File: Mathlib/Order/Interval/Set/Monotone.lean

full_name: antitone_Ici
code: theorem antitone_Ici : Antitone (Ici : α → Set α) 
goal: Antitone Set.Ici

full_name: monotone_Iic
code: theorem monotone_Iic : Monotone (Iic : α → Set α) 
goal: Monotone Set.Iic

full_name: antitone_Ioi
code: theorem antitone_Ioi : Antitone (Ioi : α → Set α) 
goal: Antitone Set.Ioi

full_name: monotone_Iio
code: theorem monotone_Iio : Monotone (Iio : α → Set α) 
goal: Monotone Set.Iio

full_name: Monotone.Ici
code: protected theorem Monotone.Ici (hf : Monotone f) : Antitone fun x => Ici (f x) 
goal: Antitone fun x => Set.Ici (f x)

full_name: MonotoneOn.Ici
code: protected theorem MonotoneOn.Ici (hf : MonotoneOn f s) : AntitoneOn (fun x => Ici (f x)) s 
goal: AntitoneOn (fun x => Set.Ici (f x)) s

full_name: Antitone.Ici
code: protected theorem Antitone.Ici (hf : Antitone f) : Monotone fun x => Ici (f x) 
goal: Monotone fun x => Set.Ici (f x)

full_name: AntitoneOn.Ici
code: protected theorem AntitoneOn.Ici (hf : AntitoneOn f s) : MonotoneOn (fun x => Ici (f x)) s 
goal: MonotoneOn (fun x => Set.Ici (f x)) s

full_name: Monotone.Iic
code: protected theorem Monotone.Iic (hf : Monotone f) : Monotone fun x => Iic (f x) 
goal: Monotone fun x => Set.Iic (f x)

full_name: MonotoneOn.Iic
code: protected theorem MonotoneOn.Iic (hf : MonotoneOn f s) : MonotoneOn (fun x => Iic (f x)) s 
goal: MonotoneOn (fun x => Set.Iic (f x)) s

full_name: Antitone.Iic
code: protected theorem Antitone.Iic (hf : Antitone f) : Antitone fun x => Iic (f x) 
goal: Antitone fun x => Set.Iic (f x)

full_name: AntitoneOn.Iic
code: protected theorem AntitoneOn.Iic (hf : AntitoneOn f s) : AntitoneOn (fun x => Iic (f x)) s 
goal: AntitoneOn (fun x => Set.Iic (f x)) s

full_name: Monotone.Ioi
code: protected theorem Monotone.Ioi (hf : Monotone f) : Antitone fun x => Ioi (f x) 
goal: Antitone fun x => Set.Ioi (f x)

full_name: MonotoneOn.Ioi
code: protected theorem MonotoneOn.Ioi (hf : MonotoneOn f s) : AntitoneOn (fun x => Ioi (f x)) s 
goal: AntitoneOn (fun x => Set.Ioi (f x)) s

full_name: Antitone.Ioi
code: protected theorem Antitone.Ioi (hf : Antitone f) : Monotone fun x => Ioi (f x) 
goal: Monotone fun x => Set.Ioi (f x)

full_name: AntitoneOn.Ioi
code: protected theorem AntitoneOn.Ioi (hf : AntitoneOn f s) : MonotoneOn (fun x => Ioi (f x)) s 
goal: MonotoneOn (fun x => Set.Ioi (f x)) s

full_name: Monotone.Iio
code: protected theorem Monotone.Iio (hf : Monotone f) : Monotone fun x => Iio (f x) 
goal: Monotone fun x => Set.Iio (f x)

full_name: MonotoneOn.Iio
code: protected theorem MonotoneOn.Iio (hf : MonotoneOn f s) : MonotoneOn (fun x => Iio (f x)) s 
goal: MonotoneOn (fun x => Set.Iio (f x)) s

full_name: Antitone.Iio
code: protected theorem Antitone.Iio (hf : Antitone f) : Antitone fun x => Iio (f x) 
goal: Antitone fun x => Set.Iio (f x)

