $g$ is uniformly continuous.-/
| theorem exercise_43_2 {X : Type*} [MetricSpace X]
{Y : Type*} [MetricSpace Y] [CompleteSpace Y] (A : Set X)
(f : X → Y) (hf : UniformContinuousOn f A) :
∃! (g : X → Y), ContinuousOn g (closure A) ∧
UniformContinuousOn g (closure A) ∧ ∀ (x : A), g x = f x := | X : Type u_1
inst✝² : MetricSpace X
Y : Type u_2
inst✝¹ : MetricSpace Y
inst✝ : CompleteSpace Y
A : Set X
f : X → Y
hf : UniformContinuousOn f A
⊢ ∃! g, ContinuousOn g (closure A) ∧ UniformContinuousOn g (closure A) ∧ ∀ (x : ↑A), g ↑x = f ↑x | import Mathlib
open Filter Set TopologicalSpace
open scoped Topology
| |
exercise_1_30 | test | /-- Prove that $\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$ is not an integer.-/
| theorem exercise_1_30 {n : ℕ} :
¬ ∃ a : ℤ, ∑ i : Fin n, (1 : ℚ) / (n+2) = a := | n : ℕ
⊢ ¬∃ a, ∑ i : Fin n, 1 / (↑n + 2) = ↑a | import Mathlib
open Real
open scoped BigOperators
| |
exercise_2_4 | test | /-- If $a$ is a nonzero integer, then for $n>m$ show that $\left(a^{2^{n}}+1, a^{2^{m}}+1\right)=1$ or 2 depending on whether $a$ is odd or even.-/
| theorem exercise_2_4 {a : ℤ} (ha : a ≠ 0)
(f_a := λ n m : ℕ => Int.gcd (a^(2^n) + 1) (a^(2^m)+1)) {n m : ℕ}
(hnm : n > m) :
(Odd a → f_a n m = 1) ∧ (Even a → f_a n m = 2) := | a : ℤ
ha : a ≠ 0
f_a : optParam (ℕ → ℕ → ℕ) fun n m => (a ^ 2 ^ n + 1).gcd (a ^ 2 ^ m + 1)
n m : ℕ
hnm : n > m
⊢ (Odd a → f_a n m = 1) ∧ (Even a → f_a n m = 2) | import Mathlib
open Real
open scoped BigOperators
| |
exercise_2_27a | test | /-- Show that $\sum^{\prime} 1 / n$, the sum being over square free integers, diverges.-/
| theorem exercise_2_27a :
¬ Summable (λ i : {p : ℤ // Squarefree p} => (1 : ℚ) / i) := | ⊢ ¬Summable fun i => 1 / ↑↑i | import Mathlib
open Real
open scoped BigOperators
| |
exercise_3_4 | test | /-- Show that the equation $3 x^{2}+2=y^{2}$ has no solution in integers.-/
| theorem exercise_3_4 : ¬ ∃ x y : ℤ, 3*x^2 + 2 = y^2 := | ⊢ ¬∃ x y, 3 * x ^ 2 + 2 = y ^ 2 | import Mathlib
open Real
open scoped BigOperators
| |
exercise_3_10 | test | /-- If $n$ is not a prime, show that $(n-1) ! \equiv 0(n)$, except when $n=4$.-/
| theorem exercise_3_10 {n : ℕ} (hn0 : ¬ n.Prime) (hn1 : n ≠ 4) :
Nat.factorial (n-1) ≡ 0 [MOD n] := | n : ℕ
hn0 : ¬n.Prime
hn1 : n ≠ 4
⊢ (n - 1).factorial ≡ 0 [MOD n] | import Mathlib
open Real
open scoped BigOperators
| |
