Upload misc_seed.jsonl with huggingface_hub
Browse files- misc_seed.jsonl +25 -25
misc_seed.jsonl
CHANGED
|
@@ -1,25 +1,25 @@
|
|
| 1 |
-
{"name": "jensen_p1", "
|
| 2 |
-
{"name": "jensen_p2", "
|
| 3 |
-
{"name": "jensen_p3", "
|
| 4 |
-
{"name": "jensen_p4", "
|
| 5 |
-
{"name": "jensen_p5", "
|
| 6 |
-
{"name": "jensen_p6", "
|
| 7 |
-
{"name": "jensen_p7", "
|
| 8 |
-
{"name": "jensen_p8", "
|
| 9 |
-
{"name": "jensen_p9", "
|
| 10 |
-
{"name": "jensen_p10", "
|
| 11 |
-
{"name": "induction_p1", "
|
| 12 |
-
{"name": "induction_p2", "
|
| 13 |
-
{"name": "induction_p3", "
|
| 14 |
-
{"name": "induction_p4", "
|
| 15 |
-
{"name": "induction_p5", "
|
| 16 |
-
{"name": "schur_p1", "
|
| 17 |
-
{"name": "schur_p2", "
|
| 18 |
-
{"name": "schur_p3", "
|
| 19 |
-
{"name": "schur_p4", "
|
| 20 |
-
{"name": "schur_p5", "
|
| 21 |
-
{"name": "sq_p1", "
|
| 22 |
-
{"name": "sq_p2", "
|
| 23 |
-
{"name": "sq_p3", "
|
| 24 |
-
{"name": "sq_p4", "
|
| 25 |
-
{"name": "sq_p5", "
|
|
|
|
| 1 |
+
{"name": "jensen_p1", "formal_statement": "theorem jensen_p1 (x y : ℝ) (h : x > 0) (g : y > 0) : ((1:ℝ)/3 * x + (2:ℝ)/3 * y) ^ 4 ≤ (1:ℝ)/3 * x^4 + (2:ℝ)/3 * y ^ 4 := by\n"}
|
| 2 |
+
{"name": "jensen_p2", "formal_statement": "theorem jensen_p2 (x y : ℝ) : Real.exp ((1:ℝ)/4 * x + (3:ℝ)/4 * y) ≤ (1:ℝ)/4 * Real.exp x + (3:ℝ)/4 * Real.exp y := by\n"}
|
| 3 |
+
{"name": "jensen_p3", "formal_statement": "theorem jensen_p3 (x y : ℝ) (h : x > 0) (g : y > 0): ((1:ℝ)/4 * x + (3:ℝ)/4 * y) * Real.log ((1:ℝ)/4 * x + (3:ℝ)/4 * y) ≤ (1:ℝ)/4 * x * Real.log x + (3:ℝ)/4 * y * Real.log y := by\n"}
|
| 4 |
+
{"name": "jensen_p4", "formal_statement": "theorem jensen_p4 (x y z: ℝ) (h : x > 0) (g : y > 0) (j : z > 0) (k : x + y + z = 3) : (1:ℝ)/3 * x^6 + (1:ℝ)/3 * y ^ 6 + (1:ℝ)/3 * z ^ 6 ≥ 1 := by\n"}
|
| 5 |
+
{"name": "jensen_p5", "formal_statement": "theorem jensen_p5 (x y z: ℝ) (h : x > 0) (g : y > 0) (j : z > 0): (1:ℝ)/4 * x ^ ((1:ℝ)/3) + (3:ℝ)/8 * y ^ ((1:ℝ)/3) + (3:ℝ)/8 * z ^ ((1:ℝ)/3) ≤ ((1:ℝ)/4 * x + (3:ℝ)/8 * y + (3:ℝ)/8 * z) ^ ((1:ℝ)/3) := by\n"}
|
| 6 |
+
{"name": "jensen_p6", "formal_statement": "theorem jensen_p6 (x y z: ℝ) (h : x > 0) (g : y > 0) (j : z > 0): (1:ℝ)/4 * Real.log x + (3:ℝ)/8 * Real.log y + (3:ℝ)/8 * Real.log z ≤ Real.log ((1:ℝ)/4 * x + (3:ℝ)/8 * y + (3:ℝ)/8 * z) := by\n"}
|
| 7 |
+
{"name": "jensen_p7", "formal_statement": "theorem jensen_p7 (x y z: ℝ) (h : x > 0) (g : y > 0) (j : z > 0): (1:ℝ)/4 * Real.sqrt x + (3:ℝ)/8 * Real.sqrt y + (3:ℝ)/8 * Real.sqrt z ≤ Real.sqrt ((1:ℝ)/4 * x + (3:ℝ)/8 * y + (3:ℝ)/8 * z) := by\n"}
|
| 8 |
+
{"name": "jensen_p8", "formal_statement": "theorem jensen_p8 (x y z: ℝ) (h : x > 0) (g : y > 0) (j : z > 0) (k : x + y + z = (π:ℝ)): (1:ℝ)/3 * Real.sin x + (1:ℝ)/3 * Real.sin y + (1:ℝ)/3 * Real.sin z ≤ √3 / 2 := by\n"}
|
| 9 |
+
{"name": "jensen_p9", "formal_statement": "theorem jensen_p9 (x y z: ℝ) (h : x > 0) (g : y > 0) (j : z > 0) (k : x + y + z = (π:ℝ)): (1:ℝ)/3 * Real.sin (x/2) + (1:ℝ)/3 * Real.sin (y/2) + (1:ℝ)/3 * Real.sin (z/2) ≤ (1:ℝ) / 2 := by\n"}
|
| 10 |
+
