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DiscreteTopology.of_t2_of_torontoSpace
Specification
Any T2, Toronto space is discrete.
Lean 4 Statement
theorem DiscreteTopology.of_t2_of_torontoSpace [T2Space X] [TorontoSpace X] :
DiscreteTopology X
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Evaluation Results
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| Model | Status | Goals Left | Submitted | Lean Snippet | Output |
|---|---|---|---|---|---|
| GPT-5.3 Codex Agent v8 | Failed | -- | Feb 25, 2026 |
infer_instance
|
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"error": null,
"provider": "openrouter_primary",
"finish_reason": "tool_calls",
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},
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"query": "DiscreteTopology.of_isClosed_singleton",
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"event": "search_mathlib",
"query": "TorontoSpace",
"results_count": 0
},
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"response_id": "gen-1772029493-lX6fpWkhVuBi42zn5vWE",
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"reasoning_text": "**Analyzing cardinality and topology interplay**",
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"summary": "**Analyzing cardinality and topology interplay**"
},
{
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"
}
]
},
{
"iteration": 4,
"event": "compile_proof",
"provider": "openrouter_primary",
"status": "failed",
"goals_remaining": null,
"timed_out": false,
"banned_tokens": [],
"proof_code": "infer_instance",
"compiler_output_preview": "/app/asi_portal/verifier/quick_1772029574_8543.lean:46:2: error(lean.synthInstanceFailed): failed to synthesize instance of type class\n DiscreteTopology X\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.\n'Toronto.DiscreteTopology.of_t2_of_torontoSpace' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
"compile_elapsed_s": 28.242,
"tactic_replay": {
"tactic_count": 10,
"successful_tactics": 9,
"first_failure_step": 10,
"goal_trajectory": [
1,
1,
1,
1,
1,
1,
1,
1,
1,
1
],
"max_goals": 1,
"final_goal_count": 1,
"goals_opened": 0,
"goals_closed": 0
},
"lsp_feedback": {
"diagnostics": [
{
"code": "lean.synthInstanceFailed",
"fullRange": {
"end": {
"character": 16,
"line": 45
},
"start": {
"character": 2,
"line": 45
}
},
"message": "failed to synthesize instance of type class\n DiscreteTopology X\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.",
"range": {
"end": {
"character": 16,
"line": 45
},
"start": {
"character": 2,
"line": 45
}
},
"severity": 1,
"source": "Lean 4"
},
{
"fullRange": {
"end": {
"character": 6,
"line": 47
},
"start": {
"character": 0,
"line": 47
}
},
"message": "'Toronto.DiscreteTopology.of_t2_of_torontoSpace' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]",
"range": {
"end": {
"character": 6,
"line": 47
},
"start": {
"character": 0,
"line": 47
}
},
"severity": 3,
"source": "Lean 4"
}
],
"error_line": 45,
"error_char": 2,
"error_msg": "failed to synthesize instance of type class\n DiscreteTopology X\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.",
"goal_before_error": "```lean\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : T2Space X\ninst\u271d : TorontoSpace X\n\u22a2 DiscreteTopology X\n```",
"full_timeline": [
