HepLean Documentation

Mathlib.Data.Multiset.Sum

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 #

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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
Instances For
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    @[simp]
    theorem Multiset.zero_disjSum {α : Type u_1} {β : Type u_2} (t : Multiset β) :
    Multiset.disjSum 0 t = Multiset.map Sum.inr t
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    @[simp]
    theorem Multiset.disjSum_zero {α : Type u_1} {β : Type u_2} (s : Multiset α) :
    s.disjSum 0 = Multiset.map Sum.inl s
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    @[simp]
    theorem Multiset.card_disjSum {α : Type u_1} {β : Type u_2} (s : Multiset α) (t : Multiset β) :
    Multiset.card (s.disjSum t) = Multiset.card s + Multiset.card t
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    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 ∈ s, Sum.inl a = x) ∨ ∃ b ∈ t, Sum.inr b = x
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    @[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
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    @[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
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    theorem Multiset.disjSum_mono {α : Type u_1} {β : Type u_2} {s₁ : Multiset α} {s₂ : Multiset α} {t₁ : Multiset β} {t₂ : Multiset β} (hs : s₁ ≤ s₂) (ht : t₁ ≤ t₂) :
    s₁.disjSum t₁ ≤ s₂.disjSum t₂
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    theorem Multiset.disjSum_mono_left {α : Type u_1} {β : Type u_2} (t : Multiset β) :
    Monotone fun (s : Multiset α) => s.disjSum t
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    theorem Multiset.disjSum_mono_right {α : Type u_1} {β : Type u_2} (s : Multiset α) :
    Monotone s.disjSum
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    theorem Multiset.disjSum_lt_disjSum_of_lt_of_le {α : Type u_1} {β : Type u_2} {s₁ : Multiset α} {s₂ : Multiset α} {t₁ : Multiset β} {t₂ : Multiset β} (hs : s₁ < s₂) (ht : t₁ ≤ t₂) :
    s₁.disjSum t₁ < s₂.disjSum t₂
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    theorem Multiset.disjSum_lt_disjSum_of_le_of_lt {α : Type u_1} {β : Type u_2} {s₁ : Multiset α} {s₂ : Multiset α} {t₁ : Multiset β} {t₂ : Multiset β} (hs : s₁ ≤ s₂) (ht : t₁ < t₂) :
    s₁.disjSum t₁ < s₂.disjSum t₂
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    theorem Multiset.disjSum_strictMono_left {α : Type u_1} {β : Type u_2} (t : Multiset β) :
    StrictMono fun (s : Multiset α) => s.disjSum t
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    theorem Multiset.disjSum_strictMono_right {α : Type u_1} {β : Type u_2} (s : Multiset α) :
    StrictMono s.disjSum
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    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