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cauchy_type1.jsonl
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{"name": "cauchy_p26", "
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{"name": "cauchy_p27", "
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{"name": "cauchy_p28", "
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{"name": "cauchy_p29", "
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{"name": "cauchy_p30", "
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{"name": "cauchy_p31", "
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{"name": "cauchy_p32", "
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{"name": "cauchy_p33", "
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{"name": "cauchy_p34", "
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{"name": "cauchy_p35", "
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{"name": "cauchy_p36", "
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{"name": "cauchy_p37", "
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{"name": "cauchy_p38", "
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{"name": "cauchy_p39", "
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{"name": "cauchy_p40", "
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{"name": "cauchy_p41", "
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{"name": "cauchy_p42", "
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{"name": "cauchy_p43", "
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{"name": "cauchy_p44", "
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{"name": "cauchy_p45", "
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{"name": "cauchy_p46", "
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{"name": "cauchy_p47", "
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{"name": "cauchy_p48", "
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{"name": "cauchy_p49", "
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{"name": "cauchy_p50", "
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{"name": "cauchy_p26", "formal_statement": "theorem cauchy_p26 (x y a b : ℝ) (h₁ : a > 0 ∧ b > 0) (h₂ : x > 0 ∧ y > 0) : ( x + y ) * ( 1 / x + 1 / y ) * ( a + b ) * ( 1 / a + 1 / b ) ≥ 16 := by\n"}
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{"name": "cauchy_p27", "formal_statement": "theorem cauchy_p2 (x y z a b c: ℝ) (h₁ : a > 0 ∧ b > 0 ∧ c > 0) (h₂ : x > 0 ∧ y > 0 ∧ z > 0 ) : ( x + y + z ) * ( 1 / x + 1 / y + 1 / z ) * ( a + b + c ) * ( 1 / a + 1 / b + 1 / c ) ≥ 81 := by\n"}
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{"name": "cauchy_p28", "formal_statement": "theorem cauchy_p28 (x y a b: ℝ) (hx : x ≥ 0) (hy : y ≥ 0) (hxy : x + y ≤ 1) (ha : a ≥ 0) (hb : b ≥ 0) (hab : a + b ≤ 1) : (4 * x^2 + 4 * y^2 + (1 - x - y)^2) * (4 * a^2 + 4 * b^2 + (1 - a - b)^2) ≥ 4 / 9 := by\n"}
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{"name": "cauchy_p29", "formal_statement": "theorem cauchy_p29 (x y a b: ℝ) (hx : x ≥ 0) (hy : y ≥ 0) (hx1 : x ≤ 1) (hy1 : y ≤ 1) (ha : a ≥ 0) (hb : b ≥ 0) (ha1 : a ≤ 1) (hb1 : b ≤ 1) : (x * √(1 - y^2) + y * √(1 - x^2)) * (a * √(1 - b^2) + b * √(1 - a^2)) ≤ 1 := by\n"}
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{"name": "cauchy_p30", "formal_statement": "theorem cauchy_p30 (x y z a b c: ℝ) (h : x > 0 ∧ y > 0 ∧ z > 0) (g : x + y + z = 3) (h1 : a > 0 ∧ b > 0 ∧ c > 0) (g1 : a + b + c = 3) : (4 / x + 1 / y + 9 / z) * (4 / a + 1 / b + 9 / c) ≥ 144 := by\n"}
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{"name": "cauchy_p31", "formal_statement": "theorem cauchy_p31 (x y z a b c: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hx : x > 0) (hy : y > 0) (hz : z > 0) : (a / (b + c) + b / (c + a) + c / (a + b)) * (x / (y + z) + y / (z + x) + z / (x + y)) ≥ 9 / 4 := by\n"}
