ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_target_aux2
theorem liftPropWithinAt_indep_chart_target_aux2 (g : H → M') {x : H} {s : Set H}
    (hf : f ∈ G'.maximalAtlas M') (xf : g x ∈ f.source) (hf' : f' ∈ G'.maximalAtlas M')
    (xf' : g x ∈ f'.source) (hgs : ContinuousWithinAt g s x) : P (f ∘ g) s x ↔ P (f' ∘ g) s x 
Arguments: {H : Type u_1} {H' : Type u_3} {M' : Type u_4} [TopologicalSpace H] [TopologicalSpace H'] [TopologicalSpace M'] [ChartedSpace H' M'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {f f' : PartialHomeomorph M' H'} {P : (H → H') → Set H → H → Prop} (hG : G.LocalInvariantProp G' P) (g : H → M') {x : H} {s : Set H} (hf : f ∈ StructureGroupoid.maximalAtlas M' G') (xf : g x ∈ f.source) (hf' : f' ∈ StructureGroupoid.maximalAtlas M' G') (xf' : g x ∈ f'.source) (hgs : ContinuousWithinAt g s x)
Goal: P (↑f ∘ g) s x ↔ P (↑f' ∘ g) s x
-- StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_target_aux
theorem liftPropWithinAt_indep_chart_target_aux {g : X → M'} {e : PartialHomeomorph X H} {x : X}
    {s : Set X} (xe : x ∈ e.source) (hf : f ∈ G'.maximalAtlas M') (xf : g x ∈ f.source)
    (hf' : f' ∈ G'.maximalAtlas M') (xf' : g x ∈ f'.source) (hgs : ContinuousWithinAt g s x) :
    P (f ∘ g ∘ e.symm) (e.symm ⁻¹' s) (e x) ↔ P (f' ∘ g ∘ e.symm) (e.symm ⁻¹' s) (e x) 
Arguments: {H : Type u_1} {H' : Type u_3} {M' : Type u_4} {X : Type u_5} [TopologicalSpace H] [TopologicalSpace H'] [TopologicalSpace M'] [ChartedSpace H' M'] [TopologicalSpace X] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {f f' : PartialHomeomorph M' H'} {P : (H → H') → Set H → H → Prop} (hG : G.LocalInvariantProp G' P) {g : X → M'} {e : PartialHomeomorph X H} {x : X} {s : Set X} (xe : x ∈ e.source) (hf : f ∈ StructureGroupoid.maximalAtlas M' G') (xf : g x ∈ f.source) (hf' : f' ∈ StructureGroupoid.maximalAtlas M' G') (xf' : g x ∈ f'.source) (hgs : ContinuousWithinAt g s x)
Goal: P (↑f ∘ g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x) ↔ P (↑f' ∘ g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)
-- StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_aux
theorem liftPropWithinAt_indep_chart_aux (he : e ∈ G.maximalAtlas M) (xe : x ∈ e.source)
    (he' : e' ∈ G.maximalAtlas M) (xe' : x ∈ e'.source) (hf : f ∈ G'.maximalAtlas M')
    (xf : g x ∈ f.source) (hf' : f' ∈ G'.maximalAtlas M') (xf' : g x ∈ f'.source)
    (hgs : ContinuousWithinAt g s x) :
    P (f ∘ g ∘ e.symm) (e.symm ⁻¹' s) (e x) ↔ P (f' ∘ g ∘ e'.symm) (e'.symm ⁻¹' s) (e' 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'} {e e' : PartialHomeomorph M H} {f f' : PartialHomeomorph M' H'} {P : (H → H') → Set H → H → Prop} {g : M → M'} {s : Set M} {x : M} (hG : G.LocalInvariantProp G' P) (he : e ∈ StructureGroupoid.maximalAtlas M G) (xe : x ∈ e.source) (he' : e' ∈ StructureGroupoid.maximalAtlas M G) (xe' : x ∈ e'.source) (hf : f ∈ StructureGroupoid.maximalAtlas M' G') (xf : g x ∈ f.source) (hf' : f' ∈ StructureGroupoid.maximalAtlas M' G') (xf' : g x ∈ f'.source) (hgs : ContinuousWithinAt g s x)
Goal: P (↑f ∘ g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x) ↔ P (↑f' ∘ g ∘ ↑e'.symm) (↑e'.symm ⁻¹' s) (↑e' x)
-- StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart
