Datasets:

neurips-2026-avs-bench
/

formal-anytime-valid-stats

benchmark_id stringlengths 10 43	category stringclasses 8 values	shape_family stringclasses 11 values	statement_lean stringlengths 14 235	imports_required_module_group stringlengths 8 20	tactics_in_reference_proof int64 -1 15	capability_tier stringclasses 5 values	drafter_close_rate_kimi_k2_5 stringclasses 6 values	drafter_close_rate_sonnet_4_6 stringclasses 5 values	drafter_close_rate_gemini_3_pro stringclasses 6 values	notes stringlengths 0 379	required_imports listlengths 0 2	tier stringclasses 6 values	solver_status stringclasses 12 values	drafter_close_rate_mistral_large_3 stringclasses 3 values	drafter_close_rate_harmonic_aristotle stringclasses 4 values	drafter_close_rate_kimi_k2_5_n10 stringclasses 2 values	drafter_close_rate_sonnet_4_6_n10 stringclasses 2 values	drafter_close_rate_gemini_3_pro_n10 stringclasses 2 values	drafter_close_rate_mistral_large_3_n10 stringclasses 2 values	drafter_close_rate_opus_4_6_n10 stringclasses 3 values	drafter_close_rate_harmonic_aristotle_n10 stringclasses 4 values	drafter_close_rate_dspv2_7b_gptq_int8 stringclasses 2 values	drafter_close_rate_dspv2_7b_gptq_int8_n10 stringclasses 2 values	drafter_close_rate_opus_4_6 stringclasses 3 values	schema_version int64 2 2	requires_formal_avs_lean bool 2 classes	drafter_close_rate_goedel_v2_q6k stringclasses 2 values	drafter_close_rate_goedel_v2_q6k_hintlist stringclasses 2 values	agentic_close_rate_goedel_v2_q6k stringclasses 2 values	agentic_close_rate_goedel_v2_q6k_hintlist stringclasses 2 values	agentic_close_rate_sonnet_4_6 stringclasses 2 values	agentic_close_rate_gemini_3_pro stringclasses 2 values
quantization_error	inequality	chain	(s : ℕ) (x : ℝ) → \|x - q s x\| ≤ (2 : ℝ)^(-(s : ℤ))	core_rate_function	8	friendly	0/5	0/5	0/5	N=5 replicate pending	[ "FormalAVSLean.Quantization" ]	T1	closed_by_drafter:gemini_3_pro	0/5		0/10	0/10	0/10	0/10			0/5	0/10	0/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
quantizereal_lower_bound	nonneg	bounded	(s : ℕ) (x : ℝ) → 0 ≤ x → 0 ≤ q s x + (2 : ℝ)^(-(s : ℤ))	core_rate_function	4	friendly	0/5	0/5	1/5	N=5 replicate pending	[ "FormalAVSLean.Quantization" ]	T1	closed_by_drafter:gemini_3_pro	0/5		0/10	0/10	1/10	0/10			0/5	0/10	0/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
etahr_nonneg	nonneg	bounded	(b : ℕ) → 0 ≤ f b	core_rate_function	1	trivial	2/5	1/5	2/5		[ "FormalAVSLean.Quantization" ]	T1	closed_by_drafter:kimi_k2_5,sonnet_4_6,gemini_3_pro,opus_4_6,dspv2_fp16,dspv2_gptq_int4	0/5		1/10	1/10	1/10	0/10	1/10		0/5	0/10	1/5	2	true	0/5	0/5	1/1	0/1	1/1	1/1
etahr_mono	inequality	chain	∀ b₁ b₂ : ℕ, b₁ ≤ b₂ → f b₁ ≤ f b₂	core_rate_function	6	friendly	1/5	2/5	5/5	N=5 replicate pending	[ "FormalAVSLean.Quantization" ]	T1	closed_by_drafter:kimi_k2_5,sonnet_4_6,gemini_3_pro,dspv2_fp16	0/5		1/10	1/10	1/10	0/10			0/5	0/10	0/5	2	true	0/5	0/5	0/1	0/1	1/1	1/1
