ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- PartialHomeomorph.piecewise_toPartialEquiv
@[simps! (config := .asFn) toPartialEquiv apply]
def piecewise (e e' : PartialHomeomorph X Y) (s : Set X) (t : Set Y) [∀ x, Decidable (x ∈ s)]
    [∀ y, Decidable (y ∈ t)] (H : e.IsImage s t) (H' : e'.IsImage s t)
    (Hs : e.source ∩ frontier s = e'.source ∩ frontier s)
    (Heq : EqOn e e' (e.source ∩ frontier s)) : PartialHomeomorph X Y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e e' : PartialHomeomorph X Y) (s : Set X) (t : Set Y) [(x : X) → Decidable (x ∈ s)] [(y : Y) → Decidable (y ∈ t)] (H : e.IsImage s t) (H' : e'.IsImage s t) (Hs : e.source ∩ frontier s = e'.source ∩ frontier s) (Heq : Set.EqOn (↑e) (↑e') (e.source ∩ frontier s))
Goal: (e.piecewise e' s t H H' Hs Heq).toPartialEquiv = e.piecewise e'.toPartialEquiv s t H H'
-- PartialHomeomorph.piecewise_apply
@[simps! (config := .asFn) toPartialEquiv apply]
def piecewise (e e' : PartialHomeomorph X Y) (s : Set X) (t : Set Y) [∀ x, Decidable (x ∈ s)]
    [∀ y, Decidable (y ∈ t)] (H : e.IsImage s t) (H' : e'.IsImage s t)
    (Hs : e.source ∩ frontier s = e'.source ∩ frontier s)
    (Heq : EqOn e e' (e.source ∩ frontier s)) : PartialHomeomorph X Y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e e' : PartialHomeomorph X Y) (s : Set X) (t : Set Y) [(x : X) → Decidable (x ∈ s)] [(y : Y) → Decidable (y ∈ t)] (H : e.IsImage s t) (H' : e'.IsImage s t) (Hs : e.source ∩ frontier s = e'.source ∩ frontier s) (Heq : Set.EqOn (↑e) (↑e') (e.source ∩ frontier s))
Goal: ↑(e.piecewise e' s t H H' Hs Heq) = s.piecewise ↑e ↑e'
-- PartialHomeomorph.symm_piecewise
@[simp]
theorem symm_piecewise (e e' : PartialHomeomorph X Y) {s : Set X} {t : Set Y}
    [∀ x, Decidable (x ∈ s)] [∀ y, Decidable (y ∈ t)] (H : e.IsImage s t) (H' : e'.IsImage s t)
    (Hs : e.source ∩ frontier s = e'.source ∩ frontier s)
    (Heq : EqOn e e' (e.source ∩ frontier s)) :
    (e.piecewise e' s t H H' Hs Heq).symm =
      e.symm.piecewise e'.symm t s H.symm H'.symm
        (H.frontier.inter_eq_of_inter_eq_of_eqOn H'.frontier Hs Heq)
        (H.frontier.symm_eqOn_of_inter_eq_of_eqOn Hs Heq) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e e' : PartialHomeomorph X Y) {s : Set X} {t : Set Y} [(x : X) → Decidable (x ∈ s)] [(y : Y) → Decidable (y ∈ t)] (H : e.IsImage s t) (H' : e'.IsImage s t) (Hs : e.source ∩ frontier s = e'.source ∩ frontier s) (Heq : Set.EqOn (↑e) (↑e') (e.source ∩ frontier s))
Goal: (e.piecewise e' s t H H' Hs Heq).symm = e.symm.piecewise e'.symm t s ⋯ ⋯ ⋯ ⋯
-- PartialHomeomorph.continuousWithinAt_iff_continuousWithinAt_comp_right
theorem continuousWithinAt_iff_continuousWithinAt_comp_right {f : Y → Z} {s : Set Y} {x : Y}
    (h : x ∈ e.target) :
    ContinuousWithinAt f s x ↔ ContinuousWithinAt (f ∘ e) (e ⁻¹' s) (e.symm x) 
Arguments: {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (e : PartialHomeomorph X Y) {f : Y → Z} {s : Set Y} {x : Y} (h : x ∈ e.target)
Goal: ContinuousWithinAt f s x ↔ ContinuousWithinAt (f ∘ ↑e) (↑e ⁻¹' s) (↑e.symm x)
-- PartialHomeomorph.continuousAt_iff_continuousAt_comp_right
