ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- range_eq_image_mulTSupport_or
@[to_additive]
theorem range_eq_image_mulTSupport_or (f : X → α) :
    range f = f '' mulTSupport f ∨ range f = insert 1 (f '' mulTSupport f) 
Arguments: {X : Type u_1} {α : Type u_2} [One α] [TopologicalSpace X] (f : X → α)
Goal: Set.range f = f '' mulTSupport f ∨ Set.range f = insert 1 (f '' mulTSupport f)
-- range_eq_image_tsupport_or
@[to_additive]
theorem range_eq_image_mulTSupport_or (f : X → α) :
    range f = f '' mulTSupport f ∨ range f = insert 1 (f '' mulTSupport f) 
Arguments: {X : Type u_1} {α : Type u_2} [Zero α] [TopologicalSpace X] (f : X → α)
Goal: Set.range f = f '' tsupport f ∨ Set.range f = insert 0 (f '' tsupport f)
-- tsupport_mul_subset_left
theorem tsupport_mul_subset_left {α : Type*} [MulZeroClass α] {f g : X → α} :
    (tsupport fun x => f x * g x) ⊆ tsupport f 
Arguments: {X : Type u_1} [TopologicalSpace X] {α : Type u_10} [MulZeroClass α] {f g : X → α}
Goal: (tsupport fun x => f x * g x) ⊆ tsupport f
-- tsupport_mul_subset_right
theorem tsupport_mul_subset_right {α : Type*} [MulZeroClass α] {f g : X → α} :
    (tsupport fun x => f x * g x) ⊆ tsupport g 
Arguments: {X : Type u_1} [TopologicalSpace X] {α : Type u_10} [MulZeroClass α] {f g : X → α}
Goal: (tsupport fun x => f x * g x) ⊆ tsupport g
-- tsupport_smul_subset_left
theorem tsupport_smul_subset_left {M α} [TopologicalSpace X] [Zero M] [Zero α] [SMulWithZero M α]
    (f : X → M) (g : X → α) : (tsupport fun x => f x • g x) ⊆ tsupport f 
Arguments: {X : Type u_1} {M : Type u_10} {α : Type u_11} [TopologicalSpace X] [Zero M] [Zero α] [SMulWithZero M α] (f : X → M) (g : X → α)
Goal: (tsupport fun x => f x • g x) ⊆ tsupport f
-- tsupport_smul_subset_right
theorem tsupport_smul_subset_right {M α} [TopologicalSpace X] [Zero α] [SMulZeroClass M α]
    (f : X → M) (g : X → α) : (tsupport fun x => f x • g x) ⊆ tsupport g 
Arguments: {X : Type u_1} {M : Type u_10} {α : Type u_11} [TopologicalSpace X] [Zero α] [SMulZeroClass M α] (f : X → M) (g : X → α)
Goal: (tsupport fun x => f x • g x) ⊆ tsupport g
-- tsupport_add
@[to_additive]
theorem mulTSupport_mul [TopologicalSpace X] [Monoid α] {f g : X → α} :
    (mulTSupport fun x ↦ f x * g x) ⊆ mulTSupport f ∪ mulTSupport g 
Arguments: {X : Type u_1} {α : Type u_2} [TopologicalSpace X] [AddMonoid α] {f g : X → α}
Goal: (tsupport fun x => f x + g x) ⊆ tsupport f ∪ tsupport g
-- mulTSupport_mul
@[to_additive]
theorem mulTSupport_mul [TopologicalSpace X] [Monoid α] {f g : X → α} :
    (mulTSupport fun x ↦ f x * g x) ⊆ mulTSupport f ∪ mulTSupport g 
Arguments: {X : Type u_1} {α : Type u_2} [TopologicalSpace X] [Monoid α] {f g : X → α}
Goal: (mulTSupport fun x => f x * g x) ⊆ mulTSupport f ∪ mulTSupport g
-- not_mem_mulTSupport_iff_eventuallyEq
@[to_additive]
theorem not_mem_mulTSupport_iff_eventuallyEq : x ∉ mulTSupport f ↔ f =ᶠ[𝓝 x] 1 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [One β] {f : α → β} {x : α}
