Hugging Face's logo

Datasets:
PAug
/
ProofNetSharp

Modalities:
Text
Formats:
parquet
Size:
< 1K
ArXiv:
Libraries:
Datasets
pandas
Dataset card Viewer Files Community
1
Dataset Viewer
Auto-converted to Parquet
Full Screen
id
stringlengths
17
29
nl_statement
stringlengths
43
377
lean4_src_header
stringclasses
14 values
lean4_formalization
stringlengths
50
426
Artin|exercise_2_2_9
Let $H$ be the subgroup generated by two elements $a, b$ of a group $G$. Prove that if $a b=b a$, then $H$ is an abelian group.
import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators
theorem exercise_2_2_9 {G : Type*} [Group G] {a b : G} (h : a * b = b * a) : βˆ€ x y : closure {x | x = a ∨ x = b}, x*y = y*x :=
Artin|exercise_2_4_19
Prove that if a group contains exactly one element of order 2 , then that element is in the center of the group.
import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators
theorem exercise_2_4_19 {G : Type*} [Group G] {x : G} (hx : orderOf x = 2) (hx1 : βˆ€ y, orderOf y = 2 β†’ y = x) : x ∈ center G :=
Artin|exercise_2_11_3
Prove that a group of even order contains an element of order $2 .$
import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators
theorem exercise_2_11_3 {G : Type*} [Group G] [Fintype G] (hG : Even (card G)) : βˆƒ x : G, orderOf x = 2 :=
Artin|exercise_3_5_6
Let $V$ be a vector space which is spanned by a countably infinite set. Prove that every linearly independent subset of $V$ is finite or countably infinite.
import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators
theorem exercise_3_5_6 {K V : Type*} [Field K] [AddCommGroup V] [Module K V] {S : Set V} (hS : Set.Countable S) (hS1 : span K S = ⊀) {ΞΉ : Type*} (R : ΞΉ β†’ V) (hR : LinearIndependent K R) : Countable ΞΉ :=
Artin|exercise_6_1_14
Let $Z$ be the center of a group $G$. Prove that if $G / Z$ is a cyclic group, then $G$ is abelian and hence $G=Z$.
import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators
theorem exercise_6_1_14 (G : Type*) [Group G] (hG : IsCyclic $ G ⧸ (center G)) : center G = ⊀ :=
Artin|exercise_6_4_3
Prove that no group of order $p^2 q$, where $p$ and $q$ are prime, is simple.
import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators
theorem exercise_6_4_3 {G : Type*} [Group G] [Fintype G] {p q : β„•} (hp : Prime p) (hq : Prime q) (hG : card G = p^2 *q) : IsSimpleGroup G β†’ false :=
Artin|exercise_6_8_1
Prove that two elements $a, b$ of a group generate the same subgroup as $b a b^2, b a b^3$.
import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators
theorem exercise_6_8_1 {G : Type*} [Group G] (a b : G) : closure ({a, b} : Set G) = Subgroup.closure {b*a*b^2, b*a*b^3} :=
Artin|exercise_10_2_4
Prove that in the ring $\mathbb{Z}[x],(2) \cap(x)=(2 x)$.
import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators
theorem exercise_10_2_4 : span ({2} : Set $ Polynomial β„€) βŠ“ (span {X}) = span ({2 * X} : Set $ Polynomial β„€) :=
Artin|exercise_10_4_6
Let $I, J$ be ideals in a ring $R$. Prove that the residue of any element of $I \cap J$ in $R / I J$ is nilpotent.
import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators
theorem exercise_10_4_6 {R : Type*} [CommRing R] (I J : Ideal R) (x : ↑(I βŠ“ J)) : IsNilpotent ((Ideal.Quotient.mk (I*J)) x) :=
Artin|exercise_10_7_10
Let $R$ be a ring, with $M$ an ideal of $R$. Suppose that every element of $R$ which is not in $M$ is a unit of $R$. Prove that $M$ is a maximal ideal and that moreover it is the only maximal ideal of $R$.
import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators
theorem exercise_10_7_10 {R : Type*} [Ring R] (M : Ideal R) (hM : βˆ€ (x : R), x βˆ‰ M β†’ IsUnit x) (hProper : βˆƒ x : R, x βˆ‰ M) : IsMaximal M ∧ βˆ€ (N : Ideal R), IsMaximal N β†’ N = M :=
Artin|exercise_11_4_1b
Prove that $x^3 + 6x + 12$ is irreducible in $\mathbb{Q}$.
import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators
theorem exercise_11_4_1b : Irreducible (12 + 6 * X + X ^ 3 : Polynomial β„š) :=
Artin|exercise_11_4_6b
Prove that $x^2+1$ is irreducible in $\mathbb{F}_7$
import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators
theorem exercise_11_4_6b {F : Type*} [Field F] [Fintype F] (hF : card F = 7) : Irreducible (X ^ 2 + 1 : Polynomial F) :=
Artin|exercise_11_4_8
Let $p$ be a prime integer. Prove that the polynomial $x^n-p$ is irreducible in $\mathbb{Q}[x]$.
import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators
theorem exercise_11_4_8 (p : β„•) (hp : Prime p) (n : β„•) (hn : n > 0) : Irreducible (X ^ n - (p : Polynomial β„š) : Polynomial β„š) :=
Artin|exercise_13_4_10
Prove that if a prime integer $p$ has the form $2^r+1$, then it actually has the form $2^{2^k}+1$.
