File: Mathlib/Data/PNat/Interval.lean

full_name: PNat.instLocallyFiniteOrder
code: instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ+ 
goal: LocallyFiniteOrder ℕ+

full_name: PNat.Icc_eq_finset_subtype
code: theorem Icc_eq_finset_subtype : Icc a b = (Icc (a : ℕ) b).subtype fun n : ℕ => 0 < n 
goal: Finset.Icc a b = Finset.subtype (fun n => 0 < n) (Finset.Icc ↑a ↑b)

full_name: PNat.Ico_eq_finset_subtype
code: theorem Ico_eq_finset_subtype : Ico a b = (Ico (a : ℕ) b).subtype fun n : ℕ => 0 < n 
goal: Finset.Ico a b = Finset.subtype (fun n => 0 < n) (Finset.Ico ↑a ↑b)

full_name: PNat.Ioc_eq_finset_subtype
code: theorem Ioc_eq_finset_subtype : Ioc a b = (Ioc (a : ℕ) b).subtype fun n : ℕ => 0 < n 
goal: Finset.Ioc a b = Finset.subtype (fun n => 0 < n) (Finset.Ioc ↑a ↑b)

full_name: PNat.Ioo_eq_finset_subtype
code: theorem Ioo_eq_finset_subtype : Ioo a b = (Ioo (a : ℕ) b).subtype fun n : ℕ => 0 < n 
goal: Finset.Ioo a b = Finset.subtype (fun n => 0 < n) (Finset.Ioo ↑a ↑b)

full_name: PNat.uIcc_eq_finset_subtype
code: theorem uIcc_eq_finset_subtype : uIcc a b = (uIcc (a : ℕ) b).subtype fun n : ℕ => 0 < n 
goal: Finset.uIcc a b = Finset.subtype (fun n => 0 < n) (Finset.uIcc ↑a ↑b)

full_name: PNat.map_subtype_embedding_Icc
code: theorem map_subtype_embedding_Icc : (Icc a b).map (Embedding.subtype _) = Icc ↑a ↑b 
goal: Finset.map (Function.Embedding.subtype fun n => 0 < n) (Finset.Icc a b) = Finset.Icc ↑a ↑b

full_name: PNat.map_subtype_embedding_Ico
code: theorem map_subtype_embedding_Ico : (Ico a b).map (Embedding.subtype _) = Ico ↑a ↑b 
goal: Finset.map (Function.Embedding.subtype fun n => 0 < n) (Finset.Ico a b) = Finset.Ico ↑a ↑b

full_name: PNat.map_subtype_embedding_Ioc
code: theorem map_subtype_embedding_Ioc : (Ioc a b).map (Embedding.subtype _) = Ioc ↑a ↑b 
goal: Finset.map (Function.Embedding.subtype fun n => 0 < n) (Finset.Ioc a b) = Finset.Ioc ↑a ↑b

full_name: PNat.map_subtype_embedding_Ioo
code: theorem map_subtype_embedding_Ioo : (Ioo a b).map (Embedding.subtype _) = Ioo ↑a ↑b 
goal: Finset.map (Function.Embedding.subtype fun n => 0 < n) (Finset.Ioo a b) = Finset.Ioo ↑a ↑b

full_name: PNat.map_subtype_embedding_uIcc
code: theorem map_subtype_embedding_uIcc : (uIcc a b).map (Embedding.subtype _) = uIcc ↑a ↑b 
goal: Finset.map (Function.Embedding.subtype fun n => 0 < n) (Finset.uIcc a b) = Finset.uIcc ↑a ↑b

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

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

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

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

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

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

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

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

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

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