ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/ModelTheory/Substructures.lean
theorem lift_card_closure_le_card_term : Cardinal.lift.{max u w} #(closure L s) ≤ #(L.Term s)
-- Mathlib/LinearAlgebra/Ray.lean
theorem exists_pos_left_iff_sameRay_and_ne_zero (hx : x ≠ 0) :
    (∃ r : R, 0 < r ∧ r • x = y) ↔ SameRay R x y ∧ y ≠ 0
-- Mathlib/Computability/Primrec.lean
theorem nat_le : PrimrecRel ((· ≤ ·) : ℕ → ℕ → Prop)
-- Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
@[to_additive]
theorem tprod_eq_mulSingle {f : β → α} (b : β) (hf : ∀ b' ≠ b, f b' = 1) :
    ∏' b, f b = f b
-- Mathlib/Topology/Constructions.lean
theorem isClosedMap_ofAdd : IsClosedMap (ofAdd : X → Multiplicative X)
-- Mathlib/FieldTheory/Minpoly/Basic.lean
@[simp]
theorem algEquiv_eq (f : B ≃ₐ[A] B') (x : B) : minpoly A (f x) = minpoly A x
-- Mathlib/GroupTheory/SchurZassenhaus.lean
/-- Do not use this lemma: It is made obsolete by `exists_right_complement'_of_coprime` -/
private theorem step6 : IsPGroup (Nat.card N).minFac N
-- .lake/packages/batteries/Batteries/Data/List/Count.lean
theorem le_count_iff_replicate_sublist {l : List α} : n ≤ count a l ↔ replicate n a <+ l
-- Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean
theorem transAssocReparamAux_zero : transAssocReparamAux 0 = 0
-- Mathlib/NumberTheory/Dioph.lean
theorem diophFn_comp2 {h : ℕ → ℕ → ℕ} (d : DiophFn fun v : Vector3 ℕ 2 => h (v &0) (v &1)) :
    DiophFn fun v => h (f v) (g v)
-- Mathlib/Analysis/Calculus/ParametricIntervalIntegral.lean
/-- Derivative under integral of `x ↦ ∫ F x a` at a given point `x₀ : 𝕜`, `𝕜 = ℝ` or `𝕜 = ℂ`,
assuming `F x₀` is integrable, `x ↦ F x a` is differentiable on an interval around `x₀` for ae `a`
(with interval radius independent of `a`) with derivative uniformly bounded by an integrable
function, and `F x` is ae-measurable for `x` in a possibly smaller neighborhood of `x₀`. -/
nonrec theorem hasDerivAt_integral_of_dominated_loc_of_deriv_le
    {F : 𝕜 → ℝ → E} {F' : 𝕜 → ℝ → E} {x₀ : 𝕜}
    (ε_pos : 0 < ε) (hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b)))
    (hF_int : IntervalIntegrable (F x₀) μ a b)
    (hF'_meas : AEStronglyMeasurable (F' x₀) (μ.restrict (Ι a b)))
    (h_bound : ∀ᵐ t ∂μ, t ∈ Ι a b → ∀ x ∈ ball x₀ ε, ‖F' x t‖ ≤ bound t)
    (bound_integrable : IntervalIntegrable bound μ a b)
    (h_diff : ∀ᵐ t ∂μ, t ∈ Ι a b → ∀ x ∈ ball x₀ ε, HasDerivAt (fun x => F x t) (F' x t) x) :
    IntervalIntegrable (F' x₀) μ a b ∧
      HasDerivAt (fun x => ∫ t in a..b, F x t ∂μ) (∫ t in a..b, F' x₀ t ∂μ) x₀
-- Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
/-- `qrSign` is symmetric when both arguments are odd. -/
protected theorem symm {m n : ℕ} (hm : Odd m) (hn : Odd n) : qrSign m n = qrSign n m
-- Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean
/-- 1 is weighted homogeneous of degree 0. -/
theorem isWeightedHomogeneous_one (w : σ → M) : IsWeightedHomogeneous w (1 : MvPolynomial σ R) 0
-- Mathlib/MeasureTheory/Measure/AEMeasurable.lean
theorem map_map_of_aemeasurable {g : β → γ} {f : α → β} (hg : AEMeasurable g (Measure.map f μ))
    (hf : AEMeasurable f μ) : (μ.map f).map g = μ.map (g ∘ f)
-- Mathlib/Analysis/Complex/LocallyUniformLimit.lean
theorem _root_.TendstoLocallyUniformlyOn.deriv (hf : TendstoLocallyUniformlyOn F f φ U)
    (hF : ∀ᶠ n in φ, DifferentiableOn ℂ (F n) U) (hU : IsOpen U) :