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{"name": "cauchy_p32", "formal_statement": "theorem cauchy_p32 (a b c d x y z w: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hx : x > 0) (hy : y > 0) (hz : z > 0) (hw : w > 0) (hd : d > 0) (h : a + b + c + d = 1) (g : x + y + z + w = 1) : (1 / (b + c + d) + 1 / (c + d + a) + 1 / (a + b + d) + 1 / (a + b + c)) * (1 / (y + z + w) + 1 / (z + w + x) + 1 / (x + y + w) + 1 / (x + y + z)) ≥ 256 / 9 := by\n"}
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{"name": "cauchy_p33", "formal_statement": "theorem cauchy_p33 (x y z a b c: ℝ) (h : x > 0 ∧ y > 0 ∧ z > 0) (h1 : a > 0 ∧ b > 0 ∧ c > 0) (g : x * (x + y) + y * (y + z) + z * (z + x) = 1) (g1 : a * (a + b) + b * (b + c) + c * (c + a) = 1) : (x / (x + y) + y / (y + z) + z / (z + x)) * (a / (a + b) + b / (b + c) + c / (c + a)) ≥ (x + y + z) ^ 2 * (a + b + c) ^ 2 := by\n"}
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{"name": "cauchy_p34", "formal_statement": "theorem cauchy_p34 (x y z a b c: ℝ) (h : x > 0 ∧ y > 0 ∧ z > 0) (h1 : a > 0 ∧ b > 0 ∧ c > 0) (g : z * (x + y) + x * (y + z) + y * (z + x) = 1) (g1 : c * (a + b) + a * (b + c) + b * (c + a) = 1) : (z / (x + y) + x / (y + z) + y / (z + x)) * (c / (a + b) + a / (b + c) + b / (c + a)) ≥ (x + y + z) ^ 2 * (a + b + c) ^ 2 := by\n"}
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{"name": "cauchy_p35", "formal_statement": "theorem cauchy_p35 (x y a b: ℝ) (hx : x > 0) (hy : y > 0) (ha : a > 0) (hb : b > 0) (g : √(2 * x + 1) + √(2 * y + 3) = 4) (h : √(2 * a + 1) + √(2 * b + 3) = 4) : (x + y) * (a + b) ≥ 4 := by\n"}
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{"name": "cauchy_p36", "formal_statement": "theorem cauchy_p36 (x y z a b c: ℝ) (h : x^2 + 2 * y^2 + 4 * z^2 > 0) (g : a^2 + 2 * b^2 + 4 * c^2 > 0) : (x + y + z)^2 / (x^2 + 2 * y^2 + 4 * z^2) * (a + b + c)^2 / (a^2 + 2 * b^2 + 4 * c^2) ≤ 49 / 16 := by\n"}
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{"name": "cauchy_p37", "formal_statement": "theorem cauchy_p37 (x y a b: ℝ) (hx : x > 0) (hy : y > 0) (ha : a > 0) (hb : b > 0) (g : 1 / (2 * x + y) + 3 / (x + y) = 2) (h : 1 / (2 * a + b) + 3 / (a + b) = 2) : (6 * x + 5 * y) * (6 * a + 5 * b) ≥ 217 / 4 + 26 * √3 := by\n"}
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{"name": "cauchy_p38", "formal_statement": "theorem cauchy_p38 (x y z a b c: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hx : x > 0) (hy : y > 0) (hz : z > 0) (h : a + b + c = 1) (g : x + y + z = 1) : (√(2 * a + 1) + √(2 * b + 1) + √(2 * c + 1)) * (√(2 * x + 1) + √(2 * y + 1) + √(2 * z + 1)) ≤ 15 := by\n"}
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{"name": "cauchy_p39", "formal_statement": "theorem cauchy_p39 (n : ℕ) (a b x y : Fin n → ℝ) (ha : ∀ i, a i > 0) (hb : ∀ i, b i > 0) (hx : ∀ i, x i > 0) (hy : ∀ i, y i > 0) : (∑ i, a i / b i) * (∑ i, x i / y i) ≥ ((∑ i, a i)^2 / ∑ i, a i * b i) * ((∑ i, x i)^2 / ∑ i, x i * y i) := by\n"}
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{"name": "cauchy_p40", "formal_statement": "theorem cauchy_p40 (n : ℕ) (a b x y : Fin n → ℝ) (ha : ∀ i, a i > 0) (hb : ∀ i, b i > 0) (hx : ∀ i, x i > 0) (hy : ∀ i, y i > 0) : (∑ i, a i / (b i)^2) * (∑ i, x i / (y i)^2) ≥ ((∑ i, a i / b i)^2 / ∑ i, a i) * ((∑ i, x i / y i)^2 / ∑ i, x i) := by\n"}
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{"name": "cauchy_p41", "formal_statement": "theorem cauchy_p41 (x y a b z w c d : ℝ) (hy : y ≠ 0) (hb : b ≠ 0) (hw : w ≠ 0) (hd : d ≠ 0) (hxy : x^2 + 1 / y^2 = 1) (hab : a^2 + 1 / b^2 = 4) (hzw : z^2 + 1 / w^2 = 1) (hcd : c^2 + 1 / d^2 = 4) : |(a / y + x / b) * (c / w + z / d)| ≤ 4 := by\n"}
