ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CondensedMod.IsSolid.isIso_solidification_map
class CondensedMod.IsSolid (A : CondensedMod.{u} R) : Prop where
  isIso_solidification_map : ∀ X : Profinite.{u}, IsIso ((yoneda.obj A).map
    ((profiniteSolidification R).app X).op)
Arguments: {R : Type (u + 1)} [Ring R] {A : CondensedMod R} [self : CondensedMod.IsSolid R A] (X : Profinite)
Goal: CategoryTheory.IsIso ((CategoryTheory.yoneda.obj A).map ((Condensed.profiniteSolidification R).app X).op)
-- SSet.leftUnitor_hom_app_apply
@[simp]
lemma leftUnitor_hom_app_apply (K : SSet.{u}) {Δ : SimplexCategoryᵒᵖ} (x : (𝟙_ _ ⊗ K).obj Δ) :
    (λ_ K).hom.app Δ x = x.2 
Arguments: (K : SSet) {Δ : SimplexCategoryᵒᵖ} (x : (𝟙_ SSet ⊗ K).obj Δ)
Goal: (λ_ K).hom.app Δ x = x.2
-- SSet.leftUnitor_inv_app_apply
@[simp]
lemma leftUnitor_inv_app_apply (K : SSet.{u}) {Δ : SimplexCategoryᵒᵖ} (x : K.obj Δ) :
    (λ_ K).inv.app Δ x = ⟨PUnit.unit, x⟩ 
Arguments: (K : SSet) {Δ : SimplexCategoryᵒᵖ} (x : K.obj Δ)
Goal: (λ_ K).inv.app Δ x = (PUnit.unit, x)
-- SSet.rightUnitor_hom_app_apply
@[simp]
lemma rightUnitor_hom_app_apply (K : SSet.{u}) {Δ : SimplexCategoryᵒᵖ} (x : (K ⊗ 𝟙_ _).obj Δ) :
    (ρ_ K).hom.app Δ x = x.1 
Arguments: (K : SSet) {Δ : SimplexCategoryᵒᵖ} (x : (K ⊗ 𝟙_ SSet).obj Δ)
Goal: (ρ_ K).hom.app Δ x = x.1
-- SSet.rightUnitor_inv_app_apply
@[simp]
lemma rightUnitor_inv_app_apply (K : SSet.{u}) {Δ : SimplexCategoryᵒᵖ} (x : K.obj Δ) :
    (ρ_ K).inv.app Δ x = ⟨x, PUnit.unit⟩ 
Arguments: (K : SSet) {Δ : SimplexCategoryᵒᵖ} (x : K.obj Δ)
Goal: (ρ_ K).inv.app Δ x = (x, PUnit.unit)
-- SSet.tensorHom_app_apply
@[simp]
lemma tensorHom_app_apply {K K' L L' : SSet.{u}} (f : K ⟶ K') (g : L ⟶ L')
    {Δ : SimplexCategoryᵒᵖ} (x : (K ⊗ L).obj Δ) :
    (f ⊗ g).app Δ x = ⟨f.app Δ x.1, g.app Δ x.2⟩ 
Arguments: {K K' L L' : SSet} (f : K ⟶ K') (g : L ⟶ L') {Δ : SimplexCategoryᵒᵖ} (x : (K ⊗ L).obj Δ)
Goal: (f ⊗ g).app Δ x = (f.app Δ x.1, g.app Δ x.2)
-- SSet.whiskerLeft_app_apply
@[simp]
lemma whiskerLeft_app_apply (K : SSet.{u}) {L L' : SSet.{u}} (g : L ⟶ L')
    {Δ : SimplexCategoryᵒᵖ} (x : (K ⊗ L).obj Δ) :
    (K ◁ g).app Δ x = ⟨x.1, g.app Δ x.2⟩ 
Arguments: (K : SSet) {L L' : SSet} (g : L ⟶ L') {Δ : SimplexCategoryᵒᵖ} (x : (K ⊗ L).obj Δ)
Goal: (K ◁ g).app Δ x = (x.1, g.app Δ x.2)
-- SSet.whiskerRight_app_apply
@[simp]
lemma whiskerRight_app_apply {K K' : SSet.{u}} (f : K ⟶ K') (L : SSet.{u})
    {Δ : SimplexCategoryᵒᵖ} (x : (K ⊗ L).obj Δ) :
    (f ▷ L).app Δ x = ⟨f.app Δ x.1, x.2⟩ 
Arguments: {K K' : SSet} (f : K ⟶ K') (L : SSet) {Δ : SimplexCategoryᵒᵖ} (x : (K ⊗ L).obj Δ)
Goal: (f ▷ L).app Δ x = (f.app Δ x.1, x.2)
-- SSet.associator_hom_app_apply
@[simp]
lemma associator_hom_app_apply (K L M : SSet.{u}) {Δ : SimplexCategoryᵒᵖ}
    (x : ((K ⊗ L) ⊗ M).obj Δ) :
    (α_ K L M).hom.app Δ x = ⟨x.1.1, x.1.2, x.2⟩ 
Arguments: (K L M : SSet) {Δ : SimplexCategoryᵒᵖ} (x : ((K ⊗ L) ⊗ M).obj Δ)
Goal: (α_ K L M).hom.app Δ x = (x.1.1, x.1.2, x.2)