full_name: AntitoneOn.Iio
code: protected theorem AntitoneOn.Iio (hf : AntitoneOn f s) : AntitoneOn (fun x => Iio (f x)) s 
goal: AntitoneOn (fun x => Set.Iio (f x)) s

full_name: Monotone.Icc
code: protected theorem Monotone.Icc (hf : Monotone f) (hg : Antitone g) :
    Antitone fun x => Icc (f x) (g x) 
goal: Antitone fun x => Set.Icc (f x) (g x)

full_name: MonotoneOn.Icc
code: protected theorem MonotoneOn.Icc (hf : MonotoneOn f s) (hg : AntitoneOn g s) :
    AntitoneOn (fun x => Icc (f x) (g x)) s 
goal: AntitoneOn (fun x => Set.Icc (f x) (g x)) s

full_name: Antitone.Icc
code: protected theorem Antitone.Icc (hf : Antitone f) (hg : Monotone g) :
    Monotone fun x => Icc (f x) (g x) 
goal: Monotone fun x => Set.Icc (f x) (g x)

full_name: AntitoneOn.Icc
code: protected theorem AntitoneOn.Icc (hf : AntitoneOn f s) (hg : MonotoneOn g s) :
    MonotoneOn (fun x => Icc (f x) (g x)) s 
goal: MonotoneOn (fun x => Set.Icc (f x) (g x)) s

full_name: Monotone.Ico
code: protected theorem Monotone.Ico (hf : Monotone f) (hg : Antitone g) :
    Antitone fun x => Ico (f x) (g x) 
goal: Antitone fun x => Set.Ico (f x) (g x)

full_name: MonotoneOn.Ico
code: protected theorem MonotoneOn.Ico (hf : MonotoneOn f s) (hg : AntitoneOn g s) :
    AntitoneOn (fun x => Ico (f x) (g x)) s 
goal: AntitoneOn (fun x => Set.Ico (f x) (g x)) s

full_name: Antitone.Ico
code: protected theorem Antitone.Ico (hf : Antitone f) (hg : Monotone g) :
    Monotone fun x => Ico (f x) (g x) 
goal: Monotone fun x => Set.Ico (f x) (g x)

full_name: AntitoneOn.Ico
code: protected theorem AntitoneOn.Ico (hf : AntitoneOn f s) (hg : MonotoneOn g s) :
    MonotoneOn (fun x => Ico (f x) (g x)) s 
goal: MonotoneOn (fun x => Set.Ico (f x) (g x)) s

full_name: Monotone.Ioc
code: protected theorem Monotone.Ioc (hf : Monotone f) (hg : Antitone g) :
    Antitone fun x => Ioc (f x) (g x) 
goal: Antitone fun x => Set.Ioc (f x) (g x)

full_name: MonotoneOn.Ioc
code: protected theorem MonotoneOn.Ioc (hf : MonotoneOn f s) (hg : AntitoneOn g s) :
    AntitoneOn (fun x => Ioc (f x) (g x)) s 
goal: AntitoneOn (fun x => Set.Ioc (f x) (g x)) s

full_name: Antitone.Ioc
code: protected theorem Antitone.Ioc (hf : Antitone f) (hg : Monotone g) :
    Monotone fun x => Ioc (f x) (g x) 
goal: Monotone fun x => Set.Ioc (f x) (g x)

full_name: AntitoneOn.Ioc
code: protected theorem AntitoneOn.Ioc (hf : AntitoneOn f s) (hg : MonotoneOn g s) :
    MonotoneOn (fun x => Ioc (f x) (g x)) s 
goal: MonotoneOn (fun x => Set.Ioc (f x) (g x)) s

full_name: Monotone.Ioo
code: protected theorem Monotone.Ioo (hf : Monotone f) (hg : Antitone g) :
    Antitone fun x => Ioo (f x) (g x) 
goal: Antitone fun x => Set.Ioo (f x) (g x)