theorem liftPropWithinAt_indep_chart [HasGroupoid M G] [HasGroupoid M' G']
    (he : e ∈ G.maximalAtlas M) (xe : x ∈ e.source) (hf : f ∈ G'.maximalAtlas M')
    (xf : g x ∈ f.source) :
    LiftPropWithinAt P g s x ↔
    ContinuousWithinAt g s x ∧ P (f ∘ g ∘ e.symm) (e.symm ⁻¹' s) (e 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'} {e : PartialHomeomorph M H} {f : PartialHomeomorph M' H'} {P : (H → H') → Set H → H → Prop} {g : M → M'} {s : Set M} {x : M} (hG : G.LocalInvariantProp G' P) [HasGroupoid M G] [HasGroupoid M' G'] (he : e ∈ StructureGroupoid.maximalAtlas M G) (xe : x ∈ e.source) (hf : f ∈ StructureGroupoid.maximalAtlas M' G') (xf : g x ∈ f.source)
Goal: ChartedSpace.LiftPropWithinAt P g s x ↔ ContinuousWithinAt g s x ∧ P (↑f ∘ g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)
-- StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_source
theorem liftPropWithinAt_indep_chart_source [HasGroupoid M G] (he : e ∈ G.maximalAtlas M)
    (xe : x ∈ e.source) :
    LiftPropWithinAt P g s x ↔ LiftPropWithinAt P (g ∘ e.symm) (e.symm ⁻¹' s) (e 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'} {e : PartialHomeomorph M H} {P : (H → H') → Set H → H → Prop} {g : M → M'} {s : Set M} {x : M} (hG : G.LocalInvariantProp G' P) [HasGroupoid M G] (he : e ∈ StructureGroupoid.maximalAtlas M G) (xe : x ∈ e.source)
Goal: ChartedSpace.LiftPropWithinAt P g s x ↔ ChartedSpace.LiftPropWithinAt P (g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)
-- StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_target
theorem liftPropWithinAt_indep_chart_target [HasGroupoid M' G'] (hf : f ∈ G'.maximalAtlas M')
    (xf : g x ∈ f.source) :
    LiftPropWithinAt P g s x ↔ ContinuousWithinAt g s x ∧ LiftPropWithinAt P (f ∘ g) s 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'} {f : PartialHomeomorph M' H'} {P : (H → H') → Set H → H → Prop} {g : M → M'} {s : Set M} {x : M} (hG : G.LocalInvariantProp G' P) [HasGroupoid M' G'] (hf : f ∈ StructureGroupoid.maximalAtlas M' G') (xf : g x ∈ f.source)
Goal: ChartedSpace.LiftPropWithinAt P g s x ↔ ContinuousWithinAt g s x ∧ ChartedSpace.LiftPropWithinAt P (↑f ∘ g) s x
-- StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart'
theorem liftPropWithinAt_indep_chart' [HasGroupoid M G] [HasGroupoid M' G']
    (he : e ∈ G.maximalAtlas M) (xe : x ∈ e.source) (hf : f ∈ G'.maximalAtlas M')
    (xf : g x ∈ f.source) :
    LiftPropWithinAt P g s x ↔
      ContinuousWithinAt g s x ∧ LiftPropWithinAt P (f ∘ g ∘ e.symm) (e.symm ⁻¹' s) (e 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'} {e : PartialHomeomorph M H} {f : PartialHomeomorph M' H'} {P : (H → H') → Set H → H → Prop} {g : M → M'} {s : Set M} {x : M} (hG : G.LocalInvariantProp G' P) [HasGroupoid M G] [HasGroupoid M' G'] (he : e ∈ StructureGroupoid.maximalAtlas M G) (xe : x ∈ e.source) (hf : f ∈ StructureGroupoid.maximalAtlas M' G') (xf : g x ∈ f.source)
Goal: ChartedSpace.LiftPropWithinAt P g s x ↔
  ContinuousWithinAt g s x ∧ ChartedSpace.LiftPropWithinAt P (↑f ∘ g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)
-- StructureGroupoid.LocalInvariantProp.liftPropOn_indep_chart
