ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Smooth.extend_one
@[to_additive]
theorem Smooth.extend_one [T2Space M] [One M'] {U : Opens M} {f : U → M'}
    (supp : HasCompactMulSupport f) (diff : Smooth I I' f) : Smooth I I' (Subtype.val.extend f 1) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] [T2Space M] [One M'] {U : TopologicalSpace.Opens M} {f : ↥U → M'} (supp : HasCompactMulSupport f) (diff : Smooth I I' f)
Goal: Smooth I I' (Function.extend Subtype.val f 1)
-- smooth_inclusion
theorem smooth_inclusion {U V : Opens M} (h : U ≤ V) : Smooth I I (Set.inclusion h : U → V) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {U V : TopologicalSpace.Opens M} (h : U ≤ V)
Goal: Smooth I I (Set.inclusion h)
-- contMDiff_openEmbedding
lemma contMDiff_openEmbedding :
    haveI := h.singletonChartedSpace; ContMDiff I I n e 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [Nonempty M] {e : M → H} (h : OpenEmbedding e) {n : WithTop ℕ}
Goal: ContMDiff I I n e
-- contMDiffOn_openEmbedding_symm
lemma contMDiffOn_openEmbedding_symm :
    haveI := h.singletonChartedSpace; ContMDiffOn I I
      n (OpenEmbedding.toPartialHomeomorph e h).symm (range e) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [Nonempty M] {e : M → H} (h : OpenEmbedding e) {n : WithTop ℕ}
Goal: ContMDiffOn I I n (↑(OpenEmbedding.toPartialHomeomorph e h).symm) (Set.range e)
-- ContMDiff.of_comp_openEmbedding
lemma ContMDiff.of_comp_openEmbedding {f : M → M'} (hf : ContMDiff I I' n (e' ∘ f)) :
    haveI := h'.singletonChartedSpace; ContMDiff I I' n f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [Nonempty M'] {e' : M' → H'} (h' : OpenEmbedding e') {n : WithTop ℕ} {f : M → M'} (hf : ContMDiff I I' n (e' ∘ f))
Goal: ContMDiff I I' n f
-- AlgebraicGeometry.LocallyRingedSpace.evaluation_eq_zero_iff_not_mem_basicOpen
@[simp]
lemma evaluation_eq_zero_iff_not_mem_basicOpen (x : U) (f : X.presheaf.obj (op U)) :
    X.evaluation x f = 0 ↔ x.val ∉ X.toRingedSpace.basicOpen f 
Arguments: (X : AlgebraicGeometry.LocallyRingedSpace) {U : TopologicalSpace.Opens ↑X.toTopCat} (x : ↥U) (f : ↑(X.presheaf.obj (Opposite.op U)))
Goal: (X.evaluation x) f = 0 ↔ ↑x ∉ X.toRingedSpace.basicOpen f
-- AlgebraicGeometry.LocallyRingedSpace.evaluation_ne_zero_iff_mem_basicOpen
lemma evaluation_ne_zero_iff_mem_basicOpen (x : U) (f : X.presheaf.obj (op U)) :
    X.evaluation x f ≠ 0 ↔ x.val ∈ X.toRingedSpace.basicOpen f 
Arguments: (X : AlgebraicGeometry.LocallyRingedSpace) {U : TopologicalSpace.Opens ↑X.toTopCat} (x : ↥U) (f : ↑(X.presheaf.obj (Opposite.op U)))
Goal: (X.evaluation x) f ≠ 0 ↔ ↑x ∈ X.toRingedSpace.basicOpen f
-- AlgebraicGeometry.LocallyRingedSpace.Γevaluation_eq_zero_iff_not_mem_basicOpen
@[simp]
lemma Γevaluation_eq_zero_iff_not_mem_basicOpen (x : X) (f : X.presheaf.obj (op ⊤)) :
    X.Γevaluation x f = 0 ↔ x ∉ X.toRingedSpace.basicOpen f 
