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regularprime_conjecture
Specification
Conjecture: The set of regular primes is infinite.
Lean 4 Statement
theorem regularprime_conjecture : RegularPrimeConjecture
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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 |
unfold RegularPrimeConjecture
have hreg : ∀ p : ℕ, ∀ hp : Nat.Prime p, @IsRegularPrime p ⟨hp⟩ := by
intro p hp
simp [IsRegularPrime]
have hprimeInf : {p : ℕ | Nat.Prime p}.Infinite := by
simpa using Nat.infinite_setOf_prime
have hEq : regularPrimes = {p : ℕ | Nat.Prime p} := by
ext p
constructor
· intro hp
rcases hp with ⟨hpp, _⟩
exact hpp
· intro hp
exact ⟨hp, hreg p hp⟩
simpa [hEq] using hprimeInf
|
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}
]
},
{
"iteration": 3,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "unfold RegularPrimeConjecture\nexact ?_",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772027906_3733.lean:35:8: error: don't know how to synthesize placeholder\ncontext:\n\u22a2 regularPrimes.Infinite\n/app/asi_portal/verifier/quick_1772027906_3733.lean:33:60: error: unsolved goals\n\u22a2 regularPrimes.Infinite\n'RegularPrimes.regularprime_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 35.624,
"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": 10,
"line": 34
},
"start": {
"character": 8,
"line": 34
}
},
"message": "don't know how to synthesize placeholder\ncontext:\n\u22a2 regularPrimes.Infinite",
"range": {
"end": {
"character": 10,
"line": 34
},
"start": {
"character": 8,
"line": 34
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 10,
"line": 34
},
"start": {
"character": 60,
"line": 32
}
},
"leanTags": [
1
],
"message": "unsolved goals\n\u22a2 regularPrimes.Infinite",
"range": {
"end": {
"character": 0,
"line": 33
},
"start": {
"character": 60,
"line": 32
}
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QOmY9HPlsmFOzebf43sQyVTtZeGCt5g7u1VAFnDeH_dtvit1E54lVQNhoTdPHus4hl3nFjqOkKk8E-SBwzoeEm06SBf9kOIw6aWN0_jSEtQYwpE4mT2wr7wT1AxiZgi2B1r3KZTJthCyUgksTqwe8tzyeOM0ykDExrgGjcxWH01zH-dUxWu2xl5jfb3ixOjobNFMJ1HUZxUAySjNBYBtg-qcI00H-MtFoONKz1_bojgwhgZBx6IEq-Zw4xGUhIbBIK4ueEM7nlWpoeyYeR7KqspRD21LAaNWTDRe_tlLPv-kY2DXOfHwi6m6vB-O55jiaWRUYvnxLI8YTK_mPqahIKwwMIk8X8n_qq6lB41Dp0f5wZt_9kZmBU6FlsCsH_eu5vfhTyMEaonF5gHrc3JsOXNS7MHRZ42WtHpWZdWyTr8cM-ViStYI22uweBR64jQSLqGfsr2D_wFNp9TgEye6rwSwp49HL8KSWjTvPjsVeUPzckUHzBMPj10lDtcoZ8z6xEkOZgrijeVGPkVHdHkARdViQzprKRWIogAk21NqC9cPFPPytGekxUvQLhqIJaXbuQ2Q05cPsVP1NbvnppvOeDktTYYqQpzirvpWysZhuBRlXLvuYqXa0pvieaqOq4W9J5_R10LAzfVfCVKQeyc1KRl9rD1jZ4azbKwp-G3utKoT-K9hPrByMfYAtxhtPuROIWWseaGHUd9GjjUOyCZNwSsl5R0ecC_lzHfuWEm7Y1tA4iKB_cYz-P662kr5usVv-1l-TiKYbTpM4GlCjAOmKLlsnxB1VZ7g3r6iWd8swh1ZU0OZ_YS1zmx_klKbCjMeA8ugHwD6qNLSoHs4XRmhYwQLrHq962AYsoCxpfxZj5TYyPESdVbdfkgAhirMPfmPLxrrk48d6ch8ce0d5oYGEHPnuiaIUYaBsYRw-vlBLzi1MEuRYDgy6bNmT0qocURbEZi4iT7ietiZfg75tFzamkgTDB7zuRdE0_Dum7YhaW49Ja6Ht0EBLhb6M7PPy_GyV0K70A_xaagXE1nbYP8LZk6pGd4TJml9nj0Q3hK31cUCiVpRh2sHoUKgu9mtaNSepNSQ82eM9TcijX6QLIh716-dB_W0QoUTTkiJFxzb1yWVEm_iiutKTdXWNsetay70n09Dbn0_WcUr--Bbm374KoAl8FOxBtBi57Dfkie_vYP-7KisP2obPwgKN7kuh_UJK01JeO16GAKqsTHbQr13bE2UuNMmAaGzeinglpDK9RP7IS8Is6pYLkht5lnQieppVhOqYw-dI7ieqexW5b3w3IejNsWgqqI_4YeOSXop7b3gFRjAjJgz6TC1-fK6Am-o-VgYLECXwM-a7nneOov8awjLIE3JPvoW4LRjh5VDS0guUmNkE0SG95rJGtW6yfzePV-ZRvmugyhXMHtatzglVQJyoz1guWiWYyaOCyr0GYQKBEsYhL3z-2ScSvXcYCTOfvMzA5Dttzm77rtM9K8BzURh3HwBkMpyv7Mr73R6Nw3A8aQ_LHRkBrxHblNit49SAs4Q6DjsKeAXdtI20zb4EyesCiHtkoJT0C1M-QZ0uMfN-2XujsIEAMEcmgV5ozRnpmBNm7qQ62PypMTcCTouT6aQbA2TUb8utNQ8X4Sk_Al_agLGZmE3oi0LsOmeAQnv2QJmnE-q82i6mz-ZOZfl7t-Rf0NSem1wW3Osy8IUMBcoKUgDaT4rXzeBRKEYF76P7qCD0mQ_hzbk1q0_Icg7hzUTNoh9gbUAj_dfO7riAsaB16VYVmJ4AJXJZKuefLNxtEJFgdwZMMbIkjrYkY-EgGR0t2YREeOxdj1OWC-AJc87BNu562nY_crUSVTl-VikHuwkFLo4DcIa8OGfrYXWqInfBAnfFs-bRKtWNxbdSJIDVgrsyOaICPXsdB8fTkJzbMDpDRZCTvYKKVaMimj2yGyBGGlioSUv35dESevoDbwojgizi5j62DfAqvamliYpE-XMUGxjeZf6nwgfhWdjXsZhXKNtUQVexAwWOpn1knM-CL6_Gwh7hhUgULBvUnWQqXlrzxRCg8sLeyqELDGm1SJf65krHG0pP8izy261R-Tg0j6Pe4sqplrvM5LpsQ3fQmnQQGzhb-JoDWMDGXobwOJ8wGF2gcDDmrUhAjzThsd2dNwNlO_lw7GZchyhNlQHxDUT73tL-UoprFhq-e5xByq-9BTL5yIq_QhgpzkLdrtS7QOm2GYzlRTktO1lwZ55kMip7ymL8Uku705yM6W5hpuCVXDRHl_6zbx8rjQ03lvvtK8jnKfZX6i3lOBsIrBPjbz2PBkgRM8W8i6gR9FLF0SqnIoyI4Li4Ugis7AvX9Y9XAewawJ1D8ObsUWF1huKQpMYMDWMv1NbUJLBwSQY5qazYp1xUH_b8af130BJrj_iOnfwnHd--Eu2fTXwvt1WBkim4bsS4oD4fSGNqiQ5hNe2anh_wU6-v-inpw47Tq8c1OubSakjo0sDZEQZFLU7dGmA6H0lGnq28Fjx0yA3TQe6DXu0N7XV_uxRs0rOgwZkCNP9NUS2xKkWxOnf9fVLjQ9XrZVo9xgXVfqkTDOLGDSP8FRDYFsNm6TKp2MbWQGdC7nT5SZ0eRcOazUyBUyLbt0ASq2Qw-cYNdpHfj_JZAlfi5acU1WVwliGzpXKLyQOPJzNyI4CwIemibEOEOJdr1cLkeyDJHIckzTSJ3NgMT-GuU_OAsr_vTYZDA27fQcvvTr7OPJp3e0db2XA26wn7cxYGhww1SZxovDDsIcwXFmIlLpfLTgQafQ9k4sbf9Z48oFhll4GgJdC1pEEYsyeHyOeIMfLpg6b4pLCMy1WqrdAv8wDx8M2VG1u0-VTlnpdlpipXuGeIzpzTvOvparu2Y3IDGEmEJMGE3WETsv_AL2YPKd2_0C820cNCLnFBrXJuH5-DCt2HkWu2jIHrPGiYmP8-jndPYpYITzHEWas5Fpy3lC0o8SOp420WtzTWtDDYHGF7BqsCYSaV06nvRhqmdWIa_sGlk7biculQLPpJDpFWfBKgVvLw0Vs9X6w-uva0eETAQqyqUm0vIAEhLLU7lQ_5HY3PP7FXgk4hmuTC7hV-HMo7_OH0Nevc2uDsAm3XWFtUMnfn68ouVYIXGWpbRgn-ktC7gvbJjwAwTAq-SwyeSFoxROplfjwlqcPzXDFXm2g4HranV0DI_Fy3pRJ1y9qGXTvqkffRY9pdMBJDDP7sA0HI2ArYCy6u8sV0DMCkWVDCpRC0MYPuz4HATfhmHO4boNwduc0Fv7lv71Rkv0JKCzaxs6VbbGpb2DalgsDps1bPDD-JReEsKQ2yI92MWAFn7-2HnPrHs_KVp-UJ2w6v-Q8JCQz88tc2j8TXduCeoCMuMGV6_quRC9gDnY73to7bHUy2eyYywEeRYvTXGQpH7XHHKTT7cZIFP7GKJ-e8idY="
}
]
},
{
"iteration": 6,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "partial",
"goals_remaining": 1,
"timed_out": false,
"banned_tokens": [],
"proof_code": "unfold RegularPrimeConjecture\nhave hreg : \u2200 p : \u2115, \u2200 hp : Nat.Prime p, @IsRegularPrime p \u27e8hp\u27e9 := by\n intro p hp\n simp [IsRegularPrime]\n\nhave hprimeInf : {p : \u2115 | Nat.Prime p}.Infinite := by\n simpa using Nat.infinite_setOf_prime\n\nhave hEq : regularPrimes = {p : \u2115 | Nat.Prime p} := by\n ext p\n constructor\n \u00b7 intro hp\n rcases hp with \u27e8hpp, _\u27e9\n exact hpp\n \u00b7 intro hp\n exact \u27e8hp, hreg p hp\u27e9\n\nsimpa [hEq] using hprimeInf",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772028033_7005.lean:35:69: error: unsolved goals\np : \u2115\nhp : Nat.Prime p\n\u22a2 p.Coprime (Fintype.card (ClassGroup (\ud835\udcde (CyclotomicField p \u211a))))\n'RegularPrimes.regularprime_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 38.943,
"tactic_replay": {
"tactic_count": 15,
"successful_tactics": 1,
"first_failure_step": 2,
"goal_trajectory": [
1,
1,
1,
1,
1,
1,
1,
1,
1,
2,
1,
1,
1,
1,
1
],
"max_goals": 2,
"final_goal_count": 1,
"goals_opened": 1,
"goals_closed": 1
},
"lsp_feedback": {
"diagnostics": [
{
"fullRange": {
"end": {
"character": 25,
"line": 36
},
"start": {
"character": 69,
"line": 34
}
},
"leanTags": [
1
],
"message": "unsolved goals\np : \u2115\nhp : Nat.Prime p\n\u22a2 p.Coprime (Fintype.card (ClassGroup (\ud835\udcde (CyclotomicField p \u211a))))",
"range": {
"end": {
"character": 0,
"line": 35
},
"start": {
"character": 69,
"line": 34
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 52
},
"start": {
"character": 0,
"line": 52
}
},
"message": "'RegularPrimes.regularprime_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
"end": {
"character": 6,
"line": 52
},
"start": {
"character": 0,
"line": 52
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 34,
"error_char": 69,
"error_msg": "unsolved goals\np : \u2115\nhp : Nat.Prime p\n\u22a2 p.Coprime (Fintype.card (ClassGroup (\ud835\udcde (CyclotomicField p \u211a))))",
"goal_before_error": "```lean\n\u22a2 regularPrimes.Infinite\n```",
"full_timeline": [
{
"line": 33,
"code": "unfold RegularPrimeConjecture",
"goal": "```lean\n\u22a2 RegularPrimeConjecture\n```",
"goal_count": 1,
"ast_complexity": 36
},
{
"line": 34,
"code": "have hreg : \u2200 p : \u2115, \u2200 hp : Nat.Prime p, @IsRegularPrime p \u27e8hp\u27e9 := by",
"goal": "```lean\n\u22a2 regularPrimes.Infinite\n```",
"goal_count": 1,
"ast_complexity": 36
},
{
"line": 35,
"code": "intro p hp",
"goal": "```lean\n\u22a2 \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\n```",
"goal_count": 1,
"ast_complexity": 60
},
{
"line": 36,
"code": "simp [IsRegularPrime]",
"goal": "```lean\np : \u2115\nhp : Nat.Prime p\n\u22a2 IsRegularPrime p\n```",
