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schanuel_conjecture
Specification
Given any set of $n$ complex numbers $\{z_1, ..., z_n\}$ that are linearly independent over $\mathbb{Q}$, the field extension $\mathbb{Q}(z_1, ..., z_n, e^{z_1}, ..., e^{z_n})$ has transcendence degree at least $n$ over $\mathbb{Q}$.
Lean 4 Statement
theorem schanuel_conjecture (n : ℕ) (z : Fin n → ℂ) (h : LinearIndependent ℚ z) :
let hinj
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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 |
dsimp
have hgoal : n ≤ Algebra.trdeg ℚ (adjoin ℚ (Set.range z ∪ Set.range (cexp ∘ z))) := ?_
exact hgoal
|
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"reasoning_text": "**Planning placeholder tactic to compile**",
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"
}
]
},
{
"iteration": 1,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "partial",
"goals_remaining": 1,
"timed_out": false,
"banned_tokens": [],
"proof_code": "dsimp\nhave hgoal : n \u2264 Algebra.trdeg \u211a (adjoin \u211a (Set.range z \u222a Set.range (cexp \u2218 z))) := ?_\nexact hgoal",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772028542_6487.lean:21:75: error: unsolved goals\ncase refine_1\nn : \u2115\nz : Fin n \u2192 \u2102\nh : LinearIndependent \u211a z\n\u22a2 \u2191n \u2264 Algebra.trdeg \u211a \u21a5(adjoin \u211a (Set.range z \u222a Set.range (cexp \u2218 z)))\n'Schanuel.schanuel_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 44.91,
"tactic_replay": {
"tactic_count": 3,
"successful_tactics": 0,
"first_failure_step": 1,
"goal_trajectory": [
1,
1,
2
],
"max_goals": 2,
"final_goal_count": 2,
"goals_opened": 1,
"goals_closed": 0
},
"lsp_feedback": {
"diagnostics": [
{
"fullRange": {
"end": {
"character": 13,
"line": 23
},
"start": {
"character": 75,
"line": 20
}
},
"leanTags": [
1
],
"message": "unsolved goals\ncase refine_1\nn : \u2115\nz : Fin n \u2192 \u2102\nh : LinearIndependent \u211a z\n\u22a2 \u2191n \u2264 Algebra.trdeg \u211a \u21a5(adjoin \u211a (Set.range z \u222a Set.range (cexp \u2218 z)))",
"range": {
"end": {
"character": 0,
"line": 21
},
"start": {
"character": 75,
"line": 20
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 25
},
"start": {
"character": 0,
"line": 25
}
},
"message": "'Schanuel.schanuel_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
"end": {
"character": 6,
"line": 25
},
"start": {
"character": 0,
"line": 25
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 20,
"error_char": 75,
"error_msg": "unsolved goals\ncase refine_1\nn : \u2115\nz : Fin n \u2192 \u2102\nh : LinearIndependent \u211a z\n\u22a2 \u2191n \u2264 Algebra.trdeg \u211a \u21a5(adjoin \u211a (Set.range z \u222a Set.range (cexp \u2218 z)))",
"goal_before_error": null,
"full_timeline": [
{
"line": 21,
"code": "dsimp",
"goal": "```lean\nn : \u2115\nz : Fin n \u2192 \u2102\nh : LinearIndependent \u211a z\n\u22a2 let hinj := \u22ef;\n \u2191n \u2264 Algebra.trdeg \u211a \u21a5(adjoin \u211a (Set.range z \u222a Set.range (cexp \u2218 z)))\n```",
"goal_count": 1,
"ast_complexity": 146
},
{
"line": 22,
"code": "have hgoal : n \u2264 Algebra.trdeg \u211a (adjoin \u211a (Set.range z \u222a Set.range (cexp \u2218 z))) := ?_",
"goal": "```lean\nn : \u2115\nz : Fin n \u2192 \u2102\nh : LinearIndependent \u211a z\n\u22a2 \u2191n \u2264 Algebra.trdeg \u211a \u21a5(adjoin \u211a (Set.range z \u222a Set.range (cexp \u2218 z)))\n```",
"goal_count": 1,
"ast_complexity": 129
},
{
"line": 23,
"code": "exact hgoal",
