| {"id": "hard1_0001", "input": "Equation 1: x = y * (y * ((x * y) * x))\nEquation 2: x * y = ((z * x) * w) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * (y * ((x * y) * x))", "equation2": "x * y = ((z * x) * w) * y", "difficulty": "hard", "index": 1}} |
| {"id": "hard1_0002", "input": "Equation 1: x * (y * z) = (z * w) * w\nEquation 2: x * (y * x) = (x * y) * z\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * (y * z) = (z * w) * w", "equation2": "x * (y * x) = (x * y) * z", "difficulty": "hard", "index": 2}} |
| {"id": "hard1_0005", "input": "Equation 1: x = ((y * z) * (z * x)) * y\nEquation 2: x * x = (y * (y * y)) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((y * z) * (z * x)) * y", "equation2": "x * x = (y * (y * y)) * y", "difficulty": "hard", "index": 5}} |
| {"id": "hard1_0008", "input": "Equation 1: x = x * (y * ((y * z) * y))\nEquation 2: x = x * ((x * (x * x)) * x)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = x * (y * ((y * z) * y))", "equation2": "x = x * ((x * (x * x)) * x)", "difficulty": "hard", "index": 8}} |
| {"id": "hard1_0009", "input": "Equation 1: x = ((x * x) * (y * z)) * y\nEquation 2: x = ((x * (y * x)) * x) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((x * x) * (y * z)) * y", "equation2": "x = ((x * (y * x)) * x) * y", "difficulty": "hard", "index": 9}} |
| {"id": "hard1_0014", "input": "Equation 1: x = (y * x) * (x * (x * z))\nEquation 2: x = y * (z * (z * (x * y)))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * x) * (x * (x * z))", "equation2": "x = y * (z * (z * (x * y)))", "difficulty": "hard", "index": 14}} |
| {"id": "hard1_0015", "input": "Equation 1: x * x = x * (y * (z * y))\nEquation 2: x * (x * x) = x * (y * x)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * x = x * (y * (z * y))", "equation2": "x * (x * x) = x * (y * x)", "difficulty": "hard", "index": 15}} |
| {"id": "hard1_0016", "input": "Equation 1: x = (x * y) * (x * (y * z))\nEquation 2: x = (x * y) * (x * (z * y))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (x * y) * (x * (y * z))", "equation2": "x = (x * y) * (x * (z * y))", "difficulty": "hard", "index": 16}} |
| {"id": "hard1_0022", "input": "Equation 1: x * y = (y * (y * x)) * x\nEquation 2: x * y = (x * y) * (x * y)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * y = (y * (y * x)) * x", "equation2": "x * y = (x * y) * (x * y)", "difficulty": "hard", "index": 22}} |
| {"id": "hard1_0024", "input": "Equation 1: x = y * (y * ((x * y) * x))\nEquation 2: x = x * ((y * z) * (z * x))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * (y * ((x * y) * x))", "equation2": "x = x * ((y * z) * (z * x))", "difficulty": "hard", "index": 24}} |
| {"id": "hard1_0025", "input": "Equation 1: x = ((y * (z * z)) * x) * y\nEquation 2: x = y * ((y * (z * x)) * z)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((y * (z * z)) * x) * y", "equation2": "x = y * ((y * (z * x)) * z)", "difficulty": "hard", "index": 25}} |
| {"id": "hard1_0026", "input": "Equation 1: x = ((x * (y * z)) * z) * x\nEquation 2: x * x = x * (y * (y * x))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((x * (y * z)) * z) * x", "equation2": "x * x = x * (y * (y * x))", "difficulty": "hard", "index": 26}} |
| {"id": "hard1_0027", "input": "Equation 1: x = x * (y * (x * z))\nEquation 2: x = ((x * y) * (z * w)) * u\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = x * (y * (x * z))", "equation2": "x = ((x * y) * (z * w)) * u", "difficulty": "hard", "index": 27}} |
| {"id": "hard1_0028", "input": "Equation 1: x * y = z * ((x * x) * y)\nEquation 2: x * (x * y) = (y * x) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * y = z * ((x * x) * y)", "equation2": "x * (x * y) = (y * x) * y", "difficulty": "hard", "index": 28}} |
| {"id": "hard1_0029", "input": "Equation 1: x = (y * y) * ((z * x) * x)\nEquation 2: x * y = (y * (z * x)) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * y) * ((z * x) * x)", "equation2": "x * y = (y * (z * x)) * y", "difficulty": "hard", "index": 29}} |
| {"id": "hard1_0030", "input": "Equation 1: x = (y * (x * (y * x))) * y\nEquation 2: x * (x * y) = z * (y * x)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * (x * (y * x))) * y", "equation2": "x * (x * y) = z * (y * x)", "difficulty": "hard", "index": 30}} |
| {"id": "hard1_0031", "input": "Equation 1: x = ((y * x) * x) * (x * z)\nEquation 2: x = y * ((x * z) * (z * w))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((y * x) * x) * (x * z)", "equation2": "x = y * ((x * z) * (z * w))", "difficulty": "hard", "index": 31}} |
| {"id": "hard1_0032", "input": "Equation 1: x = ((y * x) * x) * (x * z)\nEquation 2: x * y = (z * z) * z\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((y * x) * x) * (x * z)", "equation2": "x * y = (z * z) * z", "difficulty": "hard", "index": 32}} |
| {"id": "hard1_0035", "input": "Equation 1: x = x * ((y * z) * (w * u))\nEquation 2: x = (x * (x * (y * x))) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = x * ((y * z) * (w * u))", "equation2": "x = (x * (x * (y * x))) * x", "difficulty": "hard", "index": 35}} |
| {"id": "hard1_0037", "input": "Equation 1: x = y * (((y * z) * y) * x)\nEquation 2: x = x * (x * (y * (y * x)))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "UNPARSEABLE", "metadata": {"equation1": "x = y * (((y * z) * y) * x)", "equation2": "x = x * (x * (y * (y * x)))", "difficulty": "hard", "index": 37}} |
| {"id": "hard1_0038", "input": "Equation 1: x * y = (x * (x * y)) * z\nEquation 2: x * y = x * (z * (y * y))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * y = (x * (x * y)) * z", "equation2": "x * y = x * (z * (y * y))", "difficulty": "hard", "index": 38}} |
| {"id": "hard1_0042", "input": "Equation 1: x = (y * x) * (x * z)\nEquation 2: x = x * (y * (x * (y * z)))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * x) * (x * z)", "equation2": "x = x * (y * (x * (y * z)))", "difficulty": "hard", "index": 42}} |
