## Table: Proof State and LLM Response Analysis
### Overview
The image presents a table that analyzes the serialized proof state, stack & failure dictionary, interaction result, and LLM response across four queries. The table compares the goals, hypotheses, steps, and outcomes of each query, highlighting successes and errors.
### Components/Axes
* **Rows (Left Column):**
* Serialized Proof State
* Stack & Failure Dictionary
* Interaction Result
* LLM Response
* **Columns (Top Row):**
* Query #1
* Query #2
* Query #3
* Query #4
### Detailed Analysis
**Row 1: Serialized Proof State**
* **Query #1:**
* `[GOALS]`
* `[GOAL] 1`
* `x*x % 2 = 0`
* `[HYPOTHESES] 1`
* `[HYPOTHESIS] x : N`
* `[HYPOTHESIS] h : x % 2 = 0`
* **Query #4:**
* `[GOALS]`
* `[GOAL] 1`
* `x % 2 * (x % 2) % 2 = 0`
* `[HYPOTHESES] 1`
* `[HYPOTHESIS] x : N`
* `[HYPOTHESIS] h : x % 2 = 0`
**Row 2: Stack & Failure Dictionary**
* **Query #1:**
* `[LAST STEP]`
* `intro h,`
* **Query #2:**
* `[STEPS]`
* `[STEP] intro h,`
* `[LAST STEP]`
* `rw h,`
* **Query #3:**
* `[STEPS]`
* `[STEP] intro h,`
* `[INCORRECT STEPS]`
* `[STEP] rw h,`
* `[LAST STEP]`
* `apply nat.mul_mod_right,`
* **Query #4:**
* `[STEPS]`
* `[STEP] intro h,`
* `[LAST STEP]`
* `rw nat.mul_mod,`
**Row 3: Interaction Result**
* **Query #1:**
* `[SUCCESS]`
* `[END]`
* **Query #2:**
* `[ERROR MESSAGE]`
* `Got error in 'rw h.':`
* `error: rewrite tactic failed,`
* `did not find instance of`
* `the pattern in the`
* `target expression`
* `x % 2`
* `[END]`
* **Query #3:**
* `[ERROR MESSAGE]`
* `Got error in`
* `'apply`
* `nat.mul_mod_right,':`
* `error: invalid apply tactic,`
* `failed to unify`
* `x*x % 2 = 0`
* `with`
* `?m_1*?m_2%?m_1 = 0`
* `[END]`
* **Query #4:**
* `[SUCCESS]`
* `[END]`
**Row 4: LLM Response**
* **Query #1:**
* `[RUN TACTIC]`
* `rw h,`
* `[END]`
* **Query #2:**
* `[RUN TACTIC]`
* `apply nat.mul_mod_right,`
* `[END]`
* **Query #3:**
* `[RUN TACTIC]`
* `rw nat.mul_mod,`
* `[END]`
* **Query #4:**
* `[RUN TACTIC]`
* `rw h,`
* `[END]`
### Key Observations
* Queries #1 and #4 resulted in `[SUCCESS]`, while Queries #2 and #3 resulted in `[ERROR MESSAGE]`.
* The `Serialized Proof State` is different between Query #1 and Query #4.
* The `LLM Response` varies across all four queries, reflecting different tactic attempts.
### Interpretation
The table provides a comparative analysis of the LLM's attempts to solve a proof. The errors in Queries #2 and #3 indicate challenges in applying the correct tactics or unifying expressions. The successful outcomes in Queries #1 and #4 suggest that the LLM can sometimes find a valid proof strategy. The differences in the `Serialized Proof State` between Query #1 and Query #4, despite both being successful, indicate that there might be multiple valid paths to reach the goal. The `Stack & Failure Dictionary` provides insights into the intermediate steps and potential issues encountered during the proof process.
