ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/MeasureTheory/Function/L1Space.lean
theorem Integrable.smul_measure {f : α → β} (h : Integrable f μ) {c : ℝ≥0∞} (hc : c ≠ ∞) :
    Integrable f (c • μ)
-- Mathlib/Algebra/Order/Hom/Monoid.lean
@[to_additive (attr := simp)]
theorem comp_one (f : β →*o γ) : f.comp (1 : α →*o β) = 1
-- Mathlib/Data/Set/Card.lean
theorem Finite.finite_of_encard_le {s : Set α} {t : Set β} (hs : s.Finite)
    (h : t.encard ≤ s.encard) : t.Finite
-- Mathlib/Order/Filter/AtTopBot.lean
lemma tendsto_zpow_atTop_atTop {n : ℤ} (hn : 0 < n) : Tendsto (fun x : α ↦ x ^ n) atTop atTop
-- Mathlib/Order/SuccPred/CompleteLinearOrder.lean
lemma exists_eq_iSup_of_not_isSuccLimit (hf : ¬ IsSuccLimit (⨆ i, f i)) :
    ∃ i, f i = ⨆ i, f i
-- Mathlib/Analysis/Calculus/Deriv/Add.lean
@[simp]
theorem deriv_neg'' : deriv (fun x : 𝕜 => -x) x = -1
-- .lake/packages/batteries/Batteries/Data/String/Lemmas.lean
theorem remainingToString {it} (h : ValidFor l r it) : it.remainingToString = ⟨r⟩
-- Mathlib/Order/SupClosed.lean
lemma IsSublattice.preimage [FunLike F β α] [LatticeHomClass F β α]
    (hs : IsSublattice s) (f : F) :
    IsSublattice (f ⁻¹' s)
-- Mathlib/MeasureTheory/Measure/Hausdorff.lean
@[simp]
theorem hausdorffMeasure_preimage (e : X ≃ᵢ Y) (d : ℝ) (s : Set Y) : μH[d] (e ⁻¹' s) = μH[d] s
-- Mathlib/Analysis/Normed/Operator/LinearIsometry.lean
@[simp]
theorem mul_refl (e : E ≃ₗᵢ[R] E) : e * refl _ _ = e
-- Mathlib/Topology/PartialHomeomorph.lean
theorem map_nhdsWithin_eq (h : e.IsImage s t) (hx : x ∈ e.source) : map e (𝓝[s] x) = 𝓝[t] e x
-- Mathlib/Algebra/Polynomial/Basic.lean
@[simp high]
theorem toFinsupp_C_mul_X (a : R) : Polynomial.toFinsupp (C a * X) = Finsupp.single 1 a
-- Mathlib/Order/BoundedOrder.lean
/-- An order is an `OrderTop` if it has a greatest element.
We state this using a data mixin, holding the value of `⊤` and the greatest element constraint. -/
class OrderTop (α : Type u) [LE α] extends Top α where
  /-- `⊤` is the greatest element -/
  le_top : ∀ a : α, a ≤ ⊤
-- Mathlib/Data/DFinsupp/Basic.lean
@[to_additive (attr := simp, norm_cast)]
theorem coe_dfinsupp_prod [Monoid R] [CommMonoid S] (f : Π₀ i, β i) (g : ∀ i, β i → R →* S) :
    ⇑(f.prod g) = f.prod fun a b => ⇑(g a b)
-- Mathlib/GroupTheory/Schreier.lean
/-- **Schreier's Lemma**: If `R : Finset G` is a `rightTransversal` of `H : Subgroup G`
  with `1 ∈ R`, and if `G` is generated by `S : Finset G`, then `H` is generated by the `Finset`
  `(R * S).image (fun g ↦ g * (toFun hR g)⁻¹)`. -/
theorem closure_mul_image_eq_top' [DecidableEq G] {R S : Finset G}
    (hR : (R : Set G) ∈ rightTransversals (H : Set G)) (hR1 : (1 : G) ∈ R)
    (hS : closure (S : Set G) = ⊤) :
    closure (((R * S).image fun g => ⟨_, mul_inv_toFun_mem hR g⟩ : Finset H) : Set H) = ⊤
-- Mathlib/GroupTheory/Congruence/Basic.lean
/-- Given a congruence relation `c` on a monoid and a homomorphism `f` constant on `c`'s
    equivalence classes, `f` has the same image as the homomorphism that `f` induces on the
    quotient. -/
@[to_additive "Given an additive congruence relation `c` on an `AddMonoid` and a homomorphism `f`
constant on `c`'s equivalence classes, `f` has the same image as the homomorphism that `f` induces
on the quotient."]
