ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LocallyConstant.evalₐ_apply
@[simps!]
def evalₐ (R : Type*) [CommSemiring R] [Semiring Y] [Algebra R Y] (x : X) :
    LocallyConstant X Y →ₐ[R] Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (R : Type u_6) [CommSemiring R] [Semiring Y] [Algebra R Y] (x : X)
Goal: ∀ (a : LocallyConstant X Y), (LocallyConstant.evalₐ R x) a = a x
-- LocallyConstant.comapAddMonoidHom_apply
@[to_additive (attr := simps) "`LocallyConstant.comap` as an `AddMonoidHom`."]
def comapMonoidHom [MulOneClass Z]  (f : C(X, Y)) :
    LocallyConstant Y Z →* LocallyConstant X Z 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} [AddZeroClass Z] (f : C(X, Y)) (g : LocallyConstant Y Z)
Goal: (LocallyConstant.comapAddMonoidHom f) g = LocallyConstant.comap f g
-- LocallyConstant.comapMonoidHom_apply
@[to_additive (attr := simps) "`LocallyConstant.comap` as an `AddMonoidHom`."]
def comapMonoidHom [MulOneClass Z]  (f : C(X, Y)) :
    LocallyConstant Y Z →* LocallyConstant X Z 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} [MulOneClass Z] (f : C(X, Y)) (g : LocallyConstant Y Z)
Goal: (LocallyConstant.comapMonoidHom f) g = LocallyConstant.comap f g
-- LocallyConstant.comapₗ_apply_apply
@[simps!]
def comapₗ (R : Type*) [Semiring R] [AddCommMonoid Z] [Module R Z] (f : C(X, Y)) :
    LocallyConstant Y Z →ₗ[R] LocallyConstant X Z 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} (R : Type u_7) [Semiring R] [AddCommMonoid Z] [Module R Z] (f : C(X, Y)) (g : LocallyConstant Y Z)
Goal: ∀ (a : X), ((LocallyConstant.comapₗ R f) g) a = g (f a)
-- LocallyConstant.comapRingHom_apply_apply
@[simps!]
def comapRingHom [Semiring Z] (f : C(X, Y)) :
    LocallyConstant Y Z →+* LocallyConstant X Z 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} [Semiring Z] (f : C(X, Y)) (g : LocallyConstant Y Z)
Goal: ∀ (a : X), ((LocallyConstant.comapRingHom f) g) a = g (f a)
-- LocallyConstant.comapₐ_apply_apply
@[simps!]
def comapₐ (R : Type*) [CommSemiring R] [Semiring Z] [Algebra R Z]
    (f : C(X, Y)) : LocallyConstant Y Z →ₐ[R] LocallyConstant X Z 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} (R : Type u_7) [CommSemiring R] [Semiring Z] [Algebra R Z] (f : C(X, Y)) (g : LocallyConstant Y Z)
Goal: ∀ (a : X), ((LocallyConstant.comapₐ R f) g) a = g (f a)
-- LocallyConstant.ker_comapₗ
lemma ker_comapₗ [Semiring R] [AddCommMonoid Z] [Module R Z] (f : C(X, Y))
    (hfs : Function.Surjective f) :
    LinearMap.ker (comapₗ R f : LocallyConstant Y Z →ₗ[R] LocallyConstant X Z) = ⊥ 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {R : Type u_5} [TopologicalSpace Y] {Z : Type u_6} [Semiring R] [AddCommMonoid Z] [Module R Z] (f : C(X, Y)) (hfs : Function.Surjective ⇑f)
Goal: LinearMap.ker (LocallyConstant.comapₗ R f) = ⊥
-- LocallyConstant.congrLeftₗ_symm_apply_apply
@[simps!]
def congrLeftₗ (R : Type*) [Semiring R] [AddCommMonoid Z] [Module R Z] (e : X ≃ₜ Y) :
    LocallyConstant X Z ≃ₗ[R] LocallyConstant Y Z 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} (R : Type u_7) [Semiring R] [AddCommMonoid Z] [Module R Z] (e : X ≃ₜ Y)
Goal: ∀ (a : LocallyConstant Y Z) (a_1 : X), ((LocallyConstant.congrLeftₗ R e).symm a) a_1 = a (e a_1)
-- LocallyConstant.congrLeftₗ_apply_apply
@[simps!]
