ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Algebra/Order/Sub/Canonical.lean
protected theorem lt_tsub_iff_right_of_le (hc : AddLECancellable c) (h : c ≤ b) :
    a < b - c ↔ a + c < b
-- Mathlib/Algebra/Star/StarAlgHom.lean
@[simp]
theorem comp_assoc (f : C →⋆ₙₐ[R] D) (g : B →⋆ₙₐ[R] C) (h : A →⋆ₙₐ[R] B) :
    (f.comp g).comp h = f.comp (g.comp h)
-- Mathlib/Combinatorics/SimpleGraph/Maps.lean
@[simp]
lemma comap_symm_apply (f : V ≃ W) (G : SimpleGraph W) (w : W) :
    (SimpleGraph.Iso.comap f G).symm w = f.symm w
-- Mathlib/Data/Nat/Factorial/SuperFactorial.lean
private theorem matrixOf_eval_descPochhammer_eq_mul_matrixOf_choose {n : ℕ} (v : Fin n → ℕ) :
    (Matrix.of (fun (i j : Fin n) => (descPochhammer ℤ j).eval (v i : ℤ))).det =
    (∏ i : Fin n, Nat.factorial i) *
      (Matrix.of (fun (i j : Fin n) => (Nat.choose (v i) (j : ℕ) : ℤ))).det
-- Mathlib/Data/Set/Function.lean
theorem piecewise_eqOn (f g : α → β) : EqOn (s.piecewise f g) f s
-- Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean
theorem IsEquivalent.add_isLittleO (huv : u ~[l] v) (hwv : w =o[l] v) : u + w ~[l] v
-- Mathlib/LinearAlgebra/Basis.lean
@[simp]
theorem repr_total (v) : b.repr (Finsupp.total _ _ _ b v) = v
-- Mathlib/Order/FixedPoints.lean
theorem le_prevFixed {x : α} (hx : f x ≤ x) {y : fixedPoints f} (h : ↑y ≤ x) :
    y ≤ f.prevFixed x hx
-- Mathlib/RingTheory/HahnSeries/Addition.lean
/-- `coeff g` as a linear map -/
@[simps]
def coeff.linearMap (g : Γ) : HahnSeries Γ V →ₗ[R] V
-- Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean
/-- The submodule generated by products `Pm * Pn` of weighted homogeneous polynomials of degrees `m`
  and `n` is contained in the submodule of weighted homogeneous polynomials of degree `m + n`. -/
theorem weightedHomogeneousSubmodule_mul (w : σ → M) (m n : M) :
    weightedHomogeneousSubmodule R w m * weightedHomogeneousSubmodule R w n ≤
      weightedHomogeneousSubmodule R w (m + n)
-- Mathlib/Analysis/Normed/Group/Hom.lean
/-- The operator norm satisfies the triangle inequality. -/
theorem opNorm_add_le : ‖f + g‖ ≤ ‖f‖ + ‖g‖
-- Mathlib/RingTheory/Localization/Basic.lean
@[simp]
theorem mk'_spec_mk (x) (y : R) (hy : y ∈ M) :
    mk' S x ⟨y, hy⟩ * algebraMap R S y = algebraMap R S x
-- Mathlib/FieldTheory/Adjoin.lean
/-- The compositum of two intermediate fields is equal to the compositum of them
as subalgebras, if one of them is algebraic over the base field. -/
theorem sup_toSubalgebra_of_isAlgebraic
    (halg : Algebra.IsAlgebraic K E1 ∨ Algebra.IsAlgebraic K E2) :
    (E1 ⊔ E2).toSubalgebra = E1.toSubalgebra ⊔ E2.toSubalgebra
-- Mathlib/Topology/Constructions.lean
theorem isOpenMap_toAdd : IsOpenMap (toAdd : Multiplicative X → X)
-- Mathlib/CategoryTheory/Subobject/Limits.lean
theorem imageSubobject_le_mk {A B : C} {X : C} (g : X ⟶ B) [Mono g] (f : A ⟶ B) [HasImage f]
    (h : A ⟶ X) (w : h ≫ g = f) : imageSubobject f ≤ Subobject.mk g
-- Mathlib/Algebra/Order/ToIntervalMod.lean
@[simp]
theorem self_sub_toIcoMod (a b : α) : b - toIcoMod hp a b = toIcoDiv hp a b • p
-- Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
theorem rangeS_le : (algebraMap R A).rangeS ≤ S.toSubsemiring
-- Mathlib/NumberTheory/Pell.lean
/-- If `(x, y)` is a solution with `x` and `y` positive, then all its powers with positive
natural exponents have positive `y`. -/
theorem y_pow_succ_pos {a : Solution₁ d} (hax : 0 < a.x) (hay : 0 < a.y) (n : ℕ) :
    0 < (a ^ n.succ).y
-- Mathlib/CategoryTheory/Linear/Yoneda.lean
@[simp]
theorem whiskering_linearCoyoneda₂ :
    linearCoyoneda R C ⋙
        (whiskeringRight _ _ _).obj (forget₂ (ModuleCat.{v} R) AddCommGrp.{v}) =
      preadditiveCoyoneda
-- Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
@[measurability]
theorem measurableSet_closure : MeasurableSet (closure s)
-- Mathlib/CategoryTheory/Sites/Subsheaf.lean
theorem Subpresheaf.eq_sheafify_iff (h : Presieve.IsSheaf J F) :
    G = G.sheafify J ↔ Presieve.IsSheaf J G.toPresheaf
-- Mathlib/Topology/Algebra/Module/Basic.lean
protected theorem map_neg (f : M →SL[σ₁₂] M₂) (x : M) : f (-x) = -f x
-- Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean
theorem mapMono_δ₀' (i : Δ' ⟶ Δ) [Mono i] (hi : Isδ₀ i) : mapMono K i = K.d Δ.len Δ'.len
-- Mathlib/Algebra/MonoidAlgebra/Basic.lean
/-- Decomposition of a `k`-algebra homomorphism from `MonoidAlgebra k G` by
its values on `F (single a 1)`. -/
theorem lift_unique (F : MonoidAlgebra k G →ₐ[k] A) (f : MonoidAlgebra k G) :
    F f = f.sum fun a b => b • F (single a 1)
-- Mathlib/Data/DFinsupp/Order.lean
@[simp]
theorem support_sup : (f ⊔ g).support = f.support ∪ g.support
-- Mathlib/Order/Filter/Extr.lean
theorem IsMinOn.on_subset (hf : IsMinOn f t a) (h : s ⊆ t) : IsMinOn f s a
-- Mathlib/Analysis/Asymptotics/Asymptotics.lean
theorem IsBigO.congr' (h : f₁ =O[l] g₁) (hf : f₁ =ᶠ[l] f₂) (hg : g₁ =ᶠ[l] g₂) : f₂ =O[l] g₂
-- Mathlib/LinearAlgebra/Span.lean
theorem toSpanSingleton_isIdempotentElem_iff {e : R} :
    IsIdempotentElem (toSpanSingleton R R e) ↔ IsIdempotentElem e
-- Mathlib/GroupTheory/GroupAction/Hom.lean
/-- The inverse of a bijective equivariant map is equivariant. -/
@[simps]
def inverse' (f : X →ₑ[φ] Y) (g : Y → X) (k : Function.RightInverse φ' φ)
    (h₁ : Function.LeftInverse g f) (h₂ : Function.RightInverse g f) :
    Y →ₑ[φ'] X
-- Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean
theorem Fix.ind_rec {β : Type u} (g₁ g₂ : Fix F α → β)
    (h :
      ∀ x : F (append1 α (Fix F α)),
        appendFun id g₁ <$$> x = appendFun id g₂ <$$> x → g₁ (Fix.mk x) = g₂ (Fix.mk x)) :
    ∀ x, g₁ x = g₂ x
-- Mathlib/Analysis/InnerProductSpace/PiL2.lean
@[simp]
protected theorem coe_mk (hon : Orthonormal 𝕜 v) (hsp : ⊤ ≤ Submodule.span 𝕜 (Set.range v)) :
    ⇑(OrthonormalBasis.mk hon hsp) = v
-- Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
theorem isIrreducible_zeroLocus_iff_of_radical (I : Ideal R) (hI : I.IsRadical) :
    IsIrreducible (zeroLocus (I : Set R)) ↔ I.IsPrime
-- Mathlib/Data/Matrix/Rank.lean
@[simp]
theorem rank_self_mul_conjTranspose (A : Matrix m n R) : (A * Aᴴ).rank = A.rank
-- Mathlib/Topology/PartialHomeomorph.lean
