ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
    (hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
    HasStrictFDerivAt f (f' x) x 
Arguments: {𝕜 : Type u_3} [RCLike 𝕜] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {H : Type u_5} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {f : G → H} {f' : G → G →L[𝕜] H} {x : G} (hder : ∀ᶠ (y : G) in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x)
Goal: HasStrictFDerivAt f (f' x) x
-- hasStrictDerivAt_of_hasDerivAt_of_continuousAt
theorem hasStrictDerivAt_of_hasDerivAt_of_continuousAt {f f' : 𝕜 → G} {x : 𝕜}
    (hder : ∀ᶠ y in 𝓝 x, HasDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
    HasStrictDerivAt f (f' x) x 
Arguments: {𝕜 : Type u_3} [RCLike 𝕜] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f f' : 𝕜 → G} {x : 𝕜} (hder : ∀ᶠ (y : 𝕜) in 𝓝 x, HasDerivAt f (f' y) y) (hcont : ContinuousAt f' x)
Goal: HasStrictDerivAt f (f' x) x
-- LightCondensed.discrete_obj
@[simps!]
noncomputable def LightCondensed.discrete : C ⥤ LightCondensed.{u} C 
Arguments: (C : Type w) [CategoryTheory.Category.{u, w} C] [CategoryTheory.HasSheafify (CategoryTheory.coherentTopology LightProfinite) C] (X : C)
Goal: (LightCondensed.discrete C).obj X =
  (CategoryTheory.presheafToSheaf (CategoryTheory.coherentTopology LightProfinite) C).obj
    ((CategoryTheory.Functor.const LightProfiniteᵒᵖ).obj X)
-- LightCondensed.discrete_map
@[simps!]
noncomputable def LightCondensed.discrete : C ⥤ LightCondensed.{u} C 
Arguments: (C : Type w) [CategoryTheory.Category.{u, w} C] [CategoryTheory.HasSheafify (CategoryTheory.coherentTopology LightProfinite) C]
Goal: ∀ {X Y : C} (f : X ⟶ Y),
  (LightCondensed.discrete C).map f =
    (CategoryTheory.presheafToSheaf (CategoryTheory.coherentTopology LightProfinite) C).map
      ((CategoryTheory.Functor.const LightProfiniteᵒᵖ).map f)
-- LightCondensed.underlying_obj
@[simps!]
noncomputable def LightCondensed.underlying : LightCondensed.{u} C ⥤ C 
Arguments: (C : Type w) [CategoryTheory.Category.{u, w} C] (j : CategoryTheory.Sheaf (CategoryTheory.coherentTopology LightProfinite) C)
Goal: (LightCondensed.underlying C).obj j = j.val.obj (Opposite.op (LightProfinite.of PUnit.{u + 1}))
-- LightCondensed.underlying_map
@[simps!]
noncomputable def LightCondensed.underlying : LightCondensed.{u} C ⥤ C 
Arguments: (C : Type w) [CategoryTheory.Category.{u, w} C]
Goal: ∀ {X Y : CategoryTheory.Sheaf (CategoryTheory.coherentTopology LightProfinite) C} (f : X ⟶ Y),
  (LightCondensed.underlying C).map f = f.val.app (Opposite.op (LightProfinite.of PUnit.{u + 1}))
-- UpperHalfPlane.ext
@[ext] theorem ext {a b : ℍ} (h : (a : ℂ) = b) : a = b 
Arguments: {a b : ℍ} (h : ↑a = ↑b)
Goal: a = b
-- UpperHalfPlane.ext_iff
protected theorem ext_iff {a b : ℍ} : a = b ↔ (a : ℂ) = b 
Arguments: {a b : ℍ}
Goal: a = b ↔ ↑a = ↑b
-- UpperHalfPlane.ext_iff'
@[simp, norm_cast] theorem ext_iff' {a b : ℍ} : (a : ℂ) = b ↔ a = b 
Arguments: {a b : ℍ}
Goal: ↑a = ↑b ↔ a = b
-- UpperHalfPlane.ext'
