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zero_divisor_conjecture
Specification
**The zero-divisor conjecture** If `G` is torsion-free, then the group algebra `K[G]` has no non-trivial zero divisors.
Lean 4 Statement
theorem zero_divisor_conjecture : NoZeroDivisors (MonoidAlgebra K G)
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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 |
have h : NoZeroDivisors (MonoidAlgebra K G) := ?_
exact h
|
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"proof_code": "have h : NoZeroDivisors (MonoidAlgebra K G) := by\n infer_instance\nexact h",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772022042_9685.lean:22:4: error(lean.synthInstanceFailed): failed to synthesize instance of type class\n NoZeroDivisors (MonoidAlgebra K G)\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.\n'Kaplansky.zero_divisor_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
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"goal_before_error": "```lean\nK : Type u_1\ninst\u271d\u00b9 : Field K\nG : Type u_2\ninst\u271d : Group G\nhG : IsMulTorsionFree G\n\u22a2 NoZeroDivisors (MonoidAlgebra K G)\n```",
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{
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{
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},
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},
{
"iteration": 10,
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"query": "MonoidAlgebra noZeroSmulDivisors",
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},
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"stdout": "/app/asi_portal/verifier/quick_1772022110_7342.lean:20:72: error: unsolved goals\ncase refine_1\nK : Type u_1\ninst\u271d\u00b9 : Field K\nG : Type u_2\ninst\u271d : Group G\nhG : IsMulTorsionFree G\n\u22a2 NoZeroDivisors (MonoidAlgebra K G)\n'Kaplansky.zero_divisor_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 |
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|
|
| Gemini 3 Flash Agent v8 | Partial | 1 | Feb 17, 2026 |
constructor
intros a b hab
by_cases ha : a = 0
· left; exact ha
· right
by_contra hb
letI := Classical.decEq G
have : IsMulTorsionFree G := hG
have h_upp : ∃ y, ∃ z, y ∈ a.support ∧ z ∈ b.support ∧
∀ y' z', y' ∈ a.support → z' ∈ b.support → y' * z' = y * z → y' = y ∧ z' = z := by
refine ?_
rcases h_upp with ⟨y, z, hy, hz, h_unique⟩
let x := y * z
have h_coeff : (a * b) x = a y * b z := by
rw [MonoidAlgebra.mul_apply, Finsupp.sum, Finset.sum_eq_single y]
· rw [Finsupp.sum, Finset.sum_eq_single z]
· simp [x]
· intros z' hz' h_ne
simp only [x]
split_ifs with h_eq
· exfalso
exact h_ne (h_unique y z' hy hz' h_eq).2
· rfl
· intros hz_not; exfalso; exact hz_not hz
· intros y' hy' h_ne
rw [Finsupp.sum, Finset.sum_eq_zero]
· intros z' hz'
simp only [x]
split_ifs with h_eq
· exfalso
exact h_ne (h_unique y' z' hy' hz' h_eq).1
· rfl
· intros hy_not; exfalso; exact hy_not hy
have h_not_zero : a y * b z ≠ 0 := mul_ne_zero (Finsupp.mem_support_iff.mp hy) (Finsupp.mem_support_iff.mp hz)
rw [hab, Finsupp.zero_apply] at h_coeff
exact h_not_zero h_coeff.symm
|