theorem lift_range (H : c ≤ ker f) : MonoidHom.mrange (c.lift f H) = MonoidHom.mrange f
-- Mathlib/RingTheory/Coalgebra/Equiv.lean
@[simp]
theorem coe_toEquiv_trans : (e₁₂ : A ≃ B).trans e₂₃ = (e₁₂.trans e₂₃ : A ≃ C)
-- Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean
/-- Rotation by π is negation. -/
theorem rotation_pi_apply (x : V) : o.rotation π x = -x
-- Mathlib/Order/Height.lean
theorem cons_mem_subchain_iff :
    (a::l) ∈ s.subchain ↔ a ∈ s ∧ l ∈ s.subchain ∧ ∀ b ∈ l.head?, a < b
-- Mathlib/Topology/Algebra/UniformGroup.lean
@[to_additive]
lemma MonoidHom.tendsto_coe_cofinite_of_discrete [T2Space G] {H : Type*} [Group H] {f : H →* G}
    (hf : Function.Injective f) (hf' : DiscreteTopology f.range) :
    Tendsto f cofinite (cocompact _)
-- Mathlib/Topology/Algebra/Module/Basic.lean
@[simp]
theorem coe_toLinearEquiv (f : M₁ ≃SL[σ₁₂] M₂) : ⇑f.toLinearEquiv = f
-- Mathlib/Topology/Category/TopCat/Opens.lean
@[simp]
theorem coe_inclusion {X : TopCat} {U : Opens X} :
    (inclusion U : U → X) = Subtype.val
-- Mathlib/Algebra/Order/Hom/Basic.lean
/-- `SubmultiplicativeHomClass F α β` states that `F` is a type of submultiplicative morphisms. -/
@[to_additive SubadditiveHomClass]
class SubmultiplicativeHomClass (F α β : Type*) [Mul α] [Mul β] [LE β] [FunLike F α β] : Prop where
  /-- the image of a product is less or equal than the product of the images. -/
  map_mul_le_mul (f : F) : ∀ a b, f (a * b) ≤ f a * f b
-- Mathlib/Data/NNReal/Basic.lean
theorem mul_sup (a b c : ℝ≥0) : a * (b ⊔ c) = a * b ⊔ a * c
-- Mathlib/CategoryTheory/Equivalence.lean
@[reassoc (attr := simp)]
theorem functor_unit_comp (e : C ≌ D) (X : C) :
    e.functor.map (e.unit.app X) ≫ e.counit.app (e.functor.obj X) = 𝟙 (e.functor.obj X)
-- Mathlib/Data/Real/EReal.lean
lemma div_nonpos_of_nonneg_of_nonpos {a b : EReal} (h : 0 ≤ a) (h' : b ≤ 0) : a / b ≤ 0
-- Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean
lemma extend_finset_to_connected (Gpc : G.Preconnected) {t : Finset V} (tn : t.Nonempty) :
    ∃ (t' : Finset V), t ⊆ t' ∧ (G.induce (t' : Set V)).Connected
-- Mathlib/Data/Multiset/Basic.lean
@[simp]
theorem inf_eq_inter (s t : Multiset α) : s ⊓ t = s ∩ t
-- Mathlib/MeasureTheory/Measure/MeasureSpace.lean
@[simp]
theorem toMeasure_toOuterMeasure (m : OuterMeasure α) (h : ms ≤ m.caratheodory) :
    (m.toMeasure h).toOuterMeasure = m.trim
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
theorem neg_coe_abs_toReal_of_sign_nonpos {θ : Angle} (h : θ.sign ≤ 0) : -↑|θ.toReal| = θ
-- Mathlib/Topology/NhdsSet.lean
theorem mem_nhdsSet_iff_exists : s ∈ 𝓝ˢ t ↔ ∃ U : Set X, IsOpen U ∧ t ⊆ U ∧ U ⊆ s
-- Mathlib/CategoryTheory/Iso.lean
@[simp]
theorem symm_inv (α : X ≅ Y) : α.symm.inv = α.hom
-- Mathlib/Probability/Process/Stopping.lean
theorem progMeasurable_min_stopping_time [MetrizableSpace ι] (hτ : IsStoppingTime f τ) :
    ProgMeasurable f fun i ω => min i (τ ω)
-- Mathlib/AlgebraicGeometry/Pullbacks.lean
/-- Given `Uᵢ ×[Z] Y`, this is the glued fibered product `X ×[Z] Y`. -/
@[simps U V f t t', simps (config := .lemmasOnly) J]
def gluing : Scheme.GlueData.{u}
-- Mathlib/CategoryTheory/Adjunction/Unique.lean
@[reassoc (attr := simp)]
