ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Continuous.exists_forall_ge'
theorem Continuous.exists_forall_ge' [ClosedIciTopology α] {f : β → α} (hf : Continuous f)
    (x₀ : β) (h : ∀ᶠ x in cocompact β, f x ≤ f x₀) : ∃ x : β, ∀ y : β, f y ≤ f x 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIciTopology α] {f : β → α} (hf : Continuous f) (x₀ : β) (h : ∀ᶠ (x : β) in Filter.cocompact β, f x ≤ f x₀)
Goal: ∃ x, ∀ (y : β), f y ≤ f x
-- Continuous.exists_forall_le
theorem Continuous.exists_forall_le [ClosedIicTopology α] [Nonempty β] {f : β → α}
    (hf : Continuous f) (hlim : Tendsto f (cocompact β) atTop) : ∃ x, ∀ y, f x ≤ f y 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIicTopology α] [Nonempty β] {f : β → α} (hf : Continuous f) (hlim : Filter.Tendsto f (Filter.cocompact β) Filter.atTop)
Goal: ∃ x, ∀ (y : β), f x ≤ f y
-- Continuous.exists_forall_ge
theorem Continuous.exists_forall_ge [ClosedIciTopology α] [Nonempty β] {f : β → α}
    (hf : Continuous f) (hlim : Tendsto f (cocompact β) atBot) : ∃ x, ∀ y, f y ≤ f x 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIciTopology α] [Nonempty β] {f : β → α} (hf : Continuous f) (hlim : Filter.Tendsto f (Filter.cocompact β) Filter.atBot)
Goal: ∃ x, ∀ (y : β), f y ≤ f x
-- Continuous.exists_forall_le_of_hasCompactMulSupport
@[to_additive "A continuous function with compact support has a global minimum."]
theorem Continuous.exists_forall_le_of_hasCompactMulSupport [ClosedIicTopology α] [Nonempty β]
    [One α] {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) :
    ∃ x : β, ∀ y : β, f x ≤ f y 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIicTopology α] [Nonempty β] [One α] {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f)
Goal: ∃ x, ∀ (y : β), f x ≤ f y
-- Continuous.exists_forall_le_of_hasCompactSupport
@[to_additive "A continuous function with compact support has a global minimum."]
theorem Continuous.exists_forall_le_of_hasCompactMulSupport [ClosedIicTopology α] [Nonempty β]
    [One α] {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) :
    ∃ x : β, ∀ y : β, f x ≤ f y 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIicTopology α] [Nonempty β] [Zero α] {f : β → α} (hf : Continuous f) (h : HasCompactSupport f)
Goal: ∃ x, ∀ (y : β), f x ≤ f y
-- Continuous.exists_forall_ge_of_hasCompactMulSupport
@[to_additive "A continuous function with compact support has a global maximum."]
theorem Continuous.exists_forall_ge_of_hasCompactMulSupport [ClosedIciTopology α] [Nonempty β]
    [One α] {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) :
    ∃ x : β, ∀ y : β, f y ≤ f x 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIciTopology α] [Nonempty β] [One α] {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f)
Goal: ∃ x, ∀ (y : β), f y ≤ f x
-- Continuous.exists_forall_ge_of_hasCompactSupport
@[to_additive "A continuous function with compact support has a global maximum."]