| {"id": "hard1_0043", "input": "Equation 1: x * x = y * (x * (z * w))\nEquation 2: x * x = y * ((x * x) * z)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * x = y * (x * (z * w))", "equation2": "x * x = y * ((x * x) * z)", "difficulty": "hard", "index": 43}} |
| {"id": "hard1_0045", "input": "Equation 1: x = (y * x) * (x * (x * z))\nEquation 2: x = (x * (y * z)) * (w * w)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * x) * (x * (x * z))", "equation2": "x = (x * (y * z)) * (w * w)", "difficulty": "hard", "index": 45}} |
| {"id": "hard1_0047", "input": "Equation 1: x = (y * x) * (x * (z * z))\nEquation 2: x = y * ((y * x) * (z * y))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "UNPARSEABLE", "metadata": {"equation1": "x = (y * x) * (x * (z * z))", "equation2": "x = y * ((y * x) * (z * y))", "difficulty": "hard", "index": 47}} |
| {"id": "hard1_0053", "input": "Equation 1: x * y = (x * z) * (y * w)\nEquation 2: x * y = (x * y) * (z * x)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * y = (x * z) * (y * w)", "equation2": "x * y = (x * y) * (z * x)", "difficulty": "hard", "index": 53}} |
| {"id": "hard1_0057", "input": "Equation 1: x = y * ((z * y) * (y * x))\nEquation 2: x = y * (x * ((z * w) * x))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * ((z * y) * (y * x))", "equation2": "x = y * (x * ((z * w) * x))", "difficulty": "hard", "index": 57}} |
| {"id": "hard1_0058", "input": "Equation 1: x * (y * z) = (w * w) * z\nEquation 2: x * (y * x) = (x * z) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * (y * z) = (w * w) * z", "equation2": "x * (y * x) = (x * z) * x", "difficulty": "hard", "index": 58}} |
| {"id": "hard1_0059", "input": "Equation 1: x = ((y * z) * z) * x\nEquation 2: x = x * ((y * (y * x)) * x)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = ((y * z) * z) * x", "equation2": "x = x * ((y * (y * x)) * x)", "difficulty": "hard", "index": 59}} |
| {"id": "hard1_0061", "input": "Equation 1: x = x * (y * (z * (x * y)))\nEquation 2: x = x * (((y * x) * z) * w)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = x * (y * (z * (x * y)))", "equation2": "x = x * (((y * x) * z) * w)", "difficulty": "hard", "index": 61}} |
| {"id": "hard1_0062", "input": "Equation 1: x = ((y * x) * z) * (z * y)\nEquation 2: x = (y * (y * (x * x))) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((y * x) * z) * (z * y)", "equation2": "x = (y * (y * (x * x))) * x", "difficulty": "hard", "index": 62}} |
| {"id": "hard1_0064", "input": "Equation 1: x = x * ((y * z) * (z * z))\nEquation 2: x = (x * ((y * y) * x)) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = x * ((y * z) * (z * z))", "equation2": "x = (x * ((y * y) * x)) * x", "difficulty": "hard", "index": 64}} |
| {"id": "hard1_0065", "input": "Equation 1: x = x * (y * (z * y))\nEquation 2: x = (x * (x * (x * y))) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = x * (y * (z * y))", "equation2": "x = (x * (x * (x * y))) * x", "difficulty": "hard", "index": 65}} |
| {"id": "hard1_0066", "input": "Equation 1: x = ((y * z) * y) * (x * z)\nEquation 2: x = (((y * x) * y) * y) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((y * z) * y) * (x * z)", "equation2": "x = (((y * x) * y) * y) * y", "difficulty": "hard", "index": 66}} |
| {"id": "hard1_0067", "input": "Equation 1: x = (y * (y * x)) * (z * x)\nEquation 2: x * y = z * (y * (w * y))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = (y * (y * x)) * (z * x)", "equation2": "x * y = z * (y * (w * y))", "difficulty": "hard", "index": 67}} |
| {"id": "hard1_0068", "input": "Equation 1: x * x = y * ((x * z) * y)\nEquation 2: x * (y * x) = x * (z * x)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * x = y * ((x * z) * y)", "equation2": "x * (y * x) = x * (z * x)", "difficulty": "hard", "index": 68}} |
| {"id": "hard2_0001", "input": "Equation 1: x * (y * z) = (y * w) * w\nEquation 2: x * (y * y) = (z * w) * u\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * (y * z) = (y * w) * w", "equation2": "x * (y * y) = (z * w) * u", "difficulty": "hard", "index": 1}} |
| {"id": "hard2_0002", "input": "Equation 1: x = y * ((y * (y * x)) * x)\nEquation 2: x = (y * (z * y)) * (y * x)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * ((y * (y * x)) * x)", "equation2": "x = (y * (z * y)) * (y * x)", "difficulty": "hard", "index": 2}} |
| {"id": "hard2_0007", "input": "Equation 1: x * (x * y) = (z * y) * w\nEquation 2: x * (x * y) = y * (z * y)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * (x * y) = (z * y) * w", "equation2": "x * (x * y) = y * (z * y)", "difficulty": "hard", "index": 7}} |
| {"id": "hard2_0009", "input": "Equation 1: x = y * ((x * z) * (z * y))\nEquation 2: x = ((x * y) * (y * y)) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * ((x * z) * (z * y))", "equation2": "x = ((x * y) * (y * y)) * y", "difficulty": "hard", "index": 9}} |
| {"id": "hard2_0010", "input": "Equation 1: x = (x * y) * ((x * y) * y)\nEquation 2: x = (x * (x * (x * x))) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (x * y) * ((x * y) * y)", "equation2": "x = (x * (x * (x * x))) * y", "difficulty": "hard", "index": 10}} |
| {"id": "hard2_0011", "input": "Equation 1: x = y * ((x * (x * y)) * y)\nEquation 2: x = (y * (z * z)) * w\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * ((x * (x * y)) * y)", "equation2": "x = (y * (z * z)) * w", "difficulty": "hard", "index": 11}} |
| {"id": "hard2_0012", "input": "Equation 1: x = y * (((z * y) * x) * z)\nEquation 2: (x * x) * y = (y * z) * z\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * (((z * y) * x) * z)", "equation2": "(x * x) * y = (y * z) * z", "difficulty": "hard", "index": 12}} |
| {"id": "hard2_0013", "input": "Equation 1: x = (y * ((y * x) * x)) * y\nEquation 2: x = x * (((x * y) * x) * z)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * ((y * x) * x)) * y", "equation2": "x = x * (((x * y) * x) * z)", "difficulty": "hard", "index": 13}} |
