Documentation

Mathlib.Data.Int.Interval

Finite intervals of integers #

This file proves that ℤ is a LocallyFiniteOrder and calculates the cardinality of its intervals as finsets and fintypes.

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instance Int.instLocallyFiniteOrder :
LocallyFiniteOrder ℤ
Equations
  • One or more equations did not get rendered due to their size.
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theorem Int.Icc_eq_finset_map (a b : ℤ) :
Finset.Icc a b = Finset.map (Nat.castEmbedding.trans (addLeftEmbedding a)) (Finset.range (b + 1 - a).toNat)
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theorem Int.Ico_eq_finset_map (a b : ℤ) :
Finset.Ico a b = Finset.map (Nat.castEmbedding.trans (addLeftEmbedding a)) (Finset.range (b - a).toNat)
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theorem Int.Ioc_eq_finset_map (a b : ℤ) :
Finset.Ioc a b = Finset.map (Nat.castEmbedding.trans (addLeftEmbedding (a + 1))) (Finset.range (b - a).toNat)
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theorem Int.Ioo_eq_finset_map (a b : ℤ) :
Finset.Ioo a b = Finset.map (Nat.castEmbedding.trans (addLeftEmbedding (a + 1))) (Finset.range (b - a - 1).toNat)
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theorem Int.uIcc_eq_finset_map (a b : ℤ) :
Finset.uIcc a b = Finset.map (Nat.castEmbedding.trans (addLeftEmbedding (min a b))) (Finset.range (max a b + 1 - min a b).toNat)
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@[simp]
theorem Int.card_Icc (a b : ℤ) :
(Finset.Icc a b).card = (b + 1 - a).toNat
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@[simp]
theorem Int.card_Ico (a b : ℤ) :
(Finset.Ico a b).card = (b - a).toNat
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@[simp]
theorem Int.card_Ioc (a b : ℤ) :
(Finset.Ioc a b).card = (b - a).toNat
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@[simp]
theorem Int.card_Ioo (a b : ℤ) :
(Finset.Ioo a b).card = (b - a - 1).toNat
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@[simp]
theorem Int.card_uIcc (a b : ℤ) :
(Finset.uIcc a b).card = (b - a).natAbs + 1
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theorem Int.card_Icc_of_le (a b : ℤ) (h : a ≤ b + 1) :
↑(Finset.Icc a b).card = b + 1 - a
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theorem Int.card_Ico_of_le (a b : ℤ) (h : a ≤ b) :
↑(Finset.Ico a b).card = b - a
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theorem Int.card_Ioc_of_le (a b : ℤ) (h : a ≤ b) :
↑(Finset.Ioc a b).card = b - a
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theorem Int.card_Ioo_of_lt (a b : ℤ) (h : a < b) :
↑(Finset.Ioo a b).card = b - a - 1
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theorem Int.Icc_eq_pair (a : ℤ) :
Finset.Icc a (a + 1) = {a, a + 1}
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@[deprecated Fintype.card_Icc (since := "2025-03-28")]
theorem Int.card_fintype_Icc (a b : ℤ) :
Fintype.card ↑(Set.Icc a b) = (b + 1 - a).toNat
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@[deprecated Fintype.card_Ico (since := "2025-03-28")]
theorem Int.card_fintype_Ico (a b : ℤ) :
Fintype.card ↑(Set.Ico a b) = (b - a).toNat
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@[deprecated Fintype.card_Ioc (since := "2025-03-28")]
theorem Int.card_fintype_Ioc (a b : ℤ) :
Fintype.card ↑(Set.Ioc a b) = (b - a).toNat
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@[deprecated Fintype.card_Ioo (since := "2025-03-28")]
theorem Int.card_fintype_Ioo (a b : ℤ) :
Fintype.card ↑(Set.Ioo a b) = (b - a - 1).toNat
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@[deprecated Fintype.card_uIcc (since := "2025-03-28")]
theorem Int.card_fintype_uIcc (a b : ℤ) :
Fintype.card ↑(Set.uIcc a b) = (b - a).natAbs + 1
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theorem Int.card_fintype_Icc_of_le (a b : ℤ) (h : a ≤ b + 1) :
↑(Fintype.card ↑(Set.Icc a b)) = b + 1 - a
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theorem Int.card_fintype_Ico_of_le (a b : ℤ) (h : a ≤ b) :
↑(Fintype.card ↑(Set.Ico a b)) = b - a
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theorem Int.card_fintype_Ioc_of_le (a b : ℤ) (h : a ≤ b) :
↑(Fintype.card ↑(Set.Ioc a b)) = b - a
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theorem Int.card_fintype_Ioo_of_lt (a b : ℤ) (h : a < b) :
↑(Fintype.card ↑(Set.Ioo a b)) = b - a - 1
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theorem Int.image_Ico_emod (n a : ℤ) (h : 0 ≤ a) :
Finset.image (fun (x : ℤ) => x % a) (Finset.Ico n (n + a)) = Finset.Ico 0 a