Goal: x ∉ mulTSupport f ↔ f =ᶠ[𝓝 x] 1
-- not_mem_tsupport_iff_eventuallyEq
@[to_additive]
theorem not_mem_mulTSupport_iff_eventuallyEq : x ∉ mulTSupport f ↔ f =ᶠ[𝓝 x] 1 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [Zero β] {f : α → β} {x : α}
Goal: x ∉ tsupport f ↔ f =ᶠ[𝓝 x] 0
-- continuous_of_tsupport
@[to_additive]
theorem continuous_of_mulTSupport [TopologicalSpace β] {f : α → β}
    (hf : ∀ x ∈ mulTSupport f, ContinuousAt f x) : Continuous f 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [Zero β] [TopologicalSpace β] {f : α → β} (hf : ∀ x ∈ tsupport f, ContinuousAt f x)
Goal: Continuous f
-- continuous_of_mulTSupport
@[to_additive]
theorem continuous_of_mulTSupport [TopologicalSpace β] {f : α → β}
    (hf : ∀ x ∈ mulTSupport f, ContinuousAt f x) : Continuous f 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [One β] [TopologicalSpace β] {f : α → β} (hf : ∀ x ∈ mulTSupport f, ContinuousAt f x)
Goal: Continuous f
-- hasCompactSupport_def
@[to_additive]
theorem hasCompactMulSupport_def : HasCompactMulSupport f ↔ IsCompact (closure (mulSupport f)) 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [Zero β] {f : α → β}
Goal: HasCompactSupport f ↔ IsCompact (closure (Function.support f))
-- hasCompactMulSupport_def
@[to_additive]
theorem hasCompactMulSupport_def : HasCompactMulSupport f ↔ IsCompact (closure (mulSupport f)) 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [One β] {f : α → β}
Goal: HasCompactMulSupport f ↔ IsCompact (closure (Function.mulSupport f))
-- exists_compact_iff_hasCompactMulSupport
@[to_additive]
theorem exists_compact_iff_hasCompactMulSupport [R1Space α] :
    (∃ K : Set α, IsCompact K ∧ ∀ x, x ∉ K → f x = 1) ↔ HasCompactMulSupport f 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [One β] {f : α → β} [R1Space α]
Goal: (∃ K, IsCompact K ∧ ∀ x ∉ K, f x = 1) ↔ HasCompactMulSupport f
-- exists_compact_iff_hasCompactSupport
@[to_additive]
theorem exists_compact_iff_hasCompactMulSupport [R1Space α] :
    (∃ K : Set α, IsCompact K ∧ ∀ x, x ∉ K → f x = 1) ↔ HasCompactMulSupport f 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [Zero β] {f : α → β} [R1Space α]
Goal: (∃ K, IsCompact K ∧ ∀ x ∉ K, f x = 0) ↔ HasCompactSupport f
-- HasCompactMulSupport.intro
@[to_additive]
theorem intro [R1Space α] {K : Set α} (hK : IsCompact K)
    (hfK : ∀ x, x ∉ K → f x = 1) : HasCompactMulSupport f 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [One β] {f : α → β} [R1Space α] {K : Set α} (hK : IsCompact K) (hfK : ∀ x ∉ K, f x = 1)
Goal: HasCompactMulSupport f
-- HasCompactSupport.intro
@[to_additive]
theorem intro [R1Space α] {K : Set α} (hK : IsCompact K)
    (hfK : ∀ x, x ∉ K → f x = 1) : HasCompactMulSupport f 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [Zero β] {f : α → β} [R1Space α] {K : Set α} (hK : IsCompact K) (hfK : ∀ x ∉ K, f x = 0)
Goal: HasCompactSupport f
-- HasCompactMulSupport.intro'
@[to_additive]
theorem intro' {K : Set α} (hK : IsCompact K) (h'K : IsClosed K)