theorem ext' {a b : ℍ} (hre : a.re = b.re) (him : a.im = b.im) : a = b 
Arguments: {a b : ℍ} (hre : a.re = b.re) (him : a.im = b.im)
Goal: a = b
-- UpperHalfPlane.coe_im
@[simp]
theorem coe_im (z : ℍ) : (z : ℂ).im = z.im 
Arguments: (z : ℍ)
Goal: (↑z).im = z.im
-- UpperHalfPlane.coe_re
@[simp]
theorem coe_re (z : ℍ) : (z : ℂ).re = z.re 
Arguments: (z : ℍ)
Goal: (↑z).re = z.re
-- UpperHalfPlane.mk_re
@[simp]
theorem mk_re (z : ℂ) (h : 0 < z.im) : (mk z h).re = z.re 
Arguments: (z : ℂ) (h : 0 < z.im)
Goal: (UpperHalfPlane.mk z h).re = z.re
-- UpperHalfPlane.mk_im
@[simp]
theorem mk_im (z : ℂ) (h : 0 < z.im) : (mk z h).im = z.im 
Arguments: (z : ℂ) (h : 0 < z.im)
Goal: (UpperHalfPlane.mk z h).im = z.im
-- UpperHalfPlane.coe_mk
@[simp]
theorem coe_mk (z : ℂ) (h : 0 < z.im) : (mk z h : ℂ) = z 
Arguments: (z : ℂ) (h : 0 < z.im)
Goal: ↑(UpperHalfPlane.mk z h) = z
-- UpperHalfPlane.mk_coe
@[simp]
theorem mk_coe (z : ℍ) (h : 0 < (z : ℂ).im := z.2) : mk z h = z 
Arguments: (z : ℍ) (h : 0 < (↑z).im := ⋯)
Goal: UpperHalfPlane.mk (↑z) h = z
-- UpperHalfPlane.re_add_im
theorem re_add_im (z : ℍ) : (z.re + z.im * Complex.I : ℂ) = z 
Arguments: (z : ℍ)
Goal: ↑z.re + ↑z.im * Complex.I = ↑z
-- UpperHalfPlane.im_pos
theorem im_pos (z : ℍ) : 0 < z.im 
Arguments: (z : ℍ)
Goal: 0 < z.im
-- UpperHalfPlane.im_ne_zero
theorem im_ne_zero (z : ℍ) : z.im ≠ 0 
Arguments: (z : ℍ)
Goal: z.im ≠ 0
-- UpperHalfPlane.ne_zero
theorem ne_zero (z : ℍ) : (z : ℂ) ≠ 0 
Arguments: (z : ℍ)
Goal: ↑z ≠ 0
-- UpperHalfPlane.I_im
@[simp]
lemma I_im : I.im = 1 
Goal: UpperHalfPlane.I.im = 1
-- UpperHalfPlane.I_re
@[simp]
lemma I_re : I.re = 0 
Goal: UpperHalfPlane.I.re = 0
-- UpperHalfPlane.coe_I
@[simp, norm_cast]
lemma coe_I : I = Complex.I 
Goal: ↑UpperHalfPlane.I = Complex.I
-- UpperHalfPlane.normSq_pos
theorem normSq_pos (z : ℍ) : 0 < Complex.normSq (z : ℂ) 
Arguments: (z : ℍ)
Goal: 0 < Complex.normSq ↑z
-- UpperHalfPlane.normSq_ne_zero
theorem normSq_ne_zero (z : ℍ) : Complex.normSq (z : ℂ) ≠ 0 
Arguments: (z : ℍ)
Goal: Complex.normSq ↑z ≠ 0
-- UpperHalfPlane.im_inv_neg_coe_pos
theorem im_inv_neg_coe_pos (z : ℍ) : 0 < (-z : ℂ)⁻¹.im 
Arguments: (z : ℍ)
Goal: 0 < (-↑z)⁻¹.im
-- UpperHalfPlane.linear_ne_zero
theorem linear_ne_zero (cd : Fin 2 → ℝ) (z : ℍ) (h : cd ≠ 0) : (cd 0 : ℂ) * z + cd 1 ≠ 0 
Arguments: (cd : Fin 2 → ℝ) (z : ℍ) (h : cd ≠ 0)
Goal: ↑(cd 0) * ↑z + ↑(cd 1) ≠ 0
-- UpperHalfPlane.denom_ne_zero
theorem denom_ne_zero (g : GL(2, ℝ)⁺) (z : ℍ) : denom g z ≠ 0 
Arguments: (g : ↥GL(2, ℝ)⁺) (z : ℍ)
Goal: UpperHalfPlane.denom g z ≠ 0
-- UpperHalfPlane.normSq_denom_pos
theorem normSq_denom_pos (g : GL(2, ℝ)⁺) (z : ℍ) : 0 < Complex.normSq (denom g z) 
Arguments: (g : ↥GL(2, ℝ)⁺) (z : ℍ)
Goal: 0 < Complex.normSq (UpperHalfPlane.denom g z)
-- UpperHalfPlane.normSq_denom_ne_zero
theorem normSq_denom_ne_zero (g : GL(2, ℝ)⁺) (z : ℍ) : Complex.normSq (denom g z) ≠ 0 
Arguments: (g : ↥GL(2, ℝ)⁺) (z : ℍ)
Goal: Complex.normSq (UpperHalfPlane.denom g z) ≠ 0
-- UpperHalfPlane.smulAux'_im
theorem smulAux'_im (g : GL(2, ℝ)⁺) (z : ℍ) :