"goal_count": 1,
"ast_complexity": 53
},
{
"line": 38,
"code": "have hprimeInf : {p : \u2115 | Nat.Prime p}.Infinite := by",
"goal": "```lean\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\n\u22a2 regularPrimes.Infinite\n```",
"goal_count": 1,
"ast_complexity": 90
},
{
"line": 39,
"code": "simpa using Nat.infinite_setOf_prime",
"goal": "```lean\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\n\u22a2 {p | Nat.Prime p}.Infinite\n```",
"goal_count": 1,
"ast_complexity": 94
},
{
"line": 41,
"code": "have hEq : regularPrimes = {p : \u2115 | Nat.Prime p} := by",
"goal": "```lean\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\nhprimeInf : {p | Nat.Prime p}.Infinite\n\u22a2 regularPrimes.Infinite\n```",
"goal_count": 1,
"ast_complexity": 129
},
{
"line": 42,
"code": "ext p",
"goal": "```lean\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\nhprimeInf : {p | Nat.Prime p}.Infinite\n\u22a2 regularPrimes = {p | Nat.Prime p}\n```",
"goal_count": 1,
"ast_complexity": 140
},
{
"line": 43,
"code": "constructor",
"goal": "```lean\ncase h\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\nhprimeInf : {p | Nat.Prime p}.Infinite\np : \u2115\n\u22a2 p \u2208 regularPrimes \u2194 p \u2208 {p | Nat.Prime p}\n```",
"goal_count": 1,
"ast_complexity": 161
},
{
"line": 44,
"code": "\u00b7 intro hp",
"goal": "```lean\ncase h.mp\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\nhprimeInf : {p | Nat.Prime p}.Infinite\np : \u2115\n\u22a2 p \u2208 regularPrimes \u2192 p \u2208 {p | Nat.Prime p}\n```\n---\n```lean\ncase h.mpr\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\nhprimeInf : {p | Nat.Prime p}.Infinite\np : \u2115\n\u22a2 p \u2208 {p | Nat.Prime p} \u2192 p \u2208 regularPrimes\n```",
"goal_count": 2,
"ast_complexity": 334
},
{
"line": 45,
"code": "rcases hp with \u27e8hpp, _\u27e9",
"goal": "```lean\ncase h.mp\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\nhprimeInf : {p | Nat.Prime p}.Infinite\np : \u2115\nhp : p \u2208 regularPrimes\n\u22a2 p \u2208 {p | Nat.Prime p}\n```",
"goal_count": 1,
"ast_complexity": 167
},
{
"line": 46,
"code": "exact hpp",
"goal": "```lean\ncase h.mp\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\nhprimeInf : {p | Nat.Prime p}.Infinite\np : \u2115\nhpp : Nat.Prime p\nh\u271d : IsRegularPrime p\n\u22a2 p \u2208 {p | Nat.Prime p}\n```",
"goal_count": 1,
"ast_complexity": 184
},
{
"line": 47,
"code": "\u00b7 intro hp",
"goal": "```lean\ncase h.mpr\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\nhprimeInf : {p | Nat.Prime p}.Infinite\np : \u2115\n\u22a2 p \u2208 {p | Nat.Prime p} \u2192 p \u2208 regularPrimes\n```",
"goal_count": 1,
"ast_complexity": 165
},
{
"line": 48,
"code": "exact \u27e8hp, hreg p hp\u27e9",
"goal": "```lean\ncase h.mpr\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\nhprimeInf : {p | Nat.Prime p}.Infinite\np : \u2115\nhp : p \u2208 {p | Nat.Prime p}\n\u22a2 p \u2208 regularPrimes\n```",
"goal_count": 1,
"ast_complexity": 168
},
{
"line": 50,
"code": "simpa [hEq] using hprimeInf",
"goal": "```lean\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\nhprimeInf : {p | Nat.Prime p}.Infinite\nhEq : regularPrimes = {p | Nat.Prime p}\n\u22a2 regularPrimes.Infinite\n```",
"goal_count": 1,
"ast_complexity": 169
}
],
"tactic_count": 15,
"successful_tactics": 1,
"first_failure_step": 2,
"cps": 6.666666666666667,
"cps_peak_step": 1,
"error_code": "have hreg : \u2200 p : \u2115, \u2200 hp : Nat.Prime p, @IsRegularPrime p \u27e8hp\u27e9 := by"
}
},
{
"iteration": 7,
"event": "search_mathlib",
"query": "IsRegularPrime",
"results_count": 0
},
{
"iteration": 8,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
"response_id": "gen-1772028076-xH1pTqyEKHjsR7XB7BCj",
"api_elapsed_s": 28.508,
"usage": {
"prompt_tokens": 12600,
"completion_tokens": 1316,
"total_tokens": 13916,
"reasoning_tokens": 1072,
"cost_usd": 0.0209188
},
"openrouter_retries": 0,
"response_text_len": 0,
"used_tool_call": true,
"reasoning_text": "**Exploring classical approaches for infinitude**",
"reasoning_details": [
{