    TendstoLocallyUniformlyOn (deriv ∘ F) (deriv f) φ U
-- .lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean
protected theorem lt_ediv_of_mul_lt {a b c : Int} (H1 : 0 ≤ b) (H2 : b ∣ c) (H3 : a * b < c) :
    a < c / b
-- Mathlib/MeasureTheory/Integral/SetIntegral.lean
/-- If `s` is a countable family of compact sets, `f` is a continuous function, and the sequence
`‖f.restrict (s i)‖ * μ (s i)` is summable, then `f` is integrable on the union of the `s i`. -/
theorem integrableOn_iUnion_of_summable_norm_restrict {f : C(X, E)} {s : ι → Compacts X}
    (hf : Summable fun i : ι => ‖f.restrict (s i)‖ * ENNReal.toReal (μ <| s i)) :
    IntegrableOn f (⋃ i : ι, s i) μ
-- Mathlib/Data/Vector3.lean
@[simp]
theorem vectorAllP_nil (p : α → Prop) : VectorAllP p [] = True
-- .lake/packages/batteries/Batteries/Data/List/Lemmas.lean
theorem modifyNthTail_add (f : List α → List α) (n) (l₁ l₂ : List α) :
    modifyNthTail f (l₁.length + n) (l₁ ++ l₂) = l₁ ++ modifyNthTail f n l₂
-- Mathlib/CategoryTheory/Limits/Shapes/FunctorToTypes.lean
@[simp]
lemma inl_comp_binaryCoproductIso_inv_apply (a : C) (x : F.obj a) :
    (binaryCoproductIso F G).inv.app a (.inl x) = (Limits.coprod.inl (X := F)).app a x
-- Mathlib/Analysis/Calculus/ContDiff/Defs.lean
theorem ContDiffOn.fderiv_of_isOpen (hf : ContDiffOn 𝕜 n f s) (hs : IsOpen s) (hmn : m + 1 ≤ n) :
    ContDiffOn 𝕜 m (fun y => fderiv 𝕜 f y) s
-- .lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean
protected theorem dvd_def (a b : Int) : (a ∣ b) = Exists (fun c => b = a * c)
-- Mathlib/Data/Fin/Tuple/Basic.lean
@[simp]
theorem succAbove_cases_eq_insertNth : @succAboveCases.{u + 1} = @insertNth.{u}
-- Mathlib/Topology/Algebra/Module/Basic.lean
theorem Submodule.mapsTo_smul_closure (s : Submodule R M) (c : R) :
    Set.MapsTo (c • ·) (closure s : Set M) (closure s)
-- Mathlib/Analysis/Normed/Operator/LinearIsometry.lean
@[simp]
theorem coe_mk (e : E ≃ₛₗ[σ₁₂] E₂) (he : ∀ x, ‖e x‖ = ‖x‖) : ⇑(mk e he) = e
-- Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean
/-- `biprodComparison' F X Y` is a split mono. -/
@[simps]
def splitMonoBiprodComparison' : SplitMono (biprodComparison' F X Y)
-- Mathlib/Algebra/Group/Commute/Basic.lean
@[to_additive]
protected theorem inv_inv : Commute a b → Commute a⁻¹ b⁻¹
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Order.lean
lemma CstarRing.conjugate_le_norm_smul {a b : A} (hb : IsSelfAdjoint b := by cfc_tac) :
    star a * b * a ≤ ‖b‖ • (star a * a)
-- Mathlib/Order/Filter/CardinalInter.lean
theorem EventuallyEq.cardinal_iInter {s t : ι → Set α} (hic : #ι < c)
    (h : ∀ i, s i =ᶠ[l] t i) : ⋂ i, s i =ᶠ[l] ⋂ i, t i
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
@[simp]
theorem sin_int_mul_pi (n : ℤ) : sin (n * π) = 0
-- Mathlib/Analysis/Convex/EGauge.lean
lemma egauge_eq_top : egauge 𝕜 s x = ∞ ↔ ∀ c : 𝕜, x ∉ c • s
-- Mathlib/Algebra/Order/Sub/Basic.lean
theorem le_mul_tsub {R : Type*} [Distrib R] [Preorder R] [Sub R] [OrderedSub R]
    [CovariantClass R R (· * ·) (· ≤ ·)] {a b c : R} : a * b - a * c ≤ a * (b - c)
-- Mathlib/Algebra/Lie/Sl2.lean
lemma lie_h_pow_toEnd_f (n : ℕ) :
    ⁅h, ψ n⁆ = (μ - 2 * n) • ψ n
-- Mathlib/Algebra/MvPolynomial/Basic.lean
@[simp]
theorem eval₂_map [CommSemiring S₂] (f : R →+* S₁) (g : σ → S₂) (φ : S₁ →+* S₂)
    (p : MvPolynomial σ R) : eval₂ φ g (map f p) = eval₂ (φ.comp f) g p
