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Browse files- cauchy_seed.jsonl +25 -25
cauchy_seed.jsonl
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{"name": "cauchy_p1", "
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{"name": "cauchy_p2", "
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{"name": "cauchy_p3", "
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{"name": "cauchy_p4", "
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{"name": "cauchy_p5", "
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{"name": "cauchy_p6", "
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{"name": "cauchy_p7", "
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{"name": "cauchy_p8", "
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{"name": "cauchy_p9", "
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{"name": "cauchy_p10", "
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{"name": "cauchy_p11", "
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{"name": "cauchy_p12", "
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{"name": "cauchy_p13", "
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{"name": "cauchy_p14", "
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{"name": "cauchy_p15", "
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{"name": "cauchy_p16", "
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{"name": "cauchy_p17", "
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{"name": "cauchy_p18", "
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{"name": "cauchy_p19", "
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{"name": "cauchy_p20", "
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{"name": "cauchy_p21", "
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{"name": "cauchy_p22", "
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{"name": "cauchy_p23", "
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{"name": "cauchy_p24", "
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{"name": "cauchy_p25", "
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{"name": "cauchy_p1", "formal_statement": "theorem cauchy_p1 (x y : ℝ) (h₂ : x > 0 ∧ y > 0) : ( x + y ) * ( 1 / x + 1 / y ) ≥ 4 := by\n"}
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{"name": "cauchy_p2", "formal_statement": "theorem cauchy_p2 (x y z: ℝ) (h₂ : x > 0 ∧ y > 0 ∧ z > 0 ) : ( x + y + z ) * ( 1 / x + 1 / y + 1 / z ) ≥ 9 := by\n"}
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{"name": "cauchy_p3", "formal_statement": "theorem cauchy_p3 (x y: ℝ) (hx : x ≥ 0) (hy : y ≥ 0) (hxy : x + y ≤ 1) : 4 * x^2 + 4 * y^2 + (1 - x - y)^2 ≥ 2 / 3 := by\n"}
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{"name": "cauchy_p4", "formal_statement": "theorem cauchy_p4 (x y: ℝ) (hx : x ≥ 0) (hy : y ≥ 0) (hx1 : x ≤ 1) (hy1 : y ≤ 1) : x * √(1 - y^2) + y * √(1 - x^2) ≤ 1 := by\n"}
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{"name": "cauchy_p5", "formal_statement": "theorem cauchy_p5 (x y z: ℝ) (h : x > 0 ∧ y > 0 ∧ z > 0) (g : x + y + z = 3) : 4 / x + 1 / y + 9 / z ≥ 12 := by\n"}
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{"name": "cauchy_p6", "formal_statement": "theorem cauchy_p6 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) : a / (b + c) + b / (c + a) + c / (a + b) ≥ 3 / 2 := by\n"}
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{"name": "cauchy_p7", "formal_statement": "theorem cauchy_p7 (a b c d : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hd : d > 0) (h : a + b + c + d = 1) : 1 / (b + c + d) + 1 / (c + d + a) + 1 / (a + b + d) + 1 / (a + b + c) ≥ 16 / 3 := by\n"}
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{"name": "cauchy_p8", "formal_statement": "theorem cauchy_p8 (x y z: ℝ) (h : x > 0 ∧ y > 0 ∧ z > 0) (g : x * (x + y) + y * (y + z) + z * (z + x) = 1) : x / (x + y) + y / (y + z) + z / (z + x) ≥ (x + y + z) ^ 2 := by\n"}
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{"name": "cauchy_p9", "formal_statement": "theorem cauchy_p9 (x y z: ℝ) (h : x > 0 ∧ y > 0 ∧ z > 0) ( g : z * (x + y) + x * (y + z) + y * (z + x) = 1) : z / (x + y) + x / (y + z) + y / (z + x) ≥ (x + y + z) ^ 2 := by\n"}
