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{"name": "exercise_1_13a", "split": "valid", "informal_prefix": "/-- Suppose that $f$ is holomorphic in an open set $\\Omega$. Prove that if $\\text{Re}(f)$ is constant, then $f$ is constant.-/\n", "formal_statement": "theorem exercise_1_13a {f : ℂ → ℂ} (Ω : Set ℂ) (a b : Ω) (h : IsOpen Ω)\n (hf : DifferentiableOn ℂ f Ω) (hc : ∃ (c : ℝ), ∀ z ∈ Ω, (f z).re = c) :\n f a = f b :=", "goal": "f : ℂ → ℂ\nΩ : Set ℂ\na b : ↑Ω\nh : IsOpen Ω\nhf : DifferentiableOn ℂ f Ω\nhc : ∃ c, ∀ z ∈ Ω, (f z).re = c\n⊢ f ↑a = f ↑b", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_1_13b", "split": "test", "informal_prefix": "/-- Suppose that $f$ is holomorphic in an open set $\\Omega$. Prove that if $\\text{Im}(f)$ is constant, then $f$ is constant.-/\n", "formal_statement": "theorem exercise_1_13b {f : ℂ → ℂ} (Ω : Set ℂ) (a b : Ω) (h : IsOpen Ω)\n (hf : DifferentiableOn ℂ f Ω) (hc : ∃ (c : ℝ), ∀ z ∈ Ω, (f z).im = c) :\n f a = f b :=", "goal": "f : ℂ → ℂ\nΩ : Set ℂ\na b : ↑Ω\nh : IsOpen Ω\nhf : DifferentiableOn ℂ f Ω\nhc : ∃ c, ∀ z ∈ Ω, (f z).im = c\n⊢ f ↑a = f ↑b", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_1_13c", "split": "valid", "informal_prefix": "/-- Suppose that $f$ is holomorphic in an open set $\\Omega$. Prove that if $|f|$ is constant, then $f$ is constant.-/\n", "formal_statement": "theorem exercise_1_13c {f : ℂ → ℂ} (Ω : Set ℂ) (a b : Ω) (h : IsOpen Ω)\n (hf : DifferentiableOn ℂ f Ω) (hc : ∃ (c : ℝ), ∀ z ∈ Ω, abs (f z) = c) :\n f a = f b :=", "goal": "f : ℂ → ℂ\nΩ : Set ℂ\na b : ↑Ω\nh : IsOpen Ω\nhf : DifferentiableOn ℂ f Ω\nhc : ∃ c, ∀ z ∈ Ω, Complex.abs (f z) = c\n⊢ f ↑a = f ↑b", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_1_19a", "split": "test", "informal_prefix": "/-- Prove that the power series $\\sum nz^n$ does not converge on any point of the unit circle.-/\n", "formal_statement": "theorem exercise_1_19a (z : ℂ) (hz : abs z = 1) (s : ℕ → ℂ)\n (h : s = (λ n => ∑ i in (range n), i * z ^ i)) :\n ¬ ∃ y, Tendsto s atTop (𝓝 y) :=", "goal": "z : ℂ\nhz : Complex.abs z = 1\ns : ℕ → ℂ\nh : s = fun n => ∑ i ∈ range n, ↑i * z ^ i\n⊢ ¬∃ y, Tendsto s atTop (𝓝 y)", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_1_19b", "split": "valid", "informal_prefix": "/-- Prove that the power series $\\sum zn/n^2$ converges at every point of the unit circle.-/\n", "formal_statement": "theorem exercise_1_19b (z : ℂ) (hz : abs z = 1) (s : ℕ → ℂ)\n (h : s = (λ n => ∑ i in (range n), i * z / i ^ 2)) :\n ∃ y, Tendsto s atTop (𝓝 y) :=", "goal": "z : ℂ\nhz : Complex.abs z = 1\ns : ℕ → ℂ\nh : s = fun n => ∑ i ∈ range n, ↑i * z / ↑i ^ 2\n⊢ ∃ y, Tendsto s atTop (𝓝 y)", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_1_19c", "split": "test", "informal_prefix": "/-- Prove that the power series $\\sum zn/n$ converges at every point of the unit circle except $z = 1$.-/\n", "formal_statement": "theorem exercise_1_19c (z : ℂ) (hz : abs z = 1) (hz2 : z ≠ 1) (s : ℕ → ℂ)\n (h : s = (λ n => ∑ i in (range n), i * z / i)) :\n ∃ z, Tendsto s atTop (𝓝 z) :=", "goal": "z : ℂ\nhz : Complex.abs z = 1\nhz2 : z ≠ 1\ns : ℕ → ℂ\nh : s = fun n => ∑ i ∈ range n, ↑i * z / ↑i\n⊢ ∃ z, Tendsto s atTop (𝓝 z)", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_1_26", "split": "valid", "informal_prefix": "/-- Suppose $f$ is continuous in a region $\\Omega$. Prove that any two primitives of $f$ (if they exist) differ by a constant.-/\n", "formal_statement": "theorem exercise_1_26\n (f F₁ F₂ : ℂ → ℂ) (Ω : Set ℂ) (h1 : IsOpen Ω) (h2 : IsConnected Ω)\n (hF₁ : DifferentiableOn ℂ F₁ Ω) (hF₂ : DifferentiableOn ℂ F₂ Ω)\n (hdF₁ : ∀ x ∈ Ω, deriv F₁ x = f x) (hdF₂ : ∀ x ∈ Ω, deriv F₂ x = f x)\n : ∃ c : ℂ, ∀ x, F₁ x = F₂ x + c :=", "goal": "f F₁ F₂ : ℂ → ℂ\nΩ : Set ℂ\nh1 : IsOpen Ω\nh2 : IsConnected Ω\nhF₁ : DifferentiableOn ℂ F₁ Ω\nhF₂ : DifferentiableOn ℂ F₂ Ω\nhdF₁ : ∀ x ∈ Ω, deriv F₁ x = f x\nhdF₂ : ∀ x ∈ Ω, deriv F₂ x = f x\n⊢ ∃ c, ∀ (x : ℂ), F₁ x = F₂ x + c", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_2_2", "split": "test", "informal_prefix": "/-- Show that $\\int_{0}^{\\infty} \\frac{\\sin x}{x} d x=\\frac{\\pi}{2}$.-/\n", "formal_statement": "theorem exercise_2_2 :\n Tendsto (λ y => ∫ x in (0 : ℝ)..y, Real.sin x / x) atTop (𝓝 (Real.pi / 2)) :=", "goal": "⊢ Tendsto (fun y => ∫ (x : ℝ) in 0 ..y, x.sin / x) atTop (𝓝 (Real.pi / 2))", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_2_9", "split": "valid", "informal_prefix": "/-- Let $\\Omega$ be a bounded open subset of $\\mathbb{C}$, and $\\varphi: \\Omega \\rightarrow \\Omega$ a holomorphic function. Prove that if there exists a point $z_{0} \\in \\Omega$ such that $\\varphi\\left(z_{0}\\right)=z_{0} \\quad \\text { and } \\quad \\varphi^{\\prime}\\left(z_{0}\\right)=1$ then $\\varphi$ is linear.-/\n", "formal_statement": "theorem exercise_2_9\n {f : ℂ → ℂ} (Ω : Set ℂ) (b : Bornology.IsBounded Ω) (h : IsOpen Ω)\n (hf : DifferentiableOn ℂ f Ω) (z : Ω) (hz : f z = z) (h'z : deriv f z = 1) :\n ∃ (f_lin : ℂ →L[ℂ] ℂ), ∀ x ∈ Ω, f x = f_lin x :=", "goal": "f : ℂ → ℂ\nΩ : Set ℂ\nb : Bornology.IsBounded Ω\nh : IsOpen Ω\nhf : DifferentiableOn ℂ f Ω\nz : ↑Ω\nhz : f ↑z = ↑z\nh'z : deriv f ↑z = 1\n⊢ ∃ f_lin, ∀ x ∈ Ω, f x = f_lin x", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_2_13", "split": "test", "informal_prefix": "/-- Suppose $f$ is an analytic function defined everywhere in $\\mathbb{C}$ and such that for each $z_0 \\in \\mathbb{C}$ at least one coefficient in the expansion $f(z) = \\sum_{n=0}^\\infty c_n(z - z_0)^n$ is equal to 0. Prove that $f$ is a polynomial.-/\n", "formal_statement": "theorem exercise_2_13 {f : ℂ → ℂ}\n (hf : ∀ z₀ : ℂ, ∃ (s : Set ℂ) (c : ℕ → ℂ), IsOpen s ∧ z₀ ∈ s ∧\n ∀ z ∈ s, Tendsto (λ n => ∑ i in range n, (c i) * (z - z₀)^i) atTop (𝓝 (f z₀))\n ∧ ∃ i, c i = 0) :\n ∃ (c : ℕ → ℂ) (n : ℕ), f = λ z => ∑ i in range n, (c i) * z ^ n :=", "goal": "f : ℂ → ℂ\nhf :\n ∀ (z₀ : ℂ),\n ∃ s c,\n IsOpen s ∧ z₀ ∈ s ∧ ∀ z ∈ s, Tendsto (fun n => ∑ i ∈ range n, c i * (z - z₀) ^ i) atTop (𝓝 (f z₀)) ∧ ∃ i, c i = 0\n⊢ ∃ c n, f = fun z => ∑ i ∈ range n, c i * z ^ n", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_3_3", "split": "valid", "informal_prefix": "/-- Show that $ \\int_{-\\infty}^{\\infty} \\frac{\\cos x}{x^2 + a^2} dx = \\pi \\frac{e^{-a}}{a}$ for $a > 0$.-/\n", "formal_statement": "theorem exercise_3_3 (a : ℝ) (ha : 0 < a) :\n Tendsto (λ y => ∫ x in -y..y, Real.cos x / (x ^ 2 + a ^ 2))\n atTop (𝓝 (Real.pi * (Real.exp (-a) / a))) :=", "goal": "a : ℝ\nha : 0 < a\n⊢ Tendsto (fun y => ∫ (x : ℝ) in -y..y, x.cos / (x ^ 2 + a ^ 2)) atTop (𝓝 (Real.pi * ((-a).exp / a)))", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_3_4", "split": "test", "informal_prefix": "/-- Show that $ \\int_{-\\infty}^{\\infty} \\frac{x \\sin x}{x^2 + a^2} dx = \\pi e^{-a}$ for $a > 0$.-/\n", "formal_statement": "theorem exercise_3_4 (a : ℝ) (ha : 0 < a) :\n Tendsto (λ y => ∫ x in -y..y, x * Real.sin x / (x ^ 2 + a ^ 2))\n atTop (𝓝 (Real.pi * (Real.exp (-a)))) :=", "goal": "a : ℝ\nha : 0 < a\n⊢ Tendsto (fun y => ∫ (x : ℝ) in -y..y, x * x.sin / (x ^ 2 + a ^ 2)) atTop (𝓝 (Real.pi * (-a).exp))", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_3_9", "split": "valid", "informal_prefix": "/-- Show that $\\int_0^1 \\log(\\sin \\pi x) dx = - \\log 2$.-/\n", "formal_statement": "theorem exercise_3_9 : ∫ x in (0 : ℝ)..(1 : ℝ),\n Real.log (Real.sin (Real.pi * x)) = - Real.log 2 :=", "goal": "⊢ ∫ (x : ℝ) in 0 ..1, (Real.pi * x).sin.log = -Real.log 2", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_3_14", "split": "test", "informal_prefix": "/-- Prove that all entire functions that are also injective take the form $f(z) = az + b$, $a, b \\in \\mathbb{C}$ and $a \\neq 0$.-/\n", "formal_statement": "theorem exercise_3_14 {f : ℂ → ℂ} (hf : Differentiable ℂ f)\n (hf_inj : Function.Injective f) :\n ∃ (a b : ℂ), f = (λ z => a * z + b) ∧ a ≠ 0 :=", "goal": "f : ℂ → ℂ\nhf : Differentiable ℂ f\nhf_inj : Injective f\n⊢ ∃ a b, (f = fun z => a * z + b) ∧ a ≠ 0", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_3_22", "split": "valid", "informal_prefix": "/-- Show that there is no holomorphic function $f$ in the unit disc $D$ that extends continuously to $\\partial D$ such that $f(z) = 1/z$ for $z \\in \\partial D$.-/\n", "formal_statement": "theorem exercise_3_22 (D : Set ℂ) (hD : D = ball 0 1) (f : ℂ → ℂ)\n (hf : DifferentiableOn ℂ f D) (hfc : ContinuousOn f (closure D)) :\n ¬ ∀ z ∈ (sphere (0 : ℂ) 1), f z = 1 / z :=", "goal": "D : Set ℂ\nhD : D = ball 0 1\nf : ℂ → ℂ\nhf : DifferentiableOn ℂ f D\nhfc : ContinuousOn f (closure D)\n⊢ ¬∀ z ∈ sphere 0 1, f z = 1 / z", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_5_1", "split": "test", "informal_prefix": "/-- Prove that if $f$ is holomorphic in the unit disc, bounded and not identically zero, and $z_{1}, z_{2}, \\ldots, z_{n}, \\ldots$ are its zeros $\\left(\\left|z_{k}\\right|<1\\right)$, then $\\sum_{n}\\left(1-\\left|z_{n}\\right|\\right)<\\infty$.-/\n", "formal_statement": "theorem exercise_5_1 (f : ℂ → ℂ) (hf : DifferentiableOn ℂ f (ball 0 1))\n (hb : Bornology.IsBounded (Set.range f)) (h0 : f ≠ 0) (zeros : ℕ → ℂ) (hz : ∀ n, f (zeros n) = 0)\n (hzz : Set.range zeros = {z | f z = 0 ∧ z ∈ (ball (0 : ℂ) 1)}) :\n ∃ (z : ℂ), Tendsto (λ n => (∑ i in range n, (1 - zeros i))) atTop (𝓝 z) :=", "goal": "f : ℂ → ℂ\nhf : DifferentiableOn ℂ f (ball 0 1)\nhb : Bornology.IsBounded (Set.range f)\nh0 : f ≠ 0\nzeros : ℕ → ℂ\nhz : ∀ (n : ℕ), f (zeros n) = 0\nhzz : Set.range zeros = {z | f z = 0 ∧ z ∈ ball 0 1}\n⊢ ∃ z, Tendsto (fun n => ∑ i ∈ range n, (1 - zeros i)) atTop (𝓝 z)", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_1_1a", "split": "valid", "informal_prefix": "/-- If $r$ is rational $(r \\neq 0)$ and $x$ is irrational, prove that $r+x$ is irrational.-/\n", "formal_statement": "theorem exercise_1_1a\n (x : ℝ) (y : ℚ) :\n ( Irrational x ) -> Irrational ( x + y ) :=", "goal": "x : ℝ\ny : ℚ\n⊢ Irrational x → Irrational (x + ↑y)", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_1b", "split": "test", "informal_prefix": "/-- If $r$ is rational $(r \\neq 0)$ and $x$ is irrational, prove that $rx$ is irrational.-/\n", "formal_statement": "theorem exercise_1_1b\n(x : ℝ)\n(y : ℚ)\n(h : y ≠ 0)\n: ( Irrational x ) -> Irrational ( x * y ) :=", "goal": "x : ℝ\ny : ℚ\nh : y ≠ 0\n⊢ Irrational x → Irrational (x * ↑y)", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_2", "split": "valid", "informal_prefix": "/-- Prove that there is no rational number whose square is $12$.-/\n", "formal_statement": "theorem exercise_1_2 : ¬ ∃ (x : ℚ), ( x ^ 2 = 12 ) :=", "goal": "⊢ ¬∃ x, x ^ 2 = 12", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_4", "split": "test", "informal_prefix": "/-- Let $E$ be a nonempty subset of an ordered set; suppose $\\alpha$ is a lower bound of $E$ and $\\beta$ is an upper bound of $E$. Prove that $\\alpha \\leq \\beta$.-/\n", "formal_statement": "theorem exercise_1_4\n(α : Type*) [PartialOrder α]\n(s : Set α)\n(x y : α)\n(h₀ : Set.Nonempty s)\n(h₁ : x ∈ lowerBounds s)\n(h₂ : y ∈ upperBounds s)\n: x ≤ y :=", "goal": "α : Type u_1\ninst✝ : PartialOrder α\ns : Set α\nx y : α\nh₀ : s.Nonempty\nh₁ : x ∈ lowerBounds s\nh₂ : y ∈ upperBounds s\n⊢ x ≤ y", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_5", "split": "valid", "informal_prefix": "/-- Let $A$ be a nonempty set of real numbers which is bounded below. Let $-A$ be the set of all numbers $-x$, where $x \\in A$. Prove that $\\inf A=-\\sup (-A)$.-/\n", "formal_statement": "theorem exercise_1_5 (A minus_A : Set ℝ) (hA : A.Nonempty)\n (hA_bdd_below : BddBelow A) (hminus_A : minus_A = {x | -x ∈ A}) :\n Inf A = Sup minus_A :=", "goal": "A minus_A : Set ℝ\nhA : A.Nonempty\nhA_bdd_below : BddBelow A\nhminus_A : minus_A = {x | -x ∈ A}\n⊢ Inf ↑A = Sup ↑minus_A", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_8", "split": "test", "informal_prefix": "/-- Prove that no order can be defined in the complex field that turns it into an ordered field.-/\n", "formal_statement": "theorem exercise_1_8 : ¬ ∃ (r : ℂ → ℂ → Prop), IsLinearOrder ℂ r :=", "goal": "⊢ ¬∃ r, IsLinearOrder ℂ r", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_11a", "split": "valid", "informal_prefix": "/-- If $z$ is a complex number, prove that there exists an $r\\geq 0$ and a complex number $w$ with $| w | = 1$ such that $z = rw$.-/\n", "formal_statement": "theorem exercise_1_11a (z : ℂ) :\n ∃ (r : ℝ) (w : ℂ), abs w = 1 ∧ z = r * w :=", "goal": "z : ℂ\n⊢ ∃ r w, Complex.abs w = 1 ∧ z = ↑r * w", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_12", "split": "test", "informal_prefix": "/-- If $z_1, \\ldots, z_n$ are complex, prove that $|z_1 + z_2 + \\ldots + z_n| \\leq |z_1| + |z_2| + \\cdots + |z_n|$.-/\n", "formal_statement": "theorem exercise_1_12 (n : ℕ) (f : ℕ → ℂ) :\n abs (∑ i in range n, f i) ≤ ∑ i in range n, abs (f i) :=", "goal": "n : ℕ\nf : ℕ → ℂ\n⊢ Complex.abs (∑ i ∈ range n, f i) ≤ ∑ i ∈ range n, Complex.abs (f i)", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_13", "split": "valid", "informal_prefix": "/-- If $x, y$ are complex, prove that $||x|-|y|| \\leq |x-y|$.-/\n", "formal_statement": "theorem exercise_1_13 (x y : ℂ) :\n |(abs x) - (abs y)| ≤ abs (x - y) :=", "goal": "x y : ℂ\n⊢ |Complex.abs x - Complex.abs y| ≤ Complex.abs (x - y)", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_14", "split": "test", "informal_prefix": "/-- If $z$ is a complex number such that $|z|=1$, that is, such that $z \\bar{z}=1$, compute $|1+z|^{2}+|1-z|^{2}$.-/\n", "formal_statement": "theorem exercise_1_14\n (z : ℂ) (h : abs z = 1)\n : (abs (1 + z)) ^ 2 + (abs (1 - z)) ^ 2 = 4 :=", "goal": "z : ℂ\nh : Complex.abs z = 1\n⊢ Complex.abs (1 + z) ^ 2 + Complex.abs (1 - z) ^ 2 = 4", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_16a", "split": "valid", "informal_prefix": "/-- Suppose $k \\geq 3, x, y \\in \\mathbb{R}^k, |x - y| = d > 0$, and $r > 0$. Prove that if $2r > d$, there are infinitely many $z \\in \\mathbb{R}^k$ such that $|z-x|=|z-y|=r$.-/\n", "formal_statement": "theorem exercise_1_16a\n (n : ℕ)\n (d r : ℝ)\n (x y z : EuclideanSpace ℝ (Fin n)) -- R^n\n (h₁ : n ≥ 3)\n (h₂ : ‖x - y‖ = d)\n (h₃ : d > 0)\n (h₄ : r > 0)\n (h₅ : 2 * r > d)\n : Set.Infinite {z : EuclideanSpace ℝ (Fin n) | ‖z - x‖ = r ∧ ‖z - y‖ = r} :=", "goal": "n : ℕ\nd r : ℝ\nx y z : EuclideanSpace ℝ (Fin n)\nh₁ : n ≥ 3\nh₂ : ‖x - y‖ = d\nh₃ : d > 0\nh₄ : r > 0\nh₅ : 2 * r > d\n⊢ {z | ‖z - x‖ = r ∧ ‖z - y‖ = r}.Infinite", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_17", "split": "test", "informal_prefix": "/-- Prove that $|\\mathbf{x}+\\mathbf{y}|^{2}+|\\mathbf{x}-\\mathbf{y}|^{2}=2|\\mathbf{x}|^{2}+2|\\mathbf{y}|^{2}$ if $\\mathbf{x} \\in R^{k}$ and $\\mathbf{y} \\in R^{k}$.-/\n", "formal_statement": "theorem exercise_1_17\n (n : ℕ)\n (x y : EuclideanSpace ℝ (Fin n)) -- R^n\n : ‖x + y‖^2 + ‖x - y‖^2 = 2*‖x‖^2 + 2*‖y‖^2 :=", "goal": "n : ℕ\nx y : EuclideanSpace ℝ (Fin n)\n⊢ ‖x + y‖ ^ 2 + ‖x - y‖ ^ 2 = 2 * ‖x‖ ^ 2 + 2 * ‖y‖ ^ 2", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_18a", "split": "valid", "informal_prefix": "/-- If $k \\geq 2$ and $\\mathbf{x} \\in R^{k}$, prove that there exists $\\mathbf{y} \\in R^{k}$ such that $\\mathbf{y} \\neq 0$ but $\\mathbf{x} \\cdot \\mathbf{y}=0$-/\n", "formal_statement": "theorem exercise_1_18a\n (n : ℕ)\n (h : n > 1)\n (x : EuclideanSpace ℝ (Fin n)) -- R^n\n : ∃ (y : EuclideanSpace ℝ (Fin n)), y ≠ 0 ∧ (inner x y) = (0 : ℝ) :=", "goal": "n : ℕ\nh : n > 1\nx : EuclideanSpace ℝ (Fin n)\n⊢ ∃ y, y ≠ 0 ∧ ⟪x, y⟫_ℝ = 0", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_18b", "split": "test", "informal_prefix": "/-- If $k = 1$ and $\\mathbf{x} \\in R^{k}$, prove that there does not exist $\\mathbf{y} \\in R^{k}$ such that $\\mathbf{y} \\neq 0$ but $\\mathbf{x} \\cdot \\mathbf{y}=0$-/\n", "formal_statement": "theorem exercise_1_18b\n : ¬ ∀ (x : ℝ), ∃ (y : ℝ), y ≠ 0 ∧ x * y = 0 :=", "goal": "⊢ ¬∀ (x : ℝ), ∃ y, y ≠ 0 ∧ x * y = 0", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_19", "split": "valid", "informal_prefix": "/-- Suppose $a, b \\in R^k$. Find $c \\in R^k$ and $r > 0$ such that $|x-a|=2|x-b|$ if and only if $| x - c | = r$. Prove that $3c = 4b - a$ and $3r = 2 |b - a|$.-/\n", "formal_statement": "theorem exercise_1_19\n (n : ℕ)\n (a b c x : EuclideanSpace ℝ (Fin n))\n (r : ℝ)\n (h₁ : r > 0)\n (h₂ : 3 • c = 4 • b - a)\n (h₃ : 3 * r = 2 * ‖x - b‖)\n : ‖x - a‖ = 2 * ‖x - b‖ ↔ ‖x - c‖ = r :=", "goal": "n : ℕ\na b c x : EuclideanSpace ℝ (Fin n)\nr : ℝ\nh₁ : r > 0\nh₂ : 3 • c = 4 • b - a\nh₃ : 3 * r = 2 * ‖x - b‖\n⊢ ‖x - a‖ = 2 * ‖x - b‖ ↔ ‖x - c‖ = r", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_19a", "split": "test", "informal_prefix": "/-- If $A$ and $B$ are disjoint closed sets in some metric space $X$, prove that they are separated.-/\n", "formal_statement": "theorem exercise_2_19a {X : Type*} [MetricSpace X]\n (A B : Set X) (hA : IsClosed A) (hB : IsClosed B) (hAB : Disjoint A B) :\n SeparatedNhds A B :=", "goal": "X : Type u_1\ninst✝ : MetricSpace X\nA B : Set X\nhA : IsClosed A\nhB : IsClosed B\nhAB : Disjoint A B\n⊢ SeparatedNhds A B", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_24", "split": "valid", "informal_prefix": "/-- Let $X$ be a metric space in which every infinite subset has a limit point. Prove that $X$ is separable.-/\n", "formal_statement": "theorem exercise_2_24 {X : Type*} [MetricSpace X]\n (hX : ∀ (A : Set X), Infinite A → ∃ (x : X), x ∈ closure A) :\n SeparableSpace X :=", "goal": "X : Type u_1\ninst✝ : MetricSpace X\nhX : ∀ (A : Set X), Infinite ↑A → ∃ x, x ∈ closure A\n⊢ SeparableSpace X", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_25", "split": "test", "informal_prefix": "/-- Prove that every compact metric space $K$ has a countable base.-/\n", "formal_statement": "theorem exercise_2_25 {K : Type*} [MetricSpace K] [CompactSpace K] :\n ∃ (B : Set (Set K)), Set.Countable B ∧ IsTopologicalBasis B :=", "goal": "K : Type u_1\ninst✝¹ : MetricSpace K\ninst✝ : CompactSpace K\n⊢ ∃ B, B.Countable ∧ IsTopologicalBasis B", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_27a", "split": "valid", "informal_prefix": "/-- Suppose $E\\subset\\mathbb{R}^k$ is uncountable, and let $P$ be the set of condensation points of $E$. Prove that $P$ is perfect.-/\n", "formal_statement": "theorem exercise_2_27a (k : ℕ) (E P : Set (EuclideanSpace ℝ (Fin k)))\n (hE : E.Nonempty ∧ ¬ Set.Countable E)\n (hP : P = {x | ∀ U ∈ 𝓝 x, ¬ Set.Countable (P ∩ E)}) :\n IsClosed P ∧ P = {x | ClusterPt x (𝓟 P)} :=", "goal": "k : ℕ\nE P : Set (EuclideanSpace ℝ (Fin k))\nhE : E.Nonempty ∧ ¬E.Countable\nhP : P = {x | ∀ U ∈ 𝓝 x, ¬(P ∩ E).Countable}\n⊢ IsClosed P ∧ P = {x | ClusterPt x (𝓟 P)}", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_27b", "split": "test", "informal_prefix": "/-- Suppose $E\\subset\\mathbb{R}^k$ is uncountable, and let $P$ be the set of condensation points of $E$. Prove that at most countably many points of $E$ are not in $P$.-/\n", "formal_statement": "theorem exercise_2_27b (k : ℕ) (E P : Set (EuclideanSpace ℝ (Fin k)))\n (hE : E.Nonempty ∧ ¬ Set.Countable E)\n (hP : P = {x | ∀ U ∈ 𝓝 x, (P ∩ E).Nonempty ∧ ¬ Set.Countable (P ∩ E)}) :\n Set.Countable (E \\ P) :=", "goal": "k : ℕ\nE P : Set (EuclideanSpace ℝ (Fin k))\nhE : E.Nonempty ∧ ¬E.Countable\nhP : P = {x | ∀ U ∈ 𝓝 x, (P ∩ E).Nonempty ∧ ¬(P ∩ E).Countable}\n⊢ (E \\ P).Countable", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_28", "split": "valid", "informal_prefix": "/-- Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable.-/\n", "formal_statement": "theorem exercise_2_28 (X : Type*) [MetricSpace X] [SeparableSpace X]\n (A : Set X) (hA : IsClosed A) :\n ∃ P₁ P₂ : Set X, A = P₁ ∪ P₂ ∧\n IsClosed P₁ ∧ P₁ = {x | ClusterPt x (𝓟 P₁)} ∧\n Set.Countable P₂ :=", "goal": "X : Type u_1\ninst✝¹ : MetricSpace X\ninst✝ : SeparableSpace X\nA : Set X\nhA : IsClosed A\n⊢ ∃ P₁ P₂, A = P₁ ∪ P₂ ∧ IsClosed P₁ ∧ P₁ = {x | ClusterPt x (𝓟 P₁)} ∧ P₂.Countable", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_29", "split": "test", "informal_prefix": "/-- Prove that every open set in $\\mathbb{R}$ is the union of an at most countable collection of disjoint segments.-/\n", "formal_statement": "theorem exercise_2_29 (U : Set ℝ) (hU : IsOpen U) :\n ∃ (f : ℕ → Set ℝ), (∀ n, ∃ a b : ℝ, f n = {x | a < x ∧ x < b}) ∧ (∀ n, f n ⊆ U) ∧\n (∀ n m, n ≠ m → f n ∩ f m = ∅) ∧\n U = ⋃ n, f n :=", "goal": "U : Set ℝ\nhU : IsOpen U\n⊢ ∃ f,\n (∀ (n : ℕ), ∃ a b, f n = {x | a < x ∧ x < b}) ∧\n (∀ (n : ℕ), f n ⊆ U) ∧ (∀ (n m : ℕ), n ≠ m → f n ∩ f m = ∅) ∧ U = ⋃ n, f n", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_1a", "split": "valid", "informal_prefix": "/-- Prove that convergence of $\\left\\{s_{n}\\right\\}$ implies convergence of $\\left\\{\\left|s_{n}\\right|\\right\\}$.-/\n", "formal_statement": "theorem exercise_3_1a\n (f : ℕ → ℝ)\n (h : ∃ (a : ℝ), Tendsto (λ (n : ℕ) => f n) atTop (𝓝 a))\n : ∃ (a : ℝ), Tendsto (λ (n : ℕ) => |f n|) atTop (𝓝 a) :=", "goal": "f : ℕ → ℝ\nh : ∃ a, Tendsto (fun n => f n) atTop (𝓝 a)\n⊢ ∃ a, Tendsto (fun n => |f n|) atTop (𝓝 a)", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_2a", "split": "test", "informal_prefix": "/-- Prove that $\\lim_{n \\rightarrow \\infty}\\sqrt{n^2 + n} -n = 1/2$.-/\n", "formal_statement": "theorem exercise_3_2a\n : Tendsto (λ (n : ℝ) => (sqrt (n^2 + n) - n)) atTop (𝓝 (1/2)) :=", "goal": "⊢ Tendsto (fun n => √(n ^ 2 + n) - n) atTop (𝓝 (1 / 2))", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_3", "split": "valid", "informal_prefix": "/-- If $s_{1}=\\sqrt{2}$, and $s_{n+1}=\\sqrt{2+\\sqrt{s_{n}}} \\quad(n=1,2,3, \\ldots),$ prove that $\\left\\{s_{n}\\right\\}$ converges, and that $s_{n}<2$ for $n=1,2,3, \\ldots$.-/\n", "formal_statement": "theorem exercise_3_3\n : ∃ (x : ℝ), Tendsto f atTop (𝓝 x) ∧ ∀ n, f n < 2 :=", "goal": "⊢ ∃ x, Tendsto f atTop (𝓝 x) ∧ ∀ (n : ℕ), f n < 2", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\nnoncomputable def f : ℕ → ℝ\n| 0 => sqrt 2\n| (n + 1) => sqrt (2 + sqrt (f n))\n\n"}
{"name": "exercise_3_5", "split": "test", "informal_prefix": "/-- For any two real sequences $\\left\\{a_{n}\\right\\},\\left\\{b_{n}\\right\\}$, prove that $\\limsup _{n \\rightarrow \\infty}\\left(a_{n}+b_{n}\\right) \\leq \\limsup _{n \\rightarrow \\infty} a_{n}+\\limsup _{n \\rightarrow \\infty} b_{n},$ provided the sum on the right is not of the form $\\infty-\\infty$.-/\n", "formal_statement": "theorem exercise_3_5\n (a b : ℕ → ℝ)\n (h : limsup a + limsup b ≠ 0) :\n limsup (λ n => a n + b n) ≤ limsup a + limsup b :=", "goal": "a b : ℕ → ℝ\nh : limsup a + limsup b ≠ 0\n⊢ (limsup fun n => a n + b n) ≤ limsup a + limsup b", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_6a", "split": "valid", "informal_prefix": "/-- Prove that $\\lim_{n \\rightarrow \\infty} \\sum_{i<n} a_i = \\infty$, where $a_i = \\sqrt{i + 1} -\\sqrt{i}$.-/\n", "formal_statement": "theorem exercise_3_6a\n: Tendsto (λ (n : ℕ) => (∑ i in range n, g i)) atTop atTop :=", "goal": "⊢ Tendsto (fun n => ∑ i ∈ range n, g i) atTop atTop", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\nnoncomputable section\n\ndef g (n : ℕ) : ℝ := sqrt (n + 1) - sqrt n\n\n"}
