ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ShrinkingLemma.PartialRefinement.subset_iUnion
-- Porting note(#5171): this linter isn't ported yet. @[nolint has_nonempty_instance]
@[ext] structure PartialRefinement (u : ι → Set X) (s : Set X) where
    toFun : ι → Set X
    carrier : Set ι
    protected isOpen : ∀ i, IsOpen (toFun i)
    subset_iUnion : s ⊆ ⋃ i, toFun i
    closure_subset : ∀ {i}, i ∈ carrier → closure (toFun i) ⊆ u i
    apply_eq : ∀ {i}, i ∉ carrier → toFun i = u i
Arguments: {ι : Type u_1} {X : Type u_2} [TopologicalSpace X] {u : ι → Set X} {s : Set X} (self : ShrinkingLemma.PartialRefinement u s)
Goal: s ⊆ ⋃ i, self.toFun i
-- ShrinkingLemma.PartialRefinement.closure_subset
-- Porting note(#5171): this linter isn't ported yet. @[nolint has_nonempty_instance]
@[ext] structure PartialRefinement (u : ι → Set X) (s : Set X) where
    toFun : ι → Set X
    carrier : Set ι
    protected isOpen : ∀ i, IsOpen (toFun i)
    subset_iUnion : s ⊆ ⋃ i, toFun i
    closure_subset : ∀ {i}, i ∈ carrier → closure (toFun i) ⊆ u i
    apply_eq : ∀ {i}, i ∉ carrier → toFun i = u i
Arguments: {ι : Type u_1} {X : Type u_2} [TopologicalSpace X] {u : ι → Set X} {s : Set X} (self : ShrinkingLemma.PartialRefinement u s) {i : ι}
Goal: i ∈ self.carrier → closure (self.toFun i) ⊆ u i
-- ShrinkingLemma.PartialRefinement.apply_eq
-- Porting note(#5171): this linter isn't ported yet. @[nolint has_nonempty_instance]
@[ext] structure PartialRefinement (u : ι → Set X) (s : Set X) where
    toFun : ι → Set X
    carrier : Set ι
    protected isOpen : ∀ i, IsOpen (toFun i)
    subset_iUnion : s ⊆ ⋃ i, toFun i
    closure_subset : ∀ {i}, i ∈ carrier → closure (toFun i) ⊆ u i
    apply_eq : ∀ {i}, i ∉ carrier → toFun i = u i
Arguments: {ι : Type u_1} {X : Type u_2} [TopologicalSpace X] {u : ι → Set X} {s : Set X} (self : ShrinkingLemma.PartialRefinement u s) {i : ι}
Goal: i ∉ self.carrier → self.toFun i = u i
-- ShrinkingLemma.PartialRefinement.subset
protected theorem subset (v : PartialRefinement u s) (i : ι) : v i ⊆ u i 
Arguments: {ι : Type u_1} {X : Type u_2} [TopologicalSpace X] {u : ι → Set X} {s : Set X} (v : ShrinkingLemma.PartialRefinement u s) (i : ι)
Goal: v.toFun i ⊆ u i
-- ShrinkingLemma.PartialRefinement.apply_eq_of_chain
theorem apply_eq_of_chain {c : Set (PartialRefinement u s)} (hc : IsChain (· ≤ ·) c) {v₁ v₂}
    (h₁ : v₁ ∈ c) (h₂ : v₂ ∈ c) {i} (hi₁ : i ∈ v₁.carrier) (hi₂ : i ∈ v₂.carrier) :
    v₁ i = v₂ i 
Arguments: {ι : Type u_1} {X : Type u_2} [TopologicalSpace X] {u : ι → Set X} {s : Set X} {c : Set (ShrinkingLemma.PartialRefinement u s)} (hc : IsChain (fun x x_1 => x ≤ x_1) c) {v₁ v₂ : ShrinkingLemma.PartialRefinement u s} (h₁ : v₁ ∈ c) (h₂ : v₂ ∈ c) {i : ι} (hi₁ : i ∈ v₁.carrier) (hi₂ : i ∈ v₂.carrier)
Goal: v₁.toFun i = v₂.toFun i
-- ShrinkingLemma.PartialRefinement.find_mem
theorem find_mem {c : Set (PartialRefinement u s)} (i : ι) (ne : c.Nonempty) : find c ne i ∈ c 
