Documentation

Mathlib.Algebra.Order.Floor.Semiring

Lemmas on Nat.floor and Nat.ceil #

This file contains basic results on the natural-valued floor and ceiling functions.

TODO #

LinearOrderedSemiring can be relaxed to OrderedSemiring in many lemmas.

Tags #

rounding, floor, ceil

source
theorem Nat.floor_lt {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} {n : ℕ} (ha : 0 ≤ a) :
⌊a⌋₊ < n ↔ a < ↑n
source
theorem Nat.floor_lt_one {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} (ha : 0 ≤ a) :
⌊a⌋₊ < 1 ↔ a < 1
source
theorem Nat.lt_of_floor_lt {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} {n : ℕ} (h : ⌊a⌋₊ < n) :
a < ↑n
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theorem Nat.lt_one_of_floor_lt_one {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} (h : ⌊a⌋₊ < 1) :
a < 1
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theorem Nat.floor_le {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} (ha : 0 ≤ a) :
↑⌊a⌋₊ ≤ a
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theorem Nat.lt_succ_floor {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] (a : α) :
a < ↑⌊a⌋₊.succ
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theorem Nat.lt_floor_add_one {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] (a : α) :
a < ↑⌊a⌋₊ + 1
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@[simp]
theorem Nat.floor_natCast {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] (n : ℕ) :
⌊↑n⌋₊ = n
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@[simp]
theorem Nat.floor_zero {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] :
⌊0⌋₊ = 0
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@[simp]
theorem Nat.floor_one {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] :
⌊1⌋₊ = 1
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@[simp]
theorem Nat.floor_ofNat {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] (n : ℕ) [n.AtLeastTwo] :
⌊OfNat.ofNat n⌋₊ = OfNat.ofNat n
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theorem Nat.floor_of_nonpos {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} (ha : a ≤ 0) :
⌊a⌋₊ = 0
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theorem Nat.floor_mono {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] :
Monotone floor
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theorem Nat.floor_le_floor {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a b : α} (hab : a ≤ b) :
⌊a⌋₊ ≤ ⌊b⌋₊
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theorem Nat.le_floor_iff' {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} {n : ℕ} (hn : n ≠ 0) :
n ≤ ⌊a⌋₊ ↔ ↑n ≤ a
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@[simp]
theorem Nat.one_le_floor_iff {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] (x : α) :
1 ≤ ⌊x⌋₊ ↔ 1 ≤ x
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theorem Nat.floor_lt' {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} {n : ℕ} (hn : n ≠ 0) :
⌊a⌋₊ < n ↔ a < ↑n
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theorem Nat.floor_pos {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} :
0 < ⌊a⌋₊ ↔ 1 ≤ a
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theorem Nat.pos_of_floor_pos {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} (h : 0 < ⌊a⌋₊) :
0 < a
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theorem Nat.lt_of_lt_floor {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} {n : ℕ} (h : n < ⌊a⌋₊) :
↑n < a
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theorem Nat.floor_le_of_le {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} {n : ℕ} (h : a ≤ ↑n) :
⌊a⌋₊ ≤ n
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theorem Nat.floor_le_one_of_le_one {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} (h : a ≤ 1) :
⌊a⌋₊ ≤ 1
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@[simp]
theorem Nat.floor_eq_zero {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} :
⌊a⌋₊ = 0 ↔ a < 1
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theorem Nat.floor_eq_iff {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} {n : ℕ} (ha : 0 ≤ a) :
⌊a⌋₊ = n ↔ ↑n ≤ a ∧ a < ↑n + 1
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theorem Nat.floor_eq_iff' {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} {n : ℕ} (hn : n ≠ 0) :
⌊a⌋₊ = n ↔ ↑n ≤ a ∧ a < ↑n + 1
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theorem Nat.floor_eq_on_Ico {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] (n : ℕ) (a : α) :
a ∈ Set.Ico (↑n) (↑n + 1) → ⌊a⌋₊ = n
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theorem Nat.floor_eq_on_Ico' {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] (n : ℕ) (a : α) :
a ∈ Set.Ico (↑n) (↑n + 1) → ↑⌊a⌋₊ = ↑n
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@[simp]
theorem Nat.preimage_floor_zero {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] :
floor ⁻¹' {0} = Set.Iio 1
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theorem Nat.preimage_floor_of_ne_zero {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {n : ℕ} (hn : n ≠ 0) :
floor ⁻¹' {n} = Set.Ico (↑n) (↑n + 1)

