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infinite_isWallSunSunPrime
Specification
A prime $p$ is a Wall–Sun–Sun prime if and only if $L_p \equiv 1 \pmod{p^2}$, where $L_p$ is the $p$-th Lucas number. It is conjectured that there are infinitely many Wall-Sun-Sun primes.
Lean 4 Statement
theorem infinite_isWallSunSunPrime : {p : ℕ | IsWallSunSunPrime p}.Infinite
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Evaluation Results
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| Model | Status | Goals Left | Submitted | Lean Snippet | Output |
|---|---|---|---|---|---|
| GPT-5.3 Codex Agent v8 | Partial | 1 | Feb 25, 2026 |
classical
have h : {p : ℕ | IsWallSunSunPrime p}.Infinite := ?_
exact h
|
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"proof_code": "classical\nhave h_univ : (Set.univ : Set \u2115).Infinite := Set.infinite_univ\nhave h_target : {p : \u2115 | IsWallSunSunPrime p}.Infinite := by\n simpa using h_univ\nexact h_target",
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"proof_code": "classical\nhave h : {p : \u2115 | IsWallSunSunPrime p}.Infinite := by\n simp [IsWallSunSunPrime]\nexact h",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772031389_1763.lean:17:4: error: `simp` made no progress\n'infinite_isWallSunSunPrime' depends on axioms: [propext, sorryAx]\n\n\n",
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1
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"line": 16
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"goal": "```lean\n\u22a2 {p | IsWallSunSunPrime p}.Infinite\n```",
"goal_count": 1,
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{
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"goal": "```lean\n\u22a2 {p | IsWallSunSunPrime p}.Infinite\n```",
"goal_count": 1,
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},
{
"line": 17,
"code": "exact h",
"goal": "```lean\nh : {p | IsWallSunSunPrime p}.Infinite\n\u22a2 {p | IsWallSunSunPrime p}.Infinite\n```",
"goal_count": 1,
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}
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}
},
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"reasoning_text": "**Attempting infinite subtype proof**",
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"iteration": 9,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
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"proof_code": "classical\nlet s : Set \u2115 := {p : \u2115 | IsWallSunSunPrime p}\nhave hsub : Infinite s := by\n infer_instance\nhave hs : s.Infinite := by\n exact (Set.infinite_coe_iff).2 hsub\nsimpa [s] using hs",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772031441_8343.lean:18:4: error(lean.synthInstanceFailed): failed to synthesize instance of type class\n Infinite \u2191s\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.\n/app/asi_portal/verifier/quick_1772031441_8343.lean:20:35: error: Application type mismatch: The argument\n hsub\nhas type\n Infinite \u2191s\nbut is expected to have type\n Set.Infinite ?m.18\nin the application\n Set.infinite_coe_if\n...<truncated>",
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"first_failure_step": 4,
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1,
1,
1,
1,
1,
1,
1
],
"max_goals": 1,
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"goals_opened": 0,
"goals_closed": 0
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"fullRange": {
"end": {
"character": 18,
"line": 17
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"start": {
"character": 4,
"line": 17
}
},
"message": "failed to synthesize instance of type class\n Infinite \u2191s\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.",
"range": {
"end": {
"character": 18,
"line": 17
},
"start": {
"character": 4,
"line": 17
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},
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},
{
"fullRange": {
"end": {
"character": 39,
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"start": {
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},
"message": "Application type mismatch: The argument\n hsub\nhas type\n Infinite \u2191s\nbut is expected to have type\n Set.Infinite ?m.18\nin the application\n Set.infinite_coe_iff.mpr hsub",