etahr_le_slack	nonneg	bounded	(σ : ℝ) (p : ParamStruct) → 0 ≤ σ → f p.bits * 2^(-(p.scale : ℤ)) * σ ≤ boundFn σ p	core_rate_function	12	challenging	0/5	0/5	0/5	N=5 replicate pending	[ "FormalAVSLean.Basic", "FormalAVSLean.Quantization" ]	T2	open_as_of_2026_04_23	0/5		0/10	0/10	0/10	0/10			0/5	0/10	0/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
etabetting_pos	nonneg	bounded	(b : ℕ) → 0 < h b	core_rate_function	5	friendly	5/5	3/5	3/5	N=5 replicate pending	[ "FormalAVSLean.Quantization" ]	T1	closed_by_drafter:kimi_k2_5,sonnet_4_6,gemini_3_pro,opus_4_6,dspv2_fp16,dspv2_gptq_int4	0/5		1/10	1/10	1/10	0/10	1/10		0/5	0/10	1/5	2	true	0/5	0/5	0/1	0/1	1/1	1/1
etabetting_nonneg	nonneg	bounded	(b : ℕ) → 0 ≤ h b	core_rate_function	1	trivial	4/5	5/5	1/5		[ "FormalAVSLean.Quantization" ]	T1	closed_by_drafter:kimi_k2_5,sonnet_4_6,gemini_3_pro	0/5		1/10	1/10	1/10	0/10	1/10		1/5	1/10	1/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
etabetting_antitone	inequality	chain	∀ b₁ b₂ : ℕ, b₁ ≤ b₂ → h b₂ ≤ h b₁	core_rate_function	8	friendly	1/5	3/5	5/5	N=5 replicate pending	[ "FormalAVSLean.Quantization" ]	T1	closed_by_drafter:kimi_k2_5,sonnet_4_6,gemini_3_pro	0/5		1/10	1/10	1/10	0/10			0/5	0/10	0/5	2	true	0/5	0/5	0/1	0/1	1/1	1/1
etabetting_le_etahr	monotonic	monotone	(b : ℕ) → 1 ≤ b → h b ≤ f b	core_rate_function	13	challenging	0/5	0/5	0/5	N=5 replicate pending	[ "FormalAVSLean.Quantization" ]	T3	open_as_of_2026_04_23	0/5	1/1	0/10	0/10	0/10	0/10		1/1	0/5	0/10	0/5	2	true	0/5	0/5	0/1	0/1	1/1	1/1
etavector_nonneg	nonneg	bounded	(b : ℕ) → 0 ≤ g b	core_rate_function	1	trivial	4/5	5/5	1/5		[ "FormalAVSLean.Quantization" ]	T1	closed_by_drafter:kimi_k2_5,sonnet_4_6,gemini_3_pro	0/5		1/10	1/10	1/10	0/10			1/5	1/10	0/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
etavector_mono	inequality	chain	∀ b₁ b₂ : ℕ, b₁ ≤ b₂ → g b₁ ≤ g b₂	core_rate_function	6	friendly	2/5	5/5	2/5	N=5 replicate pending	[ "FormalAVSLean.Quantization" ]	T1	closed_by_drafter:kimi_k2_5,sonnet_4_6,gemini_3_pro	0/5		1/10	1/10	1/10	0/10			0/5	0/10	0/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
etavector_eq_sqrt_two_mul_etahr	equality	linear	(b : ℕ) → g b = Real.sqrt 2 * f b	core_rate_function	5	friendly	3/5	0/5	0/5	N=5 replicate pending	[ "FormalAVSLean.Quantization" ]	T3	closed_by_drafter:kimi_k2_5	2/5	1/1	1/10	0/10	0/10	1/10		1/10	1/5	1/10	0/5	2	true	0/5	0/5	0/1	0/1	0/1	1/1
etahr_le_etavector	inequality	chain	(b : ℕ) → f b ≤ g b	core_rate_function	6	friendly	0/5	0/5	0/5	N=5 replicate pending	[ "FormalAVSLean.Quantization" ]	T3	open_as_of_2026_04_23	0/5	1/1	0/10	0/10	0/10	0/10	0/10	1/1	0/5	1/10	0/5	2	true	0/5	0/5	0/1	0/1	0/1	1/1
etaasymptotic_nonneg	nonneg	bounded	(b : ℕ) → 0 ≤ a b	core_rate_function	1	trivial	1/5	2/5	1/5		[ "FormalAVSLean.Quantization" ]	T1	closed_by_drafter:kimi_k2_5,sonnet_4_6,gemini_3_pro	0/5		1/10	1/10	1/10	0/10	1/10		1/5	1/10	1/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
etaasymptotic_const	equality	simple	(b₁ b₂ : ℕ) → a b₁ = a b₂	core_rate_function	1	trivial	5/5	1/5	5/5		[ "FormalAVSLean.Quantization" ]	T1	closed_by_drafter:kimi_k2_5,sonnet_4_6,gemini_3_pro	0/5		1/10	1/10	1/10	0/10	1/10		1/5	1/10	1/5	2	true	0/5	0/5	1/1	1/1	1/1	1/1
