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vaught_conjecture
Specification
The Vaught conjecture states that for a countable language L and a complete L-Theory T the number of countable models of T (up to isomorphism) is finite, $\aleph_0$ or $2^{\aleph_0}$.
Lean 4 Statement
theorem vaught_conjecture {L : FirstOrder.Language} (hL : Countable L.Symbols)
{T : L.Theory} (hT : T.IsComplete) :
numberOfCountableModels T ≤ Cardinal.aleph0 ∨ numberOfCountableModels T = Cardinal.continuum
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| Model | Status | Goals Left | Submitted | Lean Snippet | Output |
|---|---|---|---|---|---|
| GPT-5.3 Codex Agent v8 | Partial | 1 | Feb 25, 2026 |
classical
by_cases h : numberOfCountableModels T ≤ Cardinal.aleph0
· exact Or.inl h
· have hEq : numberOfCountableModels T = Cardinal.continuum := ?_
exact Or.inr hEq
|
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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\nlibrary_search",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772031199_7982.lean:28:3: error: unknown tactic\n/app/asi_portal/verifier/quick_1772031199_7982.lean:26:98: error: unsolved goals\nL : FirstOrder.Language\nhL : Countable L.Symbols\nT : L.Theory\nhT : T.IsComplete\n\u22a2 numberOfCountableModels T \u2264 Cardinal.aleph0 \u2228 numberOfCountableModels T = Cardinal.continuum\n'VaughtConjecture.vaught_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 48.947,
"tactic_replay": {
"tactic_count": 2,
"successful_tactics": 0,
"first_failure_step": 1,
"goal_trajectory": [
1,
1
],
"max_goals": 1,
"final_goal_count": 1,
"goals_opened": 0,
"goals_closed": 0
},
"lsp_feedback": {
"diagnostics": [
{
"fullRange": {
"end": {
"character": 3,
"line": 27
},
"start": {
"character": 3,
"line": 27
}
},
"message": "unknown tactic",
"range": {
"end": {
"character": 3,
"line": 27
},
"start": {
"character": 3,
"line": 27
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 16,
"line": 27
},
"start": {
"character": 98,
"line": 25
}
},
"leanTags": [
1
],
"message": "unsolved goals\nL : FirstOrder.Language\nhL : Countable L.Symbols\nT : L.Theory\nhT : T.IsComplete\n\u22a2 numberOfCountableModels T \u2264 Cardinal.aleph0 \u2228 numberOfCountableModels T = Cardinal.continuum",
"range": {
"end": {
"character": 0,
"line": 26
},
"start": {
"character": 98,
"line": 25
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 29
},
"start": {
"character": 0,
"line": 29
}
},
"message": "'VaughtConjecture.vaught_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": 25,
"error_char": 98,
"error_msg": "unsolved goals\nL : FirstOrder.Language\nhL : Countable L.Symbols\nT : L.Theory\nhT : T.IsComplete\n\u22a2 numberOfCountableModels T \u2264 Cardinal.aleph0 \u2228 numberOfCountableModels T = Cardinal.continuum",
"goal_before_error": null,
"full_timeline": [
{
"line": 26,
"code": "classical",
"goal": "```lean\nL : FirstOrder.Language\nhL : Countable L.Symbols\nT : L.Theory\nhT : T.IsComplete\n\u22a2 numberOfCountableModels T \u2264 Cardinal.aleph0 \u2228 numberOfCountableModels T = Cardinal.continuum\n```",
"goal_count": 1,
"ast_complexity": 186
},
{
"line": 27,
"code": "library_search",
"goal": "```lean\nL : FirstOrder.Language\nhL : Countable L.Symbols\nT : L.Theory\nhT : T.IsComplete\n\u22a2 numberOfCountableModels T \u2264 Cardinal.aleph0 \u2228 numberOfCountableModels T = Cardinal.continuum\n```",
"goal_count": 1,
"ast_complexity": 186
}
],