"range": {
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{
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},
"message": "'infinite_isWallSunSunPrime' depends on axioms: [propext, sorryAx]",
"range": {
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"line": 22
},
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"character": 0,
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}
},
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}
],
"error_line": 17,
"error_char": 4,
"error_msg": "failed to synthesize instance of type class\n Infinite \u2191s\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.",
"goal_before_error": "```lean\ns : Set \u2115 := {p | IsWallSunSunPrime p}\n\u22a2 Infinite \u2191s\n```",
"full_timeline": [
{
"line": 14,
"code": "classical",
"goal": "```lean\n\u22a2 {p | IsWallSunSunPrime p}.Infinite\n```",
"goal_count": 1,
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},
{
"line": 15,
"code": "let s : Set \u2115 := {p : \u2115 | IsWallSunSunPrime p}",
"goal": "```lean\n\u22a2 {p | IsWallSunSunPrime p}.Infinite\n```",
"goal_count": 1,
"ast_complexity": 48
},
{
"line": 16,
"code": "have hsub : Infinite s := by",
"goal": "```lean\ns : Set \u2115 := {p | IsWallSunSunPrime p}\n\u22a2 {p | IsWallSunSunPrime p}.Infinite\n```",
"goal_count": 1,
"ast_complexity": 87
},
{
"line": 17,
"code": "infer_instance",
"goal": "```lean\ns : Set \u2115 := {p | IsWallSunSunPrime p}\n\u22a2 Infinite \u2191s\n```",
"goal_count": 1,
"ast_complexity": 64
},
{
"line": 18,
"code": "have hs : s.Infinite := by",
"goal": "```lean\ns : Set \u2115 := {p | IsWallSunSunPrime p}\nhsub : Infinite \u2191s\n\u22a2 {p | IsWallSunSunPrime p}.Infinite\n```",
"goal_count": 1,
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},
{
"line": 19,
"code": "exact (Set.infinite_coe_iff).2 hsub",
"goal": "```lean\ns : Set \u2115 := {p | IsWallSunSunPrime p}\nhsub : Infinite \u2191s\n\u22a2 s.Infinite\n```",
"goal_count": 1,
"ast_complexity": 82
},
{
"line": 20,
"code": "simpa [s] using hs",
"goal": "```lean\ns : Set \u2115 := {p | IsWallSunSunPrime p}\nhsub : Infinite \u2191s\nhs : s.Infinite\n\u22a2 {p | IsWallSunSunPrime p}.Infinite\n```",
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}
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}
},
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"summary": "**Weighing last compile attempt using library search**"
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}
]
},
{
"iteration": 10,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "classical\nhave h : {p : \u2115 | IsWallSunSunPrime p}.Infinite := by\n library_search\nexact h",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772031509_9080.lean:17:5: error: unknown tactic\n/app/asi_portal/verifier/quick_1772031509_9080.lean:16:53: error: unsolved goals\n\u22a2 {p | IsWallSunSunPrime p}.Infinite\n/app/asi_portal/verifier/quick_1772031509_9080.lean:14:79: error: unsolved goals\nh : {p | IsWallSunSunPrime p}.Infinite\n\u22a2 {p | IsWallSunSunPrime p}.Infinite\n'infinite_isWallSunSunPrime' depends on axioms: [propext, sorryAx]\n\n\n",
"compile_elapsed_s": 34.369,
"tactic_replay": {
"tactic_count": 3,
"successful_tactics": 0,
"first_failure_step": 1,
"goal_trajectory": [
1,
1,
1
],
"max_goals": 1,
"final_goal_count": 1,
"goals_opened": 0,
"goals_closed": 0
},
"lsp_feedback": {
"diagnostics": [
{
"fullRange": {
"end": {
"character": 5,
"line": 16
},
"start": {
"character": 5,
"line": 16
}
},
"message": "unknown tactic",
"range": {
"end": {
"character": 5,
"line": 16
},
"start": {
"character": 5,
"line": 16
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 18,
"line": 16
},
"start": {
"character": 53,
"line": 15
}
},
"leanTags": [
1
],
"message": "unsolved goals\n\u22a2 {p | IsWallSunSunPrime p}.Infinite",
"range": {
"end": {
"character": 0,
"line": 16
},
"start": {
"character": 53,
"line": 15
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 18,
"line": 16
},
"start": {
"character": 79,
"line": 13
}
},
"leanTags": [
1
],
"message": "unsolved goals\nh : {p | IsWallSunSunPrime p}.Infinite\n\u22a2 {p | IsWallSunSunPrime p}.Infinite",