theorem isOpen_iff (h : e.IsImage s t) : IsOpen (e.source ∩ s) ↔ IsOpen (e.target ∩ t)
-- Mathlib/FieldTheory/AbelRuffini.lean
/-- **Abel-Ruffini Theorem** (one direction): An irreducible polynomial with an
`IsSolvableByRad` root has solvable Galois group -/
theorem isSolvable' {α : E} {q : F[X]} (q_irred : Irreducible q) (q_aeval : aeval α q = 0)
    (hα : IsSolvableByRad F α) : IsSolvable q.Gal
-- Mathlib/RingTheory/HahnSeries/Basic.lean
@[simp]
theorem isPWO_support (x : HahnSeries Γ R) : x.support.IsPWO
-- Mathlib/Data/Set/NAry.lean
theorem image2_subset_iff_right : image2 f s t ⊆ u ↔ ∀ b ∈ t, (fun a => f a b) '' s ⊆ u
-- Mathlib/NumberTheory/Padics/Hensel.lean
private theorem deriv_sq_norm_pos : 0 < ‖F.derivative.eval a‖ ^ 2
-- Mathlib/Analysis/Normed/Group/Quotient.lean
/-- The operator norm of the projection is `0` if the subspace is dense. -/
theorem norm_trivial_quotient_mk (S : AddSubgroup M)
    (h : (S.topologicalClosure : Set M) = Set.univ) : ‖S.normedMk‖ = 0
-- Mathlib/Order/Fin/Tuple.lean
@[simp] lemma monotone_vecEmpty : Monotone ![a]
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
@[simp]
theorem bot_parallel_iff_eq_bot {s : AffineSubspace k P} : ⊥ ∥ s ↔ s = ⊥
-- Mathlib/Data/Fin/Tuple/Basic.lean
theorem append_assoc {p : ℕ} {α : Type*} (a : Fin m → α) (b : Fin n → α) (c : Fin p → α) :
    append (append a b) c = append a (append b c) ∘ Fin.cast (Nat.add_assoc ..)
-- Mathlib/LinearAlgebra/Quotient.lean
@[simp]
theorem restrictScalarsEquiv_mk [Ring S] [SMul S R] [Module S M] [IsScalarTower S R M]
    (P : Submodule R M) (x : M) :
    restrictScalarsEquiv S P (mk x : M ⧸ P) = (mk x : M ⧸ P)
-- Mathlib/Algebra/DirectSum/Module.lean
/-- Two `LinearMap`s out of a direct sum are equal if they agree on the generators.

See note [partially-applied ext lemmas]. -/
@[ext]
theorem linearMap_ext ⦃ψ ψ' : (⨁ i, M i) →ₗ[R] N⦄
    (H : ∀ i, ψ.comp (lof R ι M i) = ψ'.comp (lof R ι M i)) : ψ = ψ'
-- Mathlib/Data/Nat/Bitwise.lean
lemma bit_mod_two (a : Bool) (x : ℕ) :
    bit a x % 2 = if a then 1 else 0
-- Mathlib/Topology/MetricSpace/Basic.lean
@[simp] theorem dist_ofMul (a b : X) : dist (ofMul a) (ofMul b) = dist a b
-- Mathlib/FieldTheory/Adjoin.lean
theorem finiteDimensional_iSup_of_finset'
    /- Porting note: this was the mathlib3 version. Using `[h : ...]`, as in mathlib3, causes the
    error "invalid parametric local instance". -/
    {s : Finset ι} (h : ∀ i ∈ s, FiniteDimensional K (t i)) :
    FiniteDimensional K (⨆ i ∈ s, t i : IntermediateField K L)
-- Mathlib/Analysis/Calculus/FDeriv/Mul.lean
@[simp, fun_prop]
theorem Differentiable.pow (ha : Differentiable 𝕜 a) (n : ℕ) : Differentiable 𝕜 fun x => a x ^ n
-- Mathlib/Order/GaloisConnection.lean
theorem l_u_le (a) : l (u a) ≤ a
-- Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean
@[simp]
theorem unop_top : (⊤ : Subalgebra R Aᵐᵒᵖ).unop = ⊤
-- Mathlib/Data/Multiset/Lattice.lean
@[simp]
theorem inf_ndinsert (a : α) (s : Multiset α) : (ndinsert a s).inf = a ⊓ s.inf
-- Mathlib/CategoryTheory/Limits/Shapes/Pullback/Cospan.lean