def congrLeftₗ (R : Type*) [Semiring R] [AddCommMonoid Z] [Module R Z] (e : X ≃ₜ Y) :
    LocallyConstant X Z ≃ₗ[R] LocallyConstant Y Z 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} (R : Type u_7) [Semiring R] [AddCommMonoid Z] [Module R Z] (e : X ≃ₜ Y) (g : LocallyConstant X Z)
Goal: ∀ (a : Y), ((LocallyConstant.congrLeftₗ R e) g) a = g (e.symm a)
-- LocallyConstant.congrLeftRingEquiv_apply_apply
@[simps!]
def congrLeftRingEquiv [Semiring Z] (e : X ≃ₜ Y) :
    LocallyConstant X Z ≃+* LocallyConstant Y Z 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} [Semiring Z] (e : X ≃ₜ Y) (g : LocallyConstant X Z)
Goal: ∀ (a : Y), ((LocallyConstant.congrLeftRingEquiv e) g) a = g (e.symm a)
-- LocallyConstant.congrLeftRingEquiv_symm_apply_apply
@[simps!]
def congrLeftRingEquiv [Semiring Z] (e : X ≃ₜ Y) :
    LocallyConstant X Z ≃+* LocallyConstant Y Z 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} [Semiring Z] (e : X ≃ₜ Y) (g : LocallyConstant Y Z)
Goal: ∀ (a : X), ((LocallyConstant.congrLeftRingEquiv e).symm g) a = g (e a)
-- LocallyConstant.congrLeftₐ_apply_apply
@[simps!]
def congrLeftₐ (R : Type*) [CommSemiring R] [Semiring Z] [Algebra R Z] (e : X ≃ₜ Y) :
    LocallyConstant X Z ≃ₐ[R] LocallyConstant Y Z 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} (R : Type u_7) [CommSemiring R] [Semiring Z] [Algebra R Z] (e : X ≃ₜ Y) (g : LocallyConstant X Z)
Goal: ∀ (a : Y), ((LocallyConstant.congrLeftₐ R e) g) a = g (e.symm a)
-- LocallyConstant.congrLeftₐ_symm_apply_apply
@[simps!]
def congrLeftₐ (R : Type*) [CommSemiring R] [Semiring Z] [Algebra R Z] (e : X ≃ₜ Y) :
    LocallyConstant X Z ≃ₐ[R] LocallyConstant Y Z 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {Z : Type u_6} (R : Type u_7) [CommSemiring R] [Semiring Z] [Algebra R Z] (e : X ≃ₜ Y) (g : LocallyConstant Y Z)
Goal: ∀ (a : X), ((LocallyConstant.congrLeftₐ R e).symm g) a = g (e a)
-- LocallyConstant.mapMonoidHom_apply
@[to_additive (attr := simps) "`LocallyConstant.map` as an `AddMonoidHom`."]
def mapMonoidHom [MulOneClass Y] [MulOneClass Z] (f : Y →* Z) :
    LocallyConstant X Y →* LocallyConstant X Z 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} [MulOneClass Y] [MulOneClass Z] (f : Y →* Z) (g : LocallyConstant X Y)
Goal: (LocallyConstant.mapMonoidHom f) g = LocallyConstant.map (⇑f) g
-- LocallyConstant.mapAddMonoidHom_apply
@[to_additive (attr := simps) "`LocallyConstant.map` as an `AddMonoidHom`."]
def mapMonoidHom [MulOneClass Y] [MulOneClass Z] (f : Y →* Z) :
    LocallyConstant X Y →* LocallyConstant X Z 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} [AddZeroClass Y] [AddZeroClass Z] (f : Y →+ Z) (g : LocallyConstant X Y)
Goal: (LocallyConstant.mapAddMonoidHom f) g = LocallyConstant.map (⇑f) g
-- LocallyConstant.mapₗ_apply_apply
@[simps!]
def mapₗ (R : Type*) [Semiring R] [AddCommMonoid Y] [Module R Y]
    [AddCommMonoid Z] [Module R Z] (f : Y →ₗ[R] Z) :
    LocallyConstant X Y →ₗ[R] LocallyConstant X Z 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} (R : Type u_7) [Semiring R] [AddCommMonoid Y] [Module R Y] [AddCommMonoid Z] [Module R Z] (f : Y →ₗ[R] Z) (g : LocallyConstant X Y)
Goal: ∀ (a : X), ((LocallyConstant.mapₗ R f) g) a = f (g a)
-- LocallyConstant.mapRingHom_apply_apply
@[simps!]
