ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/CategoryTheory/Limits/Shapes/Reflexive.lean
/-- If `f,g` is reflexive, then `g,f` is reflexive. -/
theorem IsReflexivePair.swap [IsReflexivePair f g] : IsReflexivePair g f
-- Mathlib/Data/Sym/Sym2.lean
@[simp]
theorem coe_lift_symm_apply (F : Sym2 α → β) (a₁ a₂ : α) :
    (lift.symm F : α → α → β) a₁ a₂ = F s(a₁, a₂)
-- Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean
/-- Every functor indexing a wide (co)equalizer is naturally isomorphic (actually, equal) to a
    `parallelFamily` -/
@[simps!]
def diagramIsoParallelFamily (F : WalkingParallelFamily J ⥤ C) :
    F ≅ parallelFamily fun j => F.map (line j)
-- Mathlib/Data/Fin/Basic.lean
theorem succ_ne_last_of_lt {p i : Fin n} (h : i < p) : succ i ≠ last n
-- Mathlib/Algebra/Algebra/Equiv.lean
@[simp]
lemma one_toLinearMap :
    (1 : A₁ ≃ₐ[R] A₁).toLinearMap = 1
-- .lake/packages/batteries/Batteries/Data/String/Lemmas.lean
theorem setCurr (h : ValidFor l (c :: r) it) :
    ValidFor l (c :: r) (it.setCurr c)
-- Mathlib/Data/Complex/FiniteDimensional.lean
theorem rank_real_complex'.{u} : Cardinal.lift.{u} (Module.rank ℝ ℂ) = 2
-- Mathlib/Algebra/Symmetrized.lean
@[simp]
theorem sym_unsym (a : α) : sym (unsym a) = a
-- Mathlib/AlgebraicGeometry/StructureSheaf.lean
theorem isUnit_toStalk (x : PrimeSpectrum.Top R) (f : x.asIdeal.primeCompl) :
    IsUnit (toStalk R x (f : R))
-- Mathlib/Order/Filter/Cofinite.lean
@[simp]
theorem cofinite_eq_bot [Finite α] : @cofinite α = ⊥
-- .lake/packages/batteries/Batteries/Data/Nat/Lemmas.lean
@[simp] theorem casesDiagOn_zero_succ {motive : Nat → Nat → Sort _} (zero_zero : motive 0 0)
    (zero_succ : ∀ n, motive 0 (n+1)) (succ_zero : ∀ m, motive (m+1) 0)
    (succ_succ : ∀ m n, motive (m+1) (n+1)) (n) :
    Nat.casesDiagOn 0 (n+1) zero_zero zero_succ succ_zero succ_succ = zero_succ n
-- Mathlib/Analysis/NormedSpace/Exponential.lean
theorem expSeries_hasSum_exp_of_mem_ball (x : 𝔸)
    (hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
    HasSum (fun n => expSeries 𝕂 𝔸 n fun _ => x) (exp 𝕂 x)
-- Mathlib/Topology/SeparatedMap.lean
theorem constOn_of_comp (cont : ContinuousOn g s)
    (he : ∀ a ∈ s, ∀ a' ∈ s, p (g a) = p (g a'))
    {a a'} (ha : a ∈ s) (ha' : a' ∈ s) : g a = g a'
-- Mathlib/LinearAlgebra/Span.lean
theorem lt_sup_iff_not_mem {I : Submodule R M} {a : M} : (I < I ⊔ R ∙ a) ↔ a ∉ I
-- Mathlib/Order/Bounds/Basic.lean
theorem isLUB_le_iff (h : IsLUB s a) : a ≤ b ↔ b ∈ upperBounds s
-- Mathlib/CategoryTheory/Sites/Coherent/Basic.lean
/--
The condition `Precoherent C` is essentially the minimal condition required to define the
coherent coverage on `C`.
-/
class Precoherent : Prop where
  /--
  Given an effective epi family `π₁` over `B₁` and a morphism `f : B₂ ⟶ B₁`, there exists
  an effective epi family `π₂` over `B₂`, such that `π₂` factors through `π₁`.
