## Histogram: Proof Length Distribution ### Overview The image is a histogram showing the distribution of proof lengths, measured by the number of lines in proofs. Two different methods, "CPL" and "Simple loop", are compared. The y-axis represents the percentage of proofs, and the x-axis represents the number of lines in proofs. ### Components/Axes * **Title:** Proof length distribution * **X-axis:** Number of lines in proofs * Scale: 0 to 100, with tick marks at 0, 20, 40, 60, 80, and 100. * **Y-axis:** Percentage (%) * Scale: 0 to 6, with tick marks at 0, 1, 2, 3, 4, 5, and 6. * **Legend:** Located in the top-right corner. * CPL: Represented by light blue bars. * Simple loop: Represented by light orange bars. ### Detailed Analysis The histogram displays the frequency distribution of proof lengths for two methods: CPL (light blue) and Simple loop (light orange). The height of each bar indicates the percentage of proofs falling within a specific range of line counts. **CPL (Light Blue):** * 0-20 lines: Approximately 5.4% * 20-40 lines: Approximately 0.4% * 40-60 lines: Approximately 0.2% * 60-80 lines: Approximately 0.1% * 80-100 lines: Approximately 0.1% * 100-120 lines: Approximately 0.05% **Simple Loop (Light Orange):** * 0-20 lines: Approximately 5.9% * 20-40 lines: Approximately 0.8% * 40-60 lines: Approximately 0.0% * 60-80 lines: Approximately 0.0% * 80-100 lines: Approximately 0.0% * 100-120 lines: Approximately 0.0% ### Key Observations * Both methods show a high concentration of proofs with shorter lengths (0-20 lines). * The "Simple loop" method has a higher percentage of proofs in the 0-20 lines range compared to "CPL". * As the number of lines increases, the percentage of proofs decreases for both methods. * The "CPL" method has a slightly more gradual decrease in percentage as the number of lines increases compared to "Simple loop". ### Interpretation The histogram suggests that both the "CPL" and "Simple loop" methods tend to produce proofs with relatively short lengths. The "Simple loop" method appears to be more heavily skewed towards shorter proofs compared to "CPL". This could indicate that "Simple loop" is more efficient for simpler proofs, while "CPL" might be used or result in slightly longer proofs in some cases. The rapid decrease in percentage for both methods as the number of lines increases suggests that longer proofs are less common, regardless of the method used.

\n ## Chart: Proof Length Distribution ### Overview The image presents a histogram illustrating the distribution of proof lengths for two different proof strategies: CPL (Classical Proof Logic) and Simple Loop. The x-axis represents the number of lines in proofs, while the y-axis represents the percentage of proofs falling within specific line-count ranges. The chart visually compares the length of proofs generated by each method. ### Components/Axes * **Title:** "Proof length distribution" - positioned at the top-center of the chart. * **X-axis Label:** "Number of lines in proofs" - positioned at the bottom-center. The scale ranges from 0 to 100, with increments of 20. * **Y-axis Label:** "Percentage (%)" - positioned at the left-center. The scale ranges from 0 to 6, with increments of 1. * **Legend:** Located in the top-right corner. * "CPL" - represented by a light blue color. * "Simple loop" - represented by a light orange/brown color. ### Detailed Analysis The chart displays two overlapping histograms. **Simple Loop (Orange/Brown):** The Simple Loop histogram exhibits a strong peak between 0 and 20 lines, reaching approximately 5.8% at around the 10-line mark. The distribution slopes downward as the number of lines increases, with a noticeable drop after 20 lines. There is a small, secondary peak around 20-40 lines, reaching approximately 3.2%. The percentage of proofs with more than 40 lines decreases rapidly, approaching 0% after 80 lines. * Approximately 5.8% of Simple Loop proofs have between 0 and 20 lines. * Approximately 3.2% of Simple Loop proofs have between 20 and 40 lines. * Approximately 1% of Simple Loop proofs have between 40 and 60 lines. * Less than 0.5% of Simple Loop proofs have more than 60 lines. **CPL (Light Blue):** The CPL histogram is significantly lower in magnitude compared to the Simple Loop histogram. It shows a slight peak between 0 and 20 lines, reaching approximately 0.2%. The distribution decreases rapidly as the number of lines increases. There is a small increase around 40 lines, reaching approximately 0.4%. The percentage remains low for proofs with more than 60 lines. * Approximately 0.2% of CPL proofs have between 0 and 20 lines. * Approximately 0.4% of CPL proofs have between 40 and 60 lines. * Approximately 0.2% of CPL proofs have between 80 and 100 lines. ### Key Observations * Simple Loop proofs are generally much shorter than CPL proofs. * The majority of Simple Loop proofs are less than 40 lines long. * CPL proofs tend to be longer, with a greater proportion exceeding 40 lines. * The Simple Loop strategy produces a more concentrated distribution of proof lengths, while CPL has a more dispersed distribution. ### Interpretation The data suggests that the Simple Loop strategy is more efficient in generating shorter proofs compared to CPL. The histogram clearly demonstrates that a significantly larger percentage of Simple Loop proofs fall within the lower range of line counts. This could indicate that Simple Loop is better suited for problems where concise proofs are desired, or that CPL introduces unnecessary complexity in certain cases. The difference in distributions suggests that CPL may require more steps or a more elaborate approach to reach a proof, even for relatively simple problems. The near-zero percentage of CPL proofs under 20 lines suggests that CPL is rarely able to produce very short proofs. The slight increase in CPL proofs around 40 lines could indicate a common level of complexity encountered when using this method.

