ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- SmoothBumpFunction.smooth
protected theorem smooth : Smooth I 𝓘(ℝ) f 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c) [T2Space M] [SmoothManifoldWithCorners I M]
Goal: Smooth I 𝓘(ℝ, ℝ) ↑f
-- SmoothBumpFunction.smoothAt
protected theorem smoothAt {x} : SmoothAt I 𝓘(ℝ) f x 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c) [T2Space M] [SmoothManifoldWithCorners I M] {x : M}
Goal: SmoothAt I 𝓘(ℝ, ℝ) (↑f) x
-- SmoothBumpFunction.continuous
protected theorem continuous : Continuous f 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c) [T2Space M] [SmoothManifoldWithCorners I M]
Goal: Continuous ↑f
-- SmoothBumpFunction.smooth_smul
theorem smooth_smul {G} [NormedAddCommGroup G] [NormedSpace ℝ G] {g : M → G}
    (hg : SmoothOn I 𝓘(ℝ, G) g (chartAt H c).source) : Smooth I 𝓘(ℝ, G) fun x => f x • g x 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {c : M} (f : SmoothBumpFunction I c) [T2Space M] [SmoothManifoldWithCorners I M] {G : Type u_1} [NormedAddCommGroup G] [NormedSpace ℝ G] {g : M → G} (hg : SmoothOn I 𝓘(ℝ, G) g (chartAt H c).source)
Goal: Smooth I 𝓘(ℝ, G) fun x => ↑f x • g x
-- ModelWithCorners.source_eq
@[ext] -- Porting note(#5171): was nolint has_nonempty_instance
structure ModelWithCorners (𝕜 : Type*) [NontriviallyNormedField 𝕜] (E : Type*)
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] (H : Type*) [TopologicalSpace H] extends
    PartialEquiv H E where
  source_eq : source = univ
  unique_diff' : UniqueDiffOn 𝕜 toPartialEquiv.target
  continuous_toFun : Continuous toFun := by continuity
  continuous_invFun : Continuous invFun := by continuity
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (self : ModelWithCorners 𝕜 E H)
Goal: self.source = Set.univ
-- ModelWithCorners.unique_diff'
@[ext] -- Porting note(#5171): was nolint has_nonempty_instance
structure ModelWithCorners (𝕜 : Type*) [NontriviallyNormedField 𝕜] (E : Type*)
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] (H : Type*) [TopologicalSpace H] extends
    PartialEquiv H E where
  source_eq : source = univ
  unique_diff' : UniqueDiffOn 𝕜 toPartialEquiv.target
  continuous_toFun : Continuous toFun := by continuity
  continuous_invFun : Continuous invFun := by continuity
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (self : ModelWithCorners 𝕜 E H)
Goal: UniqueDiffOn 𝕜 self.target
-- ModelWithCorners.continuous_toFun
@[ext] -- Porting note(#5171): was nolint has_nonempty_instance
structure ModelWithCorners (𝕜 : Type*) [NontriviallyNormedField 𝕜] (E : Type*)
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] (H : Type*) [TopologicalSpace H] extends
    PartialEquiv H E where
  source_eq : source = univ
  unique_diff' : UniqueDiffOn 𝕜 toPartialEquiv.target
  continuous_toFun : Continuous toFun := by continuity
  continuous_invFun : Continuous invFun := by continuity
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (self : ModelWithCorners 𝕜 E H)
Goal: Continuous ↑self.toPartialEquiv
-- ModelWithCorners.continuous_invFun
@[ext] -- Porting note(#5171): was nolint has_nonempty_instance
structure ModelWithCorners (𝕜 : Type*) [NontriviallyNormedField 𝕜] (E : Type*)
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] (H : Type*) [TopologicalSpace H] extends
    PartialEquiv H E where
  source_eq : source = univ
  unique_diff' : UniqueDiffOn 𝕜 toPartialEquiv.target
  continuous_toFun : Continuous toFun := by continuity
  continuous_invFun : Continuous invFun := by continuity
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (self : ModelWithCorners 𝕜 E H)
Goal: Continuous self.invFun