  -/
  pullback {B₁ B₂ : C} (f : B₂ ⟶ B₁) :
    ∀ (α : Type) [Finite α] (X₁ : α → C) (π₁ : (a : α) → (X₁ a ⟶ B₁)),
      EffectiveEpiFamily X₁ π₁ →
    ∃ (β : Type) (_ : Finite β) (X₂ : β → C) (π₂ : (b : β) → (X₂ b ⟶ B₂)),
      EffectiveEpiFamily X₂ π₂ ∧
      ∃ (i : β → α) (ι : (b :  β) → (X₂ b ⟶ X₁ (i b))),
      ∀ (b : β), ι b ≫ π₁ _ = π₂ _ ≫ f
-- Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean
/-- The convergent of `GenContFract.of v` at step `n - 1` is exactly `v` if the
`IntFractPair.stream` of the corresponding continued fraction terminated at step `n`. -/
theorem of_correctness_of_nth_stream_eq_none (nth_stream_eq_none : IntFractPair.stream v n = none) :
    v = (of v).convs (n - 1)
-- Mathlib/Data/Seq/WSeq.lean
theorem LiftRel.trans (R : α → α → Prop) (H : Transitive R) : Transitive (LiftRel R)
-- Mathlib/Data/List/MinMax.lean
theorem minimum_ne_top_of_ne_nil (h : l ≠ []) : l.minimum ≠ ⊤
-- Mathlib/GroupTheory/PushoutI.lean
/-- The normal form for words in the pushout. Every element of the pushout is the product of an
element of the base group and a word made up of letters each of which is in the transversal. -/
structure _root_.Monoid.PushoutI.NormalWord (d : Transversal φ) extends CoprodI.Word G where
  /-- Every `NormalWord` is the product of an element of the base group and a word made up
  of letters each of which is in the transversal. `head` is that element of the base group. -/
  head : H
  /-- All letter in the word are in the transversal. -/
  normalized : ∀ i g, ⟨i, g⟩ ∈ toList → g ∈ d.set i
-- Mathlib/Data/Complex/Basic.lean
@[simp]
theorem conj_re (z : ℂ) : (conj z).re = z.re
-- Mathlib/Algebra/Homology/HomologicalComplex.lean
@[reassoc, elementwise] -- @[simp] -- Porting note (#10618): simp can prove this
theorem comm_from (f : Hom C₁ C₂) (i : ι) : f.f i ≫ C₂.dFrom i = C₁.dFrom i ≫ f.next i
-- Mathlib/Order/Cover.lean
@[simp]
theorem swap_wcovBy_swap : x.swap ⩿ y.swap ↔ x ⩿ y
-- Mathlib/Analysis/Normed/Operator/LinearIsometry.lean
@[simp] -- Should be replaced with `SemilinearIsometryClass.nnorm_map` when `lean4#3107` is fixed.
theorem nnnorm_map (x : E) : ‖f x‖₊ = ‖x‖₊
-- Mathlib/CategoryTheory/Shift/CommShift.lean
lemma commShiftIso_comp_hom_app [F.CommShift A] [G.CommShift A] (a : A) (X : C) :
    (commShiftIso (F ⋙ G) a).hom.app X =
      G.map ((commShiftIso F a).hom.app X) ≫ (commShiftIso G a).hom.app (F.obj X)
-- Mathlib/Algebra/Polynomial/AlgebraMap.lean
lemma aeval_apply_smul_mem_of_le_comap (f : Module.End R M) (hq : q ≤ q.comap f) :
    aeval f p m ∈ q
-- Mathlib/LinearAlgebra/Pi.lean
@[simp]
theorem pi_apply (f : (i : ι) → M₂ →ₗ[R] φ i) (c : M₂) (i : ι) : pi f c i = f i c
-- Mathlib/MeasureTheory/Integral/Bochner.lean
theorem integral_nonpos_of_ae {f : α → ℝ} (hf : f ≤ᵐ[μ] 0) : ∫ a, f a ∂μ ≤ 0
-- Mathlib/Algebra/Group/Support.lean
@[to_additive]
theorem mulSupport_mulSingle_subset : mulSupport (mulSingle a b) ⊆ {a}
-- Mathlib/Topology/MetricSpace/Isometry.lean
@[simp]
theorem symm_trans_apply (h₁ : α ≃ᵢ β) (h₂ : β ≃ᵢ γ) (x : γ) :
    (h₁.trans h₂).symm x = h₁.symm (h₂.symm x)
-- Mathlib/Data/Int/Star.lean