theorem Continuous.exists_forall_ge_of_hasCompactMulSupport [ClosedIciTopology α] [Nonempty β]
    [One α] {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) :
    ∃ x : β, ∀ y : β, f y ≤ f x 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIciTopology α] [Nonempty β] [Zero α] {f : β → α} (hf : Continuous f) (h : HasCompactSupport f)
Goal: ∃ x, ∀ (y : β), f y ≤ f x
-- IsCompact.bddBelow
theorem IsCompact.bddBelow [ClosedIicTopology α] [Nonempty α] {s : Set α} (hs : IsCompact s) :
    BddBelow s 
Arguments: {α : Type u_2} [LinearOrder α] [TopologicalSpace α] [ClosedIicTopology α] [Nonempty α] {s : Set α} (hs : IsCompact s)
Goal: BddBelow s
-- IsCompact.bddAbove
theorem IsCompact.bddAbove [ClosedIciTopology α] [Nonempty α] {s : Set α} (hs : IsCompact s) :
    BddAbove s 
Arguments: {α : Type u_2} [LinearOrder α] [TopologicalSpace α] [ClosedIciTopology α] [Nonempty α] {s : Set α} (hs : IsCompact s)
Goal: BddAbove s
-- IsCompact.bddBelow_image
theorem IsCompact.bddBelow_image [ClosedIicTopology α] [Nonempty α] {f : β → α} {K : Set β}
    (hK : IsCompact K) (hf : ContinuousOn f K) : BddBelow (f '' K) 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIicTopology α] [Nonempty α] {f : β → α} {K : Set β} (hK : IsCompact K) (hf : ContinuousOn f K)
Goal: BddBelow (f '' K)
-- IsCompact.bddAbove_image
theorem IsCompact.bddAbove_image [ClosedIciTopology α] [Nonempty α] {f : β → α} {K : Set β}
    (hK : IsCompact K) (hf : ContinuousOn f K) : BddAbove (f '' K) 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIciTopology α] [Nonempty α] {f : β → α} {K : Set β} (hK : IsCompact K) (hf : ContinuousOn f K)
Goal: BddAbove (f '' K)
-- Continuous.bddBelow_range_of_hasCompactMulSupport
@[to_additive " A continuous function with compact support is bounded below. "]
theorem Continuous.bddBelow_range_of_hasCompactMulSupport [ClosedIicTopology α] [One α]
    {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) : BddBelow (range f) 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIicTopology α] [One α] {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f)
Goal: BddBelow (Set.range f)
-- Continuous.bddBelow_range_of_hasCompactSupport
@[to_additive " A continuous function with compact support is bounded below. "]
theorem Continuous.bddBelow_range_of_hasCompactMulSupport [ClosedIicTopology α] [One α]
    {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) : BddBelow (range f) 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIicTopology α] [Zero α] {f : β → α} (hf : Continuous f) (h : HasCompactSupport f)
Goal: BddBelow (Set.range f)
-- Continuous.bddAbove_range_of_hasCompactMulSupport
@[to_additive " A continuous function with compact support is bounded above. "]
theorem Continuous.bddAbove_range_of_hasCompactMulSupport [ClosedIciTopology α] [One α]
    {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) : BddAbove (range f) 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIciTopology α] [One α] {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f)
Goal: BddAbove (Set.range f)
-- Continuous.bddAbove_range_of_hasCompactSupport
@[to_additive " A continuous function with compact support is bounded above. "]