-- ModelWithCorners.toPartialEquiv_coe
@[simp, mfld_simps]
theorem toPartialEquiv_coe : (I.toPartialEquiv : H → E) = I 
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)
Goal: ↑I.toPartialEquiv = ↑I
-- ModelWithCorners.mk_coe
@[simp, mfld_simps]
theorem mk_coe (e : PartialEquiv H E) (a b c d) :
    ((ModelWithCorners.mk e a b c d : ModelWithCorners 𝕜 E H) : H → E) = (e : H → E) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (e : PartialEquiv H E) (a : e.source = Set.univ) (b : UniqueDiffOn 𝕜 e.target) (c : Continuous ↑e) (d : Continuous e.invFun)
Goal: ↑{ toPartialEquiv := e, source_eq := a, unique_diff' := b, continuous_toFun := c, continuous_invFun := d } = ↑e
-- ModelWithCorners.toPartialEquiv_coe_symm
@[simp, mfld_simps]
theorem toPartialEquiv_coe_symm : (I.toPartialEquiv.symm : E → H) = I.symm 
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)
Goal: ↑I.symm = ↑I.symm
-- ModelWithCorners.mk_symm
@[simp, mfld_simps]
theorem mk_symm (e : PartialEquiv H E) (a b c d) :
    (ModelWithCorners.mk e a b c d : ModelWithCorners 𝕜 E H).symm = e.symm 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (e : PartialEquiv H E) (a : e.source = Set.univ) (b : UniqueDiffOn 𝕜 e.target) (c : Continuous ↑e) (d : Continuous e.invFun)
Goal: { toPartialEquiv := e, source_eq := a, unique_diff' := b, continuous_toFun := c, continuous_invFun := d }.symm = e.symm
-- ModelWithCorners.continuous
@[continuity]
protected theorem continuous : Continuous I 
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)
Goal: Continuous ↑I
-- ModelWithCorners.continuousAt
protected theorem continuousAt {x} : ContinuousAt I x 
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) {x : H}
Goal: ContinuousAt (↑I) x
-- ModelWithCorners.continuousWithinAt
protected theorem continuousWithinAt {s x} : ContinuousWithinAt I s x 
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) {s : Set H} {x : H}
Goal: ContinuousWithinAt (↑I) s x
-- ModelWithCorners.continuous_symm
@[continuity]
theorem continuous_symm : Continuous I.symm 
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)
Goal: Continuous ↑I.symm
-- ModelWithCorners.continuousAt_symm
theorem continuousAt_symm {x} : ContinuousAt I.symm x 
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) {x : E}
Goal: ContinuousAt (↑I.symm) x
-- ModelWithCorners.continuousWithinAt_symm
theorem continuousWithinAt_symm {s x} : ContinuousWithinAt I.symm s x 
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) {s : Set E} {x : E}
Goal: ContinuousWithinAt (↑I.symm) s x
-- ModelWithCorners.continuousOn_symm
theorem continuousOn_symm {s} : ContinuousOn I.symm s 
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) {s : Set E}
Goal: ContinuousOn (↑I.symm) s
-- ModelWithCorners.target_eq
@[simp, mfld_simps]
theorem target_eq : I.target = range (I : H → 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)
Goal: I.target = Set.range ↑I
-- ModelWithCorners.unique_diff
protected theorem unique_diff : UniqueDiffOn 𝕜 (range I) 
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)
Goal: UniqueDiffOn 𝕜 (Set.range ↑I)
-- ModelWithCorners.left_inv
@[simp, mfld_simps]
protected theorem left_inv (x : H) : I.symm (I x) = x 
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) (x : H)
Goal: ↑I.symm (↑I x) = x
-- ModelWithCorners.leftInverse
protected theorem leftInverse : LeftInverse I.symm I 
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)
Goal: Function.LeftInverse ↑I.symm ↑I
-- ModelWithCorners.injective
theorem injective : Injective I 
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)
Goal: Function.Injective ↑I
-- ModelWithCorners.symm_comp_self