theorem continuousAt_iff_continuousAt_comp_right {f : Y → Z} {x : Y} (h : x ∈ e.target) :
    ContinuousAt f x ↔ ContinuousAt (f ∘ e) (e.symm x) 
Arguments: {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (e : PartialHomeomorph X Y) {f : Y → Z} {x : Y} (h : x ∈ e.target)
Goal: ContinuousAt f x ↔ ContinuousAt (f ∘ ↑e) (↑e.symm x)
-- PartialHomeomorph.continuousOn_iff_continuousOn_comp_right
theorem continuousOn_iff_continuousOn_comp_right {f : Y → Z} {s : Set Y} (h : s ⊆ e.target) :
    ContinuousOn f s ↔ ContinuousOn (f ∘ e) (e.source ∩ e ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (e : PartialHomeomorph X Y) {f : Y → Z} {s : Set Y} (h : s ⊆ e.target)
Goal: ContinuousOn f s ↔ ContinuousOn (f ∘ ↑e) (e.source ∩ ↑e ⁻¹' s)
-- PartialHomeomorph.continuousWithinAt_iff_continuousWithinAt_comp_left
theorem continuousWithinAt_iff_continuousWithinAt_comp_left {f : Z → X} {s : Set Z} {x : Z}
    (hx : f x ∈ e.source) (h : f ⁻¹' e.source ∈ 𝓝[s] x) :
    ContinuousWithinAt f s x ↔ ContinuousWithinAt (e ∘ f) s x 
Arguments: {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (e : PartialHomeomorph X Y) {f : Z → X} {s : Set Z} {x : Z} (hx : f x ∈ e.source) (h : f ⁻¹' e.source ∈ 𝓝[s] x)
Goal: ContinuousWithinAt f s x ↔ ContinuousWithinAt (↑e ∘ f) s x
-- PartialHomeomorph.continuousAt_iff_continuousAt_comp_left
theorem continuousAt_iff_continuousAt_comp_left {f : Z → X} {x : Z} (h : f ⁻¹' e.source ∈ 𝓝 x) :
    ContinuousAt f x ↔ ContinuousAt (e ∘ f) x 
Arguments: {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (e : PartialHomeomorph X Y) {f : Z → X} {x : Z} (h : f ⁻¹' e.source ∈ 𝓝 x)
Goal: ContinuousAt f x ↔ ContinuousAt (↑e ∘ f) x
-- PartialHomeomorph.continuousOn_iff_continuousOn_comp_left
theorem continuousOn_iff_continuousOn_comp_left {f : Z → X} {s : Set Z} (h : s ⊆ f ⁻¹' e.source) :
    ContinuousOn f s ↔ ContinuousOn (e ∘ f) s 
Arguments: {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (e : PartialHomeomorph X Y) {f : Z → X} {s : Set Z} (h : s ⊆ f ⁻¹' e.source)
Goal: ContinuousOn f s ↔ ContinuousOn (↑e ∘ f) s
-- PartialHomeomorph.continuous_iff_continuous_comp_left
theorem continuous_iff_continuous_comp_left {f : Z → X} (h : f ⁻¹' e.source = univ) :
    Continuous f ↔ Continuous (e ∘ f) 
Arguments: {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (e : PartialHomeomorph X Y) {f : Z → X} (h : f ⁻¹' e.source = Set.univ)
Goal: Continuous f ↔ Continuous (↑e ∘ f)
-- PartialHomeomorph.homeomorphOfImageSubsetSource_symm_apply
@[simps]
def homeomorphOfImageSubsetSource {s : Set X} {t : Set Y} (hs : s ⊆ e.source) (ht : e '' s = t) :
    s ≃ₜ t 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {s : Set X} {t : Set Y} (hs : s ⊆ e.source) (ht : ↑e '' s = t)
Goal: ∀ (a : ↑t), (e.homeomorphOfImageSubsetSource hs ht).symm a = Set.MapsTo.restrict (↑e.symm) t s ⋯ a
-- PartialHomeomorph.homeomorphOfImageSubsetSource_apply
@[simps]
def homeomorphOfImageSubsetSource {s : Set X} {t : Set Y} (hs : s ⊆ e.source) (ht : e '' s = t) :
    s ≃ₜ t 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {s : Set X} {t : Set Y} (hs : s ⊆ e.source) (ht : ↑e '' s = t)
Goal: ∀ (a : ↑s), (e.homeomorphOfImageSubsetSource hs ht) a = Set.MapsTo.restrict (↑e) s t ⋯ a