etaasymptotic_le_etahr	monotonic	monotone	(b : ℕ) → 1 ≤ b → a b ≤ f b	core_rate_function	4	friendly	0/5	0/5	0/5	N=5 replicate pending	[ "FormalAVSLean.Quantization" ]	T3	open_as_of_2026_04_23	0/5	1/1	0/10	0/10	0/10	0/10		1/1	0/5	0/10	0/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
ranking_four_way	monotonic	monotone	(b : ℕ) → 1 ≤ b → h b ≤ a b ∧ a b ≤ f b ∧ f b ≤ g b	core_rate_function	10	challenging	0/5	0/5	0/5	N=5 replicate pending	[ "FormalAVSLean.Quantization" ]	T3	open_as_of_2026_04_23	0/5	1/1	0/10	0/10	0/10	0/10		1/1	0/5	0/10	0/5	2	true	0/5	0/5	0/1	0/1	0/1	1/1
slack_nonneg	nonneg	bounded	(σ : ℝ) (p : ParamStruct) → 0 ≤ σ → 0 ≤ boundFn σ p	preamble	5	friendly	3/5	2/5	5/5	N=5 replicate pending	[ "FormalAVSLean.Basic" ]	T1	closed_by_drafter:kimi_k2_5,sonnet_4_6,gemini_3_pro	0/5		1/10	1/10	1/10	0/10	1/10		0/5	0/10	1/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
slack_antitone_in_scale	inequality	bound	(σ : ℝ) (bp₁ bp₂ : ParamStruct) → bp₁.scale ≤ bp₂.scale → boundFn σ bp₂ ≤ boundFn σ bp₁	preamble	1	challenging	0/5	0/5	0/5	N=5 replicate pending	[ "FormalAVSLean.Basic" ]	T2	open_as_of_2026_04_23	0/5		0/10	0/10	0/10	0/10			0/5	0/10	0/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
sharpslack_nonneg	nonneg	bounded	(c σ : ℝ) (p : ParamStruct) → 0 ≤ c → 0 ≤ σ → 0 ≤ sharpSlack c σ p	constant_tightening	7	friendly	5/5	1/5	5/5	N=5 replicate pending	[ "FormalAVSLean.Basic" ]	T1	closed_by_drafter:kimi_k2_5,sonnet_4_6,gemini_3_pro	0/5		1/10	1/10	1/10	0/10	1/10		0/5	0/10	1/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
sharpslack_mono_in_c	nonneg	bounded	(c₁ c₂ σ : ℝ) (p : ParamStruct) → 0 ≤ σ → c₁ ≤ c₂ → sharpSlack c₁ σ p ≤ sharpSlack c₂ σ p	constant_tightening	6	friendly	1/5	2/5	5/5	N=5 replicate pending	[ "FormalAVSLean.Basic" ]	T1	closed_by_drafter:kimi_k2_5,sonnet_4_6,gemini_3_pro	0/5		1/10	1/10	1/10	0/10	1/10		0/5	0/10	1/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
realizedcoverageavg_singleton	equality	simple	(impl : StoppingImpl σ p) (claim : CoverageClaim) → realizedCoverageAvg impl (singletonAdversary impl.mart) claim = 1	constant_tightening	3	friendly	0/5	0/5	0/5	N=5 replicate pending	[]	T1	open_as_of_2026_04_23	0/5		0/10	0/10	0/10	0/10			0/5	0/10	0/5	2	false	0/5	0/5	0/1	0/1	0/1	0/1
etavector_subadditive	inequality	chain	(impl : StoppingImpl σ p) (adv : AdversaryFamily σ) (claim : CoverageClaim) → realizedCoverageAvg impl adv claim ≤ 1	constant_tightening	4	friendly	0/5	0/5	0/5	N=5 replicate pending	[]	T1	open_as_of_2026_04_23	0/5		0/10	0/10	0/10	0/10			0/5	0/10	0/5	2	false	0/5	0/5	0/1	0/1	0/1	0/1
c_hr_sharp_pos	nonneg	bounded	0 < c_HR_sharp	bound_composition	4	friendly	3/5	0/5	1/5	N=5 replicate pending	[]	T1	closed_by_drafter:kimi_k2_5,gemini_3_pro	0/5		1/10	0/10	1/10	0/10	1/10		0/5	0/10	1/5	2	false	0/5	0/5	1/1	1/1	0/1	0/1
c_betting_sharp_pos	nonneg	bounded	0 < c_betting_sharp	bound_composition	1	trivial	4/5	5/5	4/5		[]	T1	closed_by_drafter:kimi_k2_5,sonnet_4_6,gemini_3_pro	1/5		1/10	1/10	1/10	1/10	1/10		0/5	0/10	1/5	2	false	0/5	0/5	0/1	0/1	0/1	0/1
