ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- monotoneOn_toDual_comp_iff
@[simp]
theorem monotoneOn_toDual_comp_iff : MonotoneOn (toDual ∘ f : α → βᵒᵈ) s ↔ AntitoneOn f s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} {s : Set α}
Goal: MonotoneOn (⇑OrderDual.toDual ∘ f) s ↔ AntitoneOn f s
-- antitoneOn_toDual_comp_iff
@[simp]
theorem antitoneOn_toDual_comp_iff : AntitoneOn (toDual ∘ f : α → βᵒᵈ) s ↔ MonotoneOn f s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} {s : Set α}
Goal: AntitoneOn (⇑OrderDual.toDual ∘ f) s ↔ MonotoneOn f s
-- strictMono_comp_ofDual_iff
@[simp]
theorem strictMono_comp_ofDual_iff : StrictMono (f ∘ ofDual) ↔ StrictAnti f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β}
Goal: StrictMono (f ∘ ⇑OrderDual.ofDual) ↔ StrictAnti f
-- strictAnti_comp_ofDual_iff
@[simp]
theorem strictAnti_comp_ofDual_iff : StrictAnti (f ∘ ofDual) ↔ StrictMono f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β}
Goal: StrictAnti (f ∘ ⇑OrderDual.ofDual) ↔ StrictMono f
-- strictMono_toDual_comp_iff
@[simp]
theorem strictMono_toDual_comp_iff : StrictMono (toDual ∘ f : α → βᵒᵈ) ↔ StrictAnti f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β}
Goal: StrictMono (⇑OrderDual.toDual ∘ f) ↔ StrictAnti f
-- strictAnti_toDual_comp_iff
@[simp]
theorem strictAnti_toDual_comp_iff : StrictAnti (toDual ∘ f : α → βᵒᵈ) ↔ StrictMono f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β}
Goal: StrictAnti (⇑OrderDual.toDual ∘ f) ↔ StrictMono f
-- strictMonoOn_comp_ofDual_iff
@[simp]
theorem strictMonoOn_comp_ofDual_iff : StrictMonoOn (f ∘ ofDual) s ↔ StrictAntiOn f s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} {s : Set α}
Goal: StrictMonoOn (f ∘ ⇑OrderDual.ofDual) s ↔ StrictAntiOn f s
-- strictAntiOn_comp_ofDual_iff
@[simp]
theorem strictAntiOn_comp_ofDual_iff : StrictAntiOn (f ∘ ofDual) s ↔ StrictMonoOn f s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} {s : Set α}
Goal: StrictAntiOn (f ∘ ⇑OrderDual.ofDual) s ↔ StrictMonoOn f s
-- strictMonoOn_toDual_comp_iff
@[simp]
theorem strictMonoOn_toDual_comp_iff : StrictMonoOn (toDual ∘ f : α → βᵒᵈ) s ↔ StrictAntiOn f s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} {s : Set α}
Goal: StrictMonoOn (⇑OrderDual.toDual ∘ f) s ↔ StrictAntiOn f s
-- strictAntiOn_toDual_comp_iff
@[simp]
theorem strictAntiOn_toDual_comp_iff : StrictAntiOn (toDual ∘ f : α → βᵒᵈ) s ↔ StrictMonoOn f s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} {s : Set α}
Goal: StrictAntiOn (⇑OrderDual.toDual ∘ f) s ↔ StrictMonoOn f s
-- monotone_dual_iff
theorem monotone_dual_iff : Monotone (toDual ∘ f ∘ ofDual : αᵒᵈ → βᵒᵈ) ↔ Monotone f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β}
Goal: Monotone (⇑OrderDual.toDual ∘ f ∘ ⇑OrderDual.ofDual) ↔ Monotone f
-- antitone_dual_iff
theorem antitone_dual_iff : Antitone (toDual ∘ f ∘ ofDual : αᵒᵈ → βᵒᵈ) ↔ Antitone f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β}
Goal: Antitone (⇑OrderDual.toDual ∘ f ∘ ⇑OrderDual.ofDual) ↔ Antitone f
-- monotoneOn_dual_iff
theorem monotoneOn_dual_iff : MonotoneOn (toDual ∘ f ∘ ofDual : αᵒᵈ → βᵒᵈ) s ↔ MonotoneOn f s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} {s : Set α}
Goal: MonotoneOn (⇑OrderDual.toDual ∘ f ∘ ⇑OrderDual.ofDual) s ↔ MonotoneOn f s