    (smulAux' g z).im = det ↑ₘg * z.im / Complex.normSq (denom g z) 
Arguments: (g : ↥GL(2, ℝ)⁺) (z : ℍ)
Goal: (UpperHalfPlane.smulAux' g z).im = (↑↑g).det * z.im / Complex.normSq (UpperHalfPlane.denom g z)
-- UpperHalfPlane.denom_cocycle
theorem denom_cocycle (x y : GL(2, ℝ)⁺) (z : ℍ) :
    denom (x * y) z = denom x (smulAux y z) * denom y z 
Arguments: (x y : ↥GL(2, ℝ)⁺) (z : ℍ)
Goal: UpperHalfPlane.denom (x * y) z = UpperHalfPlane.denom x (UpperHalfPlane.smulAux y z) * UpperHalfPlane.denom y z
-- UpperHalfPlane.mul_smul'
theorem mul_smul' (x y : GL(2, ℝ)⁺) (z : ℍ) : smulAux (x * y) z = smulAux x (smulAux y z) 
Arguments: (x y : ↥GL(2, ℝ)⁺) (z : ℍ)
Goal: UpperHalfPlane.smulAux (x * y) z = UpperHalfPlane.smulAux x (UpperHalfPlane.smulAux y z)
-- UpperHalfPlane.ModularGroup.coe'_apply_complex
@[simp]
theorem coe'_apply_complex {g : SL(2, ℤ)} {i j : Fin 2} :
    (Units.val <| Subtype.val <| coe' g) i j = (Subtype.val g i j : ℂ) 
Arguments: {g : SL(2, ℤ)} {i j : Fin 2}
Goal: ↑(↑↑↑g i j) = ↑(↑g i j)
-- UpperHalfPlane.ModularGroup.det_coe'
@[simp]
theorem det_coe' {g : SL(2, ℤ)} : det (Units.val <| Subtype.val <| coe' g) = 1 
Arguments: {g : SL(2, ℤ)}
Goal: (↑↑↑g).det = 1
-- UpperHalfPlane.ModularGroup.SLOnGLPos_smul_apply
theorem SLOnGLPos_smul_apply (s : SL(2, ℤ)) (g : GL(2, ℝ)⁺) (z : ℍ) :
    (s • g) • z = ((s : GL(2, ℝ)⁺) * g) • z 
Arguments: (s : SL(2, ℤ)) (g : ↥GL(2, ℝ)⁺) (z : ℍ)
Goal: (s • g) • z = (↑s * g) • z
-- UpperHalfPlane.ModularGroup.subgroup_on_glpos_smul_apply
theorem subgroup_on_glpos_smul_apply (s : Γ) (g : GL(2, ℝ)⁺) (z : ℍ) :
    (s • g) • z = ((s : GL(2, ℝ)⁺) * g) • z 
Arguments: (Γ : Subgroup SL(2, ℤ)) (s : ↥Γ) (g : ↥GL(2, ℝ)⁺) (z : ℍ)
Goal: (s • g) • z = (↑↑s * g) • z
-- UpperHalfPlane.ModularGroup.subgroup_on_SL_apply
theorem subgroup_on_SL_apply (s : Γ) (g : SL(2, ℤ)) (z : ℍ) :
    (s • g) • z = ((s : SL(2, ℤ)) * g) • z 
Arguments: (Γ : Subgroup SL(2, ℤ)) (s : ↥Γ) (g : SL(2, ℤ)) (z : ℍ)
Goal: (s • g) • z = (↑s * g) • z
-- UpperHalfPlane.specialLinearGroup_apply
theorem specialLinearGroup_apply {R : Type*} [CommRing R] [Algebra R ℝ] (g : SL(2, R)) (z : ℍ) :
    g • z =
      mk
        (((algebraMap R ℝ (↑ₘ[R] g 0 0) : ℂ) * z + (algebraMap R ℝ (↑ₘ[R] g 0 1) : ℂ)) /
          ((algebraMap R ℝ (↑ₘ[R] g 1 0) : ℂ) * z + (algebraMap R ℝ (↑ₘ[R] g 1 1) : ℂ)))
        (g • z).property 
Arguments: {R : Type u_1} [CommRing R] [Algebra R ℝ] (g : SL(2, R)) (z : ℍ)
Goal: g • z =
  UpperHalfPlane.mk
    ((↑((algebraMap R ℝ) (↑↑g 0 0)) * ↑z + ↑((algebraMap R ℝ) (↑↑g 0 1))) /
      (↑((algebraMap R ℝ) (↑↑g 1 0)) * ↑z + ↑((algebraMap R ℝ) (↑↑g 1 1))))
    ⋯
-- UpperHalfPlane.coe_smul
@[simp]
theorem coe_smul (g : GL(2, ℝ)⁺) (z : ℍ) : ↑(g • z) = num g z / denom g z 
Arguments: (g : ↥GL(2, ℝ)⁺) (z : ℍ)
Goal: ↑(g • z) = UpperHalfPlane.num g z / UpperHalfPlane.denom g z
-- UpperHalfPlane.re_smul
@[simp]
theorem re_smul (g : GL(2, ℝ)⁺) (z : ℍ) : (g • z).re = (num g z / denom g z).re 
Arguments: (g : ↥GL(2, ℝ)⁺) (z : ℍ)
Goal: (g • z).re = (UpperHalfPlane.num g z / UpperHalfPlane.denom g z).re