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{"name": "cauchy_p42", "formal_statement": "theorem cauchy_p42 (a b c d e x y z w u : ℝ) (h : (a - b)^2 + (b - c)^2 + (c - d)^2 + (d - e)^2 = 1) (g : (x - y)^2 + (y - z)^2 + (z - w)^2 + (w - u)^2 = 1) : (a - 2 * b - c + 2 * e) * (x - 2 * y - z + 2 * u) ≤ 10 := by\n"}
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{"name": "cauchy_p43", "formal_statement": "theorem cauchy_p43 (n : ℕ) (hn : n > 2) (a : Fin n → ℝ) (x : Fin n → ℝ) (ha1 : ∀ i, a i < 1) (ha2 : ∀ i, a i ≥ 0) (hx1 : ∀ i, x i < 1) (hx2 : ∀ i, x i ≥ 0) (hs1 : ∑ i : Fin n, a i = n - 2) (hs2 : ∑ i : Fin n, x i = n - 2) : (∑ i : Fin n, ((a i)^2 / (1 - a i))) * (∑ i : Fin n, ((x i)^2 / (1 - x i))) ≥ ((n : ℝ) - 2)^4 / 4 := by\n"}
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{"name": "cauchy_p44", "formal_statement": "theorem cauchy_p44 (x y z a b c : ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) (ha : a > 0) (hb : b > 0) (hc : c > 0) (h : 1 / (1 + x^2) + 1 / (1 + y^2) + 1 / (1 + z^2) = 2) (g : 1 / (1 + a^2) + 1 / (1 + b^2) + 1 / (1 + c^2) = 2) : (x^2 + y^2 + z^2 + 3) * (a^2 + b^2 + c^2 + 3) ≥ (x + y + z)^2 * (a + b + c)^2 := by\n"}
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{"name": "cauchy_p45", "formal_statement": "theorem cauchy_p45 (a b c x y z : ℝ) (ha : a > 1) (hb : b > 1) (hc : c > 1) (hx : x > 1) (hy : y > 1) (hz : z > 1) (h : (a^2-1)/2 + (b^2-1)/2 + (c^2-1)/3 = 1) (g : (x^2-1)/2 + (y^2-1)/2 + (z^2-1)/3 = 1) : (a + b + c) * (x + y + z) ≤ 49 / 3 := by\n"}
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{"name": "cauchy_p46", "formal_statement": "theorem cauchy_p46 (n : ℕ) (a b x y : Fin n → ℝ) (hn : n > 0) (ha : ∀ i, a i > 0) (hb : ∀ i, b i > 0) (hx : ∀ i, x i > 0) (hy : ∀ i, y i > 0) (sum_eq1 : ∑ i, a i = ∑ i , b i) (sum_eq2 : ∑ i, x i = ∑ i , b i): (∑ i, (a i) ^ 2 / (a i + b i)) * (∑ i, (x i) ^ 2 / (x i + y i)) ≥ (∑ i, a i) * (∑ i, x i) / 4 := by\n"}
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{"name": "cauchy_p47", "formal_statement": "theorem cauchy_p47 (a b c d e s1 x y z w u s2 : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hx : x > 0) (hy : y > 0) (hz : z > 0) (hd : d > 0) (he : e > 0) (hw : w > 0) (hu : u > 0) (hs1 : s1 = a + b + c + d + e) (hs2 : s2 = x + y + z + w + u) : (a^2 / (a^2 + b * (s1 - b)) + b^2 / (b^2 + c * (s1 - c)) + c^2 / (c^2 + d * (s1 - d)) + d^2 / (d^2 + e * (s1 - e)) + e^2 / (e^2 + a * (s1 - a))) * (x^2 / (x^2 + y * (s2 - y)) + y^2 / (y^2 + z * (s2 - z)) + z^2 / (z^2 + w * (s2 - w)) + w^2 / (w^2 + u * (s2 - u)) + u^2 / (u^2 + x * (s2 - x))) ≥ 1 := by\n"}
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{"name": "cauchy_p48", "formal_statement": "theorem cauchy_p48 (x y a b: ℝ) (hx : x > 0) (hy : y > 0) (ha : a > 0) (hb : b > 0) (h : x^2 + y^2 / 2 = 1) (g : a^2 + b^2 / 2 = 1) : (x + √(2 + 3 * y^2)) * (a + √(2 + 3 * b^2)) ≤ 28 / 3 := by\n"}
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{"name": "cauchy_p49", "formal_statement": "theorem cauchy_p49 (x y z a b c: ℝ) (h : x > 0 ∧ y > 0 ∧ z > 0) (h1 : a > 0 ∧ b > 0 ∧ c > 0) (hxy : 2 * x - y^2 / x > 0) (hyz : 2 * y - z^2 / y > 0) (hzx : 2 * z - x^2 / z > 0) (hab : 2 * a - b^2 / a > 0) (hbc : 2 * b - c^2 / b > 0) (hca : 2 * c - a^2 / c > 0) : (x^3 / (2 * x - y^2 / x) + y^3 / (2 * y - z^2 / y) + z^3 / (2 * z - x^2 / z)) * (a^3 / (2 * a - b^2 / a) + b^3 / (2 * b - c^2 / b) + c^3 / (2 * c - a^2 / c)) ≥ (x^2 + y^2 + z^2) * (a^2 + b^2 + c^2) := by\n"}
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{"name": "cauchy_p50", "formal_statement": "theorem cauchy_p50 (n : ℕ) (x a : Fin n → ℝ) (sx sa : ℝ) (hn : n > 2) (hsx : sx = ∑ i, x i) (hx : ∀ i, x i < sx - x i) (hsa : sa = ∑ i, a i) (ha : ∀ i, a i < sa - a i) : (∑ i, (x i)^2 / (sx - 2 * x i)) * (∑ i, (a i)^2 / (sa - 2 * a i)) ≥ sx * sa / (n - 2)^2 := by\n"}
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