-- Mathlib/Geometry/Manifold/LocalDiffeomorph.lean
lemma isLocalDiffeomorph_iff {f : M → N} :
    IsLocalDiffeomorph I J n f ↔ ∀ x : M, IsLocalDiffeomorphAt I J n f x
-- Mathlib/LinearAlgebra/TensorProduct/Tower.lean
@[simp]
lemma baseChange_top : (⊤ : Submodule R M).baseChange A = ⊤
-- Mathlib/Algebra/Group/Indicator.lean
@[to_additive]
theorem mulIndicator_mulIndicator (s t : Set α) (f : α → M) :
    mulIndicator s (mulIndicator t f) = mulIndicator (s ∩ t) f
-- Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
/-- Two affine maps are equal if they have equal linear maps and are equal at some point. -/
theorem ext_linear {f g : P1 →ᵃ[k] P2} (h₁ : f.linear = g.linear) {p : P1} (h₂ : f p = g p) :
    f = g
-- Mathlib/Algebra/Polynomial/Eval.lean
@[simp]
theorem intCast_comp (i : ℤ) : comp (i : R[X]) p = i
-- .lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean
protected theorem le_antisymm {x y : Fin n} (h1 : x ≤ y) (h2 : y ≤ x) : x = y
-- Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean
@[measurability]
theorem measurableSet_uIcc : MeasurableSet (uIcc a b)
-- Mathlib/Algebra/Algebra/Equiv.lean
@[simp]
theorem ofLinearEquiv_symm :
    (ofLinearEquiv l map_one map_mul).symm =
      ofLinearEquiv l.symm
        (_root_.map_one <| ofLinearEquiv_symm.aux l map_one map_mul)
        (_root_.map_mul <| ofLinearEquiv_symm.aux l map_one map_mul)
-- Mathlib/Data/Ordmap/Ordset.lean
theorem Sized.dual_iff {t : Ordnode α} : Sized (.dual t) ↔ Sized t
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
@[simp]
lemma leadingCoeff_Ψ₂Sq (h : (4 : R) ≠ 0) : W.Ψ₂Sq.leadingCoeff = 4
-- Mathlib/Probability/Process/Adapted.lean
@[to_additive]
protected theorem inv [Group β] [TopologicalGroup β] (hu : Adapted f u) :
    Adapted f u⁻¹
-- Mathlib/Analysis/Calculus/Deriv/Pow.lean
theorem differentiableOn_pow : DifferentiableOn 𝕜 (fun x : 𝕜 => x ^ n) s
-- Mathlib/RingTheory/OreLocalization/Basic.lean
protected theorem add_zero (x : X[S⁻¹]) : x + 0 = x
-- Mathlib/Topology/EMetricSpace/Basic.lean
/-- A map between emetric spaces is a uniform embedding if and only if the edistance between `f x`
and `f y` is controlled in terms of the distance between `x` and `y` and conversely. -/
theorem EMetric.uniformEmbedding_iff' [EMetricSpace β] {f : γ → β} :
    UniformEmbedding f ↔
      (∀ ε > 0, ∃ δ > 0, ∀ {a b : γ}, edist a b < δ → edist (f a) (f b) < ε) ∧
        ∀ δ > 0, ∃ ε > 0, ∀ {a b : γ}, edist (f a) (f b) < ε → edist a b < δ
-- .lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean
theorem cons_ne_nil (a : α) (l : List α) : a :: l ≠ []
-- Mathlib/Topology/Algebra/Module/Basic.lean
theorem smul_apply (c : S₂) (f : M₁ →SL[σ₁₂] M₂) (x : M₁) : (c • f) x = c • f x
-- Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean
theorem normAtPlace_nonneg  (w : InfinitePlace K) (x : E K) :
    0 ≤ normAtPlace w x
-- Mathlib/MeasureTheory/Measure/MeasureSpace.lean
theorem comapₗ_apply {β} [MeasurableSpace α] {mβ : MeasurableSpace β} (f : α → β)
    (hfi : Injective f) (hf : ∀ s, MeasurableSet s → MeasurableSet (f '' s)) (μ : Measure β)
    (hs : MeasurableSet s) : comapₗ f μ s = μ (f '' s)
-- Mathlib/CategoryTheory/Sites/Coherent/Equivalence.lean
/--
The regular sheaf condition on an essentially small site can be checked after precomposing with
the equivalence with a small category.