@[simp, mfld_simps]
theorem symm_comp_self : I.symm ∘ I = id 
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)
Goal: ↑I.symm ∘ ↑I = id
-- ModelWithCorners.rightInvOn
protected theorem rightInvOn : RightInvOn I.symm I (range I) 
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)
Goal: Set.RightInvOn (↑I.symm) (↑I) (Set.range ↑I)
-- ModelWithCorners.right_inv
@[simp, mfld_simps]
protected theorem right_inv {x : E} (hx : x ∈ range I) : I (I.symm x) = x 
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) {x : E} (hx : x ∈ Set.range ↑I)
Goal: ↑I (↑I.symm x) = x
-- ModelWithCorners.preimage_image
theorem preimage_image (s : Set H) : I ⁻¹' (I '' s) = s 
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) (s : Set H)
Goal: ↑I ⁻¹' (↑I '' s) = s
-- ModelWithCorners.image_eq
protected theorem image_eq (s : Set H) : I '' s = I.symm ⁻¹' s ∩ range I 
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) (s : Set H)
Goal: ↑I '' s = ↑I.symm ⁻¹' s ∩ Set.range ↑I
-- ModelWithCorners.closedEmbedding
protected theorem closedEmbedding : ClosedEmbedding I 
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)
Goal: ClosedEmbedding ↑I
-- ModelWithCorners.isClosed_range
theorem isClosed_range : IsClosed (range I) 
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)
Goal: IsClosed (Set.range ↑I)
-- ModelWithCorners.map_nhds_eq
theorem map_nhds_eq (x : H) : map I (𝓝 x) = 𝓝[range I] I x 
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) (x : H)
Goal: Filter.map (↑I) (𝓝 x) = 𝓝[Set.range ↑I] ↑I x
-- ModelWithCorners.map_nhdsWithin_eq
theorem map_nhdsWithin_eq (s : Set H) (x : H) : map I (𝓝[s] x) = 𝓝[I '' s] I x 
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) (s : Set H) (x : H)
Goal: Filter.map (↑I) (𝓝[s] x) = 𝓝[↑I '' s] ↑I x
-- ModelWithCorners.image_mem_nhdsWithin
theorem image_mem_nhdsWithin {x : H} {s : Set H} (hs : s ∈ 𝓝 x) : I '' s ∈ 𝓝[range I] I x 
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) {x : H} {s : Set H} (hs : s ∈ 𝓝 x)
Goal: ↑I '' s ∈ 𝓝[Set.range ↑I] ↑I x
-- ModelWithCorners.symm_map_nhdsWithin_image
theorem symm_map_nhdsWithin_image {x : H} {s : Set H} : map I.symm (𝓝[I '' s] I x) = 𝓝[s] x 
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) {x : H} {s : Set H}
Goal: Filter.map (↑I.symm) (𝓝[↑I '' s] ↑I x) = 𝓝[s] x
-- ModelWithCorners.symm_map_nhdsWithin_range
theorem symm_map_nhdsWithin_range (x : H) : map I.symm (𝓝[range I] I x) = 𝓝 x 
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) (x : H)
Goal: Filter.map (↑I.symm) (𝓝[Set.range ↑I] ↑I x) = 𝓝 x
-- ModelWithCorners.unique_diff_preimage
theorem unique_diff_preimage {s : Set H} (hs : IsOpen s) :
    UniqueDiffOn 𝕜 (I.symm ⁻¹' s ∩ range I) 
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) {s : Set H} (hs : IsOpen s)
Goal: UniqueDiffOn 𝕜 (↑I.symm ⁻¹' s ∩ Set.range ↑I)
-- ModelWithCorners.unique_diff_preimage_source
theorem unique_diff_preimage_source {β : Type*} [TopologicalSpace β] {e : PartialHomeomorph H β} :
    UniqueDiffOn 𝕜 (I.symm ⁻¹' e.source ∩ range I) 
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) {β : Type u_4} [TopologicalSpace β] {e : PartialHomeomorph H β}
Goal: UniqueDiffOn 𝕜 (↑I.symm ⁻¹' e.source ∩ Set.range ↑I)
-- ModelWithCorners.unique_diff_at_image
theorem unique_diff_at_image {x : H} : UniqueDiffWithinAt 𝕜 (range I) (I x) 
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) {x : H}
Goal: UniqueDiffWithinAt 𝕜 (Set.range ↑I) (↑I x)
-- ModelWithCorners.symm_continuousWithinAt_comp_right_iff
theorem symm_continuousWithinAt_comp_right_iff {X} [TopologicalSpace X] {f : H → X} {s : Set H}
    {x : H} :