-- UpperHalfPlane.im_smul
theorem im_smul (g : GL(2, ℝ)⁺) (z : ℍ) : (g • z).im = (num g z / denom g z).im 
Arguments: (g : ↥GL(2, ℝ)⁺) (z : ℍ)
Goal: (g • z).im = (UpperHalfPlane.num g z / UpperHalfPlane.denom g z).im
-- UpperHalfPlane.im_smul_eq_div_normSq
theorem im_smul_eq_div_normSq (g : GL(2, ℝ)⁺) (z : ℍ) :
    (g • z).im = det ↑ₘg * z.im / Complex.normSq (denom g z) 
Arguments: (g : ↥GL(2, ℝ)⁺) (z : ℍ)
Goal: (g • z).im = (↑↑g).det * z.im / Complex.normSq (UpperHalfPlane.denom g z)
-- UpperHalfPlane.c_mul_im_sq_le_normSq_denom
theorem c_mul_im_sq_le_normSq_denom (z : ℍ) (g : SL(2, ℝ)) :
    ((↑ₘg 1 0 : ℝ) * z.im) ^ 2 ≤ Complex.normSq (denom g z) 
Arguments: (z : ℍ) (g : SL(2, ℝ))
Goal: (↑↑(Matrix.SpecialLinearGroup.toGLPos g) 1 0 * z.im) ^ 2 ≤
  Complex.normSq (UpperHalfPlane.denom (Matrix.SpecialLinearGroup.toGLPos g) z)
-- UpperHalfPlane.neg_smul
@[simp]
theorem neg_smul (g : GL(2, ℝ)⁺) (z : ℍ) : -g • z = g • z 
Arguments: (g : ↥GL(2, ℝ)⁺) (z : ℍ)
Goal: -g • z = g • z
-- UpperHalfPlane.ModularGroup.sl_moeb
@[simp]
theorem sl_moeb (A : SL(2, ℤ)) (z : ℍ) : A • z = (A : GL(2, ℝ)⁺) • z 
Arguments: (A : SL(2, ℤ)) (z : ℍ)
Goal: A • z = ↑A • z
-- UpperHalfPlane.ModularGroup.subgroup_moeb
theorem subgroup_moeb (A : Γ) (z : ℍ) : A • z = (A : GL(2, ℝ)⁺) • z 
Arguments: (Γ : Subgroup SL(2, ℤ)) (A : ↥Γ) (z : ℍ)
Goal: A • z = ↑↑A • z
-- UpperHalfPlane.ModularGroup.subgroup_to_sl_moeb
@[simp]
theorem subgroup_to_sl_moeb (A : Γ) (z : ℍ) : A • z = (A : SL(2, ℤ)) • z 
Arguments: (Γ : Subgroup SL(2, ℤ)) (A : ↥Γ) (z : ℍ)
Goal: A • z = ↑A • z
-- UpperHalfPlane.ModularGroup.SL_neg_smul
@[simp high]
theorem SL_neg_smul (g : SL(2, ℤ)) (z : ℍ) : -g • z = g • z 
Arguments: (g : SL(2, ℤ)) (z : ℍ)
Goal: -g • z = g • z
-- UpperHalfPlane.ModularGroup.im_smul_eq_div_normSq
nonrec theorem im_smul_eq_div_normSq :
    (g • z).im = z.im / Complex.normSq (denom g z) 
Arguments: (g : SL(2, ℤ)) (z : ℍ)
Goal: (g • z).im = z.im / Complex.normSq (UpperHalfPlane.denom (↑g) z)
-- UpperHalfPlane.ModularGroup.denom_apply
theorem denom_apply (g : SL(2, ℤ)) (z : ℍ) :
    denom g z = (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 0 * z + (↑g : Matrix (Fin 2) (Fin 2) ℤ) 1 1 
Arguments: (g : SL(2, ℤ)) (z : ℍ)
Goal: UpperHalfPlane.denom (↑g) z = ↑(↑g 1 0) * ↑z + ↑(↑g 1 1)
-- UpperHalfPlane.coe_pos_real_smul
@[simp]
theorem coe_pos_real_smul : ↑(x • z) = (x : ℝ) • (z : ℂ) 
Arguments: (x : { x // 0 < x }) (z : ℍ)
Goal: ↑(x • z) = ↑x • ↑z
-- UpperHalfPlane.pos_real_im
@[simp]
theorem pos_real_im : (x • z).im = x * z.im 
Arguments: (x : { x // 0 < x }) (z : ℍ)
Goal: (x • z).im = ↑x * z.im
-- UpperHalfPlane.pos_real_re
@[simp]
theorem pos_real_re : (x • z).re = x * z.re 
Arguments: (x : { x // 0 < x }) (z : ℍ)
Goal: (x • z).re = ↑x * z.re
-- UpperHalfPlane.coe_vadd
@[simp]
theorem coe_vadd : ↑(x +ᵥ z) = (x + z : ℂ) 
Arguments: (x : ℝ) (z : ℍ)
Goal: ↑(x +ᵥ z) = ↑x + ↑z
-- UpperHalfPlane.vadd_re
@[simp]
theorem vadd_re : (x +ᵥ z).re = x + z.re 
Arguments: (x : ℝ) (z : ℍ)
Goal: (x +ᵥ z).re = x + z.re