Arguments: (X : AlgebraicGeometry.LocallyRingedSpace) (x : ↑X.toTopCat) (f : ↑(X.presheaf.obj (Opposite.op ⊤)))
Goal: (X.Γevaluation x) f = 0 ↔ x ∉ X.toRingedSpace.basicOpen f
-- AlgebraicGeometry.LocallyRingedSpace.Γevaluation_ne_zero_iff_mem_basicOpen
lemma Γevaluation_ne_zero_iff_mem_basicOpen (x : X) (f : X.presheaf.obj (op ⊤)) :
    X.Γevaluation x f ≠ 0 ↔ x ∈ X.toRingedSpace.basicOpen f 
Arguments: (X : AlgebraicGeometry.LocallyRingedSpace) (x : ↑X.toTopCat) (f : ↑(X.presheaf.obj (Opposite.op ⊤)))
Goal: (X.Γevaluation x) f ≠ 0 ↔ x ∈ X.toRingedSpace.basicOpen f
-- AlgebraicGeometry.LocallyRingedSpace.residue_comp_residueFieldMap_eq_stalkMap_comp_residue
lemma residue_comp_residueFieldMap_eq_stalkMap_comp_residue (x : X) :
    LocalRing.residue _ ≫ residueFieldMap f x = f.stalkMap x ≫ LocalRing.residue _ 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y) (x : ↑X.toTopCat)
Goal: LocalRing.residue ↑(Y.presheaf.stalk (f.val.base x)) ≫ AlgebraicGeometry.LocallyRingedSpace.residueFieldMap f x =
  AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap f x ≫ LocalRing.residue ↑(X.presheaf.stalk x)
-- AlgebraicGeometry.LocallyRingedSpace.residueFieldMap_id
@[simp]
lemma residueFieldMap_id (x : X) :
    residueFieldMap (𝟙 X) x = 𝟙 (X.residueField x) 
Arguments: {X : AlgebraicGeometry.LocallyRingedSpace} (x : ↑X.toTopCat)
Goal: AlgebraicGeometry.LocallyRingedSpace.residueFieldMap (𝟙 X) x = 𝟙 (X.residueField x)
-- AlgebraicGeometry.LocallyRingedSpace.residueFieldMap_comp
@[simp]
lemma residueFieldMap_comp {Z : LocallyRingedSpace.{u}} (g : Y ⟶ Z) (x : X) :
    residueFieldMap (f ≫ g) x = residueFieldMap g (f.val.base x) ≫ residueFieldMap f x 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y) {Z : AlgebraicGeometry.LocallyRingedSpace} (g : Y ⟶ Z) (x : ↑X.toTopCat)
Goal: AlgebraicGeometry.LocallyRingedSpace.residueFieldMap (f ≫ g) x =
  AlgebraicGeometry.LocallyRingedSpace.residueFieldMap g (f.val.base x) ≫
    AlgebraicGeometry.LocallyRingedSpace.residueFieldMap f x
-- AlgebraicGeometry.LocallyRingedSpace.evaluation_naturality
@[reassoc]
lemma evaluation_naturality {V : Opens Y} (x : (Opens.map f.1.base).obj V) :
    Y.evaluation ⟨f.val.base x, x.property⟩ ≫ residueFieldMap f x.val =
      f.val.c.app (op V) ≫ X.evaluation x 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y) {V : TopologicalSpace.Opens ↑Y.toTopCat} (x : ↥((TopologicalSpace.Opens.map f.val.base).obj V))
Goal: Y.evaluation ⟨f.val.base ↑x, ⋯⟩ ≫ AlgebraicGeometry.LocallyRingedSpace.residueFieldMap f ↑x =
  f.val.c.app (Opposite.op V) ≫ X.evaluation x
-- AlgebraicGeometry.LocallyRingedSpace.evaluation_naturality_apply
lemma evaluation_naturality_apply {V : Opens Y} (x : (Opens.map f.1.base).obj V)
    (a : Y.presheaf.obj (op V)) :
    residueFieldMap f x.val (Y.evaluation ⟨f.val.base x, x.property⟩ a) =
      X.evaluation x (f.val.c.app (op V) a) 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y) {V : TopologicalSpace.Opens ↑Y.toTopCat} (x : ↥((TopologicalSpace.Opens.map f.val.base).obj V)) (a : ↑(Y.presheaf.obj (Opposite.op V)))
Goal: (AlgebraicGeometry.LocallyRingedSpace.residueFieldMap f ↑x) ((Y.evaluation ⟨f.val.base ↑x, ⋯⟩) a) =
  (X.evaluation x) ((f.val.c.app (Opposite.op V)) a)
-- AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality
@[reassoc]
lemma Γevaluation_naturality (x : X) :
    Y.Γevaluation (f.val.base x) ≫ residueFieldMap f x =
      f.val.c.app (op ⊤) ≫ X.Γevaluation x 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y) (x : ↑X.toTopCat)
Goal: Y.Γevaluation (f.val.base x) ≫ AlgebraicGeometry.LocallyRingedSpace.residueFieldMap f x =
  f.val.c.app (Opposite.op ⊤) ≫ X.Γevaluation x
-- AlgebraicGeometry.LocallyRingedSpace.Γevaluation_naturality_apply
lemma Γevaluation_naturality_apply (x : X) (a : Y.presheaf.obj (op ⊤)) :
    residueFieldMap f x (Y.Γevaluation (f.val.base x) a) =
      X.Γevaluation x (f.val.c.app (op ⊤) a) 
Arguments: {X Y : AlgebraicGeometry.LocallyRingedSpace} (f : X ⟶ Y) (x : ↑X.toTopCat) (a : ↑(Y.presheaf.obj (Opposite.op ⊤)))
Goal: (AlgebraicGeometry.LocallyRingedSpace.residueFieldMap f x) ((Y.Γevaluation (f.val.base x)) a) =
  (X.Γevaluation x) ((f.val.c.app (Opposite.op ⊤)) a)
-- StructureGroupoid.LocalInvariantProp.congr_of_forall
structure LocalInvariantProp (P : (H → H') → Set H → H → Prop) : Prop where
  is_local : ∀ {s x u} {f : H → H'}, IsOpen u → x ∈ u → (P f s x ↔ P f (s ∩ u) x)
  right_invariance' : ∀ {s x f} {e : PartialHomeomorph H H},
    e ∈ G → x ∈ e.source → P f s x → P (f ∘ e.symm) (e.symm ⁻¹' s) (e x)
  congr_of_forall : ∀ {s x} {f g : H → H'}, (∀ y ∈ s, f y = g y) → f x = g x → P f s x → P g s x
  left_invariance' : ∀ {s x f} {e' : PartialHomeomorph H' H'},
    e' ∈ G' → s ⊆ f ⁻¹' e'.source → f x ∈ e'.source → P f s x → P (e' ∘ f) s x
Arguments: {H : Type u_1} {H' : Type u_3} [TopologicalSpace H] [TopologicalSpace H'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {P : (H → H') → Set H → H → Prop} (self : G.LocalInvariantProp G' P) {s : Set H} {x : H} {f g : H → H'}
Goal: (∀ y ∈ s, f y = g y) → f x = g x → P f s x → P g s x
-- StructureGroupoid.LocalInvariantProp.right_invariance'
structure LocalInvariantProp (P : (H → H') → Set H → H → Prop) : Prop where
  is_local : ∀ {s x u} {f : H → H'}, IsOpen u → x ∈ u → (P f s x ↔ P f (s ∩ u) x)
  right_invariance' : ∀ {s x f} {e : PartialHomeomorph H H},
    e ∈ G → x ∈ e.source → P f s x → P (f ∘ e.symm) (e.symm ⁻¹' s) (e x)
  congr_of_forall : ∀ {s x} {f g : H → H'}, (∀ y ∈ s, f y = g y) → f x = g x → P f s x → P g s x
  left_invariance' : ∀ {s x f} {e' : PartialHomeomorph H' H'},
    e' ∈ G' → s ⊆ f ⁻¹' e'.source → f x ∈ e'.source → P f s x → P (e' ∘ f) s x
Arguments: {H : Type u_1} {H' : Type u_3} [TopologicalSpace H] [TopologicalSpace H'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {P : (H → H') → Set H → H → Prop} (self : G.LocalInvariantProp G' P) {s : Set H} {x : H} {f : H → H'} {e : PartialHomeomorph H H}