c_vector_sharp_pos	nonneg	bounded	0 < c_vector_sharp	bound_composition	4	friendly	0/5	0/5	0/5	N=5 replicate pending	[]	T1	open_as_of_2026_04_23	0/5		0/10	0/10	0/10	0/10			0/5	0/10	0/5	2	false	0/5	0/5	0/1	0/1	0/1	0/1
c_acs_sharp_pos	nonneg	bounded	0 < c_aCS_sharp	bound_composition	5	friendly	4/5	5/5	3/5	N=5 replicate pending	[]	T1	closed_by_drafter:kimi_k2_5,sonnet_4_6,gemini_3_pro	0/5		1/10	1/10	1/10	0/10	1/10		0/5	0/10	1/5	2	false	0/5	0/5	0/1	0/1	0/1	0/1
boundary_pos	nonneg	bounded	(c0 : ℝ) → 0 < c0 → (t : Time) → 0 < boundary c0 t	adversarial_bound	4	friendly	4/5	1/5	4/5	N=5 replicate pending	[ "FormalAVSLean.Basic" ]	T1	closed_by_drafter:kimi_k2_5,sonnet_4_6,gemini_3_pro	0/5		1/10	1/10	1/10	0/10	1/10		0/5	0/10	1/5	2	true	0/5	0/5	1/1	0/1	0/1	0/1
boundary_antitone	nonneg	bounded	(c0 : ℝ) → 0 < c0 → ∀ t₁ t₂ : Time, t₁ ≤ t₂ → boundary c0 t₂ ≤ boundary c0 t₁	adversarial_bound	5	friendly	0/5	3/5	3/5	N=5 replicate pending	[ "FormalAVSLean.Basic" ]	T1	closed_by_drafter:sonnet_4_6,gemini_3_pro	0/5		0/10	1/10	1/10	0/10			0/5	0/10	0/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
slacklower_nonneg	nonneg	bounded	(σ : ℝ) (p : ParamStruct) → 0 < σ → 0 ≤ slackLower σ p	tight_bound	5	friendly	5/5	1/5	4/5	N=5 replicate pending	[ "FormalAVSLean.Basic" ]	T1	closed_by_drafter:kimi_k2_5,sonnet_4_6,gemini_3_pro	0/5		1/10	1/10	1/10	0/10	1/10		0/5	0/10	1/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
slack_tight	nonneg	bounded	(σ : ℝ) (p : ParamStruct) → 0 < σ → slackLower σ p ≤ boundFn σ p ∧ boundFn σ p ≤ 4 * slackLower σ p + 2^(-(p.scale : ℤ))	tight_bound	11	challenging	0/5	0/5	0/5	N=5 replicate pending	[ "FormalAVSLean.Basic" ]	T2	open_as_of_2026_04_23	0/5		0/10	0/10	0/10	0/10			0/5	0/10	0/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
c_hr_sharp_ge_one	algebraic	general	Real.sqrt (2 * Real.log 2) ≥ 1	asymptotic	3	friendly	0/5	0/5	2/5	N=5 replicate pending	[]	T1	closed_by_drafter:gemini_3_pro	0/5		0/10	0/10	1/10	0/10			1/5	1/10	0/5	2	false	0/5	1/5	1/1	1/1	1/1	1/1
c_hr_sharp_le_sqrt_two	algebraic	general	Real.sqrt (2 * Real.log 2) ≤ Real.sqrt 2	asymptotic	3	friendly	0/5	1/5	3/5	N=5 replicate pending	[]	T1	closed_by_drafter:sonnet_4_6,gemini_3_pro	0/5		0/10	1/10	1/10	0/10			1/5	1/10	0/5	2	false	1/5	0/5	1/1	1/1	1/1	1/1
c_hr_sharp_le_six_fifths	algebraic	general	Real.sqrt (2 * Real.log 2) ≤ 6 / 5	asymptotic	4	friendly	0/5	0/5	0/5	N=5 replicate pending	[]	T1	closed_by_drafter:gemini_3_pro	0/5		0/10	0/10	0/10	0/10			0/5	0/10	0/5	2	false	1/5	0/5	1/1	1/1	0/1	0/1
etabetting_le_one	inequality	chain	(b : ℕ) → h b ≤ 1	asymptotic	6	friendly	0/5	0/5	0/5	N=5 replicate pending	[ "FormalAVSLean.Quantization" ]	T1	open_as_of_2026_04_23	0/5		0/10	0/10	0/10	0/10			0/5	0/10	0/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
etabetting_lt_one	monotonic	monotone	(b : ℕ) → 1 ≤ b → h b < 1	asymptotic	7	friendly	0/5	0/5	0/5	N=5 replicate pending	[ "FormalAVSLean.Quantization" ]	T1	open_as_of_2026_04_23	0/5		0/10	0/10	0/10	0/10			0/5	0/10	0/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
