Disjoint sum of multisets

This file defines the disjoint sum of two multisets as Multiset (α ⊕ β). Beware not to confuse with the Multiset.sum operation which computes the additive sum.

Main declarations

• Multiset.disjSum: s.disjSum t is the disjoint sum of s and t.

def Multiset.disjSum {α : Type u_1} {β : Type u_2} (s : Multiset α) (t : Multiset β) : Multiset (α ⊕ β)

Disjoint sum of multisets.

Equations

• s.disjSum t = Multiset.map Sum.inl s + Multiset.map Sum.inr t

@[simp]

theorem Multiset.zero_disjSum {α : Type u_1} {β : Type u_2} (t : Multiset β) : disjSum 0 t = map Sum.inr t

@[simp]

theorem Multiset.disjSum_zero {α : Type u_1} {β : Type u_2} (s : Multiset α) : s.disjSum 0 = map Sum.inl s

@[simp]

theorem Multiset.card_disjSum {α : Type u_1} {β : Type u_2} (s : Multiset α) (t : Multiset β) : (s.disjSum t).card = s.card + t.card

theorem Multiset.mem_disjSum {α : Type u_1} {β : Type u_2} {s : Multiset α} {t : Multiset β} {x : α ⊕ β} : x ∈ s.disjSum t ↔ (∃ (a : α), a ∈ s ∧ Sum.inl a = x) ∨ ∃ (b : β), b ∈ t ∧ Sum.inr b = x

@[simp]

theorem Multiset.inl_mem_disjSum {α : Type u_1} {β : Type u_2} {s : Multiset α} {t : Multiset β} {a : α} : Sum.inl a ∈ s.disjSum t ↔ a ∈ s

@[simp]

theorem Multiset.inr_mem_disjSum {α : Type u_1} {β : Type u_2} {s : Multiset α} {t : Multiset β} {b : β} : Sum.inr b ∈ s.disjSum t ↔ b ∈ t

theorem Multiset.disjSum_mono {α : Type u_1} {β : Type u_2} {s₁ s₂ : Multiset α} {t₁ t₂ : Multiset β} (hs : s₁ ≤ s₂) (ht : t₁ ≤ t₂) : s₁.disjSum t₁ ≤ s₂.disjSum t₂

theorem Multiset.disjSum_mono_left {α : Type u_1} {β : Type u_2} (t : Multiset β) : Monotone fun (s : Multiset α) => s.disjSum t

theorem Multiset.disjSum_mono_right {α : Type u_1} {β : Type u_2} (s : Multiset α) : Monotone s.disjSum

theorem Multiset.disjSum_lt_disjSum_of_lt_of_le {α : Type u_1} {β : Type u_2} {s₁ s₂ : Multiset α} {t₁ t₂ : Multiset β} (hs : s₁ < s₂) (ht : t₁ ≤ t₂) : s₁.disjSum t₁ < s₂.disjSum t₂

theorem Multiset.disjSum_lt_disjSum_of_le_of_lt {α : Type u_1} {β : Type u_2} {s₁ s₂ : Multiset α} {t₁ t₂ : Multiset β} (hs : s₁ ≤ s₂) (ht : t₁ < t₂) : s₁.disjSum t₁ < s₂.disjSum t₂

theorem Multiset.disjSum_strictMono_left {α : Type u_1} {β : Type u_2} (t : Multiset β) : StrictMono fun (s : Multiset α) => s.disjSum t

theorem Multiset.disjSum_strictMono_right {α : Type u_1} {β : Type u_2} (s : Multiset α) : StrictMono s.disjSum

theorem Multiset.Nodup.disjSum {α : Type u_1} {β : Type u_2} {s : Multiset α} {t : Multiset β} (hs : s.Nodup) (ht : t.Nodup) : (s.disjSum t).Nodup

theorem Multiset.map_disjSum {α : Type u_1} {β : Type u_2} {γ : Type u_3} {s : Multiset α} {t : Multiset β} (f : α ⊕ β → γ) : map f (s.disjSum t) = map (fun (x : α) => f (Sum.inl x)) s + map (fun (x : β) => f (Sum.inr x)) t