diff --git a/.gitignore b/.gitignore index a895ff0..f7e6204 100644 --- a/.gitignore +++ b/.gitignore @@ -1,4 +1,4 @@ -.venv/ +.venv*/ *.egg-info dist/ @@ -9,8 +9,13 @@ __pycache__/ *.pyd *.log *.bak +.coverage .DS_Store .idea/ +.pytest_cache/ +.mypy_cache/ +.ruff_cache/ +.ipynb_checkpoints/ self_use/ .vscode/ @@ -19,5 +24,33 @@ self_use/ CLAUDE.md .claude/ +.trae/ + +# 临时调试脚本与备份 +_debug_*.py +_debug_*.txt +_debug_output.txt +_old_*.txt +_test_*.py +_test_*.txt +_*.npy +_eval_*.json + +# 临时评测结果（保留正式的 eval_results.json） +docs/Mathematical_logic/report/eval_*_quick.json +docs/Mathematical_logic/report/eval_safe_*.json +docs/Mathematical_logic/report/eval_t*_pure.json +docs/Mathematical_logic/report/eval_task5_fix.json +docs/Mathematical_logic/report/eval_robust_*.json + +# LaTeX temporary files +*.aux +*.bbl +*.blg +*.fdb_latexmk +*.fls +*.out +*.toc +*.synctex.gz diff --git a/_debug_sprite.py b/_debug_sprite.py new file mode 100644 index 0000000..220eb56 --- /dev/null +++ b/_debug_sprite.py @@ -0,0 +1,48 @@ +from __future__ import annotations + +import sys +from pathlib import Path + +import numpy as np + +PROJECT_ROOT = Path(__file__).resolve().parent +if str(PROJECT_ROOT) not in sys.path: + sys.path.insert(0, str(PROJECT_ROOT)) + +from nesylink.env import make_env +from nesylink.core.constants import ACTION_LABELS, ACTION_LEFT, ACTION_RIGHT, ACTION_UP, ACTION_DOWN +from student_agent.baseline_policy import PLAYER_COLORS + + +def green_bounds(frame): + xs, ys = [], [] + for color in PLAYER_COLORS: + mask = np.all(frame == color, axis=2) + yy, xx = np.nonzero(mask) + if len(xx) > 0: + xs.append(xx); ys.append(yy) + xs = np.concatenate(xs); ys = np.concatenate(ys) + return int(xs.min()), int(xs.max()), int(ys.min()), int(ys.max()), int(xs.mean()), int(ys.mean()) + + +def run(): + env = make_env(task_id="mathematical_logic/task_1", observation_mode="pixels") + obs, info = env.reset(seed=0) + # move right a few ticks, then left, then up, then down - observe green bounds vs engpos + plan = [("right", 3), ("left", 6), ("up", 3), ("down", 6)] + action_map = {"right": ACTION_RIGHT, "left": ACTION_LEFT, "up": ACTION_UP, "down": ACTION_DOWN} + pos = info["agent"]["position_px"] + xmin, xmax, ymin, ymax, xmean, ymean = green_bounds(obs) + print(f"init engpos={pos} green xmin={xmin} xmax={xmax} ymin={ymin} ymax={ymax} mean=({xmean},{ymean}) left_min={xmin-4} left_mean={xmean-8}") + for direction, ticks in plan: + a = action_map[direction] + for _ in range(ticks): + obs, r, term, trunc, info = env.step(a) + pos = info["agent"]["position_px"] + xmin, xmax, ymin, ymax, xmean, ymean = green_bounds(obs) + print(f"{direction} engpos=({pos[0]:.0f},{pos[1]:.0f}) green x[{xmin},{xmax}] y[{ymin},{ymax}] mean=({xmean},{ymean}) left_min={xmin-4} left_mean={xmean-8}") + env.close() + + +if __name__ == "__main__": + run() diff --git a/_debug_task12.py b/_debug_task12.py new file mode 100644 index 0000000..81b740c --- /dev/null +++ b/_debug_task12.py @@ -0,0 +1,66 @@ +from __future__ import annotations + +import sys +from pathlib import Path + +import numpy as np + +PROJECT_ROOT = Path(__file__).resolve().parent +if str(PROJECT_ROOT) not in sys.path: + sys.path.insert(0, str(PROJECT_ROOT)) + +from nesylink.env import make_env +from nesylink.core.constants import ACTION_LABELS +from student_agent.baseline_policy import Policy + + +def run(task_id: str, seed: int = 0, max_steps: int = 1000): + env = make_env(task_id=task_id, observation_mode="pixels") + obs, info = env.reset(seed=seed) + policy = Policy() + policy.reset(seed=seed, task_id=task_id) + print(f"=== {task_id} seed={seed} ===") + prev_tile = None + for step in range(1, max_steps + 1): + action = policy.act(obs, info) + tile = policy.history.facing if False else None + # detect player tile from obs + pt = policy.detect_player_tile(obs) + keys = int(info.get("inventory", {}).get("keys", 0)) + label = ACTION_LABELS[action] if action < len(ACTION_LABELS) else str(action) + monster = policy.detect_monster_tile(obs) if task_id.endswith("task_2") else None + agent = info.get("agent", {}) + pos = agent.get("position_px", None) + det_px = policy.detect_player_px(obs) + last_msg = info.get("engine", {}).get("last_message") if "engine" in info else None + events = [r.get("name") for r in info.get("events", {}).get("records", [])] + prev_action = getattr(run, "_pa", None) + prev_pos = getattr(run, "_pp", None) + meaningful = ( + pt != prev_tile + or action != prev_action + or events + or "blocked" in str(events) + or pos != prev_pos and (pos is None or prev_pos is None or abs(pos[0] - prev_pos[0]) + abs(pos[1] - prev_pos[1]) > 15) + ) + marker = "" + if pt == prev_tile and action in (1, 2, 3, 4) and "blocked" in str(events): + marker = " <-- BLOCKED" + prev_tile = pt + run._pa = action + run._pp = pos + if meaningful or marker: + print( + f"step {step:3d} tile={pt} engpos={pos} detpx={det_px} keys={keys} facing={policy.history.facing} " + f"action={label} events={events} monster={monster}{marker}" + ) + obs, reward, terminated, truncated, info = env.step(action) + if terminated or truncated: + print(f" -> terminated={terminated} truncated={truncated} reason={info.get('terminal_reason')}") + break + env.close() + + +if __name__ == "__main__": + task = sys.argv[1] if len(sys.argv) > 1 else "mathematical_logic/task_1" + run(task) diff --git a/docs/Mathematical_logic/experiments/robustness_100_1ba1f94.json b/docs/Mathematical_logic/experiments/robustness_100_1ba1f94.json new file mode 100644 index 0000000..4b7a301 --- /dev/null +++ b/docs/Mathematical_logic/experiments/robustness_100_1ba1f94.json @@ -0,0 +1,18512 @@ +{ + "episodes": [ + { + "eval_stage": "original", + "event_counts": { + "action_blocked": 1, + "chest_opened": 1, + "door_opened": 1, + "environment_completed": 1, + "exit_reached": 1, + "key_collected": 1, + "move_left": 90, + "move_right": 82, + "move_up": 96, + "room_changed": 1, + "world_completed": 1 + }, + "map_variant": 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"button_pressed": 1.0, + "chest_opened": 1.0, + "door_opened": 1.0, + "environment_completed": 1.0, + "exit_reached": 1.0, + "gold_collected": 1.0, + "key_collected": 1.0, + "monster_killed": 1.0, + "room_changed": 1.0, + "world_completed": 1.0 + }, + "success_rate": 1.0, + "task_id": "mathematical_logic/task_5" + } + } +} \ No newline at end of file diff --git a/docs/Mathematical_logic/project_management/AGENT_INTERFACE_DRAFT.md b/docs/Mathematical_logic/project_management/AGENT_INTERFACE_DRAFT.md new file mode 100644 index 0000000..c71af73 --- /dev/null +++ b/docs/Mathematical_logic/project_management/AGENT_INTERFACE_DRAFT.md @@ -0,0 +1,149 @@ +# Agent 接口草案 + +## 1. 目的 + +本文件用于统一小组内部对 Agent 输入输出的理解，避免后续出现： + +- 调试版和提交版混在一起； +- 某些成员默认直接依赖 `info`； +- 视觉模块、策略模块和评测入口接口不一致。 + +本草案基于当前仓库现状与课程说明整理，后续可继续修订。 + +## 2. 已知事实 + +- 当前仓库的 `utils/evaluate_policy.py` 期望策略接口是 `act(obs, info) -> int` 或等价形式。 +- 当前课程说明明确要求：最终测试阶段，Agent 不能直接依赖环境内部 `info`。 +- 老师目前尚未完全确定最终提交格式，因此小组内部需要提前设计“兼容当前评测脚本，同时满足最终限制”的中间方案。 + +## 3. 建议采用的双层接口 + +### 3.1 外层兼容接口 + +为了兼容当前仓库中的评测脚本，建议保留： + +```python +def act(self, obs, info) -> int: + ... +``` + +但这里的 `info` 只作为“兼容层输入”，不能默认被策略主体直接消费。 + +### 3.2 内层真实决策接口 + +建议策略内部统一转为如下语义： + +```python +def decide(self, frame, reward, inventory, history) -> int: + ... +``` + +其中： + +- `frame`：像素观测，是最终测试阶段允许使用的核心输入。 +- `reward`：当前或上一时刻 reward，可作为历史反馈。 +- `inventory`：显式提供的物品栏信息。 +- `history`：Agent 自己维护的内部记忆，如阶段、目标、最近轨迹。 + +## 4. 调试版与提交版的边界 + +### 4.1 调试版允许内容 + +调试阶段可以为了分析或监督，读取： + +- `info["agent"]` +- `info["events"]` +- `info["entities"]` +- `info["dynamic"]` +- 其他环境内部字段 + +但这些信息只能用于： + +- 日志 +- 状态对齐 +- 监督标注 +- 结果分析 +- 训练辅助 + +### 4.2 提交版禁止内容 + +提交版策略在推理阶段不应直接读取： + +- 玩家精确坐标 +- 怪物精确坐标 +- 地图真值 +- 墙体/陷阱真值 +- 门和按钮的隐藏状态 +- 任意 `info` 中的内部符号状态 + +## 5. 推荐代码组织方式 + +建议拆成四层： + +1. `perception/` + - 输入：像素 + - 输出：对象、位置、风险区、可交互点等中间表示 +2. `planner/` + - 输入：感知结果、任务阶段、记忆 + - 输出：子目标或动作建议 +3. `controller/` + - 输入：子目标或期望移动方向 + - 输出：环境动作编号 +4. `debug_tools/` + - 输入：`info` + - 用途：训练期调试、监督、分析 + +这样可以把“最终可提交部分”和“训练期辅助部分”分离。 + +## 6. 当前建议的 Policy 类结构 + +```python +class Policy: + def __init__(self): + self.history = {} + self.last_reward = 0.0 + + def reset(self, seed=None, task_id=None): + self.history = { + "task_id": task_id, + "phase": "init", + "trace": [], + } + self.last_reward = 0.0 + + def act(self, obs, info): + # 兼容当前评测脚本，但不要把 info 直接当作最终决策输入 + inventory = self.extract_inventory(info) + action = self.decide( + frame=obs, + reward=self.last_reward, + inventory=inventory, + history=self.history, + ) + return action + + def extract_inventory(self, info): + # 后续可替换成真正的评测接口来源 + return info.get("inventory", {}) + + def decide(self, frame, reward, inventory, history): + raise NotImplementedError +``` + +## 7. 当前最值得优先统一的约定 + +建议小组尽快统一以下问题： + +1. `inventory` 的内部表示格式是什么？ +2. 视觉模块输出是否统一成 tile 坐标级别的对象表？ +3. 策略层是否采用“阶段机 + 搜索”的形式？ +4. Lean 证明对象是规划器、动作筛选器，还是目标判定器？ + +## 8. 当前结论 + +当前阶段最合理的策略不是“所有人直接围着 `act(obs, info)` 写逻辑”，而是： + +- 对外兼容当前仓库； +- 对内严格按最终测试限制设计； +- 把调试信息使用限制在辅助模块； +- 把真正提交的决策链路尽量做成可解释、可验证结构。 diff --git a/docs/Mathematical_logic/project_management/PROJECT_MEMORY.md b/docs/Mathematical_logic/project_management/PROJECT_MEMORY.md new file mode 100644 index 0000000..a817c90 --- /dev/null +++ b/docs/Mathematical_logic/project_management/PROJECT_MEMORY.md @@ -0,0 +1,132 @@ +# 数理逻辑大作业记忆清单 + +本文件记录项目过程中需要长期记住、频繁回看的事实、约束和决策。 + +## 1. 不可忘记的课程要求 + +- 最终测试阶段，Agent 不能直接读取或依赖环境内部 `info`。 +- 最终测试阶段，Agent 只能根据像素观测、reward 和显式提供的物品栏信息决策。 +- 训练、调试和监督阶段可以使用 `info`，但必须和最终推理阶段严格区分。 +- Lean 文件不允许保留未说明的 `sorry`、`admit`、`axiom`。 +- 提交材料需要包括报告、Lean 源文件、完整 Python 代码，以及能证明代码运行过的截图或附件。 +- 如果使用机器学习模型，还需要提交模型权重，并确保与最终评测接口对齐。 + +## 2. 当前确认的 Q&A 结论 + +- Q1：训练时能调用什么信息？ + - A：训练时可以根据需要使用 `pixel`、`reward`、`info` 等信息。 +- Q2：测试时能调用什么信息？ + - A：测试时只能使用像素、课程提供的 reward 和拥有物品的信息进行决策。 +- Q3：视觉模型和策略学习是否有限定算法？ + - A：没有限定，但泛化性越高越有利。 +- Q4：提交格式和评测脚本是否已经完全确定？ + - A：暂时还没有完全确定，后续需要及时关注老师通知。 + +## 3. 项目内建议长期坚持的原则 + +- 原则 1：优先做可解释、可验证、可扩展的方案。 +- 原则 2：先做最小可用 baseline，再优化复杂策略。 +- 原则 3：并行推进，避免“一个人做完下一个人才接手”的串行模式。 +- 原则 4：每天保留实验截图、日志和失败记录，避免最后补材料。 +- 原则 5：Python 实现、Lean 形式化和报告叙述三者必须保持一致。 + +## 4. 建议统一的接口观念 + +以下内容是项目推进建议，后续讨论后可细化： + +- 调试版策略输入可以包含：`obs`、reward、inventory、debug features。 +- 提交版策略输入必须限制为：像素观测、reward 历史、物品栏信息、内部记忆。 +- 视觉模块输出应尽量是“对象/位置/风险/目标”的统一表示。 +- 规划模块应尽量和视觉模块解耦，避免后期难以替换。 +- Lean 应优先证明规划器、安全层、合法动作筛选器、目标判定器等可验证层。 + +## 5. 需要持续补充的内容 + +后续请持续在本文件补充以下信息： + +- 老师新增通知或评测接口变化 +- 小组内部达成的一致接口约定 +- 每个任务的关键机制和坑点 +- 最终决定提交的技术路线 +- Lean 要证明的正式命题列表 +- 实验中已经验证有效或无效的方法 + +## 5.1 当前开发环境信息 + +- 本地已安装 `xelatex`，当前检测到版本为 `TeX Live 2026`。 +- IDE 中已安装 `LaTeX Workshop` 插件，可直接辅助查看和编译报告。 +- 仓库依赖已通过 `python -m pip install -e .` 安装完成。 +- 当前仓库评测脚本 `utils/evaluate_policy.py` 使用的调用形式仍是 `act(obs, info)`，需要通过内部接口隔离调试输入与最终提交输入。 + +## 5.2 当前已决定的项目路径 + +- 当前选择的启动路线是：先建立统一 `Policy` 骨架，再逐步把 `task_1` 到 `task_5` 填进去。 +- 自有 Agent 代码统一放在仓库根目录的 `student_agent/` 下，不直接把示例文件当主项目代码。 +- 当前自有 baseline 已在 `task_1`、`task_2`、`task_3`、`task_4` 和 `task_5` 上跑通。 +- `task_2` 当前版本是脚本化 debug baseline，来源于参考成功执行轨迹，不应被误认为最终提交版方案。 +- `task_3` 当前版本已从直接读取 `room_id` / `tile` 过渡为像素房间识别、玩家中心格定位和物品栏钥匙数驱动的规则状态机。 +- `task_4` 当前版本已从直接读取桥状态、事件和玩家 tile 过渡为像素房间识别、桥状态像素识别、玩家中心格定位、物品栏和奖励历史驱动的显式阶段机，是 Lean 形式化与证明的重点候选关卡。 +- `task_5` 当前版本采用像素房间识别、奖励历史、物品栏和显式阶段记忆驱动；关键风险不是高层路线，而是 tile 判断与碰撞箱之间的像素对齐偏差。 + +## 6. 当前已知任务难点 + +- `task_1`：单房间取钥匙开门，适合做第一关 baseline。 +- `task_2`：需要处理怪物与击杀逻辑。 +- `task_3`：多房间往返，开始考验状态记忆与路径规划。 +- `task_4`：桥和按钮机制较复杂，适合做形式化建模与子任务拆分。 +- `task_5`：综合关卡，更强调泛化与整体鲁棒性。 + +## 7. 决策记录区 + +后续按如下模板追加： + +```text +### YYYY-MM-DD 决策标题 + +- 背景： +- 决策： +- 原因： +- 影响范围： +``` + +### 2026-07-03 启动阶段的首批验证 + +- 背景：项目需要尽快确认本地运行链路和报告链路是否可用。 +- 决策：先验证 `XeLaTeX`、再安装仓库依赖、再运行参考示例。 +- 原因：这是最适合启动阶段并行推进的低风险工作，可以快速得到可记录结果。 +- 影响范围：报告、实验、环境运行和后续小组协作都能直接受益。 + +### 2026-07-03 基线代码组织 + +- 背景：继续直接改课程示例文件不利于后续统一接口、分工和提交整理。 +- 决策：创建 `student_agent/` 作为小组自有 Agent 目录，并以统一 `Policy` 骨架作为后续开发入口。 +- 原因：可以把示例代码、调试代码和最终提交代码区分开，也便于后续多人协作。 +- 影响范围：策略开发、评测调用、报告叙述和最终整理都会更清晰。 + +### 2026-07-03 Task 3 基线方案 + +- 背景：`task_3` 是第一次要求跨房间往返，需要比前两关更明确的阶段记忆。 +- 决策：先采用房间级规则状态机，不先处理视觉泛化，也不要求击杀走廊怪物。 +- 原因：该方案实现成本低、可解释性强，能快速验证“跨房间记忆”这条技术路线。 +- 影响范围：后续 `task_4` 和 `task_5` 的阶段机设计都可以复用这套思路。 + +### 2026-07-03 Task 4 基线方案 + +- 背景：`task_4` 同时涉及旋转桥、条件门、装备获取、战斗和隐藏宝箱，是最复杂的显式任务链。 +- 决策：采用显式阶段机，按“拿钥匙 -> 拿剑 -> 杀守卫 -> 开最终宝箱”的顺序推进，并单独维护桥状态与 `monster_killed` 事件记忆。 +- 原因：仅靠当前房间的 `monsters_remaining` 不能稳定判断守卫是否已被击杀，必须引入持久化阶段记忆。 +- 影响范围：后续 Lean 证明可以围绕阶段顺序正确性、桥状态可达性和关键事件触发条件展开。 + +### 2026-07-05 Task 3/4 像素感知改造 + +- 背景：课程要求最终推理阶段不能直接依赖环境内部 `info` 中的房间、玩家坐标、动态对象状态或事件真值。 +- 决策：将 `task_3` 和 `task_4` 的关键决策输入改为从图像中识别玩家中心格、房间特征、宝箱、开关、桥和怪物；物品栏和奖励历史仍作为允许的外部反馈使用。 +- 原因：这样保留了原有规则状态机的可解释性，同时减少对隐藏调试状态的依赖。 +- 影响范围：后续报告需要说明调试阶段曾用 `info` 校准像素识别，但当前策略主体不再直接读取 `info["env"]`、`info["agent"]`、`info["dynamic"]` 或 `info["events"]`。 + +### 2026-07-05 Task 5 基线方案 + +- 背景：`task_5` 需要跨起始、南、东、西四个房间完成所有宝箱目标，并同时处理按钮、钥匙门、怪物、回血宝箱和持续掉血压力。 +- 决策：采用固定子目标顺序和显式阶段记忆：起始金币宝箱 -> 按按钮 -> 南房间钥匙宝箱 -> 起始房间杀怪 -> 东门与回血宝箱 -> 西房间金币宝箱。 +- 原因：该顺序能先取得南门条件和东门钥匙，再通过东房间回血提高后续容错；实现上通过奖励信号记录宝箱、钥匙、按钮、开门、击杀等关键事件，避免直接读取隐藏事件列表。 +- 影响范围：`task_5` 后续报告应重点说明“像素 tile 已到位不等于碰撞箱已对齐”，因此策略中保留了若干行/列对齐阶段来避开墙、NPC 和出口边界。 diff --git a/docs/Mathematical_logic/project_management/PROJECT_PLAN.md b/docs/Mathematical_logic/project_management/PROJECT_PLAN.md new file mode 100644 index 0000000..58c32f7 --- /dev/null +++ b/docs/Mathematical_logic/project_management/PROJECT_PLAN.md @@ -0,0 +1,177 @@ +# 数理逻辑大作业项目总计划 + +## 1. 项目目标 + +本项目需要在一周内完成四类产出： + +1. Python Agent：能够在 `nesylink` 的数理逻辑任务中完成尽可能多的目标。 +2. Lean 形式化：对环境、状态、动作、转移或策略中的关键部分进行形式化建模。 +3. 正确性证明：证明若干关键性质，例如合法性、安全性、可达性或子策略正确性。 +4. 实验报告：说明方法设计、实验设置、结果分析、形式化内容与分工情况。 + +## 2. 已确认约束 + +- 训练和调试阶段可以使用 `pixels`、`reward`、`info`。 +- 最终测试阶段，Agent 不能直接依赖 `info`，只能根据 `obs["frame"]`、reward 和显式提供的物品栏信息决策。 +- 最终提交至少需要：报告、Lean 源文件、完整 Python 源码、可运行证据；如使用机器学习，还需提交模型权重。 +- 一周时间很紧，优先采用并行推进方案，而不是串行接力。 + +## 3. 总体技术路线 + +建议采用“可解释、可验证、可逐步扩展”的路线： + +1. 先做基线：跑通环境、评测脚本和前两关。 +2. 视觉/状态抽取：从像素观测中提取玩家、墙、钥匙、怪物、按钮、门等对象信息。 +3. 策略与规划：先实现 rule-based / search baseline，再考虑泛化增强。 +4. Lean 证明：优先证明一个可验证层，而不是试图完整证明整个学习系统。 +5. 报告与证据：同步记录实验、失败尝试、截图、曲线与最终结论。 + +## 4. 一周步骤规划 + +### Day 1：启动与基线 + +- 全组统一阅读课程说明和仓库文档。 +- 跑通仓库环境、任务和示例 Agent。 +- 明确各成员分工与接口约定。 +- 建立项目管理文档、报告框架和代码目录约定。 + +完成标志： + +- 至少 1 人能稳定运行 `task_1` 和 `task_2`。 +- 全组明确最终限制：测试时不能直接读 `info`。 +- 报告骨架和进度文档已经建立。 + +### Day 2：任务拆解与统一接口 + +- 逐一分析 `task_1` 到 `task_5` 的地图、机制和难点。 +- 统一 Agent 接口，例如 `reset(seed, task_id)`、`act(obs, inventory, reward, history)`。 +- 明确视觉模块输出格式，例如位置、对象、风险区域、可交互点。 +- 确定 Lean 中要形式化的对象和证明目标。 + +完成标志： + +- 每关形成一页简要任务分析。 +- 代码层输入输出接口冻结到第一版。 +- Lean 建模边界被明确记录。 + +### Day 3：并行开发第一轮 + +- 环境/框架方向：完成统一运行脚本和基本评测流程。 +- 视觉方向：实现最小可用状态抽取。 +- 策略方向：完成前两关可运行 baseline。 +- Lean 方向：完成状态、动作、转移与目标谓词初稿。 +- 报告方向：开始积累实验图表和方法图。 + +完成标志： + +- 至少有一个不依赖最终测试 `info` 的最小策略雏形。 +- Lean 代码中有可编译的数据类型和基本定义。 + +### Day 4：并行开发第二轮 + +- 扩展策略到多房间任务。 +- 处理机关、桥、按钮、怪物等复杂机制。 +- 建立实验记录表，比较不同策略/参数。 +- 完成第一批可提交的定理证明。 + +完成标志： + +- `task_3` 或 `task_4` 出现可重复的阶段性成功结果。 +- 有至少 1 个核心性质完成正式证明。 + +### Day 5：集成与验证 + +- 将视觉、策略、评测、Lean 与报告对齐。 +- 排查“训练能用、测试不能用”的信息泄漏问题。 +- 开始整理最终图表、结果表与截图证据。 + +完成标志： + +- 形成一版统一提交入口。 +- 明确哪些模块属于最终提交版，哪些仅用于调试。 + +### Day 6：报告与补强 + +- 集中补实验、写报告正文、润色图表。 +- 修补成功率不稳定的任务。 +- 检查 Lean 文件是否有未说明的 `sorry` 或类似占位。 + +完成标志： + +- 报告主体内容齐全。 +- 代码、证明和报告三者叙述一致。 + +### Day 7：最终封版 + +- 统一检查运行性、路径、依赖和权重文件。 +- 补齐提交说明、截图和压缩包结构。 +- 做最后一次评测与交叉检查。 + +完成标志： + +- 提交材料完整。 +- 每个文件都知道“为什么要交、如何运行、与报告哪一部分对应”。 + +## 5. 并行工作流 + +### 工作流 A：环境与评测 + +- 跑通仓库、任务、示例 Agent 与评测脚本。 +- 维护统一入口脚本。 +- 负责最终可复现性。 + +### 工作流 B：视觉与状态抽取 + +- 从像素观测中识别关键对象。 +- 输出策略可消费的状态表示。 +- 与最终测试限制保持一致。 + +### 工作流 C：策略与规划 + +- 实现基线策略。 +- 扩展到多任务、多房间与复杂机关。 +- 输出实验对比结果。 + +### 工作流 D：Lean 形式化与证明 + +- 建模状态、动作、转移、目标。 +- 证明核心性质。 +- 与 Python 策略中真正可验证的逻辑对齐。 + +### 工作流 E：报告与实验证据 + +- 同步记录实验、截图、失败案例、图表。 +- 整理报告文字和材料。 +- 确保最终叙述完整且一致。 + +## 6. 每日同步机制 + +建议每天至少同步两次： + +- 中午：汇报当前进度、卡点、当天目标是否调整。 +- 晚上：提交当天产出，包括代码、截图、结果、证明状态和明日计划。 + +每次同步至少回答四个问题： + +1. 今天做了什么？ +2. 目前卡在哪里？ +3. 产出了什么可见结果？ +4. 明天最优先做什么？ + +## 7. 风险清单 + +- 风险 1：策略训练时依赖 `info`，导致最终测试不可提交。 +- 风险 2：每个人各做各的，接口无法拼接。 +- 风险 3：Lean 证明与 Python 实现完全脱节。 +- 风险 4：只顾写代码，没有持续保留截图、日志和实验数据。 +- 风险 5：最后一天才写报告，导致叙述与代码不一致。 + +## 8. 当前执行优先级 + +当前优先级按顺序如下： + +1. 跑通环境和基线。 +2. 冻结接口和分工。 +3. 实现最小可用视觉/策略链路。 +4. 确定 Lean 证明对象。 +5. 累积实验与报告材料。 diff --git a/docs/Mathematical_logic/project_management/PROJECT_PROGRESS.md b/docs/Mathematical_logic/project_management/PROJECT_PROGRESS.md new file mode 100644 index 0000000..cc1ee7a --- /dev/null +++ b/docs/Mathematical_logic/project_management/PROJECT_PROGRESS.md @@ -0,0 +1,226 @@ +# 数理逻辑大作业进度跟踪 + +## 1. 项目状态总览 + +- 项目名称：数理逻辑课程大作业 +- 仓库：`nesylink-mathlogic-project` +- 当前阶段：最终封版与提交前检查 +- 更新时间：2026-07-17 +- 当前目标：五任务 original/spatial/color 全部通过，Task 5 最终 baseline 形式化完成 + +## 2. 当前里程碑 + +| 里程碑 | 状态 | 截止时间 | 说明 | +|---|---|---|---| +| 阅读课程要求与仓库结构 | 已完成 | Day 1 | 已确认任务要求、评分细则和测试限制 | +| 建立项目管理文档 | 已完成 | Day 1 | 已创建总计划、进度跟踪、记忆清单 | +| 建立实验报告框架 | 已完成 | Day 1 | 已创建 TeX 报告骨架 | +| 跑通环境与示例策略 | 已完成 | Day 1 | 已安装依赖、跑通示例并在自有 baseline 中覆盖前四关 | +| 冻结统一 Agent 接口 | 已完成 | Day 2 | `--task-policy` + `safe` 模式接口已验证 | +| 完成最小可用视觉/策略链路 | 已完成 | Day 3 | `task_3` / `task_4` / `task_5` 已接入像素感知 baseline | +| 完成 Lean 定义与证明 | 已完成 | Day 4 | 7 个 Lean 文件通过，Task 5 已与最终 baseline 对齐 | +| 统一实验结果并写报告主体 | 已完成 | Day 6 | 最终实验数据与证明内容已同步到报告 | +| 最终封版与交叉检查 | 进行中 | Day 7 | 待检查 PDF 后由组长提交 | + +## 3. 工作流状态 + +| 工作流 | 当前状态 | 负责人 | 下一步 | +|---|---|---|---| +| 环境与评测 | 已完成 | 小组 | robustness 500/500，六颜色变体 50/50 | +| 视觉与状态抽取 | 已完成 | 小组 | grayscale/high_contrast 与空间扰动均已覆盖 | +| 策略与规划 | 已完成 | 小组 | 五任务统一 baseline，Task 5 spatial_c 已通过 | +| Lean 形式化与证明 | 已完成 | 小组 | 环境、策略、统一框架及 Task 5 细化证明均通过 | +| 实验与报告 | 进行中 | 组长 | 最终 PDF 检查与提交 | + +## 4. 今日重点 + +- [x] 阅读课程说明和项目文档 +- [x] 规划一周项目步骤 +- [x] 创建进度与记忆文档 +- [x] 创建实验报告 TeX 骨架 +- [x] 确认小组成员方向选择 +- [x] 跑通本地环境与示例 Agent +- [x] 记录第一批实验结果 +- [x] 补任务分析与最终实验记录 + +## 5. 待办清单 + +### 高优先级 + +- 编译并检查最终报告 PDF +- 由组长检查 `git diff` 后执行 add / commit / push +- 可选：保留 Lean 与 evaluation 成功终端截图作为运行材料 + +### 中优先级 + +- 核对最终提交包不包含虚拟环境、临时地图或调试脚本 +- 核对成员姓名、分工和报告日期 + +### 低优先级 + +- 补充更多随机布局或素材变化实验 +- 继续细化像素执行层与 Lean 契约之间的证明 + +## 6. 风险与阻塞 + +| 编号 | 风险/阻塞 | 当前状态 | 应对策略 | +|---|---|---|---| +| R1 | 最终测试不能直接用 `info`，但训练可能会依赖 `info` | 已识别 | 明确区分调试版与提交版接口 | +| R2 | 只有一周，串行推进风险高 | 已识别 | 强制按并行工作流推进 | +| R3 | Lean 证明对象选得过大 | 已识别 | 先选可验证层，避免全系统证明 | +| R4 | 报告材料后补会丢失细节 | 已识别 | 每天同步记录截图、日志和结论 | + +## 7. 更新日志 + +### 2026-07-03 + +- 已通读课程说明、README 与环境说明。 +- 已确认最关键约束：测试阶段不能直接依赖 `info`。 +- 已建立项目总计划文档。 +- 已建立本进度文档，用于后续持续更新。 +- 已建立报告 TeX 骨架，用于边做边写。 +- 已完成一次基础校验，`report/main.tex` 当前无编辑器诊断报错。 +- 已在本地确认 `xelatex` 可用，版本为 `TeX Live 2026`。 +- 已成功编译 `docs/Mathematical_logic/report/main.tex`，生成 `main.pdf`。 +- 已执行 `python -m pip install -e .`，完成仓库依赖安装。 +- 已通过 `utils/evaluate_policy.py` 跑通 `docs/Mathematical_logic/examples/agent.py` 在 `task_1` 上的单轮评测： + - success_rate = 1.000 + - avg_steps = 290.0 + - avg_reward = 127.050 +- 已运行 `docs/Mathematical_logic/examples/task2_reference.py`，得到首批参考结果： + - task_id = `task_2` + - steps = 14 + - total_reward = 126.18 + - terminal_reason = `world_completed` + - world_completed = `True` +- 已新增 `AGENT_INTERFACE_DRAFT.md`，用于统一调试版与提交版策略接口边界。 +- 已新增 `TASK_ANALYSIS.md`，完成五关的目标链、关键机制和 baseline 思路整理。 +- 已创建 `student_agent/` 目录，作为小组自有 Agent 代码入口。 +- 已实现 `student_agent/baseline_policy.py` 的统一 `Policy` 骨架。 +- 已通过评测入口验证 `student_agent/baseline_policy.py` 在 `task_1` 上可成功运行： + - success_rate = 1.000 + - avg_steps = 290.0 + - avg_reward = 127.050 +- 已将 `task_2` 接入自有 baseline，并完成单轮评测： + - success_rate = 1.000 + - avg_steps = 182.0 + - avg_reward = 126.180 + - 当前实现方式：从参考成功执行轨迹中提取的脚本化 debug baseline +- 已将 `task_3` 接入自有 baseline，并完成单轮评测： + - success_rate = 1.000 + - avg_steps = 523.0 + - avg_reward = 159.770 + - 当前实现方式：房间级规则状态机，使用 `room_id`、`tile` 和 `keys` 做决策 +- 已将 `task_4` 接入自有 baseline，并完成单轮评测： + - success_rate = 1.000 + - avg_steps = 1085.0 + - avg_reward = 249.150 + - 当前实现方式：显式阶段机，维护桥状态与 `monster_killed` 事件记忆 + +### 2026-07-05 + +- 已将 `task_3` / `task_4` 的 baseline 改造为像素感知版本，策略主体不再直接读取 `info["env"]`、`info["agent"]`、`info["dynamic"]` 或 `info["events"]`。 +- 已修正玩家格子识别：从玩家精灵绿色像素反推出精灵位置，并按玩家中心点计算 tile，与环境交互语义保持一致。 +- 已补充 `task_4` 中开关按下后的像素识别，避免按下开关后 west 房间无法识别。 +- 已完成 `task_1` 到 `task_4` 的 3 个 seed 回归： + - `task_1`：success_rate = 1.000，avg_steps = 290.0，avg_reward = 127.050 + - `task_2`：success_rate = 1.000，avg_steps = 182.0，avg_reward = 126.180 + - `task_3`：success_rate = 1.000，avg_steps = 525.0，avg_reward = 159.750 + - `task_4`：success_rate = 1.000，avg_steps = 1089.0，avg_reward = 249.110 +- 已逐文件检查 `student_agent/lean/` 下 6 个 Lean 文件，均通过本地 `lean` 编译检查。 +- 已将 `task_5` 接入 `student_agent/baseline_policy.py`，采用显式阶段机推进：起始宝箱 -> 按钮 -> 南房间钥匙 -> 起始房间杀怪 -> 东门与治疗宝箱 -> 西房间金币宝箱。 +- `task_5` 路线实现中记录了多处像素碰撞对齐点：进入东门前需要在第 4 行对齐，东房间需要在第 1 行/第 1 列对齐，起始房间去西门与西房间底边也需要额外对齐，避免 tile 已到位但角色碰撞箱仍擦到墙或 NPC。 +- 已完成 `student_agent/baseline_policy.py` 的全任务单 seed 回归： + - `task_1`：success_rate = 1.000，avg_steps = 290.0，avg_reward = 127.050 + - `task_2`：success_rate = 1.000，avg_steps = 182.0，avg_reward = 126.180 + - `task_3`：success_rate = 1.000，avg_steps = 525.0，avg_reward = 159.750 + - `task_4`：success_rate = 1.000，avg_steps = 1089.0，avg_reward = 249.110 + - `task_5`：success_rate = 1.000，avg_steps = 1112.0，avg_reward = 155.730 + +## 8. 首批实验记录 + +| 日期 | 入口 | 任务 | 结果 | 备注 | +|---|---|---|---|---| +| 2026-07-03 | `utils/evaluate_policy.py` + `examples/agent.py` | `task_1` | 成功 | 1 轮评测成功，reward = 127.050 | +| 2026-07-03 | `examples/task2_reference.py` | `task_2` | 成功 | 14 步完成，终止原因为 `world_completed` | +| 2026-07-03 | `utils/evaluate_policy.py` + `student_agent/baseline_policy.py` | `task_1` | 成功 | 自有 baseline 骨架已跑通 | +| 2026-07-03 | `utils/evaluate_policy.py` + `student_agent/baseline_policy.py` | `task_2` | 成功 | 自有 baseline 已覆盖前两关 | +| 2026-07-03 | `utils/evaluate_policy.py` + `student_agent/baseline_policy.py` | `task_3` | 成功 | 自有 baseline 已覆盖前三关 | +| 2026-07-03 | `utils/evaluate_policy.py` + `student_agent/baseline_policy.py` | `task_4` | 成功 | 自有 baseline 已覆盖前四关 | +| 2026-07-05 | `utils/evaluate_policy.py` + `student_agent/baseline_policy.py` | `task_1`-`task_4` | 成功 | 3 个 seed 全部成功；`task_3` / `task_4` 已改为像素感知基线 | +| 2026-07-05 | `utils/evaluate_policy.py` + `student_agent/baseline_policy.py` | `task_1`-`task_5` | 成功 | 单 seed 全任务通过，`task_5` 用 1112 步完成 | +| 2026-07-05 | `lean` | `student_agent/lean/*.lean` | 成功 | 6 个 Lean 文件逐文件编译通过 | +| 2026-07-10 | `utils/evaluate_policy.py` + `--info-mode safe` | `task_1`-`task_5` | task_1-4 成功，task_5 失败 | safe 模式 10 seed 评测，task_5 agent_dead | +| 2026-07-11 | `utils/evaluate_policy.py` + `--robustness-suite --num-envs 30` | `task_1`-`task_4` | 成功 | original 72/72 + spatial 36/36 全通过，color 0/12 预期失败 | +| 2026-07-13 | `utils/evaluate_policy.py` + `--robustness-suite --num-envs 30` | `task_1`-`task_5` | task_1-4 成功，task_5 失败 | 完整 5 任务评测，task_1-4 original+spatial 100%，color 66.7%（dark/bright 通过），task_5 全部 agent_dead | +| 2026-07-17 | `utils/evaluate_policy.py` + 最终 baseline | `task_1`-`task_5` | 成功 | robustness 500/500（--num-envs 100）；六颜色变体 50/50；Task 5 spatial_c 1127 步完成 | + +### 2026-07-10 + +- 合并 upstream 评测脚本更新（`--info-mode safe`、`--task-policy`、spatial 变体扩展到 task_4/5）。 +- 全面适配 `safe` 信息模式：`update_memory` 与 `update_task5_memory` 重构为基于 `inventory` diff、`last_reward` 和像素感知的事件推断。 +- 在 `safe` 模式下运行 10 seed 评测：task_1-4 全部成功，task_5 事件推断正常但导航执行层仍有 bug（agent_dead）。 +- 同步更新 `Task1Formalization.lean` 注释与报告实验章节。 + +### 2026-07-11 + +- 修复 spatial 布局变体失败。 +- 根因一：`move_towards_aligned` 像素对齐阈值 `+1` 过松，在 1-tile 间隙处 sprite rect 覆盖相邻 blocking tile 导致卡住。修复为精确对齐 `+0` 并添加卡住检测（`mem_last_action_blocked` 追踪 reward ≤ -0.05）。 +- 根因二：`detect_task3_room` 硬编码 NPC/chest 位置，spatial 变体下 room 检测失败。修复为全网格扫描。 +- 修复前 spatial 通过率：task_1 1/3、task_2 1/3、task_3 0/3；修复后全部 3/3。 +- 运行 robustness suite（`--num-envs 30`）：task_1-4 的 original + spatial 共 108 个 episode 全部通过，color 变体预期失败（策略基于精确 RGB 匹配）。 +- 更新 `eval_results.json` 与报告实验章节。 + +### 2026-07-13 + +- 合并组员推送的 color 变体适配（`normalize_obs` 预处理模块，对 dark/bright/inverted 施加逆变换）和 task_2-4 Lean 策略形式化改为 BFS 模式。 +- 运行完整 5 任务 robustness suite（`--num-envs 30`，共 150 个 episode）。 +- 评测结果： + - task_1-4 original 72/72 + spatial 36/36 = 108/108 全通过（100%）。 + - task_1-4 color 8/12 通过（66.7%）：dark/bright 全通过，grayscale 有损不可逆仍失败。 + - task_5 全部失败（original 0/18、spatial 0/9、color 0/3）：agent_dead。 +- 尝试修复 task_5 导航执行层（7 个 bug）： + - Bug 1：`navigate_to_exit` 像素对齐方向检查——south/north 只处理 dx==0 的相邻 tile，east/west 只处理 dy==0，避免地图边界卡死。 + - Bug 2：`task5_button_pressed` 误标——只在 agent 在 button tile 上按 A 时标记。 + - Bug 3：button 自动触发检测——button 踩上即触发（不需 A），在 `decide_task5` 中检测 button 不可见时自动标记。 + - Bug 4：`task5_start_monster_cleared` 误标——添加 `room_id == "task5_start"` 条件。 + - Bug 5：`detect_task5_room` east/west 墙体模式错误——更新为与实际房间布局匹配的独特墙位置。 + - Bug 6：`task5_east_chest_opened` 误标——用 `chest_adjacent and not monster_adjacent` 区分开 chest 和攻击怪物。 + - Bug 7：`attack_monster` 缺少 `blocked`/`player_px` 参数——task5_east/west 分支的 attack_monster 调用未传参，导致用简单 move_towards 被墙挡住。 +- 修复后 task_5 original 评测：success_rate 仍为 0.000，但 agent 平均存活 1083 步（修复前 400 步），累计正奖励 95.020（修复前 -21.650），全部里程碑 1.000（chest_opened/key_collected/gold_collected/agent_healed/button_pressed/room_changed/door_opened/monster_killed/exit_reached）。agent 成功完成 start→south→east 三个房间的完整任务链，仅在 west 房间因 hp 耗尽死亡。 +- 验证 task_1-4 不回归：robustness suite n=30 全部通过，original+spatial 100%，color 66.7%，与修复前完全一致。 +- 更新 `eval_results.json` 为完整 5 任务结果（含 task_5 修复后数据），更新 `main.tex` 实验章节。 + +### 2026-07-17 + +- 完善 `Task5Formalization.lean`：新增 10 类 BFS 路线证书、两类战斗证书、西门举盾入场证书、24 阶段详细状态机和完成性证明。 +- 修复 Task 5 `spatial_c`：NPC/非目标出口避障、移动怪物跨 tile 目标选择、西门边界 `ACTION_B` 举盾。 +- 修复 grayscale：参考色映射到灰度均值，使用零容差避免相邻灰度调色板误并。 +- 修复 high_contrast：增加二值连通分量与玩家、怪物、宝箱、NPC、开关、按钮、墙和陷阱的局部形状识别。 +- 验证结果： + - 全部 Lean 文件无错误、无警告编译通过。 + - robustness suite：五任务共 500/500 成功（--num-envs 100）；Task 5 original 60/60、spatial 30/30、color 10/10。 + - 六颜色变体：五任务共 50/50 成功（color 阶段覆盖 grayscale/dark/bright/high_contrast/inverted 各 2 seed），default 在 original/spatial 中覆盖。 +- 更新最终报告中的 Task 5 策略形式化、实验结果、颜色修复和总结。 + +## 9. 每日更新模板 + +复制下面模板追加到本文件末尾： + +```text +### YYYY-MM-DD + +- 今日完成： +- 当前结果： +- 遇到问题： +- 解决方案 / 下一步： +- 需要组长协调的事项： +``` + +## 10. 集成前检查项 + +- [ ] 每个方向都有明确负责人 +- [ ] 统一接口已经写成文档 +- [ ] 调试信息和最终提交信息来源已明确区分 +- [ ] Lean 证明对象与代码实现已对齐 +- [ ] 报告中每个结论都能对应代码或实验结果 diff --git a/docs/Mathematical_logic/project_management/TASK_ANALYSIS.md b/docs/Mathematical_logic/project_management/TASK_ANALYSIS.md new file mode 100644 index 0000000..546259e --- /dev/null +++ b/docs/Mathematical_logic/project_management/TASK_ANALYSIS.md @@ -0,0 +1,200 @@ +# 五个任务的逐关分析 + +本文件用于帮助小组快速理解 `task_1` 到 `task_5` 的目标链、关键机制和推荐 baseline 思路。 + +## 总览 + +| 任务 | 目标链 | 关键机制 | 难度判断 | 推荐 baseline | +|---|---|---|---|---| +| `task_1` | 找钥匙 -> 开北门 | 单房间、静态路径 | 低 | 固定计划 / 简单路径规划 | +| `task_2` | 杀怪 -> 拿钥匙 -> 走西门 | 单房间、战斗、陷阱边界 | 低到中 | 规则策略 + 局部规划 | +| `task_3` | 穿过怪物房 -> 拿钥匙 -> 返回起点 -> 开东门 | 多房间、往返记忆 | 中 | 房间级状态机 + 局部规划 | +| `task_4` | 切桥 -> 拿钥匙 -> 开东门取剑 -> 去南房杀怪 -> 回中间开最终宝箱 | 动态桥、无剑起步、事件链 | 中到高 | 明确阶段机 + 动态对象状态跟踪 | +| `task_5` | 多房间探索并完成宝箱目标 | 多机制混合、HP 消耗、泛化 | 高 | 通用探索器 + 子目标规划 | + +## 1. Task 1 + +### 地图与目标 + +- 出生点在偏下方。 +- 房间内有墙体隔断，但整体是静态地图。 +- 关键宝箱在左侧，打开后获得钥匙。 +- 北侧出口是 `locked_key`，需要 1 把钥匙并完成任务。 + +### 关键机制 + +- 没有怪物。 +- 没有动态机关。 +- 主要是“走到宝箱旁 -> 交互 -> 走到门口”。 + +### baseline 思路 + +- 最简单方案是固定像素动作序列。 +- 稍微通用一点的方案是从像素中识别墙、宝箱、出口位置，再做静态路径规划。 + +### 对形式化友好度 + +- 很适合作为第一批 Lean 建模对象。 +- 可以证明：移动路径合法、不穿墙、拿到钥匙后可开启出口。 + +## 2. Task 2 + +### 地图与目标 + +- 单房间。 +- 西侧出口要求“怪物全部击败且拥有钥匙”。 +- 钥匙在左侧宝箱中。 +- 有一个 `chaser` 怪物，房间上下边界有陷阱。 + +### 关键机制 + +- 需要战斗，不再是纯路径问题。 +- 需要处理“接近怪物 -> 攻击 -> 避免无谓掉血”。 +- 陷阱只在上下边界，说明安全区域相对明确。 + +### baseline 思路 + +- 可以用规则策略： + - 先找怪物并接近； + - 相邻时攻击； + - 怪物死后去拿钥匙； + - 再去西门。 +- 如果状态抽取足够稳定，也可以用 tile 级 BFS 做导航。 + +### 对形式化友好度 + +- 可以证明：安全动作筛选器不会主动走进陷阱。 +- 可以证明：局部规划输出的路径是相邻、合法、可执行的。 + +## 3. Task 3 + +### 地图与目标 + +- 三个房间：起点房、怪物走廊、钥匙房。 +- 起点房西侧通往怪物房，东侧锁门需要钥匙。 +- 必须先往西，穿过怪物房，到最西边拿钥匙，再返回起点开东门。 + +### 关键机制 + +- 第一次真正要求“跨房间记忆”。 +- 需要记住当前在什么房间、任务进行到哪个阶段。 +- 走廊里有 `chaser` 怪物，会增加穿越难度。 + +### baseline 思路 + +- 用阶段机最合适： + - `go_west` + - `get_key` + - `return_start` + - `unlock_exit` +- 房间内继续用简单局部规划。 +- 如果视觉不稳定，训练阶段可以用 `info` 做监督标注，但提交版必须靠像素或内部记忆恢复状态。 + +### 对形式化友好度 + +- 可以证明阶段转换条件。 +- 可以证明：只要成功到达钥匙房并取得钥匙，返回起点后开门条件满足。 + +## 4. Task 4 + +### 地图与目标 + +- 五房间结构：西、中、北、东、南。 +- 出生在西房，初始只有 `shield`，没有剑。 +- 西房开关控制中央旋转桥。 +- 北房有钥匙；中间东门需要钥匙；东房有剑；南房有怪物。 +- 中央房的最终宝箱起初隐藏，要求南房怪物全灭后才显现。 + +### 关键机制 + +- 这是最适合做“显式任务链”建模的一关。 +- 中央房大面积是 `abyss`，只有桥所在的 tile 可通行。 +- 同时涉及： + - 动态桥状态 + - 条件门 + - 道具获取 + - 战斗 + - 隐藏宝箱显现 + +### baseline 思路 + +- 最推荐做成显式阶段机： + - `rotate_to_north` + - `get_key` + - `rotate_to_east` + - `get_sword` + - `rotate_to_south` + - `kill_guardian` + - `open_final_chest` +- 中央桥状态必须单独维护。 +- 这是后续 Lean 证明的优先候选关卡。 + +### 对形式化友好度 + +- 很适合证明动态对象状态机。 +- 可以证明：桥状态切换后哪些房间可达、哪些不可达。 +- 可以证明：如果阶段顺序正确执行，则最终宝箱可显现并可打开。 + +## 5. Task 5 + +### 地图与目标 + +- 四房间综合任务。 +- 起始房有： + - 金币宝箱 + - NPC + - 按钮 + - `chaser` + - 一个条件南门和一个锁门东门 +- 南房有钥匙、巡逻怪、陷阱。 +- 东房有回血宝箱和 `ambusher`。 +- 西房有金币宝箱、NPC、`chaser` 和 `ambusher`。 + +### 关键机制 + +- 任务 5 是综合探索关。 +- 奖励中还有一个特殊机制：每隔固定步数会掉 1 点血，因此不能无限拖延。 +- 需要兼顾探索、战斗、拿钥匙、开门和资源管理。 + +### baseline 思路 + +- 当前已验证 baseline 不追求最优路线，而是采用固定子目标顺序： + - 起始房金币宝箱； + - 按按钮打开南路； + - 南房拿钥匙； + - 回起始房击杀挡路 `chaser`； + - 开东门并打开回血宝箱； + - 回起始房后进入西房，打开金币宝箱。 +- 实现上需要记录已开宝箱、已按按钮、已拿钥匙、已开门、已击杀怪物等阶段事件。 +- 本关的主要工程难点不是高层顺序，而是像素 tile 与碰撞箱不完全同步： + - 到达目标 tile 后常常还需要沿行/列额外移动若干步； + - 否则角色会擦到墙、NPC 或出口边界，导致 `action_blocked`； + - 因此 baseline 中保留了多个显式像素对齐阶段。 + +### 对形式化友好度 + +- 整体全证会比较难。 +- 更适合只证明某个可验证层，如： + - 合法移动 + - 子目标切换条件 + - 安全动作屏蔽 + +## 6. 对小组开发的直接建议 + +### 适合立刻并行的方向 + +- 成员 A：把五关的房间结构、对象和门条件整理成表格。 +- 成员 B：先做 `task_1` / `task_2` 的提交版 baseline。 +- 成员 C：做视觉状态抽取最小版本，至少识别玩家、墙、宝箱、门。 +- 成员 D：以 `task_4` 为主做 Lean 建模，因为它的状态机最清晰。 +- 组长：维护统一接口和阶段机约定，避免每个人各做一套。 + +## 7. 当前建议的优先顺序 + +建议实际开发顺序如下： + +1. 用你们自己的统一代码骨架实现 `task_1` baseline。 +2. 复用同一骨架扩展到 `task_2`。 +3. 为 `task_3` 增加房间级阶段记忆。 +4. 用 `task_4` 确定最终形式化对象。 +5. 最后再把 `task_5` 当作综合展示关处理。 diff --git a/docs/Mathematical_logic/report/README.md b/docs/Mathematical_logic/report/README.md new file mode 100644 index 0000000..ce400da --- /dev/null +++ b/docs/Mathematical_logic/report/README.md @@ -0,0 +1,43 @@ +# 实验报告 TeX 项目说明 + +本目录提供数理逻辑大作业的实验报告骨架，建议在整个项目周期中边做边填，而不是最后一天集中补写。 + +## 文件说明 + +- `main.tex`：主报告文件，已包含适合本课程作业的章节框架。 +- `references.bib`：参考文献数据库，已放入仓库信息示例。 +- `figures/`：存放实验截图、结构图、结果图表。 + +## 推荐写作方式 + +建议填写顺序： + +1. 先写“任务概述”和“总体方法”。 +2. 每完成一个模块，就补“环境建模 / 策略设计 / 形式化证明 / 实验结果”。 +3. 每天把 `project_management/PROJECT_PROGRESS.md` 中的更新摘要转化为报告草稿内容。 +4. 封版前统一润色格式、图表标题和引用。 + +## 编译方式 + +优先使用 `XeLaTeX` 编译中文文档。 + +如果本地安装了 `latexmk`： + +```bash +latexmk -xelatex main.tex +``` + +如果没有安装 `latexmk`，可手动编译： + +```bash +xelatex main.tex +bibtex main +xelatex main.tex +xelatex main.tex +``` + +## 建议同步规则 + +- 每完成一项实验，就把结果图、截图和关键结论放进 `figures/` 和进度文档。 +- 每形成一个明确决策，就同步更新 `PROJECT_MEMORY.md`。 +- 报告中的每个结论都应能追溯到代码、实验或证明。 diff --git a/docs/Mathematical_logic/report/eval_results.json b/docs/Mathematical_logic/report/eval_results.json new file mode 100644 index 0000000..4b7a301 --- /dev/null +++ b/docs/Mathematical_logic/report/eval_results.json @@ -0,0 +1,18512 @@ +{ + "episodes": [ + { + "eval_stage": "original", + "event_counts": { + "action_blocked": 1, + "chest_opened": 1, + "door_opened": 1, + "environment_completed": 1, + "exit_reached": 1, + "key_collected": 1, + "move_left": 90, + "move_right": 82, + "move_up": 96, + "room_changed": 1, + "world_completed": 1 + }, + "map_variant": "default", + "milestones": {}, + "obs_variant": "default", + "seed": 0, + "steps": 270, + "success": true, + "task_id": "mathematical_logic/task_1", + "terminal_reason": "world_completed", + "terminated": true, + "total_reward": 127.25000000000003, + "truncated": false + }, + { + "eval_stage": "original", + "event_counts": { + "action_blocked": 1, + "chest_opened": 1, + 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b/docs/Mathematical_logic/report/main.tex new file mode 100644 index 0000000..4551a64 --- /dev/null +++ b/docs/Mathematical_logic/report/main.tex @@ -0,0 +1,1582 @@ +\documentclass[12pt,a4paper]{ctexart} + +\usepackage[a4paper,margin=2.5cm]{geometry} +\usepackage{amsmath,amssymb,amsthm} +\usepackage{array} +\usepackage{booktabs} +\usepackage{enumitem} +\usepackage{float} +\usepackage{graphicx} +\usepackage{hyperref} +\usepackage{longtable} +\usepackage{multirow} +\usepackage{xcolor} + +\hypersetup{ + colorlinks=true, + linkcolor=blue, + citecolor=blue, + urlcolor=blue +} + +\title{数理逻辑课程大作业实验报告} +\author{李政轩（241880177） \and 续博森（241880403） \and 吴韬（241880187） \and 陈怡琼（241880323） \and 许浩博（231880172）} +\date{\today} + +\begin{document} + +\maketitle + +\begin{abstract} +本文报告围绕 NesyLink 数理逻辑课程项目展开，目标是在 Zelda-like 地牢环境中设计可运行的 Agent，并对环境或策略中的关键部分进行 Lean 形式化建模与证明。 + +本项目已对 \texttt{task\_1} 至 \texttt{task\_5} 全部完成 Lean 环境形式化与可完成性证明， +对应文件位于 \texttt{student\_agent/lean/} 目录下。五个形式化模型均通过 Lean LSP 诊断（零错误）， +并遵循统一证明骨架：局部前置/后置引理 $\rightarrow$ 必要性引理 $\rightarrow$ +\texttt{Reachable} 归纳不变式 $\rightarrow$ \texttt{completion\_postcondition} $\rightarrow$ +witness 计划 $\rightarrow$ 可完成性。其中 \texttt{task\_4} 因涉及旋转桥、条件门、 +装备获取与事件驱动显现机制，作为重点展开对象。 +此外，本文对 \texttt{task\_1} 至 \texttt{task\_5} 进一步补充了高层符号策略形式化： +\texttt{task\_1} 至 \texttt{task\_4} 采用 BFS 证书模式 +（\texttt{symbolicPolicy} 参数化于 BFS 动作提供器，通过证书结构证明非固定路线完备性）， +\texttt{task\_5} 在房间级阶段机之上增加与最终 Python baseline 对齐的证书细化层： +10 类 BFS 路线、两类战斗契约、西门举盾入场契约和 24 个宏动作阶段均可计算检查， +并证明五个任务的符号策略从初态出发均能选择合法动作且到达对应目标。 +在此基础上，本文进一步抽取了独立的 \texttt{StrategyFramework.lean}， +形式化通用策略执行、动作 mask 安全证书、有界可达性和 BFS 访问集， +并证明“若某目标在 $n$ 步内由某个计划可达，则完整动作集上的深度 $n$ BFS 必会访问到目标态”。 +\end{abstract} + +\tableofcontents +\newpage + +\section{项目背景与任务要求} + +\subsection{课程作业目标} + +本次课程作业要求小组围绕给定游戏环境完成三部分内容： +\begin{enumerate}[leftmargin=2em] + \item 设计并实现一个能够完成任务的 Agent； + \item 使用 Lean 对环境或策略中的关键部分进行形式化建模； + \item 证明若干性质，例如动作合法性、安全性、可达性或策略正确性。 +\end{enumerate} + +\subsection{环境限制与评测要求} + +\begin{itemize}[leftmargin=2em] + \item 训练和调试阶段可以使用像素、reward 和 \texttt{info} 等信息。 + \item 最终测试阶段，Agent 不能直接依赖环境内部 \texttt{info}。 + \item 最终测试阶段，Agent 只能根据像素观测、reward 和物品栏信息进行决策。 + \item 如果使用机器学习方法，需要说明训练方式、信息来源以及与最终评测接口的一致性。 +\end{itemize} + +\subsection{本文结构} + +本文后续将依次介绍环境与任务分析、总体方法设计、视觉与状态抽取、策略与规划、 +Lean 形式化与证明（覆盖 \texttt{task\_1} 至 \texttt{task\_5}）、 +实验结果以及项目分工。 + +\section{环境与任务分析} + +\subsection{NesyLink 环境概述} + +NesyLink 是一个 Zelda-like 的离散动作地牢环境。默认任务接口采用 Gym 风格： +\texttt{obs, info = env.reset(seed)}，以及 +\texttt{obs, reward, terminated, truncated, info = env.step(action)}。 +本次课程任务要求最终提交版策略在推理阶段只能使用像素观测、reward +和显式提供的物品栏信息，不能直接读取 \texttt{info} 中的隐藏状态。 + +动作空间为 7 个离散动作： +\texttt{WAIT}、\texttt{UP}、\texttt{DOWN}、\texttt{LEFT}、\texttt{RIGHT}、 +\texttt{BUTTON\_A}、\texttt{BUTTON\_B}。默认像素控制下，房间地图区域大小为 +\texttt{160 \(\times\) 128} 像素，对应 \texttt{10 \(\times\) 8} 个 tile， +每个 tile 为 \texttt{16 \(\times\) 16} 像素。默认观测 \texttt{obs} +为 RGB 图像，shape 为 \texttt{(128, 160, 3)}。环境还提供结构化 +\texttt{info} 作为调试与监督接口，其中包含房间编号、角色坐标、事件记录、 +动态对象状态和奖励信号等信息。 + +五个任务的差异主要体现在关键机制上：\texttt{task\_1} +是静态单房间取钥匙开门；\texttt{task\_2} 引入怪物、战斗和条件门； +\texttt{task\_3} 要求跨房间记忆；\texttt{task\_4} +引入旋转桥、条件门、装备获取和隐藏宝箱，是最适合做显式环境建模的一关； +\texttt{task\_5} 则是四房间按钮门与全宝箱收集的综合任务。基于课程要求与当前项目进度， +本文对 \texttt{task\_1} 至 \texttt{task\_5} 全部进行了 Lean 环境形式化与可完成性证明， +其中 \texttt{task\_4} 因机制最复杂作为重点展开对象。 + +\subsection{任务拆解} + +五个任务可以按关键机制逐步拆解： +\begin{itemize}[leftmargin=2em] + \item Task 1：单房间取钥匙开门。 + \item Task 2：怪物处理与取钥匙。 + \item Task 3：多房间状态记忆与往返。 + \item Task 4：桥与按钮机制、复杂子任务链。 + \item Task 5：综合关卡与泛化挑战。 +\end{itemize} + +\subsection{主要难点} + +\begin{enumerate}[leftmargin=2em] + \item 最终测试不能直接读取 \texttt{info}。 + \item 需要从像素中抽取对策略有用的状态表示。 + \item 需要在短时间内兼顾实现、证明和报告。 +\end{enumerate} + +\section{总体方法设计} + +\subsection{系统框架} + +本项目采用“像素感知 + 规则状态机 + Lean 符号层验证”的整体框架。 +运行时，\texttt{student\_agent/baseline\_policy.py} 提供统一的 +\texttt{Policy} 入口，评测脚本仍按 \texttt{act(obs, info)} 调用， +但策略主体尽量只使用像素观测、reward 信号、物品栏和内部历史记忆。 +整体链路可以分为四个模块： +\begin{itemize}[leftmargin=2em] + \item \textbf{视觉/状态抽取模块}：从 \texttt{obs} 中识别玩家中心 tile、 + 宝箱、NPC、怪物、开关、按钮、墙体、桥方向和房间特征。 + \item \textbf{策略与规划模块}：按任务维护阶段记忆，将任务拆成 + “拿钥匙”“按按钮”“切桥”“开宝箱”“击杀怪物”“进入出口”等子目标。 + \item \textbf{反馈记忆模块}：在最终测评的 \texttt{safe} 信息模式下， + \texttt{info} 仅包含 \texttt{last\_reward}、\texttt{inventory} 和 \texttt{task\_id}， + 不再暴露 \texttt{reward\_signals}。策略通过物品栏 diff + （\texttt{keys}/\texttt{gold}/\texttt{items} 增量）结合 \texttt{last\_reward} 和 + 像素感知（怪物消失、按钮颜色变化）推断 \texttt{key\_collected}、 + \texttt{chest\_opened}、\texttt{monster\_killed}、\texttt{button\_pressed} 等关键事件， + 避免依赖隐藏事件列表。 + \item \textbf{形式化验证模块}：在 Lean 中抽象环境的符号层语义， + 证明关键前置条件、不变式、完成后置条件、witness 计划可完成性， + 并对 \texttt{task\_5} 的高层阶段机策略给出动作合法性与目标可达性证明。 +\end{itemize} + +\subsection{为什么选择该路线} + +选择该路线主要基于三点考虑。 +\begin{itemize}[leftmargin=2em] + \item \textbf{可解释性}：五个任务均有清晰的目标链，规则阶段机比黑盒模型更容易定位失败原因。 + \item \textbf{与 Lean 对齐}：Lean 证明适合覆盖钥匙门、按钮门、桥状态、宝箱开启、 + 怪物击杀和完成谓词这类符号语义，和阶段机策略天然一致。 + \item \textbf{实现风险可控}：课程时间有限，先做可运行 baseline， + 再把高层环境语义形式化，比从头训练复杂策略更稳健。 +\end{itemize} + +当前阶段已经选择“统一 Policy 骨架 + 分任务逐步扩展 + 可验证层优先”的路线。 +具体而言，项目先在自有代码目录中建立统一的 Agent 入口， +优先跑通 \texttt{task\_1} 的 baseline，再逐步扩展到 \texttt{task\_2} 至 \texttt{task\_5}， +并以阶段机、合法动作筛选器和动态对象状态跟踪作为后续形式化重点。 +目前自有 baseline 已覆盖 \texttt{task\_1} 至 \texttt{task\_5}。 +其中 \texttt{task\_2} 仍属于调试用途的脚本化版本， +而 \texttt{task\_3}、\texttt{task\_4} 和 \texttt{task\_5} +已经采用像素房间识别、玩家中心格定位、物品栏和 reward 历史驱动的规则状态机。 +Lean 部分首先完成五个任务的环境形式化与 witness 可完成性证明； +在此基础上，\texttt{task\_5} 还额外形式化了与 Python 阶段机对应的高层符号策略， +证明策略生成的房间级动作序列合法，并能达到全宝箱开启目标。 + +\section{视觉与状态抽取} + +\subsection{输入与输出设计} + +视觉模块输入为 RGB 像素帧 \texttt{obs}。输出不是完整地图重建， +而是服务于规则策略的少量符号状态： +\begin{itemize}[leftmargin=2em] + \item 玩家中心 tile：根据玩家绿色精灵像素反推精灵左上角，再取中心点所在 tile。 + \item 关键对象：通过颜色计数识别宝箱、NPC、怪物、墙体、开关、按钮、深渊和桥。 + \item 房间类别：针对 \texttt{task\_3}、\texttt{task\_4}、\texttt{task\_5} + 分别用房间内稳定对象和地形特征识别当前房间。 + \item 动态状态：\texttt{task\_4} 通过桥像素分布识别 + \texttt{west\_to\_north}、\texttt{west\_to\_east}、 + \texttt{west\_to\_south} 三种桥方向； + \texttt{task\_5} 通过按钮、宝箱和房间特征识别当前子任务位置。 +\end{itemize} + +\subsection{实现方法} + +实现上采用基于颜色模板和 tile 统计的轻量规则方法。 +\texttt{baseline\_policy.py} 中定义了玩家、墙、宝箱、NPC、怪物、桥、开关和按钮的颜色常量， +并提供 \texttt{count\_color}、\texttt{count\_any\_color}、 +\texttt{tile\_has\_chest}、\texttt{tile\_has\_switch}、 +\texttt{tile\_has\_button} 等辅助函数。 +\begin{itemize}[leftmargin=2em] + \item 玩家定位不直接读取 \texttt{info["agent"]}，而是从像素中恢复中心 tile。 + \item 房间识别不直接读取 \texttt{info["env"]}，而是通过房间内宝箱、 + NPC、桥、开关、墙体开口等稳定视觉特征判断。 + \item 怪物、按钮、开关和宝箱状态通过像素与物品栏/\texttt{last\_reward} 共同维护； + 例如 \texttt{task\_5} 用 \texttt{inventory} diff（\texttt{keys}/\texttt{gold} 增量）、 + \texttt{last\_reward} 正值和怪物像素消失来推断 \texttt{button\_pressed}、 + \texttt{chest\_opened}、\texttt{monster\_killed} 等事件并更新内部记忆。 + \item 物品栏只用于读取课程允许的钥匙、装备等显式状态。 +\end{itemize} + +\subsection{误差与局限} + +该视觉方案依赖当前地图素材的固定颜色和固定 tile 尺寸， +因此更适合作为课程环境内的可解释 baseline，而不是通用视觉模型。 +主要风险包括：角色精灵中心与碰撞箱位置存在偏差、tile 已到位但碰撞箱仍未越过出口边界、 +以及对象被开启或按下后像素外观变化导致房间识别失败。 +针对这些问题，策略中加入了若干行/列对齐阶段，例如 \texttt{task\_5} +进入东门、从东房返回、进入西房前都保留额外对齐动作，以减少碰撞边界误差。 + +\section{策略与规划} + +\subsection{基线策略} + +基线策略采用规则式阶段机 + BFS 路径规划，而不是学习模型。 +\begin{itemize}[leftmargin=2em] + \item \texttt{task\_1}：BFS 到宝箱邻居 \(\rightarrow\) 开箱取钥匙 \(\rightarrow\) BFS 到北门出口。 + \item \texttt{task\_2}：像素感知怪物 \(\rightarrow\) 邻接攻击 \(\rightarrow\) BFS 到宝箱 \(\rightarrow\) 开箱 \(\rightarrow\) BFS 到西出口。 + \item \texttt{task\_3}：BFS 跨房间西行到钥匙房 \(\rightarrow\) 开箱 \(\rightarrow\) BFS 返回起点 \(\rightarrow\) 开锁门。 + \item \texttt{task\_4}：维护桥方向和阶段记忆，按 + “北房钥匙 \(\rightarrow\) 切桥到东房拿剑 \(\rightarrow\) 切桥到南房杀守卫 + \(\rightarrow\) 回中心开最终宝箱”推进，房间内导航用 BFS。 + \item \texttt{task\_5}：维护按钮、钥匙、东门、怪物、四个宝箱等事件记忆， + 按“起始宝箱 \(\rightarrow\) 按按钮 \(\rightarrow\) 南房钥匙 + \(\rightarrow\) 起始房杀怪 \(\rightarrow\) 东房回血 \(\rightarrow\) 西房金币”推进， + 事件通过 \texttt{inventory} diff + \texttt{last\_reward} + 像素感知推断。 +\end{itemize} + +\subsection{多任务扩展} + +多任务扩展的核心是把每个任务拆成少量稳定子目标，并在 \texttt{AgentHistory.notes} +中保存跨步记忆。对单房间任务，固定计划即可完成；对多房间任务， +策略需要记录当前阶段、是否已开箱、是否已取得钥匙、桥是否已旋转到目标方向、 +怪物是否已经击杀等信息。路径选择主要使用 tile 级贪心移动和固定 waypoint， +在出口和交互点附近加入额外对齐动作，保证像素控制下的碰撞箱能够真正进入交互范围。 + +\subsection{最终提交版本} + +当前提交版本与调试过程的区别如下： +\begin{itemize}[leftmargin=2em] + \item 调试阶段曾使用 \texttt{info} 校准房间、坐标和事件语义， + 便于定位失败原因。 + \item 当前 baseline 已全面适配评测脚本的 \texttt{safe} 信息模式： + \texttt{reset()} 不再接收 \texttt{seed}/\texttt{task\_id} 参数， + \texttt{task\_id} 改由 \texttt{info["task\_id"]} 获取（通过 \texttt{--task-policy} 绑定）； + 原先依赖 \texttt{reward\_signals} 的 \texttt{update\_memory} 与 + \texttt{update\_task5\_memory} 已重构为基于 \texttt{inventory} diff、 + \texttt{last\_reward} 和像素感知的事件推断。 + \item 五个任务的主体决策均采用像素识别 + BFS 路径规划 + 阶段机， + 不直接读取隐藏的 \texttt{info["env"]}、\texttt{info["agent"]}、 + \texttt{info["dynamic"]} 或 \texttt{info["events"]}。 + \item Lean 验证了五个任务的符号层环境、显式 witness 计划， + 以及全部五个任务的高层符号策略（\texttt{symbolicPolicy}）的合法性与完成性 + ——\texttt{task\_1} 至 \texttt{task\_4} 采用 BFS 证书模式 + （策略参数化于 BFS 动作提供器，证书结构证明非固定路线完备性）， + \texttt{task\_5} 在原有 \texttt{PolicyStage} 房间级证明之上， + 新增 BFS、战斗和西门举盾入场证书驱动的 24 阶段 baseline 细化； + 同时新增通用策略框架文件，证明有界 BFS 搜索的框架级完备性； + 尚未证明像素感知层一定能在所有渲染扰动下恢复这些符号状态。 +\end{itemize} + +\section{Lean 形式化与证明} + +\subsection{形式化对象总览} + +本项目对 \texttt{task\_1} 至 \texttt{task\_5} 全部完成了 Lean 环境形式化与可完成性证明， +对应文件位于 \texttt{student\_agent/lean/} 目录下： +\begin{itemize}[leftmargin=2em] + \item \texttt{Task1Formalization.lean}：tile 级单房间钥匙门； + \item \texttt{Task2Formalization.lean}：zone 级战斗与条件门； + \item \texttt{Task3Formalization.lean}：room 级三房间往返取钥匙； + \item \texttt{Task4Formalization.lean}：room 级旋转桥与隐藏宝箱； + \item \texttt{Task5Formalization.lean}：room 级四房间按钮门与全宝箱收集； + \item \texttt{StrategyFramework.lean}：通用策略执行、动作 mask 安全证书、 + 有界可达性与 BFS 完备性证明。 +\end{itemize} + +任务形式化文件均通过 Lean LSP 诊断（零错误），并按照统一的设计原则建模： +\begin{enumerate}[leftmargin=2em] + \item \textbf{抽象层级与 Python 行为对齐}：根据每个任务的关键机制选择 + tile / zone / room 抽象层级，并与 + \texttt{movement.py}、\texttt{interactions.py}、\texttt{combat.py} + 及 \texttt{room\_*.json} 中的语义保持一致； + \item \textbf{保留约束而不照搬实现}：刻意忽略像素、碰撞盒、怪物 AI、击退等低层细节， + 只保留影响规划正确性的状态字段、前置条件与事件触发； + \item \textbf{统一证明骨架}：每个任务都给出 + “局部前置/后置引理 $\rightarrow$ 必要性引理 $\rightarrow$ \texttt{Reachable} 归纳不变式 + $\rightarrow$ \texttt{completion\_postcondition} $\rightarrow$ witness 计划 $\rightarrow$ 可完成性” + 的完整证明链。 +\end{enumerate} + +下面分任务展开。其中 \texttt{task\_4} 因机制最复杂（动态对象状态、条件门、 +关键道具获取和事件驱动显现），作为重点展开对象。 + +\subsection{Task 1：单房间钥匙门形式化} + +\texttt{task\_1} 是最简单的单房间取钥匙走北门任务。 +Lean 文件以 tile 级建模，对应 \texttt{room\_001.json} 的 8 行 $\times$ 10 列网格。 + +\textbf{状态与动作}： +\begin{itemize}[leftmargin=2em] + \item \texttt{EnvState}：\texttt{x, y : Nat}（tile 坐标）、\texttt{keys}、 + \texttt{chestOpened}、\texttt{completed}； + \item \texttt{Action}：\texttt{up / down / left / right / openChest / goNorth}； + \item \texttt{isWalkable} 严格复刻 \texttt{room\_001.json} 中的墙体布局 + （row 2 的 \texttt{\#\#..\#\#\#\#\#\#}、row 5 的 \texttt{\#\#\#\#\#\#\#...}）； + \item \texttt{isAdjacentToChest} 使用 Manhattan 距离 $\leq 1$， + 对齐 \texttt{state.py:is\_adjacent} 的语义（包含距离 0）。 +\end{itemize} + +\textbf{关键约束}： +\begin{itemize}[leftmargin=2em] + \item \texttt{openChest} 触发条件为邻接 chest tile 且未开过，开箱后 \texttt{keys + 1}； + \item \texttt{goNorth} 仅在 \texttt{(4, 0)} 且 \texttt{keys > 0} 时触发， + 成功后消耗一把钥匙并将 \texttt{completed} 置真（对应 \texttt{consume\_key: true}）。 +\end{itemize} + +\begin{longtable}{p{0.30\textwidth}p{0.28\textwidth}p{0.32\textwidth}} +\toprule +定理名称 & 形式化对象 & 结论说明 \\ +\midrule +\endhead +\path{step_preserves_walkable} & 安全性 & +任意动作后玩家仍位于可走 tile 上。 \\ +\path{openChest_only_at_chest} & 开箱前置条件 & +不邻接 chest 时 \texttt{openChest} 不改变状态。 \\ +\path{openChest_at_chest_increases_keys} & 开箱后置条件 & +邻接且未开过时，\texttt{keys} 增加 1。 \\ +\path{goNorth_only_at_exit} & 北门前置条件 & +不在 \texttt{(4, 0)} 时 \texttt{goNorth} 无效。 \\ +\path{goNorth_without_key} & 北门前置条件 & +在出口但无钥匙时 \texttt{goNorth} 无效。 \\ +\path{goNorth_at_exit_with_key_completes} & 北门后置条件 & +在出口且持钥匙时 \texttt{completed} 置真。 \\ +\path{only_goNorth_sets_completed} & 必要性引理 & +\texttt{goNorth} 是唯一能把 \texttt{completed} 从假翻为真的动作。 \\ +\path{witnessPlan_reaches_goal} & 执行轨迹 & +22 步 witness 计划可达目标。 \\ +\path{task1_completable} & 可完成性 & +\texttt{task\_1} 在该模型下可完成。 \\ +\bottomrule +\end{longtable} + +\textbf{witness 计划}：tile 级路径 +\[ +(4,6) \xrightarrow{\text{right}\times 3} (7,6) + \xrightarrow{\text{up}\times 3} (7,3) + \xrightarrow{\text{left}\times 7} (0,3) + \xrightarrow{\text{openChest}} \cdot + \xrightarrow{\text{right}\times 3, \text{up}\times 3, \text{right}} (4,0) + \xrightarrow{\text{goNorth}} \text{goal}. +\] +由于 22 步计划超出 \texttt{decide} 递归深度，按项目约定使用 +\texttt{native\_decide} 验证以下定理： +\begin{center} +\path{witnessPlan_executes_to_finalState} +\end{center} + +\subsection{Task 2：战斗与条件门形式化} + +\texttt{task\_2} 在单房间内引入 chaser 怪物、HP、陷阱与西向条件门。 +为突出战斗与条件门语义，采用 zone 级抽象。 + +\textbf{状态与动作}： +\begin{itemize}[leftmargin=2em] + \item \texttt{Zone}：\texttt{safeCenter / nearMonster / nearChest / westExit / topTrap / bottomTrap}； + \item \texttt{EnvState}：\texttt{zone}、\texttt{hp}、\texttt{monsterAlive}、\texttt{monsterHp}、 + \texttt{chestOpened}、\texttt{keys}、\texttt{completed}； + \item \texttt{Action}：\texttt{moveTo z / attack / openChest / useExit / wait}。 +\end{itemize} + +\textbf{关键约束}： +\begin{itemize}[leftmargin=2em] + \item \texttt{attack} 仅在 \texttt{nearMonster}、怪物存活且 \texttt{hp > 1} 时生效； + \texttt{monsterHp} 为 1 时击杀（置 \texttt{monsterAlive := false}）， + 否则仅 \texttt{monsterHp - 1}； + \item \texttt{useExit} 在 \texttt{westExit} 且怪物已死且 \texttt{keys > 0} 时触发， + 成功后\textbf{不消耗}钥匙（对应 \texttt{requires} 中未设 \texttt{consume\_key}）； + \item 走入 \texttt{topTrap / bottomTrap} 时 HP 减 1 并被重置回 \texttt{safeCenter}。 +\end{itemize} + +\begin{longtable}{p{0.32\textwidth}p{0.26\textwidth}p{0.32\textwidth}} +\toprule +定理名称 & 形式化对象 & 结论说明 \\ +\midrule +\endhead +\path{trap_move_reduces_hp} & 陷阱语义 & +踩陷阱后 \texttt{hp} 减少 1。 \\ +\path{safe_move_preserves_*} & 安全移动不变式 & +非陷阱 \texttt{moveTo} 保持 HP/怪物/宝箱/钥匙/完成状态不变。 \\ +\path{attack_reduces_monsterHp_when_adjacent} & 战斗语义 & +邻接且存活且 HP 充足时，\texttt{monsterHp} 减 1。 \\ +\path{attack_kills_monster_when_monsterHp_one} & 战斗语义 & +\texttt{monsterHp = 1} 时一击击杀。 \\ +\path{attack_no_effect_*} & 战斗前置条件 & +不在 \texttt{nearMonster}、怪物已死或 HP 过低时攻击无效。 \\ +\path{exit_blocked_if_monster_alive} & 条件门 & +怪物存活时 \texttt{useExit} 无效。 \\ +\path{exit_blocked_without_key} & 条件门 & +无钥匙时 \texttt{useExit} 无效。 \\ +\path{exit_succeeds_after_requirements} & 条件门后置 & +怪物已死且持钥匙时 \texttt{completed} 置真。 \\ +\path{only_useExit_sets_completed} & 必要性引理 & +\texttt{useExit} 是唯一完成动作。 \\ +\path{completion_postcondition} & 完成后置 & +任意可达完成态满足怪物已死 $\wedge$ 宝箱已开。 \\ +\path{hp_non_increasing} & 安全性 & +任意动作后 HP 不增。 \\ +\path{witnessPlan_reaches_goal} & 执行轨迹 & +7 步 witness 计划可达目标。 \\ +\path{task2_completable} & 可完成性 & +\texttt{task\_2} 在该模型下可完成。 \\ +\bottomrule +\end{longtable} + +\textbf{witness 计划}：zone 级路径 +\[ +\text{safeCenter} \to \text{nearMonster} + \xrightarrow{\text{attack}\times 2} \cdot + \to \text{nearChest} \xrightarrow{\text{openChest}} \cdot + \to \text{westExit} \xrightarrow{\text{useExit}} \text{goal}. +\] + +\subsection{Task 3：多房间往返形式化} + +\texttt{task\_3} 是首个多房间任务：从 \texttt{start\_room} 出发向西穿过 +\texttt{monster\_hall} 到达 \texttt{key\_room}，取钥匙后返回 +\texttt{start\_room} 走东向锁定出口。采用 room 级抽象。 + +\textbf{状态与动作}： +\begin{itemize}[leftmargin=2em] + \item \texttt{Room}：\texttt{startRoom / monsterHall / keyRoom}； + \item \texttt{EnvState}：\texttt{room}、\texttt{keys}、\texttt{keyChestOpened}、 + \texttt{hallMonsterAlive}、\texttt{completed}； + \item \texttt{Action}：\texttt{goWest / goEast / openChest / openLockedExit / wait}。 +\end{itemize} + +\textbf{关键约束}： +\begin{itemize}[leftmargin=2em] + \item \texttt{goWest}：\texttt{startRoom $\to$ monsterHall $\to$ keyRoom}； + \texttt{goEast}：反向；\texttt{keyRoom} 的 \texttt{goWest} 与 + \texttt{startRoom} 的 \texttt{goEast} 无效； + \item \texttt{openChest} 仅在 \texttt{keyRoom} 且未开过时生效，开箱后 \texttt{keys + 1}； + \item \texttt{openLockedExit} 仅在 \texttt{startRoom} 且 \texttt{keys > 0} 时触发， + 成功后消耗钥匙并完成（对应 \texttt{consume\_key: true}）。 +\end{itemize} + +\begin{longtable}{p{0.34\textwidth}p{0.24\textwidth}p{0.32\textwidth}} +\toprule +定理名称 & 形式化对象 & 结论说明 \\ +\midrule +\endhead +\path{west_then_west_reaches_keyRoom} & 导航语义 & +从 \texttt{startRoom} 连续两次 \texttt{goWest} 可达 \texttt{keyRoom}。 \\ +\path{openChest_only_in_keyRoom} & 开箱前置 & +不在 \texttt{keyRoom} 时 \texttt{openChest} 无效。 \\ +\path{openChest_in_keyRoom_increases_keys} & 开箱后置 & +在 \texttt{keyRoom} 且未开过时 \texttt{keys} 增 1。 \\ +\path{openLockedExit_blocked_without_key} & 锁门前置 & +在 \texttt{startRoom} 但无钥匙时 \texttt{openLockedExit} 无效。 \\ +\path{openLockedExit_consumes_key_and_completes} & 锁门后置 & +持钥匙时消耗一把钥匙并完成。 \\ +\path{hallMonsterAlive_never_changes} & 不变式 & +Task 3 无攻击动作，\texttt{hallMonsterAlive} 恒不变。 \\ +\path{only_openLockedExit_sets_completed} & 必要性引理 & +\texttt{openLockedExit} 是唯一完成动作。 \\ +\path{keyChestOpened_or_keys_zero} & 归纳不变式 & +任意可达状态：宝箱已开 $\vee$ 钥匙数为 0。 \\ +\path{completion_postcondition} & 完成后置 & +任意可达完成态满足 \texttt{keyChestOpened = true}。 \\ +\path{witnessPlan_reaches_goal} & 执行轨迹 & +6 步 witness 计划可达目标。 \\ +\path{task3_completable} & 可完成性 & +\texttt{task\_3} 在该模型下可完成。 \\ +\bottomrule +\end{longtable} + +\textbf{witness 计划}： +\[ +\text{startRoom} \xrightarrow{\text{goWest}\times 2} \text{keyRoom} + \xrightarrow{\text{openChest}} \cdot + \xrightarrow{\text{goEast}\times 2} \text{startRoom} + \xrightarrow{\text{openLockedExit}} \text{goal}. +\] +由于计划简单，\texttt{witnessPlan\_executes\_to\_finalState} 用 \texttt{rfl} 直接验证。 + +\subsection{Task 4：旋转桥与隐藏宝箱形式化} + +\texttt{task\_4} 引入旋转桥、条件门、装备获取与事件驱动显现机制，是机制最复杂的一关， +也是本报告的重点展开对象。对应 Lean 文件为 +\path{Task4Formalization.lean}， +与 Python 环境中以下模块对齐： +\texttt{movement.py}（门与房间切换）、 +\texttt{interactions.py}（开关与宝箱）、 +\texttt{combat.py}（击杀怪物后揭示宝箱）。 + +\textbf{具体抽象}： +\begin{itemize}[leftmargin=2em] + \item \textbf{房间状态}：使用 \texttt{Room} 枚举描述五个房间 + \texttt{west / center / north / east / south}。 + \item \textbf{动态对象状态}：使用 \texttt{BridgeState} 描述中央桥的三种连通状态 + \texttt{westToNorth / westToEast / westToSouth}。 + \item \textbf{玩家资源与关键事件}：记录 + \texttt{keys}、\texttt{hasSword}、\texttt{hasShield}， + 以及北房宝箱、东房宝箱、南房守卫和中央终点宝箱的状态。 + \item \textbf{动作}：抽象为房间级动作，包括转桥、进出中心房、开宝箱、攻击和开启最终宝箱。 + \item \textbf{转移函数}：用确定性函数 \texttt{step} 描述状态更新。 + 它保留了三类关键约束： + 东门需要钥匙但不消耗钥匙（对应 \texttt{consume\_key: false}）； + 桥状态决定中心房可达方向； + 南房守卫被击杀后才会揭示中央最终宝箱。 + \item \textbf{目标谓词}：\texttt{GoalReached} 定义为最终宝箱已开启。 +\end{itemize} + +\begin{longtable}{p{0.28\textwidth}p{0.28\textwidth}p{0.34\textwidth}} +\toprule +定理名称 & 形式化对象 & 结论说明 \\ +\midrule +\endhead +\path{rotateBridge_three_cycle} & 桥状态机 & +证明西房开关连续触发三次后，桥状态回到初始方向。 \\ +\path{goEast_blocked_without_key} & 东门条件 & +证明当玩家位于中心房、桥已连到东侧但没有钥匙时，\texttt{goEast} 不产生状态变化。 \\ +\path{goEast_succeeds_with_key} & 东门条件 & +证明在中心房、桥连东侧且已持钥匙时，\texttt{goEast} 能把玩家带入东房。 \\ +\path{goEast_preserves_keys} & 东门不变式 & +证明 \texttt{goEast} 不消耗钥匙（Python \texttt{consume\_key: false} 对齐）。 \\ +\path{attack_without_sword_has_no_effect} & 战斗语义 & +证明没有剑时，在南房执行攻击不会击杀守卫，也不会揭示最终宝箱。 \\ +\path{attack_with_sword_reveals_final_chest} & 战斗与事件触发 & +证明持剑击败南房守卫后，\texttt{guardianAlive} 变为假，且 \texttt{finalChestVisible} 变为真。 \\ +\path{openFinalChest_requires_visibility} & 终点交互语义 & +证明若最终宝箱尚未显现，则在中心房尝试开启最终宝箱不会成功。 \\ +\path{only_openFinalChest_sets_finalChestOpened} & 必要性引理 & +\texttt{openFinalChest} 是唯一完成动作。 \\ +\path{northChestOpened_or_keys_zero} & 归纳不变式 & +任意可达状态：北房宝箱已开 $\vee$ 钥匙数为 0。 \\ +\path{finalChestVisible_implies_guardianDead} & 归纳不变式 & +任意可达状态：终点宝箱显现 $\Rightarrow$ 守卫已死。 \\ +\path{completion_postcondition} & 完成后置 & +任意可达完成态满足 \texttt{guardianAlive = false}。 \\ +\path{witnessPlan_reaches_goal} & 执行轨迹 & +给出一条具体计划，证明从初始状态出发可以完成“取钥匙 $\rightarrow$ 拿剑 $\rightarrow$ 击杀守卫 $\rightarrow$ 开最终宝箱”的任务链。 \\ +\path{task4_completable} & 可达性 & +由上述执行轨迹推出：在该形式化模型下，\texttt{task\_4} 是可完成的。 \\ +\bottomrule +\end{longtable} + +\textbf{witness 计划}：17 步房间级路径 +\[ +\begin{aligned} +&\text{west} \xrightarrow{\text{goCenter, goNorth, openChest}} \text{north} + \xrightarrow{\text{goCenter, goWest, toggleBridge}} \text{west} \\ +&\xrightarrow{\text{goCenter, goEast, openChest}} \text{east} + \xrightarrow{\text{goCenter, goWest, toggleBridge}} \text{west} \\ +&\xrightarrow{\text{goCenter, goSouth, attack}} \text{south} + \xrightarrow{\text{goCenter, openFinalChest}} \text{goal}. +\end{aligned} +\] +\texttt{witnessPlan\_executes\_to\_finalState} 用 \texttt{rfl} 直接验证。 + +\subsection{Task 5：四房间按钮门与全宝箱收集形式化} + +\subsubsection{环境建模} + +Task 5 包含 4 个房间（\texttt{startRoom / southRoom / eastRoom / westRoom}）， +关键机制为： +\begin{itemize}[leftmargin=2em] + \item \textbf{按钮门}：\texttt{startRoom} 的南出口为条件门（\texttt{button\_pressed}）， + 初始 \texttt{buttonPressed = false}； + \item \textbf{钥匙门}：\texttt{startRoom} 的东出口为锁门（\texttt{consume\_key = true}）， + 初始 \texttt{keys = 0}； + \item \textbf{差异化宝箱}：4 个房间各有 1 个宝箱，效果不同—— + \texttt{startRoom}（金币，无副作用）、\texttt{southRoom}（钥匙，\texttt{keys + 1}）、 + \texttt{eastRoom}（回血，\texttt{hp + 1}）、\texttt{westRoom}（金币，无副作用）； + \item \textbf{世界完成}：与 \texttt{task\_1--4} 不同，task\_5 无 \texttt{complete\_task} 出口， + 世界完成条件为 \texttt{all\_chests\_opened}（对齐 \texttt{progress.py:36}）。 +\end{itemize} + +\texttt{GoalReached} 定义为 \texttt{allChestsOpened}，即四个宝箱全部打开。 +初始状态：\texttt{room = startRoom}，\texttt{buttonPressed = false}，\texttt{keys = 0}，\texttt{hp = 5}， +四个 \texttt{ChestOpened} 均为 \texttt{false}。 + +\subsubsection{状态空间与转移函数} + +\begin{itemize}[leftmargin=2em] + \item \texttt{pressButton}：仅在 \texttt{startRoom} 生效，\texttt{buttonPressed := true}； + \item \texttt{goSouth}：需 \texttt{room = startRoom} \(\wedge\) + \texttt{buttonPressed = true}，进入 \texttt{southRoom}； + \item \texttt{goNorth}：需 \texttt{room = southRoom}，返回 \texttt{startRoom}； + \item \texttt{goEast}：从 \texttt{startRoom}（\texttt{keys > 0}）进入 \texttt{eastRoom} 并消耗钥匙； + 从 \texttt{westRoom} 返回 \texttt{startRoom}（不消耗钥匙）； + \item \texttt{goWest}：从 \texttt{startRoom} 进入 \texttt{westRoom}；从 \texttt{eastRoom} 返回 \texttt{startRoom}； + \item \texttt{openChest}：按房间分支，每房间最多开一次，效果见上。 +\end{itemize} + +\subsubsection{核心定理} + +\begin{longtable}{p{6.5cm}p{5.5cm}} +\caption{Task 5 核心定理一览}\label{tab:task5-theorems} \\ +\toprule +\textbf{定理} & \textbf{作用} \\ +\midrule +\endfirsthead +\toprule +\textbf{定理} & \textbf{作用（续）} \\ +\midrule +\endhead +\path{pressButton_only_in_startRoom} & 按钮仅起点房生效 \\ +\path{goSouth_blocked_without_button} & 条件门前置 \\ +\path{goEast_blocked_without_key} & 锁门前置 \\ +\path{goEast_from_start_consumes_key} & 东出口消耗钥匙 \\ +\path{openChest_south_grants_key} & 南房宝箱给钥匙 \\ +\path{openChest_east_heals} & 东房宝箱回血 \\ +\path{only_pressButton_changes_buttonPressed} & 必要性：按钮 \\ +\path{non_goEast_non_openChest_preserves_keys} & 必要性：钥匙 \\ +\path{buttonPressed_false_implies_not_in_south_and_chest_closed} & 归纳不变式 \\ +\path{southChestOpened_implies_buttonPressed} & 南宝箱开 $\Rightarrow$ 按钮已按 \\ +\path{southChestClosed_implies_keys_zero} & 钥匙来源不变式 \\ +\path{completion_postcondition} & 完成态 $\Rightarrow$ buttonPressed = true \\ +\path{task5_completable} & 可完成性 \\ +\path{policyTrace_10_matches_witnessPlan} & 策略轨迹与 witness 计划一致 \\ +\path{policyTrace_10_actions_enabled} & 策略每一步动作均满足前置条件 \\ +\path{symbolicPolicy_run_10_finalState} & 策略执行 10 步到达终态 \\ +\path{task5_symbolicPolicy_completes} & 阶段机策略完成全宝箱目标 \\ +\path{verified_bfs_routes_valid} & 10 类公开布局 BFS 路线证书全部有效 \\ +\path{verified_combat_routes_valid} & 起始房与西房战斗契约全部有效 \\ +\path{verified_west_entry_valid} & 西门两次跨越与 6 tick 举盾契约有效 \\ +\path{baselineTrace_24_actions_enabled} & 24 个 baseline 宏动作逐步满足前置条件 \\ +\path{start_monster_cleared_before_east_entry} & 进入东门前已清除起始追击怪 \\ +\path{spatial_c_west_entry_is_shielded} & spatial\_c 西门跨越时盾牌处于激活状态 \\ +\path{west_monsters_cleared_before_final_chest} & 开最终宝箱前西房两只怪物均已清除 \\ +\path{task5_certified_baseline_completes} & 证书化最终 baseline 在 24 步到达详细目标 \\ +\bottomrule +\end{longtable} + +\subsubsection{归纳不变式} + +\path{buttonPressed_false_implies_not_in_south_and_chest_closed} +是核心不变式：若 \texttt{buttonPressed = false}，则玩家不在 \texttt{southRoom} 且南宝箱未开。 +其逆否命题 \path{southChestOpened_implies_buttonPressed} 说明南宝箱开启蕴含按钮已按。 + +\path{southChestClosed_implies_keys_zero} +利用归纳法证明：南宝箱是唯一的钥匙来源，而东出口（唯一消耗钥匙的动作）需要 \texttt{keys > 0}， +因此在南宝箱未开之前不可能存在任何钥匙。 + +\subsubsection{Witness 计划} + +10 步计划： +\[ +\begin{aligned} +&\text{start}: \xrightarrow{\text{openChest}} \text{start} + \xrightarrow{\text{pressButton}} \text{start} + \xrightarrow{\text{goSouth}} \text{south} \\ +&\xrightarrow{\text{openChest (keys+1)}} \text{south} + \xrightarrow{\text{goNorth}} \text{start} + \xrightarrow{\text{goEast (keys-1)}} \text{east} \\ +&\xrightarrow{\text{openChest (hp+1)}} \text{east} + \xrightarrow{\text{goWest}} \text{start} + \xrightarrow{\text{goWest}} \text{west} + \xrightarrow{\text{openChest}} \text{goal}. +\end{aligned} +\] + +终态：\texttt{room = westRoom}，\texttt{buttonPressed = true}，\texttt{keys = 0}，\texttt{hp = 6}， +四个 \texttt{ChestOpened} 均为 \texttt{true}。 +\texttt{witnessPlan\_executes\_to\_finalState} 用 \texttt{rfl} 直接验证。 + +\subsubsection{策略形式化与正确性证明} + +为覆盖评分项中的“策略形式化与证明”，五个 Task 的 Lean 文件 +在环境模型之后均进一步定义了 \texttt{symbolicPolicy}， +将 Python \texttt{baseline\_policy.py} 中的规则逻辑提升为符号策略。 +其中 \texttt{task\_1} 至 \texttt{task\_4} 采用 \textbf{BFS 证书模式} +（策略参数化于 BFS 动作提供器，通过证书结构证明非固定路线的完备性）。 +\texttt{task\_5} 保留原 \texttt{PolicyStage} 房间级证明作为抽象规范， +并新增与最终 Python baseline 对齐的 BFS、战斗和西门入场证书细化层。 +各 Task 的策略模式汇总如下： + +\begin{table}[H] +\centering +\footnotesize +\begin{tabular}{@{}ll>{\raggedright\arraybackslash}p{0.42\textwidth}l@{}} +\toprule +Task & 抽象层级 & 阶段机 / BFS 目标 & 策略模式 \\ +\midrule +Task 1 & tile & \texttt{openChest} $\to$ \texttt{goNorth} $\to$ done & BFS 证书 \\ +Task 2 & zone & \texttt{nearMonster} $\to$ \texttt{nearChest} $\to$ \texttt{westExit} & BFS 证书 \\ +Task 3 & room & \texttt{keyRoom} $\to$ \texttt{startRoom} & BFS 证书 \\ +Task 4 & room & \texttt{north} $\to$ \texttt{east} $\to$ \texttt{south} $\to$ \texttt{center} & BFS 证书 \\ +Task 5 & tile + room & 起点箱/按钮 $\to$ 南房钥匙 $\to$ 起始战斗 $\to$ 东房治疗 $\to$ 举盾进西房 $\to$ 双怪战斗/最终箱 & BFS + 战斗 + 入场证书 \\ +\bottomrule +\end{tabular} +\caption{五个任务的符号策略模式汇总} +\end{table} + +\paragraph{BFS 证书模式（Task 1--4）} +Task 1--4 的策略形式化采用统一的 BFS 证书模式，核心理念是 +\textbf{策略逻辑与具体路线解耦}：\texttt{symbolicPolicy} 参数化于一个 +\texttt{BfsAction} 类型别名（动作提供器），Lean 侧只验证阶段逻辑与 +BFS 契约的正确性，而不硬编码具体路径。该模式由以下五部分构成： + +\begin{enumerate}[leftmargin=2em] + \item \textbf{\texttt{BfsAction} 类型别名}：抽象 BFS 动作提供器， + 签名因任务抽象层级而异—— + Task 1 为 \texttt{EnvState $\to$ List Tile $\to$ Action}（tile 目标集）， + Task 2 为 \texttt{EnvState $\to$ Zone $\to$ Action}（zone 目标）， + Task 3/4 为 \texttt{EnvState $\to$ Room $\to$ Action}（room 目标）。 + \item \textbf{\texttt{symbolicPolicy} 参数化定义}：读取当前 \texttt{EnvState}， + 按阶段输出下一个高层动作；若已处于目标位置则发出交互动作（\texttt{openChest}/\texttt{attack}/\texttt{useExit}/\texttt{openFinalChest}/\texttt{up}）， + 否则委托 \texttt{bfsAction} 寻路。该结构与 Python \texttt{decide\_taskN} + 的阶段判断完全对应。 + \item \textbf{BFS 证书 \texttt{structure}}：每段 BFS 路线由一个证书结构约束， + 字段包括 \texttt{plan}（动作列表）、\texttt{enabled}（轨迹上每步动作合法）、 + \texttt{endsAt$<$Target$>$}（终点到达目标）、以及一组状态保留字段 + （\texttt{keysPreserved}/\texttt{chestOpenedPreserved}/\texttt{monsterAlivePreserved} 等）， + 确保 BFS 移动不破坏已完成阶段的关键状态。 + \item \textbf{主定理}：将多段 BFS 证书与显式交互动作链接， + 证明“若各段 BFS 证书成立，则从初态出发依策略执行可到达目标”。 + 证明中用 \texttt{let} 引入中间状态别名，以 + \texttt{simpa [GoalReached, ...] using hDone} 关闭目标 + （项目使用纯 Lean 4 无 Mathlib，\texttt{set} 战术不可用）。 + \item \textbf{\texttt{native\_decide} 回归实例}：在公开地图上用具体 \texttt{plan} + 实例化各证书字段，通过 \texttt{native\_decide} 计算验证， + 作为对非固定路线主定理的回归检查。 +\end{enumerate} + +\textbf{Python 策略与 Lean \texttt{symbolicPolicy} 对照}： +下表给出四个 BFS 证书任务中 Python \texttt{baseline\_policy.py} 的 \texttt{decide\_taskN} +与 Lean \texttt{symbolicPolicy} 的阶段对应关系，二者在阶段判定与最终交互动作上完全一致。 + +\begin{table}[H] +\centering +\footnotesize +\begin{tabular}{@{}l>{\raggedright\arraybackslash}p{0.40\textwidth}>{\raggedright\arraybackslash}p{0.40\textwidth}l@{}} +\toprule +Task & Python \texttt{decide\_taskN} & Lean \texttt{symbolicPolicy} & 对应 \\ +\midrule +Task 1 & keys$\leq$0 $\to$ BFS 到宝箱邻居 $\to$ \texttt{ACTION\_A}；keys$>$0 $\to$ BFS 到北门 $\to$ \texttt{UP} & keys$=$0 $\to$ BFS 到 chest goals $\to$ \texttt{openChest}；else $\to$ BFS 到 north exit $\to$ \texttt{up} & \checkmark \\ +Task 2 & 怪物可见 $\to$ \texttt{attack}；keys$\leq$0 $\to$ BFS 到宝箱 $\to$ \texttt{ACTION\_A}；keys$>$0 $\to$ BFS 到西出口 & \texttt{monsterAlive} $\to$ \texttt{attack}；\texttt{chestOpened$=$false} $\to$ \texttt{openChest}；else $\to$ \texttt{useExit} & \checkmark \\ +Task 3 & keys$\leq$0 + key\_room $\to$ \texttt{ACTION\_A}；keys$\leq$0 + not key\_room $\to$ west；keys$>$0 $\to$ east & \texttt{keyChestOpened$=$false} $\to$ BFS 到 keyRoom $\to$ \texttt{openChest}；else $\to$ BFS 到 startRoom $\to$ \texttt{openLockedExit} & \checkmark \\ +Task 4 & get\_key $\to$ north；get\_sword $\to$ east；kill\_guardian $\to$ south；open\_final\_chest $\to$ center & \texttt{northChestOpened$=$false} $\to$ north；\texttt{eastChestOpened$=$false} $\to$ east；\texttt{guardianAlive} $\to$ south；else $\to$ center & \checkmark \\ +\bottomrule +\end{tabular} +\caption{Task 1--4 的 Python 策略与 Lean \texttt{symbolicPolicy} 对照} +\end{table} + +\textbf{BFS 证书结构与主定理}：四个任务的证书结构与主定理命名如下。 +每段证书均要求 \texttt{enabled}（轨迹合法）与到达目标， +并保留与后续阶段相关的关键状态字段。 + +{\footnotesize +\begin{longtable}{p{0.08\textwidth}p{0.55\textwidth}p{0.27\textwidth}} +\toprule +Task & BFS 证书结构与主定理 & 显式交互动作 \\ +\midrule +\endhead +Task 1 & \path{ChestBfsCertificate}, \path{ExitBfsCertificate}； +主定理：\path{task1_bfs_certificate_completes} & \path{openChest}, \path{up} \\ +Task 2 & \path{MonsterApproachCertificate}, \path{ChestApproachCertificate}, +\path{ExitApproachCertificate}；主定理：\path{task2_bfs_certificate_completes} +& \path{attack}$\times$2, \path{openChest}, \path{useExit} \\ +Task 3 & \path{KeyRoomApproachCertificate}, \path{StartRoomApproachCertificate}； +主定理：\path{task3_bfs_certificate_completes} & \path{openChest}, \path{openLockedExit} \\ +Task 4 & \path{ApproachCertificate}（参数化于目标 Room，含 North/East/South/Center +四个别名）；主定理：\path{task4_bfs_certificate_completes} +& \path{openChest}$\times$2, \path{attack}, \path{openFinalChest} \\ +\bottomrule +\end{longtable} +} + +以 Task 2 为例，主定理 \texttt{task2\_bfs\_certificate\_completes} 的证明骨架为： +\[ +\begin{aligned} +&\text{monsterRoute}:\ \texttt{MonsterApproachCertificate initialState}\\ +&\quad \xrightarrow{\texttt{attack}\times 2}\ \text{monsterHp } 2\to 1\to 0,\ \text{monster defeated}\\ +&\text{chestRoute}:\ \texttt{ChestApproachCertificate}\ s_{A2}\\ +&\quad \xrightarrow{\texttt{openChest}}\ \text{keys}+1,\ \text{chestOpened}:=\text{true}\\ +&\text{exitRoute}:\ \texttt{ExitApproachCertificate}\ s_{O}\\ +&\quad \xrightarrow{\texttt{useExit}}\ \text{completed}:=\text{true} +\end{aligned} +\] +其中状态别名通过 \texttt{let sM / sA1 / sA2 / sC / sO / sE} 引入， +状态保留字段跨证书链式传递。例如 \path{sC.chestOpened} 依次通过 +\path{chestRoute.chestOpenedPreserved}、两次 +\path{attack_preserves_chestOpened} 和 +\path{monsterRoute.chestOpenedPreserved}，回溯到初态的宝箱关闭条件。 + +\textbf{Task 4 的参数化证书}：Task 4 将四段 BFS 路线统一为单个 +\path{ApproachCertificate} 结构，并以目标房间和起始状态参数化； +再通过 \texttt{abbrev} 派生 North、East、South、Center 四个方向实例， +避免重复定义。主定理链接四段证书与四个交互动作 +（\texttt{openChest} 取钥匙、\texttt{openChest} 取剑、\texttt{attack} 击杀守卫、 +\texttt{openFinalChest} 完成任务），并保留 \texttt{guardianAlive}、 +\texttt{hasSword}、\texttt{finalChestVisible} 等跨阶段状态。 + +\paragraph{最终 baseline 对齐的证书细化（Task 5）} +Task 5 首先保留原有 10 步 \texttt{PolicyStage}/\texttt{policyTrace} 房间级证明， +用于描述“按按钮、取钥匙、依次开四箱”的逻辑规范。在此之上， +新增 \path{BaselineState} 与 24 个 \path{BaselinePhase}，显式记录 +\path{startMonsterAlive}、\path{westMonstersRemaining}、 +\path{shieldActive} 和 \path{safe}，从而对齐 Python 最终 baseline 中 +“先清起始追击怪、跳过东房 ambusher、举盾跨西门、清除西房双怪”的真实阶段顺序。 + +细化层将执行依赖拆为三类可检查证书： +\begin{itemize}[leftmargin=2em] + \item \texttt{BfsRouteCertificate}：记录起点、目标集、障碍集和 tile 路径， + \texttt{validBfsRoute} 检查边界、首尾、四邻接连通和路径避障； + \item \texttt{CombatCertificate}：检查接近步数、攻击次数、目标已击败、 + 避开交互 tile 以及阻塞非目标出口； + \item \texttt{WestEntryCertificate}：检查两次跨门动作、6 tick 盾牌窗口和 + ambusher 接触被盾牌格挡，对应 spatial\_c 的关键修复。 +\end{itemize} + +\texttt{baselinePolicy} 根据详细阶段选择 \texttt{followBfs}/\texttt{attack}/ +\texttt{raiseShield}/\texttt{crossWest} 等宏动作；动作可执行性由 +\path{baselineActionEnabledBool} 计算，整条轨迹可由 \texttt{native\_decide} +直接回归验证。核心结论为： +\[ +\begin{aligned} +&\texttt{baselineTrace\_24\_matches\_verified\_plan};\\ +&\texttt{baselineTrace\_24\_actions\_enabled};\\ +&\texttt{spatial\_c\_west\_entry\_is\_shielded};\\ +&\texttt{baselinePolicy\_run\_24\_finalState};\\ +&\texttt{task5\_certified\_baseline\_completes}. +\end{aligned} +\] +因此 Task 5 不再仅证明“存在一个房间级 witness”，还证明与最终 baseline +阶段顺序一致的证书化策略会执行合法动作、保持安全标记，并在有限步内完成目标。 + +\subsection{统一策略框架与有界完备性证明} + +为了进一步覆盖评分细则中“采用统一策略覆盖多关卡、展现结构复用”和 +“证明基本要求之外、更具难度或价值的定理”的加分方向， +本文新增 \path{student_agent/lean/StrategyFramework.lean}。 +该文件不依赖具体任务的房间、钥匙或怪物定义，而是对任意确定性符号转移系统 +“\texttt{State} 上的状态、\texttt{Action} 上的动作、 +以及 \texttt{step : State -> Action -> State}” +抽取通用证明层： + +\begin{itemize}[leftmargin=2em] + \item \texttt{runPlan}：执行显式动作列表； + \item \texttt{policyTrace} 与 \texttt{runPolicy}：从状态驱动策略生成有限动作轨迹并执行； + \item \texttt{actionsEnabledAlong} 与 \texttt{SafeTrace}：把各任务的动作前置条件 + 抽象成 action mask 安全证书； + \item \texttt{BoundedReachable} 与 \texttt{BoundedGoalReachable}： + 刻画“存在长度不超过 \(n\) 的计划可达某状态/目标”； + \item \texttt{expandFrontier}、\texttt{bfsVisited} 与 \texttt{bfsVisitedFrom}： + 形式化深度受限 BFS 的逐层扩展和已访问状态集合。 +\end{itemize} + +核心完备性定理为 \path{bfsVisited_complete_for_bounded_goal}，其结论可写为： +\begin{quote} +\small +\texttt{BoundedGoalReachable step goal init n}\\ +\(\Rightarrow\)\\ +\texttt{BfsFindsGoal goal}\\ +\texttt{(bfsVisitedFrom step actions init n)} +\end{quote} +定理前提还要求 \texttt{actions} 是完整动作集，即 +\(\forall a,\ \texttt{a} \in \texttt{actions}\)。 +证明思路是先证明 \path{bfsVisited_frontier_invariant}： +任意长度不超过 \(n\) 的计划执行结果都属于深度 \(n\) 的 BFS 访问集； +再由 \texttt{BoundedGoalReachable} 取出目标态，推出 BFS 访问集中存在目标态。 +这正对应“若存在可行路径，则完整的有界符号搜索必能找到”的完备性命题。 + +此外，\path{policy_completion_is_bounded_goal} 证明任意完成目标的 +状态驱动策略都可以转化为 \texttt{BoundedGoalReachable} 见证； +\path{bfs_finds_goal_reached_by_policy} 进一步说明： +如果某个 \texttt{symbolicPolicy} 在 \(n\) 步内完成目标，那么同一符号层上的 +完整动作 BFS 也能在 \(n\) 步内访问到目标态。 +因此，五个 Task 文件中已经证明的 \texttt{symbolicPolicy} 完成性 +可以看作这一统一框架的具体实例，而 \texttt{StrategyFramework.lean} +则提供跨任务复用的搜索完备性证明骨架。 + +\subsection{证明统一思路} + +五个任务遵循同一套证明骨架，便于复用与审阅： + +\begin{enumerate}[leftmargin=2em] + \item \textbf{局部前置/后置引理}：对每个动作给出“何时触发”和“触发后字段如何变化”。 + 这一步对应评分细则中的“状态 / 动作 / 转移是否定义完整、是否能证明基本合法性与安全性”。 + 例如 Task 4 的 \texttt{goEast\_blocked\_without\_key} 直接对应 Python + \texttt{movement.py} 对 \texttt{locked\_key} 出口的检查； + \texttt{attack\_with\_sword\_reveals\_final\_chest} 对应 + \texttt{combat.py} 中“清怪后揭示宝箱”的事件语义； + \texttt{rotateBridge\_three\_cycle} 对应 \texttt{interactions.py} + 中 switch 对 \texttt{center\_bridge} 的循环切换。 + \item \textbf{必要性引理}（\texttt{only\_X\_sets\_completed}）： + 证明只有某个特定动作能把完成标志从假翻为真。 + 通过 \texttt{non\_X\_preserves\_completed} 这一“其它动作保持完成字段不变”的引理来支撑。 + \item \textbf{\texttt{Reachable} 归纳不变式}：定义归纳谓词 + \texttt{Reachable}，对初始状态和任意 \texttt{step} 闭合； + 证明诸如“未开箱则钥匙数为 0”“终点宝箱显现则守卫已死”等性质在所有可达状态上成立。 + \item \textbf{\texttt{completion\_postcondition}}：在 \texttt{Reachable} 归纳之上， + 证明“任意可达完成态必然满足某组前置条件”（如 Task 2 中“怪物已死 $\wedge$ 宝箱已开”、 + Task 4 中“守卫已死”）。证明中显式调用必要性引理与归纳不变式。 + \item \textbf{witness 计划}：在局部规则已经明确的基础上，构造一条显式的见证计划 + \texttt{witnessPlan}。由于 \texttt{step} 被定义为确定性函数， + 因此整条轨迹的正确性可以通过归约直接验证 + （简单计划用 \texttt{rfl}，长计划用 \texttt{native\_decide}）。 + \item \textbf{可完成性}：由 \texttt{witnessPlan\_reaches\_goal} 直接推出 + \texttt{taskN\_completable}，即“存在计划使目标谓词成立”。 +\end{enumerate} + +这使得“任务可完成”从口头分析转化为可检查的形式化结论， +并且完成态的必要前置条件也被显式记录下来。 + +\subsection{证明局限与边界} + +当前环境形式化刻画的是 \texttt{task\_1} 至 \texttt{task\_5} 的 +\textbf{tile / zone / room 级符号语义}， +而不是 Python 引擎的逐像素精确副本，因此存在以下边界： +\begin{itemize}[leftmargin=2em] + \item 未形式化逐像素移动和碰撞盒连续动力学。 + \texttt{task\_1} 保留 tile 级网格，\texttt{task\_2--4} 采用 zone / room 抽象； + \texttt{task\_5} 新增 tile 路线证书，但仍将一整段 Python BFS 跟随视为宏动作。 + \item 未逐帧形式化怪物 AI、击退和受伤时序；Task 5 已将战斗结果、 + 非目标出口阻塞和 6 tick shield 格挡抽象为可检查契约，但未证明引擎连续实现满足契约。 + \item 未证明视觉模块一定能从 \texttt{obs} 中稳定恢复这些符号状态； + 该部分属于感知层，而不是当前 Lean 环境模型的覆盖范围。 + \item 已证明抽象层面的有界 BFS 完备性： + 若某目标在 \(n\) 步内由某个计划可达，则完整动作集上的深度 \(n\) BFS + 必会访问到目标态；但尚未逐行证明 Python 具体队列、visited 集合和 parent map + 实现与该 Lean 抽象 BFS 完全等价。 +\end{itemize} + +这一边界是有意为之：课程要求强调 Lean 证明应覆盖“可验证层”。 +对于本项目而言，最具价值且最容易和实际代码保持一致的对象， +正是钥匙/门条件、战斗与事件触发、桥状态机与任务完成谓词这类符号层环境语义。 + +\section{实验设计与结果分析} + +\subsection{实验设置} + +实验使用仓库自带评测脚本，采用与最终测评一致的 \texttt{safe} 信息模式 +（\texttt{info} 仅包含 \texttt{last\_reward}、\texttt{inventory} 和 +\texttt{task\_id}，不暴露 \texttt{reward\_signals} 等隐藏事件）， +并通过 \texttt{--task-policy} 为每个任务绑定同一策略文件以获取 \texttt{task\_id}。 +为验证策略在布局扰动与颜色扰动下的鲁棒性，本次评测启用 +\texttt{--robustness-suite} 模式，每个任务的 \texttt{--num-envs 100} 个 episode +按 60\% original / 30\% spatial（a/b/c 各 1）/ 10\% color（grayscale/dark/bright/high\_contrast/inverted 五种轮换）比例分配。 +robustness suite 命令如下（\texttt{--robustness-suite} 模式下 color 部分自动轮换五种非默认变体， +\texttt{--obs-variants} 参数在此模式下被忽略）： +\begin{quote} +\small +\texttt{python utils/evaluate\_policy.py}\\ +\texttt{--tasks mathematical\_logic/task\_1 ... task\_5}\\ +\texttt{--task-policy =student\_agent/baseline\_policy.py}\\ +\texttt{--robustness-suite --num-envs 100 --info-mode safe} +\end{quote} +运行环境为 Python 3.12.10 虚拟环境。本次报告记录的是 2026 年 7 月 17 日 +在本地工作区运行的最终 baseline 结果。 +\begin{itemize}[leftmargin=2em] + \item 测试任务：\texttt{mathematical\_logic/task\_1} 至 + \texttt{mathematical\_logic/task\_5}（全部 5 个任务）。 + \item 运行次数：robustness suite 每任务 100 个 episode + （60 original + 30 spatial + 10 color），共 500 个； + color 部分覆盖 grayscale/dark/bright/high\_contrast/inverted 五种变体（各 2 seed）， + 加上 original/spatial 的 default，六种观察变体全覆盖。 + \item 成功率统计方式：评测脚本输出的 \texttt{success\_rate}，按 eval\_stage 分组。 + \item 策略版本：\texttt{student\_agent/baseline\_policy.py}（已适配 \texttt{safe} 模式， + 含 spatial\_c 举盾入场修复、grayscale 调色板映射和 high\_contrast 形状识别）。 + \item Lean 检查：逐文件运行 \texttt{lean student\_agent/lean/*.lean} + 对 7 个 Lean 文件进行编译检查，toolchain 版本为 \texttt{leanprover/lean4:v4.26.0}。 +\end{itemize} + +\subsection{实验结果} + +\begin{table}[H] +\centering +\begin{tabular}{p{0.08\textwidth}p{0.10\textwidth}p{0.08\textwidth}p{0.10\textwidth}p{0.12\textwidth}p{0.12\textwidth}p{0.14\textwidth}} +\toprule +任务 & 阶段 & 局数 & 成功率 & 平均步数 & 平均奖励 & 备注 \\ +\midrule +Task 1 & original & 60 & 1.000 & 270.0 & 127.250 & 全部通过 \\ +Task 1 & spatial & 30 & 1.000 & 175.0 & 128.183 & a/b/c 全部通过 \\ +Task 1 & color & 10 & 1.000 & 270.0 & 127.250 & 五种变体通过 \\ +\midrule +Task 2 & original & 60 & 1.000 & 150.0 & 126.450 & 全部通过 \\ +Task 2 & spatial & 30 & 1.000 & 169.7 & 127.587 & a/b/c 全部通过 \\ +Task 2 & color & 10 & 1.000 & 150.0 & 126.850 & 五种变体通过 \\ +\midrule +Task 3 & original & 60 & 1.000 & 535.0 & 159.650 & 全部通过 \\ +Task 3 & spatial & 30 & 1.000 & 597.0 & 158.997 & a/b/c 全部通过 \\ +Task 3 & color & 10 & 1.000 & 535.0 & 159.650 & 五种变体通过 \\ +\midrule +Task 4 & original & 60 & 1.000 & 1007.0 & 253.930 & 全部通过 \\ +Task 4 & spatial & 30 & 1.000 & 1221.0 & 249.373 & a/b/c 全部通过 \\ +Task 4 & color & 10 & 1.000 & 1009.4 & 253.506 & 五种变体通过 \\ +\midrule +Task 5 & original & 60 & 1.000 & 1084.0 & 234.860 & 全部通过，见 \ref{sec:task5-status} 节 \\ +Task 5 & spatial & 30 & 1.000 & 1125.7 & 178.110 & spatial\_a/b/c 全部通过 \\ +Task 5 & color & 10 & 1.000 & 1072.8 & 222.762 & 五种变体通过 \\ +\bottomrule +\end{tabular} +\caption{Robustness suite 评测结果（\texttt{--num-envs 100}，\texttt{safe} 信息模式）。 +spatial 列的 a/b/c 指布局变体 spatial\_a/spatial\_b/spatial\_c；color 列局数为五种非默认观察变体（grayscale/dark/bright/high\_contrast/inverted）合计。} +\end{table} + +\textbf{Milestone 指标}（按 readme §九要求）。milestone 为 0/1 标志， +下表为每个任务在 original/spatial/color 三阶段的平均达成率（1.00 表示全部 episode 达成）。 +\texttt{task\_1}、\texttt{task\_2} 评测脚本未定义 milestone，故未列出。 + +\begin{table}[H] +\centering +\begin{tabular}{p{0.10\textwidth}p{0.10\textwidth}p{0.16\textwidth}p{0.16\textwidth}p{0.16\textwidth}} +\toprule +任务 & 阶段 & \phantom{a} & milestone 达成率 & \phantom{a} \\ +\midrule +Task 3 & original/spatial/color & \multicolumn{3}{l}{\texttt{key\_collected}=1.00，\texttt{monster\_killed}=0.00（task 3 无需杀怪）} \\ +Task 4 & original/spatial/color & \texttt{switch\_activated}=1.00 & \texttt{key\_collected}=1.00 & \texttt{door\_opened}=1.00 \\ + & & \texttt{item\_collected}=1.00 & \texttt{monster\_killed}=1.00 & \\ +Task 5 & original/spatial/color & \texttt{chest\_opened}=1.00 & \texttt{key\_collected}=1.00 & \texttt{gold\_collected}=1.00 \\ + & & \texttt{agent\_healed}=1.00 & \texttt{button\_pressed}=1.00 & \texttt{room\_changed}=1.00 \\ + & & \texttt{door\_opened}=1.00 & \texttt{monster\_killed}=1.00 & \texttt{exit\_reached}=1.00 \\ + & & \texttt{environment\_completed}=1.00 & \texttt{world\_completed}=1.00 & \texttt{item\_collected}=0.00 \\ + & & \texttt{trap\_triggered}=0.00 & & \\ +\bottomrule +\end{tabular} +\caption{Task 3/4/5 在三阶段的 milestone 达成率。同一任务在 original/spatial/color +三阶段的 milestone 值完全一致（baseline 为无随机成分的规则策略，每个 milestone 均稳定触发）。 +\texttt{task\_5} 的 \texttt{item\_collected}=0 与 \texttt{trap\_triggered}=0 符合预期： +task 5 的目标为开启全部宝箱并完成世界，不需要拾取道具或触发陷阱。} +\end{table} + +\textbf{Progress 指标}（按 readme §九要求）。下表列出关键事件在每个 episode 的平均触发次数， +反映 agent 在每个任务中的推进路径。三阶段数值一致，故仅列 original。 + +\begin{table}[H] +\centering +\begin{tabular}{p{0.10\textwidth}p{0.10\textwidth}p{0.65\textwidth}} +\toprule +任务 & 阶段 & progress 事件（per-ep avg） \\ +\midrule +Task 1 & original & \texttt{chest\_opened}=1, \texttt{key\_collected}=1, \texttt{room\_changed}=1, \texttt{door\_opened}=1, \texttt{exit\_reached}=1, \texttt{environment\_completed}=1, \texttt{world\_completed}=1 \\ +Task 2 & original & Task 1 全部 + \texttt{monster\_killed}=1 \\ +Task 3 & original & \texttt{chest\_opened}=1, \texttt{key\_collected}=1, \texttt{room\_changed}=5, \texttt{door\_opened}=1, \texttt{exit\_reached}=5, \texttt{environment\_completed}=1, \texttt{world\_completed}=1 \\ +Task 4 & original & \texttt{chest\_opened}=3, \texttt{key\_collected}=1, \texttt{gold\_collected}=1, \texttt{item\_collected}=1, \texttt{room\_changed}=11, \texttt{door\_opened}=1, \texttt{monster\_killed}=1, \texttt{exit\_reached}=11, \texttt{environment\_completed}=1, \texttt{world\_completed}=1 \\ +Task 5 & original & \texttt{chest\_opened}=4, \texttt{key\_collected}=1, \texttt{gold\_collected}=2, \texttt{agent\_healed}=1, \texttt{button\_pressed}=1, \texttt{room\_changed}=5, \texttt{door\_opened}=1, \texttt{monster\_killed}=3, \texttt{exit\_reached}=5, \texttt{environment\_completed}=1, \texttt{world\_completed}=1 \\ +\bottomrule +\end{tabular} +\caption{Progress 事件每个 episode 的平均触发次数（original 阶段）。 +spatial/color 阶段数值与 original 完全一致，故省略。 +事件计数说明：\texttt{chest\_opened} 与任务宝箱数对应（task\_1:1, task\_2:1, task\_3:1, task\_4:3, task\_5:4）； +\texttt{room\_changed}/\texttt{exit\_reached} 与任务房间数相关（task\_3 跨 6 房间、task\_4 跨 12 房间、task\_5 跨 6 房间）； +\texttt{monster\_killed} 与各任务需击杀怪物数对应（task\_2:1, task\_4:1, task\_5:3，task\_1/3 无怪物）。 +所有任务 \texttt{environment\_completed}=1 和 \texttt{world\_completed}=1，无 \texttt{agent\_dead} 事件。} +\end{table} + +\begin{table}[H] +\centering +\begin{tabular}{p{0.40\textwidth}p{0.20\textwidth}p{0.30\textwidth}} +\toprule +检查对象 & 结果 & 说明 \\ +\midrule +\path{KillMonsterFormalization.lean} & 通过 & 怪物击杀示例证明 \\ +\path{Task1Formalization.lean} & 通过 & Task 1 环境形式化与符号策略证明 \\ +\path{Task2Formalization.lean} & 通过 & Task 2 环境形式化与符号策略证明 \\ +\path{Task3Formalization.lean} & 通过 & Task 3 环境形式化与符号策略证明 \\ +\path{Task4Formalization.lean} & 通过 & Task 4 环境形式化与符号策略证明 \\ +\path{Task5Formalization.lean} & 通过 & Task 5 环境形式化与符号策略证明 \\ +\path{StrategyFramework.lean} & 通过 & 通用策略执行与有界 BFS 完备性证明 \\ +\bottomrule +\end{tabular} +\caption{Lean 文件逐文件检查结果} +\end{table} + +\subsection{结果分析} + +本次 robustness suite 评测在 \texttt{safe} 信息模式下运行 +（\texttt{info} 仅含 \texttt{last\_reward}、\texttt{inventory}、\texttt{task\_id}）， +共 500 个 episode。五个任务在 original、spatial 和 color 阶段均 +全部成功（success\_rate = 1.000）：300 个 original、150 个 spatial 和 50 个 +color episode 无一失败。这说明策略的高层阶段机、BFS 路径规划与基于 +\texttt{inventory} diff + \texttt{last\_reward} 的事件推断逻辑在 +\texttt{safe} 模式下能稳定完成目标，且对布局扰动具有鲁棒性。 +由于 baseline 是无随机成分的规则策略，同一公开地图的多个 seed 间步数与奖励一致。 + +spatial 变体测试覆盖 \texttt{spatial\_a}、\texttt{spatial\_b}、\texttt{spatial\_c} +三种布局扰动。修复前，\texttt{task\_1} 至 \texttt{task\_3} +的 spatial 变体大量失败（\texttt{task\_1} 1/3、\texttt{task\_2} 1/3、\texttt{task\_3} 0/3）， +根因是像素对齐阈值过松（\texttt{move\_towards\_aligned} 中 \texttt{+1} 容差 +导致 1-tile 间隙处 sprite rect 覆盖相邻 blocking tile）和 +\texttt{detect\_task3\_room} 硬编码 NPC/chest 位置导致 spatial 变体下 room 检测失败。 +修复后（精确对齐 \texttt{+0}、卡住检测、全网格扫描房间识别）， +三个任务的 spatial 变体全部通过。Task 5 进一步通过阻塞 NPC 和非目标出口、 +修正移动怪物跨 tile 时的目标选择，并在西门边界主动举盾，最终使 +\texttt{spatial\_c} 与 a/b 一并完成。 + +robustness suite 的 color 部分覆盖五种非默认观察变体（grayscale/dark/ +bright/high\_contrast/inverted 各 2 seed），加上 original/spatial 的 default， +五个任务在 default、grayscale、dark、bright、high\_contrast 和 inverted 六种 +观察上均为 100\%。dark/inverted 使用逆变换， +bright 使用前向参考色；grayscale 将每个参考色映射到灰度均值并采用精确匹配； +high\_contrast 则结合二值参考色与连通分量/局部形状识别，解决颜色碰撞。 +详细方法见 \ref{sec:color-variant-fix} 节。 + +\texttt{task\_5} 在 original 阶段全部通过（success\_rate = 1.000）， +五种非默认颜色变体全部通过，spatial\_a/b/c 也全部通过， +详细分析见 \ref{sec:task5-status} 节。 + +\subsubsection{Task 5 状态说明} +\label{sec:task5-status} + +\texttt{task\_5} 在 original 阶段全部通过 +（60/60，success\_rate = 1.000，avg\_steps = 1084，avg\_reward = 234.860）， +所有里程碑（\texttt{chest\_opened}、\texttt{world\_completed} 等）均为 1.000。 +agent 成功完成全部 4 个房间（start/south/east/west）的宝箱收集， +触发 \texttt{world\_completed} 事件。 +spatial 阶段 30/30 通过，五种非默认 color 变体 10/10 通过。 + +\paragraph{color 变体表现} + +5 种 color 变体单独评测结果如下： + +\begin{table}[H] +\centering +\begin{tabular}{p{0.18\textwidth}p{0.10\textwidth}p{0.10\textwidth}p{0.12\textwidth}p{0.30\textwidth}} +\toprule +变体 & Steps & Reward & Success & 说明 \\ +\midrule +dark & 1084 & 234.860 & True & \texttt{normalize\_obs} 逆变换恢复 \\ +bright & 1084 & 234.860 & True & 前向变换参考色匹配 \\ +inverted & 1084 & 234.860 & True & \texttt{normalize\_obs} 逆变换恢复 \\ +grayscale & 1084 & 234.860 & True & 灰度参考色映射 + 精确匹配 \\ +high\_contrast & 1028 & 174.370 & True & 二值颜色 + 连通分量/局部形状识别 \\ +\bottomrule +\end{tabular} +\caption{Task 5 color 变体单独评测结果（每变体 1 episode，\texttt{safe} 信息模式）} +\label{tab:task5-color} +\end{table} + +dark/bright/inverted/grayscale 与 original 的轨迹指标一致。 +high\_contrast 因二值化后的对象边界和怪物目标选择不同，使用 1028 步完成， +但四箱、三只怪物、治疗、按钮、门和世界完成里程碑仍全部达到 1.000。 + +\paragraph{spatial 变体表现} + +3 种 spatial 布局变体全部通过（30/30，success\_rate = 1.000）。 +2026-07-15 修复了 \texttt{detect\_task5\_room} +中 start 房间检测从硬编码对象位置改为墙体特征匹配 +（spatial 变体只移对象不移墙，墙体特征仍有效）， +以及移除 south 房间宝箱位置的硬编码 fallback。 +2026-07-16 进一步修复了 spatial\_a/b 的两个关键问题 +（NPC 交互拦截和 west 房间导航效率），使二者通关。 + +\begin{table}[H] +\centering +\begin{tabular}{p{0.14\textwidth}p{0.10\textwidth}p{0.12\textwidth}p{0.12\textwidth}p{0.36\textwidth}} +\toprule +变体 & Steps & Reward & Success & 说明 \\ +\midrule +spatial\_a & 1077 & 174.030 & True & diagonal\_block 消除 start 房间接触伤害 \\ +spatial\_b & 1173 & 186.820 & True & 安全攻击 tile + NPC/出口避障 \\ +spatial\_c & 1127 & 173.480 & True & 非目标出口避障 + 西门边界举盾 \\ +\bottomrule +\end{tabular} +\caption{Task 5 spatial 变体评测详情} +\label{tab:task5-spatial} +\end{table} + +\textbf{spatial\_a 修复}：start 房间 chaser 移至 $(7,3)$，agent 导航至 east/west 出口时 +chaser 追击造成 AABB 接触伤害。在 \texttt{attack\_monster} 中新增 \texttt{diagonal\_block} +参数（默认 \texttt{True}），将怪物周围对角 tile 加入 \texttt{blocked} 集合， +BFS 绕行避免 sprite rect 重叠。最终回归中 step 1077 通关。 + +\textbf{spatial\_b 修复}：west 房间 NPC 位于 $(7,5)$，与怪物 $(6,4)$ 对角相邻。 +引擎 \texttt{try\_interaction} 在战斗前检查 NPC 邻接（无 facing 检查）， +agent 站在 $(7,4)$ 攻击怪物时 A 键被 NPC 拦截（\texttt{talked\_npc} 而非 \texttt{monster\_hit}）。 +新增 \texttt{compute\_no\_attack\_tiles} 方法检测所有 NPC/chest 邻接 tile， +\texttt{attack\_monster} 接受 \texttt{no\_attack\_tiles} 参数， +当 agent 处于不安全 tile 时通过 BFS 寻找安全攻击位置。 +同时将 west 房间的 \texttt{diagonal\_block} 设为 \texttt{False} 加速导航 +（diagonal blocking 导致 60+ 步绕行）。最终回归中 step 1173 通关。 + +\textbf{spatial\_c 修复}：该变体会使路径更容易贴近其他房间出口，且西房 ambusher +位于入口附近。最终策略将 NPC 纳入全局阻塞集，在 \texttt{navigate\_to\_exit} +中阻塞所有非目标出口，并在战斗路径中阻塞全部出口，避免误跨房和门边空转。 +同时，\texttt{detect\_closest\_monster\_tile} 优先选择非玩家 tile，消除移动怪物 +跨 tile 时把自身碎片误当目标的问题。玩家到达 start 房西门边界后先执行一次 +\texttt{ACTION\_B}，以 6 tick shield 窗口覆盖入场接触，再跨入 west 房清怪。 +最终 spatial\_c 在 step 1127 完成，\texttt{trap\_triggered = 0}， +\texttt{environment\_completed = world\_completed = 1}。 + +\paragraph{HP 预算与优化关键} + +\texttt{task\_5} 隐藏了 HP 流失机制：每 200 步扣 1 HP（初始 HP = 5）， +full drain 序列为 step 200/400/600/800/1000/1200（共 $-$6 HP）， +east 房间 heal 箱提供 $+1$ HP，总 HP 预算 = 6， +恰好等于 6 次 drain。这意味着 agent 需要在 \textbf{零接触伤害}的前提下 +于 step 1200 前开完全部 4 个宝箱（\texttt{all\_chests\_opened} 触发 +\texttt{world\_completed}）。核心策略为三项改动的协同： + +\begin{enumerate} + \item \textbf{Start room——战斗 chaser 后再导航}：start 房间 chaser 与 agent + 同速（1px/step），若跳过战斗直接导航则转向时被追上造成 AABB 接触伤害。 + 改为在 \texttt{open\_east\_chest} / \texttt{open\_west\_chest} 阶段 + 先用 \texttt{attack\_monster} 击杀 chaser，再导航至出口。 + \item \textbf{East room——跳过 ambusher 直奔宝箱}：east 房间 ambusher 位于 + $(7,5)$，\texttt{ambush\_range} = 2；宝箱位于 $(7,1)$，agent 从 $(1,4)$ + 进入走 row 1 到宝箱，路径与 ambusher 的最近 Manhattan 距离 $\geq 4 > 2$， + ambusher 不会被激活。将 ambusher tile 加入 \texttt{blocked} 让 BFS 绕行， + 节省 $\sim$130 步战斗时间，无接触伤害。 + \item \textbf{West room——杀全部怪物后导航}：west 房间含 chaser $(2,4)$ 和 + ambusher $(6,3)$ 两只怪物。原 \texttt{detect\_monster\_tile} 返回所有怪物像素 + 质心，多怪物时指向虚假位置导致 \texttt{attack\_monster} 空转。 + 修复为使用 \texttt{detect\_closest\_monster\_tile} 逐个击杀， + 全部清除后再导航至宝箱，消除所有接触伤害。 +\end{enumerate} + +三项改动后 agent 以 0 接触伤害完成全程：step 1000（第 5 次 drain）时 HP = 1， +step 1084 开完最后一个 west 宝箱触发 \texttt{world\_completed}， +远早于 step 1200 的第 6 次 drain。 + +\paragraph{优化历程} + +task\_5 original 经历多轮优化才达到 100\% 通关，关键迭代如下： + +\begin{table}[H] +\centering +\begin{tabular}{p{0.28\textwidth}p{0.10\textwidth}p{0.12\textwidth}p{0.10\textwidth}p{0.22\textwidth}} +\toprule +版本 & Steps & Reward & Success & 死亡原因 \\ +\midrule +7-bug 修复版（基线） & 1083 & 95.02 & 0 & west 接触伤害 \\ +west opt only（杀 1 怪） & 1060 & 95.25 & 0 & west 接触伤害 \\ +kill all west & 1200 & 192.85 & 0 & 第 6 次 HP drain \\ +skip all west combat & 1072 & 95.13 & 0 & west 接触伤害 \\ +skip east + kill all west & 1070 & 235.05 & 1 & — \\ +\textbf{+ spatial\_a/b 修复} & \textbf{1083} & \textbf{234.87} & \textbf{1} & — \\ +\textbf{+ spatial\_c/颜色最终版} & \textbf{1084} & \textbf{234.86} & \textbf{1} & — \\ +\bottomrule +\end{tabular} +\caption{Task 5 original 优化历程（5 episode 评测）} +\label{tab:task5-optimization} +\end{table} + +\begin{itemize}[leftmargin=2em] + \item \textbf{7-bug 修复版}：修复导航执行层 7 个 bug 后的基线版本， + agent 开 3 箱后在 west 房间被剩余怪物接触致死。 + \item \textbf{west opt only}：west 房间杀 1 怪后直奔宝箱（剩余怪物加入 + \texttt{blocked}），但 pixel 模式下 AABB 接触伤害无法通过 tile blocked 避免， + 仍在 step 1060 死亡。 + \item \textbf{kill all west}：west 房间杀全部 2 只怪物消除接触伤害， + reward 翻倍至 192.85（4 怪物击杀），但战斗耗时 $\sim$600 步， + agent 在 step 1200 死于第 6 次 HP drain，差最后 1 个宝箱。 + \item \textbf{skip all west combat}：完全跳过 west 战斗直奔宝箱， + 节省 $\sim$130 步但 2 只追击怪物造成接触伤害，step 1072 死亡。 + \item \textbf{skip east + kill all west}：组合 east 跳过 + （ambusher 不激活，省 $\sim$130 步）+ west 全杀（零接触伤害）， + agent 在 step 1070 以 HP = 1 开完最后一个宝箱，\texttt{world\_completed} = 1.000。 + \item \textbf{+ spatial\_a/b 修复}（当前版）：为修复 spatial\_a/b 变体， + 新增 \texttt{compute\_no\_attack\_tiles} 和 \texttt{diagonal\_block} 参数 + （详见 \ref{tab:task5-spatial} 节）。original 步数从 1070 微增至 1083 + （+13 步，no\_attack\_tiles 检测导致部分攻击位置绕行）， + 但仍远在 HP 预算内，success\_rate = 1.000 无回归。 + \item \textbf{+ spatial\_c/颜色最终版}（当前版）：加入 NPC/非目标出口避障、 + 西门举盾与有损颜色识别；original 仅增加 1 步并保持完成， + 同时补齐 spatial\_c、grayscale 和 high\_contrast。 +\end{itemize} + +spatial 阶段最终 30/30 通过： +2026-07-15 修复了 \texttt{detect\_task5\_room} 的 start 房间检测 +（从硬编码对象位置改为墙体特征匹配）和 south 房间宝箱硬编码 fallback， +修复后房间识别正确，agent 可进入各房间并开箱。 +2026-07-16 修复了 spatial\_a（diagonal\_block 消除接触伤害）和 +spatial\_b（no\_attack\_tiles 避免 NPC 拦截 + diagonal\_block=False 加速导航）， +二者均通关。2026-07-17 又通过非目标出口/NPC 避障、移动怪物 tile 选择和 +西门边界主动举盾修复 spatial\_c（详见 \ref{tab:task5-spatial}）。 +color 阶段 5/5 通过：dark/bright/inverted 使用逆变换或前向参考色， +grayscale/high\_contrast 分别使用灰度参考色和二值形状识别。 + +\subsubsection{颜色变体鲁棒性修复} +\label{sec:color-variant-fix} + +针对评测脚本对整帧施加的 color 变体，策略先用 \texttt{normalize\_obs} +识别观察类型。2026-07-13 完成 default/dark/bright/inverted 适配； +2026-07-17 进一步为 grayscale 增加灰度参考色映射，为 high\_contrast 增加 +连通分量和对象局部形状规则。最终五个任务在六种观察上均通过。 + +\textbf{评测脚本的 5 种 color 变体}（\texttt{apply\_obs\_variant}）： + +\begin{table}[H] +\centering +\begin{tabular}{p{0.20\textwidth}p{0.20\textwidth}p{0.13\textwidth}p{0.32\textwidth}} +\toprule +变体 & 变换公式 & 可逆性 & 修复策略 \\ +\midrule +\texttt{dark} & $p \times 0.55$ & 可逆 & 逆变换 $p / 0.55$ 恢复原始帧 \\ +\texttt{bright} & $\min(255,\, p \times 1.35)$ & 部分可逆 & 前向变换参考色 $\min(255,\, c \times 1.35)$ 匹配 \\ +\texttt{inverted} & $255 - p$ & 可逆 & 逆变换 $255 - p$ 恢复原始帧 \\ +\texttt{grayscale} & $\text{mean}(R,G,B)$ & 不可逆 & 参考色映射到通道均值，零容差匹配 \\ +\texttt{high\_contrast} & $p > 127\,?\,255\,:\,0$ & 不可逆 & 二值参考色 + 连通分量/局部形状识别 \\ +\bottomrule +\end{tabular} +\caption{5 种 color 变体的变换公式与修复策略} +\end{table} + +\textbf{检测策略}：不依赖 \texttt{info} 中的变体标识，纯帧统计检测。 +采用\textbf{参考色存在性检测}而非绝对亮度阈值（避免不同地图天然亮度差异导致误判）。 +在 \texttt{act()} 入口依次检查 \texttt{PLAYER\_COLORS} 在各变体下的变换值是否存在于帧中： + +\begin{enumerate}[leftmargin=2em] + \item 检查 default：原始参考色 $(36, 198, 72)$ 存在 $\to$ 无需变换； + \item 检查 inverted：$(255-c) = (219, 57, 183)$ 存在 $\to$ 逆变换 $255 - \text{frame}$； + \item 检查 dark：$(\lfloor c \times 0.55 \rfloor) = (19, 108, 39)$ 存在 $\to$ 逆变换 $\text{frame} / 0.55$； + \item 检查 bright：$(\min(255, \lfloor c \times 1.35 \rfloor)) = (48, 255, 97)$ 存在 $\to$ 存 variant，前向变换参考色匹配； + \item 检查 grayscale：彩色像素占比 $< 0.5\%$ $\to$ + \texttt{\_effective\_color} 将每个 RGB 参考色映射到通道均值； + \item 检查 high\_contrast：像素值仅 $\{0, 255\}$ $\to$ + 参考色逐通道阈值化，并启用二值对象形状识别。 +\end{enumerate} + +\textbf{容差匹配}：default/dark/bright/inverted 使用 $\pm 2$ 容差 +（\texttt{COLOR\_TOL = 2}），覆盖逆变换后 \texttt{uint8} 截断误差； +grayscale/high\_contrast 使用零容差，避免灰度 106/108 等相邻调色板值被合并。 + +\textbf{high\_contrast 形状消歧}：二值化后多个原始颜色会映射到同一颜色， +仅做像素计数会把玩家、墙、出口、NPC 和怪物混淆。策略缓存当前帧的 8 邻接 +连通分量，用尺寸和位置恢复玩家/怪物；宝箱、NPC、开关、按钮、墙和陷阱则用 +tile 内颜色带、核心矩形和像素数量范围识别。Task 4 的中央房改为按深渊 tile +数量判断，双 tile 出口允许其中一格被玩家遮挡。 + +\textbf{bright 变体的特殊处理}：由于 $\times 1.35$ 后 $> 255$ 的通道被 clip， +逆变换 $\div 1.35$ 无法恢复原始值（如 $G = 198 \times 1.35 = 267 \to 255 \to 255 / 1.35 = 188$， +与原始 198 差 10）。因此 bright 不逆变换帧，而是将参考色前向变换后匹配 +（\texttt{\_effective\_color} 方法返回 $\min(255, \lfloor c \times 1.35 \rfloor)$）。 + +\textbf{修复效果}： + +\begin{table}[H] +\centering +\begin{tabular}{p{0.15\textwidth}p{0.12\textwidth}p{0.12\textwidth}p{0.12\textwidth}p{0.12\textwidth}p{0.12\textwidth}p{0.12\textwidth}} +\toprule +任务 & default & dark & bright & inverted & grayscale & high\_contrast \\ +\midrule +Task 1 & 100\% & 100\% & 100\% & 100\% & 100\% & 100\% \\ +Task 2 & 100\% & 100\% & 100\% & 100\% & 100\% & 100\% \\ +Task 3 & 100\% & 100\% & 100\% & 100\% & 100\% & 100\% \\ +Task 4 & 100\% & 100\% & 100\% & 100\% & 100\% & 100\% \\ +Task 5 & 100\% & 100\% & 100\% & 100\% & 100\% & 100\% \\ +\bottomrule +\end{tabular} +\caption{颜色变体修复后各任务成功率（每个“任务 $\times$ 变体”1 episode，\texttt{safe} 信息模式）} +\end{table} + +default 无回归，dark/bright/inverted、grayscale 和 high\_contrast +在五个任务上均达到 100\%。这证明有损变换虽不能恢复原 RGB，仍可通过 +变换后的调色板与几何结构恢复策略所需的符号对象。 + +\begin{itemize}[leftmargin=2em] + \item \texttt{task\_1} 至 \texttt{task\_4} 在 \texttt{safe} 模式下 + 本轮 original + spatial 共 36 个 episode 全部成功，验证了策略对隐藏 \texttt{info} 无依赖 + 且对布局扰动具有鲁棒性。 + \item spatial 变体修复后，\texttt{task\_1} 至 \texttt{task\_3} 的 spatial 通过率 + 从 1/3、1/3、0/3 提升到 3/3，说明像素对齐和房间识别修复有效。 + \item color 变体经两轮修复后，\texttt{task\_1} 至 \texttt{task\_5} + 的六种观察矩阵 30/30 全部通过；grayscale 使用调色板映射， + high\_contrast 使用二值连通分量与形状规则。 + \item \texttt{task\_5} original、五种非默认 color 和 + spatial\_a/b/c 全部通过；spatial\_c 的西门举盾时序也在 Lean 中有对应定理。 + \item Lean 证明与实验互补：实验说明 Python baseline 可运行， + Lean 证明说明抽象后的环境语义、witness 计划、五个任务的符号策略 + 以及通用有界 BFS 搜索框架在逻辑上自洽。 +\end{itemize} + +\section{项目分工与推进过程} + +\subsection{成员分工} + +根据小组最终名单，成员分工记录如下： + +\begin{table}[H] +\centering +\begin{tabular}{p{0.18\textwidth}p{0.38\textwidth}p{0.34\textwidth}} +\toprule +成员 & 负责内容 & 主要产出 \\ +\midrule +李政轩（组长，241880177） & 项目初始化、\texttt{safe} 模式适配、spatial/color 鲁棒性修复、task5 导航攻坚 & 报告与文档骨架、task4 baseline 状态机；\texttt{safe} 模式适配（移除 \texttt{reward\_signals} 依赖）；task1-4 spatial 变体修复（像素对齐、房间识别）；task5 导航执行层 7 个 bug 修复（HP 预算分析等） \\ +续博森（241880403） & Lean 环境与策略形式化、BFS 改造、颜色变体与 task5 攻坚 & task1-5 环境形式化与策略形式化；task1-4 由硬编码改为 BFS；可逆颜色变体适配（color 3/5 $\to$ 5/5）；task5 original 优化至 success\_rate = 1.000；spatial\_a/b 修复（\texttt{no\_attack\_tiles}、\texttt{diagonal\_block}） \\ +陈怡琼（241880323） & Python baseline、策略框架完备性证明、task5 最终突破 & task3/4/5 像素 baseline；\texttt{StrategyFramework.lean} 完备性证明；task5 形式化补强与 robustness 支持（\texttt{ACTION\_B} 举盾入场解决 spatial\_c 接触伤害，500-episode 日志） \\ +许浩博（231880172） & Task1 Lean 证明对齐 & Task1 Lean 证明与 BFS 策略对齐 \\ +吴韬（241880187） & 参与项目早期方案讨论 & 参与早期方案讨论与团队协调（相关工作未直接反映在代码提交记录中） \\ +\bottomrule +\end{tabular} +\caption{小组分工} +\end{table} + +\subsection{时间安排} + +\begin{itemize}[leftmargin=2em] + \item \textbf{2026-07-03}：阅读课程说明与仓库结构，建立项目管理文档和报告骨架； + 安装依赖，跑通示例策略；创建 \texttt{student\_agent/} 和统一 baseline 入口； + 初步覆盖 \texttt{task\_1} 至 \texttt{task\_4}。 + \item \textbf{2026-07-05}：将 \texttt{task\_3}、\texttt{task\_4} + 改造为像素感知版本；修正玩家中心 tile 识别；接入 \texttt{task\_5}； + 完成五任务单 seed 回归；逐文件检查 Lean 文件。 + \item \textbf{2026-07-06}：同步团队更新，补全 \texttt{task\_1} 至 + \texttt{task\_5} 的 Lean 环境形式化与可完成性证明；重新运行 Python evaluation + 和全部 Lean 检查；进一步补充 \texttt{task\_1} 至 \texttt{task\_5} + 高层策略形式化与合法性/完成性证明； + 把最终实验结果与证明边界写入报告。 + \item \textbf{2026-07-07}：同步团队提交，将 Lean 文件统一命名为 + \texttt{TaskNFormalization.lean}，并复核报告中环境形式化、策略形式化和证明结果的一致性； + 新增通用策略框架文件，抽取通用策略执行、动作 mask 安全证书和 + 有界 BFS 完备性证明；重新编译报告 PDF。 + \item \textbf{2026-07-08}：补充 \texttt{lean-toolchain} 文件（\texttt{leanprover/lean4:v4.26.0}）， + 确保评测方可在一致的工具链下复现 Lean 编译； + 运行 10 seed 评测（seed = 0--9），五个任务全部成功； + 据此更新报告实验设置、结果表格和分析部分，并重新编译 PDF。 + \item \textbf{2026-07-10}：合并 upstream（助教仓库）的评测脚本更新 + （\texttt{--info-mode safe}、\texttt{--task-policy}、spatial 变体扩展到 task\_4/5）； + 全面适配 \texttt{safe} 信息模式，将 \texttt{update\_memory} 与 + \texttt{update\_task5\_memory} 重构为基于 \texttt{inventory} diff、 + \texttt{last\_reward} 和像素感知的事件推断； + 在 \texttt{safe} 模式下重新运行 10 seed 评测，task\_1--4 全部成功， + task\_5 事件推断正常但导航执行层仍有 bug； + 同步更新 \texttt{Task1Formalization.lean} 注释与报告实验章节。 + \item \textbf{2026-07-11}：修复 spatial 布局变体失败。 + 根因一：\texttt{move\_towards\_aligned} 像素对齐阈值 \texttt{+1} 过松， + 在 1-tile 间隙处 sprite rect 覆盖相邻 blocking tile 导致卡住， + 修复为精确对齐 \texttt{+0} 并添加卡住检测 + （\texttt{mem\_last\_action\_blocked} 追踪 reward $\leq -0.05$）； + 根因二：\texttt{detect\_task3\_room} 硬编码 NPC/chest 位置， + spatial 变体下 room 检测失败，修复为全网格扫描。 + 修复后运行 robustness suite（\texttt{--num-envs 30}）， + task\_1--4 的 original + spatial 共 108 个 episode 全部通过， + color 变体预期失败（策略基于精确 RGB 匹配）； + 更新 \texttt{eval\_results.json} 与报告实验章节。 + \item \textbf{2026-07-13}：修复 color 变体鲁棒性。 + 在 \texttt{baseline\_policy.py} 中新增 \texttt{normalize\_obs} 预处理模块 + （详见 \ref{sec:color-variant-fix} 节），采用\textbf{参考色存在性检测} + 依次识别六种观察变体， + 对可逆变体（dark/inverted）施加逆变换恢复原始帧，对部分可逆变体（bright） + 采用前向变换参考色匹配（规避 clip 不可逆问题），并引入 \texttt{COLOR\_TOL = 2} + 容差匹配覆盖 \texttt{uint8} 截断误差。 + 修复过程中排查并修正了三个关键 bug： + （1）grayscale 检测在 dark 帧地板上误判，改为彩色像素占比 $< 0.5\%$ 判据； + （2）bright 逆变换因 clip 丢失信息导致匹配失败，改为前向变换参考色； + （3）绝对亮度阈值在不同地图天然亮度下误判 default 为 dark， + 重写为参考色存在性检测彻底规避亮度依赖。 + 修复后 \texttt{task\_1} 至 \texttt{task\_4} 的 dark/bright/inverted 三种可逆变体 + 全部从 0\% 提升至 100\%，default 无回归，color 阶段通过率从 0\% 提升至 66.7\% + （grayscale/high\_contrast 因颜色信息有损不可逆仍失败，属预期）； + 同步更新报告实验设置、结果表格、结果分析与颜色变体修复小节。 + \item \textbf{2026-07-15}：修复 \texttt{task\_5} 导航执行层 7 个 bug + （navigate\_to\_exit 像素对齐、button\_pressed/monster\_cleared 事件误标、 + east/west 墙体模式修正、chest\_opened 误标、attack\_monster 缺失 blocked/player\_px 参数）， + 在此基础上进一步将 \texttt{task\_5} original 阶段从 + success\_rate = 0.000 优化至 1.000（18/18，avg\_steps = 1070，avg\_reward = 235.050）。 + 核心突破为 HP 预算分析：发现 \texttt{task\_5} 隐藏每 200 步 $-$1 HP 的流失机制 + （初始 HP = 5，6 次 drain 共 $-$6 HP，east heal 箱 $+1$ HP，总预算 = 6）， + 据此设计三项协同优化——start 房间先杀 chaser 再导航（避免转向接触伤害）、 + east 房间跳过 ambusher 直奔宝箱（ambusher 在 $(7,5)$、ambush\_range = 2， + agent 走 row 1 距离 $\geq 4$ 不会激活，省 $\sim$130 步）、 + west 房间用 \texttt{detect\_closest\_monster\_tile} 逐个击杀全部怪物 + （修复 \texttt{detect\_monster\_tile} 多怪物质心指向虚假位置的 bug）后导航（零接触伤害）， + agent 最终在 step 1070 以 HP = 1 开完全部 4 宝箱触发 \texttt{world\_completed}， + 远早于 step 1200 的第 6 次 HP drain； + 同步验证 \texttt{task\_5} color 变体 3/5 通过（dark/bright/inverted 成功）， + 修复 \texttt{detect\_task5\_room} start 房间检测（硬编码对象位置改为墙体特征匹配） + 和 south 房间宝箱硬编码 fallback，spatial 变体房间识别修复但仍因 HP 预算不足失败； + 同步更新报告结果表格、Task 5 状态说明与优化历程。 + \item \textbf{2026-07-16}：修复 \texttt{task\_5} spatial\_a/b 变体。 + 发现引擎 \texttt{try\_interaction} 在战斗前检查 NPC/chest 邻接（无 facing 检查）， + 导致 spatial\_b 中 agent 站在 NPC 邻接 tile 按下 A 时触发 \texttt{talked\_npc} + 而非攻击怪物。新增 \texttt{compute\_no\_attack\_tiles} 方法检测所有 NPC/chest + 邻接 tile，\texttt{attack\_monster} 接受 \texttt{no\_attack\_tiles} 参数 + 在不安全 tile 上回退到 BFS 寻找安全攻击位置。 + 同时新增 \texttt{diagonal\_block} 参数：start 房间保持 \texttt{True} + （spatial\_a 修复：对角 tile 加入 blocked 避免 AABB 接触伤害）， + west 房间设为 \texttt{False}（spatial\_b 修复：避免 60+ 步绕行）。 + 修复后 spatial\_a（5/5，steps=1076）和 spatial\_b（5/5，steps=1174）均以 HP = 1 通关， + original 无回归（18/18，steps=1083），color 变体无回归（3/5）。 + spatial\_c 因 south 房间导航效率过低（299 次 blocked）仍失败。 + 同步更新报告结果表格、spatial 变体小节与优化历程。 + \item \textbf{2026-07-17}：完成最终三项补强。 + 第一，Task 5 Lean 新增 10 类 \texttt{BfsRouteCertificate}、两类 + \texttt{CombatCertificate}、\texttt{WestEntryCertificate} 和 24 阶段 + \texttt{BaselineState} 细化，证明动作逐步合法、起始怪清除顺序、 + spatial\_c 西门举盾、清除西房双怪以及最终详细目标可达。 + 第二，将 NPC 与非目标出口加入导航/战斗阻塞集，修正移动怪物跨 tile 的目标选择， + 并在西门边界执行 \texttt{ACTION\_B}，使 Task 5 spatial\_c 在 1127 步通关。 + 第三，为 grayscale 增加灰度参考色映射，为 high\_contrast 增加连通分量和 + tile 局部形状识别；六颜色变体 50/50（color 阶段）、五任务 robustness 500/500 全部通过。 + 全部 Lean 文件重新编译通过，并同步更新最终报告。 +\end{itemize} + +\section{总结与展望} + +本文完成了 NesyLink 数理逻辑课程项目的三条主线工作。 +第一，构建了统一的 Python baseline，全面适配评测脚本的 \texttt{safe} 信息模式 +（不依赖隐藏 \texttt{reward\_signals}，仅用 \texttt{last\_reward}、\texttt{inventory} +和像素感知）。最终 robustness suite 的五个任务共 500 个 episode 全部通过， +覆盖 original、spatial\_a/b/c 和五种非默认颜色变体（grayscale/dark/bright/ +high\_contrast/inverted）。 +\texttt{task\_5} original 为 60/60（avg\_steps = 1084，avg\_reward = 234.860）， +spatial\_a/b/c 为 30/30，五种非默认颜色为 10/10。 +第二，使用 Lean 对五个任务的环境语义进行了符号层形式化， +覆盖状态、动作、转移、关键前置条件、不变式、完成后置条件和 witness 可完成性证明； +并对 \texttt{task\_1} 至 \texttt{task\_5} 的符号策略补充了动作合法性与目标完成性证明； +Task 5 进一步证明与最终 baseline 对齐的 BFS、战斗和举盾入场证书轨迹。 +第三，抽取了独立的 \texttt{StrategyFramework.lean}，证明通用策略执行框架、 +action mask 安全证书和有界 BFS 搜索完备性。 +第四，将实验结果、实现路线和形式化边界整理进报告，明确区分了调试信息、 +提交版策略输入和 Lean 证明对象。 + +当前工作的主要不足是：Lean 已证明五个任务的高层符号策略， +但尚未逐行证明 Python 像素 waypoint 执行代码等价于这些符号策略， +也尚未证明 Python 具体 BFS/队列实现等价于 Lean 中的抽象有界 BFS， +更没有证明视觉模块一定能在所有未来渲染扰动下稳定恢复符号状态。 +当前实测变体已全部通过，但 high\_contrast 规则依赖现有精灵的二值形状， +若素材、tile 尺寸或绘制层次变化，仍需要重新校准。 +如果时间更充足，可以进一步证明像素感知层到 Lean 符号状态之间的抽取正确性， +证明具体搜索程序与抽象 BFS 定理之间的细化关系， +将当前基于公开素材结构的对象识别扩展为更通用的视觉模型， +也可以补充更多随机布局和跨素材鲁棒性测试。 + +\newpage +\bibliographystyle{plain} +\bibliography{references} + +\newpage +\appendix +\section{运行证明}\label{sec:run-evidence} + +本附录提供 Lean 形式化验证与评测脚本实际运行的终端截图，作为代码确实运行的证据。 +完整 500-episode 评测结果见 \texttt{report/eval\_results.json}。 + +\subsection{Lean 形式化验证} + +对 \texttt{student\_agent/lean/} 下全部 7 个 \texttt{.lean} 文件执行 +\texttt{lean } 编译检查，并扫描 \texttt{sorry}/\texttt{admit}/\texttt{axiom}。 +结果：全部 7 个文件编译成功（exit code = 0），禁用关键字扫描无匹配， +符合提交要求（\texttt{lean-toolchain} 指定 \texttt{leanprover/lean4:v4.26.0}）。 + +\begin{figure}[H] +\centering +\includegraphics[width=0.92\textwidth]{figures/lean_build.png} +\caption{Lean 编译检查与 \texttt{sorry}/\texttt{admit}/\texttt{axiom} 扫描终端输出。} +\end{figure} + +\subsection{Robustness suite 评测运行} + +使用 \texttt{--robustness-suite --num-envs 10 --info-mode safe} 运行 50 个 episode +的快速复现（完整 500-episode 结果见 \texttt{eval\_results.json}）。 +命令： +\begin{quote} +\small +\texttt{python -m utils.evaluate\_policy}\\ +\texttt{--task-policy mathematical\_logic/task\_1=student\_agent/baseline\_policy.py}\\ +\texttt{...（task\_2--5 同样映射到 baseline\_policy.py）}\\ +\texttt{--info-mode safe --robustness-suite --num-envs 10}\\ +\texttt{--out-file \_final\_check.json} +\end{quote} + +五个任务在 original/spatial/color 三阶段共 50 个 episode 全部 \texttt{success=True}。 +图\ref{fig:eval-summary} 展示 Task 1 的统计汇总（success\_rate = 1.000）， +图\ref{fig:eval-final} 展示 Task 5 color 事件汇总与最终 JSON 写入确认。 + +\begin{figure}[H] +\centering +\includegraphics[width=0.92\textwidth]{figures/eval_run_part2.png} +\caption{Task 1 统计汇总：original 6/6、spatial 3/3、color 1/1 均 success\_rate = 1.000。}\label{fig:eval-summary} +\end{figure} + +\begin{figure}[H] +\centering +\includegraphics[width=0.92\textwidth]{figures/eval_run_part9.png} +\caption{Task 5 color 事件汇总（chest\_opened/monster\_killed/exit\_reached 等） +与最终 \texttt{Wrote JSON results to \_final\_check.json} 写入确认。}\label{fig:eval-final} +\end{figure} + +\end{document} diff --git a/docs/Mathematical_logic/report/references.bib b/docs/Mathematical_logic/report/references.bib new file mode 100644 index 0000000..b8850cc --- /dev/null +++ b/docs/Mathematical_logic/report/references.bib @@ -0,0 +1,7 @@ +@misc{nesylink_repo, + title = {NesyLink}, + author = {{CrazyJassBread}}, + year = {2026}, + howpublished = {\url{https://github.com/CrazyJassBread/nesylink}}, + note = {Course project environment repository} +} diff --git a/student_agent/README.md b/student_agent/README.md new file mode 100644 index 0000000..1507af2 --- /dev/null +++ b/student_agent/README.md @@ -0,0 +1,44 @@ +# 小组基线 Agent 骨架 + +本目录用于放置你们小组自己的 Agent 代码，而不是继续直接修改课程示例文件。 + +## 当前状态 + +- 已提供统一 `Policy` 类骨架。 +- 已兼容当前仓库的评测入口：`act(obs, info) -> int` +- 已将“最终可提交的决策逻辑”和“当前评测兼容层”分开。 +- 当前已实现 `task_1`、`task_2`、`task_3` 和 `task_4` 的可运行 baseline。 +- 其中 `task_2` 当前采用从参考成功执行轨迹中提取的脚本化 debug 版本。 +- `task_3` 当前采用像素房间识别 + 玩家中心格定位 + 物品栏钥匙数的规则状态机。 +- `task_4` 当前采用像素房间识别、桥状态像素识别、玩家中心格定位和阶段记忆，依次完成拿钥匙、拿剑、击杀守卫和开最终宝箱。 + +## 文件说明 + +- `baseline_policy.py`：当前可运行的基线策略入口。 +- `__init__.py`：导出 `Policy` 和 `make_policy`。 + +## 设计原则 + +- 不把课程提供的示例文件当作最终项目主代码。 +- 对外兼容现有评测脚本，对内逐步改造成符合最终测试限制的结构。 +- 先保证有统一骨架，再逐步往里面填 `task_2` 到 `task_5` 的逻辑。 + +## 当前建议的扩展顺序 + +1. 完善 `task_1` / `task_2` / `task_3` / `task_4` 的结果记录与截图。 +2. 把 `task_2` 从脚本化轨迹替换为更通用的规则策略。 +3. 继续加强 `task_3` / `task_4` 的像素识别鲁棒性，并补充截图和失败样例记录。 +4. 将 `task_4` 的阶段机和 Lean 形式化命题进一步对齐。 +5. 最后为 `task_5` 补通用探索逻辑。 + +## 运行方式 + +```bash +python utils/evaluate_policy.py --policy student_agent/baseline_policy.py --tasks mathematical_logic/task_1 --num-envs 1 +``` + +也可以写成： + +```bash +python utils/evaluate_policy.py --policy student_agent.baseline_policy:make_policy --tasks mathematical_logic/task_1 --num-envs 1 +``` diff --git a/student_agent/__init__.py b/student_agent/__init__.py new file mode 100644 index 0000000..6417c78 --- /dev/null +++ b/student_agent/__init__.py @@ -0,0 +1,3 @@ +from .baseline_policy import Policy, make_policy + +__all__ = ["Policy", "make_policy"] diff --git a/student_agent/baseline_policy.py b/student_agent/baseline_policy.py new file mode 100644 index 0000000..12c8fa9 --- /dev/null +++ b/student_agent/baseline_policy.py @@ -0,0 +1,1694 @@ +from __future__ import annotations + +from collections import deque +from dataclasses import dataclass, field +from typing import Any, cast + +import numpy as np + +from nesylink.core.constants import ( + ACTION_A, + ACTION_B, + ACTION_DOWN, + ACTION_LEFT, + ACTION_NOOP, + ACTION_RIGHT, + ACTION_UP, + GRID_HEIGHT, + GRID_WIDTH, + TILE_SIZE, +) + + +Color = tuple[int, int, int] +Tile = tuple[int, int] + + +PLAYER_COLORS: tuple[Color, ...] = ((36, 198, 72), (126, 248, 82)) +WALL_COLOR: Color = (219, 18, 82) +CHEST_WOOD: Color = (152, 82, 36) +NPC_COLOR: Color = (240, 154, 52) +MONSTER_COLORS: tuple[Color, ...] = ((238, 126, 28), (255, 180, 48), (200, 78, 16)) +ABYSS_COLOR: Color = (0, 0, 0) +BRIDGE_COLOR: Color = (172, 104, 48) +SWITCH_BODY: Color = (255, 216, 80) +SWITCH_DOWN: Color = (184, 124, 42) +BUTTON_UP: Color = (40, 190, 74) +BUTTON_DOWN_COLOR: Color = (28, 112, 52) + +# Trap spike pixels (sprites.py:39-41). Distinctive against floor/wall colors. +TRAP_SPIKE_METAL: Color = (238, 238, 236) +TRAP_SPIKE_SHADE: Color = (112, 112, 126) + +# 容差匹配：逆变换后 uint8 截断会产生 ±1 误差，设为 2 覆盖边界情况。 +COLOR_TOL: int = 2 + +# Exit tile positions per direction (schema.py:30-35) and the action that +# triggers the exit once standing on one of its tiles. +EXIT_DIRECTION_TILES: dict[str, tuple[Tile, Tile]] = { + "north": ((4, 0), (5, 0)), + "south": ((4, GRID_HEIGHT - 1), (5, GRID_HEIGHT - 1)), + "west": ((0, 3), (0, 4)), + "east": ((GRID_WIDTH - 1, 3), (GRID_WIDTH - 1, 4)), +} +EXIT_DIRECTION_ACTION: dict[str, int] = { + "north": ACTION_UP, + "south": ACTION_DOWN, + "west": ACTION_LEFT, + "east": ACTION_RIGHT, +} + + +BRIDGE_CYCLE = { + "west_to_north": "west_to_east", + "west_to_east": "west_to_south", + "west_to_south": "west_to_north", +} + + +@dataclass +class AgentHistory: + task_id: str | None = None + plan_index: int = 0 + facing: str = "down" + last_action: int = ACTION_NOOP + last_inventory: dict[str, Any] = field(default_factory=dict) + notes: dict[str, Any] = field(default_factory=dict) + + +@dataclass(frozen=True) +class PixelState: + player_tile: Tile | None + room_id: str | None + bridge_state: str | None + monster_visible: bool + + +class Policy: + """ + Group baseline policy skeleton. + + The evaluator still calls `act(obs, info)`, but the long-term goal is to keep + the real decision logic isolated from hidden debug state in `info`. + """ + + def __init__(self) -> None: + self.history = AgentHistory() + self._component_frame: np.ndarray | None = None + self._component_cache: dict[Color, list[tuple[np.ndarray, np.ndarray]]] = {} + + def reset(self) -> None: + # 新版评测脚本无参数调用 reset()。task_id 改由 info["task_id"] 在 + # act() 中获取（仅 --task-policy 模式提供）。 + self.history = AgentHistory() + self._component_frame = None + self._component_cache = {} + + def normalize_obs(self, frame: np.ndarray) -> np.ndarray: + """检测 color 变体并尽可能恢复 default 外观。 + + 使用参考色存在性检测（而非亮度统计），避免不同地图天然亮度差异导致误判。 + 依次检查 PLAYER_COLORS 在各变体下的变换值是否存在于帧中： + - default：原始参考色存在 → 无需变换； + - inverted：(255-c) 存在 → 逆变换 255-frame； + - dark：(int(c*0.55)) 存在 → 逆变换 frame/0.55； + - bright：(min(255,int(c*1.35))) 存在 → 存 variant，前向变换参考色。 + 最后检查 grayscale/high_contrast（有损不可逆，原样返回）。 + """ + img = np.asarray(frame) + if img.ndim != 3 or img.shape[2] != 3: + self.history.notes["_obs_variant"] = "default" + return frame + + int_img = img.astype(np.int16) + min_count = 3 # 至少 3 像素匹配才算检测到 + + # 1. 检查 default：原始参考色存在 + for c in PLAYER_COLORS: + diff = np.abs(int_img - np.array(c, dtype=np.int16)) + if np.count_nonzero(np.all(diff <= COLOR_TOL, axis=2)) >= min_count: + self.history.notes["_obs_variant"] = "default" + return img + + # 2. 检查 inverted：(255 - c) 存在 + for c in PLAYER_COLORS: + inv = np.array([255 - x for x in c], dtype=np.int16) + diff = np.abs(int_img - inv) + if np.count_nonzero(np.all(diff <= COLOR_TOL, axis=2)) >= min_count: + self.history.notes["_obs_variant"] = "default" + return (255 - img).astype(np.uint8) + + # 3. 检查 dark：(int(c * 0.55)) 存在 + for c in PLAYER_COLORS: + dark = np.array([int(x * 0.55) for x in c], dtype=np.int16) + diff = np.abs(int_img - dark) + if np.count_nonzero(np.all(diff <= COLOR_TOL, axis=2)) >= min_count: + self.history.notes["_obs_variant"] = "default" + return (img.astype(np.float32) / 0.55).clip(0, 255).astype(np.uint8) + + # 4. 检查 bright：(min(255, int(c * 1.35))) 存在 + for c in PLAYER_COLORS: + bright = np.array([min(255, int(x * 1.35)) for x in c], dtype=np.int16) + diff = np.abs(int_img - bright) + if np.count_nonzero(np.all(diff <= COLOR_TOL, axis=2)) >= min_count: + self.history.notes["_obs_variant"] = "bright" + return img + + # 5. 检查 grayscale：彩色像素占比极低 + r, g, b = int_img[:, :, 0], int_img[:, :, 1], int_img[:, :, 2] + max_diff = np.maximum(np.maximum(np.abs(r - g), np.abs(g - b)), np.abs(r - b)) + if float(np.mean(max_diff > 15)) < 0.005: + self.history.notes["_obs_variant"] = "grayscale" + return img + + # 6. 检查 high_contrast：像素值只有 0/255 + uniq = np.unique(img) + if uniq.size <= 6 and bool(np.all(np.isin(uniq, [0, 255]))): + self.history.notes["_obs_variant"] = "high_contrast" + return img + + # 兜底：当作 default + self.history.notes["_obs_variant"] = "default" + return img + + def _effective_color(self, color: Color) -> Color: + """根据检测到的 color 变体，将参考色前向变换到帧空间。 + + dark/inverted 已由 normalize_obs 逆变换恢复，参考色不变； + bright/grayscale 未逆变换（有损不可逆），需将参考色前向变换后匹配。 + """ + variant = self.history.notes.get("_obs_variant", "default") + if variant == "bright": + return cast(Color, tuple(min(255, int(c * 1.35)) for c in color)) + if variant == "grayscale": + gray = int(sum(color) / 3) + return (gray, gray, gray) + if variant == "high_contrast": + return cast(Color, tuple(255 if c > 127 else 0 for c in color)) + return color + + def _color_tolerance(self) -> int: + # Grayscale/high-contrast are produced by exact integer transforms. + # Using the inverse-transform tolerance here merges nearby palette + # entries (for example grayscale wall 106 and floor shadow 108). + variant = self.history.notes.get("_obs_variant", "default") + return 0 if variant in {"grayscale", "high_contrast"} else COLOR_TOL + + def extract_inventory(self, info: dict[str, Any]) -> dict[str, Any]: + inventory = info.get("inventory", {}) + if isinstance(inventory, dict): + return dict(inventory) + return {} + + def extract_last_reward(self, info: dict[str, Any]) -> float: + # safe 模式下 info["last_reward"] 为上一步标量奖励，reset 后首次为 0.0。 + return float(info.get("last_reward", 0.0) or 0.0) + + def update_memory_from_safe( + self, + inventory: dict[str, Any], + last_reward: float, + frame: np.ndarray, + ) -> None: + # safe 模式下没有 reward_signals，改为基于 inventory diff + last_reward + # 推断事件。图像感知由各 task 的 decide 函数自行处理。 + notes = self.history.notes + prev = self.history.last_inventory or {} + if int(inventory.get("keys", 0)) > int(prev.get("keys", 0) or 0): + notes["mem_keys_collected"] = int(notes.get("mem_keys_collected", 0) or 0) + 1 + notes["mem_key_collected"] = True + if int(inventory.get("gold", 0)) > int(prev.get("gold", 0) or 0): + notes["mem_gold_collected"] = True + prev_items = set(prev.get("items", []) or []) + curr_items = set(inventory.get("items", []) or []) + if curr_items - prev_items: + notes["mem_item_collected"] = True + notes["mem_last_reward"] = last_reward + notes["mem_reward_positive"] = last_reward > 0 + # Blocked actions (e.g. clipping a wall) incur reward <= -0.05. + # Used by move_towards_aligned to detect when a pixel-alignment + # correction is stuck due to detect_player_px perception offset. + notes["mem_last_action_blocked"] = last_reward <= -0.05 + + def current_keys(self, inventory: dict[str, Any]) -> int: + # safe 模式下 reward_signals 不可用，直接用 inventory.keys。 + return int(inventory.get("keys", 0)) + + def infer_task_id(self, info: dict[str, Any]) -> str | None: + # safe 模式下 task_id 通过 info["task_id"] 传入（--task-policy 模式）。 + task_id = info.get("task_id") + if task_id: + self.history.task_id = task_id + return task_id + return self.history.task_id + + def has_item(self, inventory: dict[str, Any], item_id: str) -> bool: + items = inventory.get("items", []) + if isinstance(items, (list, tuple)): + return item_id in items + return False + + def in_bounds(self, tile: Tile) -> bool: + x, y = tile + return 0 <= x < GRID_WIDTH and 0 <= y < GRID_HEIGHT + + def detect_monster_tile(self, frame: np.ndarray) -> Tile | None: + if self.history.notes.get("_obs_variant") == "high_contrast": + monsters = self._high_contrast_monster_tiles(frame) + return monsters[0] if monsters else None + if not self.has_monster(frame): + return None + x_values: list[np.ndarray] = [] + y_values: list[np.ndarray] = [] + for color in MONSTER_COLORS: + eff = self._effective_color(color) + diff = np.abs(frame.astype(np.int16) - np.array(eff, dtype=np.int16)) + mask = np.all(diff <= self._color_tolerance(), axis=2) + ys, xs = np.nonzero(mask) + if len(xs) > 0: + x_values.append(xs) + y_values.append(ys) + if not x_values: + return None + xs = np.concatenate(x_values) + ys = np.concatenate(y_values) + max_x = GRID_WIDTH - 1 + max_y = GRID_HEIGHT - 1 + center_x = int(xs.mean()) + center_y = int(ys.mean()) + return (min(max_x, center_x // TILE_SIZE), min(max_y, center_y // TILE_SIZE)) + + def detect_all_monster_tiles(self, frame: np.ndarray) -> list[Tile]: + # Detect all distinct monster tiles by clustering pixels. + if self.history.notes.get("_obs_variant") == "high_contrast": + return self._high_contrast_monster_tiles(frame) + if not self.has_monster(frame): + return [] + tiles: set[Tile] = set() + for color in MONSTER_COLORS: + eff = self._effective_color(color) + diff = np.abs(frame.astype(np.int16) - np.array(eff, dtype=np.int16)) + mask = np.all(diff <= self._color_tolerance(), axis=2) + ys, xs = np.nonzero(mask) + for x, y in zip(xs, ys): + tile = (min(GRID_WIDTH - 1, x // TILE_SIZE), min(GRID_HEIGHT - 1, y // TILE_SIZE)) + tiles.add(tile) + return list(tiles) + + def detect_closest_monster_tile(self, frame: np.ndarray, agent_tile: Tile) -> Tile | None: + monsters = self.detect_all_monster_tiles(frame) + if not monsters: + return None + ax, ay = agent_tile + # A moving 16 px monster can straddle a tile boundary, so its colored + # pixels may cover both the player's tile and an adjacent tile. The + # same-tile fragment is not a usable sword target; prefer a visible + # non-player fragment whenever one exists. + non_player_tiles = [monster for monster in monsters if monster != agent_tile] + candidates = non_player_tiles or monsters + best = None + best_dist = 999 + for mx, my in candidates: + dist = abs(mx - ax) + abs(my - ay) + if dist < best_dist: + best = (mx, my) + best_dist = dist + return best + + def facing_toward(self, dx: int, dy: int) -> str: + if dx > 0: + return "right" + if dx < 0: + return "left" + if dy > 0: + return "down" + if dy < 0: + return "up" + return self.history.facing + + def compute_no_attack_tiles(self, frame: np.ndarray) -> set[Tile]: + """Compute tiles where pressing A would trigger an NPC/chest/switch + interaction instead of attacking a monster. The engine's try_interaction + checks adjacency (not facing) and runs before combat, so standing on + these tiles makes attacks impossible.""" + no_attack: set[Tile] = set() + for y in range(GRID_HEIGHT): + for x in range(GRID_WIDTH): + tile = (x, y) + if self.tile_has_npc(frame, tile) or self.tile_has_chest(frame, tile): + for ddx, ddy in ((-1, 0), (1, 0), (0, -1), (0, 1)): + adj = (x + ddx, y + ddy) + if self.in_bounds(adj): + no_attack.add(adj) + return no_attack + + def attack_monster(self, tile: Tile, monster_tile: Tile, blocked: set[Tile] | None = None, player_px: tuple[float, float] | None = None, no_attack_tiles: set[Tile] | None = None, diagonal_block: bool = True) -> int: + # Attack a monster at monster_tile. When adjacent (distance=1), face the + # monster and press A. When farther away, BFS to the closest adjacent + # tile using the provided blocked set (if available) or an empty set. + # no_attack_tiles: tiles where pressing A triggers an NPC/chest interaction + # instead of attacking (engine checks adjacency before combat). The agent + # must avoid these as attack positions. + px, py = tile + mx, my = monster_tile + dx = mx - px + dy = my - py + manhattan = abs(dx) + abs(dy) + if manhattan <= 0: + return ACTION_NOOP + + if manhattan == 1: + # If standing on a tile where A triggers NPC/chest interaction, + # fall through to BFS to find a safe attack position. + if no_attack_tiles is not None and tile in no_attack_tiles: + pass # fall through to BFS logic below + else: + desired = self.facing_toward(dx, dy) + if self.history.facing == desired: + return ACTION_A + # Wrong facing: back off along the monster axis (away from monster) + # to re-approach on a straight line that sets the correct facing. + if desired == "right": + return ACTION_LEFT + if desired == "left": + return ACTION_RIGHT + if desired == "down": + return ACTION_UP + return ACTION_DOWN + + # manhattan >= 2 (or manhattan == 1 on unsafe tile): BFS to closest valid adjacent tile. + candidates = ( + ((mx - 1, my), False), + ((mx + 1, my), False), + ((mx, my - 1), True), + ((mx, my + 1), True), + ) + valid = [c for c in candidates if self.in_bounds(c[0])] + # 排除 blocked 中的 tile（如墙），避免选不可达的 target 导致 BFS 空转 + if blocked is not None: + valid = [c for c in valid if c[0] not in blocked] + # 排除 no_attack_tiles 中的 tile（NPC/chest 相邻格），避免按 A 被交互拦截 + if no_attack_tiles is not None: + valid = [c for c in valid if c[0] not in no_attack_tiles] + if not valid: + return ACTION_NOOP + + def dist_to(c: tuple[Tile, bool]) -> int: + ax, ay = c[0] + return abs(ax - px) + abs(ay - py) + + target, x_first = min(valid, key=dist_to) + # Block diagonal tiles around the monster to force a straight-line + # approach. When the agent passes through a tile diagonally adjacent to + # the monster, the chaser's 0.5px/step movement toward the agent can + # cause AABB sprite overlap (contact damage). Blocking diagonals forces + # BFS to route the agent along the same row/column as the monster, + # minimizing overlap risk. + if blocked is not None: + approach_blocked = set(blocked) + if diagonal_block: + for ddx, ddy in ((-1, -1), (1, -1), (-1, 1), (1, 1)): + dt = (mx + ddx, my + ddy) + if self.in_bounds(dt) and dt != target: + approach_blocked.add(dt) + # Verify target is reachable with diagonal blocks; fall back if not. + if self.bfs_path(tile, {target}, approach_blocked) is not None: + if player_px is not None: + return self.follow_bfs_aligned(player_px, tile, {target}, approach_blocked, final_action=ACTION_A) + return self.follow_bfs(tile, {target}, approach_blocked, x_first=x_first, final_action=ACTION_A) + # Fallback: target unreachable with diagonal blocks, use original set. + if player_px is not None: + return self.follow_bfs_aligned(player_px, tile, {target}, blocked, final_action=ACTION_A) + return self.follow_bfs(tile, {target}, blocked, x_first=x_first, final_action=ACTION_A) + # Fallback to simple move_towards when no blocked set is provided. + return self.move_towards(tile, target, x_first=x_first) + + def count_color(self, frame: np.ndarray, color: Color, tile: Tile | None = None) -> int: + if tile is None: + area = frame + else: + x, y = tile + area = frame[y * TILE_SIZE : (y + 1) * TILE_SIZE, x * TILE_SIZE : (x + 1) * TILE_SIZE] + # 容差匹配（±COLOR_TOL）：兼容 normalize_obs 逆变换后 uint8 截断的 ±1 误差。 + eff = self._effective_color(color) + diff = np.abs(area.astype(np.int16) - np.array(eff, dtype=np.int16)) + return int(np.count_nonzero(np.all(diff <= self._color_tolerance(), axis=2))) + + def count_any_color(self, frame: np.ndarray, colors: tuple[Color, ...], tile: Tile | None = None) -> int: + return sum(self.count_color(frame, color, tile) for color in colors) + + def _color_components(self, frame: np.ndarray, color: Color) -> list[tuple[np.ndarray, np.ndarray]]: + """Return 8-connected components for one effective map color.""" + if self._component_frame is not frame: + self._component_frame = frame + self._component_cache = {} + eff = self._effective_color(color) + cached = self._component_cache.get(eff) + if cached is not None: + return cached + + map_area = frame[: GRID_HEIGHT * TILE_SIZE] + diff = np.abs(map_area.astype(np.int16) - np.array(eff, dtype=np.int16)) + mask = np.all(diff <= self._color_tolerance(), axis=2) + remaining = {(int(y), int(x)) for y, x in np.argwhere(mask)} + components: list[tuple[np.ndarray, np.ndarray]] = [] + while remaining: + start = remaining.pop() + queue = deque([start]) + points = [start] + while queue: + y, x = queue.popleft() + for dy in (-1, 0, 1): + for dx in (-1, 0, 1): + if dx == 0 and dy == 0: + continue + neighbor = (y + dy, x + dx) + if neighbor in remaining: + remaining.remove(neighbor) + queue.append(neighbor) + points.append(neighbor) + ys = np.fromiter((point[0] for point in points), dtype=np.int16) + xs = np.fromiter((point[1] for point in points), dtype=np.int16) + components.append((ys, xs)) + self._component_cache[eff] = components + return components + + def _high_contrast_monster_tiles(self, frame: np.ndarray) -> list[Tile]: + """Recover monster centers from binary-color body components.""" + tiles: set[Tile] = set() + for color in MONSTER_COLORS: + for ys, xs in self._color_components(frame, color): + count = len(xs) + width = int(xs.max() - xs.min() + 1) + height = int(ys.max() - ys.min() + 1) + if not (20 <= count <= 90 and 6 <= width <= 13 and 6 <= height <= 11): + continue + center_x = (int(xs.min()) + int(xs.max())) / 2 + 1 + center_y = (int(ys.min()) + int(ys.max())) / 2 + 2 + tile = ( + min(GRID_WIDTH - 1, max(0, int(center_x // TILE_SIZE))), + min(GRID_HEIGHT - 1, max(0, int(center_y // TILE_SIZE))), + ) + if tile in {exit_tile for pair in EXIT_DIRECTION_TILES.values() for exit_tile in pair}: + continue + if self.tile_has_wall(frame, tile) or self.tile_has_chest(frame, tile) or self.tile_has_npc(frame, tile): + continue + tiles.add(tile) + return list(tiles) + + def detect_player_px(self, frame: np.ndarray) -> tuple[int, int] | None: + # The visible green tunic starts a few pixels inside the sprite. Recover + # the sprite's true top-left pixel position so pixel-aligned navigation + # can avoid clipping walls when squeezing through 1-tile gaps. + if self.history.notes.get("_obs_variant") == "high_contrast": + green = self._effective_color(PLAYER_COLORS[0]) + components = self._color_components(frame, green) + groups: list[list[int]] = [] + ungrouped = set(range(len(components))) + while ungrouped: + seed = ungrouped.pop() + group = [seed] + queue = deque([seed]) + while queue: + current = queue.popleft() + current_ys, current_xs = components[current] + for other in list(ungrouped): + other_ys, other_xs = components[other] + x_gap = max( + 0, + max(int(current_xs.min()), int(other_xs.min())) + - min(int(current_xs.max()), int(other_xs.max())) + - 1, + ) + y_gap = max( + 0, + max(int(current_ys.min()), int(other_ys.min())) + - min(int(current_ys.max()), int(other_ys.max())) + - 1, + ) + if x_gap <= 4 and y_gap <= 4: + ungrouped.remove(other) + group.append(other) + queue.append(other) + groups.append(group) + + candidates: list[tuple[np.ndarray, np.ndarray]] = [] + for group in groups: + ys = np.concatenate([components[index][0] for index in group]) + xs = np.concatenate([components[index][1] for index in group]) + width = int(xs.max() - xs.min() + 1) + height = int(ys.max() - ys.min() + 1) + if len(xs) >= 30 and 4 <= width <= 20 and 8 <= height <= 24: + candidates.append((ys, xs)) + if not candidates: + return None + ys, xs = max(candidates, key=lambda component: len(component[1])) + return (max(0, int(xs.min()) - 4), max(0, int(ys.min()) - 2)) + + x_values: list[np.ndarray] = [] + y_values: list[np.ndarray] = [] + for color in PLAYER_COLORS: + # 容差匹配（±COLOR_TOL）：兼容 color 变体逆变换后的截断误差。 + eff = self._effective_color(color) + diff = np.abs(frame.astype(np.int16) - np.array(eff, dtype=np.int16)) + mask = np.all(diff <= self._color_tolerance(), axis=2) + ys, xs = np.nonzero(mask) + if len(xs) > 0: + x_values.append(xs) + y_values.append(ys) + if not x_values: + return None + xs = np.concatenate(x_values) + ys = np.concatenate(y_values) + left = max(0, int(xs.min()) - 4) + top = max(0, int(ys.min()) - 2) + return (left, top) + + def detect_player_tile(self, frame: np.ndarray) -> Tile | None: + pos = self.detect_player_px(frame) + if pos is None: + return None + left, top = pos + max_x = frame.shape[1] // TILE_SIZE - 1 + max_y = frame.shape[0] // TILE_SIZE - 1 + # Use the sprite center to match the engine's tile semantics. + center_x = left + TILE_SIZE // 2 + center_y = top + TILE_SIZE // 2 + return (min(max_x, center_x // TILE_SIZE), min(max_y, center_y // TILE_SIZE)) + + def tile_has_chest(self, frame: np.ndarray, tile: Tile) -> bool: + if self.history.notes.get("_obs_variant") == "high_contrast": + x, y = tile + area = frame[y * TILE_SIZE : (y + 1) * TILE_SIZE, x * TILE_SIZE : (x + 1) * TILE_SIZE] + red = np.all(area == np.array((255, 0, 0), dtype=np.uint8), axis=2) + yellow = np.all(area == np.array((255, 255, 0), dtype=np.uint8), axis=2) + band_pixels = max( + int(np.count_nonzero(yellow[4:7, 2:14])), + int(np.count_nonzero(yellow[2:4, 3:13])), + ) + return 30 <= int(np.count_nonzero(red)) < 100 and band_pixels >= 15 + return self.count_color(frame, CHEST_WOOD, tile) >= 25 + + def tile_has_npc(self, frame: np.ndarray, tile: Tile) -> bool: + if self.history.notes.get("_obs_variant") == "high_contrast": + if tile in {exit_tile for pair in EXIT_DIRECTION_TILES.values() for exit_tile in pair}: + return False + x, y = tile + area = frame[y * TILE_SIZE : (y + 1) * TILE_SIZE, x * TILE_SIZE : (x + 1) * TILE_SIZE] + yellow = np.all(area == np.array((255, 255, 0), dtype=np.uint8), axis=2) + return int(np.count_nonzero(yellow[4:12, 5:11])) >= 40 and not self.tile_has_chest(frame, tile) + return self.count_color(frame, NPC_COLOR, tile) >= 20 + + def tile_has_switch(self, frame: np.ndarray, tile: Tile) -> bool: + if self.history.notes.get("_obs_variant") == "high_contrast": + x, y = tile + area = frame[y * TILE_SIZE : (y + 1) * TILE_SIZE, x * TILE_SIZE : (x + 1) * TILE_SIZE] + red = np.all(area == np.array((255, 0, 0), dtype=np.uint8), axis=2) + yellow = np.all(area == np.array((255, 255, 0), dtype=np.uint8), axis=2) + return max(int(np.count_nonzero(red[9, 3:13])), int(np.count_nonzero(yellow[9, 3:13]))) >= 9 and not self.tile_has_chest(frame, tile) + switch_pixels = self.count_color(frame, SWITCH_BODY, tile) + self.count_color(frame, SWITCH_DOWN, tile) + return switch_pixels >= 20 and not self.tile_has_chest(frame, tile) + + def tile_has_button(self, frame: np.ndarray, tile: Tile) -> bool: + if self.history.notes.get("_obs_variant") == "high_contrast": + x, y = tile + area = frame[y * TILE_SIZE : (y + 1) * TILE_SIZE, x * TILE_SIZE : (x + 1) * TILE_SIZE] + green = np.all(area == np.array((0, 255, 0), dtype=np.uint8), axis=2) + return int(np.count_nonzero(green[6:10, 5:11])) >= 20 + button_pixels = self.count_color(frame, BUTTON_UP, tile) + self.count_color(frame, BUTTON_DOWN_COLOR, tile) + return button_pixels >= 20 and not self.tile_has_chest(frame, tile) + + def tile_has_wall(self, frame: np.ndarray, tile: Tile) -> bool: + if self.history.notes.get("_obs_variant") == "high_contrast": + magenta = self.count_color(frame, (255, 86, 146), tile) + # A wall tile has 28 magenta highlight pixels; a normal exit frame + # has 38 (32 when partly covered by the player). Keep the upper + # bound so exits remain traversable. + return 20 <= magenta < 30 + return self.count_color(frame, WALL_COLOR, tile) >= 80 + + def tile_has_trap(self, frame: np.ndarray, tile: Tile) -> bool: + if self.history.notes.get("_obs_variant") == "high_contrast": + return self.count_color(frame, (255, 255, 255), tile) >= 8 + metal = self.count_color(frame, TRAP_SPIKE_METAL, tile) + shade = self.count_color(frame, TRAP_SPIKE_SHADE, tile) + return metal + shade >= 15 + + def tile_has_abyss(self, frame: np.ndarray, tile: Tile) -> bool: + # Abyss tiles are rendered as (0,0,0). A tile is an unwalkable abyss + # if it is mostly black AND does not have bridge pixels on top. + black = self.count_color(frame, (0, 0, 0), tile) + bridge = self.count_color(frame, BRIDGE_COLOR, tile) + return black >= 180 and bridge < 25 + + def has_monster(self, frame: np.ndarray) -> bool: + if self.history.notes.get("_obs_variant") == "high_contrast": + return bool(self._high_contrast_monster_tiles(frame)) + return self.count_any_color(frame, MONSTER_COLORS) >= 20 + + def detect_chest_tile(self, frame: np.ndarray) -> Tile | None: + for y in range(GRID_HEIGHT): + for x in range(GRID_WIDTH): + if self.tile_has_chest(frame, (x, y)): + return (x, y) + return None + + def detect_button_tile(self, frame: np.ndarray) -> Tile | None: + for y in range(GRID_HEIGHT): + for x in range(GRID_WIDTH): + if self.tile_has_button(frame, (x, y)): + return (x, y) + return None + + def detect_switch_tile(self, frame: np.ndarray) -> Tile | None: + for y in range(GRID_HEIGHT): + for x in range(GRID_WIDTH): + if self.tile_has_switch(frame, (x, y)): + return (x, y) + return None + + def neighbors_of(self, tile: Tile) -> list[Tile]: + x, y = tile + return [(x, y - 1), (x, y + 1), (x - 1, y), (x + 1, y)] + + def build_blocked_tiles(self, frame: np.ndarray, *, avoid_traps: bool) -> set[Tile]: + # Scan the visible grid for non-walkable tiles: walls, abyss (always + # blocked — bridge tiles overlay the abyss so they are NOT flagged), + # spike traps (when requested), chests, and NPCs (always blocking). + blocked: set[Tile] = set() + for y in range(GRID_HEIGHT): + for x in range(GRID_WIDTH): + tile = (x, y) + if self.tile_has_wall(frame, tile): + blocked.add(tile) + elif self.tile_has_abyss(frame, tile): + blocked.add(tile) + elif avoid_traps and self.tile_has_trap(frame, tile): + blocked.add(tile) + elif self.tile_has_chest(frame, tile): + blocked.add(tile) + elif self.tile_has_npc(frame, tile): + blocked.add(tile) + return blocked + + def detect_task3_room(self, frame: np.ndarray) -> str: + # Scan the whole grid for npc/chest instead of checking fixed tiles, + # because spatial variants move these objects to different positions. + for y in range(GRID_HEIGHT): + for x in range(GRID_WIDTH): + if self.tile_has_npc(frame, (x, y)): + return "start_room" + for y in range(GRID_HEIGHT): + for x in range(GRID_WIDTH): + if self.tile_has_chest(frame, (x, y)): + return "key_room" + return "monster_hall" + + def detect_bridge_state(self, frame: np.ndarray) -> str | None: + probes = { + "west_to_north": ((4, 1), (5, 1), (4, 2), (5, 2)), + "west_to_east": ((7, 3), (8, 3), (7, 4), (8, 4)), + "west_to_south": ((4, 5), (5, 5), (4, 6), (5, 6)), + } + scores = { + state: sum(self.count_color(frame, BRIDGE_COLOR, tile) for tile in tiles) + for state, tiles in probes.items() + } + state, score = max(scores.items(), key=lambda item: item[1]) + if score <= 0: + return None + return state + + def detect_task4_room(self, frame: np.ndarray) -> str | None: + # The center hub is the only room with a large abyss field. Counting + # whole-frame bridge-color pixels is unsound under grayscale/high + # contrast because bridge wood collides with floor shadows or walls. + abyss_tiles = sum( + self.tile_has_abyss(frame, (x, y)) + for y in range(GRID_HEIGHT) + for x in range(GRID_WIDTH) + ) + if abyss_tiles >= 8: + return "center" + # Exits are two tiles wide. The player can obscure one frame tile while + # crossing, so one visibly open tile is sufficient. + top_middle_open = not self.tile_has_wall(frame, (4, 0)) or not self.tile_has_wall(frame, (5, 0)) + bottom_middle_open = not self.tile_has_wall(frame, (4, 7)) or not self.tile_has_wall(frame, (5, 7)) + if top_middle_open and not bottom_middle_open: + return "south" + if bottom_middle_open and not top_middle_open: + return "north" + if self.detect_switch_tile(frame) is not None: + return "west" + return "east" + + def detect_task5_room(self, frame: np.ndarray) -> str | None: + # South room has a distinctive wall pattern at y=2 (x=2-7) and y=6 (x=4). + # Detect this pattern first as it's more reliable than chest pixels on + # entry. The key chest is at (8,5) which may not render immediately. + south_walls = [(2, 2), (3, 2), (4, 2), (5, 2), (6, 2), (7, 2), (4, 6)] + if all(self.tile_has_wall(frame, t) for t in south_walls): + return "task5_south" + # East room (room_1_0) walls at (2,2), (2,3), (2,4), (5,4), (6,4). + # 选取与其他房间不冲突的独特位置：(2,2) 在 south/west 也是墙，故排除。 + east_walls = [(2, 3), (2, 4), (5, 4), (6, 4)] + if all(self.tile_has_wall(frame, t) for t in east_walls): + return "task5_east" + # West room (room_-1_0) walls at (1,2), (2,2), (5,5), (4,6), (5,6). + # 选取与其他房间不冲突的独特位置：(2,2) 在 east/south 也是墙、 + # (4,6) 在 south 也是墙，故排除。 + west_walls = [(1, 2), (5, 5), (5, 6)] + if all(self.tile_has_wall(frame, t) for t in west_walls): + return "task5_west" + # Start room (room_0_0) walls at (5,1), (5,2), (3,3), (4,3), (6,5). + # (3,3) 和 (6,5) 在 south/east/west 三个房间都不是墙，是 start 独有特征。 + # 用墙体特征而非对象位置，确保 spatial 变体（对象移位、layout 不变）下仍可识别。 + start_walls = [(3, 3), (6, 5)] + if all(self.tile_has_wall(frame, t) for t in start_walls): + return "task5_start" + return None + + def perceive(self, frame: np.ndarray, task_id: str | None) -> PixelState: + player_tile = self.detect_player_tile(frame) + room_id: str | None = None + bridge_state: str | None = None + if task_id == "mathematical_logic/task_3": + room_id = self.detect_task3_room(frame) + elif task_id == "mathematical_logic/task_4": + room_id = self.detect_task4_room(frame) + if room_id == "center": + bridge_state = self.detect_bridge_state(frame) + elif task_id == "mathematical_logic/task_5": + room_id = self.detect_task5_room(frame) + # Fallback: if detection failed, use the last known room from history. + if room_id is None: + room_id = self.history.notes.get("task5_last_room") + return PixelState( + player_tile=player_tile, + room_id=room_id, + bridge_state=bridge_state, + monster_visible=self.has_monster(frame), + ) + + def remembered_bridge_state(self, pixel_state: PixelState) -> str: + if pixel_state.bridge_state is not None: + self.history.notes["task4_bridge_state"] = pixel_state.bridge_state + return pixel_state.bridge_state + state = self.history.notes.get("task4_bridge_state") + if isinstance(state, str): + return state + self.history.notes["task4_bridge_state"] = "west_to_north" + return "west_to_north" + + def cycle_remembered_bridge_state(self) -> None: + state = self.history.notes.get("task4_bridge_state", "west_to_north") + if not isinstance(state, str): + state = "west_to_north" + self.history.notes["task4_bridge_state"] = BRIDGE_CYCLE.get(state, "west_to_north") + + def decide_task1(self, pixel_state: PixelState, inventory: dict[str, Any], frame: np.ndarray) -> int: + # Task 1: navigate to the key chest, then to the north exit. Walls are + # perceived from the frame; BFS finds the shortest tile path through the + # detected gaps. A pixel-alignment guard ensures the sprite clears 1-tile + # wall gaps instead of straddling them. No hardcoded waypoints. + tile = pixel_state.player_tile + if tile is None: + return ACTION_NOOP + player_px = self.detect_player_px(frame) + if player_px is None: + return ACTION_NOOP + keys = self.current_keys(inventory) + blocked = self.build_blocked_tiles(frame, avoid_traps=False) + if keys <= 0: + chest = self.detect_chest_tile(frame) + if chest is None: + return ACTION_NOOP + goals = {n for n in self.neighbors_of(chest) if self.in_bounds(n) and n not in blocked} + return self.follow_bfs_aligned( + player_px, + tile, + goals, + blocked, + final_action=ACTION_A, + ) + exit_goals = set(EXIT_DIRECTION_TILES["north"]) + return self.follow_bfs_aligned( + player_px, + tile, + exit_goals, + blocked, + final_action=EXIT_DIRECTION_ACTION["north"], + ) + + def decide_task2( + self, + pixel_state: PixelState, + inventory: dict[str, Any], + frame: np.ndarray, + ) -> int: + # Task 2: defeat the chaser monster, then navigate to the key chest, + # then to the west exit. Traps (border rows) are perceived from the + # frame and treated as blocked; BFS finds the shortest safe tile path. + # Pixel-alignment prevents the sprite from colliding with the chest on + # an adjacent row when moving horizontally past it. No hardcoded waypoints. + tile = pixel_state.player_tile + if tile is None: + return ACTION_NOOP + player_px = self.detect_player_px(frame) + if player_px is None: + return ACTION_NOOP + monster_tile = self.detect_monster_tile(frame) + if monster_tile is not None: + return self.attack_monster(tile, monster_tile) + keys = self.current_keys(inventory) + blocked = self.build_blocked_tiles(frame, avoid_traps=True) + if keys <= 0: + chest = self.detect_chest_tile(frame) + if chest is None: + return ACTION_NOOP + goals = {n for n in self.neighbors_of(chest) if self.in_bounds(n) and n not in blocked} + return self.follow_bfs_aligned( + player_px, + tile, + goals, + blocked, + final_action=ACTION_A, + ) + exit_goals = set(EXIT_DIRECTION_TILES["west"]) + return self.follow_bfs_aligned( + player_px, + tile, + exit_goals, + blocked, + final_action=EXIT_DIRECTION_ACTION["west"], + ) + + def decide_task3(self, pixel_state: PixelState, inventory: dict[str, Any], frame: np.ndarray) -> int: + # Task 3: three rooms connected west→east: key_room ↔ monster_hall ↔ start_room. + # No keys → navigate to the key chest (in key_room). Has keys → navigate + # to the east exit (in start_room, locked_key). BFS handles within-room + # wall avoidance automatically. + room_id = pixel_state.room_id + tile = pixel_state.player_tile + keys = self.current_keys(inventory) + if room_id is None or tile is None: + return ACTION_NOOP + + player_px = self.detect_player_px(frame) + blocked = self.build_blocked_tiles(frame, avoid_traps=False) + + if keys <= 0: + if room_id == "key_room": + chest = self.detect_chest_tile(frame) + if chest is not None: + return self.navigate_to_goal(tile, chest, blocked, player_px=player_px, final_action=ACTION_A) + # In start_room or monster_hall: head west toward key_room. + return self.navigate_to_exit(tile, "west", blocked, player_px=player_px) + + # Has keys: head east toward the locked exit in start_room. + return self.navigate_to_exit(tile, "east", blocked, player_px=player_px) + + def decide_task4( + self, + pixel_state: PixelState, + inventory: dict[str, Any], + frame: np.ndarray, + ) -> int: + # Task 4: five rooms around a center hub with a rotating bridge. + # Stage management (get_key → get_sword → kill_guardian → open_final_chest) + # and bridge-cycling logic are preserved; within-room navigation is + # replaced by BFS so the agent doesn't need hardcoded direction logic. + room_id = pixel_state.room_id + tile = pixel_state.player_tile + facing = self.history.facing + bridge_state = self.remembered_bridge_state(pixel_state) + keys = int(inventory.get("keys", 0)) + has_sword = self.has_item(inventory, "sword") + if room_id is None or tile is None: + return ACTION_NOOP + + # safe 模式下无 reward_signals，用图像感知判断 guardian 是否被击杀： + # 在 south 房间有剑且怪物不可见则视为已击败。 + if room_id == "south" and has_sword and not pixel_state.monster_visible: + self.history.notes["task4_guardian_defeated"] = True + + guardian_defeated = bool(self.history.notes.get("task4_guardian_defeated", False)) + + if keys <= 0: + stage = "get_key" + elif not has_sword: + stage = "get_sword" + elif not guardian_defeated: + stage = "kill_guardian" + else: + stage = "open_final_chest" + + desired_bridge = { + "get_key": "west_to_north", + "get_sword": "west_to_east", + "kill_guardian": "west_to_south", + "open_final_chest": "west_to_south", + }[stage] + + blocked = self.build_blocked_tiles(frame, avoid_traps=False) + player_px = self.detect_player_px(frame) + + if room_id == "west": + if bridge_state != desired_bridge: + switch = self.detect_switch_tile(frame) + if switch is not None: + # Walk to a tile adjacent to the switch, face it, interact. + face_action = self.adjacent_goal_action(tile, switch) + if face_action is not None: + if self.history.facing != { + ACTION_RIGHT: "right", + ACTION_LEFT: "left", + ACTION_DOWN: "down", + ACTION_UP: "up", + }.get(face_action): + return face_action + # Facing the switch: interact once, then update bridge memory. + self.cycle_remembered_bridge_state() + return ACTION_A + goals = {n for n in self.neighbors_of(switch) if self.in_bounds(n) and n not in blocked} + if player_px is not None: + return self.follow_bfs_aligned(player_px, tile, goals, blocked, final_action=ACTION_A) + return self.follow_bfs(tile, goals, blocked, x_first=True, final_action=ACTION_A) + # Bridge is correct: exit east. + return self.navigate_to_exit(tile, "east", blocked, player_px=player_px) + + if room_id == "center": + if stage == "get_key": + return self.navigate_to_exit(tile, "north", blocked, player_px=player_px) + if stage == "get_sword": + if bridge_state != desired_bridge: + return self.navigate_to_exit(tile, "west", blocked, player_px=player_px) + return self.navigate_to_exit(tile, "east", blocked, player_px=player_px) + if stage == "kill_guardian": + if bridge_state != desired_bridge: + return self.navigate_to_exit(tile, "west", blocked, player_px=player_px) + return self.navigate_to_exit(tile, "south", blocked, player_px=player_px) + # open_final_chest: guardian defeated, chest should be visible. + chest = self.detect_chest_tile(frame) + if chest is not None: + return self.navigate_to_goal(tile, chest, blocked, player_px=player_px, final_action=ACTION_A) + # Fallback: in spatial_c the final_chest is on a non-bridge (abyss) + # tile, so it's revealed but not rendered. Probe bridge edge tiles + # by pressing A — the engine only checks adjacency for chest + # interaction, so pressing A from a bridge tile adjacent to the + # hidden chest will open it. + probe = self.probe_hidden_chest(tile, blocked, player_px) + if probe is not None: + return probe + return self.navigate_to_exit(tile, "south", blocked, player_px=player_px) + + if room_id == "north": + if stage != "get_key": + return self.navigate_to_exit(tile, "south", blocked, player_px=player_px) + chest = self.detect_chest_tile(frame) + if chest is not None: + return self.navigate_to_goal(tile, chest, blocked, player_px=player_px, final_action=ACTION_A) + return self.navigate_to_exit(tile, "south", blocked, player_px=player_px) + + if room_id == "east": + if stage != "get_sword": + return self.navigate_to_exit(tile, "west", blocked, player_px=player_px) + chest = self.detect_chest_tile(frame) + if chest is not None: + # Need to face right to pick up the sword from the chest. + face_action = self.adjacent_goal_action(tile, chest) + if face_action is not None: + if self.history.facing != "right": + return ACTION_RIGHT + return ACTION_A + goals = {n for n in self.neighbors_of(chest) if self.in_bounds(n) and n not in blocked} + if player_px is not None: + return self.follow_bfs_aligned(player_px, tile, goals, blocked, final_action=ACTION_A) + return self.follow_bfs(tile, goals, blocked, x_first=True, final_action=ACTION_A) + return self.navigate_to_exit(tile, "west", blocked, player_px=player_px) + + if room_id == "south": + if stage == "open_final_chest": + return self.navigate_to_exit(tile, "north", blocked, player_px=player_px) + # Kill the guardian monster. + monster_tile = self.detect_monster_tile(frame) + if monster_tile is not None: + return self.attack_monster(tile, monster_tile) + return self.navigate_to_exit(tile, "north", blocked, player_px=player_px) + + return ACTION_NOOP + + def move_towards(self, tile: Tile, target: Tile, *, x_first: bool = True) -> int: + x, y = tile + target_x, target_y = target + if x_first: + if x < target_x: + return ACTION_RIGHT + if x > target_x: + return ACTION_LEFT + if y < target_y: + return ACTION_DOWN + if y > target_y: + return ACTION_UP + else: + if y < target_y: + return ACTION_DOWN + if y > target_y: + return ACTION_UP + if x < target_x: + return ACTION_RIGHT + if x > target_x: + return ACTION_LEFT + return ACTION_NOOP + + def follow_waypoints( + self, + tile: Tile, + waypoints: tuple[Tile, ...], + *, + x_first: bool = True, + final_action: int = ACTION_NOOP, + ) -> int: + for waypoint in waypoints: + if tile != waypoint: + return self.move_towards(tile, waypoint, x_first=x_first) + return final_action + + def follow_route( + self, + route_key: str, + tile: Tile, + waypoints: tuple[Tile, ...], + *, + x_first: bool = True, + final_action: int = ACTION_NOOP, + ) -> int: + index_key = f"{route_key}_waypoint_index" + index = int(self.history.notes.get(index_key, 0) or 0) + while index < len(waypoints) and tile == waypoints[index]: + index += 1 + self.history.notes[index_key] = index + if index >= len(waypoints): + return final_action + return self.move_towards(tile, waypoints[index], x_first=x_first) + + def move_towards_aligned( + self, + player_px: tuple[float, float], + tile: Tile, + target: Tile, + *, + x_first: bool = True, + ) -> int: + # Perception-driven move with a pixel-alignment guard. When the + # primary axis matches the target, align pixels before moving on the + # secondary axis to avoid straddling tile boundaries. + # + # The engine moves 1px/step and the player sprite is TILE_SIZE tall, + # so even a 1px offset makes the sprite clip into the adjacent row/ + # column. At 1-tile gaps this clips a wall and blocks movement. + # Require exact alignment (py == row_top, px == col_left) instead of + # allowing a 1px overshoot. + # + # detect_player_px can be off by 1px when the facing direction changes + # the sprite's visible tunic offset. If the alignment correction is + # blocked (reward <= -0.05), the engine position is likely already + # aligned and the detected offset is a perception artifact — skip + # alignment and fall through to the secondary axis. + px, py = player_px + x, y = tile + target_x, target_y = target + last_blocked = self.history.notes.get("mem_last_action_blocked", False) + last_act = self.history.last_action + if x_first: + if x < target_x: + return ACTION_RIGHT + if x > target_x: + return ACTION_LEFT + col_left = x * TILE_SIZE + if px < col_left and not (last_blocked and last_act == ACTION_RIGHT): + return ACTION_RIGHT + if px > col_left and not (last_blocked and last_act == ACTION_LEFT): + return ACTION_LEFT + if y < target_y: + return ACTION_DOWN + if y > target_y: + return ACTION_UP + else: + if y < target_y: + return ACTION_DOWN + if y > target_y: + return ACTION_UP + row_top = y * TILE_SIZE + if py < row_top and not (last_blocked and last_act == ACTION_DOWN): + return ACTION_DOWN + # When DOWN just succeeded, detected py may read row_top + 1 due + # to facing-change perception offset (detected = engine + 1). + # Treat as aligned and fall through to the horizontal move. + skip_up = last_act == ACTION_DOWN and py == row_top + 1 + if py > row_top and not (last_blocked and last_act == ACTION_UP) and not skip_up: + return ACTION_UP + # Detected py == row_top (or offset py == row_top + 1 after DOWN). + # If the horizontal move was blocked, the engine py is likely off + # by 1px — try DOWN (common: detected = engine + 1), then UP. + if last_blocked and last_act in (ACTION_LEFT, ACTION_RIGHT): + return ACTION_DOWN + if last_blocked and last_act == ACTION_DOWN: + return ACTION_UP + if x < target_x: + return ACTION_RIGHT + if x > target_x: + return ACTION_LEFT + return ACTION_NOOP + + def follow_route_aligned( + self, + route_key: str, + player_px: tuple[float, float], + tile: Tile, + waypoints: tuple[Tile, ...], + *, + x_first: bool = True, + final_action: int = ACTION_NOOP, + ) -> int: + if tile is None: + return ACTION_NOOP + index_key = f"{route_key}_waypoint_index" + index = int(self.history.notes.get(index_key, 0) or 0) + while index < len(waypoints) and tile == waypoints[index]: + index += 1 + self.history.notes[index_key] = index + if index >= len(waypoints): + return final_action + return self.move_towards_aligned(player_px, tile, waypoints[index], x_first=x_first) + + def bfs_path( + self, + start: Tile, + goals: set[Tile], + blocked: set[Tile], + ) -> list[Tile] | None: + # Breadth-first search over the 4-connected tile grid. ``blocked`` tiles + # are not traversed unless they are also goals (e.g. exit tiles that sit + # on the border). The path includes ``start`` as the first element. + if start in goals: + return [start] + queue: deque[Tile] = deque([start]) + parent: dict[Tile, Tile | None] = {start: None} + while queue: + current = queue.popleft() + for nxt in self.neighbors_of(current): + if not self.in_bounds(nxt): + continue + if nxt in parent: + continue + if nxt in blocked and nxt not in goals: + continue + parent[nxt] = current + if nxt in goals: + path: list[Tile] = [] + node: Tile | None = nxt + while node is not None: + path.append(node) + node = parent[node] + path.reverse() + return path + queue.append(nxt) + return None + + def follow_bfs( + self, + tile: Tile, + goals: set[Tile], + blocked: set[Tile], + *, + x_first: bool = True, + final_action: int = ACTION_NOOP, + ) -> int: + # Perception-driven navigation: recompute the BFS path every step from + # the current tile, then issue a single-tile move toward the next tile + # in the path. No waypoint indices or hardcoded sequences. + path = self.bfs_path(tile, goals, blocked) + if path is None or len(path) <= 1: + return final_action + return self.move_towards(tile, path[1], x_first=x_first) + + def follow_bfs_aligned( + self, + player_px: tuple[float, float], + tile: Tile, + goals: set[Tile], + blocked: set[Tile], + *, + final_action: int = ACTION_NOOP, + ) -> int: + # Same as follow_bfs but with a pixel-alignment guard so the sprite + # clears 1-tile wall gaps and avoids pixel-level overlap with chests on + # adjacent rows. The alignment axis is chosen automatically: before a + # horizontal move the sprite is vertically aligned (so it doesn't + # collide with blocked tiles on an adjacent row), and before a vertical + # move the sprite is horizontally aligned (so it clears 1-tile wall + # gaps). + path = self.bfs_path(tile, goals, blocked) + if path is None or len(path) <= 1: + return final_action + next_tile = path[1] + dx = next_tile[0] - tile[0] + dy = next_tile[1] - tile[1] + # Vertical move → align x first (x_first=True); horizontal move → align y first (x_first=False). + x_first = dy != 0 + return self.move_towards_aligned(player_px, tile, next_tile, x_first=x_first) + + def adjacent_goal_action( + self, + tile: Tile, + goal: Tile, + ) -> int | None: + # If tile is adjacent to goal (not on goal), return the action that + # faces toward the goal. Returns None if not adjacent. + dx = goal[0] - tile[0] + dy = goal[1] - tile[1] + if abs(dx) + abs(dy) != 1: + return None + if dx == 1: + return ACTION_RIGHT + if dx == -1: + return ACTION_LEFT + if dy == 1: + return ACTION_DOWN + return ACTION_UP + + def navigate_to_goal( + self, + tile: Tile, + goal: Tile, + blocked: set[Tile], + *, + player_px: tuple[float, float] | None = None, + final_action: int = ACTION_A, + walk_onto: bool = False, + ) -> int: + # Navigate to a goal tile and interact. Two modes: + # walk_onto=False (default): goal tile is blocking (chest, switch), so + # navigate to an adjacent tile, face the goal, then interact. + # walk_onto=True: goal tile is walkable (button), so navigate onto the + # goal tile itself, then interact. + if walk_onto: + if tile == goal: + return final_action + if player_px is not None: + return self.follow_bfs_aligned(player_px, tile, {goal}, blocked, final_action=final_action) + return self.follow_bfs(tile, {goal}, blocked, x_first=True, final_action=final_action) + face_action = self.adjacent_goal_action(tile, goal) + if face_action is not None: + # Already adjacent: face the goal, then interact on the next step. + if self.history.facing != { + ACTION_RIGHT: "right", + ACTION_LEFT: "left", + ACTION_DOWN: "down", + ACTION_UP: "up", + }.get(face_action): + return face_action + return final_action + # Not adjacent yet: BFS to any walkable neighbor of the goal. + goals = {n for n in self.neighbors_of(goal) if self.in_bounds(n) and n not in blocked} + if player_px is not None: + return self.follow_bfs_aligned(player_px, tile, goals, blocked, final_action=final_action) + return self.follow_bfs(tile, goals, blocked, x_first=True, final_action=final_action) + + def navigate_to_exit( + self, + tile: Tile, + direction: str, + blocked: set[Tile], + *, + player_px: tuple[float, float] | None = None, + ) -> int: + # BFS to the exit tiles for the given direction, then issue the + # direction action to transition. Uses pixel-alignment when available. + exit_goals = set(EXIT_DIRECTION_TILES[direction]) + exit_action = EXIT_DIRECTION_ACTION[direction] + # Boundary exit tiles are walkable, but entering one can immediately + # change rooms. Exclude every non-target exit so BFS cannot use another + # room transition as a shortcut to the requested exit. + route_blocked = blocked | { + tile + for other_direction, tiles in EXIT_DIRECTION_TILES.items() + if other_direction != direction + for tile in tiles + } + # 当 agent 已在 exit tile 旁边时，先做严格像素对齐再触发 exit， + # 避免 sprite 越过 tile 边界撞到相邻墙（spatial_c 的 west east_exit）。 + # 只处理与 exit 方向一致的相邻位置（如 south 方向只处理 agent 在 + # exit tile 正上方的情况），避免在地图边界处卡死。 + if player_px is not None: + for goal in exit_goals: + if abs(goal[0] - tile[0]) + abs(goal[1] - tile[1]) == 1: + dx = goal[0] - tile[0] + dy = goal[1] - tile[1] + if direction in ("east", "west"): + # east/west: agent 必须在 exit tile 正左/右方（y 相同） + if dy != 0: + continue + else: + # north/south: agent 必须在 exit tile 正上/下方（x 相同） + if dx != 0: + continue + px, py = player_px + if direction in ("east", "west"): + row_top = tile[1] * TILE_SIZE + if py < row_top: + return ACTION_DOWN + if py > row_top: + return ACTION_UP + else: + col_left = tile[0] * TILE_SIZE + if px < col_left: + return ACTION_RIGHT + if px > col_left: + return ACTION_LEFT + return exit_action + return self.follow_bfs_aligned(player_px, tile, exit_goals, route_blocked, final_action=exit_action) + return self.follow_bfs(tile, exit_goals, route_blocked, x_first=True, final_action=exit_action) + + def probe_hidden_chest( + self, + tile: Tile, + blocked: set[Tile], + player_px: tuple[float, float] | None, + ) -> int | None: + """Probe bridge edge tiles for a hidden final_chest. + + In spatial_c the final_chest sits on a non-bridge (abyss) tile, so it's + revealed (is_visible=True) but not rendered (0 chest pixels). The engine + only checks Manhattan adjacency for chest interaction, so pressing A from + a bridge tile adjacent to the chest will open it. This method finds + walkable tiles bordering blocked (abyss) tiles, navigates to each one, + and presses A. + """ + probed = set(tuple(t) for t in self.history.notes.get("task4_probed_tiles", [])) + + # Find bridge edge tiles: walkable tiles with ≥1 adjacent blocked tile. + bridge_edges: list[Tile] = [] + for y in range(GRID_HEIGHT): + for x in range(GRID_WIDTH): + t = (x, y) + if t in blocked or not self.in_bounds(t): + continue + for dx, dy in ((-1, 0), (1, 0), (0, -1), (0, 1)): + neighbor = (x + dx, y + dy) + if self.in_bounds(neighbor) and neighbor in blocked: + bridge_edges.append(t) + break + + # If at an unprobed bridge edge tile, press A to try opening the chest. + if tile in bridge_edges and tile not in probed: + probed.add(tile) + self.history.notes["task4_probed_tiles"] = [list(t) for t in probed] + return ACTION_A + + # Navigate to the closest unprobed bridge edge tile. + unprobed = [t for t in bridge_edges if t not in probed] + if unprobed: + unprobed.sort(key=lambda t: abs(t[0] - tile[0]) + abs(t[1] - tile[1])) + goals = {unprobed[0]} + if player_px is not None: + return self.follow_bfs_aligned(player_px, tile, goals, blocked, final_action=ACTION_A) + return self.follow_bfs(tile, goals, blocked, x_first=True, final_action=ACTION_A) + + # All probed — reset and try again. + self.history.notes["task4_probed_tiles"] = [] + return ACTION_A + + def update_task5_memory( + self, + pixel_state: PixelState, + inventory: dict[str, Any], + last_reward: float, + frame: np.ndarray, + ) -> None: + # safe 模式下无 reward_signals，基于 inventory diff + last_reward + 图像感知推断事件。 + room_id = pixel_state.room_id + notes = self.history.notes + if room_id is not None: + notes["task5_last_room"] = room_id + else: + room_id = notes.get("task5_last_room") + + # key_collected: inventory.keys 增加 + prev = self.history.last_inventory or {} + if int(inventory.get("keys", 0)) > int(prev.get("keys", 0) or 0): + notes["task5_key_collected"] = True + + # 上一步按了 A 且 last_reward > 0：可能是 chest_opened / button_pressed / monster_hit + last_action = self.history.last_action + reward_positive = last_reward > 0 + if last_action == ACTION_A and reward_positive: + # chest_opened：开 chest 后 sprite 不消失，但 agent 必须在 chest 相邻 tile 上。 + # 攻击怪物时 agent 在 monster 相邻 tile 上。用 chest_adjacent 且 not + # monster_adjacent 区分（各房间 chest/monster 相邻 tile 无交集，除西房间 + # (2,5) 同时与 chest(2,6) 和 monster(2,4) 相邻，此时不标记以避免误判）。 + chest_tile = self.detect_chest_tile(frame) + monster_tile = self.detect_monster_tile(frame) + pt = pixel_state.player_tile + chest_adjacent = ( + chest_tile is not None + and pt is not None + and abs(chest_tile[0] - pt[0]) + abs(chest_tile[1] - pt[1]) == 1 + ) + monster_adjacent = ( + monster_tile is not None + and pt is not None + and abs(monster_tile[0] - pt[0]) + abs(monster_tile[1] - pt[1]) == 1 + ) + if chest_adjacent and not monster_adjacent: + if room_id == "task5_start" and not notes.get("task5_start_chest_opened"): + notes["task5_start_chest_opened"] = True + elif room_id == "task5_south" and not notes.get("task5_south_chest_opened"): + notes["task5_south_chest_opened"] = True + elif room_id == "task5_east" and not notes.get("task5_east_chest_opened"): + notes["task5_east_chest_opened"] = True + elif room_id == "task5_west" and not notes.get("task5_west_chest_opened"): + notes["task5_west_chest_opened"] = True + # button_pressed：只有当 agent 在 button tile 上按 A 时才标记 + # （开 chest 时 agent 不在 chest tile 上，以此区分） + if not notes.get("task5_button_pressed"): + button_tile = self.detect_button_tile(frame) + if button_tile is not None and pixel_state.player_tile == button_tile: + notes["task5_button_pressed"] = True + # door_opened：上一步在出口按方向键且 last_reward > 0（此处 action==A 不覆盖） + + # monster_kill：上步有怪物 + 本步无怪物 + last_reward > 0 + # 只在 task5_start 房间且上步也在 task5_start 时标记，避免跨房间切换时 + # （如从 south 返回 start）因怪物未渲染而误判 chaser 已清除。 + prev_room = notes.get("task5_prev_room") + prev_monster = notes.get("task5_prev_monster_visible", False) + curr_monster = pixel_state.monster_visible + if (prev_monster and not curr_monster and reward_positive + and room_id == "task5_start" and prev_room == "task5_start"): + notes["task5_start_monster_cleared"] = True + notes["task5_prev_monster_visible"] = curr_monster + + # monster_hit：last_reward > 0 且怪物仍可见 + if reward_positive and curr_monster: + hits = int(notes.get("task5_start_monster_hits", 0) or 0) + notes["task5_start_monster_hits"] = hits + 1 + + # door_opened / east_gate：last_reward > 0 且房间发生变化时标记 + if prev_room is not None and room_id is not None and prev_room != room_id and reward_positive: + notes["task5_east_gate_opened"] = True + if room_id is not None: + notes["task5_prev_room"] = room_id + + def task5_stage(self, inventory: dict[str, Any]) -> str: + keys = int(inventory.get("keys", 0)) + notes = self.history.notes + if not bool(notes.get("task5_start_chest_opened", False)): + return "open_start_chest" + if not bool(notes.get("task5_button_pressed", False)): + return "press_button" + if not bool(notes.get("task5_south_chest_opened", False)): + return "open_south_chest" + if not bool(notes.get("task5_east_chest_opened", False)): + return "open_east_chest" if keys > 0 or bool(notes.get("task5_east_gate_opened", False)) else "open_south_chest" + if not bool(notes.get("task5_west_chest_opened", False)): + return "open_west_chest" + return "done" + + def decide_task5( + self, + pixel_state: PixelState, + inventory: dict[str, Any], + last_reward: float, + frame: np.ndarray, + ) -> int: + # Task 5: four rooms with chests, button, locked gate, and monsters. + # Stage management (which chest to open next) is preserved; within-room + # navigation uses BFS with pixel alignment so hardcoded waypoints and + # manual alignment ticks are eliminated. + self.update_task5_memory(pixel_state, inventory, last_reward, frame) + room_id = pixel_state.room_id + tile = pixel_state.player_tile + if room_id is None or tile is None: + return ACTION_NOOP + + stage = self.task5_stage(inventory) + if stage == "done": + return ACTION_NOOP + + # button 是踩上去自动触发的（不需要 A），被踩到后从画面消失。 + # 如果 stage=press_button 但 button 不可见，说明已被踩到。 + if stage == "press_button" and self.detect_button_tile(frame) is None: + self.history.notes["task5_button_pressed"] = True + stage = self.task5_stage(inventory) + + blocked = self.build_blocked_tiles(frame, avoid_traps=True) + player_px = self.detect_player_px(frame) + no_attack_tiles = self.compute_no_attack_tiles(frame) + combat_blocked = blocked | { + exit_tile + for exit_tiles in EXIT_DIRECTION_TILES.values() + for exit_tile in exit_tiles + } + + if room_id == "task5_start": + if stage == "open_start_chest": + chest = self.detect_chest_tile(frame) + if chest is not None: + return self.navigate_to_goal(tile, chest, blocked, player_px=player_px, final_action=ACTION_A) + + if stage in {"press_button", "open_south_chest"}: + if stage == "press_button": + button = self.detect_button_tile(frame) + if button is not None: + return self.navigate_to_goal(tile, button, blocked, player_px=player_px, final_action=ACTION_A, walk_onto=True) + return self.navigate_to_exit(tile, "south", blocked, player_px=player_px) + + if stage in {"open_east_chest", "open_west_chest"}: + # 先清除 start room 的 chaser，避免导航至 east/west exit 时受接触伤害。 + if not bool(self.history.notes.get("task5_start_monster_cleared", False)): + monster_tile = self.detect_monster_tile(frame) + if monster_tile is not None: + return self.attack_monster( + tile, + monster_tile, + blocked=combat_blocked, + player_px=player_px, + no_attack_tiles=no_attack_tiles, + ) + self.history.notes["task5_start_monster_cleared"] = True + if stage == "open_east_chest": + return self.navigate_to_exit(tile, "east", blocked, player_px=player_px) + if ( + tile in EXIT_DIRECTION_TILES["west"] + and player_px is not None + and player_px[0] <= 3.5 + and not bool(self.history.notes.get("task5_west_exit_shielded", False)) + ): + # The spatial_c west-room ambusher starts next to the entry + # spawn. Raise the shield at the room boundary so + # its six-tick active window covers the unavoidable contact. + self.history.notes["task5_west_exit_shielded"] = True + return ACTION_B + return self.navigate_to_exit(tile, "west", blocked, player_px=player_px) + + if room_id == "task5_south": + if stage == "open_south_chest": + # South room's patroller is passive; ignore it and navigate to chest directly. + chest = self.detect_chest_tile(frame) + if chest is not None: + return self.navigate_to_goal(tile, chest, blocked, player_px=player_px, final_action=ACTION_A) + # Chest not yet rendered; wait one frame instead of navigating to a + # hardcoded position that may be wrong under spatial variants. + return ACTION_NOOP + return self.navigate_to_exit(tile, "north", blocked, player_px=player_px) + + if room_id == "task5_east": + if stage == "open_east_chest": + # 跳过 ambusher 战斗：ambusher 在 (7,5) ambush_range=2，chest 在 (7,1)。 + # agent 从 (1,4) 进入走 row 1 到 chest，距离 ambusher ≥ 4 tiles 不会激活。 + # ambusher 加入 blocked 以防意外接触；west 房间用 kill all 保证无接触伤害。 + self.history.notes["task5_east_monster_cleared"] = True + monster_tile = self.detect_monster_tile(frame) + if monster_tile is not None: + blocked = blocked | {monster_tile} + chest = self.detect_chest_tile(frame) + if chest is not None: + return self.navigate_to_goal(tile, chest, blocked, player_px=player_px, final_action=ACTION_A) + return self.navigate_to_exit(tile, "west", blocked, player_px=player_px) + + if room_id == "task5_west": + if stage == "open_west_chest": + # 杀全部怪物以消除导航至宝箱时的接触伤害。 + # 注意：detect_monster_tile 返回所有怪物质心，多怪物时位置不准， + # 必须用 detect_closest_monster_tile 逐个击杀。 + if not bool(self.history.notes.get("task5_west_monster_cleared", False)): + all_monsters = self.detect_all_monster_tiles(frame) + if len(all_monsters) == 0: + self.history.notes["task5_west_monster_cleared"] = True + else: + monster_tile = self.detect_closest_monster_tile(frame, tile) + if monster_tile is not None: + return self.attack_monster( + tile, + monster_tile, + blocked=combat_blocked, + player_px=player_px, + no_attack_tiles=no_attack_tiles, + diagonal_block=False, + ) + self.history.notes["task5_west_monster_cleared"] = True + chest = self.detect_chest_tile(frame) + if chest is not None: + return self.navigate_to_goal(tile, chest, blocked, player_px=player_px, final_action=ACTION_A) + return self.navigate_to_exit(tile, "east", blocked, player_px=player_px) + + return ACTION_NOOP + + def update_facing(self, action: int) -> None: + if action == ACTION_UP: + self.history.facing = "up" + elif action == ACTION_DOWN: + self.history.facing = "down" + elif action == ACTION_LEFT: + self.history.facing = "left" + elif action == ACTION_RIGHT: + self.history.facing = "right" + + def decide( + self, + frame: np.ndarray, + inventory: dict[str, Any], + last_reward: float, + info: dict[str, Any], + ) -> int: + task_id = self.infer_task_id(info) + pixel_state = self.perceive(frame, task_id) + if task_id == "mathematical_logic/task_1": + return self.decide_task1(pixel_state, inventory, frame) + if task_id == "mathematical_logic/task_2": + return self.decide_task2(pixel_state, inventory, frame) + if task_id == "mathematical_logic/task_3": + return self.decide_task3(pixel_state, inventory, frame) + if task_id == "mathematical_logic/task_4": + return self.decide_task4(pixel_state, inventory, frame) + if task_id == "mathematical_logic/task_5": + return self.decide_task5(pixel_state, inventory, last_reward, frame) + + return ACTION_NOOP + + def act(self, obs, info: dict[str, Any]) -> int: + # color 变体预处理：检测并逆转 dark/bright/inverted，恢复近似 default 外观。 + obs = self.normalize_obs(obs) + inventory = self.extract_inventory(info) + last_reward = self.extract_last_reward(info) + self.update_memory_from_safe(inventory, last_reward, obs) + action = self.decide(obs, inventory, last_reward, info) + self.history.last_inventory = dict(inventory) + self.update_facing(action) + self.history.last_action = action + return action + + +def make_policy() -> Policy: + return Policy() diff --git a/student_agent/lean/KillMonsterFormalization.lean b/student_agent/lean/KillMonsterFormalization.lean new file mode 100644 index 0000000..bc93739 --- /dev/null +++ b/student_agent/lean/KillMonsterFormalization.lean @@ -0,0 +1,742 @@ +/-! + A small Lean4 formalization corresponding to `z_new_km_fw2.py`. + + The Python file extracts a symbolic state from NesyLink observations and uses a + hand-written symbolic policy: + + * move only to safe tiles; + * if adjacent to a monster and HP is high enough, attack; + * otherwise move toward a monster/chest/exit target. + + This Lean file formalizes the symbolic layer, not the pixel-level game engine. + It is intentionally small enough to compile without Mathlib. +-/ + +namespace KillMonsterFormalization + +abbrev Position := Nat × Nat + +inductive Action where + | wait + | up + | down + | left + | right + | attack + | shield + deriving DecidableEq, Repr + +inductive GoalType where + | killMonster + | openChest + | goExit + deriving DecidableEq, Repr + +structure Goal where + kind : GoalType + target : Position + approachTiles : List Position + deriving Repr + +structure SymbolicState where + player : Position + exits : List Position + walls : List Position + traps : List Position + monsters : List Position + chests : List Position + health : Nat + keys : Nat + deriving DecidableEq, Repr + +def hasKey (s : SymbolicState) : Prop := + s.keys > 0 + +def needKey (s : SymbolicState) : Prop := + s.keys = 0 + +def manhattan (a b : Position) : Nat := + let dx := if a.1 ≤ b.1 then b.1 - a.1 else a.1 - b.1 + let dy := if a.2 ≤ b.2 then b.2 - a.2 else a.2 - b.2 + dx + dy + +def adjacent (a b : Position) : Prop := + manhattan a b = 1 + +def inBounds (p : Position) : Prop := + p.1 < 10 ∧ p.2 < 8 + +def dangerTiles (s : SymbolicState) : List Position := + s.monsters + +def isSafe (s : SymbolicState) (p : Position) : Prop := + inBounds p ∧ + p ∉ s.walls ∧ + p ∉ s.traps ∧ + p ∉ s.monsters ∧ + p ∉ s.chests ∧ + p ∉ dangerTiles s + +def nextPosition (p : Position) : Action → Position + | Action.up => (p.1, p.2 - 1) + | Action.down => (p.1, p.2 + 1) + | Action.left => (p.1 - 1, p.2) + | Action.right => (p.1 + 1, p.2) + | _ => p + +def adjacentMonster (s : SymbolicState) : Prop := + ∃ m, m ∈ s.monsters ∧ adjacent s.player m + +def adjacentChest (s : SymbolicState) : Prop := + ∃ c, c ∈ s.chests ∧ adjacent s.player c + +def canAttack (s : SymbolicState) (m : Position) : Prop := + m ∈ s.monsters ∧ adjacent s.player m ∧ s.health > 1 + +def canOpenChest (s : SymbolicState) (c : Position) : Prop := + c ∈ s.chests ∧ adjacent s.player c + +/-! + `Step s a t` is the relational version of Python `symbolic_step`. + + It is deliberately nondeterministic for attacks and chest opening because the + Python code chooses the first adjacent monster/chest from a set. In Lean, we + instead say that any adjacent attackable monster, or any adjacent openable + chest, may be selected. +-/ +inductive Step : SymbolicState → Action → SymbolicState → Prop where + | moveSafe + {s : SymbolicState} {a : Action} : + a ∈ [Action.up, Action.down, Action.left, Action.right] → + isSafe s (nextPosition s.player a) → + Step s a { s with player := nextPosition s.player a } + | moveBlocked + {s : SymbolicState} {a : Action} : + a ∈ [Action.up, Action.down, Action.left, Action.right] → + ¬ isSafe s (nextPosition s.player a) → + Step s a s + | attackMonster + {s : SymbolicState} {m : Position} : + canAttack s m → + Step s Action.attack { s with monsters := s.monsters.erase m } + | openChest + {s : SymbolicState} {c : Position} : + ¬ adjacentMonster s → + canOpenChest s c → + Step s Action.attack { s with chests := s.chests.erase c, keys := s.keys + 1 } + | attackNoEffect + {s : SymbolicState} : + ¬ adjacentMonster s → + ¬ adjacentChest s → + Step s Action.attack s + | wait + {s : SymbolicState} : + Step s Action.wait s + | shield + {s : SymbolicState} : + Step s Action.shield s + +inductive Exec : SymbolicState → List Action → SymbolicState → Prop where + | nil {s : SymbolicState} : + Exec s [] s + | cons {s t u : SymbolicState} {a : Action} {rest : List Action} : + Step s a t → + Exec t rest u → + Exec s (a :: rest) u + +def GoalReached (s : SymbolicState) : Prop := + s.monsters = [] ∧ s.keys > 0 ∧ s.player ∈ s.exits + +def TaskCompletable (s : SymbolicState) : Prop := + ∃ plan final, Exec s plan final ∧ GoalReached final + +def SafeState (s : SymbolicState) : Prop := + inBounds s.player ∧ + s.player ∉ s.walls ∧ + s.player ∉ s.traps + +theorem moveSafe_player_eq + {s t : SymbolicState} {a : Action} + (h : Step s a t) + (ha : a ∈ [Action.up, Action.down, Action.left, Action.right]) + (hsafe : isSafe s (nextPosition s.player a)) : + t.player = nextPosition s.player a := by + cases h with + | moveSafe hmove hsafe' => + rfl + | moveBlocked hmove hblocked => + exact False.elim (hblocked hsafe) + | attackMonster hattack => + cases ha <;> contradiction + | openChest hnoMonster hchest => + cases ha <;> contradiction + | attackNoEffect hnoMonster hnoChest => + cases ha <;> contradiction + | wait => + cases ha <;> contradiction + | shield => + cases ha <;> contradiction + +theorem moveBlocked_player_eq + {s t : SymbolicState} {a : Action} + (h : Step s a t) + (ha : a ∈ [Action.up, Action.down, Action.left, Action.right]) + (hunsafe : ¬ isSafe s (nextPosition s.player a)) : + t.player = s.player := by + cases h with + | moveSafe hmove hsafe => + exact False.elim (hunsafe hsafe) + | moveBlocked hmove hblocked => + rfl + | attackMonster hattack => + cases ha <;> contradiction + | openChest hnoMonster hchest => + cases ha <;> contradiction + | attackNoEffect hnoMonster hnoChest => + cases ha <;> contradiction + | wait => + cases ha <;> contradiction + | shield => + cases ha <;> contradiction + +theorem safe_move_preserves_safe_state + {s t : SymbolicState} {a : Action} + (h : Step s a t) + (ha : a ∈ [Action.up, Action.down, Action.left, Action.right]) + (hsafe : isSafe s (nextPosition s.player a)) : + SafeState t := by + cases h with + | moveSafe hmove hsafe' => + rcases hsafe' with ⟨hin, hnwall, hntrap, hnmonster, hnchest, hndanger⟩ + unfold SafeState + exact ⟨hin, hnwall, hntrap⟩ + | moveBlocked hmove hblocked => + exact False.elim (hblocked hsafe) + | attackMonster hattack => + cases ha <;> contradiction + | openChest hnoMonster hchest => + cases ha <;> contradiction + | attackNoEffect hnoMonster hnoChest => + cases ha <;> contradiction + | wait => + cases ha <;> contradiction + | shield => + cases ha <;> contradiction + +theorem attack_monster_updates_monsters + {s t : SymbolicState} {m : Position} + (_h : canAttack s m) + (_hstep : Step s Action.attack t) + (hexact : t.monsters = s.monsters.erase m) : + t.monsters = s.monsters.erase m := + hexact + +theorem open_chest_increases_keys + {s t : SymbolicState} {c : Position} + (_hnoMonster : ¬ adjacentMonster s) + (_hchest : canOpenChest s c) + (_hstep : Step s Action.attack t) + (hexact : t = { s with chests := s.chests.erase c, keys := s.keys + 1 }) : + t.keys = s.keys + 1 := by + rw [hexact] + +theorem exec_cons_inv + {s u : SymbolicState} {a : Action} {rest : List Action} + (h : Exec s (a :: rest) u) : + ∃ t, Step s a t ∧ Exec t rest u := by + cases h with + | cons hstep hexec => + exact ⟨_, hstep, hexec⟩ + +theorem exec_append + {s t u : SymbolicState} {p q : List Action} + (hp : Exec s p t) + (hq : Exec t q u) : + Exec s (p ++ q) u := by + induction hp with + | nil => + exact hq + | cons hstep hexec ih => + exact Exec.cons hstep (ih hq) + +/-! + A compositional reachability theorem for the task. + + Read it as: + + If there is a plan to reach a monster-adjacent attack state, attacking updates + the monster list, there is then a plan to reach a chest-adjacent state, opening + the chest updates the key count, and there is finally a plan to reach an exit + state with no monsters and at least one key, then the initial state is + completable. + + This does not prove BFS completeness. It proves the sound composition property + that the Python agent relies on: the three symbolic phases can be stitched into + one valid plan. +-/ +theorem task_completable_if_subplans_exist + {init nearMonster afterKill nearChest afterChest final : SymbolicState} + {toMonster toChest toExit : List Action} + {monster chest : Position} + (hToMonster : Exec init toMonster nearMonster) + (hAttackable : canAttack nearMonster monster) + (hAfterKill : afterKill = + { nearMonster with monsters := nearMonster.monsters.erase monster }) + (hToChest : Exec afterKill toChest nearChest) + (hNoAdjacentMonster : ¬ adjacentMonster nearChest) + (hOpenable : canOpenChest nearChest chest) + (hAfterChest : afterChest = + { nearChest with chests := nearChest.chests.erase chest, keys := nearChest.keys + 1 }) + (hToExit : Exec afterChest toExit final) + (hFinalMonsters : final.monsters = []) + (hFinalKeys : final.keys > 0) + (hFinalAtExit : final.player ∈ final.exits) : + TaskCompletable init := by + have hAttackStep : Step nearMonster Action.attack afterKill := by + rw [hAfterKill] + exact Step.attackMonster hAttackable + have hAttackExec : Exec nearMonster [Action.attack] afterKill := by + exact Exec.cons hAttackStep Exec.nil + + have hOpenStep : Step nearChest Action.attack afterChest := by + rw [hAfterChest] + exact Step.openChest hNoAdjacentMonster hOpenable + have hOpenExec : Exec nearChest [Action.attack] afterChest := by + exact Exec.cons hOpenStep Exec.nil + + have hPhase1 : Exec init (toMonster ++ [Action.attack]) afterKill := + exec_append hToMonster hAttackExec + have hPhase2 : Exec init ((toMonster ++ [Action.attack]) ++ toChest) nearChest := + exec_append hPhase1 hToChest + have hPhase3 : + Exec init (((toMonster ++ [Action.attack]) ++ toChest) ++ [Action.attack]) afterChest := + exec_append hPhase2 hOpenExec + have hAll : + Exec init + ((((toMonster ++ [Action.attack]) ++ toChest) ++ [Action.attack]) ++ toExit) + final := + exec_append hPhase3 hToExit + + exact ⟨ + ((((toMonster ++ [Action.attack]) ++ toChest) ++ [Action.attack]) ++ toExit), + final, + hAll, + ⟨hFinalMonsters, hFinalKeys, hFinalAtExit⟩ + ⟩ + +/-! + A lightweight BFS completeness statement. + + The Python reference implementation uses BFS as a path planner. Proving the + exact queue-and-parent implementation complete would require formalizing the + queue, visited set, parent map, and finite-state invariant. + + For the course project, the central proof obligation can be separated out: + + * `BfsFrontierComplete init n frontier` says that the BFS frontier/list has + covered every state reachable from `init` by a plan of length at most `n`. + * `BoundedGoalReachable init n` says that there exists a successful plan of + length at most `n`. + + The theorem below proves the completeness step: once the BFS frontier has this + coverage invariant, any bounded reachable goal must be detected in it. +-/ + +def BoundedReachable (init : SymbolicState) (n : Nat) (target : SymbolicState) : Prop := + ∃ plan, plan.length ≤ n ∧ Exec init plan target + +def BoundedGoalReachable (init : SymbolicState) (n : Nat) : Prop := + ∃ final, BoundedReachable init n final ∧ GoalReached final + +def BfsFrontierComplete + (init : SymbolicState) + (n : Nat) + (frontier : List SymbolicState) : Prop := + ∀ final, BoundedReachable init n final → final ∈ frontier + +def BfsFindsGoal (frontier : List SymbolicState) : Prop := + ∃ final, final ∈ frontier ∧ GoalReached final + +theorem bfs_completeness_from_frontier_invariant + {init : SymbolicState} {n : Nat} {frontier : List SymbolicState} + (hcomplete : BfsFrontierComplete init n frontier) + (hreachable : BoundedGoalReachable init n) : + BfsFindsGoal frontier := by + rcases hreachable with ⟨final, hbounded, hgoal⟩ + exact ⟨final, hcomplete final hbounded, hgoal⟩ + +/-! + An executable BFS layer for the deterministic symbolic transition function. + + This is closer to the Python planner: repeatedly expand a frontier by applying + all available actions. It omits queue ordering and parent reconstruction, but + it is a genuine BFS reachability computation by depth. +-/ + +def positionEqB (a b : Position) : Bool := + a.1 == b.1 && a.2 == b.2 + +def containsPosition (p : Position) : List Position → Bool + | [] => false + | q :: rest => if positionEqB p q then true else containsPosition p rest + +def inBoundsB (p : Position) : Bool := + decide (p.1 < 10) && decide (p.2 < 8) + +def isSafeB (s : SymbolicState) (p : Position) : Bool := + inBoundsB p && + !containsPosition p s.walls && + !containsPosition p s.traps && + !containsPosition p s.monsters && + !containsPosition p s.chests && + !containsPosition p (dangerTiles s) + +def adjacentB (a b : Position) : Bool := + manhattan a b == 1 + +def canAttackB (s : SymbolicState) (m : Position) : Bool := + containsPosition m s.monsters && + adjacentB s.player m && + decide (s.health > 1) + +def canOpenChestB (s : SymbolicState) (c : Position) : Bool := + containsPosition c s.chests && + adjacentB s.player c + +def findPosition? (xs : List Position) (p : Position → Bool) : Option Position := + match xs with + | [] => none + | x :: rest => if p x then some x else findPosition? rest p + +def symbolicStepFn (s : SymbolicState) : Action → SymbolicState + | Action.up => + let p := nextPosition s.player Action.up + if isSafeB s p then { s with player := p } else s + | Action.down => + let p := nextPosition s.player Action.down + if isSafeB s p then { s with player := p } else s + | Action.left => + let p := nextPosition s.player Action.left + if isSafeB s p then { s with player := p } else s + | Action.right => + let p := nextPosition s.player Action.right + if isSafeB s p then { s with player := p } else s + | Action.attack => + match findPosition? s.monsters (canAttackB s) with + | some monster => { s with monsters := s.monsters.erase monster } + | none => + match findPosition? s.chests (canOpenChestB s) with + | some chest => { s with chests := s.chests.erase chest, keys := s.keys + 1 } + | none => s + | Action.wait => s + | Action.shield => s + +def runPlanFn : SymbolicState → List Action → SymbolicState + | s, [] => s + | s, a :: rest => runPlanFn (symbolicStepFn s a) rest + +def allActions : List Action := + [ + Action.wait, + Action.up, + Action.down, + Action.left, + Action.right, + Action.attack, + Action.shield + ] + +theorem action_mem_allActions (a : Action) : a ∈ allActions := by + cases a <;> simp [allActions] + +def expandFrontier : List SymbolicState → List SymbolicState + | [] => [] + | s :: rest => allActions.map (symbolicStepFn s) ++ expandFrontier rest + +def bfsVisited : Nat → List SymbolicState → List SymbolicState + | 0, frontier => frontier + | n + 1, frontier => frontier ++ bfsVisited n (expandFrontier frontier) + +def bfsVisitedFrom (init : SymbolicState) (n : Nat) : List SymbolicState := + bfsVisited n [init] + +theorem mem_expandFrontier_of_mem + {s : SymbolicState} {frontier : List SymbolicState} + (a : Action) + (hs : s ∈ frontier) : + symbolicStepFn s a ∈ expandFrontier frontier := by + induction frontier with + | nil => + cases hs + | cons head tail ih => + simp [expandFrontier] at hs ⊢ + rcases hs with hEq | hTail + · subst hEq + exact Or.inl ⟨a, action_mem_allActions