| {"id": "hard2_0014", "input": "Equation 1: x = y * ((x * (x * y)) * y)\nEquation 2: (x * y) * z = (z * y) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * ((x * (x * y)) * y)", "equation2": "(x * y) * z = (z * y) * x", "difficulty": "hard", "index": 14}} |
| {"id": "hard2_0016", "input": "Equation 1: x = x * (y * ((y * z) * y))\nEquation 2: x * y = ((x * y) * x) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = x * (y * ((y * z) * y))", "equation2": "x * y = ((x * y) * x) * x", "difficulty": "hard", "index": 16}} |
| {"id": "hard2_0018", "input": "Equation 1: x = x * ((x * (y * z)) * z)\nEquation 2: x = x * (x * (y * x))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = x * ((x * (y * z)) * z)", "equation2": "x = x * (x * (y * x))", "difficulty": "hard", "index": 18}} |
| {"id": "hard2_0020", "input": "Equation 1: x * (y * z) = z * (w * u)\nEquation 2: (x * y) * y = (x * z) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * (y * z) = z * (w * u)", "equation2": "(x * y) * y = (x * z) * y", "difficulty": "hard", "index": 20}} |
| {"id": "hard2_0025", "input": "Equation 1: x * y = x * ((z * y) * x)\nEquation 2: x * y = x * (x * (y * y))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * y = x * ((z * y) * x)", "equation2": "x * y = x * (x * (y * y))", "difficulty": "hard", "index": 25}} |
| {"id": "hard2_0027", "input": "Equation 1: x = y * ((z * (y * y)) * x)\nEquation 2: x = (y * z) * ((x * z) * x)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * ((z * (y * y)) * x)", "equation2": "x = (y * z) * ((x * z) * x)", "difficulty": "hard", "index": 27}} |
| {"id": "hard2_0029", "input": "Equation 1: x * y = ((y * y) * x) * x\nEquation 2: x * y = (x * (y * y)) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * y = ((y * y) * x) * x", "equation2": "x * y = (x * (y * y)) * x", "difficulty": "hard", "index": 29}} |
| {"id": "hard2_0031", "input": "Equation 1: x = y * ((x * (x * y)) * y)\nEquation 2: x = ((y * (z * w)) * u) * z\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * ((x * (x * y)) * y)", "equation2": "x = ((y * (z * w)) * u) * z", "difficulty": "hard", "index": 31}} |
| {"id": "hard2_0032", "input": "Equation 1: x * y = (z * (x * w)) * y\nEquation 2: x * x = (y * (y * y)) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * y = (z * (x * w)) * y", "equation2": "x * x = (y * (y * y)) * x", "difficulty": "hard", "index": 32}} |
| {"id": "hard2_0034", "input": "Equation 1: x * y = (y * (x * z)) * z\nEquation 2: x * y = y * ((z * z) * x)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * y = (y * (x * z)) * z", "equation2": "x * y = y * ((z * z) * x)", "difficulty": "hard", "index": 34}} |
| {"id": "hard2_0038", "input": "Equation 1: x = y * ((x * (x * y)) * y)\nEquation 2: x = y * (y * (y * (z * w)))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * ((x * (x * y)) * y)", "equation2": "x = y * (y * (y * (z * w)))", "difficulty": "hard", "index": 38}} |
| {"id": "hard2_0041", "input": "Equation 1: x = y * ((y * (x * y)) * x)\nEquation 2: x = (x * (y * x)) * (x * x)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * ((y * (x * y)) * x)", "equation2": "x = (x * (y * x)) * (x * x)", "difficulty": "hard", "index": 41}} |
| {"id": "hard2_0042", "input": "Equation 1: x * (y * x) = (z * w) * u\nEquation 2: x * (y * z) = (w * x) * w\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * (y * x) = (z * w) * u", "equation2": "x * (y * z) = (w * x) * w", "difficulty": "hard", "index": 42}} |
| {"id": "hard2_0043", "input": "Equation 1: x = ((x * (x * y)) * y) * y\nEquation 2: x * (y * x) = x * (y * z)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((x * (x * y)) * y) * y", "equation2": "x * (y * x) = x * (y * z)", "difficulty": "hard", "index": 43}} |
| {"id": "hard2_0046", "input": "Equation 1: (x * y) * z = (w * z) * u\nEquation 2: x * (x * x) = y * (z * w)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "(x * y) * z = (w * z) * u", "equation2": "x * (x * x) = y * (z * w)", "difficulty": "hard", "index": 46}} |
| {"id": "hard2_0047", "input": "Equation 1: x * y = (y * (x * z)) * y\nEquation 2: x * x = ((x * x) * y) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * y = (y * (x * z)) * y", "equation2": "x * x = ((x * x) * y) * x", "difficulty": "hard", "index": 47}} |
| {"id": "hard2_0052", "input": "Equation 1: x = y * ((x * (x * y)) * y)\nEquation 2: x = x * (y * (z * (w * y)))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * ((x * (x * y)) * y)", "equation2": "x = x * (y * (z * (w * y)))", "difficulty": "hard", "index": 52}} |
| {"id": "hard2_0056", "input": "Equation 1: x = y * ((y * (y * x)) * x)\nEquation 2: x = ((y * z) * z) * (w * x)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * ((y * (y * x)) * x)", "equation2": "x = ((y * z) * z) * (w * x)", "difficulty": "hard", "index": 56}} |
| {"id": "hard2_0060", "input": "Equation 1: x = x * (y * ((x * z) * y))\nEquation 2: x = x * ((y * (z * w)) * w)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = x * (y * ((x * z) * y))", "equation2": "x = x * ((y * (z * w)) * w)", "difficulty": "hard", "index": 60}} |
| {"id": "hard2_0064", "input": "Equation 1: x = (y * z) * (z * (x * y))\nEquation 2: x = y * (z * (y * (z * x)))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * z) * (z * (x * y))", "equation2": "x = y * (z * (y * (z * x)))", "difficulty": "hard", "index": 64}} |
| {"id": "hard2_0067", "input": "Equation 1: x = ((y * x) * z) * (z * y)\nEquation 2: x = (y * (y * y)) * (x * y)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((y * x) * z) * (z * y)", "equation2": "x = (y * (y * y)) * (x * y)", "difficulty": "hard", "index": 67}} |
| {"id": "hard2_0068", "input": "Equation 1: x * (x * y) = (z * w) * u\nEquation 2: x * (y * z) = y * (x * z)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * (x * y) = (z * w) * u", "equation2": "x * (y * z) = y * (x * z)", "difficulty": "hard", "index": 68}} |
