mathlib-initiative/mathlib-types huggingface dataset:
-- Mathlib.GroupTheory.FinitelyPresentedGroup
/-- Being the normal closure of a finite set is invariant under surjective homomorphism.  -/
Subgroup.IsFinitelyNormallyGenerated.map : ∀ {G : Type u_1} {H : Type u_2} [inst : Group G] [inst_1 : Group H] {N : Subgroup G},
  N.IsFinitelyNormallyGenerated →
    ∀ {f : G →* H}, Function.Surjective ⇑f → (Subgroup.map f N).IsFinitelyNormallyGenerated
-- Mathlib.RingTheory.Coalgebra.Basic
/-- The counit of the coalgebra  -/
RingTheory.LinearMap.termε : Lean.ParserDescr
-- Mathlib.Algebra.Polynomial.AlgebraMap
Polynomial.aeval_eq_sum_range' : ∀ {R : Type u} {S : Type v} [inst : CommSemiring R] [inst_1 : Semiring S] [inst_2 : Algebra R S] {p : Polynomial R}
  {n : ℕ}, p.natDegree < n → ∀ (x : S), (Polynomial.aeval x) p = ∑ i ∈ Finset.range n, p.coeff i • x ^ i
-- Mathlib.Order.Filter.Pointwise
Filter.map₂_vadd : ∀ {α : Type u_2} {β : Type u_3} [inst : VAdd α β] {f : Filter α} {g : Filter β},
  Filter.map₂ (fun x1 x2 => x1 +ᵥ x2) f g = f +ᵥ g
-- Mathlib.RingTheory.PicardGroup
CommRing.Pic.ext_iff : ∀ {R : Type u} [inst : CommSemiring R] {M N : CommRing.Pic R}, M = N ↔ Nonempty (M.AsModule ≃ₗ[R] N.AsModule)
-- Mathlib.RingTheory.AdicCompletion.Algebra
AdicCompletion.instPowNat : {R : Type u_1} → [inst : CommRing R] → (I : Ideal R) → Pow (AdicCompletion I R) ℕ
-- Mathlib.Geometry.Manifold.Sheaf.Smooth
/-- The sheaf of smooth functions from `M` to `G`, for `G` an additive Lie group, as a
sheaf of additive groups.  -/
smoothSheafAddGroup : {𝕜 : Type u_1} →
  [inst : NontriviallyNormedField 𝕜] →
    {EM : Type u_2} →
      [inst_1 : NormedAddCommGroup EM] →
        [inst_2 : NormedSpace 𝕜 EM] →
          {HM : Type u_3} →
            [inst_3 : TopologicalSpace HM] →
              ModelWithCorners 𝕜 EM HM →
                {E : Type u_4} →
                  [inst_4 : NormedAddCommGroup E] →
                    [inst_5 : NormedSpace 𝕜 E] →
                      {H : Type u_5} →
                        [inst_6 : TopologicalSpace H] →
                          (I : ModelWithCorners 𝕜 E H) →
                            (M : Type u) →
                              [inst_7 : TopologicalSpace M] →
                                [ChartedSpace HM M] →
                                  (G : Type u) →
                                    [inst_9 : TopologicalSpace G] →
                                      [inst_10 : ChartedSpace H G] →
                                        [inst_11 : AddGroup G] →
                                          [LieAddGroup I (↑⊤) G] → TopCat.Sheaf AddGrpCat (TopCat.of M)