    ContinuousWithinAt (f ∘ I.symm) (I.symm ⁻¹' s ∩ range I) (I x) ↔ ContinuousWithinAt f s x 
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) {X : Type u_4} [TopologicalSpace X] {f : H → X} {s : Set H} {x : H}
Goal: ContinuousWithinAt (f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ Set.range ↑I) (↑I x) ↔ ContinuousWithinAt f s x
-- ModelWithCorners.locallyCompactSpace
protected theorem locallyCompactSpace [LocallyCompactSpace E] (I : ModelWithCorners 𝕜 E H) :
    LocallyCompactSpace H 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] [LocallyCompactSpace E] (I : ModelWithCorners 𝕜 E H)
Goal: LocallyCompactSpace H
-- ModelWithCorners.secondCountableTopology
protected theorem secondCountableTopology [SecondCountableTopology E] (I : ModelWithCorners 𝕜 E H) :
    SecondCountableTopology H 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] [SecondCountableTopology E] (I : ModelWithCorners 𝕜 E H)
Goal: SecondCountableTopology H
-- modelWithCornersSelf_partialEquiv
@[simp, mfld_simps]
theorem modelWithCornersSelf_partialEquiv : 𝓘(𝕜, E).toPartialEquiv = PartialEquiv.refl E 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (E : Type u_2) [NormedAddCommGroup E] [NormedSpace 𝕜 E]
Goal: 𝓘(𝕜, E).toPartialEquiv = PartialEquiv.refl E
-- modelWithCornersSelf_coe
@[simp, mfld_simps]
theorem modelWithCornersSelf_coe : (𝓘(𝕜, E) : E → E) = id 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (E : Type u_2) [NormedAddCommGroup E] [NormedSpace 𝕜 E]
Goal: ↑𝓘(𝕜, E) = id
-- modelWithCornersSelf_coe_symm
@[simp, mfld_simps]
theorem modelWithCornersSelf_coe_symm : (𝓘(𝕜, E).symm : E → E) = id 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] (E : Type u_2) [NormedAddCommGroup E] [NormedSpace 𝕜 E]
Goal: ↑𝓘(𝕜, E).symm = id
-- ModelWithCorners.prod_target
@[simps (config := .lemmasOnly)]
def ModelWithCorners.prod {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v}
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type w} [TopologicalSpace H]
    (I : ModelWithCorners 𝕜 E H) {E' : Type v'} [NormedAddCommGroup E'] [NormedSpace 𝕜 E']
    {H' : Type w'} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') :
    ModelWithCorners 𝕜 (E × E') (ModelProd H H') 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type w} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {E' : Type v'} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type w'} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H')
Goal: (I.prod I').target = (I.prod I'.toPartialEquiv).target
-- ModelWithCorners.prod_source
@[simps (config := .lemmasOnly)]
def ModelWithCorners.prod {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v}
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type w} [TopologicalSpace H]
    (I : ModelWithCorners 𝕜 E H) {E' : Type v'} [NormedAddCommGroup E'] [NormedSpace 𝕜 E']
    {H' : Type w'} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') :
    ModelWithCorners 𝕜 (E × E') (ModelProd H H') 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type w} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {E' : Type v'} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type w'} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H')
Goal: (I.prod I').source = {x | x.1 ∈ I.source ∧ x.2 ∈ I'.source}
-- ModelWithCorners.prod_apply
@[simps (config := .lemmasOnly)]
def ModelWithCorners.prod {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v}
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type w} [TopologicalSpace H]
    (I : ModelWithCorners 𝕜 E H) {E' : Type v'} [NormedAddCommGroup E'] [NormedSpace 𝕜 E']
    {H' : Type w'} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') :