@[simp] lemma addSubmonoid_closure_range_pow {n : ℕ} (hn : Even n) :
    closure (range fun x : ℤ ↦ x ^ n) = nonneg _
-- Mathlib/Data/Rat/Cast/Order.lean
@[mono]
theorem cast_mono : Monotone ((↑) : ℚ → K)
-- Mathlib/RingTheory/FractionalIdeal/Basic.lean
theorem eq_zero_iff {I : FractionalIdeal S P} : I = 0 ↔ ∀ x ∈ I, x = (0 : P)
-- .lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean
@[ext] theorem eq_of_getLsb_eq {x y : BitVec w}
    (pred : ∀(i : Fin w), x.getLsb i.val = y.getLsb i.val) : x = y
-- Mathlib/Computability/TuringMachine.lean
@[simp]
theorem Tape.write_mk' {Γ} [Inhabited Γ] (a b : Γ) (L R : ListBlank Γ) :
    (Tape.mk' L (R.cons a)).write b = Tape.mk' L (R.cons b)
-- Mathlib/Algebra/GroupWithZero/Defs.lean
theorem mul_eq_zero_of_right (a : M₀) {b : M₀} (h : b = 0) : a * b = 0
-- Mathlib/Topology/Compactness/Compact.lean
/-- The coproduct of the cocompact filters on two topological spaces is the cocompact filter on
their product. -/
theorem Filter.coprod_cocompact :
    (Filter.cocompact X).coprod (Filter.cocompact Y) = Filter.cocompact (X × Y)
-- Mathlib/Topology/Maps/Basic.lean
theorem nhdsSet_eq_comap (hf : Inducing f) (s : Set X) :
    𝓝ˢ s = comap f (𝓝ˢ (f '' s))
-- Mathlib/RingTheory/Polynomial/Basic.lean
/-- If `I` is an ideal, and `pᵢ` is a finite family of polynomials each satisfying
`∀ k, (pᵢ)ₖ ∈ Iⁿⁱ⁻ᵏ` for some `nᵢ`, then `p = ∏ pᵢ` also satisfies `∀ k, pₖ ∈ Iⁿ⁻ᵏ` with `n = ∑ nᵢ`.
-/
theorem _root_.Polynomial.coeff_prod_mem_ideal_pow_tsub {ι : Type*} (s : Finset ι) (f : ι → R[X])
    (I : Ideal R) (n : ι → ℕ) (h : ∀ i ∈ s, ∀ (k), (f i).coeff k ∈ I ^ (n i - k)) (k : ℕ) :
    (s.prod f).coeff k ∈ I ^ (s.sum n - k)
-- Mathlib/Topology/Category/CompactlyGenerated.lean
/-- The equivalence between isomorphisms in `CompactlyGenerated` and homeomorphisms
of topological spaces. -/
@[simps]
def isoEquivHomeo {X Y : CompactlyGenerated.{u, w}} : (X ≅ Y) ≃ (X ≃ₜ Y)
-- .lake/packages/batteries/Batteries/Classes/Order.lean
protected theorem LTCmp.compareOfLessAndEq
    [LT α] [DecidableRel (LT.lt (α := α))] [DecidableEq α]
    (lt_irrefl : ∀ x : α, ¬x < x)
    (lt_trans : ∀ {x y z : α}, x < y → y < z → x < z)
    (lt_antisymm : ∀ {x y : α}, ¬x < y → ¬y < x → x = y) :
    LTCmp (α := α) (compareOfLessAndEq · ·)
-- Mathlib/Topology/MetricSpace/Bounded.lean
theorem _root_.Bornology.IsBounded.subset_closedBall (h : IsBounded s) (c : α) :
    ∃ r, s ⊆ closedBall c r
-- Mathlib/Algebra/MonoidAlgebra/ToDirectSum.lean
@[simp]
theorem AddMonoidAlgebra.toDirectSum_single (i : ι) (m : M) :
    AddMonoidAlgebra.toDirectSum (Finsupp.single i m) = DirectSum.of _ i m
-- Mathlib/Order/Ideal.lean
theorem above_mem : D.above x ∈ D
-- Mathlib/Topology/Bornology/Hom.lean
@[simp]
theorem comp_id (f : LocallyBoundedMap α β) : f.comp (LocallyBoundedMap.id α) = f
-- Mathlib/CategoryTheory/Types.lean
theorem homOfElement_eq_iff {X : Type u} (x y : X) : homOfElement x = homOfElement y ↔ x = y
-- Mathlib/Probability/ProbabilityMassFunction/Constructions.lean
@[simp]
theorem toMeasure_ofFinset_apply [MeasurableSpace α] (ht : MeasurableSet t) :
    (ofFinset f s h h').toMeasure t = ∑' x, t.indicator f x