def mapRingHom [Semiring Y] [Semiring Z] (f : Y →+* Z) :
    LocallyConstant X Y →+* LocallyConstant X Z 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} [Semiring Y] [Semiring Z] (f : Y →+* Z) (g : LocallyConstant X Y)
Goal: ∀ (a : X), ((LocallyConstant.mapRingHom f) g) a = f (g a)
-- LocallyConstant.mapₐ_apply_apply
@[simps!]
def mapₐ (R : Type*) [CommSemiring R] [Semiring Y] [Algebra R Y] [Semiring Z] [Algebra R Z]
    (f : Y →ₐ[R] Z) : LocallyConstant X Y →ₐ[R] LocallyConstant X Z 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} (R : Type u_7) [CommSemiring R] [Semiring Y] [Algebra R Y] [Semiring Z] [Algebra R Z] (f : Y →ₐ[R] Z) (g : LocallyConstant X Y)
Goal: ∀ (a : X), ((LocallyConstant.mapₐ R f) g) a = f (g a)
-- LocallyConstant.congrRightₗ_symm_apply_apply
@[simps!]
def congrRightₗ (R : Type*) [Semiring R] [AddCommMonoid Y] [Module R Y]
    [AddCommMonoid Z] [Module R Z] (e : Y ≃ₗ[R] Z) :
    LocallyConstant X Y ≃ₗ[R] LocallyConstant X Z 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} (R : Type u_7) [Semiring R] [AddCommMonoid Y] [Module R Y] [AddCommMonoid Z] [Module R Z] (e : Y ≃ₗ[R] Z)
Goal: ∀ (a : LocallyConstant X Z) (a_1 : X), ((LocallyConstant.congrRightₗ R e).symm a) a_1 = e.symm (a a_1)
-- LocallyConstant.congrRightₗ_apply_apply
@[simps!]
def congrRightₗ (R : Type*) [Semiring R] [AddCommMonoid Y] [Module R Y]
    [AddCommMonoid Z] [Module R Z] (e : Y ≃ₗ[R] Z) :
    LocallyConstant X Y ≃ₗ[R] LocallyConstant X Z 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} (R : Type u_7) [Semiring R] [AddCommMonoid Y] [Module R Y] [AddCommMonoid Z] [Module R Z] (e : Y ≃ₗ[R] Z) (g : LocallyConstant X Y)
Goal: ∀ (a : X), ((LocallyConstant.congrRightₗ R e) g) a = e (g a)
-- LocallyConstant.congrRightRingEquiv_symm_apply_apply
@[simps!]
def congrRightRingEquiv [Semiring Y] [Semiring Z] (e : Y ≃+* Z) :
    LocallyConstant X Y ≃+* LocallyConstant X Z 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} [Semiring Y] [Semiring Z] (e : Y ≃+* Z) (g : LocallyConstant X Z)
Goal: ∀ (a : X), ((LocallyConstant.congrRightRingEquiv e).symm g) a = (↑e).symm (g a)
-- LocallyConstant.congrRightRingEquiv_apply_apply
@[simps!]
def congrRightRingEquiv [Semiring Y] [Semiring Z] (e : Y ≃+* Z) :
    LocallyConstant X Y ≃+* LocallyConstant X Z 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} [Semiring Y] [Semiring Z] (e : Y ≃+* Z) (g : LocallyConstant X Y)
Goal: ∀ (a : X), ((LocallyConstant.congrRightRingEquiv e) g) a = e (g a)
-- LocallyConstant.congrRightₐ_apply_apply
@[simps!]
def congrRightₐ (R : Type*) [CommSemiring R] [Semiring Y] [Algebra R Y] [Semiring Z] [Algebra R Z]
    (e : Y ≃ₐ[R] Z) : LocallyConstant X Y ≃ₐ[R] LocallyConstant X Z 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} (R : Type u_7) [CommSemiring R] [Semiring Y] [Algebra R Y] [Semiring Z] [Algebra R Z] (e : Y ≃ₐ[R] Z) (g : LocallyConstant X Y)
Goal: ∀ (a : X), ((LocallyConstant.congrRightₐ R e) g) a = e (g a)
-- LocallyConstant.congrRightₐ_symm_apply_apply
@[simps!]
