## Histogram: Proof Length Distribution ### Overview The image is a histogram comparing the distribution of proof lengths for "Generated proof" and "Ground truth" data. The x-axis represents the length of the proof, and the y-axis represents the number of proofs. The histogram shows the frequency of different proof lengths for both datasets. ### Components/Axes * **X-axis:** "Length", ranging from 0 to 50. * **Y-axis:** "# Proofs", ranging from 0 to 2500. * **Legend (top-right):** * "Generated proof" - Represented by red bars. * "Ground truth" - Represented by teal bars. * **Vertical dashed lines:** * Red dashed line at approximately x=5 * Teal dashed line at approximately x=12 ### Detailed Analysis * **Ground truth (teal):** * The distribution is heavily skewed to the right. * The highest frequency occurs at a length of approximately 1, with a value of approximately 2400 proofs. * The frequency decreases rapidly as the length increases. * At length 10, the number of proofs is approximately 300. * At length 20, the number of proofs is approximately 50. * At length 30, the number of proofs is approximately 20. * At length 40, the number of proofs is approximately 5. * At length 50, the number of proofs is approximately 2. * **Generated proof (red):** * The distribution is also skewed to the right, but less extreme than the "Ground truth" data. * The highest frequency occurs at a length of approximately 1, with a value of approximately 250 proofs. * The frequency decreases as the length increases, but at a slower rate than the "Ground truth" data. * The number of proofs is very low after length 10. ### Key Observations * Both "Generated proof" and "Ground truth" data show a right-skewed distribution, indicating that shorter proofs are more common. * "Ground truth" proofs are significantly more concentrated at shorter lengths compared to "Generated proof". * The red dashed line is positioned at approximately x=5, and the teal dashed line is positioned at approximately x=12. ### Interpretation The histogram suggests that the "Ground truth" data contains a larger number of very short proofs compared to the "Generated proof" data. The "Generated proof" data has a broader distribution, indicating a greater variety of proof lengths. The dashed lines may represent the mean or median proof length for each dataset, with the "Ground truth" proofs having a shorter average length. The data implies that the generated proofs tend to be longer than the ground truth proofs.

\n ## Bar Chart: Proof Length Distribution ### Overview This image presents a bar chart comparing the distribution of lengths of "Generated Proofs" and "Ground Truth" proofs. The chart displays the number of proofs (# Proofs) against their corresponding length. The chart appears to be a histogram, showing the frequency of different proof lengths. ### Components/Axes * **X-axis:** "Length" - ranging from 0 to 50, with tick marks at intervals of 5. * **Y-axis:** "# Proofs" - ranging from 0 to 2500, with tick marks at intervals of 500. * **Legend:** Located in the top-right corner. * "Generated proof" - represented by a red color. * "Ground truth" - represented by a teal/cyan color. * **Vertical Dashed Lines:** Two vertical dashed lines are present, one at approximately length 4 and another at approximately length 9. These lines may indicate specific length thresholds or points of interest. ### Detailed Analysis The chart shows two distinct distributions. **Ground Truth (Teal/Cyan):** The "Ground Truth" distribution exhibits a strong right skew. The number of proofs decreases rapidly as the length increases. * At length 0, the count is approximately 2400. * At length 1, the count is approximately 1800. * At length 2, the count is approximately 1200. * At length 3, the count is approximately 800. * At length 4, the count is approximately 500. * At length 5, the count is approximately 350. * At length 10, the count is approximately 150. * At length 20, the count is approximately 50. * At length 30, the count is approximately 20. * At length 40, the count is approximately 10. * At length 50, the count is approximately 5. **Generated Proof (Red):** The "Generated Proof" distribution is also right-skewed, but it is more concentrated at lower lengths. * At length 0, the count is approximately 100. * At length 1, the count is approximately 200. * At length 2, the count is approximately 300. * At length 3, the count is approximately 350. * At length 4, the count is approximately 250. * At length 5, the count is approximately 150. * At length 10, the count is approximately 50. * From length 15 onwards, the count is below 20 and decreases to near zero. ### Key Observations * The "Ground Truth" proofs generally have longer lengths than the "Generated Proofs". * The "Ground Truth" distribution has a significantly higher number of proofs overall compared to the "Generated Proofs". * The vertical dashed lines at lengths 4 and 9 may highlight a difference in the distributions, potentially indicating a cutoff or a region where the performance of the generated proofs is being evaluated. * The "Generated Proof" distribution is more heavily weighted towards shorter lengths. ### Interpretation The data suggests that the "Generated Proofs" are, on average, shorter than the "Ground Truth" proofs. This could indicate that the generation process is simplifying the proofs or failing to capture all the necessary information. The higher concentration of "Generated Proofs" at lower lengths suggests a potential bias in the generation algorithm towards shorter solutions. The vertical lines at 4 and 9 could be used to evaluate the percentage of generated proofs that meet a certain length requirement. The significant difference in the total number of proofs between the two distributions suggests that the "Ground Truth" dataset is much larger or more comprehensive than the set of "Generated Proofs". This could be due to the difficulty of generating proofs or the limitations of the generation algorithm. The chart provides a visual comparison of the length distributions, allowing for a quantitative assessment of the quality and characteristics of the generated proofs.