-/
theorem preregular_isSheaf_iff_of_essentiallySmall [EssentiallySmall C] (F : Cᵒᵖ ⥤ A) :
    IsSheaf (regularTopology C) F ↔ IsSheaf (regularTopology (SmallModel C))
    ((equivSmallModel C).inverse.op ⋙ F)
-- Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean
theorem comap_ofFunction {β} (f : β → α) (h : Monotone m ∨ Surjective f) :
    comap f (OuterMeasure.ofFunction m m_empty) =
      OuterMeasure.ofFunction (fun s => m (f '' s)) (by simp; simp [m_empty])
-- Mathlib/RingTheory/DedekindDomain/Factorization.lean
/-- For every nonzero ideal `I` of `v`, there are finitely many maximal ideals `v` such that
`v^(val_v(I))`, regarded as a fractional ideal, is not `(1)`. -/
theorem finite_mulSupport_coe {I : Ideal R} (hI : I ≠ 0) :
    (mulSupport fun v : HeightOneSpectrum R => (v.asIdeal : FractionalIdeal R⁰ K) ^
      ((Associates.mk v.asIdeal).count (Associates.mk I).factors : ℤ)).Finite
-- .lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean
theorem natAdd_natAdd (m n : Nat) {p : Nat} (i : Fin p) :
    natAdd m (natAdd n i) = cast (Nat.add_assoc ..) (natAdd (m + n) i)
-- Mathlib/Data/List/TFAE.lean
@[simp]
theorem tfae_cons_self {a} {l : List Prop} : TFAE (a :: a :: l) ↔ TFAE (a :: l)
-- Mathlib/Topology/ContinuousFunction/Ideals.lean
theorem ideal_isMaximal_iff (I : Ideal C(X, 𝕜)) [hI : IsClosed (I : Set C(X, 𝕜))] :
    I.IsMaximal ↔ ∃ x : X, idealOfSet 𝕜 {x}ᶜ = I
-- Mathlib/CategoryTheory/EffectiveEpi/Basic.lean
/--
A family of morphisms `π a : X a ⟶ B` indexed by `α` is effective epimorphic
provided that the `π a` exhibit `B` as a colimit of the diagram of all "relations"
`R → X a₁`, `R ⟶ X a₂` for all `a₁ a₂ : α`.
