{"name": "MonoidAlgebra.support_coeff_mul_single", "query": "{k : Type u₁} {G : Type u₂} [Semiring k] [Mul G] [IsRightCancelMul G] (f : MonoidAlgebra k G) (r : k) (hr : ∀ (y : k), y * r = 0 ↔ y = 0) (x : G) : (f * MonoidAlgebra.single x r).coeff.support = Finset.map (mulRightEmbedding x) f.coeff.support", "area": "algebra", "module": "Mathlib.Algebra.MonoidAlgebra.Support", "concrete": true, "gold": ["IsRightRegular.all", "MonoidAlgebra.support_coeff_mul_single_eq_image"], "gold_b": ["Finset.ext", "Finset.mem_image", "Finset.mem_map", "Finsupp.mem_support_iff", "IsRightRegular.all", "MonoidAlgebra.support_coeff_mul_single_eq_image"], "split": "dev"}
{"name": "Polynomial.X_mul_monomial", "query": "{R : Type u} [Semiring R] (n : ℕ) (r : R) : Polynomial.X * (Polynomial.monomial n) r = (Polynomial.monomial (n + 1)) r", "area": "algebra", "module": "Mathlib.Algebra.Polynomial.Basic", "concrete": true, "gold": ["Polynomial.X_mul", "Polynomial.monomial_mul_X"], "gold_b": ["Polynomial.X_mul", "Polynomial.monomial_mul_X"], "split": "dev"}
{"name": "AlternatingMap.alternatizeUncurryFin_curryLeft", "query": "{R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [AddCommGroup M] [AddCommGroup N] [_root_.Module R M] [_root_.Module R N] {n : ℕ} (f : M [⋀^Fin (n + 1)]→ₗ[R] N) : AlternatingMap.alternatizeUncurryFin f.curryLeft = (n + 1) • f", "area": "algebra", "module": "Mathlib.LinearAlgebra.Alternating.Uncurry.Fin", "concrete": true, "gold": ["AlternatingMap.alternatizeUncurryFin_apply", "AlternatingMap.map_insertNth"], "gold_b": ["AlternatingMap.alternatizeUncurryFin_apply", "AlternatingMap.ext", "AlternatingMap.map_insertNth", "Fin.insertNth_removeNth", "Finset.sum_const", "Fintype.card_fin", "Function.update_eq_self", "smul_apply"], "split": "dev"}
{"name": "DoubleCoset.iUnion_finset_rightRel_eq_univ_of_rightRel", "query": "{G : Type u_1} [Group G] {H K : Subgroup G} {t : Finset (DoubleCoset.Quotient ↑H ↑K)} (ht : Set.univ ⊆ ⋃ i ∈ t, Quot.mk ⇑(QuotientGroup.rightRel H) '' DoubleCoset.doubleCoset (Quotient.out i) ↑H ↑K) : ⋃ q ∈ t, DoubleCoset.doubleCoset (Quotient.out q) ↑H ↑K = Set.univ", "area": "algebra", "module": "Mathlib.GroupTheory.DoubleCoset", "concrete": true, "gold": ["DoubleCoset.doubleCoset_eq_of_mem", "DoubleCoset.mem_doubleCoset", "Quotient.eq", "Set.mem_iUnion", "Set.mem_image", "Set.ne_univ_iff_exists_notMem", "Set.univ_subset_iff", "Subgroup.one_mem", "eq_inv_mul_of_mul_eq"], "gold_b": ["DoubleCoset.doubleCoset_eq_of_mem", "DoubleCoset.mem_doubleCoset", "Quotient.eq", "Set.mem_iUnion", "Set.mem_image", "Set.ne_univ_iff_exists_notMem", "Set.univ_subset_iff", "SetLike.coe_mem", "Subgroup.one_mem", "eq_inv_mul_of_mul_eq", "inv_mem_iff"], "split": "dev"}
{"name": "Perfection.coeff_frobenius", "query": "{R : Type u_1} [CommSemiring R] {p : ℕ} [hp : Fact (Nat.Prime p)] [CharP R p] (f : Perfection R p) (n : ℕ) : (Perfection.coeff R p (n + 1)) ((frobenius (Perfection R p) p) f) = (Perfection.coeff R p n) f", "area": "algebra", "module": "Mathlib.RingTheory.Perfection", "concrete": true, "gold": ["Perfection.coeffMonoidHom_powMonoidHom"], "gold_b": ["Perfection.coeffMonoidHom_powMonoidHom"], "split": "dev"}
{"name": "CategoryTheory.Abelian.SpectralObject.opcyclesMap_threeδ₂Toδ₁_opcyclesToE", "query": "{C : Type u_1} {ι : Type u_2} [CategoryTheory.Category C] [CategoryTheory.Category ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₁₂ : i₀ ⟶ i₂) (f₂₃ : i₁ ⟶ i₃) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (h₂₃ : CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : CategoryTheory.CategoryStruct.comp (X.opcyclesMap f₁ f₂₃ f₁₂ f₃ (CategoryTheory.ComposableArrows.threeδ₂Toδ₁ f₁ f₂ f₃ f₁₂ f₂₃ h₁₂ h₂₃) n₁) (X.opcyclesToE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂) = 0", "area": "algebra", "module": "Mathlib.Algebra.Homology.SpectralObject.Page", "concrete": true, "gold": ["CategoryTheory.Limits.comp_zero", "CategoryTheory.cancel_epi"], "gold_b": ["CategoryTheory.Abelian.SpectralObject.p_opcyclesToE", "CategoryTheory.ComposableArrows.naturality'", "CategoryTheory.Limits.comp_zero", "CategoryTheory.Limits.zero_comp", "CategoryTheory.cancel_epi"], "split": "dev"}
{"name": "Ideal.absNorm_pos_iff_mem_nonZeroDivisors", "query": "{S : Type u_1} [CommRing S] [IsDedekindDomain S] [Module.Free ℤ S] [Module.Finite ℤ S] {I : Ideal S} : 0 < Ideal.absNorm I ↔ I ∈ nonZeroDivisors (Ideal S)", "area": "algebra", "module": "Mathlib.RingTheory.Ideal.Norm.AbsNorm", "concrete": true, "gold": ["Ideal.absNorm_ne_zero_iff_mem_nonZeroDivisors"], "gold_b": ["Ideal.absNorm_ne_zero_iff_mem_nonZeroDivisors"], "split": "dev"}
{"name": "Polynomial.Chebyshev.C_eq", "query": "(R : Type u_1) [CommRing R] (n : ℤ) : Polynomial.Chebyshev.C R n = Polynomial.X * Polynomial.Chebyshev.C R (n - 1) - Polynomial.Chebyshev.C R (n - 2)", "area": "algebra", "module": "Mathlib.RingTheory.Polynomial.Chebyshev", "concrete": true, "gold": ["Polynomial.Chebyshev.C_add_two"], "gold_b": ["Polynomial.Chebyshev.C_add_two", "pow_one"], "split": "dev"}
{"name": "Polynomial.map_natDegree_eq_sub", "query": "{R : Type u_1} [Semiring R] {S : Type u_2} {F : Type u_3} [Semiring S] [FunLike F (Polynomial R) (Polynomial S)] [AddMonoidHomClass F (Polynomial R) (Polynomial S)] {φ : F} {p : Polynomial R} {k : ℕ} (φ_k : ∀ (f : Polynomial R), f.natDegree < k → φ f = 0) (φ_mon : ∀ (n : ℕ) (c : R), c ≠ 0 → (φ ((Polynomial.monomial n) c)).natDegree = n - k) : (φ p).natDegree = p.natDegree - k", "area": "algebra", "module": "Mathlib.Algebra.Polynomial.EraseLead", "concrete": true, "gold": ["Polynomial.mono_map_natDegree_eq", "tsub_lt_tsub_iff_right"], "gold_b": ["Polynomial.mono_map_natDegree_eq", "tsub_lt_tsub_iff_right"], "split": "dev"}
{"name": "Polynomial.degreeLT.addLinearEquiv_symm_apply", "query": "{R : Type u_1} [Semiring R] {m n : ℕ} (PQ : ↥(Polynomial.degreeLT R m) × ↥(Polynomial.degreeLT R n)) : ↑((Polynomial.degreeLT.addLinearEquiv R m n).symm PQ) = ↑PQ.1 + ↑PQ.2 * Polynomial.X ^ m", "area": "algebra", "module": "Mathlib.RingTheory.Polynomial.DegreeLT", "concrete": true, "gold": ["LinearMap.inl_apply", "LinearMap.inr_apply", "Polynomial.degreeLT.addLinearEquiv_symm_apply_inl", "Polynomial.degreeLT.addLinearEquiv_symm_apply_inr", "Prod.add_def", "Submodule.coe_add", "map_add"], "gold_b": ["LinearMap.inl_apply", "LinearMap.inr_apply", "Polynomial.degreeLT.addLinearEquiv_symm_apply_inl", "Polynomial.degreeLT.addLinearEquiv_symm_apply_inr", "Prod.add_def", "Submodule.coe_add", "map_add"], "split": "dev"}
{"name": "Group.nilpotencyClass_pi", "query": "{η : Type u_2} {Gs : η → Type u_3} [(i : η) → Group (Gs i)] [Fintype η] [∀ (i : η), Group.IsNilpotent (Gs i)] : Group.nilpotencyClass ((i : η) → Gs i) = Finset.univ.sup fun i => Group.nilpotencyClass (Gs i)", "area": "algebra", "module": "Mathlib.GroupTheory.Nilpotent", "concrete": true, "gold": ["Finset.mem_univ", "Finset.sup_le_iff", "Subgroup.lowerCentralSeries_eq_bot_iff_nilpotencyClass_le", "Subgroup.pi_eq_bot_iff", "Subgroup.top_lowerCentralSeries_pi_of_finite", "eq_of_forall_ge_iff"], "gold_b": ["Finset.mem_univ", "Finset.sup_le_iff", "Subgroup.lowerCentralSeries_eq_bot_iff_nilpotencyClass_le", "Subgroup.pi_eq_bot_iff", "Subgroup.top_lowerCentralSeries_pi_of_finite", "eq_of_forall_ge_iff"], "split": "dev"}
{"name": "IsLocalizedModule.map_integerMultiple", "query": "{R : Type u_1} [CommSemiring R] (S : Submonoid R) {M : Type u_2} [AddCommMonoid M] [_root_.Module R M] {M' : Type u_3} [AddCommMonoid M'] [_root_.Module R M'] (f : M →ₗ[R] M') [IsLocalizedModule S f] {ι : Type u_4} (s : Finset ι) (g : ι → M') (i : ↥s) : f (IsLocalizedModule.integerMultiple S f s g i) = IsLocalizedModule.commonDenom S f s g • g ↑i", "area": "algebra", "module": "Mathlib.Algebra.Module.LocalizedModule.Int", "concrete": true, "gold": ["IsLocalizedModule.exist_integer_multiples", "Subtype.prop"], "gold_b": ["IsLocalizedModule.exist_integer_multiples", "Subtype.prop"], "split": "dev"}
{"name": "LinearMap.rTensor_pow", "query": "{R : Type u_1} [CommSemiring R] {M : Type u_4} {N : Type u_5} [AddCommMonoid M] [AddCommMonoid N] [_root_.Module R M] [_root_.Module R N] (f : M →ₗ[R] M) (n : ℕ) : LinearMap.rTensor N f ^ n = LinearMap.rTensor N (f ^ n)", "area": "algebra", "module": "Mathlib.LinearAlgebra.TensorProduct.Map", "concrete": true, "gold": ["Module.End.id_pow", "TensorProduct.map_pow"], "gold_b": ["Module.End.id_pow", "TensorProduct.map_pow"], "split": "dev"}
{"name": "SkewPolynomial.monomial_left_inj", "query": "{R : Type u_3} [Semiring R] {a : R} (ha : a ≠ 0) {i j : ℕ} : (SkewPolynomial.monomial i) a = (SkewPolynomial.monomial j) a ↔ i = j", "area": "algebra", "module": "Mathlib.Algebra.SkewPolynomial.Basic", "concrete": true, "gold": ["SkewMonoidAlgebra.single_left_inj"], "gold_b": ["SkewMonoidAlgebra.single_left_inj"], "split": "dev"}