Goal: e ∈ G → x ∈ e.source → P f s x → P (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)
-- StructureGroupoid.LocalInvariantProp.left_invariance'
structure LocalInvariantProp (P : (H → H') → Set H → H → Prop) : Prop where
  is_local : ∀ {s x u} {f : H → H'}, IsOpen u → x ∈ u → (P f s x ↔ P f (s ∩ u) x)
  right_invariance' : ∀ {s x f} {e : PartialHomeomorph H H},
    e ∈ G → x ∈ e.source → P f s x → P (f ∘ e.symm) (e.symm ⁻¹' s) (e x)
  congr_of_forall : ∀ {s x} {f g : H → H'}, (∀ y ∈ s, f y = g y) → f x = g x → P f s x → P g s x
  left_invariance' : ∀ {s x f} {e' : PartialHomeomorph H' H'},
    e' ∈ G' → s ⊆ f ⁻¹' e'.source → f x ∈ e'.source → P f s x → P (e' ∘ f) s x
Arguments: {H : Type u_1} {H' : Type u_3} [TopologicalSpace H] [TopologicalSpace H'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {P : (H → H') → Set H → H → Prop} (self : G.LocalInvariantProp G' P) {s : Set H} {x : H} {f : H → H'} {e' : PartialHomeomorph H' H'}
Goal: e' ∈ G' → s ⊆ f ⁻¹' e'.source → f x ∈ e'.source → P f s x → P (↑e' ∘ f) s x
-- StructureGroupoid.LocalInvariantProp.is_local
structure LocalInvariantProp (P : (H → H') → Set H → H → Prop) : Prop where
  is_local : ∀ {s x u} {f : H → H'}, IsOpen u → x ∈ u → (P f s x ↔ P f (s ∩ u) x)
  right_invariance' : ∀ {s x f} {e : PartialHomeomorph H H},
    e ∈ G → x ∈ e.source → P f s x → P (f ∘ e.symm) (e.symm ⁻¹' s) (e x)
  congr_of_forall : ∀ {s x} {f g : H → H'}, (∀ y ∈ s, f y = g y) → f x = g x → P f s x → P g s x
  left_invariance' : ∀ {s x f} {e' : PartialHomeomorph H' H'},
    e' ∈ G' → s ⊆ f ⁻¹' e'.source → f x ∈ e'.source → P f s x → P (e' ∘ f) s x
Arguments: {H : Type u_1} {H' : Type u_3} [TopologicalSpace H] [TopologicalSpace H'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {P : (H → H') → Set H → H → Prop} (self : G.LocalInvariantProp G' P) {s : Set H} {x : H} {u : Set H} {f : H → H'}
Goal: IsOpen u → x ∈ u → (P f s x ↔ P f (s ∩ u) x)
-- StructureGroupoid.LocalInvariantProp.congr_set
theorem congr_set {s t : Set H} {x : H} {f : H → H'} (hu : s =ᶠ[𝓝 x] t) : P f s x ↔ P f t x 
Arguments: {H : Type u_1} {H' : Type u_3} [TopologicalSpace H] [TopologicalSpace H'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {P : (H → H') → Set H → H → Prop} (hG : G.LocalInvariantProp G' P) {s t : Set H} {x : H} {f : H → H'} (hu : s =ᶠ[𝓝 x] t)
Goal: P f s x ↔ P f t x
-- StructureGroupoid.LocalInvariantProp.is_local_nhds
theorem is_local_nhds {s u : Set H} {x : H} {f : H → H'} (hu : u ∈ 𝓝[s] x) :
    P f s x ↔ P f (s ∩ u) x 
Arguments: {H : Type u_1} {H' : Type u_3} [TopologicalSpace H] [TopologicalSpace H'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {P : (H → H') → Set H → H → Prop} (hG : G.LocalInvariantProp G' P) {s u : Set H} {x : H} {f : H → H'} (hu : u ∈ 𝓝[s] x)