theorem Continuous.bddAbove_range_of_hasCompactMulSupport [ClosedIciTopology α] [One α]
    {f : β → α} (hf : Continuous f) (h : HasCompactMulSupport f) : BddAbove (range f) 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIciTopology α] [Zero α] {f : β → α} (hf : Continuous f) (h : HasCompactSupport f)
Goal: BddAbove (Set.range f)
-- IsCompact.sSup_lt_iff_of_continuous
theorem IsCompact.sSup_lt_iff_of_continuous [ClosedIciTopology α] {f : β → α} {K : Set β}
    (hK : IsCompact K) (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) :
    sSup (f '' K) < y ↔ ∀ x ∈ K, f x < y 
Arguments: {α : Type u_2} {β : Type u_3} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIciTopology α] {f : β → α} {K : Set β} (hK : IsCompact K) (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α)
Goal: sSup (f '' K) < y ↔ ∀ x ∈ K, f x < y
-- IsCompact.lt_sInf_iff_of_continuous
theorem IsCompact.lt_sInf_iff_of_continuous [ClosedIicTopology α] {f : β → α} {K : Set β}
    (hK : IsCompact K) (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α) :
    y < sInf (f '' K) ↔ ∀ x ∈ K, y < f x 
Arguments: {α : Type u_2} {β : Type u_3} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIicTopology α] {f : β → α} {K : Set β} (hK : IsCompact K) (h0K : K.Nonempty) (hf : ContinuousOn f K) (y : α)
Goal: y < sInf (f '' K) ↔ ∀ x ∈ K, y < f x
-- IsCompact.sInf_mem
theorem IsCompact.sInf_mem [ClosedIicTopology α] {s : Set α} (hs : IsCompact s)
    (ne_s : s.Nonempty) : sInf s ∈ s 
Arguments: {α : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [ClosedIicTopology α] {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty)
Goal: sInf s ∈ s
-- IsCompact.sSup_mem
theorem IsCompact.sSup_mem [ClosedIciTopology α] {s : Set α} (hs : IsCompact s)
    (ne_s : s.Nonempty) : sSup s ∈ s 
Arguments: {α : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [ClosedIciTopology α] {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty)
Goal: sSup s ∈ s
-- IsCompact.isGLB_sInf
theorem IsCompact.isGLB_sInf [ClosedIicTopology α] {s : Set α} (hs : IsCompact s)
    (ne_s : s.Nonempty) : IsGLB s (sInf s) 
Arguments: {α : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [ClosedIicTopology α] {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty)
Goal: IsGLB s (sInf s)
-- IsCompact.isLUB_sSup
theorem IsCompact.isLUB_sSup [ClosedIciTopology α] {s : Set α} (hs : IsCompact s)
    (ne_s : s.Nonempty) : IsLUB s (sSup s) 
Arguments: {α : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [ClosedIciTopology α] {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty)
Goal: IsLUB s (sSup s)
-- IsCompact.isLeast_sInf
theorem IsCompact.isLeast_sInf [ClosedIicTopology α] {s : Set α} (hs : IsCompact s)
    (ne_s : s.Nonempty) : IsLeast s (sInf s) 
Arguments: {α : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [ClosedIicTopology α] {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty)
Goal: IsLeast s (sInf s)
-- IsCompact.isGreatest_sSup
theorem IsCompact.isGreatest_sSup [ClosedIciTopology α] {s : Set α} (hs : IsCompact s)
    (ne_s : s.Nonempty) : IsGreatest s (sSup s) 
Arguments: {α : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [ClosedIciTopology α] {s : Set α} (hs : IsCompact s) (ne_s : s.Nonempty)