-- PartialHomeomorph.toHomeomorphSourceTarget_symm_apply_coe
@[simps!] -- Porting note: new `simps`
def toHomeomorphSourceTarget : e.source ≃ₜ e.target 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: ∀ (a : ↑e.target), ↑(e.toHomeomorphSourceTarget.symm a) = ↑e.symm ↑a
-- PartialHomeomorph.toHomeomorphSourceTarget_apply_coe
@[simps!] -- Porting note: new `simps`
def toHomeomorphSourceTarget : e.source ≃ₜ e.target 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: ∀ (a : ↑e.source), ↑(e.toHomeomorphSourceTarget a) = ↑e ↑a
-- PartialHomeomorph.secondCountableTopology_source
theorem secondCountableTopology_source [SecondCountableTopology Y] :
    SecondCountableTopology e.source 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) [SecondCountableTopology Y]
Goal: SecondCountableTopology ↑e.source
-- PartialHomeomorph.nhds_eq_comap_inf_principal
theorem nhds_eq_comap_inf_principal {x} (hx : x ∈ e.source) :
    𝓝 x = comap e (𝓝 (e x)) ⊓ 𝓟 e.source 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : X} (hx : x ∈ e.source)
Goal: 𝓝 x = Filter.comap (↑e) (𝓝 (↑e x)) ⊓ 𝓟 e.source
-- PartialHomeomorph.toHomeomorphOfSourceEqUnivTargetEqUniv_apply
@[simps (config := mfld_cfg) apply symm_apply]
-- Porting note (#11215): TODO: add a `PartialEquiv` version
def toHomeomorphOfSourceEqUnivTargetEqUniv (h : e.source = (univ : Set X)) (h' : e.target = univ) :
    X ≃ₜ Y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) (h : e.source = Set.univ) (h' : e.target = Set.univ)
Goal: ⇑(e.toHomeomorphOfSourceEqUnivTargetEqUniv h h') = ↑e
-- PartialHomeomorph.toHomeomorphOfSourceEqUnivTargetEqUniv_symm_apply
@[simps (config := mfld_cfg) apply symm_apply]
-- Porting note (#11215): TODO: add a `PartialEquiv` version
def toHomeomorphOfSourceEqUnivTargetEqUniv (h : e.source = (univ : Set X)) (h' : e.target = univ) :
    X ≃ₜ Y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) (h : e.source = Set.univ) (h' : e.target = Set.univ)
Goal: ⇑(e.toHomeomorphOfSourceEqUnivTargetEqUniv h h').symm = ↑e.symm
-- PartialHomeomorph.openEmbedding_restrict
theorem openEmbedding_restrict : OpenEmbedding (e.source.restrict e) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: OpenEmbedding (e.source.restrict ↑e)
-- PartialHomeomorph.to_openEmbedding
theorem to_openEmbedding (h : e.source = Set.univ) : OpenEmbedding e 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) (h : e.source = Set.univ)
Goal: OpenEmbedding ↑e
-- Homeomorph.refl_toPartialHomeomorph
@[simp, mfld_simps]
theorem refl_toPartialHomeomorph :
    (Homeomorph.refl X).toPartialHomeomorph = PartialHomeomorph.refl X 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: (Homeomorph.refl X).toPartialHomeomorph = PartialHomeomorph.refl X
-- Homeomorph.symm_toPartialHomeomorph
@[simp, mfld_simps]
theorem symm_toPartialHomeomorph : e.symm.toPartialHomeomorph = e.toPartialHomeomorph.symm 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y)
Goal: e.symm.toPartialHomeomorph = e.toPartialHomeomorph.symm
-- Homeomorph.trans_toPartialHomeomorph
@[simp, mfld_simps]
theorem trans_toPartialHomeomorph :
