{
"agent_telemetry": {
"compilations": 3,
"iterations": 10,
"duration_s": 254.9,
"best_status": "partial",
"best_goals": 2,
"max_iterations": 10,
"model": "google/gemini-3.1-pro-preview",
"error": null,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"usage": {
"prompt_tokens": 95365,
"completion_tokens": 6435,
"total_tokens": 101800,
"reasoning_tokens": 5944,
"cost_usd": 0.1734554
},
"reasoning_effort": "high",
"response_trace_count": 13
},
"response_trace": [
{
"iteration": 1,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"response_id": "gen-1771920524-on5ym0Ith2hxQmjgDaaa",
"api_elapsed_s": 33.528,
"usage": {
"prompt_tokens": 1493,
"completion_tokens": 1199,
"total_tokens": 2692,
"reasoning_tokens": 1112,
"cost_usd": 0.017374
},
"openrouter_retries": 0,
"response_text_len": 0,
"used_tool_call": true,
"reasoning_text": "**Formalizing Brocard's Conjecture**\n\nI've begun formalizing Brocard's Conjecture in Lean 4. Specifically, I'm working on translating the problem statement into Lean's syntax. I'm focusing on defining `prev` and `next` in terms of the nth prime number, which seems like a good starting point. My current hurdle is figuring out how to express the conjecture's core idea within the theorem's parameters.\n\n\n**Refining Conjecture's Premise**\n\nI've made progress refining the conjecture's premise. I've been working on expressing the relationship between the nth prime and the count of primes within the squares of the preceding and succeeding primes. I am also working on testing the `card` term. I'm carefully translating the key elements: `nth Nat.Prime`, `prev`, `next`, and `Ioo` into a suitable form within Lean 4. My main focus is accurately representing the core inequality within the Lean environment.\n\n\n**Implementing Hole-Driven Skeleton**\n\nI am now implementing a hole-driven development approach, as requested. I've drafted a skeleton of lemmas, focusing on decomposing an informal proof. I'm using Lean 4 with holes `?_` to represent the steps I need to fill in later. I am now focused on using the `?_` hole-driven method, given the instructions. I'm carefully translating each step into a Lean-compatible structure, aiming for a complete, though currently incomplete, formalization.\n\n\n**Constructing Theorem Skeleton**\n\nI'm constructing a theorem skeleton using `intro prev next`. I'm using `letI` as evaluated let variables. I'm focusing on defining `prev` and `next`, then I am going to focus on building a structure that can be compiled, and I will be using the `compile_proof` tool. I'm building it as a proof. I need to be exact, so I'm trying to follow the skeleton that I've been creating.\n\n\n",
"reasoning_details": [
{
"format": "google-gemini-v1",
"index": 0,
"type": "reasoning.text",
"text": "**Formalizing Brocard's Conjecture**\n\nI've begun formalizing Brocard's Conjecture in Lean 4. Specifically, I'm working on translating the problem statement into Lean's syntax. I'm focusing on defining `prev` and `next` in terms of the nth prime number, which seems like a good starting point. My current hurdle is figuring out how to express the conjecture's core idea within the theorem's parameters.\n\n\n**Refining Conjecture's Premise**\n\nI've made progress refining the conjecture's premise. I've been working on expressing the relationship between the nth prime and the count of primes within the squares of the preceding and succeeding primes. I am also working on testing the `card` term. I'm carefully translating the key elements: `nth Nat.Prime`, `prev`, `next`, and `Ioo` into a suitable form within Lean 4. My main focus is accurately representing the core inequality within the Lean environment.\n\n\n**Implementing Hole-Driven Skeleton**\n\nI am now implementing a hole-driven development approach, as requested. I've drafted a skeleton of lemmas, focusing on decomposing an informal proof. I'm using Lean 4 with holes `?