@[simp]
theorem cospanExt_hom_app_right :
    (cospanExt iX iY iZ wf wg).hom.app WalkingCospan.right = iY.hom
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
theorem exists_rat_pow_btwn_rat_aux (hn : n ≠ 0) (x y : ℝ) (h : x < y) (hy : 0 < y) :
    ∃ q : ℚ, 0 < q ∧ x < (q : ℝ) ^ n ∧ (q : ℝ) ^ n < y
-- Mathlib/Topology/Bases.lean
theorem IsTopologicalBasis.of_hasBasis_nhds {s : Set (Set α)}
    (h_nhds : ∀ a, (𝓝 a).HasBasis (fun t ↦ t ∈ s ∧ a ∈ t) id) : IsTopologicalBasis s
-- Mathlib/CategoryTheory/Preadditive/HomOrthogonal.lean
theorem eq_zero [HasZeroMorphisms C] (o : HomOrthogonal s) {i j : ι} (w : i ≠ j) (f : s i ⟶ s j) :
    f = 0
-- Mathlib/MeasureTheory/Measure/NullMeasurable.lean
theorem completion_apply {_ : MeasurableSpace α} (μ : Measure α) (s : Set α) :
    μ.completion s = μ s
-- Mathlib/CategoryTheory/Sites/Sieves.lean
@[simp]
lemma comp_mem_iff (i : X ⟶ Y) (f : Y ⟶ Z) [IsIso i] (S : Sieve Z) :
    S (i ≫ f) ↔ S f
-- Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean
/-- Any two morphisms to a terminal object are equal. -/
theorem IsTerminal.hom_ext {X Y : C} (t : IsTerminal X) (f g : Y ⟶ X) : f = g
-- Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean
@[simp]
lemma inl_descCochain :
    (inl φ).comp (descCochain φ α β h) (by omega) = α
-- Mathlib/Data/DFinsupp/WellFounded.lean
theorem Lex.acc_zero (hbot : ∀ ⦃i a⦄, ¬s i a 0) : Acc (DFinsupp.Lex r s) 0
-- Mathlib/Order/CompleteLattice.lean
lemma le_biSup {ι : Type*} {s : Set ι} (f : ι → α) {i : ι} (hi : i ∈ s) :
    f i ≤ ⨆ i ∈ s, f i
-- Mathlib/Order/Irreducible.lean
@[simp]
theorem infIrred_ofDual {a : αᵒᵈ} : InfIrred (ofDual a) ↔ SupIrred a
-- Mathlib/Algebra/BigOperators/Group/Multiset.lean
@[to_additive]
theorem prod_map_pow {n : ℕ} : (m.map fun i => f i ^ n).prod = (m.map f).prod ^ n
-- Mathlib/Algebra/Order/Field/Basic.lean
theorem one_div_pow_le_one_div_pow_of_le (a1 : 1 ≤ a) {m n : ℕ} (mn : m ≤ n) :
    1 / a ^ n ≤ 1 / a ^ m
-- Mathlib/Order/Lattice.lean
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
    f (x ⊓ y) = f x ⊔ f y
-- Mathlib/Order/Interval/Set/ProjIcc.lean
theorem projIic_of_le (hx : b ≤ x) : projIic b x = ⟨b, le_rfl⟩
-- Mathlib/LinearAlgebra/AffineSpace/Combination.lean
/-- The value of `weightedVSubOfPoint`, where the given points are equal. -/
@[simp (high)]
theorem weightedVSubOfPoint_apply_const (w : ι → k) (p : P) (b : P) :
    s.weightedVSubOfPoint (fun _ => p) b w = (∑ i ∈ s, w i) • (p -ᵥ b)
-- Mathlib/RingTheory/Kaehler/Basic.lean
theorem Derivation.liftKaehlerDifferential_comp (D : Derivation R S M) :
    D.liftKaehlerDifferential.compDer (KaehlerDifferential.D R S) = D
-- Mathlib/Topology/Algebra/GroupWithZero.lean
theorem ContinuousOn.div_const (hf : ContinuousOn f s) (y : G₀) :
    ContinuousOn (fun x => f x / y) s
-- Mathlib/FieldTheory/IntermediateField.lean
/-- Product of a multiset of elements in an intermediate field is in the intermediate_field. -/
protected theorem multiset_prod_mem (m : Multiset L) : (∀ a ∈ m, a ∈ S) → m.prod ∈ S
-- Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean
lemma add_of_Z_eq_zero {P Q : Fin 3 → F} (hP : W.Nonsingular P) (hQ : W.Nonsingular Q)
    (hPz : P z = 0) (hQz : Q z = 0) : W.add P Q = P x ^ 2 • ![1, 1, 0]
-- Mathlib/CategoryTheory/ComposableArrows.lean
@[simp]
lemma obj_one : obj F X 1 = F.obj' 0
-- Mathlib/RingTheory/Ideal/Cotangent.lean
theorem mem_toCotangent_ker {x : I} : x ∈ LinearMap.ker I.toCotangent ↔ (x : R) ∈ I ^ 2
-- Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean
/-- The injection of `Y` in the optimal coupling between `X` and `Y` is an isometry. -/
theorem isometry_optimalGHInjr : Isometry (optimalGHInjr X Y)
-- Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean
lemma equiv_some_of_Z_ne_zero {P : Fin 3 → F} (hPz : P z ≠ 0) :
    P ≈ ![P x / P z ^ 2, P y / P z ^ 3, 1]
-- Mathlib/Order/OmegaCompletePartialOrder.lean
@[simp]
theorem id_comp (f : β →𝒄 γ) : id.comp f = f
-- Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean
theorem condexp_of_aestronglyMeasurable' (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
    (hf : AEStronglyMeasurable' m f μ) (hfi : Integrable f μ) : μ[f|m] =ᵐ[μ] f
-- Mathlib/Combinatorics/Enumerative/Composition.lean
theorem invEmbedding_comp (i : Fin c.length) (j : Fin (c.blocksFun i)) :
    (c.invEmbedding (c.embedding i j) : ℕ) = j
-- Mathlib/Data/List/ToFinsupp.lean
@[norm_cast]
theorem coe_toFinsupp : (l.toFinsupp : ℕ → M) = (l.getD · 0)
-- Mathlib/Order/Interval/Finset/Basic.lean
theorem Icc_ssubset_Icc_left (hI : a₂ ≤ b₂) (ha : a₂ < a₁) (hb : b₁ ≤ b₂) :
    Icc a₁ b₁ ⊂ Icc a₂ b₂
-- Mathlib/Topology/ContinuousOn.lean
theorem DenseRange.nhdsWithin_neBot {ι : Type*} {f : ι → α} (h : DenseRange f) (x : α) :
    NeBot (𝓝[range f] x)
-- Mathlib/Algebra/Order/Group/Unbundled/Abs.lean
@[to_additive (attr := simp)] lemma mabs_one : |(1 : α)|ₘ = 1
-- Mathlib/NumberTheory/ModularForms/EisensteinSeries/IsBoundedAtImInfty.lean
/-- The absolute value of the restricted sum is less than the full sum of the absolute values. -/
lemma abs_le_tsum_abs (N : ℕ) (a : Fin 2 → ZMod N) (k : ℤ) (hk : 3 ≤ k) (z : ℍ) :
    Complex.abs (eisensteinSeries a k z) ≤ ∑' (x : Fin 2 → ℤ), Complex.abs (eisSummand k x z)
-- Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
private lemma pairMap_of_snd_nmem_fst {t : Finset σ × σ} (h : t.snd ∉ t.fst) :
    pairMap σ t = (t.fst.cons t.snd h, t.snd)
-- .lake/packages/batteries/Batteries/Data/List/Pairwise.lean
theorem pairwise_and_iff : l.Pairwise (fun a b => R a b ∧ S a b) ↔ Pairwise R l ∧ Pairwise S l
-- Mathlib/Data/Multiset/FinsetOps.lean
@[simp]
theorem coe_ndunion (l₁ l₂ : List α) : @ndunion α _ l₁ l₂ = (l₁ ∪ l₂ : List α)
-- Mathlib/Analysis/Calculus/ContDiff/Defs.lean
protected theorem ContDiffWithinAt.insert (h : ContDiffWithinAt 𝕜 n f s x) :
    ContDiffWithinAt 𝕜 n f (insert x s) x
-- Mathlib/Probability/Independence/Kernel.lean
lemma iIndep.iIndepSets' (hμ : iIndep m κ μ) :
    iIndepSets (fun x ↦ {s | MeasurableSet[m x] s}) κ μ
-- Mathlib/Topology/Order/LeftRightLim.lean
theorem rightLim_le_leftLim (h : x ≤ y) : rightLim f y ≤ leftLim f x