    (e.trans e').toPartialHomeomorph = e.toPartialHomeomorph.trans e'.toPartialHomeomorph 
Arguments: {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (e : X ≃ₜ Y) (e' : Y ≃ₜ Z)
Goal: (e.trans e').toPartialHomeomorph = e.toPartialHomeomorph.trans e'.toPartialHomeomorph
-- Homeomorph.transPartialHomeomorph_target
@[simps! (config := .asFn)]
def transPartialHomeomorph (e : X ≃ₜ Y) (f' : PartialHomeomorph Y Z) : PartialHomeomorph X Z 
Arguments: {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (e : X ≃ₜ Y) (f' : PartialHomeomorph Y Z)
Goal: (e.transPartialHomeomorph f').target = f'.target
-- Homeomorph.transPartialHomeomorph_source
@[simps! (config := .asFn)]
def transPartialHomeomorph (e : X ≃ₜ Y) (f' : PartialHomeomorph Y Z) : PartialHomeomorph X Z 
Arguments: {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (e : X ≃ₜ Y) (f' : PartialHomeomorph Y Z)
Goal: (e.transPartialHomeomorph f').source = ⇑e ⁻¹' f'.source
-- Homeomorph.transPartialHomeomorph_apply
@[simps! (config := .asFn)]
def transPartialHomeomorph (e : X ≃ₜ Y) (f' : PartialHomeomorph Y Z) : PartialHomeomorph X Z 
Arguments: {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (e : X ≃ₜ Y) (f' : PartialHomeomorph Y Z)
Goal: ↑(e.transPartialHomeomorph f') = ↑f' ∘ ⇑e
-- Homeomorph.transPartialHomeomorph_symm_apply
@[simps! (config := .asFn)]
def transPartialHomeomorph (e : X ≃ₜ Y) (f' : PartialHomeomorph Y Z) : PartialHomeomorph X Z 
Arguments: {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (e : X ≃ₜ Y) (f' : PartialHomeomorph Y Z)
Goal: ↑(e.transPartialHomeomorph f').symm = ⇑e.symm ∘ ↑f'.symm
-- Homeomorph.transPartialHomeomorph_eq_trans
theorem transPartialHomeomorph_eq_trans (e : X ≃ₜ Y) (f' : PartialHomeomorph Y Z) :
    e.transPartialHomeomorph f' = e.toPartialHomeomorph.trans f' 
Arguments: {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (e : X ≃ₜ Y) (f' : PartialHomeomorph Y Z)
Goal: e.transPartialHomeomorph f' = e.toPartialHomeomorph.trans f'
-- Homeomorph.transPartialHomeomorph_trans
@[simp, mfld_simps]
theorem transPartialHomeomorph_trans (e : X ≃ₜ Y) (f : PartialHomeomorph Y Z)
    (f' : PartialHomeomorph Z Z') :
    (e.transPartialHomeomorph f).trans f' = e.transPartialHomeomorph (f.trans f') 
Arguments: {X : Type u_1} {Y : Type u_3} {Z : Type u_5} {Z' : Type u_6} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [TopologicalSpace Z'] (e : X ≃ₜ Y) (f : PartialHomeomorph Y Z) (f' : PartialHomeomorph Z Z')
Goal: (e.transPartialHomeomorph f).trans f' = e.transPartialHomeomorph (f.trans f')
-- Homeomorph.trans_transPartialHomeomorph
@[simp, mfld_simps]
theorem trans_transPartialHomeomorph (e : X ≃ₜ Y) (e' : Y ≃ₜ Z) (f'' : PartialHomeomorph Z Z') :
    (e.trans e').transPartialHomeomorph f'' =
      e.transPartialHomeomorph (e'.transPartialHomeomorph f'') 
Arguments: {X : Type u_1} {Y : Type u_3} {Z : Type u_5} {Z' : Type u_6} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [TopologicalSpace Z'] (e : X ≃ₜ Y) (e' : Y ≃ₜ Z) (f'' : PartialHomeomorph Z Z')
Goal: (e.trans e').transPartialHomeomorph f'' = e.transPartialHomeomorph (e'.transPartialHomeomorph f'')
-- OpenEmbedding.toPartialHomeomorph_source