_` to represent the steps I need to fill in later. I am now focused on using the `?_` hole-driven method, given the instructions. I'm carefully translating each step into a Lean-compatible structure, aiming for a complete, though currently incomplete, formalization.\n\n\n**Constructing Theorem Skeleton**\n\nI'm constructing a theorem skeleton using `intro prev next`. I'm using `letI` as evaluated let variables. I'm focusing on defining `prev` and `next`, then I am going to focus on building a structure that can be compiled, and I will be using the `compile_proof` tool. I'm building it as a proof. I need to be exact, so I'm trying to follow the skeleton that I've been creating.\n\n\n"
},
{
"id": "tool_compile_proof_uPV0IlBRZW65ht3sXUKp",
"format": "google-gemini-v1",
"index": 0,
"type": "reasoning.encrypted",
"data": "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"
}
]
},
{
"iteration": 1,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "intro prev next\nhave h1 : prev < next := ?_\nhave h2 : prev ^ 2 + 4 \u2264 next ^ 2 := ?_\nhave h3 : 4 \u2264 ((Ioo (prev ^ 2) (next ^ 2)).filter Nat.Prime).card := ?_\nexact h3",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1771920557_2717.lean:24:8: error: Tactic `introN` failed: There are no additional binders or `let` bindings in the goal to introduce\n\nn : \u2115\nhn : 1 \u2264 n\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))\n'Brocard.brocard_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 31.062,
"tactic_replay": {
"tactic_count": 5,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"goal_trajectory": [
1,
1,
1,
1,
1
],
"max_goals": 1,
"final_goal_count": 1,
"goals_opened": 0,
"goals_closed": 0
},
"lsp_feedback": {
"diagnostics": [
{
"fullRange": {
"end": {
"character": 12,
"line": 23
},
"start": {
"character": 8,
"line": 23
}
},
"message": "Tactic `introN` failed: There are no additional binders or `let` bindings in the goal to introduce\n\nn : \u2115\nhn : 1 \u2264 n\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))",
"range": {
"end": {
"character": 12,
"line": 23
},
"start": {
"character": 8,
"line": 23
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 29
},
"start": {
"character": 0,
"line": 29
}
},
"message": "'Brocard.brocard_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
"end": {
"character": 6,
"line": 29
},
"start": {
"character": 0,
"line": 29
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 23,
"error_char": 8,
"error_msg": "Tactic `introN` failed: There are no additional binders or `let` bindings in the goal to introduce\n\nn : \u2115\nhn : 1 \u2264 n\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))",
"goal_before_error": "```lean\nn : \u2115\nhn : 1 \u2264 n\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))\n```",
"full_timeline": [
{
"line": 23,
"code": "intro prev next",
"goal": "```lean\nn : \u2115\nhn : 1 \u2264 n\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))\n```",
"goal_count": 1,
"ast_complexity": 118
},
{
"line": 24,
"code": "have h1 : prev < next := ?_",
"goal": "```lean\nn : \u2115\nhn : 1 \u2264 n\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))\n```",
"goal_count": 1,
"ast_complexity": 118
},
{
"line": 25,
"code": "have h2 : prev ^ 2 + 4 \u2264 next ^ 2 := ?_",
"goal": "```lean\nn : \u2115\nhn : 1 \u2264 n\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))\n```",
"goal_count": 1,
"ast_complexity": 118
},
{
"line": 26,
"code": "have h3 : 4 \u2264 ((Ioo (prev ^ 2) (next ^ 2)).filter Nat.Prime).card := ?_",
"goal": "```lean\nn : \u2115\nhn : 1 \u2264 n\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))\n```",
"goal_count": 1,
"ast_complexity": 118
},
{
"line": 27,
"code": "exact h3",
"goal": "```lean\nn : \u2115\nhn : 1 \u2264 n\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))\n```",
"goal_count": 1,
"ast_complexity": 118
}
],
"tactic_count": 5,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"error_code": "intro prev next"