def congrRightₐ (R : Type*) [CommSemiring R] [Semiring Y] [Algebra R Y] [Semiring Z] [Algebra R Z]
    (e : Y ≃ₐ[R] Z) : LocallyConstant X Y ≃ₐ[R] LocallyConstant X Z 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] {Z : Type u_6} (R : Type u_7) [CommSemiring R] [Semiring Y] [Algebra R Y] [Semiring Z] [Algebra R Z] (e : Y ≃ₐ[R] Z) (g : LocallyConstant X Z)
Goal: ∀ (a : X), ((LocallyConstant.congrRightₐ R e).symm g) a = e.symm (g a)
-- LocallyConstant.constₗ_apply_apply
@[simps!]
def constₗ (R : Type*) [Semiring R] [AddCommMonoid Y] [Module R Y] :
    Y →ₗ[R] LocallyConstant X Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (R : Type u_6) [Semiring R] [AddCommMonoid Y] [Module R Y] (y : Y)
Goal: ∀ (a : X), ((LocallyConstant.constₗ R) y) a = y
-- LocallyConstant.constₐ_apply_apply
@[simps!]
def constₐ (R : Type*) [CommSemiring R] [Semiring Y] [Algebra R Y] :
    Y →ₐ[R] LocallyConstant X Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] (R : Type u_6) [CommSemiring R] [Semiring Y] [Algebra R Y] (y : Y)
Goal: ∀ (a : X), ((LocallyConstant.constₐ R) y) a = y
-- ContinuousMap.idealOfSet_closed
theorem idealOfSet_closed [T2Space R] (s : Set X) :
    IsClosed (idealOfSet R s : Set C(X, R)) 
Arguments: {X : Type u_1} (R : Type u_2) [TopologicalSpace X] [Semiring R] [TopologicalSpace R] [TopologicalSemiring R] [T2Space R] (s : Set X)
Goal: IsClosed ↑(ContinuousMap.idealOfSet R s)
-- ContinuousMap.mem_idealOfSet
theorem mem_idealOfSet {s : Set X} {f : C(X, R)} :
    f ∈ idealOfSet R s ↔ ∀ ⦃x : X⦄, x ∈ sᶜ → f x = 0 
Arguments: {X : Type u_1} {R : Type u_2} [TopologicalSpace X] [Semiring R] [TopologicalSpace R] [TopologicalSemiring R] {s : Set X} {f : C(X, R)}
Goal: f ∈ ContinuousMap.idealOfSet R s ↔ ∀ ⦃x : X⦄, x ∈ sᶜ → f x = 0
-- ContinuousMap.not_mem_idealOfSet
theorem not_mem_idealOfSet {s : Set X} {f : C(X, R)} : f ∉ idealOfSet R s ↔ ∃ x ∈ sᶜ, f x ≠ 0 
Arguments: {X : Type u_1} {R : Type u_2} [TopologicalSpace X] [Semiring R] [TopologicalSpace R] [TopologicalSemiring R] {s : Set X} {f : C(X, R)}
Goal: f ∉ ContinuousMap.idealOfSet R s ↔ ∃ x ∈ sᶜ, f x ≠ 0
-- ContinuousMap.not_mem_setOfIdeal
theorem not_mem_setOfIdeal {I : Ideal C(X, R)} {x : X} :
    x ∉ setOfIdeal I ↔ ∀ ⦃f : C(X, R)⦄, f ∈ I → f x = 0 
Arguments: {X : Type u_1} {R : Type u_2} [TopologicalSpace X] [Semiring R] [TopologicalSpace R] [TopologicalSemiring R] {I : Ideal C(X, R)} {x : X}
Goal: x ∉ ContinuousMap.setOfIdeal I ↔ ∀ ⦃f : C(X, R)⦄, f ∈ I → f x = 0
-- ContinuousMap.mem_setOfIdeal
theorem mem_setOfIdeal {I : Ideal C(X, R)} {x : X} :
    x ∈ setOfIdeal I ↔ ∃ f ∈ I, (f : C(X, R)) x ≠ 0 
Arguments: {X : Type u_1} {R : Type u_2} [TopologicalSpace X] [Semiring R] [TopologicalSpace R] [TopologicalSemiring R] {I : Ideal C(X, R)} {x : X}
Goal: x ∈ ContinuousMap.setOfIdeal I ↔ ∃ f ∈ I, f x ≠ 0
-- ContinuousMap.setOfIdeal_open
theorem setOfIdeal_open [T2Space R] (I : Ideal C(X, R)) : IsOpen (setOfIdeal I) 
