Documentation

Mathlib.Data.Finset.Sum

Disjoint sum of finsets #

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

Main declarations #

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

Disjoint sum of finsets.

Equations
  • s.disjSum t = { val := s.val.disjSum t.val, nodup := ⋯ }
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@[simp]
theorem Finset.val_disjSum {α : Type u_1} {β : Type u_2} (s : Finset α) (t : Finset β) :
(s.disjSum t).val = s.val.disjSum t.val
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@[simp]
theorem Finset.empty_disjSum {α : Type u_1} {β : Type u_2} (t : Finset β) :
∅.disjSum t = Finset.map Function.Embedding.inr t
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@[simp]
theorem Finset.disjSum_empty {α : Type u_1} {β : Type u_2} (s : Finset α) :
s.disjSum ∅ = Finset.map Function.Embedding.inl s
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@[simp]
theorem Finset.card_disjSum {α : Type u_1} {β : Type u_2} (s : Finset α) (t : Finset β) :
(s.disjSum t).card = s.card + t.card
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theorem Finset.disjoint_map_inl_map_inr {α : Type u_1} {β : Type u_2} (s : Finset α) (t : Finset β) :
Disjoint (Finset.map Function.Embedding.inl s) (Finset.map Function.Embedding.inr t)
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@[simp]
theorem Finset.map_inl_disjUnion_map_inr {α : Type u_1} {β : Type u_2} (s : Finset α) (t : Finset β) :
(Finset.map Function.Embedding.inl s).disjUnion (Finset.map Function.Embedding.inr t) ⋯ = s.disjSum t
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theorem Finset.mem_disjSum {α : Type u_1} {β : Type u_2} {s : Finset α} {t : Finset 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 Finset.inl_mem_disjSum {α : Type u_1} {β : Type u_2} {s : Finset α} {t : Finset β} {a : α} :
Sum.inl a ∈ s.disjSum t ↔ a ∈ s
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@[simp]
theorem Finset.inr_mem_disjSum {α : Type u_1} {β : Type u_2} {s : Finset α} {t : Finset β} {b : β} :
Sum.inr b ∈ s.disjSum t ↔ b ∈ t
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@[simp]
theorem Finset.disjSum_eq_empty {α : Type u_1} {β : Type u_2} {s : Finset α} {t : Finset β} :
s.disjSum t = ∅ ↔ s = ∅ ∧ t = ∅
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theorem Finset.disjSum_mono {α : Type u_1} {β : Type u_2} {s₁ : Finset α} {s₂ : Finset α} {t₁ : Finset β} {t₂ : Finset β} (hs : s₁ ⊆ s₂) (ht : t₁ ⊆ t₂) :
s₁.disjSum t₁ ⊆ s₂.disjSum t₂
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theorem Finset.disjSum_mono_left {α : Type u_1} {β : Type u_2} (t : Finset β) :
Monotone fun (s : Finset α) => s.disjSum t
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theorem Finset.disjSum_mono_right {α : Type u_1} {β : Type u_2} (s : Finset α) :
Monotone s.disjSum
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theorem Finset.disjSum_ssubset_disjSum_of_ssubset_of_subset {α : Type u_1} {β : Type u_2} {s₁ : Finset α} {s₂ : Finset α} {t₁ : Finset β} {t₂ : Finset β} (hs : s₁ ⊂ s₂) (ht : t₁ ⊆ t₂) :
s₁.disjSum t₁ ⊂ s₂.disjSum t₂
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theorem Finset.disjSum_ssubset_disjSum_of_subset_of_ssubset {α : Type u_1} {β : Type u_2} {s₁ : Finset α} {s₂ : Finset α} {t₁ : Finset β} {t₂ : Finset β} (hs : s₁ ⊆ s₂) (ht : t₁ ⊂ t₂) :
s₁.disjSum t₁ ⊂ s₂.disjSum t₂
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theorem Finset.disjSum_strictMono_left {α : Type u_1} {β : Type u_2} (t : Finset β) :
StrictMono fun (s : Finset α) => s.disjSum t
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theorem Finset.disj_sum_strictMono_right {α : Type u_1} {β : Type u_2} (s : Finset α) :
StrictMono s.disjSum