import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators
theorem exercise_13_4_10 {p : β„•} {hp : Nat.Prime p} (h : βˆƒ r : β„•, p = 2 ^ r + 1) : βˆƒ (k : β„•), p = 2 ^ (2 ^ k) + 1 :=
Axler|exercise_1_3
Prove that $-(-v) = v$ for every $v \in V$.
import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators
theorem exercise_1_3 {F V : Type*} [AddCommGroup V] [Field F] [Module F V] {v : V} : -(-v) = v :=
Axler|exercise_1_6
Give an example of a nonempty subset $U$ of $\mathbf{R}^2$ such that $U$ is closed under addition and under taking additive inverses (meaning $-u \in U$ whenever $u \in U$), but $U$ is not a subspace of $\mathbf{R}^2$.
import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators
theorem exercise_1_6 : βˆƒ U : Set (ℝ Γ— ℝ), (U β‰  βˆ…) ∧ (βˆ€ (u v : ℝ Γ— ℝ), u ∈ U ∧ v ∈ U β†’ u + v ∈ U) ∧ (βˆ€ (u : ℝ Γ— ℝ), u ∈ U β†’ -u ∈ U) ∧ (βˆ€ U' : Submodule ℝ (ℝ Γ— ℝ), U β‰  ↑U') :=
Axler|exercise_1_8
Prove that the intersection of any collection of subspaces of $V$ is a subspace of $V$.
import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators
theorem exercise_1_8 {F V : Type*} [AddCommGroup V] [Field F] [Module F V] {ΞΉ : Type*} (u : ΞΉ β†’ Submodule F V) : βˆƒ U : Submodule F V, (β‹‚ (i : ΞΉ), (u i).carrier) = ↑U :=
Axler|exercise_3_1
Show that every linear map from a one-dimensional vector space to itself is multiplication by some scalar. More precisely, prove that if $\operatorname{dim} V=1$ and $T \in \mathcal{L}(V, V)$, then there exists $a \in \mathbf{F}$ such that $T v=a v$ for all $v \in V$.
import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators
theorem exercise_3_1 {F V : Type*} [AddCommGroup V] [Field F] [Module F V] [FiniteDimensional F V] (T : V β†’β‚—[F] V) (hT : finrank F V = 1) : βˆƒ c : F, βˆ€ v : V, T v = c β€’ v:=
Axler|exercise_4_4
Suppose $p \in \mathcal{P}(\mathbf{C})$ has degree $m$. Prove that $p$ has $m$ distinct roots if and only if $p$ and its derivative $p^{\prime}$ have no roots in common.
import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators
theorem exercise_4_4 (p : Polynomial β„‚) : p.degree = @card (rootSet p β„‚) (rootSetFintype p β„‚) ↔ Disjoint (@card (rootSet (derivative p) β„‚) (rootSetFintype (derivative p) β„‚)) (@card (rootSet p β„‚) (rootSetFintype p β„‚)) :=
Axler|exercise_5_4
Suppose that $S, T \in \mathcal{L}(V)$ are such that $S T=T S$. Prove that $\operatorname{null} (T-\lambda I)$ is invariant under $S$ for every $\lambda \in \mathbf{F}$.
import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators
theorem exercise_5_4 {F V : Type*} [AddCommGroup V] [Field F] [Module F V] (S T : V β†’β‚—[F] V) (hST : S ∘ T = T ∘ S) (c : F): Submodule.map S (ker (T - c β€’ LinearMap.id)) = ker (T - c β€’ LinearMap.id) :=
Axler|exercise_5_12
Suppose $T \in \mathcal{L}(V)$ is such that every vector in $V$ is an eigenvector of $T$. Prove that $T$ is a scalar multiple of the identity operator.
import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators
theorem exercise_5_12 {F V : Type*} [AddCommGroup V] [Field F] [Module F V] {S : End F V} (hS : βˆ€ v : V, βˆƒ c : F, v ∈ eigenspace S c) : βˆƒ c : F, S = c β€’ LinearMap.id :=
Axler|exercise_5_20
Suppose that $T \in \mathcal{L}(V)$ has $\operatorname{dim} V$ distinct eigenvalues and that $S \in \mathcal{L}(V)$ has the same eigenvectors as $T$ (not necessarily with the same eigenvalues). Prove that $S T=T S$.
import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators
theorem exercise_5_20 {F V : Type*} [AddCommGroup V] [Field F] [Module F V] [FiniteDimensional F V] {S T : End F V} (h1 : card (T.Eigenvalues) = finrank F V) (h2 : βˆ€ v : V, (βˆƒ c : F, v ∈ eigenspace S c) ↔ (βˆƒ c : F, v ∈ eigenspace T c)) : S * T = T * S :=
Axler|exercise_6_2
Suppose $u, v \in V$. Prove that $\langle u, v\rangle=0$ if and only if $\|u\| \leq\|u+a v\|$ for all $a \in \mathbf{F}$.
