ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean
theorem map_one_fst_of_isMulTwoCocycle {f : G × G → M} (hf : IsMulTwoCocycle f) (g : G) :
    f (1, g) = f (1, 1)
-- Mathlib/Order/Filter/Germ/Basic.lean
@[simp]
theorem const_compTendsto {l : Filter α} (b : β) {lc : Filter γ} {g : γ → α} (hg : Tendsto g lc l) :
    (↑b : Germ l β).compTendsto g hg = ↑b
-- Mathlib/Order/Interval/Set/OrdConnected.lean
@[instance]
theorem ordConnected_Iio {a : α} : OrdConnected (Iio a)
-- Mathlib/Topology/NoetherianSpace.lean
/-- In a Noetherian space, all sets are compact. -/
protected theorem NoetherianSpace.isCompact [NoetherianSpace α] (s : Set α) : IsCompact s
-- Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean
lemma polynomial_eq : W.polynomial =
    Cubic.toPoly
      ⟨0, 1, Cubic.toPoly ⟨0, 0, W.a₁, W.a₃⟩, Cubic.toPoly ⟨-1, -W.a₂, -W.a₄, -W.a₆⟩⟩
-- Mathlib/Algebra/CubicDiscriminant.lean
@[simp]
theorem natDegree_of_b_ne_zero' (hb : b ≠ 0) : (toPoly ⟨0, b, c, d⟩).natDegree = 2
-- Mathlib/NumberTheory/Padics/PadicNumbers.lean
theorem nonarchimedean (q r : ℚ_[p]) : ‖q + r‖ ≤ max ‖q‖ ‖r‖
-- Mathlib/Geometry/Manifold/MFDeriv/Basic.lean
theorem ContMDiff.mdifferentiable (hf : ContMDiff I I' n f) (hn : 1 ≤ n) : MDifferentiable I I' f
-- Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean
theorem PiToModule.fromMatrix_apply_single_one [DecidableEq ι] (A : Matrix ι ι R) (j : ι) :
    PiToModule.fromMatrix R b A (Pi.single j 1) = ∑ i : ι, A i j • b i
-- Mathlib/GroupTheory/Coxeter/Inversion.lean
@[simp]
theorem isLeftInversion_simple_iff_isLeftDescent (w : W) (i : B) :
    cs.IsLeftInversion w (s i) ↔ cs.IsLeftDescent w i
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
@[simp]
theorem sin_nat_mul_two_pi_sub (x : ℝ) (n : ℕ) : sin (n * (2 * π) - x) = -sin x
-- Mathlib/Analysis/Asymptotics/Theta.lean
theorem IsTheta.mono (h : f =Θ[l] g) (hl : l' ≤ l) : f =Θ[l'] g
-- Mathlib/RepresentationTheory/Rep.lean
@[simp]
theorem linearization_ε_hom : (linearization k G).ε.hom = Finsupp.lsingle PUnit.unit
-- Mathlib/RingTheory/Localization/Ideal.lean
theorem ideal_eq_iInf_comap_map_away {S : Finset R} (hS : Ideal.span (α := R) S = ⊤) (I : Ideal R) :
    I = ⨅ f ∈ S, (I.map (algebraMap R (Localization.Away f))).comap
    (algebraMap R (Localization.Away f))
-- Mathlib/Topology/Order/MonotoneConvergence.lean
theorem tendsto_atTop_iSup (h_mono : Monotone f) : Tendsto f atTop (𝓝 (⨆ i, f i))
-- Mathlib/Order/Hom/Basic.lean
@[simp]
theorem fst_prod_snd : (fst : α × β →o α).prod snd = id
-- Mathlib/Topology/Homotopy/Path.lean
theorem symm_trans (F : Homotopy p₀ p₁) (G : Homotopy p₁ p₂) :
    (F.trans G).symm = G.symm.trans F.symm
-- Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
/-- A finite-dimensional manifold modelled on a locally compact field
  (such as ℝ, ℂ or the `p`-adic numbers) is locally compact. -/
lemma Manifold.locallyCompact_of_finiteDimensional [LocallyCompactSpace 𝕜]
    [FiniteDimensional 𝕜 E] : LocallyCompactSpace M
-- Mathlib/Order/Atoms.lean
theorem isSimpleOrder_Ici_iff_isCoatom [PartialOrder α] [OrderTop α] {a : α} :
    IsSimpleOrder (Ici a) ↔ IsCoatom a