"tactic_count": 2,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"error_code": "numberOfCountableModels T \u2264 Cardinal.aleph0 \u2228 numberOfCountableModels T = Cardinal.continuum := by"
}
}
],
"verification": {
"banned_tokens": [],
"command": [
"[path]
"lean",
"/app/asi_portal/verifier/quick_1772031248_3847.lean"
],
"comparator_passed": null,
"duration_s": 23.599517583847046,
"goals_remaining": 1,
"is_verified": false,
"lean_file_path": "",
"problem": {
"axioms_check_name": "vaught_conjecture",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__VaughtConjecture__vaught_conjecture",
"category": "research_open",
"endings": [
"end VaughtConjecture"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 498,
"source_file": "FormalConjectures/Wikipedia/VaughtConjecture.lean",
"title": "vaught_conjecture"
},
"returncode": 1,
"status": "partial",
"stderr": "",
"stdout": "/app/asi_portal/verifier/quick_1772031248_3847.lean:30:2: error: unsolved goals\ncase neg.refine_1\nL : FirstOrder.Language\nhL : Countable L.Symbols\nT : L.Theory\nhT : T.IsComplete\nh : \u00acnumberOfCountableModels T \u2264 Cardinal.aleph0\n\u22a2 numberOfCountableModels T = Cardinal.continuum\n'VaughtConjecture.vaught_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\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": 0.05,
"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",
"finish_reason": "",
"usage": {
"prompt_tokens": 0,
"completion_tokens": 0,
"total_tokens": 0,
"reasoning_tokens": null,
"cost_usd": 0.0
},
"reasoning_effort": "high",
"response_trace_count": 0
},
"response_trace": []
}
|
|
| Gemini 3 Flash Agent v8 | Partial | 1 | Feb 17, 2026 |
unfold numberOfCountableModels
rcases le_or_gt (Cardinal.mk (Quotient (Setoid.comap (fun (model : {mt : T.ModelType // Countable mt.Carrier}) => CategoryTheory.Bundled.mk model.val.Carrier model.val.struc) equivSetoid))) Cardinal.aleph0 with h | h
· left
exact h
· right
obtain ⟨f⟩ : Nonempty (Cardinal.mk (Quotient (Setoid.comap (fun (model : {mt : T.ModelType // Countable mt.Carrier}) => CategoryTheory.Bundled.mk model.val.Carrier model.val.struc) equivSetoid)) = Cardinal.continuum) := by
cases hT
simp
apply ?_
exact f
|
View{
"agent_telemetry": {
"compilations": 8,
"iterations": 8,
"duration_s": 89.27,
"best_status": "partial",
"best_goals": 1,
"max_iterations": 8,
"model": "gemini-3-flash-preview",
"error": null,
"provider": "openrouter_primary",
"finish_reason": "stop",
"usage": {
"prompt_tokens": 37627,
"completion_tokens": 1432,
"total_tokens": 39059,
"reasoning_tokens": null,
"cost_usd": 0.01855775
},
"reasoning_effort": "",
"response_trace_count": 16
},
"response_trace": [
{
"iteration": 1,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771350942-gXdbFhZbefm0nooLVfsV",
"api_elapsed_s": 9.102,
"usage": {
"prompt_tokens": 1763,
"completion_tokens": 179,
"total_tokens": 1942,
"reasoning_tokens": null,
"cost_usd": 0.0014185
},
"openrouter_retries": 0,
"response_text_len": 542
},
{
"iteration": 1,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "partial",
"goals_remaining": 1,
"timed_out": false,
"banned_tokens": [],
"proof_code": "unfold numberOfCountableModels\nrcases le_or_gt (Cardinal.mk (Quotient (Setoid.comap (fun (model : {mt : T.ModelType // Countable mt.Carrier}) => CategoryTheory.Bundled.mk model.val.Carrier model.val.struc) equivSetoid))) Cardinal.aleph0 with h | h\n\u00b7 left\n exact h\n\u00b7 right\n obtain \u27e8f\u27e9 : Nonempty (Cardinal.mk (Quotient (Setoid.comap (fun (model : {mt : T.ModelType // Countable mt.Carrier}) => CategoryTheory.Bundled.mk model.val.Carrier model.val.struc) equivSetoid)) = Cardinal.continuum) := by\n cases hT\n simp\n apply ?_\n exact f",