-- SSet.associator_inv_app_apply
@[simp]
lemma associator_inv_app_apply (K L M : SSet.{u}) {Δ : SimplexCategoryᵒᵖ}
    (x : (K ⊗ L ⊗ M).obj Δ) :
    (α_ K L M).inv.app Δ x = ⟨⟨x.1, x.2.1⟩, x.2.2⟩ 
Arguments: (K L M : SSet) {Δ : SimplexCategoryᵒᵖ} (x : (K ⊗ L ⊗ M).obj Δ)
Goal: (α_ K L M).inv.app Δ x = ((x.1, x.2.1), x.2.2)
-- LightCondSet.continuous_coinducingCoprod
lemma continuous_coinducingCoprod {S : LightProfinite.{u}} (x : X.val.obj ⟨S⟩) :
    Continuous fun a ↦ (X.coinducingCoprod ⟨⟨S, x⟩, a⟩) 
Arguments: (X : LightCondSet) {S : LightProfinite} (x : X.val.obj (Opposite.op S))
Goal: Continuous fun a => LightCondSet.coinducingCoprod X ⟨⟨S, x⟩, a⟩
-- LightCondSet.toTopCatMap_apply
@[simps]
def toTopCatMap : X.toTopCat ⟶ Y.toTopCat 
Arguments: {X Y : LightCondSet} (f : X ⟶ Y)
Goal: ∀ (a : X.val.obj (Opposite.op (LightProfinite.of PUnit.{u + 1}))),
  (LightCondSet.toTopCatMap f) a = f.val.app (Opposite.op (LightProfinite.of PUnit.{u + 1})) a
-- lightCondSetToTopCat_map
@[simps]
def _root_.lightCondSetToTopCat : LightCondSet.{u} ⥤ TopCat.{u} 
Goal: ∀ {X Y : LightCondSet} (f : X ⟶ Y), lightCondSetToTopCat.map f = LightCondSet.toTopCatMap f
-- lightCondSetToTopCat_obj
@[simps]
def _root_.lightCondSetToTopCat : LightCondSet.{u} ⥤ TopCat.{u} 
Arguments: (X : LightCondSet)
Goal: lightCondSetToTopCat.obj X = X.toTopCat
-- LightCondSet.topCatAdjunctionCounit_bijective
lemma topCatAdjunctionCounit_bijective (X : TopCat.{u}) :
    Function.Bijective (topCatAdjunctionCounit X) 
Arguments: (X : TopCat)
Goal: Function.Bijective ⇑(LightCondSet.topCatAdjunctionCounit X)
-- LightCondSet.topCatAdjunctionUnit_val_app_apply
@[simps val_app val_app_apply]
def topCatAdjunctionUnit (X : LightCondSet.{u}) : X ⟶ X.toTopCat.toLightCondSet 
Arguments: (X : LightCondSet) (S : LightProfiniteᵒᵖ) (x : X.val.obj S) (s : ↑(LightProfinite.toTopCat.obj (Opposite.unop S)))
Goal: (X.topCatAdjunctionUnit.val.app S x) s = X.val.map (LightProfinite.const (Opposite.unop S) s).op x
-- LightCondSet.topCatAdjunctionUnit_val_app
@[simps val_app val_app_apply]
def topCatAdjunctionUnit (X : LightCondSet.{u}) : X ⟶ X.toTopCat.toLightCondSet 
Arguments: (X : LightCondSet) (S : LightProfiniteᵒᵖ) (x : X.val.obj S)
Goal: X.topCatAdjunctionUnit.val.app S x =
  { toFun := fun s => X.val.map (LightProfinite.const (Opposite.unop S) s).op x, continuous_toFun := ⋯ }
-- Complex.norm_max_aux₁
theorem norm_max_aux₁ [CompleteSpace F] {f : ℂ → F} {z w : ℂ}
    (hd : DiffContOnCl ℂ f (ball z (dist w z)))
    (hz : IsMaxOn (norm ∘ f) (closedBall z (dist w z)) z) : ‖f w‖ = ‖f z‖ 
Arguments: {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] [CompleteSpace F] {f : ℂ → F} {z w : ℂ} (hd : DiffContOnCl ℂ f (Metric.ball z (dist w z))) (hz : IsMaxOn (norm ∘ f) (Metric.closedBall z (dist w z)) z)
Goal: ‖f w‖ = ‖f z‖
-- Complex.norm_max_aux₂
theorem norm_max_aux₂ {f : ℂ → F} {z w : ℂ} (hd : DiffContOnCl ℂ f (ball z (dist w z)))