"range": {
"end": {
"character": 0,
"line": 14
},
"start": {
"character": 79,
"line": 13
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 19
},
"start": {
"character": 0,
"line": 19
}
},
"message": "'infinite_isWallSunSunPrime' depends on axioms: [propext, sorryAx]",
"range": {
"end": {
"character": 6,
"line": 19
},
"start": {
"character": 0,
"line": 19
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 13,
"error_char": 79,
"error_msg": "unsolved goals\nh : {p | IsWallSunSunPrime p}.Infinite\n\u22a2 {p | IsWallSunSunPrime p}.Infinite",
"goal_before_error": null,
"full_timeline": [
{
"line": 14,
"code": "classical",
"goal": "```lean\n\u22a2 {p | IsWallSunSunPrime p}.Infinite\n```",
"goal_count": 1,
"ast_complexity": 48
},
{
"line": 15,
"code": "have h : {p : \u2115 | IsWallSunSunPrime p}.Infinite := by",
"goal": "```lean\n\u22a2 {p | IsWallSunSunPrime p}.Infinite\n```",
"goal_count": 1,
"ast_complexity": 48
},
{
"line": 16,
"code": "library_search",
"goal": "```lean\n\u22a2 {p | IsWallSunSunPrime p}.Infinite\n```",
"goal_count": 1,
"ast_complexity": 48
}
],
"tactic_count": 3,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"error_code": "theorem infinite_isWallSunSunPrime : {p : \u2115 | IsWallSunSunPrime p}.Infinite := by"
}
}
],
"verification": {
"banned_tokens": [],
"command": [
"[path]
"lean",
"/app/asi_portal/verifier/quick_1772031544_7705.lean"
],
"comparator_passed": null,
"duration_s": 14.713806629180908,
"goals_remaining": 1,
"is_verified": false,
"lean_file_path": "",
"problem": {
"axioms_check_name": "infinite_isWallSunSunPrime",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__WallSunSun__infinite_isWallSunSunPrime",
"category": "research_open",
"endings": [],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 134,
"source_file": "FormalConjectures/Wikipedia/WallSunSun.lean",
"title": "infinite_isWallSunSunPrime"
},
"returncode": 1,
"status": "partial",
"stderr": "",
"stdout": "/app/asi_portal/verifier/quick_1772031544_7705.lean:14:79: error: unsolved goals\ncase refine_1\n\u22a2 {p | IsWallSunSunPrime p}.Infinite\n'infinite_isWallSunSunPrime' depends on axioms: [propext, sorryAx]\n\n",
"timed_out": false
},
"phase2_error": "ProgressAnalysisResult.__init__() missing 1 required positional argument: 'progress_score'"
}
|
| Gemini 3.1 Pro Agent v8 | Failed | -- | Feb 24, 2026 |
View{
"agent_telemetry": {
"compilations": 0,
"iterations": 0,
"duration_s": 0.04,
"best_status": "pending",
"best_goals": null,
"max_iterations": 10,
"model": "google/gemini-3.1-pro-preview",
"error": "OpenRouter API error 402: {\"error\":{\"message\":\"This request requires more credits, or fewer max_tokens. You requested up to 65536 tokens, but can only afford 58661. To increase, visit https://openrouter.ai/settings/keys and create a key with a higher total limit\",\"code\":402,\"metadata\":{\"provider_name\":null}},\"user_id\":\"user_2zsm3L1zzBL2yxyrvmrXwgeTimc\"}",
"provider": "openrouter_primary",
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},
"reasoning_effort": "high",
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}
|
|
| Gemini 3 Flash Agent v8 | Partial | 1 | Feb 17, 2026 |
apply Set.infinite_of_forall_exists_gt
intro n
suffices ∃ p, p > n ∧ IsWallSunSunPrime p from
let ⟨p, hp1, hp2⟩ := this
⟨p, hp2, hp1⟩
refine ?_
|
View{
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{
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},
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{
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"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
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"banned_tokens": [],
"proof_code": "apply Set.Infinite.mono (show {p | IsWallSunSunPrime p} \u2286 {p | IsWallSunSunPrime p} by intro x hx; exact hx)\nrefine ?_\nsuffices \u2200 n : \u2115, \u2203 p > n, IsWallSunSunPrime p by\n apply Set.infinite_of_forall_exists_gt\n exact this\nintro n\nlet S := {p | IsWallSunSunPrime p}\nshow \u2203 p > n, IsWallSunSunPrime p\nrefine ?_",