-- Mathlib/LinearAlgebra/Multilinear/Basic.lean
/-- The results of applying `domDomCongr` to two maps are equal if
and only if those maps are. -/
@[simp]
theorem domDomCongr_eq_iff (σ : ι₁ ≃ ι₂) (f g : MultilinearMap R (fun _ : ι₁ => M₂) M₃) :
    f.domDomCongr σ = g.domDomCongr σ ↔ f = g
-- Mathlib/Algebra/Polynomial/Basic.lean
theorem C_neg : C (-a) = -C a
-- Mathlib/Algebra/Star/NonUnitalSubalgebra.lean
@[simp]
theorem toNonUnitalSubalgebra_subtype :
    NonUnitalSubalgebraClass.subtype S = NonUnitalStarSubalgebraClass.subtype S
-- Mathlib/Analysis/SpecificLimits/Basic.lean
theorem NNReal.tendsto_algebraMap_inverse_atTop_nhds_zero_nat (𝕜 : Type*) [Semiring 𝕜]
    [Algebra ℝ≥0 𝕜] [TopologicalSpace 𝕜] [ContinuousSMul ℝ≥0 𝕜] :
    Tendsto (algebraMap ℝ≥0 𝕜 ∘ fun n : ℕ ↦ (n : ℝ≥0)⁻¹) atTop (𝓝 0)
-- Mathlib/MeasureTheory/Covering/Differentiation.lean
/-- Given a measurable set `s`, then `μ (s ∩ a) / μ a` converges when `a` shrinks to a typical
point `x` along a Vitali family. The limit is `1` for `x ∈ s` and `0` for `x ∉ s`. This shows that
almost every point of `s` is a Lebesgue density point for `s`. A version for non-measurable sets
holds, but it only gives the first conclusion, see `ae_tendsto_measure_inter_div`. -/
theorem ae_tendsto_measure_inter_div_of_measurableSet {s : Set α} (hs : MeasurableSet s) :
    ∀ᵐ x ∂μ, Tendsto (fun a => μ (s ∩ a) / μ a) (v.filterAt x) (𝓝 (s.indicator 1 x))
-- Mathlib/Data/Finset/NAry.lean
lemma sup'_image₂_right (g : γ → δ) (h : (image₂ f s t).Nonempty) :
    sup' (image₂ f s t) h g =
      sup' t h.of_image₂_right fun y ↦ sup' s h.of_image₂_left (g <| f · y)
-- Mathlib/Data/Nat/Defs.lean
lemma succ_injective : Injective Nat.succ
-- Mathlib/Analysis/NormedSpace/Real.lean
theorem dist_smul_add_one_sub_smul_le {r : ℝ} {x y : E} (h : r ∈ Icc 0 1) :
    dist (r • x + (1 - r) • y) x ≤ dist y x
-- Mathlib/Order/Filter/AtTopBot.lean
/-- If `f` tends to negative infinity along a nontrivial filter `l`, then
`fun x ↦ f x / r` tends to negative infinity if and only if `0 < r. `-/
lemma tendsto_div_const_atBot_iff_pos [NeBot l] (h : Tendsto f l atBot) :
    Tendsto (fun x ↦ f x / r) l atBot ↔ 0 < r
-- Mathlib/Order/Interval/Basic.lean
@[norm_cast] -- @[simp, norm_cast] -- Porting note: not in simpNF
theorem coe_ssubset_coe : (s : Set α) ⊂ t ↔ s < t
-- Mathlib/LinearAlgebra/Dual.lean
@[simp]
theorem finrank_range_dualMap_eq_finrank_range (f : V₁ →ₗ[K] V₂) :
    -- Porting note (#11036): broken dot notation lean4#1910
    finrank K (LinearMap.range f.dualMap) = finrank K (LinearMap.range f)
-- Mathlib/Order/Heyting/Basic.lean
theorem Disjoint.sdiff_eq_right (h : Disjoint a b) : b \ a = b
-- Mathlib/CategoryTheory/GuitartExact/Basic.lean
/-- Condition on `w : TwoSquare T L R B` expressing that it is a Guitart exact square.
It is equivalent to saying that for any `X₃ : C₃`, the induced functor
`CostructuredArrow L X₃ ⥤ CostructuredArrow R (B.obj X₃)` is final (see `guitartExact_iff_final`)
or equivalently that for any `X₂ : C₂`, the induced functor
`StructuredArrow X₂ T ⥤ StructuredArrow (R.obj X₂) B` is initial (see `guitartExact_iff_initial`).