{"name": "exercise_3_7", "split": "test", "informal_prefix": "/-- Prove that the convergence of $\\Sigma a_{n}$ implies the convergence of $\\sum \\frac{\\sqrt{a_{n}}}{n}$ if $a_n\\geq 0$.-/\n", "formal_statement": "theorem exercise_3_7\n (a : ℕ → ℝ)\n (h : ∃ y, (Tendsto (λ n => (∑ i in (range n), a i)) atTop (𝓝 y))) :\n ∃ y, Tendsto (λ n => (∑ i in (range n), sqrt (a i) / n)) atTop (𝓝 y) :=", "goal": "a : ℕ → ℝ\nh : ∃ y, Tendsto (fun n => ∑ i ∈ range n, a i) atTop (𝓝 y)\n⊢ ∃ y, Tendsto (fun n => ∑ i ∈ range n, √(a i) / ↑n) atTop (𝓝 y)", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_8", "split": "valid", "informal_prefix": "/-- If $\\Sigma a_{n}$ converges, and if $\\left\\{b_{n}\\right\\}$ is monotonic and bounded, prove that $\\Sigma a_{n} b_{n}$ converges.-/\n", "formal_statement": "theorem exercise_3_8\n (a b : ℕ → ℝ)\n (h1 : ∃ y, (Tendsto (λ n => (∑ i in (range n), a i)) atTop (𝓝 y)))\n (h2 : Monotone b)\n (h3 : Bornology.IsBounded (Set.range b)) :\n ∃ y, Tendsto (λ n => (∑ i in (range n), (a i) * (b i))) atTop (𝓝 y) :=", "goal": "a b : ℕ → ℝ\nh1 : ∃ y, Tendsto (fun n => ∑ i ∈ range n, a i) atTop (𝓝 y)\nh2 : Monotone b\nh3 : Bornology.IsBounded (Set.range b)\n⊢ ∃ y, Tendsto (fun n => ∑ i ∈ range n, a i * b i) atTop (𝓝 y)", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_13", "split": "test", "informal_prefix": "/-- Prove that the Cauchy product of two absolutely convergent series converges absolutely.-/\n", "formal_statement": "theorem exercise_3_13\n (a b : ℕ → ℝ)\n (ha : ∃ y, (Tendsto (λ n => (∑ i in (range n), |a i|)) atTop (𝓝 y)))\n (hb : ∃ y, (Tendsto (λ n => (∑ i in (range n), |b i|)) atTop (𝓝 y))) :\n ∃ y, (Tendsto (λ n => (∑ i in (range n),\n λ i => (∑ j in range (i + 1), a j * b (i - j)))) atTop (𝓝 y)) :=", "goal": "a b : ℕ → ℝ\nha : ∃ y, Tendsto (fun n => ∑ i ∈ range n, |a i|) atTop (𝓝 y)\nhb : ∃ y, Tendsto (fun n => ∑ i ∈ range n, |b i|) atTop (𝓝 y)\n⊢ ∃ y, Tendsto (fun n => ∑ i ∈ range n, fun i => ∑ j ∈ range (i + 1), a j * b (i - j)) atTop (𝓝 y)", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_20", "split": "valid", "informal_prefix": "/-- Suppose $\\left\\{p_{n}\\right\\}$ is a Cauchy sequence in a metric space $X$, and some sequence $\\left\\{p_{n l}\\right\\}$ converges to a point $p \\in X$. Prove that the full sequence $\\left\\{p_{n}\\right\\}$ converges to $p$.-/\n", "formal_statement": "theorem exercise_3_20 {X : Type*} [MetricSpace X]\n (p : ℕ → X) (l : ℕ) (r : X)\n (hp : CauchySeq p)\n (hpl : Tendsto (λ n => p (l * n)) atTop (𝓝 r)) :\n Tendsto p atTop (𝓝 r) :=", "goal": "X : Type u_1\ninst✝ : MetricSpace X\np : ℕ → X\nl : ℕ\nr : X\nhp : CauchySeq p\nhpl : Tendsto (fun n => p (l * n)) atTop (𝓝 r)\n⊢ Tendsto p atTop (𝓝 r)", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_21", "split": "test", "informal_prefix": "/-- If $\\left\\{E_{n}\\right\\}$ is a sequence of closed nonempty and bounded sets in a complete metric space $X$, if $E_{n} \\supset E_{n+1}$, and if $\\lim _{n \\rightarrow \\infty} \\operatorname{diam} E_{n}=0,$ then $\\bigcap_{1}^{\\infty} E_{n}$ consists of exactly one point.-/\n", "formal_statement": "theorem exercise_3_21\n {X : Type*} [MetricSpace X] [CompleteSpace X]\n (E : ℕ → Set X)\n (hE : ∀ n, E n ⊃ E (n + 1))\n (hE' : Tendsto (λ n => Metric.diam (E n)) atTop (𝓝 0)) :\n ∃ a, Set.iInter E = {a} :=", "goal": "X : Type u_1\ninst✝¹ : MetricSpace X\ninst✝ : CompleteSpace X\nE : ℕ → Set X\nhE : ∀ (n : ℕ), E n ⊃ E (n + 1)\nhE' : Tendsto (fun n => Metric.diam (E n)) atTop (𝓝 0)\n⊢ ∃ a, Set.iInter E = {a}", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_22", "split": "valid", "informal_prefix": "/-- Suppose $X$ is a nonempty complete metric space, and $\\left\\{G_{n}\\right\\}$ is a sequence of dense open sets of $X$. Prove Baire's theorem, namely, that $\\bigcap_{1}^{\\infty} G_{n}$ is not empty.-/\n", "formal_statement": "theorem exercise_3_22 (X : Type*) [MetricSpace X] [CompleteSpace X]\n (G : ℕ → Set X) (hG : ∀ n, IsOpen (G n) ∧ Dense (G n)) :\n ∃ x, ∀ n, x ∈ G n :=", "goal": "X : Type u_1\ninst✝¹ : MetricSpace X\ninst✝ : CompleteSpace X\nG : ℕ → Set X\nhG : ∀ (n : ℕ), IsOpen (G n) ∧ Dense (G n)\n⊢ ∃ x, ∀ (n : ℕ), x ∈ G n", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_1a", "split": "test", "informal_prefix": "/-- Suppose $f$ is a real function defined on $\\mathbb{R}$ which satisfies $\\lim_{h \\rightarrow 0} f(x + h) - f(x - h) = 0$ for every $x \\in \\mathbb{R}$. Show that $f$ does not need to be continuous.-/\n", "formal_statement": "theorem exercise_4_1a\n : ∃ (f : ℝ → ℝ), (∀ (x : ℝ), Tendsto (λ y => f (x + y) - f (x - y)) (𝓝 0) (𝓝 0)) ∧ ¬ Continuous f :=", "goal": "⊢ ∃ f, (∀ (x : ℝ), Tendsto (fun y => f (x + y) - f (x - y)) (𝓝 0) (𝓝 0)) ∧ ¬Continuous f", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_2a", "split": "valid", "informal_prefix": "/-- If $f$ is a continuous mapping of a metric space $X$ into a metric space $Y$, prove that $f(\\overline{E}) \\subset \\overline{f(E)}$ for every set $E \\subset X$. ($\\overline{E}$ denotes the closure of $E$).-/\n", "formal_statement": "theorem exercise_4_2a\n {α : Type} [MetricSpace α]\n {β : Type} [MetricSpace β]\n (f : α → β)\n (h₁ : Continuous f)\n : ∀ (x : Set α), f '' (closure x) ⊆ closure (f '' x) :=", "goal": "α : Type\ninst✝¹ : MetricSpace α\nβ : Type\ninst✝ : MetricSpace β\nf : α → β\nh₁ : Continuous f\n⊢ ∀ (x : Set α), f '' closure x ⊆ closure (f '' x)", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_3", "split": "test", "informal_prefix": "/-- Let $f$ be a continuous real function on a metric space $X$. Let $Z(f)$ (the zero set of $f$ ) be the set of all $p \\in X$ at which $f(p)=0$. Prove that $Z(f)$ is closed.-/\n", "formal_statement": "theorem exercise_4_3\n {α : Type} [MetricSpace α]\n (f : α → ℝ) (h : Continuous f) (z : Set α) (g : z = f⁻¹' {0})\n : IsClosed z :=", "goal": "α : Type\ninst✝ : MetricSpace α\nf : α → ℝ\nh : Continuous f\nz : Set α\ng : z = f ⁻¹' {0}\n⊢ IsClosed z", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_4a", "split": "valid", "informal_prefix": "/-- Let $f$ and $g$ be continuous mappings of a metric space $X$ into a metric space $Y$, and let $E$ be a dense subset of $X$. Prove that $f(E)$ is dense in $f(X)$.-/\n", "formal_statement": "theorem exercise_4_4a\n {α : Type} [MetricSpace α]\n {β : Type} [MetricSpace β]\n (f : α → β)\n (s : Set α)\n (h₁ : Continuous f)\n (h₂ : Dense s)\n : f '' Set.univ ⊆ closure (f '' s) :=", "goal": "α : Type\ninst✝¹ : MetricSpace α\nβ : Type\ninst✝ : MetricSpace β\nf : α → β\ns : Set α\nh₁ : Continuous f\nh₂ : Dense s\n⊢ f '' Set.univ ⊆ closure (f '' s)", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_4b", "split": "test", "informal_prefix": "/-- Let $f$ and $g$ be continuous mappings of a metric space $X$ into a metric space $Y$, and let $E$ be a dense subset of $X$. Prove that if $g(p) = f(p)$ for all $p \\in P$ then $g(p) = f(p)$ for all $p \\in X$.-/\n", "formal_statement": "theorem exercise_4_4b\n {α : Type} [MetricSpace α]\n {β : Type} [MetricSpace β]\n (f g : α → β)\n (s : Set α)\n (h₁ : Continuous f)\n (h₂ : Continuous g)\n (h₃ : Dense s)\n (h₄ : ∀ x ∈ s, f x = g x)\n : f = g :=", "goal": "α : Type\ninst✝¹ : MetricSpace α\nβ : Type\ninst✝ : MetricSpace β\nf g : α → β\ns : Set α\nh₁ : Continuous f\nh₂ : Continuous g\nh₃ : Dense s\nh₄ : ∀ x ∈ s, f x = g x\n⊢ f = g", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5a", "split": "valid", "informal_prefix": "/-- If $f$ is a real continuous function defined on a closed set $E \\subset \\mathbb{R}$, prove that there exist continuous real functions $g$ on $\\mathbb{R}$ such that $g(x)=f(x)$ for all $x \\in E$.-/\n", "formal_statement": "theorem exercise_4_5a\n (f : ℝ → ℝ)\n (E : Set ℝ)\n (h₁ : IsClosed E)\n (h₂ : ContinuousOn f E)\n : ∃ (g : ℝ → ℝ), Continuous g ∧ ∀ x ∈ E, f x = g x :=", "goal": "f : ℝ → ℝ\nE : Set ℝ\nh₁ : IsClosed E\nh₂ : ContinuousOn f E\n⊢ ∃ g, Continuous g ∧ ∀ x ∈ E, f x = g x", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5b", "split": "test", "informal_prefix": "/-- Show that there exist a set $E \\subset \\mathbb{R}$ and a real continuous function $f$ defined on $E$, such that there does not exist a continuous real function $g$ on $\\mathbb{R}$ such that $g(x)=f(x)$ for all $x \\in E$.-/\n", "formal_statement": "theorem exercise_4_5b\n : ∃ (E : Set ℝ) (f : ℝ → ℝ), (ContinuousOn f E) ∧\n (¬ ∃ (g : ℝ → ℝ), Continuous g ∧ ∀ x ∈ E, f x = g x) :=", "goal": "⊢ ∃ E f, ContinuousOn f E ∧ ¬∃ g, Continuous g ∧ ∀ x ∈ E, f x = g x", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_6", "split": "valid", "informal_prefix": "/-- If $f$ is defined on $E$, the graph of $f$ is the set of points $(x, f(x))$, for $x \\in E$. In particular, if $E$ is a set of real numbers, and $f$ is real-valued, the graph of $f$ is a subset of the plane. Suppose $E$ is compact, and prove that $f$ is continuous on $E$ if and only if its graph is compact.-/\n", "formal_statement": "theorem exercise_4_6\n (f : ℝ → ℝ)\n (E : Set ℝ)\n (G : Set (ℝ × ℝ))\n (h₁ : IsCompact E)\n (h₂ : G = {(x, f x) | x ∈ E})\n : ContinuousOn f E ↔ IsCompact G :=", "goal": "f : ℝ → ℝ\nE : Set ℝ\nG : Set (ℝ × ℝ)\nh₁ : IsCompact E\nh₂ : G = {x | ∃ x_1 ∈ E, (x_1, f x_1) = x}\n⊢ ContinuousOn f E ↔ IsCompact G", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_8a", "split": "test", "informal_prefix": "/-- Let $f$ be a real uniformly continuous function on the bounded set $E$ in $R^{1}$. Prove that $f$ is bounded on $E$.-/\n", "formal_statement": "theorem exercise_4_8a\n (E : Set ℝ) (f : ℝ → ℝ) (hf : UniformContinuousOn f E)\n (hE : Bornology.IsBounded E) : Bornology.IsBounded (Set.image f E) :=", "goal": "E : Set ℝ\nf : ℝ → ℝ\nhf : UniformContinuousOn f E\nhE : Bornology.IsBounded E\n⊢ Bornology.IsBounded (f '' E)", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_8b", "split": "valid", "informal_prefix": "/-- Let $E$ be a bounded set in $R^{1}$. Prove that there exists a real function $f$ such that $f$ is uniformly continuous and is not bounded on $E$.-/\n", "formal_statement": "theorem exercise_4_8b\n (E : Set ℝ) :\n ∃ f : ℝ → ℝ, UniformContinuousOn f E ∧ ¬ Bornology.IsBounded (Set.image f E) :=", "goal": "E : Set ℝ\n⊢ ∃ f, UniformContinuousOn f E ∧ ¬Bornology.IsBounded (f '' E)", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_11a", "split": "test", "informal_prefix": "/-- Suppose $f$ is a uniformly continuous mapping of a metric space $X$ into a metric space $Y$ and prove that $\\left\\{f\\left(x_{n}\\right)\\right\\}$ is a Cauchy sequence in $Y$ for every Cauchy sequence $\\{x_n\\}$ in $X$.-/\n", "formal_statement": "theorem exercise_4_11a\n {X : Type*} [MetricSpace X]\n {Y : Type*} [MetricSpace Y]\n (f : X → Y) (hf : UniformContinuous f)\n (x : ℕ → X) (hx : CauchySeq x) :\n CauchySeq (λ n => f (x n)) :=", "goal": "X : Type u_1\ninst✝¹ : MetricSpace X\nY : Type u_2\ninst✝ : MetricSpace Y\nf : X → Y\nhf : UniformContinuous f\nx : ℕ → X\nhx : CauchySeq x\n⊢ CauchySeq fun n => f (x n)", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_12", "split": "valid", "informal_prefix": "/-- A uniformly continuous function of a uniformly continuous function is uniformly continuous.-/\n", "formal_statement": "theorem exercise_4_12\n {α β γ : Type*} [UniformSpace α] [UniformSpace β] [UniformSpace γ]\n {f : α → β} {g : β → γ}\n (hf : UniformContinuous f) (hg : UniformContinuous g) :\n UniformContinuous (g ∘ f) :=", "goal": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\ng : β → γ\nhf : UniformContinuous f\nhg : UniformContinuous g\n⊢ UniformContinuous (g ∘ f)", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_15", "split": "test", "informal_prefix": "/-- Prove that every continuous open mapping of $R^{1}$ into $R^{1}$ is monotonic.-/\n", "formal_statement": "theorem exercise_4_15 {f : ℝ → ℝ}\n (hf : Continuous f) (hof : IsOpenMap f) :\n Monotone f :=", "goal": "f : ℝ → ℝ\nhf : Continuous f\nhof : IsOpenMap f\n⊢ Monotone f", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_19", "split": "valid", "informal_prefix": "/-- Suppose $f$ is a real function with domain $R^{1}$ which has the intermediate value property: if $f(a)<c<f(b)$, then $f(x)=c$ for some $x$ between $a$ and $b$. Suppose also, for every rational $r$, that the set of all $x$ with $f(x)=r$ is closed. Prove that $f$ is continuous.-/\n", "formal_statement": "theorem exercise_4_19\n {f : ℝ → ℝ} (hf : ∀ a b c, a < b → f a < c → c < f b → ∃ x, a < x ∧ x < b ∧ f x = c)\n (hg : ∀ r : ℚ, IsClosed {x | f x = r}) : Continuous f :=", "goal": "f : ℝ → ℝ\nhf : ∀ (a b c : ℝ), a < b → f a < c → c < f b → ∃ x, a < x ∧ x < b ∧ f x = c\nhg : ∀ (r : ℚ), IsClosed {x | f x = ↑r}\n⊢ Continuous f", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_21a", "split": "test", "informal_prefix": "/-- Suppose $K$ and $F$ are disjoint sets in a metric space $X, K$ is compact, $F$ is closed. Prove that there exists $\\delta>0$ such that $d(p, q)>\\delta$ if $p \\in K, q \\in F$.-/\n", "formal_statement": "theorem exercise_4_21a {X : Type*} [MetricSpace X]\n (K F : Set X) (hK : IsCompact K) (hF : IsClosed F) (hKF : Disjoint K F) :\n ∃ (δ : ℝ), δ > 0 ∧ ∀ (p q : X), p ∈ K → q ∈ F → dist p q ≥ δ :=", "goal": "X : Type u_1\ninst✝ : MetricSpace X\nK F : Set X\nhK : IsCompact K\nhF : IsClosed F\nhKF : Disjoint K F\n⊢ ∃ δ > 0, ∀ (p q : X), p ∈ K → q ∈ F → dist p q ≥ δ", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_24", "split": "valid", "informal_prefix": "/-- Assume that $f$ is a continuous real function defined in $(a, b)$ such that $f\\left(\\frac{x+y}{2}\\right) \\leq \\frac{f(x)+f(y)}{2}$ for all $x, y \\in(a, b)$. Prove that $f$ is convex.-/\n", "formal_statement": "theorem exercise_4_24 {f : ℝ → ℝ}\n (hf : Continuous f) (a b : ℝ) (hab : a < b)\n (h : ∀ x y : ℝ, a < x → x < b → a < y → y < b → f ((x + y) / 2) ≤ (f x + f y) / 2) :\n ConvexOn ℝ (Set.Ioo a b) f :=", "goal": "f : ℝ → ℝ\nhf : Continuous f\na b : ℝ\nhab : a < b\nh : ∀ (x y : ℝ), a < x → x < b → a < y → y < b → f ((x + y) / 2) ≤ (f x + f y) / 2\n⊢ ConvexOn ℝ (Set.Ioo a b) f", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_1", "split": "test", "informal_prefix": "/-- Let $f$ be defined for all real $x$, and suppose that $|f(x)-f(y)| \\leq (x-y)^{2}$ for all real $x$ and $y$. Prove that $f$ is constant.-/\n", "formal_statement": "theorem exercise_5_1\n {f : ℝ → ℝ} (hf : ∀ x y : ℝ, |(f x - f y)| ≤ (x - y) ^ 2) :\n ∃ c, f = λ x => c :=", "goal": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), |f x - f y| ≤ (x - y) ^ 2\n⊢ ∃ c, f = fun x => c", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_2", "split": "valid", "informal_prefix": "/-- Suppose $f^{\\prime}(x)>0$ in $(a, b)$. Prove that $f$ is strictly increasing in $(a, b)$, and let $g$ be its inverse function. Prove that $g$ is differentiable, and that $g^{\\prime}(f(x))=\\frac{1}{f^{\\prime}(x)} \\quad(a<x<b)$.-/\n", "formal_statement": "theorem exercise_5_2 {a b : ℝ}\n {f g : ℝ → ℝ} (hf : ∀ x ∈ Set.Ioo a b, deriv f x > 0)\n (hg : g = f⁻¹)\n (hg_diff : DifferentiableOn ℝ g (Set.Ioo a b)) :\n DifferentiableOn ℝ g (Set.Ioo a b) ∧\n ∀ x ∈ Set.Ioo a b, deriv g x = 1 / deriv f x :=", "goal": "a b : ℝ\nf g : ℝ → ℝ\nhf : ∀ x ∈ Set.Ioo a b, deriv f x > 0\nhg : g = f⁻¹\nhg_diff : DifferentiableOn ℝ g (Set.Ioo a b)\n⊢ DifferentiableOn ℝ g (Set.Ioo a b) ∧ ∀ x ∈ Set.Ioo a b, deriv g x = 1 / deriv f x", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_3", "split": "test", "informal_prefix": "/-- Suppose $g$ is a real function on $R^{1}$, with bounded derivative (say $\\left|g^{\\prime}\\right| \\leq M$ ). Fix $\\varepsilon>0$, and define $f(x)=x+\\varepsilon g(x)$. Prove that $f$ is one-to-one if $\\varepsilon$ is small enough.-/\n", "formal_statement": "theorem exercise_5_3 {g : ℝ → ℝ} (hg : Continuous g)\n (hg' : ∃ M : ℝ, ∀ x : ℝ, |deriv g x| ≤ M) :\n ∃ N, ∀ ε > 0, ε < N → Function.Injective (λ x : ℝ => x + ε * g x) :=", "goal": "g : ℝ → ℝ\nhg : Continuous g\nhg' : ∃ M, ∀ (x : ℝ), |deriv g x| ≤ M\n⊢ ∃ N, ∀ ε > 0, ε < N → Function.Injective fun x => x + ε * g x", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_4", "split": "valid", "informal_prefix": "/-- If $C_{0}+\\frac{C_{1}}{2}+\\cdots+\\frac{C_{n-1}}{n}+\\frac{C_{n}}{n+1}=0,$ where $C_{0}, \\ldots, C_{n}$ are real constants, prove that the equation $C_{0}+C_{1} x+\\cdots+C_{n-1} x^{n-1}+C_{n} x^{n}=0$ has at least one real root between 0 and 1.-/\n", "formal_statement": "theorem exercise_5_4 {n : ℕ}\n (C : ℕ → ℝ)\n (hC : ∑ i in (range (n + 1)), (C i) / (i + 1) = 0) :\n ∃ x, x ∈ (Set.Icc (0 : ℝ) 1) ∧ ∑ i in range (n + 1), (C i) * (x^i) = 0 :=", "goal": "n : ℕ\nC : ℕ → ℝ\nhC : ∑ i ∈ range (n + 1), C i / (↑i + 1) = 0\n⊢ ∃ x ∈ Set.Icc 0 1, ∑ i ∈ range (n + 1), C i * x ^ i = 0", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_5", "split": "test", "informal_prefix": "/-- Suppose $f$ is defined and differentiable for every $x>0$, and $f^{\\prime}(x) \\rightarrow 0$ as $x \\rightarrow+\\infty$. Put $g(x)=f(x+1)-f(x)$. Prove that $g(x) \\rightarrow 0$ as $x \\rightarrow+\\infty$.-/\n", "formal_statement": "theorem exercise_5_5\n {f : ℝ → ℝ}\n (hfd : Differentiable ℝ f)\n (hf : Tendsto (deriv f) atTop (𝓝 0)) :\n Tendsto (λ x => f (x + 1) - f x) atTop atTop :=", "goal": "f : ℝ → ℝ\nhfd : Differentiable ℝ f\nhf : Tendsto (deriv f) atTop (𝓝 0)\n⊢ Tendsto (fun x => f (x + 1) - f x) atTop atTop", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_6", "split": "valid", "informal_prefix": "/-- Suppose (a) $f$ is continuous for $x \\geq 0$, (b) $f^{\\prime}(x)$ exists for $x>0$, (c) $f(0)=0$, (d) $f^{\\prime}$ is monotonically increasing. Put $g(x)=\\frac{f(x)}{x} \\quad(x>0)$ and prove that $g$ is monotonically increasing.-/\n", "formal_statement": "theorem exercise_5_6\n {f : ℝ → ℝ}\n (hf1 : Continuous f)\n (hf2 : ∀ x, DifferentiableAt ℝ f x)\n (hf3 : f 0 = 0)\n (hf4 : Monotone (deriv f)) :\n MonotoneOn (λ x => f x / x) (Set.Ioi 0) :=", "goal": "f : ℝ → ℝ\nhf1 : Continuous f\nhf2 : ∀ (x : ℝ), DifferentiableAt ℝ f x\nhf3 : f 0 = 0\nhf4 : Monotone (deriv f)\n⊢ MonotoneOn (fun x => f x / x) (Set.Ioi 0)", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_7", "split": "test", "informal_prefix": "/-- Suppose $f^{\\prime}(x), g^{\\prime}(x)$ exist, $g^{\\prime}(x) \\neq 0$, and $f(x)=g(x)=0$. Prove that $\\lim _{t \\rightarrow x} \\frac{f(t)}{g(t)}=\\frac{f^{\\prime}(x)}{g^{\\prime}(x)}.$-/\n", "formal_statement": "theorem exercise_5_7\n {f g : ℝ → ℝ} {x : ℝ}\n (hf' : DifferentiableAt ℝ f 0)\n (hg' : DifferentiableAt ℝ g 0)\n (hg'_ne_0 : deriv g 0 ≠ 0)\n (f0 : f 0 = 0) (g0 : g 0 = 0) :\n Tendsto (λ x => f x / g x) (𝓝 x) (𝓝 (deriv f x / deriv g x)) :=", "goal": "f g : ℝ → ℝ\nx : ℝ\nhf' : DifferentiableAt ℝ f 0\nhg' : DifferentiableAt ℝ g 0\nhg'_ne_0 : deriv g 0 ≠ 0\nf0 : f 0 = 0\ng0 : g 0 = 0\n⊢ Tendsto (fun x => f x / g x) (𝓝 x) (𝓝 (deriv f x / deriv g x))", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_15", "split": "valid", "informal_prefix": "/-- Suppose $a \\in R^{1}, f$ is a twice-differentiable real function on $(a, \\infty)$, and $M_{0}, M_{1}, M_{2}$ are the least upper bounds of $|f(x)|,\\left|f^{\\prime}(x)\\right|,\\left|f^{\\prime \\prime}(x)\\right|$, respectively, on $(a, \\infty)$. Prove that $M_{1}^{2} \\leq 4 M_{0} M_{2} .$-/\n", "formal_statement": "theorem exercise_5_15 {f : ℝ → ℝ} (a M0 M1 M2 : ℝ)\n (hf' : DifferentiableOn ℝ f (Set.Ici a))\n (hf'' : DifferentiableOn ℝ (deriv f) (Set.Ici a))\n (hM0 : M0 = sSup {(|f x|) | x ∈ (Set.Ici a)})\n (hM1 : M1 = sSup {(|deriv f x|) | x ∈ (Set.Ici a)})\n (hM2 : M2 = sSup {(|deriv (deriv f) x|) | x ∈ (Set.Ici a)}) :\n (M1 ^ 2) ≤ 4 * M0 * M2 :=", "goal": "f : ℝ → ℝ\na M0 M1 M2 : ℝ\nhf' : DifferentiableOn ℝ f (Set.Ici a)\nhf'' : DifferentiableOn ℝ (deriv f) (Set.Ici a)\nhM0 : M0 = sSup {x | ∃ x_1 ∈ Set.Ici a, |f x_1| = x}\nhM1 : M1 = sSup {x | ∃ x_1 ∈ Set.Ici a, |deriv f x_1| = x}\nhM2 : M2 = sSup {x | ∃ x_1 ∈ Set.Ici a, |deriv (deriv f) x_1| = x}\n⊢ M1 ^ 2 ≤ 4 * M0 * M2", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_17", "split": "test", "informal_prefix": "/-- Suppose $f$ is a real, three times differentiable function on $[-1,1]$, such that $f(-1)=0, \\quad f(0)=0, \\quad f(1)=1, \\quad f^{\\prime}(0)=0 .$ Prove that $f^{(3)}(x) \\geq 3$ for some $x \\in(-1,1)$.-/\n", "formal_statement": "theorem exercise_5_17\n {f : ℝ → ℝ}\n (hf' : DifferentiableOn ℝ f (Set.Icc (-1) 1))\n (hf'' : DifferentiableOn ℝ (deriv f) (Set.Icc 1 1))\n (hf''' : DifferentiableOn ℝ (deriv (deriv f)) (Set.Icc 1 1))\n (hf0 : f (-1) = 0)\n (hf1 : f 0 = 0)\n (hf2 : f 1 = 1)\n (hf3 : deriv f 0 = 0) :\n ∃ x, x ∈ Set.Ioo (-1 : ℝ) 1 ∧ deriv (deriv (deriv f)) x ≥ 3 :=", "goal": "f : ℝ → ℝ\nhf' : DifferentiableOn ℝ f (Set.Icc (-1) 1)\nhf'' : DifferentiableOn ℝ (deriv f) (Set.Icc 1 1)\nhf''' : DifferentiableOn ℝ (deriv (deriv f)) (Set.Icc 1 1)\nhf0 : f (-1) = 0\nhf1 : f 0 = 0\nhf2 : f 1 = 1\nhf3 : deriv f 0 = 0\n⊢ ∃ x ∈ Set.Ioo (-1) 1, deriv (deriv (deriv f)) x ≥ 3", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_1_18", "split": "test", "informal_prefix": "/-- If $G$ is a finite group of even order, show that there must be an element $a \\neq e$ such that $a=a^{-1}$.-/\n", "formal_statement": "theorem exercise_2_1_18 {G : Type*} [Group G]\n [Fintype G] (hG2 : Even (card G)) :\n ∃ (a : G), a ≠ 1 ∧ a = a⁻¹ :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\nhG2 : Even (card G)\n⊢ ∃ a, a ≠ 1 ∧ a = a⁻¹", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_1_21", "split": "valid", "informal_prefix": "/-- Show that a group of order 5 must be abelian.-/\n", "formal_statement": "def exercise_2_1_21 (G : Type*) [Group G] [Fintype G]\n (hG : card G = 5) :\n CommGroup G :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\nhG : card G = 5\n⊢ CommGroup G", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_1_26", "split": "test", "informal_prefix": "/-- If $G$ is a finite group, prove that, given $a \\in G$, there is a positive integer $n$, depending on $a$, such that $a^n = e$.-/\n", "formal_statement": "theorem exercise_2_1_26 {G : Type*} [Group G]\n [Fintype G] (a : G) : ∃ (n : ℕ), a ^ n = 1 :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\na : G\n⊢ ∃ n, a ^ n = 1", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_1_27", "split": "valid", "informal_prefix": "/-- If $G$ is a finite group, prove that there is an integer $m > 0$ such that $a^m = e$ for all $a \\in G$.-/\n", "formal_statement": "theorem exercise_2_1_27 {G : Type*} [Group G]\n [Fintype G] : ∃ (m : ℕ), ∀ (a : G), a ^ m = 1 :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\n⊢ ∃ m, ∀ (a : G), a ^ m = 1", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_2_3", "split": "test", "informal_prefix": "/-- If $G$ is a group in which $(a b)^{i}=a^{i} b^{i}$ for three consecutive integers $i$, prove that $G$ is abelian.-/\n", "formal_statement": "def exercise_2_2_3 {G : Type*} [Group G]\n {P : ℕ → Prop} {hP : P = λ i => ∀ a b : G, (a*b)^i = a^i * b^i}\n (hP1 : ∃ n : ℕ, P n ∧ P (n+1) ∧ P (n+2)) : CommGroup G :=", "goal": "G : Type u_1\ninst✝ : Group G\nP : ℕ → Prop\nhP : P = fun i => ∀ (a b : G), (a * b) ^ i = a ^ i * b ^ i\nhP1 : ∃ n, P n ∧ P (n + 1) ∧ P (n + 2)\n⊢ CommGroup G", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_2_5", "split": "valid", "informal_prefix": "/-- 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.-/\n", "formal_statement": "def exercise_2_2_5 {G : Type*} [Group G]\n (h : ∀ (a b : G), (a * b) ^ 3 = a ^ 3 * b ^ 3 ∧ (a * b) ^ 5 = a ^ 5 * b ^ 5) :\n CommGroup G :=", "goal": "G : Type u_1\ninst✝ : Group G\nh : ∀ (a b : G), (a * b) ^ 3 = a ^ 3 * b ^ 3 ∧ (a * b) ^ 5 = a ^ 5 * b ^ 5\n⊢ CommGroup G", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_2_6c", "split": "test", "informal_prefix": "/-- Let $G$ be a group in which $(a b)^{n}=a^{n} b^{n}$ for some fixed integer $n>1$ for all $a, b \\in G$. For all $a, b \\in G$, prove that $\\left(a b a^{-1} b^{-1}\\right)^{n(n-1)}=e$.-/\n", "formal_statement": "theorem exercise_2_2_6c {G : Type*} [Group G] {n : ℕ} (hn : n > 1)\n (h : ∀ (a b : G), (a * b) ^ n = a ^ n * b ^ n) :\n ∀ (a b : G), (a * b * a⁻¹ * b⁻¹) ^ (n * (n - 1)) = 1 :=", "goal": "G : Type u_1\ninst✝ : Group G\nn : ℕ\nhn : n > 1\nh : ∀ (a b : G), (a * b) ^ n = a ^ n * b ^ n\n⊢ ∀ (a b : G), (a * b * a⁻¹ * b⁻¹) ^ (n * (n - 1)) = 1", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_3_17", "split": "valid", "informal_prefix": "/-- If $G$ is a group and $a, x \\in G$, prove that $C\\left(x^{-1} a x\\right)=x^{-1} C(a) x$-/\n", "formal_statement": "theorem exercise_2_3_17 {G : Type*} [Mul G] [Group G] (a x : G) :\n centralizer {x⁻¹*a*x} =\n (λ g : G => x⁻¹*g*x) '' (centralizer {a}) :=", "goal": "G : Type u_1\ninst✝¹ : Mul G\ninst✝ : Group G\na x : G\n⊢ {x⁻¹ * a * x}.centralizer = (fun g => x⁻¹ * g * x) '' {a}.centralizer", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_3_16", "split": "test", "informal_prefix": "/-- If a group $G$ has no proper subgroups, prove that $G$ is cyclic of order $p$, where $p$ is a prime number.