## Data Table: Interactive Theorem Proving Results
### Overview
This image presents a data table documenting the results of an interactive theorem proving process. The table tracks the state of a proof, the steps taken, the interaction result (success or failure), and the Large Language Model (LLM) response to each query. The table is divided into four columns, each representing a separate query/attempt to prove a mathematical statement.
### Components/Axes
The table has the following row labels:
* **Serialized Proof State:** Represents the current state of the proof, including goals and hypotheses.
* **Stack & Failure Dictionary:** Details the steps attempted and any failures encountered during the proof process.
* **Interaction Result:** Indicates whether the interaction with the theorem prover was successful or resulted in an error.
* **LLM Response:** Shows the query submitted to the LLM and the LLM's response.
The table has four columns, labeled:
* **Query #1**
* **Query #2**
* **Query #3**
* **Query #4**
### Detailed Analysis or Content Details
**Query #1:**
* **Serialized Proof State:** `x * x % 2 = 0 [GOAL] 1 [HYPOTHESES] 1 [HYPOTHESIS] x : ℕ [HYPOTHESIS] h : x % 2 = 0`
* **Stack & Failure Dictionary:** `[LAST STEP] intro h, [STEPS] [STEP] intro h, [LAST STEP] rw h`
* **Interaction Result:** `[SUCCESS] [END]`
* **LLM Response:** `[RUN TACTIC] rw h, [END]`
**Query #2:**
* **Serialized Proof State:** `x % 2 * (x % 2) % 2 = 0 [GOAL] 1 [HYPOTHESES] 1 [HYPOTHESIS] x : ℕ [HYPOTHESIS] h : x % 2 = 0`
* **Stack & Failure Dictionary:** `[STEPS] [STEP] intro h, [LAST STEP] rw h, [INCORRECT STEPS] [STEP] rw h, [LAST STEP] apply nat.mul_mod_right`
* **Interaction Result:** `[ERROR MESSAGE] Got error in 'rw h': error: rewrite tactic failed, did not find instance of the pattern in the target expression x * x % 2` [END]
* **LLM Response:** `[RUN TACTIC] apply nat.mul_mod_right, [END]`
**Query #3:**
* **Serialized Proof State:** `x % 2 * (x % 2) % 2 = 0 [GOAL] 1 [HYPOTHESES] 1 [HYPOTHESIS] x : ℕ [HYPOTHESIS] h : x % 2 = 0`
* **Stack & Failure Dictionary:** `[STEPS] [STEP] intro h, [LAST STEP] rw h, [LAST STEP] apply nat.mul_mod_mod,`
* **Interaction Result:** `[ERROR MESSAGE] Got error in 'apply nat.mul_mod_mod_right': error: invalid apply tactic, failed to unify x * x % 2 = 0 with ?m_1 * ?m_2 % ?m_1 = 0` [END]
* **LLM Response:** `[RUN TACTIC] rw nat.mul_mod, [END]`
**Query #4:**
* **Serialized Proof State:** `x % 2 * (x % 2) % 2 = 0 [GOAL] 1 [HYPOTHESES] 1 [HYPOTHESIS] x : ℕ [HYPOTHESIS] h : x % 2 = 0`
* **Stack & Failure Dictionary:** `[STEPS] [STEP] intro h, [LAST STEP] rw h`
* **Interaction Result:** `[SUCCESS] [END]`
* **LLM Response:** `[RUN TACTIC] rw h, [END]`
### Key Observations
* Query #1 and #4 were successful, while Queries #2 and #3 resulted in errors.
* The errors in Queries #2 and #3 relate to the `rw` (rewrite) and `apply` tactics failing to find matching patterns or unify expressions.
* The LLM consistently suggests `rw h` as a tactic, and it succeeds in Queries #1 and #4.
* The initial goal is `x * x % 2 = 0`, and the hypothesis is `x : ℕ` and `h : x % 2 = 0`.
* The serialized proof state remains consistent across all queries.
### Interpretation
This data table demonstrates the iterative process of interactive theorem proving, where an LLM is used to suggest tactics to advance the proof. The table highlights the challenges of automated reasoning, as the LLM's suggestions are not always successful and can lead to errors. The success of `rw h` in Queries #1 and #4 suggests that this tactic is appropriate for the given goal and hypothesis. The failures in Queries #2 and #3 indicate that the LLM sometimes struggles to apply tactics correctly or to find the right patterns for rewriting expressions. The error messages provide valuable information for debugging the proof process and understanding the limitations of the LLM's reasoning capabilities. The table provides a record of the interaction between a human prover (or automated system) and an LLM, showcasing the potential and challenges of using LLMs to assist in formal verification.