    (hfK : ∀ x, x ∉ K → f x = 1) : HasCompactMulSupport f 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [One β] {f : α → β} {K : Set α} (hK : IsCompact K) (h'K : IsClosed K) (hfK : ∀ x ∉ K, f x = 1)
Goal: HasCompactMulSupport f
-- HasCompactSupport.intro'
@[to_additive]
theorem intro' {K : Set α} (hK : IsCompact K) (h'K : IsClosed K)
    (hfK : ∀ x, x ∉ K → f x = 1) : HasCompactMulSupport f 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [Zero β] {f : α → β} {K : Set α} (hK : IsCompact K) (h'K : IsClosed K) (hfK : ∀ x ∉ K, f x = 0)
Goal: HasCompactSupport f
-- HasCompactSupport.of_support_subset_isCompact
@[to_additive]
theorem of_mulSupport_subset_isCompact [R1Space α] {K : Set α}
    (hK : IsCompact K) (h : mulSupport f ⊆ K) : HasCompactMulSupport f 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [Zero β] {f : α → β} [R1Space α] {K : Set α} (hK : IsCompact K) (h : Function.support f ⊆ K)
Goal: HasCompactSupport f
-- HasCompactMulSupport.of_mulSupport_subset_isCompact
@[to_additive]
theorem of_mulSupport_subset_isCompact [R1Space α] {K : Set α}
    (hK : IsCompact K) (h : mulSupport f ⊆ K) : HasCompactMulSupport f 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [One β] {f : α → β} [R1Space α] {K : Set α} (hK : IsCompact K) (h : Function.mulSupport f ⊆ K)
Goal: HasCompactMulSupport f
-- HasCompactSupport.isCompact
@[to_additive]
theorem isCompact (hf : HasCompactMulSupport f) : IsCompact (mulTSupport f) 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [Zero β] {f : α → β} (hf : HasCompactSupport f)
Goal: IsCompact (tsupport f)
-- HasCompactMulSupport.isCompact
@[to_additive]
theorem isCompact (hf : HasCompactMulSupport f) : IsCompact (mulTSupport f) 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [One β] {f : α → β} (hf : HasCompactMulSupport f)
Goal: IsCompact (mulTSupport f)
-- hasCompactMulSupport_iff_eventuallyEq
@[to_additive]
theorem _root_.hasCompactMulSupport_iff_eventuallyEq :
    HasCompactMulSupport f ↔ f =ᶠ[coclosedCompact α] 1 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [One β] {f : α → β}
Goal: HasCompactMulSupport f ↔ f =ᶠ[Filter.coclosedCompact α] 1
-- hasCompactSupport_iff_eventuallyEq
@[to_additive]
theorem _root_.hasCompactMulSupport_iff_eventuallyEq :
    HasCompactMulSupport f ↔ f =ᶠ[coclosedCompact α] 1 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [Zero β] {f : α → β}
Goal: HasCompactSupport f ↔ f =ᶠ[Filter.coclosedCompact α] 0
-- isCompact_range_of_support_subset_isCompact
@[to_additive]
theorem _root_.isCompact_range_of_mulSupport_subset_isCompact [TopologicalSpace β]
    (hf : Continuous f) {k : Set α} (hk : IsCompact k) (h'f : mulSupport f ⊆ k) :
    IsCompact (range f) 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [Zero β] {f : α → β} [TopologicalSpace β] (hf : Continuous f) {k : Set α} (hk : IsCompact k) (h'f : Function.support f ⊆ k)
Goal: IsCompact (Set.range f)
-- isCompact_range_of_mulSupport_subset_isCompact
@[to_additive]