theorem rightAdjointUniq_hom_app_counit {F : C ⥤ D} {G G' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F ⊣ G')
    (x : D) :
    F.map ((rightAdjointUniq adj1 adj2).hom.app x) ≫ adj2.counit.app x = adj1.counit.app x
-- Mathlib/CategoryTheory/Equivalence.lean
@[simp]
theorem pow_zero (e : C ≌ C) : e ^ (0 : ℤ) = Equivalence.refl
-- Mathlib/Algebra/Polynomial/Module/AEval.lean
@[simp] lemma mem_mapSubmodule {m : AEval R M a} :
    m ∈ mapSubmodule a hp ↔ (of R M a).symm m ∈ p
-- .lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean
@[deprecated getElem_cons_drop (since := "2024-06-12")]
theorem get_cons_drop (l : List α) (i) : get l i :: drop (i + 1) l = drop i l
-- Mathlib/GroupTheory/FreeGroup/Basic.lean
@[to_additive]
theorem lift_eq_prod_map {β : Type v} [Group β] {f : α → β} {x} : lift f x = prod (map f x)
-- Mathlib/LinearAlgebra/Matrix/IsDiag.lean
theorem IsDiag.sub [AddGroup α] {A B : Matrix n n α} (ha : A.IsDiag) (hb : B.IsDiag) :
    (A - B).IsDiag
-- Mathlib/Topology/MetricSpace/Completion.lean
@[deprecated eq_of_dist_eq_zero (since := "2024-03-10")]
protected theorem eq_of_dist_eq_zero (x y : Completion α) (h : dist x y = 0) : x = y
-- Mathlib/Geometry/Euclidean/Basic.lean
/-- The `orthogonalProjection` lies in the given subspace. -/
theorem orthogonalProjection_mem {s : AffineSubspace ℝ P} [Nonempty s]
    [HasOrthogonalProjection s.direction] (p : P) : ↑(orthogonalProjection s p) ∈ s
-- .lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean
protected theorem sub_eq_max_sub (n m : Nat) : n - m = max n m - m
-- Mathlib/Algebra/GCDMonoid/Basic.lean
theorem gcd_mk_mk {a b : α} : gcd (Associates.mk a) (Associates.mk b) = Associates.mk (gcd a b)
-- Mathlib/Algebra/Order/Module/Defs.lean
lemma smul_nonneg' [SMulPosMono α β] (ha : 0 ≤ a) (hb : 0 ≤ b₁) : 0 ≤ a • b₁
-- Mathlib/Data/List/MinMax.lean
theorem maximum_cons (a : α) (l : List α) : maximum (a :: l) = max ↑a (maximum l)
-- Mathlib/Topology/MetricSpace/Contracting.lean
theorem toLipschitzWith (hf : ContractingWith K f) : LipschitzWith K f
-- Mathlib/Computability/Language.lean
@[simp]
lemma reverse_kstar (l : Language α) : l∗.reverse = l.reverse∗
-- Mathlib/CategoryTheory/Iso.lean
/-- A functor `F : C ⥤ D` sends isomorphisms `i : X ≅ Y` to isomorphisms `F.obj X ≅ F.obj Y` -/
@[simps]
def mapIso (F : C ⥤ D) {X Y : C} (i : X ≅ Y) : F.obj X ≅ F.obj Y
-- Mathlib/Topology/ContinuousFunction/Algebra.lean
@[simp]
theorem coe_star (f : C(α, β)) : ⇑(star f) = star (⇑f)
-- Mathlib/CategoryTheory/Limits/Shapes/Countable.lean
theorem sequentialFunctor_map : Antitone (sequentialFunctor_obj J)
-- Mathlib/Data/Nat/MaxPowDiv.lean
theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1
-- Mathlib/Data/Set/Basic.lean
theorem Antitone.union [Preorder β] {f g : β → Set α} (hf : Antitone f) (hg : Antitone g) :
    Antitone fun x => f x ∪ g x
-- Mathlib/Data/Finset/Basic.lean
@[simp]
theorem toFinset_eq_empty {m : Multiset α} : m.toFinset = ∅ ↔ m = 0
-- Mathlib/Algebra/Order/Monoid/Unbundled/WithTop.lean
theorem bot_lt_add [LT α] {a b : WithBot α} : ⊥ < a + b ↔ ⊥ < a ∧ ⊥ < b
-- Mathlib/MeasureTheory/Order/Lattice.lean
@[measurability]
theorem Measurable.sup (hf : Measurable f) (hg : Measurable g) : Measurable fun a => f a ⊔ g a