{
"line": 32,
"code": "intro Y hY",
"goal": "```lean\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Finite X\n\u22a2 \u2983Y : Set X\u2984 \u2192 #\u2191Y = #X \u2192 \u2191Y \u2243\u209c X\n```",
"goal_count": 1,
"ast_complexity": 104
},
{
"line": 33,
"code": "have eq : Y = Set.univ := by",
"goal": "```lean\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Finite X\nY : Set X\nhY : #\u2191Y = #X\n\u22a2 \u2191Y \u2243\u209c X\n```",
"goal_count": 1,
"ast_complexity": 103
},
{
"line": 34,
"code": "refine (Set.eq_univ_iff_ncard Y).mpr ?_",
"goal": "```lean\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Finite X\nY : Set X\nhY : #\u2191Y = #X\n\u22a2 Y = Set.univ\n```",
"goal_count": 1,
"ast_complexity": 108
},
{
"line": 35,
"code": "have : Y.ncard = (#\u2191Y).toNat := by",
"goal": "```lean\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Finite X\nY : Set X\nhY : #\u2191Y = #X\n\u22a2 Y.ncard = Nat.card X\n```",
"goal_count": 1,
"ast_complexity": 116
},
{
"line": 36,
"code": "exact rfl",
"goal": "```lean\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Finite X\nY : Set X\nhY : #\u2191Y = #X\n\u22a2 Y.ncard = Cardinal.toNat #\u2191Y\n```",
"goal_count": 1,
"ast_complexity": 124
},
{
"line": 37,
"code": "rw [this, hY]",
"goal": "```lean\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Finite X\nY : Set X\nhY : #\u2191Y = #X\nthis : Y.ncard = Cardinal.toNat #\u2191Y\n\u22a2 Y.ncard = Nat.card X\n```",
"goal_count": 1,
"ast_complexity": 152
},
{
"line": 38,
"code": "exact rfl",
"goal": "```lean\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Finite X\nY : Set X\nhY : #\u2191Y = #X\nthis : Y.ncard = Cardinal.toNat #\u2191Y\n\u22a2 Cardinal.toNat #X = Nat.card X\n```",
"goal_count": 1,
"ast_complexity": 162
},
{
"line": 39,
"code": "rw [eq]",
"goal": "```lean\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Finite X\nY : Set X\nhY : #\u2191Y = #X\neq : Y = Set.univ\n\u22a2 \u2191Y \u2243\u209c X\n```",
"goal_count": 1,
"ast_complexity": 121
},
{
"line": 40,
"code": "exact Homeomorph.Set.univ X",
"goal": "```lean\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Finite X\nY : Set X\nhY : #\u2191Y = #X\neq : Y = Set.univ\n\u22a2 \u2191Set.univ \u2243\u209c X\n```",
"goal_count": 1,
"ast_complexity": 128
},
{
"line": 45,
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"goal": "```lean\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : T2Space X\ninst\u271d : TorontoSpace X\n\u22a2 DiscreteTopology X\n```",
"goal_count": 1,
"ast_complexity": 115
}
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"cps": 90.0,
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"compiler_output_preview": "/app/asi_portal/verifier/quick_1772029636_2357.lean:46:9: error: Function expected at\n DiscreteTopology.of_subset ?m.9 ?m.10\nbut this term has type\n DiscreteTopology \u2191?m.8\n\nNote: Expected a function because this term is being applied to the argument\n ?_\n'Toronto.DiscreteTopology.of_t2_of_torontoSpace' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n\n",
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"end": {
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"line": 45