theorem liftPropOn_indep_chart [HasGroupoid M G] [HasGroupoid M' G'] (he : e ∈ G.maximalAtlas M)
    (hf : f ∈ G'.maximalAtlas M') (h : LiftPropOn P g s) {y : H}
    (hy : y ∈ e.target ∩ e.symm ⁻¹' (s ∩ g ⁻¹' f.source)) :
    P (f ∘ g ∘ e.symm) (e.symm ⁻¹' s) y 
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'} {e : PartialHomeomorph M H} {f : PartialHomeomorph M' H'} {P : (H → H') → Set H → H → Prop} {g : M → M'} {s : Set M} (hG : G.LocalInvariantProp G' P) [HasGroupoid M G] [HasGroupoid M' G'] (he : e ∈ StructureGroupoid.maximalAtlas M G) (hf : f ∈ StructureGroupoid.maximalAtlas M' G') (h : ChartedSpace.LiftPropOn P g s) {y : H} (hy : y ∈ e.target ∩ ↑e.symm ⁻¹' (s ∩ g ⁻¹' f.source))
Goal: P (↑f ∘ g ∘ ↑e.symm) (↑e.symm ⁻¹' s) y
-- StructureGroupoid.LocalInvariantProp.liftPropWithinAt_inter'
theorem liftPropWithinAt_inter' (ht : t ∈ 𝓝[s] x) :
    LiftPropWithinAt P g (s ∩ t) x ↔ LiftPropWithinAt P g s 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} {g : M → M'} {s t : Set M} {x : M} (hG : G.LocalInvariantProp G' P) (ht : t ∈ 𝓝[s] x)
Goal: ChartedSpace.LiftPropWithinAt P g (s ∩ t) x ↔ ChartedSpace.LiftPropWithinAt P g s x
-- StructureGroupoid.LocalInvariantProp.liftPropWithinAt_inter
theorem liftPropWithinAt_inter (ht : t ∈ 𝓝 x) :
    LiftPropWithinAt P g (s ∩ t) x ↔ LiftPropWithinAt P g s 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} {g : M → M'} {s t : Set M} {x : M} (hG : G.LocalInvariantProp G' P) (ht : t ∈ 𝓝 x)
Goal: ChartedSpace.LiftPropWithinAt P g (s ∩ t) x ↔ ChartedSpace.LiftPropWithinAt P g s x
-- StructureGroupoid.LocalInvariantProp.liftPropAt_of_liftPropWithinAt
theorem liftPropAt_of_liftPropWithinAt (h : LiftPropWithinAt P g s x) (hs : s ∈ 𝓝 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'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {P : (H → H') → Set H → H → Prop} {g : M → M'} {s : Set M} {x : M} (hG : G.LocalInvariantProp G' P) (h : ChartedSpace.LiftPropWithinAt P g s x) (hs : s ∈ 𝓝 x)
Goal: ChartedSpace.LiftPropAt P g x
-- StructureGroupoid.LocalInvariantProp.liftPropWithinAt_of_liftPropAt_of_mem_nhds
theorem liftPropWithinAt_of_liftPropAt_of_mem_nhds (h : LiftPropAt P g x) (hs : s ∈ 𝓝 x) :
    LiftPropWithinAt P g s 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} {g : M → M'} {s : Set M} {x : M} (hG : G.LocalInvariantProp G' P) (h : ChartedSpace.LiftPropAt P g x) (hs : s ∈ 𝓝 x)
Goal: ChartedSpace.LiftPropWithinAt P g s x
-- StructureGroupoid.LocalInvariantProp.liftPropOn_of_locally_liftPropOn
theorem liftPropOn_of_locally_liftPropOn
    (h : ∀ x ∈ s, ∃ u, IsOpen u ∧ x ∈ u ∧ LiftPropOn P g (s ∩ u)) : LiftPropOn P g s 
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} {g : M → M'} {s : Set M} (hG : G.LocalInvariantProp G' P) (h : ∀ x ∈ s, ∃ u, IsOpen u ∧ x ∈ u ∧ ChartedSpace.LiftPropOn P g (s ∩ u))
Goal: ChartedSpace.LiftPropOn P g s
-- StructureGroupoid.LocalInvariantProp.liftProp_of_locally_liftPropOn
theorem liftProp_of_locally_liftPropOn (h : ∀ x, ∃ u, IsOpen u ∧ x ∈ u ∧ LiftPropOn P g u) :
    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'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {P : (H → H') → Set H → H → Prop} {g : M → M'} (hG : G.LocalInvariantProp G' P) (h : ∀ (x : M), ∃ u, IsOpen u ∧ x ∈ u ∧ ChartedSpace.LiftPropOn P g u)