Ceil #

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theorem Nat.add_one_le_ceil_iff {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} {n : ℕ} :
n + 1 ≤ ⌈a⌉₊ ↔ ↑n < a
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@[simp]
theorem Nat.one_le_ceil_iff {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} :
1 ≤ ⌈a⌉₊ ↔ 0 < a
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theorem Nat.ceil_le_floor_add_one {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] (a : α) :
⌈a⌉₊ ≤ ⌊a⌋₊ + 1
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theorem Nat.le_ceil {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] (a : α) :
a ≤ ↑⌈a⌉₊
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@[simp]
theorem Nat.ceil_intCast {α : Type u_2} [LinearOrderedRing α] [FloorSemiring α] (z : ℤ) :
⌈↑z⌉₊ = z.toNat
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@[simp]
theorem Nat.ceil_natCast {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] (n : ℕ) :
⌈↑n⌉₊ = n
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theorem Nat.ceil_mono {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] :
Monotone ceil
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theorem Nat.ceil_le_ceil {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a b : α} (hab : a ≤ b) :
⌈a⌉₊ ≤ ⌈b⌉₊
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@[simp]
theorem Nat.ceil_zero {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] :
⌈0⌉₊ = 0
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@[simp]
theorem Nat.ceil_one {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] :
⌈1⌉₊ = 1
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@[simp]
theorem Nat.ceil_ofNat {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] (n : ℕ) [n.AtLeastTwo] :
⌈OfNat.ofNat n⌉₊ = OfNat.ofNat n
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@[simp]
theorem Nat.ceil_eq_zero {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} :
⌈a⌉₊ = 0 ↔ a ≤ 0
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theorem Nat.lt_of_ceil_lt {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} {n : ℕ} (h : ⌈a⌉₊ < n) :
a < ↑n
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theorem Nat.le_of_ceil_le {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} {n : ℕ} (h : ⌈a⌉₊ ≤ n) :
a ≤ ↑n
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theorem Nat.floor_le_ceil {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] (a : α) :
⌊a⌋₊ ≤ ⌈a⌉₊
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theorem Nat.floor_lt_ceil_of_lt_of_pos {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a b : α} (h : a < b) (h' : 0 < b) :
⌊a⌋₊ < ⌈b⌉₊
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theorem Nat.ceil_eq_iff {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} {n : ℕ} (hn : n ≠ 0) :
⌈a⌉₊ = n ↔ ↑(n - 1) < a ∧ a ≤ ↑n
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@[simp]
theorem Nat.preimage_ceil_zero {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] :
ceil ⁻¹' {0} = Set.Iic 0
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theorem Nat.preimage_ceil_of_ne_zero {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {n : ℕ} (hn : n ≠ 0) :
ceil ⁻¹' {n} = Set.Ioc ↑(n - 1) ↑n

Intervals #

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@[simp]
theorem Nat.preimage_Ioo {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a b : α} (ha : 0 ≤ a) :
Nat.cast ⁻¹' Set.Ioo a b = Set.Ioo ⌊a⌋₊ ⌈b⌉₊
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@[simp]
theorem Nat.preimage_Ico {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a b : α} :
Nat.cast ⁻¹' Set.Ico a b = Set.Ico ⌈a⌉₊ ⌈b⌉₊
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@[simp]
theorem Nat.preimage_Ioc {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a b : α} (ha : 0 ≤ a) (hb : 0 ≤ b) :
Nat.cast ⁻¹' Set.Ioc a b = Set.Ioc ⌊a⌋₊ ⌊b⌋₊
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@[simp]
theorem Nat.preimage_Icc {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a b : α} (hb : 0 ≤ b) :
Nat.cast ⁻¹' Set.Icc a b = Set.Icc ⌈a⌉₊ ⌊b⌋₊
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@[simp]
theorem Nat.preimage_Ioi {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} (ha : 0 ≤ a) :
Nat.cast ⁻¹' Set.Ioi a = Set.Ioi ⌊a⌋₊
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@[simp]
theorem Nat.preimage_Ici {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} :
Nat.cast ⁻¹' Set.Ici a = Set.Ici ⌈a⌉₊
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@[simp]
theorem Nat.preimage_Iio {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} :
Nat.cast ⁻¹' Set.Iio a = Set.Iio ⌈a⌉₊
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@[simp]
theorem Nat.preimage_Iic {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} (ha : 0 ≤ a) :
Nat.cast ⁻¹' Set.Iic a = Set.Iic ⌊a⌋₊
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theorem Nat.floor_add_natCast {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} (ha : 0 ≤ a) (n : ℕ) :
⌊a + ↑n⌋₊ = ⌊a⌋₊ + n
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@[deprecated Nat.floor_add_natCast (since := "2025-04-01")]
theorem Nat.floor_add_nat {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} (ha : 0 ≤ a) (n : ℕ) :
⌊a + ↑n⌋₊ = ⌊a⌋₊ + n

Alias of Nat.floor_add_natCast.