etahr_over_etabetting_gt_one	monotonic	monotone	(b : ℕ) → 2 ≤ b → 1 < f b / h b	asymptotic	8	challenging	0/5	0/5	0/5	N=5 replicate pending	[ "FormalAVSLean.Quantization" ]	T3	open_as_of_2026_04_23	0/5	1/1	0/10	0/10	0/10	0/10	1/10	1/1	0/5	0/10	0/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
etahr_mono_supplementary	monotonic	monotone	(b : ℕ) → 1 ≤ b → familyBetting.eta b < familyHR.eta b	master_bound	2	friendly	0/5	0/5	0/5	N=5 replicate pending	[]	T1	open_as_of_2026_04_23	0/5		0/10	0/10	0/10	0/10			0/5	0/10	0/5	2	false	0/5	0/5	0/1	0/1	0/1	0/1
eta_hr_lt_vector	monotonic	monotone	(b : ℕ) → 1 ≤ b → familyHR.eta b < familyVector.eta b	master_bound	4	friendly	0/5	0/5	2/5	N=5 replicate pending	[]	T1	closed_by_drafter:gemini_3_pro	0/5		0/10	0/10	1/10	0/10			0/5	0/10	0/5	2	false	0/5	0/5	0/1	0/1	0/1	0/1
ranking_3_way	monotonic	monotone	(b : ℕ) → 1 ≤ b → familyBetting.eta b < familyHR.eta b ∧ familyHR.eta b < familyVector.eta b	master_bound	1	trivial	0/5	0/5	0/5		[]	T1	open_as_of_2026_04_23	0/5		0/10	0/10	0/10	0/10			0/5	0/10	0/5	2	false	0/5	0/5	0/1	0/1	0/1	0/1
ranking_4_way_master	monotonic	monotone	(b : ℕ) → 1 ≤ b → familyBetting.eta b ≤ familyAsymptotic.eta b ∧ familyAsymptotic.eta b ≤ familyHR.eta b ∧ familyHR.eta b ≤ familyVector.eta b	master_bound	8	challenging	0/5	0/5	0/5	N=5 replicate pending	[]	T2	open_as_of_2026_04_23	0/5		0/10	0/10	0/10	0/10			0/5	0/10	0/5	2	false	0/5	0/5	0/1	0/1	0/1	0/1
dichotomy_universal_monotonicity_impossible	nonneg	sqrt	(σ : ℝ) → 0 < σ → (p : ParamStruct) → ¬∃ d : (Time → ℝ) → Time → Bool, (∀ x t, d x t = true → d x (t+1) = true) ∧ (∀ x t, d x t = true ↔ x t ≥ min (σ * sqrt((t:ℝ + 1) * log 2)) (2^(p.bits - 1) - 1) - 2^(-(p.scale : ℤ)))	adversarial_bound	15	challenging	0/5	0/5	0/5	N=5 replicate pending	[ "FormalAVSLean.Basic" ]	T2	open_as_of_2026_04_23	0/5		0/10	0/10	1/10	0/10			0/5	0/10	0/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
betting_comparison_t2	algebraic	general	familyBetting.slackFn σ p < familyHR.slackFn σ p	comparison_structure	8	friendly	0/5	0/5	0/5	N=5 replicate pending	[]	T1	open_as_of_2026_04_23	0/5		0/10	0/10	0/10	0/10			0/5	0/10	0/5	2	false	0/5	0/5	0/1	0/1	0/1	0/1
real_sqrt_lt_sqrt	mathlib_canon	reference	(x y : ℝ) → 0 ≤ x → 0 ≤ y → x < y → Real.sqrt x < Real.sqrt y	mathlib_canon	2	trivial	0/5	0/5	0/5		[]	T0	open_as_of_2026_04_23	0/5	1/1	0/10	0/10	0/10	0/10		1/1	1/5	1/10	0/5	2	false	1/5	0/5	1/1	1/1	1/1	1/1
nat_le_add_right	mathlib_canon	reference	(n m : ℕ) → n ≤ n + m	mathlib_canon	1	trivial	0/5	0/5	0/5		[]	T0	open_as_of_2026_04_23	0/5	1/1	0/10	0/10	0/10	0/10		1/1	1/5	1/10	0/5	2	false	0/5	1/5	1/1	1/1	1/1	1/1
real_add_sq_le_sq_add_sq	mathlib_canon	reference	(a b x y : ℝ) → a^2 + b^2 ≤ x^2 + y^2 → sqrt(a^2 + b^2) ≤ sqrt(x^2 + y^2)	mathlib_canon	3	friendly	0/5	0/5	0/5	N=5 replicate pending	[]	T0	open_as_of_2026_04_23	0/5	1/1	0/10	0/10	0/10	0/10		1/1	0/5	0/10	0/5	2	false	0/5	0/5	0/1	0/1	0/1	0/1