{"name": "UniqueProds.of_same", "query": "{G : Type u_1} [Semigroup G] [IsCancelMul G] (h : ∀ {A : Finset G}, A.Nonempty → ∃ a1 ∈ A, ∃ a2 ∈ A, UniqueMul A A a1 a2) : UniqueProds G", "area": "algebra", "module": "Mathlib.Algebra.Group.UniqueProds.Basic", "concrete": true, "gold": ["Finset.Nonempty.mul", "Finset.mem_mul", "Finset.mul_mem_mul", "mul_left_cancel", "mul_right_cancel"], "gold_b": ["Finset.Nonempty.mul", "Finset.mem_mul", "Finset.mul_mem_mul", "mul_left_cancel", "mul_right_cancel"], "split": "dev"}
{"name": "CategoryTheory.Abelian.SpectralObject.Ψ_fromOpcycles", "query": "{C : Type u_1} {ι : Type u_2} [CategoryTheory.Category C] [CategoryTheory.Category ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (f₁₂ : i ⟶ k) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) : CategoryTheory.CategoryStruct.comp (X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁) (X.fromOpcycles f₁ f₂ f₁₂ h₁₂ n₁) = CategoryTheory.CategoryStruct.comp (X.iCycles f₂ f₃ n₀) (X.δ f₁₂ f₃ n₀ n₁ hn₁)", "area": "algebra", "module": "Mathlib.Algebra.Homology.SpectralObject.Differentials", "concrete": true, "gold": ["CategoryTheory.Abelian.SpectralObject.p_fromOpcycles", "CategoryTheory.Abelian.SpectralObject.δ_naturality", "CategoryTheory.cancel_epi"], "gold_b": ["CategoryTheory.Abelian.SpectralObject.p_fromOpcycles", "CategoryTheory.Abelian.SpectralObject.δ_naturality", "CategoryTheory.cancel_epi"], "split": "dev"}
{"name": "WithBot.natCast_ne_bot", "query": "{α : Type u} [AddMonoidWithOne α] (n : ℕ) : ↑n ≠ ⊥", "area": "algebra", "module": "Mathlib.Algebra.Order.Monoid.Unbundled.WithTop", "concrete": true, "gold": ["WithBot.coe_ne_bot"], "gold_b": ["WithBot.coe_ne_bot"], "split": "dev"}
{"name": "ringChar.eq_iff", "query": "{R : Type u_1} [NonAssocSemiring R] {p : ℕ} : ringChar R = p ↔ CharP R p", "area": "algebra", "module": "Mathlib.Algebra.CharP.Defs", "concrete": true, "gold": ["ringChar.eq", "ringChar.of_eq"], "gold_b": ["ringChar.eq", "ringChar.of_eq"], "split": "dev"}
{"name": "ENNReal.rpow_add_le_mul_rpow_add_rpow", "query": "(z₁ z₂ : ENNReal) {p : ℝ} (hp : 1 ≤ p) : (z₁ + z₂) ^ p ≤ 2 ^ (p - 1) * (z₁ ^ p + z₂ ^ p)", "area": "algebra", "module": "Mathlib.Analysis.MeanInequalitiesPow", "concrete": true, "gold": ["ENNReal.add_halves", "ENNReal.div_eq_inv_mul", "ENNReal.inv_mul_cancel", "ENNReal.mul_rpow_of_nonneg", "ENNReal.ofNat_ne_top", "ENNReal.rpow_arith_mean_le_arith_mean2_rpow", "ENNReal.rpow_one", "ENNReal.rpow_sub", "two_ne_zero", "zero_le_one"], "gold_b": ["ENNReal.add_halves", "ENNReal.div_eq_inv_mul", "ENNReal.inv_mul_cancel", "ENNReal.mul_rpow_of_nonneg", "ENNReal.ofNat_ne_top", "ENNReal.rpow_arith_mean_le_arith_mean2_rpow", "ENNReal.rpow_one", "ENNReal.rpow_sub", "one_div", "two_ne_zero", "zero_le_one"], "split": "dev"}
{"name": "IsAlgebraic.natDenominator_ne_zero", "query": "{S : Type u_2} [CommRing S] {x : S} (hx : IsAlgebraic ℤ x) : Algebra.natDenominator x ≠ 0", "area": "algebra", "module": "Mathlib.RingTheory.Algebraic.Denominator", "concrete": true, "gold": ["Algebra.natDenominator_def", "IsAlgebraic.denominator_ne_zero"], "gold_b": ["Algebra.natDenominator_def", "IsAlgebraic.denominator_ne_zero"], "split": "dev"}
{"name": "Finset.sum_centroidWeights_eq_one_of_card_eq_add_one", "query": "(k : Type u_1) [DivisionRing k] {ι : Type u_4} (s : Finset ι) [CharZero k] {n : ℕ} (h : s.card = n + 1) : ∑ i ∈ s, Finset.centroidWeights k s i = 1", "area": "algebra", "module": "Mathlib.LinearAlgebra.AffineSpace.Centroid", "concrete": true, "gold": ["Finset.sum_centroidWeights_eq_one_of_card_ne_zero"], "gold_b": ["Finset.sum_centroidWeights_eq_one_of_card_ne_zero"], "split": "dev"}
{"name": "WithBot.sum_eq_bot_iff", "query": "{ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {s : Finset ι} {f : ι → WithBot M} : ∑ i ∈ s, f i = ⊥ ↔ ∃ i ∈ s, f i = ⊥", "area": "algebra", "module": "Mathlib.Algebra.BigOperators.WithTop", "concrete": true, "gold": ["Finset.cons_induction"], "gold_b": ["Finset.cons_induction", "Finset.mem_cons", "Finset.notMem_empty", "Finset.sum_cons", "WithBot.add_eq_bot", "WithBot.zero_ne_bot"], "split": "dev"}
{"name": "Finset.HasMulAntidiagonal.mulAntidiagonal_congr'", "query": "{A : Type u_1} [CancelCommMonoid A] [Finset.HasMulAntidiagonal A] {p q : A × A} {n : A} (hp : p ∈ Finset.HasMulAntidiagonal.mulAntidiagonal n) (hq : q ∈ Finset.HasMulAntidiagonal.mulAntidiagonal n) : p = q ↔ p.2 = q.2", "area": "algebra", "module": "Mathlib.Algebra.Order.Antidiag.Prod", "concrete": true, "gold": ["Finset.HasMulAntidiagonal.mulAntidiagonal_congr", "Finset.HasMulAntidiagonal.swap_mem_mulAntidiagonal"], "gold_b": ["Finset.HasMulAntidiagonal.mulAntidiagonal_congr", "Finset.HasMulAntidiagonal.swap_mem_mulAntidiagonal"], "split": "dev"}
{"name": "AntivaryOn.sum_smul_lt_sum_comp_perm_smul_iff", "query": "{ι : Type u_1} {α : Type u_2} {β : Type u_3} [Semiring α] [LinearOrder α] [IsStrictOrderedRing α] [ExistsAddOfLE α] [AddCommMonoid β] [LinearOrder β] [IsOrderedCancelAddMonoid β] [_root_.Module α β] [PosSMulStrictMono α β] {s : Finset ι} {σ : Equiv.Perm ι} {f : ι → α} {g : ι → β} (hfg : AntivaryOn f g ↑s) (hσ : {x | σ x ≠ x} ⊆ ↑s) : ∑ i ∈ s, f i • g i < ∑ i ∈ s, f (σ i) • g i ↔ ¬AntivaryOn (f ∘ ⇑σ) g ↑s", "area": "algebra", "module": "Mathlib.Algebra.Order.Rearrangement", "concrete": true, "gold": ["AntivaryOn.sum_comp_perm_smul_eq_sum_smul_iff", "AntivaryOn.sum_smul_le_sum_comp_perm_smul", "lt_iff_le_and_ne"], "gold_b": ["AntivaryOn.sum_comp_perm_smul_eq_sum_smul_iff", "AntivaryOn.sum_smul_le_sum_comp_perm_smul", "lt_iff_le_and_ne"], "split": "dev"}
{"name": "SetLike.prod_pow_mem_graded", "query": "{ι : Type u_2} {R : Type u_3} {S : Type u_4} [SetLike S R] [CommMonoid R] [AddCommMonoid ι] (A : ι → S) [SetLike.GradedMonoid A] {κ : Type u_5} (i : κ → ι) (g : κ → R) {F : Finset κ} (n : κ → ℕ) (hF : ∀ k ∈ F, g k ∈ A (i k)) : ∏ k ∈ F, g k ^ n k ∈ A (∑ k ∈ F, n k • i k)", "area": "algebra", "module": "Mathlib.Algebra.GradedMonoid", "concrete": true, "gold": ["SetLike.pow_mem_graded", "SetLike.prod_mem_graded"], "gold_b": ["SetLike.pow_mem_graded", "SetLike.prod_mem_graded"], "split": "dev"}
{"name": "Matrix.det_vandermonde_add", "query": "{R : Type u_1} [CommRing R] {n : ℕ} (v : Fin n → R) (a : R) : (Matrix.vandermonde fun i => v i + a).det = (Matrix.vandermonde v).det", "area": "algebra", "module": "Mathlib.LinearAlgebra.Vandermonde", "concrete": true, "gold": ["Matrix.det_vandermonde"], "gold_b": ["Finset.prod_congr", "Matrix.det_vandermonde", "add_sub_add_right_eq_sub"], "split": "dev"}
{"name": "Int.natCast_mul_floor_div_cancel", "query": "{R : Type u_4} [Ring R] [LinearOrder R] [IsOrderedAddMonoid R] [FloorRing R] {n : ℕ} (hn : n ≠ 0) (a : R) : ⌊↑n * a⌋ / ↑n = ⌊a⌋", "area": "algebra", "module": "Mathlib.Algebra.Order.Floor.Ring", "concrete": true, "gold": ["Int.mul_natCast_floor_div_cancel", "Nat.cast_comm"], "gold_b": ["Int.mul_natCast_floor_div_cancel", "Nat.cast_comm"], "split": "dev"}
{"name": "Finset.Nonempty.smul", "query": "{α : Type u_1} {β : Type u_2} [DecidableEq β] [SMul α β] {s : Finset α} {t : Finset β} : s.Nonempty → t.Nonempty → (s • t).Nonempty", "area": "algebra", "module": "Mathlib.Algebra.Group.Pointwise.Finset.Scalar", "concrete": true, "gold": ["Finset.Nonempty.image₂"], "gold_b": ["Finset.Nonempty.image₂"], "split": "dev"}
{"name": "Module.Basis.flag_strictMono", "query": "{R : Type u_1} {M : Type u_2} [Semiring R] [AddCommMonoid M] [_root_.Module R M] {n : ℕ} [Nontrivial R] (b : Module.Basis (Fin n) R M) : StrictMono b.flag", "area": "algebra", "module": "Mathlib.LinearAlgebra.Basis.Flag", "concrete": true, "gold": ["Fin.strictMono_iff_lt_succ", "Module.Basis.flag_succ"], "gold_b": ["Fin.strictMono_iff_lt_succ", "Module.Basis.flag_succ", "Module.Basis.self_mem_flag_iff", "Submodule.span_singleton_le_iff_mem", "lt_self_iff_false", "right_lt_sup"], "split": "dev"}
{"name": "Finset.Iic_sub_one_eq_Iio", "query": "{α : Type u_1} [LinearOrder α] [One α] [LocallyFiniteOrderBot α] [Sub α] [PredSubOrder α] [NoMinOrder α] (b : α) : Finset.Iic (b - 1) = Finset.Iio b", "area": "algebra", "module": "Mathlib.Algebra.Order.Interval.Finset.SuccPred", "concrete": true, "gold": ["Finset.Iic_pred_eq_Iio", "Order.pred_eq_sub_one"], "gold_b": ["Finset.Iic_pred_eq_Iio", "Order.pred_eq_sub_one"], "split": "dev"}
{"name": "LinearMap.BilinForm.sum_left", "query": "{R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [_root_.Module R M] (B : LinearMap.BilinForm R M) {α : Type u_7} (t : Finset α) (g : α → M) (w : M) : (B (∑ i ∈ t, g i)) w = ∑ i ∈ t, (B (g i)) w", "area": "algebra", "module": "Mathlib.LinearAlgebra.BilinearForm.Hom", "concrete": true, "gold": ["LinearMap.map_sum₂"], "gold_b": ["LinearMap.map_sum₂"], "split": "dev"}