See also  `guitartExact_iff_isConnected_rightwards`, `guitartExact_iff_isConnected_downwards`
for characterizations in terms of the connectedness of auxiliary categories. -/
class GuitartExact : Prop where
  isConnected_rightwards {X₂ : C₂} {X₃ : C₃} (g : R.obj X₂ ⟶ B.obj X₃) :
    IsConnected (w.StructuredArrowRightwards g)
-- Mathlib/NumberTheory/GaussSum.lean
private theorem gaussSum_mul_aux {χ : MulChar R R'} (hχ : χ ≠ 1) (ψ : AddChar R R')
    (b : R) :
    ∑ a, χ (a * b⁻¹) * ψ (a - b) = ∑ c, χ c * ψ (b * (c - 1))
-- Mathlib/Analysis/Normed/Group/Basic.lean
@[to_additive (attr := simp) coe_comp_nnnorm]
theorem coe_comp_nnnorm' : (toReal : ℝ≥0 → ℝ) ∘ (nnnorm : E → ℝ≥0) = norm
-- Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
@[simp]
theorem norm_mkPiAlgebraFin [NormOneClass A] :
    ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n A‖ = 1
-- Mathlib/Topology/MetricSpace/HausdorffDimension.lean
@[simp]
theorem dimH_singleton (x : X) : dimH ({x} : Set X) = 0
-- Mathlib/Topology/Homeomorph.lean
@[simp]
theorem preimage_image (h : X ≃ₜ Y) (s : Set X) : h ⁻¹' (h '' s) = s
-- Mathlib/Analysis/BoxIntegral/Partition/Measure.lean
@[simp]
theorem volume_apply' (I : Box ι) :
    ((volume : Measure (ι → ℝ)) I).toReal = ∏ i, (I.upper i - I.lower i)
-- Mathlib/FieldTheory/SeparableDegree.lean
/-- The minimal polynomial of an element of `E / F` of exponential characteristic `q` has
separable degree one if and only if the minimal polynomial is of the form
`Polynomial.expand F (q ^ n) (X - C y)` for some `n : ℕ` and `y : F`. -/
theorem natSepDegree_eq_one_iff_eq_expand_X_sub_C : (minpoly F x).natSepDegree = 1 ↔
    ∃ (n : ℕ) (y : F), minpoly F x = expand F (q ^ n) (X - C y)
-- Mathlib/Logic/Lemmas.lean
theorem dite_dite_distrib_right {a : p → q → α} {b : p → ¬q → α} {c : ¬p → α} :
    dite p (fun hp ↦ dite q (a hp) (b hp)) c =
      dite q (fun hq ↦ dite p (fun hp ↦ a hp hq) c) fun hq ↦ dite p (fun hp ↦ b hp hq) c
-- Mathlib/Topology/Order/LocalExtr.lean
nonrec theorem IsLocalMin.comp_antitone (hf : IsLocalMin f a) {g : β → γ} (hg : Antitone g) :
    IsLocalMax (g ∘ f) a
-- Mathlib/LinearAlgebra/BilinearMap.lean
theorem map_neg₂ (f : M' →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P') (x y) : f (-x) y = -f x y
-- Mathlib/Order/CompleteLattice.lean
theorem iSup_lt_iff : iSup f < a ↔ ∃ b, b < a ∧ ∀ i, f i ≤ b
-- Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean
theorem boundedBy_caratheodory {m : Set α → ℝ≥0∞} {s : Set α}
    (hs : ∀ t, m (t ∩ s) + m (t \ s) ≤ m t) : MeasurableSet[(boundedBy m).caratheodory] s
-- .lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean
theorem eq_of_toFin_eq : ∀ {x y : BitVec w}, x.toFin = y.toFin → x = y
-- Mathlib/LinearAlgebra/Prod.lean
@[deprecated (since := "2024-06-05")]
theorem tailings_disjoint_tunnel (f : M × N →ₗ[R] M) (i : Injective f) (n : ℕ) :
    Disjoint (tailings f i n) (OrderDual.ofDual (α := Submodule R M) <| tunnel f i (n + 1))
-- .lake/packages/batteries/Batteries/Data/List/Lemmas.lean
theorem Sublist.append (hl : l₁ <+ l₂) (hr : r₁ <+ r₂) : l₁ ++ r₁ <+ l₂ ++ r₂