## 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.

## Bar Chart: Proof Length Distribution ### Overview The bar chart displays the distribution of proof lengths in terms of the number of lines. It compares two types of proofs: CPL (Complex Proof) and Simple loop. ### Components/Axes - **X-axis**: Number of lines in proofs, ranging from 0 to 100. - **Y-axis**: Percentage (%), ranging from 0 to 6. - **Legend**: Two categories are represented by different colors: blue for CPL and orange for Simple loop. ### Detailed Analysis or ### Content Details - **CPL (Complex Proof)**: The majority of proofs have between 0 and 20 lines, with a peak at 5 lines. - **Simple loop**: The majority of proofs have between 0 and 20 lines, with a peak at 5 lines. - **Distribution**: Both categories show a similar distribution, with a slight peak at 5 lines. ### Key Observations - **Similarity**: Both CPL and Simple loop proofs have a similar distribution, with a peak at 5 lines. - **No significant outliers**: There are no significant outliers in the data. ### Interpretation The data suggests that both CPL and Simple loop proofs are relatively short, with a peak at 5 lines. This could indicate that these proof types are commonly used in mathematical or logical contexts where brevity is preferred. The similarity in distribution between the two categories suggests that they may be used in similar contexts or that the data is representative of a broader trend.

## Bar Chart: Proof length distribution ### Overview The chart compares the distribution of proof lengths (number of lines) for two methods: CPL and Simple loop. It shows the percentage of proofs falling into specific line ranges (0-20, 20-40, 40-60, 60-80, 80-100). CPL dominates in shorter proof lengths, while Simple loop has significantly lower percentages across all ranges. ### Components/Axes - **Title**: "Proof length distribution" - **X-axis**: "Number of lines in proofs" (categories: 0-20, 20-40, 40-60, 60-80, 80-100) - **Y-axis**: "Percentage (%)" (0% to 6%) - **Legend**: - Blue: CPL - Orange: Simple loop - **Bar Groups**: Each x-axis category contains two adjacent bars (CPL and Simple loop). ### Detailed Analysis - **0-20 lines**: - CPL: ~5.5% (blue bar) - Simple loop: ~0.5% (orange bar) - **20-40 lines**: - CPL: ~2.5% (blue bar) - Simple loop: ~0.3% (orange bar) - **40-60 lines**: - CPL: ~0.5% (blue bar) - Simple loop: ~0.1% (orange bar) - **60-80 lines**: - CPL: ~0.2% (blue bar) - Simple loop: ~0.05% (orange bar) - **80-100 lines**: - CPL: ~0.1% (blue bar) - Simple loop: ~0.02% (orange bar) ### Key Observations 1. **CPL Dominance**: CPL accounts for over 95% of proofs in the 0-40 line range, while Simple loop contributes less than 1%. 2. **Rapid Decline**: Both methods show exponential decay in proof length frequency, but CPL has a steeper drop-off after 20 lines. 3. **Simple Loop Efficiency**: Simple loop proofs are consistently shorter, with 99.98% of its proofs under 100 lines. 4. **CPL Outliers**: CPL has a small but notable presence in the 40-100 line range (~1% total), suggesting occasional longer proofs. ### Interpretation The data suggests CPL is more effective at generating concise proofs, with the majority of its proofs under 20 lines. Simple loop appears even more efficient in brevity but may sacrifice coverage or complexity, as it lacks significant representation in longer proof ranges. The near-absence of Simple loop proofs beyond 20 lines implies it might be optimized for trivial cases or specific problem types. CPL's tail distribution (40-100 lines) could indicate its use in handling more complex proofs where additional lines are necessary, though these cases remain rare. The stark contrast in distributions highlights fundamental differences in the methodologies' design philosophies or problem-solving approaches.