ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Data/Set/Countable.lean
theorem Countable.preimage_of_injOn {s : Set β} (hs : s.Countable) {f : α → β}
    (hf : InjOn f (f ⁻¹' s)) : (f ⁻¹' s).Countable
-- Mathlib/Data/Fin/Basic.lean
theorem lt_succ_castPred {a : Fin (n + 1)} (ha : a ≠ last n) :
    a < succ (a.castPred ha)
-- Mathlib/Analysis/RCLike/Basic.lean
@[rclike_simps, norm_cast]
theorem ofReal_pow (r : ℝ) (n : ℕ) : ((r ^ n : ℝ) : K) = (r : K) ^ n
-- Mathlib/Algebra/Module/Basic.lean
lemma support_const_smul_of_ne_zero [Zero R] [Zero M] [SMulWithZero R M] [NoZeroSMulDivisors R M]
    (c : R) (g : α → M) (hc : c ≠ 0) : support (c • g) = support g
-- .lake/packages/lean4/src/lean/Init/Data/Int/Order.lean
theorem natAbs_negOfNat (n : Nat) : natAbs (negOfNat n) = n
-- Mathlib/Data/Fin/Basic.lean
@[simp, norm_cast] lemma coe_castSuccEmb : (castSuccEmb : Fin n → Fin (n + 1)) = Fin.castSucc
-- Mathlib/Algebra/Homology/SingleHomology.lean
/-- The computation of the homology of single complexes, as a natural isomorphism
`single C c j ⋙ homologyFunctor C c j ≅ 𝟭 C`. -/
@[simps!]
noncomputable def homologyFunctorSingleIso [CategoryWithHomology C] :
    single C c j ⋙ homologyFunctor C c j ≅ 𝟭 _
-- Mathlib/Order/Interval/Set/Monotone.lean
protected theorem Antitone.Ico (hf : Antitone f) (hg : Monotone g) :
    Monotone fun x => Ico (f x) (g x)
-- Mathlib/Topology/Semicontinuous.lean
theorem Continuous.comp_upperSemicontinuous {g : γ → δ} {f : α → γ} (hg : Continuous g)
    (hf : UpperSemicontinuous f) (gmon : Monotone g) : UpperSemicontinuous (g ∘ f)
-- Mathlib/ModelTheory/Bundled.lean
/-- The reduct of any model of `φ.onTheory T` is a model of `T`. -/
@[simps]
def reduct {L' : Language} (φ : L →ᴸ L') (M : (φ.onTheory T).ModelType) : T.ModelType
-- Mathlib/Computability/Primrec.lean
theorem nat_rec' {f : α → ℕ} {g : α → β} {h : α → ℕ × β → β}
    (hf : Primrec f) (hg : Primrec g) (hh : Primrec₂ h) :
    Primrec fun a => (f a).rec (motive := fun _ => β) (g a) fun n IH => h a (n, IH)
-- Mathlib/RingTheory/Ideal/Basic.lean
theorem coe_subset_nonunits [Semiring α] {I : Ideal α} (h : I ≠ ⊤) : (I : Set α) ⊆ nonunits α
-- Mathlib/Order/Filter/AtTopBot.lean
/-- If `f` tends to negative infinity along a nontrivial filter `l`,
then `fun x ↦ f x * r` tends to infinity if and only if `r < 0. `-/
theorem tendsto_mul_const_atTop_iff_neg [NeBot l] (h : Tendsto f l atBot) :
    Tendsto (fun x => f x * r) l atTop ↔ r < 0
-- Mathlib/Data/Set/Pointwise/Basic.lean
@[to_additive]
theorem union_mul_inter_subset : (s ∪ t) * (s ∩ t) ⊆ s * t
-- Mathlib/Analysis/NormedSpace/OperatorNorm/Mul.lean
@[simp]
theorem lsmul_apply (c : 𝕜') (x : E) : lsmul 𝕜 𝕜' c x = c • x
-- Mathlib/Analysis/Complex/CauchyIntegral.lean
/-- Suppose that a function `f : ℂ → E` is *real* differentiable on a closed rectangle with opposite
corners at `z w : ℂ` and $\frac{\partial f}{\partial \bar z}$ is integrable on this rectangle. Then
the integral of `f` over the boundary of the rectangle is equal to the integral of
$2i\frac{\partial f}{\partial \bar z}=i\frac{\partial f}{\partial x}-\frac{\partial f}{\partial y}$