    (hz : IsMaxOn (norm ∘ f) (closedBall z (dist w z)) z) : ‖f w‖ = ‖f z‖ 
Arguments: {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] {f : ℂ → F} {z w : ℂ} (hd : DiffContOnCl ℂ f (Metric.ball z (dist w z))) (hz : IsMaxOn (norm ∘ f) (Metric.closedBall z (dist w z)) z)
Goal: ‖f w‖ = ‖f z‖
-- Complex.norm_max_aux₃
theorem norm_max_aux₃ {f : ℂ → F} {z w : ℂ} {r : ℝ} (hr : dist w z = r)
    (hd : DiffContOnCl ℂ f (ball z r)) (hz : IsMaxOn (norm ∘ f) (ball z r) z) : ‖f w‖ = ‖f z‖ 
Arguments: {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] {f : ℂ → F} {z w : ℂ} {r : ℝ} (hr : dist w z = r) (hd : DiffContOnCl ℂ f (Metric.ball z r)) (hz : IsMaxOn (norm ∘ f) (Metric.ball z r) z)
Goal: ‖f w‖ = ‖f z‖
-- Complex.norm_eqOn_closedBall_of_isMaxOn
theorem norm_eqOn_closedBall_of_isMaxOn {f : E → F} {z : E} {r : ℝ}
    (hd : DiffContOnCl ℂ f (ball z r)) (hz : IsMaxOn (norm ∘ f) (ball z r) z) :
    EqOn (norm ∘ f) (const E ‖f z‖) (closedBall z r) 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] {f : E → F} {z : E} {r : ℝ} (hd : DiffContOnCl ℂ f (Metric.ball z r)) (hz : IsMaxOn (norm ∘ f) (Metric.ball z r) z)
Goal: Set.EqOn (norm ∘ f) (Function.const E ‖f z‖) (Metric.closedBall z r)
-- Complex.norm_eq_norm_of_isMaxOn_of_ball_subset
theorem norm_eq_norm_of_isMaxOn_of_ball_subset {f : E → F} {s : Set E} {z w : E}
    (hd : DiffContOnCl ℂ f s) (hz : IsMaxOn (norm ∘ f) s z) (hsub : ball z (dist w z) ⊆ s) :
    ‖f w‖ = ‖f z‖ 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] {f : E → F} {s : Set E} {z w : E} (hd : DiffContOnCl ℂ f s) (hz : IsMaxOn (norm ∘ f) s z) (hsub : Metric.ball z (dist w z) ⊆ s)
Goal: ‖f w‖ = ‖f z‖
-- Complex.norm_eventually_eq_of_isLocalMax
theorem norm_eventually_eq_of_isLocalMax {f : E → F} {c : E}
    (hd : ∀ᶠ z in 𝓝 c, DifferentiableAt ℂ f z) (hc : IsLocalMax (norm ∘ f) c) :
    ∀ᶠ y in 𝓝 c, ‖f y‖ = ‖f c‖ 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] {f : E → F} {c : E} (hd : ∀ᶠ (z : E) in 𝓝 c, DifferentiableAt ℂ f z) (hc : IsLocalMax (norm ∘ f) c)
Goal: ∀ᶠ (y : E) in 𝓝 c, ‖f y‖ = ‖f c‖
-- Complex.isOpen_setOf_mem_nhds_and_isMaxOn_norm
theorem isOpen_setOf_mem_nhds_and_isMaxOn_norm {f : E → F} {s : Set E}
    (hd : DifferentiableOn ℂ f s) : IsOpen {z | s ∈ 𝓝 z ∧ IsMaxOn (norm ∘ f) s z} 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] {f : E → F} {s : Set E} (hd : DifferentiableOn ℂ f s)
Goal: IsOpen {z | s ∈ 𝓝 z ∧ IsMaxOn (norm ∘ f) s z}
-- Complex.norm_eqOn_of_isPreconnected_of_isMaxOn
theorem norm_eqOn_of_isPreconnected_of_isMaxOn {f : E → F} {U : Set E} {c : E}
    (hc : IsPreconnected U) (ho : IsOpen U) (hd : DifferentiableOn ℂ f U) (hcU : c ∈ U)
    (hm : IsMaxOn (norm ∘ f) U c) : EqOn (norm ∘ f) (const E ‖f c‖) U 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] {f : E → F} {U : Set E} {c : E} (hc : IsPreconnected U) (ho : IsOpen U) (hd : DifferentiableOn ℂ f U) (hcU : c ∈ U) (hm : IsMaxOn (norm ∘ f) U c)
Goal: Set.EqOn (norm ∘ f) (Function.const E ‖f c‖) U