{"name": "Ideal.mul_add_mem_pow_succ_unique", "query": "{S : Type u_1} [CommRing S] {P : Ideal S} [P_prime : P.IsPrime] [IsDedekindDomain S] (hP : P ≠ ⊥) {i : ℕ} (a d d' e e' : S) (a_notMem : a ∉ P ^ (i + 1)) (e_mem : e ∈ P ^ (i + 1)) (e'_mem : e' ∈ P ^ (i + 1)) (h : a * d + e - (a * d' + e') ∈ P ^ (i + 1)) : d - d' ∈ P", "area": "algebra", "module": "Mathlib.RingTheory.Ideal.Norm.AbsNorm", "concrete": true, "gold": ["Ideal.add_mem", "Ideal.mem_prime_of_mul_mem_pow", "Ideal.sub_mem"], "gold_b": ["Ideal.add_mem", "Ideal.mem_prime_of_mul_mem_pow", "Ideal.sub_mem"], "split": "dev"}
{"name": "Finset.prod_mulIndicator_subset_of_eq_one", "query": "{ι : Type u_1} {α : Type u_3} {β : Type u_4} [CommMonoid β] [One α] (f : ι → α) (g : ι → α → β) {s t : Finset ι} (h : s ⊆ t) (hg : ∀ (a : ι), g a 1 = 1) : ∏ i ∈ t, g i ((↑s).mulIndicator f i) = ∏ i ∈ s, g i (f i)", "area": "algebra", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Indicator", "concrete": true, "gold": ["Finset.prod_congr", "Finset.prod_subset", "Set.mulIndicator_of_mem"], "gold_b": ["Finset.prod_congr", "Finset.prod_subset", "Set.mulIndicator_of_mem", "Set.mulIndicator_of_notMem", "SetLike.mem_coe"], "split": "dev"}
{"name": "IsCoprime.ne_zero", "query": "{R : Type u} [CommSemiring R] [Nontrivial R] {p : Fin 2 → R} (h : IsCoprime (p 0) (p 1)) : p ≠ 0", "area": "algebra", "module": "Mathlib.RingTheory.Coprime.Basic", "concrete": true, "gold": ["not_isCoprime_zero_zero"], "gold_b": ["not_isCoprime_zero_zero"], "split": "dev"}
{"name": "WithTop.natCast_ne_top", "query": "{α : Type u} [AddMonoidWithOne α] (n : ℕ) : ↑n ≠ ⊤", "area": "algebra", "module": "Mathlib.Algebra.Order.Monoid.Unbundled.WithTop", "concrete": true, "gold": ["WithTop.coe_ne_top"], "gold_b": ["WithTop.coe_ne_top"], "split": "dev"}
{"name": "Ideal.IsPrime.prod_le", "query": "{R : Type u} {ι : Type u_1} [CommSemiring R] {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : P.IsPrime) : s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P", "area": "algebra", "module": "Mathlib.RingTheory.Ideal.Operations", "concrete": true, "gold": ["Ideal.IsPrime.multiset_prod_map_le"], "gold_b": ["Ideal.IsPrime.multiset_prod_map_le"], "split": "dev"}
{"name": "Subgroup.zpowers_le_zpowers_iff", "query": "{G : Type u_2} [Group G] (g : G) (i j : ℤ) : Subgroup.zpowers (g ^ i) ≤ Subgroup.zpowers (g ^ j) ↔ j.gcd ↑(orderOf g) ∣ i.gcd ↑(orderOf g)", "area": "algebra", "module": "Mathlib.GroupTheory.SpecificGroups.Cyclic", "concrete": true, "gold": ["Nat.gcd_gcd_gcd_right", "Subgroup.relIndex_zpowers_zpow_zpow", "mul_left_eq_self₀"], "gold_b": ["Nat.gcd_gcd_gcd_right", "OneMemClass.one_mem", "Subgroup.relIndex_eq_one", "Subgroup.relIndex_zpowers_zpow_zpow", "Subgroup.zpowers_le", "Subgroup.zpowers_one_eq_bot", "mul_left_eq_self₀", "orderOf_eq_zero_iff", "zpow_ofNat"], "split": "dev"}
{"name": "Polynomial.cyclotomic.eval_apply_ofReal", "query": "(q : ℝ) (n : ℕ) : Polynomial.eval (↑q) (Polynomial.cyclotomic n ℂ) = ↑(Polynomial.eval q (Polynomial.cyclotomic n ℝ))", "area": "algebra", "module": "Mathlib.RingTheory.Polynomial.Cyclotomic.Basic", "concrete": true, "gold": ["Polynomial.cyclotomic.eval_apply"], "gold_b": ["Polynomial.cyclotomic.eval_apply"], "split": "dev"}
{"name": "Set.pow_subset_pow_right", "query": "{α : Type u_2} [Monoid α] {s : Set α} {m n : ℕ} (hs : 1 ∈ s) (hmn : m ≤ n) : s ^ m ⊆ s ^ n", "area": "algebra", "module": "Mathlib.Algebra.Group.Pointwise.Set.Basic", "concrete": true, "gold": ["Set.pow_right_monotone"], "gold_b": ["Set.pow_right_monotone"], "split": "dev"}
{"name": "DirectSum.decompose_symm_sum", "query": "{ι : Type u_1} {M : Type u_3} {σ : Type u_4} [DecidableEq ι] [AddCommMonoid M] [SetLike σ M] [AddSubmonoidClass σ M] (ℳ : ι → σ) [DirectSum.Decomposition ℳ] {ι' : Type u_5} (s : Finset ι') (f : ι' → DirectSum ι fun i => ↥(ℳ i)) : (DirectSum.decompose ℳ).symm (∑ i ∈ s, f i) = ∑ i ∈ s, (DirectSum.decompose ℳ).symm (f i)", "area": "algebra", "module": "Mathlib.Algebra.DirectSum.Decomposition", "concrete": true, "gold": ["map_sum"], "gold_b": ["map_sum"], "split": "dev"}
{"name": "GradedTensorProduct.tmul_coe_mul_algebraMap_tmul", "query": "{R : Type u_1} {ι : Type u_2} {A : Type u_3} {B : Type u_4} [CommSemiring ι] [DecidableEq ι] [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B] (𝒜 : ι → Submodule R A) (ℬ : ι → Submodule R B) [GradedAlgebra 𝒜] [GradedAlgebra ℬ] [_root_.Module ι (Additive ℤˣ)] {j₁ : ι} (a₁ : A) (b₁ : ↥(ℬ j₁)) (r : R) (b₂ : B) : a₁ ᵍ⊗ₜ[R] ↑b₁ * (algebraMap R A) r ᵍ⊗ₜ[R] b₂ = (a₁ * (algebraMap R A) r) ᵍ⊗ₜ[R] (↑b₁ * b₂)", "area": "algebra", "module": "Mathlib.LinearAlgebra.TensorProduct.Graded.Internal", "concrete": true, "gold": ["GradedTensorProduct.tmul_coe_mul_zero_coe_tmul"], "gold_b": ["GradedTensorProduct.tmul_coe_mul_zero_coe_tmul"], "split": "dev"}
{"name": "DividedPowers.RatAlgebra.dividedPowers_unique", "query": "{R : Type u_1} [CommSemiring R] {I : Ideal R} [DecidablePred fun x => x ∈ I] [Algebra ℚ R] (hI : DividedPowers I) : hI = DividedPowers.RatAlgebra.dividedPowers I", "area": "algebra", "module": "Mathlib.RingTheory.DividedPowers.RatAlgebra", "concrete": true, "gold": ["DividedPowers.RatAlgebra.dpow_apply", "DividedPowers.ext", "DividedPowers.factorial_mul_dpow_eq_pow", "IsUnit.natCast_factorial_of_algebra", "Ring.inverse_mul_eq_iff_eq_mul"], "gold_b": ["DividedPowers.RatAlgebra.dpow_apply", "DividedPowers.ext", "DividedPowers.factorial_mul_dpow_eq_pow", "IsUnit.natCast_factorial_of_algebra", "Ring.inverse_mul_eq_iff_eq_mul"], "split": "dev"}
{"name": "CochainComplex.HomComplex.Cochain.rightUnshift_smul", "query": "{C : Type u} [CategoryTheory.Category C] [CategoryTheory.Preadditive C] {R : Type u_1} [Ring R] [CategoryTheory.Linear R C] {K L : CochainComplex C ℤ} {n' a : ℤ} (γ : CochainComplex.HomComplex.Cochain K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj L) n') (n : ℤ) (hn : n' + a = n) (x : R) : (x • γ).rightUnshift n hn = x • γ.rightUnshift n hn", "area": "algebra", "module": "Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift", "concrete": true, "gold": ["map_smul"], "gold_b": ["map_smul"], "split": "dev"}
{"name": "UniqueMul.iff_existsUnique", "query": "{G : Type u_1} [Mul G] {A B : Finset G} {a0 b0 : G} (aA : a0 ∈ A) (bB : b0 ∈ B) : UniqueMul A B a0 b0 ↔ ∃! ab, ab ∈ A ×ˢ B ∧ ab.1 * ab.2 = a0 * b0", "area": "algebra", "module": "Mathlib.Algebra.Group.UniqueProds.Basic", "concrete": true, "gold": ["ExistsUnique.elim", "Finset.mk_mem_product", "Prod.mk_inj"], "gold_b": ["ExistsUnique.elim", "Finset.mem_product", "Finset.mk_mem_product", "Prod.mk_inj"], "split": "dev"}
{"name": "AddCommGroup.zsmul_add_modEq", "query": "{G : Type u_1} [AddCommGroup G] {p a : G} (z : ℤ) : z • p + a ≡ a [PMOD p]", "area": "algebra", "module": "Mathlib.Algebra.Group.ModEq", "concrete": true, "gold": ["AddCommGroup.modEq_iff_zsmul"], "gold_b": ["AddCommGroup.modEq_iff_zsmul", "neg_zsmul", "sub_add_cancel_right"], "split": "dev"}
{"name": "Nat.smoothNumbersUpTo_card_le", "query": "(N k : ℕ) : (N.smoothNumbersUpTo k).card ≤ 2 ^ k.primesBelow.card * N.sqrt", "area": "number_theory", "module": "Mathlib.NumberTheory.SmoothNumbers", "concrete": true, "gold": ["Finset.card_erase_of_mem", "Finset.card_image_le", "Finset.card_le_card", "Finset.card_powerset", "Finset.card_product", "Finset.card_range", "Finset.mem_range", "Nat.smoothNumbersUpTo_subset_image"], "gold_b": ["Finset.card_erase_of_mem", "Finset.card_image_le", "Finset.card_le_card", "Finset.card_powerset", "Finset.card_product", "Finset.card_range", "Finset.mem_range", "Nat.smoothNumbersUpTo_subset_image"], "split": "dev"}
{"name": "AddCommGroup.intCast_modEq_intCast'", "query": "{G : Type u_1} [AddCommGroupWithOne G] [CharZero G] {a b : ℤ} {n : ℕ} : ↑a ≡ ↑b [PMOD ↑n] ↔ a ≡ b [PMOD ↑n]", "area": "number_theory", "module": "Mathlib.Data.Int.ModEq", "concrete": true, "gold": ["AddCommGroup.intCast_modEq_intCast"], "gold_b": ["AddCommGroup.intCast_modEq_intCast", "AddCommGroup.modEq_iff_intModEq", "Int.cast_natCast"], "split": "dev"}
{"name": "IsNonarchimedean.iSup_abv_linearMap_apply_le", "query": "{K : Type u_1} [Field K] {ι : Type u_2} {ι' : Type u_3} [Fintype ι] [Finite ι'] {v : AbsoluteValue K ℝ} (hv : IsNonarchimedean ⇑v) (A : ι' × ι → K) (x : ι → K) : ⨆ j, v (∑ i, A (j, i) * x i) ≤ (⨆ ji, v (A ji)) * ⨆ i, v (x i)", "area": "number_theory", "module": "Mathlib.NumberTheory.Height.MvPolynomial", "concrete": true, "gold": ["Finite.le_ciSup_of_le", "Finset.mem_univ", "Function.Surjective.iSup_comp", "IsNonarchimedean.apply_sum_le", "Real.iSup_nonneg_of_nonnegHomClass", "ciSup_le", "isEmpty_or_nonempty", "le_rfl"], "gold_b": ["AbsoluteValue.map_zero", "Finite.le_ciSup_of_le", "Finset.mem_univ", "Finset.univ_eq_empty", "Function.Surjective.iSup_comp", "IsNonarchimedean.apply_sum_le", "NonnegHomClass.apply_nonneg", "Real.iSup_const_zero", "Real.iSup_nonneg_of_nonnegHomClass", "Real.sSup_empty", "ciSup_le", "iSup_of_empty'", "isEmpty_or_nonempty", "le_imp_le_of_le_of_le", "le_rfl", "mul_le_mul"], "split": "dev"}