Arguments: {ι : Type u_1} {X : Type u_2} [TopologicalSpace X] {u : ι → Set X} {s : Set X} {c : Set (ShrinkingLemma.PartialRefinement u s)} (i : ι) (ne : c.Nonempty)
Goal: ShrinkingLemma.PartialRefinement.find c ne i ∈ c
-- ShrinkingLemma.PartialRefinement.mem_find_carrier_iff
theorem mem_find_carrier_iff {c : Set (PartialRefinement u s)} {i : ι} (ne : c.Nonempty) :
    i ∈ (find c ne i).carrier ↔ i ∈ chainSupCarrier c 
Arguments: {ι : Type u_1} {X : Type u_2} [TopologicalSpace X] {u : ι → Set X} {s : Set X} {c : Set (ShrinkingLemma.PartialRefinement u s)} {i : ι} (ne : c.Nonempty)
Goal: i ∈ (ShrinkingLemma.PartialRefinement.find c ne i).carrier ↔ i ∈ ShrinkingLemma.PartialRefinement.chainSupCarrier c
-- ShrinkingLemma.PartialRefinement.find_apply_of_mem
theorem find_apply_of_mem {c : Set (PartialRefinement u s)} (hc : IsChain (· ≤ ·) c)
    (ne : c.Nonempty) {i v} (hv : v ∈ c) (hi : i ∈ carrier v) : find c ne i i = v i 
Arguments: {ι : Type u_1} {X : Type u_2} [TopologicalSpace X] {u : ι → Set X} {s : Set X} {c : Set (ShrinkingLemma.PartialRefinement u s)} (hc : IsChain (fun x x_1 => x ≤ x_1) c) (ne : c.Nonempty) {i : ι} {v : ShrinkingLemma.PartialRefinement u s} (hv : v ∈ c) (hi : i ∈ v.carrier)
Goal: (ShrinkingLemma.PartialRefinement.find c ne i).toFun i = v.toFun i
-- ShrinkingLemma.PartialRefinement.le_chainSup
theorem le_chainSup {c : Set (PartialRefinement u s)} (hc : IsChain (· ≤ ·) c) (ne : c.Nonempty)
    (hfin : ∀ x ∈ s, { i | x ∈ u i }.Finite) (hU : s ⊆ ⋃ i, u i) {v} (hv : v ∈ c) :
    v ≤ chainSup c hc ne hfin hU 
Arguments: {ι : Type u_1} {X : Type u_2} [TopologicalSpace X] {u : ι → Set X} {s : Set X} {c : Set (ShrinkingLemma.PartialRefinement u s)} (hc : IsChain (fun x x_1 => x ≤ x_1) c) (ne : c.Nonempty) (hfin : ∀ x ∈ s, {i | x ∈ u i}.Finite) (hU : s ⊆ ⋃ i, u i) {v : ShrinkingLemma.PartialRefinement u s} (hv : v ∈ c)
Goal: v ≤ ShrinkingLemma.PartialRefinement.chainSup c hc ne hfin hU
-- ShrinkingLemma.PartialRefinement.exists_gt
theorem exists_gt (v : PartialRefinement u s) (hs : IsClosed s) (i : ι) (hi : i ∉ v.carrier) :
    ∃ v' : PartialRefinement u s, v < v' 
Arguments: {ι : Type u_1} {X : Type u_2} [TopologicalSpace X] [NormalSpace X] {u : ι → Set X} {s : Set X} (v : ShrinkingLemma.PartialRefinement u s) (hs : IsClosed s) (i : ι) (hi : i ∉ v.carrier)
Goal: ∃ v', v < v'
-- exists_subset_iUnion_closure_subset
theorem exists_subset_iUnion_closure_subset (hs : IsClosed s) (uo : ∀ i, IsOpen (u i))
    (uf : ∀ x ∈ s, { i | x ∈ u i }.Finite) (us : s ⊆ ⋃ i, u i) :
    ∃ v : ι → Set X, s ⊆ iUnion v ∧ (∀ i, IsOpen (v i)) ∧ ∀ i, closure (v i) ⊆ u i 
Arguments: {ι : Type u_1} {X : Type u_2} [TopologicalSpace X] [NormalSpace X] {u : ι → Set X} {s : Set X} (hs : IsClosed s) (uo : ∀ (i : ι), IsOpen (u i)) (uf : ∀ x ∈ s, {i | x ∈ u i}.Finite) (us : s ⊆ ⋃ i, u i)
Goal: ∃ v, s ⊆ Set.iUnion v ∧ (∀ (i : ι), IsOpen (v i)) ∧ ∀ (i : ι), closure (v i) ⊆ u i
-- exists_subset_iUnion_closed_subset