Goal: P f s x ↔ P f (s ∩ u) x
-- StructureGroupoid.LocalInvariantProp.congr_iff_nhdsWithin
theorem congr_iff_nhdsWithin {s : Set H} {x : H} {f g : H → H'} (h1 : f =ᶠ[𝓝[s] x] g)
    (h2 : f x = g x) : P f s x ↔ P g s x 
Arguments: {H : Type u_1} {H' : Type u_3} [TopologicalSpace H] [TopologicalSpace H'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {P : (H → H') → Set H → H → Prop} (hG : G.LocalInvariantProp G' P) {s : Set H} {x : H} {f g : H → H'} (h1 : f =ᶠ[𝓝[s] x] g) (h2 : f x = g x)
Goal: P f s x ↔ P g s x
-- StructureGroupoid.LocalInvariantProp.congr_nhdsWithin
theorem congr_nhdsWithin {s : Set H} {x : H} {f g : H → H'} (h1 : f =ᶠ[𝓝[s] x] g) (h2 : f x = g x)
    (hP : P f s x) : P g s x 
Arguments: {H : Type u_1} {H' : Type u_3} [TopologicalSpace H] [TopologicalSpace H'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {P : (H → H') → Set H → H → Prop} (hG : G.LocalInvariantProp G' P) {s : Set H} {x : H} {f g : H → H'} (h1 : f =ᶠ[𝓝[s] x] g) (h2 : f x = g x) (hP : P f s x)
Goal: P g s x
-- StructureGroupoid.LocalInvariantProp.congr_nhdsWithin'
theorem congr_nhdsWithin' {s : Set H} {x : H} {f g : H → H'} (h1 : f =ᶠ[𝓝[s] x] g) (h2 : f x = g x)
    (hP : P g s x) : P f s x 
Arguments: {H : Type u_1} {H' : Type u_3} [TopologicalSpace H] [TopologicalSpace H'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {P : (H → H') → Set H → H → Prop} (hG : G.LocalInvariantProp G' P) {s : Set H} {x : H} {f g : H → H'} (h1 : f =ᶠ[𝓝[s] x] g) (h2 : f x = g x) (hP : P g s x)
Goal: P f s x
-- StructureGroupoid.LocalInvariantProp.congr_iff
theorem congr_iff {s : Set H} {x : H} {f g : H → H'} (h : f =ᶠ[𝓝 x] g) : P f s x ↔ P g s x 
Arguments: {H : Type u_1} {H' : Type u_3} [TopologicalSpace H] [TopologicalSpace H'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {P : (H → H') → Set H → H → Prop} (hG : G.LocalInvariantProp G' P) {s : Set H} {x : H} {f g : H → H'} (h : f =ᶠ[𝓝 x] g)
Goal: P f s x ↔ P g s x
-- StructureGroupoid.LocalInvariantProp.congr
theorem congr {s : Set H} {x : H} {f g : H → H'} (h : f =ᶠ[𝓝 x] g) (hP : P f s x) : P g s x 
Arguments: {H : Type u_1} {H' : Type u_3} [TopologicalSpace H] [TopologicalSpace H'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {P : (H → H') → Set H → H → Prop} (hG : G.LocalInvariantProp G' P) {s : Set H} {x : H} {f g : H → H'} (h : f =ᶠ[𝓝 x] g) (hP : P f s x)
Goal: P g s x
-- StructureGroupoid.LocalInvariantProp.congr'
theorem congr' {s : Set H} {x : H} {f g : H → H'} (h : f =ᶠ[𝓝 x] g) (hP : P g s x) : P f s x 
Arguments: {H : Type u_1} {H' : Type u_3} [TopologicalSpace H] [TopologicalSpace H'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {P : (H → H') → Set H → H → Prop} (hG : G.LocalInvariantProp G' P) {s : Set H} {x : H} {f g : H → H'} (h : f =ᶠ[𝓝 x] g) (hP : P g s x)
Goal: P f s x
-- StructureGroupoid.LocalInvariantProp.left_invariance
theorem left_invariance {s : Set H} {x : H} {f : H → H'} {e' : PartialHomeomorph H' H'}