-- UpperHalfPlane.vadd_im
@[simp]
theorem vadd_im : (x +ᵥ z).im = z.im 
Arguments: (x : ℝ) (z : ℍ)
Goal: (x +ᵥ z).im = z.im
-- UpperHalfPlane.modular_S_smul
theorem modular_S_smul (z : ℍ) : ModularGroup.S • z = mk (-z : ℂ)⁻¹ z.im_inv_neg_coe_pos 
Arguments: (z : ℍ)
Goal: ModularGroup.S • z = UpperHalfPlane.mk (-↑z)⁻¹ ⋯
-- UpperHalfPlane.modular_T_zpow_smul
theorem modular_T_zpow_smul (z : ℍ) (n : ℤ) : ModularGroup.T ^ n • z = (n : ℝ) +ᵥ z 
Arguments: (z : ℍ) (n : ℤ)
Goal: ModularGroup.T ^ n • z = ↑n +ᵥ z
-- UpperHalfPlane.modular_T_smul
theorem modular_T_smul (z : ℍ) : ModularGroup.T • z = (1 : ℝ) +ᵥ z 
Arguments: (z : ℍ)
Goal: ModularGroup.T • z = 1 +ᵥ z
-- UpperHalfPlane.exists_SL2_smul_eq_of_apply_zero_one_eq_zero
theorem exists_SL2_smul_eq_of_apply_zero_one_eq_zero (g : SL(2, ℝ)) (hc : ↑ₘ[ℝ] g 1 0 = 0) :
    ∃ (u : { x : ℝ // 0 < x }) (v : ℝ), (g • · : ℍ → ℍ) = (v +ᵥ ·) ∘ (u • ·) 
Arguments: (g : SL(2, ℝ)) (hc : ↑↑g 1 0 = 0)
Goal: ∃ u v, (fun x => g • x) = (fun x => v +ᵥ x) ∘ fun x => u • x
-- UpperHalfPlane.exists_SL2_smul_eq_of_apply_zero_one_ne_zero
theorem exists_SL2_smul_eq_of_apply_zero_one_ne_zero (g : SL(2, ℝ)) (hc : ↑ₘ[ℝ] g 1 0 ≠ 0) :
    ∃ (u : { x : ℝ // 0 < x }) (v w : ℝ),
      (g • · : ℍ → ℍ) =
        (w +ᵥ ·) ∘ (ModularGroup.S • · : ℍ → ℍ) ∘ (v +ᵥ · : ℍ → ℍ) ∘ (u • · : ℍ → ℍ) 
Arguments: (g : SL(2, ℝ)) (hc : ↑↑g 1 0 ≠ 0)
Goal: ∃ u v w, (fun x => g • x) = (fun x => w +ᵥ x) ∘ (fun x => ModularGroup.S • x) ∘ (fun x => v +ᵥ x) ∘ fun x => u • x
-- CondensedSet.continuous_coinducingCoprod
lemma continuous_coinducingCoprod {S : CompHaus.{u}} (x : X.val.obj ⟨S⟩) :
    Continuous fun a ↦ (X.coinducingCoprod ⟨⟨S, x⟩, a⟩) 
Arguments: (X : CondensedSet) {S : CompHaus} (x : X.val.obj (Opposite.op S))
Goal: Continuous fun a => CondensedSet.coinducingCoprod X ⟨⟨S, x⟩, a⟩
-- CondensedSet.toTopCatMap_apply
@[simps]
def toTopCatMap : X.toTopCat ⟶ Y.toTopCat 
Arguments: {X Y : CondensedSet} (f : X ⟶ Y)
Goal: ∀ (a : X.val.obj (Opposite.op (CompHaus.of PUnit.{u + 1}))),
  (CondensedSet.toTopCatMap f) a = f.val.app (Opposite.op (CompHaus.of PUnit.{u + 1})) a
-- condensedSetToTopCat_map
@[simps]
def condensedSetToTopCat : CondensedSet.{u} ⥤ TopCat.{u+1} 
Goal: ∀ {X Y : CondensedSet} (f : X ⟶ Y), condensedSetToTopCat.map f = CondensedSet.toTopCatMap f
-- condensedSetToTopCat_obj
@[simps]
def condensedSetToTopCat : CondensedSet.{u} ⥤ TopCat.{u+1} 
Arguments: (X : CondensedSet)
Goal: condensedSetToTopCat.obj X = X.toTopCat
-- CondensedSet.topCatAdjunctionCounit_apply
@[simps]
def topCatAdjunctionCounit (X : TopCat.{u+1}) : X.toCondensedSet.toTopCat ⟶ X 
Arguments: (X : TopCat) (x : ↑X.toCondensedSet.toTopCat)
Goal: (CondensedSet.topCatAdjunctionCounit X) x = x.toFun PUnit.unit
-- CondensedSet.topCatAdjunctionCounit_bijective
lemma topCatAdjunctionCounit_bijective (X : TopCat.{u+1}) :
    Function.Bijective (topCatAdjunctionCounit X) 
Arguments: (X : TopCat)
Goal: Function.Bijective ⇑(CondensedSet.topCatAdjunctionCounit X)
-- CondensedSet.topCatAdjunctionUnit_val_app