-- antitoneOn_dual_iff
theorem antitoneOn_dual_iff : AntitoneOn (toDual ∘ f ∘ ofDual : αᵒᵈ → βᵒᵈ) s ↔ AntitoneOn f s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} {s : Set α}
Goal: AntitoneOn (⇑OrderDual.toDual ∘ f ∘ ⇑OrderDual.ofDual) s ↔ AntitoneOn f s
-- strictMono_dual_iff
theorem strictMono_dual_iff : StrictMono (toDual ∘ f ∘ ofDual : αᵒᵈ → βᵒᵈ) ↔ StrictMono f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β}
Goal: StrictMono (⇑OrderDual.toDual ∘ f ∘ ⇑OrderDual.ofDual) ↔ StrictMono f
-- strictAnti_dual_iff
theorem strictAnti_dual_iff : StrictAnti (toDual ∘ f ∘ ofDual : αᵒᵈ → βᵒᵈ) ↔ StrictAnti f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β}
Goal: StrictAnti (⇑OrderDual.toDual ∘ f ∘ ⇑OrderDual.ofDual) ↔ StrictAnti f
-- strictMonoOn_dual_iff
theorem strictMonoOn_dual_iff :
    StrictMonoOn (toDual ∘ f ∘ ofDual : αᵒᵈ → βᵒᵈ) s ↔ StrictMonoOn f s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} {s : Set α}
Goal: StrictMonoOn (⇑OrderDual.toDual ∘ f ∘ ⇑OrderDual.ofDual) s ↔ StrictMonoOn f s
-- strictAntiOn_dual_iff
theorem strictAntiOn_dual_iff :
    StrictAntiOn (toDual ∘ f ∘ ofDual : αᵒᵈ → βᵒᵈ) s ↔ StrictAntiOn f s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} {s : Set α}
Goal: StrictAntiOn (⇑OrderDual.toDual ∘ f ∘ ⇑OrderDual.ofDual) s ↔ StrictAntiOn f s
-- Monotone.comp_le_comp_left
theorem Monotone.comp_le_comp_left
    [Preorder β] {f : β → α} {g h : γ → β} (hf : Monotone f) (le_gh : g ≤ h) :
    LE.le.{max w u} (f ∘ g) (f ∘ h) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] {f : β → α} {g h : γ → β} (hf : Monotone f) (le_gh : g ≤ h)
Goal: f ∘ g ≤ f ∘ h
-- monotone_lam
theorem monotone_lam {f : α → β → γ} (hf : ∀ b, Monotone fun a ↦ f a b) : Monotone f 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder γ] {f : α → β → γ} (hf : ∀ (b : β), Monotone fun a => f a b)
Goal: Monotone f
-- monotone_app
theorem monotone_app (f : β → α → γ) (b : β) (hf : Monotone fun a b ↦ f b a) : Monotone (f b) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder γ] (f : β → α → γ) (b : β) (hf : Monotone fun a b => f b a)
Goal: Monotone (f b)
-- antitone_lam
theorem antitone_lam {f : α → β → γ} (hf : ∀ b, Antitone fun a ↦ f a b) : Antitone f 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder γ] {f : α → β → γ} (hf : ∀ (b : β), Antitone fun a => f a b)
Goal: Antitone f
-- antitone_app
theorem antitone_app (f : β → α → γ) (b : β) (hf : Antitone fun a b ↦ f b a) : Antitone (f b) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder γ] (f : β → α → γ) (b : β) (hf : Antitone fun a b => f b a)
Goal: Antitone (f b)
-- Function.monotone_eval
theorem Function.monotone_eval {ι : Type u} {α : ι → Type v} [∀ i, Preorder (α i)] (i : ι) :
    Monotone (Function.eval i : (∀ i, α i) → α i) 
Arguments: {ι : Type u} {α : ι → Type v} [(i : ι) → Preorder (α i)] (i : ι)
Goal: Monotone (Function.eval i)
-- Monotone.imp
theorem Monotone.imp (hf : Monotone f) (h : a ≤ b) : f a ≤ f b 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} {a b : α} (hf : Monotone f) (h : a ≤ b)
Goal: f a ≤ f b
-- Antitone.imp
theorem Antitone.imp (hf : Antitone f) (h : a ≤ b) : f b ≤ f a 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} {a b : α} (hf : Antitone f) (h : a ≤ b)
Goal: f b ≤ f a