"compiler_output_preview": "[path]:32:224: error: unsolved goals\nL : FirstOrder.Language\nhL : Countable L.Symbols\nT : L.Theory\nh :\n Cardinal.aleph0 <\n Cardinal.mk (Quotient (Setoid.comap (fun model \u21a6 { \u03b1 := \u2191\u2191model, str := (\u2191model).struc }) equivSetoid))\nleft\u271d : T.IsSatisfiable\nright\u271d : \u2200 (\u03c6 : L.Sentence), T \u22a8\u1d47 \u03c6 \u2228 T \u22a8\u1d47 Formula.not \u03c6\n\u22a2 Cardinal.mk (Quotient (Setoid.comap (fun model \u21a6 { \u03b1 := \u2191\u2191model, st\n...<truncated>"
},
{
"iteration": 2,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771350959-G1TNjNThCvGKm5QOA53w",
"api_elapsed_s": 5.369,
"usage": {
"prompt_tokens": 2256,
"completion_tokens": 285,
"total_tokens": 2541,
"reasoning_tokens": null,
"cost_usd": 0.001983
},
"openrouter_retries": 0,
"response_text_len": 838
},
{
"iteration": 2,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "unfold numberOfCountableModels\nrcases le_or_gt (Cardinal.mk (Quotient (Setoid.comap (fun model \u21a6 CategoryTheory.Bundled.mk model.val.Carrier model.val.struc) equivSetoid))) Cardinal.aleph0 with h | h\n\u00b7 left\n exact h\n\u00b7 right\n obtain \u27e8is_sat, compl\u27e9 := hT\n let S := Quotient (Setoid.comap (fun model \u21a6 CategoryTheory.Bundled.mk model.val.Carrier model.val.struc) equivSetoid)\n have : \u2135\u2080 < Cardinal.mk S := h\n let countable_models := {mt : T.ModelType // Countable mt.Carrier}\n let \u03a6 : countable_models \u2192 CategoryTheory.Bundled (Model L) := fun model \u21a6 CategoryTheory.Bundled.mk model.val.Carrier model.val.struc\n have : Cardinal.mk S \u2264 Cardinal.continuum := by\n apply Cardinal.mk_le_continuum_of_countable_language_and_theory hL\n apply le_antisymm\n \u00b7 exact this\n \u00b7 apply Cardinal.le_of_aleph0_lt_mk_of_vaught_conjecture hL hT h",
"compiler_output_preview": "[path]:28:2: error: tactic 'generalize' failed, result is not type correct\n Cardinal.mk (Quotient (Setoid.comap (fun model \u21a6 { \u03b1 := ?m.225, str := ?m.226 }) equivSetoid)) \u2264 Cardinal.aleph0 \u2228\n Cardinal.aleph0 < Cardinal.mk (Quotient (Setoid.comap (fun model \u21a6 { \u03b1 := ?m.225, str := ?m.226 }) equivSetoid)) \u2192\n Cardinal.mk (Quotient (Setoid.comap (fun model \u21a6 { \u03b1 := \u2191\u2191model,\n...<truncated>"
},
{
"iteration": 3,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771350972-m3N5Psh5xSpkxXV2bCty",
"api_elapsed_s": 5.838,
"usage": {
"prompt_tokens": 3017,
"completion_tokens": 158,
"total_tokens": 3175,
"reasoning_tokens": null,
"cost_usd": 0.0019825
},
"openrouter_retries": 0,
"response_text_len": 442
},
{
"iteration": 3,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
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"banned_tokens": [],
"proof_code": "unfold numberOfCountableModels\nset n := Cardinal.mk (Quotient (Setoid.comap (fun model \u21a6 CategoryTheory.Bundled.mk model.val.Carrier model.val.struc) equivSetoid))\ncases le_or_gt n Cardinal.aleph0 with\n| inl h =>\n left\n exact h\n| inr h =>\n right\n have h_le : n \u2264 Cardinal.continuum := by\n apply ?le_cont\n have h_ge : Cardinal.continuum \u2264 n := by\n apply ?ge_cont\n exact le_antisymm h_le h_ge\n \u00b7 refine ?le_cont\n \u00b7 refine ?ge_cont",