-/\n", "formal_statement": "theorem exercise_2_3_16 {G : Type*} [Group G]\n (hG : ∀ H : Subgroup G, H = ⊤ ∨ H = ⊥) :\n IsCyclic G ∧ ∃ (p : ℕ) (Fin : Fintype G), Nat.Prime p ∧ @card G Fin = p :=", "goal": "G : Type u_1\ninst✝ : Group G\nhG : ∀ (H : Subgroup G), H = ⊤ ∨ H = ⊥\n⊢ IsCyclic G ∧ ∃ p Fin, p.Prime ∧ card G = p", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_4_36", "split": "valid", "informal_prefix": "/-- If $a > 1$ is an integer, show that $n \\mid \\varphi(a^n - 1)$, where $\\phi$ is the Euler $\\varphi$-function.-/\n", "formal_statement": "theorem exercise_2_4_36 {a n : ℕ} (h : a > 1) :\n n ∣ (a ^ n - 1).totient :=", "goal": "a n : ℕ\nh : a > 1\n⊢ n ∣ (a ^ n - 1).totient", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_5_23", "split": "test", "informal_prefix": "/-- Let $G$ be a group such that all subgroups of $G$ are normal in $G$. If $a, b \\in G$, prove that $ba = a^jb$ for some $j$.-/\n", "formal_statement": "theorem exercise_2_5_23 {G : Type*} [Group G]\n (hG : ∀ (H : Subgroup G), H.Normal) (a b : G) :\n ∃ (j : ℤ) , b*a = a^j * b :=", "goal": "G : Type u_1\ninst✝ : Group G\nhG : ∀ (H : Subgroup G), H.Normal\na b : G\n⊢ ∃ j, b * a = a ^ j * b", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_5_30", "split": "valid", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_2_5_30 {G : Type*} [Group G] [Fintype G]\n {p m : ℕ} (hp : Nat.Prime p) (hp1 : ¬ p ∣ m) (hG : card G = p*m)\n {H : Subgroup G} [Fintype H] [H.Normal] (hH : card H = p):\n Subgroup.Characteristic H :=", "goal": "G : Type u_1\ninst✝³ : Group G\ninst✝² : Fintype G\np m : ℕ\nhp : p.Prime\nhp1 : ¬p ∣ m\nhG : card G = p * m\nH : Subgroup G\ninst✝¹ : Fintype ↥H\ninst✝ : H.Normal\nhH : card ↥H = p\n⊢ H.Characteristic", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_5_31", "split": "test", "informal_prefix": "/-- Suppose that $G$ is an abelian group of order $p^nm$ where $p \\nmid m$ is a prime. If $H$ is a subgroup of $G$ of order $p^n$, prove that $H$ is a characteristic subgroup of $G$.-/\n", "formal_statement": "theorem exercise_2_5_31 {G : Type*} [CommGroup G] [Fintype G]\n {p m n : ℕ} (hp : Nat.Prime p) (hp1 : ¬ p ∣ m) (hG : card G = p^n*m)\n {H : Subgroup G} [Fintype H] (hH : card H = p^n) :\n Subgroup.Characteristic H :=", "goal": "G : Type u_1\ninst✝² : CommGroup G\ninst✝¹ : Fintype G\np m n : ℕ\nhp : p.Prime\nhp1 : ¬p ∣ m\nhG : card G = p ^ n * m\nH : Subgroup G\ninst✝ : Fintype ↥H\nhH : card ↥H = p ^ n\n⊢ H.Characteristic", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_5_37", "split": "valid", "informal_prefix": "/-- If $G$ is a nonabelian group of order 6, prove that $G \\simeq S_3$.-/\n", "formal_statement": "def exercise_2_5_37 (G : Type*) [Group G] [Fintype G]\n (hG : card G = 6) (hG' : IsEmpty (CommGroup G)) :\n G ≃* Equiv.Perm (Fin 3) :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\nhG : card G = 6\nhG' : IsEmpty (CommGroup G)\n⊢ G ≃* Equiv.Perm (Fin 3)", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_5_43", "split": "test", "informal_prefix": "/-- Prove that a group of order 9 must be abelian.-/\n", "formal_statement": "def exercise_2_5_43 (G : Type*) [Group G] [Fintype G]\n (hG : card G = 9) :\n CommGroup G :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\nhG : card G = 9\n⊢ CommGroup G", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_5_44", "split": "valid", "informal_prefix": "/-- Prove that a group of order $p^2$, $p$ a prime, has a normal subgroup of order $p$.-/\n", "formal_statement": "theorem exercise_2_5_44 {G : Type*} [Group G] [Fintype G] {p : ℕ}\n (hp : Nat.Prime p) (hG : card G = p^2) :\n ∃ (N : Subgroup G) (Fin : Fintype N), @card N Fin = p ∧ N.Normal :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\np : ℕ\nhp : p.Prime\nhG : card G = p ^ 2\n⊢ ∃ N Fin, card ↥N = p ∧ N.Normal", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_5_52", "split": "test", "informal_prefix": "/-- Let $G$ be a finite group and $\\varphi$ an automorphism of $G$ such that $\\varphi(x) = x^{-1}$ for more than three-fourths of the elements of $G$. Prove that $\\varphi(y) = y^{-1}$ for all $y \\in G$, and so $G$ is abelian.-/\n", "formal_statement": "theorem exercise_2_5_52 {G : Type*} [Group G] [Fintype G]\n (φ : G ≃* G) {I : Finset G} (hI : ∀ x ∈ I, φ x = x⁻¹)\n (hI1 : (0.75 : ℚ) * card G ≤ card I) :\n ∀ x : G, φ x = x⁻¹ ∧ ∀ x y : G, x*y = y*x :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\nφ : G ≃* G\nI : Finset G\nhI : ∀ x ∈ I, φ x = x⁻¹\nhI1 : 0.75 * ↑(card G) ≤ ↑(card { x // x ∈ I })\n⊢ ∀ (x : G), φ x = x⁻¹ ∧ ∀ (x y : G), x * y = y * x", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_6_15", "split": "valid", "informal_prefix": "/-- 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 $mn$.-/\n", "formal_statement": "theorem exercise_2_6_15 {G : Type*} [CommGroup G] {m n : ℕ}\n (hm : ∃ (g : G), orderOf g = m)\n (hn : ∃ (g : G), orderOf g = n)\n (hmn : m.Coprime n) :\n ∃ (g : G), orderOf g = m * n :=", "goal": "G : Type u_1\ninst✝ : CommGroup G\nm n : ℕ\nhm : ∃ g, orderOf g = m\nhn : ∃ g, orderOf g = n\nhmn : m.Coprime n\n⊢ ∃ g, orderOf g = m * n", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_7_7", "split": "test", "informal_prefix": "/-- If $\\varphi$ is a homomorphism of $G$ onto $G'$ and $N \\triangleleft G$, show that $\\varphi(N) \\triangleleft G'$.-/\n", "formal_statement": "theorem exercise_2_7_7 {G : Type*} [Group G] {G' : Type*} [Group G']\n (φ : G →* G') (N : Subgroup G) [N.Normal] :\n (Subgroup.map φ N).Normal :=", "goal": "G : Type u_1\ninst✝² : Group G\nG' : Type u_2\ninst✝¹ : Group G'\nφ : G →* G'\nN : Subgroup G\ninst✝ : N.Normal\n⊢ (Subgroup.map φ N).Normal", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_8_12", "split": "valid", "informal_prefix": "/-- Prove that any two nonabelian groups of order 21 are isomorphic.-/\n", "formal_statement": "def exercise_2_8_12 {G H : Type*} [Fintype G] [Fintype H]\n [Group G] [Group H] (hG : card G = 21) (hH : card H = 21)\n (hG1 : IsEmpty (CommGroup G)) (hH1 : IsEmpty (CommGroup H)) :\n G ≃* H :=", "goal": "G : Type u_1\nH : Type u_2\ninst✝³ : Fintype G\ninst✝² : Fintype H\ninst✝¹ : Group G\ninst✝ : Group H\nhG : card G = 21\nhH : card H = 21\nhG1 : IsEmpty (CommGroup G)\nhH1 : IsEmpty (CommGroup H)\n⊢ G ≃* H", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_8_15", "split": "test", "informal_prefix": "/-- Prove that if $p > q$ are two primes such that $q \\mid p - 1$, then any two nonabelian groups of order $pq$ are isomorphic.-/\n", "formal_statement": "def exercise_2_8_15 {G H: Type*} [Fintype G] [Group G] [Fintype H]\n [Group H] {p q : ℕ} (hp : Nat.Prime p) (hq : Nat.Prime q)\n (h : p > q) (h1 : q ∣ p - 1) (hG : card G = p*q) (hH : card G = p*q) :\n G ≃* H :=", "goal": "G : Type u_1\nH : Type u_2\ninst✝³ : Fintype G\ninst✝² : Group G\ninst✝¹ : Fintype H\ninst✝ : Group H\np q : ℕ\nhp : p.Prime\nhq : q.Prime\nh : p > q\nh1 : q ∣ p - 1\nhG hH : card G = p * q\n⊢ G ≃* H", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_9_2", "split": "valid", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_2_9_2 {G H : Type*} [Fintype G] [Fintype H] [Group G]\n [Group H] (hG : IsCyclic G) (hH : IsCyclic H) :\n IsCyclic (G × H) ↔ (card G).Coprime (card H) :=", "goal": "G : Type u_1\nH : Type u_2\ninst✝³ : Fintype G\ninst✝² : Fintype H\ninst✝¹ : Group G\ninst✝ : Group H\nhG : IsCyclic G\nhH : IsCyclic H\n⊢ IsCyclic (G × H) ↔ (card G).Coprime (card H)", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_10_1", "split": "test", "informal_prefix": "/-- Let $A$ be a normal subgroup of a group $G$, and suppose that $b \\in G$ is an element of prime order $p$, and that $b \\not\\in A$. Show that $A \\cap (b) = (e)$.-/\n", "formal_statement": "theorem exercise_2_10_1 {G : Type*} [Group G] (A : Subgroup G)\n [A.Normal] {b : G} (hp : Nat.Prime (orderOf b)) :\n A ⊓ (Subgroup.closure {b}) = ⊥ :=", "goal": "G : Type u_1\ninst✝¹ : Group G\nA : Subgroup G\ninst✝ : A.Normal\nb : G\nhp : (orderOf b).Prime\n⊢ A ⊓ Subgroup.closure {b} = ⊥", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_11_6", "split": "valid", "informal_prefix": "/-- If $P$ is a $p$-Sylow subgroup of $G$ and $P \\triangleleft G$, prove that $P$ is the only $p$-Sylow subgroup of $G$.-/\n", "formal_statement": "theorem exercise_2_11_6 {G : Type*} [Group G] {p : ℕ} (hp : Nat.Prime p)\n {P : Sylow p G} (hP : P.Normal) :\n ∀ (Q : Sylow p G), P = Q :=", "goal": "G : Type u_1\ninst✝ : Group G\np : ℕ\nhp : p.Prime\nP : Sylow p G\nhP : (↑P).Normal\n⊢ ∀ (Q : Sylow p G), P = Q", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_11_7", "split": "test", "informal_prefix": "/-- If $P \\triangleleft G$, $P$ a $p$-Sylow subgroup of $G$, prove that $\\varphi(P) = P$ for every automorphism $\\varphi$ of $G$.-/\n", "formal_statement": "theorem exercise_2_11_7 {G : Type*} [Group G] {p : ℕ} (hp : Nat.Prime p)\n {P : Sylow p G} (hP : P.Normal) :\n Subgroup.Characteristic (P : Subgroup G) :=", "goal": "G : Type u_1\ninst✝ : Group G\np : ℕ\nhp : p.Prime\nP : Sylow p G\nhP : (↑P).Normal\n⊢ (↑P).Characteristic", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_11_22", "split": "valid", "informal_prefix": "/-- Show that any subgroup of order $p^{n-1}$ in a group $G$ of order $p^n$ is normal in $G$.-/\n", "formal_statement": "theorem exercise_2_11_22 {p : ℕ} {n : ℕ} {G : Type*} [Fintype G]\n [Group G] (hp : Nat.Prime p) (hG : card G = p ^ n) {K : Subgroup G}\n [Fintype K] (hK : card K = p ^ (n-1)) :\n K.Normal :=", "goal": "p n : ℕ\nG : Type u_1\ninst✝² : Fintype G\ninst✝¹ : Group G\nhp : p.Prime\nhG : card G = p ^ n\nK : Subgroup G\ninst✝ : Fintype ↥K\nhK : card ↥K = p ^ (n - 1)\n⊢ K.Normal", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_2_21", "split": "test", "informal_prefix": "/-- If $\\sigma, \\tau$ are two permutations that disturb no common element and $\\sigma \\tau = e$, prove that $\\sigma = \\tau = e$.-/\n", "formal_statement": "theorem exercise_3_2_21 {α : Type*} [Fintype α] {σ τ: Equiv.Perm α}\n (h1 : ∀ a : α, σ a = a ↔ τ a ≠ a) (h2 : τ ∘ σ = id) :\n σ = 1 ∧ τ = 1 :=", "goal": "α : Type u_1\ninst✝ : Fintype α\nσ τ : Equiv.Perm α\nh1 : ∀ (a : α), σ a = a ↔ τ a ≠ a\nh2 : ⇑τ ∘ ⇑σ = id\n⊢ σ = 1 ∧ τ = 1", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_1_19", "split": "valid", "informal_prefix": "/-- Show that there is an infinite number of solutions to $x^2 = -1$ in the quaternions.-/\n", "formal_statement": "theorem exercise_4_1_19 : Infinite {x : Quaternion ℝ | x^2 = -1} :=", "goal": "⊢ Infinite ↑{x | x ^ 2 = -1}", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_1_34", "split": "test", "informal_prefix": "/-- Let $T$ be the group of $2\\times 2$ matrices $A$ with entries in the field $\\mathbb{Z}_2$ such that $\\det A$ is not equal to 0. Prove that $T$ is isomorphic to $S_3$, the symmetric group of degree 3.-/\n", "formal_statement": "def exercise_4_1_34 : Equiv.Perm (Fin 3) ≃* Matrix.GeneralLinearGroup (Fin 2) (ZMod 2) :=", "goal": "⊢ Equiv.Perm (Fin 3) ≃* GL (Fin 2) (ZMod 2)", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_2_5", "split": "valid", "informal_prefix": "/-- Let $R$ be a ring in which $x^3 = x$ for every $x \\in R$. Prove that $R$ is commutative.-/\n", "formal_statement": "def exercise_4_2_5 {R : Type*} [Ring R]\n (h : ∀ x : R, x ^ 3 = x) : CommRing R :=", "goal": "R : Type u_1\ninst✝ : Ring R\nh : ∀ (x : R), x ^ 3 = x\n⊢ CommRing R", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_2_6", "split": "test", "informal_prefix": "/-- If $a^2 = 0$ in $R$, show that $ax + xa$ commutes with $a$.-/\n", "formal_statement": "theorem exercise_4_2_6 {R : Type*} [Ring R] (a x : R)\n (h : a ^ 2 = 0) : a * (a * x + x * a) = (x + x * a) * a :=", "goal": "R : Type u_1\ninst✝ : Ring R\na x : R\nh : a ^ 2 = 0\n⊢ a * (a * x + x * a) = (x + x * a) * a", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_2_9", "split": "valid", "informal_prefix": "/-- 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 $p \\mid a$.-/\n", "formal_statement": "theorem exercise_4_2_9 {p : ℕ} (hp : Nat.Prime p) (hp1 : Odd p) :\n ∃ (a b : ℤ), (a / b : ℚ) = ∑ i in Finset.range p, 1 / (i + 1) → ↑p ∣ a :=", "goal": "p : ℕ\nhp : p.Prime\nhp1 : Odd p\n⊢ ∃ a b, ↑a / ↑b = ↑(∑ i ∈ Finset.range p, 1 / (i + 1)) → ↑p ∣ a", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_3_1", "split": "test", "informal_prefix": "/-- If $R$ is a commutative ring and $a \\in R$, let $L(a) = \\{x \\in R \\mid xa = 0\\}$. Prove that $L(a)$ is an ideal of $R$.-/\n", "formal_statement": "theorem exercise_4_3_1 {R : Type*} [CommRing R] (a : R) :\n ∃ I : Ideal R, {x : R | x*a=0} = I :=", "goal": "R : Type u_1\ninst✝ : CommRing R\na : R\n⊢ ∃ I, {x | x * a = 0} = ↑I", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_3_25", "split": "valid", "informal_prefix": "/-- 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 $I = R$.-/\n", "formal_statement": "theorem exercise_4_3_25 (I : Ideal (Matrix (Fin 2) (Fin 2) ℝ)) :\n I = ⊥ ∨ I = ⊤ :=", "goal": "I : Ideal (Matrix (Fin 2) (Fin 2) ℝ)\n⊢ I = ⊥ ∨ I = ⊤", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_4_9", "split": "test", "informal_prefix": "/-- Show that $(p - 1)/2$ of the numbers $1, 2, \\ldots, p - 1$ are quadratic residues and $(p - 1)/2$ are quadratic nonresidues $\\mod p$.-/\n", "formal_statement": "theorem exercise_4_4_9 (p : ℕ) (hp : Nat.Prime p) :\n (∃ S : Finset (ZMod p), S.card = (p-1)/2 ∧ ∃ x : ZMod p, x^2 = p) ∧\n (∃ S : Finset (ZMod p), S.card = (p-1)/2 ∧ ¬ ∃ x : ZMod p, x^2 = p) :=", "goal": "p : ℕ\nhp : p.Prime\n⊢ (∃ S, S.card = (p - 1) / 2 ∧ ∃ x, x ^ 2 = ↑p) ∧ ∃ S, S.card = (p - 1) / 2 ∧ ¬∃ x, x ^ 2 = ↑p", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_16", "split": "valid", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_4_5_16 {p n: ℕ} (hp : Nat.Prime p)\n {q : Polynomial (ZMod p)} (hq : Irreducible q) (hn : q.degree = n) :\n ∃ is_fin : Fintype $ Polynomial (ZMod p) ⧸ span ({q} : Set (Polynomial $ ZMod p)),\n @card (Polynomial (ZMod p) ⧸ span {q}) is_fin = p ^ n ∧\n IsField (Polynomial $ ZMod p) :=", "goal": "p n : ℕ\nhp : p.Prime\nq : (ZMod p)[X]\nhq : Irreducible q\nhn : q.degree = ↑n\n⊢ ∃ is_fin, card ((ZMod p)[X] ⧸ span {q}) = p ^ n ∧ IsField (ZMod p)[X]", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_23", "split": "test", "informal_prefix": "/-- Let $F = \\mathbb{Z}_7$ and let $p(x) = x^3 - 2$ and $q(x) = x^3 + 2$ be in $F[x]$. Show that $p(x)$ and $q(x)$ are irreducible in $F[x]$ and that the fields $F[x]/(p(x))$ and $F[x]/(q(x))$ are isomorphic.-/\n", "formal_statement": "theorem exercise_4_5_23 {p q: Polynomial (ZMod 7)}\n (hp : p = X^3 - 2) (hq : q = X^3 + 2) :\n Irreducible p ∧ Irreducible q ∧\n (Nonempty $ Polynomial (ZMod 7) ⧸ span ({p} : Set $ Polynomial $ ZMod 7) ≃+*\n Polynomial (ZMod 7) ⧸ span ({q} : Set $ Polynomial $ ZMod 7)) :=", "goal": "p q : (ZMod 7)[X]\nhp : p = X ^ 3 - 2\nhq : q = X ^ 3 + 2\n⊢ Irreducible p ∧ Irreducible q ∧ Nonempty ((ZMod 7)[X] ⧸ span {p} ≃+* (ZMod 7)[X] ⧸ span {q})", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_25", "split": "valid", "informal_prefix": "/-- If $p$ is a prime, show that $q(x) = 1 + x + x^2 + \\cdots x^{p - 1}$ is irreducible in $Q[x]$.-/\n", "formal_statement": "theorem exercise_4_5_25 {p : ℕ} (hp : Nat.Prime p) :\n Irreducible (∑ i : Finset.range p, X ^ p : Polynomial ℚ) :=", "goal": "p : ℕ\nhp : p.Prime\n⊢ Irreducible (∑ i : { x // x ∈ Finset.range p }, X ^ p)", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_6_2", "split": "test", "informal_prefix": "/-- Prove that $f(x) = x^3 + 3x + 2$ is irreducible in $Q[x]$.-/\n", "formal_statement": "theorem exercise_4_6_2 : Irreducible (X^3 + 3*X + 2 : Polynomial ℚ) :=", "goal": "⊢ Irreducible (X ^ 3 + 3 * X + 2)", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_6_3", "split": "valid", "informal_prefix": "/-- Show that there is an infinite number of integers a such that $f(x) = x^7 + 15x^2 - 30x + a$ is irreducible in $Q[x]$.-/\n", "formal_statement": "theorem exercise_4_6_3 :\n Infinite {a : ℤ | Irreducible (X^7 + 15*X^2 - 30*X + (a : Polynomial ℚ) : Polynomial ℚ)} :=", "goal": "⊢ Infinite ↑{a | Irreducible (X ^ 7 + 15 * X ^ 2 - 30 * X + ↑a)}", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_1_8", "split": "test", "informal_prefix": "/-- If $F$ is a field of characteristic $p \\neq 0$, show that $(a + b)^m = a^m + b^m$, where $m = p^n$, for all $a, b \\in F$ and any positive integer $n$.-/\n", "formal_statement": "theorem exercise_5_1_8 {p m n: ℕ} {F : Type*} [Field F]\n (hp : Nat.Prime p) (hF : CharP F p) (a b : F) (hm : m = p ^ n) :\n (a + b) ^ m = a^m + b^m :=", "goal": "p m n : ℕ\nF : Type u_1\ninst✝ : Field F\nhp : p.Prime\nhF : CharP F p\na b : F\nhm : m = p ^ n\n⊢ (a + b) ^ m = a ^ m + b ^ m", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_2_20", "split": "valid", "informal_prefix": "/-- 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 $V$.-/\n", "formal_statement": "theorem exercise_5_2_20 {F V ι: Type*} [Infinite F] [Field F]\n [AddCommGroup V] [Module F V] {u : ι → Submodule F V}\n (hu : ∀ i : ι, u i ≠ ⊤) :\n (⋃ i : ι, (u i : Set V)) ≠ ⊤ :=", "goal": "F : Type u_1\nV : Type u_2\nι : Type u_3\ninst✝³ : Infinite F\ninst✝² : Field F\ninst✝¹ : AddCommGroup V\ninst✝ : Module F V\nu : ι → Submodule F V\nhu : ∀ (i : ι), u i ≠ ⊤\n⊢ ⋃ i, ↑(u i) ≠ ⊤", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_3_7", "split": "test", "informal_prefix": "/-- If $a \\in K$ is such that $a^2$ is algebraic over the subfield $F$ of $K$, show that a is algebraic over $F$.-/\n", "formal_statement": "theorem exercise_5_3_7 {K : Type*} [Field K] {F : Subfield K}\n {a : K} (ha : IsAlgebraic F (a ^ 2)) : IsAlgebraic F a :=", "goal": "K : Type u_1\ninst✝ : Field K\nF : Subfield K\na : K\nha : IsAlgebraic (↥F) (a ^ 2)\n⊢ IsAlgebraic (↥F) a", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_3_10", "split": "valid", "informal_prefix": "/-- Prove that $\\cos 1^{\\circ}$ is algebraic over $\\mathbb{Q}$.-/\n", "formal_statement": "theorem exercise_5_3_10 : IsAlgebraic ℚ (cos (Real.pi / 180)) :=", "goal": "⊢ IsAlgebraic ℚ (π / 180).cos", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_4_3", "split": "test", "informal_prefix": "/-- If $a \\in C$ is such that $p(a) = 0$, where $p(x) = x^5 + \\sqrt{2}x^3 + \\sqrt{5}x^2 + \\sqrt{7}x + \\sqrt{11}$, show that $a$ is algebraic over $\\mathbb{Q}$ of degree at most 80.-/\n", "formal_statement": "theorem exercise_5_4_3 {a : ℂ} {p : ℂ → ℂ}\n (hp : p = λ (x : ℂ) => x^5 + sqrt 2 * x^3 + sqrt 5 * x^2 + sqrt 7 * x + 11)\n (ha : p a = 0) :\n ∃ p : Polynomial ℂ , p.degree < 80 ∧ a ∈ p.roots ∧\n ∀ n : p.support, ∃ a b : ℤ, p.coeff n = a / b :=", "goal": "a : ℂ\np : ℂ → ℂ\nhp : p = fun x => x ^ 5 + ↑√2 * x ^ 3 + ↑√5 * x ^ 2 + ↑√7 * x + 11\nha : p a = 0\n⊢ ∃ p, p.degree < 80 ∧ a ∈ p.roots ∧ ∀ (n : { x // x ∈ p.support }), ∃ a b, p.coeff ↑n = ↑a / ↑b", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_5_2", "split": "valid", "informal_prefix": "/-- Prove that $x^3 - 3x - 1$ is irreducible over $\\mathbb{Q}$.-/\n", "formal_statement": "theorem exercise_5_5_2 : Irreducible (X^3 - 3*X - 1 : Polynomial ℚ) :=", "goal": "⊢ Irreducible (X ^ 3 - 3 * X - 1)", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_6_14", "split": "test", "informal_prefix": "/-- If $F$ is of characteristic $p \\neq 0$, show that all the roots of $x^m - x$, where $m = p^n$, are distinct.-/\n", "formal_statement": "theorem exercise_5_6_14 {p m n: ℕ} (hp : Nat.Prime p) {F : Type*}\n [Field F] [CharP F p] (hm : m = p ^ n) :\n card (rootSet (X ^ m - X : Polynomial F) F) = m :=", "goal": "p m n : ℕ\nhp : p.Prime\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : CharP F p\nhm : m = p ^ n\n⊢ card ↑((X ^ m - X).rootSet F) = m", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_12a", "split": "valid", "informal_prefix": "/-- Let $(p_n)$ be a sequence and $f:\\mathbb{N}\\to\\mathbb{N}$. The sequence $(q_k)_{k\\in\\mathbb{N}}$ with $q_k=p_{f(k)}$ is called a rearrangement of $(p_n)$. Show that if $f$ is an injection, the limit of a sequence is unaffected by rearrangement.-/\n", "formal_statement": "theorem exercise_2_12a (f : ℕ → ℕ) (p : ℕ → ℝ) (a : ℝ)\n (hf : Injective f) (hp : Tendsto p atTop (𝓝 a)) :\n Tendsto (λ n => p (f n)) atTop (𝓝 a) :=", "goal": "f : ℕ → ℕ\np : ℕ → ℝ\na : ℝ\nhf : Injective f\nhp : Tendsto p atTop (𝓝 a)\n⊢ Tendsto (fun n => p (f n)) atTop (𝓝 a)", "header": "import Mathlib\n\nopen Filter Real Function\nopen scoped Topology\n\n"}
{"name": "exercise_2_26", "split": "test", "informal_prefix": "/-- Prove that a set $U \\subset M$ is open if and only if none of its points are limits of its complement.-/\n", "formal_statement": "theorem exercise_2_26 {M : Type*} [TopologicalSpace M]\n (U : Set M) : IsOpen U ↔ ∀ x ∈ U, ¬ ClusterPt x (𝓟 Uᶜ) :=", "goal": "M : Type u_1\ninst✝ : TopologicalSpace M\nU : Set M\n⊢ IsOpen U ↔ ∀ x ∈ U, ¬ClusterPt x (𝓟 Uᶜ)", "header": "import Mathlib\n\nopen Filter Real Function\nopen scoped Topology\n\n"}
{"name": "exercise_2_29", "split": "valid", "informal_prefix": "/-- Let $\\mathcal{T}$ be the collection of open subsets of a metric space $\\mathrm{M}$, and $\\mathcal{K}$ the collection of closed subsets. Show that there is a bijection from $\\mathcal{T}$ onto $\\mathcal{K}$.-/\n", "formal_statement": "theorem exercise_2_29 (M : Type*) [MetricSpace M]\n (O C : Set (Set M))\n (hO : O = {s | IsOpen s})\n (hC : C = {s | IsClosed s}) :\n ∃ f : O → C, Bijective f :=", "goal": "M : Type u_1\ninst✝ : MetricSpace M\nO C : Set (Set M)\nhO : O = {s | IsOpen s}\nhC : C = {s | IsClosed s}\n⊢ ∃ f, Bijective f", "header": "import Mathlib\n\nopen Filter Real Function\nopen scoped Topology\n\n"}
{"name": "exercise_2_32a", "split": "test", "informal_prefix": "/-- Show that every subset of $\\mathbb{N}$ is clopen.-/\n", "formal_statement": "theorem exercise_2_32a (A : Set ℕ) : IsClopen A :=", "goal": "A : Set ℕ\n⊢ IsClopen A", "header": "import Mathlib\n\nopen Filter Real Function\nopen scoped Topology\n\n"}
{"name": "exercise_2_41", "split": "valid", "informal_prefix": "/-- Let $\\|\\cdot\\|$ be any norm on $\\mathbb{R}^{m}$ and let $B=\\left\\{x \\in \\mathbb{R}^{m}:\\|x\\| \\leq 1\\right\\}$. Prove that $B$ is compact.-/\n", "formal_statement": "theorem exercise_2_41 (m : ℕ) {X : Type*} [NormedSpace ℝ ((Fin m) → ℝ)] :\n IsCompact (Metric.closedBall 0 1) :=", "goal": "m : ℕ\nX : Type u_1\ninst✝ : NormedSpace ℝ (Fin m → ℝ)\n⊢ IsCompact (Metric.closedBall 0 1)", "header": "import Mathlib\n\nopen Filter Real Function\nopen scoped Topology\n\n"}
{"name": "exercise_2_46", "split": "test", "informal_prefix": "/-- Assume that $A, B$ are compact, disjoint, nonempty subsets of $M$. Prove that there are $a_0 \\in A$ and $b_0 \\in B$ such that for all $a \\in A$ and $b \\in B$ we have $d(a_0, b_0) \\leq d(a, b)$.-/\n", "formal_statement": "theorem exercise_2_46 {M : Type*} [MetricSpace M]\n {A B : Set M} (hA : IsCompact A) (hB : IsCompact B)\n (hAB : Disjoint A B) (hA₀ : A ≠ ∅) (hB₀ : B ≠ ∅) :\n ∃ a₀ b₀, a₀ ∈ A ∧ b₀ ∈ B ∧ ∀ (a : M) (b : M),\n a ∈ A → b ∈ B → dist a₀ b₀ ≤ dist a b :=", "goal": "M : Type u_1\ninst✝ : MetricSpace M\nA B : Set M\nhA : IsCompact A\nhB : IsCompact B\nhAB : Disjoint A B\nhA₀ : A ≠ ∅\nhB₀ : B ≠ ∅\n⊢ ∃ a₀ b₀, a₀ ∈ A ∧ b₀ ∈ B ∧ ∀ (a b : M), a ∈ A → b ∈ B → dist a₀ b₀ ≤ dist a b", "header": "import Mathlib\n\nopen Filter Real Function\nopen scoped Topology\n\n"}
{"name": "exercise_2_57", "split": "valid", "informal_prefix": "/-- Show that if $S$ is connected, it is not true in general that its interior is connected.-/\n", "formal_statement": "theorem exercise_2_57 {X : Type*} [TopologicalSpace X]\n : ∃ (S : Set X), IsConnected S ∧ ¬ IsConnected (interior S) :=", "goal": "X : Type u_1\ninst✝ : TopologicalSpace X\n⊢ ∃ S, IsConnected S ∧ ¬IsConnected (interior S)", "header": "import Mathlib\n\nopen Filter Real Function\nopen scoped Topology\n\n"}
{"name": "exercise_2_92", "split": "test", "informal_prefix": "/-- Give a direct proof that the nested decreasing intersection of nonempty covering compact sets is nonempty.-/\n", "formal_statement": "theorem exercise_2_92 {α : Type*} [TopologicalSpace α]\n {s : ℕ → Set α}\n (hs : ∀ i, IsCompact (s i))\n (hs : ∀ i, (s i).Nonempty)\n (hs : ∀ i, (s i) ⊃ (s (i + 1))) :\n (⋂ i, s i).Nonempty :=", "goal": "α : Type u_1\ninst✝ : TopologicalSpace α\ns : ℕ → Set α\nhs✝¹ : ∀ (i : ℕ), IsCompact (s i)\nhs✝ : ∀ (i : ℕ), (s i).Nonempty\nhs : ∀ (i : ℕ), s i ⊃ s (i + 1)\n⊢ (⋂ i, s i).Nonempty", "header": "import Mathlib\n\nopen Filter Real Function\nopen scoped Topology\n\n"}