-/
class EffectiveEpiFamily {B : C} {α : Type*} (X : α → C) (π : (a : α) → (X a ⟶ B)) : Prop where
  /-- `π` is an effective epimorphic family if there exists an `EffectiveEpiFamilyStruct` for `π` -/
  effectiveEpiFamily : Nonempty (EffectiveEpiFamilyStruct X π)
-- Mathlib/LinearAlgebra/Ray.lean
/-- If a vector is in the same ray as its negation, that vector is zero. -/
theorem eq_zero_of_sameRay_self_neg [NoZeroSMulDivisors R M] (h : SameRay R x (-x)) : x = 0
-- Mathlib/NumberTheory/Divisors.lean
theorem divisors_subset_properDivisors {m : ℕ} (hzero : n ≠ 0) (h : m ∣ n) (hdiff : m ≠ n) :
    divisors m ⊆ properDivisors n
-- Mathlib/RingTheory/JacobsonIdeal.lean
@[simp]
theorem jacobson_top : jacobson (⊤ : Ideal R) = ⊤
-- Mathlib/CategoryTheory/Functor/KanExtension/Basic.lean
/-- Left Kan extensions of isomorphic functors are isomorphic. -/
@[simps]
noncomputable def leftKanExtensionUniqueOfIso {G : C ⥤ H} (i : F ≅ G) (G' : D ⥤ H)
    (β : G ⟶ L ⋙ G') [G'.IsLeftKanExtension β] : F' ≅ G'
-- Mathlib/Order/Sublattice.lean
@[simp] lemma infClosed (L : Sublattice α) : InfClosed (L : Set α)
-- Mathlib/Data/Fin/Basic.lean
/-- By moving `succ` to the outside of this expression, we create opportunities for further
simplification using `succAbove_zero` or `succ_succAbove_zero`. -/
@[simp] lemma succ_succAbove_one {n : ℕ} [NeZero n] (i : Fin (n + 1)) :
    i.succ.succAbove 1 = (i.succAbove 0).succ
-- Mathlib/Algebra/Algebra/Hom.lean
@[simp]
theorem coe_id : ⇑(AlgHom.id R A) = id
-- Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean
@[simp]
theorem TrailingMonic.trailingCoeff {p : R[X]} (hp : p.TrailingMonic) : trailingCoeff p = 1
-- Mathlib/Order/Filter/Basic.lean
@[simp]
theorem frequently_sSup {p : α → Prop} {fs : Set (Filter α)} :
    (∃ᶠ x in sSup fs, p x) ↔ ∃ f ∈ fs, ∃ᶠ x in f, p x
-- Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
theorem integral_comp_sub_mul (hc : c ≠ 0) (d) :
    (∫ x in a..b, f (d - c * x)) = c⁻¹ • ∫ x in d - c * b..d - c * a, f x
-- Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
/-- An integral domain is a *discrete valuation ring* (DVR) if it's a local PID which
  is not a field. -/
class DiscreteValuationRing (R : Type u) [CommRing R] [IsDomain R]
    extends IsPrincipalIdealRing R, LocalRing R : Prop where
  not_a_field' : maximalIdeal R ≠ ⊥
-- Mathlib/FieldTheory/ChevalleyWarning.lean
/-- The **Chevalley–Warning theorem**, `Fintype` version.
Let `(f i)` be a finite family of multivariate polynomials
in finitely many variables (`X s`, `s : σ`) over a finite field of characteristic `p`.
Assume that the sum of the total degrees of the `f i` is less than the cardinality of `σ`.
Then the number of common solutions of the `f i` is divisible by `p`. -/
theorem char_dvd_card_solutions_of_fintype_sum_lt [Fintype ι] {f : ι → MvPolynomial σ K}
    (h : (∑ i, (f i).totalDegree) < Fintype.card σ) :
    p ∣ Fintype.card { x : σ → K // ∀ i, eval x (f i) = 0 }
-- Mathlib/CategoryTheory/Limits/Cones.lean
/-- Postcomposing by the identity does not change the cone up to isomorphism. -/
@[simps!]