@[simps val_app val_app_apply]
def topCatAdjunctionUnit (X : CondensedSet.{u}) : X ⟶ X.toTopCat.toCondensedSet 
Arguments: (X : CondensedSet) (S : CompHausᵒᵖ) (x : X.val.obj S)
Goal: X.topCatAdjunctionUnit.val.app S x =
  { toFun := fun s => X.val.map (CompHaus.const (Opposite.unop S) s).op x, continuous_toFun := ⋯ }
-- CondensedSet.topCatAdjunctionUnit_val_app_apply
@[simps val_app val_app_apply]
def topCatAdjunctionUnit (X : CondensedSet.{u}) : X ⟶ X.toTopCat.toCondensedSet 
Arguments: (X : CondensedSet) (S : CompHausᵒᵖ) (x : X.val.obj S) (s : ↑(compHausToTop.obj (Opposite.unop S)))
Goal: (X.topCatAdjunctionUnit.val.app S x) s = X.val.map (CompHaus.const (Opposite.unop S) s).op x
-- Condensed.epi_iff_locallySurjective_on_compHaus
lemma epi_iff_locallySurjective_on_compHaus : Epi f ↔
    ∀ (S : CompHaus) (y : Y.val.obj ⟨S⟩),
      (∃ (S' : CompHaus) (φ : S' ⟶ S) (_ : Function.Surjective φ) (x : X.val.obj ⟨S'⟩),
        f.val.app ⟨S'⟩ x = Y.val.map ⟨φ⟩ y) 
Arguments: (A : Type u') [CategoryTheory.Category.{v', u'} A] [CategoryTheory.ConcreteCategory A] [CategoryTheory.ConcreteCategory.HasFunctorialSurjectiveInjectiveFactorization A] [(CategoryTheory.coherentTopology CompHaus).WEqualsLocallyBijective A] [CategoryTheory.HasSheafify (CategoryTheory.coherentTopology CompHaus) A] [(CategoryTheory.coherentTopology CompHaus).HasSheafCompose (CategoryTheory.forget A)] [CategoryTheory.Balanced (CategoryTheory.Sheaf (CategoryTheory.coherentTopology CompHaus) A)] [CategoryTheory.Limits.PreservesFiniteProducts (CategoryTheory.forget A)] {X Y : Condensed A} (f : X ⟶ Y)
Goal: CategoryTheory.Epi f ↔
  ∀ (S : CompHaus) (y : (CategoryTheory.forget A).obj (Y.val.obj (Opposite.op S))),
    ∃ S' φ, ∃ (_ : Function.Surjective ⇑φ), ∃ x, (f.val.app (Opposite.op S')) x = (Y.val.map (Opposite.op φ)) y
-- Condensed.epi_iff_surjective_on_stonean
lemma epi_iff_surjective_on_stonean : Epi f ↔
    ∀ (S : Stonean), Function.Surjective (f.val.app (op S.compHaus)) 
Arguments: (A : Type u') [CategoryTheory.Category.{v', u'} A] [CategoryTheory.ConcreteCategory A] [CategoryTheory.ConcreteCategory.HasFunctorialSurjectiveInjectiveFactorization A] [CategoryTheory.Limits.PreservesFiniteProducts (CategoryTheory.forget A)] [∀ (X : CompHausᵒᵖ), CategoryTheory.Limits.HasLimitsOfShape (CategoryTheory.StructuredArrow X Stonean.toCompHaus.op) A] {X Y : Condensed A} (f : X ⟶ Y) [(CategoryTheory.extensiveTopology Stonean).WEqualsLocallyBijective A] [CategoryTheory.HasSheafify (CategoryTheory.extensiveTopology Stonean) A] [(CategoryTheory.extensiveTopology Stonean).HasSheafCompose (CategoryTheory.forget A)] [CategoryTheory.Balanced (CategoryTheory.Sheaf (CategoryTheory.extensiveTopology Stonean) A)]
Goal: CategoryTheory.Epi f ↔ ∀ (S : Stonean), Function.Surjective ⇑(f.val.app (Opposite.op S.compHaus))
-- CondensedSet.epi_iff_locallySurjective_on_compHaus
lemma epi_iff_locallySurjective_on_compHaus : Epi f ↔
    ∀ (S : CompHaus) (y : Y.val.obj ⟨S⟩),
      (∃ (S' : CompHaus) (φ : S' ⟶ S) (_ : Function.Surjective φ) (x : X.val.obj ⟨S'⟩),