{"name": "exercise_2_126", "split": "valid", "informal_prefix": "/-- Suppose that $E$ is an uncountable subset of $\\mathbb{R}$. Prove that there exists a point $p \\in \\mathbb{R}$ at which $E$ condenses.-/\n", "formal_statement": "theorem exercise_2_126 {E : Set ℝ}\n (hE : ¬ Set.Countable E) : ∃ (p : ℝ), ClusterPt p (𝓟 E) :=", "goal": "E : Set ℝ\nhE : ¬E.Countable\n⊢ ∃ p, ClusterPt p (𝓟 E)", "header": "import Mathlib\n\nopen Filter Real Function\nopen scoped Topology\n\n"}
{"name": "exercise_3_1", "split": "test", "informal_prefix": "/-- Assume that $f \\colon \\mathbb{R} \\rightarrow \\mathbb{R}$ satisfies $|f(t)-f(x)| \\leq|t-x|^{2}$ for all $t, x$. Prove that $f$ is constant.-/\n", "formal_statement": "theorem exercise_3_1 {f : ℝ → ℝ}\n (hf : ∀ x y, |f x - f y| ≤ |x - y| ^ 2) :\n ∃ c, f = λ x => c :=", "goal": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), |f x - f y| ≤ |x - y| ^ 2\n⊢ ∃ c, f = fun x => c", "header": "import Mathlib\n\nopen Filter Real Function\nopen scoped Topology\n\n"}
{"name": "exercise_3_4", "split": "valid", "informal_prefix": "/-- Prove that $\\sqrt{n+1}-\\sqrt{n} \\rightarrow 0$ as $n \\rightarrow \\infty$.-/\n", "formal_statement": "theorem exercise_3_4 (n : ℕ) :\n Tendsto (λ n => (sqrt (n + 1) - sqrt n)) atTop (𝓝 0) :=", "goal": "n : ℕ\n⊢ Tendsto (fun n => √(n + 1) - √n) atTop (𝓝 0)", "header": "import Mathlib\n\nopen Filter Real Function\nopen scoped Topology\n\n"}
{"name": "exercise_3_63a", "split": "test", "informal_prefix": "/-- Prove that $\\sum 1/k(\\log(k))^p$ converges when $p > 1$.-/\n", "formal_statement": "theorem exercise_3_63a (p : ℝ) (f : ℕ → ℝ) (hp : p > 1)\n (h : f = λ (k : ℕ) => (1 : ℝ) / (k * (log k) ^ p)) :\n ∃ l, Tendsto f atTop (𝓝 l) :=", "goal": "p : ℝ\nf : ℕ → ℝ\nhp : p > 1\nh : f = fun k => 1 / (↑k * (↑k).log ^ p)\n⊢ ∃ l, Tendsto f atTop (𝓝 l)", "header": "import Mathlib\n\nopen Filter Real Function\nopen scoped Topology\n\n"}
{"name": "exercise_3_63b", "split": "valid", "informal_prefix": "/-- Prove that $\\sum 1/k(\\log(k))^p$ diverges when $p \\leq 1$.-/\n", "formal_statement": "theorem exercise_3_63b (p : ℝ) (f : ℕ → ℝ) (hp : p ≤ 1)\n (h : f = λ (k : ℕ) => (1 : ℝ) / (k * (log k) ^ p)) :\n ¬ ∃ l, Tendsto f atTop (𝓝 l) :=", "goal": "p : ℝ\nf : ℕ → ℝ\nhp : p ≤ 1\nh : f = fun k => 1 / (↑k * (↑k).log ^ p)\n⊢ ¬∃ l, Tendsto f atTop (𝓝 l)", "header": "import Mathlib\n\nopen Filter Real Function\nopen scoped Topology\n\n"}
{"name": "exercise_4_15a", "split": "test", "informal_prefix": "/-- A continuous, strictly increasing function $\\mu \\colon (0, \\infty) \\rightarrow (0, \\infty)$ is a modulus of continuity if $\\mu(s) \\rightarrow 0$ as $s \\rightarrow 0$. A function $f \\colon [a, b] \\rightarrow \\mathbb{R}$ has modulus of continuity $\\mu$ if $|f(s) - f(t)| \\leq \\mu(|s - t|)$ for all $s, t \\in [a, b]$. Prove that a function is uniformly continuous if and only if it has a modulus of continuity.-/\n", "formal_statement": "theorem exercise_4_15a {α : Type*}\n (a b : ℝ) (F : Set (ℝ → ℝ)) :\n (∀ x : ℝ, ∀ ε > 0, ∃ U ∈ (𝓝 x),\n (∀ y z : U, ∀ f : ℝ → ℝ, f ∈ F → (dist (f y) (f z) < ε)))\n ↔\n ∃ (μ : ℝ → ℝ), ∀ (x : ℝ), (0 : ℝ) ≤ μ x ∧ Tendsto μ (𝓝 0) (𝓝 0) ∧\n (∀ (s t : ℝ) (f : ℝ → ℝ), f ∈ F → |(f s) - (f t)| ≤ μ (|s - t|)) :=", "goal": "α : Type u_1\na b : ℝ\nF : Set (ℝ → ℝ)\n⊢ (∀ (x ε : ℝ), ε > 0 → ∃ U ∈ 𝓝 x, ∀ (y z : ↑U), ∀ f ∈ F, dist (f ↑y) (f ↑z) < ε) ↔\n ∃ μ, ∀ (x : ℝ), 0 ≤ μ x ∧ Tendsto μ (𝓝 0) (𝓝 0) ∧ ∀ (s t : ℝ), ∀ f ∈ F, |f s - f t| ≤ μ |s - t|", "header": "import Mathlib\n\nopen Filter Real Function\nopen scoped Topology\n\n"}
{"name": "exercise_2_2_9", "split": "valid", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_2_2_9 {G : Type} [Group G] {a b : G} (h : a * b = b * a) :\n ∀ x y : closure {x| x = a ∨ x = b}, x * y = y * x :=", "goal": "G : Type\ninst✝ : Group G\na b : G\nh : a * b = b * a\n⊢ ∀ (x y : ↥(Subgroup.closure {x | x = a ∨ x = b})), x * y = y * x", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_3_2", "split": "test", "informal_prefix": "/-- Prove that the products $a b$ and $b a$ are conjugate elements in a group.-/\n", "formal_statement": "theorem exercise_2_3_2 {G : Type*} [Group G] (a b : G) :\n ∃ g : G, b* a = g * a * b * g⁻¹ :=", "goal": "G : Type u_1\ninst✝ : Group G\na b : G\n⊢ ∃ g, b * a = g * a * b * g⁻¹", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_4_19", "split": "valid", "informal_prefix": "/-- Prove that if a group contains exactly one element of order 2 , then that element is in the center of the group.-/\n", "formal_statement": "theorem exercise_2_4_19 {G : Type*} [Group G] {x : G}\n (hx : orderOf x = 2) (hx1 : ∀ y, orderOf y = 2 → y = x) :\n x ∈ center G :=", "goal": "G : Type u_1\ninst✝ : Group G\nx : G\nhx : orderOf x = 2\nhx1 : ∀ (y : G), orderOf y = 2 → y = x\n⊢ x ∈ center G", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_8_6", "split": "test", "informal_prefix": "/-- Prove that the center of the product of two groups is the product of their centers.-/\n", "formal_statement": "noncomputable def exercise_2_8_6 {G H : Type*} [Group G] [Group H] :\n center (G × H) ≃* (center G) × (center H) :=", "goal": "G : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\n⊢ ↥(center (G × H)) ≃* ↥(center G) × ↥(center H)", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n--center of (G × H) equivalent, preserves multiplication with (center G) × (center H)\n"}
{"name": "exercise_2_11_3", "split": "valid", "informal_prefix": "/-- Prove that a group of even order contains an element of order $2 .$-/\n", "formal_statement": "theorem exercise_2_11_3 {G : Type*} [Group G] [Fintype G](hG : Even (card G)) :\n ∃ x : G, orderOf x = 2 :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\nhG : Even (card G)\n⊢ ∃ x, orderOf x = 2", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_2_7", "split": "test", "informal_prefix": "/-- Prove that every homomorphism of fields is injective.-/\n", "formal_statement": "theorem exercise_3_2_7 {F : Type*} [Field F] {G : Type*} [Field G]\n (φ : F →+* G) : Injective φ :=", "goal": "F : Type u_1\ninst✝¹ : Field F\nG : Type u_2\ninst✝ : Field G\nφ : F →+* G\n⊢ Injective ⇑φ", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\nopen RingHom\n"}
{"name": "exercise_3_5_6", "split": "valid", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_3_5_6 {K V : Type*} [Field K] [AddCommGroup V]\n [Module K V] {S : Set V} (hS : Set.Countable S)\n (hS1 : span K S = ⊤) {ι : Type*} (R : ι → V)\n (hR : LinearIndependent K R) : Countable ι :=", "goal": "K : Type u_1\nV : Type u_2\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nS : Set V\nhS : S.Countable\nhS1 : Submodule.span K S = ⊤\nι : Type u_3\nR : ι → V\nhR : LinearIndependent K R\n⊢ Countable ι", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_7_2", "split": "test", "informal_prefix": "/-- Let $V$ be a vector space over an infinite field $F$. Prove that $V$ is not the union of finitely many proper subspaces.-/\n", "formal_statement": "theorem exercise_3_7_2 {K V : Type*} [Field K] [AddCommGroup V]\n [Module K V] {ι : Type*} [Fintype ι] (γ : ι → Submodule K V)\n (h : ∀ i : ι, γ i ≠ ⊤) :\n (⋂ (i : ι), (γ i : Set V)) ≠ ⊤ :=", "goal": "K : Type u_1\nV : Type u_2\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_3\ninst✝ : Fintype ι\nγ : ι → Submodule K V\nh : ∀ (i : ι), γ i ≠ ⊤\n⊢ ⋂ i, ↑(γ i) ≠ ⊤", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_6_1_14", "split": "valid", "informal_prefix": "/-- Let $Z$ be the center of a group $G$. Prove that if $G / Z$ is a cyclic group, then $G$ is abelian and hence $G=Z$.-/\n", "formal_statement": "theorem exercise_6_1_14 (G : Type*) [Group G]\n (hG : IsCyclic $ G ⧸ (center G)) :\n center G = ⊤ :=", "goal": "G : Type u_1\ninst✝ : Group G\nhG : IsCyclic (G ⧸ center G)\n⊢ center G = ⊤", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_6_4_2", "split": "test", "informal_prefix": "/-- Prove that no group of order $p q$, where $p$ and $q$ are prime, is simple.-/\n", "formal_statement": "theorem exercise_6_4_2 {G : Type*} [Group G] [Fintype G] {p q : ℕ}\n (hp : Prime p) (hq : Prime q) (hG : card G = p*q) :\n IsSimpleGroup G → false :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\np q : ℕ\nhp : Prime p\nhq : Prime q\nhG : card G = p * q\n⊢ IsSimpleGroup G → false = true", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_6_4_3", "split": "valid", "informal_prefix": "/-- Prove that no group of order $p^2 q$, where $p$ and $q$ are prime, is simple.-/\n", "formal_statement": "theorem exercise_6_4_3 {G : Type*} [Group G] [Fintype G] {p q : ℕ}\n (hp : Prime p) (hq : Prime q) (hG : card G = p^2 *q) :\n IsSimpleGroup G → false :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\np q : ℕ\nhp : Prime p\nhq : Prime q\nhG : card G = p ^ 2 * q\n⊢ IsSimpleGroup G → false = true", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_6_4_12", "split": "test", "informal_prefix": "/-- Prove that no group of order 224 is simple.-/\n", "formal_statement": "theorem exercise_6_4_12 {G : Type*} [Group G] [Fintype G]\n (hG : card G = 224) :\n IsSimpleGroup G → false :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\nhG : card G = 224\n⊢ IsSimpleGroup G → false = true", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_6_8_1", "split": "valid", "informal_prefix": "/-- Prove that two elements $a, b$ of a group generate the same subgroup as $b a b^2, b a b^3$.-/\n", "formal_statement": "theorem exercise_6_8_1 {G : Type*} [Group G]\n (a b : G) : closure ({a, b} : Set G) = Subgroup.closure {b*a*b^2, b*a*b^3} :=", "goal": "G : Type u_1\ninst✝ : Group G\na b : G\n⊢ Subgroup.closure {a, b} = Subgroup.closure {b * a * b ^ 2, b * a * b ^ 3}", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_10_1_13", "split": "test", "informal_prefix": "/-- An element $x$ of a ring $R$ is called nilpotent if some power of $x$ is zero. Prove that if $x$ is nilpotent, then $1+x$ is a unit in $R$.-/\n", "formal_statement": "theorem exercise_10_1_13 {R : Type*} [Ring R] {x : R}\n (hx : IsNilpotent x) : IsUnit (1 + x) :=", "goal": "R : Type u_1\ninst✝ : Ring R\nx : R\nhx : IsNilpotent x\n⊢ IsUnit (1 + x)", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_10_2_4", "split": "valid", "informal_prefix": "/-- Prove that in the ring $\\mathbb{Z}[x],(2) \\cap(x)=(2 x)$.-/\n", "formal_statement": "theorem exercise_10_2_4 :\n span ({2} : Set $ Polynomial ℤ) ⊓ (span {X}) =\n span ({2 * X} : Set $ Polynomial ℤ) :=", "goal": "⊢ Ideal.span {2} ⊓ Ideal.span {X} = Ideal.span {2 * X}", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_10_6_7", "split": "test", "informal_prefix": "/-- Prove that every nonzero ideal in the ring of Gauss integers contains a nonzero integer.-/\n", "formal_statement": "theorem exercise_10_6_7 {I : Ideal GaussianInt}\n (hI : I ≠ ⊥) : ∃ (z : I), z ≠ 0 ∧ (z : GaussianInt).im = 0 :=", "goal": "I : Ideal GaussianInt\nhI : I ≠ ⊥\n⊢ ∃ z, z ≠ 0 ∧ (↑z).im = 0", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_10_4_6", "split": "valid", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_10_4_6 {R : Type*} [CommRing R]\n [NoZeroDivisors R] (I J : Ideal R) (x : ↑(I ⊓ J)) :\n IsNilpotent ((Ideal.Quotient.mk (I*J)) x) :=", "goal": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\nI J : Ideal R\nx : ↥(I ⊓ J)\n⊢ IsNilpotent ((Ideal.Quotient.mk (I * J)) ↑x)", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_10_4_7a", "split": "test", "informal_prefix": "/-- Let $I, J$ be ideals of a ring $R$ such that $I+J=R$. Prove that $I J=I \\cap J$.-/\n", "formal_statement": "theorem exercise_10_4_7a {R : Type*} [CommRing R] [NoZeroDivisors R]\n (I J : Ideal R) (hIJ : I + J = ⊤) : I * J = I ⊓ J :=", "goal": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\nI J : Ideal R\nhIJ : I + J = ⊤\n⊢ I * J = I ⊓ J", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_10_7_10", "split": "valid", "informal_prefix": "/-- 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 $R$.-/\n", "formal_statement": "theorem exercise_10_7_10 {R : Type*} [Ring R]\n (M : Ideal R) (hM : ∀ (x : R), x ∉ M → IsUnit x)\n (hProper : ∃ x : R, x ∉ M) :\n IsMaximal M ∧ ∀ (N : Ideal R), IsMaximal N → N = M :=", "goal": "R : Type u_1\ninst✝ : Ring R\nM : Ideal R\nhM : ∀ x ∉ M, IsUnit x\nhProper : ∃ x, x ∉ M\n⊢ M.IsMaximal ∧ ∀ (N : Ideal R), N.IsMaximal → N = M", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_11_2_13", "split": "test", "informal_prefix": "/-- If $a, b$ are integers and if $a$ divides $b$ in the ring of Gauss integers, then $a$ divides $b$ in $\\mathbb{Z}$.-/\n", "formal_statement": "theorem exercise_11_2_13 (a b : ℤ) :\n (ofInt a : GaussianInt) ∣ ofInt b → a ∣ b :=", "goal": "a b : ℤ\n⊢ ofInt a ∣ ofInt b → a ∣ b", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_11_4_1b", "split": "valid", "informal_prefix": "/-- Prove that $x^3 + 6x + 12$ is irreducible in $\\mathbb{Q}$.-/\n", "formal_statement": "theorem exercise_11_4_1b {F : Type*} [Field F] [Fintype F] (hF : card F = 2) :\n Irreducible (12 + 6 * X + X ^ 3 : Polynomial F) :=", "goal": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : card F = 2\n⊢ Irreducible (12 + 6 * X + X ^ 3)", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_11_4_6a", "split": "test", "informal_prefix": "/-- Prove that $x^2+x+1$ is irreducible in the field $\\mathbb{F}_2$.-/\n", "formal_statement": "theorem exercise_11_4_6a {F : Type*} [Field F] [Fintype F] (hF : card F = 7) :\n Irreducible (X ^ 2 + 1 : Polynomial F) :=", "goal": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : card F = 7\n⊢ Irreducible (X ^ 2 + 1)", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_11_4_6b", "split": "valid", "informal_prefix": "/-- Prove that $x^2+1$ is irreducible in $\\mathbb{F}_7$-/\n", "formal_statement": "theorem exercise_11_4_6b {F : Type*} [Field F] [Fintype F] (hF : card F = 31) :\n Irreducible (X ^ 3 - 9 : Polynomial F) :=", "goal": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : card F = 31\n⊢ Irreducible (X ^ 3 - 9)", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_11_4_6c", "split": "test", "informal_prefix": "/-- Prove that $x^3 - 9$ is irreducible in $\\mathbb{F}_{31}$.-/\n", "formal_statement": "theorem exercise_11_4_6c : Irreducible (X^3 - 9 : Polynomial (ZMod 31)) :=", "goal": "⊢ Irreducible (X ^ 3 - 9)", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_11_4_8", "split": "valid", "informal_prefix": "/-- Let $p$ be a prime integer. Prove that the polynomial $x^n-p$ is irreducible in $\\mathbb{Q}[x]$.-/\n", "formal_statement": "theorem exercise_11_4_8 (p : ℕ) (hp : Prime p) (n : ℕ) :\n -- p ∈ ℕ can be written as p ∈ ℚ[X]\n Irreducible (X ^ n - (p : Polynomial ℚ) : Polynomial ℚ) :=", "goal": "p : ℕ\nhp : Prime p\nn : ℕ\n⊢ Irreducible (X ^ n - ↑p)", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_11_13_3", "split": "test", "informal_prefix": "/-- Prove that there are infinitely many primes congruent to $-1$ (modulo $4$).-/\n", "formal_statement": "theorem exercise_11_13_3 (N : ℕ):\n ∃ p ≥ N, Nat.Prime p ∧ p + 1 ≡ 0 [MOD 4] :=", "goal": "N : ℕ\n⊢ ∃ p ≥ N, p.Prime ∧ p + 1 ≡ 0 [MOD 4]", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_13_4_10", "split": "valid", "informal_prefix": "/-- Prove that if a prime integer $p$ has the form $2^r+1$, then it actually has the form $2^{2^k}+1$.-/\n", "formal_statement": "theorem exercise_13_4_10\n {p : ℕ} {hp : Nat.Prime p} (h : ∃ r : ℕ, p = 2 ^ r + 1) :\n ∃ (k : ℕ), p = 2 ^ (2 ^ k) + 1 :=", "goal": "p : ℕ\nhp : p.Prime\nh : ∃ r, p = 2 ^ r + 1\n⊢ ∃ k, p = 2 ^ 2 ^ k + 1", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_13_6_10", "split": "test", "informal_prefix": "/-- Let $K$ be a finite field. Prove that the product of the nonzero elements of $K$ is $-1$.-/\n", "formal_statement": "theorem exercise_13_6_10 {K : Type*} [Field K] [Fintype Kˣ] :\n (∏ x : Kˣ, x) = -1 :=", "goal": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : Fintype Kˣ\n⊢ ∏ x : Kˣ, x = -1", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_2", "split": "test", "informal_prefix": "/-- Show that $\\frac{-1 + \\sqrt{3}i}{2}$ is a cube root of 1 (meaning that its cube equals 1).-/\n", "formal_statement": "theorem exercise_1_2 :\n (⟨-1/2, Real.sqrt 3 / 2⟩ : ℂ) ^ 3 = -1 :=", "goal": "⊢ { re := -1 / 2, im := √3 / 2 } ^ 3 = -1", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_3", "split": "valid", "informal_prefix": "/-- Prove that $-(-v) = v$ for every $v \\in V$.-/\n", "formal_statement": "theorem exercise_1_3 {F V : Type*} [AddCommGroup V] [Field F]\n [Module F V] {v : V} : -(-v) = v :=", "goal": "F : Type u_1\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Field F\ninst✝ : Module F V\nv : V\n⊢ - -v = v", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_4", "split": "test", "informal_prefix": "/-- Prove that if $a \\in \\mathbf{F}$, $v \\in V$, and $av = 0$, then $a = 0$ or $v = 0$.-/\n", "formal_statement": "theorem exercise_1_4 {F V : Type*} [AddCommGroup V] [Field F]\n [Module F V] (v : V) (a : F): a • v = 0 ↔ a = 0 ∨ v = 0 :=", "goal": "F : Type u_1\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Field F\ninst✝ : Module F V\nv : V\na : F\n⊢ a • v = 0 ↔ a = 0 ∨ v = 0", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_6", "split": "valid", "informal_prefix": "/-- 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 $\\mathbf{R}^2$.-/\n", "formal_statement": "theorem exercise_1_6 : ∃ U : Set (ℝ × ℝ),\n (U ≠ ∅) ∧\n (∀ (u v : ℝ × ℝ), u ∈ U ∧ v ∈ U → u + v ∈ U) ∧\n (∀ (u : ℝ × ℝ), u ∈ U → -u ∈ U) ∧\n (∀ U' : Submodule ℝ (ℝ × ℝ), U ≠ ↑U') :=", "goal": "⊢ ∃ U, U ≠ ∅ ∧ (∀ (u v : ℝ × ℝ), u ∈ U ∧ v ∈ U → u + v ∈ U) ∧ (∀ u ∈ U, -u ∈ U) ∧ ∀ (U' : Submodule ℝ (ℝ × ℝ)), U ≠ ↑U'", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_7", "split": "test", "informal_prefix": "/-- Give an example of a nonempty subset $U$ of $\\mathbf{R}^2$ such that $U$ is closed under scalar multiplication, but $U$ is not a subspace of $\\mathbf{R}^2$.-/\n", "formal_statement": "theorem exercise_1_7 : ∃ U : Set (ℝ × ℝ),\n (U ≠ ∅) ∧\n (∀ (c : ℝ) (u : ℝ × ℝ), u ∈ U → c • u ∈ U) ∧\n (∀ U' : Submodule ℝ (ℝ × ℝ), U ≠ ↑U') :=", "goal": "⊢ ∃ U, U ≠ ∅ ∧ (∀ (c : ℝ), ∀ u ∈ U, c • u ∈ U) ∧ ∀ (U' : Submodule ℝ (ℝ × ℝ)), U ≠ ↑U'", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_8", "split": "valid", "informal_prefix": "/-- Prove that the intersection of any collection of subspaces of $V$ is a subspace of $V$.-/\n", "formal_statement": "theorem exercise_1_8 {F V : Type*} [AddCommGroup V] [Field F]\n [Module F V] {ι : Type*} (u : ι → Submodule F V) :\n ∃ U : Submodule F V, (⋂ (i : ι), (u i).carrier) = ↑U :=", "goal": "F : Type u_1\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Field F\ninst✝ : Module F V\nι : Type u_3\nu : ι → Submodule F V\n⊢ ∃ U, ⋂ i, (u i).carrier = ↑U", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_9", "split": "test", "informal_prefix": "/-- Prove that the union of two subspaces of $V$ is a subspace of $V$ if and only if one of the subspaces is contained in the other.-/\n", "formal_statement": "theorem exercise_1_9 {F V : Type*} [AddCommGroup V] [Field F]\n [Module F V] (U W : Submodule F V):\n ∃ U' : Submodule F V, (U'.carrier = ↑U ∩ ↑W ↔ (U ≤ W ∨ W ≤ U)) :=", "goal": "F : Type u_1\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Field F\ninst✝ : Module F V\nU W : Submodule F V\n⊢ ∃ U', U'.carrier = ↑U ∩ ↑W ↔ U ≤ W ∨ W ≤ U", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_1", "split": "valid", "informal_prefix": "/-- 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 $v \\in V$.-/\n", "formal_statement": "theorem exercise_3_1 {F V : Type*}\n [AddCommGroup V] [Field F] [Module F V] [FiniteDimensional F V]\n (T : V →ₗ[F] V) (hT : finrank F V = 1) :\n ∃ c : F, ∀ v : V, T v = c • v :=", "goal": "F : Type u_1\nV : Type u_2\ninst✝³ : AddCommGroup V\ninst✝² : Field F\ninst✝¹ : Module F V\ninst✝ : FiniteDimensional F V\nT : V →ₗ[F] V\nhT : finrank F V = 1\n⊢ ∃ c, ∀ (v : V), T v = c • v", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_8", "split": "test", "informal_prefix": "/-- Suppose that $V$ is finite dimensional and that $T \\in \\mathcal{L}(V, W)$. Prove that there exists a subspace $U$ of $V$ such that $U \\cap \\operatorname{null} T=\\{0\\}$ and range $T=\\{T u: u \\in U\\}$.-/\n", "formal_statement": "theorem exercise_3_8 {F V W : Type*} [AddCommGroup V]\n [AddCommGroup W] [Field F] [Module F V] [Module F W]\n (L : V →ₗ[F] W) :\n ∃ U : Submodule F V, U ⊓ (ker L) = ⊥ ∧\n (range L = range (domRestrict L U)) :=", "goal": "F : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AddCommGroup W\ninst✝² : Field F\ninst✝¹ : Module F V\ninst✝ : Module F W\nL : V →ₗ[F] W\n⊢ ∃ U, U ⊓ ker L = ⊥ ∧ range L = range (L.domRestrict U)", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_4", "split": "valid", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_4_4 (p : Polynomial ℂ) :\n p.degree = @card (rootSet p ℂ) (rootSetFintype p ℂ) ↔\n Disjoint\n (@card (rootSet (derivative p) ℂ) (rootSetFintype (derivative p) ℂ))\n (@card (rootSet p ℂ) (rootSetFintype p ℂ)) :=", "goal": "p : ℂ[X]\n⊢ p.degree = ↑(card ↑(p.rootSet ℂ)) ↔ Disjoint (card ↑((derivative p).rootSet ℂ)) (card ↑(p.rootSet ℂ))", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_1", "split": "test", "informal_prefix": "/-- Suppose $T \\in \\mathcal{L}(V)$. Prove that if $U_{1}, \\ldots, U_{m}$ are subspaces of $V$ invariant under $T$, then $U_{1}+\\cdots+U_{m}$ is invariant under $T$.-/\n", "formal_statement": "theorem exercise_5_1 {F V : Type*} [AddCommGroup V] [Field F]\n [Module F V] {L : V →ₗ[F] V} {n : ℕ} (U : Fin n → Submodule F V)\n (hU : ∀ i : Fin n, Submodule.map L (U i) = U i) :\n Submodule.map L (∑ i : Fin n, U i : Submodule F V) =\n (∑ i : Fin n, U i : Submodule F V) :=", "goal": "F : Type u_1\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Field F\ninst✝ : Module F V\nL : V →ₗ[F] V\nn : ℕ\nU : Fin n → Submodule F V\nhU : ∀ (i : Fin n), Submodule.map L (U i) = U i\n⊢ Submodule.map L (∑ i : Fin n, U i) = ∑ i : Fin n, U i", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_4", "split": "valid", "informal_prefix": "/-- 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 $\\lambda \\in \\mathbf{F}$.-/\n", "formal_statement": "theorem exercise_5_4 {F V : Type*} [AddCommGroup V] [Field F]\n [Module F V] (S T : V →ₗ[F] V) (hST : S ∘ T = T ∘ S) (c : F):\n Submodule.map S (ker (T - c • LinearMap.id)) = ker (T - c • LinearMap.id) :=", "goal": "F : Type u_1\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Field F\ninst✝ : Module F V\nS T : V →ₗ[F] V\nhST : ⇑S ∘ ⇑T = ⇑T ∘ ⇑S\nc : F\n⊢ Submodule.map S (ker (T - c • LinearMap.id)) = ker (T - c • LinearMap.id)", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_11", "split": "test", "informal_prefix": "/-- Suppose $S, T \\in \\mathcal{L}(V)$. Prove that $S T$ and $T S$ have the same eigenvalues.-/\n", "formal_statement": "theorem exercise_5_11 {F V : Type*} [AddCommGroup V] [Field F]\n [Module F V] (S T : End F V) :\n (S * T).Eigenvalues = (T * S).Eigenvalues :=", "goal": "F : Type u_1\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Field F\ninst✝ : Module F V\nS T : End F V\n⊢ (S * T).Eigenvalues = (T * S).Eigenvalues", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_12", "split": "valid", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_5_12 {F V : Type*} [AddCommGroup V] [Field F]\n [Module F V] {S : End F V}\n (hS : ∀ v : V, ∃ c : F, v ∈ eigenspace S c) :\n ∃ c : F, S = c • LinearMap.id :=", "goal": "F : Type u_1\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Field F\ninst✝ : Module F V\nS : End F V\nhS : ∀ (v : V), ∃ c, v ∈ S.eigenspace c\n⊢ ∃ c, S = c • LinearMap.id", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_13", "split": "test", "informal_prefix": "/-- Suppose $T \\in \\mathcal{L}(V)$ is such that every subspace of $V$ with dimension $\\operatorname{dim} V-1$ is invariant under $T$. Prove that $T$ is a scalar multiple of the identity operator.