\n ## Histogram: Comparison of Generated Proof vs. Ground Truth Lengths ### Overview The image is a histogram comparing the frequency distribution of proof lengths for two categories: "Generated proof" and "Ground truth." The chart illustrates that proofs in the "Ground truth" dataset are significantly more numerous and tend to be longer than those in the "Generated proof" dataset. ### Components/Axes * **Chart Type:** Histogram (Bar Chart). * **X-Axis:** Labeled **"Length"**. It represents the length of a proof, with a scale from 0 to 50. Major tick marks are at 0, 10, 20, 30, 40, and 50. * **Y-Axis:** Labeled **"# Proofs"**. It represents the count or frequency of proofs, with a scale from 0 to 2500. Major tick marks are at 0, 500, 1000, 1500, 2000, and 2500. * **Legend:** Located in the **top-right corner** of the chart area. * A **red square** is labeled **"Generated proof"**. * A **teal (blue-green) square** is labeled **"Ground truth"**. * **Vertical Reference Lines:** * A **red dashed vertical line** is positioned at approximately **Length = 6**. * A **teal dashed vertical line** is positioned at approximately **Length = 13**. ### Detailed Analysis The data is presented as two overlapping histograms with bars for each integer length value from 1 to 50. **1. Ground Truth (Teal Bars):** * **Trend:** The distribution is strongly right-skewed. The frequency is highest at the shortest lengths and decreases rapidly as length increases. * **Key Data Points (Approximate):** * Length 1: ~2500 proofs (the global peak). * Length 2: ~1600 proofs. * Length 3: ~1300 proofs. * Length 4: ~900 proofs. * Length 5: ~700 proofs. * The frequency continues to decline steadily. By Length 20, the count is below 200. By Length 50, the count is near zero. * The **teal dashed line at Length ~13** likely represents a central tendency measure (e.g., median or mean) for the Ground Truth distribution. **2. Generated Proof (Red Bars):** * **Trend:** Also right-skewed, but with a much lower overall frequency and a shorter effective range. * **Key Data Points (Approximate):** * Length 1: ~200 proofs (the peak for this series). * Length 2: ~150 proofs. * Length 3: ~100 proofs. * Length 4: ~70 proofs. * The frequency drops off quickly. By Length 10, the count is very low (likely <20). The bars become negligible or invisible beyond approximately Length 15. * The **red dashed line at Length ~6** likely represents a central tendency measure for the Generated Proof distribution. ### Key Observations 1. **Magnitude Disparity:** The "Ground truth" dataset contains orders of magnitude more proofs than the "Generated proof" dataset, especially at short lengths. 2. **Length Disparity:** The central tendency for "Ground truth" proofs (Length ~13) is more than double that of "Generated proof" proofs (Length ~6). The generated proofs are systematically shorter. 3. **Distribution Shape:** Both distributions follow a similar decaying pattern, but the "Generated proof" distribution is truncated, failing to produce proofs of longer lengths present in the ground truth. 4. **Overlap:** The red bars are visible only at the very beginning of the x-axis (Lengths 1-~12), sitting in front of the much taller teal bars. ### Interpretation This histogram suggests a significant gap between the capabilities of a proof generation system and the actual complexity (as measured by length) of real-world proofs. * **The data demonstrates** that the generation system primarily produces very short proofs (peaking at length 1) and its output frequency drops to near zero well before the ground truth distribution does. The system appears unable to replicate the longer, more complex proofs found in the ground truth dataset. * **The elements relate** by showing a direct comparison on the same scale. The vertical dashed lines provide a quick visual summary of the core disparity: the "average" generated proof is less than half the length of the "average" real proof. * **Notable anomaly/trend:** The most striking trend is the **complete absence of generated proofs beyond a certain length (~15)**, while the ground truth continues with measurable frequency up to length 50. This indicates a potential limitation or bias in the generation model towards simplicity. The system's output is not just less frequent but also fundamentally less complex than the data it is presumably trained on or meant to emulate.