| {"id": "hard2_0077", "input": "Equation 1: x = (y * ((y * x) * x)) * y\nEquation 2: x = (((x * y) * x) * x) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * ((y * x) * x)) * y", "equation2": "x = (((x * y) * x) * x) * x", "difficulty": "hard", "index": 77}} |
| {"id": "hard2_0084", "input": "Equation 1: x * (y * z) = (z * y) * y\nEquation 2: x * (y * z) = (y * z) * w\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * (y * z) = (z * y) * y", "equation2": "x * (y * z) = (y * z) * w", "difficulty": "hard", "index": 84}} |
| {"id": "hard2_0088", "input": "Equation 1: x = (x * (x * y)) * y\nEquation 2: x * (y * z) = x * (z * y)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (x * (x * y)) * y", "equation2": "x * (y * z) = x * (z * y)", "difficulty": "hard", "index": 88}} |
| {"id": "hard2_0090", "input": "Equation 1: x * y = (y * (x * z)) * y\nEquation 2: x * y = ((x * y) * x) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * y = (y * (x * z)) * y", "equation2": "x * y = ((x * y) * x) * y", "difficulty": "hard", "index": 90}} |
| {"id": "hard2_0092", "input": "Equation 1: x * (y * z) = (w * u) * u\nEquation 2: x * (y * z) = (y * w) * u\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * (y * z) = (w * u) * u", "equation2": "x * (y * z) = (y * w) * u", "difficulty": "hard", "index": 92}} |
| {"id": "hard2_0093", "input": "Equation 1: x = (y * y) * (x * (y * x))\nEquation 2: x * y = (x * (x * x)) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * y) * (x * (y * x))", "equation2": "x * y = (x * (x * x)) * y", "difficulty": "hard", "index": 93}} |
| {"id": "hard2_0095", "input": "Equation 1: x = y * ((z * (z * y)) * x)\nEquation 2: x * y = (y * (y * x)) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * ((z * (z * y)) * x)", "equation2": "x * y = (y * (y * x)) * y", "difficulty": "hard", "index": 95}} |
| {"id": "hard2_0096", "input": "Equation 1: x = y * ((z * y) * (y * x))\nEquation 2: x * y = ((z * x) * x) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * ((z * y) * (y * x))", "equation2": "x * y = ((z * x) * x) * y", "difficulty": "hard", "index": 96}} |
| {"id": "hard2_0097", "input": "Equation 1: x = (x * (x * (y * y))) * z\nEquation 2: x * x = ((x * y) * z) * w\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = (x * (x * (y * y))) * z", "equation2": "x * x = ((x * y) * z) * w", "difficulty": "hard", "index": 97}} |
| {"id": "hard2_0098", "input": "Equation 1: x = (y * (x * (y * z))) * y\nEquation 2: x = y * ((y * (x * x)) * x)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = (y * (x * (y * z))) * y", "equation2": "x = y * ((y * (x * x)) * x)", "difficulty": "hard", "index": 98}} |
| {"id": "hard2_0100", "input": "Equation 1: x = (x * ((x * y) * y)) * y\nEquation 2: x * y = ((x * y) * y) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (x * ((x * y) * y)) * y", "equation2": "x * y = ((x * y) * y) * x", "difficulty": "hard", "index": 100}} |
| {"id": "hard2_0102", "input": "Equation 1: x = (y * ((y * x) * x)) * y\nEquation 2: x = ((y * (z * x)) * w) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * ((y * x) * x)) * y", "equation2": "x = ((y * (z * x)) * w) * x", "difficulty": "hard", "index": 102}} |
| {"id": "hard2_0105", "input": "Equation 1: x = y * ((y * (y * x)) * x)\nEquation 2: x * (x * y) = y * (x * y)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * ((y * (y * x)) * x)", "equation2": "x * (x * y) = y * (x * y)", "difficulty": "hard", "index": 105}} |
| {"id": "hard2_0108", "input": "Equation 1: x * x = ((y * z) * w) * u\nEquation 2: x * (y * z) = (w * u) * z\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * x = ((y * z) * w) * u", "equation2": "x * (y * z) = (w * u) * z", "difficulty": "hard", "index": 108}} |
| {"id": "hard2_0109", "input": "Equation 1: x = (y * x) * (x * (y * z))\nEquation 2: x * (y * y) = (x * y) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * x) * (x * (y * z))", "equation2": "x * (y * y) = (x * y) * x", "difficulty": "hard", "index": 109}} |
| {"id": "hard2_0110", "input": "Equation 1: x = y * (((z * x) * x) * x)\nEquation 2: x = x * ((x * (x * y)) * x)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = y * (((z * x) * x) * x)", "equation2": "x = x * ((x * (x * y)) * x)", "difficulty": "hard", "index": 110}} |
| {"id": "hard2_0112", "input": "Equation 1: x = (((x * y) * y) * x) * y\nEquation 2: x * x = x * ((y * z) * x)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (((x * y) * y) * x) * y", "equation2": "x * x = x * ((y * z) * x)", "difficulty": "hard", "index": 112}} |
| {"id": "hard2_0118", "input": "Equation 1: x = (x * ((x * y) * y)) * y\nEquation 2: x = (x * y) * (x * (y * z))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (x * ((x * y) * y)) * y", "equation2": "x = (x * y) * (x * (y * z))", "difficulty": "hard", "index": 118}} |
| {"id": "hard2_0122", "input": "Equation 1: x = y * ((y * (y * x)) * x)\nEquation 2: x = y * (((z * x) * x) * x)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * ((y * (y * x)) * x)", "equation2": "x = y * (((z * x) * x) * x)", "difficulty": "hard", "index": 122}} |
| {"id": "hard2_0123", "input": "Equation 1: x = y * (((z * y) * x) * z)\nEquation 2: x * (y * z) = x * (z * y)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * (((z * y) * x) * z)", "equation2": "x * (y * z) = x * (z * y)", "difficulty": "hard", "index": 123}} |
| {"id": "hard2_0124", "input": "Equation 1: x = ((x * x) * y) * (y * z)\nEquation 2: x * x = x * (x * (y * z))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((x * x) * y) * (y * z)", "equation2": "x * x = x * (x * (y * z))", "difficulty": "hard", "index": 124}} |
| {"id": "hard2_0125", "input": "Equation 1: x = ((x * (y * z)) * z) * x\nEquation 2: x = ((x * x) * (y * y)) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((x * (y * z)) * z) * x", "equation2": "x = ((x * x) * (y * y)) * x", "difficulty": "hard", "index": 125}} |