-- Mathlib/Topology/Instances/ENNReal.lean
theorem tendsto_toReal_iff {ι} {fi : Filter ι} {f : ι → ℝ≥0∞} (hf : ∀ i, f i ≠ ∞) {x : ℝ≥0∞}
    (hx : x ≠ ∞) : Tendsto (fun n => (f n).toReal) fi (𝓝 x.toReal) ↔ Tendsto f fi (𝓝 x)
-- Mathlib/Analysis/Matrix.lean
theorem nnnorm_lt_iff {r : ℝ≥0} (hr : 0 < r) {A : Matrix m n α} :
    ‖A‖₊ < r ↔ ∀ i j, ‖A i j‖₊ < r
-- Mathlib/RingTheory/FiniteType.lean
theorem id : FiniteType (RingHom.id A)
-- Mathlib/Topology/FiberBundle/Basic.lean
@[simp, mfld_simps]
theorem localTriv_symm_apply (p : B × F) :
    (Z.localTriv i).toPartialHomeomorph.symm p = ⟨p.1, Z.coordChange i (Z.indexAt p.1) p.1 p.2⟩
-- Mathlib/Algebra/Star/StarAlgHom.lean
theorem ext_iff {f g : A ≃⋆ₐ[R] B} : f = g ↔ ∀ a, f a = g a
-- Mathlib/RingTheory/Artinian.lean
theorem isArtinian_iff_wellFounded :
    IsArtinian R M ↔ WellFounded ((· < ·) : Submodule R M → Submodule R M → Prop)
-- Mathlib/GroupTheory/OrderOfElement.lean
@[to_additive]
lemma isOfFinOrder_iff_zpow_eq_one {G} [Group G] {x : G} :
    IsOfFinOrder x ↔ ∃ (n : ℤ), n ≠ 0 ∧ x ^ n = 1
-- Mathlib/MeasureTheory/Function/UnifTight.lean
/-- Intermediate lemma for `unifTight_of_tendsto_Lp`. -/
private theorem unifTight_of_tendsto_Lp_zero (hp' : p ≠ ∞) (hf : ∀ n, Memℒp (f n) p μ)
    (hf_tendsto : Tendsto (fun n ↦ eLpNorm (f n) p μ) atTop (𝓝 0)) : UnifTight f p μ
-- Mathlib/NumberTheory/EllipticDivisibilitySequence.lean
@[simp]
lemma preNormEDS'_four : preNormEDS' b c d 4 = d
-- Mathlib/GroupTheory/OrderOfElement.lean
@[to_additive]
lemma Nat.Coprime.orderOf_pow (h : (orderOf y).Coprime m) : orderOf (y ^ m) = orderOf y
-- Mathlib/RingTheory/HahnSeries/Summable.lean
@[simp]
theorem smul_apply {x : HahnSeries Γ R} {s : SummableFamily Γ R α} {a : α} : (x • s) a = x * s a
-- Mathlib/CategoryTheory/Sites/Plus.lean
@[reassoc (attr := simp)]
theorem toPlus_naturality {P Q : Cᵒᵖ ⥤ D} (η : P ⟶ Q) :
    η ≫ J.toPlus Q = J.toPlus _ ≫ J.plusMap η
-- .lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean
protected theorem eq_iff_le_and_ge : ∀{a b : Nat}, a = b ↔ a ≤ b ∧ b ≤ a
-- Mathlib/Data/Matroid/IndepAxioms.lean
@[simp] theorem ofFinset_indep [DecidableEq α] (E : Set α) Indep indep_empty indep_subset indep_aug
    subset_ground {I : Finset α} : (IndepMatroid.ofFinset
      E Indep indep_empty indep_subset indep_aug subset_ground).Indep I ↔ Indep I
-- .lake/packages/lean4/src/lean/Init/Data/Int/Order.lean
@[simp] theorem sign_of_add_one (x : Nat) : Int.sign (x + 1) = 1
-- Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
theorem coe_center : ↑(center R) = Set.center R
-- Mathlib/Data/Real/ENatENNReal.lean
@[simp, norm_cast]
theorem toENNReal_mul (m n : ℕ∞) : ↑(m * n) = (m * n : ℝ≥0∞)
-- Mathlib/SetTheory/Ordinal/NaturalOps.lean
theorem nmul_lt_nmul_of_pos_right (h₁ : a < b) (h₂ : 0 < c) : a ⨳ c < b ⨳ c
-- Mathlib/Analysis/Matrix.lean
@[simp]
theorem frobenius_norm_diagonal [DecidableEq n] (v : n → α) :
    ‖diagonal v‖ = ‖(WithLp.equiv 2 _).symm v‖
-- Mathlib/Analysis/InnerProductSpace/Symmetric.lean
/-- A symmetric linear map `T` is zero if and only if `⟪T x, x⟫_ℝ = 0` for all `x`.