## Bar Chart: Proof Distribution ### Overview The bar chart displays the distribution of proof lengths, categorized into two groups: "Generated proof" and "Ground truth." The x-axis represents the length of the proof in units, while the y-axis shows the number of proofs. ### Components/Axes - **X-Axis**: Labeled "Length," it ranges from 0 to 50 units. - **Y-Axis**: Labeled "# Proofs," it ranges from 0 to 2500. - **Legend**: Located at the top right, it differentiates between "Generated proof" (red) and "Ground truth" (blue). ### Detailed Analysis or ### Content Details - The majority of proofs are short, with most falling between 0 and 10 units. - There is a significant peak in the "Generated proof" category at around 5 units. - The "Ground truth" category shows a more even distribution, with fewer proofs of any length. - The "Generated proof" category has a slightly higher number of proofs at longer lengths compared to the "Ground truth" category. ### Key Observations - The "Generated proof" category has a higher number of proofs at shorter lengths. - The "Ground truth" category has a more even distribution across all lengths. - There is a noticeable difference in the number of proofs between the two categories, with the "Generated proof" category having a higher count. ### Interpretation The data suggests that the generated proofs tend to be shorter than the ground truth proofs. This could indicate that the generated proofs are more concise or that the ground truth proofs are more complex. The even distribution in the "Ground truth" category suggests that the generated proofs are more consistent in length. The difference in the number of proofs between the two categories could be due to the nature of the proofs being generated or the complexity of the ground truth proofs.

## Bar Chart: Comparison of Generated Proofs vs Ground Truth by Length ### Overview The chart compares the distribution of "Generated proof" (red bars) and "Ground truth" (blue bars) across lengths from 0 to 50. The y-axis represents the number of proofs, while the x-axis represents length. The legend is positioned in the top-right corner, with red corresponding to "Generated proof" and blue to "Ground truth." ### Components/Axes - **X-axis (Length)**: Labeled "Length," with markers at intervals of 0, 10, 20, 30, 40, and 50. - **Y-axis (# Proofs)**: Labeled "# Proofs," scaled from 0 to 2500 in increments of 500. - **Legend**: Located in the top-right corner, with red for "Generated proof" and blue for "Ground truth." ### Detailed Analysis - **Ground truth (blue)**: - Dominates at shorter lengths, peaking at ~2500 proofs at length 0. - Declines sharply, reaching ~1000 proofs at length 10, ~500 at length 15, and ~100 at length 20. - Near-zero values observed from length 25 onward. - **Generated proof (red)**: - Starts at ~300 proofs at length 0, peaking slightly at ~400 at length 5. - Declines gradually, maintaining ~100–200 proofs until length 20. - Drops to near-zero by length 30, with minimal values thereafter. ### Key Observations 1. **Ground truth** exhibits a steep decline, with most proofs concentrated at shorter lengths (0–15). 2. **Generated proof** declines more gradually, persisting at non-zero values up to length 30. 3. No overlap between the two series except at length 0, where "Generated proof" is ~10% of "Ground truth." 4. Both series show near-zero values beyond length 30, suggesting a cutoff in proof generation or validation. ### Interpretation The data suggests that "Ground truth" proofs are predominantly short, with a rapid drop-off in frequency as length increases. In contrast, "Generated proof" maintains a more sustained distribution, indicating potential differences in generation or validation criteria. The stark divergence at shorter lengths (e.g., length 0) may reflect methodological differences, such as automated generation favoring longer proofs or ground truth being manually curated for brevity. The near-zero values beyond length 30 imply a practical limit to proof complexity or scope in the dataset.