    (he' : e' ∈ G') (hfs : ContinuousWithinAt f s x) (hxe' : f x ∈ e'.source) :
    P (e' ∘ f) s x ↔ P f s x 
Arguments: {H : Type u_1} {H' : Type u_3} [TopologicalSpace H] [TopologicalSpace H'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {P : (H → H') → Set H → H → Prop} (hG : G.LocalInvariantProp G' P) {s : Set H} {x : H} {f : H → H'} {e' : PartialHomeomorph H' H'} (he' : e' ∈ G') (hfs : ContinuousWithinAt f s x) (hxe' : f x ∈ e'.source)
Goal: P (↑e' ∘ f) s x ↔ P f s x
-- StructureGroupoid.LocalInvariantProp.right_invariance
theorem right_invariance {s : Set H} {x : H} {f : H → H'} {e : PartialHomeomorph H H} (he : e ∈ G)
    (hxe : x ∈ e.source) : P (f ∘ e.symm) (e.symm ⁻¹' s) (e x) ↔ P f s x 
Arguments: {H : Type u_1} {H' : Type u_3} [TopologicalSpace H] [TopologicalSpace H'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {P : (H → H') → Set H → H → Prop} (hG : G.LocalInvariantProp G' P) {s : Set H} {x : H} {f : H → H'} {e : PartialHomeomorph H H} (he : e ∈ G) (hxe : x ∈ e.source)
Goal: P (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x) ↔ P f s x
-- ChartedSpace.LiftPropWithinAt.continuousWithinAt
@[mk_iff liftPropWithinAt_iff']
structure LiftPropWithinAt (P : (H → H') → Set H → H → Prop) (f : M → M') (s : Set M) (x : M) :
    Prop where
  continuousWithinAt : ContinuousWithinAt f s x
  prop : P (chartAt H' (f x) ∘ f ∘ (chartAt H x).symm) ((chartAt H x).symm ⁻¹' s) (chartAt H x x)
Arguments: {H : Type u_1} {M : Type u_2} {H' : Type u_3} {M' : Type u_4} [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] [TopologicalSpace H'] [TopologicalSpace M'] [ChartedSpace H' M'] {P : (H → H') → Set H → H → Prop} {f : M → M'} {s : Set M} {x : M} (self : ChartedSpace.LiftPropWithinAt P f s x)
Goal: ContinuousWithinAt f s x
-- ChartedSpace.liftPropWithinAt_iff'
@[mk_iff liftPropWithinAt_iff']
structure LiftPropWithinAt (P : (H → H') → Set H → H → Prop) (f : M → M') (s : Set M) (x : M) :
    Prop where
  continuousWithinAt : ContinuousWithinAt f s x
  prop : P (chartAt H' (f x) ∘ f ∘ (chartAt H x).symm) ((chartAt H x).symm ⁻¹' s) (chartAt H x x)
Arguments: {H : Type u_1} {M : Type u_2} {H' : Type u_3} {M' : Type u_4} [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] [TopologicalSpace H'] [TopologicalSpace M'] [ChartedSpace H' M'] (P : (H → H') → Set H → H → Prop) (f : M → M') (s : Set M) (x : M)
Goal: ChartedSpace.LiftPropWithinAt P f s x ↔
  ContinuousWithinAt f s x ∧
    P (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x)
-- ChartedSpace.LiftPropWithinAt.prop
@[mk_iff liftPropWithinAt_iff']
structure LiftPropWithinAt (P : (H → H') → Set H → H → Prop) (f : M → M') (s : Set M) (x : M) :
    Prop where
  continuousWithinAt : ContinuousWithinAt f s x
  prop : P (chartAt H' (f x) ∘ f ∘ (chartAt H x).symm) ((chartAt H x).symm ⁻¹' s) (chartAt H x x)