-- Mathlib/MeasureTheory/Measure/FiniteMeasureProd.lean
@[simp] lemma prod_zero : μ.prod (0 : FiniteMeasure β) = 0
-- Mathlib/Topology/UniformSpace/Basic.lean
theorem UniformSpace.isClosed_ball (x : α) {V : Set (α × α)} (hV : IsClosed V) :
    IsClosed (ball x V)
-- Mathlib/CategoryTheory/Monoidal/FunctorCategory.lean
@[simp]
theorem whiskerRight_app {F G F' : C ⥤ D} {α : F ⟶ G} {X} :
    (α ▷ F').app X = α.app X ▷ F'.obj X
-- Mathlib/LinearAlgebra/TensorProduct/Submodule.lean
theorem mulMap_comp_map_inclusion {M' N' : Submodule R S} (hM : M' ≤ M) (hN : N' ≤ N) :
    mulMap M N ∘ₗ TensorProduct.map (inclusion hM) (inclusion hN) = mulMap M' N'
-- Mathlib/Analysis/Convex/Side.lean
theorem wOppSide_smul_vsub_vadd_left {s : AffineSubspace R P} {p₁ p₂ : P} (x : P) (hp₁ : p₁ ∈ s)
    (hp₂ : p₂ ∈ s) {t : R} (ht : t ≤ 0) : s.WOppSide (t • (x -ᵥ p₁) +ᵥ p₂) x
-- Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean
@[reassoc (attr := simp)]
theorem inr_comp_pushoutComparison (f : X ⟶ Y) (g : X ⟶ Z) [HasPushout f g]
    [HasPushout (G.map f) (G.map g)] : pushout.inr _ _ ≫ pushoutComparison G f g =
      G.map (pushout.inr _ _)
-- Mathlib/Data/Nat/Prime/Defs.lean
theorem dvd_prime {p m : ℕ} (pp : Prime p) : m ∣ p ↔ m = 1 ∨ m = p
-- Mathlib/Analysis/Complex/CauchyIntegral.lean
/-- If `f : ℂ → E` is complex differentiable on a closed disc of positive radius, then it is
analytic on the corresponding open disc, and the coefficients of the power series are given by
Cauchy integral formulas. See also
`Complex.hasFPowerSeriesOnBall_of_differentiable_off_countable` for a version of this lemma with
weaker assumptions. -/
protected theorem _root_.DifferentiableOn.hasFPowerSeriesOnBall {R : ℝ≥0} {c : ℂ} {f : ℂ → E}
    (hd : DifferentiableOn ℂ f (closedBall c R)) (hR : 0 < R) :
    HasFPowerSeriesOnBall f (cauchyPowerSeries f c R) c R
-- Mathlib/Topology/UniformSpace/Basic.lean
theorem continuousWithinAt_iff'_left [TopologicalSpace β] {f : β → α} {b : β} {s : Set β} :
    ContinuousWithinAt f s b ↔ Tendsto (fun x => (f x, f b)) (𝓝[s] b) (𝓤 α)
-- Mathlib/Data/Set/Finite.lean
@[simp]
theorem not_infinite {s : Set α} : ¬s.Infinite ↔ s.Finite
-- Mathlib/Analysis/NormedSpace/AffineIsometry.lean
theorem ediam_image (s : Set P) : EMetric.diam (f '' s) = EMetric.diam s
-- Mathlib/Topology/ContinuousFunction/Bounded.lean
@[simp]
theorem dist_extend_extend (f : α ↪ δ) (g₁ g₂ : α →ᵇ β) (h₁ h₂ : δ →ᵇ β) :
    dist (g₁.extend f h₁) (g₂.extend f h₂) =
      max (dist g₁ g₂) (dist (h₁.restrict (range f)ᶜ) (h₂.restrict (range f)ᶜ))
-- Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
@[norm_cast]
theorem WithBot.coe_sInf' [InfSet α] {s : Set α} (hs : BddBelow s) :
    ↑(sInf s) = (sInf ((fun (a : α) ↦ ↑a) '' s) : WithBot α)
-- Mathlib/Analysis/InnerProductSpace/Spectrum.lean
/-- The eigenspaces of a self-adjoint operator on a finite-dimensional inner product space `E` gives
an internal direct sum decomposition of `E`. -/
theorem direct_sum_isInternal : DirectSum.IsInternal fun μ : Eigenvalues T => eigenspace T μ