-/\n", "formal_statement": "theorem exercise_5_13 {F V : Type*} [AddCommGroup V] [Field F]\n [Module F V] [FiniteDimensional F V] {T : End F V}\n (hS : ∀ U : Submodule F V, finrank F U = finrank F V - 1 →\n Submodule.map T U = U) : ∃ c : F, T = c • LinearMap.id :=", "goal": "F : Type u_1\nV : Type u_2\ninst✝³ : AddCommGroup V\ninst✝² : Field F\ninst✝¹ : Module F V\ninst✝ : FiniteDimensional F V\nT : End F V\nhS : ∀ (U : Submodule F V), finrank F ↥U = finrank F V - 1 → Submodule.map T U = U\n⊢ ∃ c, T = c • LinearMap.id", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_20", "split": "valid", "informal_prefix": "/-- 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 $S T=T S$.-/\n", "formal_statement": "theorem exercise_5_20 {F V : Type*} [AddCommGroup V] [Field F]\n [Module F V] [FiniteDimensional F V] {S T : End F V}\n (h1 : card (T.Eigenvalues) = finrank F V)\n (h2 : ∀ v : V, ∃ c : F, v ∈ eigenspace S c ↔ ∃ c : F, v ∈ eigenspace T c) :\n S * T = T * S :=", "goal": "F : Type u_1\nV : Type u_2\ninst✝³ : AddCommGroup V\ninst✝² : Field F\ninst✝¹ : Module F V\ninst✝ : FiniteDimensional F V\nS T : End F V\nh1 : card T.Eigenvalues = finrank F V\nh2 : ∀ (v : V), ∃ c, v ∈ S.eigenspace c ↔ ∃ c, v ∈ T.eigenspace c\n⊢ S * T = T * S", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_24", "split": "test", "informal_prefix": "/-- Suppose $V$ is a real vector space and $T \\in \\mathcal{L}(V)$ has no eigenvalues. Prove that every subspace of $V$ invariant under $T$ has even dimension.-/\n", "formal_statement": "theorem exercise_5_24 {V : Type*} [AddCommGroup V]\n [Module ℝ V] [FiniteDimensional ℝ V] {T : End ℝ V}\n (hT : ∀ c : ℝ, eigenspace T c = ⊥) {U : Submodule ℝ V}\n (hU : Submodule.map T U = U) : Even (finrank U) :=", "goal": "V : Type u_1\ninst✝² : AddCommGroup V\ninst✝¹ : Module ℝ V\ninst✝ : FiniteDimensional ℝ V\nT : End ℝ V\nhT : ∀ (c : ℝ), T.eigenspace c = ⊥\nU : Submodule ℝ V\nhU : Submodule.map T U = U\n⊢ Even (finrank ↥U)", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_6_2", "split": "valid", "informal_prefix": "/-- Suppose $u, v \\in V$. Prove that $\\langle u, v\\rangle=0$ if and only if $\\|u\\| \\leq\\|u+a v\\|$ for all $a \\in \\mathbf{F}$.-/\n", "formal_statement": "theorem exercise_6_2 {V : Type*} [NormedAddCommGroup V] [Module ℂ V]\n[InnerProductSpace ℂ V] (u v : V) :\n ⟪u, v⟫_ℂ = 0 ↔ ∀ (a : ℂ), ‖u‖ ≤ ‖u + a • v‖ :=", "goal": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : Module ℂ V\ninst✝ : InnerProductSpace ℂ V\nu v : V\n⊢ ⟪u, v⟫_ℂ = 0 ↔ ∀ (a : ℂ), ‖u‖ ≤ ‖u + a • v‖", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_6_3", "split": "test", "informal_prefix": "/-- Prove that $\\left(\\sum_{j=1}^{n} a_{j} b_{j}\\right)^{2} \\leq\\left(\\sum_{j=1}^{n} j a_{j}{ }^{2}\\right)\\left(\\sum_{j=1}^{n} \\frac{b_{j}{ }^{2}}{j}\\right)$ for all real numbers $a_{1}, \\ldots, a_{n}$ and $b_{1}, \\ldots, b_{n}$.-/\n", "formal_statement": "theorem exercise_6_3 {n : ℕ} (a b : Fin n → ℝ) :\n (∑ i, a i * b i) ^ 2 ≤ (∑ i : Fin n, i * a i ^ 2) * (∑ i, b i ^ 2 / i) :=", "goal": "n : ℕ\na b : Fin n → ℝ\n⊢ (∑ i : Fin n, a i * b i) ^ 2 ≤ (∑ i : Fin n, ↑↑i * a i ^ 2) * ∑ i : Fin n, b i ^ 2 / ↑↑i", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_6_7", "split": "valid", "informal_prefix": "/-- 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 $u, v \\in V$.-/\n", "formal_statement": "theorem exercise_6_7 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] (u v : V) :\n ⟪u, v⟫_ℂ = (‖u + v‖^2 - ‖u - v‖^2 + I*‖u + I•v‖^2 - I*‖u-I•v‖^2) / 4 :=", "goal": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℂ V\nu v : V\n⊢ ⟪u, v⟫_ℂ = (↑‖u + v‖ ^ 2 - ↑‖u - v‖ ^ 2 + I * ↑‖u + I • v‖ ^ 2 - I * ↑‖u - I • v‖ ^ 2) / 4", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_6_13", "split": "test", "informal_prefix": "/-- Suppose $\\left(e_{1}, \\ldots, e_{m}\\right)$ is an or thonormal list of vectors in $V$. Let $v \\in V$. Prove that $\\|v\\|^{2}=\\left|\\left\\langle v, e_{1}\\right\\rangle\\right|^{2}+\\cdots+\\left|\\left\\langle v, e_{m}\\right\\rangle\\right|^{2}$ if and only if $v \\in \\operatorname{span}\\left(e_{1}, \\ldots, e_{m}\\right)$.-/\n", "formal_statement": "theorem exercise_6_13 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] {n : ℕ}\n {e : Fin n → V} (he : Orthonormal ℂ e) (v : V) :\n ‖v‖^2 = ∑ i : Fin n, ‖⟪v, e i⟫_ℂ‖^2 ↔ v ∈ Submodule.span ℂ (e '' Set.univ) :=", "goal": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℂ V\nn : ℕ\ne : Fin n → V\nhe : Orthonormal ℂ e\nv : V\n⊢ ‖v‖ ^ 2 = ∑ i : Fin n, ‖⟪v, e i⟫_ℂ‖ ^ 2 ↔ v ∈ Submodule.span ℂ (e '' Set.univ)", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_6_16", "split": "valid", "informal_prefix": "/-- Suppose $U$ is a subspace of $V$. Prove that $U^{\\perp}=\\{0\\}$ if and only if $U=V$-/\n", "formal_statement": "theorem exercise_6_16 {K V : Type*} [RCLike K] [NormedAddCommGroup V] [InnerProductSpace K V]\n {U : Submodule K V} :\n U.orthogonal = ⊥ ↔ U = ⊤ :=", "goal": "K : Type u_1\nV : Type u_2\ninst✝² : RCLike K\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace K V\nU : Submodule K V\n⊢ Uᗮ = ⊥ ↔ U = ⊤", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_7_5", "split": "test", "informal_prefix": "/-- Show that if $\\operatorname{dim} V \\geq 2$, then the set of normal operators on $V$ is not a subspace of $\\mathcal{L}(V)$.-/\n", "formal_statement": "theorem exercise_7_5 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]\n [FiniteDimensional ℂ V] (hV : finrank V ≥ 2) :\n ∀ U : Submodule ℂ (End ℂ V), U.carrier ≠\n {T | T * adjoint T = adjoint T * T} :=", "goal": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℂ V\ninst✝ : FiniteDimensional ℂ V\nhV : finrank V ≥ 2\n⊢ ∀ (U : Submodule ℂ (End ℂ V)), U.carrier ≠ {T | T * adjoint T = adjoint T * T}", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_7_6", "split": "valid", "informal_prefix": "/-- Prove that if $T \\in \\mathcal{L}(V)$ is normal, then $\\operatorname{range} T=\\operatorname{range} T^{*}.$-/\n", "formal_statement": "theorem exercise_7_6 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]\n [FiniteDimensional ℂ V] (T : End ℂ V)\n (hT : T * adjoint T = adjoint T * T) :\n range T = range (adjoint T) :=", "goal": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℂ V\ninst✝ : FiniteDimensional ℂ V\nT : End ℂ V\nhT : T * adjoint T = adjoint T * T\n⊢ range T = range (adjoint T)", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_7_9", "split": "test", "informal_prefix": "/-- Prove that a normal operator on a complex inner-product space is self-adjoint if and only if all its eigenvalues are real.-/\n", "formal_statement": "theorem exercise_7_9 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]\n [FiniteDimensional ℂ V] (T : End ℂ V)\n (hT : T * adjoint T = adjoint T * T) :\n IsSelfAdjoint T ↔ ∀ e : T.Eigenvalues, (e : ℂ).im = 0 :=", "goal": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℂ V\ninst✝ : FiniteDimensional ℂ V\nT : End ℂ V\nhT : T * adjoint T = adjoint T * T\n⊢ IsSelfAdjoint T ↔ ∀ (e : T.Eigenvalues), (↑T e).im = 0", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_7_10", "split": "valid", "informal_prefix": "/-- 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 $T^{2}=T$.-/\n", "formal_statement": "theorem exercise_7_10 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]\n [FiniteDimensional ℂ V] (T : End ℂ V)\n (hT : T * adjoint T = adjoint T * T) (hT1 : T^9 = T^8) :\n IsSelfAdjoint T ∧ T^2 = T :=", "goal": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℂ V\ninst✝ : FiniteDimensional ℂ V\nT : End ℂ V\nhT : T * adjoint T = adjoint T * T\nhT1 : T ^ 9 = T ^ 8\n⊢ IsSelfAdjoint T ∧ T ^ 2 = T", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_7_11", "split": "test", "informal_prefix": "/-- Suppose $V$ is a complex inner-product space. Prove that every normal operator on $V$ has a square root. (An operator $S \\in \\mathcal{L}(V)$ is called a square root of $T \\in \\mathcal{L}(V)$ if $S^{2}=T$.)-/\n", "formal_statement": "theorem exercise_7_11 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]\n [FiniteDimensional ℂ V] {T : End ℂ V} (hT : T*adjoint T = adjoint T*T) :\n ∃ (S : End ℂ V), S ^ 2 = T :=", "goal": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℂ V\ninst✝ : FiniteDimensional ℂ V\nT : End ℂ V\nhT : T * adjoint T = adjoint T * T\n⊢ ∃ S, S ^ 2 = T", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_7_14", "split": "valid", "informal_prefix": "/-- 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 $\\left|\\lambda-\\lambda^{\\prime}\\right|<\\epsilon$.-/\n", "formal_statement": "theorem exercise_7_14 {𝕜 V : Type*} [RCLike 𝕜] [NormedAddCommGroup V]\n [InnerProductSpace 𝕜 V] [FiniteDimensional 𝕜 V]\n {T : Module.End 𝕜 V} (hT : IsSelfAdjoint T)\n {l : 𝕜} {ε : ℝ} (he : ε > 0) : ∃ v : V, ‖v‖= 1 ∧ (‖T v - l • v‖ < ε →\n (∃ l' : T.Eigenvalues, ‖l - l'‖ < ε)) :=", "goal": "𝕜 : Type u_1\nV : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace 𝕜 V\ninst✝ : FiniteDimensional 𝕜 V\nT : End 𝕜 V\nhT : IsSelfAdjoint T\nl : 𝕜\nε : ℝ\nhe : ε > 0\n⊢ ∃ v, ‖v‖ = 1 ∧ (‖T v - l • v‖ < ε → ∃ l', ‖l - ↑T l'‖ < ε)", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_1_2a", "split": "test", "informal_prefix": "/-- Prove the the operation $\\star$ on $\\mathbb{Z}$ defined by $a\\star b=a-b$ is not commutative.-/\n", "formal_statement": "theorem exercise_1_1_2a : ∃ a b : ℤ, a - b ≠ b - a :=", "goal": "⊢ ∃ a b, a - b ≠ b - a", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_1_3", "split": "valid", "informal_prefix": "/-- Prove that the addition of residue classes $\\mathbb{Z}/n\\mathbb{Z}$ is associative.-/\n", "formal_statement": "theorem exercise_1_1_3 (n : ℤ) :\n ∀ (a b c : ℤ), (a+b)+c ≡ a+(b+c) [ZMOD n] :=", "goal": "n : ℤ\n⊢ ∀ (a b c : ℤ), a + b + c ≡ a + (b + c) [ZMOD n]", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_1_4", "split": "test", "informal_prefix": "/-- Prove that the multiplication of residue class $\\mathbb{Z}/n\\mathbb{Z}$ is associative.-/\n", "formal_statement": "theorem exercise_1_1_4 (n : ℕ) :\n ∀ (a b c : ℕ), (a * b) * c ≡ a * (b * c) [ZMOD n] :=", "goal": "n : ℕ\n⊢ ∀ (a b c : ℕ), ↑a * ↑b * ↑c ≡ ↑a * (↑b * ↑c) [ZMOD ↑n]", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_1_5", "split": "valid", "informal_prefix": "/-- Prove that for all $n>1$ that $\\mathbb{Z}/n\\mathbb{Z}$ is not a group under multiplication of residue classes.-/\n", "formal_statement": "theorem exercise_1_1_5 (n : ℕ) (hn : 1 < n) :\n IsEmpty (Group (ZMod n)) :=", "goal": "n : ℕ\nhn : 1 < n\n⊢ IsEmpty (Group (ZMod n))", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_1_15", "split": "test", "informal_prefix": "/-- Prove that $(a_1a_2\\dots a_n)^{-1} = a_n^{-1}a_{n-1}^{-1}\\dots a_1^{-1}$ for all $a_1, a_2, \\dots, a_n\\in G$.-/\n", "formal_statement": "theorem exercise_1_1_15 {G : Type*} [Group G] (as : List G) :\n as.prod⁻¹ = (as.reverse.map (λ x => x⁻¹)).prod :=", "goal": "G : Type u_1\ninst✝ : Group G\nas : List G\n⊢ as.prod⁻¹ = (List.map (fun x => x⁻¹) as.reverse).prod", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_1_16", "split": "valid", "informal_prefix": "/-- Let $x$ be an element of $G$. Prove that $x^2=1$ if and only if $|x|$ is either $1$ or $2$.-/\n", "formal_statement": "theorem exercise_1_1_16 {G : Type*} [Group G]\n (x : G) (hx : x ^ 2 = 1) :\n orderOf x = 1 ∨ orderOf x = 2 :=", "goal": "G : Type u_1\ninst✝ : Group G\nx : G\nhx : x ^ 2 = 1\n⊢ orderOf x = 1 ∨ orderOf x = 2", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_1_17", "split": "test", "informal_prefix": "/-- Let $x$ be an element of $G$. Prove that if $|x|=n$ for some positive integer $n$ then $x^{-1}=x^{n-1}$.-/\n", "formal_statement": "theorem exercise_1_1_17 {G : Type*} [Group G] {x : G} {n : ℕ}\n (hxn: orderOf x = n) :\n x⁻¹ = x ^ (n - 1 : ℤ) :=", "goal": "G : Type u_1\ninst✝ : Group G\nx : G\nn : ℕ\nhxn : orderOf x = n\n⊢ x⁻¹ = x ^ (↑n - 1)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_1_18", "split": "valid", "informal_prefix": "/-- Let $x$ and $y$ be elements of $G$. Prove that $xy=yx$ if and only if $y^{-1}xy=x$ if and only if $x^{-1}y^{-1}xy=1$.-/\n", "formal_statement": "theorem exercise_1_1_18 {G : Type*} [Group G]\n (x y : G) : (x * y = y * x ↔ y⁻¹ * x * y = x) ↔ (x⁻¹ * y⁻¹ * x * y = 1) :=", "goal": "G : Type u_1\ninst✝ : Group G\nx y : G\n⊢ (x * y = y * x ↔ y⁻¹ * x * y = x) ↔ x⁻¹ * y⁻¹ * x * y = 1", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_1_20", "split": "test", "informal_prefix": "/-- For $x$ an element in $G$ show that $x$ and $x^{-1}$ have the same order.-/\n", "formal_statement": "theorem exercise_1_1_20 {G : Type*} [Group G] {x : G} :\n orderOf x = orderOf x⁻¹ :=", "goal": "G : Type u_1\ninst✝ : Group G\nx : G\n⊢ orderOf x = orderOf x⁻¹", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_1_22a", "split": "valid", "informal_prefix": "/-- If $x$ and $g$ are elements of the group $G$, prove that $|x|=\\left|g^{-1} x g\\right|$.-/\n", "formal_statement": "theorem exercise_1_1_22a {G : Type*} [Group G] (x g : G) :\n orderOf x = orderOf (g⁻¹ * x * g) :=", "goal": "G : Type u_1\ninst✝ : Group G\nx g : G\n⊢ orderOf x = orderOf (g⁻¹ * x * g)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_1_22b", "split": "test", "informal_prefix": "/-- Deduce that $|a b|=|b a|$ for all $a, b \\in G$.-/\n", "formal_statement": "theorem exercise_1_1_22b {G: Type*} [Group G] (a b : G) :\n orderOf (a * b) = orderOf (b * a) :=", "goal": "G : Type u_1\ninst✝ : Group G\na b : G\n⊢ orderOf (a * b) = orderOf (b * a)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_1_25", "split": "valid", "informal_prefix": "/-- Prove that if $x^{2}=1$ for all $x \\in G$ then $G$ is abelian.-/\n", "formal_statement": "theorem exercise_1_1_25 {G : Type*} [Group G]\n (h : ∀ x : G, x ^ 2 = 1) : ∀ a b : G, a*b = b*a :=", "goal": "G : Type u_1\ninst✝ : Group G\nh : ∀ (x : G), x ^ 2 = 1\n⊢ ∀ (a b : G), a * b = b * a", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_1_29", "split": "test", "informal_prefix": "/-- Prove that $A \\times B$ is an abelian group if and only if both $A$ and $B$ are abelian.-/\n", "formal_statement": "theorem exercise_1_1_29 {A B : Type*} [Group A] [Group B] :\n ∀ x y : A × B, x*y = y*x ↔ (∀ x y : A, x*y = y*x) ∧\n (∀ x y : B, x*y = y*x) :=", "goal": "A : Type u_1\nB : Type u_2\ninst✝¹ : Group A\ninst✝ : Group B\n⊢ ∀ (x y : A × B), x * y = y * x ↔ (∀ (x y : A), x * y = y * x) ∧ ∀ (x y : B), x * y = y * x", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_1_34", "split": "valid", "informal_prefix": "/-- If $x$ is an element of infinite order in $G$, prove that the elements $x^{n}, n \\in \\mathbb{Z}$ are all distinct.-/\n", "formal_statement": "theorem exercise_1_1_34 {G : Type*} [Group G] {x : G}\n (hx_inf : orderOf x = 0) (n m : ℤ) :\n x ^ n ≠ x ^ m :=", "goal": "G : Type u_1\ninst✝ : Group G\nx : G\nhx_inf : orderOf x = 0\nn m : ℤ\n⊢ x ^ n ≠ x ^ m", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_3_8", "split": "test", "informal_prefix": "/-- Prove that if $\\Omega=\\{1,2,3, \\ldots\\}$ then $S_{\\Omega}$ is an infinite group-/\n", "formal_statement": "theorem exercise_1_3_8 : Infinite (Equiv.Perm ℕ) :=", "goal": "⊢ Infinite (Equiv.Perm ℕ)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_6_4", "split": "valid", "informal_prefix": "/-- Prove that the multiplicative groups $\\mathbb{R}-\\{0\\}$ and $\\mathbb{C}-\\{0\\}$ are not isomorphic.-/\n", "formal_statement": "theorem exercise_1_6_4 :\n IsEmpty (Multiplicative ℝ ≃* Multiplicative ℂ) :=", "goal": "⊢ IsEmpty (Multiplicative ℝ ≃* Multiplicative ℂ)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_6_11", "split": "test", "informal_prefix": "/-- Let $A$ and $B$ be groups. Prove that $A \\times B \\cong B \\times A$.-/\n", "formal_statement": "noncomputable def exercise_1_6_11 {A B : Type*} [Group A] [Group B] :\n A × B ≃* B × A :=", "goal": "A : Type u_1\nB : Type u_2\ninst✝¹ : Group A\ninst✝ : Group B\n⊢ A × B ≃* B × A", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_6_17", "split": "valid", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_1_6_17 {G : Type*} [Group G] (f : G → G)\n (hf : f = λ g => g⁻¹) :\n ∀ x y : G, f x * f y = f (x*y) ↔ ∀ x y : G, x*y = y*x :=", "goal": "G : Type u_1\ninst✝ : Group G\nf : G → G\nhf : f = fun g => g⁻¹\n⊢ ∀ (x y : G), f x * f y = f (x * y) ↔ ∀ (x y : G), x * y = y * x", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_6_23", "split": "test", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_1_6_23 {G : Type*}\n [Group G] (σ : MulAut G) (hs : ∀ g : G, σ g = 1 → g = 1)\n (hs2 : ∀ g : G, σ (σ g) = g) :\n ∀ x y : G, x*y = y*x :=", "goal": "G : Type u_1\ninst✝ : Group G\nσ : MulAut G\nhs : ∀ (g : G), σ g = 1 → g = 1\nhs2 : ∀ (g : G), σ (σ g) = g\n⊢ ∀ (x y : G), x * y = y * x", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_1_5", "split": "valid", "informal_prefix": "/-- Prove that $G$ cannot have a subgroup $H$ with $|H|=n-1$, where $n=|G|>2$.-/\n", "formal_statement": "theorem exercise_2_1_5 {G : Type*} [Group G] [Fintype G]\n (hG : card G > 2) (H : Subgroup G) [Fintype H] :\n card H ≠ card G - 1 :=", "goal": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\nhG : card G > 2\nH : Subgroup G\ninst✝ : Fintype ↥H\n⊢ card ↥H ≠ card G - 1", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_1_13", "split": "test", "informal_prefix": "/-- 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 $\\mathbb{Q}$.-/\n", "formal_statement": "theorem exercise_2_1_13 (H : AddSubgroup ℚ) {x : ℚ}\n (hH : x ∈ H → (1 / x) ∈ H):\n H = ⊥ ∨ H = ⊤ :=", "goal": "H : AddSubgroup ℚ\nx : ℚ\nhH : x ∈ H → 1 / x ∈ H\n⊢ H = ⊥ ∨ H = ⊤", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_4_4", "split": "valid", "informal_prefix": "/-- Prove that if $H$ is a subgroup of $G$ then $H$ is generated by the set $H-\\{1\\}$.-/\n", "formal_statement": "theorem exercise_2_4_4 {G : Type*} [Group G] (H : Subgroup G) :\n closure ((H : Set G) \\ {1}) = ⊤ :=", "goal": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ Subgroup.closure (↑H \\ {1}) = ⊤", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_4_16a", "split": "test", "informal_prefix": "/-- 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 $H$.-/\n", "formal_statement": "theorem exercise_2_4_16a {G : Type*} [Group G] {H : Subgroup G}\n (hH : H ≠ ⊤) :\n ∃ M : Subgroup G, M ≠ ⊤ ∧\n ∀ K : Subgroup G, M ≤ K → K = M ∨ K = ⊤ ∧\n H ≤ M :=", "goal": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhH : H ≠ ⊤\n⊢ ∃ M, M ≠ ⊤ ∧ ∀ (K : Subgroup G), M ≤ K → K = M ∨ K = ⊤ ∧ H ≤ M", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_4_16b", "split": "valid", "informal_prefix": "/-- Show that the subgroup of all rotations in a dihedral group is a maximal subgroup.-/\n", "formal_statement": "theorem exercise_2_4_16b {n : ℕ} {hn : n ≠ 0}\n {R : Subgroup (DihedralGroup n)}\n (hR : R = Subgroup.closure {DihedralGroup.r 1}) :\n R ≠ ⊤ ∧\n ∀ K : Subgroup (DihedralGroup n), R ≤ K → K = R ∨ K = ⊤ :=", "goal": "n : ℕ\nhn : n ≠ 0\nR : Subgroup (DihedralGroup n)\nhR : R = Subgroup.closure {DihedralGroup.r 1}\n⊢ R ≠ ⊤ ∧ ∀ (K : Subgroup (DihedralGroup n)), R ≤ K → K = R ∨ K = ⊤", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_4_16c", "split": "test", "informal_prefix": "/-- 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 $n$.-/\n", "formal_statement": "theorem exercise_2_4_16c {n : ℕ} (H : AddSubgroup (ZMod n)) :\n ∃ p : (ZMod n), Prime p ∧ H = AddSubgroup.closure {p} ↔\n (H ≠ ⊤ ∧ ∀ K : AddSubgroup (ZMod n), H ≤ K → K = H ∨ K = ⊤) :=", "goal": "n : ℕ\nH : AddSubgroup (ZMod n)\n⊢ ∃ p, Prime p ∧ H = AddSubgroup.closure {p} ↔ H ≠ ⊤ ∧ ∀ (K : AddSubgroup (ZMod n)), H ≤ K → K = H ∨ K = ⊤", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_1_3a", "split": "valid", "informal_prefix": "/-- Let $A$ be an abelian group and let $B$ be a subgroup of $A$. Prove that $A / B$ is abelian.-/\n", "formal_statement": "theorem exercise_3_1_3a {A : Type*} [CommGroup A] (B : Subgroup A) :\n ∀ a b : A ⧸ B, a*b = b*a :=", "goal": "A : Type u_1\ninst✝ : CommGroup A\nB : Subgroup A\n⊢ ∀ (a b : A ⧸ B), a * b = b * a", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_1_22a", "split": "test", "informal_prefix": "/-- 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 $G$.-/\n", "formal_statement": "theorem exercise_3_1_22a (G : Type*) [Group G] (H K : Subgroup G)\n [Normal H] [Normal K] :\n Normal (H ⊓ K) :=", "goal": "G : Type u_1\ninst✝² : Group G\nH K : Subgroup G\ninst✝¹ : H.Normal\ninst✝ : K.Normal\n⊢ (H ⊓ K).Normal", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_1_22b", "split": "valid", "informal_prefix": "/-- 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).-/\n", "formal_statement": "theorem exercise_3_1_22b {G : Type*} [Group G] (I : Type*)\n (H : I → Subgroup G) (hH : ∀ i : I, Normal (H i)) :\n Normal (⨅ (i : I), H i) :=", "goal": "G : Type u_1\ninst✝ : Group G\nI : Type u_2\nH : I → Subgroup G\nhH : ∀ (i : I), (H i).Normal\n⊢ (⨅ i, H i).Normal", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_2_8", "split": "test", "informal_prefix": "/-- Prove that if $H$ and $K$ are finite subgroups of $G$ whose orders are relatively prime then $H \\cap K=1$.-/\n", "formal_statement": "theorem exercise_3_2_8 {G : Type*} [Group G] (H K : Subgroup G)\n [Fintype H] [Fintype K]\n (hHK : Nat.Coprime (card H) (card K)) :\n H ⊓ K = ⊥ :=", "goal": "G : Type u_1\ninst✝² : Group G\nH K : Subgroup G\ninst✝¹ : Fintype ↥H\ninst✝ : Fintype ↥K\nhHK : (card ↥H).Coprime (card ↥K)\n⊢ H ⊓ K = ⊥", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_2_11", "split": "valid", "informal_prefix": "/-- Let $H \\leq K \\leq G$. Prove that $|G: H|=|G: K| \\cdot|K: H|$ (do not assume $G$ is finite).-/\n", "formal_statement": "theorem exercise_3_2_11 {G : Type*} [Group G] {H K : Subgroup G}\n (hHK : H ≤ K) :\n H.index = K.index * H.relindex K :=", "goal": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nhHK : H ≤ K\n⊢ H.index = K.index * H.relindex K", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_2_16", "split": "test", "informal_prefix": "/-- 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 $a \\in \\mathbb{Z}$.-/\n", "formal_statement": "theorem exercise_3_2_16 (p : ℕ) (hp : Nat.Prime p) (a : ℕ) :\n Nat.Coprime a p → a ^ p ≡ a [ZMOD p] :=", "goal": "p : ℕ\nhp : p.Prime\na : ℕ\n⊢ a.Coprime p → ↑a ^ p ≡ ↑a [ZMOD ↑p]", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_2_21a", "split": "valid", "informal_prefix": "/-- Prove that $\\mathbb{Q}$ has no proper subgroups of finite index.-/\n", "formal_statement": "theorem exercise_3_2_21a (H : AddSubgroup ℚ) (hH : H ≠ ⊤) : H.index = 0 :=", "goal": "H : AddSubgroup ℚ\nhH : H ≠ ⊤\n⊢ H.index = 0", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_3_3", "split": "test", "informal_prefix": "/-- 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 $|K: K \\cap H|=p$.-/\n", "formal_statement": "theorem exercise_3_3_3 {p : Nat.Primes} {G : Type*} [Group G]\n {H : Subgroup G} [hH : H.Normal] (hH1 : H.index = p) :\n ∀ K : Subgroup G, K ≤ H ∨ H ⊔ K = ⊤ ∨ (K ⊓ H).relindex K = p :=", "goal": "p : Nat.Primes\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhH : H.Normal\nhH1 : H.index = ↑p\n⊢ ∀ (K : Subgroup G), K ≤ H ∨ H ⊔ K = ⊤ ∨ (K ⊓ H).relindex K = ↑p", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_4_1", "split": "valid", "informal_prefix": "/-- 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).-/\n", "formal_statement": "theorem exercise_3_4_1 (G : Type*) [CommGroup G] [IsSimpleGroup G] :\n IsCyclic G ∧ ∃ G_fin : Fintype G, Nat.Prime (@card G G_fin) :=", "goal": "G : Type u_1\ninst✝¹ : CommGroup G\ninst✝ : IsSimpleGroup G\n⊢ IsCyclic G ∧ ∃ G_fin, (card G).Prime", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_4_4", "split": "test", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_3_4_4 {G : Type*} [CommGroup G] [Fintype G] {n : ℕ}\n (hn : n ∣ (card G)) :\n ∃ (H : Subgroup G) (H_fin : Fintype H), @card H H_fin = n :=", "goal": "G : Type u_1\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\nn : ℕ\nhn : n ∣ card G\n⊢ ∃ H H_fin, card ↥H = n", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_4_5a", "split": "valid", "informal_prefix": "/-- Prove that subgroups of a solvable group are solvable.