Arguments: {H : Type u_1} {M : Type u_2} {H' : Type u_3} {M' : Type u_4} [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] [TopologicalSpace H'] [TopologicalSpace M'] [ChartedSpace H' M'] {P : (H → H') → Set H → H → Prop} {f : M → M'} {s : Set M} {x : M} (self : ChartedSpace.LiftPropWithinAt P f s x)
Goal: P (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x)
-- ChartedSpace.liftPropAt_iff
theorem liftPropAt_iff {P : (H → H') → Set H → H → Prop} {f : M → M'} {x : M} :
    LiftPropAt P f x ↔
      ContinuousAt f x ∧ P (chartAt H' (f x) ∘ f ∘ (chartAt H x).symm) univ (chartAt H x x) 
Arguments: {H : Type u_1} {M : Type u_2} {H' : Type u_3} {M' : Type u_4} [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] [TopologicalSpace H'] [TopologicalSpace M'] [ChartedSpace H' M'] {P : (H → H') → Set H → H → Prop} {f : M → M'} {x : M}
Goal: ChartedSpace.LiftPropAt P f x ↔
  ContinuousAt f x ∧ P (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm) Set.univ (↑(chartAt H x) x)
-- ChartedSpace.liftProp_iff
theorem liftProp_iff {P : (H → H') → Set H → H → Prop} {f : M → M'} :
    LiftProp P f ↔
      Continuous f ∧ ∀ x, P (chartAt H' (f x) ∘ f ∘ (chartAt H x).symm) univ (chartAt H x x) 
Arguments: {H : Type u_1} {M : Type u_2} {H' : Type u_3} {M' : Type u_4} [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] [TopologicalSpace H'] [TopologicalSpace M'] [ChartedSpace H' M'] {P : (H → H') → Set H → H → Prop} {f : M → M'}
Goal: ChartedSpace.LiftProp P f ↔
  Continuous f ∧ ∀ (x : M), P (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm) Set.univ (↑(chartAt H x) x)
-- StructureGroupoid.liftPropWithinAt_univ
theorem liftPropWithinAt_univ : LiftPropWithinAt P g univ x ↔ LiftPropAt P g x 
Arguments: {H : Type u_1} {M : Type u_2} {H' : Type u_3} {M' : Type u_4} [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] [TopologicalSpace H'] [TopologicalSpace M'] [ChartedSpace H' M'] {P : (H → H') → Set H → H → Prop} {g : M → M'} {x : M}
Goal: ChartedSpace.LiftPropWithinAt P g Set.univ x ↔ ChartedSpace.LiftPropAt P g x
-- StructureGroupoid.liftPropOn_univ
theorem liftPropOn_univ : LiftPropOn P g univ ↔ LiftProp P g 
Arguments: {H : Type u_1} {M : Type u_2} {H' : Type u_3} {M' : Type u_4} [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] [TopologicalSpace H'] [TopologicalSpace M'] [ChartedSpace H' M'] {P : (H → H') → Set H → H → Prop} {g : M → M'}
Goal: ChartedSpace.LiftPropOn P g Set.univ ↔ ChartedSpace.LiftProp P g
-- StructureGroupoid.liftPropWithinAt_self
theorem liftPropWithinAt_self {f : H → H'} {s : Set H} {x : H} :
    LiftPropWithinAt P f s x ↔ ContinuousWithinAt f s x ∧ P f s x 
Arguments: {H : Type u_1} {H' : Type u_3} [TopologicalSpace H] [TopologicalSpace H'] {P : (H → H') → Set H → H → Prop} {f : H → H'} {s : Set H} {x : H}
Goal: ChartedSpace.LiftPropWithinAt P f s x ↔ ContinuousWithinAt f s x ∧ P f s x
-- StructureGroupoid.liftPropWithinAt_self_source