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.summary",
"summary": "**Exploring classical approaches for infinitude**"
},
{
"id": "rs_0fb1ae943183039a01699f00ac76688194a350c1a74ea6b612",
"format": "openai-responses-v1",
"index": 0,
"type": "reasoning.encrypted",
"data": 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oYPewf_4B535S6rceP7WK_shVI7IG0P56OwaNf9zjLACtTTFS0aJMIkT3J3fcUBdAGhtdIaeQGlGvsPpRXv9iSJob-19hcP_9EWiprYl1Re-a11XV0xK-3sV9KNR4VRz7rchZ0YlZfmLGs6z4oQt3dvOjUGVCeI3G1g3R9jMqSr9DNcFHd_og3Focu-ouSu8E4blHPbEtQ_dzH7Hr3rDydL13vIQHYRg7MgRCmNq850wQP1Ai2G5zr5fMKaviSNmPn1gj8lWVGCmPlKNHUtMSvJQE7V6XskhisAbN_uwrUySYd_60gb1pcFTgbzfP60pEfGef6XG4w6S-5R7h7NUfNsMd3MM8PSXlTpX0M1h6dxhCFW7ATtpFLmhiWB5rKF4ShlsawsRI4Vpobnh6qvvWa-VM5nVziYEjKzZwH7u9xrybrnmYKWDi-WNIy9kfW9qlctjZSRuW0sFOjbHl444rxBUXllq2fu0RGVat-f9UNnw9vRsYtpO7B0lPT1dNzQcGBMDgNSiBPAffYg8-nklLwhJew_qZhNJZwCSJ4HKAs6Oc-pmkbDtLblIbel_MaZW2qgQsBv28kEKhGU8TjMLaAromuamBDJiX_Yx4RbQzHO6HTefWV12-KlhJ6qnkB0Fj1jaUKP7MDewm_65HeXn2Zxmzep0rFDPMTcwe2j6I9n8reLVdK08wtINn4R1ljc6vuj_KoEXUx1RvmFKlxjMUUfp5N4xwhCro4bLS4uo1KlHKLWgKZ9h9FE-Wf_-aELhqrapACLzfuNX9HT3vQcuDFTfYyCxnw4-pj0UEIDFJx50ZskzSdMEGDrQvUtgapGFDC9hxwd7VTMcQufYciJI1kKh3GkI7j49NyxVHkkbdJ6u8r7QeljWVxMUvbm9qxFYG10Iu97UL9qeRri2INsEvIh1DR_Ooc6-c9pHagwBZ4DqMtbKqUKagfaMypJTgd5ZEyO_qVVEjCtM26o_4uvqgRM8qHKGYePKTG-jRrYt6IK44d_qG_C0giV6Svd_WhgG_xL4Gcr2AZEsDJjK2fiuzputvGoEokO9PkXbpti0T_kHK3LksipV2X6-NcZP2j3Hqf-oQfUeczSimWcoHvUyYqLt8DzlncFPsAAEUZcs3K36mAbPm-2bAe3jP0PqzjWD9cZkb0OhM6wLpZ6ImFvkv0XlBYp8ake3uUYcJ76BKm0EPdog23ILPdH1opgbn4UmMOmkUI48LWzKc4DgnvpfGskqZXEO4fQKheaTgyf-k-CrmxA0f561HmZgajDb6Yg1tdlCmOWO2I_BhMg1GrxKpbqmm-XreoB_E9J4yHlwrD3NaMIIReDxqN9U4TVoe2Y3zzdBJLcSPsBLU3DMkfT255VeTnBBJ_8KotfoHgCntguk7GH8JNqHdlJx2TJsXGYtn2JipBOqpmCHTYIiWrNKB6LaM2GxTmVoSmuwtmm9ASDk_HVt6wVkQximPs4C003MD8amKa7KIyDZLcatIxJtUOOpJlIzfT_TQcAAUfKe6qqeoyn6vHwepYVnI2WP6tm1eUamAc7XgJy9h1sBok6gXUYu0XcTnTydNroSb_cSg7jkLvFef-AUt8rcOQSLtkks46qUkKBduoKkeuOUYkIQJ9fk-Wrrwz7DbFaF1VFDL448ieXqzUi8U_FR7NE8EnziezeINDdhQXu7l4-5yDeeBvVDtIZyp3_gdVcXd3T2_ltWzIqc32zWvfT_Br7I-FaJFyIje__C7fZSDQQBsy9dxI91jhQL9H1K4Fb_C9snfY62gjvYwo1T6cPEHgP2BV5f0kkje"
}
]
},
{
"iteration": 8,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "unfold RegularPrimeConjecture\nhave hreg : \u2200 p : \u2115, \u2200 hp : Nat.Prime p, @IsRegularPrime p \u27e8hp\u27e9 := by\n intro p hp\n unfold IsRegularPrime\n have hc : Fintype.card (ClassGroup (\ud835\udcde (CyclotomicField p \u211a))) = 1 := by\n simp\n rw [hc]\n simpa using hp.coprime_one\n\nhave hprimeInf : {p : \u2115 | Nat.Prime p}.Infinite := by\n simpa using Nat.infinite_setOf_prime\n\nhave hEq : regularPrimes = {p : \u2115 | Nat.Prime p} := by\n ext p\n constructor\n \u00b7 intro hp\n rcases hp with \u27e8hpp, _\u27e9\n exact hpp\n \u00b7 intro hp\n exact \u27e8hp, hreg p hp\u27e9\n\nsimpa [hEq] using hprimeInf",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772028104_8095.lean:39:6: error: `simp` made no progress\n/app/asi_portal/verifier/quick_1772028104_8095.lean:41:4: error: try 'simp' instead of 'simpa'\n\nNote: This linter can be disabled with `set_option linter.unnecessarySimpa false`\n'RegularPrimes.regularprime_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 36.363,
"tactic_replay": {
"tactic_count": 19,
"successful_tactics": 5,
"first_failure_step": 6,
"goal_trajectory": [
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
2,
1,
1,
1,
1,
1
],
"max_goals": 2,
"final_goal_count": 1,
"goals_opened": 1,
"goals_closed": 1
},
"lsp_feedback": {
"diagnostics": [
{
"fullRange": {
"end": {
"character": 10,
"line": 38
},
"start": {
"character": 6,
"line": 38
}
},
"message": "`simp` made no progress",
"range": {
"end": {
"character": 10,
"line": 38
},
"start": {
"character": 6,
"line": 38
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 30,
"line": 40
},
"start": {
"character": 4,
"line": 40
}
},
"message": "try 'simp' instead of 'simpa'\n\nNote: This linter can be disabled with `set_option linter.unnecessarySimpa false`",
"range": {
"end": {
"character": 30,
"line": 40
},
"start": {
"character": 4,
"line": 40