@[simps! (config := mfld_cfg) apply source target]
noncomputable def toPartialHomeomorph [Nonempty X] : PartialHomeomorph X Y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) (h : OpenEmbedding f) [Nonempty X]
Goal: (OpenEmbedding.toPartialHomeomorph f h).source = Set.univ
-- OpenEmbedding.toPartialHomeomorph_apply
@[simps! (config := mfld_cfg) apply source target]
noncomputable def toPartialHomeomorph [Nonempty X] : PartialHomeomorph X Y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) (h : OpenEmbedding f) [Nonempty X]
Goal: ↑(OpenEmbedding.toPartialHomeomorph f h) = f
-- OpenEmbedding.toPartialHomeomorph_target
@[simps! (config := mfld_cfg) apply source target]
noncomputable def toPartialHomeomorph [Nonempty X] : PartialHomeomorph X Y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) (h : OpenEmbedding f) [Nonempty X]
Goal: (OpenEmbedding.toPartialHomeomorph f h).target = Set.range f
-- OpenEmbedding.toPartialHomeomorph_left_inv
lemma toPartialHomeomorph_left_inv {x : X} : (h.toPartialHomeomorph f).symm (f x) = x 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) (h : OpenEmbedding f) [Nonempty X] {x : X}
Goal: ↑(OpenEmbedding.toPartialHomeomorph f h).symm (f x) = x
-- OpenEmbedding.toPartialHomeomorph_right_inv
lemma toPartialHomeomorph_right_inv {x : Y} (hx : x ∈ Set.range f) :
    f ((h.toPartialHomeomorph f).symm x) = x 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) (h : OpenEmbedding f) [Nonempty X] {x : Y} (hx : x ∈ Set.range f)
Goal: f (↑(OpenEmbedding.toPartialHomeomorph f h).symm x) = x
-- TopologicalSpace.Opens.partialHomeomorphSubtypeCoe_coe
@[simp, mfld_simps]
theorem partialHomeomorphSubtypeCoe_coe : (s.partialHomeomorphSubtypeCoe hs : s → X) = (↑) 
Arguments: {X : Type u_1} [TopologicalSpace X] (s : TopologicalSpace.Opens X) (hs : Nonempty ↥s)
Goal: ↑(s.partialHomeomorphSubtypeCoe hs) = Subtype.val
-- TopologicalSpace.Opens.partialHomeomorphSubtypeCoe_source
@[simp, mfld_simps]
theorem partialHomeomorphSubtypeCoe_source : (s.partialHomeomorphSubtypeCoe hs).source = Set.univ 
Arguments: {X : Type u_1} [TopologicalSpace X] (s : TopologicalSpace.Opens X) (hs : Nonempty ↥s)
Goal: (s.partialHomeomorphSubtypeCoe hs).source = Set.univ
-- TopologicalSpace.Opens.partialHomeomorphSubtypeCoe_target
@[simp, mfld_simps]
theorem partialHomeomorphSubtypeCoe_target : (s.partialHomeomorphSubtypeCoe hs).target = s 
Arguments: {X : Type u_1} [TopologicalSpace X] (s : TopologicalSpace.Opens X) (hs : Nonempty ↥s)
Goal: (s.partialHomeomorphSubtypeCoe hs).target = ↑s
-- PartialHomeomorph.transHomeomorph_symm_apply
@[simps! (config := .asFn)]
def transHomeomorph (e : PartialHomeomorph X Y) (f' : Y ≃ₜ Z) : PartialHomeomorph X Z 
Arguments: {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (e : PartialHomeomorph X Y) (f' : Y ≃ₜ Z)
Goal: ↑(e.transHomeomorph f').symm = ↑e.symm ∘ ⇑f'.symm
-- PartialHomeomorph.transHomeomorph_apply
@[simps! (config := .asFn)]
def transHomeomorph (e : PartialHomeomorph X Y) (f' : Y ≃ₜ Z) : PartialHomeomorph X Z 
Arguments: {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (e : PartialHomeomorph X Y) (f' : Y ≃ₜ Z)
Goal: ↑(e.transHomeomorph f') = ⇑f' ∘ ↑e