-- Complex.norm_eqOn_closure_of_isPreconnected_of_isMaxOn
theorem norm_eqOn_closure_of_isPreconnected_of_isMaxOn {f : E → F} {U : Set E} {c : E}
    (hc : IsPreconnected U) (ho : IsOpen U) (hd : DiffContOnCl ℂ f U) (hcU : c ∈ U)
    (hm : IsMaxOn (norm ∘ f) U c) : EqOn (norm ∘ f) (const E ‖f c‖) (closure U) 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] {f : E → F} {U : Set E} {c : E} (hc : IsPreconnected U) (ho : IsOpen U) (hd : DiffContOnCl ℂ f U) (hcU : c ∈ U) (hm : IsMaxOn (norm ∘ f) U c)
Goal: Set.EqOn (norm ∘ f) (Function.const E ‖f c‖) (closure U)
-- Complex.eqOn_of_isPreconnected_of_isMaxOn_norm
theorem eqOn_of_isPreconnected_of_isMaxOn_norm {f : E → F} {U : Set E} {c : E}
    (hc : IsPreconnected U) (ho : IsOpen U) (hd : DifferentiableOn ℂ f U) (hcU : c ∈ U)
    (hm : IsMaxOn (norm ∘ f) U c) : EqOn f (const E (f c)) U 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] [StrictConvexSpace ℝ F] {f : E → F} {U : Set E} {c : E} (hc : IsPreconnected U) (ho : IsOpen U) (hd : DifferentiableOn ℂ f U) (hcU : c ∈ U) (hm : IsMaxOn (norm ∘ f) U c)
Goal: Set.EqOn f (Function.const E (f c)) U
-- Complex.eqOn_closure_of_isPreconnected_of_isMaxOn_norm
theorem eqOn_closure_of_isPreconnected_of_isMaxOn_norm {f : E → F} {U : Set E} {c : E}
    (hc : IsPreconnected U) (ho : IsOpen U) (hd : DiffContOnCl ℂ f U) (hcU : c ∈ U)
    (hm : IsMaxOn (norm ∘ f) U c) : EqOn f (const E (f c)) (closure U) 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] [StrictConvexSpace ℝ F] {f : E → F} {U : Set E} {c : E} (hc : IsPreconnected U) (ho : IsOpen U) (hd : DiffContOnCl ℂ f U) (hcU : c ∈ U) (hm : IsMaxOn (norm ∘ f) U c)
Goal: Set.EqOn f (Function.const E (f c)) (closure U)
-- Complex.eq_of_isMaxOn_of_ball_subset
theorem eq_of_isMaxOn_of_ball_subset {f : E → F} {s : Set E} {z w : E} (hd : DiffContOnCl ℂ f s)
    (hz : IsMaxOn (norm ∘ f) s z) (hsub : ball z (dist w z) ⊆ s) : f w = f z 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] [StrictConvexSpace ℝ F] {f : E → F} {s : Set E} {z w : E} (hd : DiffContOnCl ℂ f s) (hz : IsMaxOn (norm ∘ f) s z) (hsub : Metric.ball z (dist w z) ⊆ s)
Goal: f w = f z
-- Complex.eqOn_closedBall_of_isMaxOn_norm
theorem eqOn_closedBall_of_isMaxOn_norm {f : E → F} {z : E} {r : ℝ}
    (hd : DiffContOnCl ℂ f (ball z r)) (hz : IsMaxOn (norm ∘ f) (ball z r) z) :
    EqOn f (const E (f z)) (closedBall z r) 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] [StrictConvexSpace ℝ F] {f : E → F} {z : E} {r : ℝ} (hd : DiffContOnCl ℂ f (Metric.ball z r)) (hz : IsMaxOn (norm ∘ f) (Metric.ball z r) z)
Goal: Set.EqOn f (Function.const E (f z)) (Metric.closedBall z r)
-- Complex.eventually_eq_of_isLocalMax_norm
theorem eventually_eq_of_isLocalMax_norm {f : E → F} {c : E}
    (hd : ∀ᶠ z in 𝓝 c, DifferentiableAt ℂ f z) (hc : IsLocalMax (norm ∘ f) c) :
    ∀ᶠ y in 𝓝 c, f y = f c 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] [StrictConvexSpace ℝ F] {f : E → F} {c : E} (hd : ∀ᶠ (z : E) in 𝓝 c, DifferentiableAt ℂ f z) (hc : IsLocalMax (norm ∘ f) c)