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"proof_code": "constructor\nintros a b hab\nby_cases ha : a = 0\n\u00b7 left; exact ha\n\u00b7 right\n by_contra hb\n letI := Classical.decEq G\n obtain \u27e8y, z, hy, hz, h_unique\u27e9 : \u2203 y, \u2203 z, y \u2208 a.support \u2227 z \u2208 b.support \u2227\n \u2200 y' z', y' \u2208 a.support \u2192 z' \u2208 b.support \u2192 y' * z' = y * z \u2192 y' = y \u2227 z' = z := ?_\n let x := y * z\n have h_coeff : (a * b) x = a y * b z := by\n rw [MonoidAlgebra.mul_apply, Finsupp.sum, Finset.sum_eq_single y]\n \u00b7 rw [Finsupp.sum, Finset.sum_eq_single z]\n \u00b7 simp [x]\n \u00b7 intros z' hz' h_ne\n simp only [x]\n split_ifs with h_eq\n \u00b7 exfalso\n exact h_ne (h_unique y z' hy hz' h_eq).2\n \u00b7 rfl\n \u00b7 intros hz_not; exfalso; exact hz_not hz\n \u00b7 intros y' hy' h_ne\n rw [Finsupp.sum, Finset.sum_eq_zero]\n \u00b7 intros z' hz'\n simp only [x]\n split_ifs with h_eq\n \u00b7 exfalso\n exact h_ne (h_unique y' z' hy' hz' h_eq).1\n \u00b7 rfl\n \u00b7 intros hy_not; exfalso; exact hy_not hy\n have h_not_zero : a y * b z \u2260 0 := mul_ne_zero (Finsupp.mem_support_iff.mp hy) (Finsupp.mem_support_iff.mp hz)\n rw [hab, Finsupp.zero_apply] at h_coeff\n exact h_not_zero h_coeff.symm",
"compiler_output_preview": "[path]:20:0: error: automatically included section variable(s) unused in theorem 'Kaplansky.zero_divisor_conjecture':\n hG\nconsider restructuring your `variable` declarations so that the variables are not in scope or explicitly omit them:\n omit hG in theorem ...\n\nNote: This linter can be disabled with `set_option linter.unusedSectionVars false`\n'Kaplansky.zero_divisor_conjecture\n...<truncated>"
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"proof_code": "constructor\nintros a b hab\nby_cases ha : a = 0\n\u00b7 left; exact ha\n\u00b7 right\n by_contra hb\n letI := Classical.decEq G\n have hG' : IsMulTorsionFree G := hG\n have h_upp : \u2203 y, \u2203 z, y \u2208 a.support \u2227 z \u2208 b.support \u2227\n \u2200 y' z', y' \u2208 a.support \u2192 z' \u2208 b.support \u2192 y' * z' = y * z \u2192 y' = y \u2227 z' = z := by\n clear ha hb a b hab hG' -- Use something but still leave a hole\n refine ?_\n rcases h_upp with \u27e8y, \u27e8z, hy, hz, h_unique\u27e9\u27e9\n let x := y * z\n have h_coeff : (a * b) x = a y * b z := by\n rw [MonoidAlgebra.mul_apply, Finsupp.sum, Finset.sum_eq_single y]\n \u00b7 rw [Finsupp.sum, Finset.sum_eq_single z]\n \u00b7 simp [x]\n \u00b7 intros z' hz' h_ne\n simp only [x]\n split_ifs with h_eq\n \u00b7 exfalso\n exact h_ne (h_unique y z' hy hz' h_eq).2\n \u00b7 rfl\n \u00b7 intros hz_not; exfalso; exact hz_not hz\n \u00b7 intros y' hy' h_ne\n rw [Finsupp.sum, Finset.sum_eq_zero]\n \u00b7 intros z' hz'\n simp only [x]\n split_ifs with h_eq\n \u00b7 exfalso\n exact h_ne (h_unique y' z' hy' hz' h_eq).1\n \u00b7 rfl\n \u00b7 intros hy_not; exfalso; exact hy_not hy\n have h_not_zero : a y * b z \u2260 0 := mul_ne_zero (Finsupp.mem_support_iff.mp hy) (Finsupp.mem_support_iff.mp hz)\n rw [hab, Finsupp.zero_apply] at h_coeff\n exact h_not_zero h_coeff.symm",