{"name": "Nat.add_one_sqrt_le_of_ne_zero", "query": "{n : ℕ} (hn : n ≠ 0) : (n + 1).sqrt ≤ n", "area": "number_theory", "module": "Mathlib.Data.Nat.Sqrt", "concrete": true, "gold": ["Nat.le_induction", "Nat.sqrt_succ_le_succ_sqrt", "le_trans"], "gold_b": ["Nat.le_induction", "Nat.sqrt_succ_le_succ_sqrt", "Nat.sqrt_two", "le_trans"], "split": "dev"}
{"name": "MulChar.IsQuadratic.gaussSum_frob_iter", "query": "{R : Type u} [CommRing R] [Fintype R] {R' : Type v} [CommRing R'] (p : ℕ) [fp : Fact (Nat.Prime p)] [hch : CharP R' p] (n : ℕ) (hp : IsUnit ↑p) {χ : MulChar R R'} (hχ : χ.IsQuadratic) (ψ : AddChar R R') : gaussSum χ ψ ^ p ^ n = χ (↑p ^ n) * gaussSum χ ψ", "area": "number_theory", "module": "Mathlib.NumberTheory.GaussSum", "concrete": true, "gold": ["MulChar.IsQuadratic.gaussSum_frob", "MulChar.IsQuadratic.pow_char", "MulChar.map_one", "MulChar.pow_apply'", "Nat.Prime.ne_zero", "mul_pow", "pow_mul", "pow_one", "pow_succ"], "gold_b": ["Fact.out", "MulChar.IsQuadratic.gaussSum_frob", "MulChar.IsQuadratic.pow_char", "MulChar.map_one", "MulChar.pow_apply'", "Nat.Prime.ne_zero", "mul_pow", "pow_mul", "pow_one", "pow_succ"], "split": "dev"}
{"name": "ModularGroup.bottom_row_coprime", "query": "{R : Type u_1} [CommRing R] (g : Matrix.SpecialLinearGroup (Fin 2) R) : IsCoprime (↑g 1 0) (↑g 1 1)", "area": "number_theory", "module": "Mathlib.NumberTheory.Modular", "concrete": true, "gold": ["Matrix.SpecialLinearGroup.isCoprime_row"], "gold_b": ["Matrix.SpecialLinearGroup.isCoprime_row"], "split": "dev"}
{"name": "ZMod.natAbs_valMinAbs_eq_natAbs_valMinAbs", "query": "{n : ℕ} {a b : ZMod n} : a.valMinAbs.natAbs = b.valMinAbs.natAbs ↔ a = b ∨ a = -b", "area": "number_theory", "module": "Mathlib.Data.ZMod.ValMinAbs", "concrete": true, "gold": ["ZMod.eq_neg_of_valMinAbs_eq_neg_valMinAbs", "ZMod.natAbs_valMinAbs_neg", "ZMod.valMinAbs_inj"], "gold_b": ["ZMod.eq_neg_of_valMinAbs_eq_neg_valMinAbs", "ZMod.natAbs_valMinAbs_neg", "ZMod.valMinAbs_inj"], "split": "dev"}
{"name": "ModularForm.one_sub_eta_q_ne_zero", "query": "(n : ℕ) {z : ℂ} (hz : z ∈ UpperHalfPlane.upperHalfPlaneSet) : 1 - ModularForm.eta_q n z ≠ 0", "area": "number_theory", "module": "Mathlib.NumberTheory.ModularForms.DedekindEta", "concrete": true, "gold": ["ModularForm.eta_q_eq_cexp", "UpperHalfPlane.norm_exp_two_pi_I_lt_one", "sub_ne_zero"], "gold_b": ["Complex.mul_im", "ModularForm.eta_q_eq_cexp", "Nat.cast_add", "Nat.cast_nonneg'", "NormOneClass.norm_one", "Right.add_pos_of_nonneg_of_pos", "UpperHalfPlane.norm_exp_two_pi_I_lt_one", "lt_self_iff_false", "mul_pos_iff_of_pos_left", "nsmul_eq_mul", "sub_ne_zero"], "split": "dev"}
{"name": "Nat.Prime.sub_one_mul_multiplicity_factorial", "query": "{n p : ℕ} (hp : Nat.Prime p) : (p - 1) * multiplicity p n.factorial = n - (p.digits n).sum", "area": "number_theory", "module": "Mathlib.Data.Nat.Multiplicity", "concrete": true, "gold": ["Finset.sum_Ico_add'", "Nat.Ico_zero_eq_range", "Nat.Prime.emultiplicity_factorial", "Nat.sub_one_mul_sum_log_div_pow_eq_sub_sum_digits", "multiplicity_eq_of_emultiplicity_eq_some"], "gold_b": ["Finset.sum_Ico_add'", "Nat.Ico_zero_eq_range", "Nat.Prime.emultiplicity_factorial", "Nat.sub_one_mul_sum_log_div_pow_eq_sub_sum_digits", "multiplicity_eq_of_emultiplicity_eq_some"], "split": "dev"}
{"name": "EisensteinSeries.summable_norm_eisSummand", "query": "{k : ℤ} (hk : 3 ≤ k) (z : UpperHalfPlane) : Summable fun x => ‖EisensteinSeries.eisSummand k x z‖", "area": "number_theory", "module": "Mathlib.NumberTheory.ModularForms.EisensteinSeries.IsBoundedAtImInfty", "concrete": true, "gold": ["EisensteinSeries.summable_one_div_norm_rpow", "EisensteinSeries.summand_bound", "Summable.mul_left", "Summable.of_nonneg_of_le", "norm_nonneg", "norm_zpow"], "gold_b": ["EisensteinSeries.summable_one_div_norm_rpow", "EisensteinSeries.summand_bound", "Int.cast_lt", "Int.cast_neg", "Int.cast_ofNat", "Int.cast_pos", "Real.rpow_intCast", "Summable.mul_left", "Summable.of_nonneg_of_le", "le_of_lt", "lt_of_lt_of_le", "norm_nonneg", "norm_zpow"], "split": "dev"}
{"name": "PadicInt.p_nonunit", "query": "{p : ℕ} [hp : Fact (Nat.Prime p)] : ↑p ∈ nonunits ℤ_[p]", "area": "number_theory", "module": "Mathlib.NumberTheory.Padics.PadicIntegers", "concrete": true, "gold": ["Fact.out", "Nat.Prime.one_lt", "PadicInt.mem_nonunits", "PadicInt.norm_p", "inv_lt_one_of_one_lt₀"], "gold_b": ["Fact.out", "Nat.Prime.one_lt", "Nat.cast_lt", "PadicInt.mem_nonunits", "PadicInt.norm_p", "inv_lt_one_of_one_lt₀"], "split": "dev"}
{"name": "EisensteinSeries.tsum_symmetricIco_tsum_eq_S_act", "query": "(z : UpperHalfPlane) : ∑'[SummationFilter.symmetricIco ℤ] (n : ℤ), ∑' (m : ℤ), 1 / (↑m * ↑z + ↑n) ^ 2 = (↑z ^ 2)⁻¹ * EisensteinSeries.G2 (ModularGroup.S • z)", "area": "number_theory", "module": "Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable", "concrete": true, "gold": ["EisensteinSeries.linear_left_summable", "EisensteinSeries.tendsto_double_sum_S_act", "Filter.Tendsto.congr", "HasSum.tsum_eq", "Summable.tsum_finsetSum", "SummationFilter.hasSum_symmetricIco_int_iff", "UpperHalfPlane.ne_zero", "le_rfl"], "gold_b": ["EisensteinSeries.linear_left_summable", "EisensteinSeries.tendsto_double_sum_S_act", "Filter.Tendsto.congr", "HasSum.tsum_eq", "Summable.tsum_finsetSum", "SummationFilter.hasSum_symmetricIco_int_iff", "UpperHalfPlane.ne_zero", "le_rfl", "one_div", "zpow_ofNat"], "split": "dev"}
{"name": "Nat.dvd_of_primeFactorsList_subperm", "query": "{a b : ℕ} (ha : a ≠ 0) (h : a.primeFactorsList.Subperm b.primeFactorsList) : a ∣ b", "area": "number_theory", "module": "Mathlib.Data.Nat.Factors", "concrete": true, "gold": ["LT.lt.ne'", "List.Perm.prod_eq", "Nat.prod_primeFactorsList", "dvd_zero", "one_dvd"], "gold_b": ["LT.lt.ne'", "List.Perm.prod_eq", "Nat.prod_primeFactorsList", "dvd_zero", "one_dvd"], "split": "dev"}
{"name": "PNat.find_min'", "query": "{p : ℕ+ → Prop} [DecidablePred p] (h : ∃ n, p n) {m : ℕ+} (hm : p m) : PNat.find h ≤ m", "area": "number_theory", "module": "Mathlib.Data.PNat.Find", "concrete": true, "gold": ["PNat.find_min", "le_of_not_gt"], "gold_b": ["PNat.find_min", "le_of_not_gt"], "split": "dev"}
{"name": "CuspForm.hasSum_L", "query": "{Γ : Subgroup (GL (Fin 2) ℝ)} [Γ.IsArithmetic] {k : ℤ} (hk : 0 < k) {F : Type u_1} [FunLike F UpperHalfPlane ℂ] (f : F) {s : ℂ} [CuspFormClass F Γ k] (hs : ↑k / 2 + 1 < s.re) : HasSum (fun n => (PowerSeries.coeff n) (UpperHalfPlane.qExpansion Γ.strictWidthInfty ⇑f) / ↑n ^ s) (↑Γ.strictWidthInfty ^ (-s) * ModularForm.L hk f s)", "area": "number_theory", "module": "Mathlib.NumberTheory.ModularForms.LFunction", "concrete": true, "gold": ["CuspForm.hasSum_Λ"], "gold_b": ["Complex.Gamma_ne_zero_of_re_pos", "Complex.cpow_eq_zero_iff", "Complex.cpow_ne_zero_iff", "Complex.cpow_neg", "Complex.div_cpow_ofReal_nonneg", "Complex.mul_cpow_ofReal_nonneg", "Complex.ne_zero_of_re_pos", "Complex.ofReal_div", "Complex.ofReal_eq_zero", "Complex.ofReal_mul", "Complex.ofReal_natCast", "Complex.ofReal_ne_zero", "Complex.ofReal_ofNat", "CuspForm.hasSum_Λ", "HasSum.mul_right", "Int.cast_lt", "Int.cast_zero", "LT.lt.le", "LT.lt.ne'", "ModularForm.L_apply_of_ne_zero", "Nat.cast_nonneg", "Nat.cast_zero", "Real.pi_ne_zero", "Real.pi_pos", "Subgroup.strictWidthInfty_nonneg", "Subgroup.strictWidthInfty_pos", "div_div", "div_one", "div_self", "inv_inv", "lt_of_not_ge", "mul_rotate'", "neg_neg_of_pos", "one_div", "two_ne_zero", "two_pos", "zero_le_two"], "split": "dev"}
{"name": "Nat.factorizationLCMRight_pos", "query": "(a b : ℕ) : 0 < a.factorizationLCMRight b", "area": "number_theory", "module": "Mathlib.Data.Nat.Factorization.LCM", "concrete": true, "gold": ["Finsupp.prod_ne_zero_iff", "pow_eq_zero_iff'"], "gold_b": ["Finsupp.prod_ne_zero_iff", "Nat.factorization_zero_right", "pow_eq_zero_iff'"], "split": "dev"}
{"name": "Nat.pairwise_divisorsAntidiagonalList_fst", "query": "{n : ℕ} : List.Pairwise (fun x1 x2 => x1.1 < x2.1) n.divisorsAntidiagonalList", "area": "number_theory", "module": "Mathlib.NumberTheory.Divisors", "concrete": true, "gold": ["List.SortedLT.pairwise", "List.sortedLT_range'"], "gold_b": ["List.SortedLT.pairwise", "List.sortedLT_range'"], "split": "dev"}
{"name": "CongruenceSubgroup.conjGL_coe", "query": "(Γ : Subgroup (Matrix.SpecialLinearGroup (Fin 2) ℤ)) (g : Matrix.SpecialLinearGroup (Fin 2) ℤ) : CongruenceSubgroup.conjGL Γ (Matrix.SpecialLinearGroup.toGL ((Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) = ConjAct.toConjAct g⁻¹ • Γ", "area": "number_theory", "module": "Mathlib.NumberTheory.ModularForms.CongruenceSubgroups", "concrete": true, "gold": ["CongruenceSubgroup.mem_conjGL", "Matrix.SpecialLinearGroup.map_intCast_injective", "Matrix.SpecialLinearGroup.toGL_injective", "Subgroup.mem_inv_pointwise_smul_iff", "map_inv"], "gold_b": ["CongruenceSubgroup.mem_conjGL", "Matrix.SpecialLinearGroup.map_intCast_injective", "Matrix.SpecialLinearGroup.toGL_injective", "Subgroup.ext", "Subgroup.mem_inv_pointwise_smul_iff", "map_inv"], "split": "dev"}