"goal": "```lean\ncase refine_2\nn : \u2115\nz : Fin n \u2192 \u2102\nh : LinearIndependent \u211a z\nhgoal : \u2191n \u2264 Algebra.trdeg \u211a \u21a5(adjoin \u211a (Set.range z \u222a Set.range (cexp \u2218 z)))\n\u22a2 \u2191n \u2264 Algebra.trdeg \u211a \u21a5(adjoin \u211a (Set.range z \u222a Set.range (cexp \u2218 z)))\n```\n---\n```lean\ncase refine_1\nn : \u2115\nz : Fin n \u2192 \u2102\nh : LinearIndependent \u211a z\n\u22a2 \u2191n \u2264 Algebra.trdeg \u211a \u21a5(adjoin \u211a (Set.range z \u222a Set.range (cexp \u2218 z)))\n```",
"goal_count": 2,
"ast_complexity": 369
}
],
"tactic_count": 3,
"successful_tactics": 0,
"first_failure_step": 1,
"cps": 0.0,
"cps_peak_step": 0,
"error_code": "n \u2264 Algebra.trdeg \u211a (adjoin \u211a (Set.range z \u222a Set.range (cexp \u2218 z))) := by"
}
},
{
"iteration": 2,
"event": "search_mathlib",
"query": "schanuel",
"results_count": 0
},
{
"iteration": 3,
"event": "search_mathlib",
"query": "linearIndependent_exp",
"results_count": 0
},
{
"iteration": 4,
"event": "search_mathlib",
"query": "LinearIndependent exp",
"results_count": 0
},
{
"iteration": 5,
"event": "search_mathlib",
"query": "Algebra.trdeg",
"results_count": 1
},
{
"iteration": 6,
"event": "search_mathlib",
"query": "trdeg_le",
"results_count": 8
},
{
"iteration": 7,
"event": "search_mathlib",
"query": "le_trdeg",
"results_count": 8
},
{
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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 |
let L := adjoin ℚ (Set.range z ∪ Set.range (Complex.exp ∘ z))
have htr : Algebra.trdeg ℚ L = Algebra.trdeg ℚ (adjoin ℚ (Set.range z ∪ Set.range (Complex.exp ∘ z))) := rfl
have h_lin : LinearIndependent ℚ z := h
let s := Set.range z
let exp_s := Set.range (Complex.exp ∘ z)
let K := adjoin ℚ (s ∪ exp_s)
show n ≤ Algebra.trdeg ℚ K
refine ?_
|
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"proof_code": "set L := adjoin \u211a (Set.range z \u222a Set.range (Complex.exp \u2218 z))\nlet S1 := Set.range z\nlet S2 := Set.range (Complex.exp \u2218 z)\nhave hS : L = adjoin \u211a (S1 \u222a S2) := rfl\nhave h_trdeg : Algebra.trdeg \u211a L = RelSecondary.trdeg \u211a L := rfl\nhave h_size : Set.ncard (S1 \u222a S2) \u2264 2 * n := by\n have h1 : Set.ncard S1 \u2264 n := by\n rw [Set.ncard_range]\n exact Fintype.card_fin n\n have h2 : Set.ncard S2 \u2264 n := by\n rw [Set.ncard_range]\n exact Fintype.card_fin n\n exact (Set.ncard_union_le S1 S2).trans (by linarith)\nrw [hS]\napply @RelSecondary.trdeg_ge_of_linear_independent \u211a _ _ _ _ z h\n\u00b7 intro i\n apply subset_adjoin\n left\n exact Set.mem_range_self i\n\u00b7 intro i\n apply subset_adjoin\n right\n exact Set.mem_range_self i",
"compiler_output_preview": "[path]:26:37: error: unknown identifier 'RelSecondary.trdeg'\n'Schanuel.schanuel_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
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"proof_code": "intro n\nintro z\nintro h\nlet L := adjoin \u211a (Set.range z \u222a Set.range (Complex.exp \u2218 z))\nlet S1 := Set.range z\nlet S2 := Set.range (Complex.exp \u2218 z)\nlet S := S1 \u222a S2\nhave hL : L = adjoin \u211a S := rfl\nhave h_trdeg : Algebra.trdeg \u211a L = Algebra.trdeg \u211a (adjoin \u211a S) := rfl\nrefine ?_\n-- Schanuel's conjecture is an open problem in transcendental number theory.\n-- The requested proof is for a conjecture, so we provide an outline of the goal state.\n-- We use refine to leave the main inequality as a goal.",
"compiler_output_preview": "[path]:23:8: error: tactic 'introN' failed, insufficient number of binders\nn\u271d : \u2115\nz : Fin n\u271d \u2192 \u2102\nh : LinearIndependent \u211a z\nn : Function.Injective \u21d1(algebraMap \u211a \u21a5(adjoin \u211a (Set.range z \u222a Set.range (cexp \u2218 z)))) :=\n RingHom.injective (algebraMap \u211a \u21a5(adjoin \u211a (Set.range z \u222a Set.range (cexp \u2218 z))))\n\u22a2 \u2191n\u271d \u2264 Algebra.trdeg \u211a \u21a5(adjoin \u211a (Set.range z \u222a Set.range (cexp \u2218 z)))\n'Schanuel.schanu\n...<truncated>"