-- Mathlib/GroupTheory/CoprodI.lean
theorem of_leftInverse [DecidableEq ι] (i : ι) :
    Function.LeftInverse (lift <| Pi.mulSingle i (MonoidHom.id (M i))) of
-- Mathlib/Topology/UniformSpace/Basic.lean
theorem comp_comp_symm_mem_uniformity_sets {s : Set (α × α)} (hs : s ∈ 𝓤 α) :
    ∃ t ∈ 𝓤 α, SymmetricRel t ∧ t ○ t ○ t ⊆ s
-- Mathlib/Dynamics/BirkhoffSum/Basic.lean
theorem map_birkhoffSum {F N : Type*} [AddCommMonoid N] [FunLike F M N] [AddMonoidHomClass F M N]
    (g' : F) (f : α → α) (g : α → M) (n : ℕ) (x : α) :
    g' (birkhoffSum f g n x) = birkhoffSum f (g' ∘ g) n x
-- Mathlib/Data/List/Enum.lean
theorem fst_lt_add_of_mem_enumFrom {x : ℕ × α} {n : ℕ} {l : List α} (h : x ∈ enumFrom n l) :
    x.1 < n + length l
-- Mathlib/Data/Set/Image.lean
@[simp]
theorem range_inl_inter_range_inr : range (Sum.inl : α → α ⊕ β) ∩ range Sum.inr = ∅
-- Mathlib/Algebra/Homology/ImageToKernel.lean
@[simp]
theorem imageToKernel_epi_comp {Z : V} (h : Z ⟶ A) [Epi h] (w) :
    imageToKernel (h ≫ f) g w =
      Subobject.ofLE _ _ (imageSubobject_comp_le h f) ≫
        imageToKernel f g ((cancel_epi h).mp (by simpa using w : h ≫ f ≫ g = h ≫ 0))
-- Mathlib/CategoryTheory/Limits/FilteredColimitCommutesFiniteLimit.lean
/-- This follows this proof from
* Borceux, Handbook of categorical algebra 1, Theorem 2.13.4
although with different names.
-/
theorem colimitLimitToLimitColimit_surjective :
    Function.Surjective (colimitLimitToLimitColimit F)
-- Mathlib/Data/Real/EReal.lean
theorem addLECancellable_coe (x : ℝ) : AddLECancellable (x : EReal)
-- Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
theorem measure_union_ne_top (hs : μ s ≠ ∞) (ht : μ t ≠ ∞) : μ (s ∪ t) ≠ ∞
-- Mathlib/RingTheory/FinitePresentation.lean
/-- An algebra over a commutative semiring is `Algebra.FinitePresentation` if it is the quotient of
a polynomial ring in `n` variables by a finitely generated ideal. -/
class Algebra.FinitePresentation [CommSemiring R] [Semiring A] [Algebra R A] : Prop where
  out : ∃ (n : ℕ) (f : MvPolynomial (Fin n) R →ₐ[R] A), Surjective f ∧ f.toRingHom.ker.FG
-- Mathlib/Probability/Kernel/Basic.lean
theorem setLIntegral_deterministic' {f : β → ℝ≥0∞} {g : α → β} {a : α} (hg : Measurable g)
    (hf : Measurable f) {s : Set β} (hs : MeasurableSet s) [Decidable (g a ∈ s)] :
    ∫⁻ x in s, f x ∂deterministic g hg a = if g a ∈ s then f (g a) else 0
-- Mathlib/Analysis/CstarAlgebra/Multiplier.lean
@[simp]
theorem nnnorm_fst (a : 𝓜(𝕜, A)) : ‖a.fst‖₊ = ‖a‖₊
-- Mathlib/Data/Set/Finite.lean
theorem finite_isBot (α : Type*) [PartialOrder α] : { x : α | IsBot x }.Finite
-- Mathlib/Data/ENNReal/Operations.lean
/-- This lemma has an abbreviated name because it is used frequently. -/
theorem cancel_of_ne {a : ℝ≥0∞} (h : a ≠ ∞) : AddLECancellable a
-- Mathlib/Topology/Compactness/SigmaCompact.lean
@[simp]
theorem find_shiftr (x : X) : K.shiftr.find x = K.find x + 1
-- Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
theorem mul_floor_map_zero_le_floor_iterate_zero (n : ℕ) : ↑n * ⌊f 0⌋ ≤ ⌊f^[n] 0⌋
