## Line Chart: pass@1-with-n-queries ### Overview The image is a line chart comparing the performance of two theorem proving systems, COPRA (GPT-4-turbo) with retrieval and Proverbot9001, based on the number of theorems proved as a function of the number of queries. The x-axis represents the number of queries (n), and the y-axis represents the number of theorems proved. ### Components/Axes * **Title:** pass@1-with-n-queries * **X-axis:** * Label: Number of Queries (n) * Scale: 0 to 60, with tick marks at intervals of 10 (0, 10, 20, 30, 40, 50, 60) * **Y-axis:** * Label: Number of Theorems Proved * Scale: 0 to 50, with tick marks at intervals of 10 (0, 10, 20, 30, 40, 50) * **Legend:** Located in the bottom-right corner of the chart. * COPRA (GPT-4-turbo) (with retrieval): Represented by a goldenrod line. * Proverbot9001: Represented by a dark blue line. ### Detailed Analysis * **COPRA (GPT-4-turbo) (with retrieval) (Goldenrod Line):** * Trend: The line generally slopes upward, indicating an increase in the number of theorems proved as the number of queries increases. The line plateaus around 57 theorems proved. * Data Points: * At approximately 2 queries, around 20 theorems are proved. * At approximately 8 queries, around 40 theorems are proved. * At approximately 18 queries, around 55 theorems are proved. * The line plateaus at approximately 57 theorems proved after 20 queries. * **Proverbot9001 (Dark Blue Line):** * Trend: The line generally slopes upward, indicating an increase in the number of theorems proved as the number of queries increases. The line appears to be approaching a plateau around 54 theorems proved. * Data Points: * At 2 queries, approximately 2 theorems are proved. * At 10 queries, approximately 25 theorems are proved. * At 20 queries, approximately 35 theorems are proved. * At 40 queries, approximately 50 theorems are proved. * At 60 queries, approximately 54 theorems are proved. ### Key Observations * COPRA (GPT-4-turbo) (with retrieval) proves theorems more rapidly with fewer queries compared to Proverbot9001. * COPRA (GPT-4-turbo) (with retrieval) plateaus at a higher number of theorems proved compared to Proverbot9001 within the observed query range. * Proverbot9001 shows a more gradual increase in the number of theorems proved as the number of queries increases. ### Interpretation The chart demonstrates that COPRA (GPT-4-turbo) (with retrieval) is more efficient at proving theorems compared to Proverbot9001, especially with a smaller number of queries. The plateauing of both lines suggests that there may be a limit to the number of theorems that can be proved with the given query setup or that additional queries beyond a certain point do not significantly improve performance. The difference in the initial slope and plateau level indicates that COPRA (GPT-4-turbo) (with retrieval) has a more effective strategy for theorem proving within the tested parameters.

## Line Chart: Theorem Proving Performance ### Overview This line chart compares the performance of two theorem proving systems, COPRAGPT-4-turbo (with retrieval) and Proverbot9001, based on the number of theorems proved as a function of the number of queries submitted. The chart title indicates the metric being evaluated is "pass@1-with-n-queries". ### Components/Axes * **X-axis:** "Number of Queries (n)", ranging from 0 to 60, with tick marks at intervals of 10. * **Y-axis:** "Number of Theorems Proved", ranging from 0 to 60, with tick marks at intervals of 10. * **Lines:** * COPRAGPT-4-turbo (with retrieval) - represented by a solid orange line. * Proverbot9001 - represented by a solid blue line. * **Legend:** Located in the bottom-right corner of the chart. * **Title:** "pass@1-with-n-queries" positioned at the top-center of the chart. ### Detailed Analysis **COPRAGPT-4-turbo (with retrieval) - Orange Line:** The orange line starts at approximately 0 theorems proved at 0 queries. It exhibits a steep upward slope initially, reaching approximately 15 theorems proved at around 8 queries. The slope gradually decreases, leveling off around 55-58 theorems proved between 25 and 60 queries. There is a plateau between approximately 20 and 40 queries where the number of theorems proved remains relatively constant. * (0, 0) * (5, ~2) * (10, ~15) * (15, ~25) * (20, ~35) * (25, ~45) * (30, ~48) * (35, ~50) * (40, ~52) * (45, ~54) * (50, ~55) * (60, ~58) **Proverbot9001 - Blue Line:** The blue line also starts at approximately 0 theorems proved at 0 queries. It has a more gradual initial slope compared to the orange line, reaching approximately 8 theorems proved at