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"proof_code": "let L := adjoin \u211a (Set.range z \u222a Set.range (Complex.exp \u2218 z))\nhave htr : Algebra.trdeg \u211a L = Algebra.trdeg \u211a (adjoin \u211a (Set.range z \u222a Set.range (Complex.exp \u2218 z))) := rfl\nhave h_lin : LinearIndependent \u211a z := h\nlet s := Set.range z\nlet exp_s := Set.range (Complex.exp \u2218 z)\nlet K := adjoin \u211a (s \u222a exp_s)\nshow n \u2264 Algebra.trdeg \u211a K\nrefine ?_",
"compiler_output_preview": "[path]:21:75: error: unsolved goals\nn : \u2115\nz : Fin n \u2192 \u2102\nh : LinearIndependent \u211a z\nL : IntermediateField \u211a \u2102 := adjoin \u211a (Set.range z \u222a Set.range (cexp \u2218 z))\nhtr : Algebra.trdeg \u211a \u21a5L = Algebra.trdeg \u211a \u21a5(adjoin \u211a (Set.range z \u222a Set.range (cexp \u2218 z)))\nh_lin : LinearIndependent \u211a z\ns : Set \u2102 := Set.range z\nexp_s : Set \u2102 := Set.range (cexp \u2218 z)\nK : IntermediateField \u211a \u2102 := adjoin \u211a (s \u222a ex\n...<truncated>"
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"proof_code": "let s := Set.range z\nlet exp_s := Set.range (Complex.exp \u2218 z)\nlet K := adjoin \u211a (s \u222a exp_s)\nhave h_tr : Algebra.trdeg \u211a K = Algebra.trdeg \u211a (adjoin \u211a (s \u222a exp_s)) := rfl\nhave h_sub : s \u2286 K := (subset_adjoin \u211a _) \u2218 Set.subset_union_left s exp_s\nhave h_exp_sub : exp_s \u2286 K := (subset_adjoin \u211a _) \u2218 Set.subset_union_right s exp_s\nhave : n = Set.ncard (Set.range z) := by\n rw [Set.ncard_range_of_injective]\n \u00b7 exact Fintype.card_fin n\n \u00b7 exact h.injective\nrefine ?_",
"compiler_output_preview": "[path]:26:46: error: Function expected at\n Set.subset_union_left ?m.13849\nbut this term has type\n ?m.13827 \u2208 ?m.13825 \u222a ?m.13826\n\nNote: Expected a function because this term is being applied to the argument\n exp_s\n[path]:26:68: error: Application type mismatch: In the ap\n...<truncated>"
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"proof_code": "let s := Set.range z\nlet exp_s := Set.range (Complex.exp \u2218 z)\nlet K := adjoin \u211a (s \u222a exp_s)\nhave hn : n = (Finset.univ : Finset (Fin n)).card := by simp\nhave h_subset : s \u222a exp_s \u2286 K := subset_adjoin \u211a (s \u222a exp_s)\nrefine ?_",
"compiler_output_preview": "[path]:21:75: error: unsolved goals\nn : \u2115\nz : Fin n \u2192 \u2102\nh : LinearIndependent \u211a z\ns : Set \u2102 := Set.range z\nexp_s : Set \u2102 := Set.range (cexp \u2218 z)\nK : IntermediateField \u211a \u2102 := adjoin \u211a (s \u222a exp_s)\nhn : n = Finset.univ.card\nh_subset : s \u222a exp_s \u2286 \u2191K\n\u22a2 let hinj := \u22ef;\n \u2191n \u2264 Algebra.trdeg \u211a \u21a5(adjoin \u211a (Set.range z \u222a Set.range (cexp \u2218 z)))\n'Schanuel.schanuel_conjecture' depends on axioms: [\n...<truncated>"
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"proof_code": "intro n\nintro z\nintro h\nlet S1 := Set.range z\nlet S2 := Set.range (Complex.exp \u2218 z)\nlet S := S1 \u222a S2\nlet K := adjoin \u211a S\nhave h_lin : LinearIndependent \u211a z := h\nhave h_trdeg : Algebra.trdeg \u211a K = Algebra.trdeg \u211a (adjoin \u211a S) := rfl\nhave h_card : Set.ncard S1 = n := by\n rw [Set.ncard_range_of_injective]\n \u00b7 exact Fintype.card_fin n\n \u00b7 exact h_lin.injective\nrefine ?_",