theorem liftPropWithinAt_self_source {f : H → M'} {s : Set H} {x : H} :
    LiftPropWithinAt P f s x ↔ ContinuousWithinAt f s x ∧ P (chartAt H' (f x) ∘ f) s x 
Arguments: {H : Type u_1} {H' : Type u_3} {M' : Type u_4} [TopologicalSpace H] [TopologicalSpace H'] [TopologicalSpace M'] [ChartedSpace H' M'] {P : (H → H') → Set H → H → Prop} {f : H → M'} {s : Set H} {x : H}
Goal: ChartedSpace.LiftPropWithinAt P f s x ↔ ContinuousWithinAt f s x ∧ P (↑(chartAt H' (f x)) ∘ f) s x
-- StructureGroupoid.liftPropWithinAt_self_target
theorem liftPropWithinAt_self_target {f : M → H'} :
    LiftPropWithinAt P f s x ↔ ContinuousWithinAt f s x ∧
      P (f ∘ (chartAt H x).symm) ((chartAt H x).symm ⁻¹' s) (chartAt H x x) 
Arguments: {H : Type u_1} {M : Type u_2} {H' : Type u_3} [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] [TopologicalSpace H'] {P : (H → H') → Set H → H → Prop} {s : Set M} {x : M} {f : M → H'}
Goal: ChartedSpace.LiftPropWithinAt P f s x ↔
  ContinuousWithinAt f s x ∧ P (f ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x)
-- StructureGroupoid.LocalInvariantProp.liftPropWithinAt_iff
theorem liftPropWithinAt_iff {f : M → M'} :
    LiftPropWithinAt P f s x ↔
      ContinuousWithinAt f s x ∧
        P (chartAt H' (f x) ∘ f ∘ (chartAt H x).symm)
          ((chartAt H x).target ∩ (chartAt H x).symm ⁻¹' (s ∩ f ⁻¹' (chartAt H' (f x)).source))
          (chartAt H x x) 
Arguments: {H : Type u_1} {M : Type u_2} {H' : Type u_3} {M' : Type u_4} [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] [TopologicalSpace H'] [TopologicalSpace M'] [ChartedSpace H' M'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {P : (H → H') → Set H → H → Prop} {s : Set M} {x : M} (hG : G.LocalInvariantProp G' P) {f : M → M'}
Goal: ChartedSpace.LiftPropWithinAt P f s x ↔
  ContinuousWithinAt f s x ∧
    P (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm)
      ((chartAt H x).target ∩ ↑(chartAt H x).symm ⁻¹' (s ∩ f ⁻¹' (chartAt H' (f x)).source)) (↑(chartAt H x) x)
-- StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_source_aux
theorem liftPropWithinAt_indep_chart_source_aux (g : M → H') (he : e ∈ G.maximalAtlas M)
    (xe : x ∈ e.source) (he' : e' ∈ G.maximalAtlas M) (xe' : x ∈ e'.source) :
    P (g ∘ e.symm) (e.symm ⁻¹' s) (e x) ↔ P (g ∘ e'.symm) (e'.symm ⁻¹' s) (e' x) 
Arguments: {H : Type u_1} {M : Type u_2} {H' : Type u_3} [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] [TopologicalSpace H'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {e e' : PartialHomeomorph M H} {P : (H → H') → Set H → H → Prop} {s : Set M} {x : M} (hG : G.LocalInvariantProp G' P) (g : M → H') (he : e ∈ StructureGroupoid.maximalAtlas M G) (xe : x ∈ e.source) (he' : e' ∈ StructureGroupoid.maximalAtlas M G) (xe' : x ∈ e'.source)
Goal: P (g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x) ↔ P (g ∘ ↑e'.symm) (↑e'.symm ⁻¹' s) (↑e' x)