around 8 queries. The slope increases more noticeably between 8 and 20 queries, reaching approximately 32 theorems proved at 20 queries. The slope then decreases, and the line plateaus around 52-54 theorems proved between 40 and 60 queries. The line has more pronounced step-like increases than the orange line. * (0, 0) * (5, ~3) * (10, ~8) * (15, ~18) * (20, ~32) * (25, ~38) * (30, ~42) * (35, ~46) * (40, ~50) * (45, ~52) * (50, ~53) * (60, ~54) ### Key Observations * COPRAGPT-4-turbo (with retrieval) consistently proves more theorems than Proverbot9001 across the entire range of queries. * Both systems exhibit diminishing returns as the number of queries increases, with the rate of theorem proving slowing down. * Proverbot9001 shows more discrete jumps in the number of theorems proved, suggesting it may solve theorems in batches or have a more step-wise learning process. * The orange line (COPRAGPT-4-turbo) has a smoother curve, indicating a more continuous improvement in performance. ### Interpretation The chart demonstrates that COPRAGPT-4-turbo (with retrieval) outperforms Proverbot9001 in theorem proving, as measured by the number of theorems proved per query. The "pass@1-with-n-queries" metric suggests that the systems are evaluated on their ability to correctly prove theorems given a single attempt per query. The diminishing returns observed in both systems indicate that there is a limit to the effectiveness of simply increasing the number of queries. The differences in the smoothness of the curves suggest that the two systems employ different strategies for theorem proving. The step-like increases in Proverbot9001's performance could be due to the system learning from its mistakes and making significant progress in batches. The plateauing of both lines suggests that the systems are approaching their maximum performance level within the tested query range. This data could be used to compare the efficiency and effectiveness of different theorem proving approaches and to identify areas for improvement.

## Line Chart: pass@1-with-n-queries ### Overview The image is a line chart comparing the performance of two automated theorem-proving systems. The chart plots the cumulative number of theorems successfully proved against the number of queries (attempts) made by each system. The title is "pass@1-with-n-queries". ### Components/Axes * **Chart Title:** `pass@1-with-n-queries` * **X-Axis:** * **Label:** `Number of Queries (n)` * **Scale:** Linear, ranging from 0 to 60, with major tick marks every 10 units (0, 10, 20, 30, 40, 50, 60). * **Y-Axis:** * **Label:** `Number of Theorems Proved` * **Scale:** Linear, ranging from 0 to approximately 58, with major tick marks every 10 units (0, 10, 20, 30, 40, 50). * **Legend:** Located in the bottom-right corner of the plot area. * **Orange Line:** `COPRA (GPT-4-turbo) (with retrieval)` * **Blue Line:** `Proverbot9001` ### Detailed Analysis The chart displays two step-like, monotonically increasing lines, indicating cumulative success. **1. COPRA (GPT-4-turbo) (with retrieval) - Orange Line:** * **Trend:** Exhibits a very steep initial ascent, followed by a rapid deceleration and eventual plateau. * **Data Points (Approximate):** * Starts at (0, 0). * Shows a dramatic increase between 0 and 5 queries, reaching approximately 35 theorems proved. * Continues to climb steeply until about 10 queries, reaching ~45 theorems. * The rate of increase slows significantly after 10 queries. * Reaches a plateau of approximately 56-57 theorems proved by around 20 queries. * The line remains flat (plateaued) from ~20 queries to the end of the chart at 60 queries. **2. Proverbot9001 - Blue Line:** * **Trend:** Shows a steadier, more gradual, and sustained increase compared to the orange line. * **Data Points (Approximate):** * Starts at (0, 0). * Increases steadily, reaching ~10 theorems by 5 queries and ~25 theorems by 10 queries. * Continues a consistent upward climb, crossing 40 theorems at approximately 30 queries. * Reaches approximately 50 theorems by 40 queries. * The line ends at approximately 54 theorems proved at 60 queries. A short dotted blue line extends horizontally to the right from the final point, suggesting the process may continue beyond the plotted range. ### Key Observations 1. **Performance Divergence:** The two systems have markedly different performance profiles. COPRA demonstrates high early efficiency, proving a large number of theorems with very few queries. Proverbot9001 is slower to start but maintains a more consistent rate of proving over a longer sequence of queries. 