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"message": "Function expected at\n DiscreteTopology.of_subset ?m.9 ?m.10\nbut this term has type\n DiscreteTopology \u2191?m.8\n\nNote: Expected a function because this term is being applied to the argument\n ?_",
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},
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"goal_before_error": "```lean\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : T2Space X\ninst\u271d : TorontoSpace X\n\u22a2 DiscreteTopology X\n```",
"full_timeline": [
{
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"code": "intro Y hY",
"goal": "```lean\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Finite X\n\u22a2 \u2983Y : Set X\u2984 \u2192 #\u2191Y = #X \u2192 \u2191Y \u2243\u209c X\n```",
"goal_count": 1,
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},
{
"line": 33,
"code": "have eq : Y = Set.univ := by",
"goal": "```lean\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Finite X\nY : Set X\nhY : #\u2191Y = #X\n\u22a2 \u2191Y \u2243\u209c X\n```",
"goal_count": 1,
"ast_complexity": 103
},
{
"line": 34,
"code": "refine (Set.eq_univ_iff_ncard Y).mpr ?_",
"goal": "```lean\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Finite X\nY : Set X\nhY : #\u2191Y = #X\n\u22a2 Y = Set.univ\n```",
"goal_count": 1,
"ast_complexity": 108
},
{
"line": 35,
"code": "have : Y.ncard = (#\u2191Y).toNat := by",
"goal": "```lean\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Finite X\nY : Set X\nhY : #\u2191Y = #X\n\u22a2 Y.ncard = Nat.card X\n```",
"goal_count": 1,
"ast_complexity": 116
},
{
"line": 36,
"code": "exact rfl",
"goal": "```lean\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Finite X\nY : Set X\nhY : #\u2191Y = #X\n\u22a2 Y.ncard = Cardinal.toNat #\u2191Y\n```",
"goal_count": 1,
"ast_complexity": 124
},
{
"line": 37,
"code": "rw [this, hY]",
"goal": "```lean\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Finite X\nY : Set X\nhY : #\u2191Y = #X\nthis : Y.ncard = Cardinal.toNat #\u2191Y\n\u22a2 Y.ncard = Nat.card X\n```",
"goal_count": 1,
"ast_complexity": 152
},
{
"line": 38,
"code": "exact rfl",
"goal": "```lean\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Finite X\nY : Set X\nhY : #\u2191Y = #X\nthis : Y.ncard = Cardinal.toNat #\u2191Y\n\u22a2 Cardinal.toNat #X = Nat.card X\n```",
"goal_count": 1,
"ast_complexity": 162
},
{
"line": 39,
"code": "rw [eq]",
"goal": "```lean\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Finite X\nY : Set X\nhY : #\u2191Y = #X\neq : Y = Set.univ\n\u22a2 \u2191Y \u2243\u209c X\n```",
"goal_count": 1,
"ast_complexity": 121
},
{
"line": 40,
"code": "exact Homeomorph.Set.univ X",
"goal": "```lean\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Finite X\nY : Set X\nhY : #\u2191Y = #X\neq : Y = Set.univ\n\u22a2 \u2191Set.univ \u2243\u209c X\n```",
"goal_count": 1,
"ast_complexity": 128
},
{
"line": 45,
"code": "refine DiscreteTopology.of_subset ?_ ?_ ?_",
"goal": "```lean\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : T2Space X\ninst\u271d : TorontoSpace X\n\u22a2 DiscreteTopology X\n```",
"goal_count": 1,
"ast_complexity": 115
}
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"cps": 90.0,
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],
"comparator_passed": null,
"duration_s": 12.816506147384644,
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"lean_file_path": "",