-- StrictMono.imp
theorem StrictMono.imp (hf : StrictMono f) (h : a < b) : f a < f b 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} {a b : α} (hf : StrictMono f) (h : a < b)
Goal: f a < f b
-- StrictAnti.imp
theorem StrictAnti.imp (hf : StrictAnti f) (h : a < b) : f b < f a 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} {a b : α} (hf : StrictAnti f) (h : a < b)
Goal: f b < f a
-- Monotone.monotoneOn
protected theorem Monotone.monotoneOn (hf : Monotone f) (s : Set α) : MonotoneOn f s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} (hf : Monotone f) (s : Set α)
Goal: MonotoneOn f s
-- Antitone.antitoneOn
protected theorem Antitone.antitoneOn (hf : Antitone f) (s : Set α) : AntitoneOn f s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} (hf : Antitone f) (s : Set α)
Goal: AntitoneOn f s
-- monotoneOn_univ
@[simp] theorem monotoneOn_univ : MonotoneOn f Set.univ ↔ Monotone f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β}
Goal: MonotoneOn f Set.univ ↔ Monotone f
-- antitoneOn_univ
@[simp] theorem antitoneOn_univ : AntitoneOn f Set.univ ↔ Antitone f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β}
Goal: AntitoneOn f Set.univ ↔ Antitone f
-- StrictMono.strictMonoOn
protected theorem StrictMono.strictMonoOn (hf : StrictMono f) (s : Set α) : StrictMonoOn f s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} (hf : StrictMono f) (s : Set α)
Goal: StrictMonoOn f s
-- StrictAnti.strictAntiOn
protected theorem StrictAnti.strictAntiOn (hf : StrictAnti f) (s : Set α) : StrictAntiOn f s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} (hf : StrictAnti f) (s : Set α)
Goal: StrictAntiOn f s
-- strictMonoOn_univ
@[simp] theorem strictMonoOn_univ : StrictMonoOn f Set.univ ↔ StrictMono f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β}
Goal: StrictMonoOn f Set.univ ↔ StrictMono f
-- strictAntiOn_univ
@[simp] theorem strictAntiOn_univ : StrictAntiOn f Set.univ ↔ StrictAnti f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β}
Goal: StrictAntiOn f Set.univ ↔ StrictAnti f
-- Monotone.strictMono_of_injective
theorem Monotone.strictMono_of_injective (h₁ : Monotone f) (h₂ : Injective f) : StrictMono f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [PartialOrder β] {f : α → β} (h₁ : Monotone f) (h₂ : Function.Injective f)
Goal: StrictMono f
-- Antitone.strictAnti_of_injective
theorem Antitone.strictAnti_of_injective (h₁ : Antitone f) (h₂ : Injective f) : StrictAnti f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [PartialOrder β] {f : α → β} (h₁ : Antitone f) (h₂ : Function.Injective f)
Goal: StrictAnti f
-- monotone_iff_forall_lt
theorem monotone_iff_forall_lt : Monotone f ↔ ∀ ⦃a b⦄, a < b → f a ≤ f b 
Arguments: {α : Type u} {β : Type v} [PartialOrder α] [Preorder β] {f : α → β}
Goal: Monotone f ↔ ∀ ⦃a b : α⦄, a < b → f a ≤ f b
-- antitone_iff_forall_lt
theorem antitone_iff_forall_lt : Antitone f ↔ ∀ ⦃a b⦄, a < b → f b ≤ f a 
Arguments: {α : Type u} {β : Type v} [PartialOrder α] [Preorder β] {f : α → β}
Goal: Antitone f ↔ ∀ ⦃a b : α⦄, a < b → f b ≤ f a
-- monotoneOn_iff_forall_lt
theorem monotoneOn_iff_forall_lt :
    MonotoneOn f s ↔ ∀ ⦃a⦄ (_ : a ∈ s) ⦃b⦄ (_ : b ∈ s), a < b → f a ≤ f b 
Arguments: {α : Type u} {β : Type v} [PartialOrder α] [Preorder β] {f : α → β} {s : Set α}