{"name": "Pell.yn_modEq_a_sub_one", "query": "{a : ℕ} (a1 : 1 < a) (n : ℕ) : Pell.yn a1 n ≡ n [MOD a - 1]", "area": "number_theory", "module": "Mathlib.NumberTheory.PellMatiyasevic", "concrete": true, "gold": ["LT.lt.le", "Nat.ModEq.add_right_cancel", "Nat.ModEq.mul", "Nat.ModEq.mul_left", "Nat.ModEq.refl", "Nat.modEq_sub", "Pell.yn_succ_succ"], "gold_b": ["LT.lt.le", "Nat.ModEq.add_right_cancel", "Nat.ModEq.mul", "Nat.ModEq.mul_left", "Nat.ModEq.refl", "Nat.modEq_sub", "Pell.yn_succ_succ"], "split": "dev"}
{"name": "AbsoluteValue.IsAdmissible.exists_approx_aux", "query": "{R : Type u_1} [EuclideanDomain R] {abv : AbsoluteValue R ℤ} (n : ℕ) (h : abv.IsAdmissible) {ε : ℝ} (_hε : 0 < ε) {b : R} (_hb : b ≠ 0) (A : Fin (h.card ε ^ n).succ → Fin n → R) : ∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ (k : Fin n), ↑(abv (A i₁ k % b - A i₀ k % b)) < abv b • ε", "area": "number_theory", "module": "Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue", "concrete": true, "gold": ["AbsoluteValue.IsAdmissible.exists_partition", "Finset.length_toList", "Finset.mem_filter", "Finset.mem_toList", "Finset.nodup_toList", "Fintype.card_fin", "Fintype.exists_lt_card_fiber_of_mul_lt_card", "List.Nodup.getElem_inj_iff", "pow_succ'"], "gold_b": ["AbsoluteValue.IsAdmissible.exists_partition", "Finset.length_toList", "Finset.mem_filter", "Finset.mem_toList", "Finset.nodup_toList", "Fintype.card_congr'", "Fintype.card_fin", "Fintype.exists_lt_card_fiber_of_mul_lt_card", "List.Nodup.getElem_inj_iff", "OfNat.ofNat_ne_one", "pow_succ'"], "split": "dev"}
{"name": "Padic.valuation_p_lt_one", "query": "{p : ℕ} [hp : Fact (Nat.Prime p)] {Γ₀ : Type u_1} [LinearOrderedCommMonoidWithZero Γ₀] (v : Valuation ℚ_[p] Γ₀) [v.Compatible] : v ↑p < 1", "area": "number_theory", "module": "Mathlib.NumberTheory.Padics.ValuativeRel", "concrete": true, "gold": ["Fact.out", "Nat.Prime.ne_zero", "Valuation.IsEquiv.lt_one_iff_lt_one", "ValuativeRel.isEquiv"], "gold_b": ["Fact.out", "Nat.Prime.ne_zero", "Nat.cast_eq_zero", "Padic.valuation_natCast", "Valuation.IsEquiv.lt_one_iff_lt_one", "ValuativeRel.isEquiv", "WithZero.exp_lt_one_iff", "padicValNat_self", "zero_lt_one"], "split": "dev"}
{"name": "Nat.totient_gcd_mul_totient_mul", "query": "(a b : ℕ) : (a.gcd b).totient * (a * b).totient = a.totient * b.totient * a.gcd b", "area": "number_theory", "module": "Mathlib.Data.Nat.Totient", "concrete": true, "gold": ["Nat.prod_primeFactors_dvd", "Nat.prod_primeFactors_gcd_mul_prod_primeFactors_mul", "Nat.totient_eq_div_primeFactors_mul", "mul_dvd_mul", "mul_mul_mul_comm"], "gold_b": ["Nat.prod_primeFactors_dvd", "Nat.prod_primeFactors_gcd_mul_prod_primeFactors_mul", "Nat.totient_eq_div_primeFactors_mul", "mul_dvd_mul", "mul_mul_mul_comm"], "split": "dev"}
{"name": "Nat.totient_eq_one_iff", "query": "{n : ℕ} : n.totient = 1 ↔ n = 1 ∨ n = 2", "area": "number_theory", "module": "Mathlib.Data.Nat.Totient", "concrete": true, "gold": ["Nat.not_even_one", "Nat.totient_even", "le_add_self"], "gold_b": ["Nat.not_even_one", "Nat.totient_even", "Nat.totient_two", "OfNat.ofNat_ne_one", "OfNat.one_ne_ofNat", "OfNat.zero_ne_ofNat", "le_add_self", "zero_ne_one"], "split": "dev"}
{"name": "Int.exists_mul_self", "query": "(x : ℤ) : (∃ n, n * n = x) ↔ Int.sqrt x * Int.sqrt x = x", "area": "number_theory", "module": "Mathlib.Data.Int.Sqrt", "concrete": true, "gold": ["Int.sqrt_eq"], "gold_b": ["Int.sqrt_eq"], "split": "dev"}
{"name": "HurwitzZeta.sinKernel_undef", "query": "(a : UnitAddCircle) {x : ℝ} (hx : x ≤ 0) : HurwitzZeta.sinKernel a x = 0", "area": "number_theory", "module": "Mathlib.NumberTheory.LSeries.HurwitzZetaOdd", "concrete": true, "gold": ["Complex.ofReal_eq_zero", "HurwitzZeta.sinKernel_def", "QuotientAddGroup.induction_on", "jacobiTheta₂'_undef", "zero_div"], "gold_b": ["Complex.mul_im", "Complex.ofReal_eq_zero", "HurwitzZeta.sinKernel_def", "QuotientAddGroup.induction_on", "jacobiTheta₂'_undef", "zero_div"], "split": "dev"}
{"name": "Int.padicValuation_lt_one_iff", "query": "{p : ℕ} [Fact (Nat.Prime p)] {x : ℤ} : (Int.padicValuation p) x < 1 ↔ ↑p ∣ x", "area": "number_theory", "module": "Mathlib.NumberTheory.Padics.PadicNumbers", "concrete": true, "gold": ["Int.padicValuation_eq_one_iff", "Int.padicValuation_le_one", "lt_iff_le_and_ne"], "gold_b": ["Int.padicValuation_eq_one_iff", "Int.padicValuation_le_one", "lt_iff_le_and_ne"], "split": "dev"}
{"name": "ModularFormClass.tendsto_atImInfty", "query": "{k : ℤ} {F : Type u_1} [FunLike F UpperHalfPlane ℂ] {Γ : Subgroup (GL (Fin 2) ℝ)} {h : ℝ} (f : F) [ModularFormClass F Γ k] (hh : 0 < h) (hΓ : h ∈ Γ.strictPeriods) : Filter.Tendsto (⇑f) UpperHalfPlane.atImInfty (nhds ((PowerSeries.coeff 0) (UpperHalfPlane.qExpansion h ⇑f)))", "area": "number_theory", "module": "Mathlib.NumberTheory.ModularForms.QExpansion", "concrete": true, "gold": ["ModularFormClass.bdd_at_infty", "ModularFormClass.holo", "SlashInvariantFormClass.periodic_comp_ofComplex", "Subgroup.isCusp_of_mem_strictPeriods", "UpperHalfPlane.hasSum_qExpansion", "UpperHalfPlane.tendsto_atImInfty_of_hasSum_qExpansion"], "gold_b": ["ModularFormClass.bdd_at_infty", "ModularFormClass.holo", "SlashInvariantFormClass.periodic_comp_ofComplex", "Subgroup.isCusp_of_mem_strictPeriods", "UpperHalfPlane.hasSum_qExpansion", "UpperHalfPlane.tendsto_atImInfty_of_hasSum_qExpansion"], "split": "dev"}
{"name": "PNat.Coprime.factor_eq_gcd_left", "query": "{a b m n : ℕ+} (cop : m.Coprime n) (am : a ∣ m) (bn : b ∣ n) : a = (a * b).gcd m", "area": "number_theory", "module": "Mathlib.Data.PNat.Prime", "concrete": true, "gold": ["PNat.Coprime.coprime_dvd_left", "PNat.Coprime.gcd_mul_right_cancel", "PNat.Coprime.symm", "PNat.gcd_eq_left_iff_dvd"], "gold_b": ["PNat.Coprime.coprime_dvd_left", "PNat.Coprime.gcd_mul_right_cancel", "PNat.Coprime.symm", "PNat.gcd_eq_left_iff_dvd"], "split": "dev"}
{"name": "ModularForm.gnpow_eq_pow", "query": "{Γ : Subgroup (GL (Fin 2) ℝ)} [Γ.HasDetPlusMinusOne] {k : ℤ} (f : ModularForm Γ k) (n : ℕ) : ⟨n • k, GradedMonoid.GMonoid.gnpow n f⟩ = ⟨↑n * k, f.pow n⟩", "area": "number_theory", "module": "Mathlib.NumberTheory.ModularForms.Basic", "concrete": true, "gold": ["GradedMonoid.GMonoid.gnpow_succ'", "GradedMonoid.GMonoid.gnpow_zero'", "ModularForm.ext", "ModularForm.gradedMonoid_eq_of_cast"], "gold_b": ["GradedMonoid.GMonoid.gnpow_succ'", "GradedMonoid.GMonoid.gnpow_zero'", "ModularForm.ext", "ModularForm.gradedMonoid_eq_of_cast"], "split": "dev"}
{"name": "nat_log_eq_padicValNat_iff", "query": "{p n : ℕ} [hp : Fact (Nat.Prime p)] (hn : n ≠ 0) : Nat.log p n = padicValNat p n ↔ n < p ^ (padicValNat p n + 1)", "area": "number_theory", "module": "Mathlib.NumberTheory.Padics.PadicVal.Basic", "concrete": true, "gold": ["Fact.out", "Nat.log_eq_iff", "pow_padicValNat_dvd"], "gold_b": ["Fact.out", "Nat.log_eq_iff", "pow_padicValNat_dvd"], "split": "dev"}
{"name": "IsEllipticNet.atomRel_abs₁", "query": "{R : Type u_1} [CommRing R] {W : ℤ → R} (neg : Function.Odd W) (a b c d : ℤ) : IsEllipticNet.atomRel W |a| b c d = IsEllipticNet.atomRel W a b c d", "area": "number_theory", "module": "Mathlib.NumberTheory.EllipticDivisibilitySequence", "concrete": true, "gold": ["IsEllipticNet.atom_abs_left"], "gold_b": ["IsEllipticNet.atom_abs_left"], "split": "dev"}
{"name": "LSeries_def₀", "query": "{f : ℕ → ℂ} (hf : f 0 = 0) (s : ℂ) : LSeries f s = ∑' (n : ℕ), f n / ↑n ^ s", "area": "number_theory", "module": "Mathlib.NumberTheory.LSeries.Basic", "concrete": true, "gold": ["Complex.cpow_neg", "LSeries.term_def₀", "div_eq_mul_inv"], "gold_b": ["Complex.cpow_neg", "LSeries.term_def₀", "div_eq_mul_inv"], "split": "dev"}
{"name": "riemannZeta_def_of_ne_zero", "query": "{s : ℂ} (hs : s ≠ 0) : riemannZeta s = completedRiemannZeta s / s.Gammaℝ", "area": "number_theory", "module": "Mathlib.NumberTheory.LSeries.RiemannZeta", "concrete": true, "gold": ["Function.update_of_ne", "HurwitzZeta.completedHurwitzZetaEven_zero"], "gold_b": ["Function.update_of_ne", "HurwitzZeta.completedHurwitzZetaEven_zero"], "split": "dev"}
{"name": "ZMod.ringEquivCongr_refl_apply", "query": "{a : ℕ} (x : ZMod a) : (ZMod.ringEquivCongr ⋯) x = x", "area": "number_theory", "module": "Mathlib.Data.ZMod.Basic", "concrete": true, "gold": ["ZMod.ringEquivCongr_refl"], "gold_b": ["ZMod.ringEquivCongr_refl"], "split": "dev"}
{"name": "Nat.pow_le_of_le_log", "query": "{b x y : ℕ} (hy : y ≠ 0) (h : x ≤ Nat.log b y) : b ^ x ≤ y", "area": "number_theory", "module": "Mathlib.Data.Nat.Log", "concrete": true, "gold": ["Nat.le_log_iff_pow_le", "Nat.log_of_left_le_one", "le_or_gt"], "gold_b": ["Nat.le_log_iff_pow_le", "Nat.log_of_left_le_one", "le_or_gt"], "split": "dev"}