import Mathlib open InnerProductSpace RCLike ContinuousLinearMap Complex open scoped BigOperators
theorem exercise_6_2 {V : Type*} [NormedAddCommGroup V] [NormedField F] [RCLike F] [Module F V] [InnerProductSpace F V] (u v : V) : βŸͺu, v⟫_F = 0 ↔ βˆ€ (a : F), β€–uβ€– ≀ β€–u + a β€’ vβ€– :=
Axler|exercise_6_7
Prove that if $V$ is a complex inner-product space, then $\langle u, v\rangle=\frac{\|u+v\|^{2}-\|u-v\|^{2}+\|u+i v\|^{2} i-\|u-i v\|^{2} i}{4}$ for all $u, v \in V$.
import Mathlib open InnerProductSpace ContinuousLinearMap Complex open scoped BigOperators
theorem exercise_6_7 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace β„‚ V] (u v : V) : βŸͺu, v⟫_β„‚ = (β€–u + vβ€–^2 - β€–u - vβ€–^2 + I*β€–u + Iβ€’vβ€–^2 - I*β€–u-Iβ€’vβ€–^2) / 4 :=
Axler|exercise_6_16
Suppose $U$ is a subspace of $V$. Prove that $U^{\perp}=\{0\}$ if and only if $U=V$
import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators
theorem exercise_6_16 {K V : Type*} [RCLike K] [NormedAddCommGroup V] [InnerProductSpace K V] {U : Submodule K V} : U.orthogonal = βŠ₯ ↔ U = ⊀ :=
Axler|exercise_7_6
Prove that if $T \in \mathcal{L}(V)$ is normal, then $\operatorname{range} T=\operatorname{range} T^{*}.$
import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators
theorem exercise_7_6 {V : Type*} [NormedAddCommGroup V] [RCLike F] [InnerProductSpace F V] [FiniteDimensional F V] (T : End F V) (hT : T * adjoint T = adjoint T * T) : range T = range (adjoint T) :=
Axler|exercise_7_10
Suppose $V$ is a complex inner-product space and $T \in \mathcal{L}(V)$ is a normal operator such that $T^{9}=T^{8}$. Prove that $T$ is self-adjoint and $T^{2}=T$.
import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators
theorem exercise_7_10 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace β„‚ V] [FiniteDimensional β„‚ V] (T : End β„‚ V) (hT : T * adjoint T = adjoint T * T) (hT1 : T^9 = T^8) : IsSelfAdjoint T ∧ T^2 = T :=
Axler|exercise_7_14
Suppose $T \in \mathcal{L}(V)$ is self-adjoint, $\lambda \in \mathbf{F}$, and $\epsilon>0$. Prove that if there exists $v \in V$ such that $\|v\|=1$ and $\|T v-\lambda v\|<\epsilon,$ then $T$ has an eigenvalue $\lambda^{\prime}$ such that $\left|\lambda-\lambda^{\prime}\right|<\epsilon$.
import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators
theorem exercise_7_14 {π•œ V : Type*} [RCLike π•œ] [NormedAddCommGroup V] [InnerProductSpace π•œ V] [FiniteDimensional π•œ V] {T : Module.End π•œ V} (hT : IsSelfAdjoint T) {l : π•œ} {Ξ΅ : ℝ} (he : Ξ΅ > 0) : (βˆƒ v : V, β€–vβ€–= 1 ∧ β€–T v - l β€’ vβ€– < Ξ΅) β†’ (βˆƒ l' : T.Eigenvalues, β€–l - l'β€– < Ξ΅) :=
Dummit-Foote|exercise_1_1_3
Prove that the addition of residue classes $\mathbb{Z}/n\mathbb{Z}$ is associative.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_1_1_3 (n : β„•) : βˆ€ (x y z : ZMod n), (x + y) + z = x + (y + z) :=
Dummit-Foote|exercise_1_1_5
Prove that for all $n>1$ that $\mathbb{Z}/n\mathbb{Z}$ is not a group under multiplication of residue classes.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_1_1_5 (n : β„•) (hn : 1 < n) : IsEmpty (Group (ZMod n)) :=
Dummit-Foote|exercise_1_1_16
Let $x$ be an element of $G$. Prove that $x^2=1$ if and only if $|x|$ is either $1$ or $2$.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_1_1_16 {G : Type*} [Group G] (x : G) : x ^ 2 = 1 ↔ (orderOf x = 1 ∨ orderOf x = 2) :=
Dummit-Foote|exercise_1_1_18
Let $x$ and $y$ be elements of $G$. Prove that $xy=yx$ if and only if $y^{-1}xy=x$ if and only if $x^{-1}y^{-1}xy=1$.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_1_1_18 {G : Type*} [Group G] (x y : G) : (x * y = y * x ↔ y⁻¹ * x * y = x) ∧ (y⁻¹ * x * y = x ↔ x⁻¹ * y⁻¹ * x * y = 1) :=
Dummit-Foote|exercise_1_1_22a
If $x$ and $g$ are elements of the group $G$, prove that $|x|=\left|g^{-1} x g\right|$.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_1_1_22a {G : Type*} [Group G] (x g : G) : orderOf x = orderOf (g⁻¹ * x * g) :=
Dummit-Foote|exercise_1_1_25
Prove that if $x^{2}=1$ for all $x \in G$ then $G$ is abelian.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_1_1_25 {G : Type*} [Group G] (h : βˆ€ x : G, x ^ 2 = 1) : βˆ€ a b : G, a*b = b*a :=
Dummit-Foote|exercise_1_1_34
If $x$ is an element of infinite order in $G$, prove that the elements $x^{n}, n \in \mathbb{Z}$ are all distinct.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_1_1_34 {G : Type*} [Group G] {x : G} (hx_inf : orderOf x = 0) (n m : β„€) (hnm : n β‰  m) : x ^ n β‰  x ^ m :=
Dummit-Foote|exercise_1_6_4
Prove that the multiplicative groups $\mathbb{R}-\{0\}$ and $\mathbb{C}-\{0\}$ are not isomorphic.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_1_6_4 : IsEmpty (Multiplicative ℝ ≃* Multiplicative β„‚) :=
Dummit-Foote|exercise_1_6_17