theorem exists_subset_iUnion_closed_subset (hs : IsClosed s) (uo : ∀ i, IsOpen (u i))
    (uf : ∀ x ∈ s, { i | x ∈ u i }.Finite) (us : s ⊆ ⋃ i, u i) :
    ∃ v : ι → Set X, s ⊆ iUnion v ∧ (∀ i, IsClosed (v i)) ∧ ∀ i, v i ⊆ u i 
Arguments: {ι : Type u_1} {X : Type u_2} [TopologicalSpace X] [NormalSpace X] {u : ι → Set X} {s : Set X} (hs : IsClosed s) (uo : ∀ (i : ι), IsOpen (u i)) (uf : ∀ x ∈ s, {i | x ∈ u i}.Finite) (us : s ⊆ ⋃ i, u i)
Goal: ∃ v, s ⊆ Set.iUnion v ∧ (∀ (i : ι), IsClosed (v i)) ∧ ∀ (i : ι), v i ⊆ u i
-- exists_iUnion_eq_closure_subset
theorem exists_iUnion_eq_closure_subset (uo : ∀ i, IsOpen (u i)) (uf : ∀ x, { i | x ∈ u i }.Finite)
    (uU : ⋃ i, u i = univ) :
    ∃ v : ι → Set X, iUnion v = univ ∧ (∀ i, IsOpen (v i)) ∧ ∀ i, closure (v i) ⊆ u i 
Arguments: {ι : Type u_1} {X : Type u_2} [TopologicalSpace X] [NormalSpace X] {u : ι → Set X} (uo : ∀ (i : ι), IsOpen (u i)) (uf : ∀ (x : X), {i | x ∈ u i}.Finite) (uU : ⋃ i, u i = Set.univ)
Goal: ∃ v, Set.iUnion v = Set.univ ∧ (∀ (i : ι), IsOpen (v i)) ∧ ∀ (i : ι), closure (v i) ⊆ u i
-- exists_iUnion_eq_closed_subset
theorem exists_iUnion_eq_closed_subset (uo : ∀ i, IsOpen (u i)) (uf : ∀ x, { i | x ∈ u i }.Finite)
    (uU : ⋃ i, u i = univ) :
    ∃ v : ι → Set X, iUnion v = univ ∧ (∀ i, IsClosed (v i)) ∧ ∀ i, v i ⊆ u i 
Arguments: {ι : Type u_1} {X : Type u_2} [TopologicalSpace X] [NormalSpace X] {u : ι → Set X} (uo : ∀ (i : ι), IsOpen (u i)) (uf : ∀ (x : X), {i | x ∈ u i}.Finite) (uU : ⋃ i, u i = Set.univ)
Goal: ∃ v, Set.iUnion v = Set.univ ∧ (∀ (i : ι), IsClosed (v i)) ∧ ∀ (i : ι), v i ⊆ u i
-- IsLocalExtrOn.elim
theorem IsLocalExtrOn.elim {p : Prop} :
    IsLocalExtrOn f s a → (IsLocalMinOn f s a → p) → (IsLocalMaxOn f s a → p) → p 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {f : α → β} {s : Set α} {a : α} {p : Prop}
Goal: IsLocalExtrOn f s a → (IsLocalMinOn f s a → p) → (IsLocalMaxOn f s a → p) → p
-- IsLocalExtr.elim
theorem IsLocalExtr.elim {p : Prop} :
    IsLocalExtr f a → (IsLocalMin f a → p) → (IsLocalMax f a → p) → p 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {f : α → β} {a : α} {p : Prop}
Goal: IsLocalExtr f a → (IsLocalMin f a → p) → (IsLocalMax f a → p) → p
-- IsLocalMin.on
theorem IsLocalMin.on (h : IsLocalMin f a) (s) : IsLocalMinOn f s a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {f : α → β} {a : α} (h : IsLocalMin f a) (s : Set α)
Goal: IsLocalMinOn f s a
-- IsLocalMax.on
theorem IsLocalMax.on (h : IsLocalMax f a) (s) : IsLocalMaxOn f s a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {f : α → β} {a : α} (h : IsLocalMax f a) (s : Set α)
Goal: IsLocalMaxOn f s a
-- IsLocalExtr.on
theorem IsLocalExtr.on (h : IsLocalExtr f a) (s) : IsLocalExtrOn f s a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {f : α → β} {a : α} (h : IsLocalExtr f a) (s : Set α)
Goal: IsLocalExtrOn f s a
-- IsLocalMinOn.on_subset
theorem IsLocalMinOn.on_subset {t : Set α} (hf : IsLocalMinOn f t a) (h : s ⊆ t) :