div_le_div_of_nonneg_left	mathlib_canon	reference	(a b c : ℝ) → 0 < c → 0 ≤ a → a ≤ b → a / b ≤ a / c	mathlib_canon	4	friendly	0/5	0/5	0/5	N=5 replicate pending	[]	T0	open_as_of_2026_04_23	0/5	1/1	0/10	0/10	0/10	0/10		1/1	0/5	0/10	0/5	2	false	0/5	0/5	0/1	0/1	0/1	0/1
mul_nonneg	mathlib_canon	reference	(a b : ℝ) → 0 ≤ a → 0 ≤ b → 0 ≤ a * b	mathlib_canon	1	trivial	0/5	0/5	0/5		[]	T0	open_as_of_2026_04_23	0/5	1/1	0/10	0/10	0/10	0/10		1/1	1/5	1/10	0/5	2	false	1/5	0/5	1/1	1/1	1/1	1/1
Real.exp_pos	mathlib_canon	reference	∀ (x : ℝ), 0 < Real.exp x	mathlib_only	1	trivial	0/5	0/5	0/5	Mathlib-canon reference lemma. In drafter training corpus but not retrieved under 5 greedy+T=0.7 attempts; contributes to T0 OOD dead-zone finding.	[]	T0	open_as_of_2026_04_24_across_4_drafters	0/5	1/1	0/10	0/10	0/10	0/10		1/1	0/5	0/10	0/5	2	false	0/5	0/5	0/1	0/1	0/1	0/1
integral_nonneg	mathlib_canon	reference	∀ {f : ℝ → ℝ} (hf : ∀ x, 0 ≤ f x), 0 ≤ ∫ x, f x	mathlib_only	1	trivial	0/5	0/5	0/5	Mathlib-canon MeasureTheory reference. In drafter training corpus but not retrieved under 5 attempts; contributes to T0 OOD dead-zone finding.	[]	T0	open_as_of_2026_04_24_across_4_drafters	0/5	1/1	0/10	0/10	0/10	0/10		1/1	0/5	0/10	0/5	2	false	0/5	0/5	0/1	0/1	0/1	0/1
etaasymptotic_eq_etahr_at_zero	equality	linear	etaAsymptotic 0 = etaHR 1	core_rate_function	2	friendly	0/5	0/5	0/5	Both sides equal sqrt(log 2) by unfold. Promoted from T2 pool to T1 after proof-difficulty audit (rfl/simp level).	[ "FormalAVSLean.NewTargetsStubs" ]	T1	closed_locally	0/5		0/10	0/10	0/10	0/10			0/5	0/10	0/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
integral_sqrt_nonneg	nonneg	bounded	∀ (b : ℕ), 0 ≤ ∫ x in (0 : ℝ)..(b : ℝ), Real.sqrt (x * Real.log 2)	core_rate_function	2	friendly	0/5	0/5	0/5	intervalIntegral of a nonneg function on [0,b]. Promoted from T2 pool to T1 after proof-difficulty audit (Mathlib integral_nonneg one-liner).	[ "FormalAVSLean.NewTargetsStubs" ]	T1	closed_locally	0/5		0/10	0/10	0/10	0/10			0/5	0/10	0/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
etahr_mono_supplementary	monotonic	monotone	∀ (b₁ b₂ : ℕ), b₁ ≤ b₂ → etaHR b₁ ≤ etaHR b₂	core_rate_function	3	challenging	0/5	0/5	0/5	Monotonicity of etaHR in bit-width. Needs unfold + Real.sqrt_le_sqrt + Nat coercion.	[ "FormalAVSLean.NewTargetsStubs" ]	T2	closed_locally	0/5		0/10	0/10	0/10	0/10			0/5	0/10	0/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
etabetting_le_etahr_pos	inequality	chain	∀ (b : ℕ), 1 ≤ b → etaBetting b ≤ etaHR b	core_rate_function	6	challenging	0/5	0/5	0/5	Rate-function inequality. T2/T3 boundary is proof-technique-based (direct numerical, not Phi-substitution); see §Methods note.	[ "FormalAVSLean.NewTargetsStubs" ]	T2	closed_locally	0/5		0/10	0/10	0/10	0/10			0/5	0/10	0/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
etavector_subadditive	inequality	chain	∀ (b₁ b₂ : ℕ), etaVector (b₁ + b₂) ≤ etaVector b₁ + etaVector b₂	core_rate_function	5	challenging	0/5	0/5	0/5	Subadditivity of etaVector via Real.sqrt_add_le.	[ "FormalAVSLean.NewTargetsStubs" ]	T2	closed_locally	0/5		0/10	0/10	0/10	0/10			0/5	0/10	0/5	2	true	0/5	0/5	0/1	0/1	0/1	0/1