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.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_1_6_17 {G : Type*} [Group G] (f : G β†’ G) (hf : f = Ξ» g => g⁻¹) : (βˆ€ x y : G, f x * f y = f (x*y)) ↔ βˆ€ x y : G, x*y = y*x :=
Dummit-Foote|exercise_2_1_5
Prove that $G$ cannot have a subgroup $H$ with $|H|=n-1$, where $n=|G|>2$.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
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 :=
Dummit-Foote|exercise_2_4_4
Prove that if $H$ is a subgroup of $G$ then $H$ is generated by the set $H-\{1\}$.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_2_4_4 {G : Type*} [Group G] (H : Subgroup G) : closure ((H : Set G) \ {1}) = H :=
Dummit-Foote|exercise_2_4_16b
Show that the subgroup of all rotations in a dihedral group is a maximal subgroup.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_2_4_16b {n : β„•} {hn : n β‰  0} {R : Subgroup (DihedralGroup n)} (hR : R = Subgroup.closure {DihedralGroup.r 1}) : R β‰  ⊀ ∧ βˆ€ K : Subgroup (DihedralGroup n), R ≀ K β†’ K = R ∨ K = ⊀ :=
Dummit-Foote|exercise_3_1_3a
Let $A$ be an abelian group and let $B$ be a subgroup of $A$. Prove that $A / B$ is abelian.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_3_1_3a {A : Type*} [CommGroup A] (B : Subgroup A) : βˆ€ a b : A β§Έ B, a*b = b*a :=
Dummit-Foote|exercise_3_1_22b
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).
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_3_1_22b {G : Type*} [Group G] (I : Type*) [Nonempty I] (H : I β†’ Subgroup G) (hH : βˆ€ i : I, Normal (H i)) : Normal (β¨… (i : I), H i):=
Dummit-Foote|exercise_3_2_11
Let $H \leq K \leq G$. Prove that $|G: H|=|G: K| \cdot|K: H|$ (do not assume $G$ is finite).
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_3_2_11 {G : Type*} [Group G] {H K : Subgroup G} (hHK : H ≀ K) : H.index = K.index * H.relindex K :=
Dummit-Foote|exercise_3_2_21a
Prove that $\mathbb{Q}$ has no proper subgroups of finite index.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_3_2_21a (H : AddSubgroup β„š) (hH : H β‰  ⊀) : H.index = 0 :=
Dummit-Foote|exercise_3_4_1
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).
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_3_4_1 (G : Type*) [CommGroup G] [IsSimpleGroup G] : IsCyclic G ∧ βˆƒ G_fin : Fintype G, Nat.Prime (@card G G_fin) :=
Dummit-Foote|exercise_3_4_5a
Prove that subgroups of a solvable group are solvable.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_3_4_5a {G : Type*} [Group G] (H : Subgroup G) [IsSolvable G] : IsSolvable H :=
Dummit-Foote|exercise_3_4_11
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.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_3_4_11 {G : Type*} [Group G] [IsSolvable G] {H : Subgroup G} (hH : H β‰  βŠ₯) [H.Normal] : βˆƒ A ≀ H, A β‰  βŠ₯ ∧ A.Normal ∧ βˆ€ a b : A, a*b = b*a :=
Dummit-Foote|exercise_4_2_14
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.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_4_2_14 {G : Type*} [Fintype G] [Group G] (hG : Β¬ (card G).Prime) (hG1 : βˆ€ k : β„•, k ∣ card G β†’ βˆƒ (H : Subgroup G) (fH : Fintype H), @card H fH = k) : Β¬ IsSimpleGroup G :=
Dummit-Foote|exercise_4_3_26
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 $a \in A$.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_4_3_26 {Ξ± : Type*} [Fintype Ξ±] (ha : card Ξ± > 1) (h_tran : βˆ€ a b: Ξ±, βˆƒ Οƒ : Equiv.Perm Ξ±, Οƒ a = b) : βˆƒ Οƒ : Equiv.Perm Ξ±, βˆ€ a : Ξ±, Οƒ a β‰  a :=
Dummit-Foote|exercise_4_4_6a
Prove that characteristic subgroups are normal.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_4_4_6a {G : Type*} [Group G] (H : Subgroup G) [Characteristic H] : Normal H :=
Dummit-Foote|exercise_4_4_7
If $H$ is the unique subgroup of a given order in a group $G$ prove $H$ is characteristic in $G$.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_4_4_7 {G : Type*} [Group G] {H : Subgroup G} [Fintype H] (hH : βˆ€ (K : Subgroup G) (fK : Fintype K), card H = @card K fK β†’ H = K) : H.Characteristic :=
Dummit-Foote|exercise_4_5_1a
Prove that if $P \in \operatorname{Syl}_{p}(G)$ and $H$ is a subgroup of $G$ containing $P$ then $P \in \operatorname{Syl}_{p}(H)$.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_4_5_1a {p : β„•} {G : Type*} [Group G] {P : Sylow p G} (H : Subgroup G) (hH : P ≀ H) : IsPGroup p (P.subgroupOf H) ∧ βˆ€ (Q : Subgroup H), IsPGroup p Q β†’ (P.subgroupOf H) ≀ Q β†’ Q = (P.subgroupOf H) :=
Dummit-Foote|exercise_4_5_14
Prove that a group of order 312 has a normal Sylow $p$-subgroup for some prime $p$ dividing its order.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_4_5_14 {G : Type*} [Group G] [Fintype G] (hG : card G = 312) : βˆƒ (p : β„•) (P : Sylow p G), p.Prime ∧ (p ∣ card G) ∧ P.Normal :=
Dummit-Foote|exercise_4_5_16