        f.val.app ⟨S'⟩ x = Y.val.map ⟨φ⟩ y) 
Arguments: {X Y : CondensedSet} (f : X ⟶ Y)
Goal: CategoryTheory.Epi f ↔
  ∀ (S : CompHaus) (y : Y.val.obj (Opposite.op S)),
    ∃ S' φ, ∃ (_ : Function.Surjective ⇑φ), ∃ x, f.val.app (Opposite.op S') x = Y.val.map (Opposite.op φ) y
-- CondensedSet.epi_iff_surjective_on_stonean
lemma epi_iff_surjective_on_stonean : Epi f ↔
    ∀ (S : Stonean), Function.Surjective (f.val.app (op S.compHaus)) 
Arguments: {X Y : CondensedSet} (f : X ⟶ Y)
Goal: CategoryTheory.Epi f ↔ ∀ (S : Stonean), Function.Surjective (f.val.app (Opposite.op S.compHaus))
-- CondensedMod.epi_iff_locallySurjective_on_compHaus
lemma epi_iff_locallySurjective_on_compHaus : Epi f ↔
    ∀ (S : CompHaus) (y : Y.val.obj ⟨S⟩),
      (∃ (S' : CompHaus) (φ : S' ⟶ S) (_ : Function.Surjective φ) (x : X.val.obj ⟨S'⟩),
        f.val.app ⟨S'⟩ x = Y.val.map ⟨φ⟩ y) 
Arguments: (R : Type (u + 1)) [Ring R] {X Y : CondensedMod R} (f : X ⟶ Y)
Goal: CategoryTheory.Epi f ↔
  ∀ (S : CompHaus) (y : ↑(Y.val.obj (Opposite.op S))),
    ∃ S' φ, ∃ (_ : Function.Surjective ⇑φ), ∃ x, (f.val.app (Opposite.op S')) x = (Y.val.map (Opposite.op φ)) y
-- CondensedMod.epi_iff_surjective_on_stonean
lemma epi_iff_surjective_on_stonean : Epi f ↔
    ∀ (S : Stonean), Function.Surjective (f.val.app (op S.compHaus)) 
Arguments: (R : Type (u + 1)) [Ring R] {X Y : CondensedMod R} (f : X ⟶ Y)
Goal: CategoryTheory.Epi f ↔ ∀ (S : Stonean), Function.Surjective ⇑(f.val.app (Opposite.op S.compHaus))
-- LightCondensed.ofSheafLightProfinite_val
@[simps]
noncomputable def ofSheafLightProfinite (F : LightProfinite.{u}ᵒᵖ ⥤ A) [PreservesFiniteProducts F]
    (hF : EqualizerCondition F) : LightCondensed A 
Arguments: {A : Type u_1} [CategoryTheory.Category.{u_2, u_1} A] (F : LightProfiniteᵒᵖ ⥤ A) [CategoryTheory.Limits.PreservesFiniteProducts F] (hF : CategoryTheory.regularTopology.EqualizerCondition F)
Goal: (LightCondensed.ofSheafLightProfinite F hF).val = F
-- LightCondensed.ofSheafForgetLightProfinite_val
@[simps]
noncomputable def ofSheafForgetLightProfinite
    [ConcreteCategory A] [ReflectsFiniteLimits (CategoryTheory.forget A)]
    (F : LightProfinite.{u}ᵒᵖ ⥤ A) [PreservesFiniteProducts (F ⋙ CategoryTheory.forget A)]
    (hF : EqualizerCondition (F ⋙ CategoryTheory.forget A)) : LightCondensed A 
Arguments: {A : Type u_1} [CategoryTheory.Category.{u_2, u_1} A] [CategoryTheory.ConcreteCategory A] [CategoryTheory.Limits.ReflectsFiniteLimits (CategoryTheory.forget A)] (F : LightProfiniteᵒᵖ ⥤ A) [CategoryTheory.Limits.PreservesFiniteProducts (F ⋙ CategoryTheory.forget A)] (hF : CategoryTheory.regularTopology.EqualizerCondition (F ⋙ CategoryTheory.forget A))
Goal: (LightCondensed.ofSheafForgetLightProfinite F hF).val = F
-- LightCondensed.equalizerCondition
theorem equalizerCondition (X : LightCondensed A) : EqualizerCondition X.val 
Arguments: {A : Type u_1} [CategoryTheory.Category.{u_3, u_1} A] (X : LightCondensed A)
Goal: CategoryTheory.regularTopology.EqualizerCondition X.val