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
@[simp]
theorem exp_one_rpow (x : ℝ) : exp 1 ^ x = exp x
-- Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
lemma coherentTopology.isLocallySurjective_iff [Preregular C] [FinitaryExtensive C]
    {F G : Sheaf (coherentTopology C) D} (f : F ⟶ G) [PreservesFiniteProducts (forget D)] :
      IsLocallySurjective f ↔ Presheaf.IsLocallySurjective (regularTopology C) f.val
-- Mathlib/Algebra/Polynomial/Roots.lean
lemma eq_zero_of_forall_eval_zero_of_natDegree_lt_card
    (f : R[X]) (hf : ∀ r, f.eval r = 0) (hfR : f.natDegree < #R) : f = 0
-- Mathlib/Data/PNat/Prime.lean
@[simp]
theorem one_coprime {n : ℕ+} : (1 : ℕ+).Coprime n
-- Mathlib/Analysis/InnerProductSpace/Basic.lean
/-- The sum defined in Bessel's inequality is summable. -/
theorem Orthonormal.inner_products_summable (hv : Orthonormal 𝕜 v) :
    Summable fun i => ‖⟪v i, x⟫‖ ^ 2
-- Mathlib/Topology/ContinuousOn.lean
theorem ContinuousWithinAt.preimage_mem_nhdsWithin''
    {f : α → β} {x : α} {y : β} {s t : Set β}
    (h : ContinuousWithinAt f (f ⁻¹' s) x) (ht : t ∈ 𝓝[s] y) (hxy : y = f x) :
    f ⁻¹' t ∈ 𝓝[f ⁻¹' s] x
-- Mathlib/Topology/ExtremallyDisconnected.lean
/-- Theorem 2.5 in [Gleason, *Projective topological spaces*][gleason1958]:
in the category of compact spaces and continuous maps,
the projective spaces are precisely the extremally disconnected spaces. -/
protected theorem CompactT2.ExtremallyDisconnected.projective [ExtremallyDisconnected A]
    [CompactSpace A] [T2Space A] : CompactT2.Projective A
-- Mathlib/SetTheory/Cardinal/Basic.lean
theorem powerlt_mono_left (a) : Monotone fun c => a ^< c
-- Mathlib/Algebra/Periodic.lean
theorem Antiperiodic.div_inv [DivisionSemiring α] [Neg β] (h : Antiperiodic f c) {a : α}
    (ha : a ≠ 0) : Antiperiodic (fun x => f (x / a)) (c * a)
-- Mathlib/CategoryTheory/Preadditive/Opposite.lean
@[simp]
theorem op_neg {X Y : C} (f : X ⟶ Y) : (-f).op = -f.op
-- Mathlib/Order/Interval/Finset/Nat.lean
@[simp]
theorem Ioc_succ_singleton : Ioc b (b + 1) = {b + 1}
-- Mathlib/Algebra/Group/Invertible/Defs.lean
theorem invOf_eq_right_inv [Monoid α] {a b : α} [Invertible a] (hac : a * b = 1) : ⅟ a = b
-- Mathlib/Data/Finsupp/Basic.lean
/-- `Finsupp.comapDomain` is an `AddMonoidHom`. -/
@[simps]
def comapDomain.addMonoidHom (hf : Function.Injective f) : (β →₀ M) →+ α →₀ M
-- Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean
@[simp]
theorem ofDirectSum_of_tprod {n} (x : Fin n → M) :
    ofDirectSum (DirectSum.of _ n (PiTensorProduct.tprod R x)) = tprod R M n x
-- Mathlib/Data/Fin/Basic.lean
protected lemma lt_or_lt_of_ne (h : a ≠ b) : a < b ∨ b < a
-- Mathlib/GroupTheory/GroupAction/Quotient.lean
theorem normalCore_eq_ker : H.normalCore = (MulAction.toPermHom G (G ⧸ H)).ker
-- Mathlib/Topology/Irreducible.lean
theorem isPreirreducible_empty : IsPreirreducible (∅ : Set X)
-- Mathlib/NumberTheory/VonMangoldt.lean
theorem vonMangoldt_le_log : ∀ {n : ℕ}, Λ n ≤ Real.log (n : ℝ)
-- Mathlib/Topology/UniformSpace/Separation.lean