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 $r$.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_4_5_16 {p q r : β„•} {G : Type*} [Group G] [Fintype G] (hpqr : p < q ∧ q < r) (hpqr1 : p.Prime ∧ q.Prime ∧ r.Prime)(hG : card G = p*q*r) : (βˆƒ (P : Sylow p G), P.Normal) ∨ (βˆƒ (P : Sylow q G), P.Normal) ∨ (βˆƒ (P : Sylow r G), P.Normal) :=
Dummit-Foote|exercise_4_5_18
Prove that a group of order 200 has a normal Sylow 5-subgroup.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_4_5_18 {G : Type*} [Fintype G] [Group G] (hG : card G = 200) : βˆƒ N : Sylow 5 G, N.Normal :=
Dummit-Foote|exercise_4_5_20
Prove that if $|G|=1365$ then $G$ is not simple.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_4_5_20 {G : Type*} [Fintype G] [Group G] (hG : card G = 1365) : Β¬ IsSimpleGroup G :=
Dummit-Foote|exercise_4_5_22
Prove that if $|G|=132$ then $G$ is not simple.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_4_5_22 {G : Type*} [Fintype G] [Group G] (hG : card G = 132) : Β¬ IsSimpleGroup G :=
Dummit-Foote|exercise_4_5_28
Let $G$ be a group of order 105. Prove that if a Sylow 3-subgroup of $G$ is normal then $G$ is abelian.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_4_5_28 {G : Type*} [Group G] [Fintype G] (hG : card G = 105) (P : Sylow 3 G) [hP : P.Normal] : βˆ€ a b : G, a*b = b*a :=
Dummit-Foote|exercise_5_4_2
Prove that a subgroup $H$ of $G$ is normal if and only if $[G, H] \leq H$.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_5_4_2 {G : Type*} [Group G] (H : Subgroup G) : H.Normal ↔ ⁅(⊀ : Subgroup G), H⁆ ≀ H :=
Dummit-Foote|exercise_7_1_11
Prove that if $R$ is an integral domain and $x^{2}=1$ for some $x \in R$ then $x=\pm 1$.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_7_1_11 {R : Type*} [CommRing R] [IsDomain R] {x : R} (hx : x^2 = 1) : x = 1 ∨ x = -1 :=
Dummit-Foote|exercise_7_1_15
A ring $R$ is called a Boolean ring if $a^{2}=a$ for all $a \in R$. Prove that every Boolean ring is commutative.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_7_1_15 {R : Type*} [Ring R] (hR : βˆ€ a : R, a^2 = a) : βˆ€ a b : R, a*b = b*a :=
Dummit-Foote|exercise_7_2_12
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 $R G$.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_7_2_12 {R G : Type*} [Ring R] [Group G] [Fintype G] : βˆ‘ g : G, MonoidAlgebra.of R G g ∈ center (MonoidAlgebra R G) :=
Dummit-Foote|exercise_7_3_37
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 $\mathbb{Z} / p^{m} \mathbb{Z}$.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_7_3_37 {p m : β„•} (hp : p.Prime) : IsNilpotent (span ({↑p} : Set $ ZMod $ p^m) : Ideal $ ZMod $ p^m) :=
Dummit-Foote|exercise_8_1_12
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 $d \bmod \varphi(N)$: $d d^{\prime} \equiv 1 \pmod {\varphi(N)}$.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
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' ≑ M^d [ZMOD N]) : βˆƒ d' : β„•, d' * d ≑ 1 [ZMOD N.totient] ∧ M ≑ M'^d' [ZMOD N] :=
Dummit-Foote|exercise_8_3_4
Prove that if an integer is the sum of two rational squares, then it is the sum of two integer squares.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_8_3_4 {n : β„€} {r s : β„š} (h : r^2 + s^2 = n) : βˆƒ a b : β„€, a^2 + b^2 = n :=
Dummit-Foote|exercise_8_3_6a
Prove that the quotient ring $\mathbb{Z}[i] /(1+i)$ is a field of order 2.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_8_3_6a {R : Type} [Ring R] (hR : R = (GaussianInt β§Έ span ({⟨1, 1⟩} : Set GaussianInt))) : IsField R ∧ βˆƒ finR : Fintype R, @card R finR = 2 :=
Dummit-Foote|exercise_9_1_6
Prove that $(x, y)$ is not a principal ideal in $\mathbb{Q}[x, y]$.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_9_1_6 : Β¬ Submodule.IsPrincipal (span ({MvPolynomial.X 0, MvPolynomial.X 1} : Set (MvPolynomial (Fin 2) β„š))) :=
Dummit-Foote|exercise_9_3_2
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.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_9_3_2 {f g : Polynomial β„š} (i j : β„•) (hfg : βˆ€ n : β„•, βˆƒ a : β„€, (f*g).coeff = a) : βˆƒ a : β„€, f.coeff i * g.coeff j = a :=
Dummit-Foote|exercise_9_4_2b
Prove that $x^6+30x^5-15x^3 + 6x-120$ is irreducible in $\mathbb{Z}[x]$.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_9_4_2b : Irreducible (X^6 + 30*X^5 - 15*X^3 + 6*X - 120 : Polynomial β„€) :=
Dummit-Foote|exercise_9_4_2d
Prove that $\frac{(x+2)^p-2^p}{x}$, where $p$ is an odd prime, is irreducible in $\mathbb{Z}[x]$.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_9_4_2d {p : β„•} (hp : p.Prime ∧ p > 2) {f : Polynomial β„€} (hf : f = (X + 2)^p): Irreducible (βˆ‘ n in (f.support \ {0}), (f.coeff n : Polynomial β„€) * X ^ (n-1) : Polynomial β„€) :=
Dummit-Foote|exercise_9_4_11
Prove that $x^2+y^2-1$ is irreducible in $\mathbb{Q}[x,y]$.