Goal: MonotoneOn f s ↔ ∀ ⦃a : α⦄, a ∈ s → ∀ ⦃b : α⦄, b ∈ s → a < b → f a ≤ f b
-- antitoneOn_iff_forall_lt
theorem antitoneOn_iff_forall_lt :
    AntitoneOn f s ↔ ∀ ⦃a⦄ (_ : a ∈ s) ⦃b⦄ (_ : b ∈ s), a < b → f b ≤ f a 
Arguments: {α : Type u} {β : Type v} [PartialOrder α] [Preorder β] {f : α → β} {s : Set α}
Goal: AntitoneOn f s ↔ ∀ ⦃a : α⦄, a ∈ s → ∀ ⦃b : α⦄, b ∈ s → a < b → f b ≤ f a
-- StrictMonoOn.monotoneOn
protected theorem StrictMonoOn.monotoneOn (hf : StrictMonoOn f s) : MonotoneOn f s 
Arguments: {α : Type u} {β : Type v} [PartialOrder α] [Preorder β] {f : α → β} {s : Set α} (hf : StrictMonoOn f s)
Goal: MonotoneOn f s
-- StrictAntiOn.antitoneOn
protected theorem StrictAntiOn.antitoneOn (hf : StrictAntiOn f s) : AntitoneOn f s 
Arguments: {α : Type u} {β : Type v} [PartialOrder α] [Preorder β] {f : α → β} {s : Set α} (hf : StrictAntiOn f s)
Goal: AntitoneOn f s
-- StrictMono.monotone
protected theorem StrictMono.monotone (hf : StrictMono f) : Monotone f 
Arguments: {α : Type u} {β : Type v} [PartialOrder α] [Preorder β] {f : α → β} (hf : StrictMono f)
Goal: Monotone f
-- StrictAnti.antitone
protected theorem StrictAnti.antitone (hf : StrictAnti f) : Antitone f 
Arguments: {α : Type u} {β : Type v} [PartialOrder α] [Preorder β] {f : α → β} (hf : StrictAnti f)
Goal: Antitone f
-- Subsingleton.monotone
protected theorem monotone [Subsingleton α] (f : α → β) : Monotone f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] [Subsingleton α] (f : α → β)
Goal: Monotone f
-- Subsingleton.antitone
protected theorem antitone [Subsingleton α] (f : α → β) : Antitone f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] [Subsingleton α] (f : α → β)
Goal: Antitone f
-- Subsingleton.monotone'
theorem monotone' [Subsingleton β] (f : α → β) : Monotone f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] [Subsingleton β] (f : α → β)
Goal: Monotone f
-- Subsingleton.antitone'
theorem antitone' [Subsingleton β] (f : α → β) : Antitone f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] [Subsingleton β] (f : α → β)
Goal: Antitone f
-- Subsingleton.strictMono
protected theorem strictMono [Subsingleton α] (f : α → β) : StrictMono f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] [Subsingleton α] (f : α → β)
Goal: StrictMono f
-- Subsingleton.strictAnti
protected theorem strictAnti [Subsingleton α] (f : α → β) : StrictAnti f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] [Subsingleton α] (f : α → β)
Goal: StrictAnti f
-- monotone_id
theorem monotone_id [Preorder α] : Monotone (id : α → α) 
Arguments: {α : Type u} [Preorder α]
Goal: Monotone id
-- monotoneOn_id
theorem monotoneOn_id [Preorder α] {s : Set α} : MonotoneOn id s 
Arguments: {α : Type u} [Preorder α] {s : Set α}
Goal: MonotoneOn id s
-- strictMono_id
theorem strictMono_id [Preorder α] : StrictMono (id : α → α) 
Arguments: {α : Type u} [Preorder α]
Goal: StrictMono id
-- strictMonoOn_id
theorem strictMonoOn_id [Preorder α] {s : Set α} : StrictMonoOn id s 
Arguments: {α : Type u} [Preorder α] {s : Set α}
Goal: StrictMonoOn id s
-- monotone_const
theorem monotone_const [Preorder α] [Preorder β] {c : β} : Monotone fun _ : α ↦ c 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {c : β}