Arguments: {X : Type u_1} {R : Type u_2} [TopologicalSpace X] [Semiring R] [TopologicalSpace R] [TopologicalSemiring R] [T2Space R] (I : Ideal C(X, R))
Goal: IsOpen (ContinuousMap.setOfIdeal I)
-- ContinuousMap.opensOfIdeal_coe
@[simps]
def opensOfIdeal [T2Space R] (I : Ideal C(X, R)) : Opens X 
Arguments: {X : Type u_1} {R : Type u_2} [TopologicalSpace X] [Semiring R] [TopologicalSpace R] [TopologicalSemiring R] [T2Space R] (I : Ideal C(X, R))
Goal: ↑(ContinuousMap.opensOfIdeal I) = ContinuousMap.setOfIdeal I
-- ContinuousMap.setOfTop_eq_univ
@[simp]
theorem setOfTop_eq_univ [Nontrivial R] : setOfIdeal (⊤ : Ideal C(X, R)) = Set.univ 
Arguments: {X : Type u_1} {R : Type u_2} [TopologicalSpace X] [Semiring R] [TopologicalSpace R] [TopologicalSemiring R] [Nontrivial R]
Goal: ContinuousMap.setOfIdeal ⊤ = Set.univ
-- ContinuousMap.idealOfEmpty_eq_bot
@[simp]
theorem idealOfEmpty_eq_bot : idealOfSet R (∅ : Set X) = ⊥ 
Arguments: {X : Type u_1} {R : Type u_2} [TopologicalSpace X] [Semiring R] [TopologicalSpace R] [TopologicalSemiring R]
Goal: ContinuousMap.idealOfSet R ∅ = ⊥
-- ContinuousMap.mem_idealOfSet_compl_singleton
@[simp]
theorem mem_idealOfSet_compl_singleton (x : X) (f : C(X, R)) :
    f ∈ idealOfSet R ({x}ᶜ : Set X) ↔ f x = 0 
Arguments: {X : Type u_1} {R : Type u_2} [TopologicalSpace X] [Semiring R] [TopologicalSpace R] [TopologicalSemiring R] (x : X) (f : C(X, R))
Goal: f ∈ ContinuousMap.idealOfSet R {x}ᶜ ↔ f x = 0
-- ContinuousMap.ideal_gc
theorem ideal_gc : GaloisConnection (setOfIdeal : Ideal C(X, R) → Set X) (idealOfSet R) 
Arguments: (X : Type u_1) (R : Type u_2) [TopologicalSpace X] [Semiring R] [TopologicalSpace R] [TopologicalSemiring R]
Goal: GaloisConnection ContinuousMap.setOfIdeal (ContinuousMap.idealOfSet R)
-- ContinuousMap.exists_mul_le_one_eqOn_ge
theorem exists_mul_le_one_eqOn_ge (f : C(X, ℝ≥0)) {c : ℝ≥0} (hc : 0 < c) :
    ∃ g : C(X, ℝ≥0), (∀ x : X, (g * f) x ≤ 1) ∧ {x : X | c ≤ f x}.EqOn (g * f) 1 
Arguments: {X : Type u_1} [TopologicalSpace X] (f : C(X, ℝ≥0)) {c : ℝ≥0} (hc : 0 < c)
Goal: ∃ g, (∀ (x : X), (g * f) x ≤ 1) ∧ Set.EqOn (⇑(g * f)) 1 {x | c ≤ f x}
-- ContinuousMap.idealOfSet_ofIdeal_eq_closure
@[simp]
theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
    idealOfSet 𝕜 (setOfIdeal I) = I.closure 
Arguments: {X : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] [TopologicalSpace X] [CompactSpace X] [T2Space X] (I : Ideal C(X, 𝕜))
Goal: ContinuousMap.idealOfSet 𝕜 (ContinuousMap.setOfIdeal I) = I.closure
-- ContinuousMap.idealOfSet_ofIdeal_isClosed
theorem idealOfSet_ofIdeal_isClosed {I : Ideal C(X, 𝕜)} (hI : IsClosed (I : Set C(X, 𝕜))) :
    idealOfSet 𝕜 (setOfIdeal I) = I 
Arguments: {X : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] [TopologicalSpace X] [CompactSpace X] [T2Space X] {I : Ideal C(X, 𝕜)} (hI : IsClosed ↑I)
Goal: ContinuousMap.idealOfSet 𝕜 (ContinuousMap.setOfIdeal I) = I
-- ContinuousMap.setOfIdeal_ofSet_eq_interior