import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators
theorem exercise_9_4_11 : Irreducible ((MvPolynomial.X 0)^2 + (MvPolynomial.X 1)^2 - 1 : MvPolynomial (Fin 2) β„š) :=
Herstein|exercise_2_1_21
Show that a group of order 5 must be abelian.
import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators
theorem exercise_2_1_21 (G : Type*) [Group G] [Fintype G] (hG : card G = 5) : βˆ€ a b : G, a*b = b*a :=
Herstein|exercise_2_1_27
If $G$ is a finite group, prove that there is an integer $m > 0$ such that $a^m = e$ for all $a \in G$.
import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators
theorem exercise_2_1_27 {G : Type*} [Group G] [Fintype G] : βˆƒ (m : β„•), m > 0 ∧ βˆ€ (a : G), a ^ m = 1 :=
Herstein|exercise_2_2_5
Let $G$ be a group in which $(a b)^{3}=a^{3} b^{3}$ and $(a b)^{5}=a^{5} b^{5}$ for all $a, b \in G$. Show that $G$ is abelian.
import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators
theorem exercise_2_2_5 {G : Type*} [Group G] (h : βˆ€ (a b : G), (a * b) ^ 3 = a ^ 3 * b ^ 3 ∧ (a * b) ^ 5 = a ^ 5 * b ^ 5) : βˆ€ a b : G, a*b = b*a :=
Herstein|exercise_2_3_17
If $G$ is a group and $a, x \in G$, prove that $C\left(x^{-1} a x\right)=x^{-1} C(a) x$
import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators
theorem exercise_2_3_17 {G : Type*} [Mul G] [Group G] (a x : G) : centralizer {x⁻¹*a*x} = (λ g : G => x⁻¹*g*x) '' (centralizer {a}) :=
Herstein|exercise_2_4_36
If $a > 1$ is an integer, show that $n \mid \varphi(a^n - 1)$, where $\phi$ is the Euler $\varphi$-function.
import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators
theorem exercise_2_4_36 {a n : β„•} (h : a > 1) : n ∣ (a ^ n - 1).totient :=
Herstein|exercise_2_5_30
Suppose that $|G| = pm$, where $p \nmid m$ and $p$ is a prime. If $H$ is a normal subgroup of order $p$ in $G$, prove that $H$ is characteristic.
import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators
theorem exercise_2_5_30 {G : Type*} [Group G] [Fintype G] {p m : β„•} (hp : Nat.Prime p) (hp1 : Β¬ p ∣ m) (hG : card G = p*m) {H : Subgroup G} [Fintype H] [H.Normal] (hH : card H = p): Subgroup.Characteristic H :=
Herstein|exercise_2_5_37
If $G$ is a nonabelian group of order 6, prove that $G \simeq S_3$.
import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators
theorem exercise_2_5_37 (G : Type*) [Group G] [Fintype G] (hG : card G = 6) (hG' : IsEmpty (CommGroup G)) : Nonempty (G ≃* Equiv.Perm (Fin 3)) :=
Herstein|exercise_2_5_44
Prove that a group of order $p^2$, $p$ a prime, has a normal subgroup of order $p$.
import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators
theorem exercise_2_5_44 {G : Type*} [Group G] [Fintype G] {p : β„•} (hp : Nat.Prime p) (hG : card G = p^2) : βˆƒ (N : Subgroup G) (Fin : Fintype N), @card N Fin = p ∧ N.Normal :=
Herstein|exercise_2_6_15
If $G$ is an abelian group and if $G$ has an element of order $m$ and one of order $n$, where $m$ and $n$ are relatively prime, prove that $G$ has an element of order $mn$.
import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators
theorem exercise_2_6_15 {G : Type*} [CommGroup G] {m n : β„•} (hm : βˆƒ (g : G), orderOf g = m) (hn : βˆƒ (g : G), orderOf g = n) (hmn : m.Coprime n) : βˆƒ (g : G), orderOf g = m * n :=
Herstein|exercise_2_8_12
Prove that any two nonabelian groups of order 21 are isomorphic.