See `inner_map_self_eq_zero` for the complex version without the symmetric assumption. -/
theorem IsSymmetric.inner_map_self_eq_zero {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) :
    (∀ x, ⟪T x, x⟫ = 0) ↔ T = 0
-- Mathlib/Order/Sublattice.lean
@[simp, norm_cast] lemma coe_comap (L : Sublattice β) (f : LatticeHom α β) : L.comap f = f ⁻¹' L
-- Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
@[simp]
theorem nnnorm_ofSubsingleton [Subsingleton ι] (i : ι) (f : G →L[𝕜] G') :
    ‖ofSubsingleton 𝕜 G G' i f‖₊ = ‖f‖₊
-- Mathlib/RingTheory/FractionalIdeal/Operations.lean
@[simp]
theorem div_spanSingleton (J : FractionalIdeal R₁⁰ K) (d : K) :
    J / spanSingleton R₁⁰ d = spanSingleton R₁⁰ d⁻¹ * J
-- Mathlib/CategoryTheory/Monoidal/Opposite.lean
@[simp] lemma op_whiskerLeft (X : C) {Y Z : C} (f : Y ⟶ Z) :
    (X ◁ f).op = op X ◁ f.op
-- Mathlib/MeasureTheory/Integral/Bochner.lean
theorem SimpleFunc.coe_toLargerSpace_eq (hm : m ≤ m0) (f : @SimpleFunc β m γ) :
    ⇑(f.toLargerSpace hm) = f
-- Mathlib/Algebra/Polynomial/FieldDivision.lean
theorem lt_rootMultiplicity_iff_isRoot_iterate_derivative_of_mem_nonZeroDivisors'
    {p : R[X]} {t : R} {n : ℕ} (h : p ≠ 0)
    (hnzd : ∀ m ≤ n, m ≠ 0 → (m : R) ∈ nonZeroDivisors R) :
    n < p.rootMultiplicity t ↔ ∀ m ≤ n, (derivative^[m] p).IsRoot t
-- Mathlib/Algebra/Module/Submodule/Bilinear.lean
theorem map₂_sup_left (f : M →ₗ[R] N →ₗ[R] P) (p₁ p₂ : Submodule R M) (q : Submodule R N) :
    map₂ f (p₁ ⊔ p₂) q = map₂ f p₁ q ⊔ map₂ f p₂ q
-- Mathlib/LinearAlgebra/TensorProduct/Prod.lean
@[simp] theorem prodRight_symm_tmul (m₁ : M₁) (m₂ : M₂) (m₃ : M₃) :
    (prodRight R M₁ M₂ M₃).symm (m₁ ⊗ₜ m₂, m₁ ⊗ₜ m₃) = (m₁ ⊗ₜ (m₂, m₃))
-- Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean
theorem natTrailingDegree_eq_of_trailingDegree_eq [Semiring S] {q : S[X]}
    (h : trailingDegree p = trailingDegree q) : natTrailingDegree p = natTrailingDegree q
-- Mathlib/Data/Real/Cardinality.lean
/-- The cardinality of the interval (a, ∞). -/
theorem mk_Ioi_real (a : ℝ) : #(Ioi a) = 𝔠
-- Mathlib/RingTheory/WittVector/Truncated.lean
@[simp]
theorem coeff_zero (i : Fin n) : (0 : TruncatedWittVector p n R).coeff i = 0
-- Mathlib/CategoryTheory/Monoidal/Opposite.lean
@[simp] lemma unmop_inv_associator (X Y Z : Cᴹᵒᵖ) :
    (α_ X Y Z).inv.unmop = (α_ (unmop Z) (unmop Y) (unmop X)).hom
-- Mathlib/Computability/PartrecCode.lean
/-- Roger's fixed-point theorem: Any total, computable `f` has a fixed point: That is, under the
interpretation given by `Nat.Partrec.Code.eval`, there is a code `c` such that `c` and `f c` have
the same evaluation.