Goal: IsGreatest s (sSup s)
-- IsCompact.exists_sInf_image_eq_and_le
theorem IsCompact.exists_sInf_image_eq_and_le [ClosedIicTopology α] {s : Set β}
    (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) :
    ∃ x ∈ s, sInf (f '' s) = f x ∧ ∀ y ∈ s, f x ≤ f y 
Arguments: {α : Type u_2} {β : Type u_3} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIicTopology α] {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s)
Goal: ∃ x ∈ s, sInf (f '' s) = f x ∧ ∀ y ∈ s, f x ≤ f y
-- IsCompact.exists_sSup_image_eq_and_ge
theorem IsCompact.exists_sSup_image_eq_and_ge [ClosedIciTopology α] {s : Set β}
    (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) :
    ∃ x ∈ s, sSup (f '' s) = f x ∧ ∀ y ∈ s, f y ≤ f x 
Arguments: {α : Type u_2} {β : Type u_3} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIciTopology α] {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s)
Goal: ∃ x ∈ s, sSup (f '' s) = f x ∧ ∀ y ∈ s, f y ≤ f x
-- IsCompact.exists_sInf_image_eq
theorem IsCompact.exists_sInf_image_eq [ClosedIicTopology α] {s : Set β} (hs : IsCompact s)
    (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, sInf (f '' s) = f x 
Arguments: {α : Type u_2} {β : Type u_3} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIicTopology α] {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s)
Goal: ∃ x ∈ s, sInf (f '' s) = f x
-- IsCompact.exists_sSup_image_eq
theorem IsCompact.exists_sSup_image_eq [ClosedIciTopology α] {s : Set β} (hs : IsCompact s)
    (ne_s : s.Nonempty) : ∀ {f : β → α}, ContinuousOn f s → ∃ x ∈ s, sSup (f '' s) = f x 
Arguments: {α : Type u_2} {β : Type u_3} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIciTopology α] {s : Set β} (hs : IsCompact s) (ne_s : s.Nonempty) {f : β → α}
Goal: ContinuousOn f s → ∃ x ∈ s, sSup (f '' s) = f x
-- IsCompact.exists_isMinOn_mem_subset
theorem IsCompact.exists_isMinOn_mem_subset [ClosedIicTopology α] {f : β → α} {s t : Set β}
    {z : β} (ht : IsCompact t) (hf : ContinuousOn f t) (hz : z ∈ t)
    (hfz : ∀ z' ∈ t \ s, f z < f z') : ∃ x ∈ s, IsMinOn f t x 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIicTopology α] {f : β → α} {s t : Set β} {z : β} (ht : IsCompact t) (hf : ContinuousOn f t) (hz : z ∈ t) (hfz : ∀ z' ∈ t \ s, f z < f z')
Goal: ∃ x ∈ s, IsMinOn f t x
-- IsCompact.exists_isMaxOn_mem_subset
theorem IsCompact.exists_isMaxOn_mem_subset [ClosedIciTopology α] {f : β → α} {s t : Set β}
    {z : β} (ht : IsCompact t) (hf : ContinuousOn f t) (hz : z ∈ t)
    (hfz : ∀ z' ∈ t \ s, f z' < f z) : ∃ x ∈ s, IsMaxOn f t x 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIciTopology α] {f : β → α} {s t : Set β} {z : β} (ht : IsCompact t) (hf : ContinuousOn f t) (hz : z ∈ t) (hfz : ∀ z' ∈ t \ s, f z' < f z)
Goal: ∃ x ∈ s, IsMaxOn f t x
-- IsCompact.exists_isLocalMinOn_mem_subset
@[deprecated IsCompact.exists_isMinOn_mem_subset (since := "2023-02-06")]
theorem IsCompact.exists_isLocalMinOn_mem_subset [ClosedIicTopology α] {f : β → α} {s t : Set β}
    {z : β} (ht : IsCompact t) (hf : ContinuousOn f t) (hz : z ∈ t)