"compiler_output_preview": "[path]:31:6: error: tactic 'clear' failed, target depends on 'b'\nK : Type u_1\ninst\u271d\u00b9 : Field K\nG : Type u_2\ninst\u271d : Group G\nhG : IsMulTorsionFree G\na b : MonoidAlgebra K G\nthis : DecidableEq G := Classical.decEq G\n\u22a2 \u2203 y z,\n y \u2208 a.support \u2227 z \u2208 b.support \u2227 \u2200 (y' z' : G), y' \u2208 a.support \u2192 z' \u2208 b.support \u2192 y' * z' = y * z \u2192 y' = y \u2227 z' = z\n'Kaplansky.zero_divisor_conjecture' depe\n...<truncated>"
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"proof_code": "constructor\nintros a b hab\nby_cases ha : a = 0\n\u00b7 left; exact ha\n\u00b7 right\n by_contra hb\n letI := Classical.decEq G\n have : IsMulTorsionFree G := hG\n have h_upp : \u2203 y, \u2203 z, y \u2208 a.support \u2227 z \u2208 b.support \u2227\n \u2200 y' z', y' \u2208 a.support \u2192 z' \u2208 b.support \u2192 y' * z' = y * z \u2192 y' = y \u2227 z' = z := by\n refine ?_\n rcases h_upp with \u27e8y, z, hy, hz, h_unique\u27e9\n let x := y * z\n have h_coeff : (a * b) x = a y * b z := by\n rw [MonoidAlgebra.mul_apply, Finsupp.sum, Finset.sum_eq_single y]\n \u00b7 rw [Finsupp.sum, Finset.sum_eq_single z]\n \u00b7 simp [x]\n \u00b7 intros z' hz' h_ne\n simp only [x]\n split_ifs with h_eq\n \u00b7 exfalso\n exact h_ne (h_unique y z' hy hz' h_eq).2\n \u00b7 rfl\n \u00b7 intros hz_not; exfalso; exact hz_not hz\n \u00b7 intros y' hy' h_ne\n rw [Finsupp.sum, Finset.sum_eq_zero]\n \u00b7 intros z' hz'\n simp only [x]\n split_ifs with h_eq\n \u00b7 exfalso\n exact h_ne (h_unique y' z' hy' hz' h_eq).1\n \u00b7 rfl\n \u00b7 intros hy_not; exfalso; exact hy_not hy\n have h_not_zero : a y * b z \u2260 0 := mul_ne_zero (Finsupp.mem_support_iff.mp hy) (Finsupp.mem_support_iff.mp hz)\n rw [hab, Finsupp.zero_apply] at h_coeff\n exact h_not_zero h_coeff.symm",
"compiler_output_preview": "[path]:30:88: error: unsolved goals\nK : Type u_1\ninst\u271d\u00b9 : Field K\nG : Type u_2\ninst\u271d : Group G\nhG : IsMulTorsionFree G\na b : MonoidAlgebra K G\nhab : a * b = 0\nha : \u00aca = 0\nhb : \u00acb = 0\nthis\u271d : DecidableEq G := Classical.decEq G\nthis : IsMulTorsionFree G\n\u22a2 \u2203 y z,\n y \u2208 a.support \u2227 z \u2208 b.support \u2227 \u2200 (y' z' : G), y' \u2208 a.support \u2192 z' \u2208 b.support \u2192 y' * z' = y * z \u2192 y' = y \u2227 z' = z\n'K\n...<truncated>"
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"proof_code": "constructor\nintros a b hab\nby_cases ha : a = 0\n\u00b7 left; exact ha\n\u00b7 right\n by_contra hb\n letI := Classical.decEq G\n have : IsMulTorsionFree G := hG\n have h_upp : \u2203 y, \u2203 z, y \u2208 a.support \u2227 z \u2208 b.support \u2227\n \u2200 y' z', y' \u2208 a.support \u2192 z' \u2208 b.support \u2192 y' * z' = y * z \u2192 y' = y \u2227 z' = z := by\n refine ?_\n rcases h_upp with \u27e8y, z, hy, hz, h_unique\u27e9\n let x := y * z\n have h_coeff : (a * b) x = a y * b z := by\n rw [MonoidAlgebra.mul_apply, Finsupp.sum, Finset.sum_eq_single y]\n \u00b7 rw [Finsupp.sum, Finset.sum_eq_single z]\n \u00b7 simp [x]\n \u00b7 intros z' hz' h_ne\n simp only [x]\n split_ifs with h_eq\n \u00b7 exfalso\n exact h_ne (h_unique y z' hy hz' h_eq).2\n \u00b7 rfl\n \u00b7 intros hz_not; exfalso; exact hz_not hz\n \u00b7 intros y' hy' h_ne\n rw [Finsupp.sum, Finset.sum_eq_zero]\n \u00b7 intros z' hz'\n simp only [x]\n split_ifs with h_eq\n \u00b7 exfalso\n exact h_ne (h_unique y' z' hy' hz' h_eq).1\n \u00b7 rfl\n \u00b7 intros hy_not; exfalso; exact hy_not hy\n have h_not_zero : a y * b z \u2260 0 := mul_ne_zero (Finsupp.mem_support_iff.mp hy) (Finsupp.mem_support_iff.mp hz)\n rw [hab, Finsupp.zero_apply] at h_coeff\n exact h_not_zero h_coeff.symm",