| {"id": "hard2_0129", "input": "Equation 1: x = y * (z * ((z * y) * x))\nEquation 2: x = y * (x * ((y * x) * x))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * (z * ((z * y) * x))", "equation2": "x = y * (x * ((y * x) * x))", "difficulty": "hard", "index": 129}} |
| {"id": "hard2_0133", "input": "Equation 1: x = y * ((x * z) * (z * y))\nEquation 2: x = y * ((z * x) * (y * z))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * ((x * z) * (z * y))", "equation2": "x = y * ((z * x) * (y * z))", "difficulty": "hard", "index": 133}} |
| {"id": "hard2_0134", "input": "Equation 1: x = (y * (y * x)) * (y * x)\nEquation 2: x * y = ((z * y) * w) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * (y * x)) * (y * x)", "equation2": "x * y = ((z * y) * w) * y", "difficulty": "hard", "index": 134}} |
| {"id": "hard2_0139", "input": "Equation 1: x = (x * y) * ((z * w) * x)\nEquation 2: x * y = ((x * z) * x) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (x * y) * ((z * w) * x)", "equation2": "x * y = ((x * z) * x) * y", "difficulty": "hard", "index": 139}} |
| {"id": "hard2_0146", "input": "Equation 1: x * x = y * (z * (w * u))\nEquation 2: x * (x * y) = y * (y * x)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * x = y * (z * (w * u))", "equation2": "x * (x * y) = y * (y * x)", "difficulty": "hard", "index": 146}} |
| {"id": "hard2_0148", "input": "Equation 1: (x * y) * z = (z * w) * u\nEquation 2: x * (y * z) = (y * z) * z\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "(x * y) * z = (z * w) * u", "equation2": "x * (y * z) = (y * z) * z", "difficulty": "hard", "index": 148}} |
| {"id": "hard2_0150", "input": "Equation 1: x = (y * ((y * x) * x)) * y\nEquation 2: x * y = z * (w * (x * z))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * ((y * x) * x)) * y", "equation2": "x * y = z * (w * (x * z))", "difficulty": "hard", "index": 150}} |
| {"id": "hard2_0154", "input": "Equation 1: x = ((y * (z * y)) * x) * w\nEquation 2: x = x * (((y * y) * x) * z)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = ((y * (z * y)) * x) * w", "equation2": "x = x * (((y * y) * x) * z)", "difficulty": "hard", "index": 154}} |
| {"id": "hard2_0157", "input": "Equation 1: x = (x * ((x * y) * y)) * y\nEquation 2: x * y = ((x * z) * x) * w\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (x * ((x * y) * y)) * y", "equation2": "x * y = ((x * z) * x) * w", "difficulty": "hard", "index": 157}} |
| {"id": "hard2_0163", "input": "Equation 1: x = (x * (y * z)) * (w * x)\nEquation 2: (x * y) * z = (x * w) * z\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (x * (y * z)) * (w * x)", "equation2": "(x * y) * z = (x * w) * z", "difficulty": "hard", "index": 163}} |
| {"id": "hard2_0165", "input": "Equation 1: x = ((y * x) * z) * (z * y)\nEquation 2: x = y * (x * ((z * y) * z))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((y * x) * z) * (z * y)", "equation2": "x = y * (x * ((z * y) * z))", "difficulty": "hard", "index": 165}} |
| {"id": "hard2_0169", "input": "Equation 1: x = (y * (y * (x * y))) * y\nEquation 2: x = (x * ((x * x) * x)) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * (y * (x * y))) * y", "equation2": "x = (x * ((x * x) * x)) * x", "difficulty": "hard", "index": 169}} |
| {"id": "hard2_0171", "input": "Equation 1: x = y * ((x * (x * y)) * y)\nEquation 2: x * y = (x * (z * z)) * w\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * ((x * (x * y)) * y)", "equation2": "x * y = (x * (z * z)) * w", "difficulty": "hard", "index": 171}} |
| {"id": "hard2_0172", "input": "Equation 1: x = y * ((x * (x * y)) * y)\nEquation 2: x = ((y * y) * z) * (z * x)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * ((x * (x * y)) * y)", "equation2": "x = ((y * y) * z) * (z * x)", "difficulty": "hard", "index": 172}} |
| {"id": "hard2_0175", "input": "Equation 1: x = ((y * z) * (z * x)) * y\nEquation 2: x = ((x * (x * y)) * x) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((y * z) * (z * x)) * y", "equation2": "x = ((x * (x * y)) * x) * y", "difficulty": "hard", "index": 175}} |
| {"id": "hard2_0176", "input": "Equation 1: x = ((x * (y * y)) * z) * x\nEquation 2: x = ((x * (y * z)) * w) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((x * (y * y)) * z) * x", "equation2": "x = ((x * (y * z)) * w) * x", "difficulty": "hard", "index": 176}} |
| {"id": "hard2_0177", "input": "Equation 1: x = (y * ((y * x) * x)) * y\nEquation 2: x = (y * (x * (z * y))) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * ((y * x) * x)) * y", "equation2": "x = (y * (x * (z * y))) * x", "difficulty": "hard", "index": 177}} |
| {"id": "hard2_0179", "input": "Equation 1: x = (y * x) * (x * (z * y))\nEquation 2: (x * y) * x = (y * x) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * x) * (x * (z * y))", "equation2": "(x * y) * x = (y * x) * x", "difficulty": "hard", "index": 179}} |
| {"id": "hard2_0180", "input": "Equation 1: x * x = (y * (z * w)) * u\nEquation 2: x * (y * z) = (z * x) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * x = (y * (z * w)) * u", "equation2": "x * (y * z) = (z * x) * y", "difficulty": "hard", "index": 180}} |
| {"id": "hard2_0181", "input": "Equation 1: x = x * (y * ((y * z) * y))\nEquation 2: x = ((x * (x * y)) * z) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = x * (y * ((y * z) * y))", "equation2": "x = ((x * (x * y)) * z) * x", "difficulty": "hard", "index": 181}} |
| {"id": "hard2_0182", "input": "Equation 1: x = y * ((x * (x * y)) * y)\nEquation 2: x * y = z * (y * (z * w))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * ((x * (x * y)) * y)", "equation2": "x * y = z * (y * (z * w))", "difficulty": "hard", "index": 182}} |
| {"id": "hard2_0186", "input": "Equation 1: x = y * (x * ((x * x) * z))\nEquation 2: x = y * (y * (x * (x * z)))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = y * (x * ((x * x) * z))", "equation2": "x = y * (y * (x * (x * z)))", "difficulty": "hard", "index": 186}} |