"compiler_output_preview": "[path]:28:86: error: Invalid field notation: Type of\n model\nis not known; cannot resolve field `val`\n[path]:28:104: error: Invalid field notation: Type of\n model\nis not known; cannot resolve field `val`\n[path]
},
{
"iteration": 4,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771350985-CnsjchCOHed9R9JbWtxs",
"api_elapsed_s": 1.6,
"usage": {
"prompt_tokens": 4051,
"completion_tokens": 125,
"total_tokens": 4176,
"reasoning_tokens": null,
"cost_usd": 0.0024005
},
"openrouter_retries": 0,
"response_text_len": 366
},
{
"iteration": 4,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "partial",
"goals_remaining": 2,
"timed_out": false,
"banned_tokens": [],
"proof_code": "unfold numberOfCountableModels\nset n := Cardinal.mk (Quotient (Setoid.comap (fun model : {mt : T.ModelType // Countable mt.Carrier} \u21a6 CategoryTheory.Bundled.mk model.val.Carrier model.val.struc) equivSetoid))\ncases le_or_gt n Cardinal.aleph0 with\n| inl h =>\n left; exact h\n| inr h =>\n right\n apply le_antisymm\n \u00b7 refine ?upper_bound\n \u00b7 refine ?vaught_core_logic",
"compiler_output_preview": "[path]:35:4: error: unsolved goals\ncase upper_bound\nL : FirstOrder.Language\nhL : Countable L.Symbols\nT : L.Theory\nhT : T.IsComplete\nn : Cardinal.{max (max u_1 u_2) (u_3 + 1)} :=\n Cardinal.mk (Quotient (Setoid.comap (fun model \u21a6 { \u03b1 := \u2191\u2191model, str := (\u2191model).struc }) equivSetoid))\nh : Cardinal.aleph0 < n\n\u22a2 Cardinal.mk (Quotient (Setoid.comap (fun model \u21a6 { \u03b1 := \u2191\u2191model, str :=\n...<truncated>"
},
{
"iteration": 5,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771350994-8OOrMbr8SsVaVs1xhyjK",
"api_elapsed_s": 1.849,
"usage": {
"prompt_tokens": 4701,
"completion_tokens": 138,
"total_tokens": 4839,
"reasoning_tokens": null,
"cost_usd": 0.00094695
},
"openrouter_retries": 0,
"response_text_len": 399
},
{
"iteration": 5,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "unfold numberOfCountableModels\nset n := Cardinal.mk (Quotient (Setoid.comap (fun model : {mt : T.ModelType // Countable mt.Carrier} \u21a6 CategoryTheory.Bundled.mk model.val.Carrier model.val.struc) equivSetoid))\ncases le_or_gt n Cardinal.aleph0 with\n| inl h => left; exact h\n| inr h =>\n right\n apply le_antisymm\n \u00b7 have : n \u2264 Cardinal.mk (Set T.ModelType) := by refine ?_\n refine ?_\n \u00b7 refine ?_",
"compiler_output_preview": "[path]:34:6: error: stuck at solving universe constraint\n max (max (u_1+2) (u_2+2)) (u_3+3) =?= max (max (u_1+2) (u_2+2)) (?u.564+3)\nwhile trying to unify\n CoeT.{(max (max u_1 u_2) (?u.564 + 1)) + 2, (max (max u_1 u_2) (u_3 + 1)) + 2}\n Cardinal.{max (max u_1 u_2) (?u.564 + 1)} x\n Cardinal.{max (max u_1 u_2) (u_3 + 1)} : Type ((max (max u_1 u_2) (u_3 + 1)) + 1)\nwith\n Coe\n...<truncated>"
},
{
"iteration": 6,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771351003-QKxUMbY3gTyxfdtvZQlL",
"api_elapsed_s": 2.383,
"usage": {
"prompt_tokens": 6482,
"completion_tokens": 218,
"total_tokens": 6700,
"reasoning_tokens": null,
"cost_usd": 0.003895
},
"openrouter_retries": 0,
"response_text_len": 724
},
{