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 56
},
"start": {
"character": 0,
"line": 56
}
},
"message": "'RegularPrimes.regularprime_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
"end": {
"character": 6,
"line": 56
},
"start": {
"character": 0,
"line": 56
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 38,
"error_char": 6,
"error_msg": "`simp` made no progress",
"goal_before_error": "```lean\np : \u2115\nhp : Nat.Prime p\n\u22a2 Fintype.card (ClassGroup (\ud835\udcde (CyclotomicField p \u211a))) = 1\n```",
"full_timeline": [
{
"line": 33,
"code": "unfold RegularPrimeConjecture",
"goal": "```lean\n\u22a2 RegularPrimeConjecture\n```",
"goal_count": 1,
"ast_complexity": 36
},
{
"line": 34,
"code": "have hreg : \u2200 p : \u2115, \u2200 hp : Nat.Prime p, @IsRegularPrime p \u27e8hp\u27e9 := by",
"goal": "```lean\n\u22a2 regularPrimes.Infinite\n```",
"goal_count": 1,
"ast_complexity": 36
},
{
"line": 35,
"code": "intro p hp",
"goal": "```lean\n\u22a2 \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\n```",
"goal_count": 1,
"ast_complexity": 60
},
{
"line": 36,
"code": "unfold IsRegularPrime",
"goal": "```lean\np : \u2115\nhp : Nat.Prime p\n\u22a2 IsRegularPrime p\n```",
"goal_count": 1,
"ast_complexity": 53
},
{
"line": 37,
"code": "have hc : Fintype.card (ClassGroup (\ud835\udcde (CyclotomicField p \u211a))) = 1 := by",
"goal": "```lean\np : \u2115\nhp : Nat.Prime p\n\u22a2 p.Coprime (Fintype.card (ClassGroup (\ud835\udcde (CyclotomicField p \u211a))))\n```",
"goal_count": 1,
"ast_complexity": 100
},
{
"line": 38,
"code": "simp",
"goal": "```lean\np : \u2115\nhp : Nat.Prime p\n\u22a2 Fintype.card (ClassGroup (\ud835\udcde (CyclotomicField p \u211a))) = 1\n```",
"goal_count": 1,
"ast_complexity": 92
},
{
"line": 39,
"code": "rw [hc]",
"goal": "```lean\np : \u2115\nhp : Nat.Prime p\nhc : Fintype.card (ClassGroup (\ud835\udcde (CyclotomicField p \u211a))) = 1\n\u22a2 p.Coprime (Fintype.card (ClassGroup (\ud835\udcde (CyclotomicField p \u211a))))\n```",
"goal_count": 1,
"ast_complexity": 161
},
{
"line": 40,
"code": "simpa using hp.coprime_one",
"goal": "```lean\np : \u2115\nhp : Nat.Prime p\nhc : Fintype.card (ClassGroup (\ud835\udcde (CyclotomicField p \u211a))) = 1\n\u22a2 p.Coprime 1\n```",
"goal_count": 1,
"ast_complexity": 109
},
{
"line": 42,
"code": "have hprimeInf : {p : \u2115 | Nat.Prime p}.Infinite := by",
"goal": "```lean\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\n\u22a2 regularPrimes.Infinite\n```",
"goal_count": 1,
"ast_complexity": 90
},
{
"line": 43,
"code": "simpa using Nat.infinite_setOf_prime",
"goal": "```lean\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\n\u22a2 {p | Nat.Prime p}.Infinite\n```",
"goal_count": 1,
"ast_complexity": 94
},
{
"line": 45,
"code": "have hEq : regularPrimes = {p : \u2115 | Nat.Prime p} := by",
"goal": "```lean\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\nhprimeInf : {p | Nat.Prime p}.Infinite\n\u22a2 regularPrimes.Infinite\n```",
"goal_count": 1,
"ast_complexity": 129
},
{
"line": 46,
"code": "ext p",
"goal": "```lean\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\nhprimeInf : {p | Nat.Prime p}.Infinite\n\u22a2 regularPrimes = {p | Nat.Prime p}\n```",
"goal_count": 1,
"ast_complexity": 140
},
{
"line": 47,
"code": "constructor",
"goal": "```lean\ncase h\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\nhprimeInf : {p | Nat.Prime p}.Infinite\np : \u2115\n\u22a2 p \u2208 regularPrimes \u2194 p \u2208 {p | Nat.Prime p}\n```",
"goal_count": 1,
"ast_complexity": 161
},
{
"line": 48,
"code": "\u00b7 intro hp",
"goal": "```lean\ncase h.mp\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\nhprimeInf : {p | Nat.Prime p}.Infinite\np : \u2115\n\u22a2 p \u2208 regularPrimes \u2192 p \u2208 {p | Nat.Prime p}\n```\n---\n```lean\ncase h.mpr\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\nhprimeInf : {p | Nat.Prime p}.Infinite\np : \u2115\n\u22a2 p \u2208 {p | Nat.Prime p} \u2192 p \u2208 regularPrimes\n```",
"goal_count": 2,
"ast_complexity": 334
},
{
"line": 49,
"code": "rcases hp with \u27e8hpp, _\u27e9",