-/
theorem fixed_point {f : Code → Code} (hf : Computable f) : ∃ c : Code, eval (f c) = eval c
-- Mathlib/Topology/Support.lean
@[to_additive]
theorem comp₂_left (hf : HasCompactMulSupport f)
    (hf₂ : HasCompactMulSupport f₂) (hm : m 1 1 = 1) :
    HasCompactMulSupport fun x => m (f x) (f₂ x)
-- Mathlib/FieldTheory/SeparableClosure.lean
@[simp]
theorem insepDegree_bot : insepDegree F (⊥ : IntermediateField F E) = 1
-- Mathlib/Algebra/Ring/Ext.lean
@[ext] theorem ext ⦃inst₁ inst₂ : NonUnitalCommSemiring R⦄
    (h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
    (h_mul : local_hMul[R, inst₁] = local_hMul[R, inst₂]) :
    inst₁ = inst₂
-- Mathlib/Topology/Instances/ENNReal.lean
@[simp, norm_cast]
theorem tendsto_coe_nhds_top {f : α → ℝ≥0} {l : Filter α} :
    Tendsto (fun x => (f x : ℝ≥0∞)) l (𝓝 ∞) ↔ Tendsto f l atTop
-- Mathlib/Probability/Kernel/CondDistrib.lean
theorem integrable_comp_snd_map_prod_mk_iff {Ω} {_ : MeasurableSpace Ω} {X : Ω → β} {μ : Measure Ω}
    (hX : Measurable X) {f : Ω → F} :
    Integrable (fun x : β × Ω => f x.2) (μ.map fun ω => (X ω, ω)) ↔ Integrable f μ
-- Mathlib/CategoryTheory/Limits/HasLimits.lean
@[reassoc (attr := simp)]
theorem limit.lift_π {F : J ⥤ C} [HasLimit F] (c : Cone F) (j : J) :
    limit.lift F c ≫ limit.π F j = c.π.app j
-- Mathlib/Algebra/Group/Defs.lean
/-- A `RightCancelSemigroup` is a semigroup such that `a * b = c * b` implies `a = c`. -/
@[ext]
class RightCancelSemigroup (G : Type u) extends Semigroup G where
  protected mul_right_cancel : ∀ a b c : G, a * b = c * b → a = c
-- Mathlib/Algebra/Ring/Subsemiring/MulOpposite.lean
theorem op_sSup (S : Set (Subsemiring R)) : (sSup S).op = sSup (.unop ⁻¹' S)
-- Mathlib/Topology/Separation.lean
protected theorem disjoint (h : SeparatedNhds s t) : Disjoint s t
-- Mathlib/Data/Finset/Basic.lean
@[simp, norm_cast]
theorem coe_union (s₁ s₂ : Finset α) : ↑(s₁ ∪ s₂) = (s₁ ∪ s₂ : Set α)
-- Mathlib/MeasureTheory/Integral/Lebesgue.lean
theorem lintegral_add_right' (f : α → ℝ≥0∞) {g : α → ℝ≥0∞} (hg : AEMeasurable g μ) :
    ∫⁻ a, f a + g a ∂μ = ∫⁻ a, f a ∂μ + ∫⁻ a, g a ∂μ
-- Mathlib/Analysis/Normed/Operator/LinearIsometry.lean
protected theorem map_neg (x : E) : f (-x) = -f x
-- Mathlib/Topology/Semicontinuous.lean
theorem LowerSemicontinuousWithinAt.mono (h : LowerSemicontinuousWithinAt f s x) (hst : t ⊆ s) :
    LowerSemicontinuousWithinAt f t x
-- Mathlib/SetTheory/Game/Basic.lean
@[simp]
theorem mul_moveRight_inl {x y : PGame} {i j} :
    (x * y).moveRight (toRightMovesMul (Sum.inl (i, j))) =
      x.moveLeft i * y + x * y.moveRight j - x.moveLeft i * y.moveRight j
-- Mathlib/LinearAlgebra/Trace.lean
/-- Auxiliary lemma for `trace_eq_matrix_trace`. -/
theorem trace_eq_matrix_trace_of_finset {s : Finset M} (b : Basis s R M) (f : M →ₗ[R] M) :
    trace R M f = Matrix.trace (LinearMap.toMatrix b b f)
-- Mathlib/Computability/RegularExpressions.lean
@[simp]
theorem rmatch_iff_matches' (P : RegularExpression α) (x : List α) :
    P.rmatch x ↔ x ∈ P.matches'
-- Mathlib/Analysis/BoxIntegral/Partition/SubboxInduction.lean
@[simp]
theorem unionComplToSubordinate_boxes (π₁ : TaggedPrepartition I) (π₂ : Prepartition I)