theorem _root_.isCompact_range_of_mulSupport_subset_isCompact [TopologicalSpace β]
    (hf : Continuous f) {k : Set α} (hk : IsCompact k) (h'f : mulSupport f ⊆ k) :
    IsCompact (range f) 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [One β] {f : α → β} [TopologicalSpace β] (hf : Continuous f) {k : Set α} (hk : IsCompact k) (h'f : Function.mulSupport f ⊆ k)
Goal: IsCompact (Set.range f)
-- HasCompactMulSupport.isCompact_range
@[to_additive]
theorem isCompact_range [TopologicalSpace β] (h : HasCompactMulSupport f)
    (hf : Continuous f) : IsCompact (range f) 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [One β] {f : α → β} [TopologicalSpace β] (h : HasCompactMulSupport f) (hf : Continuous f)
Goal: IsCompact (Set.range f)
-- HasCompactSupport.isCompact_range
@[to_additive]
theorem isCompact_range [TopologicalSpace β] (h : HasCompactMulSupport f)
    (hf : Continuous f) : IsCompact (range f) 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [Zero β] {f : α → β} [TopologicalSpace β] (h : HasCompactSupport f) (hf : Continuous f)
Goal: IsCompact (Set.range f)
-- HasCompactSupport.mono'
@[to_additive]
theorem mono' {f' : α → γ} (hf : HasCompactMulSupport f)
    (hff' : mulSupport f' ⊆ mulTSupport f) : HasCompactMulSupport f' 
Arguments: {α : Type u_2} {β : Type u_4} {γ : Type u_5} [TopologicalSpace α] [Zero β] [Zero γ] {f : α → β} {f' : α → γ} (hf : HasCompactSupport f) (hff' : Function.support f' ⊆ tsupport f)
Goal: HasCompactSupport f'
-- HasCompactMulSupport.mono'
@[to_additive]
theorem mono' {f' : α → γ} (hf : HasCompactMulSupport f)
    (hff' : mulSupport f' ⊆ mulTSupport f) : HasCompactMulSupport f' 
Arguments: {α : Type u_2} {β : Type u_4} {γ : Type u_5} [TopologicalSpace α] [One β] [One γ] {f : α → β} {f' : α → γ} (hf : HasCompactMulSupport f) (hff' : Function.mulSupport f' ⊆ mulTSupport f)
Goal: HasCompactMulSupport f'
-- HasCompactSupport.mono
@[to_additive]
theorem mono {f' : α → γ} (hf : HasCompactMulSupport f)
    (hff' : mulSupport f' ⊆ mulSupport f) : HasCompactMulSupport f' 
Arguments: {α : Type u_2} {β : Type u_4} {γ : Type u_5} [TopologicalSpace α] [Zero β] [Zero γ] {f : α → β} {f' : α → γ} (hf : HasCompactSupport f) (hff' : Function.support f' ⊆ Function.support f)
Goal: HasCompactSupport f'
-- HasCompactMulSupport.mono
@[to_additive]
theorem mono {f' : α → γ} (hf : HasCompactMulSupport f)
    (hff' : mulSupport f' ⊆ mulSupport f) : HasCompactMulSupport f' 
Arguments: {α : Type u_2} {β : Type u_4} {γ : Type u_5} [TopologicalSpace α] [One β] [One γ] {f : α → β} {f' : α → γ} (hf : HasCompactMulSupport f) (hff' : Function.mulSupport f' ⊆ Function.mulSupport f)
Goal: HasCompactMulSupport f'
-- HasCompactSupport.comp_left
@[to_additive]
theorem comp_left (hf : HasCompactMulSupport f) (hg : g 1 = 1) :
    HasCompactMulSupport (g ∘ f) 
Arguments: {α : Type u_2} {β : Type u_4} {γ : Type u_5} [TopologicalSpace α] [Zero β] [Zero γ] {g : β → γ} {f : α → β} (hf : HasCompactSupport f) (hg : g 0 = 0)