theorem comap_mk_uniformity : (𝓤 (SeparationQuotient α)).comap (Prod.map mk mk) = 𝓤 α
-- Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean
theorem denom_cocycle (x y : GL(2, ℝ)⁺) (z : ℍ) :
    denom (x * y) z = denom x (smulAux y z) * denom y z
-- Mathlib/Topology/UniformSpace/Cauchy.lean
theorem CauchySeq.subseq_subseq_mem {V : ℕ → Set (α × α)} (hV : ∀ n, V n ∈ 𝓤 α) {u : ℕ → α}
    (hu : CauchySeq u) {f g : ℕ → ℕ} (hf : Tendsto f atTop atTop) (hg : Tendsto g atTop atTop) :
    ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ n, ((u ∘ f ∘ φ) n, (u ∘ g ∘ φ) n) ∈ V n
-- Mathlib/CategoryTheory/Limits/Fubini.lean
@[simp]
theorem colimitCurrySwapCompColimIsoColimitCurryCompColim_ι_ι_hom {j} {k} :
    colimit.ι _ j ≫ colimit.ι (curry.obj (Prod.swap K J ⋙ G) ⋙ colim) k ≫
      (colimitCurrySwapCompColimIsoColimitCurryCompColim G).hom =
        (colimit.ι _ k ≫ colimit.ι (curry.obj G ⋙ colim) j : _ ⟶ colimit (curry.obj G⋙ colim))
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
theorem cof_le_card (o) : cof o ≤ card o
-- Mathlib/Topology/Compactness/Lindelof.lean
/-- Type class for Hereditarily Lindelöf spaces.  -/
class HereditarilyLindelofSpace (X : Type*) [TopologicalSpace X] : Prop where
  /-- In a Hereditarily Lindelöf space, `Set.univ` is a Hereditarily Lindelöf set. -/
  isHereditarilyLindelof_univ : IsHereditarilyLindelof (univ : Set X)
-- Mathlib/Geometry/Manifold/PartitionOfUnity.lean
theorem nonneg (i : ι) (x : M) : 0 ≤ f i x
-- Mathlib/Data/Set/Finite.lean
theorem card_image_of_inj_on {s : Set α} [Fintype s] {f : α → β} [Fintype (f '' s)]
    (H : ∀ x ∈ s, ∀ y ∈ s, f x = f y → x = y) : Fintype.card (f '' s) = Fintype.card s
-- Mathlib/Order/Lattice.lean
@[simp]
theorem snd_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd
-- Mathlib/Data/Nat/GCD/Basic.lean
@[simp]
theorem coprime_add_self_left {m n : ℕ} : Coprime (m + n) n ↔ Coprime m n
-- Mathlib/Algebra/Star/SelfAdjoint.lean
protected lemma isSelfAdjoint : IsSelfAdjoint f ↔ ∀ i, IsSelfAdjoint (f i)
-- Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean
@[simp] lemma add_τ₃ (φ φ' : S₁ ⟶ S₂) : (φ + φ').τ₃ = φ.τ₃ + φ'.τ₃
-- Mathlib/CategoryTheory/MorphismProperty/Basic.lean
lemma inverseImage_equivalence_inverse_eq_map_functor
    (P : MorphismProperty D) [RespectsIso P] (E : C ≌ D) :
    P.inverseImage E.functor = P.map E.inverse
-- Mathlib/Data/Seq/Computation.lean
@[simp]
theorem map_pure (f : α → β) (a) : map f (pure a) = pure (f a)
-- Mathlib/Topology/MetricSpace/HausdorffDimension.lean
theorem dimH_preimage_le (hf : AntilipschitzWith K f) (s : Set Y) : dimH (f ⁻¹' s) ≤ dimH s
-- Mathlib/Algebra/MvPolynomial/Monad.lean
theorem bind₂_comp_bind₂ (f : R →+* MvPolynomial σ S) (g : S →+* MvPolynomial σ T) :
    (bind₂ g).comp (bind₂ f) = bind₂ ((bind₂ g).comp f)
-- Mathlib/Analysis/InnerProductSpace/Projection.lean
/-- Given a finite-dimensional subspace `K₂`, and a subspace `K₁`
contained in it, the dimensions of `K₁` and the intersection of its
orthogonal subspace with `K₂` add to that of `K₂`. -/