    IsLocalMinOn f s a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {f : α → β} {s : Set α} {a : α} {t : Set α} (hf : IsLocalMinOn f t a) (h : s ⊆ t)
Goal: IsLocalMinOn f s a
-- IsLocalMaxOn.on_subset
theorem IsLocalMaxOn.on_subset {t : Set α} (hf : IsLocalMaxOn f t a) (h : s ⊆ t) :
    IsLocalMaxOn f s a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {f : α → β} {s : Set α} {a : α} {t : Set α} (hf : IsLocalMaxOn f t a) (h : s ⊆ t)
Goal: IsLocalMaxOn f s a
-- IsLocalExtrOn.on_subset
theorem IsLocalExtrOn.on_subset {t : Set α} (hf : IsLocalExtrOn f t a) (h : s ⊆ t) :
    IsLocalExtrOn f s a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {f : α → β} {s : Set α} {a : α} {t : Set α} (hf : IsLocalExtrOn f t a) (h : s ⊆ t)
Goal: IsLocalExtrOn f s a
-- IsLocalMinOn.inter
theorem IsLocalMinOn.inter (hf : IsLocalMinOn f s a) (t) : IsLocalMinOn f (s ∩ t) a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {f : α → β} {s : Set α} {a : α} (hf : IsLocalMinOn f s a) (t : Set α)
Goal: IsLocalMinOn f (s ∩ t) a
-- IsLocalMaxOn.inter
theorem IsLocalMaxOn.inter (hf : IsLocalMaxOn f s a) (t) : IsLocalMaxOn f (s ∩ t) a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {f : α → β} {s : Set α} {a : α} (hf : IsLocalMaxOn f s a) (t : Set α)
Goal: IsLocalMaxOn f (s ∩ t) a
-- IsLocalExtrOn.inter
theorem IsLocalExtrOn.inter (hf : IsLocalExtrOn f s a) (t) : IsLocalExtrOn f (s ∩ t) a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {f : α → β} {s : Set α} {a : α} (hf : IsLocalExtrOn f s a) (t : Set α)
Goal: IsLocalExtrOn f (s ∩ t) a
-- IsMinOn.localize
theorem IsMinOn.localize (hf : IsMinOn f s a) : IsLocalMinOn f s a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {f : α → β} {s : Set α} {a : α} (hf : IsMinOn f s a)
Goal: IsLocalMinOn f s a
-- IsMaxOn.localize
theorem IsMaxOn.localize (hf : IsMaxOn f s a) : IsLocalMaxOn f s a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {f : α → β} {s : Set α} {a : α} (hf : IsMaxOn f s a)
Goal: IsLocalMaxOn f s a
-- IsExtrOn.localize
theorem IsExtrOn.localize (hf : IsExtrOn f s a) : IsLocalExtrOn f s a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {f : α → β} {s : Set α} {a : α} (hf : IsExtrOn f s a)
Goal: IsLocalExtrOn f s a
-- IsLocalMinOn.isLocalMin
theorem IsLocalMinOn.isLocalMin (hf : IsLocalMinOn f s a) (hs : s ∈ 𝓝 a) : IsLocalMin f a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {f : α → β} {s : Set α} {a : α} (hf : IsLocalMinOn f s a) (hs : s ∈ 𝓝 a)
Goal: IsLocalMin f a
-- IsLocalMaxOn.isLocalMax
theorem IsLocalMaxOn.isLocalMax (hf : IsLocalMaxOn f s a) (hs : s ∈ 𝓝 a) : IsLocalMax f a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {f : α → β} {s : Set α} {a : α} (hf : IsLocalMaxOn f s a) (hs : s ∈ 𝓝 a)
Goal: IsLocalMax f a
-- IsLocalExtrOn.isLocalExtr
theorem IsLocalExtrOn.isLocalExtr (hf : IsLocalExtrOn f s a) (hs : s ∈ 𝓝 a) : IsLocalExtr f a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {f : α → β} {s : Set α} {a : α} (hf : IsLocalExtrOn f s a) (hs : s ∈ 𝓝 a)
