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idempotent_conjecture
Specification
**The idempotent conjecture** If `G` is torsion-free, then `K[G]` has no non-trivial idempotents.
Lean 4 Statement
theorem idempotent_conjecture (a : MonoidAlgebra K G) (h : IsIdempotentElem a) :
a = 0 ∨ a = 1
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Evaluation Results
Recent Submissions
| Model | Status | Goals Left | Submitted | Lean Snippet | Output |
|---|---|---|---|---|---|
| GPT-5.3 Codex Agent v8 | Partial | 1 | Feb 25, 2026 |
classical
have hcases : a = 0 ∨ a = 1 := ?_
exact hcases
|
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|
| Gemini 3.1 Pro Agent v8 | Failed | -- | Feb 24, 2026 |
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|
|
| Gemini 3 Flash Agent v8 | Partial | 1 | Feb 17, 2026 |
by_cases ha : a = 0
· left; exact ha
· right
have h1 : a * (a - 1) = 0 := by
rw [mul_sub, mul_one, h, sub_self]
|
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"proof_code": "by_cases ha : a = 0\n\u00b7 left; exact ha\n\u00b7 right\n have h1 : a * (a - 1) = 0 := by\n rw [mul_sub, mul_one, h, sub_self]",
"compiler_output_preview": "[path]:24:2: error: unsolved goals\ncase neg.h\nK : Type u_1\ninst\u271d\u00b9 : Field K\nG : Type u_2\ninst\u271d : Group G\nhG : IsMulTorsionFree G\na : MonoidAlgebra K G\nh : IsIdempotentElem a\nha : \u00aca = 0\nh1 : a * (a - 1) = 0\n\u22a2 a = 1\n'Kaplansky.idempotent_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"proof_code": "by_cases ha : a = 0\n\u00b7 left; exact ha\n\u00b7 right\n rcases Classical.em (\u2203 g k, a = Finsupp.single g k) with (\u27e8g, k, rfl\u27e9 | h_not_single)\n \u00b7 have h_k_nz : k \u2260 0 := by\n intro h_k0\n rw [h_k0, Finsupp.single_zero] at ha\n exact ha rfl\n have h_id : Finsupp.single g k * Finsupp.single g k = Finsupp.single g k := h\n rw [MonoidAlgebra.single_mul_single, Finsupp.single_eq_single_iff] at h_id\n rcases h_id with \u27e8-, h_k0\u27e9 | \u27e8h_gg, h_kk\u27e9\n \u00b7 exact (h_k_nz h_k0).elim\n \u00b7 have hg : g = 1 := mul_left_cancel (h_gg.trans (mul_one g).symm)\n have hk : k = 1 := (mul_left_cancel\u2080 h_k_nz (h_kk.trans (mul_one k).symm)).symm\n rw [hg, hk]\n rfl\n \u00b7",
"compiler_output_preview": "[path]:40:0: error: expected '{' or indented tactic sequence\n[path]:30:82: error: type mismatch\n h\nhas type\n IsIdempotentElem fun\u2080 | g => k : Prop\nbut is expected to have type\n ((fun\u2080 | g => k) * fun\u2080 | g => k) = fun\u2080 | g => k : Prop\n[path]
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"proof_code": "by_cases ha : a = 0\n\u00b7 left; exact ha\n\u00b7 right\n have h1 : a * (a - 1) = 0 := by\n rw [mul_sub, mul_one, h, sub_self]",
"compiler_output_preview": "[path]:24:2: error: unsolved goals\ncase neg.h\nK : Type u_1\ninst\u271d\u00b9 : Field K\nG : Type u_2\ninst\u271d : Group G\nhG : IsMulTorsionFree G\na : MonoidAlgebra K G\nh : IsIdempotentElem a\nha : \u00aca = 0\nh1 : a * (a - 1) = 0\n\u22a2 a = 1\n'Kaplansky.idempotent_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"proof_code": "by_cases ha : a = 0\n\u00b7 left; exact ha\n\u00b7 right\n by_cases htriv : \u2200 g : G, g = 1\n \u00b7 have h_a : a = Finsupp.single 1 (a 1) := by\n ext g\n rw [htriv g, Finsupp.single_apply_self]\n rw [h_a] at *\n have h_id : Finsupp.single 1 (a 1) * Finsupp.single 1 (a 1) = Finsupp.single 1 (a 1) := h\n rw [MonoidAlgebra.single_mul_single, mul_one, Finsupp.single_eq_single_iff] at h_id\n rcases h_id with \u27e8-, hk0\u27e9 | \u27e8-, hkk\u27e9\n \u00b7 rw [hk0, Finsupp.single_zero] at ha; contradiction\n \u00b7 have h_nz : a 1 \u2260 0 := by\n intro hk0; rw [hk0, Finsupp.single_zero] at ha; contradiction\n have hk1 : a 1 = 1 := (mul_left_cancel\u2080 h_nz (hkk.trans (mul_one (a 1)).symm)).symm\n rw [hk1]\n rfl\n \u00b7 have h1 : a * (a - 1) = 0 := by\n rw [mul_sub, mul_one, h, sub_self]",