    (hfz : ∀ z' ∈ t \ s, f z < f z') : ∃ x ∈ s, IsLocalMinOn f t x 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIicTopology α] {f : β → α} {s t : Set β} {z : β} (ht : IsCompact t) (hf : ContinuousOn f t) (hz : z ∈ t) (hfz : ∀ z' ∈ t \ s, f z < f z')
Goal: ∃ x ∈ s, IsLocalMinOn f t x
-- IsCompact.exists_isLocalMin_mem_open
theorem IsCompact.exists_isLocalMin_mem_open [ClosedIicTopology α] {f : β → α} {s t : Set β}
    {z : β} (ht : IsCompact t) (hst : s ⊆ t) (hf : ContinuousOn f t) (hz : z ∈ t)
    (hfz : ∀ z' ∈ t \ s, f z < f z') (hs : IsOpen s) : ∃ x ∈ s, IsLocalMin f x 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIicTopology α] {f : β → α} {s t : Set β} {z : β} (ht : IsCompact t) (hst : s ⊆ t) (hf : ContinuousOn f t) (hz : z ∈ t) (hfz : ∀ z' ∈ t \ s, f z < f z') (hs : IsOpen s)
Goal: ∃ x ∈ s, IsLocalMin f x
-- IsCompact.exists_isLocalMax_mem_open
theorem IsCompact.exists_isLocalMax_mem_open [ClosedIciTopology α] {f : β → α} {s t : Set β}
    {z : β} (ht : IsCompact t) (hst : s ⊆ t) (hf : ContinuousOn f t) (hz : z ∈ t)
    (hfz : ∀ z' ∈ t \ s, f z' < f z) (hs : IsOpen s) : ∃ x ∈ s, IsLocalMax f x 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [TopologicalSpace α] [TopologicalSpace β] [ClosedIciTopology α] {f : β → α} {s t : Set β} {z : β} (ht : IsCompact t) (hst : s ⊆ t) (hf : ContinuousOn f t) (hz : z ∈ t) (hfz : ∀ z' ∈ t \ s, f z' < f z) (hs : IsOpen s)
Goal: ∃ x ∈ s, IsLocalMax f x
-- eq_Icc_of_connected_compact
theorem eq_Icc_of_connected_compact {s : Set α} (h₁ : IsConnected s) (h₂ : IsCompact s) :
    s = Icc (sInf s) (sSup s) 
Arguments: {α : Type u_2} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] {s : Set α} (h₁ : IsConnected s) (h₂ : IsCompact s)
Goal: s = Set.Icc (sInf s) (sSup s)
-- IsCompact.continuous_sSup
/- Porting note (#11215): TODO: generalize. The following version seems to be true:
```
theorem IsCompact.tendsto_sSup {f : γ → β → α} {g : β → α} {K : Set β} {l : Filter γ}
    (hK : IsCompact K) (hf : ∀ y ∈ K, Tendsto ↿f (l ×ˢ 𝓝[K] y) (𝓝 (g y)))
    (hgc : ContinuousOn g K) :
    Tendsto (fun x => sSup (f x '' K)) l (𝓝 (sSup (g '' K))) := _
```
Moreover, it seems that `hgc` follows from `hf` (Yury Kudryashov). -/
theorem IsCompact.continuous_sSup {f : γ → β → α} {K : Set β} (hK : IsCompact K)
    (hf : Continuous ↿f) : Continuous fun x => sSup (f x '' K) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] [TopologicalSpace γ] {f : γ → β → α} {K : Set β} (hK : IsCompact K) (hf : Continuous ↿f)
Goal: Continuous fun x => sSup (f x '' K)
-- IsCompact.continuous_sInf
theorem IsCompact.continuous_sInf {f : γ → β → α} {K : Set β} (hK : IsCompact K)
    (hf : Continuous ↿f) : Continuous fun x => sInf (f x '' K) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] [TopologicalSpace γ] {f : γ → β → α} {K : Set β} (hK : IsCompact K) (hf : Continuous ↿f)
Goal: Continuous fun x => sInf (f x '' K)
-- ContinuousOn.image_Icc
theorem image_Icc (hab : a ≤ b) (h : ContinuousOn f <| Icc a b) :
    f '' Icc a b = Icc (sInf <| f '' Icc a b) (sSup <| f '' Icc a b) 
Arguments: {α : Type u_2} {β : Type u_3} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] [DenselyOrdered α] [ConditionallyCompleteLinearOrder β] [OrderTopology β] {f : α → β} {a b : α} (hab : a ≤ b) (h : ContinuousOn f (Set.Icc a b))