"problem": {
"axioms_check_name": "DiscreteTopology.of_t2_of_torontoSpace",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__Toronto__DiscreteTopology.of_t2_of_torontoSpace",
"category": "research_open",
"endings": [
"end Toronto"
],
"imports": [
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],
"preamble_chars": 1007,
"source_file": "FormalConjectures/Wikipedia/Toronto.lean",
"title": "DiscreteTopology.of_t2_of_torontoSpace"
},
"returncode": 1,
"status": "failed",
"stderr": "",
"stdout": "/app/asi_portal/verifier/quick_1772029664_6185.lean:46:2: error(lean.synthInstanceFailed): failed to synthesize instance of type class\n DiscreteTopology X\n\nHint: Type class instance resolution failures can be inspected with the `set_option trace.Meta.synthInstance true` command.\n'Toronto.DiscreteTopology.of_t2_of_torontoSpace' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n",
"timed_out": false
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"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 | Failed | -- | Feb 17, 2026 |
constructor
intro U
obtain h_fin | h_inf := em (Finite X)
· haveI : DiscreteTopology X := by
cases h_fin
have h_homeo := TorontoSpace.toronto (Y := Set.univ) (by simp)
apply discreteTopology_iff_singleton_mem_nhds.mpr
intro x
exact T2Space.t2_separation x
exact IsOpen.union (Set.univ_subset_iff.mp (by simp)) (by simp)
· haveI h_t2 : T2Space X := ‹_›
haveI h_tor : TorontoSpace X := ‹_›
by_contra h_not_discrete
have h_not_bot : 𝓣 ≠ ⊥ := by
intro h_bot
subst h_bot
exact h_not_discrete (DiscreteTopology.mk rfl)
set s := {x | IsOpen {x}}
have h_s_open : ∀ x ∈ s, IsOpen {x} := fun x hx => hx
let Y := (Set.univ : Set X)
have h_equiv := h_tor.toronto (Y := Y) (by simp)
exact h_not_discrete (DiscreteTopology.mk (by
refine eq_bot_of_singletons_open (fun x => ?_)
by_contra h_nhd
let Z : Set X := Set.univ
have hZ : #Z = #X := by simp
have h_homeo := h_tor.toronto hZ
aesop))
|
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"proof_code": "constructor\n\u00b7 apply topologicalSpace_eq_bot_of_countably_generated_at\n intro x\n obtain \u27e8h_fin | h_inf\u27e9 := em (Finite X)\n \u00b7 have : DiscreteTopology X := by\n letI : TorontoSpace X := Finite.torontoSpace X\n have : Subsingleton (TopologicalSpace X) := by\n refine \u27e8\u03bb t1 t2 => ?_\u27e9\n admit\n admit\n exact (DiscreteTopology.discrete X).nhds x\n \u00b7 let s := {x}\n have : \u00ac #X \u2264 #s := by\n rw [Cardinal.mk_singleton]\n apply Cardinal.one_lt_iff_ne_zero.mpr\n simp [h_inf]\n admit\n\u00b7 intro x\n have h_t2 := \u2039T2Space X\u203a\n have h_tor := \u2039TorontoSpace X\u203a\n obtain \u27e8h_fin | h_inf\u27e9 := em (Finite X)\n \u00b7 exact (Finite.torontoSpace X).toronto (Set.eq_univ_iff_ncard _ |>.mpr rfl)\n \u00b7 admit",
"compiler_output_preview": "\nBanned tokens found: admit"