"compiler_output_preview": "[path]:28:21: error: unknown constant 'Finsupp.single_apply_self'\n[path]:30:94: error: type mismatch\n h\nhas type\n IsIdempotentElem a : Prop\nbut is expected to have type\n ((fun\u2080 | 1 => a 1) * fun\u2080 | 1 => a 1) = fun\u2080 | 1 => a 1 : Prop\n[path]
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"proof_code": "by_cases ha : a = 0\n\u00b7 left; exact ha\n\u00b7 right\n have h1 : a * (a - 1) = 0 := by\n rw [mul_sub, mul_one, h, sub_self]",
"compiler_output_preview": "[path]:24:2: error: unsolved goals\ncase neg.h\nK : Type u_1\ninst\u271d\u00b9 : Field K\nG : Type u_2\ninst\u271d : Group G\nhG : IsMulTorsionFree G\na : MonoidAlgebra K G\nh : IsIdempotentElem a\nha : \u00aca = 0\nh1 : a * (a - 1) = 0\n\u22a2 a = 1\n'Kaplansky.idempotent_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"compiler_output_preview": "[path]:26:4: error: unsolved goals\ncase neg\nK : Type u_1\ninst\u271d\u00b9 : Field K\nG : Type u_2\ninst\u271d : Group G\nhG : IsMulTorsionFree G\na : MonoidAlgebra K G\nh : IsIdempotentElem a\nha0 : \u00aca = 0\nha1 : \u00aca = 1\nh1 : a * (a - 1) = 0\n\u22a2 False\n'Kaplansky.idempotent_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"proof_code": "by_cases ha0 : a = 0\n\u00b7 left; exact ha0\n\u00b7 by_cases ha1 : a = 1\n \u00b7 right; exact ha1\n \u00b7 exfalso\n have h1 : a * (a - 1) = 0 := by\n rw [mul_sub, mul_one, h, sub_self]",
"compiler_output_preview": "[path]:26:4: error: unsolved goals\ncase neg\nK : Type u_1\ninst\u271d\u00b9 : Field K\nG : Type u_2\ninst\u271d : Group G\nhG : IsMulTorsionFree G\na : MonoidAlgebra K G\nh : IsIdempotentElem a\nha0 : \u00aca = 0\nha1 : \u00aca = 1\nh1 : a * (a - 1) = 0\n\u22a2 False\n'Kaplansky.idempotent_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"proof_code": "by_cases ha0 : a = 0\n\u00b7 left; exact ha0\n\u00b7 by_cases ha1 : a = 1\n \u00b7 right; exact ha1\n \u00b7 exfalso\n have h1 : a * (a - 1) = 0 := by\n rw [mul_sub, mul_one, h, sub_self]",
"compiler_output_preview": "[path]:26:4: error: unsolved goals\ncase neg\nK : Type u_1\ninst\u271d\u00b9 : Field K\nG : Type u_2\ninst\u271d : Group G\nhG : IsMulTorsionFree G\na : MonoidAlgebra K G\nh : IsIdempotentElem a\nha0 : \u00aca = 0\nha1 : \u00aca = 1\nh1 : a * (a - 1) = 0\n\u22a2 False\n'Kaplansky.idempotent_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"problem": {
"axioms_check_name": "idempotent_conjecture",
"axioms_check_skipped": false,
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"category": "research_open",
"endings": [
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"stdout": "[path]:24:2: error: unsolved goals\ncase neg.h\nK : Type u_1\ninst\u271d\u00b9 : Field K\nG : Type u_2\ninst\u271d : Group G\nhG : IsMulTorsionFree G\na : MonoidAlgebra K G\nh : IsIdempotentElem a\nha : \u00aca = 0\nh1 : a * (a - 1) = 0\n\u22a2 a = 1\n'Kaplansky.idempotent_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
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|
| Gemini 3 Flash Preview | Failed | -- | Feb 14, 2026 |
rw [← map_mul]
rw [h_id]
have h_f_val : f a = 0 ∨ f a = 1 := by
have : f a * (f a - 1) = 0 := by
rw [mul_sub, mul_one, h_f_id, sub_self]
exact mul_eq_zero.mp this
rcases h_f_val with h0 | h1
· left
ext g
simp [h0]
aesop
· right
ext g
simp [h1]
aesop
|
View{
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"status": "failed",
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"stdout": "[path]:22:6: error: tactic 'rewrite' failed, did not find instance of the pattern in the target expression\n ?f ?x * ?f ?y\nK : Type u_1\ninst\u271d\u00b9 : Field K\nG : Type u_2\ninst\u271d : Group G\nhG : IsMulTorsionFree G\na : MonoidAlgebra K G\nh : IsIdempotentElem a\n\u22a2 a = 0 \u2228 a = 1\n[path]:23:4: error: unexpected identifier; expected command\n",
"timed_out": false
}
}
|