Upload cauchy_type2.jsonl with huggingface_hub
Browse files- cauchy_type2.jsonl +25 -25
cauchy_type2.jsonl
CHANGED
|
@@ -1,25 +1,25 @@
|
|
| 1 |
-
{"name": "cauchy_p51", "
|
| 2 |
-
{"name": "cauchy_p52", "
|
| 3 |
-
{"name": "cauchy_p53", "
|
| 4 |
-
{"name": "cauchy_p54", "
|
| 5 |
-
{"name": "cauchy_p55", "
|
| 6 |
-
{"name": "cauchy_p56", "
|
| 7 |
-
{"name": "cauchy_p57", "
|
| 8 |
-
{"name": "cauchy_p58", "
|
| 9 |
-
{"name": "cauchy_p59", "
|
| 10 |
-
{"name": "cauchy_p60", "
|
| 11 |
-
{"name": "cauchy_p61", "
|
| 12 |
-
{"name": "cauchy_p62", "
|
| 13 |
-
{"name": "cauchy_p63", "
|
| 14 |
-
{"name": "cauchy_p64", "
|
| 15 |
-
{"name": "cauchy_p65", "
|
| 16 |
-
{"name": "cauchy_p66", "
|
| 17 |
-
{"name": "cauchy_p67", "
|
| 18 |
-
{"name": "cauchy_p68", "
|
| 19 |
-
{"name": "cauchy_p69", "
|
| 20 |
-
{"name": "cauchy_p70", "
|
| 21 |
-
{"name": "cauchy_p71", "
|
| 22 |
-
{"name": "cauchy_p72", "
|
| 23 |
-
{"name": "cauchy_p73", "
|
| 24 |
-
{"name": "cauchy_p74", "
|
| 25 |
-
{"name": "cauchy_p75", "
|
|
|
|
| 1 |
+
{"name": "cauchy_p51", "formal_statement": "theorem cauchy_p51 (x y : ℝ) (h₂ : x < 1 ∧ y < 1) : ( 2 - x - y ) * ( 1 / (1 - x) + 1 / (1 - y) ) ≥ 4 := by\n"}
|
| 2 |
+
{"name": "cauchy_p52", "formal_statement": "theorem cauchy_p52 (x y z: ℝ) (h₂ : x > 0 ∧ y > 0 ∧ z > 0 ) : 2 * ( x + y + z ) * ( 1 / (x + y) + 1 / (y + z) + 1 / (z + x) ) ≥ 9 := by\n"}
|
| 3 |
+
{"name": "cauchy_p53", "formal_statement": "theorem cauchy_p53 (x y: ℝ) (hx : x ≥ 0) (hy : y ≥ 0) (hxy : √x + √y ≤ 1) : 4 * x + 4 * y + (1 - √x - √y)^2 ≥ 2 / 3 := by\n"}
|
| 4 |
+
{"name": "cauchy_p54", "formal_statement": "theorem cauchy_p54 (x y: ℝ) (hx : x ≥ 0) (hy : y ≥ 0) (hx1 : x ≤ 1) (hy1 : y ≤ 1) : (1 - x) * √(2 * y - y^2) + (1 - y) * √(2 * x - x^2) ≤ 1 := by\n"}
|
| 5 |
+
{"name": "cauchy_p55", "formal_statement": "theorem cauchy_p55 (x y z: ℝ) (h : x > 1 ∧ y > 1 ∧ z > 1) (g : x * y * z = exp 3) : 4 / log x + 1 / log y + 9 / log z ≥ 12 := by\n"}
|
| 6 |
+
{"name": "cauchy_p56", "formal_statement": "theorem cauchy_p56 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) : a^5 / (b^5 + c^5) + b^5 / (c^5 + a^5) + c^5 / (a^5 + b^5) ≥ 3 / 2 := by\n"}
|
| 7 |
+
{"name": "cauchy_p57", "formal_statement": "theorem cauchy_p57 (a b c d : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hd : d > 0) (h : 1 / a + 1 / b + 1 / c + 1 / d = 1) : (b * c * d) / (b * c + c * d + d * b) + (c * d * a) / (c * d + d * a + a * c) + (a * b * d) / (a * b + b * d + d * a) + (a * b * c) / (a * b + b * c + c * a) ≥ 16 / 3 := by\n"}
|
| 8 |
+
{"name": "cauchy_p58", "formal_statement": "theorem cauchy_p58 (x y z: ℝ) (h : x + y > z ∧ y + z > x ∧ z + x > y) (g : (y + z - x) * x + (x + y - z) * z + (z + x - y) * y = 2) : (y + z - x) / x + (z + x - y) / y + (x + y - z) / z ≥ 2 * (x + y + z) ^ 2 := by\n"}
|
| 9 |
+
{"name": "cauchy_p59", "formal_statement": "theorem cauchy_p59 (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 * (√(x * y) + √(y * z) + √(z * x)) := by\n"}
|
| 10 |
+
{"name": "cauchy_p60", "formal_statement": "theorem cauchy_p60 (x y: ℝ) (hx : x > √2) (hy : y > √2) (g : √(x^2 - 1) + √(y^2 + 1) = 4) : (x^2 - 2) / 2 + (y^2 - 2) / 2 ≥ 2 := by\n"}
|
| 11 |
+
{"name": "cauchy_p61", "formal_statement": "theorem cauchy_p61 (x y z: ℝ) (h : x^2 + 2 * y^2 + 4 * z^2 > 0) : (x + y + z) / √(x^2 + 2 * y^2 + 4 * z^2) ≤ √7 / 2 := by\n"}