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"proof_code": "constructor\nintro U\nobtain h_fin | h_inf := em (Finite X)\n\u00b7 haveI : DiscreteTopology X := by\n cases h_fin\n have h_homeo := TorontoSpace.toronto (Y := Set.univ) (by simp)\n apply discreteTopology_iff_singleton_mem_nhds.mpr\n intro x\n exact T2Space.t2_separation x\n exact IsOpen.union (Set.univ_subset_iff.mp (by simp)) (by simp)\n\u00b7 haveI h_t2 : T2Space X := \u2039_\u203a\n haveI h_tor : TorontoSpace X := \u2039_\u203a\n by_contra h_not_discrete\n have h_not_bot : \ud835\udce3 \u2260 \u22a5 := by\n intro h_bot\n subst h_bot\n exact h_not_discrete (DiscreteTopology.mk rfl)\n set s := {x | IsOpen {x}}\n have h_s_open : \u2200 x \u2208 s, IsOpen {x} := fun x hx => hx\n let Y := (Set.univ : Set X)\n have h_equiv := h_tor.toronto (Y := Y) (by simp)\n exact h_not_discrete (DiscreteTopology.mk (by\n refine eq_bot_of_singletons_open (fun x => ?_)\n by_contra h_nhd\n let Z : Set X := Set.univ\n have hZ : #Z = #X := by simp\n have h_homeo := h_tor.toronto hZ\n aesop))",
"compiler_output_preview": "[path]:47:8: error: tactic 'introN' failed, insufficient number of binders\ncase eq_bot\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : T2Space X\ninst\u271d : TorontoSpace X\n\u22a2 inst\u271d\u00b2 = \u22a5\n'Toronto.DiscreteTopology.of_t2_of_torontoSpace' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
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"proof_code": "constructor\nunfold DiscreteTopology\nrefine eq_bot_of_singletons_open (fun x => ?_)\nby_contra h_not_open\nobtain h_fin | h_inf := em (Finite X)\n\u00b7 letI : TorontoSpace X := Finite.torontoSpace X\n obtain \u27e8y, hy\u27e9 := (Set.univ : Set X).nonempty\n have h_univ : #(Set.univ : Set X) = #X := by simp\n have h_homeo := TorontoSpace.toronto h_univ\n exact h_not_open (IsOpen.singleton x)\n\u00b7 have h_tor := \u2039TorontoSpace X\u203a\n have h_t2 := \u2039T2Space X\u203a\n let s := {z | IsOpen {z}}\n have h_s_open : \u2200 z \u2208 s, IsOpen {z} := fun z hz => hz\n by_cases h_s_card : #s = #X\n \u00b7 have h_homeo : s \u2243\u209c X := h_tor.toronto h_s_card\n have : DiscreteTopology s := \u27e8by\n refine eq_bot_of_singletons_open (fun \u27e8z, hz\u27e9 => ?_)\n have : IsOpen ({z} : Set X) := hz\n rw [isOpen_induced_iff]\n refine \u27e8{z}, hz, by ext; simp\u27e9\u27e9\n have : DiscreteTopology X := h_homeo.discreteTopology_iff.mp this\n exact h_not_open (IsOpen.singleton x)\n \u00b7 let s_comp := s\u1d9c\n have h_sc_card : #s_comp = #X := by\n refine Cardinal.eq_of_add_eq_of_not_lt ?_ (by\n intro h_lt\n have := Cardinal.mk_add_mk_compl s\n rw [h_lt.le.mk_add_mk_compl_eq_left (Cardinal.infinite_iff.mp h_inf)] at this\n exact h_s_card (this.symm))\n rw [Cardinal.mk_add_mk_compl s]\n have h_homeo_sc : s_comp \u2243\u209c X := h_tor.toronto h_sc_card\n have : x \u2208 s_comp := (Set.mem_compl_iff _ _).mpr h_not_open\n aesop",
"compiler_output_preview": "[path]:47:9: error: tactic 'unfold' failed to unfold 'DiscreteTopology' at\n inst\u271d\u00b2 = \u22a5\n'Toronto.DiscreteTopology.of_t2_of_torontoSpace' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n\n"
},
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"proof_code": "constructor\nhaveI h_t2 : T2Space X := \u2039_\u203a\nhaveI h_tor : TorontoSpace X := \u2039_\u203a\nrefine TopologicalSpace.ext_iff.mpr (\u03bb U => ?_)\nconstructor\n\u00b7 intro hU\n refine eq_bot_of_singletons_open (\u03bb x => ?