}
},
{
"iteration": 2,
"event": "search_mathlib",
"query": "brocard",
"results_count": 0
},
{
"iteration": 3,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"response_id": "gen-1771920609-fWbKdgkrTjlU3Ov5EoCC",
"api_elapsed_s": 18.497,
"usage": {
"prompt_tokens": 5635,
"completion_tokens": 984,
"total_tokens": 6619,
"reasoning_tokens": 881,
"cost_usd": 0.023078
},
"openrouter_retries": 0,
"response_text_len": 0,
"used_tool_call": true,
"reasoning_text": "**Locating a Theorem**\n\nI've hit a snag. The mathlib library doesn't contain a theorem named `brocard`. Furthermore, the instructions explicitly caution against relying on unproven prior theorems, which is frustrating. I'm essentially starting from scratch with this.\n\n\n**Reconsidering Constraints**\n\nOkay, so Brocard's Conjecture is off the table, and I need to prove something *without* directly applying existing, potentially unproven theorems. I realize that \"not depending on them\" might mean I can *look* at them for inspiration. I've rephrased the goal and begun exploring decomposition. Creating holes didn't work. Using well-founded recursion seems like a non-starter. I need to break this down further, to something easier to attack directly.\n\n\n**Exploring Recursion Limitations**\n\nOkay, so I tried `let rec` within the tactic, but I realized it necessitates a termination proof, which I don't have. I'm pivoting to Hole-Driven Development and aiming for sub-lemmas. My goal is now to prove `4 \u2264 card (filter Nat.Prime (Ioo p_n^2 p_{n+1}^2))`. I've defined $p_n$ as the $n$-th prime, and used Bertrand's postulate to relate $p_n$ and $p_{n+1}$. I'm attempting to derive inequalities, attempting to close them using tactics, and hitting a wall.\n\n\n",
"reasoning_details": [
{
"format": "google-gemini-v1",
"index": 0,
"type": "reasoning.text",
"text": "**Locating a Theorem**\n\nI've hit a snag. The mathlib library doesn't contain a theorem named `brocard`. Furthermore, the instructions explicitly caution against relying on unproven prior theorems, which is frustrating. I'm essentially starting from scratch with this.\n\n\n**Reconsidering Constraints**\n\nOkay, so Brocard's Conjecture is off the table, and I need to prove something *without* directly applying existing, potentially unproven theorems. I realize that \"not depending on them\" might mean I can *look* at them for inspiration. I've rephrased the goal and begun exploring decomposition. Creating holes didn't work. Using well-founded recursion seems like a non-starter. I need to break this down further, to something easier to attack directly.\n\n\n**Exploring Recursion Limitations**\n\nOkay, so I tried `let rec` within the tactic, but I realized it necessitates a termination proof, which I don't have. I'm pivoting to Hole-Driven Development and aiming for sub-lemmas. My goal is now to prove `4 \u2264 card (filter Nat.Prime (Ioo p_n^2 p_{n+1}^2))`. I've defined $p_n$ as the $n$-th prime, and used Bertrand's postulate to relate $p_n$ and $p_{n+1}$. I'm attempting to derive inequalities, attempting to close them using tactics, and hitting a wall.\n\n\n"
},
{
"id": "tool_compile_proof_Mkwhjfatj184Cc6WnCL5",
"format": "google-gemini-v1",
"index": 0,
"type": "reasoning.encrypted",
"data": "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"
}
]
},
{
"iteration": 3,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "partial",
"goals_remaining": 2,
"timed_out": false,
"banned_tokens": [],
"proof_code": "have h1 : Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2 := ?_\nhave h2 : 4 \u2264 ((Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)).filter Nat.Prime).card := ?_\nexact h2",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1771920628_3818.lean:23:59: error: unsolved goals\ncase refine_2.refine_1\nn : \u2115\nhn : 1 \u2264 n\nh1 : Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))\n\ncase refine_1\nn : \u2115\nhn : 1 \u2264 n\n\u22a2 Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2\n'Brocard.brocard_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 28.339,