Goal: HasCompactSupport (g ∘ f)
-- HasCompactMulSupport.comp_left
@[to_additive]
theorem comp_left (hf : HasCompactMulSupport f) (hg : g 1 = 1) :
    HasCompactMulSupport (g ∘ f) 
Arguments: {α : Type u_2} {β : Type u_4} {γ : Type u_5} [TopologicalSpace α] [One β] [One γ] {g : β → γ} {f : α → β} (hf : HasCompactMulSupport f) (hg : g 1 = 1)
Goal: HasCompactMulSupport (g ∘ f)
-- hasCompactSupport_comp_left
@[to_additive]
theorem _root_.hasCompactMulSupport_comp_left (hg : ∀ {x}, g x = 1 ↔ x = 1) :
    HasCompactMulSupport (g ∘ f) ↔ HasCompactMulSupport f 
Arguments: {α : Type u_2} {β : Type u_4} {γ : Type u_5} [TopologicalSpace α] [Zero β] [Zero γ] {g : β → γ} {f : α → β} (hg : ∀ {x : β}, g x = 0 ↔ x = 0)
Goal: HasCompactSupport (g ∘ f) ↔ HasCompactSupport f
-- hasCompactMulSupport_comp_left
@[to_additive]
theorem _root_.hasCompactMulSupport_comp_left (hg : ∀ {x}, g x = 1 ↔ x = 1) :
    HasCompactMulSupport (g ∘ f) ↔ HasCompactMulSupport f 
Arguments: {α : Type u_2} {β : Type u_4} {γ : Type u_5} [TopologicalSpace α] [One β] [One γ] {g : β → γ} {f : α → β} (hg : ∀ {x : β}, g x = 1 ↔ x = 1)
Goal: HasCompactMulSupport (g ∘ f) ↔ HasCompactMulSupport f
-- HasCompactMulSupport.comp_closedEmbedding
@[to_additive]
theorem comp_closedEmbedding (hf : HasCompactMulSupport f) {g : α' → α}
    (hg : ClosedEmbedding g) : HasCompactMulSupport (f ∘ g) 
Arguments: {α : Type u_2} {α' : Type u_3} {β : Type u_4} [TopologicalSpace α] [TopologicalSpace α'] [One β] {f : α → β} (hf : HasCompactMulSupport f) {g : α' → α} (hg : ClosedEmbedding g)
Goal: HasCompactMulSupport (f ∘ g)
-- HasCompactSupport.comp_closedEmbedding
@[to_additive]
theorem comp_closedEmbedding (hf : HasCompactMulSupport f) {g : α' → α}
    (hg : ClosedEmbedding g) : HasCompactMulSupport (f ∘ g) 
Arguments: {α : Type u_2} {α' : Type u_3} {β : Type u_4} [TopologicalSpace α] [TopologicalSpace α'] [Zero β] {f : α → β} (hf : HasCompactSupport f) {g : α' → α} (hg : ClosedEmbedding g)
Goal: HasCompactSupport (f ∘ g)
-- HasCompactMulSupport.comp₂_left
@[to_additive]
theorem comp₂_left (hf : HasCompactMulSupport f)
    (hf₂ : HasCompactMulSupport f₂) (hm : m 1 1 = 1) :
    HasCompactMulSupport fun x => m (f x) (f₂ x) 
Arguments: {α : Type u_2} {β : Type u_4} {γ : Type u_5} {δ : Type u_6} [TopologicalSpace α] [One β] [One γ] [One δ] {f : α → β} {f₂ : α → γ} {m : β → γ → δ} (hf : HasCompactMulSupport f) (hf₂ : HasCompactMulSupport f₂) (hm : m 1 1 = 1)
Goal: HasCompactMulSupport fun x => m (f x) (f₂ x)
-- HasCompactSupport.comp₂_left
@[to_additive]
theorem comp₂_left (hf : HasCompactMulSupport f)
    (hf₂ : HasCompactMulSupport f₂) (hm : m 1 1 = 1) :
    HasCompactMulSupport fun x => m (f x) (f₂ x) 
Arguments: {α : Type u_2} {β : Type u_4} {γ : Type u_5} {δ : Type u_6} [TopologicalSpace α] [Zero β] [Zero γ] [Zero δ] {f : α → β} {f₂ : α → γ} {m : β → γ → δ} (hf : HasCompactSupport f) (hf₂ : HasCompactSupport f₂) (hm : m 0 0 = 0)
Goal: HasCompactSupport fun x => m (f x) (f₂ x)