_)\n obtain h_fin | h_inf := em (Finite X)\n \u00b7 cases h_fin\n have h_homeo := h_tor.toronto (Y := Set.univ) (by simp)\n obtain \u27e8y, hy\u27e9 := (Set.univ : Set X).nonempty\n let s_y : Set X := {y}\n have : IsOpen s_y := by\n letI : TorontoSpace X := Finite.torontoSpace X\n aesop\n aesop\n \u00b7 let S := {z | IsOpen {z}}\n by_cases h_S : #S = #X\n \u00b7 have h_home : S \u2243\u209c X := h_tor.toronto h_S\n haveI : DiscreteTopology S := \u27e8by\n ext V\n constructor\n \u00b7 intro _\n apply IsOpen.induced\n have h_open_V : \u2200 v \u2208 V, IsOpen {v.val} := \u03bb v _ => v.property\n exact isOpen_iff_forall_mem_open.mpr (\u03bb v hv => \u27e8{v.val}, Set.subset_singleton_iff.mpr rfl, v.property, rfl\u27e9)\n \u00b7 intro; exact isOpen_induced_iff.mpr \u27e8V, by aesop, by aesop\u27e9\u27e9\n exact h_home.discreteTopology_iff.mp inferInstance\n \u00b7 let S_c := S\u1d9c\n have h_card : #S_c = #X := by\n have := Cardinal.mk_add_mk_compl S\n rw [Cardinal.add_eq_max] at this\n aesop\n have h_home_c : S_c \u2243\u209c X := h_tor.toronto h_card\n aesop\n\u00b7 intro h_bot\n rw [h_bot]\n exact isOpen_bot",
"compiler_output_preview": "[path]:52:4: error: type mismatch\n eq_bot_of_singletons_open fun x \u21a6 ?m.1458\nhas type\n ?m.1452 = \u22a5 : Prop\nbut is expected to have type\n IsOpen U : Prop\n[path]:82:8: error: tactic 'rewrite' failed, equality or iff proof expected\n IsOpe\n...<truncated>"
},
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"proof_code": "constructor\nhaveI : T2Space X := \u2039_\u203a\nhaveI : TorontoSpace X := \u2039_\u203a\nhave h_bot : \ud835\udce3 = \u22a5 := by\n refine eq_bot_of_singletons_open (\u03bb x => ?_)\n obtain h_fin | h_inf := em (Finite X)\n \u00b7 cases h_fin\n have h_homeo := TorontoSpace.toronto (Y := (Set.univ : Set X)) (by simp)\n let s := {z | IsOpen {z}}\n have : x \u2208 s := by\n have : s.Nonempty := by\n obtain \u27e8y\u27e9 := (inferInstance : Nonempty X)\n have := T2Space.t2_separation y\n aesop\n have h_s_card : #s = #X := by\n rw [Cardinal.mk_set_eq_nat_iff_ncard]\n aesop\n have h_homeo_s := TorontoSpace.toronto h_s_card\n aesop\n exact this\n \u00b7 let s := {z | IsOpen {z}}\n by_cases h_s_card : #s = #X\n \u00b7 have h_homeo := TorontoSpace.toronto h_s_card\n have h_disc_s : DiscreteTopology s := \u27e8by\n ext V\n refine \u27e8\u03bb _ => IsOpen.induced V, \u03bb _ => ?_\u27e9\n have : \u2200 v \u2208 V, IsOpen ({v.val} : Set X) := \u03bb v _ => v.2\n rw [isOpen_induced_iff]\n refine \u27e8\u22c3 v \u2208 V, {v.val}, isOpen_biUnion this, by ext; simp\u27e9\u27e9\n have : DiscreteTopology X := h_homeo.discreteTopology_iff.mp h_disc_s\n exact IsOpen.singleton x\n \u00b7 let s_c := s\u1d9c\n have h_sc_card : #s_c = #X := by\n have := Cardinal.mk_add_mk_compl s\n have h_max := Cardinal.add_eq_max (Cardinal.infinite_iff.mp h_inf)\n rw [h_max] at this\n aesop\n have h_homeo_sc := TorontoSpace.toronto h_sc_card\n aesop\nexact h_bot",
"compiler_output_preview": "[path]:49:15: error: unknown identifier '\ud835\udce3'\n[path]:45:26: error: unsolved goals\ncase eq_bot\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : T2Space X\ninst\u271d : TorontoSpace X\nthis\u271d : T2Space X\nthis : TorontoSpace X\n\u22a2 inst\u271d\u00b2 = \u22a5\n'Toronto.D\n...<truncated>"
},