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{"name": "cauchy_p10", "formal_statement": "theorem cauchy_p10 (x y: ℝ) (hx : x > 0) (hy : y > 0) (g : √(2 * x + 1) + √(2 * y + 3) = 4) : x + y ≥ 2 := by\n"}
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{"name": "cauchy_p11", "formal_statement": "theorem cauchy_p11 (x y z: ℝ) (h : x^2 + 2 * y^2 + 4 * z^2 > 0) : (x + y + z)^2 / (x^2 + 2 * y^2 + 4 * z^2) ≤ 7 / 4 := by\n"}
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{"name": "cauchy_p12", "formal_statement": "theorem cauchy_p12 (x y: ℝ) (hx : x > 0) (hy : y > 0) (g : 1 / (2 * x + y) + 3 / (x + y) = 2) : 6 * x + 5 * y ≥ 13 / 2 + 2 * √3 := by\n"}
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{"name": "cauchy_p13", "formal_statement": "theorem cauchy_p13 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (h : a + b + c = 1) : √(2 * a + 1) + √(2 * b + 1) + √(2 * c + 1) ≤ √15 := by\n"}
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{"name": "cauchy_p14", "formal_statement": "theorem cauchy_p14 (n : ℕ) (a b : Fin n → ℝ) (ha : ∀ i, a i > 0) (hb : ∀ i, b i > 0) : ∑ i, a i / b i ≥ (∑ i, a i)^2 / ∑ i, a i * b i := by\n"}
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{"name": "cauchy_p15", "formal_statement": "theorem cauchy_p15 (n : ℕ) (a b : Fin n → ℝ) (ha : ∀ i, a i > 0) (hb : ∀ i, b i > 0) : ∑ i, a i / (b i)^2 ≥ (∑ i, a i / b i)^2 / ∑ i, a i := by\n"}
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{"name": "cauchy_p16", "formal_statement": "theorem cauchy_p16 (x y a b: ℝ) (hy : y ≠ 0) (hb : b ≠ 0) (hxy : x^2 + 1 / y^2 = 1) (hab : a^2 + 1 / b^2 = 4) : |a / y + x / b| ≤ 2 := by\n"}
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{"name": "cauchy_p17", "formal_statement": "theorem cauchy_p17 (a b c d e : ℝ) (h : (a - b)^2 + (b - c)^2 + (c - d)^2 + (d - e)^2 = 1) : a - 2 * b - c + 2 * e ≤ √10 := by\n"}
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{"name": "cauchy_p18", "formal_statement": "theorem cauchy_p18 (n : ℕ) (hn : n > 2) (a : Fin n → ℝ) (ha1 : ∀ i, a i < 1) (ha2 : ∀ i, a i �� 0) (hs : ∑ i : Fin n, a i = n - 2) : ∑ i : Fin n, ((a i)^2 / (1 - a i)) ≥ ((n : ℝ) - 2)^2 / 2 := by\n"}
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{"name": "cauchy_p19", "formal_statement": "theorem cauchy_p19 (x y z : ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) (h : 1 / (1 + x^2) + 1 / (1 + y^2) + 1 / (1 + z^2) = 2) : x^2 + y^2 + z^2 + 3 ≥ (x + y + z)^2 := by\n"}
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{"name": "cauchy_p20", "formal_statement": "theorem cauchy_p20 (a b c : ℝ) (ha : a > 1) (hb : b > 1) (hc : c > 1) (h : (a^2-1)/2 + (b^2-1)/2 + (c^2-1)/3 = 1) : a + b + c ≤ 7 * √3 / 3 := by\n"}
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{"name": "cauchy_p21", "formal_statement": "theorem cauchy_p21 (n : ℕ) (a b : Fin n → ℝ) (hn : n > 0) (ha : ∀ i, a i > 0) (hb : ∀ i, b i > 0) (sum_eq : ∑ i, a i = ∑ i , b i): ∑ i, (a i) ^ 2 / (a i + b i) ≥ (∑ i, a i) / 2 := by\n"}
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{"name": "cauchy_p22", "formal_statement": "theorem cauchy_p22 (a b c d e s : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hd : d > 0) (he : e > 0) (hs : s = a + b + c + d + e) : a^2 / (a^2 + b * (s - b)) + b^2 / (b^2 + c * (s - c)) + c^2 / (c^2 + d * (s - d)) + d^2 / (d^2 + e * (s - e)) + e^2 / (e^2 + a * (s - a)) ≥ 1 := by\n"}
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{"name": "cauchy_p23", "formal_statement": "theorem cauchy_p23 (x y: ℝ) (hx : x > 0) (hy : y > 0) (g : x^2 + y^2 / 2 = 1) : x + √(2 + 3 * y^2) ≤ 2 * √21 / 3 := by\n"}
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{"name": "cauchy_p24", "formal_statement": "theorem cauchy_p24 (x y z: ℝ) (h : x > 0 ∧ y > 0 ∧ z > 0) (hxy : 2 * x - y^2 / x > 0) (hyz : 2 * y - z^2 / y > 0) (hzx : 2 * z - x^2 / z > 0) : x^3 / (2 * x - y^2 / x) + y^3 / (2 * y - z^2 / y) + z^3 / (2 * z - x^2 / z) ≥ x^2 + y^2 + z^2 := by\n"}
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{"name": "cauchy_p25", "formal_statement": "theorem cauchy_p25 (n : ℕ) (x : Fin n → ℝ) (s : ℝ) (hn : n > 2) (hs : s = ∑ i, x i) (hx : ∀ i, x i < s - x i) : ∑ i, (x i)^2 / (s - 2 * x i) ≥ s / (n - 2) := by\n"}
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