over the rectangle. -/
theorem integral_boundary_rect_of_differentiableOn_real (f : ℂ → E) (z w : ℂ)
    (Hd : DifferentiableOn ℝ f ([[z.re, w.re]] ×ℂ [[z.im, w.im]]))
    (Hi : IntegrableOn (fun z => I • fderiv ℝ f z 1 - fderiv ℝ f z I)
      ([[z.re, w.re]] ×ℂ [[z.im, w.im]])) :
    (∫ x : ℝ in z.re..w.re, f (x + z.im * I)) - (∫ x : ℝ in z.re..w.re, f (x + w.im * I)) +
      I • (∫ y : ℝ in z.im..w.im, f (re w + y * I)) -
      I • (∫ y : ℝ in z.im..w.im, f (re z + y * I)) =
      ∫ x : ℝ in z.re..w.re, ∫ y : ℝ in z.im..w.im,
        I • fderiv ℝ f (x + y * I) 1 - fderiv ℝ f (x + y * I) I
-- Mathlib/Data/Set/Sups.lean
lemma infs_self_subset : s ⊼ s ⊆ s ↔ InfClosed s
-- Mathlib/Order/Filter/Bases.lean
theorem isCountablyGenerated_iff_exists_antitone_basis {f : Filter α} :
    IsCountablyGenerated f ↔ ∃ x : ℕ → Set α, f.HasAntitoneBasis x
-- Mathlib/Data/QPF/Univariate/Basic.lean
theorem Fix.dest_mk (x : F (Fix F)) : Fix.dest (Fix.mk x) = x
-- .lake/packages/batteries/Batteries/Data/String/Lemmas.lean
theorem prev : ∀ {s}, Valid s → s.toString.1 = m₁ ++ c :: m₂ →
    s.prev ⟨utf8Len m₁ + c.utf8Size⟩ = ⟨utf8Len m₁⟩
-- Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean
@[reassoc (attr := simp)]
lemma map_hom_ofInv_id (s : Y ⟶ X) (hs : W s) (L : C ⥤ D) (hL : W.IsInvertedBy L) :
    L.map s ≫ (ofInv s hs).map L hL = 𝟙 _
-- Mathlib/Algebra/Lie/Weights/Basic.lean
lemma zero_apply [Nontrivial (weightSpace M (0 : L → R))] (x) : (0 : Weight R L M) x = 0
-- Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean
@[simp] theorem toBaseChange_ι (Q : QuadraticForm R V) (z : A) (v : V) :
    toBaseChange A Q (ι (Q.baseChange A) (z ⊗ₜ v)) = z ⊗ₜ ι Q v
-- Mathlib/MeasureTheory/Measure/Typeclasses.lean
theorem abs_toReal_measure_sub_le_measure_symmDiff [IsFiniteMeasure μ]
    (hs : MeasurableSet s) (ht : MeasurableSet t) :
    |(μ s).toReal - (μ t).toReal| ≤ (μ (s ∆ t)).toReal
-- Mathlib/Analysis/Calculus/FDeriv/Linear.lean
@[fun_prop]
protected theorem ContinuousLinearMap.hasStrictFDerivAt {x : E} : HasStrictFDerivAt e e x
-- Mathlib/Data/TypeVec.lean
theorem appendFun_comp_splitFun {α γ : TypeVec n} {β δ : Type*} {ε : TypeVec (n + 1)}
    (f₀ : drop ε ⟹ α) (f₁ : α ⟹ γ) (g₀ : last ε → β) (g₁ : β → δ) :
    appendFun f₁ g₁ ⊚ splitFun f₀ g₀ = splitFun (α' := γ.append1 δ) (f₁ ⊚ f₀) (g₁ ∘ g₀)
-- Mathlib/Algebra/Homology/ShortComplex/Homology.lean
/-- A homology data for a short complex `S` induces a homology data for `S.op`. -/
@[simps]
def op (h : S.HomologyData) : S.op.HomologyData
-- Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean
theorem mapLMultilinear_opNorm : ‖mapLMultilinear 𝕜 E E'‖ ≤ 1
-- Mathlib/Algebra/Group/Pi/Basic.lean
@[to_additive (attr := simp)]
lemma _root_.Function.const_mul [Mul β] (a b : β) : const α a * const α b = const α (a * b)
-- Mathlib/MeasureTheory/Function/Jacobian.lean
/-- Change of variable formula for differentiable functions, set version: if a function `f` is
injective and differentiable on a measurable set `s`, then the measure of `f '' s` is given by the
integral of `|(f' x).det|` on `s`.