"compiler_output_preview": "[path]:19:4: error: type mismatch\n this\nhas type\n \u2200 (n : \u2115), \u2203 p > n, IsWallSunSunPrime p : Prop\nbut is expected to have type\n \u2200 (a : \u2115), \u2203 b \u2208 {p | IsWallSunSunPrime p}, a < b : Prop\n[path]:14:79: error: unsolved goals\nn : \u2115\nS : Set \u2115 := {p | IsWallSun\n...<truncated>"
},
{
"iteration": 2,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771351051-XOzRF4Zg4XR32KF5n6JO",
"api_elapsed_s": 1.479,
"usage": {
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},
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{
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"provider": "openrouter_primary",
"status": "partial",
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"banned_tokens": [],
"proof_code": "apply Set.infinite_of_forall_exists_gt\nintro n\nsuffices \u2203 p, p > n \u2227 IsWallSunSunPrime p from\n let \u27e8p, hp1, hp2\u27e9 := this\n \u27e8p, hp2, hp1\u27e9\nrefine ?_",
"compiler_output_preview": "[path]:14:79: error: unsolved goals\ncase h\nn : \u2115\n\u22a2 \u2203 p > n, IsWallSunSunPrime p\n'infinite_isWallSunSunPrime' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
{
"iteration": 3,
"provider": "openrouter_primary",
"finish_reason": "stop",
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"api_elapsed_s": 1.43,
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},
{
"iteration": 3,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "partial",
"goals_remaining": 1,
"timed_out": false,
"banned_tokens": [],
"proof_code": "apply Set.infinite_of_forall_exists_gt\nintro n\nsuffices \u2203 p > n, IsWallSunSunPrime p from\n let \u27e8p, hp\u27e9 := this\n let \u27e8hp1, hp2\u27e9 := hp\n \u27e8p, hp2, hp1\u27e9\nrefine ?_",
"compiler_output_preview": "[path]:14:79: error: unsolved goals\ncase h\nn : \u2115\n\u22a2 \u2203 p > n, IsWallSunSunPrime p\n'infinite_isWallSunSunPrime' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
{
"iteration": 4,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771351068-gJzVphlw0NHCTXAzw7y6",
"api_elapsed_s": 1.387,
"usage": {
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"cost_usd": 0.001477
},
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},
{
"iteration": 4,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "partial",
"goals_remaining": 1,
"timed_out": false,
"banned_tokens": [],
"proof_code": "apply Set.infinite_of_forall_exists_gt\nintro n\nsuffices \u2203 p > n, IsWallSunSunPrime p by\n rcases this with \u27e8p, hp_gt, hp_prime\u27e9\n exact \u27e8p, hp_prime, hp_gt\u27e9\nrefine ?_",
"compiler_output_preview": "[path]:14:79: error: unsolved goals\ncase h\nn : \u2115\n\u22a2 \u2203 p > n, IsWallSunSunPrime p\n'infinite_isWallSunSunPrime' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
{
"iteration": 5,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771351077-L30iQTlzhEjTK00qvIQ8",
"api_elapsed_s": 1.078,
"usage": {
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},
"openrouter_retries": 0,
"response_text_len": 100
},
{
"iteration": 5,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "partial",
"goals_remaining": 1,
"timed_out": false,
"banned_tokens": [],
"proof_code": "apply Set.infinite_of_forall_exists_gt\nintro n\nshow \u2203 p \u2208 {p | IsWallSunSunPrime p}, n < p\nrefine ?_",
"compiler_output_preview": "[path]:14:79: error: unsolved goals\ncase h\nn : \u2115\n\u22a2 \u2203 p \u2208 {p | IsWallSunSunPrime p}, n < p\n'infinite_isWallSunSunPrime' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
{
"iteration": 6,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771351085-fXGGo7juw4DxYE1mLOKH",
"api_elapsed_s": 1.473,
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},
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},
{
"iteration": 6,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "partial",
"goals_remaining": 1,
"timed_out": false,
"banned_tokens": [],
"proof_code": "apply Set.infinite_of_forall_exists_gt\nintro n\nsuffices \u2203 p, p > n \u2227 IsWallSunSunPrime p from\n let \u27e8p, hgt, hWSS\u27e9 := this\n \u27e8p, hWSS, hgt\u27e9\nrefine ?_",
"compiler_output_preview": "[path]:14:79: error: unsolved goals\ncase h\nn : \u2115\n\u22a2 \u2203 p > n, IsWallSunSunPrime p\n'infinite_isWallSunSunPrime' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