-- PartialHomeomorph.transHomeomorph_target
@[simps! (config := .asFn)]
def transHomeomorph (e : PartialHomeomorph X Y) (f' : Y ≃ₜ Z) : PartialHomeomorph X Z 
Arguments: {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (e : PartialHomeomorph X Y) (f' : Y ≃ₜ Z)
Goal: (e.transHomeomorph f').target = ⇑f'.symm ⁻¹' e.target
-- PartialHomeomorph.transHomeomorph_source
@[simps! (config := .asFn)]
def transHomeomorph (e : PartialHomeomorph X Y) (f' : Y ≃ₜ Z) : PartialHomeomorph X Z 
Arguments: {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (e : PartialHomeomorph X Y) (f' : Y ≃ₜ Z)
Goal: (e.transHomeomorph f').source = e.source
-- PartialHomeomorph.transHomeomorph_eq_trans
theorem transHomeomorph_eq_trans (e : PartialHomeomorph X Y) (f' : Y ≃ₜ Z) :
    e.transHomeomorph f' = e.trans f'.toPartialHomeomorph 
Arguments: {X : Type u_1} {Y : Type u_3} {Z : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (e : PartialHomeomorph X Y) (f' : Y ≃ₜ Z)
Goal: e.transHomeomorph f' = e.trans f'.toPartialHomeomorph
-- PartialHomeomorph.transHomeomorph_transHomeomorph
@[simp, mfld_simps]
theorem transHomeomorph_transHomeomorph (e : PartialHomeomorph X Y) (f' : Y ≃ₜ Z) (f'' : Z ≃ₜ Z') :
    (e.transHomeomorph f').transHomeomorph f'' = e.transHomeomorph (f'.trans f'') 
Arguments: {X : Type u_1} {Y : Type u_3} {Z : Type u_5} {Z' : Type u_6} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [TopologicalSpace Z'] (e : PartialHomeomorph X Y) (f' : Y ≃ₜ Z) (f'' : Z ≃ₜ Z')
Goal: (e.transHomeomorph f').transHomeomorph f'' = e.transHomeomorph (f'.trans f'')
-- PartialHomeomorph.trans_transHomeomorph
@[simp, mfld_simps]
theorem trans_transHomeomorph (e : PartialHomeomorph X Y) (e' : PartialHomeomorph Y Z)
    (f'' : Z ≃ₜ Z') :
    (e.trans e').transHomeomorph f'' = e.trans (e'.transHomeomorph f'') 
Arguments: {X : Type u_1} {Y : Type u_3} {Z : Type u_5} {Z' : Type u_6} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] [TopologicalSpace Z'] (e : PartialHomeomorph X Y) (e' : PartialHomeomorph Y Z) (f'' : Z ≃ₜ Z')
Goal: (e.trans e').transHomeomorph f'' = e.trans (e'.transHomeomorph f'')
-- PartialHomeomorph.subtypeRestr_def
theorem subtypeRestr_def : e.subtypeRestr hs = (s.partialHomeomorphSubtypeCoe hs).trans e 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {s : TopologicalSpace.Opens X} (hs : Nonempty ↥s)
Goal: e.subtypeRestr hs = (s.partialHomeomorphSubtypeCoe hs).trans e
-- PartialHomeomorph.subtypeRestr_coe
@[simp, mfld_simps]
theorem subtypeRestr_coe :
    ((e.subtypeRestr hs : PartialHomeomorph s Y) : s → Y) = Set.restrict ↑s (e : X → Y) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {s : TopologicalSpace.Opens X} (hs : Nonempty ↥s)
Goal: ↑(e.subtypeRestr hs) = (↑s).restrict ↑e
-- PartialHomeomorph.subtypeRestr_source
@[simp, mfld_simps]
theorem subtypeRestr_source : (e.subtypeRestr hs).source = (↑) ⁻¹' e.source 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {s : TopologicalSpace.Opens X} (hs : Nonempty ↥s)
Goal: (e.subtypeRestr hs).source = Subtype.val ⁻¹' e.source
-- PartialHomeomorph.map_subtype_source
theorem map_subtype_source {x : s} (hxe : (x : X) ∈ e.source) :