    (hU : π₂.iUnion = ↑I \ π₁.iUnion) (r : (ι → ℝ) → Ioi (0 : ℝ)) :
    (π₁.unionComplToSubordinate π₂ hU r).boxes = π₁.boxes ∪ (π₂.toSubordinate r).boxes
-- Mathlib/Analysis/Convex/GaugeRescale.lean
theorem continuous_gaugeRescale {s t : Set E} (hs : Convex ℝ s) (hs₀ : s ∈ 𝓝 0)
    (ht : Convex ℝ t) (ht₀ : t ∈ 𝓝 0) (htb : IsVonNBounded ℝ t) :
    Continuous (gaugeRescale s t)
-- Mathlib/GroupTheory/Congruence/Basic.lean
@[to_additive]
theorem smul {α M : Type*} [MulOneClass M] [SMul α M] [IsScalarTower α M M] (c : Con M) (a : α)
    {w x : M} (h : c w x) : c (a • w) (a • x)
-- Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean
theorem natTrailingDegree_le_of_ne_zero (h : coeff p n ≠ 0) : natTrailingDegree p ≤ n
-- Mathlib/Topology/MetricSpace/Infsep.lean
theorem einfsep_lt_top_iff : s.einfsep < ∞ ↔ s.Nontrivial
-- Mathlib/Data/Finsupp/Defs.lean
@[simp]
theorem equivFunOnFinite_symm_single [DecidableEq α] [Finite α] (x : α) (m : M) :
    Finsupp.equivFunOnFinite.symm (Pi.single x m) = Finsupp.single x m
-- Mathlib/RingTheory/Jacobson.lean
theorem radical_eq_jacobson [H : IsJacobson R] (I : Ideal R) : I.radical = I.jacobson
-- Mathlib/Analysis/LocallyConvex/Bounded.lean
theorem isBounded_iff_subset_smul_ball {s : Set E} :
    Bornology.IsBounded s ↔ ∃ a : 𝕜, s ⊆ a • Metric.ball (0 : E) 1
-- Mathlib/CategoryTheory/IsConnected.lean
/-- Any equivalence relation containing (⟶) holds for all pairs of a connected category. -/
theorem equiv_relation [IsPreconnected J] (r : J → J → Prop) (hr : _root_.Equivalence r)
    (h : ∀ {j₁ j₂ : J} (_ : j₁ ⟶ j₂), r j₁ j₂) : ∀ j₁ j₂ : J, r j₁ j₂
-- Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
/-- If a multilinear map satisfies an inequality `‖f m‖ ≤ C * ∏ i, ‖m i‖`, then it is
continuous. -/
theorem continuous_of_bound (C : ℝ) (H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) : Continuous f
-- Mathlib/Data/ZMod/Basic.lean
@[simp]
theorem lift_coe (x : ℤ) : lift n f (x : ZMod n) = f.val x
-- Mathlib/Topology/Inseparable.lean
theorem Specializes.snd {a b : X × Y} (h : a ⤳ b) : a.2 ⤳ b.2
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Unital.lean
lemma cfc_map_polynomial (q : R[X]) (f : R → R) (a : A) (ha : p a := by cfc_tac)
    (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) :
    cfc (fun x ↦ q.eval (f x)) a = aeval (cfc f a) q
-- Mathlib/GroupTheory/CoprodI.lean
theorem of_injective (i : ι) : Function.Injective (of : M i →* _)
-- Mathlib/NumberTheory/Liouville/Basic.lean
theorem exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f : ℤ[X]} (f0 : f ≠ 0)
    (fa : eval α (map (algebraMap ℤ ℝ) f) = 0) :
    ∃ A : ℝ, 0 < A ∧ ∀ a : ℤ, ∀ b : ℕ,
      (1 : ℝ) ≤ ((b : ℝ) + 1) ^ f.natDegree * (|α - a / (b + 1)| * A)
-- .lake/packages/lean4/src/lean/Init/Data/List/Basic.lean
@[simp] theorem intersperse_cons₂ (sep : α) :
    (x::y::zs).intersperse sep = x::sep::((y::zs).intersperse sep)
-- Mathlib/Order/Hom/Basic.lean
theorem le_iff_le (e : α ≃o β) {x y : α} : e x ≤ e y ↔ x ≤ y
-- Mathlib/Algebra/BigOperators/Group/Finset.lean
@[to_additive]
theorem prod_coe_sort : ∏ i : s, f i = ∏ i ∈ s, f i
-- Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
theorem rpow_inv_le_iff {x y : ℝ≥0} {z : ℝ} (hz : 0 < z) : x ^ z⁻¹ ≤ y ↔ x ≤ y ^ z