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"proof_code": "constructor\nletI h_t2 : T2Space X := \u2039_\u203a\nletI h_tor : TorontoSpace X := \u2039_\u203a\nhave h_bot : \u2039TopologicalSpace X\u203a = \u22a5 := by\n refine eq_bot_of_singletons_open (\u03bb x => ?_)\n obtain \u27e8h_fin\u27e9 | h_inf := em (Finite X)\n \u00b7 let s := {z | IsOpen ({z} : Set X)}\n have h_s_nonempty : s.Nonempty := by\n obtain \u27e8y\u27e9 := (inferInstance : Nonempty X)\n cases \u2039Finite X\u203a\n have : \u2200 (A : Set X), IsOpen A := by\n intro A\n induction A using Set.Finite.induction_on\n \u00b7 simp\n \u00b7 rw [Set.insert_eq]\n apply IsOpen.union\n letI : TorontoSpace X := Finite.torontoSpace X\n aesop\n exact \u27e8y, this {y}\u27e9\n have h_s_card : #s = #X := by\n simp [Cardinal.mk_set_eq_nat_iff_ncard, Finite.cast_ncard_eq_mk]\n congr\n ext z; constructor <;> intro\n \u00b7 trivial\n \u00b7 apply IsOpen.singleton\n have h_homeo := h_tor.toronto h_s_card\n exact IsOpen.singleton x\n \u00b7 let s := {z | IsOpen ({z} : Set X)}\n by_cases h_s : #s = #X\n \u00b7 have h_homeo := h_tor.toronto h_s\n haveI : DiscreteTopology s := \u27e8by\n ext V\n constructor\n \u00b7 intro _; apply IsOpen.induced V\n \u00b7 intro hV\n obtain \u27e8U, hU, hUV\u27e9 := isOpen_induced_iff.mp hV\n rw [\u2190 hUV]\n apply IsOpen.preimage _ hU\n exact continuous_subtype_val\u27e9\n exact h_homeo.discreteTopology_iff.mp inferInstance |>.is_open_singleton x\n \u00b7 let s_c := s\u1d9c\n have h_sc_card : #s_c = #X := by\n have := Cardinal.mk_add_mk_compl s\n rw [Cardinal.add_eq_max (Cardinal.infinite_iff.mp h_inf)] at this\n aesop\n have h_homeo := h_tor.toronto h_sc_card\n aesop\nexact h_bot",
"compiler_output_preview": "[path]:54:23: error: failed to synthesize\n Nonempty X\n\nHint: Additional diagnostic information may be available using the `set_option diagnostics true` command.\n[path]:58:10: error: insufficient number of targets for '@Set.Finite.inducti\n...<truncated>"
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"proof_code": "constructor\ncases isEmpty_or_nonempty X\n\u00b7 exact \u27e8Subsingleton.elim _ \u22a5\u27e9\n\u00b7 letI : T2Space X := \u2039_\u203a\n letI : TorontoSpace X := \u2039_\u203a\n refine \u27e8eq_bot_of_singletons_open (fun x => ?_)\u27e9\n obtain h_fin | h_inf := em (Finite X)\n \u00b7 letI := h_fin\n have : DiscreteTopology X := by\n let s := {z | IsOpen ({z} : Set X)}\n have hs : #s = #X := by\n suffices s = Set.univ by rw [this]; simp\n refine Set.eq_univ_of_forall (fun z => ?_)\n by_contra hz\n let sc := s\u1d9c\n have : sc.Nonempty := \u27e8z, hz\u27e9\n obtain \u27e8y, hy\u27e9 := TorontoSpace.toronto (Y := (Set.univ : Set X)) (by simp) |>.symm.surjective \u27e8z, hz\u27e9\n aesop\n have h_homeo := TorontoSpace.toronto hs\n exact h_homeo.discreteTopology_iff.mp (DiscreteTopology.mk (eq_bot_of_singletons_open (fun \u27e8z, hz\u27e9 => by\n rw [isOpen_induced_iff]; refine \u27e8{z}, hz, by ext; simp\u27e9)))\n exact IsOpen.singleton x\n \u00b7 let s := {z | IsOpen ({z} : Set X)}\n by_cases h_s : #s = #X\n \u00b7 have h_homeo := TorontoSpace.toronto h_s\n haveI : DiscreteTopology s := \u27e8eq_bot_of_singletons_open (fun \u27e8z, hz\u27e9 => by\n rw [isOpen_induced_iff]; refine \u27e8{z}, hz, by ext; simp\u27e9)\u27e9\n exact h_homeo.discreteTopology_iff.mp \u2039_\u203a |>.isOpen_singleton x\n \u00b7 let sc := s\u1d9c\n have h_sc : #sc = #X := by\n have h_sum := Cardinal.mk_add_mk_compl s\n rw [Cardinal.add_eq_max (Cardinal.infinite_iff.mp h_inf)] at h_sum\n aesop\n have h_homeo := TorontoSpace.toronto h_sc\n aesop",