"tactic_replay": {
"tactic_count": 3,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"goal_trajectory": [
1,
2,
3
],
"max_goals": 3,
"final_goal_count": 3,
"goals_opened": 2,
"goals_closed": 0
},
"lsp_feedback": {
"diagnostics": [
{
"fullRange": {
"end": {
"character": 10,
"line": 25
},
"start": {
"character": 59,
"line": 22
}
},
"leanTags": [
1
],
"message": "unsolved goals\ncase refine_2.refine_1\nn : \u2115\nhn : 1 \u2264 n\nh1 : Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))\n\ncase refine_1\nn : \u2115\nhn : 1 \u2264 n\n\u22a2 Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2",
"range": {
"end": {
"character": 0,
"line": 23
},
"start": {
"character": 59,
"line": 22
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 27
},
"start": {
"character": 0,
"line": 27
}
},
"message": "'Brocard.brocard_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
"end": {
"character": 6,
"line": 27
},
"start": {
"character": 0,
"line": 27
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 22,
"error_char": 59,
"error_msg": "unsolved goals\ncase refine_2.refine_1\nn : \u2115\nhn : 1 \u2264 n\nh1 : Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))\n\ncase refine_1\nn : \u2115\nhn : 1 \u2264 n\n\u22a2 Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2",
"goal_before_error": null,
"full_timeline": [
{
"line": 23,
"code": "have h1 : Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2 := ?_",
"goal": "```lean\nn : \u2115\nhn : 1 \u2264 n\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))\n```",
"goal_count": 1,
"ast_complexity": 118
},
{
"line": 24,
"code": "have h2 : 4 \u2264 ((Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)).filter Nat.Prime).card := ?_",
"goal": "```lean\ncase refine_2\nn : \u2115\nhn : 1 \u2264 n\nh1 : Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))\n```\n---\n```lean\ncase refine_1\nn : \u2115\nhn : 1 \u2264 n\n\u22a2 Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2\n```",
"goal_count": 2,
"ast_complexity": 298
},
{
"line": 25,
"code": "exact h2",
"goal": "```lean\ncase refine_2.refine_2\nn : \u2115\nhn : 1 \u2264 n\nh1 : Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2\nh2 : 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))\n```\n---\n```lean\ncase refine_2.refine_1\nn : \u2115\nhn : 1 \u2264 n\nh1 : Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))\n```\n---\n```lean\ncase refine_1\nn : \u2115\nhn : 1 \u2264 n\n\u22a2 Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2\n```",
"goal_count": 3,
"ast_complexity": 607
}
],
"tactic_count": 3,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"error_code": "4 \u2264 ((Ioo (prev^2) (next^2)).filter Nat.Prime).card := by"
}
},
{
"iteration": 4,
"event": "search_mathlib",
"query": "Nat.nth",
"results_count": 1
},
{
"iteration": 5,
"event": "search_mathlib",
"query": "Nat.nth_lt",
"results_count": 8
},
{
"iteration": 6,
"event": "search_mathlib",
"query": "Nat.Prime.nth",
"results_count": 8
},
{
"iteration": 7,
"event": "search_mathlib",
"query": "Nat.nth",
"results_count": 1
},
{
"iteration": 8,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"response_id": "gen-1771920714-SPSU0BdcduiF6R2SicB6",
"api_elapsed_s": 18.828,
"usage": {
"prompt_tokens": 11654,
"completion_tokens": 689,
"total_tokens": 12343,
"reasoning_tokens": 532,
"cost_usd": 0.0170572
},
"openrouter_retries": 0,
"response_text_len": 0,
"used_tool_call": true,