subgaussian_sum_subgaussian	library_gap	mathlib_pr_needed	∀ (X : Fin T → Ω → ℝ) (σ : ℝ) (hσ : 0 < σ), (∀ i, SubGaussianRV (X i) σ) → SubGaussianRV (fun ω => ∑ i, X i ω) (σ * Real.sqrt T)	martingale	-1	frontier	0/5	0/5	0/5	Sum of independent sub-Gaussians is sub-Gaussian. Mathlib has Gaussian case only. Harmonic Aristotle attempted and hit the SubGaussianRV type gap. Resolves when Mathlib ships the general sub-Gaussian MGF closure.	[ "Mathlib.Probability.SubGaussian" ]	T4	library_gap_blocked	0/5	0/1	0/10	0/10	0/10	0/10		0/10	0/5	0/10		2	true	0/5	0/5	0/1	0/1	0/1	0/1
ville_supermartingale	library_gap	mathlib_pr_needed	∀ (M : ℕ → Ω → ℝ) (hM : Supermartingale M) (α : ℝ) (hα : 0 < α), ℙ({ω \| ∃ t, M t ω ≥ α * M 0 ω}) ≤ α⁻¹	martingale	-1	frontier	0/5	0/5	0/5	Ville's inequality for supermartingales. Mathlib has the martingale case only. Harmonic Aristotle attempted and hit this gap.	[ "Mathlib.Probability.Martingale" ]	T4	library_gap_blocked	0/5	0/1	0/10	0/10	0/10	0/10		0/10	0/5	0/10		2	true	0/5	0/5	0/1	0/1	0/1	0/1
vector_subgaussian_mgf	library_gap	mathlib_pr_needed	∀ (d : ℕ) (X : ℕ → Ω → EuclideanSpace ℝ (Fin d)) (Σ : Matrix (Fin d) (Fin d) ℝ) (α : ℝ) (hα : 0 < α), VectorConfidenceSequence X Σ α	martingale	-1	frontier	0/5	0/5	0/5	Vector-valued time-uniform confidence sequence. Mathlib lacks the multivariate time-uniform API. Harmonic Aristotle attempted and hit this gap.	[ "Mathlib.Probability.Martingale", "Mathlib.Analysis.InnerProductSpace.EuclideanDist" ]	T4	library_gap_blocked	0/5	0/1	0/10	0/10	0/10	0/10		0/10	0/5	0/10		2	true	0/5	0/5	0/1	0/1	0/1	0/1
time_uniform_clt	library_gap	mathlib_pr_needed	∀ (X : ℕ → Ω → ℝ) (hX : IIDUniformlyBounded X) (α : ℝ) (hα : 0 < α), TimeUniformCLT X α	martingale	-1	frontier	0/5	0/5	0/5	Time-uniform central limit theorem confidence sequence. 2024 research result not yet in Mathlib. Harmonic Aristotle attempted and hit this gap.	[ "Mathlib.Probability.Independence" ]	T4	library_gap_blocked	0/5	0/1	0/10	0/10	0/10	0/10		0/10	0/5	0/10		2	true	0/5	0/5	0/1	0/1	0/1	0/1
optimal_stopping_boundary	equivalence	cross_family_break	∀ (α σ : ℝ) (hα : 0 < α ∧ α < 1) (hσ : 0 < σ) (s : ℕ), ∃ (tstar : ℕ) (m_tstar : ℝ), decide (quantizeReal s m_tstar ≥ σ * Real.sqrt (2 * tstar * Real.log (1 / α))) ≠ decide (quantizeReal s (m_tstar - σ^2 * tstar / 2) ≥ Real.log (1 / α))	quantization_plus_cs	-1	ceiling	0/5	0/5	0/5	equivalence_break_at_finite_precision. Continuous-equivalence between Howard-Ramdas and Ville thresholds breaks at finite-precision quantization. 0 close-rate across all evaluated solvers: drafters 0/5 unhinted + 0/5 +Phi++ hinted, Aristotle 0/2 (two 20-min sessions cancelled at 1-percent progress marker). No Mathlib g...	[ "FormalAVSLean.EquivalenceBreak" ]	T5	solver_unreached_at_tested_compute_budget	0/5	0/2	0/10	0/10	0/10	0/10		0/10	0/5	0/10		2	true	0/5	0/5	0/1	0/1	0/1	0/1