"compiler_output_preview": "[path]:30:88: error: unsolved goals\nK : Type u_1\ninst\u271d\u00b9 : Field K\nG : Type u_2\ninst\u271d : Group G\nhG : IsMulTorsionFree G\na b : MonoidAlgebra K G\nhab : a * b = 0\nha : \u00aca = 0\nhb : \u00acb = 0\nthis\u271d : DecidableEq G := Classical.decEq G\nthis : IsMulTorsionFree G\n\u22a2 \u2203 y z,\n y \u2208 a.support \u2227 z \u2208 b.support \u2227 \u2200 (y' z' : G), y' \u2208 a.support \u2192 z' \u2208 b.support \u2192 y' * z' = y * z \u2192 y' = y \u2227 z' = z\n'K\n...<truncated>"
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"proof_code": "constructor\nintros a b hab\nby_cases ha : a = 0\n\u00b7 left; exact ha\n\u00b7 right\n by_contra hb\n letI := Classical.decEq G\n have : IsMulTorsionFree G := hG\n have h_upp : \u2203 y, \u2203 z, y \u2208 a.support \u2227 z \u2208 b.support \u2227\n \u2200 y' z', y' \u2208 a.support \u2192 z' \u2208 b.support \u2192 y' * z' = y * z \u2192 y' = y \u2227 z' = z := by\n refine ?_\n rcases h_upp with \u27e8y, z, hy, hz, h_unique\u27e9\n let x := y * z\n have h_coeff : (a * b) x = a y * b z := by\n rw [MonoidAlgebra.mul_apply, Finsupp.sum, Finset.sum_eq_single y]\n \u00b7 rw [Finsupp.sum, Finset.sum_eq_single z]\n \u00b7 simp [x]\n \u00b7 intros z' hz' h_ne\n simp only [x]\n split_ifs with h_eq\n \u00b7 exfalso\n exact h_ne (h_unique y z' hy hz' h_eq).2\n \u00b7 rfl\n \u00b7 intros hz_not; exfalso; exact hz_not hz\n \u00b7 intros y' hy' h_ne\n rw [Finsupp.sum, Finset.sum_eq_zero]\n \u00b7 intros z' hz'\n simp only [x]\n split_ifs with h_eq\n \u00b7 exfalso\n exact h_ne (h_unique y' z' hy' hz' h_eq).1\n \u00b7 rfl\n \u00b7 intros hy_not; exfalso; exact hy_not hy\n have h_not_zero : a y * b z \u2260 0 := mul_ne_zero (Finsupp.mem_support_iff.mp hy) (Finsupp.mem_support_iff.mp hz)\n rw [hab, Finsupp.zero_apply] at h_coeff\n exact h_not_zero h_coeff.symm",
"compiler_output_preview": "[path]:30:88: error: unsolved goals\nK : Type u_1\ninst\u271d\u00b9 : Field K\nG : Type u_2\ninst\u271d : Group G\nhG : IsMulTorsionFree G\na b : MonoidAlgebra K G\nhab : a * b = 0\nha : \u00aca = 0\nhb : \u00acb = 0\nthis\u271d : DecidableEq G := Classical.decEq G\nthis : IsMulTorsionFree G\n\u22a2 \u2203 y z,\n y \u2208 a.support \u2227 z \u2208 b.support \u2227 \u2200 (y' z' : G), y' \u2208 a.support \u2192 z' \u2208 b.support \u2192 y' * z' = y * z \u2192 y' = y \u2227 z' = z\n'K\n...<truncated>"