"goal": "```lean\ncase h.mp\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\nhprimeInf : {p | Nat.Prime p}.Infinite\np : \u2115\nhp : p \u2208 regularPrimes\n\u22a2 p \u2208 {p | Nat.Prime p}\n```",
"goal_count": 1,
"ast_complexity": 167
},
{
"line": 50,
"code": "exact hpp",
"goal": "```lean\ncase h.mp\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\nhprimeInf : {p | Nat.Prime p}.Infinite\np : \u2115\nhpp : Nat.Prime p\nh\u271d : IsRegularPrime p\n\u22a2 p \u2208 {p | Nat.Prime p}\n```",
"goal_count": 1,
"ast_complexity": 184
},
{
"line": 51,
"code": "\u00b7 intro hp",
"goal": "```lean\ncase h.mpr\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\nhprimeInf : {p | Nat.Prime p}.Infinite\np : \u2115\n\u22a2 p \u2208 {p | Nat.Prime p} \u2192 p \u2208 regularPrimes\n```",
"goal_count": 1,
"ast_complexity": 165
},
{
"line": 52,
"code": "exact \u27e8hp, hreg p hp\u27e9",
"goal": "```lean\ncase h.mpr\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\nhprimeInf : {p | Nat.Prime p}.Infinite\np : \u2115\nhp : p \u2208 {p | Nat.Prime p}\n\u22a2 p \u2208 regularPrimes\n```",
"goal_count": 1,
"ast_complexity": 168
},
{
"line": 54,
"code": "simpa [hEq] using hprimeInf",
"goal": "```lean\nhreg : \u2200 (p : \u2115) (hp : Nat.Prime p), IsRegularPrime p\nhprimeInf : {p | Nat.Prime p}.Infinite\nhEq : regularPrimes = {p | Nat.Prime p}\n\u22a2 regularPrimes.Infinite\n```",
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|
| Gemini 3.1 Pro Agent v8 | Failed | -- | Feb 24, 2026 |
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|
|
| Gemini 3 Flash Agent v8 | Failed | -- | Feb 17, 2026 |
unfold RegularPrimeConjecture regularPrimes
constructor
· intro h
have : Set.Finite {p | ∃ (hp : Nat.Prime p), @IsRegularPrime p ⟨hp⟩} := h
rw [Set.finite_def] at this
obtain ⟨s, hs⟩ := this
refine ?_
· refine ?_
|
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"proof_code": "unfold RegularPrimeConjecture regularPrimes\nintro h\nlet S := { p | \u2203 (hp : Nat.Prime p), @IsRegularPrime p \u27e8hp\u27e9 }\nhave h_inf : Set.Infinite S := h\nobtain \u27e8p, hp_prime, hp_reg\u27e9 : \u2203 p, Nat.Prime p \u2227 IsRegularPrime p := by\n obtain \u27e8p, hp_in\u27e9 := h_inf.nonempty\n use p\n exact hp_in\nrefine ?_\nintro f\nlet P := { p : \u2115 | p.Prime }\nhave : S \u2286 P := by\n intro x hx\n rcases hx with \u27e8hp, _\u27e9\n exact hp\nexact Set.Infinite.mono this ?_\nadmit",
"compiler_output_preview": "\nBanned tokens found: admit"
},
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"proof_code": "unfold RegularPrimeConjecture regularPrimes\nconstructor\n\u00b7 intro h\n have : Set.Finite {p | \u2203 (hp : Nat.Prime p), @IsRegularPrime p \u27e8hp\u27e9} := h\n rw [Set.finite_def] at this\n obtain \u27e8s, hs\u27e9 := this\n refine ?_\n\u00b7 refine ?_",
"compiler_output_preview": "[path]:35:2: error: tactic 'constructor' failed, target is not an inductive datatype\n\u22a2 {p | \u2203 (hp : Nat.Prime p), IsRegularPrime p}.Infinite\n'RegularPrimes.regularprime_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
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"proof_code": "unfold RegularPrimeConjecture regularPrimes\nrefine Set.infinite_of_not_finite ?_\nintro (h : Set.Finite {p | \u2203 (hp : Nat.Prime p), IsRegularPrime p})\nhave h_primes : Set.Infinite {p | Nat.Prime p} := Nat.Primes.infinite\nlet odd_primes := {p | Nat.Prime p \u2227 p \u2260 2}\nhave h_odd : Set.Infinite odd_primes := by\n have : {p | Nat.Prime p} \\ {2} = odd_primes := by\n ext p; simp; aesop\n apply Set.Infinite.diff h_primes (Set.finite_singleton 2)\nlet irregular := {p | \u2203 (hp : Nat.Prime p), \u00ac IsRegularPrime p}\nhave h_union : {p | Nat.Prime p} = {p | \u2203 hp, IsRegularPrime p} \u222a irregular := by\n ext p; constructor\n \u00b7 intro hp; by_cases hr : IsRegularPrime p\n \u00b7 left; exact \u27e8hp, hr\u27e9\n \u00b7 right; exact \u27e8hp, hr\u27e9\n \u00b7 rintro (\u27e8hp, _\u27e9|\u27e8hp, _\u27e9); exact hp\nhave h_irr : Set.Infinite irregular := ?_\nrefine ?_",