-- HasCompactSupport.isCompact_preimage
@[to_additive]
lemma isCompact_preimage [TopologicalSpace β]
    (h'f : HasCompactMulSupport f) (hf : Continuous f) {k : Set β} (hk : IsClosed k)
    (h'k : 1 ∉ k) : IsCompact (f ⁻¹' k) 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [Zero β] {f : α → β} [TopologicalSpace β] (h'f : HasCompactSupport f) (hf : Continuous f) {k : Set β} (hk : IsClosed k) (h'k : 0 ∉ k)
Goal: IsCompact (f ⁻¹' k)
-- HasCompactMulSupport.isCompact_preimage
@[to_additive]
lemma isCompact_preimage [TopologicalSpace β]
    (h'f : HasCompactMulSupport f) (hf : Continuous f) {k : Set β} (hk : IsClosed k)
    (h'k : 1 ∉ k) : IsCompact (f ⁻¹' k) 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [One β] {f : α → β} [TopologicalSpace β] (h'f : HasCompactMulSupport f) (hf : Continuous f) {k : Set β} (hk : IsClosed k) (h'k : 1 ∉ k)
Goal: IsCompact (f ⁻¹' k)
-- HasCompactMulSupport.mulTSupport_extend_one_subset
@[to_additive]
theorem mulTSupport_extend_one_subset :
    mulTSupport (g.extend f 1) ⊆ g '' mulTSupport f 
Arguments: {α : Type u_2} {α' : Type u_3} {β : Type u_4} [TopologicalSpace α] [TopologicalSpace α'] [One β] {f : α → β} [T2Space α'] (hf : HasCompactMulSupport f) {g : α → α'} (cont : Continuous g)
Goal: mulTSupport (Function.extend g f 1) ⊆ g '' mulTSupport f
-- HasCompactSupport.tsupport_extend_zero_subset
@[to_additive]
theorem mulTSupport_extend_one_subset :
    mulTSupport (g.extend f 1) ⊆ g '' mulTSupport f 
Arguments: {α : Type u_2} {α' : Type u_3} {β : Type u_4} [TopologicalSpace α] [TopologicalSpace α'] [Zero β] {f : α → β} [T2Space α'] (hf : HasCompactSupport f) {g : α → α'} (cont : Continuous g)
Goal: tsupport (Function.extend g f 0) ⊆ g '' tsupport f
-- HasCompactSupport.extend_zero
@[to_additive]
theorem extend_one : HasCompactMulSupport (g.extend f 1) 
Arguments: {α : Type u_2} {α' : Type u_3} {β : Type u_4} [TopologicalSpace α] [TopologicalSpace α'] [Zero β] {f : α → β} [T2Space α'] (hf : HasCompactSupport f) {g : α → α'} (cont : Continuous g)
Goal: HasCompactSupport (Function.extend g f 0)
-- HasCompactMulSupport.extend_one
@[to_additive]
theorem extend_one : HasCompactMulSupport (g.extend f 1) 
Arguments: {α : Type u_2} {α' : Type u_3} {β : Type u_4} [TopologicalSpace α] [TopologicalSpace α'] [One β] {f : α → β} [T2Space α'] (hf : HasCompactMulSupport f) {g : α → α'} (cont : Continuous g)
Goal: HasCompactMulSupport (Function.extend g f 1)
-- HasCompactMulSupport.mulTSupport_extend_one
@[to_additive]
theorem mulTSupport_extend_one (inj : g.Injective) :
    mulTSupport (g.extend f 1) = g '' mulTSupport f 
Arguments: {α : Type u_2} {α' : Type u_3} {β : Type u_4} [TopologicalSpace α] [TopologicalSpace α'] [One β] {f : α → β} [T2Space α'] (hf : HasCompactMulSupport f) {g : α → α'} (cont : Continuous g) (inj : Function.Injective g)
Goal: mulTSupport (Function.extend g f 1) = g '' mulTSupport f
-- HasCompactSupport.tsupport_extend_zero
@[to_additive]
theorem mulTSupport_extend_one (inj : g.Injective) :