-/\n", "formal_statement": "theorem exercise_3_4_5a {G : Type*} [Group G]\n (H : Subgroup G) [IsSolvable G] : IsSolvable H :=", "goal": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : IsSolvable G\n⊢ IsSolvable ↥H", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_4_5b", "split": "test", "informal_prefix": "/-- Prove that quotient groups of a solvable group are solvable.-/\n", "formal_statement": "theorem exercise_3_4_5b {G : Type*} [Group G] [IsSolvable G]\n (H : Subgroup G) [Normal H] :\n IsSolvable (G ⧸ H) :=", "goal": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : IsSolvable G\nH : Subgroup G\ninst✝ : H.Normal\n⊢ IsSolvable (G ⧸ H)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_4_11", "split": "valid", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_3_4_11 {G : Type*} [Group G] [IsSolvable G]\n {H : Subgroup G} (hH : H ≠ ⊥) [H.Normal] :\n ∃ A ≤ H, A.Normal ∧ ∀ a b : A, a*b = b*a :=", "goal": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : IsSolvable G\nH : Subgroup G\nhH : H ≠ ⊥\ninst✝ : H.Normal\n⊢ ∃ A ≤ H, A.Normal ∧ ∀ (a b : ↥A), a * b = b * a", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_2_8", "split": "test", "informal_prefix": "/-- Prove that if $H$ has finite index $n$ then there is a normal subgroup $K$ of $G$ with $K \\leq H$ and $|G: K| \\leq n!$.-/\n", "formal_statement": "theorem exercise_4_2_8 {G : Type*} [Group G] {H : Subgroup G}\n {n : ℕ} (hn : n > 0) (hH : H.index = n) :\n ∃ K ≤ H, K.Normal ∧ K.index ≤ n.factorial :=", "goal": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nn : ℕ\nhn : n > 0\nhH : H.index = n\n⊢ ∃ K ≤ H, K.Normal ∧ K.index ≤ n.factorial", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_3_26", "split": "valid", "informal_prefix": "/-- 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 $a \\in A$.-/\n", "formal_statement": "theorem exercise_4_3_26 {α : Type*} [Fintype α] (ha : card α > 1)\n (h_tran : ∀ a b: α, ∃ σ : Equiv.Perm α, σ a = b) :\n ∃ σ : Equiv.Perm α, ∀ a : α, σ a ≠ a :=", "goal": "α : Type u_1\ninst✝ : Fintype α\nha : card α > 1\nh_tran : ∀ (a b : α), ∃ σ, σ a = b\n⊢ ∃ σ, ∀ (a : α), σ a ≠ a", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_2_9a", "split": "test", "informal_prefix": "/-- 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 $G$.-/\n", "formal_statement": "theorem exercise_4_2_9a {G : Type*} [Fintype G] [Group G] {p α : ℕ}\n (hp : p.Prime) (ha : α > 0) (hG : card G = p ^ α) :\n ∀ H : Subgroup G, H.index = p → H.Normal :=", "goal": "G : Type u_1\ninst✝¹ : Fintype G\ninst✝ : Group G\np α : ℕ\nhp : p.Prime\nha : α > 0\nhG : card G = p ^ α\n⊢ ∀ (H : Subgroup G), H.index = p → H.Normal", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_2_14", "split": "valid", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_4_2_14 {G : Type*} [Fintype G] [Group G]\n (hG : ¬ (card G).Prime) (hG1 : ∀ k : ℕ, k ∣ card G →\n ∃ (H : Subgroup G) (fH : Fintype H), @card H fH = k) :\n ¬ IsSimpleGroup G :=", "goal": "G : Type u_1\ninst✝¹ : Fintype G\ninst✝ : Group G\nhG : ¬(card G).Prime\nhG1 : ∀ (k : ℕ), k ∣ card G → ∃ H fH, card ↥H = k\n⊢ ¬IsSimpleGroup G", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_4_2", "split": "test", "informal_prefix": "/-- Prove that if $G$ is an abelian group of order $p q$, where $p$ and $q$ are distinct primes, then $G$ is cyclic.-/\n", "formal_statement": "theorem exercise_4_4_2 {G : Type*} [Fintype G] [Group G]\n {p q : Nat.Primes} (hpq : p ≠ q) (hG : card G = p*q) :\n IsCyclic G :=", "goal": "G : Type u_1\ninst✝¹ : Fintype G\ninst✝ : Group G\np q : Nat.Primes\nhpq : p ≠ q\nhG : card G = ↑p * ↑q\n⊢ IsCyclic G", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_4_6a", "split": "valid", "informal_prefix": "/-- Prove that characteristic subgroups are normal.-/\n", "formal_statement": "theorem exercise_4_4_6a {G : Type*} [Group G] (H : Subgroup G)\n [Characteristic H] : Normal H :=", "goal": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : H.Characteristic\n⊢ H.Normal", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_4_6b", "split": "test", "informal_prefix": "/-- Prove that there exists a normal subgroup that is not characteristic.-/\n", "formal_statement": "theorem exercise_4_4_6b :\n ∃ (G : Type*) (hG : Group G) (H : @Subgroup G hG), @Characteristic G hG H ∧ ¬ @Normal G hG H :=", "goal": "⊢ ∃ G hG H, H.Characteristic ∧ ¬H.Normal", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_4_7", "split": "valid", "informal_prefix": "/-- If $H$ is the unique subgroup of a given order in a group $G$ prove $H$ is characteristic in $G$.-/\n", "formal_statement": "theorem exercise_4_4_7 {G : Type*} [Group G] {H : Subgroup G} [Fintype H]\n (hH : ∀ (K : Subgroup G) (fK : Fintype K), card H = @card K fK → H = K) :\n H.Characteristic :=", "goal": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Fintype ↥H\nhH : ∀ (K : Subgroup G) (fK : Fintype ↥K), card ↥H = card ↥K → H = K\n⊢ H.Characteristic", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_4_8a", "split": "test", "informal_prefix": "/-- 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 $G$.-/\n", "formal_statement": "theorem exercise_4_4_8a {G : Type*} [Group G] (H K : Subgroup G)\n (hHK : H ≤ K) [hHK1 : (H.subgroupOf K).Normal] [hK : K.Normal] :\n H.Normal :=", "goal": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nhHK : H ≤ K\nhHK1 : (H.subgroupOf K).Normal\nhK : K.Normal\n⊢ H.Normal", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_1a", "split": "valid", "informal_prefix": "/-- Prove that if $P \\in \\operatorname{Syl}_{p}(G)$ and $H$ is a subgroup of $G$ containing $P$ then $P \\in \\operatorname{Syl}_{p}(H)$.-/\n", "formal_statement": "theorem exercise_4_5_1a {p : ℕ} {G : Type*} [Group G]\n {P : Subgroup G} (hP : IsPGroup p P) (H : Subgroup G)\n (hH : P ≤ H) : IsPGroup p H :=", "goal": "p : ℕ\nG : Type u_1\ninst✝ : Group G\nP : Subgroup G\nhP : IsPGroup p ↥P\nH : Subgroup G\nhH : P ≤ H\n⊢ IsPGroup p ↥H", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_13", "split": "test", "informal_prefix": "/-- Prove that a group of order 56 has a normal Sylow $p$-subgroup for some prime $p$ dividing its order.-/\n", "formal_statement": "theorem exercise_4_5_13 {G : Type*} [Group G] [Fintype G]\n (hG : card G = 56) :\n ∃ (p : ℕ) (P : Sylow p G), P.Normal :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\nhG : card G = 56\n⊢ ∃ p P, (↑P).Normal", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_14", "split": "valid", "informal_prefix": "/-- Prove that a group of order 312 has a normal Sylow $p$-subgroup for some prime $p$ dividing its order.-/\n", "formal_statement": "theorem exercise_4_5_14 {G : Type*} [Group G] [Fintype G]\n (hG : card G = 312) :\n ∃ (p : ℕ) (P : Sylow p G), P.Normal :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\nhG : card G = 312\n⊢ ∃ p P, (↑P).Normal", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_15", "split": "test", "informal_prefix": "/-- Prove that a group of order 351 has a normal Sylow $p$-subgroup for some prime $p$ dividing its order.-/\n", "formal_statement": "theorem exercise_4_5_15 {G : Type*} [Group G] [Fintype G]\n (hG : card G = 351) :\n ∃ (p : ℕ) (P : Sylow p G), P.Normal :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\nhG : card G = 351\n⊢ ∃ p P, (↑P).Normal", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_16", "split": "valid", "informal_prefix": "/-- 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 $r$.-/\n", "formal_statement": "theorem exercise_4_5_16 {p q r : ℕ} {G : Type*} [Group G]\n [Fintype G] (hpqr : p < q ∧ q < r)\n (hpqr1 : p.Prime ∧ q.Prime ∧ r.Prime)(hG : card G = p*q*r) :\n Nonempty (Sylow p G) ∨ Nonempty (Sylow q G) ∨ Nonempty (Sylow r G) :=", "goal": "p q r : ℕ\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\nhpqr : p < q ∧ q < r\nhpqr1 : p.Prime ∧ q.Prime ∧ r.Prime\nhG : card G = p * q * r\n⊢ Nonempty (Sylow p G) ∨ Nonempty (Sylow q G) ∨ Nonempty (Sylow r G)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_17", "split": "test", "informal_prefix": "/-- Prove that if $|G|=105$ then $G$ has a normal Sylow 5 -subgroup and a normal Sylow 7-subgroup.-/\n", "formal_statement": "theorem exercise_4_5_17 {G : Type*} [Fintype G] [Group G]\n (hG : card G = 105) :\n Nonempty (Sylow 5 G) ∧ Nonempty (Sylow 7 G) :=", "goal": "G : Type u_1\ninst✝¹ : Fintype G\ninst✝ : Group G\nhG : card G = 105\n⊢ Nonempty (Sylow 5 G) ∧ Nonempty (Sylow 7 G)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_18", "split": "valid", "informal_prefix": "/-- Prove that a group of order 200 has a normal Sylow 5-subgroup.-/\n", "formal_statement": "theorem exercise_4_5_18 {G : Type*} [Fintype G] [Group G]\n (hG : card G = 200) :\n ∃ N : Sylow 5 G, N.Normal :=", "goal": "G : Type u_1\ninst✝¹ : Fintype G\ninst✝ : Group G\nhG : card G = 200\n⊢ ∃ N, (↑N).Normal", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_19", "split": "test", "informal_prefix": "/-- Prove that if $|G|=6545$ then $G$ is not simple.-/\n", "formal_statement": "theorem exercise_4_5_19 {G : Type*} [Fintype G] [Group G]\n (hG : card G = 6545) : ¬ IsSimpleGroup G :=", "goal": "G : Type u_1\ninst✝¹ : Fintype G\ninst✝ : Group G\nhG : card G = 6545\n⊢ ¬IsSimpleGroup G", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_20", "split": "valid", "informal_prefix": "/-- Prove that if $|G|=1365$ then $G$ is not simple.-/\n", "formal_statement": "theorem exercise_4_5_20 {G : Type*} [Fintype G] [Group G]\n (hG : card G = 1365) : ¬ IsSimpleGroup G :=", "goal": "G : Type u_1\ninst✝¹ : Fintype G\ninst✝ : Group G\nhG : card G = 1365\n⊢ ¬IsSimpleGroup G", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_21", "split": "test", "informal_prefix": "/-- Prove that if $|G|=2907$ then $G$ is not simple.-/\n", "formal_statement": "theorem exercise_4_5_21 {G : Type*} [Fintype G] [Group G]\n (hG : card G = 2907) : ¬ IsSimpleGroup G :=", "goal": "G : Type u_1\ninst✝¹ : Fintype G\ninst✝ : Group G\nhG : card G = 2907\n⊢ ¬IsSimpleGroup G", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_22", "split": "valid", "informal_prefix": "/-- Prove that if $|G|=132$ then $G$ is not simple.-/\n", "formal_statement": "theorem exercise_4_5_22 {G : Type*} [Fintype G] [Group G]\n (hG : card G = 132) : ¬ IsSimpleGroup G :=", "goal": "G : Type u_1\ninst✝¹ : Fintype G\ninst✝ : Group G\nhG : card G = 132\n⊢ ¬IsSimpleGroup G", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_23", "split": "test", "informal_prefix": "/-- Prove that if $|G|=462$ then $G$ is not simple.-/\n", "formal_statement": "theorem exercise_4_5_23 {G : Type*} [Fintype G] [Group G]\n (hG : card G = 462) : ¬ IsSimpleGroup G :=", "goal": "G : Type u_1\ninst✝¹ : Fintype G\ninst✝ : Group G\nhG : card G = 462\n⊢ ¬IsSimpleGroup G", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_28", "split": "valid", "informal_prefix": "/-- Let $G$ be a group of order 105. Prove that if a Sylow 3-subgroup of $G$ is normal then $G$ is abelian.-/\n", "formal_statement": "def exercise_4_5_28 {G : Type*} [Group G] [Fintype G]\n (hG : card G = 105) (P : Sylow 3 G) [hP : P.Normal] :\n CommGroup G :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\nhG : card G = 105\nP : Sylow 3 G\nhP : (↑P).Normal\n⊢ CommGroup G", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_33", "split": "test", "informal_prefix": "/-- 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 $H$.-/\n", "formal_statement": "theorem exercise_4_5_33 {G : Type*} [Group G] [Fintype G] {p : ℕ}\n (P : Sylow p G) [hP : P.Normal] (H : Subgroup G) [Fintype H] :\n ∀ R : Sylow p H, R.toSubgroup = (H ⊓ P.toSubgroup).subgroupOf H ∧\n Nonempty (Sylow p H) :=", "goal": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\np : ℕ\nP : Sylow p G\nhP : (↑P).Normal\nH : Subgroup G\ninst✝ : Fintype ↥H\n⊢ ∀ (R : Sylow p ↥H), ↑R = (H ⊓ ↑P).subgroupOf H ∧ Nonempty (Sylow p ↥H)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_4_2", "split": "valid", "informal_prefix": "/-- Prove that a subgroup $H$ of $G$ is normal if and only if $[G, H] \\leq H$.-/\n", "formal_statement": "theorem exercise_5_4_2 {G : Type*} [Group G] (H : Subgroup G) :\n H.Normal ↔ ⁅(⊤ : Subgroup G), H⁆ ≤ H :=", "goal": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ H.Normal ↔ ⁅⊤, H⁆ ≤ H", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_7_1_2", "split": "test", "informal_prefix": "/-- Prove that if $u$ is a unit in $R$ then so is $-u$.-/\n", "formal_statement": "theorem exercise_7_1_2 {R : Type*} [Ring R] {u : R}\n (hu : IsUnit u) : IsUnit (-u) :=", "goal": "R : Type u_1\ninst✝ : Ring R\nu : R\nhu : IsUnit u\n⊢ IsUnit (-u)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_7_1_11", "split": "valid", "informal_prefix": "/-- Prove that if $R$ is an integral domain and $x^{2}=1$ for some $x \\in R$ then $x=\\pm 1$.-/\n", "formal_statement": "theorem exercise_7_1_11 {R : Type*} [Ring R] [IsDomain R]\n {x : R} (hx : x^2 = 1) : x = 1 ∨ x = -1 :=", "goal": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : IsDomain R\nx : R\nhx : x ^ 2 = 1\n⊢ x = 1 ∨ x = -1", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_7_1_12", "split": "test", "informal_prefix": "/-- Prove that any subring of a field which contains the identity is an integral domain.-/\n", "formal_statement": "theorem exercise_7_1_12 {F : Type*} [Field F] {K : Subring F}\n (hK : (1 : F) ∈ K) : IsDomain K :=", "goal": "F : Type u_1\ninst✝ : Field F\nK : Subring F\nhK : 1 ∈ K\n⊢ IsDomain ↥K", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_7_1_15", "split": "valid", "informal_prefix": "/-- A ring $R$ is called a Boolean ring if $a^{2}=a$ for all $a \\in R$. Prove that every Boolean ring is commutative.-/\n", "formal_statement": "def exercise_7_1_15 {R : Type*} [Ring R] (hR : ∀ a : R, a^2 = a) :\n CommRing R :=", "goal": "R : Type u_1\ninst✝ : Ring R\nhR : ∀ (a : R), a ^ 2 = a\n⊢ CommRing R", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_7_2_2", "split": "test", "informal_prefix": "/-- 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 $b p(x)=0$.-/\n", "formal_statement": "theorem exercise_7_2_2 {R : Type*} [Ring R] (p : Polynomial R) :\n p ∣ 0 ↔ ∃ b : R, b ≠ 0 ∧ b • p = 0 :=", "goal": "R : Type u_1\ninst✝ : Ring R\np : R[X]\n⊢ p ∣ 0 ↔ ∃ b, b ≠ 0 ∧ b • p = 0", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_7_2_12", "split": "valid", "informal_prefix": "/-- 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 $R G$.-/\n", "formal_statement": "theorem exercise_7_2_12 {R G : Type*} [Ring R] [Group G] [Fintype G] :\n ∑ g : G, MonoidAlgebra.of R G g ∈ center (MonoidAlgebra R G) :=", "goal": "R : Type u_1\nG : Type u_2\ninst✝² : Ring R\ninst✝¹ : Group G\ninst✝ : Fintype G\n⊢ ∑ g : G, (MonoidAlgebra.of R G) g ∈ Set.center (MonoidAlgebra R G)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_7_3_16", "split": "test", "informal_prefix": "/-- 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 $S$.-/\n", "formal_statement": "theorem exercise_7_3_16 {R S : Type*} [Ring R] [Ring S]\n {φ : R →+* S} (hf : Function.Surjective φ) :\n φ '' (center R) ⊂ center S :=", "goal": "R : Type u_1\nS : Type u_2\ninst✝¹ : Ring R\ninst✝ : Ring S\nφ : R →+* S\nhf : Function.Surjective ⇑φ\n⊢ ⇑φ '' Set.center R ⊂ Set.center S", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_7_3_37", "split": "valid", "informal_prefix": "/-- 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 $\\mathbb{Z} / p^{m} \\mathbb{Z}$.-/\n", "formal_statement": "theorem exercise_7_3_37 {p m : ℕ} (hp : p.Prime) :\n IsNilpotent (span ({↑p} : Set $ ZMod $ p^m) : Ideal $ ZMod $ p^m) :=", "goal": "p m : ℕ\nhp : p.Prime\n⊢ IsNilpotent (span {↑p})", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_7_4_27", "split": "test", "informal_prefix": "/-- 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 $b \\in R$.-/\n", "formal_statement": "theorem exercise_7_4_27 {R : Type*} [CommRing R] (hR : (0 : R) ≠ 1)\n {a : R} (ha : IsNilpotent a) (b : R) :\n IsUnit (1-a*b) :=", "goal": "R : Type u_1\ninst✝ : CommRing R\nhR : 0 ≠ 1\na : R\nha : IsNilpotent a\nb : R\n⊢ IsUnit (1 - a * b)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_8_1_12", "split": "valid", "informal_prefix": "/-- 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 $d \\bmod \\varphi(N)$: $d d^{\\prime} \\equiv 1 \\pmod {\\varphi(N)}$.-/\n", "formal_statement": "theorem exercise_8_1_12 {N : ℕ} (hN : N > 0) {M M': ℤ} {d : ℕ}\n (hMN : M.gcd N = 1) (hMd : d.gcd N.totient = 1)\n (hM' : M' ≡ M^d [ZMOD N]) :\n ∃ d' : ℕ, d' * d ≡ 1 [ZMOD N.totient] ∧\n M ≡ M'^d' [ZMOD N] :=", "goal": "N : ℕ\nhN : N > 0\nM M' : ℤ\nd : ℕ\nhMN : M.gcd ↑N = 1\nhMd : d.gcd N.totient = 1\nhM' : M' ≡ M ^ d [ZMOD ↑N]\n⊢ ∃ d', ↑d' * ↑d ≡ 1 [ZMOD ↑N.totient] ∧ M ≡ M' ^ d' [ZMOD ↑N]", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_8_2_4", "split": "test", "informal_prefix": "/-- 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 $n \\geq N$.-/\n", "formal_statement": "theorem exercise_8_2_4 {R : Type*} [Ring R][NoZeroDivisors R]\n [CancelCommMonoidWithZero R] [GCDMonoid R]\n (h1 : ∀ a b : R, a ≠ 0 → b ≠ 0 → ∃ r s : R, gcd a b = r*a + s*b)\n (h2 : ∀ a : ℕ → R, (∀ i j : ℕ, i < j → a i ∣ a j) →\n ∃ N : ℕ, ∀ n ≥ N, ∃ u : R, IsUnit u ∧ a n = u * a N) :\n IsPrincipalIdealRing R :=", "goal": "R : Type u_1\ninst✝³ : Ring R\ninst✝² : NoZeroDivisors R\ninst✝¹ : CancelCommMonoidWithZero R\ninst✝ : GCDMonoid R\nh1 : ∀ (a b : R), a ≠ 0 → b ≠ 0 → ∃ r s, gcd a b = r * a + s * b\nh2 : ∀ (a : ℕ → R), (∀ (i j : ℕ), i < j → a i ∣ a j) → ∃ N, ∀ n ≥ N, ∃ u, IsUnit u ∧ a n = u * a N\n⊢ IsPrincipalIdealRing R", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_8_3_4", "split": "valid", "informal_prefix": "/-- Prove that if an integer is the sum of two rational squares, then it is the sum of two integer squares.-/\n", "formal_statement": "theorem exercise_8_3_4 {R : Type*} {n : ℤ} {r s : ℚ}\n (h : r^2 + s^2 = n) :\n ∃ a b : ℤ, a^2 + b^2 = n :=", "goal": "R : Type u_1\nn : ℤ\nr s : ℚ\nh : r ^ 2 + s ^ 2 = ↑n\n⊢ ∃ a b, a ^ 2 + b ^ 2 = n", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_8_3_5a", "split": "test", "informal_prefix": "/-- 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 $R$.-/\n", "formal_statement": "theorem exercise_8_3_5a {n : ℤ} (hn0 : n > 3) (hn1 : Squarefree n) :\n Irreducible (2 : Zsqrtd $ -n) ∧\n Irreducible (⟨0, 1⟩ : Zsqrtd $ -n) ∧\n Irreducible (1 + ⟨0, 1⟩ : Zsqrtd $ -n) :=", "goal": "n : ℤ\nhn0 : n > 3\nhn1 : Squarefree n\n⊢ Irreducible 2 ∧ Irreducible { re := 0, im := 1 } ∧ Irreducible (1 + { re := 0, im := 1 })", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_8_3_6a", "split": "valid", "informal_prefix": "/-- Prove that the quotient ring $\\mathbb{Z}[i] /(1+i)$ is a field of order 2.-/\n", "formal_statement": "theorem exercise_8_3_6a {R : Type} [Ring R]\n (hR : R = (GaussianInt ⧸ span ({⟨0, 1⟩} : Set GaussianInt))) :\n IsField R ∧ ∃ finR : Fintype R, @card R finR = 2 :=", "goal": "R : Type\ninst✝ : Ring R\nhR : R = (GaussianInt ⧸ span {{ re := 0, im := 1 }})\n⊢ IsField R ∧ ∃ finR, card R = 2", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_8_3_6b", "split": "test", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_8_3_6b {q : ℕ} (hq0 : q.Prime)\n (hq1 : q ≡ 3 [ZMOD 4]) {R : Type} [Ring R]\n (hR : R = (GaussianInt ⧸ span ({↑q} : Set GaussianInt))) :\n IsField R ∧ ∃ finR : Fintype R, @card R finR = q^2 :=", "goal": "q : ℕ\nhq0 : q.Prime\nhq1 : ↑q ≡ 3 [ZMOD 4]\nR : Type\ninst✝ : Ring R\nhR : R = (GaussianInt ⧸ span {↑q})\n⊢ IsField R ∧ ∃ finR, card R = q ^ 2", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_9_1_6", "split": "valid", "informal_prefix": "/-- Prove that $(x, y)$ is not a principal ideal in $\\mathbb{Q}[x, y]$.-/\n", "formal_statement": "theorem exercise_9_1_6 : ¬ Submodule.IsPrincipal\n (span ({MvPolynomial.X 0, MvPolynomial.X 1} : Set (MvPolynomial (Fin 2) ℚ))) :=", "goal": "⊢ ¬Submodule.IsPrincipal (span {MvPolynomial.X 0, MvPolynomial.X 1})", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_9_1_10", "split": "test", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_9_1_10 {f : ℕ → MvPolynomial ℕ ℤ}\n (hf : f = λ i => MvPolynomial.X i * MvPolynomial.X (i+1)):\n Infinite (minimalPrimes (MvPolynomial ℕ ℤ ⧸ span (range f))) :=", "goal": "f : ℕ → MvPolynomial ℕ ℤ\nhf : f = fun i => MvPolynomial.X i * MvPolynomial.X (i + 1)\n⊢ Infinite ↑(minimalPrimes (MvPolynomial ℕ ℤ ⧸ span (range f)))", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_9_3_2", "split": "valid", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_9_3_2 {f g : Polynomial ℚ} (i j : ℕ)\n (hfg : ∀ n : ℕ, ∃ a : ℤ, (f*g).coeff = a) :\n ∃ a : ℤ, f.coeff i * g.coeff j = a :=", "goal": "f g : ℚ[X]\ni j : ℕ\nhfg : ℕ → ∃ a, (f * g).coeff = ↑a\n⊢ ∃ a, f.coeff i * g.coeff j = ↑a", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_9_4_2a", "split": "test", "informal_prefix": "/-- Prove that $x^4-4x^3+6$ is irreducible in $\\mathbb{Z}[x]$.-/\n", "formal_statement": "theorem exercise_9_4_2a : Irreducible (X^4 - 4*X^3 + 6 : Polynomial ℤ) :=", "goal": "⊢ Irreducible (X ^ 4 - 4 * X ^ 3 + 6)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_9_4_2b", "split": "valid", "informal_prefix": "/-- Prove that $x^6+30x^5-15x^3 + 6x-120$ is irreducible in $\\mathbb{Z}[x]$.-/\n", "formal_statement": "theorem exercise_9_4_2b : Irreducible\n (X^6 + 30*X^5 - 15*X^3 + 6*X - 120 : Polynomial ℤ) :=", "goal": "⊢ Irreducible (X ^ 6 + 30 * X ^ 5 - 15 * X ^ 3 + 6 * X - 120)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_9_4_2c", "split": "test", "informal_prefix": "/-- Prove that $x^4+4x^3+6x^2+2x+1$ is irreducible in $\\mathbb{Z}[x]$.-/\n", "formal_statement": "theorem exercise_9_4_2c : Irreducible\n (X^4 + 4*X^3 + 6*X^2 + 2*X + 1 : Polynomial ℤ) :=", "goal": "⊢ Irreducible (X ^ 4 + 4 * X ^ 3 + 6 * X ^ 2 + 2 * X + 1)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_9_4_2d", "split": "valid", "informal_prefix": "/-- Prove that $\\frac{(x+2)^p-2^p}{x}$, where $p$ is an odd prime, is irreducible in $\\mathbb{Z}[x]$.-/\n", "formal_statement": "theorem exercise_9_4_2d {p : ℕ} (hp : p.Prime ∧ p > 2)\n {f : Polynomial ℤ} (hf : f = (X + 2)^p):\n Irreducible (∑ n in (f.support \\ {0}), (f.coeff n : Polynomial ℤ) * X ^ (n-1) :\n Polynomial ℤ) :=", "goal": "p : ℕ\nhp : p.Prime ∧ p > 2\nf : ℤ[X]\nhf : f = (X + 2) ^ p\n⊢ Irreducible (∑ n ∈ f.support \\ {0}, ↑(f.coeff n) * X ^ (n - 1))", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_9_4_9", "split": "test", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_9_4_9 :\n Irreducible (X^2 - C Zsqrtd.sqrtd : Polynomial (Zsqrtd 2)) :=", "goal": "⊢ Irreducible (X ^ 2 - C Zsqrtd.sqrtd)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_9_4_11", "split": "valid", "informal_prefix": "/-- Prove that $x^2+y^2-1$ is irreducible in $\\mathbb{Q}[x,y]$.-/\n", "formal_statement": "theorem exercise_9_4_11 :\n Irreducible ((MvPolynomial.X 0)^2 + (MvPolynomial.X 1)^2 - 1 : MvPolynomial (Fin 2) ℚ) :=", "goal": "⊢ Irreducible (MvPolynomial.X 0 ^ 2 + MvPolynomial.X 1 ^ 2 - 1)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_11_1_13", "split": "test", "informal_prefix": "/-- Prove that as vector spaces over $\\mathbb{Q}, \\mathbb{R}^n \\cong \\mathbb{R}$, for all $n \\in \\mathbb{Z}^{+}$.-/\n", "formal_statement": "def exercise_11_1_13 {ι : Type*} [Fintype ι] :\n (ι → ℝ) ≃ₗ[ℚ] ℝ :=", "goal": "ι : Type u_1\ninst✝ : Fintype ι\n⊢ (ι → ℝ) ≃ₗ[ℚ] ℝ", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_13_1", "split": "valid", "informal_prefix": "/-- 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 $X$.-/\n", "formal_statement": "theorem exercise_13_1 (X : Type*) [TopologicalSpace X] (A : Set X)\n (h1 : ∀ x ∈ A, ∃ U : Set X, x ∈ U ∧ IsOpen U ∧ U ⊆ A) :\n IsOpen A :=", "goal": "X : Type u_1\ninst✝ : TopologicalSpace X\nA : Set X\nh1 : ∀ x ∈ A, ∃ U, x ∈ U ∧ IsOpen U ∧ U ⊆ A\n⊢ IsOpen A", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_13_3b", "split": "test", "informal_prefix": "/-- 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 $X$.-/\n", "formal_statement": "theorem exercise_13_3b : ¬ ∀ X : Type, ∀s : Set (Set X),\n (∀ t : Set X, t ∈ s → (Set.Infinite tᶜ ∨ t = ∅ ∨ t = ⊤)) →\n (Set.Infinite (⋃₀ s)ᶜ ∨ (⋃₀ s) = ∅ ∨ (⋃₀ s) = ⊤) :=", "goal": "⊢ ¬∀ (X : Type) (s : Set (Set X)), (∀ t ∈ s, tᶜ.Infinite ∨ t = ∅ ∨ t = ⊤) → (⋃₀ s)ᶜ.Infinite ∨ ⋃₀ s = ∅ ∨ ⋃₀ s = ⊤", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_13_4a1", "split": "valid", "informal_prefix": "/-- If $\\mathcal{T}_\\alpha$ is a family of topologies on $X$, show that $\\bigcap \\mathcal{T}_\\alpha$ is a topology on $X$.-/\n", "formal_statement": "theorem exercise_13_4a1 (X I : Type*) (T : I → Set (Set X)) (h : ∀ i, is_topology X (T i)) :\n is_topology X (⋂ i : I, T i) :=", "goal": "X : Type u_1\nI : Type u_2\nT : I → Set (Set X)\nh : ∀ (i : I), is_topology X (T i)\n⊢ is_topology X (⋂ i, T i)", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\ndef is_topology (X : Type*) (T : Set (Set X)) :=\n univ ∈ T ∧\n (∀ s t, s ∈ T → t ∈ T → s ∩ t ∈ T) ∧\n (∀s, (∀t ∈ s, t ∈ T) → sUnion s ∈ T)\n\n"}