Goal: Monotone fun x => c
-- monotoneOn_const
theorem monotoneOn_const [Preorder α] [Preorder β] {c : β} {s : Set α} :
    MonotoneOn (fun _ : α ↦ c) s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {c : β} {s : Set α}
Goal: MonotoneOn (fun x => c) s
-- antitone_const
theorem antitone_const [Preorder α] [Preorder β] {c : β} : Antitone fun _ : α ↦ c 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {c : β}
Goal: Antitone fun x => c
-- antitoneOn_const
theorem antitoneOn_const [Preorder α] [Preorder β] {c : β} {s : Set α} :
    AntitoneOn (fun _ : α ↦ c) s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {c : β} {s : Set α}
Goal: AntitoneOn (fun x => c) s
-- strictMono_of_le_iff_le
theorem strictMono_of_le_iff_le [Preorder α] [Preorder β] {f : α → β}
    (h : ∀ x y, x ≤ y ↔ f x ≤ f y) : StrictMono f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} (h : ∀ (x y : α), x ≤ y ↔ f x ≤ f y)
Goal: StrictMono f
-- strictAnti_of_le_iff_le
theorem strictAnti_of_le_iff_le [Preorder α] [Preorder β] {f : α → β}
    (h : ∀ x y, x ≤ y ↔ f y ≤ f x) : StrictAnti f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} (h : ∀ (x y : α), x ≤ y ↔ f y ≤ f x)
Goal: StrictAnti f
-- injective_of_lt_imp_ne
theorem injective_of_lt_imp_ne [LinearOrder α] {f : α → β} (h : ∀ x y, x < y → f x ≠ f y) :
    Injective f 
Arguments: {α : Type u} {β : Type v} [LinearOrder α] {f : α → β} (h : ∀ (x y : α), x < y → f x ≠ f y)
Goal: Function.Injective f
-- injective_of_le_imp_le
theorem injective_of_le_imp_le [PartialOrder α] [Preorder β] (f : α → β)
    (h : ∀ {x y}, f x ≤ f y → x ≤ y) : Injective f 
Arguments: {α : Type u} {β : Type v} [PartialOrder α] [Preorder β] (f : α → β) (h : ∀ {x y : α}, f x ≤ f y → x ≤ y)
Goal: Function.Injective f
-- StrictMono.isMax_of_apply
theorem StrictMono.isMax_of_apply (hf : StrictMono f) (ha : IsMax (f a)) : IsMax a 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} {a : α} (hf : StrictMono f) (ha : IsMax (f a))
Goal: IsMax a
-- StrictMono.isMin_of_apply
theorem StrictMono.isMin_of_apply (hf : StrictMono f) (ha : IsMin (f a)) : IsMin a 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} {a : α} (hf : StrictMono f) (ha : IsMin (f a))
Goal: IsMin a
-- StrictAnti.isMax_of_apply
theorem StrictAnti.isMax_of_apply (hf : StrictAnti f) (ha : IsMin (f a)) : IsMax a 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} {a : α} (hf : StrictAnti f) (ha : IsMin (f a))
Goal: IsMax a
-- StrictAnti.isMin_of_apply
theorem StrictAnti.isMin_of_apply (hf : StrictAnti f) (ha : IsMax (f a)) : IsMin a 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} {a : α} (hf : StrictAnti f) (ha : IsMax (f a))
Goal: IsMin a
-- StrictMono.add_le_nat
lemma StrictMono.add_le_nat {f : ℕ → ℕ} (hf : StrictMono f) (m n : ℕ) : m + f n ≤ f (m + n)  
Arguments: {f : ℕ → ℕ} (hf : StrictMono f) (m n : ℕ)
Goal: m + f n ≤ f (m + n)
-- StrictMono.ite'
protected theorem StrictMono.ite' (hf : StrictMono f) (hg : StrictMono g) {p : α → Prop}
    [DecidablePred p]
    (hp : ∀ ⦃x y⦄, x < y → p y → p x) (hfg : ∀ ⦃x y⦄, p x → ¬p y → x < y → f x < g y) :
    StrictMono fun x ↦ if p x then f x else g x 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f g : α → β} (hf : StrictMono f) (hg : StrictMono g) {p : α → Prop} [DecidablePred p] (hp : ∀ ⦃x y : α⦄, x < y → p y → p x) (hfg : ∀ ⦃x y : α⦄, p x → ¬p y → x < y → f x < g y)