Goal: ChartedSpace.LiftProp P g
-- StructureGroupoid.LocalInvariantProp.liftPropWithinAt_congr_of_eventuallyEq
theorem liftPropWithinAt_congr_of_eventuallyEq (h : LiftPropWithinAt P g s x) (h₁ : g' =ᶠ[𝓝[s] x] g)
    (hx : g' x = g x) : LiftPropWithinAt P g' s 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} {g g' : M → M'} {s : Set M} {x : M} (hG : G.LocalInvariantProp G' P) (h : ChartedSpace.LiftPropWithinAt P g s x) (h₁ : g' =ᶠ[𝓝[s] x] g) (hx : g' x = g x)
Goal: ChartedSpace.LiftPropWithinAt P g' s x
-- StructureGroupoid.LocalInvariantProp.liftPropWithinAt_congr_iff_of_eventuallyEq
theorem liftPropWithinAt_congr_iff_of_eventuallyEq (h₁ : g' =ᶠ[𝓝[s] x] g) (hx : g' x = g x) :
    LiftPropWithinAt P g' s x ↔ LiftPropWithinAt P g s 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} {g g' : M → M'} {s : Set M} {x : M} (hG : G.LocalInvariantProp G' P) (h₁ : g' =ᶠ[𝓝[s] x] g) (hx : g' x = g x)
Goal: ChartedSpace.LiftPropWithinAt P g' s x ↔ ChartedSpace.LiftPropWithinAt P g s x
-- StructureGroupoid.LocalInvariantProp.liftPropWithinAt_congr_iff
theorem liftPropWithinAt_congr_iff (h₁ : ∀ y ∈ s, g' y = g y) (hx : g' x = g x) :
    LiftPropWithinAt P g' s x ↔ LiftPropWithinAt P g s 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} {g g' : M → M'} {s : Set M} {x : M} (hG : G.LocalInvariantProp G' P) (h₁ : ∀ y ∈ s, g' y = g y) (hx : g' x = g x)
Goal: ChartedSpace.LiftPropWithinAt P g' s x ↔ ChartedSpace.LiftPropWithinAt P g s x
-- StructureGroupoid.LocalInvariantProp.liftPropWithinAt_congr
theorem liftPropWithinAt_congr (h : LiftPropWithinAt P g s x) (h₁ : ∀ y ∈ s, g' y = g y)
    (hx : g' x = g x) : LiftPropWithinAt P g' s 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} {g g' : M → M'} {s : Set M} {x : M} (hG : G.LocalInvariantProp G' P) (h : ChartedSpace.LiftPropWithinAt P g s x) (h₁ : ∀ y ∈ s, g' y = g y) (hx : g' x = g x)
Goal: ChartedSpace.LiftPropWithinAt P g' s x
-- StructureGroupoid.LocalInvariantProp.liftPropAt_congr_iff_of_eventuallyEq
theorem liftPropAt_congr_iff_of_eventuallyEq (h₁ : g' =ᶠ[𝓝 x] g) :
    LiftPropAt P g' 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'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {P : (H → H') → Set H → H → Prop} {g g' : M → M'} {x : M} (hG : G.LocalInvariantProp G' P) (h₁ : g' =ᶠ[𝓝 x] g)
Goal: ChartedSpace.LiftPropAt P g' x ↔ ChartedSpace.LiftPropAt P g x
-- StructureGroupoid.LocalInvariantProp.liftPropAt_congr_of_eventuallyEq
theorem liftPropAt_congr_of_eventuallyEq (h : LiftPropAt P g x) (h₁ : g' =ᶠ[𝓝 x] g) :
    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'] {G : StructureGroupoid H} {G' : StructureGroupoid H'} {P : (H → H') → Set H → H → Prop} {g g' : M → M'} {x : M} (hG : G.LocalInvariantProp G' P) (h : ChartedSpace.LiftPropAt P g x) (h₁ : g' =ᶠ[𝓝 x] g)
Goal: ChartedSpace.LiftPropAt P g' x
-- StructureGroupoid.LocalInvariantProp.liftPropOn_congr
theorem liftPropOn_congr (h : LiftPropOn P g s) (h₁ : ∀ y ∈ s, g' y = g y) : LiftPropOn P g' s 
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} {g g' : M → M'} {s : Set M} (hG : G.LocalInvariantProp G' P) (h : ChartedSpace.LiftPropOn P g s) (h₁ : ∀ y ∈ s, g' y = g y)