theorem Submodule.finrank_add_inf_finrank_orthogonal' {K₁ K₂ : Submodule 𝕜 E}
    [FiniteDimensional 𝕜 K₂] (h : K₁ ≤ K₂) {n : ℕ} (h_dim : finrank 𝕜 K₁ + n = finrank 𝕜 K₂) :
    finrank 𝕜 (K₁ᗮ ⊓ K₂ : Submodule 𝕜 E) = n
-- Mathlib/RingTheory/AlgebraicIndependent.lean
theorem AlgebraicIndependent.restrict_of_comp_subtype {s : Set ι}
    (hs : AlgebraicIndependent R (x ∘ (↑) : s → A)) : AlgebraicIndependent R (s.restrict x)
-- Mathlib/Data/NNRat/Defs.lean
theorem toNNRat_le_iff_le_coe {p : ℚ≥0} : toNNRat q ≤ p ↔ q ≤ ↑p
-- Mathlib/Topology/Maps/Proper/Basic.lean
/-- An injective and continuous function is proper if and only if it is closed. -/
lemma isProperMap_iff_isClosedMap_of_inj (f_cont : Continuous f) (f_inj : f.Injective) :
    IsProperMap f ↔ IsClosedMap f
-- Mathlib/Algebra/Algebra/Equiv.lean
@[simp]
theorem autCongr_symm (ϕ : A₁ ≃ₐ[R] A₂) : (autCongr ϕ).symm = autCongr ϕ.symm
-- Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
lemma image_exp_Ioc_eq_sphere : (fun θ : ℝ ↦ exp (θ * I)) '' Set.Ioc (-π) π = sphere 0 1
-- Mathlib/Algebra/Star/StarAlgHom.lean
@[simp]
theorem comp_id (f : A →⋆ₐ[R] B) : f.comp (StarAlgHom.id _ _) = f
-- Mathlib/Data/ULift.lean
@[simp] lemma map_injective : Injective (PLift.map f) ↔ Injective f
-- Mathlib/Data/Ordmap/Ordset.lean
theorem Valid'.dual_iff {t : Ordnode α} {o₁ o₂} : Valid' o₁ t o₂ ↔ @Valid' αᵒᵈ _ o₂ (.dual t) o₁
-- Mathlib/Probability/Martingale/Basic.lean
protected theorem adapted [LE E] (hf : Submartingale f ℱ μ) : Adapted ℱ f
-- Mathlib/RingTheory/Localization/Basic.lean
/-- A `CommRing` `S` which is the localization of a ring `R` without zero divisors at a subset of
non-zero elements does not have zero divisors. -/
theorem noZeroDivisors_of_le_nonZeroDivisors [Algebra A S] {M : Submonoid A} [IsLocalization M S]
    (hM : M ≤ nonZeroDivisors A) : NoZeroDivisors S
-- Mathlib/RingTheory/Ideal/Norm.lean
@[simp]
theorem cardQuot_bot [Infinite M] : cardQuot (⊥ : Submodule R M) = 0
-- Mathlib/FieldTheory/IsPerfectClosure.lean
/-- Composition of `p`-radical ring homomorphisms is also `p`-radical. -/
theorem IsPRadical.trans [IsPRadical i p] [IsPRadical f p] :
    IsPRadical (f.comp i) p
-- Mathlib/Order/Filter/Prod.lean
@[simp]
theorem pure_prod {a : α} {f : Filter β} : pure a ×ˢ f = map (Prod.mk a) f
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Unital.lean
/-- The **spectral mapping theorem** for the continuous functional calculus. -/
lemma cfcHom_map_spectrum (f : C(spectrum R a, R)) :
    spectrum R (cfcHom ha f) = Set.range f
-- Mathlib/CategoryTheory/Limits/IndYoneda.lean
@[reassoc (attr := simp)]
lemma limitRightOpIsoOpColimit_hom_comp_ι (F : Iᵒᵖ ⥤ C) [HasColimit F] (i : I) :
    (limitRightOpIsoOpColimit F).hom ≫ (colimit.ι F ⟨i⟩).op = limit.π F.rightOp i
-- Mathlib/Order/WithBot.lean
@[simp]
theorem map_bot (f : α → β) : map f ⊥ = ⊥
-- Mathlib/Data/Set/Finite.lean
/-- See also `finite_or_infinite`, `fintypeOrInfinite`. -/
protected theorem finite_or_infinite (s : Set α) : s.Finite ∨ s.Infinite
-- Mathlib/Algebra/Order/Field/Unbundled/Basic.lean