-- Mathlib/Topology/UnitInterval.lean
theorem symm_bijective : Function.Bijective (symm : I → I)
-- Mathlib/Algebra/Order/Ring/Basic.lean
lemma Even.pow_pos_iff (hn : Even n) (h₀ : n ≠ 0) : 0 < a ^ n ↔ a ≠ 0
-- Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
@[to_additive]
theorem Function.Injective.multipliable_iff {g : γ → β} (hg : Injective g)
    (hf : ∀ x ∉ Set.range g, f x = 1) : Multipliable (f ∘ g) ↔ Multipliable f
-- Mathlib/Algebra/Category/Grp/Basic.lean
@[to_additive (attr := simp)]
theorem coe_id' {G : Type _} [CommGroup G] :
    @DFunLike.coe (G →* G) G (fun _ ↦ G) MonoidHom.instFunLike
      (CategoryStruct.id (X := CommGrp.of G)) = id
-- Mathlib/Algebra/Ring/Hom/Basic.lean
protected theorem map_dvd (f : α →+* β) {a b : α} : a ∣ b → f a ∣ f b
-- Mathlib/NumberTheory/Pell.lean
/-- If there is a fundamental solution, it is unique. -/
theorem subsingleton {a b : Solution₁ d} (ha : IsFundamental a) (hb : IsFundamental b) : a = b
-- Mathlib/AlgebraicTopology/CechNerve.lean
/-- The augmented Čech conerve construction, as a functor from `Arrow C`. -/
@[simps]
def augmentedCechConerve : Arrow C ⥤ CosimplicialObject.Augmented C
-- Mathlib/CategoryTheory/Limits/Presheaf.lean
lemma yonedaEquiv_ι_presheafHom (P : Cᵒᵖ ⥤ Type v₁) {X : C} (f : yoneda.obj X ⟶ P) :
    yonedaEquiv (f ≫ presheafHom φ P) =
      (G.map f).app (Opposite.op (F.obj X)) ((φ.app X).app _ (𝟙 _))
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean
/-- The power series expansion of `Complex.cosh`. -/
lemma hasSum_cosh (z : ℂ) : HasSum (fun n ↦ z ^ (2 * n) / ↑(2 * n)!) (cosh z)
-- Mathlib/LinearAlgebra/Quotient.lean
@[simp]
theorem liftQ_mkQ (f : M →ₛₗ[τ₁₂] M₂) (h) : (p.liftQ f h).comp p.mkQ = f
-- Mathlib/Data/Set/Pointwise/Interval.lean
@[simp]
theorem preimage_add_const_Ioo : (fun x => x + a) ⁻¹' Ioo b c = Ioo (b - a) (c - a)
-- Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
lemma isLocalAtTarget [P.IsMultiplicative]
    (hP : ∀ {X Y Z : Scheme.{u}} (f : X ⟶ Y) (g : Y ⟶ Z) [IsOpenImmersion g], P (f ≫ g) → P f) :
    IsLocalAtTarget P
-- Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean
lemma Y_eq_negY_of_Y_eq [NoZeroDivisors R] {P Q : Fin 3 → R} (hQz : Q z ≠ 0)
    (hx : P x * Q z ^ 2 = Q x * P z ^ 2) (hy : P y * Q z ^ 3 = Q y * P z ^ 3)
    (hy' : P y * Q z ^ 3 = W'.negY Q * P z ^ 3) : P y = W'.negY P
-- Mathlib/Order/Interval/Set/Basic.lean
theorem not_mem_Iio : c ∉ Iio b ↔ b ≤ c
-- Mathlib/MeasureTheory/Integral/IntegrableOn.lean
@[simp]
theorem integrableOn_add_measure :
    IntegrableOn f s (μ + ν) ↔ IntegrableOn f s μ ∧ IntegrableOn f s ν
-- Mathlib/Geometry/Manifold/MFDeriv/Basic.lean
theorem MDifferentiableAt.hasMFDerivAt (h : MDifferentiableAt I I' f x) :
    HasMFDerivAt I I' f x (mfderiv I I' f x)
-- Mathlib/Geometry/Euclidean/Inversion/ImageHyperplane.lean
/-- The inversion with center `c` and radius `R` maps a sphere passing through the center to a
hyperplane. -/
theorem inversion_mem_perpBisector_inversion_iff (hR : R ≠ 0) (hx : x ≠ c) (hy : y ≠ c) :
    inversion c R x ∈ perpBisector c (inversion c R y) ↔ dist x y = dist y c
-- Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean
/-- The image of a binary bicone under a functor. -/
@[simps!]