    mulTSupport (g.extend f 1) = g '' mulTSupport f 
Arguments: {α : Type u_2} {α' : Type u_3} {β : Type u_4} [TopologicalSpace α] [TopologicalSpace α'] [Zero β] {f : α → β} [T2Space α'] (hf : HasCompactSupport f) {g : α → α'} (cont : Continuous g) (inj : Function.Injective g)
Goal: tsupport (Function.extend g f 0) = g '' tsupport f
-- HasCompactSupport.continuous_extend_zero
@[to_additive]
theorem continuous_extend_one [TopologicalSpace β] {U : Set α'} (hU : IsOpen U) {f : U → β}
    (cont : Continuous f) (supp : HasCompactMulSupport f) :
    Continuous (Subtype.val.extend f 1) 
Arguments: {α' : Type u_3} {β : Type u_4} [TopologicalSpace α'] [Zero β] [T2Space α'] [TopologicalSpace β] {U : Set α'} (hU : IsOpen U) {f : ↑U → β} (cont : Continuous f) (supp : HasCompactSupport f)
Goal: Continuous (Function.extend Subtype.val f 0)
-- HasCompactMulSupport.continuous_extend_one
@[to_additive]
theorem continuous_extend_one [TopologicalSpace β] {U : Set α'} (hU : IsOpen U) {f : U → β}
    (cont : Continuous f) (supp : HasCompactMulSupport f) :
    Continuous (Subtype.val.extend f 1) 
Arguments: {α' : Type u_3} {β : Type u_4} [TopologicalSpace α'] [One β] [T2Space α'] [TopologicalSpace β] {U : Set α'} (hU : IsOpen U) {f : ↑U → β} (cont : Continuous f) (supp : HasCompactMulSupport f)
Goal: Continuous (Function.extend Subtype.val f 1)
-- HasCompactMulSupport.is_one_at_infty
@[to_additive "If `f` has compact support, then `f` tends to zero at infinity."]
theorem is_one_at_infty {f : α → γ} [TopologicalSpace γ] [One γ]
    (h : HasCompactMulSupport f) : Tendsto f (cocompact α) (𝓝 1) 
Arguments: {α : Type u_2} {γ : Type u_5} [TopologicalSpace α] {f : α → γ} [TopologicalSpace γ] [One γ] (h : HasCompactMulSupport f)
Goal: Filter.Tendsto f (Filter.cocompact α) (𝓝 1)
-- HasCompactSupport.is_zero_at_infty
@[to_additive "If `f` has compact support, then `f` tends to zero at infinity."]
theorem is_one_at_infty {f : α → γ} [TopologicalSpace γ] [One γ]
    (h : HasCompactMulSupport f) : Tendsto f (cocompact α) (𝓝 1) 
Arguments: {α : Type u_2} {γ : Type u_5} [TopologicalSpace α] {f : α → γ} [TopologicalSpace γ] [Zero γ] (h : HasCompactSupport f)
Goal: Filter.Tendsto f (Filter.cocompact α) (𝓝 0)
-- HasCompactSupport.of_compactSpace
@[to_additive]
theorem HasCompactMulSupport.of_compactSpace (f : α → γ) :
    HasCompactMulSupport f 
Arguments: {α : Type u_2} {γ : Type u_5} [TopologicalSpace α] [CompactSpace α] [Zero γ] (f : α → γ)
Goal: HasCompactSupport f
-- HasCompactMulSupport.of_compactSpace
@[to_additive]
theorem HasCompactMulSupport.of_compactSpace (f : α → γ) :
    HasCompactMulSupport f 
Arguments: {α : Type u_2} {γ : Type u_5} [TopologicalSpace α] [CompactSpace α] [One γ] (f : α → γ)
Goal: HasCompactMulSupport f
-- HasCompactSupport.add
@[to_additive]
theorem HasCompactMulSupport.mul (hf : HasCompactMulSupport f) (hf' : HasCompactMulSupport f') :
    HasCompactMulSupport (f * f') 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [AddZeroClass β] {f f' : α → β} (hf : HasCompactSupport f) (hf' : HasCompactSupport f')