Note that the measurability of `f '' s` is given by `measurable_image_of_fderivWithin`. -/
theorem lintegral_abs_det_fderiv_eq_addHaar_image (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) :
    (∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ) = μ (f '' s)
-- Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean
/-- A morphism is `QuasiSeparated` if diagonal map is quasi-compact. -/
@[mk_iff]
class QuasiSeparated (f : X ⟶ Y) : Prop where
  /-- A morphism is `QuasiSeparated` if diagonal map is quasi-compact. -/
  diagonalQuasiCompact : QuasiCompact (pullback.diagonal f) := by infer_instance
-- Mathlib/Order/Max.lean
@[simp]
theorem not_isBot [NoBotOrder α] (a : α) : ¬IsBot a
-- Mathlib/RingTheory/WittVector/Defs.lean
@[simp]
theorem constantCoeff_wittSub (n : ℕ) : constantCoeff (wittSub p n) = 0
-- Mathlib/Analysis/Calculus/Deriv/Inv.lean
theorem DifferentiableOn.div (hc : DifferentiableOn 𝕜 c s) (hd : DifferentiableOn 𝕜 d s)
    (hx : ∀ x ∈ s, d x ≠ 0) : DifferentiableOn 𝕜 (fun x => c x / d x) s
-- Mathlib/RingTheory/UniqueFactorizationDomain.lean
theorem eq_of_eq_counts {a b : Associates α} (ha : a ≠ 0) (hb : b ≠ 0)
    (h : ∀ p : Associates α, Irreducible p → p.count a.factors = p.count b.factors) : a = b
-- Mathlib/Algebra/Field/Basic.lean
theorem div_sub_div_same (a b c : K) : a / c - b / c = (a - b) / c
-- Mathlib/Algebra/Homology/Homotopy.lean
/-- homotopy is a transitive relation. -/
@[simps!, trans]
def trans {e f g : C ⟶ D} (h : Homotopy e f) (k : Homotopy f g) : Homotopy e g
-- Mathlib/LinearAlgebra/TensorProduct/Submodule.lean
@[simp]
theorem comm_trans_rTensorOne :
    (TensorProduct.comm R _ _).trans M.rTensorOne = M.lTensorOne
-- Mathlib/Data/Real/Irrational.lean
theorem ne_one (h : Irrational x) : x ≠ 1
-- Mathlib/Topology/EMetricSpace/Basic.lean
@[simp] theorem SeparationQuotient.edist_mk [PseudoEMetricSpace X] (x y : X) :
    edist (mk x) (mk y) = edist x y
-- Mathlib/Order/Disjoint.lean
theorem inf_eq_bot (h : IsCompl x y) : x ⊓ y = ⊥
-- Mathlib/RingTheory/FractionalIdeal/Basic.lean
theorem mul_mem_mul {I J : FractionalIdeal S P} {i j : P} (hi : i ∈ I) (hj : j ∈ J) :
    i * j ∈ I * J
-- Mathlib/Order/Filter/Basic.lean
theorem comap_iSup {ι} {f : ι → Filter β} {m : α → β} : comap m (iSup f) = ⨆ i, comap m (f i)
-- Mathlib/MeasureTheory/Measure/Stieltjes.lean
@[simp]
theorem length_Ioc (a b : ℝ) : f.length (Ioc a b) = ofReal (f b - f a)
-- Mathlib/Order/Hom/Basic.lean
protected theorem injective (e : α ≃o β) : Function.Injective e
-- Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean
theorem mdifferentiableOn_fst {s : Set (M × M')} : MDifferentiableOn (I.prod I') I Prod.fst s
-- Mathlib/NumberTheory/Padics/PadicNumbers.lean
/-- An auxiliary lemma for manipulating sequence indices. -/
theorem lift_index_right {f : PadicSeq p} (hf : ¬f ≈ 0) (v1 v2 : ℕ) :
    padicNorm p (f (stationaryPoint hf)) =
    padicNorm p (f (max v1 (max v2 (stationaryPoint hf))))
-- Mathlib/NumberTheory/Cyclotomic/Rat.lean
/-- In a `p ^ (k + 1)`-th cyclotomic extension of `ℚ`, we have that