-- Mathlib/RingTheory/Localization/Basic.lean
/-- If `CommSemiring` homs `g : R →+* P, l : P →+* A` induce maps of localizations, the composition
of the induced maps equals the map of localizations induced by `l ∘ g`. -/
theorem map_comp_map {A : Type*} [CommSemiring A] {U : Submonoid A} {W} [CommSemiring W]
    [Algebra A W] [IsLocalization U W] {l : P →+* A} (hl : T ≤ U.comap l) :
    (map W l hl).comp (map Q g hy : S →+* _) = map W (l.comp g) fun _ hx => hl (hy hx)
-- Mathlib/Data/Real/EReal.lean
@[simp] lemma bot_mul_top : (⊥ : EReal) * ⊤ = ⊥
-- Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean
theorem HasStrictFDerivAt.exp (hf : HasStrictFDerivAt f f' x) :
    HasStrictFDerivAt (fun x => Real.exp (f x)) (Real.exp (f x) • f') x
-- Mathlib/Algebra/Star/SelfAdjoint.lean
@[simp]
theorem star_mul_self [Mul R] [StarMul R] (x : R) : IsSelfAdjoint (star x * x)
-- Mathlib/Algebra/Field/Basic.lean
theorem div_neg (a : K) : a / -b = -(a / b)
-- Mathlib/Data/ENNReal/Basic.lean
theorem lt_iff_exists_real_btwn :
    a < b ↔ ∃ r : ℝ, 0 ≤ r ∧ a < ENNReal.ofReal r ∧ (ENNReal.ofReal r : ℝ≥0∞) < b
-- Mathlib/Algebra/GCDMonoid/Basic.lean
theorem lcm_dvd_iff [GCDMonoid α] {a b c : α} : lcm a b ∣ c ↔ a ∣ c ∧ b ∣ c
-- Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean
/-- The derivative of `log ∘ f` is the logarithmic derivative provided `f` is differentiable and
`f x  ≠ 0`. -/
lemma Real.deriv_log_comp_eq_logDeriv {f : ℝ → ℝ} {x : ℝ} (h₁ : DifferentiableAt ℝ f x)
    (h₂ : f x ≠ 0) : deriv (log ∘ f) x = logDeriv f x
-- Mathlib/Algebra/Star/Pointwise.lean
@[simp]
theorem star_empty [Star α] : (∅ : Set α)⋆ = ∅
-- Mathlib/Probability/Moments.lean
@[simp]
theorem centralMoment_one [IsProbabilityMeasure μ] : centralMoment X 1 μ = 0
-- Mathlib/Data/Multiset/Basic.lean
@[simp]
theorem not_mem_zero (a : α) : a ∉ (0 : Multiset α)
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
@[simp]
theorem comap_top {f : P₁ →ᵃ[k] P₂} : (⊤ : AffineSubspace k P₂).comap f = ⊤
-- Mathlib/Algebra/Order/Field/Basic.lean
theorem inv_mul_lt_iff (h : 0 < b) : b⁻¹ * a < c ↔ a < b * c
-- Mathlib/AlgebraicTopology/SimplexCategory.lean
/-- A monomorphism in `SimplexCategory` must increase lengths-/
theorem len_le_of_mono {x y : SimplexCategory} {f : x ⟶ y} : Mono f → x.len ≤ y.len
-- Mathlib/Probability/IdentDistrib.lean
protected theorem nnnorm [NormedAddCommGroup γ] [BorelSpace γ] (h : IdentDistrib f g μ ν) :
    IdentDistrib (fun x => ‖f x‖₊) (fun x => ‖g x‖₊) μ ν
-- Mathlib/Algebra/Group/Int.lean
@[local simp] lemma emod_two_ne_zero : ¬n % 2 = 0 ↔ n % 2 = 1
-- Mathlib/Geometry/Manifold/MFDeriv/FDeriv.lean
/-- For maps between vector spaces, `MDifferentiableOn` and `DifferentiableOn` coincide -/
theorem mdifferentiableOn_iff_differentiableOn :
    MDifferentiableOn 𝓘(𝕜, E) 𝓘(𝕜, E') f s ↔ DifferentiableOn 𝕜 f s
-- Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean
/-- If there is a diagram where the morphisms `W ⟶ Y` and `X ⟶ Z` are open embeddings, and `S ⟶ T`
is mono, then the induced morphism `W ×ₛ X ⟶ Y ×ₜ Z` is also an open embedding.