Goal: IsLocalExtr f a
-- IsMinOn.isLocalMin
theorem IsMinOn.isLocalMin (hf : IsMinOn f s a) (hs : s ∈ 𝓝 a) : IsLocalMin f a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {f : α → β} {s : Set α} {a : α} (hf : IsMinOn f s a) (hs : s ∈ 𝓝 a)
Goal: IsLocalMin f a
-- IsMaxOn.isLocalMax
theorem IsMaxOn.isLocalMax (hf : IsMaxOn f s a) (hs : s ∈ 𝓝 a) : IsLocalMax f a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {f : α → β} {s : Set α} {a : α} (hf : IsMaxOn f s a) (hs : s ∈ 𝓝 a)
Goal: IsLocalMax f a
-- IsExtrOn.isLocalExtr
theorem IsExtrOn.isLocalExtr (hf : IsExtrOn f s a) (hs : s ∈ 𝓝 a) : IsLocalExtr f a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {f : α → β} {s : Set α} {a : α} (hf : IsExtrOn f s a) (hs : s ∈ 𝓝 a)
Goal: IsLocalExtr f a
-- IsLocalMinOn.not_nhds_le_map
theorem IsLocalMinOn.not_nhds_le_map [TopologicalSpace β] (hf : IsLocalMinOn f s a)
    [NeBot (𝓝[<] f a)] : ¬𝓝 (f a) ≤ map f (𝓝[s] a) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {f : α → β} {s : Set α} {a : α} [TopologicalSpace β] (hf : IsLocalMinOn f s a) [(𝓝[<] f a).NeBot]
Goal: ¬𝓝 (f a) ≤ Filter.map f (𝓝[s] a)
-- IsLocalMaxOn.not_nhds_le_map
theorem IsLocalMaxOn.not_nhds_le_map [TopologicalSpace β] (hf : IsLocalMaxOn f s a)
    [NeBot (𝓝[>] f a)] : ¬𝓝 (f a) ≤ map f (𝓝[s] a) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {f : α → β} {s : Set α} {a : α} [TopologicalSpace β] (hf : IsLocalMaxOn f s a) [(𝓝[>] f a).NeBot]
Goal: ¬𝓝 (f a) ≤ Filter.map f (𝓝[s] a)
-- IsLocalExtrOn.not_nhds_le_map
theorem IsLocalExtrOn.not_nhds_le_map [TopologicalSpace β] (hf : IsLocalExtrOn f s a)
    [NeBot (𝓝[<] f a)] [NeBot (𝓝[>] f a)] : ¬𝓝 (f a) ≤ map f (𝓝[s] a) 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {f : α → β} {s : Set α} {a : α} [TopologicalSpace β] (hf : IsLocalExtrOn f s a) [(𝓝[<] f a).NeBot] [(𝓝[>] f a).NeBot]
Goal: ¬𝓝 (f a) ≤ Filter.map f (𝓝[s] a)
-- isLocalMinOn_const
theorem isLocalMinOn_const {b : β} : IsLocalMinOn (fun _ => b) s a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {s : Set α} {a : α} {b : β}
Goal: IsLocalMinOn (fun x => b) s a
-- isLocalMaxOn_const
theorem isLocalMaxOn_const {b : β} : IsLocalMaxOn (fun _ => b) s a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {s : Set α} {a : α} {b : β}
Goal: IsLocalMaxOn (fun x => b) s a
-- isLocalExtrOn_const
theorem isLocalExtrOn_const {b : β} : IsLocalExtrOn (fun _ => b) s a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {s : Set α} {a : α} {b : β}
Goal: IsLocalExtrOn (fun x => b) s a
-- isLocalMin_const
theorem isLocalMin_const {b : β} : IsLocalMin (fun _ => b) a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {a : α} {b : β}
Goal: IsLocalMin (fun x => b) a
-- isLocalMax_const
theorem isLocalMax_const {b : β} : IsLocalMax (fun _ => b) a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {a : α} {b : β}
Goal: IsLocalMax (fun x => b) a
-- isLocalExtr_const