| {"id": "hard2_0187", "input": "Equation 1: x = (y * z) * (z * x)\nEquation 2: x = y * ((y * (z * y)) * x)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * z) * (z * x)", "equation2": "x = y * ((y * (z * y)) * x)", "difficulty": "hard", "index": 187}} |
| {"id": "hard2_0188", "input": "Equation 1: (x * y) * z = (z * w) * x\nEquation 2: (x * x) * y = (z * w) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "(x * y) * z = (z * w) * x", "equation2": "(x * x) * y = (z * w) * y", "difficulty": "hard", "index": 188}} |
| {"id": "hard2_0194", "input": "Equation 1: x = (y * ((y * x) * x)) * y\nEquation 2: x = (y * (x * (y * x))) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * ((y * x) * x)) * y", "equation2": "x = (y * (x * (y * x))) * y", "difficulty": "hard", "index": 194}} |
| {"id": "hard2_0195", "input": "Equation 1: x * (x * y) = (z * y) * w\nEquation 2: x * (y * x) = (z * z) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * (x * y) = (z * y) * w", "equation2": "x * (y * x) = (z * z) * x", "difficulty": "hard", "index": 195}} |
| {"id": "hard2_0196", "input": "Equation 1: x = (y * ((y * x) * x)) * y\nEquation 2: x = x * ((y * y) * (z * y))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * ((y * x) * x)) * y", "equation2": "x = x * ((y * y) * (z * y))", "difficulty": "hard", "index": 196}} |
| {"id": "hard2_0197", "input": "Equation 1: x * x = y * (z * (y * w))\nEquation 2: x * x = y * (z * (x * z))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * x = y * (z * (y * w))", "equation2": "x * x = y * (z * (x * z))", "difficulty": "hard", "index": 197}} |
| {"id": "hard2_0199", "input": "Equation 1: x = ((x * y) * (x * z)) * w\nEquation 2: x = ((x * (x * x)) * x) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = ((x * y) * (x * z)) * w", "equation2": "x = ((x * (x * x)) * x) * y", "difficulty": "hard", "index": 199}} |
| {"id": "normal_0029", "input": "Equation 1: x = x * (x * ((y * z) * z))\nEquation 2: x = (x * x) * (y * (z * y))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = x * (x * ((y * z) * z))", "equation2": "x = (x * x) * (y * (z * y))", "difficulty": "normal", "index": 29}} |
| {"id": "normal_0050", "input": "Equation 1: x = (y * x) * (y * (z * y))\nEquation 2: x = ((y * z) * w) * (y * w)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = (y * x) * (y * (z * y))", "equation2": "x = ((y * z) * w) * (y * w)", "difficulty": "normal", "index": 50}} |
| {"id": "normal_0053", "input": "Equation 1: x = (y * x) * (x * (z * x))\nEquation 2: x = (((y * z) * w) * w) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * x) * (x * (z * x))", "equation2": "x = (((y * z) * w) * w) * x", "difficulty": "normal", "index": 53}} |
| {"id": "normal_0062", "input": "Equation 1: x = ((y * x) * (y * z)) * y\nEquation 2: x = ((y * z) * (y * w)) * u\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = ((y * x) * (y * z)) * y", "equation2": "x = ((y * z) * (y * w)) * u", "difficulty": "normal", "index": 62}} |
| {"id": "normal_0094", "input": "Equation 1: x * (x * y) = z * (x * z)\nEquation 2: x * y = (z * w) * w\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * (x * y) = z * (x * z)", "equation2": "x * y = (z * w) * w", "difficulty": "normal", "index": 94}} |
| {"id": "normal_0161", "input": "Equation 1: x = y * (y * ((x * z) * z))\nEquation 2: x * y = y * ((z * x) * z)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = y * (y * ((x * z) * z))", "equation2": "x * y = y * ((z * x) * z)", "difficulty": "normal", "index": 161}} |
| {"id": "normal_0176", "input": "Equation 1: x = ((y * x) * x) * (y * y)\nEquation 2: (x * y) * x = (z * x) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((y * x) * x) * (y * y)", "equation2": "(x * y) * x = (z * x) * y", "difficulty": "normal", "index": 176}} |
| {"id": "normal_0183", "input": "Equation 1: x = ((y * (z * w)) * y) * x\nEquation 2: x * y = ((y * x) * x) * z\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((y * (z * w)) * y) * x", "equation2": "x * y = ((y * x) * x) * z", "difficulty": "normal", "index": 183}} |
| {"id": "normal_0204", "input": "Equation 1: x = (y * (y * (x * y))) * y\nEquation 2: x * y = (y * x) * (z * z)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * (y * (x * y))) * y", "equation2": "x * y = (y * x) * (z * z)", "difficulty": "normal", "index": 204}} |
| {"id": "normal_0219", "input": "Equation 1: x = y * (x * (y * (x * z)))\nEquation 2: x * x = y * (x * (z * y))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = y * (x * (y * (x * z)))", "equation2": "x * x = y * (x * (z * y))", "difficulty": "normal", "index": 219}} |
| {"id": "normal_0223", "input": "Equation 1: x = ((y * (x * y)) * x) * y\nEquation 2: x = ((y * (z * w)) * u) * z\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((y * (x * y)) * x) * y", "equation2": "x = ((y * (z * w)) * u) * z", "difficulty": "normal", "index": 223}} |
| {"id": "normal_0251", "input": "Equation 1: x = ((y * x) * (x * y)) * y\nEquation 2: x * x = x * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((y * x) * (x * y)) * y", "equation2": "x * x = x * y", "difficulty": "normal", "index": 251}} |
| {"id": "normal_0272", "input": "Equation 1: x = (y * ((x * x) * y)) * y\nEquation 2: (x * y) * z = (x * z) * w\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * ((x * x) * y)) * y", "equation2": "(x * y) * z = (x * z) * w", "difficulty": "normal", "index": 272}} |
| {"id": "normal_0303", "input": "Equation 1: x = (y * (z * x)) * z\nEquation 2: x = y * (((z * x) * x) * y)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = (y * (z * x)) * z", "equation2": "x = y * (((z * x) * x) * y)", "difficulty": "normal", "index": 303}} |
| {"id": "normal_0305", "input": "Equation 1: x = (y * y) * ((x * z) * z)\nEquation 2: x = x * (y * (z * (y * z)))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * y) * ((x * z) * z)", "equation2": "x = x * (y * (z * (y * z)))", "difficulty": "normal", "index": 305}} |