    e x ∈ (e.subtypeRestr hs).target 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {s : TopologicalSpace.Opens X} (hs : Nonempty ↥s) {x : ↥s} (hxe : ↑x ∈ e.source)
Goal: ↑e ↑x ∈ (e.subtypeRestr hs).target
-- PartialHomeomorph.subtypeRestr_symm_trans_subtypeRestr
theorem subtypeRestr_symm_trans_subtypeRestr (f f' : PartialHomeomorph X Y) :
    (f.subtypeRestr hs).symm.trans (f'.subtypeRestr hs) ≈
      (f.symm.trans f').restr (f.target ∩ f.symm ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {s : TopologicalSpace.Opens X} (hs : Nonempty ↥s) (f f' : PartialHomeomorph X Y)
Goal: (f.subtypeRestr hs).symm.trans (f'.subtypeRestr hs) ≈ (f.symm.trans f').restr (f.target ∩ ↑f.symm ⁻¹' ↑s)
-- PartialHomeomorph.subtypeRestr_symm_eqOn
theorem subtypeRestr_symm_eqOn {U : Opens X} (hU : Nonempty U) :
    EqOn e.symm (Subtype.val ∘ (e.subtypeRestr hU).symm) (e.subtypeRestr hU).target 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {U : TopologicalSpace.Opens X} (hU : Nonempty ↥U)
Goal: Set.EqOn (↑e.symm) (Subtype.val ∘ ↑(e.subtypeRestr hU).symm) (e.subtypeRestr hU).target
-- PartialHomeomorph.subtypeRestr_symm_eqOn_of_le
theorem subtypeRestr_symm_eqOn_of_le {U V : Opens X} (hU : Nonempty U) (hV : Nonempty V)
    (hUV : U ≤ V) : EqOn (e.subtypeRestr hV).symm (Set.inclusion hUV ∘ (e.subtypeRestr hU).symm)
      (e.subtypeRestr hU).target 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {U V : TopologicalSpace.Opens X} (hU : Nonempty ↥U) (hV : Nonempty ↥V) (hUV : U ≤ V)
Goal: Set.EqOn (↑(e.subtypeRestr hV).symm) (Set.inclusion hUV ∘ ↑(e.subtypeRestr hU).symm) (e.subtypeRestr hU).target
-- ContinuousMap.attachBound_apply_coe
@[simp]
theorem attachBound_apply_coe (f : C(X, ℝ)) (x : X) : ((attachBound f) x : ℝ) = f x 
Arguments: {X : Type u_1} [TopologicalSpace X] [CompactSpace X] (f : C(X, ℝ)) (x : X)
Goal: ↑(f.attachBound x) = f x
-- ContinuousMap.polynomial_comp_attachBound
theorem polynomial_comp_attachBound (A : Subalgebra ℝ C(X, ℝ)) (f : A) (g : ℝ[X]) :
    (g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (f : C(X, ℝ)).attachBound =
      Polynomial.aeval f g 
Arguments: {X : Type u_1} [TopologicalSpace X] [CompactSpace X] (A : Subalgebra ℝ C(X, ℝ)) (f : ↥A) (g : Polynomial ℝ)
Goal: (g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (↑f).attachBound = ↑((Polynomial.aeval f) g)
-- ContinuousMap.polynomial_comp_attachBound_mem
theorem polynomial_comp_attachBound_mem (A : Subalgebra ℝ C(X, ℝ)) (f : A) (g : ℝ[X]) :
    (g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (f : C(X, ℝ)).attachBound ∈ A 
Arguments: {X : Type u_1} [TopologicalSpace X] [CompactSpace X] (A : Subalgebra ℝ C(X, ℝ)) (f : ↥A) (g : Polynomial ℝ)
Goal: (g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (↑f).attachBound ∈ A
-- ContinuousMap.comp_attachBound_mem_closure
theorem comp_attachBound_mem_closure (A : Subalgebra ℝ C(X, ℝ)) (f : A)
    (p : C(Set.Icc (-‖f‖) ‖f‖, ℝ)) : p.comp (attachBound (f : C(X, ℝ))) ∈ A.topologicalClosure 
Arguments: {X : Type u_1} [TopologicalSpace X] [CompactSpace X] (A : Subalgebra ℝ C(X, ℝ)) (f : ↥A) (p : C(↑(Set.Icc (-‖f‖) ‖f‖), ℝ))
Goal: p.comp (↑f).attachBound ∈ A.topologicalClosure