theorem isLocalExtr_const {b : β} : IsLocalExtr (fun _ => b) a 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [Preorder β] {a : α} {b : β}
Goal: IsLocalExtr (fun x => b) a
-- IsLocalMin.comp_mono
nonrec theorem IsLocalMin.comp_mono (hf : IsLocalMin f a) {g : β → γ} (hg : Monotone g) :
    IsLocalMin (g ∘ f) a 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [Preorder β] [Preorder γ] {f : α → β} {a : α} (hf : IsLocalMin f a) {g : β → γ} (hg : Monotone g)
Goal: IsLocalMin (g ∘ f) a
-- IsLocalMax.comp_mono
nonrec theorem IsLocalMax.comp_mono (hf : IsLocalMax f a) {g : β → γ} (hg : Monotone g) :
    IsLocalMax (g ∘ f) a 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [Preorder β] [Preorder γ] {f : α → β} {a : α} (hf : IsLocalMax f a) {g : β → γ} (hg : Monotone g)
Goal: IsLocalMax (g ∘ f) a
-- IsLocalExtr.comp_mono
nonrec theorem IsLocalExtr.comp_mono (hf : IsLocalExtr f a) {g : β → γ} (hg : Monotone g) :
    IsLocalExtr (g ∘ f) a 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [Preorder β] [Preorder γ] {f : α → β} {a : α} (hf : IsLocalExtr f a) {g : β → γ} (hg : Monotone g)
Goal: IsLocalExtr (g ∘ f) a
-- IsLocalMin.comp_antitone
nonrec theorem IsLocalMin.comp_antitone (hf : IsLocalMin f a) {g : β → γ} (hg : Antitone g) :
    IsLocalMax (g ∘ f) a 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [Preorder β] [Preorder γ] {f : α → β} {a : α} (hf : IsLocalMin f a) {g : β → γ} (hg : Antitone g)
Goal: IsLocalMax (g ∘ f) a
-- IsLocalMax.comp_antitone
nonrec theorem IsLocalMax.comp_antitone (hf : IsLocalMax f a) {g : β → γ} (hg : Antitone g) :
    IsLocalMin (g ∘ f) a 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [Preorder β] [Preorder γ] {f : α → β} {a : α} (hf : IsLocalMax f a) {g : β → γ} (hg : Antitone g)
Goal: IsLocalMin (g ∘ f) a
-- IsLocalExtr.comp_antitone
nonrec theorem IsLocalExtr.comp_antitone (hf : IsLocalExtr f a) {g : β → γ} (hg : Antitone g) :
    IsLocalExtr (g ∘ f) a 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [Preorder β] [Preorder γ] {f : α → β} {a : α} (hf : IsLocalExtr f a) {g : β → γ} (hg : Antitone g)
Goal: IsLocalExtr (g ∘ f) a
-- IsLocalMinOn.comp_mono
nonrec theorem IsLocalMinOn.comp_mono (hf : IsLocalMinOn f s a) {g : β → γ} (hg : Monotone g) :
    IsLocalMinOn (g ∘ f) s a 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [Preorder β] [Preorder γ] {f : α → β} {s : Set α} {a : α} (hf : IsLocalMinOn f s a) {g : β → γ} (hg : Monotone g)
Goal: IsLocalMinOn (g ∘ f) s a
-- IsLocalMaxOn.comp_mono
nonrec theorem IsLocalMaxOn.comp_mono (hf : IsLocalMaxOn f s a) {g : β → γ} (hg : Monotone g) :
    IsLocalMaxOn (g ∘ f) s a 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [Preorder β] [Preorder γ] {f : α → β} {s : Set α} {a : α} (hf : IsLocalMaxOn f s a) {g : β → γ} (hg : Monotone g)
Goal: IsLocalMaxOn (g ∘ f) s a
-- IsLocalExtrOn.comp_mono
nonrec theorem IsLocalExtrOn.comp_mono (hf : IsLocalExtrOn f s a) {g : β → γ} (hg : Monotone g) :