| {"id": "normal_0307", "input": "Equation 1: x = (y * (x * x)) * (y * y)\nEquation 2: (x * y) * y = (z * y) * w\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * (x * x)) * (y * y)", "equation2": "(x * y) * y = (z * y) * w", "difficulty": "normal", "index": 307}} |
| {"id": "normal_0324", "input": "Equation 1: x * x = ((y * x) * x) * z\nEquation 2: x * x = ((y * z) * z) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x * x = ((y * x) * x) * z", "equation2": "x * x = ((y * z) * z) * x", "difficulty": "normal", "index": 324}} |
| {"id": "normal_0326", "input": "Equation 1: x = x * (y * ((y * z) * y))\nEquation 2: x = ((x * (x * y)) * y) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = x * (y * ((y * z) * y))", "equation2": "x = ((x * (x * y)) * y) * x", "difficulty": "normal", "index": 326}} |
| {"id": "normal_0335", "input": "Equation 1: x = y * (x * (x * y))\nEquation 2: x = (x * (y * (y * z))) * w\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * (x * (x * y))", "equation2": "x = (x * (y * (y * z))) * w", "difficulty": "normal", "index": 335}} |
| {"id": "normal_0343", "input": "Equation 1: x = (y * (x * y)) * (x * y)\nEquation 2: x = y * (y * (z * (z * z)))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * (x * y)) * (x * y)", "equation2": "x = y * (y * (z * (z * z)))", "difficulty": "normal", "index": 343}} |
| {"id": "normal_0356", "input": "Equation 1: x * x = ((y * x) * y) * z\nEquation 2: x * y = (y * (z * x)) * z\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * x = ((y * x) * y) * z", "equation2": "x * y = (y * (z * x)) * z", "difficulty": "normal", "index": 356}} |
| {"id": "normal_0364", "input": "Equation 1: x = ((y * x) * x) * (x * x)\nEquation 2: x = y * (((y * z) * z) * x)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((y * x) * x) * (x * x)", "equation2": "x = y * (((y * z) * z) * x)", "difficulty": "normal", "index": 364}} |
| {"id": "normal_0381", "input": "Equation 1: x = (y * ((y * x) * x)) * x\nEquation 2: x = (y * ((z * y) * y)) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * ((y * x) * x)) * x", "equation2": "x = (y * ((z * y) * y)) * x", "difficulty": "normal", "index": 381}} |
| {"id": "normal_0440", "input": "Equation 1: x * x = y * (z * w)\nEquation 2: x * x = (y * (z * y)) * z\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * x = y * (z * w)", "equation2": "x * x = (y * (z * y)) * z", "difficulty": "normal", "index": 440}} |
| {"id": "normal_0453", "input": "Equation 1: x = ((y * y) * x) * (x * x)\nEquation 2: (x * y) * z = (x * w) * z\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((y * y) * x) * (x * x)", "equation2": "(x * y) * z = (x * w) * z", "difficulty": "normal", "index": 453}} |
| {"id": "normal_0512", "input": "Equation 1: x * (x * y) = (z * y) * x\nEquation 2: (x * y) * x = (z * y) * z\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * (x * y) = (z * y) * x", "equation2": "(x * y) * x = (z * y) * z", "difficulty": "normal", "index": 512}} |
| {"id": "normal_0522", "input": "Equation 1: x = (y * y) * (y * (z * x))\nEquation 2: x * y = (y * y) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = (y * y) * (y * (z * x))", "equation2": "x * y = (y * y) * y", "difficulty": "normal", "index": 522}} |
| {"id": "normal_0543", "input": "Equation 1: x * (y * x) = (z * z) * w\nEquation 2: x * y = (y * y) * (z * z)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * (y * x) = (z * z) * w", "equation2": "x * y = (y * y) * (z * z)", "difficulty": "normal", "index": 543}} |
| {"id": "normal_0565", "input": "Equation 1: x = (y * (x * x)) * (y * y)\nEquation 2: x = (y * y) * (x * (z * x))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * (x * x)) * (y * y)", "equation2": "x = (y * y) * (x * (z * x))", "difficulty": "normal", "index": 565}} |
| {"id": "normal_0592", "input": "Equation 1: x = (y * y) * ((x * z) * z)\nEquation 2: x = (x * y) * (y * (y * y))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * y) * ((x * z) * z)", "equation2": "x = (x * y) * (y * (y * y))", "difficulty": "normal", "index": 592}} |
| {"id": "normal_0619", "input": "Equation 1: x * y = ((z * x) * z) * y\nEquation 2: x * (x * x) = x * (y * x)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * y = ((z * x) * z) * y", "equation2": "x * (x * x) = x * (y * x)", "difficulty": "normal", "index": 619}} |
| {"id": "normal_0628", "input": "Equation 1: x = ((y * y) * (x * x)) * y\nEquation 2: x = y * (z * ((z * y) * w))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((y * y) * (x * x)) * y", "equation2": "x = y * (z * ((z * y) * w))", "difficulty": "normal", "index": 628}} |
| {"id": "normal_0632", "input": "Equation 1: x * y = y * ((z * z) * x)\nEquation 2: x * x = (y * (x * x)) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "UNPARSEABLE", "metadata": {"equation1": "x * y = y * ((z * z) * x)", "equation2": "x * x = (y * (x * x)) * x", "difficulty": "normal", "index": 632}} |
| {"id": "normal_0700", "input": "Equation 1: x = (y * ((x * y) * z)) * z\nEquation 2: x * (y * y) = (x * z) * z\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * ((x * y) * z)) * z", "equation2": "x * (y * y) = (x * z) * z", "difficulty": "normal", "index": 700}} |
| {"id": "normal_0703", "input": "Equation 1: x * (y * z) = (z * x) * x\nEquation 2: x * y = z * ((w * w) * w)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * (y * z) = (z * x) * x", "equation2": "x * y = z * ((w * w) * w)", "difficulty": "normal", "index": 703}} |
| {"id": "normal_0717", "input": "Equation 1: x * x = y * (x * (z * w))\nEquation 2: x * y = (y * (z * x)) * z\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * x = y * (x * (z * w))", "equation2": "x * y = (y * (z * x)) * z", "difficulty": "normal", "index": 717}} |
