File: Mathlib/Order/SuccPred/IntervalSucc.lean

full_name: Monotone.biUnion_Ico_Ioc_map_succ
code: /-- If `α` is a linear archimedean succ order and `β` is a linear order, then for any monotone
function `f` and `m n : α`, the union of intervals `Set.Ioc (f i) (f (Order.succ i))`, `m ≤ i < n`,
is equal to `Set.Ioc (f m) (f n)` -/
theorem biUnion_Ico_Ioc_map_succ [SuccOrder α] [IsSuccArchimedean α] [LinearOrder β] {f : α → β}
    (hf : Monotone f) (m n : α) : ⋃ i ∈ Ico m n, Ioc (f i) (f (succ i)) = Ioc (f m) (f n) 
goal: ⋃ i ∈ Set.Ico m n, Set.Ioc (f i) (f (Order.succ i)) = Set.Ioc (f m) (f n)

full_name: Monotone.pairwise_disjoint_on_Ioc_succ
code: /-- If `α` is a linear succ order, `β` is a preorder, and `f : α → β` is a monotone function, then
the intervals `Set.Ioc (f n) (f (Order.succ n))` are pairwise disjoint. -/
theorem pairwise_disjoint_on_Ioc_succ [SuccOrder α] [Preorder β] {f : α → β} (hf : Monotone f) :
    Pairwise (Disjoint on fun n => Ioc (f n) (f (succ n))) 
goal: Pairwise (Disjoint on fun n => Set.Ioc (f n) (f (Order.succ n)))

full_name: Monotone.pairwise_disjoint_on_Ico_succ
code: /-- If `α` is a linear succ order, `β` is a preorder, and `f : α → β` is a monotone function, then
the intervals `Set.Ico (f n) (f (Order.succ n))` are pairwise disjoint. -/
theorem pairwise_disjoint_on_Ico_succ [SuccOrder α] [Preorder β] {f : α → β} (hf : Monotone f) :
    Pairwise (Disjoint on fun n => Ico (f n) (f (succ n))) 
goal: Pairwise (Disjoint on fun n => Set.Ico (f n) (f (Order.succ n)))

full_name: Monotone.pairwise_disjoint_on_Ioo_succ
code: /-- If `α` is a linear succ order, `β` is a preorder, and `f : α → β` is a monotone function, then
the intervals `Set.Ioo (f n) (f (Order.succ n))` are pairwise disjoint. -/
theorem pairwise_disjoint_on_Ioo_succ [SuccOrder α] [Preorder β] {f : α → β} (hf : Monotone f) :
    Pairwise (Disjoint on fun n => Ioo (f n) (f (succ n))) 
goal: Pairwise (Disjoint on fun n => Set.Ioo (f n) (f (Order.succ n)))

full_name: Monotone.pairwise_disjoint_on_Ioc_pred
code: /-- If `α` is a linear pred order, `β` is a preorder, and `f : α → β` is a monotone function, then
the intervals `Set.Ioc (f Order.pred n) (f n)` are pairwise disjoint. -/
theorem pairwise_disjoint_on_Ioc_pred [PredOrder α] [Preorder β] {f : α → β} (hf : Monotone f) :
    Pairwise (Disjoint on fun n => Ioc (f (pred n)) (f n)) 
goal: Pairwise (Disjoint on fun n => Set.Ioc (f (Order.pred n)) (f n))

full_name: Monotone.pairwise_disjoint_on_Ico_pred
code: /-- If `α` is a linear pred order, `β` is a preorder, and `f : α → β` is a monotone function, then
the intervals `Set.Ico (f Order.pred n) (f n)` are pairwise disjoint. -/
theorem pairwise_disjoint_on_Ico_pred [PredOrder α] [Preorder β] {f : α → β} (hf : Monotone f) :
    Pairwise (Disjoint on fun n => Ico (f (pred n)) (f n)) 
goal: Pairwise (Disjoint on fun n => Set.Ico (f (Order.pred n)) (f n))