@[simp]
theorem setOfIdeal_ofSet_eq_interior (s : Set X) : setOfIdeal (idealOfSet 𝕜 s) = interior s 
Arguments: {X : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] [TopologicalSpace X] [CompactSpace X] [T2Space X] (s : Set X)
Goal: ContinuousMap.setOfIdeal (ContinuousMap.idealOfSet 𝕜 s) = interior s
-- ContinuousMap.setOfIdeal_ofSet_of_isOpen
theorem setOfIdeal_ofSet_of_isOpen {s : Set X} (hs : IsOpen s) : setOfIdeal (idealOfSet 𝕜 s) = s 
Arguments: {X : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] [TopologicalSpace X] [CompactSpace X] [T2Space X] {s : Set X} (hs : IsOpen s)
Goal: ContinuousMap.setOfIdeal (ContinuousMap.idealOfSet 𝕜 s) = s
-- ContinuousMap.idealOpensGI_choice
@[simps]
def idealOpensGI :
    GaloisInsertion (opensOfIdeal : Ideal C(X, 𝕜) → Opens X) fun s => idealOfSet 𝕜 s 
Arguments: (X : Type u_1) (𝕜 : Type u_2) [RCLike 𝕜] [TopologicalSpace X] [CompactSpace X] [T2Space X] (I : Ideal C(X, 𝕜))
Goal: ∀ (x : ContinuousMap.idealOfSet 𝕜 ↑(ContinuousMap.opensOfIdeal I) ≤ I),
  (ContinuousMap.idealOpensGI X 𝕜).choice I x = ContinuousMap.opensOfIdeal I.closure
-- ContinuousMap.idealOfSet_isMaximal_iff
theorem idealOfSet_isMaximal_iff (s : Opens X) :
    (idealOfSet 𝕜 (s : Set X)).IsMaximal ↔ IsCoatom s 
Arguments: {X : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] [TopologicalSpace X] [CompactSpace X] [T2Space X] (s : TopologicalSpace.Opens X)
Goal: (ContinuousMap.idealOfSet 𝕜 ↑s).IsMaximal ↔ IsCoatom s
-- ContinuousMap.idealOf_compl_singleton_isMaximal
theorem idealOf_compl_singleton_isMaximal (x : X) : (idealOfSet 𝕜 ({x}ᶜ : Set X)).IsMaximal 
Arguments: {X : Type u_1} (𝕜 : Type u_2) [RCLike 𝕜] [TopologicalSpace X] [CompactSpace X] [T2Space X] (x : X)
Goal: (ContinuousMap.idealOfSet 𝕜 {x}ᶜ).IsMaximal
-- ContinuousMap.setOfIdeal_eq_compl_singleton
theorem setOfIdeal_eq_compl_singleton (I : Ideal C(X, 𝕜)) [hI : I.IsMaximal] :
    ∃ x : X, setOfIdeal I = {x}ᶜ 
Arguments: {X : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] [TopologicalSpace X] [CompactSpace X] [T2Space X] (I : Ideal C(X, 𝕜)) [hI : I.IsMaximal]
Goal: ∃ x, ContinuousMap.setOfIdeal I = {x}ᶜ
-- ContinuousMap.ideal_isMaximal_iff
theorem ideal_isMaximal_iff (I : Ideal C(X, 𝕜)) [hI : IsClosed (I : Set C(X, 𝕜))] :
    I.IsMaximal ↔ ∃ x : X, idealOfSet 𝕜 {x}ᶜ = I 
Arguments: {X : Type u_1} {𝕜 : Type u_2} [RCLike 𝕜] [TopologicalSpace X] [CompactSpace X] [T2Space X] (I : Ideal C(X, 𝕜)) [hI : IsClosed ↑I]
Goal: I.IsMaximal ↔ ∃ x, ContinuousMap.idealOfSet 𝕜 {x}ᶜ = I
-- WeakDual.CharacterSpace.continuousMapEval_apply_apply
@[simp]
theorem continuousMapEval_apply_apply (x : X) (f : C(X, 𝕜)) : continuousMapEval X 𝕜 x f = f x 
Arguments: (X : Type u_1) (𝕜 : Type u_2) [TopologicalSpace X] [CommRing 𝕜] [TopologicalSpace 𝕜] [TopologicalRing 𝕜] [Nontrivial 𝕜] [NoZeroDivisors 𝕜] (x : X) (f : C(X, 𝕜))
Goal: ((WeakDual.CharacterSpace.continuousMapEval X 𝕜) x) f = f x