Goal: ChartedSpace.LiftPropOn P g' s
-- StructureGroupoid.LocalInvariantProp.liftPropOn_congr_iff
theorem liftPropOn_congr_iff (h₁ : ∀ y ∈ s, g' y = g y) : LiftPropOn P g' s ↔ LiftPropOn P g s 
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} {g g' : M → M'} {s : Set M} (hG : G.LocalInvariantProp G' P) (h₁ : ∀ y ∈ s, g' y = g y)
Goal: ChartedSpace.LiftPropOn P g' s ↔ ChartedSpace.LiftPropOn P g s
-- StructureGroupoid.LocalInvariantProp.liftPropWithinAt_mono_of_mem
theorem liftPropWithinAt_mono_of_mem
    (mono_of_mem : ∀ ⦃s x t⦄ ⦃f : H → H'⦄, s ∈ 𝓝[t] x → P f s x → P f t x)
    (h : LiftPropWithinAt P g s x) (hst : s ∈ 𝓝[t] x) : LiftPropWithinAt P g t 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'} {s t : Set M} {x : M} (mono_of_mem : ∀ ⦃s : Set H⦄ ⦃x : H⦄ ⦃t : Set H⦄ ⦃f : H → H'⦄, s ∈ 𝓝[t] x → P f s x → P f t x) (h : ChartedSpace.LiftPropWithinAt P g s x) (hst : s ∈ 𝓝[t] x)
Goal: ChartedSpace.LiftPropWithinAt P g t x
-- StructureGroupoid.LocalInvariantProp.liftPropWithinAt_mono
theorem liftPropWithinAt_mono (mono : ∀ ⦃s x t⦄ ⦃f : H → H'⦄, t ⊆ s → P f s x → P f t x)
    (h : LiftPropWithinAt P g s x) (hts : t ⊆ s) : LiftPropWithinAt P g t 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'} {s t : Set M} {x : M} (mono : ∀ ⦃s : Set H⦄ ⦃x : H⦄ ⦃t : Set H⦄ ⦃f : H → H'⦄, t ⊆ s → P f s x → P f t x) (h : ChartedSpace.LiftPropWithinAt P g s x) (hts : t ⊆ s)
Goal: ChartedSpace.LiftPropWithinAt P g t x
-- StructureGroupoid.LocalInvariantProp.liftPropWithinAt_of_liftPropAt
theorem liftPropWithinAt_of_liftPropAt (mono : ∀ ⦃s x t⦄ ⦃f : H → H'⦄, t ⊆ s → P f s x → P f t x)
    (h : LiftPropAt P g x) : LiftPropWithinAt P g s 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'} {s : Set M} {x : M} (mono : ∀ ⦃s : Set H⦄ ⦃x : H⦄ ⦃t : Set H⦄ ⦃f : H → H'⦄, t ⊆ s → P f s x → P f t x) (h : ChartedSpace.LiftPropAt P g x)
Goal: ChartedSpace.LiftPropWithinAt P g s x
-- StructureGroupoid.LocalInvariantProp.liftPropOn_mono
theorem liftPropOn_mono (mono : ∀ ⦃s x t⦄ ⦃f : H → H'⦄, t ⊆ s → P f s x → P f t x)
    (h : LiftPropOn P g t) (hst : s ⊆ t) : LiftPropOn P g s 
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'} {s t : Set M} (mono : ∀ ⦃s : Set H⦄ ⦃x : H⦄ ⦃t : Set H⦄ ⦃f : H → H'⦄, t ⊆ s → P f s x → P f t x) (h : ChartedSpace.LiftPropOn P g t) (hst : s ⊆ t)
Goal: ChartedSpace.LiftPropOn P g s
-- StructureGroupoid.LocalInvariantProp.liftPropOn_of_liftProp
theorem liftPropOn_of_liftProp (mono : ∀ ⦃s x t⦄ ⦃f : H → H'⦄, t ⊆ s → P f s x → P f t x)
    (h : LiftProp P g) : LiftPropOn P g s 
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'} {s : Set M} (mono : ∀ ⦃s : Set H⦄ ⦃x : H⦄ ⦃t : Set H⦄ ⦃f : H → H'⦄, t ⊆ s → P f s x → P f t x) (h : ChartedSpace.LiftProp P g)
Goal: ChartedSpace.LiftPropOn P g s
-- StructureGroupoid.LocalInvariantProp.liftPropAt_of_mem_maximalAtlas
theorem liftPropAt_of_mem_maximalAtlas [HasGroupoid M G] (hG : G.LocalInvariantProp G Q)
    (hQ : ∀ y, Q id univ y) (he : e ∈ maximalAtlas M G) (hx : x ∈ e.source) : LiftPropAt Q e x 