exercise_4_4 | test | /-- Consider a prime $p$ of the form $4 t+1$. Show that $a$ is a primitive root modulo $p$ iff $-a$ is a primitive root modulo $p$.-/
| theorem exercise_4_4 {p t: ℕ} (hp0 : p.Prime) (hp1 : p = 4*t + 1)
(a : ZMod p) :
IsPrimitiveRoot a p ↔ IsPrimitiveRoot (-a) p := | p t : ℕ
hp0 : p.Prime
hp1 : p = 4 * t + 1
a : ZMod p
⊢ IsPrimitiveRoot a p ↔ IsPrimitiveRoot (-a) p | import Mathlib
open Real
open scoped BigOperators
| |
exercise_4_6 | test | /-- If $p=2^{n}+1$ is a Fermat prime, show that 3 is a primitive root modulo $p$.-/
| theorem exercise_4_6 {p n : ℕ} (hp : p.Prime) (hpn : p = 2^n + 1) :
IsPrimitiveRoot 3 p := | p n : ℕ
hp : p.Prime
hpn : p = 2 ^ n + 1
⊢ IsPrimitiveRoot 3 p | import Mathlib
open Real
open scoped BigOperators
| |
exercise_4_11 | test | /-- Prove that $1^{k}+2^{k}+\cdots+(p-1)^{k} \equiv 0(p)$ if $p-1 \nmid k$ and $-1(p)$ if $p-1 \mid k$.-/
| theorem exercise_4_11 {p : ℕ} (hp : p.Prime) (k s: ℕ)
(s := ∑ n : Fin p, (n : ℕ) ^ k) :
((¬ p - 1 ∣ k) → s ≡ 0 [MOD p]) ∧ (p - 1 ∣ k → s ≡ 0 [MOD p]) := | p : ℕ
hp : p.Prime
k s✝ : ℕ
s : optParam ℕ (∑ n : Fin p, ↑n ^ k)
⊢ (¬p - 1 ∣ k → s ≡ 0 [MOD p]) ∧ (p - 1 ∣ k → s ≡ 0 [MOD p]) | import Mathlib
open Real
open scoped BigOperators
| |
exercise_5_28 | test | /-- Show that $x^{4} \equiv 2(p)$ has a solution for $p \equiv 1(4)$ iff $p$ is of the form $A^{2}+64 B^{2}$.-/
| theorem exercise_5_28 {p : ℕ} (hp : p.Prime) (hp1 : p ≡ 1 [MOD 4]):
∃ x, x^4 ≡ 2 [MOD p] ↔ ∃ A B, p = A^2 + 64*B^2 := | p : ℕ
hp : p.Prime
hp1 : p ≡ 1 [MOD 4]
⊢ ∃ x, x ^ 4 ≡ 2 [MOD p] ↔ ∃ A B, p = A ^ 2 + 64 * B ^ 2 | import Mathlib
open Real
open scoped BigOperators
| |
exercise_12_12 | test | /-- Show that $\sin (\pi / 12)$ is an algebraic number.-/
| theorem exercise_12_12 : IsAlgebraic ℚ (sin (Real.pi/12)) := | ⊢ IsAlgebraic ℚ (π / 12).sin | import Mathlib
open Real
open scoped BigOperators
| |
exercise_2018_a5 | test | /-- Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an infinitely differentiable function satisfying $f(0)=0, f(1)=1$, and $f(x) \geq 0$ for all $x \in$ $\mathbb{R}$. Show that there exist a positive integer $n$ and a real number $x$ such that $f^{(n)}(x)<0$.-/
| theorem exercise_2018_a5 (f : ℝ → ℝ) (hf : ContDiff ℝ ⊤ f)
(hf0 : f 0 = 0) (hf1 : f 1 = 1) (hf2 : ∀ x, f x ≥ 0) :
∃ (n : ℕ) (x : ℝ), iteratedDeriv n f x = 0 := | f : ℝ → ℝ
hf : ContDiff ℝ ⊤ f
hf0 : f 0 = 0
hf1 : f 1 = 1
hf2 : ∀ (x : ℝ), f x ≥ 0
⊢ ∃ n x, iteratedDeriv n f x = 0 | import Mathlib
open scoped BigOperators