{"name": "Nat.Primes.tprod_eq_tprod_ite", "query": "{M : Type u_1} [CommMonoid M] [TopologicalSpace M] (f : ℕ → M) : ∏' (p : Nat.Primes), f ↑p = ∏' (n : ℕ), if Nat.Prime n then f n else 1", "area": "number_theory", "module": "Mathlib.NumberTheory.SumPrimeReciprocals", "concrete": true, "gold": ["tprod_subtype"], "gold_b": ["Set.mulIndicator_apply", "tprod_subtype"], "split": "dev"}
{"name": "DirichletCharacter.delta_mul", "query": "{n : ℕ} (χ : DirichletCharacter ℂ n) : (LSeries.delta * fun n_1 => χ ↑n_1) = LSeries.delta", "area": "number_theory", "module": "Mathlib.NumberTheory.LSeries.Dirichlet", "concrete": true, "gold": ["DirichletCharacter.mul_delta"], "gold_b": ["DirichletCharacter.mul_delta"], "split": "dev"}
{"name": "PadicInt.exists_mem_range_of_norm_rat_le_one", "query": "{p : ℕ} [hp_prime : Fact (Nat.Prime p)] (r : ℚ) (h : ‖↑r‖ ≤ 1) : ∃ n, 0 ≤ n ∧ n < ↑p ∧ ‖⟨↑r, h⟩ - ↑n‖ < 1", "area": "number_theory", "module": "Mathlib.NumberTheory.Padics.RingHoms", "concrete": true, "gold": ["PadicInt.modPart_lt_p", "PadicInt.modPart_nonneg", "PadicInt.norm_sub_modPart"], "gold_b": ["PadicInt.modPart_lt_p", "PadicInt.modPart_nonneg", "PadicInt.norm_sub_modPart"], "split": "dev"}
{"name": "Nat.psub_eq_some", "query": "{m n k : ℕ} : m.psub n = some k ↔ k + n = m", "area": "number_theory", "module": "Mathlib.Data.Nat.PSub", "concrete": true, "gold": ["Nat.ppred_eq_some", "add_left_comm"], "gold_b": ["Nat.ppred_eq_some", "add_left_comm"], "split": "dev"}
{"name": "EisensteinSeries.summable_linear_sub_mul_linear_add", "query": "(z : ℂ) (c₁ c₂ : ℤ) : Summable fun n => ((↑c₁ * z - ↑n) * (↑c₂ * z + ↑n))⁻¹", "area": "number_theory", "module": "Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable", "concrete": true, "gold": ["Asymptotics.IsBigO.comp_neg_int", "Asymptotics.IsBigO.mul", "EisensteinSeries.linear_inv_isBigO_right", "EisensteinSeries.summable_inv_of_isBigO_rpow_inv", "pow_two"], "gold_b": ["Asymptotics.IsBigO.comp_neg_int", "Asymptotics.IsBigO.mul", "Asymptotics.isBigO_neg_right", "Asymptotics.isBigO_sup", "EisensteinSeries.linear_inv_isBigO_right", "EisensteinSeries.summable_inv_of_isBigO_rpow_inv", "Int.cast_neg", "Int.cofinite_eq", "Nat.cast_lt", "Real.rpow_ofNat", "abs_mul_abs_self", "inv_neg", "mul_inv_rev", "mul_neg", "pow_two"], "split": "dev"}
{"name": "SemiconjBy.intCast_mul_right", "query": "{α : Type u_2} [Ring α] {a x y : α} (h : SemiconjBy a x y) (n : ℤ) : SemiconjBy a (↑n * x) (↑n * y)", "area": "number_theory", "module": "Mathlib.Data.Int.Cast.Lemmas", "concrete": true, "gold": ["Int.commute_cast", "SemiconjBy.mul_right"], "gold_b": ["Int.commute_cast", "SemiconjBy.mul_right"], "split": "dev"}
{"name": "Nat.Prime.mem_primeFactors'", "query": "{n p : ℕ} (hp : Nat.Prime p) (hdvd : p ∣ n) [NeZero n] : p ∈ n.primeFactors", "area": "number_theory", "module": "Mathlib.Data.Nat.PrimeFin", "concrete": true, "gold": ["Nat.Prime.mem_primeFactors"], "gold_b": ["Nat.Prime.mem_primeFactors"], "split": "dev"}
{"name": "Nat.pow_sub_lt_descFactorial", "query": "{n k : ℕ} : 2 ≤ k → k ≤ n → (n + 1 - k) ^ k < n.descFactorial k", "area": "number_theory", "module": "Mathlib.Data.Nat.Factorial.Basic", "concrete": true, "gold": ["Nat.pow_sub_lt_descFactorial'"], "gold_b": ["Nat.pow_sub_lt_descFactorial'"], "split": "dev"}
{"name": "DirichletCharacter.LFunction_apply_one_ne_zero", "query": "{N : ℕ} {χ : DirichletCharacter ℂ N} [NeZero N] (hχ : χ ≠ 1) : DirichletCharacter.LFunction χ 1 ≠ 0", "area": "number_theory", "module": "Mathlib.NumberTheory.LSeries.Nonvanishing", "concrete": true, "gold": ["Complex.one_re", "DirichletCharacter.LFunction_ne_zero_of_one_le_re", "le_rfl"], "gold_b": ["Complex.one_re", "DirichletCharacter.LFunction_ne_zero_of_one_le_re", "le_rfl"], "split": "dev"}
{"name": "Nat.sqrt_lt", "query": "{m n : ℕ} : m.sqrt < n ↔ m < n * n", "area": "number_theory", "module": "Mathlib.Data.Nat.Sqrt", "concrete": true, "gold": ["Nat.le_sqrt", "not_le"], "gold_b": ["Nat.le_sqrt", "not_le"], "split": "dev"}
{"name": "Finset.map_perm", "query": "{α : Type u_1} {s : Finset α} {σ : Equiv.Perm α} (hs : {a | σ a ≠ a} ⊆ ↑s) : Finset.map (Equiv.toEmbedding σ) s = s", "area": "combinatorics", "module": "Mathlib.Data.Finset.Image", "concrete": true, "gold": ["Finset.coe_injective", "Finset.coe_map", "Set.image_perm"], "gold_b": ["Finset.coe_injective", "Finset.coe_map", "Set.image_perm"], "split": "dev"}
{"name": "Finset.sup'_comp_eq_image", "query": "{α : Type u_2} {β : Type u_3} {γ : Type u_4} [SemilatticeSup α] [DecidableEq β] {s : Finset γ} {f : γ → β} (hs : s.Nonempty) (g : β → α) : s.sup' hs (g ∘ f) = (Finset.image f s).sup' ⋯ g", "area": "combinatorics", "module": "Mathlib.Data.Finset.Lattice.Fold", "concrete": true, "gold": ["Finset.sup'_image"], "gold_b": ["Finset.Nonempty.image", "Finset.sup'_image"], "split": "dev"}
{"name": "Finset.Nat.antidiagonal_filter_le_fst_of_le", "query": "{n k : ℕ} (h : k ≤ n) : {a ∈ Finset.HasAntidiagonal.antidiagonal n | k ≤ a.1} = Finset.map ({ toFun := fun x => x + k, inj' := ⋯ }.prodMap (Function.Embedding.refl ℕ)) (Finset.HasAntidiagonal.antidiagonal (n - k))", "area": "combinatorics", "module": "Mathlib.Data.Finset.NatAntidiagonal", "concrete": true, "gold": ["add_right_comm", "tsub_add_cancel_of_le"], "gold_b": ["Finset.HasAntidiagonal.mem_antidiagonal", "Finset.ext", "Finset.mem_filter", "Finset.mem_map", "add_left_injective", "add_right_comm", "tsub_add_cancel_of_le"], "split": "dev"}
{"name": "Finset.image_biUnion", "query": "{α : Type u_1} {β : Type u_2} {γ : Type u_3} [DecidableEq β] [DecidableEq γ] {f : α → β} {s : Finset α} {t : β → Finset γ} : (Finset.image f s).biUnion t = s.biUnion fun a => t (f a)", "area": "combinatorics", "module": "Mathlib.Data.Finset.Union", "concrete": true, "gold": ["Finset.biUnion_insert", "Finset.image_insert", "Finset.induction_on"], "gold_b": ["Finset.biUnion_insert", "Finset.image_insert", "Finset.induction_on"], "split": "dev"}
{"name": "catalan_eq_centralBinom_div", "query": "(n : ℕ) : catalan n = n.centralBinom / (n + 1)", "area": "combinatorics", "module": "Mathlib.Combinatorics.Enumerative.Catalan.Basic", "concrete": true, "gold": ["Finset.sum_range", "Finset.sum_range_sub", "Nat.cast_mul", "Nat.cast_sum", "Nat.succ_dvd_centralBinom", "catalan_succ", "mul_div", "tsub_le_self"], "gold_b": ["Finset.sum_range", "Finset.sum_range_sub", "Int.cast_div_ofNat_charZero", "Int.cast_inj", "Int.cast_natCast", "Nat.cast_add", "Nat.cast_inj", "Nat.cast_mul", "Nat.cast_nonneg'", "Nat.cast_sub", "Nat.cast_sum", "Nat.cast_zero", "Nat.centralBinom_zero", "Nat.succ_dvd_centralBinom", "Nat.succ_mul_centralBinom_succ", "Right.add_pos_of_nonneg_of_pos", "add_pos'", "catalan_succ", "catalan_zero", "div_one", "div_self", "mul_div", "mul_div_cancel_left₀", "mul_neg", "ne_of_gt", "neg_add_rev", "tsub_le_self", "tsub_self", "zero_sub"], "split": "dev"}
{"name": "SimpleGraph.Walk.getVert_mem_tail_support", "query": "{V : Type u} {G : SimpleGraph V} {u v : V} {p : G.Walk u v} (hp : ¬p.Nil) {i : ℕ} : i ≠ 0 → p.getVert i ∈ p.support.tail", "area": "combinatorics", "module": "Mathlib.Combinatorics.SimpleGraph.Walk.Operations", "concrete": true, "gold": ["SimpleGraph.Walk.getVert_mem_support", "SimpleGraph.Walk.getVert_tail", "SimpleGraph.Walk.support_tail_of_not_nil"], "gold_b": ["SimpleGraph.Walk.getVert_mem_support", "SimpleGraph.Walk.getVert_tail", "SimpleGraph.Walk.support_tail_of_not_nil"], "split": "dev"}
{"name": "Finset.subset_coe_filter_of_subset_forall", "query": "{α : Type u_1} (p : α → Prop) [DecidablePred p] (s : Finset α) {t : Set α} (h₁ : t ⊆ ↑s) (h₂ : ∀ x ∈ t, p x) : t ⊆ ↑(Finset.filter p s)", "area": "combinatorics", "module": "Mathlib.Data.Finset.Filter", "concrete": true, "gold": ["Finset.coe_filter"], "gold_b": ["Finset.coe_filter"], "split": "dev"}
{"name": "SimpleGraph.Walk.drop_of_length_le", "query": "{V : Type u} {G : SimpleGraph V} {u v : V} {n : ℕ} {p : G.Walk u v} (h : p.length ≤ n) : p.drop n = SimpleGraph.Walk.nil.copy ⋯ ⋯", "area": "combinatorics", "module": "Mathlib.Combinatorics.SimpleGraph.Walk.Operations", "concrete": true, "gold": ["SimpleGraph.Walk.Nil.eq_copy_nil", "SimpleGraph.Walk.nil_drop_of_length_le"], "gold_b": ["SimpleGraph.Walk.Nil.eq_copy_nil", "SimpleGraph.Walk.nil_drop_of_length_le"], "split": "dev"}
{"name": "Fin.univ_succAbove", "query": "(n : ℕ) (p : Fin (n + 1)) : Finset.univ = Finset.cons p (Finset.map p.succAboveEmb Finset.univ) ⋯", "area": "combinatorics", "module": "Mathlib.Data.Fintype.Basic", "concrete": true, "gold": ["Finset.map_eq_image"], "gold_b": ["Fin.image_succAbove_univ", "Fin.succAbove_ne", "Finset.cons_eq_insert", "Finset.insert_compl_self", "Finset.map_eq_image", "Finset.mem_map", "Finset.mem_univ"], "split": "dev"}