"reasoning_text": "**Decomposing the Proof**\n\nI verified that `Nat.nth` is available. Banned tokens prevent `exact?` or `apply?`, so I will decompose the second hole into more holes. I plan to decompose the informal proof into a sequence of strictly sequential lemmas.\n\n\n**Refining the Inequality**\n\nI'm now focusing on proving `Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2`. I've constructed two subgoals: `h1` and `h2`, where `h1` addresses the inequality and `h2` deals with the cardinality condition. I'm exploring the best approach to close these, and considering the limitations on attempts. I'm avoiding any monolithic approaches. I am not even going to try `sorryAx`. I am at round 4.\n\n\n",
"reasoning_details": [
{
"format": "google-gemini-v1",
"index": 0,
"type": "reasoning.text",
"text": "**Decomposing the Proof**\n\nI verified that `Nat.nth` is available. Banned tokens prevent `exact?` or `apply?`, so I will decompose the second hole into more holes. I plan to decompose the informal proof into a sequence of strictly sequential lemmas.\n\n\n**Refining the Inequality**\n\nI'm now focusing on proving `Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2`. I've constructed two subgoals: `h1` and `h2`, where `h1` addresses the inequality and `h2` deals with the cardinality condition. I'm exploring the best approach to close these, and considering the limitations on attempts. I'm avoiding any monolithic approaches. I am not even going to try `sorryAx`. I am at round 4.\n\n\n"
},
{
"id": "tool_compile_proof_9V6aXdc0VcDxrMkam32e",
"format": "google-gemini-v1",
"index": 0,
"type": "reasoning.encrypted",
"data": "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"
}
]
},
{
"iteration": 8,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "partial",
"goals_remaining": 2,
"timed_out": false,
"banned_tokens": [],
"proof_code": "have h1 : Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2 := by\n have l1 : n < n + 1 := Nat.lt_succ_self n\n have l2 : Nat.nth Nat.Prime n < Nat.nth Nat.Prime (n + 1) := ?_\n gcongr\nhave h2 : 4 \u2264 ((Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)).filter Nat.Prime).card := ?_\nexact h2",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1771920732_7914.lean:24:71: error: unsolved goals\ncase refine_1\nn : \u2115\nhn : 1 \u2264 n\nl1 : n < n + 1\n\u22a2 Nat.nth Nat.Prime n < Nat.nth Nat.Prime (n + 1)\n/app/asi_portal/verifier/quick_1771920732_7914.lean:23:59: error: unsolved goals\ncase refine_1\nn : \u2115\nhn : 1 \u2264 n\nh1 : Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))\n'Brocard.brocard_conjecture' depends on axioms: [propext, so\n...<truncated>",
"compile_elapsed_s": 31.968,
"tactic_replay": {
"tactic_count": 6,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"goal_trajectory": [
1,
1,
1,
2,
1,
2
],
"max_goals": 2,
"final_goal_count": 2,
"goals_opened": 2,
"goals_closed": 1
},
"lsp_feedback": {
"diagnostics": [
{
"fullRange": {
"end": {
"character": 10,
"line": 26
},
"start": {
"character": 71,
"line": 23
}
},
"leanTags": [
1
],
"message": "unsolved goals\ncase refine_1\nn : \u2115\nhn : 1 \u2264 n\nl1 : n < n + 1\n\u22a2 Nat.nth Nat.Prime n < Nat.nth Nat.Prime (n + 1)",
"range": {
"end": {
"character": 0,
"line": 24
},
"start": {
"character": 71,
"line": 23
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 10,
"line": 28
},
"start": {
"character": 59,
"line": 22
}
},
"leanTags": [
1
],
"message": "unsolved goals\ncase refine_1\nn : \u2115\nhn : 1 \u2264 n\nh1 : Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))",
"range": {
"end": {
"character": 0,
"line": 23
},
"start": {
"character": 59,
"line": 22
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 30
},
"start": {
"character": 0,
"line": 30
}
},
"message": "'Brocard.brocard_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
"end": {
"character": 6,
"line": 30
},
"start": {
"character": 0,
"line": 30
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 22,
"error_char": 59,
"error_msg": "unsolved goals\ncase refine_1\nn : \u2115\nhn : 1 \u2264 n\nh1 : Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))",
"goal_before_error": null,
"full_timeline": [
{
"line": 23,
"code": "have h1 : Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2 := by",