import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators
theorem exercise_2_8_12 {G H : Type*} [Fintype G] [Fintype H] [Group G] [Group H] (hG : card G = 21) (hH : card H = 21) (hG1 : IsEmpty (CommGroup G)) (hH1 : IsEmpty (CommGroup H)) : Nonempty (G ≃* H) :=
Herstein|exercise_2_9_2
If $G_1$ and $G_2$ are cyclic groups of orders $m$ and $n$, respectively, prove that $G_1 \times G_2$ is cyclic if and only if $m$ and $n$ are relatively prime.
import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators
theorem exercise_2_9_2 {G H : Type*} [Fintype G] [Fintype H] [Group G] [Group H] (hG : IsCyclic G) (hH : IsCyclic H) : IsCyclic (G Γ— H) ↔ (card G).Coprime (card H) :=
Herstein|exercise_2_11_6
If $P$ is a $p$-Sylow subgroup of $G$ and $P \triangleleft G$, prove that $P$ is the only $p$-Sylow subgroup of $G$.
import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators
theorem exercise_2_11_6 {G : Type*} [Group G] {p : β„•} (hp : Nat.Prime p) {P : Sylow p G} (hP : P.Normal) : βˆ€ (Q : Sylow p G), P = Q :=
Herstein|exercise_2_11_22
Show that any subgroup of order $p^{n-1}$ in a group $G$ of order $p^n$ is normal in $G$.
import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators
theorem exercise_2_11_22 {p : β„•} {n : β„•} {G : Type*} [Fintype G] [Group G] (hp : Nat.Prime p) (hG : card G = p ^ n) {K : Subgroup G} [Fintype K] (hK : card K = p ^ (n-1)) : K.Normal :=
Herstein|exercise_4_1_19
Show that there is an infinite number of solutions to $x^2 = -1$ in the quaternions.
import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators
theorem exercise_4_1_19 : Infinite {x : Quaternion ℝ | x^2 = -1} :=
Herstein|exercise_4_2_5
Let $R$ be a ring in which $x^3 = x$ for every $x \in R$. Prove that $R$ is commutative.
import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators
theorem exercise_4_2_5 {R : Type*} [Ring R] (h : βˆ€ x : R, x ^ 3 = x) : Nonempty (CommRing R) :=
Herstein|exercise_4_2_9
Let $p$ be an odd prime and let $1 + \frac{1}{2} + ... + \frac{1}{p - 1} = \frac{a}{b}$, where $a, b$ are integers. Show that $p \mid a$.
import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators
theorem exercise_4_2_9 {p : β„•} (hp : Nat.Prime p) (hp1 : Odd p) : βˆ€ (a b : β„€), (a / b : β„š) = βˆ‘ i in Finset.range (p-1), (1 / (i + 1) : β„š) β†’ ↑p ∣ a :=
Herstein|exercise_4_3_25
Let $R$ be the ring of $2 \times 2$ matrices over the real numbers; suppose that $I$ is an ideal of $R$. Show that $I = (0)$ or $I = R$.
import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators
theorem exercise_4_3_25 (I : Ideal (Matrix (Fin 2) (Fin 2) ℝ)) : I = βŠ₯ ∨ I = ⊀ :=
Herstein|exercise_4_5_16
Let $F = \mathbb{Z}_p$ be the field of integers $\mod p$, where $p$ is a prime, and let $q(x) \in F[x]$ be irreducible of degree $n$. Show that $F[x]/(q(x))$ is a field having at exactly $p^n$ elements.
import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators
theorem exercise_4_5_16 {p n: β„•} (hp : Nat.Prime p) {q : Polynomial (ZMod p)} (hq : Irreducible q) (hn : q.degree = n) : (βˆƒ is_fin : Fintype $ Polynomial (ZMod p) β§Έ span ({q}), @card (Polynomial (ZMod p) β§Έ span {q}) is_fin = p ^ n) ∧ IsField (Polynomial (ZMod p) β§Έ span {q}) :=
Herstein|exercise_4_5_25
If $p$ is a prime, show that $q(x) = 1 + x + x^2 + \cdots x^{p - 1}$ is irreducible in $Q[x]$.
import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators
theorem exercise_4_5_25 {p : β„•} (hp : Nat.Prime p) : Irreducible (βˆ‘ i in Finset.range p, X ^ i : Polynomial β„š) :=
Herstein|exercise_4_6_3
Show that there is an infinite number of integers a such that $f(x) = x^7 + 15x^2 - 30x + a$ is irreducible in $Q[x]$.
import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators
theorem exercise_4_6_3 : Infinite {a : β„€ | Irreducible (X^7 + 15*X^2 - 30*X + (a : Polynomial β„š) : Polynomial β„š)} :=
Herstein|exercise_5_2_20
Let $V$ be a vector space over an infinite field $F$. Show that $V$ cannot be the set-theoretic union of a finite number of proper subspaces of $V$.