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"proof_code": "constructor\nintros a b hab\nby_cases ha : a = 0\n\u00b7 left; exact ha\n\u00b7 right\n by_contra hb\n letI := Classical.decEq G\n have : IsMulTorsionFree G := hG\n have h_upp : \u2203 y, \u2203 z, y \u2208 a.support \u2227 z \u2208 b.support \u2227\n \u2200 y' z', y' \u2208 a.support \u2192 z' \u2208 b.support \u2192 y' * z' = y * z \u2192 y' = y \u2227 z' = z := by\n refine ?_\n rcases h_upp with \u27e8y, z, hy, hz, h_unique\u27e9\n let x := y * z\n have h_coeff : (a * b) x = a y * b z := by\n rw [MonoidAlgebra.mul_apply, Finsupp.sum, Finset.sum_eq_single y]\n \u00b7 rw [Finsupp.sum, Finset.sum_eq_single z]\n \u00b7 simp [x]\n \u00b7 intros z' hz' h_ne\n simp only [x]\n split_ifs with h_eq\n \u00b7 exfalso\n exact h_ne (h_unique y z' hy hz' h_eq).2\n \u00b7 rfl\n \u00b7 intros hz_not; exfalso; exact hz_not hz\n \u00b7 intros y' hy' h_ne\n rw [Finsupp.sum, Finset.sum_eq_zero]\n \u00b7 intros z' hz'\n simp only [x]\n split_ifs with h_eq\n \u00b7 exfalso\n exact h_ne (h_unique y' z' hy' hz' h_eq).1\n \u00b7 rfl\n \u00b7 intros hy_not; exfalso; exact hy_not hy\n have h_not_zero : a y * b z \u2260 0 := mul_ne_zero (Finsupp.mem_support_iff.mp hy) (Finsupp.mem_support_iff.mp hz)\n rw [hab, Finsupp.zero_apply] at h_coeff\n exact h_not_zero h_coeff.symm",
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"proof_code": "constructor\nintros a b hab\nby_cases ha : a = 0\n\u00b7 left; exact ha\n\u00b7 right\n by_contra hb\n letI := Classical.decEq G\n have : IsMulTorsionFree G := hG\n have h_upp : \u2203 y, \u2203 z, y \u2208 a.support \u2227 z \u2208 b.support \u2227\n \u2200 y' z', y' \u2208 a.support \u2192 z' \u2208 b.support \u2192 y' * z' = y * z \u2192 y' = y \u2227 z' = z := by\n refine ?_\n rcases h_upp with \u27e8y, z, hy, hz, h_unique\u27e9\n let x := y * z\n have h_coeff : (a * b) x = a y * b z := by\n rw [MonoidAlgebra.mul_apply, Finsupp.sum, Finset.sum_eq_single y]\n \u00b7 rw [Finsupp.sum, Finset.sum_eq_single z]\n \u00b7 simp [x]\n \u00b7 intros z' hz' h_ne\n simp only [x]\n split_ifs with h_eq\n \u00b7 exfalso\n exact h_ne (h_unique y z' hy hz' h_eq).2\n \u00b7 rfl\n \u00b7 intros hz_not; exfalso; exact hz_not hz\n \u00b7 intros y' hy' h_ne\n rw [Finsupp.sum, Finset.sum_eq_zero]\n \u00b7 intros z' hz'\n simp only [x]\n split_ifs with h_eq\n \u00b7 exfalso\n exact h_ne (h_unique y' z' hy' hz' h_eq).1\n \u00b7 rfl\n \u00b7 intros hy_not; exfalso; exact hy_not hy\n have h_not_zero : a y * b z \u2260 0 := mul_ne_zero (Finsupp.mem_support_iff.mp hy) (Finsupp.mem_support_iff.mp hz)\n rw [hab, Finsupp.zero_apply] at h_coeff\n exact h_not_zero h_coeff.symm",
"compiler_output_preview": "[path]:30:88: error: unsolved goals\nK : Type u_1\ninst\u271d\u00b9 : Field K\nG : Type u_2\ninst\u271d : Group G\nhG : IsMulTorsionFree G\na b : MonoidAlgebra K G\nhab : a * b = 0\nha : \u00aca = 0\nhb : \u00acb = 0\nthis\u271d : DecidableEq G := Classical.decEq G\nthis : IsMulTorsionFree G\n\u22a2 \u2203 y z,\n y \u2208 a.support \u2227 z \u2208 b.support \u2227 \u2200 (y' z' : G), y' \u2208 a.support \u2192 z' \u2208 b.support \u2192 y' * z' = y * z \u2192 y' = y \u2227 z' = z\n'K\n...<truncated>"
