Documentation

Mathlib.Data.Rat.Floor

Floor Function for Rational Numbers #

Summary #

We define the FloorRing instance on ℚ. Some technical lemmas relating floor to integer division and modulo arithmetic are derived as well as some simple inequalities.

Tags #

rat, rationals, ℚ, floor

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theorem Rat.floor_def' (a : ℚ) :
a.floor = a.num / ↑a.den
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theorem Rat.le_floor {z : ℤ} {r : ℚ} :
z ≤ r.floor ↔ ↑z ≤ r
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instance Rat.instFloorRing :
FloorRing ℚ
Equations
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theorem Rat.floor_def {q : ℚ} :
⌊q⌋ = q.num / ↑q.den
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theorem Rat.ceil_def (q : ℚ) :
⌈q⌉ = -(-q.num / ↑q.den)
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theorem Rat.floor_intCast_div_natCast (n : ℤ) (d : ℕ) :
⌊↑n / ↑d⌋ = n / ↑d
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theorem Rat.ceil_intCast_div_natCast (n : ℤ) (d : ℕ) :
⌈↑n / ↑d⌉ = -(-n / ↑d)
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theorem Rat.floor_natCast_div_natCast (n d : ℕ) :
⌊↑n / ↑d⌋ = ↑n / ↑d
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theorem Rat.ceil_natCast_div_natCast (n d : ℕ) :
⌈↑n / ↑d⌉ = -(-↑n / ↑d)
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theorem Rat.natFloor_natCast_div_natCast (n d : ℕ) :
⌊↑n / ↑d⌋₊ = n / d
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@[simp]
theorem Rat.floor_cast {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] [FloorRing α] (x : ℚ) :
⌊↑x⌋ = ⌊x⌋
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@[simp]
theorem Rat.ceil_cast {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] [FloorRing α] (x : ℚ) :
⌈↑x⌉ = ⌈x⌉
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@[simp]
theorem Rat.round_cast {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] [FloorRing α] (x : ℚ) :
round ↑x = round x
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@[simp]
theorem Rat.cast_fract {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] [FloorRing α] (x : ℚ) :
↑(Int.fract x) = Int.fract ↑x
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theorem Rat.isNat_intFloor {R : Type u_2} [Ring R] [LinearOrder R] [IsStrictOrderedRing R] [FloorRing R] (r : R) (m : ℕ) :
Mathlib.Meta.NormNum.IsNat r m → Mathlib.Meta.NormNum.IsNat ⌊r⌋ m
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theorem Rat.isInt_intFloor {R : Type u_2} [Ring R] [LinearOrder R] [IsStrictOrderedRing R] [FloorRing R] (r : R) (m : ℤ) :
Mathlib.Meta.NormNum.IsInt r m → Mathlib.Meta.NormNum.IsInt ⌊r⌋ m
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theorem Rat.isInt_intFloor_ofIsRat {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] [FloorRing α] (r : α) (n : ℤ) (d : ℕ) :
Mathlib.Meta.NormNum.IsRat r n d → Mathlib.Meta.NormNum.IsInt ⌊r⌋ (n / ↑d)
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def Rat.evalIntFloor :
Mathlib.Meta.NormNum.NormNumExt

norm_num extension for Int.floor

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theorem Rat.isNat_intCeil {R : Type u_2} [Ring R] [LinearOrder R] [IsStrictOrderedRing R] [FloorRing R] (r : R) (m : ℕ) :
Mathlib.Meta.NormNum.IsNat r m → Mathlib.Meta.NormNum.IsNat ⌈r⌉ m
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theorem Rat.isInt_intCeil {R : Type u_2} [Ring R] [LinearOrder R] [IsStrictOrderedRing R] [FloorRing R] (r : R) (m : ℤ) :
Mathlib.Meta.NormNum.IsInt r m → Mathlib.Meta.NormNum.IsInt ⌈r⌉ m
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theorem Rat.isInt_intCeil_ofIsRat {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] [FloorRing α] (r : α) (n : ℤ) (d : ℕ) :
Mathlib.Meta.NormNum.IsRat r n d → Mathlib.Meta.NormNum.IsInt ⌈r⌉ (-(-n / ↑d))
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def Rat.evalIntCeil :
Mathlib.Meta.NormNum.NormNumExt

norm_num extension for Int.ceil

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theorem Int.mod_nat_eq_sub_mul_floor_rat_div {n : ℤ} {d : ℕ} :
n % ↑d = n - ↑d * ⌊↑n / ↑d⌋
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theorem Nat.coprime_sub_mul_floor_rat_div_of_coprime {n d : ℕ} (n_coprime_d : n.Coprime d) :
(↑n - ↑d * ⌊↑n / ↑d⌋).natAbs.Coprime d
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theorem Rat.num_lt_succ_floor_mul_den (q : ℚ) :
q.num < (⌊q⌋ + 1) * ↑q.den
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theorem Rat.fract_inv_num_lt_num_of_pos {q : ℚ} (q_pos : 0 < q) :
(Int.fract q⁻¹).num < q.num