2. **Crossover Point:** The orange line (COPRA) is above the blue line (Proverbot9001) for the entire visible range of the chart. COPRA maintains a lead in theorems proved at every query count from 1 to 60. 3. **Plateau vs. Continued Growth:** COPRA's performance plateaus after ~20 queries, suggesting it may have exhausted the "easier" theorems in the dataset or reached a capability limit. Proverbot9001 shows no sign of plateauing within the 60-query window, indicating it might continue to prove more theorems if given more queries. 4. **Final Values:** At the 60-query mark, COPRA has proved approximately 56-57 theorems, while Proverbot9001 has proved approximately 54. The gap between them narrows significantly from a peak of ~20 theorems (at ~10 queries) to only ~2-3 theorems at 60 queries. ### Interpretation This chart illustrates a classic trade-off between **initial efficiency** and **sustained performance** in automated reasoning tasks. * **COPRA (with retrieval)** appears to be highly optimized for quickly solving a subset of problems, likely leveraging its retrieval mechanism and the powerful GPT-4-turbo model to identify and prove theorems that are more straightforward or closely match patterns in its knowledge base. Its rapid plateau suggests it may struggle with more complex or novel theorems that require deeper, sequential reasoning beyond its initial retrieval-augmented approach. * **Proverbot9001** exhibits the behavior of a more traditional, possibly search-based or iterative theorem prover. It makes slower but steady progress, systematically working through problems. Its lack of a plateau within the observed window implies it has a more robust mechanism for tackling difficult problems, albeit at a higher computational cost (more queries). The data suggests that for scenarios requiring quick results on a limited budget of attempts, COPRA is superior. However, for maximizing the total number of proven theorems given a larger allowance of queries, Proverbot9001 may ultimately be more effective, as indicated by its continuing upward trend and the narrowing performance gap. The "pass@1" metric in the title implies this measures success on the first attempt per query, highlighting COPRA's strength in single-shot accuracy.

## Line Graph: pass@1-with-n-queries ### Overview The image is a line graph comparing the performance of two systems, COPRA (GPT-4-turbo) with retrieval and Proverbot9001, in terms of the number of theorems proved as the number of queries increases. The graph spans 0 to 60 queries on the x-axis and 0 to 55 theorems proved on the y-axis. ### Components/Axes - **Title**: "pass@1-with-n-queries" (centered at the top). - **X-axis**: "Number of Queries (n)" with values from 0 to 60 in increments of 10. - **Y-axis**: "Number of Theorems Proved" with values from 0 to 55 in increments of 10. - **Legend**: Located in the bottom-right corner, with: - **Orange solid line**: "COPRA (GPT-4-turbo) (with retrieval)". - **Blue dotted line**: "Proverbot9001". ### Detailed Analysis 1. **COPRA (GPT-4-turbo) (with retrieval)**: - Starts at (0, 0) and rises sharply, reaching ~55 theorems proved by ~20 queries. - Plateaus at ~55 theorems proved for queries ≥20. - Key data points: - ~15 theorems at 10 queries. - ~35 theorems at 20 queries. - ~55 theorems at 30+ queries. 2. **Proverbot9001**: - Starts at (0, 0) and rises gradually, reaching ~50 theorems proved by ~50 queries. - Plateaus at ~50 theorems proved for queries ≥50. - Key data points: - ~10 theorems at 10 queries. - ~30 theorems at 30 queries. - ~50 theorems at 50+ queries. ### Key Observations - **COPRA outperforms Proverbot9001** in the early stages, achieving ~55 theorems by 20 queries compared to Proverbot9001’s ~30 theorems at the same query count. - **Diminishing returns** are evident for both systems after ~20 (COPRA) and ~50 (Proverbot9001) queries. - **Proverbot9001’s slower ascent** suggests higher computational/resource costs per query compared to COPRA. ### Interpretation The graph demonstrates that **COPRA (GPT-4-turbo) with retrieval** is significantly more efficient at proving theorems with fewer queries, making it preferable for scenarios requiring rapid results. However, **Proverbot9001** maintains a steadier performance over a larger query range, potentially indicating robustness in handling complex or extended tasks. The plateauing trends imply that both systems face inherent limitations in scalability beyond certain query thresholds. This data could guide decisions in resource allocation, system selection, or algorithm optimization for theorem-proving applications.