```
W ⟶ Y
 ↘   ↘
  S ⟶ T
 ↗   ↗
X ⟶ Z
```
-/
theorem pullback_map_openEmbedding_of_open_embeddings {W X Y Z S T : TopCat.{u}} (f₁ : W ⟶ S)
    (f₂ : X ⟶ S) (g₁ : Y ⟶ T) (g₂ : Z ⟶ T) {i₁ : W ⟶ Y} {i₂ : X ⟶ Z} (H₁ : OpenEmbedding i₁)
    (H₂ : OpenEmbedding i₂) (i₃ : S ⟶ T) [H₃ : Mono i₃] (eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁)
    (eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂) : OpenEmbedding (pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂)
-- Mathlib/Geometry/Manifold/MFDeriv/Basic.lean
theorem HasMFDerivAt.congr_of_eventuallyEq (h : HasMFDerivAt I I' f x f') (h₁ : f₁ =ᶠ[𝓝 x] f) :
    HasMFDerivAt I I' f₁ x f'
-- Mathlib/CategoryTheory/Filtered/Basic.lean
/-- For every universe `w`, `C` is filtered if and only if every finite diagram in `C` with shape
    in `w` admits a cocone. -/
theorem iff_cocone_nonempty : IsFiltered C ↔
    ∀ {J : Type w} [SmallCategory J] [FinCategory J] (F : J ⥤ C), Nonempty (Cocone F)
-- Mathlib/CategoryTheory/Opposites.lean
theorem Quiver.Hom.unop_inj {X Y : Cᵒᵖ} :
    Function.Injective (Quiver.Hom.unop : (X ⟶ Y) → (Opposite.unop Y ⟶ Opposite.unop X))
-- Mathlib/Algebra/Polynomial/RingDivision.lean
theorem natDegree_eq_zero_of_isUnit (h : IsUnit p) : natDegree p = 0
-- Mathlib/Data/Sigma/Order.lean
@[simp]
theorem mk_le_mk_iff [∀ i, LE (α i)] {i : ι} {a b : α i} : (⟨i, a⟩ : Sigma α) ≤ ⟨i, b⟩ ↔ a ≤ b
-- Mathlib/ModelTheory/Basic.lean
@[simp]
theorem map_fun (φ : M ↪[L] N) {n : ℕ} (f : L.Functions n) (x : Fin n → M) :
    φ (funMap f x) = funMap f (φ ∘ x)
-- Mathlib/Order/Minimal.lean
theorem RelEmbedding.minimals_preimage_eq (f : r ↪r s) (y : Set β) :
    minimals r (f ⁻¹' y) = f ⁻¹' minimals s (y ∩ range f)
-- Mathlib/ModelTheory/Satisfiability.lean
theorem models_formula_iff {φ : L.Formula α} :
    T ⊨ᵇ φ ↔ ∀ (M : ModelType.{u, v, max u v} T) (v : α → M), φ.Realize v
-- Mathlib/Computability/Primrec.lean
theorem nat_max : Primrec₂ (@max ℕ _)
-- Mathlib/Algebra/Associated/Basic.lean
@[simp]
lemma prime_pow_iff [CancelCommMonoidWithZero α] {p : α} {n : ℕ} :
    Prime (p ^ n) ↔ Prime p ∧ n = 1
-- Mathlib/Combinatorics/SimpleGraph/Basic.lean
@[simp]
theorem completeGraph_eq_top (V : Type u) : completeGraph V = ⊤
-- Mathlib/Logic/Encodable/Basic.lean
@[simp]
theorem encode_inr (b : β) : @encode (α ⊕ β) _ (Sum.inr b) = 2 * (encode b) + 1
-- .lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean
@[simp] protected theorem fdiv_zero : ∀ a : Int, fdiv a 0 = 0
-- Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
@[simp] -- Porting note (#10618): simp can prove this
theorem intervalIntegrable_const [IsLocallyFiniteMeasure μ] {c : E} :
    IntervalIntegrable (fun _ => c) μ a b
-- Mathlib/Analysis/InnerProductSpace/PiL2.lean
@[simp]
theorem basisFun_repr (x : EuclideanSpace 𝕜 ι) (i : ι) : (basisFun ι 𝕜).repr x i = x i
-- Mathlib/Topology/Bornology/Absorbs.lean
lemma absorbent_iff_inv_smul {s : Set α} :
    Absorbent G₀ s ↔ ∀ x, ∀ᶠ c in cobounded G₀, c⁻¹ • x ∈ s
-- Mathlib/Algebra/Order/CauSeq/Basic.lean
@[simp, norm_cast]
theorem neg_apply (f : CauSeq β abv) (i) : (-f) i = -f i
-- Mathlib/Algebra/Star/SelfAdjoint.lean
@[simp, norm_cast] lemma val_ratCast (q : ℚ) : (q : selfAdjoint R) = (q : R)