  `ζ - 1` divides `p` in `𝓞 K`. -/
lemma toInteger_sub_one_dvd_prime [hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
    (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) : ((hζ.toInteger - 1)) ∣ p
-- Mathlib/Algebra/Homology/ComplexShapeSigns.lean
lemma rel_π₁ {i₁ i₁' : I₁} (h : c₁.Rel i₁ i₁') (i₂ : I₂) :
    c₁₂.Rel (π c₁ c₂ c₁₂ ⟨i₁, i₂⟩) (π c₁ c₂ c₁₂ ⟨i₁', i₂⟩)
-- Mathlib/SetTheory/Cardinal/ENat.lean
lemma toENatAux_ofENat : ∀ n : ℕ∞, toENatAux n = n
-- Mathlib/Order/Filter/Prod.lean
theorem prod_map_left (f : α → β) (F : Filter α) (G : Filter γ) :
    map f F ×ˢ G = map (Prod.map f id) (F ×ˢ G)
-- Mathlib/Analysis/SpecialFunctions/Exponential.lean
/-- The exponential map in a complete normed field `𝕂` of characteristic zero has derivative
`exp 𝕂 x` at any point `x` in the disk of convergence. -/
theorem hasDerivAt_exp_of_mem_ball [CharZero 𝕂] {x : 𝕂}
    (hx : x ∈ EMetric.ball (0 : 𝕂) (expSeries 𝕂 𝕂).radius) : HasDerivAt (exp 𝕂) (exp 𝕂 x) x
-- Mathlib/Algebra/Order/Hom/Monoid.lean
@[to_additive (attr := simp)]
theorem cancel_right {g₁ g₂ : β →*o γ} {f : α →*o β} (hf : Function.Surjective f) :
    g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- Mathlib/Analysis/Calculus/LocalExtr/Basic.lean
/-- **Fermat's Theorem**: the derivative of a function at a local minimum equals zero. -/
theorem IsLocalMin.hasFDerivAt_eq_zero (h : IsLocalMin f a) (hf : HasFDerivAt f f' a) : f' = 0
-- Mathlib/LinearAlgebra/SModEq.lean
lemma pow {I : Ideal A} {x y : A} (n : ℕ) (hxy : x ≡ y [SMOD I]) :
    x ^ n ≡ y ^ n [SMOD I]
-- Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean
/-- Recurrence for the `GenContFract.of` an element `v` of `K` in terms of that of the inverse of
the fractional part of `v`.
-/
theorem of_s_succ (n : ℕ) : (of v).s.get? (n + 1) = (of (fract v)⁻¹).s.get? n
-- Mathlib/Analysis/Normed/Module/Complemented.lean
@[simp]
theorem coe_continuous_linearProjOfClosedCompl (h : IsCompl p q) (hp : IsClosed (p : Set E))
    (hq : IsClosed (q : Set E)) :
    (p.linearProjOfClosedCompl q h hp hq : E →ₗ[𝕜] p) = p.linearProjOfIsCompl q h
-- Mathlib/Analysis/Calculus/FDeriv/Prod.lean
theorem fderivWithin_snd {s : Set (E × F)} (hs : UniqueDiffWithinAt 𝕜 s p) :
    fderivWithin 𝕜 Prod.snd s p = snd 𝕜 E F
-- Mathlib/Data/Complex/Exponential.lean
theorem cos_ofReal_re (x : ℝ) : (cos x).re = Real.cos x
-- Mathlib/Data/Setoid/Partition.lean
/-- A family of injective functions with pairwise disjoint
domains and pairwise disjoint ranges can be glued together
to form an injective function. -/
theorem piecewise_inj {β : Type*} {f : ι → α → β}
    (h_injOn : ∀ i, InjOn (f i) (s i))
    (h_disjoint : PairwiseDisjoint (univ : Set ι) fun i => (f i) '' (s i)) :
    Injective (piecewise hs f)
-- Mathlib/Analysis/SpecialFunctions/Complex/Arctan.lean
/-- The power series expansion of `Complex.arctan`, valid on the open unit disc. -/
theorem hasSum_arctan {z : ℂ} (hz : ‖z‖ < 1) :
    HasSum (fun n : ℕ ↦ (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1)) (arctan z)