{
"iteration": 7,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771351094-WDmJg5tgbChA3Oh8YLVy",
"api_elapsed_s": 2.167,
"usage": {
"prompt_tokens": 3262,
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"total_tokens": 3299,
"reasoning_tokens": null,
"cost_usd": 0.00084335
},
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"response_text_len": 94
},
{
"iteration": 7,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "partial",
"goals_remaining": 1,
"timed_out": false,
"banned_tokens": [],
"proof_code": "apply Set.infinite_of_forall_exists_gt\nintro n\nshow \u2203 p, IsWallSunSunPrime p \u2227 p > n\nrefine ?_",
"compiler_output_preview": "[path]:14:79: error: unsolved goals\ncase h\nn : \u2115\n\u22a2 \u2203 p, IsWallSunSunPrime p \u2227 p > n\n'infinite_isWallSunSunPrime' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
{
"iteration": 8,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771351103-luc3TETLWTTbDKoF5sVL",
"api_elapsed_s": 1.466,
"usage": {
"prompt_tokens": 3480,
"completion_tokens": 60,
"total_tokens": 3540,
"reasoning_tokens": null,
"cost_usd": 0.0010218
},
"openrouter_retries": 0,
"response_text_len": 139
},
{
"iteration": 8,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "partial",
"goals_remaining": 1,
"timed_out": false,
"banned_tokens": [],
"proof_code": "apply Set.infinite_of_forall_exists_gt\nintro n\nsuffices \u2203 p > n, IsWallSunSunPrime p from\n let \u27e8p, hp\u27e9 := this\n \u27e8p, hp.2, hp.1\u27e9\nrefine ?_",
"compiler_output_preview": "[path]:14:79: error: unsolved goals\ncase h\nn : \u2115\n\u22a2 \u2203 p > n, IsWallSunSunPrime p\n'infinite_isWallSunSunPrime' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
}
],
"verification": {
"banned_tokens": [],
"command": [
"[path]
"lean",
"[path]
],
"duration_s": 6.604700565338135,
"goals_remaining": 1,
"is_verified": false,
"lean_file_path": "[path]
"problem": {
"axioms_check_name": "infinite_isWallSunSunPrime",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__WallSunSun__infinite_isWallSunSunPrime",
"category": "research_open",
"endings": [],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 134,
"source_file": "FormalConjectures/Wikipedia/WallSunSun.lean",
"title": "infinite_isWallSunSunPrime"
},
"returncode": 1,
"status": "partial",
"stderr": "",
"stdout": "[path]:14:79: error: unsolved goals\ncase h\nn : \u2115\n\u22a2 \u2203 p > n, IsWallSunSunPrime p\n'infinite_isWallSunSunPrime' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
"timed_out": false
}
}
|
| Gemini 3 Flash Preview | Partial | 1 | Feb 14, 2026 |
by_contra h_le
have h_div : p_next ∣ Nat.factorial (m + 1) := Nat.dvd_factorial (Nat.Prime.pos h_p_next) (le_of_not_gt h_le)
have h_div_one : p_next ∣ 1 := (Nat.dvd_add_right h_div).mp (Nat.minFac_dvd (Nat.factorial (m + 1) + 1))
exact Nat.Prime.not_dvd_one h_p_next h_div_one
by_contra
apply h_fin.not_infinite
exact Set.infinite_of_forall_exists_gt (λ x => ⟨p_next, ⟨h_gt, trivial⟩⟩)
|
View{
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"max_tokens": 8192,
"model_id": "google/gemini-3-flash-preview",
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"reasoning_effort": "",
"structured_output": true,
"temperature": 0.0,
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"cost": 0.002082,
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},
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"[path]
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],
"duration_s": 7.25892186164856,
"goals_remaining": 1,
"is_verified": false,
"lean_file_path": "[path]
"problem": {
"benchmark_id": "Wikipedia__WallSunSun__infinite_isWallSunSunPrime",
"category": "research_open",
"endings": [],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 134,
"source_file": "FormalConjectures/Wikipedia/WallSunSun.lean",
"title": "infinite_isWallSunSunPrime"
},
"returncode": 1,
"status": "partial",
"stderr": "",
"stdout": "[path]:14:79: error: unsolved goals\nh_le : \u00ac{p | IsWallSunSunPrime p}.Infinite\n\u22a2 False\n[path]:16:4: error: unexpected token 'have'; expected command\n",
"timed_out": false
}
}
|