Goal: f '' Set.Icc a b = Set.Icc (sInf (f '' Set.Icc a b)) (sSup (f '' Set.Icc a b))
-- ContinuousOn.image_uIcc_eq_Icc
theorem image_uIcc_eq_Icc (h : ContinuousOn f [[a, b]]) :
    f '' [[a, b]] = Icc (sInf (f '' [[a, b]])) (sSup (f '' [[a, b]])) 
Arguments: {α : Type u_2} {β : Type u_3} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] [DenselyOrdered α] [ConditionallyCompleteLinearOrder β] [OrderTopology β] {f : α → β} {a b : α} (h : ContinuousOn f (Set.uIcc a b))
Goal: f '' Set.uIcc a b = Set.Icc (sInf (f '' Set.uIcc a b)) (sSup (f '' Set.uIcc a b))
-- ContinuousOn.image_uIcc
theorem image_uIcc (h : ContinuousOn f <| [[a, b]]) :
    f '' [[a, b]] = [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]] 
Arguments: {α : Type u_2} {β : Type u_3} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] [DenselyOrdered α] [ConditionallyCompleteLinearOrder β] [OrderTopology β] {f : α → β} {a b : α} (h : ContinuousOn f (Set.uIcc a b))
Goal: f '' Set.uIcc a b = Set.uIcc (sInf (f '' Set.uIcc a b)) (sSup (f '' Set.uIcc a b))
-- ContinuousOn.sInf_image_Icc_le
theorem sInf_image_Icc_le (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
    sInf (f '' Icc a b) ≤ f c 
Arguments: {α : Type u_2} {β : Type u_3} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] [DenselyOrdered α] [ConditionallyCompleteLinearOrder β] [OrderTopology β] {f : α → β} {a b c : α} (h : ContinuousOn f (Set.Icc a b)) (hc : c ∈ Set.Icc a b)
Goal: sInf (f '' Set.Icc a b) ≤ f c
-- ContinuousOn.le_sSup_image_Icc
theorem le_sSup_image_Icc (h : ContinuousOn f <| Icc a b) (hc : c ∈ Icc a b) :
    f c ≤ sSup (f '' Icc a b) 
Arguments: {α : Type u_2} {β : Type u_3} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [TopologicalSpace β] [DenselyOrdered α] [ConditionallyCompleteLinearOrder β] [OrderTopology β] {f : α → β} {a b c : α} (h : ContinuousOn f (Set.Icc a b)) (hc : c ∈ Set.Icc a b)
Goal: f c ≤ sSup (f '' Set.Icc a b)
-- ContinuousMap.coe_add
@[to_additive (attr := norm_cast, simp)]
theorem coe_mul [Mul β] [ContinuousMul β] (f g : C(α, β)) : ⇑(f * g) = f * g 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Add β] [ContinuousAdd β] (f g : C(α, β))
Goal: ⇑(f + g) = ⇑f + ⇑g
-- ContinuousMap.coe_mul
@[to_additive (attr := norm_cast, simp)]
theorem coe_mul [Mul β] [ContinuousMul β] (f g : C(α, β)) : ⇑(f * g) = f * g 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Mul β] [ContinuousMul β] (f g : C(α, β))
Goal: ⇑(f * g) = ⇑f * ⇑g
-- ContinuousMap.add_apply
@[to_additive (attr := simp)]
theorem mul_apply [Mul β] [ContinuousMul β] (f g : C(α, β)) (x : α) : (f * g) x = f x * g x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Add β] [ContinuousAdd β] (f g : C(α, β)) (x : α)
Goal: (f + g) x = f x + g x
-- ContinuousMap.mul_apply
@[to_additive (attr := simp)]
theorem mul_apply [Mul β] [ContinuousMul β] (f g : C(α, β)) (x : α) : (f * g) x = f x * g x 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Mul β] [ContinuousMul β] (f g : C(α, β)) (x : α)
Goal: (f * g) x = f x * g x
-- ContinuousMap.mul_comp
@[to_additive (attr := simp)]
theorem mul_comp [Mul γ] [ContinuousMul γ] (f₁ f₂ : C(β, γ)) (g : C(α, β)) :