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"proof_code": "constructor\nintros a b hab\nby_cases ha : a = 0\n\u00b7 left; exact ha\n\u00b7 right\n by_contra hb\n letI := Classical.decEq G\n have : IsMulTorsionFree G := hG\n have h_upp : \u2203 y, \u2203 z, y \u2208 a.support \u2227 z \u2208 b.support \u2227\n \u2200 y' z', y' \u2208 a.support \u2192 z' \u2208 b.support \u2192 y' * z' = y * z \u2192 y' = y \u2227 z' = z := by\n refine ?_\n rcases h_upp with \u27e8y, z, hy, hz, h_unique\u27e9\n let x := y * z\n have h_coeff : (a * b) x = a y * b z := by\n rw [MonoidAlgebra.mul_apply, Finsupp.sum, Finset.sum_eq_single y]\n \u00b7 rw [Finsupp.sum, Finset.sum_eq_single z]\n \u00b7 simp [x]\n \u00b7 intros z' hz' h_ne\n simp only [x]\n split_ifs with h_eq\n \u00b7 exfalso\n exact h_ne (h_unique y z' hy hz' h_eq).2\n \u00b7 rfl\n \u00b7 intros hz_not; exfalso; exact hz_not hz\n \u00b7 intros y' hy' h_ne\n rw [Finsupp.sum, Finset.sum_eq_zero]\n \u00b7 intros z' hz'\n simp only [x]\n split_ifs with h_eq\n \u00b7 exfalso\n exact h_ne (h_unique y' z' hy' hz' h_eq).1\n \u00b7 rfl\n \u00b7 intros hy_not; exfalso; exact hy_not hy\n have h_not_zero : a y * b z \u2260 0 := mul_ne_zero (Finsupp.mem_support_iff.mp hy) (Finsupp.mem_support_iff.mp hz)\n rw [hab, Finsupp.zero_apply] at h_coeff\n exact h_not_zero h_coeff.symm",
"compiler_output_preview": "[path]:30:88: error: unsolved goals\nK : Type u_1\ninst\u271d\u00b9 : Field K\nG : Type u_2\ninst\u271d : Group G\nhG : IsMulTorsionFree G\na b : MonoidAlgebra K G\nhab : a * b = 0\nha : \u00aca = 0\nhb : \u00acb = 0\nthis\u271d : DecidableEq G := Classical.decEq G\nthis : IsMulTorsionFree G\n\u22a2 \u2203 y z,\n y \u2208 a.support \u2227 z \u2208 b.support \u2227 \u2200 (y' z' : G), y' \u2208 a.support \u2192 z' \u2208 b.support \u2192 y' * z' = y * z \u2192 y' = y \u2227 z' = z\n'K\n...<truncated>"
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"stdout": "[path]:30:88: error: unsolved goals\nK : Type u_1\ninst\u271d\u00b9 : Field K\nG : Type u_2\ninst\u271d : Group G\nhG : IsMulTorsionFree G\na b : MonoidAlgebra K G\nhab : a * b = 0\nha : \u00aca = 0\nhb : \u00acb = 0\nthis\u271d : DecidableEq G := Classical.decEq G\nthis : IsMulTorsionFree G\n\u22a2 \u2203 y z,\n y \u2208 a.support \u2227 z \u2208 b.support \u2227 \u2200 (y' z' : G), y' \u2208 a.support \u2192 z' \u2208 b.support \u2192 y' * z' = y * z \u2192 y' = y \u2227 z' = z\n'Kaplansky.zero_divisor_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
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|
| Gemini 3 Flash Preview | Failed | -- | Feb 14, 2026 |
constructor
intro f g hf hg
contrapose! hf
by_cases hf0 : f = 0
· exact hf0
by_cases hg0 : g = 0
· exfalso
exact hg hg0
let sf := f.support
let sg := g.support
have hsf : sf.Nonempty := Finsupp.support_nonempty_iff.mpr hf0
have hsg : sg.Nonempty := Finsupp.support_nonempty_iff.mpr hg0
apply Finsupp.ne_zero_iff.mpr
let x := sf.min' hsf
let y := sg.min' hsg
use x * y
simp only [MonoidAlgebra.mul_apply]
apply hG
repeat assumption
|
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"stdout": "[path]:22:15: error: tactic 'introN' failed, insufficient number of binders\ncase eq_zero_or_eq_zero_of_mul_eq_zero\nK : Type u_1\ninst\u271d\u00b9 : Field K\nG : Type u_2\ninst\u271d : Group G\nhG : IsMulTorsionFree G\nf g : MonoidAlgebra K G\nhf : f * g = 0\n\u22a2 f = 0 \u2228 g = 0\n",
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|