"compiler_output_preview": "[path]:35:9: error: unknown constant 'Set.infinite_of_not_finite'\n[path]:36:2: error: no goals to be solved\n'RegularPrimes.regularprime_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
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"proof_code": "unfold RegularPrimeConjecture regularPrimes\nby_contra h_fin\nreplace h_fin : {p | \u2203 (hp : Nat.Prime p), IsRegularPrime p}.Finite := h_fin\nhave h_primes : {p | Nat.Prime p}.Infinite := Nat.Primes.infinite\nlet irregular := {p | \u2203 (hp : Nat.Prime p), \u00ac IsRegularPrime p}\nhave h_union : {p | Nat.Prime p} = {p | \u2203 hp, IsRegularPrime p} \u222a irregular := by\n ext p; constructor\n \u00b7 intro hp\n by_cases hr : IsRegularPrime p\n \u00b7 left; use hp\n \u00b7 right; use hp\n \u00b7 rintro (\u27e8hp, _\u27e9 | \u27e8hp, _\u27e9)\n exact hp\nhave h_irr_inf : irregular.Infinite := by\n refine Set.Infinite.of_union_left ?_ h_fin\n rw [\u2190 h_union]\n exact h_primes\nexact h_irr_inf.ne_finite ?_",
"compiler_output_preview": "[path]:36:45: error: failed to synthesize\n Fact (Nat.Prime p)\n\nHint: Additional diagnostic information may be available using the `set_option diagnostics true` command.\n[path]:37:48: error: type mismatch\n Nat.Primes.infinite\nhas type\n Infinite Nat.Prime\n...<truncated>"
},
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"proof_code": "unfold RegularPrimeConjecture regularPrimes\nlet S := { p | \u2203 (hp : Nat.Prime p), @IsRegularPrime p \u27e8hp\u27e9 }\nshow S.Infinite\nlet irregular := { p | \u2203 (hp : Nat.Prime p), \u00ac @IsRegularPrime p \u27e8hp\u27e9 }\nhave h_total : {p | Nat.Prime p} = S \u222a irregular := by\n ext p; constructor\n \u00b7 intro hp\n by_cases hr : @IsRegularPrime p \u27e8hp\u27e9\n \u00b7 left; exact \u27e8hp, hr\u27e9\n \u00b7 right; exact \u27e8hp, hr\u27e9\n \u00b7 rintro (\u27e8hp, _\u27e9 | \u27e8hp, _\u27e9)\n exact hp\nhave h_prime_inf : Set.Infinite {p | Nat.Prime p} := by\n rw [Set.infinite_coe_iff]\n exact Nat.Primes.infinite\nrefine @Set.Infinite.of_union_left _ S irregular ?_\nrw [\u2190 h_total]\nexact h_prime_inf",
"compiler_output_preview": "[path]:44:4: error: unsolved goals\ncase h.mpr.inr.intro\nS : Set \u2115 := {p | \u2203 (hp : Nat.Prime p), IsRegularPrime p}\nirregular : Set \u2115 := {p | \u2203 (hp : Nat.Prime p), \u00acIsRegularPrime p}\np : \u2115\nhp : Nat.Prime p\nh\u271d : \u00acIsRegularPrime p\n\u22a2 p \u2208 {p | Nat.Prime p}\n[path]
},
{
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"proof_code": "unfold RegularPrimeConjecture regularPrimes\nlet S := { p | \u2203 (hp : Nat.Prime p), @IsRegularPrime p \u27e8hp\u27e9 }\nlet I := { p | \u2203 (hp : Nat.Prime p), \u00ac @IsRegularPrime p \u27e8hp\u27e9 }\nlet P := { p | Nat.Prime p }\nhave hP : P = S \u222a I := by\n ext p\n simp only [Set.mem_setOf_eq, Set.mem_union]\n constructor\n \u00b7 intro hp\n by_cases h : @IsRegularPrime p \u27e8hp\u27e9\n \u00b7 left; use hp\n \u00b7 right; use hp\n \u00b7 rintro (\u27e8hp, _\u27e9 | \u27e8hp, _\u27e9)\n exact hp\nhave hP_inf : P.Infinite := by\n refine Set.infinite_of_not_finite ?_\n intro hf\n have : Finite {p // Nat.Prime p} := Set.finite_coe_iff.mpr hf\n exact (not_finite_index_primes)\nrw [hP] at hP_inf\nby_cases hS : S.Infinite\n\u00b7 exact hS\n\u00b7 have hS_fin : S.Finite := Set.not_infinite.mp hS\n have hI_inf : I.Infinite := by\n apply Set.Infinite.of_union_left (s := I) (t := S)\n \u00b7 rwa [Set.union_comm]\n \u00b7 exact hS_fin\n refine ?_",
"compiler_output_preview": "[path]:46:4: error: unsolved goals\ncase h.mpr.inr.intro\nS : Set \u2115 := {p | \u2203 (hp : Nat.Prime p), IsRegularPrime p}\nI : Set \u2115 := {p | \u2203 (hp : Nat.Prime p), \u00acIsRegularPrime p}\nP : Set \u2115 := {p | Nat.Prime p}\np : \u2115\nhp : Nat.Prime p\nh\u271d : \u00acIsRegularPrime p\n\u22a2 p \u2208 P\n[path]
},