import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators
theorem exercise_5_2_20 {F V ΞΉ: Type*} [Infinite F] [Field F] [AddCommGroup V] [Module F V] [Finite ΞΉ] {u : ΞΉ β†’ Submodule F V} (hu : βˆ€ i : ΞΉ, u i β‰  ⊀) : (⋃ i : ΞΉ, (u i : Set V)) β‰  ⊀ :=
Herstein|exercise_5_3_10
Prove that $\cos 1^{\circ}$ is algebraic over $\mathbb{Q}$.
import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators
theorem exercise_5_3_10 : IsAlgebraic β„š (cos (Real.pi / 180)) :=
Herstein|exercise_5_5_2
Prove that $x^3 - 3x - 1$ is irreducible over $\mathbb{Q}$.
import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators
theorem exercise_5_5_2 : Irreducible (X^3 - 3*X - 1 : Polynomial β„š) :=
Ireland-Rosen|exercise_1_27
For all odd $n$ show that $8 \mid n^{2}-1$.
import Mathlib open Real open scoped BigOperators
theorem exercise_1_27 {n : β„•} (hn : Odd n) : 8 ∣ (n^2 - 1) :=
Ireland-Rosen|exercise_1_31
Show that 2 is divisible by $(1+i)^{2}$ in $\mathbb{Z}[i]$.
import Mathlib open Real open scoped BigOperators
theorem exercise_1_31 : (⟨1, 1⟩ : GaussianInt) ^ 2 ∣ 2 :=
Ireland-Rosen|exercise_2_21
Define $\wedge(n)=\log p$ if $n$ is a power of $p$ and zero otherwise. Prove that $\sum_{A \mid n} \mu(n / d) \log d$ $=\wedge(n)$.
import Mathlib open Real open scoped BigOperators
theorem exercise_2_21 {l : β„• β†’ ℝ} (hl : βˆ€ p n : β„•, p.Prime β†’ l (p^n) = log p ) (hl1 : βˆ€ m : β„•, Β¬ IsPrimePow m β†’ l m = 0) : l = Ξ» n => βˆ‘ d : Nat.divisors n, ArithmeticFunction.moebius (n/d) * log d :=
Ireland-Rosen|exercise_3_1
Show that there are infinitely many primes congruent to $-1$ modulo 6 .
import Mathlib open Real open scoped BigOperators
theorem exercise_3_1 : Infinite {p : Nat.Primes // p ≑ -1 [ZMOD 6]} :=
Ireland-Rosen|exercise_3_5
Show that the equation $7 x^{3}+2=y^{3}$ has no solution in integers.
import Mathlib open Real open scoped BigOperators
theorem exercise_3_5 : Β¬ βˆƒ x y : β„€, 7*x^3 + 2 = y^3 :=
Ireland-Rosen|exercise_3_14
Let $p$ and $q$ be distinct odd primes such that $p-1$ divides $q-1$. If $(n, p q)=1$, show that $n^{q-1} \equiv 1(p q)$.
import Mathlib open Real open scoped BigOperators
theorem exercise_3_14 {p q n : β„•} (hp0 : p.Prime ∧ p > 2) (hq0 : q.Prime ∧ q > 2) (hpq0 : p β‰  q) (hpq1 : p - 1 ∣ q - 1) (hn : n.gcd (p*q) = 1) : n^(q-1) ≑ 1 [MOD p*q] :=
End of preview. Expand in Dataset Viewer.

ProofNet#

ProofNet# is a Lean 4 port of the ProofNet benchmark including fixes. This benchmark has been made compatible for all Lean versions between v4.7.0 and v4.16.0.

A comparison with some of the other existing ports can be found at: https://proofnet4-fix.streamlit.app/.

Original Dataset Summary

ProofNet is a benchmark for autoformalization and formal proving of undergraduate-level mathematics. The ProofNet benchmarks consists of 371 examples, each consisting of a formal theorem statement in Lean 3, a natural language theorem statement, and a natural language proof. The problems are primarily drawn from popular undergraduate pure mathematics textbooks and cover topics such as real and complex analysis, linear algebra, abstract algebra, and topology. We intend for ProofNet to be a challenging benchmark that will drive progress in autoformalization and automatic theorem proving.

Citation

ProofNet# is introduced in Improving Autoformalization using Type Checking.

@misc{poiroux2024improvingautoformalizationusingtype,
    title={Improving Autoformalization using Type Checking}, 
    author={Auguste Poiroux and Gail Weiss and Viktor Kunčak and Antoine Bosselut},
    year={2024},
    eprint={2406.07222},
    archivePrefix={arXiv},
    primaryClass={cs.CL},
    url={https://arxiv.org/abs/2406.07222}, 
}

Original work where ProofNet has been introduced:

@misc{azerbayev2023proofnet,
      title={ProofNet: Autoformalizing and Formally Proving Undergraduate-Level Mathematics}, 
      author={Zhangir Azerbayev and Bartosz Piotrowski and Hailey Schoelkopf and Edward W. Ayers and Dragomir Radev and Jeremy Avigad},
      year={2023},
      eprint={2302.12433},
      archivePrefix={arXiv},
      primaryClass={cs.CL}
}

Data Fields 

id: Unique string identifier for the problem.
nl_statement: Natural language theorem statement.
lean4_src_header: File header including imports, namespaces, and locales required for the formal statement.
lean4_formalization: Formal theorem statement in Lean 4.
Downloads last month
11
Size of downloaded dataset files:
68.8 kB
Size of the auto-converted Parquet files:
68.8 kB
Number of rows:
371