"iteration": 6,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "unfold numberOfCountableModels\nset n := Cardinal.mk (Quotient (Setoid.comap (fun model : {mt : T.ModelType // Countable mt.Carrier} \u21a6 CategoryTheory.Bundled.mk model.val.Carrier model.val.struc) equivSetoid))\ncases le_or_gt n Cardinal.aleph0 with\n| inl h => left; exact h\n| inr h =>\n right\n apply le_antisymm\n \u00b7 refine ?upper_bound\n \u00b7 refine ?vaught_conjecture_core\n\n case upper_bound =>\n -- A countable theory in a countable language has at most continuum many models.\n -- This is a standard result in model theory.\n refine ?_\n\n case vaught_conjecture_core =>\n -- Morley's Theorem or the main Vaught Conjecture claim.\n -- If a countable theory has > aleph_0 models, it has continuum many.\n refine ?_",
"compiler_output_preview": "[path]:34:4: error: unsolved goals\ncase upper_bound\nL : FirstOrder.Language\nhL : Countable L.Symbols\nT : L.Theory\nhT : T.IsComplete\nn : Cardinal.{max (max u_1 u_2) (u_3 + 1)} :=\n Cardinal.mk (Quotient (Setoid.comap (fun model \u21a6 { \u03b1 := \u2191\u2191model, str := (\u2191model).struc }) equivSetoid))\nh : Cardinal.aleph0 < n\n\u22a2 Cardinal.mk (Quotient (Setoid.comap (fun model \u21a6 { \u03b1 := \u2191\u2191model, str :=\n...<truncated>"
},
{
"iteration": 7,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771351013-x0MpDz9oVJqUe2GXNY2e",
"api_elapsed_s": 1.621,
"usage": {
"prompt_tokens": 7301,
"completion_tokens": 154,
"total_tokens": 7455,
"reasoning_tokens": null,
"cost_usd": 0.0041125
},
"openrouter_retries": 0,
"response_text_len": 455
},
{
"iteration": 7,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "unfold numberOfCountableModels\nset n := Cardinal.mk (Quotient (Setoid.comap (fun model : {mt : T.ModelType // Countable mt.Carrier} \u21a6 CategoryTheory.Bundled.mk model.val.Carrier model.val.struc) equivSetoid))\ncases le_or_gt n Cardinal.aleph0 with\n| inl h => left; exact h\n| inr h =>\n right\n apply le_antisymm\n \u00b7 refine ?upper_bound\n \u00b7 refine ?vaught_conjecture_logic\n\n case upper_bound =>\n refine ?_\n case vaught_conjecture_logic =>\n refine ?_",
"compiler_output_preview": "[path]:34:4: error: unsolved goals\ncase upper_bound\nL : FirstOrder.Language\nhL : Countable L.Symbols\nT : L.Theory\nhT : T.IsComplete\nn : Cardinal.{max (max u_1 u_2) (u_3 + 1)} :=\n Cardinal.mk (Quotient (Setoid.comap (fun model \u21a6 { \u03b1 := \u2191\u2191model, str := (\u2191model).struc }) equivSetoid))\nh : Cardinal.aleph0 < n\n\u22a2 Cardinal.mk (Quotient (Setoid.comap (fun model \u21a6 { \u03b1 := \u2191\u2191model, str :=\n...<truncated>"
},
{
"iteration": 8,
"provider": "openrouter_primary",
"finish_reason": "stop",
"response_id": "gen-1771351022-Dh1PgEAgE69ndcYwTxai",
"api_elapsed_s": 2.056,
"usage": {
"prompt_tokens": 8056,
"completion_tokens": 175,
"total_tokens": 8231,
"reasoning_tokens": null,
"cost_usd": 0.0018188
},
"openrouter_retries": 0,
"response_text_len": 511
},
{
"iteration": 8,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "unfold numberOfCountableModels\nset n := Cardinal.mk (Quotient (Setoid.comap (fun model : {mt : T.ModelType // Countable mt.Carrier} \u21a6 CategoryTheory.Bundled.mk model.val.Carrier model.val.struc) equivSetoid))\ncases le_or_gt n Cardinal.aleph0 with\n| inl h => left; exact h\n| inr h =>\n right\n obtain \u27e8compl_sat, compl_theory\u27e9 := hT\n obtain \u27e8ge_cont, le_cont\u27e9 : n \u2265 Cardinal.continuum \u2227 n \u2264 Cardinal.continuum := by\n constructor\n \u00b7 refine ?ge_cont\n \u00b7 refine ?le_cont\n exact le_antisymm le_cont ge_cont",