Goal: ∀ᶠ (y : E) in 𝓝 c, f y = f c
-- Complex.eventually_eq_or_eq_zero_of_isLocalMin_norm
theorem eventually_eq_or_eq_zero_of_isLocalMin_norm {f : E → ℂ} {c : E}
    (hf : ∀ᶠ z in 𝓝 c, DifferentiableAt ℂ f z) (hc : IsLocalMin (norm ∘ f) c) :
    (∀ᶠ z in 𝓝 c, f z = f c) ∨ f c = 0 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {c : E} (hf : ∀ᶠ (z : E) in 𝓝 c, DifferentiableAt ℂ f z) (hc : IsLocalMin (norm ∘ f) c)
Goal: (∀ᶠ (z : E) in 𝓝 c, f z = f c) ∨ f c = 0
-- Complex.exists_mem_frontier_isMaxOn_norm
theorem exists_mem_frontier_isMaxOn_norm [FiniteDimensional ℂ E] {f : E → F} {U : Set E}
    (hb : IsBounded U) (hne : U.Nonempty) (hd : DiffContOnCl ℂ f U) :
    ∃ z ∈ frontier U, IsMaxOn (norm ∘ f) (closure U) z 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] [Nontrivial E] [FiniteDimensional ℂ E] {f : E → F} {U : Set E} (hb : Bornology.IsBounded U) (hne : U.Nonempty) (hd : DiffContOnCl ℂ f U)
Goal: ∃ z ∈ frontier U, IsMaxOn (norm ∘ f) (closure U) z
-- Complex.norm_le_of_forall_mem_frontier_norm_le
theorem norm_le_of_forall_mem_frontier_norm_le {f : E → F} {U : Set E} (hU : IsBounded U)
    (hd : DiffContOnCl ℂ f U) {C : ℝ} (hC : ∀ z ∈ frontier U, ‖f z‖ ≤ C) {z : E}
    (hz : z ∈ closure U) : ‖f z‖ ≤ C 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] [Nontrivial E] {f : E → F} {U : Set E} (hU : Bornology.IsBounded U) (hd : DiffContOnCl ℂ f U) {C : ℝ} (hC : ∀ z ∈ frontier U, ‖f z‖ ≤ C) {z : E} (hz : z ∈ closure U)
Goal: ‖f z‖ ≤ C
-- Complex.eqOn_closure_of_eqOn_frontier
theorem eqOn_closure_of_eqOn_frontier {f g : E → F} {U : Set E} (hU : IsBounded U)
    (hf : DiffContOnCl ℂ f U) (hg : DiffContOnCl ℂ g U) (hfg : EqOn f g (frontier U)) :
    EqOn f g (closure U) 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] [Nontrivial E] {f g : E → F} {U : Set E} (hU : Bornology.IsBounded U) (hf : DiffContOnCl ℂ f U) (hg : DiffContOnCl ℂ g U) (hfg : Set.EqOn f g (frontier U))
Goal: Set.EqOn f g (closure U)
-- Complex.eqOn_of_eqOn_frontier
theorem eqOn_of_eqOn_frontier {f g : E → F} {U : Set E} (hU : IsBounded U) (hf : DiffContOnCl ℂ f U)
    (hg : DiffContOnCl ℂ g U) (hfg : EqOn f g (frontier U)) : EqOn f g U 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type v} [NormedAddCommGroup F] [NormedSpace ℂ F] [Nontrivial E] {f g : E → F} {U : Set E} (hU : Bornology.IsBounded U) (hf : DiffContOnCl ℂ f U) (hg : DiffContOnCl ℂ g U) (hfg : Set.EqOn f g (frontier U))
Goal: Set.EqOn f g U
-- CondensedMod.hom_naturality_apply
@[simp]
lemma hom_naturality_apply {X Y : CondensedMod.{u} R} (f : X ⟶ Y)  {S T : CompHausᵒᵖ} (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 + 1)) [Ring R] {X Y : CondensedMod R} (f : X ⟶ Y) {S T : CompHausᵒᵖ} (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)
-- HasFPowerSeriesAt.hasStrictFDerivAt
theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
    HasStrictFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p 1)) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} {f : E → F} {x : E} (h : HasFPowerSeriesAt f p x)
Goal: HasStrictFDerivAt f ((continuousMultilinearCurryFin1 𝕜 E F) (p 1)) x
-- HasFPowerSeriesAt.hasFDerivAt
theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
    HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p 1)) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} {f : E → F} {x : E} (h : HasFPowerSeriesAt f p x)
Goal: HasFDerivAt f ((continuousMultilinearCurryFin1 𝕜 E F) (p 1)) x
-- HasFPowerSeriesAt.differentiableAt
theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt 𝕜 f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} {f : E → F} {x : E} (h : HasFPowerSeriesAt f p x)
Goal: DifferentiableAt 𝕜 f x
-- AnalyticAt.differentiableAt
theorem AnalyticAt.differentiableAt : AnalyticAt 𝕜 f x → DifferentiableAt 𝕜 f x
  
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E}
Goal: AnalyticAt 𝕜 f x → DifferentiableAt 𝕜 f x
-- AnalyticAt.differentiableWithinAt
theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt 𝕜 f x) : DifferentiableWithinAt 𝕜 f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} {s : Set E} (h : AnalyticAt 𝕜 f x)
Goal: DifferentiableWithinAt 𝕜 f s x
-- HasFPowerSeriesAt.fderiv_eq
theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
    fderiv 𝕜 f x = continuousMultilinearCurryFin1 𝕜 E F (p 1) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} {f : E → F} {x : E} (h : HasFPowerSeriesAt f p x)
Goal: fderiv 𝕜 f x = (continuousMultilinearCurryFin1 𝕜 E F) (p 1)
-- HasFPowerSeriesOnBall.differentiableOn
theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
    (h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn 𝕜 f (EMetric.ball x r) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} {r : ℝ≥0∞} {f : E → F} {x : E} [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
Goal: DifferentiableOn 𝕜 f (EMetric.ball x r)
-- AnalyticOn.differentiableOn
theorem AnalyticOn.differentiableOn (h : AnalyticOn 𝕜 f s) : DifferentiableOn 𝕜 f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} (h : AnalyticOn 𝕜 f s)
Goal: DifferentiableOn 𝕜 f s
-- HasFPowerSeriesOnBall.hasFDerivAt
theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
    {y : E} (hy : (‖y‖₊ : ℝ≥0∞) < r) :
    HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p.changeOrigin y 1)) (x + y) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} {r : ℝ≥0∞} {f : E → F} {x : E} [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) {y : E} (hy : ↑‖y‖₊ < r)
Goal: HasFDerivAt f ((continuousMultilinearCurryFin1 𝕜 E F) (p.changeOrigin y 1)) (x + y)
-- HasFPowerSeriesOnBall.fderiv_eq
theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
    {y : E} (hy : (‖y‖₊ : ℝ≥0∞) < r) :
    fderiv 𝕜 f (x + y) = continuousMultilinearCurryFin1 𝕜 E F (p.changeOrigin y 1) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} {r : ℝ≥0∞} {f : E → F} {x : E} [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) {y : E} (hy : ↑‖y‖₊ < r)
