Documentation

Mathlib.Data.Finset.Fin

Finsets in Fin n #

A few constructions for Finsets in Fin n.

Main declarations #

source
def Finset.attachFin (s : Finset ℕ) {n : ℕ} (h : ∀ m ∈ s, m < n) :
Finset (Fin n)

Given a Finset s of ℕ contained in {0,..., n-1}, the corresponding Finset in Fin n is s.attachFin h where h is a proof that all elements of s are less than n.

Equations
    Instances For
      source
      @[simp]
      theorem Finset.mem_attachFin {n : ℕ} {s : Finset ℕ} (h : ∀ m ∈ s, m < n) {a : Fin n} :
      a ∈ s.attachFin h ↔ ↑a ∈ s
      source
      @[simp]
      theorem Finset.coe_attachFin {n : ℕ} {s : Finset ℕ} (h : ∀ m ∈ s, m < n) :
      ↑(s.attachFin h) = Fin.val ⁻¹' ↑s
      source
      @[simp]
      theorem Finset.card_attachFin {n : ℕ} (s : Finset ℕ) (h : ∀ m ∈ s, m < n) :
      (s.attachFin h).card = s.card
      source
      @[simp]
      theorem Finset.image_val_attachFin {n : ℕ} {s : Finset ℕ} (h : ∀ m ∈ s, m < n) :
      image Fin.val (s.attachFin h) = s
      source
      @[simp]
      theorem Finset.map_valEmbedding_attachFin {n : ℕ} {s : Finset ℕ} (h : ∀ m ∈ s, m < n) :
      map Fin.valEmbedding (s.attachFin h) = s
      source
      @[simp]
      theorem Finset.attachFin_subset_attachFin_iff {n : ℕ} {s t : Finset ℕ} (hs : ∀ m ∈ s, m < n) (ht : ∀ m ∈ t, m < n) :
      s.attachFin hs ⊆ t.attachFin ht ↔ s ⊆ t
      source
      theorem Finset.attachFin_subset_attachFin {n : ℕ} {s t : Finset ℕ} (hst : s ⊆ t) (ht : ∀ m ∈ t, m < n) :
      s.attachFin ⋯ ⊆ t.attachFin ht
      source
      @[simp]
      theorem Finset.attachFin_ssubset_attachFin_iff {n : ℕ} {s t : Finset ℕ} (hs : ∀ m ∈ s, m < n) (ht : ∀ m ∈ t, m < n) :
      s.attachFin hs ⊂ t.attachFin ht ↔ s ⊂ t
      source
      theorem Finset.attachFin_ssubset_attachFin {n : ℕ} {s t : Finset ℕ} (hst : s ⊂ t) (ht : ∀ m ∈ t, m < n) :
      s.attachFin ⋯ ⊂ t.attachFin ht