"goal": "```lean\nn : \u2115\nhn : 1 \u2264 n\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))\n```",
"goal_count": 1,
"ast_complexity": 118
},
{
"line": 24,
"code": "have l1 : n < n + 1 := Nat.lt_succ_self n",
"goal": "```lean\nn : \u2115\nhn : 1 \u2264 n\n\u22a2 Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2\n```",
"goal_count": 1,
"ast_complexity": 86
},
{
"line": 25,
"code": "have l2 : Nat.nth Nat.Prime n < Nat.nth Nat.Prime (n + 1) := ?_",
"goal": "```lean\nn : \u2115\nhn : 1 \u2264 n\nl1 : n < n + 1\n\u22a2 Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2\n```",
"goal_count": 1,
"ast_complexity": 101
},
{
"line": 26,
"code": "gcongr",
"goal": "```lean\ncase refine_2\nn : \u2115\nhn : 1 \u2264 n\nl1 : n < n + 1\nl2 : Nat.nth Nat.Prime n < Nat.nth Nat.Prime (n + 1)\n\u22a2 Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2\n```\n---\n```lean\ncase refine_1\nn : \u2115\nhn : 1 \u2264 n\nl1 : n < n + 1\n\u22a2 Nat.nth Nat.Prime n < Nat.nth Nat.Prime (n + 1)\n```",
"goal_count": 2,
"ast_complexity": 280
},
{
"line": 27,
"code": "have h2 : 4 \u2264 ((Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)).filter Nat.Prime).card := ?_",
"goal": "```lean\nn : \u2115\nhn : 1 \u2264 n\nh1 : Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))\n```",
"goal_count": 1,
"ast_complexity": 179
},
{
"line": 28,
"code": "exact h2",
"goal": "```lean\ncase refine_2\nn : \u2115\nhn : 1 \u2264 n\nh1 : Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2\nh2 : 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))\n```\n---\n```lean\ncase refine_1\nn : \u2115\nhn : 1 \u2264 n\nh1 : Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))\n```",
"goal_count": 2,
"ast_complexity": 484
}
],
"tactic_count": 6,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"error_code": "4 \u2264 ((Ioo (prev^2) (next^2)).filter Nat.Prime).card := by"
}
},
{
"iteration": 9,
"event": "search_mathlib",
"query": "Nat.nth_strictMono",
"results_count": 1
},
{
"iteration": 10,
"event": "search_mathlib",
"query": "Nat.infinite_setOf_prime",
"results_count": 1
}
],
"verification": {
"banned_tokens": [],
"command": [
"[path]
"lean",
"/app/asi_portal/verifier/quick_1771920779_1422.lean"
],
"comparator_passed": null,
"duration_s": 12.941914558410645,
"goals_remaining": 2,
"is_verified": false,
"lean_file_path": "",
"problem": {
"axioms_check_name": "brocard_conjecture",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__BrocardConjecture__brocard_conjecture",
"category": "research_open",
"endings": [
"end Brocard"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 263,
"source_file": "FormalConjectures/Wikipedia/BrocardConjecture.lean",
"title": "brocard_conjecture"
},
"returncode": 1,
"status": "partial",
"stderr": "",
"stdout": "/app/asi_portal/verifier/quick_1771920779_1422.lean:23:59: error: unsolved goals\ncase refine_2.refine_1\nn : \u2115\nhn : 1 \u2264 n\nh1 : Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2\n\u22a2 4 \u2264 #(filter Nat.Prime (Ioo (Nat.nth Nat.Prime n ^ 2) (Nat.nth Nat.Prime (n + 1) ^ 2)))\n\ncase refine_1\nn : \u2115\nhn : 1 \u2264 n\n\u22a2 Nat.nth Nat.Prime n ^ 2 < Nat.nth Nat.Prime (n + 1) ^ 2\n'Brocard.brocard_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n",
"timed_out": false
}
}
Gemini 3 Flash Agent v8
Partial
4
Feb 17, 2026
set p := Nat.nth Nat.Prime n
set q := Nat.nth Nat.Prime (n + 1)
have h_inf : (setOf Nat.Prime).Infinite := by refine ?_
have hp : Nat.Prime p := Nat.nth_mem_of_infinite h_inf n
have hq : Nat.Prime q := Nat.nth_mem_of_infinite h_inf (n + 1)
have hp_lt_hq : p < q := (Nat.nth_strictMono h_inf) (Nat.lt_succ_self n)
have hp_ge_3 : p ≥ 3 := by
have := (Nat.nth_le_nth h_inf).mpr hn
refine ?_
have hq_ge_p2 : q ≥ p + 2 := by refine ?_
have h_sq_ge : q^2 ≥ p^2 + 4 * p + 4 := by
calc q^2 ≥ (p + 2)^2 := Nat.pow_le_pow_left hq_ge_p2 2
_ = p^2 + 4 * p + 4 := by ring
refine ?_