    (f₁ * f₂).comp g = f₁.comp g * f₂.comp g 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [Mul γ] [ContinuousMul γ] (f₁ f₂ : C(β, γ)) (g : C(α, β))
Goal: (f₁ * f₂).comp g = f₁.comp g * f₂.comp g
-- ContinuousMap.add_comp
@[to_additive (attr := simp)]
theorem mul_comp [Mul γ] [ContinuousMul γ] (f₁ f₂ : C(β, γ)) (g : C(α, β)) :
    (f₁ * f₂).comp g = f₁.comp g * f₂.comp g 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [Add γ] [ContinuousAdd γ] (f₁ f₂ : C(β, γ)) (g : C(α, β))
Goal: (f₁ + f₂).comp g = f₁.comp g + f₂.comp g
-- ContinuousMap.coe_zero
@[to_additive (attr := norm_cast, simp)]
theorem coe_one [One β] : ⇑(1 : C(α, β)) = 1 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Zero β]
Goal: ⇑0 = 0
-- ContinuousMap.coe_one
@[to_additive (attr := norm_cast, simp)]
theorem coe_one [One β] : ⇑(1 : C(α, β)) = 1 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [One β]
Goal: ⇑1 = 1
-- ContinuousMap.zero_apply
@[to_additive (attr := simp)]
theorem one_apply [One β] (x : α) : (1 : C(α, β)) x = 1 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Zero β] (x : α)
Goal: 0 x = 0
-- ContinuousMap.one_apply
@[to_additive (attr := simp)]
theorem one_apply [One β] (x : α) : (1 : C(α, β)) x = 1 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [One β] (x : α)
Goal: 1 x = 1
-- ContinuousMap.one_comp
@[to_additive (attr := simp)]
theorem one_comp [One γ] (g : C(α, β)) : (1 : C(β, γ)).comp g = 1 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [One γ] (g : C(α, β))
Goal: ContinuousMap.comp 1 g = 1
-- ContinuousMap.zero_comp
@[to_additive (attr := simp)]
theorem one_comp [One γ] (g : C(α, β)) : (1 : C(β, γ)).comp g = 1 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [Zero γ] (g : C(α, β))
Goal: ContinuousMap.comp 0 g = 0
-- ContinuousMap.coe_natCast
@[simp, norm_cast]
theorem coe_natCast [NatCast β] (n : ℕ) : ((n : C(α, β)) : α → β) = n 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [NatCast β] (n : ℕ)
Goal: ⇑↑n = ↑n
-- ContinuousMap.natCast_apply
@[simp]
theorem natCast_apply [NatCast β] (n : ℕ) (x : α) : (n : C(α, β)) x = n 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [NatCast β] (n : ℕ) (x : α)
Goal: ↑n x = ↑n
-- ContinuousMap.coe_intCast
@[simp, norm_cast]
theorem coe_intCast [IntCast β] (n : ℤ) : ((n : C(α, β)) : α → β) = n 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [IntCast β] (n : ℤ)
Goal: ⇑↑n = ↑n
-- ContinuousMap.intCast_apply
@[simp]
theorem intCast_apply [IntCast β] (n : ℤ) (x : α) : (n : C(α, β)) x = n 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [IntCast β] (n : ℤ) (x : α)
Goal: ↑n x = ↑n
-- ContinuousMap.coe_nsmul
@[to_additive (attr := norm_cast) (reorder := 7 8)]
theorem coe_pow [Monoid β] [ContinuousMul β] (f : C(α, β)) (n : ℕ) : ⇑(f ^ n) = (⇑f) ^ n 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [AddMonoid β] [ContinuousAdd β] (n : ℕ) (f : C(α, β))
Goal: ⇑(n • f) = n • ⇑f
-- ContinuousMap.coe_pow
@[to_additive (attr := norm_cast) (reorder := 7 8)]
theorem coe_pow [Monoid β] [ContinuousMul β] (f : C(α, β)) (n : ℕ) : ⇑(f ^ n) = (⇑f) ^ n 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [Monoid β] [ContinuousMul β] (f : C(α, β)) (n : ℕ)
Goal: ⇑(f ^ n) = ⇑f ^ n