"compiler_output_preview": "[path]:48:10: error: invalid constructor \u27e8...\u27e9, insufficient number of arguments, constructs 'Eq.refl' does not have explicit fields, but #1 provided\n[path]:51:11: error: invalid constructor \u27e8...\u27e9, insufficient number of arguments, constr\n...<truncated>"
},
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"banned_tokens": [
"sorry"
],
"proof_code": "constructor\nobtain h_fin | h_inf := em (Finite X)\n\u00b7 cases h_fin\n have h_univ : #(Set.univ : Set X) = #X := by simp\n have h_homeo := TorontoSpace.toronto h_univ\n refine \u27e8eq_bot_of_singletons_open (fun x => ?_)\u27e9\n let s := {z | IsOpen ({z} : Set X)}\n have h_s : s = Set.univ := by\n refine Set.eq_univ_of_forall (fun z => ?_)\n by_contra hz\n have : #s < #X := by\n rw [Cardinal.mk_set_eq_nat_iff_finset]\n simp\n sorry\n aesop\n exact h_s.symm.subset (Set.mem_univ x)\n\u00b7 haveI : Infinite X := Cardinal.infinite_iff.mp h_inf\n refine \u27e8eq_bot_of_singletons_open (fun x => ?_)\u27e9\n let s := {z | IsOpen ({z} : Set X)}\n by_cases h_scard : #s = #X\n \u00b7 have h_homeo := TorontoSpace.toronto h_scard\n haveI : DiscreteTopology s := \u27e8eq_bot_of_singletons_open (fun \u27e8z, hz\u27e9 => by\n rw [isOpen_induced_iff]\n refine \u27e8{z}, hz, by ext; simp\u27e9)\u27e9\n exact h_homeo.discreteTopology_iff.mp \u2039_\u203a |>.isOpen_singleton x\n \u00b7 let sc := s\u1d9c\n have h_sc_card : #sc = #X := by\n have := Cardinal.mk_add_mk_compl s\n rw [Cardinal.add_eq_max (Cardinal.infinite_iff.mp h_inf)] at this\n simp at this\n aesop\n have h_homeo := TorontoSpace.toronto h_sc_card\n aesop",
"compiler_output_preview": "\nBanned tokens found: sorry"
}
],
"verification": {
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"command": [
"[path]
"lean",
"[path]
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"lean_file_path": "[path]
"problem": {
"axioms_check_name": "DiscreteTopology.of_t2_of_torontoSpace",
"axioms_check_skipped": false,
"benchmark_id": "Wikipedia__Toronto__DiscreteTopology.of_t2_of_torontoSpace",
"category": "research_open",
"endings": [
"end Toronto"
],
"imports": [
"FormalConjectures.Util.ProblemImports"
],
"preamble_chars": 1007,
"source_file": "FormalConjectures/Wikipedia/Toronto.lean",
"title": "DiscreteTopology.of_t2_of_torontoSpace"
},
"returncode": 1,
"status": "failed",
"stderr": "",
"stdout": "[path]:47:8: error: tactic 'introN' failed, insufficient number of binders\ncase eq_bot\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : T2Space X\ninst\u271d : TorontoSpace X\n\u22a2 inst\u271d\u00b2 = \u22a5\n'Toronto.DiscreteTopology.of_t2_of_torontoSpace' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]\n",
"timed_out": false
}
}
|
| Gemini 3 Flash Preview | Failed | -- | Feb 14, 2026 |
by_cases hf : Finite X
· exact finite_discrete_of_t2_of_toronto hf
· constructor
rw [eq_bot_iff]
intro U hU
rw [isOpen_discrete]
apply TorontoSpace.toronto
have : #X = #X := rfl
let f := TorontoSpace.toronto (Y := Set.univ) (by simp)
have h_iso : ∀ x : X, IsIsolated x
· intro x
let S := {x}ᶜ
by_contra h_not
have h_card : #(Set.univ : Set X) = #X := by simp
have h_homeo := TorontoSpace.toronto h_card
exact h_not (isIsolated_of_homeo h_homeo x)
exact h_iso
|
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