def mapBinaryBicone {X Y : C} (b : BinaryBicone X Y) : BinaryBicone (F.obj X) (F.obj Y)
-- Mathlib/Data/Matroid/Closure.lean
lemma Basis'.basis_closure_right (h : M.Basis' I X) : M.Basis I (M.closure X)
-- Mathlib/Algebra/Quandle.lean
@[simp]
theorem act'_symm_apply (x y : R) : (act' x).symm y = x ◃⁻¹ y
-- Mathlib/Data/Nat/Find.lean
lemma findGreatest_spec (hmb : m ≤ n) (hm : P m) : P (Nat.findGreatest P n)
-- Mathlib/Topology/Algebra/Group/Basic.lean
@[to_additive]
theorem map_mul_right_nhds_one (x : G) : map (· * x) (𝓝 1) = 𝓝 x
-- Mathlib/Analysis/Convex/Basic.lean
/-- The standard one-dimensional simplex in `Fin 2 → 𝕜` is equivalent to the unit interval. -/
@[simps (config := .asFn)]
def stdSimplexEquivIcc : stdSimplex 𝕜 (Fin 2) ≃ Icc (0 : 𝕜) 1
-- Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
theorem measurable_of_countable [Countable α] [MeasurableSingletonClass α] (f : α → β) :
    Measurable f
-- Mathlib/GroupTheory/GroupAction/Blocks.lean
/-- Alternate definition of a block -/
theorem IsBlock.mk_notempty {B : Set X} :
    IsBlock G B ↔ ∀ g g' : G, g • B ∩ g' • B ≠ ∅ → g • B = g' • B
-- Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean
@[reassoc (attr := simp)]
theorem diagonal_fst : diagonal f ≫ pullback.fst _ _ = 𝟙 _
-- Mathlib/RingTheory/Ideal/IdempotentFG.lean
/-- A finitely generated idempotent ideal is generated by an idempotent element -/
theorem isIdempotentElem_iff_of_fg {R : Type*} [CommRing R] (I : Ideal R) (h : I.FG) :
    IsIdempotentElem I ↔ ∃ e : R, IsIdempotentElem e ∧ I = R ∙ e
-- Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean
@[simp]
lemma algebraMap_galRestrictHom_apply (σ : L →ₐ[K] L) (x : B) :
    algebraMap B L (galRestrictHom A K L B σ x) = σ (algebraMap B L x)
-- Mathlib/Algebra/Ring/Subsemiring/Basic.lean
theorem mem_sSup_of_directedOn {S : Set (Subsemiring R)} (Sne : S.Nonempty)
    (hS : DirectedOn (· ≤ ·) S) {x : R} : x ∈ sSup S ↔ ∃ s ∈ S, x ∈ s
-- .lake/packages/lean4/src/lean/Init/Data/Int/Order.lean
theorem not_ofNat_neg (n : Nat) : ¬((n : Int) < 0)
-- Mathlib/Order/Minimal.lean
theorem IsAntichain.minimals_eq (h : IsAntichain r s) : minimals r s = s
-- Mathlib/Order/Lattice.lean
@[simp]
theorem right_eq_sup : b = a ⊔ b ↔ a ≤ b
-- Mathlib/Data/TypeVec.lean
theorem appendFun_id_id {α : TypeVec n} {β : Type*} :
    (@TypeVec.id n α ::: @_root_.id β) = TypeVec.id
-- Mathlib/RingTheory/TensorProduct/Basic.lean
protected theorem mul_one (x : A ⊗[R] B) : mul x (1 ⊗ₜ 1) = x
-- Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
/-- Compose a quadratic map with a linear function on the left. -/
@[simps! (config := { simpRhs := true })]
def _root_.LinearMap.compQuadraticMap' [CommSemiring S] [Algebra S R] [Module S N] [Module S M]
    [IsScalarTower S R N] [IsScalarTower S R M] [Module S P]
    (f : N →ₗ[S] P) (Q : QuadraticMap R M N) : QuadraticMap S M P
-- Mathlib/Algebra/BigOperators/Group/Finset.lean
@[to_additive (attr := simp)]
theorem prod_dite_irrel (p : Prop) [Decidable p] (s : Finset α) (f : p → α → β) (g : ¬p → α → β) :