Goal: HasCompactSupport (f + f')
-- HasCompactMulSupport.mul
@[to_additive]
theorem HasCompactMulSupport.mul (hf : HasCompactMulSupport f) (hf' : HasCompactMulSupport f') :
    HasCompactMulSupport (f * f') 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [MulOneClass β] {f f' : α → β} (hf : HasCompactMulSupport f) (hf' : HasCompactMulSupport f')
Goal: HasCompactMulSupport (f * f')
-- HasCompactSupport.smul_left
theorem HasCompactSupport.smul_left (hf : HasCompactSupport f') : HasCompactSupport (f • f') 
Arguments: {α : Type u_2} {M : Type u_7} {R : Type u_9} [TopologicalSpace α] [Zero M] [SMulZeroClass R M] {f : α → R} {f' : α → M} (hf : HasCompactSupport f')
Goal: HasCompactSupport (f • f')
-- HasCompactSupport.smul_right
theorem HasCompactSupport.smul_right (hf : HasCompactSupport f) : HasCompactSupport (f • f') 
Arguments: {α : Type u_2} {M : Type u_7} {R : Type u_9} [TopologicalSpace α] [Zero R] [Zero M] [SMulWithZero R M] {f : α → R} {f' : α → M} (hf : HasCompactSupport f)
Goal: HasCompactSupport (f • f')
-- HasCompactSupport.mul_right
theorem HasCompactSupport.mul_right (hf : HasCompactSupport f) : HasCompactSupport (f * f') 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [MulZeroClass β] {f f' : α → β} (hf : HasCompactSupport f)
Goal: HasCompactSupport (f * f')
-- HasCompactSupport.mul_left
theorem HasCompactSupport.mul_left (hf : HasCompactSupport f') : HasCompactSupport (f * f') 
Arguments: {α : Type u_2} {β : Type u_4} [TopologicalSpace α] [MulZeroClass β] {f f' : α → β} (hf : HasCompactSupport f')
Goal: HasCompactSupport (f * f')
-- HasCompactSupport.abs
protected theorem HasCompactSupport.abs {f : α → β} (hf : HasCompactSupport f) :
    HasCompactSupport |f| 
Arguments: {α : Type u_10} {β : Type u_11} [TopologicalSpace α] [AddGroup β] [Lattice β] [CovariantClass β β (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {f : α → β} (hf : HasCompactSupport f)
Goal: HasCompactSupport |f|
-- LocallyFinite.exists_finset_nhd_support_subset
@[to_additive " If a family of functions `f` has locally-finite support, subordinate to a family of
open sets, then for any point we can find a neighbourhood on which only finitely-many members of `f`
are non-zero. "]
theorem LocallyFinite.exists_finset_nhd_mulSupport_subset {U : ι → Set X} [One R] {f : ι → X → R}
    (hlf : LocallyFinite fun i => mulSupport (f i)) (hso : ∀ i, mulTSupport (f i) ⊆ U i)
    (ho : ∀ i, IsOpen (U i)) (x : X) :
    ∃ (is : Finset ι), ∃ n, n ∈ 𝓝 x ∧ (n ⊆ ⋂ i ∈ is, U i) ∧
      ∀ z ∈ n, (mulSupport fun i => f i z) ⊆ is 
Arguments: {X : Type u_1} {R : Type u_9} {ι : Type u_10} [TopologicalSpace X] {U : ι → Set X} [Zero R] {f : ι → X → R} (hlf : LocallyFinite fun i => Function.support (f i)) (hso : ∀ (i : ι), tsupport (f i) ⊆ U i) (ho : ∀ (i : ι), IsOpen (U i)) (x : X)
Goal: ∃ is, ∃ n ∈ 𝓝 x, n ⊆ ⋂ i ∈ is, U i ∧ ∀ z ∈ n, (Function.support fun i => f i z) ⊆ ↑is