## Heatmap: Performance Metrics ### Overview The heatmap illustrates the performance metrics of different methods over time, with a focus on the number of theorems proven and the number of tactics applied. The methods compared are "ours," "ours+hammer," "ours+hammer+intuition," "ours+hammer+intuition+auto," and "ours+hammer+intuition+auto+easy." ### Components/Axes - **X-axis**: Time (seconds) - **Y-axis**: Number of theorems proven - **Color Scale**: Theorems proven range from 0 to 4000, with darker shades indicating higher numbers. - **Legend**: The legend on the right side of the heatmap indicates the methods used. - **Tactics**: The number of tactics applied is shown on the bottom right, with a scale from 0 to 300. ### Detailed Analysis or ### Content Details - **ours**: The method "ours" shows a steady increase in the number of theorems proven over time, peaking at around 3500 theorems proven after 600 seconds. - **ours+hammer**: The method "ours+hammer" also shows an increase, but at a slower rate compared to "ours." It peaks at around 3800 theorems proven after 600 seconds. - **ours+hammer+intuition**: This method combines "ours," "hammer," and "intuition." It shows a significant increase in the number of theorems proven, peaking at around 4000 theorems proven after 600 seconds. - **ours+hammer+intuition+auto**: This method includes "ours," "hammer," "intuition," and "auto." It shows the highest increase in the number of theorems proven, peaking at around 4200 theorems proven after 600 seconds. - **ours+hammer+intuition+auto+easy**: This method includes "ours," "hammer," "intuition," "auto," and "easy." It shows the highest increase in the number of theorems proven, peaking at around 4400 theorems proven after 600 seconds. ### Key Observations - The method "ours+hammer+intuition+auto+easy" consistently outperforms the other methods in terms of the number of theorems proven. - The method "ours+hammer+intuition" shows the second highest performance, followed by "ours+hammer+intuition+auto." - The method "ours+hammer" shows the third highest performance, followed by "ours+hammer+intuition+auto+easy." - The method "ours" shows the lowest performance, followed by "ours+hammer+intuition+auto+easy." ### Interpretation The heatmap suggests that the combination of "ours," "hammer," "intuition," "auto," and "easy" tactics leads to the highest performance in terms of the number of theorems proven. The method "ours+hammer+intuition+auto+easy" appears to be the most effective in this scenario. The other methods show varying degrees of performance, with "ours+hammer+intuition" and "ours+hammer+intuition+auto" showing significant improvements over "ours." The method "ours" shows the lowest performance, indicating that it may not be the most effective method for this task.