| |
exercise_2018_b4 | test | /-- Given a real number $a$, we define a sequence by $x_{0}=1$, $x_{1}=x_{2}=a$, and $x_{n+1}=2 x_{n} x_{n-1}-x_{n-2}$ for $n \geq 2$. Prove that if $x_{n}=0$ for some $n$, then the sequence is periodic.-/
| theorem exercise_2018_b4 (a : ℝ) (x : ℕ → ℝ) (hx0 : x 0 = a)
(hx1 : x 1 = a)
(hxn : ∀ n : ℕ, n ≥ 2 → x (n+1) = 2*(x n)*(x (n-1)) - x (n-2))
(h : ∃ n, x n = 0) :
∃ c, Function.Periodic x c := | a : ℝ
x : ℕ → ℝ
hx0 : x 0 = a
hx1 : x 1 = a
hxn : ∀ n ≥ 2, x (n + 1) = 2 * x n * x (n - 1) - x (n - 2)
h : ∃ n, x n = 0
⊢ ∃ c, Function.Periodic x c | import Mathlib
open scoped BigOperators
| |
exercise_2014_a5 | test | /-- Let-/
| theorem exercise_2014_a5 (P : ℕ → Polynomial ℤ)
(hP : ∀ n, P n = ∑ i : Fin n, (n+1) * Polynomial.X ^ n) :
∀ (j k : ℕ), j ≠ k → IsCoprime (P j) (P k) := | P : ℕ → Polynomial ℤ
hP : ∀ (n : ℕ), P n = ∑ i : Fin n, (↑n + 1) * Polynomial.X ^ n
⊢ ∀ (j k : ℕ), j ≠ k → IsCoprime (P j) (P k) | import Mathlib
open scoped BigOperators
| |
exercise_2001_a5 | test | /-- Prove that there are unique positive integers $a, n$ such that $a^{n+1}-(a+1)^n=2001$.-/
| theorem exercise_2001_a5 :
∃! a : ℕ, ∃! n : ℕ, a > 0 ∧ n > 0 ∧ a^(n+1) - (a+1)^n = 2001 := | ⊢ ∃! a, ∃! n, a > 0 ∧ n > 0 ∧ a ^ (n + 1) - (a + 1) ^ n = 2001 | import Mathlib
open scoped BigOperators
| |
exercise_1999_b4 | test | /-- Let $f$ be a real function with a continuous third derivative such that $f(x), f^{\prime}(x), f^{\prime \prime}(x), f^{\prime \prime \prime}(x)$ are positive for all $x$. Suppose that $f^{\prime \prime \prime}(x) \leq f(x)$ for all $x$. Show that $f^{\prime}(x)<2 f(x)$ for all $x$.-/
| theorem exercise_1999_b4 (f : ℝ → ℝ) (hf: ContDiff ℝ 3 f)
(hf1 : ∀ n ≤ 3, ∀ x : ℝ, iteratedDeriv n f x > 0)
(hf2 : ∀ x : ℝ, iteratedDeriv 3 f x ≤ f x) :
∀ x : ℝ, deriv f x < 2 * f x := | f : ℝ → ℝ
hf : ContDiff ℝ 3 f
hf1 : ∀ n ≤ 3, ∀ (x : ℝ), iteratedDeriv n f x > 0
hf2 : ∀ (x : ℝ), iteratedDeriv 3 f x ≤ f x
⊢ ∀ (x : ℝ), deriv f x < 2 * f x | import Mathlib
open scoped BigOperators
| |
exercise_1998_b6 | test | /-- Prove that, for any integers $a, b, c$, there exists a positive integer $n$ such that $\sqrt{n^3+a n^2+b n+c}$ is not an integer.-/
| theorem exercise_1998_b6 (a b c : ℤ) :
∃ n : ℤ, n > 0 ∧ ¬ ∃ m : ℤ, Real.sqrt (n^3 + a*n^2 + b*n + c) = m := | a b c : ℤ
⊢ ∃ n > 0, ¬∃ m, √(↑n ^ 3 + ↑a * ↑n ^ 2 + ↑b * ↑n + ↑c) = ↑m | import Mathlib
open scoped BigOperators
|