## Histogram: Proof Length Distribution
### Overview
The image is a histogram comparing the distribution of proof lengths (measured in number of lines) for two categories: "CPL" and "Simple loop". The chart visualizes the percentage of proofs that fall within specific line-count ranges.
### Components/Axes
* **Title:** "Proof length distribution" (centered at the top).
* **X-axis:** Labeled "Number of lines in proofs". The axis has major tick marks at 0, 20, 40, 60, 80, and 100.
* **Y-axis:** Labeled "Percentage (%)". The axis has major tick marks at 0, 1, 2, 3, 4, 5, and 6.
* **Legend:** Located in the top-right corner of the plot area.
* A blue square is labeled "CPL".
* An orange square is labeled "Simple loop".
* **Data Series:** Two overlapping histograms.
* **CPL (Blue):** Bars are a light blue color.
* **Simple loop (Orange):** Bars are a light orange color. Where the two series overlap, the color appears as a brownish blend.
### Detailed Analysis
The histogram bins proofs into ranges of 10 lines each (0-10, 10-20, etc.). The height of each bar represents the percentage of proofs in that length range for each category.
**Trend Verification:** Both distributions are strongly right-skewed. The vast majority of proofs are short, with the frequency dropping sharply as the number of lines increases. The "Simple loop" distribution appears to be contained within the shorter proof lengths, while the "CPL" distribution has a longer tail extending to higher line counts.
**Approximate Data Points (by bin):**
* **Bin 0-10 lines:**
* Simple loop: ~6.0%
* CPL: ~5.5%
* **Bin 10-20 lines:**
* Simple loop: ~2.8%
* CPL: ~2.4%
* **Bin 20-30 lines:**
* Simple loop: ~0.8%
* CPL: ~0.7%
* **Bin 30-40 lines:**
* Simple loop: ~0.2% (very small bar)
* CPL: ~0.5%
* **Bin 40-50 lines:**
* Simple loop: Not visible (0% or negligible).
* CPL: ~0.4%
* **Bin 50-60 lines:**
* Simple loop: Not visible.
* CPL: ~0.2%
* **Bin 60-70 lines:**
* Simple loop: Not visible.
* CPL: ~0.1% (very small bar)
* **Bin 70-80 lines:**
* Simple loop: Not visible.
* CPL: ~0.1% (very small bar)
* **Bin 80-90 lines:**
* Simple loop: Not visible.
* CPL: ~0.2%
* **Bin 90-100 lines:**
* Simple loop: Not visible.
* CPL: ~0.1% (very small bar)
* **Bin 100+ lines:**
* Simple loop: Not visible.
* CPL: ~0.1% (very small bar)
### Key Observations
1. **Dominance of Short Proofs:** Over 80% of the visible distribution for both categories consists of proofs with fewer than 20 lines.
2. **Category Comparison:** In the shortest bin (0-10 lines), "Simple loop" has a slightly higher percentage than "CPL". For bins beyond 20 lines, "CPL" consistently shows a higher percentage than "Simple loop".
3. **Range Limit:** The "Simple loop" data appears to be absent (or at 0%) for proofs longer than approximately 40 lines. The "CPL" data has a long, thin tail extending to over 100 lines.
4. **Overlap:** The most significant overlap and blending of colors occurs in the first three bins (0-30 lines), indicating both categories are most frequently found in this range.
### Interpretation
This histogram suggests a fundamental difference in the nature of proofs generated by or for "CPL" versus "Simple loop". The "Simple loop" proofs are almost exclusively short, implying they are generated from a more constrained or simpler process. The "CPL" proofs, while still predominantly short, have a non-trivial probability of being much longer. This could indicate that CPL is used for more complex problem-solving tasks that occasionally require lengthy, detailed proofs, or that it is a more expressive system capable of generating longer derivations. The data demonstrates that proof length is a heavy-tailed distribution, a common characteristic in computational and logical systems where most instances are simple, but a few complex outliers exist.