-- Mathlib/Tactic/LinearCombination.lean
theorem eq_of_add [AddGroup α] (p : (a:α) = b) (H : (a' - b') - (a - b) = 0) : a' = b'
-- Mathlib/NumberTheory/Primorial.lean
theorem primorial_le_4_pow (n : ℕ) : n# ≤ 4 ^ n
-- Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean
theorem op_sInf (S : Set (Subalgebra R A)) : (sInf S).op = sInf (.unop ⁻¹' S)
-- Mathlib/MeasureTheory/Integral/SetToL1.lean
theorem smul_measure_iff (c : ℝ≥0∞) (hc_ne_zero : c ≠ 0) (hc_ne_top : c ≠ ∞) :
    FinMeasAdditive (c • μ) T ↔ FinMeasAdditive μ T
-- Mathlib/Order/Filter/Basic.lean
protected theorem ext_iff : f = g ↔ ∀ s, s ∈ f ↔ s ∈ g
-- Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean
lemma lift_f (p q : ℤ) (hpq : p + 1 = q) :
    (lift φ α β eq).f p = α.1.v p q hpq ≫
      (inl φ).v q p (by omega) + β.v p p (add_zero p) ≫ (inr φ).f p
-- .lake/packages/lean4/src/lean/Init/Core.lean
theorem Iff.comm: (a ↔ b) ↔ (b ↔ a)
-- Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean
theorem differentiable_exp : Differentiable ℝ exp
-- Mathlib/Algebra/MonoidAlgebra/Degree.lean
theorem le_inf_support_list_prod (degt0 : 0 ≤ degt 0)
    (degtm : ∀ a b, degt a + degt b ≤ degt (a + b)) (l : List R[A]) :
    (l.map fun f : R[A] => f.support.inf degt).sum ≤ l.prod.support.inf degt
-- Mathlib/Topology/Algebra/UniformField.lean
/-- A topological field is completable if it is separated and the image under
the mapping x ↦ x⁻¹ of every Cauchy filter (with respect to the additive uniform structure)
which does not have a cluster point at 0 is a Cauchy filter
(with respect to the additive uniform structure). This ensures the completion is
a field.
-/
class CompletableTopField extends T0Space K : Prop where
  nice : ∀ F : Filter K, Cauchy F → 𝓝 0 ⊓ F = ⊥ → Cauchy (map (fun x => x⁻¹) F)
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean
lemma ψ_odd (m : ℕ) : W.ψ (2 * (m + 2) + 1) =
    W.ψ (m + 4) * W.ψ (m + 2) ^ 3 - W.ψ (m + 1) * W.ψ (m + 3) ^ 3
-- Mathlib/SetTheory/Ordinal/Topology.lean
theorem nhds_right' (a : Ordinal) : 𝓝[>] a = ⊥
-- Mathlib/Data/Set/Lattice.lean
theorem union_iUnion_nat_succ (u : ℕ → Set α) : (u 0 ∪ ⋃ i, u (i + 1)) = ⋃ i, u i
-- .lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
theorem foldrM_eq_reverse_foldlM_data.aux [Monad m]
    (f : α → β → m β) (arr : Array α) (init : β) (i h) :
    (arr.data.take i).reverse.foldlM (fun x y => f y x) init = foldrM.fold f arr 0 i h init
-- Mathlib/Combinatorics/SimpleGraph/Prod.lean
protected theorem Preconnected.ofBoxProdRight [Nonempty α] (h : (G □ H).Preconnected) :
    H.Preconnected
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
theorem contDiff_sin {n} : ContDiff ℂ n sin
-- Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean
theorem ae_nonneg_restrict_of_forall_setIntegral_nonneg {f : α → ℝ}
    (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
    (hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) {t : Set α}
    (ht : MeasurableSet t) (hμt : μ t ≠ ∞) : 0 ≤ᵐ[μ.restrict t] f
-- Mathlib/LinearAlgebra/Eigenspace/Basic.lean