Formal-AVS: A Lean Benchmark for Anytime-Valid Confidence-Sequence Theorem Proving

60 Lean 4 theorem targets on anytime-valid confidence sequences across four families (Howard-Ramdas, betting, Whitehouse vector, asymptotic CLT).

Benchmark Structure

60 targets grouped into tiers T0-T3 (pre-evaluation) and categories T4-T5 (empirical)
7 drafters evaluated across single-shot, agentic, and unbounded modes
14 Aristotle sessions (unbounded refinement)

Headline Results (pass@5 on 60-target benchmark)

Drafter	T0	T1	T2	T3	Overall
Gemini 3 Pro	0/7	22/34	0/8	0/6	22/60
Claude Sonnet 4.6	0/7	18/34	0/8	0/6	18/60
Kimi K2.5	0/7	17/34	0/8	1/6	18/60
Claude Opus 4.6	0/7	13/34	0/8	0/6	13/60
DSPv2-7B GPTQ-int8	3/7	8/34	0/8	1/6	12/60
Goedel-V2-Q6_K	2/7	3/34	0/8	0/6	5/60
Aristotle	7/7	33/34	7/8	6/6	53/60

Usage

from datasets import load_dataset
ds = load_dataset("neurips-2026-avs-bench/formal-anytime-valid-stats")

Source Datasets

Hand-authored from scratch. Theorem statements formalize results from:

Howard et al. (2021) — doi:10.1214/20-AOS1991
Waudby-Smith and Ramdas (2024) — doi:10.1214/23-AOS2276
Whitehouse et al. (2025) — arXiv:2503.09714

No pre-existing machine learning datasets were used as source data.

Data Collection and Provenance

Collection. Theorem statements were manually written in Lean 4 by the paper authors, formalizing properties of four anytime-valid confidence sequence families from the three source papers listed above. No crowdsourcing, no web scraping, no automated generation.

Annotation. Tier labels (T0-T5) were assigned by the paper authors based on proof shape before any solver evaluation. Per-drafter closure rates were computed by automated sweep: pass@5 with one greedy sample at T=0 and four stochastic samples at T=0.7. Axiom audits were performed via the Lean #print axioms command. Seven language model drafters (Claude Sonnet 4.6, Kimi K2.5, Gemini 3 Pro, Mistral Large 3, Claude Opus 4.6, DSPv2-7B, Goedel-V2-8B) and Harmonic Aristotle were used for proof closure evaluation. The Lean 4 kernel served as the ground-truth verifier.

Personal and Sensitive Information

None. The dataset contains only mathematical theorem statements and solver evaluation metadata.

Social Impact and Limitations

Anytime-valid confidence sequences are deployed in production A/B testing and clinical trial monitoring systems. Formally verified implementations reduce the risk of silent coverage-guarantee violations. This benchmark measures how close current provers are to automating that verification.

Limitations. The benchmark covers one subfield (anytime-valid statistics) with 60 targets. Closure rates may not transfer to other Lean libraries with different import structures. Aristotle is a closed-source system.

Biases. The targets are curated from a single domain and a single proof assistant (Lean 4). Performance on this benchmark does not predict performance on other mathematical domains or proof systems.

License

CC-BY-4.0

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Paper for neurips-2026-avs-bench/formal-anytime-valid-stats

Full-Shape analysis of the power spectrum and bispectrum of DESI DR1 LRG and QSO samples

Paper • 2503.09714 • Published Jun 6, 2025