{"name": "Finset.pluennecke_ruzsa_inequality_pow_div", "query": "{G : Type u_1} [DecidableEq G] [CommGroup G] {A : Finset G} (hA : A.Nonempty) (B : Finset G) (n : ℕ) : ↑(B ^ n).card ≤ (↑(A / B).card / ↑A.card) ^ n * ↑A.card", "area": "combinatorics", "module": "Mathlib.Combinatorics.Additive.PluenneckeRuzsa", "concrete": true, "gold": ["Finset.pluennecke_ruzsa_inequality_pow_div_pow_div", "div_one"], "gold_b": ["Finset.pluennecke_ruzsa_inequality_pow_div_pow_div", "div_one"], "split": "dev"}
{"name": "Finset.sup_preimage_val_id", "query": "{α : Type u} [Lattice α] [OrderBot α] {P : α → Prop} (Psup : ∀ ⦃s t : α⦄, P s → P t → P (s ⊔ t)) (Pbot : P ⊥) {t : Finset α} (ht : ∀ x ∈ t, P x) : (t.preimage Subtype.val ⋯).sup id = ⟨t.sup id, ⋯⟩", "area": "combinatorics", "module": "Mathlib.Data.Finset.Preimage", "concrete": true, "gold": ["Finset.sup_coe", "Finset.sup_induction", "Finset.sup_preimage_self", "Function.Injective.injOn", "Set.injOn_of_injective", "Set.mapsTo_preimage", "Subtype.val_injective"], "gold_b": ["Finset.sup_coe", "Finset.sup_induction", "Finset.sup_preimage_self", "Function.Injective.injOn", "Set.injOn_of_injective", "Set.mapsTo_preimage", "Set.mem_image", "Set.mem_preimage", "SetLike.mem_coe", "Subtype.val_injective"], "split": "dev"}
{"name": "SimpleGraph.zarankiewicz_le_iff_of_nonneg", "query": "{m n s t : ℕ} {V : Type u_1} {W : Type u_2} {α : Type u_3} {β : Type u_4} [Fintype V] [Fintype W] [Fintype α] [Fintype β] {R : Type u_5} [Semiring R] [LinearOrder R] [FloorSemiring R] (hm : Fintype.card V = m) (hn : Fintype.card W = n) (hs : Fintype.card α = s) (ht : Fintype.card β = t) {x : R} (h : 0 ≤ x) : ↑(SimpleGraph.zarankiewicz m n s t) ≤ x ↔ ∀ ⦃G : SimpleGraph (V ⊕ W)⦄ [inst : DecidableRel G.Adj], G ≤ completeBipartiteGraph V W → (completeBipartiteGraph α β).Free G → ↑G.edgeFinset.card ≤ x", "area": "combinatorics", "module": "Mathlib.Combinatorics.SimpleGraph.Extremal.Zarankiewicz", "concrete": true, "gold": ["Nat.le_floor_iff", "SimpleGraph.zarankiewicz_le_iff"], "gold_b": ["Nat.le_floor_iff", "SimpleGraph.zarankiewicz_le_iff"], "split": "dev"}
{"name": "Finset.image_inter", "query": "{α : Type u_1} {β : Type u_2} [DecidableEq β] {f : α → β} [DecidableEq α] (s₁ s₂ : Finset α) (hf : Function.Injective f) : Finset.image f (s₁ ∩ s₂) = Finset.image f s₁ ∩ Finset.image f s₂", "area": "combinatorics", "module": "Mathlib.Data.Finset.Image", "concrete": true, "gold": ["Finset.image_inter_of_injOn", "Function.Injective.injOn"], "gold_b": ["Finset.image_inter_of_injOn", "Function.Injective.injOn"], "split": "dev"}
{"name": "Nat.two_le_centralBinom", "query": "(n : ℕ) (n_pos : 0 < n) : 2 ≤ n.centralBinom", "area": "combinatorics", "module": "Mathlib.Data.Nat.Choose.Central", "concrete": true, "gold": ["Nat.choose_le_centralBinom", "Nat.choose_one_right"], "gold_b": ["Nat.choose_le_centralBinom", "Nat.choose_one_right"], "split": "dev"}
{"name": "Fintype.all_card_le_rel_image_card_iff_exists_injective", "query": "{α : Type u} {β : Type v} [DecidableEq β] (R : SetRel α β) [(a : α) → Fintype ↑(R.image {a})] : (∀ (A : Finset α), A.card ≤ Fintype.card ↑(R.image ↑A)) ↔ ∃ f, Function.Injective f ∧ ∀ (x : α), (x, f x) ∈ R", "area": "combinatorics", "module": "Mathlib.Combinatorics.Hall.Basic", "concrete": true, "gold": ["Finset.all_card_le_biUnion_card_iff_exists_injective", "Set.toFinset_card"], "gold_b": ["Finset.all_card_le_biUnion_card_iff_exists_injective", "Finset.ext", "Finset.mem_biUnion", "Set.mem_singleton_iff", "Set.mem_toFinset", "Set.toFinset_card", "SetLike.mem_coe"], "split": "dev"}
{"name": "Finset.image₂_subset_iff_right", "query": "{α : Type u_1} {β : Type u_3} {γ : Type u_5} [DecidableEq γ] {f : α → β → γ} {s : Finset α} {t : Finset β} {u : Finset γ} : Finset.image₂ f s t ⊆ u ↔ ∀ b ∈ t, Finset.image (fun a => f a b) s ⊆ u", "area": "combinatorics", "module": "Mathlib.Data.Finset.NAry", "concrete": true, "gold": ["Finset.image_subset_iff", "Finset.image₂_subset_iff", "forall₂_comm"], "gold_b": ["Finset.image_subset_iff", "Finset.image₂_subset_iff", "forall₂_comm"], "split": "dev"}
{"name": "SimpleGraph.is3Clique_iff", "query": "{α : Type u_1} {G : SimpleGraph α} {s : Finset α} [DecidableEq α] : G.IsNClique 3 s ↔ ∃ a b c, G.Adj a b ∧ G.Adj a c ∧ G.Adj b c ∧ s = {a, b, c}", "area": "combinatorics", "module": "Mathlib.Combinatorics.SimpleGraph.Clique", "concrete": true, "gold": ["Finset.card_eq_three", "SimpleGraph.IsNClique.card_eq", "SimpleGraph.is3Clique_triple_iff"], "gold_b": ["Finset.card_eq_three", "SimpleGraph.IsNClique.card_eq", "SimpleGraph.is3Clique_triple_iff"], "split": "dev"}
{"name": "Nat.sum_range_choose", "query": "(n : ℕ) : ∑ m ∈ Finset.range (n + 1), n.choose m = 2 ^ n", "area": "combinatorics", "module": "Mathlib.Data.Nat.Choose.Sum", "concrete": true, "gold": ["add_pow", "one_add_one_eq_two"], "gold_b": ["add_pow", "one_add_one_eq_two", "one_pow"], "split": "dev"}
{"name": "Finset.singleton_ne_univ", "query": "{α : Type u_1} [Fintype α] [Nontrivial α] (a : α) : {a} ≠ Finset.univ", "area": "combinatorics", "module": "Mathlib.Data.Finset.BooleanAlgebra", "concrete": true, "gold": ["SetLike.coe_ne_coe"], "gold_b": ["Finset.coe_singleton", "Finset.coe_univ", "Set.singleton_ne_univ", "SetLike.coe_ne_coe"], "split": "dev"}
{"name": "Finset.max_mono", "query": "{α : Type u_1} [LinearOrder α] {s t : Finset α} (st : s ⊆ t) : s.max ≤ t.max", "area": "combinatorics", "module": "Mathlib.Data.Finset.Max", "concrete": true, "gold": ["Finset.sup_mono"], "gold_b": ["Finset.sup_mono"], "split": "dev"}
{"name": "Finset.image₂_singleton_left'", "query": "{α : Type u_1} {β : Type u_3} {γ : Type u_5} [DecidableEq γ] {f : α → β → γ} {t : Finset β} {a : α} : Finset.image₂ f {a} t = Finset.image (f a) t", "area": "combinatorics", "module": "Mathlib.Data.Finset.NAry", "concrete": true, "gold": ["Finset.image₂_singleton_left"], "gold_b": ["Finset.image₂_singleton_left"], "split": "dev"}
{"name": "Finset.erase_subset_erase", "query": "{α : Type u_1} [DecidableEq α] (a : α) {s t : Finset α} (h : s ⊆ t) : s.erase a ⊆ t.erase a", "area": "combinatorics", "module": "Mathlib.Data.Finset.Erase", "concrete": true, "gold": ["Finset.val_le_iff", "Multiset.erase_le_erase"], "gold_b": ["Finset.val_le_iff", "Multiset.erase_le_erase"], "split": "dev"}
{"name": "Finset.sup_product_right", "query": "{α : Type u_1} {β : Type u_2} {γ : Type u_3} [SemilatticeSup α] [OrderBot α] (s : Finset β) (t : Finset γ) (f : β × γ → α) : (s ×ˢ t).sup f = t.sup fun i' => s.sup fun i => f (i, i')", "area": "combinatorics", "module": "Mathlib.Data.Finset.Lattice.Prod", "concrete": true, "gold": ["Finset.sup_comm", "Finset.sup_product_left"], "gold_b": ["Finset.sup_comm", "Finset.sup_product_left"], "split": "dev"}
{"name": "Set.toFinset_ofFinset", "query": "{α : Type u_1} {p : Set α} (s : Finset α) (H : ∀ (x : α), x ∈ s ↔ x ∈ p) : p.toFinset = s", "area": "combinatorics", "module": "Mathlib.Data.Fintype.Sets", "concrete": true, "gold": ["Finset.ext", "Set.mem_toFinset"], "gold_b": ["Finset.ext", "Set.mem_toFinset"], "split": "dev"}
{"name": "SzemerediRegularity.pow_mul_m_le_card_part", "query": "{α : Type u_1} [DecidableEq α] [Fintype α] {P : Finpartition Finset.univ} {u : Finset α} (hP : P.IsEquipartition) (hu : u ∈ P.parts) : 4 ^ P.parts.card * ↑(Fintype.card α / SzemerediRegularity.stepBound P.parts.card) ≤ ↑u.card", "area": "combinatorics", "module": "Mathlib.Combinatorics.SimpleGraph.Regularity.Bound", "concrete": true, "gold": ["Finpartition.IsEquipartition.average_le_card_part"], "gold_b": ["Finpartition.IsEquipartition.average_le_card_part", "Nat.cast_le", "Nat.cast_mul", "Nat.cast_pow"], "split": "dev"}
{"name": "Finset.isLowerSet_shatterer", "query": "{α : Type u_1} [DecidableEq α] (𝒜 : Finset (Finset α)) : IsLowerSet ↑𝒜.shatterer", "area": "combinatorics", "module": "Mathlib.Combinatorics.SetFamily.Shatter", "concrete": true, "gold": ["Finset.Shatters.mono_right"], "gold_b": ["Finset.Shatters.mono_right", "Finset.mem_shatterer", "SetLike.mem_coe"], "split": "dev"}
{"name": "SimpleGraph.IsTuranMaximal.degree_eq_of_not_adj", "query": "{V : Type u_1} [Fintype V] {G : SimpleGraph V} [DecidableRel G.Adj] {r : ℕ} {s t : V} (h : G.IsTuranMaximal r) (hn : ¬G.Adj s t) : G.degree s = G.degree t", "area": "combinatorics", "module": "Mathlib.Combinatorics.SimpleGraph.Extremal.Turan", "concrete": true, "gold": ["LT.lt.ne'", "SimpleGraph.CliqueFree.replaceVertex", "SimpleGraph.adj_comm", "SimpleGraph.card_edgeFinset_replaceVertex_of_not_adj", "le_of_not_gt", "lt_of_le_of_ne"], "gold_b": ["LT.lt.ne'", "SimpleGraph.CliqueFree.replaceVertex", "SimpleGraph.adj_comm", "SimpleGraph.card_edgeFinset_replaceVertex_of_not_adj", "le_of_not_gt", "lt_of_le_of_ne", "not_le"], "split": "dev"}