{"name": "exercise_13_4a2", "split": "test", "informal_prefix": "/-- 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 $X$.-/\n", "formal_statement": "theorem exercise_13_4a2 :\n ∃ (X I : Type*) (T : I → Set (Set X)),\n (∀ i, is_topology X (T i)) ∧ ¬ is_topology X (⋂ i : I, T i) :=", "goal": "⊢ ∃ X I T, (∀ (i : I), is_topology X (T i)) ∧ ¬is_topology X (⋂ i, T i)", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\ndef is_topology (X : Type*) (T : Set (Set X)) :=\n univ ∈ T ∧\n (∀ s t, s ∈ T → t ∈ T → s ∩ t ∈ T) ∧\n (∀s, (∀t ∈ s, t ∈ T) → sUnion s ∈ T)\n\n"}
{"name": "exercise_13_4b1", "split": "valid", "informal_prefix": "/-- 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 $\\mathcal{T}_\\alpha$.-/\n", "formal_statement": "theorem exercise_13_4b1 (X I : Type*) (T : I → Set (Set X)) (h : ∀ i, is_topology X (T i)) :\n ∃! T', is_topology X T' ∧ (∀ i, T i ⊆ T') ∧\n ∀ T'', is_topology X T'' → (∀ i, T i ⊆ T'') → T'' ⊆ T' :=", "goal": "X : Type u_1\nI : Type u_2\nT : I → Set (Set X)\nh : ∀ (i : I), is_topology X (T i)\n⊢ ∃! T',\n is_topology X T' ∧\n (∀ (i : I), T i ⊆ T') ∧ ∀ (T'' : Set (Set X)), is_topology X T'' → (∀ (i : I), T i ⊆ T'') → T'' ⊆ T'", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\ndef is_topology (X : Type*) (T : Set (Set X)) :=\n univ ∈ T ∧\n (∀ s t, s ∈ T → t ∈ T → s ∩ t ∈ T) ∧\n (∀s, (∀t ∈ s, t ∈ T) → sUnion s ∈ T)\n\n"}
{"name": "exercise_13_4b2", "split": "test", "informal_prefix": "/-- 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 $\\mathcal{T}_\\alpha$.-/\n", "formal_statement": "theorem exercise_13_4b2 (X I : Type*) (T : I → Set (Set X)) (h : ∀ i, is_topology X (T i)) :\n ∃! T', is_topology X T' ∧ (∀ i, T' ⊆ T i) ∧\n ∀ T'', is_topology X T'' → (∀ i, T'' ⊆ T i) → T' ⊆ T'' :=", "goal": "X : Type u_1\nI : Type u_2\nT : I → Set (Set X)\nh : ∀ (i : I), is_topology X (T i)\n⊢ ∃! T',\n is_topology X T' ∧\n (∀ (i : I), T' ⊆ T i) ∧ ∀ (T'' : Set (Set X)), is_topology X T'' → (∀ (i : I), T'' ⊆ T i) → T' ⊆ T''", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\ndef is_topology (X : Type*) (T : Set (Set X)) :=\n univ ∈ T ∧\n (∀ s t, s ∈ T → t ∈ T → s ∩ t ∈ T) ∧\n (∀s, (∀t ∈ s, t ∈ T) → sUnion s ∈ T)\n\n"}
{"name": "exercise_13_5a", "split": "valid", "informal_prefix": "/-- 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 $\\mathcal{A}$.-/\n", "formal_statement": "theorem exercise_13_5a {X : Type*}\n [TopologicalSpace X] (A : Set (Set X)) (hA : IsTopologicalBasis A) :\n generateFrom A = generateFrom (sInter {T | is_topology X T ∧ A ⊆ T}) :=", "goal": "X : Type u_1\ninst✝ : TopologicalSpace X\nA : Set (Set X)\nhA : IsTopologicalBasis A\n⊢ generateFrom A = generateFrom (⋂₀ {T | is_topology X T ∧ A ⊆ T})", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\ndef is_topology (X : Type*) (T : Set (Set X)) :=\n univ ∈ T ∧\n (∀ s t, s ∈ T → t ∈ T → s ∩ t ∈ T) ∧\n (∀s, (∀t ∈ s, t ∈ T) → sUnion s ∈ T)\n\n"}
{"name": "exercise_13_5b", "split": "test", "informal_prefix": "/-- 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 $\\mathcal{A}$.-/\n", "formal_statement": "theorem exercise_13_5b {X : Type*}\n [t : TopologicalSpace X] (A : Set (Set X)) (hA : t = generateFrom A) :\n generateFrom A = generateFrom (sInter {T | is_topology X T ∧ A ⊆ T}) :=", "goal": "X : Type u_1\nt : TopologicalSpace X\nA : Set (Set X)\nhA : t = generateFrom A\n⊢ generateFrom A = generateFrom (⋂₀ {T | is_topology X T ∧ A ⊆ T})", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\ndef is_topology (X : Type*) (T : Set (Set X)) :=\n univ ∈ T ∧\n (∀ s t, s ∈ T → t ∈ T → s ∩ t ∈ T) ∧\n (∀s, (∀t ∈ s, t ∈ T) → sUnion s ∈ T)\n\n"}
{"name": "exercise_13_6", "split": "valid", "informal_prefix": "/-- Show that the lower limit topology $\\mathbb{R}_l$ and $K$-topology $\\mathbb{R}_K$ are not comparable.-/\n", "formal_statement": "theorem exercise_13_6 :\n ¬ (∀ U, Rl.IsOpen U → K_topology.IsOpen U) ∧ ¬ (∀ U, K_topology.IsOpen U → Rl.IsOpen U) :=", "goal": "⊢ (¬∀ (U : Set ℝ), TopologicalSpace.IsOpen U → TopologicalSpace.IsOpen U) ∧\n ¬∀ (U : Set ℝ), TopologicalSpace.IsOpen U → TopologicalSpace.IsOpen U", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\ndef lower_limit_topology (X : Type) [Preorder X] :=\n generateFrom {S : Set X | ∃ a b, a < b ∧ S = Ico a b}\n\ndef Rl := lower_limit_topology ℝ\n\ndef K : Set ℝ := {r | ∃ n : ℕ, r = 1 / n}\n\ndef K_topology := generateFrom\n ({S : Set ℝ | ∃ a b, a < b ∧ S = Ioo a b} ∪ {S : Set ℝ | ∃ a b, a < b ∧ S = Ioo a b \\ K})\n\n"}
{"name": "exercise_13_8a", "split": "test", "informal_prefix": "/-- Show that the collection $\\{(a,b) \\mid a < b, a \\text{ and } b \\text{ rational}\\}$ is a basis that generates the standard topology on $\\mathbb{R}$.-/\n", "formal_statement": "theorem exercise_13_8a :\n IsTopologicalBasis {S : Set ℝ | ∃ a b : ℚ, a < b ∧ S = Ioo ↑a ↑b} :=", "goal": "⊢ IsTopologicalBasis {S | ∃ a b, a < b ∧ S = Ioo ↑a ↑b}", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_13_8b", "split": "valid", "informal_prefix": "/-- 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 $\\mathbb{R}$.-/\n", "formal_statement": "theorem exercise_13_8b :\n (generateFrom {S : Set ℝ | ∃ a b : ℚ, a < b ∧ S = Ico ↑a ↑b}).IsOpen ≠\n (lower_limit_topology ℝ).IsOpen :=", "goal": "⊢ TopologicalSpace.IsOpen ≠ TopologicalSpace.IsOpen", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\ndef lower_limit_topology (X : Type) [Preorder X] :=\n generateFrom {S : Set X | ∃ a b, a < b ∧ S = Ico a b}\n\n"}
{"name": "exercise_16_1", "split": "test", "informal_prefix": "/-- 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 $X$.-/\n", "formal_statement": "theorem exercise_16_1 {X : Type*} [TopologicalSpace X]\n (Y : Set X)\n (A : Set Y) :\n ∀ U : Set A, IsOpen U ↔ IsOpen (Subtype.val '' U) :=", "goal": "X : Type u_1\ninst✝ : TopologicalSpace X\nY : Set X\nA : Set ↑Y\n⊢ ∀ (U : Set ↑A), IsOpen U ↔ IsOpen (Subtype.val '' U)", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_16_4", "split": "valid", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_16_4 {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n (π₁ : X × Y → X)\n (π₂ : X × Y → Y)\n (h₁ : π₁ = Prod.fst)\n (h₂ : π₂ = Prod.snd) :\n IsOpenMap π₁ ∧ IsOpenMap π₂ :=", "goal": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nπ₁ : X × Y → X\nπ₂ : X × Y → Y\nh₁ : π₁ = Prod.fst\nh₂ : π₂ = Prod.snd\n⊢ IsOpenMap π₁ ∧ IsOpenMap π₂", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_16_6", "split": "test", "informal_prefix": "/-- 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 $\\mathbb{R}^2$.-/\n", "formal_statement": "theorem exercise_16_6\n (S : Set (Set (ℝ × ℝ)))\n (hS : ∀ s, s ∈ S → ∃ a b c d, (rational a ∧ rational b ∧ rational c ∧ rational d\n ∧ s = {x | ∃ x₁ x₂, x = (x₁, x₂) ∧ a < x₁ ∧ x₁ < b ∧ c < x₂ ∧ x₂ < d})) :\n IsTopologicalBasis S :=", "goal": "S : Set (Set (ℝ × ℝ))\nhS :\n ∀ s ∈ S,\n ∃ a b c d,\n rational a ∧\n rational b ∧ rational c ∧ rational d ∧ s = {x | ∃ x₁ x₂, x = (x₁, x₂) ∧ a < x₁ ∧ x₁ < b ∧ c < x₂ ∧ x₂ < d}\n⊢ IsTopologicalBasis S", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\ndef rational (x : ℝ) := x ∈ range ((↑) : ℚ → ℝ)\n\n"}
{"name": "exercise_17_4", "split": "valid", "informal_prefix": "/-- 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 $X$.-/\n", "formal_statement": "theorem exercise_17_4 {X : Type*} [TopologicalSpace X]\n (U A : Set X) (hU : IsOpen U) (hA : IsClosed A) :\n IsOpen (U \\ A) ∧ IsClosed (A \\ U) :=", "goal": "X : Type u_1\ninst✝ : TopologicalSpace X\nU A : Set X\nhU : IsOpen U\nhA : IsClosed A\n⊢ IsOpen (U \\ A) ∧ IsClosed (A \\ U)", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_18_8a", "split": "test", "informal_prefix": "/-- 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 $X$.-/\n", "formal_statement": "theorem exercise_18_8a {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n [LinearOrder Y] [OrderTopology Y] {f g : X → Y}\n (hf : Continuous f) (hg : Continuous g) :\n IsClosed {x | f x ≤ g x} :=", "goal": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : LinearOrder Y\ninst✝ : OrderTopology Y\nf g : X → Y\nhf : Continuous f\nhg : Continuous g\n⊢ IsClosed {x | f x ≤ g x}", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_18_8b", "split": "valid", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_18_8b {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n [LinearOrder Y] [OrderTopology Y] {f g : X → Y}\n (hf : Continuous f) (hg : Continuous g) :\n Continuous (λ x => min (f x) (g x)) :=", "goal": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : LinearOrder Y\ninst✝ : OrderTopology Y\nf g : X → Y\nhf : Continuous f\nhg : Continuous g\n⊢ Continuous fun x => min (f x) (g x)", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_18_13", "split": "test", "informal_prefix": "/-- 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 $f$.-/\n", "formal_statement": "theorem exercise_18_13\n {X : Type*} [TopologicalSpace X] {Y : Type*} [TopologicalSpace Y]\n [T2Space Y] {A : Set X} {f : A → Y} (hf : Continuous f)\n (g : closure A → Y)\n (g_con : Continuous g) :\n ∀ (g' : closure A → Y), Continuous g' → (∀ (x : closure A), g x = g' x) :=", "goal": "X : Type u_1\ninst✝² : TopologicalSpace X\nY : Type u_2\ninst✝¹ : TopologicalSpace Y\ninst✝ : T2Space Y\nA : Set X\nf : ↑A → Y\nhf : Continuous f\ng : ↑(closure A) → Y\ng_con : Continuous g\n⊢ ∀ (g' : ↑(closure A) → Y), Continuous g' → ∀ (x : ↑(closure A)), g x = g' x", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_19_6a", "split": "valid", "informal_prefix": "/-- 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 $\\alpha$.-/\n", "formal_statement": "theorem exercise_19_6a\n {n : ℕ}\n {f : Fin n → Type*} {x : ℕ → Πa, f a}\n (y : Πi, f i)\n [Πa, TopologicalSpace (f a)] :\n Tendsto x atTop (𝓝 y) ↔ ∀ i, Tendsto (λ j => (x j) i) atTop (𝓝 (y i)) :=", "goal": "n : ℕ\nf : Fin n → Type u_1\nx : ℕ → (a : Fin n) → f a\ny : (i : Fin n) → f i\ninst✝ : (a : Fin n) → TopologicalSpace (f a)\n⊢ Tendsto x atTop (𝓝 y) ↔ ∀ (i : Fin n), Tendsto (fun j => x j i) atTop (𝓝 (y i))", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_20_2", "split": "test", "informal_prefix": "/-- Show that $\\mathbb{R} \\times \\mathbb{R}$ in the dictionary order topology is metrizable.-/\n", "formal_statement": "theorem exercise_20_2\n [TopologicalSpace (ℝ ×ₗ ℝ)] [OrderTopology (ℝ ×ₗ ℝ)]\n : MetrizableSpace (ℝ ×ₗ ℝ) :=", "goal": "inst✝¹ : TopologicalSpace (Lex (ℝ × ℝ))\ninst✝ : OrderTopology (Lex (ℝ × ℝ))\n⊢ MetrizableSpace (Lex (ℝ × ℝ))", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_21_6a", "split": "valid", "informal_prefix": "/-- 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 $x \\in[0,1]$.-/\n", "formal_statement": "theorem exercise_21_6a\n (f : ℕ → I → ℝ )\n (h : ∀ x n, f n x = x ^ n) :\n ∀ x, ∃ y, Tendsto (λ n => f n x) atTop (𝓝 y) :=", "goal": "f : ℕ → ↑I → ℝ\nh : ∀ (x : ↑I) (n : ℕ), f n x = ↑x ^ n\n⊢ ∀ (x : ↑I), ∃ y, Tendsto (fun n => f n x) atTop (𝓝 y)", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\nabbrev I : Set ℝ := Icc 0 1\n\n"}
{"name": "exercise_21_6b", "split": "test", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_21_6b\n (f : ℕ → I → ℝ )\n (h : ∀ x n, f n x = x ^ n) :\n ¬ ∃ f₀, TendstoUniformly f f₀ atTop :=", "goal": "f : ℕ → ↑I → ℝ\nh : ∀ (x : ↑I) (n : ℕ), f n x = ↑x ^ n\n⊢ ¬∃ f₀, TendstoUniformly f f₀ atTop", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\nabbrev I : Set ℝ := Icc 0 1\n\n"}
{"name": "exercise_21_8", "split": "valid", "informal_prefix": "/-- 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 $f(x)$.-/\n", "formal_statement": "theorem exercise_21_8\n {X : Type*} [TopologicalSpace X] {Y : Type*} [MetricSpace Y]\n {f : ℕ → X → Y} {x : ℕ → X}\n (hf : ∀ n, Continuous (f n))\n (x₀ : X)\n (hx : Tendsto x atTop (𝓝 x₀))\n (f₀ : X → Y)\n (hh : TendstoUniformly f f₀ atTop) :\n Tendsto (λ n => f n (x n)) atTop (𝓝 (f₀ x₀)) :=", "goal": "X : Type u_1\ninst✝¹ : TopologicalSpace X\nY : Type u_2\ninst✝ : MetricSpace Y\nf : ℕ → X → Y\nx : ℕ → X\nhf : ∀ (n : ℕ), Continuous (f n)\nx₀ : X\nhx : Tendsto x atTop (𝓝 x₀)\nf₀ : X → Y\nhh : TendstoUniformly f f₀ atTop\n⊢ Tendsto (fun n => f n (x n)) atTop (𝓝 (f₀ x₀))", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_22_2a", "split": "test", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_22_2a {X Y : Type*} [TopologicalSpace X]\n [TopologicalSpace Y] (p : X → Y) (h : Continuous p) :\n QuotientMap p ↔ ∃ (f : Y → X), Continuous f ∧ p ∘ f = id :=", "goal": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\np : X → Y\nh : Continuous p\n⊢ QuotientMap p ↔ ∃ f, Continuous f ∧ p ∘ f = id", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_22_2b", "split": "valid", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_22_2b {X : Type*} [TopologicalSpace X]\n {A : Set X} (r : X → A) (hr : Continuous r) (h : ∀ x : A, r x = x) :\n QuotientMap r :=", "goal": "X : Type u_1\ninst✝ : TopologicalSpace X\nA : Set X\nr : X → ↑A\nhr : Continuous r\nh : ∀ (x : ↑A), r ↑x = x\n⊢ QuotientMap r", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_22_5", "split": "test", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_22_5 {X Y : Type*} [TopologicalSpace X]\n [TopologicalSpace Y] (p : X → Y) (hp : IsOpenMap p)\n (A : Set X) (hA : IsOpen A) : IsOpenMap (p ∘ Subtype.val : A → Y) :=", "goal": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\np : X → Y\nhp : IsOpenMap p\nA : Set X\nhA : IsOpen A\n⊢ IsOpenMap (p ∘ Subtype.val)", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_23_2", "split": "valid", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_23_2 {X : Type*}\n [TopologicalSpace X] {A : ℕ → Set X} (hA : ∀ n, IsConnected (A n))\n (hAn : ∀ n, A n ∩ A (n + 1) ≠ ∅) :\n IsConnected (⋃ n, A n) :=", "goal": "X : Type u_1\ninst✝ : TopologicalSpace X\nA : ℕ → Set X\nhA : ∀ (n : ℕ), IsConnected (A n)\nhAn : ∀ (n : ℕ), A n ∩ A (n + 1) ≠ ∅\n⊢ IsConnected (⋃ n, A n)", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_23_3", "split": "test", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_23_3 {X : Type*} [TopologicalSpace X]\n [TopologicalSpace X] {A : ℕ → Set X}\n (hAn : ∀ n, IsConnected (A n))\n (A₀ : Set X)\n (hA : IsConnected A₀)\n (h : ∀ n, A₀ ∩ A n ≠ ∅) :\n IsConnected (A₀ ∪ (⋃ n, A n)) :=", "goal": "X : Type u_1\ninst✝¹ inst✝ : TopologicalSpace X\nA : ℕ → Set X\nhAn : ∀ (n : ℕ), IsConnected (A n)\nA₀ : Set X\nhA : IsConnected A₀\nh : ∀ (n : ℕ), A₀ ∩ A n ≠ ∅\n⊢ IsConnected (A₀ ∪ ⋃ n, A n)", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_23_4", "split": "valid", "informal_prefix": "/-- Show that if $X$ is an infinite set, it is connected in the finite complement topology.-/\n", "formal_statement": "theorem exercise_23_4 {X : Type*} [TopologicalSpace X] [CofiniteTopology X]\n (s : Set X) : Infinite s → IsConnected s :=", "goal": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CofiniteTopology X\ns : Set X\n⊢ Infinite ↑s → IsConnected s", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\nset_option checkBinderAnnotations false\n\n"}
{"name": "exercise_23_6", "split": "test", "informal_prefix": "/-- Let $A \\subset X$. Show that if $C$ is a connected subspace of $X$ that intersects both $A$ and $X-A$, then $C$ intersects $\\operatorname{Bd} A$.-/\n", "formal_statement": "theorem exercise_23_6 {X : Type*}\n [TopologicalSpace X] {A C : Set X} (hc : IsConnected C)\n (hCA : C ∩ A ≠ ∅) (hCXA : C ∩ Aᶜ ≠ ∅) :\n C ∩ (frontier A) ≠ ∅ :=", "goal": "X : Type u_1\ninst✝ : TopologicalSpace X\nA C : Set X\nhc : IsConnected C\nhCA : C ∩ A ≠ ∅\nhCXA : C ∩ Aᶜ ≠ ∅\n⊢ C ∩ frontier A ≠ ∅", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_23_9", "split": "valid", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_23_9 {X Y : Type*}\n [TopologicalSpace X] [TopologicalSpace Y]\n (A₁ A₂ : Set X)\n (B₁ B₂ : Set Y)\n (hA : A₁ ⊂ A₂)\n (hB : B₁ ⊂ B₂)\n (hA : IsConnected A₂)\n (hB : IsConnected B₂) :\n IsConnected ({x | ∃ a b, x = (a, b) ∧ a ∈ A₂ ∧ b ∈ B₂} \\\n {x | ∃ a b, x = (a, b) ∧ a ∈ A₁ ∧ b ∈ B₁}) :=", "goal": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nA₁ A₂ : Set X\nB₁ B₂ : Set Y\nhA✝ : A₁ ⊂ A₂\nhB✝ : B₁ ⊂ B₂\nhA : IsConnected A₂\nhB : IsConnected B₂\n⊢ IsConnected ({x | ∃ a b, x = (a, b) ∧ a ∈ A₂ ∧ b ∈ B₂} \\ {x | ∃ a b, x = (a, b) ∧ a ∈ A₁ ∧ b ∈ B₁})", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_23_11", "split": "test", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_23_11 {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n (p : X → Y) (hq : QuotientMap p)\n (hY : ConnectedSpace Y) (hX : ∀ y : Y, IsConnected (p ⁻¹' {y})) :\n ConnectedSpace X :=", "goal": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\np : X → Y\nhq : QuotientMap p\nhY : ConnectedSpace Y\nhX : ∀ (y : Y), IsConnected (p ⁻¹' {y})\n⊢ ConnectedSpace X", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_24_2", "split": "valid", "informal_prefix": "/-- Let $f: S^{1} \\rightarrow \\mathbb{R}$ be a continuous map. Show there exists a point $x$ of $S^{1}$ such that $f(x)=f(-x)$.-/\n", "formal_statement": "theorem exercise_24_2 {f : (Metric.sphere 0 1 : Set ℝ) → ℝ}\n (hf : Continuous f) : ∃ x, f x = f (-x) :=", "goal": "f : ↑(Metric.sphere 0 1) → ℝ\nhf : Continuous f\n⊢ ∃ x, f x = f (-x)", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_24_3a", "split": "test", "informal_prefix": "/-- 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 $f$.)-/\n", "formal_statement": "theorem exercise_24_3a [TopologicalSpace I] [CompactSpace I]\n (f : I → I) (hf : Continuous f) :\n ∃ (x : I), f x = x :=", "goal": "I : Type u_1\ninst✝¹ : TopologicalSpace I\ninst✝ : CompactSpace I\nf : I → I\nhf : Continuous f\n⊢ ∃ x, f x = x", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_25_4", "split": "valid", "informal_prefix": "/-- Let $X$ be locally path connected. Show that every connected open set in $X$ is path connected.-/\n", "formal_statement": "theorem exercise_25_4 {X : Type*} [TopologicalSpace X]\n [LocPathConnectedSpace X] (U : Set X) (hU : IsOpen U)\n (hcU : IsConnected U) : IsPathConnected U :=", "goal": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : LocPathConnectedSpace X\nU : Set X\nhU : IsOpen U\nhcU : IsConnected U\n⊢ IsPathConnected U", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_25_9", "split": "test", "informal_prefix": "/-- 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 $G$.-/\n", "formal_statement": "theorem exercise_25_9 {G : Type*} [TopologicalSpace G] [Group G]\n [TopologicalGroup G] (C : Set G) (h : C = connectedComponent 1) :\n IsNormalSubgroup C :=", "goal": "G : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : TopologicalGroup G\nC : Set G\nh : C = connectedComponent 1\n⊢ IsNormalSubgroup C", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_26_11", "split": "valid", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_26_11\n {X : Type*} [TopologicalSpace X] [CompactSpace X] [T2Space X]\n (A : Set (Set X)) (hA : ∀ (a b : Set X), a ∈ A → b ∈ A → a ⊆ b ∨ b ⊆ a)\n (hA' : ∀ a ∈ A, IsClosed a) (hA'' : ∀ a ∈ A, IsConnected a) :\n IsConnected (⋂₀ A) :=", "goal": "X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nA : Set (Set X)\nhA : ∀ (a b : Set X), a ∈ A → b ∈ A → a ⊆ b ∨ b ⊆ a\nhA' : ∀ a ∈ A, IsClosed a\nhA'' : ∀ a ∈ A, IsConnected a\n⊢ IsConnected (⋂₀ A)", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_26_12", "split": "test", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_26_12 {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n (p : X → Y) (h : Function.Surjective p) (hc : Continuous p) (hp : ∀ y, IsCompact (p ⁻¹' {y}))\n (hY : CompactSpace Y) : CompactSpace X :=", "goal": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\np : X → Y\nh : Function.Surjective p\nhc : Continuous p\nhp : ∀ (y : Y), IsCompact (p ⁻¹' {y})\nhY : CompactSpace Y\n⊢ CompactSpace X", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_27_4", "split": "valid", "informal_prefix": "/-- Show that a connected metric space having more than one point is uncountable.-/\n", "formal_statement": "theorem exercise_27_4\n {X : Type*} [MetricSpace X] [ConnectedSpace X] (hX : ∃ x y : X, x ≠ y) :\n ¬ Countable (univ : Set X) :=", "goal": "X : Type u_1\ninst✝¹ : MetricSpace X\ninst✝ : ConnectedSpace X\nhX : ∃ x y, x ≠ y\n⊢ ¬Countable ↑univ", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_28_4", "split": "test", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_28_4 {X : Type*}\n [TopologicalSpace X] (hT1 : T1Space X) :\n countably_compact X ↔ limit_point_compact X :=", "goal": "X : Type u_1\ninst✝ : TopologicalSpace X\nhT1 : T1Space X\n⊢ countably_compact X ↔ limit_point_compact X", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\ndef countably_compact (X : Type*) [TopologicalSpace X] :=\n ∀ U : ℕ → Set X,\n (∀ i, IsOpen (U i)) ∧ ((univ : Set X) ⊆ ⋃ i, U i) →\n (∃ t : Finset ℕ, (univ : Set X) ⊆ ⋃ i ∈ t, U i)\n\ndef limit_point_compact (X : Type*) [TopologicalSpace X] :=\n ∀ U : Set X, Infinite U → ∃ x ∈ U, ClusterPt x (𝓟 U)\n\n"}
{"name": "exercise_28_5", "split": "valid", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_28_5\n (X : Type*) [TopologicalSpace X] :\n countably_compact X ↔ ∀ (C : ℕ → Set X), (∀ n, IsClosed (C n)) ∧\n (∀ n, C n ≠ ∅) ∧ (∀ n, C n ⊆ C (n + 1)) → ∃ x, ∀ n, x ∈ C n :=", "goal": "X : Type u_1\ninst✝ : TopologicalSpace X\n⊢ countably_compact X ↔\n ∀ (C : ℕ → Set X),\n ((∀ (n : ℕ), IsClosed (C n)) ∧ (∀ (n : ℕ), C n ≠ ∅) ∧ ∀ (n : ℕ), C n ⊆ C (n + 1)) → ∃ x, ∀ (n : ℕ), x ∈ C n", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\ndef countably_compact (X : Type*) [TopologicalSpace X] :=\n ∀ U : ℕ → Set X,\n (∀ i, IsOpen (U i)) ∧ ((univ : Set X) ⊆ ⋃ i, U i) →\n (∃ t : Finset ℕ, (univ : Set X) ⊆ ⋃ i ∈ t, U i)\n\n"}
{"name": "exercise_28_6", "split": "test", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_28_6 {X : Type*} [MetricSpace X]\n [CompactSpace X] {f : X → X} (hf : Isometry f) :\n Function.Bijective f :=", "goal": "X : Type u_1\ninst✝¹ : MetricSpace X\ninst✝ : CompactSpace X\nf : X → X\nhf : Isometry f\n⊢ Function.Bijective f", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_29_1", "split": "valid", "informal_prefix": "/-- Show that the rationals $\\mathbb{Q}$ are not locally compact.-/\n", "formal_statement": "theorem exercise_29_1 : ¬ LocallyCompactSpace ℚ :=", "goal": "⊢ ¬LocallyCompactSpace ℚ", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_29_4", "split": "test", "informal_prefix": "/-- Show that $[0, 1]^\\omega$ is not locally compact in the uniform topology.-/\n", "formal_statement": "theorem exercise_29_4 [TopologicalSpace (ℕ → I)] :\n ¬ LocallyCompactSpace (ℕ → I) :=", "goal": "inst✝ : TopologicalSpace (ℕ → ↑I)\n⊢ ¬LocallyCompactSpace (ℕ → ↑I)", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\nabbrev I : Set ℝ := Icc 0 1\n\n"}