{"name": "jensen_p10", "formal_statement": "theorem jensen_p10 (x y z: ℝ) (h : x > 0) (g : y > 0) (j : z > 0) (k : x + y + z = (π:ℝ)): (1:ℝ)/3 * Real.cos (x/2) + (1:ℝ)/3 * Real.cos (y/2) + (1:ℝ)/3 * Real.cos (z/2) ≤ √3 / 2 := by\n"}
|
| 11 |
+
{"name": "induction_p1", "formal_statement": "theorem induction_p1 (n : ℕ) (h : n ≥ 4) : 2 ^ n ≥ n + 1 := by\n"}
|
| 12 |
+
{"name": "induction_p2", "formal_statement": "theorem induction_p2 (x : ℝ) (n : ℕ) (h₀ : -1 < x) (h₁ : 0 < n) : 1 + ↑n * x ≤ (1 + x) ^ (n : ℕ) := by\n"}
|
| 13 |
+
{"name": "induction_p3", "formal_statement": "theorem induction_p3 (n : ℕ) (h₀ : 4 ≤ n) : n ^ 2 ≤ n ! := by\n"}
|
| 14 |
+
{"name": "induction_p4", "formal_statement": "theorem induction_p4 (n : ℕ) (h₀ : 3 ≤ n) : n ! < n ^ (n - 1) := by\n"}
|
| 15 |
+
{"name": "induction_p5", "formal_statement": "theorem induction_p5 (n : ℕ) (h₀ : 0 < n) : (∏ k in Finset.Icc 1 n, (1 + (1 : ℝ) / k ^ 3)) ≤ (3 : ℝ) - 1 / ↑n := by\n"}
|
| 16 |
+
{"name": "schur_p1", "formal_statement": "theorem schur_p1 (a b c: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (h : a * b * c = 1) : (a - 1 + 1 / b) * (b - 1 + 1 / c) * (c - 1 + 1 / a) ≤ 1 := by\n"}
|
| 17 |
+
{"name": "schur_p2", "formal_statement": "theorem schur_p2 (a b c: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (h : a * b * c = 1) : 3 + a / b + b / c + c / a ≥ a + b + c + 1 / a + 1 / b + 1 / c := by\n"}
|
| 18 |
+
{"name": "schur_p3", "formal_statement": "theorem schur_p3 (a b c t: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hab : a ≥ b)(hbc : b ≥ c) (ht : t > 0) : a^t * (a - b) * (a - c) + b^t * (b - c) * (b - a) + c^t * (c - a) * (c - b) ≥ 0 := by\n"}
|
| 19 |
+
{"name": "schur_p4", "formal_statement": "theorem schur_p4 (a b c: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (h : a + b + c = 1): 5 * (a^2 + b^2 + c^2) ≤ 6 * (a^3 + b^3 + c^3) + 1 := by\n"}
|
| 20 |
+
{"name": "schur_p5", "formal_statement": "theorem schur_p5 (a b c: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hab : a + b > c) (hbc : b + c > a) (hca : c + a > b) : 2 * a^2 * (b + c) + 2 * b^2 * (c + a) + 2 * c^2 * (a + b) ≥ a^3 + b^3 + c^3 + 9 * a * b * c := by\n"}
|
| 21 |
+
{"name": "sq_p1", "formal_statement": "theorem sq_p1 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) : (a+b) * (b+c) * (c+a) ≥ 8 * a * b * c := by\n"}
|
| 22 |
+
{"name": "sq_p2", "formal_statement": "theorem sq_p2 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) : a^2 * b^2 + b^2 * c^2 + c^2 * a^2 ≥ a * b * c * (a + b + c) := by\n"}
|
| 23 |
+
{"name": "sq_p3", "formal_statement": "theorem sq_p3 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) : a ^ 4 + b^4 + c^4 ≥ a * b * c * (a + b + c) := by\n"}
|
| 24 |
+
{"name": "sq_p4", "formal_statement": "theorem sq_p4 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) : (a+b+c)^3 ≥ 27 * a * b * c := by\n"}
|
| 25 |
+
{"name": "sq_p5", "formal_statement": "theorem sq_p5 (a b c d: ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hd : d > 0) : a^2 + b^2 + c^2 + d^2 ≥ a*b + b*c + c*d + d*a := by\n"}
|