"compiler_output_preview": "[path]:23:8: error: tactic 'introN' failed, insufficient number of binders\nn\u271d : \u2115\nz : Fin n\u271d \u2192 \u2102\nh : LinearIndependent \u211a z\nn : Function.Injective \u21d1(algebraMap \u211a \u21a5(adjoin \u211a (Set.range z \u222a Set.range (cexp \u2218 z)))) :=\n RingHom.injective (algebraMap \u211a \u21a5(adjoin \u211a (Set.range z \u222a Set.range (cexp \u2218 z))))\n\u22a2 \u2191n\u271d \u2264 Algebra.trdeg \u211a \u21a5(adjoin \u211a (Set.range z \u222a Set.range (cexp \u2218 z)))\n'Schanuel.schanu\n...<truncated>"
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"compiler_output_preview": "[path]:29:8: error: unknown constant 'Set.ncard_range_of_injective'\n[path]:21:75: error: unsolved goals\nn : \u2115\nz : Fin n \u2192 \u2102\nh : LinearIndependent \u211a z\nS1 : Set \u2102 := Set.range z\nS2 : Set \u2102 := Set.range (cexp \u2218 z)\nS : Set \u2102 := S1 \u222a S2\nK : IntermediateField \u211a \u2102 := adjoin \u211a S\nh_\n...<truncated>"
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"compiler_output_preview": "[path]:21:75: error: unsolved goals\nn : \u2115\nz : Fin n \u2192 \u2102\nh : LinearIndependent \u211a z\ns : Set \u2102 := Set.range z\nexp_s : Set \u2102 := Set.range (cexp \u2218 z)\nS : Set \u2102 := s \u222a exp_s\nK : IntermediateField \u211a \u2102 := adjoin \u211a S\nh_tr : Algebra.trdeg \u211a \u21a5K = Algebra.trdeg \u211a \u21a5(adjoin \u211a S)\nh_inj : Function.Injective z\nh_card : Nat.card \u2191s = n\n\u22a2 let hinj := \u22ef;\n \u2191n \u2264 Algebra.trdeg \u211a \u21a5(adjoin \u211a (Set.range z \u222a S\n...<truncated>"
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"stdout": "[path]:21:75: error: unsolved goals\nn : \u2115\nz : Fin n \u2192 \u2102\nh : LinearIndependent \u211a z\nL : IntermediateField \u211a \u2102 := adjoin \u211a (Set.range z \u222a Set.range (cexp \u2218 z))\nhtr : Algebra.trdeg \u211a \u21a5L = Algebra.trdeg \u211a \u21a5(adjoin \u211a (Set.range z \u222a Set.range (cexp \u2218 z)))\nh_lin : LinearIndependent \u211a z\ns : Set \u2102 := Set.range z\nexp_s : Set \u2102 := Set.range (cexp \u2218 z)\nK : IntermediateField \u211a \u2102 := adjoin \u211a (s \u222a exp_s)\n\u22a2 \u2191n \u2264 Algebra.trdeg \u211a \u21a5K\n'Schanuel.schanuel_conjecture' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
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|
| Gemini 3 Flash Preview | Partial | 1 | Feb 14, 2026 |
simp only [Set.range_comp]
trivial
have h_card : Nat.card (Set.range z) = n := by
rw [Set.card_range_of_injective]
exact h.injective
have h_union_card : Nat.card S ≥ n := by
apply le_trans (m := Nat.card (Set.range z))
· exact Eq.ge h_card
· apply Nat.card_mono
· exact Set.toFinite S
· exact Set.subset_union_left _ _
apply le_trans (m := Algebra.trdeg ℚ K)
· skip
· rfl
apply le_of_not_lt
intro h_lt
let B := Algebra.trdeg ℚ K
apply absurd h
intro h_dep
apply h_lt
apply Nat.le_of_lt
exact h_lt
|
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"status": "partial",
"stderr": "",
"stdout": "[path]:21:75: error: unsolved goals\nn : \u2115\nz : Fin n \u2192 \u2102\nh : LinearIndependent \u211a z\n\u22a2 \u2191n \u2264 Algebra.trdeg \u211a \u21a5(adjoin \u211a (Set.range z \u222a Set.range (cexp \u2218 z)))\n[path]:22:13: warning: This simp argument is unused:\n Set.range_comp\n\nHint: Omit it from the simp argument list.\n simp only \u0335[\u0335S\u0335e\u0335t\u0335.\u0335r\u0335a\u0335n\u0335g\u0335e\u0335_\u0335c\u0335o\u0335m\u0335p\u0335]\u0335\n\nNote: This linter can be disabled with `set_option linter.unusedSimpArgs false`\n[path]:23:4: error: unexpected identifier; expected command\n",
"timed_out": false
}
}
|