a, rfl⟩ + · exact Or.inr (ih hTail) + +theorem runPlanFn_mem_bfsVisited_of_mem + {s : SymbolicState} {frontier : List SymbolicState} + (plan : List Action) + {n : Nat} + (hs : s ∈ frontier) + (hlen : plan.length ≤ n) : + runPlanFn s plan ∈ bfsVisited n frontier := by + induction n generalizing s frontier plan with + | zero => + cases plan with + | nil => + simpa [runPlanFn, bfsVisited] using hs + | cons a rest => + cases hlen + | succ n ih => + cases plan with + | nil => + simp [runPlanFn, bfsVisited, hs] + | cons a rest => + have hRestLen : rest.length ≤ n := Nat.succ_le_succ_iff.mp hlen + have hStepMem : symbolicStepFn s a ∈ expandFrontier frontier := + mem_expandFrontier_of_mem a hs + have hRest : + runPlanFn (symbolicStepFn s a) rest ∈ bfsVisited n (expandFrontier frontier) := + ih rest hStepMem hRestLen + simp [runPlanFn, bfsVisited] + exact Or.inr hRest + +def FnBoundedReachable (init : SymbolicState) (n : Nat) (target : SymbolicState) : Prop := + ∃ plan, plan.length ≤ n ∧ runPlanFn init plan = target + +def FnBfsFrontierComplete + (init : SymbolicState) + (n : Nat) + (visited : List SymbolicState) : Prop := + ∀ target, FnBoundedReachable init n target → target ∈ visited + +theorem bfsVisited_frontier_invariant + (init : SymbolicState) + (n : Nat) : + FnBfsFrontierComplete init n (bfsVisitedFrom init n) := by + intro target hreachable + rcases hreachable with ⟨plan, hlen, hrun⟩ + rw [← hrun] + exact runPlanFn_mem_bfsVisited_of_mem + (s := init) + (frontier := [init]) + plan + (by simp) + hlen + +def FnBoundedGoalReachable (init : SymbolicState) (n : Nat) : Prop := + ∃ final, FnBoundedReachable init n final ∧ GoalReached final + +def FnBfsFindsGoal (visited : List SymbolicState) : Prop := + ∃ final, final ∈ visited ∧ GoalReached final + +theorem bfsVisited_complete_for_bounded_goal + {init : SymbolicState} {n : Nat} + (hreachable : FnBoundedGoalReachable init n) : + FnBfsFindsGoal (bfsVisitedFrom init n) := by + rcases hreachable with ⟨final, hbounded, hgoal⟩ + exact ⟨final, bfsVisited_frontier_invariant init n final hbounded, hgoal⟩ + +/-! + A concrete witness for the simplified `task_2` model. + + This abstracts the map in `nesylink/map_data/mathematical_logic/task_2/room_001.json`. + The real environment gives the monster HP 2, while `Step.attackMonster` above + follows `z_new_km_fw2.py`'s simplified symbolic model where one attack removes + an adjacent monster. +-/ + +def task2Monster : Position := (2, 2) + +def task2Chest : Position := (1, 3) + +def task2Exits : List Position := [(0, 3), (0, 4)] + +def task2Traps : List Position := + [ + (1, 0), (2, 0), (3, 0), (4, 0), (5, 0), (6, 0), (7, 0), (8, 0), + (1, 7), (2, 7), (3, 7), (4, 7), (5, 7), (6, 7), (7, 7), (8, 7) + ] + +def task2Init : SymbolicState := + { + player := (7, 3), + exits := task2Exits, + walls := [], + traps := task2Traps, + monsters := [task2Monster], + chests := [task2Chest], + health := 5, + keys := 0 + } + +def task2ToMonster : List Action := + [Action.left, Action.left, Action.left, Action.left, Action.up] + +def task2M1 : SymbolicState := + { task2Init with player := (6, 3) } + +def task2M2 : SymbolicState := + { task2M1 with player := (5, 3) } + +def task2M3 : SymbolicState := + { task2M2 with player := (4, 3) } + +def task2M4 : SymbolicState := + { task2M3 with player := (3, 3) } + +def task2NearMonster : SymbolicState := + { task2M4 with player := (3, 2) } + +def task2AfterKill : SymbolicState := + { task2NearMonster with monsters := task2NearMonster.monsters.erase task2Monster } + +def task2ToChest : List Action := + [Action.left, Action.left] + +def task2C1 : SymbolicState := + { task2AfterKill with player := (2, 2) } + +def task2NearChest : SymbolicState := + { task2C1 with player := (1, 2) } + +def task2AfterChest : SymbolicState := + { task2NearChest with chests := task2NearChest.chests.erase task2Chest, keys := task2NearChest.keys + 1 } + +def task2ToExit : List Action := + [Action.left, Action.down] + +def task2E1 : SymbolicState := + { task2AfterChest with player := (0, 2) } + +def task2Final : SymbolicState := + { task2E1 with player := (0, 3) } + +theorem task2_exec_to_monster : + Exec task2Init task2ToMonster task2NearMonster := by + unfold task2ToMonster + apply Exec.cons (t := task2M1) + · simpa [task2M1, task2Init, task2Exits, task2Traps, task2Monster, task2Chest] + using (Step.moveSafe (s := task2Init) (a := Action.left) (by decide) + (by simp [isSafe, inBounds, dangerTiles, nextPosition, task2Init, task2Exits, task2Traps, task2Monster, task2Chest])) + apply Exec.cons (t := task2M2) + · simpa [task2M2, task2M1, task2Init, task2Exits, task2Traps, task2Monster, task2Chest] + using (Step.moveSafe (s := task2M1) (a := Action.left) (by decide) + (by simp [isSafe, inBounds, dangerTiles, nextPosition, task2M1, task2Init, task2Exits, task2Traps, task2Monster, task2Chest])) + apply Exec.cons (t := task2M3) + · simpa [task2M3, task2M2, task2M1, task2Init, task2Exits, task2Traps, task2Monster, task2Chest] + using (Step.moveSafe (s := task2M2) (a := Action.left) (by decide) + (by simp [isSafe, inBounds, dangerTiles, nextPosition, task2M2, task2M1, task2Init, task2Exits, task2Traps, task2Monster, task2Chest])) + apply Exec.cons (t := task2M4) + · simpa [task2M4, task2M3, task2M2, task2M1, task2Init, task2Exits, task2Traps, task2Monster, task2Chest] + using (Step.moveSafe (s := task2M3) (a := Action.left) (by decide) + (by simp [isSafe, inBounds, dangerTiles, nextPosition, task2M3, task2M2, task2M1, task2Init, task2Exits, task2Traps, task2Monster, task2Chest])) + apply Exec.cons (t := task2NearMonster) + · simpa [task2NearMonster, task2M4, task2M3, task2M2, task2M1, task2Init, task2Exits, task2Traps, task2Monster, task2Chest] + using (Step.moveSafe (s := task2M4) (a := Action.up) (by decide) + (by simp [isSafe, inBounds, dangerTiles, nextPosition, task2M4, task2M3, task2M2, task2M1, task2Init, task2Exits, task2Traps, task2Monster, task2Chest])) + exact Exec.nil + +theorem task2_can_attack : + canAttack task2NearMonster task2Monster := by + unfold canAttack task2NearMonster task2M4 task2M3 task2M2 task2M1 task2Init task2Monster adjacent manhattan + decide + +theorem task2_exec_to_chest : + Exec task2AfterKill task2ToChest task2NearChest := by + unfold task2ToChest + apply Exec.cons (t := task2C1) + · simpa [task2C1, task2AfterKill, task2NearMonster, task2M4, task2M3, task2M2, task2M1, + task2Init, task2Exits, task2Traps, task2Monster, task2Chest] + using (Step.moveSafe (s := task2AfterKill) (a := Action.left) (by decide) + (by simp [isSafe, inBounds, dangerTiles, nextPosition, task2AfterKill, task2NearMonster, task2M4, task2M3, + task2M2, task2M1, task2Init, task2Exits, task2Traps, task2Monster, task2Chest])) + apply Exec.cons (t := task2NearChest) + · simpa [task2NearChest, task2C1, task2AfterKill, task2NearMonster, task2M4, task2M3, task2M2, task2M1, + task2Init, task2Exits, task2Traps, task2Monster, task2Chest] + using (Step.moveSafe (s := task2C1) (a := Action.left) (by decide) + (by simp [isSafe, inBounds, dangerTiles, nextPosition, task2C1, task2AfterKill, task2NearMonster, task2M4, + task2M3, task2M2, task2M1, task2Init, task2Exits, task2Traps, task2Monster, task2Chest])) + exact Exec.nil + +theorem task2_no_adjacent_monster_near_chest : + ¬ adjacentMonster task2NearChest := by + unfold adjacentMonster task2NearChest task2C1 task2AfterKill task2NearMonster task2M4 task2M3 task2M2 task2M1 task2Init + unfold task2Monster + intro h + rcases h with ⟨m, hm, hadj⟩ + simp at hm + +theorem task2_can_open_chest : + canOpenChest task2NearChest task2Chest := by + unfold canOpenChest task2NearChest task2C1 task2AfterKill task2NearMonster task2M4 task2M3 task2M2 task2M1 task2Init + unfold task2Chest adjacent manhattan + decide + +theorem task2_exec_to_exit : + Exec task2AfterChest task2ToExit task2Final := by + unfold task2ToExit + apply Exec.cons (t := task2E1) + · simpa [task2E1, task2AfterChest, task2NearChest, task2C1, task2AfterKill, task2NearMonster, + task2M4, task2M3, task2M2, task2M1, task2Init, task2Exits, task2Traps, task2Monster, task2Chest] + using (Step.moveSafe (s := task2AfterChest) (a := Action.left) (by decide) + (by simp [isSafe, inBounds, dangerTiles, nextPosition, task2AfterChest, task2NearChest, task2C1, task2AfterKill, + task2NearMonster, task2M4, task2M3, task2M2, task2M1, task2Init, task2Exits, task2Traps, task2Monster, task2Chest])) + apply Exec.cons (t := task2Final) + · simpa [task2Final, task2E1, task2AfterChest, task2NearChest, task2C1, task2AfterKill, task2NearMonster, + task2M4, task2M3, task2M2, task2M1, task2Init, task2Exits, task2Traps, task2Monster, task2Chest] + using (Step.moveSafe (s := task2E1) (a := Action.down) (by decide) + (by simp [isSafe, inBounds, dangerTiles, nextPosition, task2E1, task2AfterChest, task2NearChest, task2C1, + task2AfterKill, task2NearMonster, task2M4, task2M3, task2M2, task2M1, task2Init, task2Exits, task2Traps, + task2Monster, task2Chest])) + exact Exec.nil + +theorem task2_final_monsters : + task2Final.monsters = [] := by + unfold task2Final task2E1 task2AfterChest task2NearChest task2C1 task2AfterKill task2NearMonster + unfold task2M4 task2M3 task2M2 task2M1 task2Init + unfold task2Monster + decide + +theorem task2_final_keys : + task2Final.keys > 0 := by + unfold task2Final task2E1 task2AfterChest task2NearChest task2C1 task2AfterKill task2NearMonster + unfold task2M4 task2M3 task2M2 task2M1 task2Init + decide + +theorem task2_final_at_exit : + task2Final.player ∈ task2Final.exits := by + unfold task2Final task2E1 task2AfterChest task2NearChest task2C1 task2AfterKill task2NearMonster + unfold task2M4 task2M3 task2M2 task2M1 task2Init + unfold task2Exits + decide + +theorem task2_concrete_completable : + TaskCompletable task2Init := by + exact task_completable_if_subplans_exist + (init := task2Init) + (nearMonster := task2NearMonster) + (afterKill := task2AfterKill) + (nearChest := task2NearChest) + (afterChest := task2AfterChest) + (final := task2Final) + (toMonster := task2ToMonster) + (toChest := task2ToChest) + (toExit := task2ToExit) + (monster := task2Monster) + (chest := task2Chest) + task2_exec_to_monster + task2_can_attack + rfl + task2_exec_to_chest + task2_no_adjacent_monster_near_chest + task2_can_open_chest + rfl + task2_exec_to_exit + task2_final_monsters + task2_final_keys + task2_final_at_exit + +end KillMonsterFormalization diff --git a/student_agent/lean/StrategyFramework.lean b/student_agent/lean/StrategyFramework.lean new file mode 100644 index 0000000..e54be33 --- /dev/null +++ b/student_agent/lean/StrategyFramework.lean @@ -0,0 +1,300 @@ +/-! + A reusable symbolic strategy framework for the NesyLink mathematical logic + project. + + The task-specific Lean files prove concrete stage-machine policies for + task_1 through task_5. This file factors out the common proof layer: + + * executing an explicit plan; + * extracting a finite trace from a state-driven policy; + * checking that all actions along a trace satisfy an action mask; + * proving bounded completeness of breadth-first symbolic search. + + The BFS theorem is intentionally stated for an abstract deterministic + transition function. This matches the project's verified layer: each task + supplies its own symbolic state, action type, transition function, action + mask, and goal predicate. +-/ + +namespace StrategyFramework + +section Generic + +variable {State Action : Type} + +/-! ### Plan and policy execution -/ + +/-- Execute a finite list of actions from an initial symbolic state. -/ +def runPlan (step : State → Action → State) : State → List Action → State + | s, [] => s + | s, a :: rest => runPlan step (step s a) rest + +/-- Extract the first `n` actions generated by a state-driven policy. -/ +def policyTrace + (policy : State → Action) + (step : State → Action → State) : + Nat → State → List Action + | 0, _ => [] + | n + 1, s => + let a := policy s + a :: policyTrace policy step n (step s a) + +/-- Run a state-driven policy for exactly `n` symbolic steps. -/ +def runPolicy + (policy : State → Action) + (step : State → Action → State) : + Nat → State → State + | 0, s => s + | n + 1, s => runPolicy policy step n (step s (policy s)) + +theorem policyTrace_length + (policy : State → Action) + (step : State → Action → State) : + ∀ n s, (policyTrace policy step n s).length = n := by + intro n + induction n with + | zero => + intro s + rfl + | succ n ih => + intro s + simp [policyTrace, ih] + +theorem runPolicy_eq_runPlan_policyTrace + (policy : State → Action) + (step : State → Action → State) : + ∀ n s, + runPolicy policy step n s = + runPlan step s (policyTrace policy step n s) := by + intro n + induction n with + | zero => + intro s + rfl + | succ n ih => + intro s + simp [runPolicy, policyTrace, runPlan, ih] + +/-! ### Action-mask certificates -/ + +/-- +Check that every action in a finite plan is enabled at the state where it is +executed. The task files instantiate `enabled` with their symbolic precondition +checks, such as key, bridge, button, and chest conditions. +-/ +def actionsEnabledAlong + (enabled : State → Action → Bool) + (step : State → Action → State) : + State → List Action → Bool + | _, [] => true + | s, a :: rest => + enabled s a && actionsEnabledAlong enabled step (step s a) rest + +def SafeTrace + (enabled : State → Action → Bool) + (step : State → Action → State) + (init : State) + (plan : List Action) : Prop := + actionsEnabledAlong enabled step init plan = true + +theorem policyTrace_safety_certificate + (enabled : State → Action → Bool) + (policy : State → Action) + (step : State → Action → State) + (init : State) + (n : Nat) + (h : + actionsEnabledAlong enabled step init + (policyTrace policy step n init) = true) : + SafeTrace enabled step init (policyTrace policy step n init) := by + exact h + +/-! ### Bounded reachability and BFS completeness -/ + +/-- A target state is reachable by some plan of length at most `n`. -/ +def BoundedReachable + (step : State → Action → State) + (init : State) + (n : Nat) + (target : State) : Prop := + ∃ plan, plan.length ≤ n ∧ runPlan step init plan = target + +/-- A goal state is reachable by some plan of length at most `n`. -/ +def BoundedGoalReachable + (step : State → Action → State) + (goal : State → Prop) + (init : State) + (n : Nat) : Prop := + ∃ final, BoundedReachable step init n final ∧ goal final + +/-- One BFS expansion layer from a frontier, using the complete action list. -/ +def expandFrontier + (step : State → Action → State) + (actions : List Action) : + List State → List State + | [] => [] + | s :: rest => actions.map (step s) ++ expandFrontier step actions rest + +/-- +All states visited by a depth-limited BFS. The definition keeps earlier +frontiers in the list, so a goal found at depth less than `n` remains visible +at depth `n`. +-/ +def bfsVisited + (step : State → Action → State) + (actions : List Action) : + Nat → List State → List State + | 0, frontier => frontier + | n + 1, frontier => + frontier ++ bfsVisited step actions n (expandFrontier step actions frontier) + +def bfsVisitedFrom + (step : State → Action → State) + (actions : List Action) + (init : State) + (n : Nat) : List State := + bfsVisited step actions n [init] + +def BfsFrontierComplete + (step : State → Action → State) + (init : State) + (n : Nat) + (visited : List State) : Prop := + ∀ target, BoundedReachable step init n target → target ∈ visited + +def BfsFindsGoal + (goal : State → Prop) + (visited : List State) : Prop := + ∃ final, final ∈ visited ∧ goal final + +theorem mem_expandFrontier_of_mem + (step : State → Action → State) + (actions : List Action) + (hActions : ∀ a : Action, a ∈ actions) + (a : Action) + {s : State} + {frontier : List State} + (hs : s ∈ frontier) : + step s a ∈ expandFrontier step actions frontier := by + induction frontier with + | nil => + cases hs + | cons head tail ih => + simp [expandFrontier] at hs ⊢ + rcases hs with hEq | hTail + · subst hEq + exact Or.inl ⟨a, hActions a, rfl⟩ + · exact Or.inr (ih hTail) + +theorem runPlan_mem_bfsVisited_of_mem + (step : State → Action → State) + (actions : List Action) + (hActions : ∀ a : Action, a ∈ actions) + (plan : List Action) + {n : Nat} + {s : State} + {frontier : List State} + (hs : s ∈ frontier) + (hlen : plan.length ≤ n) : + runPlan step s plan ∈ bfsVisited step actions n frontier := by + induction n generalizing s frontier plan with + | zero => + cases plan with + | nil => + simpa [runPlan, bfsVisited] using hs + | cons _ _ => + cases hlen + | succ n ih => + cases plan with + | nil => + simp [runPlan, bfsVisited, hs] + | cons a rest => + have hRestLen : rest.length ≤ n := Nat.succ_le_succ_iff.mp hlen + have hStepMem : + step s a ∈ expandFrontier step actions frontier := + mem_expandFrontier_of_mem step actions hActions a hs + have hRest : + runPlan step (step s a) rest ∈ + bfsVisited step actions n + (expandFrontier step actions frontier) := + ih rest hStepMem hRestLen + simp [runPlan, bfsVisited] + exact Or.inr hRest + +theorem bfsVisited_frontier_invariant + (step : State → Action → State) + (actions : List Action) + (hActions : ∀ a : Action, a ∈ actions) + (init : State) + (n : Nat) : + BfsFrontierComplete step init n (bfsVisitedFrom step actions init n) := by + intro target hreachable + rcases hreachable with ⟨plan, hlen, hrun⟩ + rw [← hrun] + exact runPlan_mem_bfsVisited_of_mem + step actions hActions plan + (s := init) + (frontier := [init]) + (by simp) + hlen + +/-- +Bounded BFS completeness: if a goal can be reached by some plan of length at +most `n`, and the action list covers every action, then depth-`n` BFS will have +visited a goal state. +-/ +theorem bfsVisited_complete_for_bounded_goal + (step : State → Action → State) + (actions : List Action) + (hActions : ∀ a : Action, a ∈ actions) + (goal : State → Prop) + {init : State} + {n : Nat} + (hreachable : BoundedGoalReachable step goal init n) : + BfsFindsGoal goal (bfsVisitedFrom step actions init n) := by + rcases hreachable with ⟨final, hbounded, hgoal⟩ + exact ⟨ + final, + bfsVisited_frontier_invariant step actions hActions init n final hbounded, + hgoal + ⟩ + +/-- +A completed deterministic policy induces a bounded reachable goal. This bridges +the task-specific `symbolicPolicy` proofs to the generic BFS theorem above. +-/ +theorem policy_completion_is_bounded_goal + (policy : State → Action) + (step : State → Action → State) + (goal : State → Prop) + (init : State) + (n : Nat) + (hgoal : goal (runPolicy policy step n init)) : + BoundedGoalReachable step goal init n := by + refine ⟨runPolicy policy step n init, ?_, hgoal⟩ + refine ⟨policyTrace policy step n init, ?_, ?_⟩ + · rw [policyTrace_length] + exact Nat.le_refl n + · exact (runPolicy_eq_runPlan_policyTrace policy step n init).symm + +/-- +If a state-driven policy reaches a goal in `n` steps, then a complete bounded +BFS over the same symbolic transition layer also finds a goal within `n` steps. +-/ +theorem bfs_finds_goal_reached_by_policy + (policy : State → Action) + (step : State → Action → State) + (actions : List Action) + (hActions : ∀ a : Action, a ∈ actions) + (goal : State → Prop) + (init : State) + (n : Nat) + (hgoal : goal (runPolicy policy step n init)) : + BfsFindsGoal goal (bfsVisitedFrom step actions init n) := by + exact bfsVisited_complete_for_bounded_goal + step actions hActions goal + (policy_completion_is_bounded_goal policy step goal init n hgoal) + +end Generic + +end StrategyFramework diff --git a/student_agent/lean/Task1Formalization.lean b/student_agent/lean/Task1Formalization.lean new file mode 100644 index 0000000..8da0279 --- /dev/null +++ b/student_agent/lean/Task1Formalization.lean @@ -0,0 +1,408 @@ +/-! + Task 1 formalization, rewritten to match the current BFS-based + `student_agent/baseline_policy.py` implementation. + + Important alignment with Python policy: + + * `decide_task1` does NOT follow a fixed action list. + * It reads `keys = current_keys(inventory)`. + * If `keys <= 0`, it detects the visible chest, builds `blocked` from the + current frame, sets goals to the walkable neighbors of that chest, and calls + `follow_bfs_aligned(..., final_action = ACTION_A)`. + * If `keys > 0`, it sets goals to `EXIT_DIRECTION_TILES["north"]` and calls + `follow_bfs_aligned(..., final_action = ACTION_UP)`. + * `follow_bfs_aligned` recomputes BFS at every step. This Lean model proves + the symbolic BFS contract rather than proving one fixed route. + + Pixel alignment is intentionally abstracted away: the proof layer reasons at + tile level. Alignment micro-actions are implementation details whose only + specification obligation is that they eventually realize the next BFS tile + transition without changing the high-level stage. +-/ + +namespace Task1Formalization + +/-! ### Tile and action model -/ + +abbrev Tile := Nat × Nat + +def GRID_WIDTH : Nat := 10 +def GRID_HEIGHT : Nat := 8 + +def inBounds (t : Tile) : Bool := + decide (t.1 < GRID_WIDTH ∧ t.2 < GRID_HEIGHT) + +def chestTile : Tile := (0, 3) + +def northExitTiles : List Tile := [(4, 0), (5, 0)] + +def isNorthExitTile (x y : Nat) : Bool := + decide (y = 0 ∧ (x = 4 ∨ x = 5)) + +def manhattan (a b : Tile) : Nat := + (if a.1 ≤ b.1 then b.1 - a.1 else a.1 - b.1) + + (if a.2 ≤ b.2 then b.2 - a.2 else a.2 - b.2) + +def isAdjacent (a b : Tile) : Bool := + decide (manhattan a b = 1) + +/-- +Four-neighbor order matches Python `neighbors_of`: +`[(x, y - 1), (x, y + 1), (x - 1, y), (x + 1, y)]`, except that this symbolic +version omits out-of-range underflow candidates instead of generating saturated +natural-number duplicates. Later `inBounds` checks play the same role as the +Python filter. +-/ +def neighborsOf (t : Tile) : List Tile := + (if t.2 = 0 then [] else [(t.1, t.2 - 1)]) ++ + [(t.1, t.2 + 1)] ++ + (if t.1 = 0 then [] else [(t.1 - 1, t.2)]) ++ + [(t.1 + 1, t.2)] + +inductive Action where + | noop + | up | down | left | right + | openChest + deriving DecidableEq, Repr + + + +/-! ### Perceived room abstraction -/ + +/-- +`RoomModel` is the symbolic counterpart of the current visual frame. + +`blocked` corresponds to `build_blocked_tiles(frame, avoid_traps=False)` for +Task 1: walls, abysses if any, and currently visible chest tiles are treated as +not traversable by BFS. `detectedChest` corresponds to +`detect_chest_tile(frame)`. +-/ +structure RoomModel where + blocked : Tile → Bool + detectedChest : Option Tile + +/-- BFS may step on a tile iff it is within the grid and not blocked. -/ +def freeTile (r : RoomModel) (t : Tile) : Bool := + inBounds t && !r.blocked t + +/-- Python task1 chest goals: walkable neighbors of the detected chest. -/ +def chestNeighborGoals (r : RoomModel) : List Tile := + match r.detectedChest with + | none => [] + | some c => (neighborsOf c).filter (fun n => freeTile r n) + +def isAdjacentToDetectedChest (r : RoomModel) (spos : Tile) : Bool := + match r.detectedChest with + | none => false + | some c => isAdjacent spos c + +/-! ### Environment state and transition -/ + +structure EnvState where + x : Nat + y : Nat + keys : Nat + chestOpened : Bool + completed : Bool + deriving DecidableEq, Repr + +-- Public task-1 spawn tile. +def initialState : EnvState := + { x := 4, y := 6, keys := 0, chestOpened := false, completed := false } + +def pos (s : EnvState) : Tile := (s.x, s.y) + +def GoalReached (s : EnvState) : Prop := + s.completed = true + +/-- Movement target with boundary underflow represented as `none`. -/ +def moveTarget (s : EnvState) : Action → Option Tile + | Action.up => if s.y = 0 then none else some (s.x, s.y - 1) + | Action.down => some (s.x, s.y + 1) + | Action.left => if s.x = 0 then none else some (s.x - 1, s.y) + | Action.right => some (s.x + 1, s.y) + | _ => none + +def moveStep (r : RoomModel) (s : EnvState) (target : Option Tile) : EnvState := + match target with + | none => s + | some t => if freeTile r t then { s with x := t.1, y := t.2 } else s + +/-- +Tile-level transition. The only high-level special actions are exactly the +ones used by Task 1: `ACTION_A` for opening the detected key chest and +`ACTION_UP` on a north exit tile for leaving through the locked door. +-/ +def step (r : RoomModel) (s : EnvState) : Action → EnvState + | Action.noop => s + | Action.up => + if isNorthExitTile s.x s.y && decide (s.keys > 0) then + { s with keys := s.keys - 1, completed := true } + else + moveStep r s (moveTarget s Action.up) + | Action.down => moveStep r s (moveTarget s Action.down) + | Action.left => moveStep r s (moveTarget s Action.left) + | Action.right => moveStep r s (moveTarget s Action.right) + | Action.openChest => + if isAdjacentToDetectedChest r (pos s) && !s.chestOpened then + { s with chestOpened := true, keys := s.keys + 1 } + else + s + + +def run (r : RoomModel) : EnvState → List Action → EnvState + | s, [] => s + | s, a :: rest => run r (step r s a) rest + +/-! ### Action-mask/safety layer -/ + +def actionEnabled (r : RoomModel) (s : EnvState) : Action → Bool + | Action.noop => true + | Action.up => + match moveTarget s Action.up with + | none => false + | some t => freeTile r t + | Action.down => + match moveTarget s Action.down with + | none => false + | some t => freeTile r t + | Action.left => + match moveTarget s Action.left with + | none => false + | some t => freeTile r t + | Action.right => + match moveTarget s Action.right with + | none => false + | some t => freeTile r t + | Action.openChest => isAdjacentToDetectedChest r (pos s) && !s.chestOpened + +def actionsEnabledAlong (r : RoomModel) : EnvState → List Action → Bool + | _, [] => true + | s, a :: rest => + actionEnabled r s a && actionsEnabledAlong r (step r s a) rest + +/-! ### Baseline-aligned symbolic policy shape -/ + +/-- +A symbolic BFS action provider. In Python this is implemented by +`bfs_path(...)` plus `follow_bfs_aligned(...)`, which returns one movement +action toward the first tile of the current shortest path, or the supplied +`final_action` when already at a goal. +-/ +abbrev BfsAction := EnvState → List Tile → Action + +/-- +Policy shape matching `decide_task1`. + +This definition is intentionally parameterized by `bfsAction`: Lean verifies the +stage logic and the BFS contract separately, instead of hardcoding a route. +-/ +def symbolicPolicy (r : RoomModel) (bfsAction : BfsAction) (s : EnvState) : Action := + if s.completed then + Action.noop + else if s.keys = 0 then + match r.detectedChest with + | none => Action.noop + | some _ => + let goals := chestNeighborGoals r + if pos s ∈ goals then Action.openChest else bfsAction s goals + else + if isNorthExitTile s.x s.y then Action.up else bfsAction s northExitTiles + +/-- When no key is held and the BFS has reached a chest-neighbor goal, the + symbolic policy emits the same final action as Python: `ACTION_A`. -/ +theorem symbolicPolicy_getKey_goal_emits_openChest + (hDone : s.completed = false) + (hKeys : s.keys = 0) + (hChest : r.detectedChest ≠ none) + (hGoal : pos s ∈ chestNeighborGoals r) : + symbolicPolicy r bfsAction s = Action.openChest := by + unfold symbolicPolicy + simp [hDone, hKeys, hGoal] + + cases hDet : r.detectedChest with + | none => + exfalso + exact hChest hDet + | some c => + simp + + +/-- Once a key is held and the agent is on a north-exit tile, the symbolic + policy emits the same final action as Python: `ACTION_UP` toward the north + locked exit, represented here as `goNorth`. -/ +theorem symbolicPolicy_exit_goal_emits_up + (r : RoomModel) (bfsAction : BfsAction) {s : EnvState} + (hDone : s.completed = false) + (hKeys : s.keys > 0) + (hExit : isNorthExitTile s.x s.y = true) : + symbolicPolicy r bfsAction s = Action.up := by + unfold symbolicPolicy + have hNotZero : ¬ s.keys = 0 := Nat.ne_of_gt hKeys + simp [hDone, hNotZero, hExit] + + +/-! ### BFS route certificates -/ + +/-- +Soundness certificate for the first BFS phase: from the current state, BFS has +found an enabled movement plan that ends adjacent to the detected chest. This +is the formal counterpart of: + +``` +goals = {n for n in neighbors_of(chest) if in_bounds(n) and n not in blocked} +follow_bfs_aligned(..., goals, blocked, final_action=ACTION_A) +``` +-/ +structure ChestBfsCertificate (r : RoomModel) (start : EnvState) where + plan : List Action + enabled : actionsEnabledAlong r start plan = true + endsAdjacent : isAdjacentToDetectedChest r (pos (run r start plan)) = true + chestStillClosed : (run r start plan).chestOpened = false + +/-- +Soundness certificate for the second BFS phase: after opening the chest, BFS has +found an enabled movement plan that ends on a north-exit tile while a key is +still held. This is the formal counterpart of: + +``` +exit_goals = set(EXIT_DIRECTION_TILES["north"]) +follow_bfs_aligned(..., exit_goals, blocked, final_action=ACTION_UP) +``` +-/ +structure ExitBfsCertificate (r : RoomModel) (start : EnvState) where + plan : List Action + enabled : actionsEnabledAlong r start plan = true + endsAtNorthExit : isNorthExitTile (run r start plan).x (run r start plan).y = true + keyPositiveAtGoal : (run r start plan).keys > 0 + +/-! ### Local interaction correctness lemmas -/ + +theorem openChest_at_detected_chest_neighbor_gets_key + (r : RoomModel) {s : EnvState} + (hAdj : isAdjacentToDetectedChest r (pos s) = true) + (hClosed : s.chestOpened = false) : + (step r s Action.openChest).keys = s.keys + 1 ∧ + (step r s Action.openChest).chestOpened = true := by + simp [step, hAdj, hClosed] + +theorem up_at_exit_with_key_completes + (r : RoomModel) {s : EnvState} + (hExit : isNorthExitTile s.x s.y = true) + (hKey : s.keys > 0) : + (step r s Action.up).completed = true := by + simp [step, hExit, hKey] + + +/-! ### Main non-fixed-route Task 1 theorem -/ + +/-- +Task 1 completion theorem aligned with the BFS policy. + +This theorem is not a fixed-route proof. It says: for any perceived Task-1 +room model `r`, if the first BFS phase supplies a sound route to a walkable +neighbor of the detected chest, and the second BFS phase supplies a sound route +to a north-exit tile after the chest is opened, then the baseline Task-1 policy +structure completes the task. +-/ +theorem task1_bfs_certificate_completes + (r : RoomModel) + (chestRoute : ChestBfsCertificate r initialState) + (exitRoute : + ExitBfsCertificate r + (step r (run r initialState chestRoute.plan) Action.openChest)) : + GoalReached + (step r + (run r + (step r (run r initialState chestRoute.plan) Action.openChest) + exitRoute.plan) + Action.up) := by + let sOpen := step r (run r initialState chestRoute.plan) Action.openChest + let sExit := run r sOpen exitRoute.plan + have hDone : (step r sExit Action.up).completed = true := + up_at_exit_with_key_completes r + (s := sExit) + exitRoute.endsAtNorthExit + exitRoute.keyPositiveAtGoal + simpa [GoalReached, sOpen, sExit] using hDone + +/-- The first BFS phase followed by `ACTION_A` really produces a key. -/ +theorem task1_chest_phase_gets_key + (r : RoomModel) + (chestRoute : ChestBfsCertificate r initialState) : + (step r (run r initialState chestRoute.plan) Action.openChest).keys = + (run r initialState chestRoute.plan).keys + 1 := by + have h := openChest_at_detected_chest_neighbor_gets_key r + (s := run r initialState chestRoute.plan) + chestRoute.endsAdjacent + chestRoute.chestStillClosed + exact h.1 + +/-! ### Public map regression instance + +The following closed certificates instantiate the generic BFS theorem on the +public Task-1 layout. They are included only as regression checks for the known +training map; the main theorem above is the non-fixed-route proof used to align +with the current BFS policy. +-/ + +def publicWall (t : Tile) : Bool := + decide ( + (t.2 = 2 ∧ (t.1 = 0 ∨ t.1 = 1 ∨ t.1 = 4 ∨ t.1 = 5 ∨ + t.1 = 6 ∨ t.1 = 7 ∨ t.1 = 8 ∨ t.1 = 9)) ∨ + (t.2 = 5 ∧ (t.1 = 0 ∨ t.1 = 1 ∨ t.1 = 2 ∨ t.1 = 3 ∨ + t.1 = 4 ∨ t.1 = 5 ∨ t.1 = 6)) + ) + +def publicBlocked (t : Tile) : Bool := + publicWall t || decide (t = chestTile) + +def publicRoom : RoomModel := + { blocked := publicBlocked, detectedChest := some chestTile } + +def publicChestPlan : List Action := [ + Action.right, Action.right, Action.right, + Action.up, Action.up, Action.up, + Action.left, Action.left, Action.left, Action.left, Action.left, Action.left +] + +def publicExitPlan : List Action := [ + Action.right, Action.right, + Action.up, Action.up, Action.up, + Action.right +] + +def publicChestRoute : ChestBfsCertificate publicRoom initialState := + { plan := publicChestPlan + enabled := by native_decide + endsAdjacent := by native_decide + chestStillClosed := by native_decide } + +def publicExitRoute : + ExitBfsCertificate publicRoom + (step publicRoom (run publicRoom initialState publicChestRoute.plan) Action.openChest) := + { plan := publicExitPlan + enabled := by native_decide + endsAtNorthExit := by native_decide + keyPositiveAtGoal := by native_decide } + +theorem public_task1_bfs_certificate_completes : + GoalReached + (step publicRoom + (run publicRoom + (step publicRoom (run publicRoom initialState publicChestRoute.plan) Action.openChest) + publicExitRoute.plan) + Action.up) := by + exact task1_bfs_certificate_completes publicRoom publicChestRoute publicExitRoute + +/-- Optional closed-map trace for debugging only. This is not the main proof. -/ +def publicTask1Trace : List Action := + publicChestRoute.plan ++ [Action.openChest] ++ publicExitRoute.plan ++ [Action.up] + +theorem publicTask1Trace_reaches_goal : + GoalReached (run publicRoom initialState publicTask1Trace) := by + unfold GoalReached + native_decide + + +end Task1Formalization diff --git a/student_agent/lean/Task2Formalization.lean b/student_agent/lean/Task2Formalization.lean new file mode 100644 index 0000000..2004799 --- /dev/null +++ b/student_agent/lean/Task2Formalization.lean @@ -0,0 +1,840 @@ +/- + Environment formalization for `mathematical_logic/task_2`. + + Task 2 is a single-room combat-and-exit task: + + * defeat the chaser monster; + * open the chest to get one key; + * leave through the west conditional exit. + + The real engine also contains trap rows on the top and bottom boundaries. This + Lean model keeps a minimal trap effect by letting a "move into trap zone" + action reduce HP and send the player back to the safe center zone. + + Python correspondence: + * `nesylink/map_data/mathematical_logic/task_2/room_001.json` + * `nesylink/core/mechanics/interactions.py` + * `nesylink/core/mechanics/movement.py` + * `nesylink/core/mechanics/combat.py` +-/ + +namespace Task2Formalization + +/-! ### Zones and actions -/ + +inductive Zone where + | safeCenter + | nearMonster + | nearChest + | westExit + | topTrap + | bottomTrap + deriving DecidableEq, Repr + +inductive Action where + | moveTo (z : Zone) + | attack + | openChest + | useExit + | wait + deriving DecidableEq, Repr + +/-! ### Environment state -/ + +structure EnvState where + zone : Zone + hp : Nat + monsterAlive : Bool + monsterHp : Nat + chestOpened : Bool + keys : Nat + completed : Bool + deriving DecidableEq, Repr + +-- Spawn at tile (7, 3) per room_001.json, modelled as safeCenter. +def initialState : EnvState := + { + zone := Zone.safeCenter + hp := 5 + monsterAlive := true + monsterHp := 2 + chestOpened := false + keys := 0 + completed := false + } + +def GoalReached (s : EnvState) : Prop := + s.completed = true + +/-! ### Transition function -/ + +def step (s : EnvState) : Action → EnvState + | Action.moveTo z => + match z with + | Zone.topTrap => + { s with zone := Zone.safeCenter, hp := s.hp - 1 } + | Zone.bottomTrap => + { s with zone := Zone.safeCenter, hp := s.hp - 1 } + | _ => + { s with zone := z } + | Action.attack => + if s.zone = Zone.nearMonster ∧ s.monsterAlive = true ∧ s.hp > 1 then + if s.monsterHp = 1 then + { s with monsterAlive := false, monsterHp := 0 } + else + { s with monsterHp := s.monsterHp - 1 } + else + s + | Action.openChest => + if s.zone = Zone.nearChest ∧ s.chestOpened = false then + { s with chestOpened := true, keys := s.keys + 1 } + else + s + | Action.useExit => + if s.zone = Zone.westExit ∧ s.monsterAlive = false ∧ s.keys > 0 then + { s with completed := true } + else + s + | Action.wait => s + +def run : EnvState → List Action → EnvState + | s, [] => s + | s, a :: rest => run (step s a) rest + +/-! ### Basic move lemmas -/ + +theorem move_to_zone + {s : EnvState} {z : Zone} + (hz : z ≠ Zone.topTrap) (hz' : z ≠ Zone.bottomTrap) : + (step s (Action.moveTo z)).zone = z := by + simp [step] + +theorem move_to_zone_not_trap + {s : EnvState} (z : Zone) : (step s (Action.moveTo z)).zone = + match z with + | Zone.topTrap => Zone.safeCenter + | Zone.bottomTrap => Zone.safeCenter + | _ => z := by + cases z <;> simp [step] + +/-! ### Trap lemmas -/ + +theorem trap_move_reduces_hp + {s : EnvState} (_ : s.hp > 0) : + (step s (Action.moveTo Zone.topTrap)).hp = s.hp - 1 := by + simp [step] + +theorem trap_move_resets_to_safe_center + {s : EnvState} : + (step s (Action.moveTo Zone.bottomTrap)).zone = Zone.safeCenter := by + simp [step] + +theorem trap_move_preserves_monster + {s : EnvState} : + (step s (Action.moveTo Zone.topTrap)).monsterAlive = s.monsterAlive := by + simp [step] + +theorem trap_move_preserves_chestOpened + {s : EnvState} : + (step s (Action.moveTo Zone.bottomTrap)).chestOpened = s.chestOpened := by + simp [step] + +theorem trap_move_preserves_keys + {s : EnvState} : + (step s (Action.moveTo Zone.topTrap)).keys = s.keys := by + simp [step] + +theorem trap_move_preserves_completed + {s : EnvState} : + (step s (Action.moveTo Zone.bottomTrap)).completed = s.completed := by + simp [step] + +theorem trap_move_preserves_monsterHp + {s : EnvState} : + (step s (Action.moveTo Zone.topTrap)).monsterHp = s.monsterHp := by + simp [step] + +/-! ### Safe move lemmas (non-trap) -/ + +theorem safe_move_preserves_hp + {s : EnvState} {z : Zone} + (hz : z ≠ Zone.topTrap) (hz' : z ≠ Zone.bottomTrap) : + (step s (Action.moveTo z)).hp = s.hp := by + simp [step] + +theorem safe_move_preserves_monsterAlive + {s : EnvState} {z : Zone} + (hz : z ≠ Zone.topTrap) (hz' : z ≠ Zone.bottomTrap) : + (step s (Action.moveTo z)).monsterAlive = s.monsterAlive := by + simp [step] + +theorem safe_move_preserves_chestOpened + {s : EnvState} {z : Zone} + (hz : z ≠ Zone.topTrap) (hz' : z ≠ Zone.bottomTrap) : + (step s (Action.moveTo z)).chestOpened = s.chestOpened := by + simp [step] + +theorem safe_move_preserves_keys + {s : EnvState} {z : Zone} + (hz : z ≠ Zone.topTrap) (hz' : z ≠ Zone.bottomTrap) : + (step s (Action.moveTo z)).keys = s.keys := by + simp [step] + +theorem safe_move_preserves_completed + {s : EnvState} {z : Zone} + (hz : z ≠ Zone.topTrap) (hz' : z ≠ Zone.bottomTrap) : + (step s (Action.moveTo z)).completed = s.completed := by + simp [step] + +theorem safe_move_preserves_monsterHp + {s : EnvState} {z : Zone} + (hz : z ≠ Zone.topTrap) (hz' : z ≠ Zone.bottomTrap) : + (step s (Action.moveTo z)).monsterHp = s.monsterHp := by + simp [step] + +/-! ### Attack lemmas -/ + +theorem attack_reduces_monsterHp_when_adjacent + {s : EnvState} + (hZone : s.zone = Zone.nearMonster) + (hMonster : s.monsterAlive = true) + (hHp : s.hp > 1) : + (step s Action.attack).monsterHp = s.monsterHp - 1 := by + by_cases hMhp : s.monsterHp = 1 + · simp [step, hZone, hMonster, hHp, hMhp] + · simp [step, hZone, hMonster, hHp, hMhp] + +theorem attack_kills_monster_when_monsterHp_one + {s : EnvState} + (hZone : s.zone = Zone.nearMonster) + (hMonster : s.monsterAlive = true) + (hHp : s.hp > 1) + (hMonsterHp : s.monsterHp = 1) : + (step s Action.attack).monsterAlive = false := by + simp [step, hZone, hMonster, hHp, hMonsterHp] + +theorem attack_preserves_monsterAlive_when_monsterHp_gt_one + {s : EnvState} + (hZone : s.zone = Zone.nearMonster) + (hMonster : s.monsterAlive = true) + (hHp : s.hp > 1) + (hMonsterHp : s.monsterHp > 1) : + (step s Action.attack).monsterAlive = true := by + simp [step, hZone, hMonster, hHp] + split + · omega + · rfl + +theorem attack_no_effect_without_monster + {s : EnvState} (hMonster : s.monsterAlive = false) : + step s Action.attack = s := by + simp [step, hMonster] + +theorem attack_no_effect_not_adjacent + {s : EnvState} (hZone : s.zone ≠ Zone.nearMonster) : + step s Action.attack = s := by + simp [step, hZone] + +theorem attack_no_effect_low_hp + {s : EnvState} (hHp : s.hp ≤ 1) : + step s Action.attack = s := by + by_cases h1 : s.zone = Zone.nearMonster ∧ s.monsterAlive = true ∧ s.hp > 1 + · omega + · simp [step, h1] + +theorem attack_preserves_zone + {s : EnvState} : (step s Action.attack).zone = s.zone := by + by_cases h1 : s.zone = Zone.nearMonster ∧ s.monsterAlive = true ∧ s.hp > 1 + · simp [step, h1] + by_cases h2 : s.monsterHp = 1 <;> simp [h2] + · simp [step, h1] + +theorem attack_preserves_chestOpened + {s : EnvState} : (step s Action.attack).chestOpened = s.chestOpened := by + by_cases h1 : s.zone = Zone.nearMonster ∧ s.monsterAlive = true ∧ s.hp > 1 + · simp [step, h1] + by_cases h2 : s.monsterHp = 1 <;> simp [h2] + · simp [step, h1] + +theorem attack_preserves_keys + {s : EnvState} : (step s Action.attack).keys = s.keys := by + by_cases h1 : s.zone = Zone.nearMonster ∧ s.monsterAlive = true ∧ s.hp > 1 + · simp [step, h1] + by_cases h2 : s.monsterHp = 1 <;> simp [h2] + · simp [step, h1] + +theorem attack_preserves_hp + {s : EnvState} : (step s Action.attack).hp = s.hp := by + by_cases h1 : s.zone = Zone.nearMonster ∧ s.monsterAlive = true ∧ s.hp > 1 + · simp [step, h1] + by_cases h2 : s.monsterHp = 1 <;> simp [h2] + · simp [step, h1] + +theorem attack_preserves_completed + {s : EnvState} : (step s Action.attack).completed = s.completed := by + by_cases h1 : s.zone = Zone.nearMonster ∧ s.monsterAlive = true ∧ s.hp > 1 + · simp [step, h1] + by_cases h2 : s.monsterHp = 1 <;> simp [h2] + · simp [step, h1] + +/-! ### Chest lemmas -/ + +theorem chest_open_only_at_chest + {s : EnvState} (hZone : s.zone ≠ Zone.nearChest) : + step s Action.openChest = s := by + simp [step, hZone] + +theorem chest_open_already_opened + {s : EnvState} (hChest : s.chestOpened = true) : + step s Action.openChest = s := by + simp [step, hChest] + +theorem chest_open_increases_keys + {s : EnvState} + (hZone : s.zone = Zone.nearChest) + (hChest : s.chestOpened = false) : + (step s Action.openChest).keys = s.keys + 1 := by + simp [step, hZone, hChest] + +theorem chest_open_marks_opened + {s : EnvState} + (hZone : s.zone = Zone.nearChest) + (hChest : s.chestOpened = false) : + (step s Action.openChest).chestOpened = true := by + simp [step, hZone, hChest] + +theorem chest_open_preserves_zone + {s : EnvState} : (step s Action.openChest).zone = s.zone := by + simp [step]; split <;> rfl + +theorem chest_open_preserves_monsterAlive + {s : EnvState} : (step s Action.openChest).monsterAlive = s.monsterAlive := by + simp [step]; split <;> rfl + +theorem chest_open_preserves_hp + {s : EnvState} : (step s Action.openChest).hp = s.hp := by + simp [step]; split <;> rfl + +theorem chest_open_preserves_completed + {s : EnvState} : (step s Action.openChest).completed = s.completed := by + simp [step]; split <;> rfl + +theorem chest_open_preserves_monsterHp + {s : EnvState} : (step s Action.openChest).monsterHp = s.monsterHp := by + simp [step]; split <;> rfl + +/-! ### Exit lemmas -/ + +theorem use_exit_only_at_exit + {s : EnvState} (hZone : s.zone ≠ Zone.westExit) : + step s Action.useExit = s := by + simp [step, hZone] + +theorem exit_blocked_if_monster_alive + {s : EnvState} + (hZone : s.zone = Zone.westExit) + (hMonster : s.monsterAlive = true) : + step s Action.useExit = s := by + simp [step, hZone, hMonster] + +theorem exit_blocked_without_key + {s : EnvState} + (hZone : s.zone = Zone.westExit) + (hMonster : s.monsterAlive = false) + (hKeys : s.keys = 0) : + step s Action.useExit = s := by + simp [step, hZone, hMonster, hKeys] + +theorem exit_succeeds_after_requirements + {s : EnvState} + (hZone : s.zone = Zone.westExit) + (hMonster : s.monsterAlive = false) + (hKeys : s.keys > 0) : + (step s Action.useExit).completed = true := by + simp [step, hZone, hMonster, hKeys] + +theorem use_exit_preserves_zone + {s : EnvState} : (step s Action.useExit).zone = s.zone := by + simp [step]; split <;> rfl + +theorem use_exit_preserves_monsterAlive + {s : EnvState} : (step s Action.useExit).monsterAlive = s.monsterAlive := by + simp [step]; split <;> rfl + +theorem use_exit_preserves_chestOpened + {s : EnvState} : (step s Action.useExit).chestOpened = s.chestOpened := by + simp [step]; split <;> rfl + +theorem use_exit_preserves_keys + {s : EnvState} : (step s Action.useExit).keys = s.keys := by + simp [step]; split <;> rfl + +theorem use_exit_preserves_monsterHp + {s : EnvState} : (step s Action.useExit).monsterHp = s.monsterHp := by + simp [step]; split <;> rfl + +/-! ### Necessity lemmas -/ + +/-- Non-useExit actions preserve the `completed` field. -/ +theorem non_useExit_preserves_completed + {s : EnvState} (a : Action) (ha : a ≠ Action.useExit) : + (step s a).completed = s.completed := by + cases a with + | moveTo z => cases z <;> simp [step] + | attack => + by_cases h1 : s.zone = Zone.nearMonster ∧ s.monsterAlive = true ∧ s.hp > 1 + · simp [step, h1] + by_cases h2 : s.monsterHp = 1 + · simp [h2] + · simp [h2] + · simp [step, h1] + | openChest => + by_cases h1 : s.zone = Zone.nearChest ∧ s.chestOpened = false + · simp [step, h1] + · simp [step, h1] + | useExit => exact absurd rfl ha + | wait => simp [step] + +/-- `useExit` is the only action that can flip `completed` from `false` to + `true`. All other actions leave `completed` unchanged. -/ +theorem only_useExit_sets_completed + {s : EnvState} (hs : s.completed = false) (a : Action) + (h : (step s a).completed = true) : a = Action.useExit := by + by_cases ha : a = Action.useExit + · exact ha + · have := non_useExit_preserves_completed (s := s) a ha + rw [this, hs] at h + simp at h + +/-! ### Action-wise invariant lemmas -/ + +/-- Attack is the only action that can change `monsterAlive`. -/ +theorem only_attack_changes_monsterAlive + {s : EnvState} (a : Action) (ha : a ≠ Action.attack) : + (step s a).monsterAlive = s.monsterAlive := by + cases a with + | moveTo z => cases z <;> simp [step] + | attack => exact absurd rfl ha + | openChest => + by_cases h1 : s.zone = Zone.nearChest ∧ s.chestOpened = false + · simp [step, h1] + · simp [step, h1] + | useExit => + by_cases h1 : s.zone = Zone.westExit ∧ s.monsterAlive = false ∧ s.keys > 0 + · simp [step, h1] + · simp [step, h1] + | wait => simp [step] + +/-- `openChest` is the only action that can change `chestOpened`. -/ +theorem only_openChest_changes_chestOpened + {s : EnvState} (a : Action) (ha : a ≠ Action.openChest) : + (step s a).chestOpened = s.chestOpened := by + cases a with + | moveTo z => cases z <;> simp [step] + | attack => + by_cases h1 : s.zone = Zone.nearMonster ∧ s.monsterAlive = true ∧ s.hp > 1 + · simp [step, h1] + by_cases h2 : s.monsterHp = 1 + · simp [h2] + · simp [h2] + · simp [step, h1] + | openChest => exact absurd rfl ha + | useExit => + by_cases h1 : s.zone = Zone.westExit ∧ s.monsterAlive = false ∧ s.keys > 0 + · simp [step, h1] + · simp [step, h1] + | wait => simp [step] + +/-- Attack is the only action that can change `monsterHp`. -/ +theorem only_attack_changes_monsterHp + {s : EnvState} (a : Action) (ha : a ≠ Action.attack) : + (step s a).monsterHp = s.monsterHp := by + cases a with + | moveTo z => cases z <;> simp [step] + | attack => exact absurd rfl ha + | openChest => + by_cases h1 : s.zone = Zone.nearChest ∧ s.chestOpened = false + · simp [step, h1] + · simp [step, h1] + | useExit => + by_cases h1 : s.zone = Zone.westExit ∧ s.monsterAlive = false ∧ s.keys > 0 + · simp [step, h1] + · simp [step, h1] + | wait => simp [step] + +/-- Only `openChest` and `useExit` can change `keys`. -/ +theorem only_openChest_or_useExit_changes_keys + {s : EnvState} (a : Action) (ha : a ≠ Action.openChest) (ha' : a ≠ Action.useExit) : + (step s a).keys = s.keys := by + cases a with + | moveTo z => cases z <;> simp [step] + | attack => + by_cases h1 : s.zone = Zone.nearMonster ∧ s.monsterAlive = true ∧ s.hp > 1 + · simp [step, h1] + by_cases h2 : s.monsterHp = 1 + · simp [h2] + · simp [h2] + · simp [step, h1] + | openChest => exact absurd rfl ha + | useExit => exact absurd rfl ha' + | wait => simp [step] + +/-- Reachability: states reachable from `initialState` via arbitrary actions. -/ +inductive Reachable : EnvState → Prop where + | init : Reachable initialState + | step (s : EnvState) (a : Action) : Reachable s → Reachable (step s a) + +/-- Invariant: if the chest is not opened then no keys have been collected. -/ +theorem chestOpened_or_keys_zero {s : EnvState} (hr : Reachable s) : + s.chestOpened = true ∨ s.keys = 0 := by + induction hr with + | init => simp [initialState] + | step s a hr ih => + cases a with + | moveTo z => cases z <;> simp [step] <;> exact ih + | attack => + by_cases h1 : s.zone = Zone.nearMonster ∧ s.monsterAlive = true ∧ s.hp > 1 + · simp [step, h1] + by_cases h2 : s.monsterHp = 1 <;> simp [h2] <;> exact ih + · simp [step, h1]; exact ih + | openChest => + by_cases h1 : s.zone = Zone.nearChest ∧ s.chestOpened = false + · simp [step, h1]; + · simp [step, h1]; exact ih + | useExit => + by_cases h1 : s.zone = Zone.westExit ∧ s.monsterAlive = false ∧ s.keys > 0 + · simp [step, h1]; exact ih + · simp [step, h1]; exact ih + | wait => simp [step]; exact ih + +/-- If a reachable state has `completed = true`, then the monster is defeated and + the chest has been opened. -/ +theorem completion_postcondition {s : EnvState} (hr : Reachable s) (hc : s.completed = true) : + s.monsterAlive = false ∧ s.chestOpened = true := by + induction hr with + | init => simp [initialState] at hc + | step s' a hr ih => + by_cases hs : s'.completed = true + · -- s' already completed: use ih, then show step preserves both fields + rcases ih hs with ⟨hm, hch⟩ + have hm' : (step s' a).monsterAlive = s'.monsterAlive := by + by_cases ha : a = Action.attack + · subst ha + have : step s' Action.attack = s' := attack_no_effect_without_monster hm + simp [this, hm] + · exact only_attack_changes_monsterAlive a ha + have hc' : (step s' a).chestOpened = s'.chestOpened := by + by_cases ha : a = Action.openChest + · subst ha + have : step s' Action.openChest = s' := chest_open_already_opened hch + simp [this, hch] + · exact only_openChest_changes_chestOpened a ha + simp [hm', hc', hm, hch] + · -- s'.completed ≠ true ⇒ s'.completed = false (Bool exhaust) + have hf : s'.completed = false := by + cases h : s'.completed with + | true => exact absurd h hs + | false => rfl + have ha : a = Action.useExit := only_useExit_sets_completed hf a hc + subst ha + by_cases hcond : s'.zone = Zone.westExit ∧ s'.monsterAlive = false ∧ s'.keys > 0 + · have hchest : s'.chestOpened = true := by + have h_cases := chestOpened_or_keys_zero hr + rcases h_cases with (h | h) + · exact h + · rw [h] at hcond; simp at hcond + have hm' : (step s' Action.useExit).monsterAlive = s'.monsterAlive := + use_exit_preserves_monsterAlive + have hc' : (step s' Action.useExit).chestOpened = s'.chestOpened := + use_exit_preserves_chestOpened + simp [hm', hc', hcond.2.1, hchest] + · have hstep : step s' Action.useExit = s' := by simp [step, hcond] + rw [hstep, hf] at hc + simp at hc + +/-- HP never increases after any single step. -/ +theorem hp_non_increasing + {s : EnvState} (a : Action) : (step s a).hp ≤ s.hp := by + cases a with + | moveTo z => + cases z <;> simp [step] <;> (first | apply Nat.le_refl | apply Nat.sub_le_self) + | attack => + have hph : (step s Action.attack).hp = s.hp := by + simp [step] + split + · split <;> rfl + · rfl + calc + (step s Action.attack).hp = s.hp := hph + _ ≤ s.hp := Nat.le_refl _ + | openChest => + have hph : (step s Action.openChest).hp = s.hp := by + simp [step]; split <;> rfl + rw [hph]; apply Nat.le_refl + | useExit => + have hph : (step s Action.useExit).hp = s.hp := by + simp [step]; split <;> rfl + rw [hph]; apply Nat.le_refl + | wait => simp [step]; + +/-! ### Action-mask/safety layer -/ + +/-- Legal-action predicate for the symbolic transition layer (Bool version + for computable checking via `native_decide`). -/ +def actionEnabled (s : EnvState) : Action → Bool + | Action.moveTo z => z ≠ Zone.topTrap && z ≠ Zone.bottomTrap + | Action.attack => s.zone = Zone.nearMonster && s.monsterAlive = true && s.hp > 1 + | Action.openChest => s.zone = Zone.nearChest && s.chestOpened = false + | Action.useExit => s.zone = Zone.westExit && !s.monsterAlive && s.keys > 0 + | Action.wait => true + +def actionsEnabledAlong : EnvState → List Action → Bool + | _, [] => true + | s, a :: rest => actionEnabled s a && actionsEnabledAlong (step s a) rest + +/-! ### Baseline-aligned symbolic policy (BFS certificate mode) -/ + +/-- +A symbolic BFS action provider. In Python this is implemented by +`bfs_path(...)` plus `follow_bfs_aligned(...)`, which returns one movement +action toward the first tile of the current shortest path, or the supplied +`final_action` when already at a goal zone. +-/ +abbrev BfsAction := EnvState → Zone → Action + +/-- +Policy shape matching `decide_task2`. + +1. If a monster is alive: if at `nearMonster`, attack; else BFS to `nearMonster`. +2. Else if chest not opened: if at `nearChest`, openChest; else BFS to `nearChest`. +3. Else (has keys): if at `westExit`, useExit; else BFS to `westExit`. + +This definition is parameterized by `bfsAction`: Lean verifies the stage logic +and the BFS contract separately, instead of hardcoding a route. +-/ +def symbolicPolicy (bfsAction : BfsAction) (s : EnvState) : Action := + if s.completed then Action.wait + else if s.monsterAlive = true then + if s.zone = Zone.nearMonster then Action.attack + else bfsAction s Zone.nearMonster + else if s.chestOpened = false then + if s.zone = Zone.nearChest then Action.openChest + else bfsAction s Zone.nearChest + else + if s.zone = Zone.westExit then Action.useExit + else bfsAction s Zone.westExit + +/-! ### BFS route certificates -/ + +/-- +Soundness certificate for the monster-approach BFS phase. The plan must end at +`nearMonster` zone while preserving combat-relevant state (monster still alive, +monsterHp unchanged, hp unchanged, chest still closed, keys still zero). +-/ +structure MonsterApproachCertificate (start : EnvState) where + plan : List Action + enabled : actionsEnabledAlong start plan = true + endsAtMonster : (run start plan).zone = Zone.nearMonster + monsterAlivePreserved : (run start plan).monsterAlive = start.monsterAlive + monsterHpPreserved : (run start plan).monsterHp = start.monsterHp + hpPreserved : (run start plan).hp = start.hp + keysPreserved : (run start plan).keys = start.keys + chestOpenedPreserved : (run start plan).chestOpened = start.chestOpened + completedPreserved : (run start plan).completed = start.completed + +/-- +Soundness certificate for the chest-approach BFS phase. The plan must end at +`nearChest` zone while preserving state (monster defeated, keys unchanged, chest +still closed). +-/ +structure ChestApproachCertificate (start : EnvState) where + plan : List Action + enabled : actionsEnabledAlong start plan = true + endsAtChest : (run start plan).zone = Zone.nearChest + monsterAlivePreserved : (run start plan).monsterAlive = start.monsterAlive + monsterHpPreserved : (run start plan).monsterHp = start.monsterHp + hpPreserved : (run start plan).hp = start.hp + keysPreserved : (run start plan).keys = start.keys + chestOpenedPreserved : (run start plan).chestOpened = start.chestOpened + completedPreserved : (run start plan).completed = start.completed + +/-- +Soundness certificate for the exit-approach BFS phase. The plan must end at +`westExit` zone while preserving exit-relevant state (monster defeated, +keys > 0, chest opened). +-/ +structure ExitApproachCertificate (start : EnvState) where + plan : List Action + enabled : actionsEnabledAlong start plan = true + endsAtExit : (run start plan).zone = Zone.westExit + monsterAlivePreserved : (run start plan).monsterAlive = start.monsterAlive + monsterHpPreserved : (run start plan).monsterHp = start.monsterHp + hpPreserved : (run start plan).hp = start.hp + keysPreserved : (run start plan).keys = start.keys + chestOpenedPreserved : (run start plan).chestOpened = start.chestOpened + completedPreserved : (run start plan).completed = start.completed + +/-! ### Main non-fixed-route Task 2 theorem -/ + +/-- +Task 2 completion theorem aligned with the BFS policy. + +The policy structure is: +1. BFS to monster zone → attack × 2 (chaser has hp=2 per room_001.json) +2. BFS to chest zone → openChest +3. BFS to west exit zone → useExit (triggers conditional exit) + +This theorem chains the BFS certificates and the interaction actions. The two +attacks are explicit because the map fixes `monsterHp = 2`. +-/ +theorem task2_bfs_certificate_completes + (monsterRoute : MonsterApproachCertificate initialState) + (hMonsterHp2 : (run initialState monsterRoute.plan).monsterHp = 2) + (chestRoute : + ChestApproachCertificate + (step (step (run initialState monsterRoute.plan) Action.attack) Action.attack)) + (exitRoute : + ExitApproachCertificate + (step + (run + (step (step (run initialState monsterRoute.plan) Action.attack) Action.attack) + chestRoute.plan) + Action.openChest)) : + GoalReached + (step + (run + (step + (run + (step (step (run initialState monsterRoute.plan) Action.attack) Action.attack) + chestRoute.plan) + Action.openChest) + exitRoute.plan) + Action.useExit) := by + let sM := run initialState monsterRoute.plan + let sA1 := step sM Action.attack + let sA2 := step sA1 Action.attack + let sC := run sA2 chestRoute.plan + let sO := step sC Action.openChest + let sE := run sO exitRoute.plan + -- Monster-approach BFS: ends at nearMonster, state preserved. + have hZoneM : sM.zone = Zone.nearMonster := monsterRoute.endsAtMonster + have hAliveM : sM.monsterAlive = true := monsterRoute.monsterAlivePreserved + have hHpM : sM.hp > 1 := by + have hp_eq : sM.hp = initialState.hp := monsterRoute.hpPreserved + have : initialState.hp = 5 := by simp [initialState] + omega + have hMhpM : sM.monsterHp = 2 := hMonsterHp2 + have hMhpMgt1 : sM.monsterHp > 1 := by omega + -- First attack: monsterHp 2 → 1, monster still alive, zone/hp preserved. + have hA1Mhp : sA1.monsterHp = 1 := by + have h := attack_reduces_monsterHp_when_adjacent hZoneM hAliveM hHpM + show (step sM Action.attack).monsterHp = 1 + rw [h]; omega + have hA1Alive : sA1.monsterAlive = true := + attack_preserves_monsterAlive_when_monsterHp_gt_one hZoneM hAliveM hHpM hMhpMgt1 + have hA1Hp : sA1.hp > 1 := by + have h : (step sM Action.attack).hp = sM.hp := attack_preserves_hp + show (step sM Action.attack).hp > 1 + rw [h]; exact hHpM + have hA1Zone : sA1.zone = Zone.nearMonster := by + have h : (step sM Action.attack).zone = sM.zone := attack_preserves_zone + show (step sM Action.attack).zone = Zone.nearMonster + rw [h]; exact hZoneM + -- Second attack: monsterHp 1 → 0, monster defeated. + have hA2Defeated : sA2.monsterAlive = false := + attack_kills_monster_when_monsterHp_one hA1Zone hA1Alive hA1Hp hA1Mhp + -- Chest-approach BFS: ends at nearChest, chest still closed. + have hChestZone : sC.zone = Zone.nearChest := chestRoute.endsAtChest + have hChestClosed : sC.chestOpened = false := by + have h1 : sC.chestOpened = sA2.chestOpened := chestRoute.chestOpenedPreserved + have h2 : sA2.chestOpened = sA1.chestOpened := attack_preserves_chestOpened + have h3 : sA1.chestOpened = sM.chestOpened := attack_preserves_chestOpened + have h4 : sM.chestOpened = initialState.chestOpened := monsterRoute.chestOpenedPreserved + rw [h1, h2, h3, h4]; simp [initialState] + -- Open chest: keys + 1, monster defeated preserved. + have hOKeys : sO.keys > 0 := by + have hInc : (step sC Action.openChest).keys = sC.keys + 1 := + chest_open_increases_keys hChestZone hChestClosed + show (step sC Action.openChest).keys > 0 + rw [hInc]; omega + have hODefeated : sO.monsterAlive = false := by + have h1 : (step sC Action.openChest).monsterAlive = sC.monsterAlive := chest_open_preserves_monsterAlive + have h2 : sC.monsterAlive = sA2.monsterAlive := chestRoute.monsterAlivePreserved + show (step sC Action.openChest).monsterAlive = false + rw [h1, h2]; exact hA2Defeated + -- Exit-approach BFS: ends at westExit. + have hExitZone : sE.zone = Zone.westExit := exitRoute.endsAtExit + have hEDefeated : sE.monsterAlive = false := by + have h1 : sE.monsterAlive = sO.monsterAlive := exitRoute.monsterAlivePreserved + rw [h1]; exact hODefeated + have hEKeys : sE.keys > 0 := by + have h1 : sE.keys = sO.keys := exitRoute.keysPreserved + rw [h1]; exact hOKeys + -- useExit on westExit with requirements → completed. + have hDone : (step sE Action.useExit).completed = true := + exit_succeeds_after_requirements hExitZone hEDefeated hEKeys + simpa [GoalReached, sM, sA1, sA2, sC, sO, sE] using hDone + +/-! ### Public map regression instance -/ + +-- Zone-level BFS plans for the public Task-2 layout. +def publicMonsterPlan : List Action := [Action.moveTo Zone.nearMonster] +def publicChestPlan : List Action := [Action.moveTo Zone.nearChest] +def publicExitPlan : List Action := [Action.moveTo Zone.westExit] + +def publicMonsterRoute : MonsterApproachCertificate initialState := + { plan := publicMonsterPlan + enabled := by native_decide + endsAtMonster := by native_decide + monsterAlivePreserved := by native_decide + monsterHpPreserved := by native_decide + hpPreserved := by native_decide + keysPreserved := by native_decide + chestOpenedPreserved := by native_decide + completedPreserved := by native_decide } + +def publicChestRoute : + ChestApproachCertificate + (step (step (run initialState publicMonsterPlan) Action.attack) Action.attack) := + { plan := publicChestPlan + enabled := by native_decide + endsAtChest := by native_decide + monsterAlivePreserved := by native_decide + monsterHpPreserved := by native_decide + hpPreserved := by native_decide + keysPreserved := by native_decide + chestOpenedPreserved := by native_decide + completedPreserved := by native_decide } + +def publicExitRoute : + ExitApproachCertificate + (step + (run + (step (step (run initialState publicMonsterPlan) Action.attack) Action.attack) + publicChestPlan) + Action.openChest) := + { plan := publicExitPlan + enabled := by native_decide + endsAtExit := by native_decide + monsterAlivePreserved := by native_decide + monsterHpPreserved := by native_decide + hpPreserved := by native_decide + keysPreserved := by native_decide + chestOpenedPreserved := by native_decide + completedPreserved := by native_decide } + +theorem public_task2_bfs_certificate_completes : + GoalReached + (step + (run + (step + (run + (step (step (run initialState publicMonsterPlan) Action.attack) Action.attack) + publicChestPlan) + Action.openChest) + publicExitPlan) + Action.useExit) := by + exact task2_bfs_certificate_completes publicMonsterRoute (by native_decide) + publicChestRoute publicExitRoute + +end Task2Formalization diff --git a/student_agent/lean/Task3Formalization.lean b/student_agent/lean/Task3Formalization.lean new file mode 100644 index 0000000..430ba1e --- /dev/null +++ b/student_agent/lean/Task3Formalization.lean @@ -0,0 +1,440 @@ +/- + Environment formalization for `mathematical_logic/task_3`. + + Task 3 is the first multi-room return task: + + * start in `start_room`; + * move west through `monster_hall`; + * reach `key_room` and open the chest; + * return east to `start_room`; + * use the locked east exit with the collected key. + + The Python engine contains a chaser monster in `monster_hall`, but the task's + high-level requirement is a room-to-room subgoal chain rather than monster + elimination. This Lean model therefore keeps the monster as part of the state + but abstracts away combat and continuous motion. + + Python correspondence: + * `nesylink/map_data/mathematical_logic/task_3/*.json` + * `nesylink/core/mechanics/interactions.py` + * `nesylink/core/mechanics/movement.py` +-/ + +namespace Task3Formalization + +inductive Room where + | startRoom + | monsterHall + | keyRoom + deriving DecidableEq, Repr + +inductive Action where + | goWest + | goEast + | openChest + | openLockedExit + | wait + deriving DecidableEq, Repr + +structure EnvState where + room : Room + keys : Nat + keyChestOpened : Bool + hallMonsterAlive : Bool + completed : Bool + deriving DecidableEq, Repr + +def initialState : EnvState := + { + room := Room.startRoom + keys := 0 + keyChestOpened := false + hallMonsterAlive := true + completed := false + } + +def GoalReached (s : EnvState) : Prop := + s.completed = true + +def step (s : EnvState) : Action → EnvState + | Action.goWest => + match s.room with + | Room.startRoom => { s with room := Room.monsterHall } + | Room.monsterHall => { s with room := Room.keyRoom } + | Room.keyRoom => s + | Action.goEast => + match s.room with + | Room.keyRoom => { s with room := Room.monsterHall } + | Room.monsterHall => { s with room := Room.startRoom } + | Room.startRoom => s + | Action.openChest => + if s.room = Room.keyRoom ∧ s.keyChestOpened = false then + { s with keyChestOpened := true, keys := s.keys + 1 } + else + s + | Action.openLockedExit => + if s.room = Room.startRoom ∧ s.keys > 0 then + { s with keys := s.keys - 1, completed := true } + else + s + | Action.wait => s + +def run : EnvState → List Action → EnvState + | s, [] => s + | s, a :: rest => run (step s a) rest + +theorem west_then_west_reaches_keyRoom : + (run initialState [Action.goWest, Action.goWest]).room = Room.keyRoom := by + rfl + +theorem openChest_in_keyRoom_increases_keys + {s : EnvState} + (hRoom : s.room = Room.keyRoom) + (hChest : s.keyChestOpened = false) : + (step s Action.openChest).keys = s.keys + 1 := by + simp [step, hRoom, hChest] + +theorem openLockedExit_blocked_without_key + {s : EnvState} + (hRoom : s.room = Room.startRoom) + (hKeys : s.keys = 0) : + step s Action.openLockedExit = s := by + simp [step, hRoom, hKeys] + +theorem openLockedExit_consumes_key_and_completes + {s : EnvState} + (hRoom : s.room = Room.startRoom) + (hKeys : s.keys > 0) : + let t := step s Action.openLockedExit + t.keys = s.keys - 1 ∧ t.completed = true := by + simp [step, hRoom, hKeys] + +/-! ### Chest lemmas -/ + +theorem openChest_only_in_keyRoom + {s : EnvState} (hRoom : s.room ≠ Room.keyRoom) : + step s Action.openChest = s := by + simp [step, hRoom] + +theorem openChest_already_opened + {s : EnvState} (hChest : s.keyChestOpened = true) : + step s Action.openChest = s := by + simp [step, hChest] + +theorem openChest_in_keyRoom_marks_opened + {s : EnvState} + (hRoom : s.room = Room.keyRoom) + (hChest : s.keyChestOpened = false) : + (step s Action.openChest).keyChestOpened = true := by + simp [step, hRoom, hChest] + +/-! ### Locked exit lemmas -/ + +theorem openLockedExit_blocked_not_in_startRoom + {s : EnvState} (hRoom : s.room ≠ Room.startRoom) : + step s Action.openLockedExit = s := by + simp [step, hRoom] + +theorem openLockedExit_preserves_room + {s : EnvState} : (step s Action.openLockedExit).room = s.room := by + by_cases h1 : s.room = Room.startRoom ∧ s.keys > 0 + · simp [step, h1] + · simp [step, h1] + +theorem openLockedExit_preserves_keyChestOpened + {s : EnvState} : (step s Action.openLockedExit).keyChestOpened = s.keyChestOpened := by + by_cases h1 : s.room = Room.startRoom ∧ s.keys > 0 + · simp [step, h1] + · simp [step, h1] + +/-! ### Necessity lemmas -/ + +/-- Non-openLockedExit actions preserve the `completed` field. -/ +theorem non_openLockedExit_preserves_completed + {s : EnvState} (a : Action) (ha : a ≠ Action.openLockedExit) : + (step s a).completed = s.completed := by + cases a with + | goWest => unfold step; cases s.room <;> rfl + | goEast => unfold step; cases s.room <;> rfl + | openChest => + by_cases h1 : s.room = Room.keyRoom ∧ s.keyChestOpened = false + · simp [step, h1] + · simp [step, h1] + | openLockedExit => exact absurd rfl ha + | wait => simp [step] + +/-- `openLockedExit` is the only action that can flip `completed` from `false` + to `true`. -/ +theorem only_openLockedExit_sets_completed + {s : EnvState} (hs : s.completed = false) (a : Action) + (h : (step s a).completed = true) : a = Action.openLockedExit := by + by_cases ha : a = Action.openLockedExit + · exact ha + · have := non_openLockedExit_preserves_completed (s := s) a ha + rw [this, hs] at h + simp at h + +/-! ### Action-wise invariant lemmas -/ + +/-- `hallMonsterAlive` never changes (no attack action in Task 3). -/ +theorem hallMonsterAlive_never_changes + {s : EnvState} (a : Action) : + (step s a).hallMonsterAlive = s.hallMonsterAlive := by + cases a with + | goWest => unfold step; cases s.room <;> rfl + | goEast => unfold step; cases s.room <;> rfl + | openChest => + by_cases h1 : s.room = Room.keyRoom ∧ s.keyChestOpened = false + · simp [step, h1] + · simp [step, h1] + | openLockedExit => + by_cases h1 : s.room = Room.startRoom ∧ s.keys > 0 + · simp [step, h1] + · simp [step, h1] + | wait => simp [step] + +/-- `openChest` is the only action that can change `keyChestOpened`. -/ +theorem only_openChest_changes_keyChestOpened + {s : EnvState} (a : Action) (ha : a ≠ Action.openChest) : + (step s a).keyChestOpened = s.keyChestOpened := by + cases a with + | goWest => unfold step; cases s.room <;> rfl + | goEast => unfold step; cases s.room <;> rfl + | openChest => exact absurd rfl ha + | openLockedExit => + by_cases h1 : s.room = Room.startRoom ∧ s.keys > 0 + · simp [step, h1] + · simp [step, h1] + | wait => simp [step] + +/-- Only `openChest` (gains) and `openLockedExit` (consumes) can change `keys`. -/ +theorem only_openChest_or_openLockedExit_changes_keys + {s : EnvState} (a : Action) + (ha : a ≠ Action.openChest) (ha' : a ≠ Action.openLockedExit) : + (step s a).keys = s.keys := by + cases a with + | goWest => unfold step; cases s.room <;> rfl + | goEast => unfold step; cases s.room <;> rfl + | openChest => exact absurd rfl ha + | openLockedExit => exact absurd rfl ha' + | wait => simp [step] + +/-! ### Reachable states and invariants -/ + +/-- Reachability: states reachable from `initialState` via arbitrary actions. -/ +inductive Reachable : EnvState → Prop where + | init : Reachable initialState + | step (s : EnvState) (a : Action) : Reachable s → Reachable (step s a) + +/-- Invariant: if the key chest is not opened then no keys have been collected. -/ +theorem keyChestOpened_or_keys_zero {s : EnvState} (hr : Reachable s) : + s.keyChestOpened = true ∨ s.keys = 0 := by + induction hr with + | init => simp [initialState] + | step s a hr ih => + cases a with + | goWest => unfold step; cases s.room <;> simp <;> exact ih + | goEast => unfold step; cases s.room <;> simp <;> exact ih + | openChest => + by_cases h1 : s.room = Room.keyRoom ∧ s.keyChestOpened = false + · left; simp [step, h1] + · simp [step, h1]; exact ih + | openLockedExit => + by_cases h1 : s.room = Room.startRoom ∧ s.keys > 0 + · rcases ih with (h | h) + · left; simp [step, h1, h] + · right; omega + · simp [step, h1]; exact ih + | wait => simp [step]; exact ih + +/-- If a reachable state has `completed = true`, then the key chest has been + opened. (We cannot conclude `room = startRoom` because the player may + navigate away after completing the task.) -/ +theorem completion_postcondition {s : EnvState} (hr : Reachable s) (hc : s.completed = true) : + s.keyChestOpened = true := by + induction hr with + | init => simp [initialState] at hc + | step s' a hr ih => + by_cases hs : s'.completed = true + · have hch := ih hs + have hch' : (step s' a).keyChestOpened = s'.keyChestOpened := by + by_cases ha : a = Action.openChest + · subst ha + have hstep : step s' Action.openChest = s' := openChest_already_opened hch + simp [hstep, hch] + · exact only_openChest_changes_keyChestOpened a ha + simp [hch', hch] + · have hf : s'.completed = false := by + cases h : s'.completed with + | true => exact absurd h hs + | false => rfl + have ha : a = Action.openLockedExit := only_openLockedExit_sets_completed hf a hc + subst ha + by_cases hcond : s'.room = Room.startRoom ∧ s'.keys > 0 + · have hchest : s'.keyChestOpened = true := by + have h_cases := keyChestOpened_or_keys_zero hr + rcases h_cases with (h | h) + · exact h + · rw [h] at hcond; simp at hcond + have hch' : (step s' Action.openLockedExit).keyChestOpened = s'.keyChestOpened := + openLockedExit_preserves_keyChestOpened + simp [hch', hchest] + · have hstep : step s' Action.openLockedExit = s' := by simp [step, hcond] + rw [hstep, hf] at hc + simp at hc + +/-! ### Action-mask/safety layer -/ + +/-- Legal-action predicate for the symbolic transition layer (Bool version + for computable checking via `native_decide`). -/ +def actionEnabled (s : EnvState) : Action → Bool + | Action.goWest => s.room = Room.startRoom || s.room = Room.monsterHall + | Action.goEast => s.room = Room.keyRoom || s.room = Room.monsterHall + | Action.openChest => s.room = Room.keyRoom && !s.keyChestOpened + | Action.openLockedExit => s.room = Room.startRoom && s.keys > 0 + | Action.wait => true + +def actionsEnabledAlong : EnvState → List Action → Bool + | _, [] => true + | s, a :: rest => actionEnabled s a && actionsEnabledAlong (step s a) rest + +/-! ### Baseline-aligned symbolic policy (BFS certificate mode) -/ + +/-- +A symbolic BFS action provider. In Python this is implemented by +`bfs_path(...)` plus `follow_bfs_aligned(...)`, which returns one movement +action toward the first tile of the current shortest path, or the supplied +`final_action` when already at a goal room. +-/ +abbrev BfsAction := EnvState → Room → Action + +/-- +Policy shape matching `decide_task3`. + +1. If the key chest is not yet opened: if at `keyRoom`, openChest; else BFS to + `keyRoom`. +2. Else (key held): if at `startRoom`, openLockedExit; else BFS to `startRoom`. + +This definition is parameterized by `bfsAction`: Lean verifies the stage logic +and the BFS contract separately, instead of hardcoding a route. +-/ +def symbolicPolicy (bfsAction : BfsAction) (s : EnvState) : Action := + if s.completed then Action.wait + else if s.keyChestOpened = false then + if s.room = Room.keyRoom then Action.openChest + else bfsAction s Room.keyRoom + else + if s.room = Room.startRoom then Action.openLockedExit + else bfsAction s Room.startRoom + +/-! ### BFS route certificates -/ + +/-- +Soundness certificate for the key-room-approach BFS phase. The plan must end at +`keyRoom` while preserving state (keys unchanged, chest still closed, hall +monster unchanged, not completed). +-/ +structure KeyRoomApproachCertificate (start : EnvState) where + plan : List Action + enabled : actionsEnabledAlong start plan = true + endsAtKeyRoom : (run start plan).room = Room.keyRoom + keysPreserved : (run start plan).keys = start.keys + keyChestOpenedPreserved : (run start plan).keyChestOpened = start.keyChestOpened + hallMonsterAlivePreserved : (run start plan).hallMonsterAlive = start.hallMonsterAlive + completedPreserved : (run start plan).completed = start.completed + +/-- +Soundness certificate for the start-room-approach BFS phase. The plan must end +at `startRoom` while preserving state (keys > 0, chest opened, hall monster +unchanged, not completed). +-/ +structure StartRoomApproachCertificate (start : EnvState) where + plan : List Action + enabled : actionsEnabledAlong start plan = true + endsAtStartRoom : (run start plan).room = Room.startRoom + keysPreserved : (run start plan).keys = start.keys + keyChestOpenedPreserved : (run start plan).keyChestOpened = start.keyChestOpened + hallMonsterAlivePreserved : (run start plan).hallMonsterAlive = start.hallMonsterAlive + completedPreserved : (run start plan).completed = start.completed + +/-! ### Main non-fixed-route Task 3 theorem -/ + +/-- +Task 3 completion theorem aligned with the BFS policy. + +The policy structure is: +1. BFS to keyRoom → openChest (gains one key) +2. BFS back to startRoom → openLockedExit (consumes key, completes task) + +This theorem chains the BFS certificates and the interaction actions. +-/ +theorem task3_bfs_certificate_completes + (keyRoute : KeyRoomApproachCertificate initialState) + (startRoute : + StartRoomApproachCertificate + (step (run initialState keyRoute.plan) Action.openChest)) : + GoalReached + (step + (run + (step (run initialState keyRoute.plan) Action.openChest) + startRoute.plan) + Action.openLockedExit) := by + let sK := run initialState keyRoute.plan + let sO := step sK Action.openChest + let sS := run sO startRoute.plan + -- Key-room-approach BFS: ends at keyRoom, chest still closed. + have hRoomK : sK.room = Room.keyRoom := keyRoute.endsAtKeyRoom + have hChestClosed : sK.keyChestOpened = false := by + have h := keyRoute.keyChestOpenedPreserved + rw [h]; simp [initialState] + -- Open chest: keys + 1, chest marked opened. + have hOKeys : sO.keys > 0 := by + have hInc : (step sK Action.openChest).keys = sK.keys + 1 := + openChest_in_keyRoom_increases_keys hRoomK hChestClosed + show (step sK Action.openChest).keys > 0 + rw [hInc]; omega + -- Start-room-approach BFS: ends at startRoom, keys preserved > 0. + have hRoomS : sS.room = Room.startRoom := startRoute.endsAtStartRoom + have hSKeys : sS.keys > 0 := by + have h1 : sS.keys = sO.keys := startRoute.keysPreserved + rw [h1]; exact hOKeys + -- openLockedExit on startRoom with keys > 0 → completed. + have h := openLockedExit_consumes_key_and_completes hRoomS hSKeys + simpa [GoalReached, sK, sO, sS] using h.2 + +/-! ### Public map regression instance -/ + +-- Room-level BFS plans for the public Task-3 layout. +def publicKeyPlan : List Action := [Action.goWest, Action.goWest] +def publicStartPlan : List Action := [Action.goEast, Action.goEast] + +def publicKeyRoute : KeyRoomApproachCertificate initialState := + { plan := publicKeyPlan + enabled := by native_decide + endsAtKeyRoom := by native_decide + keysPreserved := by native_decide + keyChestOpenedPreserved := by native_decide + hallMonsterAlivePreserved := by native_decide + completedPreserved := by native_decide } + +def publicStartRoute : + StartRoomApproachCertificate + (step (run initialState publicKeyPlan) Action.openChest) := + { plan := publicStartPlan + enabled := by native_decide + endsAtStartRoom := by native_decide + keysPreserved := by native_decide + keyChestOpenedPreserved := by native_decide + hallMonsterAlivePreserved := by native_decide + completedPreserved := by native_decide } + +theorem public_task3_bfs_certificate_completes : + GoalReached + (step + (run + (step (run initialState publicKeyPlan) Action.openChest) + publicStartPlan) + Action.openLockedExit) := by + exact task3_bfs_certificate_completes publicKeyRoute publicStartRoute + +end Task3Formalization diff --git a/student_agent/lean/Task4Formalization.lean b/student_agent/lean/Task4Formalization.lean new file mode 100644 index 0000000..9c206d5 --- /dev/null +++ b/student_agent/lean/Task4Formalization.lean @@ -0,0 +1,1028 @@ +/- + Environment formalization for `mathematical_logic/task_4`. + + This file models the task-4 environment layer that is most useful for the + course report: + + * room-level navigation; + * the rotating bridge in the center room; + * the locked east exit that requires a key but does not consume it; + * the north chest that grants a key; + * the east chest that grants a sword; + * the south guardian whose defeat reveals the final chest; + * the final chest as the task completion condition. + + It intentionally abstracts away pixel motion, monster AI, knockback and HUD + rendering. Those details exist in the Python engine, but the report's + environment-formalization section only needs the symbolic transition layer that + the planner and proofs rely on. + + Python correspondence: + * `nesylink/core/mechanics/movement.py` + * `nesylink/core/mechanics/interactions.py` + * `nesylink/core/mechanics/combat.py` + * `nesylink/map_data/mathematical_logic/task_4/*.json` +-/ + +namespace Task4Formalization + +inductive Room where + | west + | center + | north + | east + | south + deriving DecidableEq, Repr + +inductive BridgeState where + | westToNorth + | westToEast + | westToSouth + deriving DecidableEq, Repr + +inductive Action where + | toggleBridge + | goCenter + | goWest + | goNorth + | goEast + | goSouth + | openChest + | attack + | openFinalChest + | wait + deriving DecidableEq, Repr + +structure EnvState where + room : Room + bridge : BridgeState + keys : Nat + hasShield : Bool + hasSword : Bool + northChestOpened : Bool + eastChestOpened : Bool + guardianAlive : Bool + finalChestVisible : Bool + finalChestOpened : Bool + deriving DecidableEq, Repr + +def initialState : EnvState := + { + room := Room.west + bridge := BridgeState.westToNorth + keys := 0 + hasShield := true + hasSword := false + northChestOpened := false + eastChestOpened := false + guardianAlive := true + finalChestVisible := false + finalChestOpened := false + } + +def rotateBridge : BridgeState → BridgeState + | BridgeState.westToNorth => BridgeState.westToEast + | BridgeState.westToEast => BridgeState.westToSouth + | BridgeState.westToSouth => BridgeState.westToNorth + +def bridgeAllowsNorth (b : BridgeState) : Prop := + b = BridgeState.westToNorth + +def bridgeAllowsEast (b : BridgeState) : Prop := + b = BridgeState.westToEast + +def bridgeAllowsSouth (b : BridgeState) : Prop := + b = BridgeState.westToSouth + +def hasKey (s : EnvState) : Prop := + s.keys > 0 + +instance (b : BridgeState) : Decidable (bridgeAllowsNorth b) := + inferInstanceAs (Decidable (b = BridgeState.westToNorth)) + +instance (b : BridgeState) : Decidable (bridgeAllowsEast b) := + inferInstanceAs (Decidable (b = BridgeState.westToEast)) + +instance (b : BridgeState) : Decidable (bridgeAllowsSouth b) := + inferInstanceAs (Decidable (b = BridgeState.westToSouth)) + +instance (s : EnvState) : Decidable (hasKey s) := + inferInstanceAs (Decidable (s.keys > 0)) + +def GoalReached (s : EnvState) : Prop := + s.finalChestOpened = true + +def step (s : EnvState) : Action → EnvState + | Action.toggleBridge => + if s.room = Room.west then + { s with bridge := rotateBridge s.bridge } + else + s + | Action.goCenter => + if s.room = Room.west ∨ s.room = Room.north ∨ s.room = Room.east ∨ s.room = Room.south then + { s with room := Room.center } + else + s + | Action.goWest => + if s.room = Room.center then + { s with room := Room.west } + else + s + | Action.goNorth => + if s.room = Room.center ∧ bridgeAllowsNorth s.bridge then + { s with room := Room.north } + else + s + | Action.goEast => + if s.room = Room.center ∧ bridgeAllowsEast s.bridge ∧ hasKey s then + { s with room := Room.east } + else + s + | Action.goSouth => + if s.room = Room.center ∧ bridgeAllowsSouth s.bridge then + { s with room := Room.south } + else + s + | Action.openChest => + if s.room = Room.north ∧ s.northChestOpened = false then + { s with northChestOpened := true, keys := s.keys + 1 } + else if s.room = Room.east ∧ s.eastChestOpened = false then + { s with eastChestOpened := true, hasSword := true } + else + s + | Action.attack => + if s.room = Room.south ∧ s.guardianAlive = true ∧ s.hasSword = true then + { s with guardianAlive := false, finalChestVisible := true } + else + s + | Action.openFinalChest => + if s.room = Room.center ∧ s.finalChestVisible = true then + { s with finalChestOpened := true } + else + s + | Action.wait => s + +def run : EnvState → List Action → EnvState + | s, [] => s + | s, a :: rest => run (step s a) rest + +theorem rotateBridge_three_cycle : + rotateBridge (rotateBridge (rotateBridge BridgeState.westToNorth)) = + BridgeState.westToNorth := by + rfl + +theorem goEast_blocked_without_key + {s : EnvState} + (hRoom : s.room = Room.center) + (hBridge : s.bridge = BridgeState.westToEast) + (hKeys : s.keys = 0) : + step s Action.goEast = s := by + simp [step, hRoom, hBridge, bridgeAllowsEast, hasKey, hKeys] + +theorem goEast_succeeds_with_key + {s : EnvState} + (hRoom : s.room = Room.center) + (hBridge : s.bridge = BridgeState.westToEast) + (hKeys : s.keys > 0) : + (step s Action.goEast).room = Room.east := by + simp [step, hRoom, hBridge, bridgeAllowsEast, hasKey, hKeys] + +theorem attack_without_sword_has_no_effect + {s : EnvState} + (hRoom : s.room = Room.south) + (hGuardian : s.guardianAlive = true) + (hSword : s.hasSword = false) : + step s Action.attack = s := by + simp [step, hRoom, hGuardian, hSword] + +theorem attack_with_sword_reveals_final_chest + {s : EnvState} + (hRoom : s.room = Room.south) + (hGuardian : s.guardianAlive = true) + (hSword : s.hasSword = true) : + let t := step s Action.attack + t.guardianAlive = false ∧ t.finalChestVisible = true := by + simp [step, hRoom, hGuardian, hSword] + +theorem openFinalChest_requires_visibility + {s : EnvState} + (hRoom : s.room = Room.center) + (hVisible : s.finalChestVisible = false) : + step s Action.openFinalChest = s := by + simp [step, hRoom, hVisible] + +theorem openFinalChest_blocked_not_in_center + {s : EnvState} (hRoom : s.room ≠ Room.center) : + step s Action.openFinalChest = s := by + simp [step, hRoom] + +theorem openFinalChest_succeeds_when_visible + {s : EnvState} + (hRoom : s.room = Room.center) + (hVisible : s.finalChestVisible = true) : + (step s Action.openFinalChest).finalChestOpened = true := by + simp [step, hRoom, hVisible] + +/-! ### toggleBridge lemmas -/ + +theorem toggleBridge_only_in_west + {s : EnvState} (hRoom : s.room ≠ Room.west) : + step s Action.toggleBridge = s := by + simp [step, hRoom] + +/-! ### Navigation lemmas -/ + +/-- `goEast` does NOT consume keys (Python conditional exit behavior). + This is an explicit invariant per project convention. -/ +theorem goEast_preserves_keys + {s : EnvState} : (step s Action.goEast).keys = s.keys := by + by_cases h1 : s.room = Room.center ∧ bridgeAllowsEast s.bridge ∧ hasKey s + · simp [step, h1] + · simp [step, h1] + +/-! ### Chest lemmas (north and east independent) -/ + +theorem openChest_north_only + {s : EnvState} + (hRoom : s.room ≠ Room.north) + (hRoom' : s.room ≠ Room.east) : + step s Action.openChest = s := by + show (if s.room = Room.north ∧ s.northChestOpened = false then + { s with northChestOpened := true, keys := s.keys + 1 } + else if s.room = Room.east ∧ s.eastChestOpened = false then + { s with eastChestOpened := true, hasSword := true } + else s) = s + by_cases h1 : s.room = Room.north ∧ s.northChestOpened = false + · exact absurd h1.1 hRoom + · by_cases h2 : s.room = Room.east ∧ s.eastChestOpened = false + · exact absurd h2.1 hRoom' + · simp [h1, h2] + +theorem openChest_north_increases_keys + {s : EnvState} + (hRoom : s.room = Room.north) + (hChest : s.northChestOpened = false) : + (step s Action.openChest).keys = s.keys + 1 := by + simp [step, hRoom, hChest] + +theorem openChest_north_marks_opened + {s : EnvState} + (hRoom : s.room = Room.north) + (hChest : s.northChestOpened = false) : + (step s Action.openChest).northChestOpened = true := by + simp [step, hRoom, hChest] + +theorem openChest_north_already_opened + {s : EnvState} + (hRoom : s.room = Room.north) + (hChest : s.northChestOpened = true) : + step s Action.openChest = s := by + simp [step, hRoom, hChest] + +theorem openChest_east_grants_sword + {s : EnvState} + (hRoom : s.room = Room.east) + (hChest : s.eastChestOpened = false) : + (step s Action.openChest).hasSword = true := by + have hNorth : ¬(s.room = Room.north ∧ s.northChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + simp [step, hRoom, hChest] + +theorem openChest_east_marks_opened + {s : EnvState} + (hRoom : s.room = Room.east) + (hChest : s.eastChestOpened = false) : + (step s Action.openChest).eastChestOpened = true := by + have hNorth : ¬(s.room = Room.north ∧ s.northChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + simp [step, hRoom, hChest] + +theorem openChest_east_already_opened + {s : EnvState} + (hRoom : s.room = Room.east) + (hChest : s.eastChestOpened = true) : + step s Action.openChest = s := by + have hNorth : ¬(s.room = Room.north ∧ s.northChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + simp [step, hRoom, hChest] + +/-! ### openFinalChest preserves lemmas -/ + +theorem openFinalChest_preserves_guardianAlive + {s : EnvState} : (step s Action.openFinalChest).guardianAlive = s.guardianAlive := by + by_cases h1 : s.room = Room.center ∧ s.finalChestVisible = true + · simp [step, h1] + · simp [step, h1] + +/-! ### Necessity lemmas -/ + +/-- Non-openFinalChest actions preserve the `finalChestOpened` field. -/ +theorem non_openFinalChest_preserves_finalChestOpened + {s : EnvState} (a : Action) (ha : a ≠ Action.openFinalChest) : + (step s a).finalChestOpened = s.finalChestOpened := by + cases a with + | toggleBridge => + by_cases h1 : s.room = Room.west + · simp [step, h1] + · simp [step, h1] + | goCenter => + by_cases h1 : s.room = Room.west ∨ s.room = Room.north ∨ s.room = Room.east ∨ s.room = Room.south + · simp [step, h1] + · simp [step, h1] + | goWest => + by_cases h1 : s.room = Room.center + · simp [step, h1] + · simp [step, h1] + | goNorth => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsNorth s.bridge + · simp [step, h1] + · simp [step, h1] + | goEast => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsEast s.bridge ∧ hasKey s + · simp [step, h1] + · simp [step, h1] + | goSouth => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsSouth s.bridge + · simp [step, h1] + · simp [step, h1] + | openChest => + by_cases h1 : s.room = Room.north ∧ s.northChestOpened = false + · simp [step, h1] + · by_cases h2 : s.room = Room.east ∧ s.eastChestOpened = false + · simp [step, h2] + · simp [step, h1, h2] + | attack => + by_cases h1 : s.room = Room.south ∧ s.guardianAlive = true ∧ s.hasSword = true + · simp [step, h1] + · simp [step, h1] + | openFinalChest => exact absurd rfl ha + | wait => simp [step] + +/-- `openFinalChest` is the only action that can flip `finalChestOpened` from + `false` to `true`. -/ +theorem only_openFinalChest_sets_finalChestOpened + {s : EnvState} (hs : s.finalChestOpened = false) (a : Action) + (h : (step s a).finalChestOpened = true) : a = Action.openFinalChest := by + by_cases ha : a = Action.openFinalChest + · exact ha + · have := non_openFinalChest_preserves_finalChestOpened (s := s) a ha + rw [this, hs] at h + simp at h + +/-! ### Action-wise invariant lemmas -/ + +/-- `attack` is the only action that can change `guardianAlive`. -/ +theorem only_attack_changes_guardianAlive + {s : EnvState} (a : Action) (ha : a ≠ Action.attack) : + (step s a).guardianAlive = s.guardianAlive := by + cases a with + | toggleBridge => + by_cases h1 : s.room = Room.west + · simp [step, h1] + · simp [step, h1] + | goCenter => + by_cases h1 : s.room = Room.west ∨ s.room = Room.north ∨ s.room = Room.east ∨ s.room = Room.south + · simp [step, h1] + · simp [step, h1] + | goWest => + by_cases h1 : s.room = Room.center + · simp [step, h1] + · simp [step, h1] + | goNorth => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsNorth s.bridge + · simp [step, h1] + · simp [step, h1] + | goEast => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsEast s.bridge ∧ hasKey s + · simp [step, h1] + · simp [step, h1] + | goSouth => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsSouth s.bridge + · simp [step, h1] + · simp [step, h1] + | openChest => + by_cases h1 : s.room = Room.north ∧ s.northChestOpened = false + · simp [step, h1] + · by_cases h2 : s.room = Room.east ∧ s.eastChestOpened = false + · simp [step, h2] + · simp [step, h1, h2] + | attack => exact absurd rfl ha + | openFinalChest => + by_cases h1 : s.room = Room.center ∧ s.finalChestVisible = true + · simp [step, h1] + · simp [step, h1] + | wait => simp [step] + +/-- `attack` is the only action that can change `finalChestVisible` (it reveals + the final chest when the guardian is defeated). -/ +theorem only_attack_changes_finalChestVisible + {s : EnvState} (a : Action) (ha : a ≠ Action.attack) : + (step s a).finalChestVisible = s.finalChestVisible := by + cases a with + | toggleBridge => + by_cases h1 : s.room = Room.west + · simp [step, h1] + · simp [step, h1] + | goCenter => + by_cases h1 : s.room = Room.west ∨ s.room = Room.north ∨ s.room = Room.east ∨ s.room = Room.south + · simp [step, h1] + · simp [step, h1] + | goWest => + by_cases h1 : s.room = Room.center + · simp [step, h1] + · simp [step, h1] + | goNorth => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsNorth s.bridge + · simp [step, h1] + · simp [step, h1] + | goEast => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsEast s.bridge ∧ hasKey s + · simp [step, h1] + · simp [step, h1] + | goSouth => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsSouth s.bridge + · simp [step, h1] + · simp [step, h1] + | openChest => + by_cases h1 : s.room = Room.north ∧ s.northChestOpened = false + · simp [step, h1] + · by_cases h2 : s.room = Room.east ∧ s.eastChestOpened = false + · simp [step, h2] + · simp [step, h1, h2] + | attack => exact absurd rfl ha + | openFinalChest => + by_cases h1 : s.room = Room.center ∧ s.finalChestVisible = true + · simp [step, h1] + · simp [step, h1] + | wait => simp [step] + +/-- `openChest` is the only action that can change `northChestOpened`. -/ +theorem only_openChest_changes_northChestOpened + {s : EnvState} (a : Action) (ha : a ≠ Action.openChest) : + (step s a).northChestOpened = s.northChestOpened := by + cases a with + | toggleBridge => + by_cases h1 : s.room = Room.west + · simp [step, h1] + · simp [step, h1] + | goCenter => + by_cases h1 : s.room = Room.west ∨ s.room = Room.north ∨ s.room = Room.east ∨ s.room = Room.south + · simp [step, h1] + · simp [step, h1] + | goWest => + by_cases h1 : s.room = Room.center + · simp [step, h1] + · simp [step, h1] + | goNorth => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsNorth s.bridge + · simp [step, h1] + · simp [step, h1] + | goEast => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsEast s.bridge ∧ hasKey s + · simp [step, h1] + · simp [step, h1] + | goSouth => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsSouth s.bridge + · simp [step, h1] + · simp [step, h1] + | openChest => exact absurd rfl ha + | attack => + by_cases h1 : s.room = Room.south ∧ s.guardianAlive = true ∧ s.hasSword = true + · simp [step, h1] + · simp [step, h1] + | openFinalChest => + by_cases h1 : s.room = Room.center ∧ s.finalChestVisible = true + · simp [step, h1] + · simp [step, h1] + | wait => simp [step] + +/-- `openChest` is the only action that can change `eastChestOpened`. -/ +theorem only_openChest_changes_eastChestOpened + {s : EnvState} (a : Action) (ha : a ≠ Action.openChest) : + (step s a).eastChestOpened = s.eastChestOpened := by + cases a with + | toggleBridge => + by_cases h1 : s.room = Room.west + · simp [step, h1] + · simp [step, h1] + | goCenter => + by_cases h1 : s.room = Room.west ∨ s.room = Room.north ∨ s.room = Room.east ∨ s.room = Room.south + · simp [step, h1] + · simp [step, h1] + | goWest => + by_cases h1 : s.room = Room.center + · simp [step, h1] + · simp [step, h1] + | goNorth => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsNorth s.bridge + · simp [step, h1] + · simp [step, h1] + | goEast => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsEast s.bridge ∧ hasKey s + · simp [step, h1] + · simp [step, h1] + | goSouth => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsSouth s.bridge + · simp [step, h1] + · simp [step, h1] + | openChest => exact absurd rfl ha + | attack => + by_cases h1 : s.room = Room.south ∧ s.guardianAlive = true ∧ s.hasSword = true + · simp [step, h1] + · simp [step, h1] + | openFinalChest => + by_cases h1 : s.room = Room.center ∧ s.finalChestVisible = true + · simp [step, h1] + · simp [step, h1] + | wait => simp [step] + +/-- Only `openChest` (north branch) can change `keys`. -/ +theorem only_openChest_changes_keys + {s : EnvState} (a : Action) (ha : a ≠ Action.openChest) : + (step s a).keys = s.keys := by + cases a with + | toggleBridge => + by_cases h1 : s.room = Room.west + · simp [step, h1] + · simp [step, h1] + | goCenter => + by_cases h1 : s.room = Room.west ∨ s.room = Room.north ∨ s.room = Room.east ∨ s.room = Room.south + · simp [step, h1] + · simp [step, h1] + | goWest => + by_cases h1 : s.room = Room.center + · simp [step, h1] + · simp [step, h1] + | goNorth => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsNorth s.bridge + · simp [step, h1] + · simp [step, h1] + | goEast => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsEast s.bridge ∧ hasKey s + · simp [step, h1] + · simp [step, h1] + | goSouth => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsSouth s.bridge + · simp [step, h1] + · simp [step, h1] + | openChest => exact absurd rfl ha + | attack => + by_cases h1 : s.room = Room.south ∧ s.guardianAlive = true ∧ s.hasSword = true + · simp [step, h1] + · simp [step, h1] + | openFinalChest => + by_cases h1 : s.room = Room.center ∧ s.finalChestVisible = true + · simp [step, h1] + · simp [step, h1] + | wait => simp [step] + +/-- `hasShield` never changes (shield is in initial state and never modified). -/ +theorem hasShield_never_changes + {s : EnvState} (a : Action) : (step s a).hasShield = s.hasShield := by + cases a with + | toggleBridge => + by_cases h1 : s.room = Room.west + · simp [step, h1] + · simp [step, h1] + | goCenter => + by_cases h1 : s.room = Room.west ∨ s.room = Room.north ∨ s.room = Room.east ∨ s.room = Room.south + · simp [step, h1] + · simp [step, h1] + | goWest => + by_cases h1 : s.room = Room.center + · simp [step, h1] + · simp [step, h1] + | goNorth => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsNorth s.bridge + · simp [step, h1] + · simp [step, h1] + | goEast => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsEast s.bridge ∧ hasKey s + · simp [step, h1] + · simp [step, h1] + | goSouth => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsSouth s.bridge + · simp [step, h1] + · simp [step, h1] + | openChest => + by_cases h1 : s.room = Room.north ∧ s.northChestOpened = false + · simp [step, h1] + · by_cases h2 : s.room = Room.east ∧ s.eastChestOpened = false + · simp [step, h2] + · simp [step, h1, h2] + | attack => + by_cases h1 : s.room = Room.south ∧ s.guardianAlive = true ∧ s.hasSword = true + · simp [step, h1] + · simp [step, h1] + | openFinalChest => + by_cases h1 : s.room = Room.center ∧ s.finalChestVisible = true + · simp [step, h1] + · simp [step, h1] + | wait => simp [step] + +/-! ### Reachable states and invariants -/ + +/-- Reachability: states reachable from `initialState` via arbitrary actions. -/ +inductive Reachable : EnvState → Prop where + | init : Reachable initialState + | step (s : EnvState) (a : Action) : Reachable s → Reachable (step s a) + +/-- Invariant: if the north chest is not opened then no keys have been collected. -/ +theorem northChestOpened_or_keys_zero {s : EnvState} (hr : Reachable s) : + s.northChestOpened = true ∨ s.keys = 0 := by + induction hr with + | init => simp [initialState] + | step s a hr ih => + cases a with + | toggleBridge => + by_cases h1 : s.room = Room.west + · simp [step, h1]; exact ih + · simp [step, h1]; exact ih + | goCenter => + by_cases h1 : s.room = Room.west ∨ s.room = Room.north ∨ s.room = Room.east ∨ s.room = Room.south + · simp [step, h1]; exact ih + · simp [step, h1]; exact ih + | goWest => + by_cases h1 : s.room = Room.center + · simp [step, h1]; exact ih + · simp [step, h1]; exact ih + | goNorth => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsNorth s.bridge + · simp [step, h1]; exact ih + · simp [step, h1]; exact ih + | goEast => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsEast s.bridge ∧ hasKey s + · simp [step, h1]; exact ih + · simp [step, h1]; exact ih + | goSouth => + by_cases h1 : s.room = Room.center ∧ bridgeAllowsSouth s.bridge + · simp [step, h1]; exact ih + · simp [step, h1]; exact ih + | openChest => + by_cases h1 : s.room = Room.north ∧ s.northChestOpened = false + · left; simp [step, h1] + · by_cases h2 : s.room = Room.east ∧ s.eastChestOpened = false + · simp [step, h2]; exact ih + · simp [step, h1, h2]; exact ih + | attack => + by_cases h1 : s.room = Room.south ∧ s.guardianAlive = true ∧ s.hasSword = true + · simp [step, h1]; exact ih + · simp [step, h1]; exact ih + | openFinalChest => + by_cases h1 : s.room = Room.center ∧ s.finalChestVisible = true + · simp [step, h1]; exact ih + · simp [step, h1]; exact ih + | wait => simp [step]; exact ih + +/-- Safety invariant: `keys` never decreases below zero (trivially true for + `Nat`, but stated for documentation). -/ +theorem keys_non_negative {s : EnvState} (_ : Reachable s) : s.keys ≥ 0 := by + omega + +/-- Helper invariant: if `finalChestVisible = true` then `guardianAlive = false` + (the final chest is only revealed when the guardian is defeated by attack). -/ +theorem finalChestVisible_implies_guardianDead {s : EnvState} (hr : Reachable s) + (hv : s.finalChestVisible = true) : s.guardianAlive = false := by + induction hr with + | init => simp [initialState] at hv + | step s' a hr ih => + by_cases hs : s'.finalChestVisible = true + · have hguard' : (step s' a).guardianAlive = s'.guardianAlive := by + by_cases ha : a = Action.attack + · subst ha + have hcond : ¬(s'.room = Room.south ∧ s'.guardianAlive = true ∧ s'.hasSword = true) := by + intro h; have := ih hs; rw [this] at h; simp at h + simp [step, hcond] + · exact only_attack_changes_guardianAlive a ha + rw [hguard'] + exact ih hs + · have hf : s'.finalChestVisible = false := by + cases h : s'.finalChestVisible with + | true => exact absurd h hs + | false => rfl + have ha : a = Action.attack := by + by_cases h' : a = Action.attack + · exact h' + · have := only_attack_changes_finalChestVisible (s := s') a h' + rw [this, hf] at hv + simp at hv + subst ha + by_cases hcond : s'.room = Room.south ∧ s'.guardianAlive = true ∧ s'.hasSword = true + · simp [step, hcond] + · have hstep : step s' Action.attack = s' := by simp [step, hcond] + rw [hstep, hf] at hv + simp at hv + +/-- If a reachable state has `finalChestOpened = true`, then the guardian has + been defeated. (Room position is not part of the postcondition because the + player may move after opening the final chest.) -/ +theorem completion_postcondition {s : EnvState} (hr : Reachable s) + (hc : s.finalChestOpened = true) : + s.guardianAlive = false := by + induction hr with + | init => simp [initialState] at hc + | step s' a hr ih => + by_cases hs : s'.finalChestOpened = true + · have hguard' : (step s' a).guardianAlive = s'.guardianAlive := by + by_cases ha : a = Action.attack + · subst ha + have hcond : ¬(s'.room = Room.south ∧ s'.guardianAlive = true ∧ s'.hasSword = true) := by + intro h; have := ih hs; rw [this] at h; simp at h + simp [step, hcond] + · exact only_attack_changes_guardianAlive a ha + rw [hguard'] + exact ih hs + · have hf : s'.finalChestOpened = false := by + cases h : s'.finalChestOpened with + | true => exact absurd h hs + | false => rfl + have ha : a = Action.openFinalChest := only_openFinalChest_sets_finalChestOpened hf a hc + subst ha + by_cases hcond : s'.room = Room.center ∧ s'.finalChestVisible = true + · have hguardDead : s'.guardianAlive = false := + finalChestVisible_implies_guardianDead hr hcond.2 + have hguard' : (step s' Action.openFinalChest).guardianAlive = s'.guardianAlive := + openFinalChest_preserves_guardianAlive + rw [hguard']; exact hguardDead + · have hstep : step s' Action.openFinalChest = s' := by simp [step, hcond] + rw [hstep, hf] at hc + simp at hc + +/-! ### Action-mask/safety layer -/ + +/-- Legal-action predicate for the symbolic transition layer (Bool version + for computable checking via `native_decide`). -/ +def actionEnabled (s : EnvState) : Action → Bool + | Action.toggleBridge => s.room = Room.west + | Action.goCenter => + s.room = Room.west || s.room = Room.north || + s.room = Room.east || s.room = Room.south + | Action.goWest => s.room = Room.center + | Action.goNorth => s.room = Room.center && decide (bridgeAllowsNorth s.bridge) + | Action.goEast => s.room = Room.center && decide (bridgeAllowsEast s.bridge) && s.keys > 0 + | Action.goSouth => s.room = Room.center && decide (bridgeAllowsSouth s.bridge) + | Action.openChest => + (s.room = Room.north && s.northChestOpened = false) || + (s.room = Room.east && s.eastChestOpened = false) + | Action.attack => + s.room = Room.south && s.guardianAlive = true && s.hasSword = true + | Action.openFinalChest => + s.room = Room.center && s.finalChestVisible = true + | Action.wait => true + +def actionsEnabledAlong : EnvState → List Action → Bool + | _, [] => true + | s, a :: rest => actionEnabled s a && actionsEnabledAlong (step s a) rest + +/-! ### Interaction preserves lemmas (for BFS certificate proofs) -/ + +theorem openChest_north_preserves_guardianAlive + {s : EnvState} (hRoom : s.room = Room.north) (hChest : s.northChestOpened = false) : + (step s Action.openChest).guardianAlive = s.guardianAlive := by + simp [step, hRoom, hChest] + +theorem openChest_north_preserves_eastChestOpened + {s : EnvState} (hRoom : s.room = Room.north) (hChest : s.northChestOpened = false) : + (step s Action.openChest).eastChestOpened = s.eastChestOpened := by + simp [step, hRoom, hChest] + +theorem openChest_east_preserves_guardianAlive + {s : EnvState} (hRoom : s.room = Room.east) (hChest : s.eastChestOpened = false) : + (step s Action.openChest).guardianAlive = s.guardianAlive := by + simp [step, hRoom, hChest] + +/-! ### Baseline-aligned symbolic policy (BFS certificate mode) -/ + +/-- +A symbolic BFS action provider. In Python this is implemented by +`bfs_path(...)` plus `follow_bfs_aligned(...)`, which returns one movement +action toward the first tile of the current shortest path, or the supplied +`final_action` when already at a goal room. +-/ +abbrev BfsAction := EnvState → Room → Action + +/-- +Policy shape matching `decide_task4`. + +1. If north chest not opened: if at north, openChest; else BFS to north. +2. Else if east chest not opened: if at east, openChest; else BFS to east. +3. Else if guardian alive: if at south, attack; else BFS to south. +4. Else: if at center, openFinalChest; else BFS to center. + +This definition is parameterized by `bfsAction`: Lean verifies the stage logic +and the BFS contract separately, instead of hardcoding a route. +-/ +def symbolicPolicy (bfsAction : BfsAction) (s : EnvState) : Action := + if s.finalChestOpened then Action.wait + else if s.northChestOpened = false then + if s.room = Room.north then Action.openChest + else bfsAction s Room.north + else if s.eastChestOpened = false then + if s.room = Room.east then Action.openChest + else bfsAction s Room.east + else if s.guardianAlive = true then + if s.room = Room.south then Action.attack + else bfsAction s Room.south + else + if s.room = Room.center then Action.openFinalChest + else bfsAction s Room.center + +/-! ### BFS route certificates -/ + +/-- +Soundness certificate for a room-approach BFS phase. The plan must end at the +target room while preserving all interaction-relevant state (keys, sword, +chests, guardian, visibility, completion). +-/ +structure ApproachCertificate (target : Room) (start : EnvState) where + plan : List Action + enabled : actionsEnabledAlong start plan = true + endsAtRoom : (run start plan).room = target + keysPreserved : (run start plan).keys = start.keys + hasShieldPreserved : (run start plan).hasShield = start.hasShield + hasSwordPreserved : (run start plan).hasSword = start.hasSword + northChestOpenedPreserved : (run start plan).northChestOpened = start.northChestOpened + eastChestOpenedPreserved : (run start plan).eastChestOpened = start.eastChestOpened + guardianAlivePreserved : (run start plan).guardianAlive = start.guardianAlive + finalChestVisiblePreserved : (run start plan).finalChestVisible = start.finalChestVisible + finalChestOpenedPreserved : (run start plan).finalChestOpened = start.finalChestOpened + +abbrev NorthApproachCertificate (start : EnvState) := ApproachCertificate Room.north start +abbrev EastApproachCertificate (start : EnvState) := ApproachCertificate Room.east start +abbrev SouthApproachCertificate (start : EnvState) := ApproachCertificate Room.south start +abbrev CenterApproachCertificate (start : EnvState) := ApproachCertificate Room.center start + +/-! ### Main non-fixed-route Task 4 theorem -/ + +/-- +Task 4 completion theorem aligned with the BFS policy. + +The policy structure is: +1. BFS to north → openChest (gains one key) +2. BFS to east → openChest (gains sword) +3. BFS to south → attack (defeats guardian, reveals final chest) +4. BFS to center → openFinalChest (completes task) + +This theorem chains the BFS certificates and the interaction actions. +-/ +theorem task4_bfs_certificate_completes + (northRoute : NorthApproachCertificate initialState) + (eastRoute : EastApproachCertificate + (step (run initialState northRoute.plan) Action.openChest)) + (southRoute : SouthApproachCertificate + (step (run (step (run initialState northRoute.plan) Action.openChest) eastRoute.plan) Action.openChest)) + (centerRoute : CenterApproachCertificate + (step (run (step (run (step (run initialState northRoute.plan) Action.openChest) eastRoute.plan) Action.openChest) southRoute.plan) Action.attack)) : + GoalReached + (step + (run + (step + (run + (step + (run + (step (run initialState northRoute.plan) Action.openChest) + eastRoute.plan) + Action.openChest) + southRoute.plan) + Action.attack) + centerRoute.plan) + Action.openFinalChest) := by + let sN := run initialState northRoute.plan + let sNK := step sN Action.openChest + let sE := run sNK eastRoute.plan + let sEK := step sE Action.openChest + let sS := run sEK southRoute.plan + let sSK := step sS Action.attack + let sC := run sSK centerRoute.plan + -- North-approach BFS: ends at north, northChestOpened still false, guardian alive. + have hRoomN : sN.room = Room.north := northRoute.endsAtRoom + have hNChestClosed : sN.northChestOpened = false := by + have h := northRoute.northChestOpenedPreserved; rw [h]; simp [initialState] + have hNGuardian : sN.guardianAlive = true := by + have h := northRoute.guardianAlivePreserved; rw [h]; simp [initialState] + have hNEastChest : sN.eastChestOpened = false := by + have h := northRoute.eastChestOpenedPreserved; rw [h]; simp [initialState] + -- Open chest (north): guardian/eastChest preserved. + have hNKGuardian : sNK.guardianAlive = sN.guardianAlive := + openChest_north_preserves_guardianAlive hRoomN hNChestClosed + have hNKEastChest : sNK.eastChestOpened = sN.eastChestOpened := + openChest_north_preserves_eastChestOpened hRoomN hNChestClosed + -- East-approach BFS: ends at east, eastChestOpened still false, guardian preserved. + have hRoomE : sE.room = Room.east := eastRoute.endsAtRoom + have hEEastChest : sE.eastChestOpened = false := by + have h1 : sE.eastChestOpened = sNK.eastChestOpened := eastRoute.eastChestOpenedPreserved + rw [h1, hNKEastChest, hNEastChest] + have hEGuardian : sE.guardianAlive = true := by + have h1 : sE.guardianAlive = sNK.guardianAlive := eastRoute.guardianAlivePreserved + rw [h1, hNKGuardian, hNGuardian] + -- Open chest (east): guardian preserved. + have hEKGuardian : sEK.guardianAlive = sE.guardianAlive := + openChest_east_preserves_guardianAlive hRoomE hEEastChest + -- South-approach BFS: ends at south, guardian preserved, hasSword preserved. + have hRoomS : sS.room = Room.south := southRoute.endsAtRoom + have hSGuardian : sS.guardianAlive = true := by + have h1 : sS.guardianAlive = sEK.guardianAlive := southRoute.guardianAlivePreserved + rw [h1, hEKGuardian, hEGuardian] + have hSSword : sS.hasSword = true := by + have h1 : sS.hasSword = sEK.hasSword := southRoute.hasSwordPreserved + have h2 : sEK.hasSword = true := openChest_east_grants_sword hRoomE hEEastChest + rw [h1]; exact h2 + -- Attack: guardian defeated, final chest visible. + have hAttack := attack_with_sword_reveals_final_chest hRoomS hSGuardian hSSword + have hSKVisible : sSK.finalChestVisible = true := hAttack.2 + -- Center-approach BFS: ends at center, finalChestVisible preserved. + have hRoomC : sC.room = Room.center := centerRoute.endsAtRoom + have hCVisible : sC.finalChestVisible = true := by + have h1 : sC.finalChestVisible = sSK.finalChestVisible := centerRoute.finalChestVisiblePreserved + rw [h1]; exact hSKVisible + -- openFinalChest on center with visibility → finalChestOpened. + have hDone : (step sC Action.openFinalChest).finalChestOpened = true := + openFinalChest_succeeds_when_visible hRoomC hCVisible + simpa [GoalReached, sN, sNK, sE, sEK, sS, sSK, sC] using hDone + +/-! ### Public map regression instance -/ + +-- Room-level BFS plans for the public Task-4 layout. +def publicNorthPlan : List Action := + [Action.goCenter, Action.goNorth] +def publicEastPlan : List Action := + [Action.goCenter, Action.goWest, Action.toggleBridge, Action.goCenter, Action.goEast] +def publicSouthPlan : List Action := + [Action.goCenter, Action.goWest, Action.toggleBridge, Action.goCenter, Action.goSouth] +def publicCenterPlan : List Action := + [Action.goCenter] + +def publicNorthRoute : NorthApproachCertificate initialState := + { plan := publicNorthPlan + enabled := by native_decide + endsAtRoom := by native_decide + keysPreserved := by native_decide + hasShieldPreserved := by native_decide + hasSwordPreserved := by native_decide + northChestOpenedPreserved := by native_decide + eastChestOpenedPreserved := by native_decide + guardianAlivePreserved := by native_decide + finalChestVisiblePreserved := by native_decide + finalChestOpenedPreserved := by native_decide } + +def publicEastRoute : EastApproachCertificate + (step (run initialState publicNorthPlan) Action.openChest) := + { plan := publicEastPlan + enabled := by native_decide + endsAtRoom := by native_decide + keysPreserved := by native_decide + hasShieldPreserved := by native_decide + hasSwordPreserved := by native_decide + northChestOpenedPreserved := by native_decide + eastChestOpenedPreserved := by native_decide + guardianAlivePreserved := by native_decide + finalChestVisiblePreserved := by native_decide + finalChestOpenedPreserved := by native_decide } + +def publicSouthRoute : SouthApproachCertificate + (step (run (step (run initialState publicNorthPlan) Action.openChest) publicEastPlan) Action.openChest) := + { plan := publicSouthPlan + enabled := by native_decide + endsAtRoom := by native_decide + keysPreserved := by native_decide + hasShieldPreserved := by native_decide + hasSwordPreserved := by native_decide + northChestOpenedPreserved := by native_decide + eastChestOpenedPreserved := by native_decide + guardianAlivePreserved := by native_decide + finalChestVisiblePreserved := by native_decide + finalChestOpenedPreserved := by native_decide } + +def publicCenterRoute : CenterApproachCertificate + (step (run (step (run (step (run initialState publicNorthPlan) Action.openChest) publicEastPlan) Action.openChest) publicSouthPlan) Action.attack) := + { plan := publicCenterPlan + enabled := by native_decide + endsAtRoom := by native_decide + keysPreserved := by native_decide + hasShieldPreserved := by native_decide + hasSwordPreserved := by native_decide + northChestOpenedPreserved := by native_decide + eastChestOpenedPreserved := by native_decide + guardianAlivePreserved := by native_decide + finalChestVisiblePreserved := by native_decide + finalChestOpenedPreserved := by native_decide } + +theorem public_task4_bfs_certificate_completes : + GoalReached + (step + (run + (step + (run + (step + (run + (step (run initialState publicNorthPlan) Action.openChest) + publicEastPlan) + Action.openChest) + publicSouthPlan) + Action.attack) + publicCenterPlan) + Action.openFinalChest) := by + exact task4_bfs_certificate_completes publicNorthRoute publicEastRoute + publicSouthRoute publicCenterRoute + +end Task4Formalization diff --git a/student_agent/lean/Task5Formalization.lean b/student_agent/lean/Task5Formalization.lean new file mode 100644 index 0000000..3920b75 --- /dev/null +++ b/student_agent/lean/Task5Formalization.lean @@ -0,0 +1,1627 @@ +/- + Environment formalization for `mathematical_logic/task_5`. + + Task 5 is a four-room exploration task with mixed mechanics: + + * press the start-room button to unlock the south conditional exit; + * go south and open the key chest; + * return to the start room and use the east locked exit (consumes key); + * explore west and east rooms; + * open every visible chest in the dungeon. + + In the Python engine, world completion is triggered when all visible chests are + open and the map does not use an explicit `complete_task` exit. This file + formalizes that room-level logic (see `progress.py:all_chests_opened` and + `engine.py:106`). + + It intentionally abstracts away monster AI, trap collision geometry and the + reward-side periodic HP drain. The focus is the symbolic environment layer that + the planner must satisfy: button gating, key gating, chest opening and global + completion. + + Python correspondence: + * `nesylink/map_data/mathematical_logic/task_5/*.json` + * `nesylink/core/mechanics/interactions.py` + * `nesylink/core/mechanics/movement.py` + * `nesylink/core/mechanics/progress.py` + * `nesylink/core/mechanics/engine.py` +-/ + +namespace Task5Formalization + +/-! ### Rooms, actions, state -/ + +inductive Room where + | startRoom + | southRoom + | eastRoom + | westRoom + deriving DecidableEq, Repr + +inductive Action where + | pressButton + | goSouth + | goNorth + | goEast + | goWest + | openChest + | wait + deriving DecidableEq, Repr + +structure EnvState where + room : Room + buttonPressed : Bool + keys : Nat + hp : Nat + startChestOpened : Bool + southChestOpened : Bool + eastChestOpened : Bool + westChestOpened : Bool + deriving DecidableEq, Repr + +-- Spawn in room_0_0 (startRoom) per dungeon.json + room_0_0.json:default_spawn. +def initialState : EnvState := + { + room := Room.startRoom + buttonPressed := false + keys := 0 + hp := 5 + startChestOpened := false + southChestOpened := false + eastChestOpened := false + westChestOpened := false + } + +/-- World completion predicate: every visible chest has been opened. + Aligns with `progress.py:all_chests_opened` (no hidden chests in task_5). -/ +def allChestsOpened (s : EnvState) : Prop := + s.startChestOpened = true ∧ + s.southChestOpened = true ∧ + s.eastChestOpened = true ∧ + s.westChestOpened = true + +def GoalReached (s : EnvState) : Prop := + allChestsOpened s + +/-! ### Transition function -/ + +def step (s : EnvState) : Action → EnvState + | Action.pressButton => + if s.room = Room.startRoom then + { s with buttonPressed := true } + else + s + | Action.goSouth => + -- room_0_0 south_exit: type=conditional, requires button_pressed=button_1. + if s.room = Room.startRoom ∧ s.buttonPressed = true then + { s with room := Room.southRoom } + else + s + | Action.goNorth => + -- room_0_1 north_exit: type=normal. + if s.room = Room.southRoom then + { s with room := Room.startRoom } + else + s + | Action.goEast => + -- startRoom east_exit: locked_key, consume_key=true. + -- eastRoom west_exit: normal (returns to startRoom). + if s.room = Room.startRoom ∧ s.keys > 0 then + { s with room := Room.eastRoom, keys := s.keys - 1 } + else if s.room = Room.westRoom then + { s with room := Room.startRoom } + else + s + | Action.goWest => + -- startRoom west_exit: normal. + -- eastRoom west_exit: normal (returns to startRoom). + if s.room = Room.startRoom then + { s with room := Room.westRoom } + else if s.room = Room.eastRoom then + { s with room := Room.startRoom } + else + s + | Action.openChest => + -- Each room has exactly one chest with distinct loot: + -- startRoom: gold (no side effect) + -- southRoom: key (keys + 1) + -- eastRoom: heal (hp + 1) + -- westRoom: gold (no side effect) + if s.room = Room.startRoom ∧ s.startChestOpened = false then + { s with startChestOpened := true } + else if s.room = Room.southRoom ∧ s.southChestOpened = false then + { s with southChestOpened := true, keys := s.keys + 1 } + else if s.room = Room.eastRoom ∧ s.eastChestOpened = false then + { s with eastChestOpened := true, hp := s.hp + 1 } + else if s.room = Room.westRoom ∧ s.westChestOpened = false then + { s with westChestOpened := true } + else + s + | Action.wait => s + +def run : EnvState → List Action → EnvState + | s, [] => s + | s, a :: rest => run (step s a) rest + +/-! ### Navigation lemmas -/ + +theorem pressButton_only_in_startRoom + {s : EnvState} (hRoom : s.room ≠ Room.startRoom) : + step s Action.pressButton = s := by + simp [step, hRoom] + +theorem pressButton_marks_pressed + {s : EnvState} (hRoom : s.room = Room.startRoom) : + (step s Action.pressButton).buttonPressed = true := by + simp [step, hRoom] + +theorem pressButton_preserves_room + {s : EnvState} : (step s Action.pressButton).room = s.room := by + by_cases h : s.room = Room.startRoom + · simp [step, h] + · simp [step, h] + +theorem pressButton_preserves_keys + {s : EnvState} : (step s Action.pressButton).keys = s.keys := by + by_cases h : s.room = Room.startRoom + · simp [step, h] + · simp [step, h] + +theorem pressButton_preserves_chests + {s : EnvState} : + (step s Action.pressButton).startChestOpened = s.startChestOpened ∧ + (step s Action.pressButton).southChestOpened = s.southChestOpened ∧ + (step s Action.pressButton).eastChestOpened = s.eastChestOpened ∧ + (step s Action.pressButton).westChestOpened = s.westChestOpened := by + by_cases h : s.room = Room.startRoom + · simp [step, h] + · simp [step, h] + +theorem goSouth_blocked_without_button + {s : EnvState} + (hRoom : s.room = Room.startRoom) + (hButton : s.buttonPressed = false) : + step s Action.goSouth = s := by + simp [step, hRoom, hButton] + +theorem goSouth_succeeds_after_button + {s : EnvState} + (hRoom : s.room = Room.startRoom) + (hButton : s.buttonPressed = true) : + (step s Action.goSouth).room = Room.southRoom := by + simp [step, hRoom, hButton] + +theorem goSouth_preserves_buttonPressed + {s : EnvState} : (step s Action.goSouth).buttonPressed = s.buttonPressed := by + by_cases h : s.room = Room.startRoom ∧ s.buttonPressed = true + · simp [step, h] + · simp [step, h] + +theorem goSouth_preserves_keys + {s : EnvState} : (step s Action.goSouth).keys = s.keys := by + by_cases h : s.room = Room.startRoom ∧ s.buttonPressed = true + · simp [step, h] + · simp [step, h] + +theorem goSouth_preserves_chests + {s : EnvState} : + (step s Action.goSouth).startChestOpened = s.startChestOpened ∧ + (step s Action.goSouth).southChestOpened = s.southChestOpened ∧ + (step s Action.goSouth).eastChestOpened = s.eastChestOpened ∧ + (step s Action.goSouth).westChestOpened = s.westChestOpened := by + by_cases h : s.room = Room.startRoom ∧ s.buttonPressed = true + · simp [step, h] + · simp [step, h] + +theorem goNorth_only_in_southRoom + {s : EnvState} (hRoom : s.room ≠ Room.southRoom) : + step s Action.goNorth = s := by + simp [step, hRoom] + +theorem goNorth_succeeds_in_southRoom + {s : EnvState} (hRoom : s.room = Room.southRoom) : + (step s Action.goNorth).room = Room.startRoom := by + simp [step, hRoom] + +theorem goNorth_preserves_buttonPressed + {s : EnvState} : (step s Action.goNorth).buttonPressed = s.buttonPressed := by + by_cases h : s.room = Room.southRoom + · simp [step, h] + · simp [step, h] + +theorem goNorth_preserves_keys + {s : EnvState} : (step s Action.goNorth).keys = s.keys := by + by_cases h : s.room = Room.southRoom + · simp [step, h] + · simp [step, h] + +theorem goNorth_preserves_chests + {s : EnvState} : + (step s Action.goNorth).startChestOpened = s.startChestOpened ∧ + (step s Action.goNorth).southChestOpened = s.southChestOpened ∧ + (step s Action.goNorth).eastChestOpened = s.eastChestOpened ∧ + (step s Action.goNorth).westChestOpened = s.westChestOpened := by + by_cases h : s.room = Room.southRoom + · simp [step, h] + · simp [step, h] + +theorem goEast_blocked_without_key + {s : EnvState} + (hRoom : s.room = Room.startRoom) + (hKeys : s.keys = 0) : + step s Action.goEast = s := by + simp [step, hRoom, hKeys] + +theorem goEast_from_start_consumes_key + {s : EnvState} + (hRoom : s.room = Room.startRoom) + (hKeys : s.keys > 0) : + (step s Action.goEast).room = Room.eastRoom ∧ + (step s Action.goEast).keys = s.keys - 1 := by + simp [step, hRoom, hKeys] + +theorem goEast_from_west_returns_to_start + {s : EnvState} + (hRoom : s.room = Room.westRoom) : + (step s Action.goEast).room = Room.startRoom ∧ + (step s Action.goEast).keys = s.keys := by + have h : ¬(s.room = Room.startRoom ∧ s.keys > 0) := by + intro h; rw [hRoom] at h; simp at h + simp [step, hRoom] + +theorem goEast_preserves_buttonPressed + {s : EnvState} : (step s Action.goEast).buttonPressed = s.buttonPressed := by + by_cases h1 : s.room = Room.startRoom ∧ s.keys > 0 + · simp [step, h1] + · by_cases h2 : s.room = Room.westRoom + · simp [step, h2] + · simp [step, h1, h2] + +theorem goEast_preserves_chests + {s : EnvState} : + (step s Action.goEast).startChestOpened = s.startChestOpened ∧ + (step s Action.goEast).southChestOpened = s.southChestOpened ∧ + (step s Action.goEast).eastChestOpened = s.eastChestOpened ∧ + (step s Action.goEast).westChestOpened = s.westChestOpened := by + by_cases h1 : s.room = Room.startRoom ∧ s.keys > 0 + · simp [step, h1] + · by_cases h2 : s.room = Room.westRoom + · simp [step, h2] + · simp [step, h1, h2] + +theorem goWest_from_start_to_west + {s : EnvState} + (hRoom : s.room = Room.startRoom) : + (step s Action.goWest).room = Room.westRoom ∧ + (step s Action.goWest).keys = s.keys := by + simp [step, hRoom] + +theorem goWest_from_east_returns_to_start + {s : EnvState} + (hRoom : s.room = Room.eastRoom) : + (step s Action.goWest).room = Room.startRoom ∧ + (step s Action.goWest).keys = s.keys := by + have h : ¬s.room = Room.startRoom := by + intro hh; rw [hh] at hRoom; simp at hRoom + simp [step, hRoom] + +theorem goWest_preserves_buttonPressed + {s : EnvState} : (step s Action.goWest).buttonPressed = s.buttonPressed := by + by_cases h1 : s.room = Room.startRoom + · simp [step, h1] + · by_cases h2 : s.room = Room.eastRoom + · simp [step, h2] + · simp [step, h1, h2] + +theorem goWest_preserves_keys + {s : EnvState} : (step s Action.goWest).keys = s.keys := by + by_cases h1 : s.room = Room.startRoom + · simp [step, h1] + · by_cases h2 : s.room = Room.eastRoom + · simp [step, h2] + · simp [step, h1, h2] + +theorem goWest_preserves_chests + {s : EnvState} : + (step s Action.goWest).startChestOpened = s.startChestOpened ∧ + (step s Action.goWest).southChestOpened = s.southChestOpened ∧ + (step s Action.goWest).eastChestOpened = s.eastChestOpened ∧ + (step s Action.goWest).westChestOpened = s.westChestOpened := by + by_cases h1 : s.room = Room.startRoom + · simp [step, h1] + · by_cases h2 : s.room = Room.eastRoom + · simp [step, h2] + · simp [step, h1, h2] + +/-! ### Chest lemmas -/ + +theorem openChest_start_marks_opened + {s : EnvState} + (hRoom : s.room = Room.startRoom) + (hChest : s.startChestOpened = false) : + (step s Action.openChest).startChestOpened = true ∧ + (step s Action.openChest).room = Room.startRoom ∧ + (step s Action.openChest).keys = s.keys ∧ + (step s Action.openChest).hp = s.hp := by + have h2 : ¬(s.room = Room.southRoom ∧ s.southChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + have h3 : ¬(s.room = Room.eastRoom ∧ s.eastChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + have h4 : ¬(s.room = Room.westRoom ∧ s.westChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + simp [step, hRoom, hChest] + +theorem openChest_start_already_opened + {s : EnvState} + (hRoom : s.room = Room.startRoom) + (hChest : s.startChestOpened = true) : + step s Action.openChest = s := by + have h1 : ¬(s.room = Room.startRoom ∧ s.startChestOpened = false) := by + intro h; rw [hChest] at h; simp at h + have h2 : ¬(s.room = Room.southRoom ∧ s.southChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + have h3 : ¬(s.room = Room.eastRoom ∧ s.eastChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + have h4 : ¬(s.room = Room.westRoom ∧ s.westChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + simp [step, h1, h2, h3, h4] + +theorem openChest_south_grants_key + {s : EnvState} + (hRoom : s.room = Room.southRoom) + (hChest : s.southChestOpened = false) : + (step s Action.openChest).southChestOpened = true ∧ + (step s Action.openChest).keys = s.keys + 1 ∧ + (step s Action.openChest).room = Room.southRoom := by + have h1 : ¬(s.room = Room.startRoom ∧ s.startChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + have h3 : ¬(s.room = Room.eastRoom ∧ s.eastChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + have h4 : ¬(s.room = Room.westRoom ∧ s.westChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + simp [step, hRoom, hChest] + +theorem openChest_south_already_opened + {s : EnvState} + (hRoom : s.room = Room.southRoom) + (hChest : s.southChestOpened = true) : + step s Action.openChest = s := by + have h1 : ¬(s.room = Room.startRoom ∧ s.startChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + have h2 : ¬(s.room = Room.southRoom ∧ s.southChestOpened = false) := by + intro h; rw [hChest] at h; simp at h + have h3 : ¬(s.room = Room.eastRoom ∧ s.eastChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + have h4 : ¬(s.room = Room.westRoom ∧ s.westChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + simp [step, h1, h2, h3, h4] + +theorem openChest_east_heals + {s : EnvState} + (hRoom : s.room = Room.eastRoom) + (hChest : s.eastChestOpened = false) : + (step s Action.openChest).eastChestOpened = true ∧ + (step s Action.openChest).hp = s.hp + 1 ∧ + (step s Action.openChest).room = Room.eastRoom ∧ + (step s Action.openChest).keys = s.keys := by + have h1 : ¬(s.room = Room.startRoom ∧ s.startChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + have h2 : ¬(s.room = Room.southRoom ∧ s.southChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + have h4 : ¬(s.room = Room.westRoom ∧ s.westChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + simp [step, hRoom, hChest] + +theorem openChest_east_already_opened + {s : EnvState} + (hRoom : s.room = Room.eastRoom) + (hChest : s.eastChestOpened = true) : + step s Action.openChest = s := by + have h1 : ¬(s.room = Room.startRoom ∧ s.startChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + have h2 : ¬(s.room = Room.southRoom ∧ s.southChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + have h3 : ¬(s.room = Room.eastRoom ∧ s.eastChestOpened = false) := by + intro h; rw [hChest] at h; simp at h + have h4 : ¬(s.room = Room.westRoom ∧ s.westChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + simp [step, h1, h2, h3, h4] + +theorem openChest_west_marks_opened + {s : EnvState} + (hRoom : s.room = Room.westRoom) + (hChest : s.westChestOpened = false) : + (step s Action.openChest).westChestOpened = true ∧ + (step s Action.openChest).room = Room.westRoom ∧ + (step s Action.openChest).keys = s.keys ∧ + (step s Action.openChest).hp = s.hp := by + have h1 : ¬(s.room = Room.startRoom ∧ s.startChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + have h2 : ¬(s.room = Room.southRoom ∧ s.southChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + have h3 : ¬(s.room = Room.eastRoom ∧ s.eastChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + simp [step, hRoom, hChest] + +theorem openChest_west_already_opened + {s : EnvState} + (hRoom : s.room = Room.westRoom) + (hChest : s.westChestOpened = true) : + step s Action.openChest = s := by + have h1 : ¬(s.room = Room.startRoom ∧ s.startChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + have h2 : ¬(s.room = Room.southRoom ∧ s.southChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + have h3 : ¬(s.room = Room.eastRoom ∧ s.eastChestOpened = false) := by + intro h; rw [hRoom] at h; simp at h + have h4 : ¬(s.room = Room.westRoom ∧ s.westChestOpened = false) := by + intro h; rw [hChest] at h; simp at h + simp [step, h1, h2, h3, h4] + +theorem openChest_preserves_buttonPressed + {s : EnvState} : + (step s Action.openChest).buttonPressed = s.buttonPressed := by + by_cases h1 : s.room = Room.startRoom ∧ s.startChestOpened = false + · simp [step, h1] + · by_cases h2 : s.room = Room.southRoom ∧ s.southChestOpened = false + · simp [step, h2] + · by_cases h3 : s.room = Room.eastRoom ∧ s.eastChestOpened = false + · simp [step, h3] + · by_cases h4 : s.room = Room.westRoom ∧ s.westChestOpened = false + · simp [step, h4] + · simp [step, h1, h2, h3, h4] + +theorem openChest_preserves_room + {s : EnvState} : (step s Action.openChest).room = s.room := by + by_cases h1 : s.room = Room.startRoom ∧ s.startChestOpened = false + · simp [step, h1] + · by_cases h2 : s.room = Room.southRoom ∧ s.southChestOpened = false + · simp [step, h2] + · by_cases h3 : s.room = Room.eastRoom ∧ s.eastChestOpened = false + · simp [step, h3] + · by_cases h4 : s.room = Room.westRoom ∧ s.westChestOpened = false + · simp [step, h4] + · simp [step, h1, h2, h3, h4] + +/-! ### Necessity lemmas (action-wise invariants) -/ + +/-- Only `pressButton` can change `buttonPressed`. -/ +theorem only_pressButton_changes_buttonPressed + {s : EnvState} (a : Action) (ha : a ≠ Action.pressButton) : + (step s a).buttonPressed = s.buttonPressed := by + cases a with + | pressButton => exact absurd rfl ha + | goSouth => exact goSouth_preserves_buttonPressed + | goNorth => exact goNorth_preserves_buttonPressed + | goEast => exact goEast_preserves_buttonPressed + | goWest => exact goWest_preserves_buttonPressed + | openChest => exact openChest_preserves_buttonPressed + | wait => simp [step] + +/-- Only `openChest` can change `startChestOpened`. -/ +theorem only_openChest_changes_startChestOpened + {s : EnvState} (a : Action) (ha : a ≠ Action.openChest) : + (step s a).startChestOpened = s.startChestOpened := by + cases a with + | pressButton => + have := pressButton_preserves_chests (s := s) + rcases this with ⟨h, _, _, _⟩; exact h + | goSouth => + have := goSouth_preserves_chests (s := s) + rcases this with ⟨h, _, _, _⟩; exact h + | goNorth => + have := goNorth_preserves_chests (s := s) + rcases this with ⟨h, _, _, _⟩; exact h + | goEast => + have := goEast_preserves_chests (s := s) + rcases this with ⟨h, _, _, _⟩; exact h + | goWest => + have := goWest_preserves_chests (s := s) + rcases this with ⟨h, _, _, _⟩; exact h + | openChest => exact absurd rfl ha + | wait => simp [step] + +/-- Only `openChest` can change `southChestOpened`. -/ +theorem only_openChest_changes_southChestOpened + {s : EnvState} (a : Action) (ha : a ≠ Action.openChest) : + (step s a).southChestOpened = s.southChestOpened := by + cases a with + | pressButton => + have := pressButton_preserves_chests (s := s) + rcases this with ⟨_, h, _, _⟩; exact h + | goSouth => + have := goSouth_preserves_chests (s := s) + rcases this with ⟨_, h, _, _⟩; exact h + | goNorth => + have := goNorth_preserves_chests (s := s) + rcases this with ⟨_, h, _, _⟩; exact h + | goEast => + have := goEast_preserves_chests (s := s) + rcases this with ⟨_, h, _, _⟩; exact h + | goWest => + have := goWest_preserves_chests (s := s) + rcases this with ⟨_, h, _, _⟩; exact h + | openChest => exact absurd rfl ha + | wait => simp [step] + +/-- Only `openChest` can change `eastChestOpened`. -/ +theorem only_openChest_changes_eastChestOpened + {s : EnvState} (a : Action) (ha : a ≠ Action.openChest) : + (step s a).eastChestOpened = s.eastChestOpened := by + cases a with + | pressButton => + have := pressButton_preserves_chests (s := s) + rcases this with ⟨_, _, h, _⟩; exact h + | goSouth => + have := goSouth_preserves_chests (s := s) + rcases this with ⟨_, _, h, _⟩; exact h + | goNorth => + have := goNorth_preserves_chests (s := s) + rcases this with ⟨_, _, h, _⟩; exact h + | goEast => + have := goEast_preserves_chests (s := s) + rcases this with ⟨_, _, h, _⟩; exact h + | goWest => + have := goWest_preserves_chests (s := s) + rcases this with ⟨_, _, h, _⟩; exact h + | openChest => exact absurd rfl ha + | wait => simp [step] + +/-- Only `openChest` can change `westChestOpened`. -/ +theorem only_openChest_changes_westChestOpened + {s : EnvState} (a : Action) (ha : a ≠ Action.openChest) : + (step s a).westChestOpened = s.westChestOpened := by + cases a with + | pressButton => + have := pressButton_preserves_chests (s := s) + rcases this with ⟨_, _, _, h⟩; exact h + | goSouth => + have := goSouth_preserves_chests (s := s) + rcases this with ⟨_, _, _, h⟩; exact h + | goNorth => + have := goNorth_preserves_chests (s := s) + rcases this with ⟨_, _, _, h⟩; exact h + | goEast => + have := goEast_preserves_chests (s := s) + rcases this with ⟨_, _, _, h⟩; exact h + | goWest => + have := goWest_preserves_chests (s := s) + rcases this with ⟨_, _, _, h⟩; exact h + | openChest => exact absurd rfl ha + | wait => simp [step] + +/-- Only `goEast` (from startRoom) and `openChest` (in southRoom) can change `keys`. -/ +theorem non_goEast_non_openChest_preserves_keys + {s : EnvState} (a : Action) + (ha : a ≠ Action.goEast) (ha' : a ≠ Action.openChest) : + (step s a).keys = s.keys := by + cases a with + | pressButton => exact pressButton_preserves_keys + | goSouth => exact goSouth_preserves_keys + | goNorth => exact goNorth_preserves_keys + | goEast => exact absurd rfl ha + | goWest => exact goWest_preserves_keys + | openChest => exact absurd rfl ha' + | wait => simp [step] + +/-- Only `openChest` (in eastRoom) can change `hp`. -/ +theorem non_openChest_preserves_hp + {s : EnvState} (a : Action) (ha : a ≠ Action.openChest) : + (step s a).hp = s.hp := by + cases a with + | pressButton => + by_cases h : s.room = Room.startRoom + · simp [step, h] + · simp [step, h] + | goSouth => + by_cases h : s.room = Room.startRoom ∧ s.buttonPressed = true + · simp [step, h] + · simp [step, h] + | goNorth => + by_cases h : s.room = Room.southRoom + · simp [step, h] + · simp [step, h] + | goEast => + by_cases h1 : s.room = Room.startRoom ∧ s.keys > 0 + · simp [step, h1] + · by_cases h2 : s.room = Room.westRoom + · simp [step, h2] + · simp [step, h1, h2] + | goWest => + by_cases h1 : s.room = Room.startRoom + · simp [step, h1] + · by_cases h2 : s.room = Room.eastRoom + · simp [step, h2] + · simp [step, h1, h2] + | openChest => exact absurd rfl ha + | wait => simp [step] + +/-! ### Reachable states and invariants -/ + +/-- Reachability: states reachable from `initialState` via arbitrary actions. -/ +inductive Reachable : EnvState → Prop where + | init : Reachable initialState + | step (s : EnvState) (a : Action) : Reachable s → Reachable (step s a) + +/-- Invariant: if `buttonPressed = false`, then the player is not in southRoom + and the south chest has not been opened. Equivalently (contrapositive): + being in southRoom or having opened the south chest implies the button was + pressed. This captures the gating effect of the conditional south exit. -/ +theorem buttonPressed_false_implies_not_in_south_and_chest_closed + {s : EnvState} (hr : Reachable s) : + s.buttonPressed = false → + s.room ≠ Room.southRoom ∧ s.southChestOpened = false := by + induction hr with + | init => simp [initialState] + | step s a hr ih => + intro hb + cases a with + | pressButton => + by_cases h : s.room = Room.startRoom + · simp [step, h] at hb + · have hstep : step s Action.pressButton = s := by simp [step, h] + rw [hstep] at hb ⊢; exact ih hb + | goSouth => + by_cases h : s.room = Room.startRoom ∧ s.buttonPressed = true + · simp [step, h] at hb + · have hstep : step s Action.goSouth = s := by simp [step, h] + rw [hstep] at hb ⊢; exact ih hb + | goNorth => + by_cases h : s.room = Room.southRoom + · have hp := goNorth_preserves_buttonPressed (s := s) + rw [hp] at hb + have ⟨hnot, _⟩ := ih hb + exact absurd h hnot + · have hstep : step s Action.goNorth = s := by simp [step, h] + rw [hstep] at hb ⊢; exact ih hb + | goEast => + have hp := goEast_preserves_buttonPressed (s := s) + rw [hp] at hb + have ⟨hnot, hchest⟩ := ih hb + by_cases h1 : s.room = Room.startRoom ∧ s.keys > 0 + · constructor + · simp [step, h1]; + · have hp2 := goEast_preserves_chests (s := s) + rcases hp2 with ⟨_, hsc, _, _⟩ + rw [hsc]; exact hchest + · by_cases h2 : s.room = Room.westRoom + · constructor + · simp [step, h2]; + · have hp2 := goEast_preserves_chests (s := s) + rcases hp2 with ⟨_, hsc, _, _⟩ + rw [hsc]; exact hchest + · have hstep : step s Action.goEast = s := by simp [step, h1, h2] + rw [hstep]; exact ⟨hnot, hchest⟩ + | goWest => + have hp := goWest_preserves_buttonPressed (s := s) + rw [hp] at hb + have ⟨hnot, hchest⟩ := ih hb + by_cases h1 : s.room = Room.startRoom + · constructor + · simp [step, h1]; + · have hp2 := goWest_preserves_chests (s := s) + rcases hp2 with ⟨_, hsc, _, _⟩ + rw [hsc]; exact hchest + · by_cases h2 : s.room = Room.eastRoom + · constructor + · simp [step, h2]; + · have hp2 := goWest_preserves_chests (s := s) + rcases hp2 with ⟨_, hsc, _, _⟩ + rw [hsc]; exact hchest + · have hstep : step s Action.goWest = s := by simp [step, h1, h2] + rw [hstep]; exact ⟨hnot, hchest⟩ + | openChest => + have hp := openChest_preserves_buttonPressed (s := s) + rw [hp] at hb + have ⟨hnot, hchest⟩ := ih hb + constructor + · have hp2 := openChest_preserves_room (s := s) + rw [hp2]; exact hnot + · by_cases hs : s.room = Room.southRoom ∧ s.southChestOpened = false + · exact absurd hs.1 hnot + · have hp2 : (step s Action.openChest).southChestOpened = s.southChestOpened := by + by_cases h1 : s.room = Room.startRoom ∧ s.startChestOpened = false + · simp [step, h1] + · by_cases h2 : s.room = Room.southRoom ∧ s.southChestOpened = false + · exact absurd h2 hs + · by_cases h3 : s.room = Room.eastRoom ∧ s.eastChestOpened = false + · simp [step, h3] + · by_cases h4 : s.room = Room.westRoom ∧ s.westChestOpened = false + · simp [step, h4] + · simp [step, h1, h2, h3, h4] + rw [hp2]; exact hchest + | wait => + simp [step] at hb ⊢; exact ih hb + +/-- If a reachable state has `southChestOpened = true`, then `buttonPressed = true`. -/ +theorem southChestOpened_implies_buttonPressed + {s : EnvState} (hr : Reachable s) : + s.southChestOpened = true → s.buttonPressed = true := by + intro hc + by_cases hb : s.buttonPressed = false + · have := buttonPressed_false_implies_not_in_south_and_chest_closed hr hb + rcases this with ⟨_, hchest⟩ + rw [hc] at hchest; simp at hchest + · simp [hb] + +/-- Invariant: if `southChestOpened = false`, then `keys = 0`. + The south chest is the only source of keys, and `goEast` (the only key + consumer) requires `keys > 0` to fire, so before the chest is opened no + key can exist. -/ +theorem southChestClosed_implies_keys_zero + {s : EnvState} (hr : Reachable s) : + s.southChestOpened = false → s.keys = 0 := by + induction hr with + | init => simp [initialState] + | step s a hr ih => + intro hc + cases a with + | pressButton => + have hpres : (step s Action.pressButton).southChestOpened = s.southChestOpened := by + have := pressButton_preserves_chests (s := s) + rcases this with ⟨_, h, _, _⟩; exact h + rw [hpres] at hc + have hpresk : (step s Action.pressButton).keys = s.keys := pressButton_preserves_keys + rw [hpresk] + exact ih hc + | goSouth => + have hpres : (step s Action.goSouth).southChestOpened = s.southChestOpened := by + have := goSouth_preserves_chests (s := s) + rcases this with ⟨_, h, _, _⟩; exact h + rw [hpres] at hc + have hpresk : (step s Action.goSouth).keys = s.keys := goSouth_preserves_keys + rw [hpresk] + exact ih hc + | goNorth => + have hpres : (step s Action.goNorth).southChestOpened = s.southChestOpened := by + have := goNorth_preserves_chests (s := s) + rcases this with ⟨_, h, _, _⟩; exact h + rw [hpres] at hc + have hpresk : (step s Action.goNorth).keys = s.keys := goNorth_preserves_keys + rw [hpresk] + exact ih hc + | goEast => + have hpres : (step s Action.goEast).southChestOpened = s.southChestOpened := by + have := goEast_preserves_chests (s := s) + rcases this with ⟨_, h, _, _⟩; exact h + rw [hpres] at hc + -- goEast changes keys only when fired from startRoom with keys>0. + -- If fired, then s.keys > 0, so by ih s.southChestOpened = true, + -- contradicting hc. + by_cases h1 : s.room = Room.startRoom ∧ s.keys > 0 + · have hkey : s.keys > 0 := h1.2 + have hsc : s.southChestOpened = true := by + by_cases hclosed : s.southChestOpened = false + · have hzero := ih hclosed + rw [hzero] at hkey; simp at hkey + · simp [hclosed] + rw [hsc] at hc; simp at hc + · -- goEast was a no-op or westRoom→startRoom; keys unchanged. + have hpresk : (step s Action.goEast).keys = s.keys := by + by_cases h2 : s.room = Room.westRoom + · have := goEast_from_west_returns_to_start (hRoom := h2) + rcases this with ⟨_, hk⟩; exact hk + · simp [step, h1, h2] + rw [hpresk] + exact ih hc + | goWest => + have hpres : (step s Action.goWest).southChestOpened = s.southChestOpened := by + have := goWest_preserves_chests (s := s) + rcases this with ⟨_, h, _, _⟩; exact h + rw [hpres] at hc + have hpresk : (step s Action.goWest).keys = s.keys := goWest_preserves_keys + rw [hpresk] + exact ih hc + | openChest => + by_cases hs : s.room = Room.southRoom ∧ s.southChestOpened = false + · -- south branch fires: southChestOpened becomes true, contradicting hc. + have hfires : (step s Action.openChest).southChestOpened = true := by + by_cases h1 : s.room = Room.startRoom ∧ s.startChestOpened = false + · rw [hs.1] at h1; simp at h1 + · simp [step, hs] + rw [hfires] at hc; simp at hc + · -- non-south branch: southChestOpened and keys both unchanged. + have hpres_sc : (step s Action.openChest).southChestOpened = s.southChestOpened := by + by_cases h1 : s.room = Room.startRoom ∧ s.startChestOpened = false + · simp [step, h1] + · by_cases h2 : s.room = Room.southRoom ∧ s.southChestOpened = false + · exact absurd h2 hs + · by_cases h3 : s.room = Room.eastRoom ∧ s.eastChestOpened = false + · simp [step, h3] + · by_cases h4 : s.room = Room.westRoom ∧ s.westChestOpened = false + · simp [step, h4] + · simp [step, h1, h2, h3, h4] + have hpres_k : (step s Action.openChest).keys = s.keys := by + by_cases h1 : s.room = Room.startRoom ∧ s.startChestOpened = false + · simp [step, h1] + · by_cases h2 : s.room = Room.southRoom ∧ s.southChestOpened = false + · exact absurd h2 hs + · by_cases h3 : s.room = Room.eastRoom ∧ s.eastChestOpened = false + · simp [step, h3] + · by_cases h4 : s.room = Room.westRoom ∧ s.westChestOpened = false + · simp [step, h4] + · simp [step, h1, h2, h3, h4] + rw [hpres_sc] at hc + rw [hpres_k] + exact ih hc + | wait => + simp [step] at hc + simp [step] + exact ih hc + +/-- If a reachable state has `GoalReached`, then the button was pressed. -/ +theorem completion_postcondition + {s : EnvState} (hr : Reachable s) (hc : GoalReached s) : + s.buttonPressed = true := by + unfold GoalReached allChestsOpened at hc + rcases hc with ⟨_, hsc, _, _⟩ + exact southChestOpened_implies_buttonPressed hr hsc + +/-! ### Witness plan -/ + +-- Room-level route: +-- startRoom: openChest (start chest) +-- → pressButton +-- → goSouth (button gating) +-- southRoom: openChest (key chest, keys + 1) +-- → goNorth +-- startRoom: goEast (locked_key, consume_key=true, keys - 1) +-- eastRoom: openChest (heal chest, hp + 1) +-- → goWest (back to startRoom) +-- startRoom: goWest (normal exit to westRoom) +-- westRoom: openChest (gold chest) +def witnessPlan : List Action := + [ + Action.openChest, + Action.pressButton, + Action.goSouth, + Action.openChest, + Action.goNorth, + Action.goEast, + Action.openChest, + Action.goWest, + Action.goWest, + Action.openChest + ] + +def finalState : EnvState := + { + room := Room.westRoom + buttonPressed := true + keys := 0 + hp := 6 + startChestOpened := true + southChestOpened := true + eastChestOpened := true + westChestOpened := true + } + +theorem witnessPlan_executes_to_finalState : + run initialState witnessPlan = finalState := by + rfl + +theorem witnessPlan_reaches_goal : + GoalReached (run initialState witnessPlan) := by + rw [witnessPlan_executes_to_finalState] + simp [GoalReached, allChestsOpened, finalState] + +theorem task5_completable : + ∃ plan, GoalReached (run initialState plan) := by + exact ⟨witnessPlan, witnessPlan_reaches_goal⟩ + +/-! ### Symbolic policy formalization -/ + +/-- +The abstract stage machine behind the task 5 baseline. This is the symbolic +counterpart of the Python `task5_stage` routine: it ignores pixel-level +waypoints and keeps only the high-level subgoal ordering that matters for the +environment proof. +-/ +inductive PolicyStage where + | openStartChest + | pressStartButton + | openSouthChest + | openEastChest + | openWestChest + | done + deriving DecidableEq, Repr + +def policyStage (s : EnvState) : PolicyStage := + if s.startChestOpened = false then + PolicyStage.openStartChest + else if s.buttonPressed = false then + PolicyStage.pressStartButton + else if s.southChestOpened = false then + PolicyStage.openSouthChest + else if s.eastChestOpened = false then + PolicyStage.openEastChest + else if s.westChestOpened = false then + PolicyStage.openWestChest + else + PolicyStage.done + +/-- +State-driven room-level policy. It maps the current symbolic state to the next +high-level action, including navigation actions that return to the start room +when the current stage needs a different room. +-/ +def symbolicPolicy (s : EnvState) : Action := + match policyStage s with + | PolicyStage.openStartChest => + match s.room with + | Room.startRoom => Action.openChest + | Room.southRoom => Action.goNorth + | Room.eastRoom => Action.goWest + | Room.westRoom => Action.goEast + | PolicyStage.pressStartButton => + match s.room with + | Room.startRoom => Action.pressButton + | Room.southRoom => Action.goNorth + | Room.eastRoom => Action.goWest + | Room.westRoom => Action.goEast + | PolicyStage.openSouthChest => + match s.room with + | Room.startRoom => + if s.buttonPressed = true then Action.goSouth else Action.pressButton + | Room.southRoom => Action.openChest + | Room.eastRoom => Action.goWest + | Room.westRoom => Action.goEast + | PolicyStage.openEastChest => + match s.room with + | Room.startRoom => + if s.keys > 0 then Action.goEast else Action.wait + | Room.southRoom => Action.goNorth + | Room.eastRoom => Action.openChest + | Room.westRoom => Action.goEast + | PolicyStage.openWestChest => + match s.room with + | Room.startRoom => Action.goWest + | Room.southRoom => Action.goNorth + | Room.eastRoom => Action.goWest + | Room.westRoom => Action.openChest + | PolicyStage.done => Action.wait + +def policyStep (s : EnvState) : EnvState := + step s (symbolicPolicy s) + +def runPolicy : Nat → EnvState → EnvState + | 0, s => s + | n + 1, s => runPolicy n (policyStep s) + +def policyTrace : Nat → EnvState → List Action + | 0, _ => [] + | n + 1, s => + let a := symbolicPolicy s + a :: policyTrace n (step s a) + +/-- Legal-action predicate for the symbolic transition layer. -/ +def ActionEnabled (s : EnvState) : Action → Prop + | Action.pressButton => s.room = Room.startRoom + | Action.goSouth => s.room = Room.startRoom ∧ s.buttonPressed = true + | Action.goNorth => s.room = Room.southRoom + | Action.goEast => + (s.room = Room.startRoom ∧ s.keys > 0) ∨ s.room = Room.westRoom + | Action.goWest => s.room = Room.startRoom ∨ s.room = Room.eastRoom + | Action.openChest => + (s.room = Room.startRoom ∧ s.startChestOpened = false) ∨ + (s.room = Room.southRoom ∧ s.southChestOpened = false) ∨ + (s.room = Room.eastRoom ∧ s.eastChestOpened = false) ∨ + (s.room = Room.westRoom ∧ s.westChestOpened = false) + | Action.wait => True + +def actionsEnabledAlong : EnvState → List Action → Prop + | _, [] => True + | s, a :: rest => ActionEnabled s a ∧ actionsEnabledAlong (step s a) rest + +theorem policyTrace_10_matches_witnessPlan : + policyTrace 10 initialState = witnessPlan := by + native_decide + +theorem policyTrace_10_actions_enabled : + actionsEnabledAlong initialState (policyTrace 10 initialState) := by + simp [policyTrace, symbolicPolicy, policyStage, actionsEnabledAlong, + ActionEnabled, initialState, step] + +theorem symbolicPolicy_run_10_finalState : + runPolicy 10 initialState = finalState := by + native_decide + +theorem symbolicPolicy_reaches_goal : + GoalReached (runPolicy 10 initialState) := by + rw [symbolicPolicy_run_10_finalState] + simp [GoalReached, allChestsOpened, finalState] + +theorem task5_symbolicPolicy_completes : + ∃ n, GoalReached (runPolicy n initialState) := by + exact ⟨10, symbolicPolicy_reaches_goal⟩ + +/-! ### Baseline-aligned certified refinement -/ + +/- +The room-level policy above proves the logical objective, but the final Python +baseline contains additional verifiable control structure: + +* every approach phase is delegated to BFS over the visible blocked-tile map; +* the start-room chaser is cleared before taking the locked east exit; +* the east ambusher is avoided while approaching the heal chest; +* the shield is raised at the west boundary before the spatial_c entry contact; +* both west-room monsters are cleared before the final chest is approached. + +The definitions below formalize that control layer as certified macro steps. +Pixel motion and monster dynamics remain abstracted behind route/combat +certificates, but the certificate checks and their ordering are explicit. +-/ + +abbrev RouteTile := Nat × Nat + +def routeInBounds (tile : RouteTile) : Bool := + decide (tile.1 < 10) && decide (tile.2 < 8) + +def routeAdjacent (a b : RouteTile) : Bool := + ((a.1 + 1 == b.1) && (a.2 == b.2)) || + ((b.1 + 1 == a.1) && (a.2 == b.2)) || + ((a.1 == b.1) && (a.2 + 1 == b.2)) || + ((a.1 == b.1) && (b.2 + 1 == a.2)) + +def routeConnected : List RouteTile → Bool + | [] => false + | [_] => true + | a :: b :: rest => routeAdjacent a b && routeConnected (b :: rest) + +def routeStartsAt (path : List RouteTile) (start : RouteTile) : Bool := + match path with + | [] => false + | first :: _ => first == start + +def routeEndsIn (path goals : List RouteTile) : Bool := + match path with + | [] => false + | [last] => goals.contains last + | _ :: rest => routeEndsIn rest goals + +inductive BfsTarget where + | startChest + | startButton + | southExit + | southChest + | northExit + | eastExit + | eastChest + | eastReturnExit + | westExit + | westChest + deriving DecidableEq, Repr + +/-- +A checkable contract for one Python `bfs_path` / `follow_bfs_aligned` macro. +`blocked` is the union of visible walls, abyss, avoided traps, chests, NPCs, +and non-target exits used by the Python policy for that phase. +-/ +structure BfsRouteCertificate (target : BfsTarget) where + start : RouteTile + goals : List RouteTile + blocked : List RouteTile + path : List RouteTile + deriving Repr + +def validBfsRoute {target : BfsTarget} (certificate : BfsRouteCertificate target) : Bool := + routeStartsAt certificate.path certificate.start && + routeEndsIn certificate.path certificate.goals && + routeConnected certificate.path && + certificate.path.all routeInBounds && + certificate.path.all (fun tile => !(certificate.blocked.contains tile)) + +abbrev BfsOracle := (target : BfsTarget) → BfsRouteCertificate target + +def mkRoute + (target : BfsTarget) + (start : RouteTile) + (goals blocked path : List RouteTile) : BfsRouteCertificate target := + { start := start, goals := goals, blocked := blocked, path := path } + +/- Public-layout route witnesses. Spatial variants may provide different + certificates without changing the stage or completion proof. -/ +def verifiedBfsOracle : BfsOracle + | BfsTarget.startChest => + mkRoute BfsTarget.startChest (1, 1) [(3, 2)] + [(5, 1), (5, 2), (3, 3), (4, 3), (6, 5), (4, 2), (7, 6)] + [(1, 1), (1, 2), (2, 2), (3, 2)] + | BfsTarget.startButton => + mkRoute BfsTarget.startButton (3, 2) [(2, 6)] + [(5, 1), (5, 2), (3, 3), (4, 3), (6, 5), (4, 2), (7, 6)] + [(3, 2), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)] + | BfsTarget.southExit => + mkRoute BfsTarget.southExit (2, 6) [(4, 7), (5, 7)] + [(5, 1), (5, 2), (3, 3), (4, 3), (6, 5), (4, 2), (7, 6)] + [(2, 6), (3, 6), (4, 6), (4, 7)] + | BfsTarget.southChest => + mkRoute BfsTarget.southChest (4, 1) [(8, 4)] + [(2, 2), (3, 2), (4, 2), (5, 2), (6, 2), (7, 2), (4, 6), (8, 5)] + [(4, 1), (5, 1), (6, 1), (7, 1), (8, 1), (8, 2), (8, 3), (8, 4)] + | BfsTarget.northExit => + mkRoute BfsTarget.northExit (8, 4) [(4, 0), (5, 0)] + [(2, 2), (3, 2), (4, 2), (5, 2), (6, 2), (7, 2), (4, 6), (8, 5)] + [(8, 4), (8, 3), (8, 2), (8, 1), (8, 0), (7, 0), (6, 0), (5, 0)] + | BfsTarget.eastExit => + mkRoute BfsTarget.eastExit (4, 6) [(9, 3), (9, 4)] + [(5, 1), (5, 2), (3, 3), (4, 3), (6, 5), (4, 2), (7, 6)] + [(4, 6), (5, 6), (5, 5), (5, 4), (6, 4), (7, 4), (8, 4), (9, 4)] + | BfsTarget.eastChest => + mkRoute BfsTarget.eastChest (1, 4) [(6, 1)] + [(2, 2), (2, 3), (2, 4), (5, 4), (6, 4), (7, 1)] + [(1, 4), (1, 3), (1, 2), (1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1)] + | BfsTarget.eastReturnExit => + mkRoute BfsTarget.eastReturnExit (6, 1) [(0, 3), (0, 4)] + [(2, 2), (2, 3), (2, 4), (5, 4), (6, 4), (7, 1)] + [(6, 1), (5, 1), (4, 1), (3, 1), (2, 1), (1, 1), (1, 2), (1, 3), (0, 3)] + | BfsTarget.westExit => + mkRoute BfsTarget.westExit (8, 3) [(0, 3), (0, 4)] + [(5, 1), (5, 2), (3, 3), (4, 3), (6, 5), (4, 2), (7, 6)] + [(8, 3), (8, 4), (7, 4), (6, 4), (5, 4), (4, 4), (3, 4), (2, 4), (1, 4), (0, 4)] + | BfsTarget.westChest => + mkRoute BfsTarget.westChest (8, 4) [(3, 6)] + [(1, 2), (2, 2), (5, 5), (4, 6), (5, 6), (2, 6), (7, 6)] + [(8, 4), (7, 4), (6, 4), (5, 4), (4, 4), (3, 4), (3, 5), (3, 6)] + +inductive CombatTarget where + | startChaser + | westMonsters + deriving DecidableEq, Repr + +/-- Contract for the repeated approach/face/attack loop in `attack_monster`. -/ +structure CombatCertificate (target : CombatTarget) where + approachSteps : Nat + attackBursts : Nat + targetDefeated : Bool + avoidsInteractionTiles : Bool + blocksNonTargetExits : Bool + deriving Repr + +def validCombat {target : CombatTarget} (certificate : CombatCertificate target) : Bool := + decide (0 < certificate.approachSteps) && + decide (0 < certificate.attackBursts) && + certificate.targetDefeated && + certificate.avoidsInteractionTiles && + certificate.blocksNonTargetExits + +abbrev CombatOracle := (target : CombatTarget) → CombatCertificate target + +def verifiedCombatOracle : CombatOracle + | CombatTarget.startChaser => + { approachSteps := 120 + attackBursts := 2 + targetDefeated := true + avoidsInteractionTiles := true + blocksNonTargetExits := true } + | CombatTarget.westMonsters => + { approachSteps := 70 + attackBursts := 2 + targetDefeated := true + avoidsInteractionTiles := true + blocksNonTargetExits := true } + +/-- Certificate for the spatial_c west-room boundary contact. -/ +structure WestEntryCertificate where + crossingTicks : Nat + shieldTicks : Nat + raisedAtBoundary : Bool + contactBlocked : Bool + deriving Repr + +def validWestEntry (certificate : WestEntryCertificate) : Bool := + decide (certificate.crossingTicks < certificate.shieldTicks) && + certificate.raisedAtBoundary && certificate.contactBlocked + +def verifiedWestEntry : WestEntryCertificate := + { crossingTicks := 2 + shieldTicks := 6 + raisedAtBoundary := true + contactBlocked := true } + +structure BaselineCertificates where + bfs : BfsOracle + combat : CombatOracle + westEntry : WestEntryCertificate + +def verifiedCertificates : BaselineCertificates := + { bfs := verifiedBfsOracle + combat := verifiedCombatOracle + westEntry := verifiedWestEntry } + +inductive BaselinePhase where + | approachStartChest + | openStartChest + | approachButton + | pressStartButton + | approachSouthExit + | enterSouth + | approachSouthChest + | openSouthChest + | approachNorthExit + | returnStartFromSouth + | clearStartMonster + | approachEastExit + | enterEast + | approachEastChest + | openEastChest + | approachEastReturn + | returnStartFromEast + | approachWestExit + | raiseWestShield + | enterWest + | clearWestMonsters + | approachWestChest + | openWestChest + | done + deriving DecidableEq, Repr + +inductive BaselineAction where + | followBfs (target : BfsTarget) + | openChest + | pressButton + | crossSouth + | crossNorth + | crossEast + | crossWest + | attack + | raiseShield + | wait + deriving DecidableEq, Repr + +structure BaselineState where + env : EnvState + phase : BaselinePhase + startMonsterAlive : Bool + westMonstersRemaining : Nat + shieldActive : Bool + safe : Bool + deriving DecidableEq, Repr + +def initialBaselineState : BaselineState := + { env := initialState + phase := BaselinePhase.approachStartChest + startMonsterAlive := true + westMonstersRemaining := 2 + shieldActive := false + safe := true } + +def baselineActionEnabledBool + (certificates : BaselineCertificates) + (s : BaselineState) : BaselineAction → Bool + | BaselineAction.followBfs target => + validBfsRoute (certificates.bfs target) && + match s.phase, target with + | BaselinePhase.approachStartChest, BfsTarget.startChest => true + | BaselinePhase.approachButton, BfsTarget.startButton => true + | BaselinePhase.approachSouthExit, BfsTarget.southExit => true + | BaselinePhase.approachSouthChest, BfsTarget.southChest => true + | BaselinePhase.approachNorthExit, BfsTarget.northExit => true + | BaselinePhase.approachEastExit, BfsTarget.eastExit => true + | BaselinePhase.approachEastChest, BfsTarget.eastChest => true + | BaselinePhase.approachEastReturn, BfsTarget.eastReturnExit => true + | BaselinePhase.approachWestExit, BfsTarget.westExit => true + | BaselinePhase.approachWestChest, BfsTarget.westChest => true + | _, _ => false + | BaselineAction.openChest => + match s.phase with + | BaselinePhase.openStartChest => + decide (s.env.room = Room.startRoom) && !s.env.startChestOpened + | BaselinePhase.openSouthChest => + decide (s.env.room = Room.southRoom) && !s.env.southChestOpened + | BaselinePhase.openEastChest => + decide (s.env.room = Room.eastRoom) && !s.env.eastChestOpened + | BaselinePhase.openWestChest => + decide (s.env.room = Room.westRoom) && !s.env.westChestOpened && + decide (s.westMonstersRemaining = 0) + | _ => false + | BaselineAction.pressButton => + decide (s.phase = BaselinePhase.pressStartButton) && + decide (s.env.room = Room.startRoom) + | BaselineAction.crossSouth => + decide (s.phase = BaselinePhase.enterSouth) && + decide (s.env.room = Room.startRoom) && s.env.buttonPressed + | BaselineAction.crossNorth => + decide (s.phase = BaselinePhase.returnStartFromSouth) && + decide (s.env.room = Room.southRoom) + | BaselineAction.crossEast => + decide (s.phase = BaselinePhase.enterEast) && + decide (s.env.room = Room.startRoom) && + decide (s.env.keys > 0) && !s.startMonsterAlive + | BaselineAction.crossWest => + (decide (s.phase = BaselinePhase.returnStartFromEast) && + decide (s.env.room = Room.eastRoom)) || + (decide (s.phase = BaselinePhase.enterWest) && + decide (s.env.room = Room.startRoom) && s.shieldActive && + validWestEntry certificates.westEntry) + | BaselineAction.attack => + (decide (s.phase = BaselinePhase.clearStartMonster) && + decide (s.env.room = Room.startRoom) && s.startMonsterAlive && + validCombat (certificates.combat CombatTarget.startChaser)) || + (decide (s.phase = BaselinePhase.clearWestMonsters) && + decide (s.env.room = Room.westRoom) && + decide (s.westMonstersRemaining > 0) && + validCombat (certificates.combat CombatTarget.westMonsters)) + | BaselineAction.raiseShield => + decide (s.phase = BaselinePhase.raiseWestShield) && + decide (s.env.room = Room.startRoom) && + validWestEntry certificates.westEntry + | BaselineAction.wait => decide (s.phase = BaselinePhase.done) + +def BaselineActionEnabled + (certificates : BaselineCertificates) + (s : BaselineState) + (action : BaselineAction) : Prop := + baselineActionEnabledBool certificates s action = true + +def unsafeBaselineState (s : BaselineState) : BaselineState := + { s with safe := false } + +def advanceBfs + (certificates : BaselineCertificates) + (s : BaselineState) + (target : BfsTarget) + (next : BaselinePhase) : BaselineState := + if validBfsRoute (certificates.bfs target) then + { s with phase := next } + else + unsafeBaselineState s + +def baselineStep + (certificates : BaselineCertificates) + (s : BaselineState) + (action : BaselineAction) : BaselineState := + match s.phase, action with + | BaselinePhase.approachStartChest, BaselineAction.followBfs BfsTarget.startChest => + advanceBfs certificates s BfsTarget.startChest BaselinePhase.openStartChest + | BaselinePhase.openStartChest, BaselineAction.openChest => + if s.env.room = Room.startRoom ∧ s.env.startChestOpened = false then + { s with env := step s.env Action.openChest, phase := BaselinePhase.approachButton } + else unsafeBaselineState s + | BaselinePhase.approachButton, BaselineAction.followBfs BfsTarget.startButton => + advanceBfs certificates s BfsTarget.startButton BaselinePhase.pressStartButton + | BaselinePhase.pressStartButton, BaselineAction.pressButton => + if s.env.room = Room.startRoom then + { s with env := step s.env Action.pressButton, phase := BaselinePhase.approachSouthExit } + else unsafeBaselineState s + | BaselinePhase.approachSouthExit, BaselineAction.followBfs BfsTarget.southExit => + advanceBfs certificates s BfsTarget.southExit BaselinePhase.enterSouth + | BaselinePhase.enterSouth, BaselineAction.crossSouth => + if s.env.room = Room.startRoom ∧ s.env.buttonPressed = true then + { s with env := step s.env Action.goSouth, phase := BaselinePhase.approachSouthChest } + else unsafeBaselineState s + | BaselinePhase.approachSouthChest, BaselineAction.followBfs BfsTarget.southChest => + advanceBfs certificates s BfsTarget.southChest BaselinePhase.openSouthChest + | BaselinePhase.openSouthChest, BaselineAction.openChest => + if s.env.room = Room.southRoom ∧ s.env.southChestOpened = false then + { s with env := step s.env Action.openChest, phase := BaselinePhase.approachNorthExit } + else unsafeBaselineState s + | BaselinePhase.approachNorthExit, BaselineAction.followBfs BfsTarget.northExit => + advanceBfs certificates s BfsTarget.northExit BaselinePhase.returnStartFromSouth + | BaselinePhase.returnStartFromSouth, BaselineAction.crossNorth => + if s.env.room = Room.southRoom then + { s with env := step s.env Action.goNorth, phase := BaselinePhase.clearStartMonster } + else unsafeBaselineState s + | BaselinePhase.clearStartMonster, BaselineAction.attack => + if s.env.room = Room.startRoom ∧ s.startMonsterAlive = true ∧ + validCombat (certificates.combat CombatTarget.startChaser) = true then + { s with startMonsterAlive := false, phase := BaselinePhase.approachEastExit } + else unsafeBaselineState s + | BaselinePhase.approachEastExit, BaselineAction.followBfs BfsTarget.eastExit => + advanceBfs certificates s BfsTarget.eastExit BaselinePhase.enterEast + | BaselinePhase.enterEast, BaselineAction.crossEast => + if s.env.room = Room.startRoom ∧ s.env.keys > 0 ∧ s.startMonsterAlive = false then + { s with env := step s.env Action.goEast, phase := BaselinePhase.approachEastChest } + else unsafeBaselineState s + | BaselinePhase.approachEastChest, BaselineAction.followBfs BfsTarget.eastChest => + advanceBfs certificates s BfsTarget.eastChest BaselinePhase.openEastChest + | BaselinePhase.openEastChest, BaselineAction.openChest => + if s.env.room = Room.eastRoom ∧ s.env.eastChestOpened = false then + { s with env := step s.env Action.openChest, phase := BaselinePhase.approachEastReturn } + else unsafeBaselineState s + | BaselinePhase.approachEastReturn, BaselineAction.followBfs BfsTarget.eastReturnExit => + advanceBfs certificates s BfsTarget.eastReturnExit BaselinePhase.returnStartFromEast + | BaselinePhase.returnStartFromEast, BaselineAction.crossWest => + if s.env.room = Room.eastRoom then + { s with env := step s.env Action.goWest, phase := BaselinePhase.approachWestExit } + else unsafeBaselineState s + | BaselinePhase.approachWestExit, BaselineAction.followBfs BfsTarget.westExit => + advanceBfs certificates s BfsTarget.westExit BaselinePhase.raiseWestShield + | BaselinePhase.raiseWestShield, BaselineAction.raiseShield => + if s.env.room = Room.startRoom ∧ validWestEntry certificates.westEntry = true then + { s with shieldActive := true, phase := BaselinePhase.enterWest } + else unsafeBaselineState s + | BaselinePhase.enterWest, BaselineAction.crossWest => + if s.env.room = Room.startRoom ∧ s.shieldActive = true ∧ + validWestEntry certificates.westEntry = true then + { s with env := step s.env Action.goWest + shieldActive := false + phase := BaselinePhase.clearWestMonsters } + else unsafeBaselineState s + | BaselinePhase.clearWestMonsters, BaselineAction.attack => + if s.env.room = Room.westRoom ∧ s.westMonstersRemaining > 0 ∧ + validCombat (certificates.combat CombatTarget.westMonsters) = true then + let remaining := s.westMonstersRemaining - 1 + { s with westMonstersRemaining := remaining + phase := if remaining = 0 then + BaselinePhase.approachWestChest + else + BaselinePhase.clearWestMonsters } + else unsafeBaselineState s + | BaselinePhase.approachWestChest, BaselineAction.followBfs BfsTarget.westChest => + advanceBfs certificates s BfsTarget.westChest BaselinePhase.openWestChest + | BaselinePhase.openWestChest, BaselineAction.openChest => + if s.env.room = Room.westRoom ∧ s.env.westChestOpened = false ∧ + s.westMonstersRemaining = 0 then + { s with env := step s.env Action.openChest, phase := BaselinePhase.done } + else unsafeBaselineState s + | BaselinePhase.done, BaselineAction.wait => s + | _, _ => unsafeBaselineState s + +def certifiedBfsAction + (certificates : BaselineCertificates) + (target : BfsTarget) : BaselineAction := + if validBfsRoute (certificates.bfs target) then + BaselineAction.followBfs target + else + BaselineAction.wait + +def certifiedCombatAction + (certificates : BaselineCertificates) + (target : CombatTarget) : BaselineAction := + if validCombat (certificates.combat target) then + BaselineAction.attack + else + BaselineAction.wait + +/-- State-driven macro policy matching the final Python task-5 ordering. -/ +def baselinePolicy + (certificates : BaselineCertificates) + (s : BaselineState) : BaselineAction := + match s.phase with + | BaselinePhase.approachStartChest => certifiedBfsAction certificates BfsTarget.startChest + | BaselinePhase.openStartChest => BaselineAction.openChest + | BaselinePhase.approachButton => certifiedBfsAction certificates BfsTarget.startButton + | BaselinePhase.pressStartButton => BaselineAction.pressButton + | BaselinePhase.approachSouthExit => certifiedBfsAction certificates BfsTarget.southExit + | BaselinePhase.enterSouth => BaselineAction.crossSouth + | BaselinePhase.approachSouthChest => certifiedBfsAction certificates BfsTarget.southChest + | BaselinePhase.openSouthChest => BaselineAction.openChest + | BaselinePhase.approachNorthExit => certifiedBfsAction certificates BfsTarget.northExit + | BaselinePhase.returnStartFromSouth => BaselineAction.crossNorth + | BaselinePhase.clearStartMonster => certifiedCombatAction certificates CombatTarget.startChaser + | BaselinePhase.approachEastExit => certifiedBfsAction certificates BfsTarget.eastExit + | BaselinePhase.enterEast => BaselineAction.crossEast + | BaselinePhase.approachEastChest => certifiedBfsAction certificates BfsTarget.eastChest + | BaselinePhase.openEastChest => BaselineAction.openChest + | BaselinePhase.approachEastReturn => certifiedBfsAction certificates BfsTarget.eastReturnExit + | BaselinePhase.returnStartFromEast => BaselineAction.crossWest + | BaselinePhase.approachWestExit => certifiedBfsAction certificates BfsTarget.westExit + | BaselinePhase.raiseWestShield => + if validWestEntry certificates.westEntry then BaselineAction.raiseShield else BaselineAction.wait + | BaselinePhase.enterWest => BaselineAction.crossWest + | BaselinePhase.clearWestMonsters => certifiedCombatAction certificates CombatTarget.westMonsters + | BaselinePhase.approachWestChest => certifiedBfsAction certificates BfsTarget.westChest + | BaselinePhase.openWestChest => BaselineAction.openChest + | BaselinePhase.done => BaselineAction.wait + +def runBaseline + (certificates : BaselineCertificates) : Nat → BaselineState → BaselineState + | 0, s => s + | n + 1, s => + let action := baselinePolicy certificates s + runBaseline certificates n (baselineStep certificates s action) + +def baselineTrace + (certificates : BaselineCertificates) : Nat → BaselineState → List BaselineAction + | 0, _ => [] + | n + 1, s => + let action := baselinePolicy certificates s + action :: baselineTrace certificates n (baselineStep certificates s action) + +def baselineActionsEnabledAlong + (certificates : BaselineCertificates) : BaselineState → List BaselineAction → Bool + | _, [] => true + | s, action :: rest => + baselineActionEnabledBool certificates s action && + baselineActionsEnabledAlong certificates (baselineStep certificates s action) rest + +def detailedFinalState : BaselineState := + { env := finalState + phase := BaselinePhase.done + startMonsterAlive := false + westMonstersRemaining := 0 + shieldActive := false + safe := true } + +def verifiedBaselinePlan : List BaselineAction := + [ BaselineAction.followBfs BfsTarget.startChest + , BaselineAction.openChest + , BaselineAction.followBfs BfsTarget.startButton + , BaselineAction.pressButton + , BaselineAction.followBfs BfsTarget.southExit + , BaselineAction.crossSouth + , BaselineAction.followBfs BfsTarget.southChest + , BaselineAction.openChest + , BaselineAction.followBfs BfsTarget.northExit + , BaselineAction.crossNorth + , BaselineAction.attack + , BaselineAction.followBfs BfsTarget.eastExit + , BaselineAction.crossEast + , BaselineAction.followBfs BfsTarget.eastChest + , BaselineAction.openChest + , BaselineAction.followBfs BfsTarget.eastReturnExit + , BaselineAction.crossWest + , BaselineAction.followBfs BfsTarget.westExit + , BaselineAction.raiseShield + , BaselineAction.crossWest + , BaselineAction.attack + , BaselineAction.attack + , BaselineAction.followBfs BfsTarget.westChest + , BaselineAction.openChest ] + +def DetailedGoalReached (s : BaselineState) : Prop := + GoalReached s.env ∧ + s.startMonsterAlive = false ∧ + s.westMonstersRemaining = 0 ∧ + s.safe = true + +theorem verified_bfs_routes_valid : + ∀ target, validBfsRoute (verifiedCertificates.bfs target) = true := by + intro target + cases target <;> native_decide + +theorem verified_combat_routes_valid : + ∀ target, validCombat (verifiedCertificates.combat target) = true := by + intro target + cases target <;> native_decide + +theorem verified_west_entry_valid : + validWestEntry verifiedCertificates.westEntry = true := by + native_decide + +theorem baselineTrace_24_matches_verified_plan : + baselineTrace verifiedCertificates 24 initialBaselineState = verifiedBaselinePlan := by + native_decide + +theorem baselineTrace_24_actions_enabled : + baselineActionsEnabledAlong verifiedCertificates initialBaselineState + (baselineTrace verifiedCertificates 24 initialBaselineState) = true := by + native_decide + +theorem start_monster_cleared_before_east_entry : + let s := runBaseline verifiedCertificates 12 initialBaselineState + s.phase = BaselinePhase.enterEast ∧ s.startMonsterAlive = false := by + native_decide + +theorem spatial_c_west_entry_is_shielded : + let s := runBaseline verifiedCertificates 19 initialBaselineState + s.phase = BaselinePhase.enterWest ∧ + s.shieldActive = true ∧ + BaselineActionEnabled verifiedCertificates s BaselineAction.crossWest := by + simp only [BaselineActionEnabled] + native_decide + +theorem west_monsters_cleared_before_final_chest : + let s := runBaseline verifiedCertificates 23 initialBaselineState + s.phase = BaselinePhase.openWestChest ∧ s.westMonstersRemaining = 0 := by + native_decide + +theorem baselinePolicy_run_24_finalState : + runBaseline verifiedCertificates 24 initialBaselineState = detailedFinalState := by + native_decide + +theorem baselinePolicy_refines_room_level_finalState : + (runBaseline verifiedCertificates 24 initialBaselineState).env = finalState := by + rw [baselinePolicy_run_24_finalState] + rfl + +theorem baselinePolicy_reaches_detailed_goal : + DetailedGoalReached (runBaseline verifiedCertificates 24 initialBaselineState) := by + rw [baselinePolicy_run_24_finalState] + simp [DetailedGoalReached, detailedFinalState, GoalReached, allChestsOpened, finalState] + +theorem task5_certified_baseline_completes : + ∃ n, DetailedGoalReached (runBaseline verifiedCertificates n initialBaselineState) := by + exact ⟨24, baselinePolicy_reaches_detailed_goal⟩ + +end Task5Formalization diff --git a/student_agent/lean/lean-toolchain b/student_agent/lean/lean-toolchain new file mode 100644 index 0000000..e59446d --- /dev/null +++ b/student_agent/lean/lean-toolchain @@ -0,0 +1 @@ +leanprover/lean4:v4.26.0