{"name": "Finset.map_disjSum", "query": "{α : Type u_1} {β : Type u_2} {γ : Type u_3} {s : Finset α} {t : Finset β} (f : α ⊕ β ↪ γ) : Finset.map f (s.disjSum t) = (Finset.map (Function.Embedding.inl.trans f) s).disjUnion (Finset.map (Function.Embedding.inr.trans f) t) ⋯", "area": "combinatorics", "module": "Mathlib.Data.Finset.Sum", "concrete": true, "gold": ["Finset.val_injective", "Multiset.map_disjSum"], "gold_b": ["Finset.disjoint_map", "Finset.disjoint_map_inl_map_inr", "Finset.map_map", "Finset.val_injective", "Multiset.map_disjSum"], "split": "dev"}
{"name": "Finset.diffs_subset_left", "query": "{α : Type u_2} [DecidableEq α] [GeneralizedBooleanAlgebra α] {s t₁ t₂ : Finset α} : t₁ ⊆ t₂ → s.diffs t₁ ⊆ s.diffs t₂", "area": "combinatorics", "module": "Mathlib.Data.Finset.Sups", "concrete": true, "gold": ["Finset.image₂_subset_left"], "gold_b": ["Finset.image₂_subset_left"], "split": "dev"}
{"name": "Rel.edgeDensity_add_edgeDensity_compl", "query": "{α : Type u_4} {β : Type u_5} (r : α → β → Prop) [(a : α) → DecidablePred (r a)] {s : Finset α} {t : Finset β} (hs : s.Nonempty) (ht : t.Nonempty) : Rel.edgeDensity r s t + Rel.edgeDensity (fun x y => ¬r x y) s t = 1", "area": "combinatorics", "module": "Mathlib.Combinatorics.SimpleGraph.Density", "concrete": true, "gold": ["Finset.Nonempty.card_pos", "LT.lt.ne'", "Rel.card_interedges_add_card_interedges_compl", "add_div", "div_eq_one_iff_eq", "mul_pos"], "gold_b": ["Finset.Nonempty.card_pos", "LT.lt.ne'", "Nat.cast_add", "Nat.cast_inj", "Nat.cast_mul", "Nat.cast_zero", "Rel.card_interedges_add_card_interedges_compl", "add_div", "div_eq_one_iff_eq", "mul_pos"], "split": "dev"}
{"name": "Finset.max'_eq_sorted_last", "query": "{α : Type u_1} [LinearOrder α] {s : Finset α} {h : s.Nonempty} : s.max' h = (s.sort fun a b => a ≤ b)[(s.sort fun a b => a ≤ b).length - 1]", "area": "combinatorics", "module": "Mathlib.Data.Finset.Sort", "concrete": true, "gold": ["Finset.card_pos", "Finset.sorted_last_eq_max'_aux"], "gold_b": ["Finset.card_pos", "Finset.length_sort", "Finset.sorted_last_eq_max'_aux"], "split": "dev"}
{"name": "Finset.Colex.toColex_sdiff_lt_toColex_sdiff", "query": "{α : Type u_1} [PartialOrder α] {s t u : Finset α} [DecidableEq α] (hus : u ⊆ s) (hut : u ⊆ t) : toColex (s \\ u) < toColex (t \\ u) ↔ toColex s < toColex t", "area": "combinatorics", "module": "Mathlib.Combinatorics.Colex", "concrete": true, "gold": ["Finset.Colex.toColex_sdiff_le_toColex_sdiff", "lt_iff_lt_of_le_iff_le'"], "gold_b": ["Finset.Colex.toColex_sdiff_le_toColex_sdiff", "lt_iff_lt_of_le_iff_le'"], "split": "dev"}
{"name": "Finset.covBy_iff_exists_erase", "query": "{α : Type u_1} {s t : Finset α} [DecidableEq α] : s ⋖ t ↔ ∃ a ∈ t, t.erase a = s", "area": "combinatorics", "module": "Mathlib.Data.Finset.Grade", "concrete": true, "gold": ["Finset.coe_covBy_coe", "Finset.coe_erase", "Finset.coe_inj", "Finset.mem_coe", "Set.covBy_iff_exists_sdiff_singleton"], "gold_b": ["Finset.coe_covBy_coe", "Finset.coe_erase", "Finset.coe_inj", "Finset.mem_coe", "Set.covBy_iff_exists_sdiff_singleton"], "split": "dev"}
{"name": "Finset.image_subset_sups_left", "query": "{α : Type u_2} [DecidableEq α] [SemilatticeSup α] {s t : Finset α} {b : α} : b ∈ t → Finset.image (fun x => x ⊔ b) s ⊆ s ⊻ t", "area": "combinatorics", "module": "Mathlib.Data.Finset.Sups", "concrete": true, "gold": ["Finset.image_subset_image₂_left"], "gold_b": ["Finset.image_subset_image₂_left"], "split": "dev"}
{"name": "SimpleGraph.le_chromaticNumber_iff_forall_surjective", "query": "{V : Type u} {G : SimpleGraph V} {n : ℕ} : ↑n ≤ G.chromaticNumber ↔ ∀ (C : G.Coloring (Fin n)), Function.Surjective ⇑C", "area": "combinatorics", "module": "Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex", "concrete": true, "gold": ["SimpleGraph.card_le_chromaticNumber_iff_forall_surjective"], "gold_b": ["Fintype.card_fin", "SimpleGraph.card_le_chromaticNumber_iff_forall_surjective"], "split": "dev"}
{"name": "Finset.three_lt_card_iff", "query": "{α : Type u_1} {s : Finset α} : 3 < s.card ↔ ∃ a b c d, a ∈ s ∧ b ∈ s ∧ c ∈ s ∧ d ∈ s ∧ a ≠ b ∧ a ≠ c ∧ a ≠ d ∧ b ≠ c ∧ b ≠ d ∧ c ≠ d", "area": "combinatorics", "module": "Mathlib.Data.Finset.Card", "concrete": true, "gold": ["Finset.card_eq_four", "Finset.insert_subset_iff", "Finset.le_card_iff_exists_subset_card"], "gold_b": ["Finset.card_eq_four", "Finset.insert_subset_iff", "Finset.le_card_iff_exists_subset_card", "Finset.singleton_subset_iff"], "split": "dev"}
{"name": "Finset.smulAntidiagonal_min_smul_min", "query": "{G : Type u_1} {P : Type u_2} [LinearOrder G] [LinearOrder P] [SMul G P] [IsOrderedCancelSMul G P] {s : Set G} {t : Set P} (hs : s.IsWF) (ht : t.IsWF) (hns : s.Nonempty) (hnt : t.Nonempty) : Finset.SMulAntidiagonal (hs.min hns • ht.min hnt) ⋯ = {(hs.min hns, ht.min hnt)}", "area": "combinatorics", "module": "Mathlib.Data.Finset.SMulAntidiagonal", "concrete": true, "gold": ["Finset.mem_singleton", "Finset.mem_smulAntidiagonal", "IsCancelSMul.left_cancel", "LE.le.eq_of_not_lt", "LT.lt.ne'", "SMul.smul_lt_smul_of_lt_of_le", "Set.IsWF.min_le", "Set.IsWF.min_mem"], "gold_b": ["Finset.ext", "Finset.mem_singleton", "Finset.mem_smulAntidiagonal", "IsCancelSMul.left_cancel", "LE.le.eq_of_not_lt", "LT.lt.ne'", "SMul.smul_lt_smul_of_lt_of_le", "Set.IsWF.isPWO", "Set.IsWF.min_le", "Set.IsWF.min_mem", "Set.SMulAntidiagonal.finite_of_isPWO"], "split": "dev"}
{"name": "Finset.sup_bot", "query": "{α : Type u_2} {β : Type u_3} [SemilatticeSup α] [OrderBot α] (s : Finset β) : (s.sup fun x => ⊥) = ⊥", "area": "combinatorics", "module": "Mathlib.Data.Finset.Lattice.Fold", "concrete": true, "gold": ["Finset.eq_empty_or_nonempty", "Finset.sup_const", "Finset.sup_empty"], "gold_b": ["Finset.eq_empty_or_nonempty", "Finset.sup_const", "Finset.sup_empty"], "split": "dev"}
{"name": "Finset.truncatedSup_union_of_notMem", "query": "{α : Type u_1} [SemilatticeSup α] {s t : Finset α} {a : α} [DecidableLE α] [OrderTop α] [DecidableEq α] (hs : a ∉ lowerClosure ↑s) (ht : a ∉ lowerClosure ↑t) : (s ∪ t).truncatedSup a = ⊤", "area": "combinatorics", "module": "Mathlib.Combinatorics.SetFamily.AhlswedeZhang", "concrete": true, "gold": ["Finset.truncatedSup_of_notMem"], "gold_b": ["Finset.coe_union", "Finset.truncatedSup_of_notMem", "LowerSet.mem_sup_iff", "lowerClosure_union"], "split": "dev"}
{"name": "SimpleGraph.isBipartiteWith_neighborFinset", "query": "{V : Type u_1} {v : V} {G : SimpleGraph V} {s t : Finset V} [Fintype ↑(G.neighborSet v)] [DecidableRel G.Adj] (h : G.IsBipartiteWith ↑s ↑t) (hv : v ∈ s) : G.neighborFinset v = {w ∈ t | G.Adj v w}", "area": "combinatorics", "module": "Mathlib.Combinatorics.SimpleGraph.Bipartite", "concrete": true, "gold": ["Finset.mem_filter", "SimpleGraph.IsBipartiteWith.mem_of_mem_adj", "SimpleGraph.mem_neighborFinset"], "gold_b": ["Finset.ext", "Finset.mem_filter", "SimpleGraph.IsBipartiteWith.mem_of_mem_adj", "SimpleGraph.mem_neighborFinset"], "split": "dev"}
{"name": "Finset.Nonempty.exists_eq_singleton_or_nontrivial", "query": "{α : Type u_1} {s : Finset α} : s.Nonempty → (∃ a, s = {a}) ∨ s.Nontrivial", "area": "combinatorics", "module": "Mathlib.Data.Finset.Insert", "concrete": true, "gold": ["Finset.eq_singleton_or_nontrivial"], "gold_b": ["Finset.eq_singleton_or_nontrivial"], "split": "dev"}
{"name": "Fin.card_filter_univ_succ", "query": "{n : ℕ} (p : Fin (n + 1) → Prop) [DecidablePred p] : {x | p x}.card = if p 0 then {x | p x.succ}.card + 1 else {x | p x.succ}.card", "area": "combinatorics", "module": "Mathlib.Data.Fintype.Fin", "concrete": true, "gold": ["Fin.univ_succ", "Finset.card_cons", "Finset.card_map", "Finset.filter_cons", "Finset.filter_map"], "gold_b": ["Fin.image_succ_univ", "Fin.succ_injective", "Fin.univ_succ", "Finset.card_cons", "Finset.card_map", "Finset.filter_cons", "Finset.filter_map", "Finset.map_eq_image", "Finset.mem_compl", "Finset.mem_of_mem_filter", "Finset.mem_singleton", "Function.mt"], "split": "dev"}
{"name": "SimpleGraph.card_cliqueFinset_le", "query": "{α : Type u_1} {G : SimpleGraph α} [Fintype α] [DecidableEq α] [DecidableRel G.Adj] {n : ℕ} : (G.cliqueFinset n).card ≤ (Fintype.card α).choose n", "area": "combinatorics", "module": "Mathlib.Combinatorics.SimpleGraph.Clique", "concrete": true, "gold": ["Finset.card_mono", "Finset.card_powersetCard", "Finset.card_univ", "SimpleGraph.IsNClique.card_eq"], "gold_b": ["Finset.card_mono", "Finset.card_powersetCard", "Finset.card_univ", "Finset.mem_powersetCard", "Finset.subset_univ", "SimpleGraph.IsNClique.card_eq", "SimpleGraph.mem_cliqueFinset_iff"], "split": "dev"}
{"name": "Finset.disjoint_filter_filter_not", "query": "{α : Type u_1} (s t : Finset α) (p : α → Prop) [DecidablePred p] [(x : α) → Decidable ¬p x] : Disjoint (Finset.filter p s) ({a ∈ t | ¬p a})", "area": "combinatorics", "module": "Mathlib.Data.Finset.Basic", "concrete": true, "gold": ["Finset.disjoint_filter_filter'", "disjoint_compl_right"], "gold_b": ["Finset.disjoint_filter_filter'", "disjoint_compl_right"], "split": "dev"}