Goal: StrictMono fun x => if p x then f x else g x
-- StrictMono.ite
protected theorem StrictMono.ite (hf : StrictMono f) (hg : StrictMono g) {p : α → Prop}
    [DecidablePred p] (hp : ∀ ⦃x y⦄, x < y → p y → p x) (hfg : ∀ x, f x ≤ g x) :
    StrictMono fun x ↦ if p x then f x else g x 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f g : α → β} (hf : StrictMono f) (hg : StrictMono g) {p : α → Prop} [DecidablePred p] (hp : ∀ ⦃x y : α⦄, x < y → p y → p x) (hfg : ∀ (x : α), f x ≤ g x)
Goal: StrictMono fun x => if p x then f x else g x
-- StrictAnti.ite'
protected theorem StrictAnti.ite' (hf : StrictAnti f) (hg : StrictAnti g) {p : α → Prop}
    [DecidablePred p]
    (hp : ∀ ⦃x y⦄, x < y → p y → p x) (hfg : ∀ ⦃x y⦄, p x → ¬p y → x < y → g y < f x) :
    StrictAnti fun x ↦ if p x then f x else g x 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f g : α → β} (hf : StrictAnti f) (hg : StrictAnti g) {p : α → Prop} [DecidablePred p] (hp : ∀ ⦃x y : α⦄, x < y → p y → p x) (hfg : ∀ ⦃x y : α⦄, p x → ¬p y → x < y → g y < f x)
Goal: StrictAnti fun x => if p x then f x else g x
-- StrictAnti.ite
protected theorem StrictAnti.ite (hf : StrictAnti f) (hg : StrictAnti g) {p : α → Prop}
    [DecidablePred p] (hp : ∀ ⦃x y⦄, x < y → p y → p x) (hfg : ∀ x, g x ≤ f x) :
    StrictAnti fun x ↦ if p x then f x else g x 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f g : α → β} (hf : StrictAnti f) (hg : StrictAnti g) {p : α → Prop} [DecidablePred p] (hp : ∀ ⦃x y : α⦄, x < y → p y → p x) (hfg : ∀ (x : α), g x ≤ f x)
Goal: StrictAnti fun x => if p x then f x else g x
-- Monotone.comp
protected theorem Monotone.comp (hg : Monotone g) (hf : Monotone f) : Monotone (g ∘ f) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {g : β → γ} {f : α → β} (hg : Monotone g) (hf : Monotone f)
Goal: Monotone (g ∘ f)
-- Monotone.comp_antitone
theorem Monotone.comp_antitone (hg : Monotone g) (hf : Antitone f) : Antitone (g ∘ f) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {g : β → γ} {f : α → β} (hg : Monotone g) (hf : Antitone f)
Goal: Antitone (g ∘ f)
-- Antitone.comp
protected theorem Antitone.comp (hg : Antitone g) (hf : Antitone f) : Monotone (g ∘ f) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {g : β → γ} {f : α → β} (hg : Antitone g) (hf : Antitone f)
Goal: Monotone (g ∘ f)
-- Antitone.comp_monotone
theorem Antitone.comp_monotone (hg : Antitone g) (hf : Monotone f) : Antitone (g ∘ f) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {g : β → γ} {f : α → β} (hg : Antitone g) (hf : Monotone f)
Goal: Antitone (g ∘ f)
-- Monotone.iterate
protected theorem Monotone.iterate {f : α → α} (hf : Monotone f) (n : ℕ) : Monotone f^[n] 
Arguments: {α : Type u} [Preorder α] {f : α → α} (hf : Monotone f) (n : ℕ)
Goal: Monotone f^[n]
-- Monotone.comp_monotoneOn
protected theorem Monotone.comp_monotoneOn (hg : Monotone g) (hf : MonotoneOn f s) :
    MonotoneOn (g ∘ f) s 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {g : β → γ} {f : α → β} {s : Set α} (hg : Monotone g) (hf : MonotoneOn f s)
Goal: MonotoneOn (g ∘ f) s
-- Monotone.comp_antitoneOn
