File: Mathlib/Data/Int/Interval.lean

full_name: Int.instLocallyFiniteOrder
code: instance instLocallyFiniteOrder : LocallyFiniteOrder ℤ 
goal: LocallyFiniteOrder ℤ

full_name: Int.Icc_eq_finset_map
code: theorem Icc_eq_finset_map :
    Icc a b =
      (Finset.range (b + 1 - a).toNat).map (Nat.castEmbedding.trans <| addLeftEmbedding a) 
goal: Finset.Icc a b = Finset.map (Nat.castEmbedding.trans (addLeftEmbedding a)) (Finset.range (b + 1 - a).toNat)

full_name: Int.Ico_eq_finset_map
code: theorem Ico_eq_finset_map :
    Ico a b = (Finset.range (b - a).toNat).map (Nat.castEmbedding.trans <| addLeftEmbedding a) 
goal: Finset.Ico a b = Finset.map (Nat.castEmbedding.trans (addLeftEmbedding a)) (Finset.range (b - a).toNat)

full_name: Int.Ioc_eq_finset_map
code: theorem Ioc_eq_finset_map :
    Ioc a b =
      (Finset.range (b - a).toNat).map (Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)) 
goal: Finset.Ioc a b = Finset.map (Nat.castEmbedding.trans (addLeftEmbedding (a + 1))) (Finset.range (b - a).toNat)

full_name: Int.Ioo_eq_finset_map
code: theorem Ioo_eq_finset_map :
    Ioo a b =
      (Finset.range (b - a - 1).toNat).map (Nat.castEmbedding.trans <| addLeftEmbedding (a + 1)) 
goal: Finset.Ioo a b = Finset.map (Nat.castEmbedding.trans (addLeftEmbedding (a + 1))) (Finset.range (b - a - 1).toNat)

full_name: Int.uIcc_eq_finset_map
code: theorem uIcc_eq_finset_map :
    uIcc a b = (range (max a b + 1 - min a b).toNat).map
      (Nat.castEmbedding.trans <| addLeftEmbedding <| min a b) 
goal: Finset.uIcc a b =
  Finset.map (Nat.castEmbedding.trans (addLeftEmbedding (min a b))) (Finset.range (max a b + 1 - min a b).toNat)

full_name: Int.card_Icc
code: @[simp]
theorem card_Icc : (Icc a b).card = (b + 1 - a).toNat 
goal: (Finset.Icc a b).card = (b + 1 - a).toNat

full_name: Int.card_Ico
code: @[simp]
theorem card_Ico : (Ico a b).card = (b - a).toNat 
goal: (Finset.Ico a b).card = (b - a).toNat

full_name: Int.card_Ioc
code: @[simp]
theorem card_Ioc : (Ioc a b).card = (b - a).toNat 
goal: (Finset.Ioc a b).card = (b - a).toNat

full_name: Int.card_Ioo
code: @[simp]
theorem card_Ioo : (Ioo a b).card = (b - a - 1).toNat 
goal: (Finset.Ioo a b).card = (b - a - 1).toNat

full_name: Int.card_uIcc
code: @[simp]
theorem card_uIcc : (uIcc a b).card = (b - a).natAbs + 1 
goal: (Finset.uIcc a b).card = (b - a).natAbs + 1

full_name: Int.card_Icc_of_le
code: theorem card_Icc_of_le (h : a ≤ b + 1) : ((Icc a b).card : ℤ) = b + 1 - a 
goal: ↑(Finset.Icc a b).card = b + 1 - a

full_name: Int.card_Ico_of_le
code: theorem card_Ico_of_le (h : a ≤ b) : ((Ico a b).card : ℤ) = b - a 
goal: ↑(Finset.Ico a b).card = b - a

full_name: Int.card_Ioc_of_le
code: theorem card_Ioc_of_le (h : a ≤ b) : ((Ioc a b).card : ℤ) = b - a 
goal: ↑(Finset.Ioc a b).card = b - a

full_name: Int.card_Ioo_of_lt
code: theorem card_Ioo_of_lt (h : a < b) : ((Ioo a b).card : ℤ) = b - a - 1 
goal: ↑(Finset.Ioo a b).card = b - a - 1

full_name: Int.card_fintype_Icc
code: theorem card_fintype_Icc : Fintype.card (Set.Icc a b) = (b + 1 - a).toNat 
goal: Fintype.card ↑(Set.Icc a b) = (b + 1 - a).toNat

full_name: Int.card_fintype_Ico
code: theorem card_fintype_Ico : Fintype.card (Set.Ico a b) = (b - a).toNat 
goal: Fintype.card ↑(Set.Ico a b) = (b - a).toNat

full_name: Int.card_fintype_Ioc
code: theorem card_fintype_Ioc : Fintype.card (Set.Ioc a b) = (b - a).toNat 
goal: Fintype.card ↑(Set.Ioc a b) = (b - a).toNat

full_name: Int.card_fintype_Ioo
code: theorem card_fintype_Ioo : Fintype.card (Set.Ioo a b) = (b - a - 1).toNat 
goal: Fintype.card ↑(Set.Ioo a b) = (b - a - 1).toNat

full_name: Int.card_fintype_uIcc
code: theorem card_fintype_uIcc : Fintype.card (Set.uIcc a b) = (b - a).natAbs + 1 
goal: Fintype.card ↑(Set.uIcc a b) = (b - a).natAbs + 1

full_name: Int.card_fintype_Icc_of_le
code: theorem card_fintype_Icc_of_le (h : a ≤ b + 1) : (Fintype.card (Set.Icc a b) : ℤ) = b + 1 - a 
goal: ↑(Fintype.card ↑(Set.Icc a b)) = b + 1 - a

full_name: Int.card_fintype_Ico_of_le
code: theorem card_fintype_Ico_of_le (h : a ≤ b) : (Fintype.card (Set.Ico a b) : ℤ) = b - a 
goal: ↑(Fintype.card ↑(Set.Ico a b)) = b - a

full_name: Int.card_fintype_Ioc_of_le
code: theorem card_fintype_Ioc_of_le (h : a ≤ b) : (Fintype.card (Set.Ioc a b) : ℤ) = b - a 
goal: ↑(Fintype.card ↑(Set.Ioc a b)) = b - a

full_name: Int.card_fintype_Ioo_of_lt
code: theorem card_fintype_Ioo_of_lt (h : a < b) : (Fintype.card (Set.Ioo a b) : ℤ) = b - a - 1 
goal: ↑(Fintype.card ↑(Set.Ioo a b)) = b - a - 1

full_name: Int.image_Ico_emod
code: theorem image_Ico_emod (n a : ℤ) (h : 0 ≤ a) : (Ico n (n + a)).image (· % a) = Ico 0 a 
goal: Finset.image (fun x => x % a) (Finset.Ico n (n + a)) = Finset.Ico 0 a