full_name: Monotone.pairwise_disjoint_on_Ioo_pred
code: /-- If `α` is a linear pred order, `β` is a preorder, and `f : α → β` is a monotone function, then
the intervals `Set.Ioo (f Order.pred n) (f n)` are pairwise disjoint. -/
theorem pairwise_disjoint_on_Ioo_pred [PredOrder α] [Preorder β] {f : α → β} (hf : Monotone f) :
    Pairwise (Disjoint on fun n => Ioo (f (pred n)) (f n)) 
goal: Pairwise (Disjoint on fun n => Set.Ioo (f (Order.pred n)) (f n))

full_name: Antitone.pairwise_disjoint_on_Ioc_succ
code: /-- If `α` is a linear succ order, `β` is a preorder, and `f : α → β` is an antitone function, then
the intervals `Set.Ioc (f (Order.succ n)) (f n)` are pairwise disjoint. -/
theorem pairwise_disjoint_on_Ioc_succ [SuccOrder α] [Preorder β] {f : α → β} (hf : Antitone f) :
    Pairwise (Disjoint on fun n => Ioc (f (succ n)) (f n)) 
goal: Pairwise (Disjoint on fun n => Set.Ioc (f (Order.succ n)) (f n))

full_name: Antitone.pairwise_disjoint_on_Ico_succ
code: /-- If `α` is a linear succ order, `β` is a preorder, and `f : α → β` is an antitone function, then
the intervals `Set.Ico (f (Order.succ n)) (f n)` are pairwise disjoint. -/
theorem pairwise_disjoint_on_Ico_succ [SuccOrder α] [Preorder β] {f : α → β} (hf : Antitone f) :
    Pairwise (Disjoint on fun n => Ico (f (succ n)) (f n)) 
goal: Pairwise (Disjoint on fun n => Set.Ico (f (Order.succ n)) (f n))

full_name: Antitone.pairwise_disjoint_on_Ioo_succ
code: /-- If `α` is a linear succ order, `β` is a preorder, and `f : α → β` is an antitone function, then
the intervals `Set.Ioo (f (Order.succ n)) (f n)` are pairwise disjoint. -/
theorem pairwise_disjoint_on_Ioo_succ [SuccOrder α] [Preorder β] {f : α → β} (hf : Antitone f) :
    Pairwise (Disjoint on fun n => Ioo (f (succ n)) (f n)) 
goal: Pairwise (Disjoint on fun n => Set.Ioo (f (Order.succ n)) (f n))

full_name: Antitone.pairwise_disjoint_on_Ioc_pred
code: /-- If `α` is a linear pred order, `β` is a preorder, and `f : α → β` is an antitone function, then
the intervals `Set.Ioc (f n) (f (Order.pred n))` are pairwise disjoint. -/
theorem pairwise_disjoint_on_Ioc_pred [PredOrder α] [Preorder β] {f : α → β} (hf : Antitone f) :
    Pairwise (Disjoint on fun n => Ioc (f n) (f (pred n))) 
goal: Pairwise (Disjoint on fun n => Set.Ioc (f n) (f (Order.pred n)))

full_name: Antitone.pairwise_disjoint_on_Ico_pred
code: /-- If `α` is a linear pred order, `β` is a preorder, and `f : α → β` is an antitone function, then
the intervals `Set.Ico (f n) (f (Order.pred n))` are pairwise disjoint. -/
theorem pairwise_disjoint_on_Ico_pred [PredOrder α] [Preorder β] {f : α → β} (hf : Antitone f) :
    Pairwise (Disjoint on fun n => Ico (f n) (f (pred n))) 
goal: Pairwise (Disjoint on fun n => Set.Ico (f n) (f (Order.pred n)))

full_name: Antitone.pairwise_disjoint_on_Ioo_pred
code: /-- If `α` is a linear pred order, `β` is a preorder, and `f : α → β` is an antitone function, then
the intervals `Set.Ioo (f n) (f (Order.pred n))` are pairwise disjoint. -/
theorem pairwise_disjoint_on_Ioo_pred [PredOrder α] [Preorder β] {f : α → β} (hf : Antitone f) :
    Pairwise (Disjoint on fun n => Ioo (f n) (f (pred n))) 
goal: Pairwise (Disjoint on fun n => Set.Ioo (f n) (f (Order.pred n)))