theorem Monotone.comp_antitoneOn (hg : Monotone g) (hf : AntitoneOn f s) : AntitoneOn (g ∘ f) s 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {g : β → γ} {f : α → β} {s : Set α} (hg : Monotone g) (hf : AntitoneOn f s)
Goal: AntitoneOn (g ∘ f) s
-- Antitone.comp_antitoneOn
protected theorem Antitone.comp_antitoneOn (hg : Antitone g) (hf : AntitoneOn f s) :
    MonotoneOn (g ∘ f) s 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {g : β → γ} {f : α → β} {s : Set α} (hg : Antitone g) (hf : AntitoneOn f s)
Goal: MonotoneOn (g ∘ f) s
-- Antitone.comp_monotoneOn
theorem Antitone.comp_monotoneOn (hg : Antitone g) (hf : MonotoneOn f s) : AntitoneOn (g ∘ f) s 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {g : β → γ} {f : α → β} {s : Set α} (hg : Antitone g) (hf : MonotoneOn f s)
Goal: AntitoneOn (g ∘ f) s
-- StrictMono.comp
protected theorem StrictMono.comp (hg : StrictMono g) (hf : StrictMono f) : StrictMono (g ∘ f) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {g : β → γ} {f : α → β} (hg : StrictMono g) (hf : StrictMono f)
Goal: StrictMono (g ∘ f)
-- StrictMono.comp_strictAnti
theorem StrictMono.comp_strictAnti (hg : StrictMono g) (hf : StrictAnti f) : StrictAnti (g ∘ f) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {g : β → γ} {f : α → β} (hg : StrictMono g) (hf : StrictAnti f)
Goal: StrictAnti (g ∘ f)
-- StrictAnti.comp
protected theorem StrictAnti.comp (hg : StrictAnti g) (hf : StrictAnti f) : StrictMono (g ∘ f) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {g : β → γ} {f : α → β} (hg : StrictAnti g) (hf : StrictAnti f)
Goal: StrictMono (g ∘ f)
-- StrictAnti.comp_strictMono
theorem StrictAnti.comp_strictMono (hg : StrictAnti g) (hf : StrictMono f) : StrictAnti (g ∘ f) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {g : β → γ} {f : α → β} (hg : StrictAnti g) (hf : StrictMono f)
Goal: StrictAnti (g ∘ f)
-- StrictMono.iterate
protected theorem StrictMono.iterate {f : α → α} (hf : StrictMono f) (n : ℕ) : StrictMono f^[n] 
Arguments: {α : Type u} [Preorder α] {f : α → α} (hf : StrictMono f) (n : ℕ)
Goal: StrictMono f^[n]
-- StrictMono.comp_strictMonoOn
protected theorem StrictMono.comp_strictMonoOn (hg : StrictMono g) (hf : StrictMonoOn f s) :
    StrictMonoOn (g ∘ f) s 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {g : β → γ} {f : α → β} {s : Set α} (hg : StrictMono g) (hf : StrictMonoOn f s)
Goal: StrictMonoOn (g ∘ f) s
-- StrictMono.comp_strictAntiOn
theorem StrictMono.comp_strictAntiOn (hg : StrictMono g) (hf : StrictAntiOn f s) :
    StrictAntiOn (g ∘ f) s 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {g : β → γ} {f : α → β} {s : Set α} (hg : StrictMono g) (hf : StrictAntiOn f s)
Goal: StrictAntiOn (g ∘ f) s
-- StrictAnti.comp_strictAntiOn
protected theorem StrictAnti.comp_strictAntiOn (hg : StrictAnti g) (hf : StrictAntiOn f s) :
    StrictMonoOn (g ∘ f) s 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {g : β → γ} {f : α → β} {s : Set α} (hg : StrictAnti g) (hf : StrictAntiOn f s)
Goal: StrictMonoOn (g ∘ f) s
-- StrictAnti.comp_strictMonoOn
theorem StrictAnti.comp_strictMonoOn (hg : StrictAnti g) (hf : StrictMonoOn f s) :
    StrictAntiOn (g ∘ f) s 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {g : β → γ} {f : α → β} {s : Set α} (hg : StrictAnti g) (hf : StrictMonoOn f s)
Goal: StrictAntiOn (g ∘ f) s