def postcomposeId : postcompose (𝟙 F) ≅ 𝟭 (Cone F)
-- Mathlib/Algebra/Group/Submonoid/Units.lean
@[to_additive]
lemma coe_inv_val_mul_coe_val (S : Submonoid M) {x : Sˣ} :
    ((x⁻¹ : Sˣ) : M) * ((x : Sˣ) : M) = 1
-- Mathlib/Probability/Independence/Kernel.lean
theorem IndepFun.neg_left {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'} [Neg β]
    [MeasurableNeg β] (hfg : IndepFun f g κ μ) :
    IndepFun (-f) g κ μ
-- Mathlib/MeasureTheory/Measure/Lebesgue/Complex.lean
@[simp]
theorem measurableEquivPi_symm_apply (p : (Fin 2) → ℝ) :
    measurableEquivPi.symm p = (p 0) + (p 1) * I
-- Mathlib/Topology/MetricSpace/DilationEquiv.lean
@[simp] theorem refl_symm : (refl X).symm = refl X
-- Mathlib/Data/Bool/Basic.lean
theorem or_le : ∀ {x y z}, x ≤ z → y ≤ z → (x || y) ≤ z
-- Mathlib/Order/UpperLower/Basic.lean
lemma fibration_iff_image_Iic [Preorder α] [Preorder β] (hf : Monotone f) :
    Fibration (· ≤ ·) (· ≤ ·) f ↔ ∀ x, f '' Iic x = Iic (f x)
-- Mathlib/Data/Rat/Cast/Order.lean
@[simp] lemma cast_pos : (0 : K) < q ↔ 0 < q
-- Mathlib/GroupTheory/PushoutI.lean
/-- The intersection of the images of the maps from any two distinct groups in the diagram
into the amalgamated product is the image of the map from the base group in the diagram. -/
theorem inf_of_range_eq_base_range (hφ : ∀ i, Injective (φ i)) {i j : ι} (hij : i ≠ j) :
    (of i).range ⊓ (of j).range = (base φ).range
-- Mathlib/Order/Bounds/Basic.lean
@[simp]
theorem lowerBounds_singleton : lowerBounds {a} = Iic a
-- Mathlib/Data/Fintype/Pi.lean
lemma eval_image_piFinset_subset (t : ∀ a, Finset (δ a)) (a : α) [DecidableEq (δ a)] :
    ((piFinset t).image fun f ↦ f a) ⊆ t a
-- Mathlib/Order/Filter/Basic.lean
@[simp]
theorem map_compose : Filter.map m' ∘ Filter.map m = Filter.map (m' ∘ m)
-- Mathlib/Topology/FiberBundle/Trivialization.lean
theorem mk_mem_target {x : B} {y : F} : (x, y) ∈ e'.target ↔ x ∈ e'.baseSet
-- Mathlib/Topology/ContinuousFunction/Sigma.lean
theorem embedding_sigmaMk_comp [Nonempty X] :
    Embedding (fun g : Σ i, C(X, Y i) ↦ (sigmaMk g.1).comp g.2)
-- Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean
@[simp]
theorem iUnion_mk (π : Prepartition I) (f h) : (mk π f h).iUnion = π.iUnion
-- Mathlib/Algebra/Polynomial/Eval.lean
@[simp]
theorem eval₂_neg {S} [Ring S] (f : R →+* S) {x : S} : (-p).eval₂ f x = -p.eval₂ f x
-- Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean
@[to_additive]
theorem eq_one_or_one_lt (a : α) : a = 1 ∨ 1 < a
-- Mathlib/SetTheory/Ordinal/Basic.lean
@[simp]
theorem lift.initialSeg_coe : (lift.initialSeg.{u,v} : Ordinal → Ordinal) = lift.{v,u}
-- Mathlib/Algebra/Polynomial/Roots.lean
lemma nthRootsFinset_def (n : ℕ) (R : Type*) [CommRing R] [IsDomain R] [DecidableEq R] :
    nthRootsFinset n R = Multiset.toFinset (nthRoots n (1 : R))
-- Mathlib/Data/Set/Basic.lean
@[simp]
theorem union_diff_self {s t : Set α} : s ∪ t \ s = s ∪ t
-- Mathlib/NumberTheory/ModularForms/EisensteinSeries/Defs.lean
/-- How the `eisSummand` function changes under the Moebius action. -/
theorem eisSummand_SL2_apply (k : ℤ) (i : (Fin 2 → ℤ)) (A : SL(2, ℤ)) (z : ℍ) :
    eisSummand k i (A • z) = (z.denom A) ^ k * eisSummand k (i ᵥ* A) z
-- Mathlib/Algebra/Group/Hom/Basic.lean
/-- A homomorphism from a group to a monoid is injective iff its kernel is trivial.