|
| 12 |
+
{"name": "cauchy_p62", "formal_statement": "theorem cauchy_p62 (x y: ℝ) (hx : x + y > 0) (hy : x - y > 0) (g : 1 / (2 * (x - y)^2 + (x + y)^2) + 3 / ((x - y)^2 + (x + y)^2) = 2) : 6 * (x - y)^2 + 5 * (x + y)^2 ≥ 13 / 2 + 2 * √3 := by\n"}
|
| 13 |
+
{"name": "cauchy_p63", "formal_statement": "theorem cauchy_p63 (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"}
|
| 14 |
+
{"name": "cauchy_p64", "formal_statement": "theorem cauchy_p64 (n : ℕ) (a b c : Fin n → ℝ) (ha : ∀ i, a i > 0) (hb : ∀ i, b i > 0) (hc : ∀ i, c i > 0) : ∑ i, a i * c i / b i ≥ (∑ i, a i)^2 / ∑ i, a i * b i / c i := by\n"}
|
| 15 |
+
{"name": "cauchy_p65", "formal_statement": "theorem cauchy_p65 (n : ℕ) (a b : Fin n → ℝ) (ha : ∀ i, a i > 0) (hb : ∀ i, b i > 0) : ∑ i, 1 / (a i * (b i)^2) ≥ (∑ i, 1 / b i)^2 / ∑ i, a i := by\n"}
|
| 16 |
+
{"name": "cauchy_p66", "formal_statement": "theorem cauchy_p66 (x y a b: ℝ) (hx : x > 0) (hy : y > 0) (ha : a > 0) (hb : b > 0) (hxy : x + 1 / y = 1) (hab : a + 1 / b = 4) : √a / √y + √x / √b ≤ 2 := by\n"}
|
| 17 |
+
{"name": "cauchy_p67", "formal_statement": "theorem cauchy_p67 (a b c d e : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hd : d > 0) (he : e > 0) (h : (log (a / b))^2 + (log (b / c))^2 + (log (c / d))^2 + (log (d / e))^2 = 1) : log (a * e^2 / (b^2 * c)) ≤ √10 := by\n"}
|
| 18 |
+
{"name": "cauchy_p68", "formal_statement": "theorem cauchy_p68 (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)^2 = n - 2) : ∑ i : Fin n, ((a i)^4 / (1 - (a i)^2)) ≥ ((n : ℝ) - 2)^2 / 2 := by\n"}
|
| 19 |
+
{"name": "cauchy_p69", "formal_statement": "theorem cauchy_p69 (x y z : ℝ) (hx : x > 1) (hy : y > 1) (hz : z > 1) (h : 1/x + 1/y + 1/z = 2) : Real.sqrt (x + y + z) ≥ Real.sqrt (x - 1) + Real.sqrt (y - 1) + Real.sqrt (z - 1) := by\n"}
|
| 20 |
+
{"name": "cauchy_p70", "formal_statement": "theorem cauchy_p70 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (h : a + b + c = 1) : √(2 * a + 1) + √(2 * b + 1) + √(3 * c + 1) ≤ 7 * √3 / 3 := by\n"}
|
| 21 |
+
{"name": "cauchy_p71", "formal_statement": "theorem cauchy_p71 (n : ℕ) (a b : Fin n → ℝ) (hn : n > 0) (ha : ∀ i, a i > 0) (hb : ∀ i, b i > 0) (sum_eq : ∑ i, (1 / a i) = ∑ i , b i): ∑ i, 1 / (a i * (1 + a i * b i)) ≥ (∑ i, 1 / a i) / 2 := by\n"}
|
| 22 |
+
{"name": "cauchy_p72", "formal_statement": "theorem cauchy_p72 (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 / (a + √b * (s - √b)) + b / (b + √c * (s - √c)) + c / (c + √d * (s - √d)) + d / (d + √e * (s - √e)) + e / (e + √a * (s - √a)) ≥ 1 := by\n"}
|
| 23 |
+
{"name": "cauchy_p73", "formal_statement": "theorem cauchy_p73 (x y: ℝ) (hx : x > 0) (hy : y > 0) (g : x^2 + y^2 / 2 = 2) : √(x^2 - 1) + √(2 + 3 * y^2) ≤ 2 * √21 / 3 := by\n"}
|
| 24 |
+
{"name": "cauchy_p74", "formal_statement": "theorem cauchy_p74 (x y z: ℝ) (h : x > 0 ∧ y > 0 ∧ z > 0) (hxy : 2 / x - x / y^2 > 0) (hyz : 2 / y - y / z^2 > 0) (hzx : 2 / z - z / x^2 > 0) : 1 / (2 * x^2 - x^4 / y^2) + 1 / (2 * y^2 - y^4 / z^2) + 1 / (2 * z^2 - z^4 / x^2) ≥ 1 / x^2 + 1 / y^2 + 1 / z^2 := by\n"}
|
| 25 |
+
{"name": "cauchy_p75", "formal_statement": "theorem cauchy_p75 (n : ℕ) (x : Fin n → ℝ) (s : ℝ) (hn : n > 2) (hs : s = n - ∑ i, x i) (hx : ∀ i, 2 < s + 2 * x i) : ∑ i, (1 - x i)^2 / (s + 2 * x i - 2) ≥ s / (n - 2) := by\n"}
|