-- WeakDual.CharacterSpace.continuousMapEval_bijective
theorem continuousMapEval_bijective : Bijective (continuousMapEval X 𝕜) 
Arguments: (X : Type u_1) (𝕜 : Type u_2) [TopologicalSpace X] [CompactSpace X] [T2Space X] [RCLike 𝕜]
Goal: Function.Bijective ⇑(WeakDual.CharacterSpace.continuousMapEval X 𝕜)
-- GroupFilterBasis.inv'
class GroupFilterBasis (G : Type u) [Group G] extends FilterBasis G where
  one' : ∀ {U}, U ∈ sets → (1 : G) ∈ U
  mul' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V * V ⊆ U
  inv' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (fun x ↦ x⁻¹) ⁻¹' U
  conj' : ∀ x₀, ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (fun x ↦ x₀ * x * x₀⁻¹) ⁻¹' U
Arguments: {G : Type u} [Group G] [self : GroupFilterBasis G] {U : Set G}
Goal: U ∈ GroupFilterBasis.toFilterBasis.sets → ∃ V ∈ GroupFilterBasis.toFilterBasis.sets, V ⊆ (fun x => x⁻¹) ⁻¹' U
-- GroupFilterBasis.one'
class GroupFilterBasis (G : Type u) [Group G] extends FilterBasis G where
  one' : ∀ {U}, U ∈ sets → (1 : G) ∈ U
  mul' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V * V ⊆ U
  inv' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (fun x ↦ x⁻¹) ⁻¹' U
  conj' : ∀ x₀, ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (fun x ↦ x₀ * x * x₀⁻¹) ⁻¹' U
Arguments: {G : Type u} [Group G] [self : GroupFilterBasis G] {U : Set G}
Goal: U ∈ GroupFilterBasis.toFilterBasis.sets → 1 ∈ U
-- GroupFilterBasis.mul'
class GroupFilterBasis (G : Type u) [Group G] extends FilterBasis G where
  one' : ∀ {U}, U ∈ sets → (1 : G) ∈ U
  mul' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V * V ⊆ U
  inv' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (fun x ↦ x⁻¹) ⁻¹' U
  conj' : ∀ x₀, ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (fun x ↦ x₀ * x * x₀⁻¹) ⁻¹' U
Arguments: {G : Type u} [Group G] [self : GroupFilterBasis G] {U : Set G}
Goal: U ∈ GroupFilterBasis.toFilterBasis.sets → ∃ V ∈ GroupFilterBasis.toFilterBasis.sets, V * V ⊆ U
-- GroupFilterBasis.conj'
class GroupFilterBasis (G : Type u) [Group G] extends FilterBasis G where
  one' : ∀ {U}, U ∈ sets → (1 : G) ∈ U
  mul' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V * V ⊆ U
  inv' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (fun x ↦ x⁻¹) ⁻¹' U
  conj' : ∀ x₀, ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (fun x ↦ x₀ * x * x₀⁻¹) ⁻¹' U
Arguments: {G : Type u} [Group G] [self : GroupFilterBasis G] (x₀ : G) {U : Set G}
Goal: U ∈ GroupFilterBasis.toFilterBasis.sets → ∃ V ∈ GroupFilterBasis.toFilterBasis.sets, V ⊆ (fun x => x₀ * x * x₀⁻¹) ⁻¹' U
-- AddGroupFilterBasis.conj'
class AddGroupFilterBasis (A : Type u) [AddGroup A] extends FilterBasis A where
  zero' : ∀ {U}, U ∈ sets → (0 : A) ∈ U
  add' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V + V ⊆ U
  neg' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (fun x ↦ -x) ⁻¹' U
  conj' : ∀ x₀, ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (fun x ↦ x₀ + x + -x₀) ⁻¹' U