/-- The exponent of a generalized eigenvalue is never 0. -/
theorem exp_ne_zero_of_hasGenEigenvalue {f : End R M} {μ : R} {k : ℕ}
    (h : f.HasGenEigenvalue μ k) : k ≠ 0
-- Mathlib/NumberTheory/Ostrowski.lean
/-- There exists a minimal positive integer with absolute value smaller than 1. -/
lemma exists_minimal_nat_zero_lt_mulRingNorm_lt_one : ∃ p : ℕ, (0 < f p ∧ f p < 1) ∧
    ∀ m : ℕ, 0 < f m ∧ f m < 1 → p ≤ m
-- Mathlib/SetTheory/Ordinal/NaturalOps.lean
theorem nmul_le_nmul_of_nonneg_right (h₁ : a ≤ b) (h₂ : 0 ≤ c) : a ⨳ c ≤ b ⨳ c
-- Mathlib/NumberTheory/Padics/PadicVal.lean
/-- A rewrite lemma for `padicValRat p (q * r)` with conditions `q ≠ 0`, `r ≠ 0`. -/
protected theorem mul {q r : ℚ} (hq : q ≠ 0) (hr : r ≠ 0) :
    padicValRat p (q * r) = padicValRat p q + padicValRat p r
-- Mathlib/Topology/Algebra/Module/Determinant.lean
@[simp]
theorem det_coe_symm {R : Type*} [Field R] {M : Type*} [TopologicalSpace M] [AddCommGroup M]
    [Module R M] (A : M ≃L[R] M) : (A.symm : M →L[R] M).det = (A : M →L[R] M).det⁻¹
-- Mathlib/Algebra/Module/LocalizedModule.lean
@[simp]
theorem mk_cancel_common_left (s' s : S) (m : M) : mk (s' • m) (s' * s) = mk m s
-- Mathlib/Data/Nat/PartENat.lean
@[simp]
theorem get_add {x y : PartENat} (h : (x + y).Dom) : get (x + y) h = x.get h.1 + y.get h.2
-- Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
/-- If a function coincides with a measurable function outside of a countable set, it is
measurable. -/
theorem Measurable.measurable_of_countable_ne [MeasurableSingletonClass α] (hf : Measurable f)
    (h : Set.Countable { x | f x ≠ g x }) : Measurable g
-- .lake/packages/batteries/Batteries/Data/List/Lemmas.lean
theorem cons_subset_cons {l₁ l₂ : List α} (a : α) (s : l₁ ⊆ l₂) : a :: l₁ ⊆ a :: l₂
-- Mathlib/Order/Interval/Set/Basic.lean
@[simp]
theorem Ioo_insert_right (h : a < b) : insert b (Ioo a b) = Ioc a b
-- Mathlib/Topology/Order/Lattice.lean
lemma ContinuousAt.finset_inf' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousAt (f i) x) :
    ContinuousAt (s.inf' hne f) x
-- Mathlib/MeasureTheory/Group/Arithmetic.lean
@[measurability]
lemma measurableSet_eq_fun' {β : Type*} [CanonicallyOrderedAddCommMonoid β] [Sub β] [OrderedSub β]
    {_ : MeasurableSpace β} [MeasurableSub₂ β] [MeasurableSingletonClass β]
    {f g : α → β} (hf : Measurable f) (hg : Measurable g) :
    MeasurableSet {x | f x = g x}
-- Mathlib/Algebra/Order/Chebyshev.lean
/-- Special case of **Chebyshev's Sum Inequality** or the **Cauchy-Schwarz Inequality**: The square
of the sum is less than the size of the set times the sum of the squares. -/
theorem sq_sum_le_card_mul_sum_sq : (∑ i ∈ s, f i) ^ 2 ≤ s.card * ∑ i ∈ s, f i ^ 2
-- Mathlib/Algebra/MvPolynomial/Basic.lean
@[simp]
theorem C_inj {σ : Type*} (R : Type*) [CommSemiring R] (r s : R) :
    (C r : MvPolynomial σ R) = C s ↔ r = s
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean
lemma cfcₙ_id : cfcₙ (id : R → R) a = a
-- Mathlib/CategoryTheory/Sites/Whiskering.lean
lemma Presheaf.IsSheaf.isSeparated {F : Cᵒᵖ ⥤ A} [ConcreteCategory A]