    ∏ x ∈ s, (if h : p then f h x else g h x) =
      if h : p then ∏ x ∈ s, f h x else ∏ x ∈ s, g h x
-- Mathlib/Data/Nat/Factors.lean
theorem primeFactorsList_subset_of_dvd {n k : ℕ} (h : n ∣ k) (h' : k ≠ 0) :
    n.primeFactorsList ⊆ k.primeFactorsList
-- Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean
lemma logTaylor_zero : logTaylor 0 = fun _ ↦ 0
-- Mathlib/Algebra/Group/Subgroup/Basic.lean
@[to_additive (attr := simp)]
theorem comap_inclusion_subgroupOf {K₁ K₂ : Subgroup G} (h : K₁ ≤ K₂) (H : Subgroup G) :
    (H.subgroupOf K₂).comap (inclusion h) = H.subgroupOf K₁
-- Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean
@[simp]
theorem reverse.map_one : reverse (1 : CliffordAlgebra Q) = 1
-- Mathlib/Data/Real/EReal.lean
@[simp]
theorem sign_mul (x y : EReal) : sign (x * y) = sign x * sign y
-- Mathlib/Data/Nat/GCD/Basic.lean
@[simp]
theorem coprime_sub_self_left {m n : ℕ} (h : m ≤ n) : Coprime (n - m) m ↔ Coprime n m
-- Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
theorem ContDiffWithinAt.rpow_const_of_ne (hf : ContDiffWithinAt ℝ n f s x) (h : f x ≠ 0) :
    ContDiffWithinAt ℝ n (fun x => f x ^ p) s x
-- Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean
/-- The measure equivalence between `EuclideanSpace ℝ ι` and `ι → ℝ` is volume preserving. -/
theorem EuclideanSpace.volume_preserving_measurableEquiv :
    MeasurePreserving (EuclideanSpace.measurableEquiv ι)
-- Mathlib/Algebra/Order/Archimedean.lean
theorem le_of_forall_lt_rat_imp_le (h : ∀ q : ℚ, y < q → x ≤ q) : x ≤ y
-- Mathlib/Topology/Basic.lean
theorem isOpen_setOf_eventually_nhds {p : X → Prop} : IsOpen { x | ∀ᶠ y in 𝓝 x, p y }
-- Mathlib/CategoryTheory/Preadditive/Biproducts.lean
/-- The existence of binary biproducts implies that there is at most one preadditive structure. -/
theorem biprod.add_eq_lift_id_desc [HasBinaryBiproduct X X] :
    f + g = biprod.lift (𝟙 X) (𝟙 X) ≫ biprod.desc f g
-- Mathlib/RingTheory/MvPowerSeries/Inverse.lean
protected theorem eq_mul_inv_iff_mul_eq {φ₁ φ₂ φ₃ : MvPowerSeries σ k}
    (h : constantCoeff σ k φ₃ ≠ 0) : φ₁ = φ₂ * φ₃⁻¹ ↔ φ₁ * φ₃ = φ₂
-- Mathlib/Data/Finset/Basic.lean
nonrec lemma Nontrivial.exists_ne (hs : s.Nontrivial) (a : α) : ∃ b ∈ s, b ≠ a
-- Mathlib/Data/QPF/Univariate/Basic.lean
theorem lawfulFunctor
    (h : ∀ α β : Type u, @Functor.mapConst F _ α _ = Functor.map ∘ Function.const β) :
    LawfulFunctor F
-- Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean
lemma leftFractionRel_op_iff
    {X Y : C} (z₁ z₂ : W.RightFraction X Y) :
    LeftFractionRel z₁.op z₂.op ↔ RightFractionRel z₁ z₂
-- Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean
theorem hurwitzZetaEven_one_sub_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc (0 : ℝ) 1) :
    hurwitzZetaEven x (1 - 2 * k) =
      -1 / (2 * k) * ((Polynomial.bernoulli (2 * k)).map (algebraMap ℚ ℂ)).eval (x : ℂ)
-- Mathlib/CategoryTheory/Monoidal/Internal/FunctorCategory.lean
/-- The counit for the equivalence `CommMon_ (C ⥤ D) ≌ C ⥤ CommMon_ D`.