Arguments: {A : Type u} [AddGroup A] [self : AddGroupFilterBasis A] (x₀ : A) {U : Set A}
Goal: U ∈ AddGroupFilterBasis.toFilterBasis.sets →
  ∃ V ∈ AddGroupFilterBasis.toFilterBasis.sets, V ⊆ (fun x => x₀ + x + -x₀) ⁻¹' U
-- AddGroupFilterBasis.add'
class AddGroupFilterBasis (A : Type u) [AddGroup A] extends FilterBasis A where
  zero' : ∀ {U}, U ∈ sets → (0 : A) ∈ U
  add' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V + V ⊆ U
  neg' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (fun x ↦ -x) ⁻¹' U
  conj' : ∀ x₀, ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (fun x ↦ x₀ + x + -x₀) ⁻¹' U
Arguments: {A : Type u} [AddGroup A] [self : AddGroupFilterBasis A] {U : Set A}
Goal: U ∈ AddGroupFilterBasis.toFilterBasis.sets → ∃ V ∈ AddGroupFilterBasis.toFilterBasis.sets, V + V ⊆ U
-- AddGroupFilterBasis.neg'
class AddGroupFilterBasis (A : Type u) [AddGroup A] extends FilterBasis A where
  zero' : ∀ {U}, U ∈ sets → (0 : A) ∈ U
  add' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V + V ⊆ U
  neg' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (fun x ↦ -x) ⁻¹' U
  conj' : ∀ x₀, ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (fun x ↦ x₀ + x + -x₀) ⁻¹' U
Arguments: {A : Type u} [AddGroup A] [self : AddGroupFilterBasis A] {U : Set A}
Goal: U ∈ AddGroupFilterBasis.toFilterBasis.sets → ∃ V ∈ AddGroupFilterBasis.toFilterBasis.sets, V ⊆ (fun x => -x) ⁻¹' U
-- AddGroupFilterBasis.zero'
class AddGroupFilterBasis (A : Type u) [AddGroup A] extends FilterBasis A where
  zero' : ∀ {U}, U ∈ sets → (0 : A) ∈ U
  add' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V + V ⊆ U
  neg' : ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (fun x ↦ -x) ⁻¹' U
  conj' : ∀ x₀, ∀ {U}, U ∈ sets → ∃ V ∈ sets, V ⊆ (fun x ↦ x₀ + x + -x₀) ⁻¹' U
Arguments: {A : Type u} [AddGroup A] [self : AddGroupFilterBasis A] {U : Set A}
Goal: U ∈ AddGroupFilterBasis.toFilterBasis.sets → 0 ∈ U
-- GroupFilterBasis.one
@[to_additive]
theorem one {U : Set G} : U ∈ B → (1 : G) ∈ U 
Arguments: {G : Type u} [Group G] {B : GroupFilterBasis G} {U : Set G}
Goal: U ∈ B → 1 ∈ U
-- AddGroupFilterBasis.zero
@[to_additive]
theorem one {U : Set G} : U ∈ B → (1 : G) ∈ U 
Arguments: {G : Type u} [AddGroup G] {B : AddGroupFilterBasis G} {U : Set G}
Goal: U ∈ B → 0 ∈ U
-- GroupFilterBasis.mul
@[to_additive]
theorem mul {U : Set G} : U ∈ B → ∃ V ∈ B, V * V ⊆ U 
Arguments: {G : Type u} [Group G] {B : GroupFilterBasis G} {U : Set G}
Goal: U ∈ B → ∃ V ∈ B, V * V ⊆ U
-- AddGroupFilterBasis.add
@[to_additive]
theorem mul {U : Set G} : U ∈ B → ∃ V ∈ B, V * V ⊆ U 
Arguments: {G : Type u} [AddGroup G] {B : AddGroupFilterBasis G} {U : Set G}
Goal: U ∈ B → ∃ V ∈ B, V + V ⊆ U
-- AddGroupFilterBasis.neg
@[to_additive]
theorem inv {U : Set G} : U ∈ B → ∃ V ∈ B, V ⊆ (fun x ↦ x⁻¹) ⁻¹' U 
Arguments: {G : Type u} [AddGroup G] {B : AddGroupFilterBasis G} {U : Set G}
Goal: U ∈ B → ∃ V ∈ B, V ⊆ (fun x => -x) ⁻¹' U