    [J.HasSheafCompose (forget A)] (hF : Presheaf.IsSheaf J F) :
    Presheaf.IsSeparated J F
-- Mathlib/RingTheory/NonUnitalSubring/Basic.lean
theorem mem_prod {s : NonUnitalSubring R} {t : NonUnitalSubring S} {p : R × S} :
    p ∈ s.prod t ↔ p.1 ∈ s ∧ p.2 ∈ t
-- Mathlib/RingTheory/Ideal/QuotientOperations.lean
@[simp]
theorem quotQuotEquivQuotOfLE_symm_mk (x : R) (h : I ≤ J) :
    (quotQuotEquivQuotOfLE h).symm ((Ideal.Quotient.mk J) x) = quotQuotMk I J x
-- Mathlib/CategoryTheory/Limits/EpiMono.lean
lemma epi_iff_isPushout : Epi f ↔ IsPushout f f (𝟙 Y) (𝟙 Y)
-- Mathlib/RingTheory/LaurentSeries.lean
@[norm_cast] -- Porting note (#10618): simp can prove this
theorem coe_zero : ((0 : PowerSeries R) : LaurentSeries R) = 0
-- Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean
lemma _root_.MeasurableEmbedding.singularPart_map (hf : MeasurableEmbedding f)
    (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν] :
    (μ.map f).singularPart (ν.map f) = (μ.singularPart ν).map f
-- .lake/packages/lean4/src/lean/Init/Data/Bool.lean
theorem xor_left_comm : ∀ (x y z : Bool), xor x (xor y z) = xor y (xor x z)
-- Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean
theorem Set.OrdConnected.measurableSet (h : OrdConnected s) : MeasurableSet s
-- Mathlib/Analysis/Convex/Cone/InnerDual.lean
/-- The inner dual of inner dual of a non-empty, closed convex cone is itself.  -/
theorem ConvexCone.innerDualCone_of_innerDualCone_eq_self (K : ConvexCone ℝ H)
    (ne : (K : Set H).Nonempty) (hc : IsClosed (K : Set H)) :
    ((K : Set H).innerDualCone : Set H).innerDualCone = K
-- Mathlib/Algebra/CharP/ExpChar.lean
theorem list_sum_pow_char_pow (l : List R) : l.sum ^ p ^ n = (l.map (· ^ p ^ n : R → R)).sum
-- Mathlib/MeasureTheory/Function/LpSpace.lean
theorem cauchy_tendsto_of_tendsto {f : ℕ → α → E} (hf : ∀ n, AEStronglyMeasurable (f n) μ)
    (f_lim : α → E) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
    (h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N)
    (h_lim : ∀ᵐ x : α ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))) :
    atTop.Tendsto (fun n => eLpNorm (f n - f_lim) p μ) (𝓝 0)
-- Mathlib/Data/Matrix/Basic.lean
theorem mulVec_add [Fintype n] (A : Matrix m n α) (x y : n → α) :
    A *ᵥ (x + y) = A *ᵥ x + A *ᵥ y
-- Mathlib/MeasureTheory/Integral/SetToL1.lean
theorem setToFun_mono_left {T T' : Set α → E →L[ℝ] G''} {C C' : ℝ}
    (hT : DominatedFinMeasAdditive μ T C) (hT' : DominatedFinMeasAdditive μ T' C')
    (hTT' : ∀ s x, T s x ≤ T' s x) (f : α →₁[μ] E) : setToFun μ T hT f ≤ setToFun μ T' hT' f
-- Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean
/-- The angle at the apex of an isosceles triangle is `π` minus twice a base angle, oriented
vector angle form. -/
theorem oangle_eq_pi_sub_two_zsmul_oangle_sub_of_norm_eq {x y : V} (hn : x ≠ y) (h : ‖x‖ = ‖y‖) :
    o.oangle y x = π - (2 : ℤ) • o.oangle (y - x) y
-- Mathlib/Data/Finset/Image.lean
@[simp]
theorem finsetCongr_apply (e : α ≃ β) (s : Finset α) : e.finsetCongr s = s.map e.toEmbedding
-- Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean
theorem eLpNorm_le_add_measure_left (f : α → F) (μ ν : Measure α) {p : ℝ≥0∞} :
    eLpNorm f p ν ≤ eLpNorm f p (μ + ν)
-- Mathlib/Data/Set/Prod.lean
theorem image_prod_mk_subset_prod {f : α → β} {g : α → γ} {s : Set α} :
    (fun x => (f x, g x)) '' s ⊆ (f '' s) ×ˢ (g '' s)
-- Mathlib/SetTheory/Surreal/Basic.lean
/-- Definition of `x ≤ y` on numeric pre-games, in terms of `<` -/
theorem le_iff_forall_lt {x y : PGame} (ox : x.Numeric) (oy : y.Numeric) :
    x ≤ y ↔ (∀ i, x.moveLeft i < y) ∧ ∀ j, x < y.moveRight j
-- Mathlib/Algebra/BigOperators/Group/Finset.lean
@[to_additive]
theorem prod_multiset_count_of_subset [DecidableEq α] [CommMonoid α] (m : Multiset α) (s : Finset α)
    (hs : m.toFinset ⊆ s) : m.prod = ∏ i ∈ s, i ^ m.count i
-- Mathlib/Algebra/Quaternion.lean
@[simp]
theorem im_re : a.im.re = 0
-- Mathlib/Algebra/Lie/Free.lean
theorem liftAux_map_smul (f : X → L) (t : R) (a : lib R X) :
    liftAux R f (t • a) = t • liftAux R f a
-- Mathlib/Topology/Algebra/Order/Field.lean
theorem tendsto_const_mul_pow_nhds_iff' {n : ℕ} {c d : 𝕜} :
    Tendsto (fun x : 𝕜 => c * x ^ n) atTop (𝓝 d) ↔ (c = 0 ∨ n = 0) ∧ c = d
-- Mathlib/Data/Finsupp/Defs.lean
@[simp] theorem update_idem (f : α →₀ M) (a : α) (b c : M) :
    update (update f a b) a c = update f a c
-- Mathlib/NumberTheory/DiophantineApproximation.lean
/-- The set of good rational approximations to a real number `ξ` is infinite if and only if
`ξ` is irrational. -/
theorem Real.infinite_rat_abs_sub_lt_one_div_den_sq_iff_irrational (ξ : ℝ) :
    {q : ℚ | |ξ - q| < 1 / (q.den : ℝ) ^ 2}.Infinite ↔ Irrational ξ
-- Mathlib/CategoryTheory/Idempotents/Basic.lean
theorem isIdempotentComplete_iff_opposite : IsIdempotentComplete Cᵒᵖ ↔ IsIdempotentComplete C
-- Mathlib/Algebra/DirectLimit.lean
theorem lift_unique [IsDirected ι (· ≤ ·)] (F : DirectLimit G f →+* P) (x) :
    F x = lift G f P (fun i => F.comp <| of G f i) (fun i j hij x => by simp [of_f]) x
-- Mathlib/GroupTheory/PushoutI.lean
theorem base_injective (hφ : ∀ i, Function.Injective (φ i)) :
    Function.Injective (base φ)
-- Mathlib/Tactic/Ring/Basic.lean
theorem single_pow {b : ℕ} (_ : (a : R) ^ b = c) : (a + 0) ^ b = c + 0
-- Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
theorem finite_approx_of_totallyBounded {s : Set α} (hs : TotallyBounded s) :
    ∀ ε > 0, ∃ t, t ⊆ s ∧ Set.Finite t ∧ s ⊆ ⋃ y ∈ t, ball y ε
-- Mathlib/Algebra/EuclideanDomain/Basic.lean
protected theorem mul_div_cancel' {a b : R} (hb : b ≠ 0) (hab : b ∣ a) : b * (a / b) = a
-- Mathlib/Data/Nat/GCD/Basic.lean
@[simp]
theorem coprime_mul_left_add_right (m n k : ℕ) : Coprime m (m * k + n) ↔ Coprime m n
-- .lake/packages/lean4/src/lean/Init/Data/List/Impl.lean
@[csimp] theorem unzip_eq_unzipTR : @unzip = @unzipTR
-- Mathlib/Analysis/RCLike/Basic.lean
@[continuity]
theorem continuous_normSq : Continuous (normSq : K → ℝ)
-- Mathlib/Data/Fintype/Basic.lean
@[simp]
theorem toFinset_ssubset [Fintype s] {t : Finset α} : s.toFinset ⊂ t ↔ s ⊂ t