-/
@[simps!]
def counitIso : inverse ⋙ functor ≅ 𝟭 (C ⥤ CommMon_ D)
-- Mathlib/RingTheory/Ideal/MinimalPrime.lean
lemma minimalPrimes_eq_minimals : minimalPrimes R = minimals (· ≤ ·) (setOf Ideal.IsPrime)
-- Mathlib/Algebra/Lie/TraceForm.lean
/-- A nilpotent Lie algebra with a representation whose trace form is non-singular is Abelian. -/
lemma isLieAbelian_of_ker_traceForm_eq_bot (h : LinearMap.ker (traceForm R L M) = ⊥) :
    IsLieAbelian L
-- Mathlib/Analysis/SpecialFunctions/Integrals.lean
@[simp]
theorem intervalIntegrable_inv_one_add_sq :
    IntervalIntegrable (fun x : ℝ => (↑1 + x ^ 2)⁻¹) μ a b
-- Mathlib/NumberTheory/Padics/PadicVal.lean
lemma padicValRat_def (p : ℕ) (q : ℚ) :
    padicValRat p q = padicValInt p q.num - padicValNat p q.den
-- Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean
/-- The inverse of a homotopy equivalence. -/
@[simps]
def symm (e : HomotopyEquiv S₁ S₂) : HomotopyEquiv S₂ S₁
-- Mathlib/LinearAlgebra/FinsuppVectorSpace.lean
@[simp]
theorem coe_basis {φ : ι → Type*} (b : ∀ i, Basis (φ i) R M) :
    ⇑(Finsupp.basis b) = fun ix : Σi, φ i => single ix.1 (b ix.1 ix.2)
-- Mathlib/Algebra/Polynomial/Expand.lean
theorem expand_contract' [NoZeroDivisors R] {f : R[X]} (hf : Polynomial.derivative f = 0) :
    expand R p (contract p f) = f
-- Mathlib/Algebra/Group/Support.lean
@[to_additive]
theorem mulSupport_eq_preimage (f : α → M) : mulSupport f = f ⁻¹' {1}ᶜ
-- Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean
@[to_additive (attr := simp) lt_add_iff_pos_left]
theorem lt_mul_iff_one_lt_left' [CovariantClass α α (swap (· * ·)) (· < ·)]
    [ContravariantClass α α (swap (· * ·)) (· < ·)] (a : α) {b : α} : a < b * a ↔ 1 < b
-- Mathlib/Data/Matrix/Basic.lean
theorem map_mulVec [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (M : Matrix m n R)
    (v : n → R) (i : m) : f ((M *ᵥ v) i) = (M.map f *ᵥ (f ∘ v)) i
-- Mathlib/CategoryTheory/Equivalence.lean
@[simp]
theorem Equivalence_mk'_counit (functor inverse unit_iso counit_iso f) :
    (⟨functor, inverse, unit_iso, counit_iso, f⟩ : C ≌ D).counit = counit_iso.hom
-- Mathlib/RingTheory/Derivation/Basic.lean
protected theorem map_neg : D (-a) = -D a
-- Mathlib/RingTheory/JacobsonIdeal.lean
theorem isRadical_of_eq_jacobson (h : jacobson I = I) : I.IsRadical
-- Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean
@[simp]
theorem den_zero : ((0 : NumDenSameDeg 𝒜 x).den : A) = 1
-- Mathlib/Data/Part.lean
@[elab_as_elim]
protected theorem induction_on {P : Part α → Prop} (a : Part α) (hnone : P none)
    (hsome : ∀ a : α, P (some a)) : P a
-- Mathlib/Data/Nat/Bits.lean
/--
The same as `binaryRec_eq`,
but that one unfortunately requires `f` to be the identity when appending `false` to `0`.