-- TopCat.toLightCondSet_val_obj
@[simps! val_obj val_map]
noncomputable def TopCat.toLightCondSet (X : TopCat.{u}) : LightCondSet.{u} 
Arguments: (X : TopCat)
Goal: ∀ (X_1 : LightProfiniteᵒᵖ), X.toLightCondSet.val.obj X_1 = C(↑(Opposite.unop X_1).toTop, ↑X)
-- TopCat.toLightCondSet_val_map
@[simps! val_obj val_map]
noncomputable def TopCat.toLightCondSet (X : TopCat.{u}) : LightCondSet.{u} 
Arguments: (X : TopCat)
Goal: ∀ {X_1 Y : LightProfiniteᵒᵖ} (f : X_1 ⟶ Y) (g : C(↑(LightProfinite.toTopCat.obj (Opposite.unop X_1)), ↑X)),
  X.toLightCondSet.val.map f g = g.comp f.unop
-- topCatToLightCondSet_obj
@[simps]
noncomputable def topCatToLightCondSet : TopCat.{u} ⥤ LightCondSet.{u} 
Arguments: (X : TopCat)
Goal: topCatToLightCondSet.obj X = X.toLightCondSet
-- topCatToLightCondSet_map_val_app
@[simps]
noncomputable def topCatToLightCondSet : TopCat.{u} ⥤ LightCondSet.{u} 
Goal: ∀ {X Y : TopCat} (f : X ⟶ Y) (x : LightProfiniteᵒᵖ) (g : ((fun X => X.toLightCondSet) X).val.obj x),
  (topCatToLightCondSet.map f).val.app x g = ContinuousMap.comp f g
-- LightCondMod.hom_naturality_apply
@[simp]
lemma hom_naturality_apply {X Y : LightCondMod.{u} R} (f : X ⟶ Y)  {S T : LightProfiniteᵒᵖ}
    (g : S ⟶ T) (x : X.val.obj S) : f.val.app T (X.val.map g x) = Y.val.map g (f.val.app S x) 
Arguments: (R : Type u) [Ring R] {X Y : LightCondMod R} (f : X ⟶ Y) {S T : LightProfiniteᵒᵖ} (g : S ⟶ T) (x : ↑(X.val.obj S))
Goal: (f.val.app T) ((X.val.map g) x) = (Y.val.map g) ((f.val.app S) x)
-- Condensed.discrete_map
@[simps!]
noncomputable def Condensed.discrete : C ⥤ Condensed.{u} C 
Arguments: (C : Type w) [CategoryTheory.Category.{u + 1, w} C] [CategoryTheory.HasWeakSheafify (CategoryTheory.coherentTopology CompHaus) C]
Goal: ∀ {X Y : C} (f : X ⟶ Y),
  (Condensed.discrete C).map f =
    (CategoryTheory.presheafToSheaf (CategoryTheory.coherentTopology CompHaus) C).map
      ((CategoryTheory.Functor.const CompHausᵒᵖ).map f)
-- Condensed.discrete_obj
@[simps!]
noncomputable def Condensed.discrete : C ⥤ Condensed.{u} C 
Arguments: (C : Type w) [CategoryTheory.Category.{u + 1, w} C] [CategoryTheory.HasWeakSheafify (CategoryTheory.coherentTopology CompHaus) C] (X : C)
Goal: (Condensed.discrete C).obj X =
  (CategoryTheory.presheafToSheaf (CategoryTheory.coherentTopology CompHaus) C).obj
    ((CategoryTheory.Functor.const CompHausᵒᵖ).obj X)
-- Condensed.underlying_obj
@[simps!]
noncomputable def Condensed.underlying : Condensed.{u} C ⥤ C 
Arguments: (C : Type w) [CategoryTheory.Category.{u + 1, w} C] (j : CategoryTheory.Sheaf (CategoryTheory.coherentTopology CompHaus) C)
Goal: (Condensed.underlying C).obj j = j.val.obj (Opposite.op (CompHaus.of PUnit.{u + 1}))
-- Condensed.underlying_map
@[simps!]
noncomputable def Condensed.underlying : Condensed.{u} C ⥤ C 
Arguments: (C : Type w) [CategoryTheory.Category.{u + 1, w} C]
Goal: ∀ {X Y : CategoryTheory.Sheaf (CategoryTheory.coherentTopology CompHaus) C} (f : X ⟶ Y),
  (Condensed.underlying C).map f = f.val.app (Opposite.op (CompHaus.of PUnit.{u + 1}))