Goal: fderiv 𝕜 f (x + y) = (continuousMultilinearCurryFin1 𝕜 E F) (p.changeOrigin y 1)
-- HasFPowerSeriesOnBall.fderiv
theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
    HasFPowerSeriesOnBall (fderiv 𝕜 f) p.derivSeries x r 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} {r : ℝ≥0∞} {f : E → F} {x : E} [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
Goal: HasFPowerSeriesOnBall (fderiv 𝕜 f) p.derivSeries x r
-- AnalyticOn.fderiv
theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) :
    AnalyticOn 𝕜 (fderiv 𝕜 f) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} [CompleteSpace F] (h : AnalyticOn 𝕜 f s)
Goal: AnalyticOn 𝕜 (fderiv 𝕜 f) s
-- AnalyticOn.iteratedFDeriv
theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
    AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ)
Goal: AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s
-- AnalyticOn.contDiffOn
theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
    ContDiffOn 𝕜 n f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞}
Goal: ContDiffOn 𝕜 n f s
-- AnalyticAt.contDiffAt
theorem AnalyticAt.contDiffAt [CompleteSpace F] (h : AnalyticAt 𝕜 f x) {n : ℕ∞} :
    ContDiffAt 𝕜 n f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} [CompleteSpace F] (h : AnalyticAt 𝕜 f x) {n : ℕ∞}
Goal: ContDiffAt 𝕜 n f x
-- HasFPowerSeriesAt.hasStrictDerivAt
protected theorem HasFPowerSeriesAt.hasStrictDerivAt (h : HasFPowerSeriesAt f p x) :
    HasStrictDerivAt f (p 1 fun _ => 1) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : FormalMultilinearSeries 𝕜 𝕜 F} {f : 𝕜 → F} {x : 𝕜} (h : HasFPowerSeriesAt f p x)
Goal: HasStrictDerivAt f ((p 1) fun x => 1) x
-- HasFPowerSeriesAt.hasDerivAt
protected theorem HasFPowerSeriesAt.hasDerivAt (h : HasFPowerSeriesAt f p x) :
    HasDerivAt f (p 1 fun _ => 1) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : FormalMultilinearSeries 𝕜 𝕜 F} {f : 𝕜 → F} {x : 𝕜} (h : HasFPowerSeriesAt f p x)
Goal: HasDerivAt f ((p 1) fun x => 1) x
-- HasFPowerSeriesAt.deriv
protected theorem HasFPowerSeriesAt.deriv (h : HasFPowerSeriesAt f p x) :
    deriv f x = p 1 fun _ => 1 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : FormalMultilinearSeries 𝕜 𝕜 F} {f : 𝕜 → F} {x : 𝕜} (h : HasFPowerSeriesAt f p x)
Goal: deriv f x = (p 1) fun x => 1
-- AnalyticOn.deriv
theorem AnalyticOn.deriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) : AnalyticOn 𝕜 (deriv f) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {s : Set 𝕜} [CompleteSpace F] (h : AnalyticOn 𝕜 f s)
Goal: AnalyticOn 𝕜 (deriv f) s
-- AnalyticOn.iterated_deriv
theorem AnalyticOn.iterated_deriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
    AnalyticOn 𝕜 (_root_.deriv^[n] f) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {s : Set 𝕜} [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ)
Goal: AnalyticOn 𝕜 (deriv^[n] f) s