Here, we allow you to explicitly say that that case is not happening,
i.e. supplying `n = 0 → b = true`. -/
theorem binaryRec_eq' {C : ℕ → Sort*} {z : C 0} {f : ∀ b n, C n → C (bit b n)} (b n)
    (h : f false 0 z = z ∨ (n = 0 → b = true)) :
    binaryRec z f (bit b n) = f b n (binaryRec z f n)
-- Mathlib/CategoryTheory/EqToHom.lean
theorem postcomp_map_heq (H : D ⥤ E) (hx : F.obj X = G.obj X) (hy : F.obj Y = G.obj Y)
    (hmap : HEq (F.map f) (G.map f)) : HEq ((F ⋙ H).map f) ((G ⋙ H).map f)
-- Mathlib/Data/Nat/Prime/Basic.lean
theorem prime_two : Prime (2 : ℤ)
-- Mathlib/Analysis/CstarAlgebra/GelfandDuality.lean
/-- The Gelfand transform as a `StarAlgEquiv` between a commutative unital C⋆-algebra over `ℂ`
and the continuous functions on its `characterSpace`. -/
@[simps!]
noncomputable def gelfandStarTransform : A ≃⋆ₐ[ℂ] C(characterSpace ℂ A, ℂ)
-- Mathlib/RingTheory/Ideal/QuotientOperations.lean
@[simp]
theorem quotient_map_mkₐ {I : Ideal A} (J : Ideal B) (f : A →ₐ[R₁] B) (H : I ≤ J.comap f) {x : A} :
    quotientMapₐ J f H (Quotient.mk I x) = Quotient.mkₐ R₁ J (f x)
-- Mathlib/Topology/MetricSpace/Contracting.lean
theorem efixedPoint_isFixedPt' {s : Set α} (hsc : IsComplete s) (hsf : MapsTo f s s)
    (hf : ContractingWith K <| hsf.restrict f s s) {x : α} (hxs : x ∈ s) (hx : edist x (f x) ≠ ∞) :
    IsFixedPt f (efixedPoint' f hsc hsf hf x hxs hx)
-- Mathlib/RingTheory/Regular/RegularSequence.lean
@[simp] lemma ofList_nil : (ofList [] : Ideal R) = ⊥
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean
theorem derivWithin_arctan (hf : DifferentiableWithinAt ℝ f s x) (hxs : UniqueDiffWithinAt ℝ s x) :
    derivWithin (fun x => arctan (f x)) s x = 1 / (1 + f x ^ 2) * derivWithin f s x
-- Mathlib/RingTheory/HahnSeries/Basic.lean
theorem forallLTEqZero_supp_BddBelow (f : Γ → R) (n : Γ) (hn : ∀(m : Γ), m < n → f m = 0) :
    BddBelow (Function.support f)
-- Mathlib/Algebra/Category/Grp/FilteredColimits.lean
@[to_additive]
theorem colimitInvAux_eq_of_rel (x y : Σ j, F.obj j)
    (h : Types.FilteredColimit.Rel (F ⋙ forget Grp) x y) :
    colimitInvAux.{v, u} F x = colimitInvAux F y
-- Mathlib/Algebra/Polynomial/Basic.lean
@[simp]
theorem toFinsupp_sub {R : Type u} [Ring R] (a b : R[X]) :
    (a - b).toFinsupp = a.toFinsupp - b.toFinsupp
-- Mathlib/Logic/Equiv/Defs.lean
protected theorem congr_fun {f g : Equiv α β} (h : f = g) (x : α) : f x = g x
-- Mathlib/Data/ENNReal/Basic.lean
@[simp] theorem ofReal_one : ENNReal.ofReal (1 : ℝ) = (1 : ℝ≥0∞)
-- Mathlib/Topology/Connected/PathConnected.lean
@[simp]
theorem coe_reparam (γ : Path x y) {f : I → I} (hfcont : Continuous f) (hf₀ : f 0 = 0)
    (hf₁ : f 1 = 1) : ⇑(γ.reparam f hfcont hf₀ hf₁) = γ ∘ f
-- Mathlib/Algebra/Homology/ShortComplex/Linear.lean
@[simp]
lemma homologyMap_smul [S₁.HasHomology] [S₂.HasHomology] :
    homologyMap (a • φ) = a • homologyMap φ