    IsLocalExtrOn (g ∘ f) s a 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [Preorder β] [Preorder γ] {f : α → β} {s : Set α} {a : α} (hf : IsLocalExtrOn f s a) {g : β → γ} (hg : Monotone g)
Goal: IsLocalExtrOn (g ∘ f) s a
-- IsLocalMinOn.comp_antitone
nonrec theorem IsLocalMinOn.comp_antitone (hf : IsLocalMinOn f s a) {g : β → γ} (hg : Antitone g) :
    IsLocalMaxOn (g ∘ f) s a 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [Preorder β] [Preorder γ] {f : α → β} {s : Set α} {a : α} (hf : IsLocalMinOn f s a) {g : β → γ} (hg : Antitone g)
Goal: IsLocalMaxOn (g ∘ f) s a
-- IsLocalMaxOn.comp_antitone
nonrec theorem IsLocalMaxOn.comp_antitone (hf : IsLocalMaxOn f s a) {g : β → γ} (hg : Antitone g) :
    IsLocalMinOn (g ∘ f) s a 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [Preorder β] [Preorder γ] {f : α → β} {s : Set α} {a : α} (hf : IsLocalMaxOn f s a) {g : β → γ} (hg : Antitone g)
Goal: IsLocalMinOn (g ∘ f) s a
-- IsLocalExtrOn.comp_antitone
nonrec theorem IsLocalExtrOn.comp_antitone (hf : IsLocalExtrOn f s a) {g : β → γ}
    (hg : Antitone g) : IsLocalExtrOn (g ∘ f) s a 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [TopologicalSpace α] [Preorder β] [Preorder γ] {f : α → β} {s : Set α} {a : α} (hf : IsLocalExtrOn f s a) {g : β → γ} (hg : Antitone g)
Goal: IsLocalExtrOn (g ∘ f) s a
-- IsLocalMin.bicomp_mono
nonrec theorem IsLocalMin.bicomp_mono [Preorder δ] {op : β → γ → δ}
    (hop : ((· ≤ ·) ⇒ (· ≤ ·) ⇒ (· ≤ ·)) op op) (hf : IsLocalMin f a) {g : α → γ}
    (hg : IsLocalMin g a) : IsLocalMin (fun x => op (f x) (g x)) a 
Arguments: {α : Type u} {β : Type v} {γ : Type w} {δ : Type x} [TopologicalSpace α] [Preorder β] [Preorder γ] {f : α → β} {a : α} [Preorder δ] {op : β → γ → δ} (hop : ((fun x x_1 => x ≤ x_1) ⇒ (fun x x_1 => x ≤ x_1) ⇒ fun x x_1 => x ≤ x_1) op op) (hf : IsLocalMin f a) {g : α → γ} (hg : IsLocalMin g a)
Goal: IsLocalMin (fun x => op (f x) (g x)) a
-- IsLocalMax.bicomp_mono
nonrec theorem IsLocalMax.bicomp_mono [Preorder δ] {op : β → γ → δ}
    (hop : ((· ≤ ·) ⇒ (· ≤ ·) ⇒ (· ≤ ·)) op op) (hf : IsLocalMax f a) {g : α → γ}
    (hg : IsLocalMax g a) : IsLocalMax (fun x => op (f x) (g x)) a 
Arguments: {α : Type u} {β : Type v} {γ : Type w} {δ : Type x} [TopologicalSpace α] [Preorder β] [Preorder γ] {f : α → β} {a : α} [Preorder δ] {op : β → γ → δ} (hop : ((fun x x_1 => x ≤ x_1) ⇒ (fun x x_1 => x ≤ x_1) ⇒ fun x x_1 => x ≤ x_1) op op) (hf : IsLocalMax f a) {g : α → γ} (hg : IsLocalMax g a)
Goal: IsLocalMax (fun x => op (f x) (g x)) a
-- IsLocalMinOn.bicomp_mono
nonrec theorem IsLocalMinOn.bicomp_mono [Preorder δ] {op : β → γ → δ}
    (hop : ((· ≤ ·) ⇒ (· ≤ ·) ⇒ (· ≤ ·)) op op) (hf : IsLocalMinOn f s a) {g : α → γ}
    (hg : IsLocalMinOn g s a) : IsLocalMinOn (fun x => op (f x) (g x)) s a 
Arguments: {α : Type u} {β : Type v} {γ : Type w} {δ : Type x} [TopologicalSpace α] [Preorder β] [Preorder γ] {f : α → β} {s : Set α} {a : α} [Preorder δ] {op : β → γ → δ} (hop : ((fun x x_1 => x ≤ x_1) ⇒ (fun x x_1 => x ≤ x_1) ⇒ fun x x_1 => x ≤ x_1) op op) (hf : IsLocalMinOn f s a) {g : α → γ} (hg : IsLocalMinOn g s a)