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"proof_code": "unfold RegularPrimeConjecture regularPrimes\nlet S := { p | \u2203 (hp : Nat.Prime p), @IsRegularPrime p \u27e8hp\u27e9 }\nlet I := { p | \u2203 (hp : Nat.Prime p), \u00ac @IsRegularPrime p \u27e8hp\u27e9 }\nlet P := { p | Nat.Prime p }\nhave hP : P = S \u222a I := by\n ext p\n constructor\n \u00b7 intro hp\n by_cases h : @IsRegularPrime p \u27e8hp\u27e9\n \u00b7 left; exact \u27e8hp, h\u27e9\n \u00b7 right; exact \u27e8hp, h\u27e9\n \u00b7 rintro (\u27e8hp, _\u27e9 | \u27e8hp, _\u27e9)\n exact hp\nhave hP_inf : P.Infinite := by\n let f : P \u2192 \u2115 := Subtype.val\n have : Function.Injective f := Subtype.val_injective\n refine Set.infinite_coe_iff.mp ?_\n exact Nat.Primes.infinite\nrw [hP] at hP_inf\nby_cases hS : S.Infinite\n\u00b7 exact hS\n\u00b7 have hS_fin : S.Finite := Set.not_infinite.mp hS\n have hI_inf : I.Infinite := by\n apply Set.Infinite.of_union\n rw [Set.union_comm] at hP_inf\n exact hP_inf\n exact hS_fin\n suffices : False\n \u00b7 contradiction\n refine ?_",
"compiler_output_preview": "[path]:45:4: error: unsolved goals\ncase h.mpr.inr.intro\nS : Set \u2115 := {p | \u2203 (hp : Nat.Prime p), IsRegularPrime p}\nI : Set \u2115 := {p | \u2203 (hp : Nat.Prime p), \u00acIsRegularPrime p}\nP : Set \u2115 := {p | Nat.Prime p}\np : \u2115\nhp : Nat.Prime p\nh\u271d : \u00acIsRegularPrime p\n\u22a2 p \u2208 P\n[path]
},
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"proof_code": "unfold RegularPrimeConjecture regularPrimes\nlet S := { p | \u2203 (hp : Nat.Prime p), @IsRegularPrime p \u27e8hp\u27e9 }\nlet I := { p | \u2203 (hp : Nat.Prime p), \u00ac @IsRegularPrime p \u27e8hp\u27e9 }\nlet P := { p | Nat.Prime p }\nhave hP : P = S \u222a I := by\n ext p; constructor\n \u00b7 intro hp\n by_cases h : @IsRegularPrime p \u27e8hp\u27e9\n \u00b7 left; exact \u27e8hp, h\u27e9\n \u00b7 right; exact \u27e8hp, h\u27e9\n \u00b7 rintro (\u27e8hp, _\u27e9 | \u27e8hp, _\u27e9)\n exact hp\nhave hP_inf : P.Infinite := by\n letI : Infinite {p // Nat.Prime p} := Nat.Primes.infinite\n exact Set.infinite_coe_iff.mp inferInstance\nrw [hP] at hP_inf\nby_cases hS : S.Infinite\n\u00b7 exact hS\n\u00b7 have hS_fin : S.Finite := Set.not_infinite.mp hS\n have hI_inf : I.Infinite := by\n apply Set.Finite.infinite_complement hS_fin\n refine ?_\n refine ?_",
"compiler_output_preview": "[path]:44:4: error: unsolved goals\ncase h.mpr.inr.intro\nS : Set \u2115 := {p | \u2203 (hp : Nat.Prime p), IsRegularPrime p}\nI : Set \u2115 := {p | \u2203 (hp : Nat.Prime p), \u00acIsRegularPrime p}\nP : Set \u2115 := {p | Nat.Prime p}\np : \u2115\nhp : Nat.Prime p\nh\u271d : \u00acIsRegularPrime p\n\u22a2 p \u2208 P\n[path]
}
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"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__RegularPrimes__regularprime_conjecture",
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},
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"status": "failed",
"stderr": "",
"stdout": "[path]:35:2: error: tactic 'constructor' failed, target is not an inductive datatype\n\u22a2 {p | \u2203 (hp : Nat.Prime p), IsRegularPrime p}.Infinite\n'RegularPrimes.regularprime_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
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}
|
| Gemini 3 Flash Preview | Failed | -- | Feb 14, 2026 |
unfold RegularPrimeConjecture
unfold regularPrimes
apply Set.Infinite.mono (s := {p | Nat.Prime p ∧ p > 2})
· intro p hp
simp at hp
use hp.1
unfold IsRegularPrime
simp
contrapose!
intro h
have : p ∣ Fintype.card (ClassGroup (𝓞 (CyclotomicField p ℚ))) := h
trivial
· apply Set.infinite_of_forall_exists_gt
intro n
obtain ⟨p, hp1, hp2⟩ := Nat.exists_infinite_primes (n + 1)
use p
refine ⟨⟨hp1, ?_⟩, hp2⟩
exact Nat.Prime.one_lt hp1
|
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"stdout": "[path]:41:4: error: simp made no progress\n[path]:50:13: error: Application type mismatch: In the application\n And.intro hp1\nthe argument\n hp1\nhas type\n n + 1 \u2264 p : Prop\nbut is expected to have type\n Nat.Prime p : Prop\n[path]:51:27: error: Application type mismatch: In the application\n Nat.Prime.one_lt hp1\nthe argument\n hp1\nhas type\n n + 1 \u2264 p : Prop\nbut is expected to have type\n Nat.Prime ?m.3543 : Prop\n",
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|