-- Mathlib/AlgebraicGeometry/OpenImmersion.lean
theorem scheme_eq_of_locallyRingedSpace_eq {X Y : Scheme.{u}}
    (H : X.toLocallyRingedSpace = Y.toLocallyRingedSpace) : X = Y
-- Mathlib/MeasureTheory/Function/EssSup.lean
theorem essSup_map_measure (hg : AEMeasurable g (Measure.map f μ)) (hf : AEMeasurable f μ) :
    essSup g (Measure.map f μ) = essSup (g ∘ f) μ
-- Mathlib/Analysis/SpecialFunctions/Integrals.lean
@[simp]
theorem integral_cos_sq : ∫ x in a..b, cos x ^ 2 = (cos b * sin b - cos a * sin a + b - a) / 2
-- Mathlib/CategoryTheory/Limits/Shapes/Products.lean
lemma Sigma.map'_id_id {f : α → C} [HasCoproduct f] :
    Sigma.map' id (fun a => 𝟙 (f a)) = 𝟙 (∐ f)
-- Mathlib/CategoryTheory/Preadditive/Yoneda/Projective.lean
theorem projective_iff_preservesEpimorphisms_preadditiveCoyoneda_obj (P : C) :
    Projective P ↔ (preadditiveCoyoneda.obj (op P)).PreservesEpimorphisms
-- Mathlib/Algebra/Category/HopfAlgebraCat/Basic.lean
@[simp] theorem toHopfAlgebraCatIso_refl :
    toHopfAlgebraCatIso (BialgEquiv.refl R X) = .refl _
-- Mathlib/Analysis/BoxIntegral/Partition/Split.lean
/-- For every prepartition `π` of `I` there exists a prepartition that covers exactly
`I \ π.iUnion`. -/
theorem exists_iUnion_eq_diff (π : Prepartition I) :
    ∃ π' : Prepartition I, π'.iUnion = ↑I \ π.iUnion
-- Mathlib/CategoryTheory/CommSq.lean
theorem of_arrow {f g : Arrow C} (h : f ⟶ g) : CommSq f.hom h.left h.right g.hom
-- Mathlib/Order/Interval/Basic.lean
@[simp]
theorem map_map (g : β →o γ) (f : α →o β) (s : Interval α) : (s.map f).map g = s.map (g.comp f)
-- Mathlib/Data/Finset/Grade.lean
lemma _root_.CovBy.exists_finset_cons (h : s ⋖ t) : ∃ a, ∃ ha : a ∉ s, s.cons a ha = t
-- Mathlib/Algebra/Module/LocalizedModule.lean
theorem smul_injective (s : S) : Function.Injective fun m : M' => s • m
-- Mathlib/Data/Nat/Cast/Defs.lean
@[norm_cast 500]
theorem cast_succ (n : ℕ) : ((succ n : ℕ) : R) = n + 1
-- Mathlib/MeasureTheory/Measure/MeasureSpace.lean
lemma measure_symmDiff_le (s t u : Set α) :
    μ (s ∆ u) ≤ μ (s ∆ t) + μ (t ∆ u)
-- Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
theorem singletonSubgraph_eq_induce {v : V} :
    G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v}
-- Mathlib/Tactic/NoncommRing.lean
lemma nat_lit_mul_eq_nsmul [n.AtLeastTwo] : no_index (OfNat.ofNat n) * r = n • r
-- Mathlib/RingTheory/FractionalIdeal/Operations.lean
@[simp]
theorem mapEquiv_refl : mapEquiv AlgEquiv.refl = RingEquiv.refl (FractionalIdeal S P)
-- Mathlib/Analysis/Normed/Lp/PiLp.lean
theorem infty_equiv_isometry [∀ i, PseudoEMetricSpace (β i)] :
    Isometry (WithLp.equiv ∞ (∀ i, β i))
-- Mathlib/Algebra/Order/Monovary.lean
/-- Two functions monovary iff the rearrangement inequality holds. -/
lemma monovary_iff_mul_rearrangement :
    Monovary f g ↔ ∀ i j, f i * g j + f j * g i ≤ f i * g i + f j * g j
-- Mathlib/RingTheory/UniqueFactorizationDomain.lean
theorem prime_pow_coprime_prod_of_coprime_insert [DecidableEq α] {s : Finset α} (i : α → ℕ) (p : α)
    (hps : p ∉ s) (is_prime : ∀ q ∈ insert p s, Prime q)
    (is_coprime : ∀ᵉ (q ∈ insert p s) (q' ∈ insert p s), q ∣ q' → q = q') :
    IsRelPrime (p ^ i p) (∏ p' ∈ s, p' ^ i p')