Goal: IsLocalMinOn (fun x => op (f x) (g x)) s a
-- IsLocalMaxOn.bicomp_mono
nonrec theorem IsLocalMaxOn.bicomp_mono [Preorder δ] {op : β → γ → δ}
    (hop : ((· ≤ ·) ⇒ (· ≤ ·) ⇒ (· ≤ ·)) op op) (hf : IsLocalMaxOn f s a) {g : α → γ}
    (hg : IsLocalMaxOn g s a) : IsLocalMaxOn (fun x => op (f x) (g x)) s a 
Arguments: {α : Type u} {β : Type v} {γ : Type w} {δ : Type x} [TopologicalSpace α] [Preorder β] [Preorder γ] {f : α → β} {s : Set α} {a : α} [Preorder δ] {op : β → γ → δ} (hop : ((fun x x_1 => x ≤ x_1) ⇒ (fun x x_1 => x ≤ x_1) ⇒ fun x x_1 => x ≤ x_1) op op) (hf : IsLocalMaxOn f s a) {g : α → γ} (hg : IsLocalMaxOn g s a)
Goal: IsLocalMaxOn (fun x => op (f x) (g x)) s a
-- IsLocalMin.comp_continuous
theorem IsLocalMin.comp_continuous [TopologicalSpace δ] {g : δ → α} {b : δ}
    (hf : IsLocalMin f (g b)) (hg : ContinuousAt g b) : IsLocalMin (f ∘ g) b 
Arguments: {α : Type u} {β : Type v} {δ : Type x} [TopologicalSpace α] [Preorder β] {f : α → β} [TopologicalSpace δ] {g : δ → α} {b : δ} (hf : IsLocalMin f (g b)) (hg : ContinuousAt g b)
Goal: IsLocalMin (f ∘ g) b
-- IsLocalMax.comp_continuous
theorem IsLocalMax.comp_continuous [TopologicalSpace δ] {g : δ → α} {b : δ}
    (hf : IsLocalMax f (g b)) (hg : ContinuousAt g b) : IsLocalMax (f ∘ g) b 
Arguments: {α : Type u} {β : Type v} {δ : Type x} [TopologicalSpace α] [Preorder β] {f : α → β} [TopologicalSpace δ] {g : δ → α} {b : δ} (hf : IsLocalMax f (g b)) (hg : ContinuousAt g b)
Goal: IsLocalMax (f ∘ g) b
-- IsLocalExtr.comp_continuous
theorem IsLocalExtr.comp_continuous [TopologicalSpace δ] {g : δ → α} {b : δ}
    (hf : IsLocalExtr f (g b)) (hg : ContinuousAt g b) : IsLocalExtr (f ∘ g) b 
Arguments: {α : Type u} {β : Type v} {δ : Type x} [TopologicalSpace α] [Preorder β] {f : α → β} [TopologicalSpace δ] {g : δ → α} {b : δ} (hf : IsLocalExtr f (g b)) (hg : ContinuousAt g b)
Goal: IsLocalExtr (f ∘ g) b
-- IsLocalMin.comp_continuousOn
theorem IsLocalMin.comp_continuousOn [TopologicalSpace δ] {s : Set δ} {g : δ → α} {b : δ}
    (hf : IsLocalMin f (g b)) (hg : ContinuousOn g s) (hb : b ∈ s) : IsLocalMinOn (f ∘ g) s b 
Arguments: {α : Type u} {β : Type v} {δ : Type x} [TopologicalSpace α] [Preorder β] {f : α → β} [TopologicalSpace δ] {s : Set δ} {g : δ → α} {b : δ} (hf : IsLocalMin f (g b)) (hg : ContinuousOn g s) (hb : b ∈ s)
Goal: IsLocalMinOn (f ∘ g) s b
-- IsLocalMax.comp_continuousOn
theorem IsLocalMax.comp_continuousOn [TopologicalSpace δ] {s : Set δ} {g : δ → α} {b : δ}
    (hf : IsLocalMax f (g b)) (hg : ContinuousOn g s) (hb : b ∈ s) : IsLocalMaxOn (f ∘ g) s b 
Arguments: {α : Type u} {β : Type v} {δ : Type x} [TopologicalSpace α] [Preorder β] {f : α → β} [TopologicalSpace δ] {s : Set δ} {g : δ → α} {b : δ} (hf : IsLocalMax f (g b)) (hg : ContinuousOn g s) (hb : b ∈ s)
Goal: IsLocalMaxOn (f ∘ g) s b