@[simp] lemma inv_lt_zero : a⁻¹ < 0 ↔ a < 0
-- Mathlib/MeasureTheory/OuterMeasure/AE.lean
@[simp high]
theorem ae_eq_univ : s =ᵐ[μ] (univ : Set α) ↔ μ sᶜ = 0
-- Mathlib/Topology/UnitInterval.lean
@[simp]
theorem symm_symm (x : I) : σ (σ x) = x
-- Mathlib/RingTheory/RingHom/Integral.lean
theorem isIntegral_stableUnderBaseChange : StableUnderBaseChange fun f => f.IsIntegral
-- Mathlib/RingTheory/WittVector/Truncated.lean
@[simp]
theorem coeff_truncate (x : 𝕎 R) (i : Fin n) : (truncate n x).coeff i = x.coeff i
-- Mathlib/Analysis/Normed/Operator/BoundedLinearMaps.lean
theorem isBigO_comp {E : Type*} {g : F → G} (hg : IsBoundedLinearMap 𝕜 g) {f : E → F}
    (l : Filter E) : (fun x' => g (f x')) =O[l] f
-- Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
protected theorem continuousWithinAt {s x} : ContinuousWithinAt I s x
-- Mathlib/Analysis/Complex/Liouville.lean
/-- If `f` is complex differentiable on an open disc of radius `R > 0`, is continuous on its
closure, and its values on the boundary circle of this disc are bounded from above by `C`, then the
norm of its derivative at the center is at most `C / R`. -/
theorem norm_deriv_le_of_forall_mem_sphere_norm_le {c : ℂ} {R C : ℝ} {f : ℂ → F} (hR : 0 < R)
    (hd : DiffContOnCl ℂ f (ball c R)) (hC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C) :
    ‖deriv f c‖ ≤ C / R
-- Mathlib/Data/ENNReal/Real.lean
theorem toReal_inv (a : ℝ≥0∞) : a⁻¹.toReal = a.toReal⁻¹
-- Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean
/-- If the angle between three points is `π`, the first two points are not equal. -/
theorem left_ne_of_oangle_eq_pi {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = π) : p₁ ≠ p₂
-- Mathlib/Order/SuccPred/Basic.lean
theorem le_pred_iff_eq_bot : a ≤ pred a ↔ a = ⊥
-- Mathlib/Topology/Separation.lean
theorem t2_separation_compact_nhds [LocallyCompactSpace X] [T2Space X] {x y : X} (h : x ≠ y) :
    ∃ u v, u ∈ 𝓝 x ∧ v ∈ 𝓝 y ∧ IsCompact u ∧ IsCompact v ∧ Disjoint u v
-- Mathlib/Analysis/Complex/UpperHalfPlane/Topology.lean
theorem mem_verticalStrip_iff (A B : ℝ) (z : ℍ) : z ∈ verticalStrip A B ↔ |z.re| ≤ A ∧ B ≤ z.im
-- Mathlib/Topology/EMetricSpace/Basic.lean
@[simp]
theorem edist_ofAdd (a b : X) : edist (ofAdd a) (ofAdd b) = edist a b
-- Mathlib/LinearAlgebra/FiniteDimensional.lean
@[simp]
theorem coe_basisOfLinearIndependentOfCardEqFinrank {ι : Type*} [Nonempty ι] [Fintype ι]
    {b : ι → V} (lin_ind : LinearIndependent K b) (card_eq : Fintype.card ι = finrank K V) :
    ⇑(basisOfLinearIndependentOfCardEqFinrank lin_ind card_eq) = b
-- Mathlib/Topology/Separation.lean
theorem Function.LeftInverse.isClosed_range [T2Space X] {f : X → Y} {g : Y → X}
    (h : Function.LeftInverse f g) (hf : Continuous f) (hg : Continuous g) : IsClosed (range g)
-- Mathlib/Algebra/Group/Units.lean
/-- Used for `field_simp` to deal with inverses of units. This form of the lemma
is essential since `field_simp` likes to use `inv_eq_one_div` to rewrite
`↑u⁻¹ = ↑(1 / u)`.
-/
@[field_simps]
theorem inv_eq_one_divp' (u : αˣ) : ((1 / u : αˣ) : α) = 1 /ₚ u