Arguments: {H : Type u_1} {M : Type u_2} [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] {G : StructureGroupoid H} {e : PartialHomeomorph M H} {x : M} {Q : (H → H) → Set H → H → Prop} [HasGroupoid M G] (hG : G.LocalInvariantProp G Q) (hQ : ∀ (y : H), Q id Set.univ y) (he : e ∈ StructureGroupoid.maximalAtlas M G) (hx : x ∈ e.source)
Goal: ChartedSpace.LiftPropAt Q (↑e) x
-- StructureGroupoid.LocalInvariantProp.liftPropOn_of_mem_maximalAtlas
theorem liftPropOn_of_mem_maximalAtlas [HasGroupoid M G] (hG : G.LocalInvariantProp G Q)
    (hQ : ∀ y, Q id univ y) (he : e ∈ maximalAtlas M G) : LiftPropOn Q e e.source 
Arguments: {H : Type u_1} {M : Type u_2} [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] {G : StructureGroupoid H} {e : PartialHomeomorph M H} {Q : (H → H) → Set H → H → Prop} [HasGroupoid M G] (hG : G.LocalInvariantProp G Q) (hQ : ∀ (y : H), Q id Set.univ y) (he : e ∈ StructureGroupoid.maximalAtlas M G)
Goal: ChartedSpace.LiftPropOn Q (↑e) e.source
-- StructureGroupoid.LocalInvariantProp.liftPropAt_symm_of_mem_maximalAtlas
theorem liftPropAt_symm_of_mem_maximalAtlas [HasGroupoid M G] {x : H}
    (hG : G.LocalInvariantProp G Q) (hQ : ∀ y, Q id univ y) (he : e ∈ maximalAtlas M G)
    (hx : x ∈ e.target) : LiftPropAt Q e.symm x 
Arguments: {H : Type u_1} {M : Type u_2} [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] {G : StructureGroupoid H} {e : PartialHomeomorph M H} {Q : (H → H) → Set H → H → Prop} [HasGroupoid M G] {x : H} (hG : G.LocalInvariantProp G Q) (hQ : ∀ (y : H), Q id Set.univ y) (he : e ∈ StructureGroupoid.maximalAtlas M G) (hx : x ∈ e.target)
Goal: ChartedSpace.LiftPropAt Q (↑e.symm) x
-- StructureGroupoid.LocalInvariantProp.liftPropOn_symm_of_mem_maximalAtlas
theorem liftPropOn_symm_of_mem_maximalAtlas [HasGroupoid M G] (hG : G.LocalInvariantProp G Q)
    (hQ : ∀ y, Q id univ y) (he : e ∈ maximalAtlas M G) : LiftPropOn Q e.symm e.target 
Arguments: {H : Type u_1} {M : Type u_2} [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] {G : StructureGroupoid H} {e : PartialHomeomorph M H} {Q : (H → H) → Set H → H → Prop} [HasGroupoid M G] (hG : G.LocalInvariantProp G Q) (hQ : ∀ (y : H), Q id Set.univ y) (he : e ∈ StructureGroupoid.maximalAtlas M G)
Goal: ChartedSpace.LiftPropOn Q (↑e.symm) e.target
-- StructureGroupoid.LocalInvariantProp.liftPropAt_chart
theorem liftPropAt_chart [HasGroupoid M G] (hG : G.LocalInvariantProp G Q) (hQ : ∀ y, Q id univ y) :
    LiftPropAt Q (chartAt (H := H) x) x 
Arguments: {H : Type u_1} {M : Type u_2} [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] {G : StructureGroupoid H} {x : M} {Q : (H → H) → Set H → H → Prop} [HasGroupoid M G] (hG : G.LocalInvariantProp G Q) (hQ : ∀ (y : H), Q id Set.univ y)
Goal: ChartedSpace.LiftPropAt Q (↑(chartAt H x)) x
-- StructureGroupoid.LocalInvariantProp.liftPropOn_chart
theorem liftPropOn_chart [HasGroupoid M G] (hG : G.LocalInvariantProp G Q) (hQ : ∀ y, Q id univ y) :
    LiftPropOn Q (chartAt (H := H) x) (chartAt (H := H) x).source 
Arguments: {H : Type u_1} {M : Type u_2} [TopologicalSpace H] [TopologicalSpace M] [ChartedSpace H M] {G : StructureGroupoid H} {x : M} {Q : (H → H) → Set H → H → Prop} [HasGroupoid M G] (hG : G.LocalInvariantProp G Q) (hQ : ∀ (y : H), Q id Set.univ y)
Goal: ChartedSpace.LiftPropOn Q (↑(chartAt H x)) (chartAt H x).source