full_name: MonotoneOn.Ioo
code: protected theorem MonotoneOn.Ioo (hf : MonotoneOn f s) (hg : AntitoneOn g s) :
    AntitoneOn (fun x => Ioo (f x) (g x)) s 
goal: AntitoneOn (fun x => Set.Ioo (f x) (g x)) s

full_name: Antitone.Ioo
code: protected theorem Antitone.Ioo (hf : Antitone f) (hg : Monotone g) :
    Monotone fun x => Ioo (f x) (g x) 
goal: Monotone fun x => Set.Ioo (f x) (g x)

full_name: AntitoneOn.Ioo
code: protected theorem AntitoneOn.Ioo (hf : AntitoneOn f s) (hg : MonotoneOn g s) :
    MonotoneOn (fun x => Ioo (f x) (g x)) s 
goal: MonotoneOn (fun x => Set.Ioo (f x) (g x)) s

full_name: iUnion_Ioo_of_mono_of_isGLB_of_isLUB
code: theorem iUnion_Ioo_of_mono_of_isGLB_of_isLUB (hf : Antitone f) (hg : Monotone g)
    (ha : IsGLB (range f) a) (hb : IsLUB (range g) b) : ⋃ x, Ioo (f x) (g x) = Ioo a b 
goal: ⋃ x, Set.Ioo (f x) (g x) = Set.Ioo a b

full_name: StrictMonoOn.Iic_id_le
code: theorem StrictMonoOn.Iic_id_le [SuccOrder α] [IsSuccArchimedean α] [OrderBot α] {n : α} {φ : α → α}
    (hφ : StrictMonoOn φ (Set.Iic n)) : ∀ m ≤ n, m ≤ φ m 
goal: m ≤ n → m ≤ φ m

full_name: StrictMonoOn.Ici_le_id
code: theorem StrictMonoOn.Ici_le_id [PredOrder α] [IsPredArchimedean α] [OrderTop α] {n : α} {φ : α → α}
    (hφ : StrictMonoOn φ (Set.Ici n)) : ∀ m, n ≤ m → φ m ≤ m 
goal: n ≤ m → φ m ≤ m

full_name: strictMonoOn_Iic_of_lt_succ
code: /-- A function `ψ` on a `SuccOrder` is strictly monotone before some `n` if for all `m` such that
`m < n`, we have `ψ m < ψ (succ m)`. -/
theorem strictMonoOn_Iic_of_lt_succ [SuccOrder α] [IsSuccArchimedean α] {n : α}
    (hψ : ∀ m, m < n → ψ m < ψ (succ m)) : StrictMonoOn ψ (Set.Iic n) 
goal: StrictMonoOn ψ (Set.Iic n)

full_name: strictAntiOn_Iic_of_succ_lt
code: theorem strictAntiOn_Iic_of_succ_lt [SuccOrder α] [IsSuccArchimedean α] {n : α}
    (hψ : ∀ m, m < n → ψ (succ m) < ψ m) : StrictAntiOn ψ (Set.Iic n) 
goal: StrictAntiOn ψ (Set.Iic n)

full_name: strictMonoOn_Ici_of_pred_lt
code: theorem strictMonoOn_Ici_of_pred_lt [PredOrder α] [IsPredArchimedean α] {n : α}
    (hψ : ∀ m, n < m → ψ (pred m) < ψ m) : StrictMonoOn ψ (Set.Ici n) 
goal: StrictMonoOn ψ (Set.Ici n)

full_name: strictAntiOn_Ici_of_lt_pred
code: theorem strictAntiOn_Ici_of_lt_pred [PredOrder α] [IsPredArchimedean α] {n : α}
    (hψ : ∀ m, n < m → ψ m < ψ (pred m)) : StrictAntiOn ψ (Set.Ici n) 
goal: StrictAntiOn ψ (Set.Ici n)