{"name": "exercise_29_10", "split": "valid", "informal_prefix": "/-- 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 $\\bar{V} \\subset U$.-/\n", "formal_statement": "theorem exercise_29_10 {X : Type*}\n [TopologicalSpace X] [T2Space X] (x : X)\n (hx : ∃ U : Set X, x ∈ U ∧ IsOpen U ∧ (∃ K : Set X, U ⊂ K ∧ IsCompact K))\n (U : Set X) (hU : IsOpen U) (hxU : x ∈ U) :\n ∃ (V : Set X), IsOpen V ∧ x ∈ V ∧ IsCompact (closure V) ∧ closure V ⊆ U :=", "goal": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : T2Space X\nx : X\nhx : ∃ U, x ∈ U ∧ IsOpen U ∧ ∃ K, U ⊂ K ∧ IsCompact K\nU : Set X\nhU : IsOpen U\nhxU : x ∈ U\n⊢ ∃ V, IsOpen V ∧ x ∈ V ∧ IsCompact (closure V) ∧ closure V ⊆ U", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_30_10", "split": "test", "informal_prefix": "/-- Show that if $X$ is a countable product of spaces having countable dense subsets, then $X$ has a countable dense subset.-/\n", "formal_statement": "theorem exercise_30_10\n {X : ℕ → Type*} [∀ i, TopologicalSpace (X i)]\n (h : ∀ i, ∃ (s : Set (X i)), Countable s ∧ Dense s) :\n ∃ (s : Set (Π i, X i)), Countable s ∧ Dense s :=", "goal": "X : ℕ → Type u_1\ninst✝ : (i : ℕ) → TopologicalSpace (X i)\nh : ∀ (i : ℕ), ∃ s, Countable ↑s ∧ Dense s\n⊢ ∃ s, Countable ↑s ∧ Dense s", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_30_13", "split": "valid", "informal_prefix": "/-- Show that if $X$ has a countable dense subset, every collection of disjoint open sets in $X$ is countable.-/\n", "formal_statement": "theorem exercise_30_13 {X : Type*} [TopologicalSpace X]\n (h : ∃ (s : Set X), Countable s ∧ Dense s) (U : Set (Set X))\n (hU : ∀ (x y : Set X), x ∈ U → y ∈ U → x ≠ y → x ∩ y = ∅) :\n Countable U :=", "goal": "X : Type u_1\ninst✝ : TopologicalSpace X\nh : ∃ s, Countable ↑s ∧ Dense s\nU : Set (Set X)\nhU : ∀ (x y : Set X), x ∈ U → y ∈ U → x ≠ y → x ∩ y = ∅\n⊢ Countable ↑U", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_31_1", "split": "test", "informal_prefix": "/-- Show that if $X$ is regular, every pair of points of $X$ have neighborhoods whose closures are disjoint.-/\n", "formal_statement": "theorem exercise_31_1 {X : Type*} [TopologicalSpace X]\n (hX : RegularSpace X) (x y : X) :\n ∃ (U V : Set X), IsOpen U ∧ IsOpen V ∧ x ∈ U ∧ y ∈ V ∧ closure U ∩ closure V = ∅ :=", "goal": "X : Type u_1\ninst✝ : TopologicalSpace X\nhX : RegularSpace X\nx y : X\n⊢ ∃ U V, IsOpen U ∧ IsOpen V ∧ x ∈ U ∧ y ∈ V ∧ closure U ∩ closure V = ∅", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_31_2", "split": "valid", "informal_prefix": "/-- Show that if $X$ is normal, every pair of disjoint closed sets have neighborhoods whose closures are disjoint.-/\n", "formal_statement": "theorem exercise_31_2 {X : Type*}\n [TopologicalSpace X] [NormalSpace X] {A B : Set X}\n (hA : IsClosed A) (hB : IsClosed B) (hAB : Disjoint A B) :\n ∃ (U V : Set X), IsOpen U ∧ IsOpen V ∧ A ⊆ U ∧ B ⊆ V ∧ closure U ∩ closure V = ∅ :=", "goal": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nA B : Set X\nhA : IsClosed A\nhB : IsClosed B\nhAB : Disjoint A B\n⊢ ∃ U V, IsOpen U ∧ IsOpen V ∧ A ⊆ U ∧ B ⊆ V ∧ closure U ∩ closure V = ∅", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_31_3", "split": "test", "informal_prefix": "/-- Show that every order topology is regular.-/\n", "formal_statement": "theorem exercise_31_3 {α : Type*} [PartialOrder α]\n [TopologicalSpace α] (h : OrderTopology α) : RegularSpace α :=", "goal": "α : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : TopologicalSpace α\nh : OrderTopology α\n⊢ RegularSpace α", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_32_1", "split": "valid", "informal_prefix": "/-- Show that a closed subspace of a normal space is normal.-/\n", "formal_statement": "theorem exercise_32_1 {X : Type*} [TopologicalSpace X]\n (hX : NormalSpace X) (A : Set X) (hA : IsClosed A) :\n NormalSpace {x // x ∈ A} :=", "goal": "X : Type u_1\ninst✝ : TopologicalSpace X\nhX : NormalSpace X\nA : Set X\nhA : IsClosed A\n⊢ NormalSpace { x // x ∈ A }", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_32_2a", "split": "test", "informal_prefix": "/-- Show that if $\\prod X_\\alpha$ is Hausdorff, then so is $X_\\alpha$. Assume that each $X_\\alpha$ is nonempty.-/\n", "formal_statement": "theorem exercise_32_2a\n {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)]\n (h : ∀ i, Nonempty (X i)) (h2 : T2Space (Π i, X i)) :\n ∀ i, T2Space (X i) :=", "goal": "ι : Type u_1\nX : ι → Type u_2\ninst✝ : (i : ι) → TopologicalSpace (X i)\nh : ∀ (i : ι), Nonempty (X i)\nh2 : T2Space ((i : ι) → X i)\n⊢ ∀ (i : ι), T2Space (X i)", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_32_2b", "split": "valid", "informal_prefix": "/-- Show that if $\\prod X_\\alpha$ is regular, then so is $X_\\alpha$. Assume that each $X_\\alpha$ is nonempty.-/\n", "formal_statement": "theorem exercise_32_2b\n {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)]\n (h : ∀ i, Nonempty (X i)) (h2 : RegularSpace (Π i, X i)) :\n ∀ i, RegularSpace (X i) :=", "goal": "ι : Type u_1\nX : ι → Type u_2\ninst✝ : (i : ι) → TopologicalSpace (X i)\nh : ∀ (i : ι), Nonempty (X i)\nh2 : RegularSpace ((i : ι) → X i)\n⊢ ∀ (i : ι), RegularSpace (X i)", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_32_2c", "split": "test", "informal_prefix": "/-- Show that if $\\prod X_\\alpha$ is normal, then so is $X_\\alpha$. Assume that each $X_\\alpha$ is nonempty.-/\n", "formal_statement": "theorem exercise_32_2c\n {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)]\n (h : ∀ i, Nonempty (X i)) (h2 : NormalSpace (Π i, X i)) :\n ∀ i, NormalSpace (X i) :=", "goal": "ι : Type u_1\nX : ι → Type u_2\ninst✝ : (i : ι) → TopologicalSpace (X i)\nh : ∀ (i : ι), Nonempty (X i)\nh2 : NormalSpace ((i : ι) → X i)\n⊢ ∀ (i : ι), NormalSpace (X i)", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_32_3", "split": "valid", "informal_prefix": "/-- Show that every locally compact Hausdorff space is regular.-/\n", "formal_statement": "theorem exercise_32_3 {X : Type*} [TopologicalSpace X]\n (hX : LocallyCompactSpace X) (hX' : T2Space X) :\n RegularSpace X :=", "goal": "X : Type u_1\ninst✝ : TopologicalSpace X\nhX : LocallyCompactSpace X\nhX' : T2Space X\n⊢ RegularSpace X", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_33_7", "split": "test", "informal_prefix": "/-- Show that every locally compact Hausdorff space is completely regular.-/\n", "formal_statement": "theorem exercise_33_7 {X : Type*} [TopologicalSpace X]\n (hX : LocallyCompactSpace X) (hX' : T2Space X) :\n ∀ x A, IsClosed A ∧ ¬ x ∈ A →\n ∃ (f : X → I), Continuous f ∧ f x = 1 ∧ f '' A = {0} :=", "goal": "X : Type u_1\ninst✝ : TopologicalSpace X\nhX : LocallyCompactSpace X\nhX' : T2Space X\n⊢ ∀ (x : X) (A : Set X), IsClosed A ∧ x ∉ A → ∃ f, Continuous f ∧ f x = 1 ∧ f '' A = {0}", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\nabbrev I : Set ℝ := Icc 0 1\n\n"}
{"name": "exercise_33_8", "split": "valid", "informal_prefix": "/-- 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 $f(B) = \\{1\\}$.-/\n", "formal_statement": "theorem exercise_33_8\n (X : Type*) [TopologicalSpace X] [RegularSpace X]\n (h : ∀ x A, IsClosed A ∧ ¬ x ∈ A →\n ∃ (f : X → I), Continuous f ∧ f x = (1 : I) ∧ f '' A = {0})\n (A B : Set X) (hA : IsClosed A) (hB : IsClosed B)\n (hAB : Disjoint A B)\n (hAc : IsCompact A) :\n ∃ (f : X → I), Continuous f ∧ f '' A = {0} ∧ f '' B = {1} :=", "goal": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : RegularSpace X\nh : ∀ (x : X) (A : Set X), IsClosed A ∧ x ∉ A → ∃ f, Continuous f ∧ f x = 1 ∧ f '' A = {0}\nA B : Set X\nhA : IsClosed A\nhB : IsClosed B\nhAB : Disjoint A B\nhAc : IsCompact A\n⊢ ∃ f, Continuous f ∧ f '' A = {0} ∧ f '' B = {1}", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\nabbrev I : Set ℝ := Icc 0 1\n\n"}
{"name": "exercise_34_9", "split": "test", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_34_9\n (X : Type*) [TopologicalSpace X] [CompactSpace X]\n (X1 X2 : Set X) (hX1 : IsClosed X1) (hX2 : IsClosed X2)\n (hX : X1 ∪ X2 = univ) (hX1m : MetrizableSpace X1)\n (hX2m : MetrizableSpace X2) : MetrizableSpace X :=", "goal": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nX1 X2 : Set X\nhX1 : IsClosed X1\nhX2 : IsClosed X2\nhX : X1 ∪ X2 = univ\nhX1m : MetrizableSpace ↑X1\nhX2m : MetrizableSpace ↑X2\n⊢ MetrizableSpace X", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_38_6", "split": "valid", "informal_prefix": "/-- Let $X$ be completely regular. Show that $X$ is connected if and only if the Stone-Čech compactification of $X$ is connected.-/\n", "formal_statement": "theorem exercise_38_6 {X : Type*}\n (X : Type*) [TopologicalSpace X] [RegularSpace X]\n (h : ∀ x A, IsClosed A ∧ ¬ x ∈ A →\n ∃ (f : X → I), Continuous f ∧ f x = (1 : I) ∧ f '' A = {0}) :\n IsConnected (univ : Set X) ↔ IsConnected (univ : Set (StoneCech X)) :=", "goal": "X✝ : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : RegularSpace X\nh : ∀ (x : X) (A : Set X), IsClosed A ∧ x ∉ A → ∃ f, Continuous f ∧ f x = 1 ∧ f '' A = {0}\n⊢ IsConnected univ ↔ IsConnected univ", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\nabbrev I : Set ℝ := Icc 0 1\n\n"}
{"name": "exercise_43_2", "split": "test", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_43_2 {X : Type*} [MetricSpace X]\n {Y : Type*} [MetricSpace Y] [CompleteSpace Y] (A : Set X)\n (f : X → Y) (hf : UniformContinuousOn f A) :\n ∃! (g : X → Y), ContinuousOn g (closure A) ∧\n UniformContinuousOn g (closure A) ∧ ∀ (x : A), g x = f x :=", "goal": "X : Type u_1\ninst✝² : MetricSpace X\nY : Type u_2\ninst✝¹ : MetricSpace Y\ninst✝ : CompleteSpace Y\nA : Set X\nf : X → Y\nhf : UniformContinuousOn f A\n⊢ ∃! g, ContinuousOn g (closure A) ∧ UniformContinuousOn g (closure A) ∧ ∀ (x : ↑A), g ↑x = f ↑x", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_1_27", "split": "valid", "informal_prefix": "/-- For all odd $n$ show that $8 \\mid n^{2}-1$.-/\n", "formal_statement": "theorem exercise_1_27 {n : ℕ} (hn : Odd n) : 8 ∣ (n^2 - 1) :=", "goal": "n : ℕ\nhn : Odd n\n⊢ 8 ∣ n ^ 2 - 1", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_30", "split": "test", "informal_prefix": "/-- Prove that $\\frac{1}{2}+\\frac{1}{3}+\\cdots+\\frac{1}{n}$ is not an integer.-/\n", "formal_statement": "theorem exercise_1_30 {n : ℕ} :\n ¬ ∃ a : ℤ, ∑ i : Fin n, (1 : ℚ) / (n+2) = a :=", "goal": "n : ℕ\n⊢ ¬∃ a, ∑ i : Fin n, 1 / (↑n + 2) = ↑a", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_31", "split": "valid", "informal_prefix": "/-- Show that 2 is divisible by $(1+i)^{2}$ in $\\mathbb{Z}[i]$.-/\n", "formal_statement": "theorem exercise_1_31 : (⟨1, 1⟩ : GaussianInt) ^ 2 ∣ 2 :=", "goal": "⊢ { re := 1, im := 1 } ^ 2 ∣ 2", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_4", "split": "test", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_2_4 {a : ℤ} (ha : a ≠ 0)\n (f_a := λ n m : ℕ => Int.gcd (a^(2^n) + 1) (a^(2^m)+1)) {n m : ℕ}\n (hnm : n > m) :\n (Odd a → f_a n m = 1) ∧ (Even a → f_a n m = 2) :=", "goal": "a : ℤ\nha : a ≠ 0\nf_a : optParam (ℕ → ℕ → ℕ) fun n m => (a ^ 2 ^ n + 1).gcd (a ^ 2 ^ m + 1)\nn m : ℕ\nhnm : n > m\n⊢ (Odd a → f_a n m = 1) ∧ (Even a → f_a n m = 2)", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_21", "split": "valid", "informal_prefix": "/-- Define $\\wedge(n)=\\log p$ if $n$ is a power of $p$ and zero otherwise. Prove that $\\sum_{A \\mid n} \\mu(n / d) \\log d$ $=\\wedge(n)$.-/\n", "formal_statement": "theorem exercise_2_21 {l : ℕ → ℝ}\n (hl : ∀ p n : ℕ, p.Prime → l (p^n) = log p )\n (hl1 : ∀ m : ℕ, ¬ IsPrimePow m → l m = 0) :\n l = λ n => ∑ d : Nat.divisors n, ArithmeticFunction.moebius (n/d) * log d :=", "goal": "l : ℕ → ℝ\nhl : ∀ (p n : ℕ), p.Prime → l (p ^ n) = (↑p).log\nhl1 : ∀ (m : ℕ), ¬IsPrimePow m → l m = 0\n⊢ l = fun n => ∑ d : { x // x ∈ n.divisors }, ↑(ArithmeticFunction.moebius (n / ↑d)) * (↑↑d).log", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_27a", "split": "test", "informal_prefix": "/-- Show that $\\sum^{\\prime} 1 / n$, the sum being over square free integers, diverges.-/\n", "formal_statement": "theorem exercise_2_27a :\n ¬ Summable (λ i : {p : ℤ // Squarefree p} => (1 : ℚ) / i) :=", "goal": "⊢ ¬Summable fun i => 1 / ↑↑i", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_1", "split": "valid", "informal_prefix": "/-- Show that there are infinitely many primes congruent to $-1$ modulo 6 .-/\n", "formal_statement": "theorem exercise_3_1 : Infinite {p : Nat.Primes // p ≡ -1 [ZMOD 6]} :=", "goal": "⊢ Infinite { p // ↑↑p ≡ -1 [ZMOD 6] }", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_4", "split": "test", "informal_prefix": "/-- Show that the equation $3 x^{2}+2=y^{2}$ has no solution in integers.-/\n", "formal_statement": "theorem exercise_3_4 : ¬ ∃ x y : ℤ, 3*x^2 + 2 = y^2 :=", "goal": "⊢ ¬∃ x y, 3 * x ^ 2 + 2 = y ^ 2", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_5", "split": "valid", "informal_prefix": "/-- Show that the equation $7 x^{3}+2=y^{3}$ has no solution in integers.-/\n", "formal_statement": "theorem exercise_3_5 : ¬ ∃ x y : ℤ, 7*x^3 + 2 = y^3 :=", "goal": "⊢ ¬∃ x y, 7 * x ^ 3 + 2 = y ^ 3", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_10", "split": "test", "informal_prefix": "/-- If $n$ is not a prime, show that $(n-1) ! \\equiv 0(n)$, except when $n=4$.-/\n", "formal_statement": "theorem exercise_3_10 {n : ℕ} (hn0 : ¬ n.Prime) (hn1 : n ≠ 4) :\n Nat.factorial (n-1) ≡ 0 [MOD n] :=", "goal": "n : ℕ\nhn0 : ¬n.Prime\nhn1 : n ≠ 4\n⊢ (n - 1).factorial ≡ 0 [MOD n]", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_14", "split": "valid", "informal_prefix": "/-- Let $p$ and $q$ be distinct odd primes such that $p-1$ divides $q-1$. If $(n, p q)=1$, show that $n^{q-1} \\equiv 1(p q)$.-/\n", "formal_statement": "theorem exercise_3_14 {p q n : ℕ} (hp0 : p.Prime ∧ p > 2)\n (hq0 : q.Prime ∧ q > 2) (hpq0 : p ≠ q) (hpq1 : p - 1 ∣ q - 1)\n (hn : n.gcd (p*q) = 1) :\n n^(q-1) ≡ 1 [MOD p*q] :=", "goal": "p q n : ℕ\nhp0 : p.Prime ∧ p > 2\nhq0 : q.Prime ∧ q > 2\nhpq0 : p ≠ q\nhpq1 : p - 1 ∣ q - 1\nhn : n.gcd (p * q) = 1\n⊢ n ^ (q - 1) ≡ 1 [MOD p * q]", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_4", "split": "test", "informal_prefix": "/-- 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 $p$.-/\n", "formal_statement": "theorem exercise_4_4 {p t: ℕ} (hp0 : p.Prime) (hp1 : p = 4*t + 1)\n (a : ZMod p) :\n IsPrimitiveRoot a p ↔ IsPrimitiveRoot (-a) p :=", "goal": "p t : ℕ\nhp0 : p.Prime\nhp1 : p = 4 * t + 1\na : ZMod p\n⊢ IsPrimitiveRoot a p ↔ IsPrimitiveRoot (-a) p", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5", "split": "valid", "informal_prefix": "/-- Consider a prime $p$ of the form $4 t+3$. Show that $a$ is a primitive root modulo $p$ iff $-a$ has order $(p-1) / 2$.-/\n", "formal_statement": "theorem exercise_4_5 {p t : ℕ} (hp0 : p.Prime) (hp1 : p = 4*t + 3)\n (a : ZMod p) :\n IsPrimitiveRoot a p ↔ ((-a) ^ ((p-1)/2) = 1 ∧ ∀ (k : ℕ), k < (p-1)/2 → (-a)^k ≠ 1) :=", "goal": "p t : ℕ\nhp0 : p.Prime\nhp1 : p = 4 * t + 3\na : ZMod p\n⊢ IsPrimitiveRoot a p ↔ (-a) ^ ((p - 1) / 2) = 1 ∧ ∀ k < (p - 1) / 2, (-a) ^ k ≠ 1", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_6", "split": "test", "informal_prefix": "/-- If $p=2^{n}+1$ is a Fermat prime, show that 3 is a primitive root modulo $p$.-/\n", "formal_statement": "theorem exercise_4_6 {p n : ℕ} (hp : p.Prime) (hpn : p = 2^n + 1) :\n IsPrimitiveRoot 3 p :=", "goal": "p n : ℕ\nhp : p.Prime\nhpn : p = 2 ^ n + 1\n⊢ IsPrimitiveRoot 3 p", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_8", "split": "valid", "informal_prefix": "/-- 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 $p-1$.-/\n", "formal_statement": "theorem exercise_4_8 {p a : ℕ} (hp : Odd p) :\n IsPrimitiveRoot a p ↔ (∀ q : ℕ, q ∣ (p-1) → q.Prime → ¬ a^(p-1) ≡ 1 [MOD p]) :=", "goal": "p a : ℕ\nhp : Odd p\n⊢ IsPrimitiveRoot a p ↔ ∀ (q : ℕ), q ∣ p - 1 → q.Prime → ¬a ^ (p - 1) ≡ 1 [MOD p]", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_11", "split": "test", "informal_prefix": "/-- Prove that $1^{k}+2^{k}+\\cdots+(p-1)^{k} \\equiv 0(p)$ if $p-1 \\nmid k$ and $-1(p)$ if $p-1 \\mid k$.-/\n", "formal_statement": "theorem exercise_4_11 {p : ℕ} (hp : p.Prime) (k s: ℕ)\n (s := ∑ n : Fin p, (n : ℕ) ^ k) :\n ((¬ p - 1 ∣ k) → s ≡ 0 [MOD p]) ∧ (p - 1 ∣ k → s ≡ 0 [MOD p]) :=", "goal": "p : ℕ\nhp : p.Prime\nk s✝ : ℕ\ns : optParam ℕ (∑ n : Fin p, ↑n ^ k)\n⊢ (¬p - 1 ∣ k → s ≡ 0 [MOD p]) ∧ (p - 1 ∣ k → s ≡ 0 [MOD p])", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_13", "split": "valid", "informal_prefix": "/-- Show that any prime divisor of $x^{4}-x^{2}+1$ is congruent to 1 modulo 12 .-/\n", "formal_statement": "theorem exercise_5_13 {p x: ℤ} (hp : Prime p)\n (hpx : p ∣ (x^4 - x^2 + 1)) : p ≡ 1 [ZMOD 12] :=", "goal": "p x : ℤ\nhp : Prime p\nhpx : p ∣ x ^ 4 - x ^ 2 + 1\n⊢ p ≡ 1 [ZMOD 12]", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_28", "split": "test", "informal_prefix": "/-- Show that $x^{4} \\equiv 2(p)$ has a solution for $p \\equiv 1(4)$ iff $p$ is of the form $A^{2}+64 B^{2}$.-/\n", "formal_statement": "theorem exercise_5_28 {p : ℕ} (hp : p.Prime) (hp1 : p ≡ 1 [MOD 4]):\n ∃ x, x^4 ≡ 2 [MOD p] ↔ ∃ A B, p = A^2 + 64*B^2 :=", "goal": "p : ℕ\nhp : p.Prime\nhp1 : p ≡ 1 [MOD 4]\n⊢ ∃ x, x ^ 4 ≡ 2 [MOD p] ↔ ∃ A B, p = A ^ 2 + 64 * B ^ 2", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_37", "split": "valid", "informal_prefix": "/-- Show that if $a$ is negative then $p \\equiv q(4 a) together with p\\not | a$ imply $(a / p)=(a / q)$.-/\n", "formal_statement": "theorem exercise_5_37 {p q : ℕ} [Fact (p.Prime)] [Fact (q.Prime)] {a : ℤ}\n (ha : a < 0) (h0 : p ≡ q [ZMOD 4*a]) (h1 : ¬ ((p : ℤ) ∣ a)) :\n legendreSym p a = legendreSym q a :=", "goal": "p q : ℕ\ninst✝¹ : Fact p.Prime\ninst✝ : Fact q.Prime\na : ℤ\nha : a < 0\nh0 : ↑p ≡ ↑q [ZMOD 4 * a]\nh1 : ¬↑p ∣ a\n⊢ legendreSym p a = legendreSym q a", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_12_12", "split": "test", "informal_prefix": "/-- Show that $\\sin (\\pi / 12)$ is an algebraic number.-/\n", "formal_statement": "theorem exercise_12_12 : IsAlgebraic ℚ (sin (Real.pi/12)) :=", "goal": "⊢ IsAlgebraic ℚ (π / 12).sin", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_18_4", "split": "valid", "informal_prefix": "/-- Show that 1729 is the smallest positive integer expressible as the sum of two different integral cubes in two ways.-/\n", "formal_statement": "theorem exercise_18_4 {n : ℕ} (hn : ∃ x y z w : ℤ,\n x^3 + y^3 = n ∧ z^3 + w^3 = n ∧ x ≠ z ∧ x ≠ w ∧ y ≠ z ∧ y ≠ w) :\n n ≥ 1729 :=", "goal": "n : ℕ\nhn : ∃ x y z w, x ^ 3 + y ^ 3 = ↑n ∧ z ^ 3 + w ^ 3 = ↑n ∧ x ≠ z ∧ x ≠ w ∧ y ≠ z ∧ y ≠ w\n⊢ n ≥ 1729", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_2020_b5", "split": "valid", "informal_prefix": "/-- 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 $3-z_{1}-z_{2}-z_{3}-z_{4}+z_{1} z_{2} z_{3} z_{4} \\neq 0 .$-/\n", "formal_statement": "theorem exercise_2020_b5 (z : Fin 4 → ℂ) (hz0 : ∀ n, ‖z n‖ < 1)\n (hz1 : ∀ n : Fin 4, z n ≠ 1) :\n 3 - z 0 - z 1 - z 2 - z 3 + (z 0) * (z 1) * (z 2) * (z 3) ≠ 0 :=", "goal": "z : Fin 4 → ℂ\nhz0 : ∀ (n : Fin 4), ‖z n‖ < 1\nhz1 : ∀ (n : Fin 4), z n ≠ 1\n⊢ 3 - z 0 - z 1 - z 2 - z 3 + z 0 * z 1 * z 2 * z 3 ≠ 0", "header": "import Mathlib\n\nopen scoped BigOperators\n\nopen scoped BigOperators\n\n"}
{"name": "exercise_2018_a5", "split": "test", "informal_prefix": "/-- 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 $f^{(n)}(x)<0$.-/\n", "formal_statement": "theorem exercise_2018_a5 (f : ℝ → ℝ) (hf : ContDiff ℝ ⊤ f)\n (hf0 : f 0 = 0) (hf1 : f 1 = 1) (hf2 : ∀ x, f x ≥ 0) :\n ∃ (n : ℕ) (x : ℝ), iteratedDeriv n f x = 0 :=", "goal": "f : ℝ → ℝ\nhf : ContDiff ℝ ⊤ f\nhf0 : f 0 = 0\nhf1 : f 1 = 1\nhf2 : ∀ (x : ℝ), f x ≥ 0\n⊢ ∃ n x, iteratedDeriv n f x = 0", "header": "import Mathlib\n\nopen scoped BigOperators\n\n"}
{"name": "exercise_2018_b2", "split": "valid", "informal_prefix": "/-- 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 $\\{z \\in \\mathbb{C}:|z| \\leq 1\\}$.-/\n", "formal_statement": "theorem exercise_2018_b2 (n : ℕ) (hn : n > 0) (f : ℕ → ℂ → ℂ)\n (hf : ∀ n : ℕ, f n = λ (z : ℂ) => (∑ i : Fin n, (n-i)* z^(i : ℕ))) :\n ¬ (∃ z : ℂ, ‖z‖ ≤ 1 ∧ f n z = 0) :=", "goal": "n : ℕ\nhn : n > 0\nf : ℕ → ℂ → ℂ\nhf : ∀ (n : ℕ), f n = fun z => ∑ i : Fin n, (↑n - ↑↑i) * z ^ ↑i\n⊢ ¬∃ z, ‖z‖ ≤ 1 ∧ f n z = 0", "header": "import Mathlib\n\nopen scoped BigOperators\n\n"}
{"name": "exercise_2018_b4", "split": "test", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_2018_b4 (a : ℝ) (x : ℕ → ℝ) (hx0 : x 0 = a)\n (hx1 : x 1 = a)\n (hxn : ∀ n : ℕ, n ≥ 2 → x (n+1) = 2*(x n)*(x (n-1)) - x (n-2))\n (h : ∃ n, x n = 0) :\n ∃ c, Function.Periodic x c :=", "goal": "a : ℝ\nx : ℕ → ℝ\nhx0 : x 0 = a\nhx1 : x 1 = a\nhxn : ∀ n ≥ 2, x (n + 1) = 2 * x n * x (n - 1) - x (n - 2)\nh : ∃ n, x n = 0\n⊢ ∃ c, Function.Periodic x c", "header": "import Mathlib\n\nopen scoped BigOperators\n\n"}
{"name": "exercise_2017_b3", "split": "valid", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_2017_b3 (f : ℝ → ℝ) (c : ℕ → ℝ)\n (hf : f = λ x => (∑' (i : ℕ), (c i) * x^i))\n (hc : ∀ n, c n = 0 ∨ c n = 1)\n (hf1 : f (2/3) = 3/2) :\n Irrational (f (1/2)) :=", "goal": "f : ℝ → ℝ\nc : ℕ → ℝ\nhf : f = fun x => ∑' (i : ℕ), c i * x ^ i\nhc : ∀ (n : ℕ), c n = 0 ∨ c n = 1\nhf1 : f (2 / 3) = 3 / 2\n⊢ Irrational (f (1 / 2))", "header": "import Mathlib\n\nopen scoped BigOperators\n\n"}
{"name": "exercise_2014_a5", "split": "test", "informal_prefix": "/-- Let-/\n", "formal_statement": "theorem exercise_2014_a5 (P : ℕ → Polynomial ℤ)\n (hP : ∀ n, P n = ∑ i : Fin n, (n+1) * Polynomial.X ^ n) :\n ∀ (j k : ℕ), j ≠ k → IsCoprime (P j) (P k) :=", "goal": "P : ℕ → Polynomial ℤ\nhP : ∀ (n : ℕ), P n = ∑ i : Fin n, (↑n + 1) * Polynomial.X ^ n\n⊢ ∀ (j k : ℕ), j ≠ k → IsCoprime (P j) (P k)", "header": "import Mathlib\n\nopen scoped BigOperators\n\n"}
{"name": "exercise_2010_a4", "split": "valid", "informal_prefix": "/-- Prove that for each positive integer $n$, the number $10^{10^{10^n}}+10^{10^n}+10^n-1$ is not prime.-/\n", "formal_statement": "theorem exercise_2010_a4 (n : ℕ) :\n ¬ Nat.Prime (10^10^10^n + 10^10^n + 10^n - 1) :=", "goal": "n : ℕ\n⊢ ¬(10 ^ 10 ^ 10 ^ n + 10 ^ 10 ^ n + 10 ^ n - 1).Prime", "header": "import Mathlib\n\nopen scoped BigOperators\n\n"}
{"name": "exercise_2001_a5", "split": "test", "informal_prefix": "/-- Prove that there are unique positive integers $a, n$ such that $a^{n+1}-(a+1)^n=2001$.-/\n", "formal_statement": "theorem exercise_2001_a5 :\n ∃! a : ℕ, ∃! n : ℕ, a > 0 ∧ n > 0 ∧ a^(n+1) - (a+1)^n = 2001 :=", "goal": "⊢ ∃! a, ∃! n, a > 0 ∧ n > 0 ∧ a ^ (n + 1) - (a + 1) ^ n = 2001", "header": "import Mathlib\n\nopen scoped BigOperators\n\n"}
{"name": "exercise_2000_a2", "split": "valid", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_2000_a2 :\n ∀ N : ℕ, ∃ n : ℕ, n > N ∧ ∃ i : Fin 6 → ℕ, n = (i 0)^2 + (i 1)^2 ∧\n n + 1 = (i 2)^2 + (i 3)^2 ∧ n + 2 = (i 4)^2 + (i 5)^2 :=", "goal": "⊢ ∀ (N : ℕ), ∃ n > N, ∃ i, n = i 0 ^ 2 + i 1 ^ 2 ∧ n + 1 = i 2 ^ 2 + i 3 ^ 2 ∧ n + 2 = i 4 ^ 2 + i 5 ^ 2", "header": "import Mathlib\n\nopen scoped BigOperators\n\n"}
{"name": "exercise_1999_b4", "split": "test", "informal_prefix": "/-- 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 $x$.-/\n", "formal_statement": "theorem exercise_1999_b4 (f : ℝ → ℝ) (hf: ContDiff ℝ 3 f)\n (hf1 : ∀ n ≤ 3, ∀ x : ℝ, iteratedDeriv n f x > 0)\n (hf2 : ∀ x : ℝ, iteratedDeriv 3 f x ≤ f x) :\n ∀ x : ℝ, deriv f x < 2 * f x :=", "goal": "f : ℝ → ℝ\nhf : ContDiff ℝ 3 f\nhf1 : ∀ n ≤ 3, ∀ (x : ℝ), iteratedDeriv n f x > 0\nhf2 : ∀ (x : ℝ), iteratedDeriv 3 f x ≤ f x\n⊢ ∀ (x : ℝ), deriv f x < 2 * f x", "header": "import Mathlib\n\nopen scoped BigOperators\n\n"}
{"name": "exercise_1998_a3", "split": "valid", "informal_prefix": "/-- Let $f$ be a real function on the real line with continuous third derivative. Prove that there exists a point $a$ such that-/\n", "formal_statement": "theorem exercise_1998_a3 (f : ℝ → ℝ) (hf : ContDiff ℝ 3 f) :\n ∃ a : ℝ, (f a) * (deriv f a) * (iteratedDeriv 2 f a) * (iteratedDeriv 3 f a) ≥ 0 :=", "goal": "f : ℝ → ℝ\nhf : ContDiff ℝ 3 f\n⊢ ∃ a, f a * deriv f a * iteratedDeriv 2 f a * iteratedDeriv 3 f a ≥ 0", "header": "import Mathlib\n\nopen scoped BigOperators\n\n"}
{"name": "exercise_1998_b6", "split": "test", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_1998_b6 (a b c : ℤ) :\n ∃ n : ℤ, n > 0 ∧ ¬ ∃ m : ℤ, Real.sqrt (n^3 + a*n^2 + b*n + c) = m :=", "goal": "a b c : ℤ\n⊢ ∃ n > 0, ¬∃ m, √(↑n ^ 3 + ↑a * ↑n ^ 2 + ↑b * ↑n + ↑c) = ↑m", "header": "import Mathlib\n\nopen scoped BigOperators\n\n"}
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