| {"id": "normal_0745", "input": "Equation 1: (x * y) * x = (y * z) * w\nEquation 2: x * y = x * (x * (x * y))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "(x * y) * x = (y * z) * w", "equation2": "x * y = x * (x * (x * y))", "difficulty": "normal", "index": 745}} |
| {"id": "normal_0762", "input": "Equation 1: x = y * ((y * x) * (x * z))\nEquation 2: x = (y * x) * ((x * z) * y)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = y * ((y * x) * (x * z))", "equation2": "x = (y * x) * ((x * z) * y)", "difficulty": "normal", "index": 762}} |
| {"id": "normal_0802", "input": "Equation 1: x = (((y * y) * z) * w) * x\nEquation 2: x = ((y * (x * y)) * x) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = (((y * y) * z) * w) * x", "equation2": "x = ((y * (x * y)) * x) * x", "difficulty": "normal", "index": 802}} |
| {"id": "normal_0830", "input": "Equation 1: x = y * (((x * x) * x) * y)\nEquation 2: x = (y * x) * ((z * z) * y)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * (((x * x) * x) * y)", "equation2": "x = (y * x) * ((z * z) * y)", "difficulty": "normal", "index": 830}} |
| {"id": "normal_0831", "input": "Equation 1: x * (y * z) = (y * y) * x\nEquation 2: x * y = (z * x) * (y * x)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * (y * z) = (y * y) * x", "equation2": "x * y = (z * x) * (y * x)", "difficulty": "normal", "index": 831}} |
| {"id": "normal_0832", "input": "Equation 1: x = ((x * x) * y) * (y * z)\nEquation 2: x * x = ((x * x) * y) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((x * x) * y) * (y * z)", "equation2": "x * x = ((x * x) * y) * x", "difficulty": "normal", "index": 832}} |
| {"id": "normal_0837", "input": "Equation 1: x = ((y * (x * x)) * y) * y\nEquation 2: x * y = z * ((z * y) * z)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((y * (x * x)) * y) * y", "equation2": "x * y = z * ((z * y) * z)", "difficulty": "normal", "index": 837}} |
| {"id": "normal_0846", "input": "Equation 1: x = ((x * (x * y)) * z) * w\nEquation 2: x = (y * ((y * x) * z)) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((x * (x * y)) * z) * w", "equation2": "x = (y * ((y * x) * z)) * y", "difficulty": "normal", "index": 846}} |
| {"id": "normal_0848", "input": "Equation 1: x = (y * x) * (y * (z * y))\nEquation 2: x = ((y * z) * w) * (x * x)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = (y * x) * (y * (z * y))", "equation2": "x = ((y * z) * w) * (x * x)", "difficulty": "normal", "index": 848}} |
| {"id": "normal_0852", "input": "Equation 1: x = y * (y * (x * (y * x)))\nEquation 2: x = y * ((z * (x * w)) * x)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = y * (y * (x * (y * x)))", "equation2": "x = y * ((z * (x * w)) * x)", "difficulty": "normal", "index": 852}} |
| {"id": "normal_0882", "input": "Equation 1: x * x = (y * (z * y)) * y\nEquation 2: x * x = ((y * x) * y) * z\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * x = (y * (z * y)) * y", "equation2": "x * x = ((y * x) * y) * z", "difficulty": "normal", "index": 882}} |
| {"id": "normal_0883", "input": "Equation 1: x = ((y * z) * (w * y)) * x\nEquation 2: x * (y * z) = (x * y) * y\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((y * z) * (w * y)) * x", "equation2": "x * (y * z) = (x * y) * y", "difficulty": "normal", "index": 883}} |
| {"id": "normal_0898", "input": "Equation 1: x = (y * ((y * y) * z)) * x\nEquation 2: x = (y * x) * (y * y)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * ((y * y) * z)) * x", "equation2": "x = (y * x) * (y * y)", "difficulty": "normal", "index": 898}} |
| {"id": "normal_0903", "input": "Equation 1: x * y = y * (x * (x * x))\nEquation 2: x * y = ((z * y) * z) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * y = y * (x * (x * x))", "equation2": "x * y = ((z * y) * z) * x", "difficulty": "normal", "index": 903}} |
| {"id": "normal_0906", "input": "Equation 1: x = ((y * x) * (x * x)) * z\nEquation 2: x = y * ((z * x) * (x * y))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "True", "answer": "False", "metadata": {"equation1": "x = ((y * x) * (x * x)) * z", "equation2": "x = y * ((z * x) * (x * y))", "difficulty": "normal", "index": 906}} |
| {"id": "normal_0907", "input": "Equation 1: x = ((x * (y * x)) * x) * z\nEquation 2: x = (x * (y * (z * y))) * w\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = ((x * (y * x)) * x) * z", "equation2": "x = (x * (y * (z * y))) * w", "difficulty": "normal", "index": 907}} |
| {"id": "normal_0956", "input": "Equation 1: x = (y * x) * ((x * x) * y)\nEquation 2: x = x * ((x * y) * (y * x))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * x) * ((x * x) * y)", "equation2": "x = x * ((x * y) * (y * x))", "difficulty": "normal", "index": 956}} |
| {"id": "normal_0971", "input": "Equation 1: x = (y * (x * z)) * (x * x)\nEquation 2: x * y = y * (x * (z * y))\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * (x * z)) * (x * x)", "equation2": "x * y = y * (x * (z * y))", "difficulty": "normal", "index": 971}} |
| {"id": "normal_0987", "input": "Equation 1: x * (y * z) = (w * z) * w\nEquation 2: x * y = z * ((w * z) * u)\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * (y * z) = (w * z) * w", "equation2": "x * y = z * ((w * z) * u)", "difficulty": "normal", "index": 987}} |
| {"id": "normal_0990", "input": "Equation 1: x * (y * y) = (z * x) * x\nEquation 2: x * x = ((x * y) * z) * w\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x * (y * y) = (z * x) * x", "equation2": "x * x = ((x * y) * z) * w", "difficulty": "normal", "index": 990}} |
| {"id": "normal_0997", "input": "Equation 1: x = (y * z) * (x * (z * x))\nEquation 2: x * x = (y * (y * y)) * x\n\nDoes Equation 1 imply Equation 2 over all magmas?\nAnswer True or False.", "target": "False", "answer": "True", "metadata": {"equation1": "x = (y * z) * (x * (z * x))", "equation2": "x * x = (y * (y * y)) * x", "difficulty": "normal", "index": 997}} |
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