For the iff statement on the triviality of the kernel, see `injective_iff_map_eq_one'`.  -/
@[to_additive
  "A homomorphism from an additive group to an additive monoid is injective iff
  its kernel is trivial. For the iff statement on the triviality of the kernel,
  see `injective_iff_map_eq_zero'`."]
theorem _root_.injective_iff_map_eq_one {G H} [Group G] [MulOneClass H]
    [FunLike F G H] [MonoidHomClass F G H]
    (f : F) : Function.Injective f ↔ ∀ a, f a = 1 → a = 1
-- Mathlib/RingTheory/DedekindDomain/Different.lean
@[simp]
lemma dual_zero :
    dual A K (0 : FractionalIdeal B⁰ L) = 0
-- Mathlib/CategoryTheory/Triangulated/Opposite.lean
lemma complete_distinguished_triangle_morphism (T₁ T₂ : Triangle Cᵒᵖ)
    (hT₁ : T₁ ∈ distinguishedTriangles C) (hT₂ : T₂ ∈ distinguishedTriangles C)
    (a : T₁.obj₁ ⟶ T₂.obj₁) (b : T₁.obj₂ ⟶ T₂.obj₂) (comm : T₁.mor₁ ≫ b = a ≫ T₂.mor₁) :
    ∃ (c : T₁.obj₃ ⟶ T₂.obj₃), T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧
      T₁.mor₃ ≫ a⟦1⟧' = c ≫ T₂.mor₃
-- Mathlib/Order/Interval/Set/Monotone.lean
theorem StrictMonoOn.Iic_id_le [SuccOrder α] [IsSuccArchimedean α] [OrderBot α] {n : α} {φ : α → α}
    (hφ : StrictMonoOn φ (Set.Iic n)) : ∀ m ≤ n, m ≤ φ m
-- Mathlib/GroupTheory/SemidirectProduct.lean
@[simp]
theorem right_inr (g : G) : (inr g : N ⋊[φ] G).right = g
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
/-- The set of vectors in the direction of a nonempty affine subspace is given by `vsub_set`. -/
theorem coe_direction_eq_vsub_set {s : AffineSubspace k P} (h : (s : Set P).Nonempty) :
    (s.direction : Set V) = (s : Set P) -ᵥ s
-- Mathlib/Algebra/Algebra/Pi.lean
theorem algebraMap_def {_ : CommSemiring R} [_s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)]
    (a : R) : algebraMap R (∀ i, f i) a = fun i => algebraMap R (f i) a
-- Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean
@[simp]
theorem toIdeal_sSup (ℐ : Set (HomogeneousIdeal 𝒜)) : (sSup ℐ).toIdeal = ⨆ s ∈ ℐ, toIdeal s
-- Mathlib/Data/List/AList.lean
theorem erase_erase (a a' : α) (s : AList β) : (s.erase a).erase a' = (s.erase a').erase a
-- Mathlib/Data/Set/Basic.lean
theorem not_mem_of_mem_compl {s : Set α} {x : α} (h : x ∈ sᶜ) : x ∉ s
-- Mathlib/Data/Finite/Defs.lean
protected theorem Infinite.false [Finite α] (_ : Infinite α) : False
-- Mathlib/Algebra/CharP/Quotient.lean
/-- `CharP.quotient'` as an `Iff`. -/
theorem quotient_iff {R : Type*} [CommRing R] (n : ℕ) [CharP R n] (I : Ideal R) :
    CharP (R ⧸ I) n ↔ ∀ x : ℕ, ↑x ∈ I → (x : R) = 0
-- Mathlib/LinearAlgebra/Matrix/ToLin.lean
theorem toMatrix_lmul' (x : S) (i j) :
    LinearMap.toMatrix b b (lmul R S x) i j = b.repr (x * b j) i
-- Mathlib/CategoryTheory/Monoidal/Opposite.lean
@[simp] lemma unop_associator (X Y Z : Cᵒᵖ) :
    (α_ X Y Z).unop = (α_ (unop X) (unop Y) (unop Z)).symm
-- Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean
theorem RingHom.isIntegral_of_surjective (hf : Function.Surjective f) : f.IsIntegral
-- Mathlib/Order/Interval/Set/Monotone.lean
protected theorem Antitone.Iio (hf : Antitone f) : Antitone fun x => Iio (f x)
-- Mathlib/Analysis/SpecialFunctions/Log/Monotone.lean
theorem log_div_self_antitoneOn : AntitoneOn (fun x : ℝ => log x / x) { x | exp 1 ≤ x }