"compiler_output_preview": "[path]:36:6: error: unsolved goals\ncase ge_cont\nL : FirstOrder.Language\nhL : Countable L.Symbols\nT : L.Theory\nn : Cardinal.{max (max u_1 u_2) (u_3 + 1)} :=\n Cardinal.mk (Quotient (Setoid.comap (fun model \u21a6 { \u03b1 := \u2191\u2191model, str := (\u2191model).struc }) equivSetoid))\nh : Cardinal.aleph0 < n\ncompl_sat : T.IsSatisfiable\ncompl_theory : \u2200 (\u03c6 : L.Sentence), T \u22a8\u1d47 \u03c6 \u2228 T \u22a8\u1d47 Formula.not \u03c6\n\u22a2 n \n...<truncated>"
}
],
"verification": {
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"command": [
"[path]
"lean",
"[path]
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"duration_s": 8.588222026824951,
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"lean_file_path": "[path]
"problem": {
"axioms_check_name": "vaught_conjecture",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__VaughtConjecture__vaught_conjecture",
"category": "research_open",
"endings": [
"end VaughtConjecture"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 498,
"source_file": "FormalConjectures/Wikipedia/VaughtConjecture.lean",
"title": "vaught_conjecture"
},
"returncode": 1,
"status": "partial",
"stderr": "",
"stdout": "[path]:32:224: error: unsolved goals\nL : FirstOrder.Language\nhL : Countable L.Symbols\nT : L.Theory\nh :\n Cardinal.aleph0 <\n Cardinal.mk (Quotient (Setoid.comap (fun model \u21a6 { \u03b1 := \u2191\u2191model, str := (\u2191model).struc }) equivSetoid))\nleft\u271d : T.IsSatisfiable\nright\u271d : \u2200 (\u03c6 : L.Sentence), T \u22a8\u1d47 \u03c6 \u2228 T \u22a8\u1d47 Formula.not \u03c6\n\u22a2 Cardinal.mk (Quotient (Setoid.comap (fun model \u21a6 { \u03b1 := \u2191\u2191model, str := (\u2191model).struc }) equivSetoid)) =\n Cardinal.continuum\n'VaughtConjecture.vaught_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
"timed_out": false
}
}
|
| Gemini 3 Flash Preview | Failed | -- | Feb 14, 2026 |
apply Cardinal.mk_quotient_le_size
have : Cardinal.mk models ≤ Cardinal.continuum := by
trans Cardinal.mk (Σ (α : Type), L.Structure α)
· apply Cardinal.mk_subtype_le
|
View{
"generation": {
"attempts": 2,
"banned_tokens_found": [],
"elapsed_s": 4.7,
"finish_reason": "stop",
"max_tokens": 8192,
"model_id": "google/gemini-3-flash-preview",
"proof_code_length": 176,
"raw_content_length": 1270,
"reasoning_effort": "",
"structured_output": true,
"temperature": 0.0,
"usage": {
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"completion_tokens_details": {
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"cost": 0.0018985,
"cost_details": {
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"prompt_tokens_details": {
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"command": [
"[path]
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"[path]
],
"duration_s": 7.354097366333008,
"goals_remaining": null,
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"lean_file_path": "[path]
"problem": {
"benchmark_id": "Wikipedia__VaughtConjecture__vaught_conjecture",
"category": "research_open",
"endings": [
"end VaughtConjecture"
],
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"FormalConjectures.Util.ProblemImports"
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"preamble_chars": 498,
"source_file": "FormalConjectures/Wikipedia/VaughtConjecture.lean",
"title": "vaught_conjecture"
},
"returncode": 1,
"status": "failed",
"stderr": "",
"stdout": "[path]:27:8: error: unknown constant 'Cardinal.mk_quotient_le_size'\n[path]:28:2: error: no goals to be solved\n",
"timed_out": false
}
}
|