    ModelWithCorners 𝕜 (E × E') (ModelProd H H') 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type w} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {E' : Type v'} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type w'} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') (x : ModelProd H H')
Goal: ↑(I.prod I') x = (↑I x.1, ↑I' x.2)
-- ModelWithCorners.prod_symm_apply
@[simps (config := .lemmasOnly)]
def ModelWithCorners.prod {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v}
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type w} [TopologicalSpace H]
    (I : ModelWithCorners 𝕜 E H) {E' : Type v'} [NormedAddCommGroup E'] [NormedSpace 𝕜 E']
    {H' : Type w'} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') :
    ModelWithCorners 𝕜 (E × E') (ModelProd H H') 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type w} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {E' : Type v'} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type w'} [TopologicalSpace H'] (I' : ModelWithCorners 𝕜 E' H') (x : E × E')
Goal: ↑(I.prod I').symm x = (↑I.symm x.1, ↑I'.symm x.2)
-- modelWithCorners_prod_toPartialEquiv
@[simp, mfld_simps]
theorem modelWithCorners_prod_toPartialEquiv :
    (I.prod J).toPartialEquiv = I.toPartialEquiv.prod J.toPartialEquiv 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_6} [TopologicalSpace H] {G : Type u_8} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G}
Goal: (I.prod J).toPartialEquiv = I.prod J.toPartialEquiv
-- modelWithCorners_prod_coe
@[simp, mfld_simps]
theorem modelWithCorners_prod_coe (I : ModelWithCorners 𝕜 E H) (I' : ModelWithCorners 𝕜 E' H') :
    (I.prod I' : _ × _ → _ × _) = Prod.map I I' 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_6} [TopologicalSpace H] {H' : Type u_7} [TopologicalSpace H'] (I : ModelWithCorners 𝕜 E H) (I' : ModelWithCorners 𝕜 E' H')
Goal: ↑(I.prod I') = Prod.map ↑I ↑I'
-- modelWithCorners_prod_coe_symm
@[simp, mfld_simps]
theorem modelWithCorners_prod_coe_symm (I : ModelWithCorners 𝕜 E H)
    (I' : ModelWithCorners 𝕜 E' H') :
    ((I.prod I').symm : _ × _ → _ × _) = Prod.map I.symm I'.symm 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_6} [TopologicalSpace H] {H' : Type u_7} [TopologicalSpace H'] (I : ModelWithCorners 𝕜 E H) (I' : ModelWithCorners 𝕜 E' H')
Goal: ↑(I.prod I').symm = Prod.map ↑I.symm ↑I'.symm
-- modelWithCornersSelf_prod
theorem modelWithCornersSelf_prod : 𝓘(𝕜, E × F) = 𝓘(𝕜, E).prod 𝓘(𝕜, F) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
Goal: 𝓘(𝕜, E × F) = 𝓘(𝕜, E).prod 𝓘(𝕜, F)
-- ModelWithCorners.range_prod
theorem ModelWithCorners.range_prod : range (I.prod J) = range I ×ˢ range J 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_6} [TopologicalSpace H] {G : Type u_8} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G}
Goal: Set.range ↑(I.prod J) = Set.range ↑I ×ˢ Set.range ↑J
-- ModelWithCorners.Boundaryless.range_eq_univ
class ModelWithCorners.Boundaryless {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*}
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H]
    (I : ModelWithCorners 𝕜 E H) : Prop where
  range_eq_univ : range I = univ
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} [self : I.Boundaryless]
Goal: Set.range ↑I = Set.univ
-- ModelWithCorners.range_eq_univ
theorem ModelWithCorners.range_eq_univ {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*}
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H]
    (I : ModelWithCorners 𝕜 E H) [I.Boundaryless] :
    range I = univ 
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) [I.Boundaryless]
Goal: Set.range ↑I = Set.univ