-- Mathlib/NumberTheory/EllipticDivisibilitySequence.lean
@[simp]
lemma preNormEDS_neg (n : ℤ) : preNormEDS b c d (-n) = -preNormEDS b c d n
-- Mathlib/Algebra/Polynomial/Coeff.lean
theorem coeff_one_add_X_pow (R : Type*) [Semiring R] (n k : ℕ) :
    ((1 + X) ^ n).coeff k = (n.choose k : R)
-- Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
private theorem lintegral_tendsto_of_monotone_of_nat {φ : ℕ → Set α} (hφ : AECover μ atTop φ)
    (hmono : Monotone φ) {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) :
    Tendsto (fun i => ∫⁻ x in φ i, f x ∂μ) atTop (𝓝 <| ∫⁻ x, f x ∂μ)
-- .lake/packages/lean4/src/lean/Init/Data/Bool.lean
@[simp] theorem bne_self_right : ∀(a b : Bool), ((a != b) != b) = a
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean
theorem inner_condexpL2_left_eq_right (hm : m ≤ m0) {f g : α →₂[μ] E} :
    ⟪(condexpL2 E 𝕜 hm f : α →₂[μ] E), g⟫₂ = ⟪f, (condexpL2 E 𝕜 hm g : α →₂[μ] E)⟫₂
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
@[simp] lemma sinh_eq_zero : sinh x = 0 ↔ x = 0
-- Mathlib/Data/List/Sublists.lean
theorem sublistsAux_eq_array_foldl :
    sublistsAux = fun (a : α) (r : List (List α)) =>
      (r.toArray.foldl (init := #[])
        fun r l => (r.push l).push (a :: l)).toList
-- Mathlib/Data/Finset/NoncommProd.lean
@[to_additive noncommSum_eq_card_nsmul]
theorem noncommProd_eq_pow_card (s : Multiset α) (comm) (m : α) (h : ∀ x ∈ s, x = m) :
    s.noncommProd comm = m ^ Multiset.card s
-- Mathlib/Order/Interval/Multiset.lean
@[simp] lemma mem_Iio : x ∈ Iio b ↔ x < b
-- Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean
@[simp] lemma ofIsLimitKernelFork_f' (hf : S.f = 0) (c : KernelFork S.g) (hc : IsLimit c) :
    (ofIsLimitKernelFork S hf c hc).f' = 0
-- Mathlib/Data/Nat/Choose/Basic.lean
theorem add_choose (i j : ℕ) : (i + j).choose j = (i + j)! / (i ! * j !)
-- Mathlib/Order/Filter/AtTopBot.lean
@[nontriviality]
theorem Subsingleton.atBot_eq (α) [Subsingleton α] [Preorder α] : (atBot : Filter α) = ⊤
-- Mathlib/Topology/MetricSpace/Bounded.lean
/-- Spheres are bounded -/
theorem isBounded_sphere : IsBounded (sphere x r)
-- Mathlib/Data/Matroid/Basic.lean
@[simp] theorem not_indep_iff (hX : X ⊆ M.E := by aesop_mat) : ¬ M.Indep X ↔ M.Dep X
-- Mathlib/Data/Set/Finite.lean
theorem exists_upper_bound_image [Nonempty α] [LinearOrder β] (s : Set α) (f : α → β)
    (h : s.Finite) : ∃ a : α, ∀ b ∈ s, f b ≤ f a
-- Mathlib/Data/Nat/Multiplicity.lean
theorem multiplicity_mul {p m n : ℕ} (hp : p.Prime) :
    multiplicity p (m * n) = multiplicity p m + multiplicity p n
-- Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
@[simp]
theorem induce_empty : G'.induce ∅ = ⊥
-- Mathlib/RingTheory/Polynomial/Pochhammer.lean
theorem descPochhammer_int_eq_ascFactorial (a b : ℕ) :
    (descPochhammer ℤ b).eval (a + b : ℤ) = (a + 1).ascFactorial b
-- Mathlib/Analysis/Normed/Operator/LinearIsometry.lean
theorem ediam_range [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) :
    EMetric.diam (range f) = EMetric.diam (univ : Set E)
-- Mathlib/RingTheory/GradedAlgebra/Basic.lean
@[simp]
theorem decompose_symm_one : (decompose 𝒜).symm 1 = (1 : A)
-- Mathlib/Order/Defs.lean
theorem le_total : ∀ a b : α, a ≤ b ∨ b ≤ a