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theorem Nat.floor_add_one {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} (ha : 0 ≤ a) :
⌊a + 1⌋₊ = ⌊a⌋₊ + 1
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theorem Nat.floor_add_ofNat {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} (ha : 0 ≤ a) (n : ℕ) [n.AtLeastTwo] :
⌊a + OfNat.ofNat n⌋₊ = ⌊a⌋₊ + OfNat.ofNat n
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@[simp]
theorem Nat.floor_sub_natCast {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] [Sub α] [OrderedSub α] [ExistsAddOfLE α] (a : α) (n : ℕ) :
⌊a - ↑n⌋₊ = ⌊a⌋₊ - n
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@[deprecated Nat.floor_sub_natCast (since := "2025-04-01")]
theorem Nat.floor_sub_nat {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] [Sub α] [OrderedSub α] [ExistsAddOfLE α] (a : α) (n : ℕ) :
⌊a - ↑n⌋₊ = ⌊a⌋₊ - n

Alias of Nat.floor_sub_natCast.

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@[simp]
theorem Nat.floor_sub_one {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] [Sub α] [OrderedSub α] [ExistsAddOfLE α] (a : α) :
⌊a - 1⌋₊ = ⌊a⌋₊ - 1
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@[simp]
theorem Nat.floor_sub_ofNat {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] [Sub α] [OrderedSub α] [ExistsAddOfLE α] (a : α) (n : ℕ) [n.AtLeastTwo] :
⌊a - OfNat.ofNat n⌋₊ = ⌊a⌋₊ - OfNat.ofNat n
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theorem Nat.ceil_add_natCast {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} (ha : 0 ≤ a) (n : ℕ) :
⌈a + ↑n⌉₊ = ⌈a⌉₊ + n
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@[deprecated Nat.ceil_add_natCast (since := "2025-04-01")]
theorem Nat.ceil_add_nat {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} (ha : 0 ≤ a) (n : ℕ) :
⌈a + ↑n⌉₊ = ⌈a⌉₊ + n

Alias of Nat.ceil_add_natCast.

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theorem Nat.ceil_add_one {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} (ha : 0 ≤ a) :
⌈a + 1⌉₊ = ⌈a⌉₊ + 1
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theorem Nat.ceil_add_ofNat {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} (ha : 0 ≤ a) (n : ℕ) [n.AtLeastTwo] :
⌈a + OfNat.ofNat n⌉₊ = ⌈a⌉₊ + OfNat.ofNat n
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theorem Nat.ceil_lt_add_one {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} (ha : 0 ≤ a) :
↑⌈a⌉₊ < a + 1
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theorem Nat.ceil_add_le {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] (a b : α) :
⌈a + b⌉₊ ≤ ⌈a⌉₊ + ⌈b⌉₊
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theorem Nat.sub_one_lt_floor {α : Type u_1} [LinearOrderedRing α] [FloorSemiring α] (a : α) :
a - 1 < ↑⌊a⌋₊
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theorem Nat.floor_div_natCast {α : Type u_1} [LinearOrderedSemifield α] [FloorSemiring α] (a : α) (n : ℕ) :
⌊a / ↑n⌋₊ = ⌊a⌋₊ / n
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@[deprecated Nat.floor_div_natCast (since := "2025-04-01")]
theorem Nat.floor_div_nat {α : Type u_1} [LinearOrderedSemifield α] [FloorSemiring α] (a : α) (n : ℕ) :
⌊a / ↑n⌋₊ = ⌊a⌋₊ / n

Alias of Nat.floor_div_natCast.

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theorem Nat.floor_div_ofNat {α : Type u_1} [LinearOrderedSemifield α] [FloorSemiring α] (a : α) (n : ℕ) [n.AtLeastTwo] :
⌊a / OfNat.ofNat n⌋₊ = ⌊a⌋₊ / OfNat.ofNat n
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theorem Nat.floor_div_eq_div {α : Type u_1} [LinearOrderedSemifield α] [FloorSemiring α] (m n : ℕ) :
⌊↑m / ↑n⌋₊ = m / n

Natural division is the floor of field division.

source
theorem Nat.mul_lt_floor {α : Type u_1} [LinearOrderedField α] [FloorSemiring α] {a b : α} (hb₀ : 0 < b) (hb : b < 1) (hba : ↑⌈b / (1 - b)⌉₊ ≤ a) :
b * a < ↑⌊a⌋₊
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theorem Nat.ceil_lt_mul {α : Type u_1} [LinearOrderedField α] [FloorSemiring α] {a b : α} (hb : 1 < b) (hba : ↑⌈(b - 1)⁻¹⌉₊ / b < a) :
↑⌈a⌉₊ < b * a
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theorem Nat.ceil_le_mul {α : Type u_1} [LinearOrderedField α] [FloorSemiring α] {a b : α} (hb : 1 < b) (hba : ↑⌈(b - 1)⁻¹⌉₊ / b ≤ a) :
↑⌈a⌉₊ ≤ b * a
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theorem Nat.div_two_lt_floor {α : Type u_1} [LinearOrderedField α] [FloorSemiring α] {a : α} (ha : 1 ≤ a) :
a / 2 < ↑⌊a⌋₊
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theorem Nat.ceil_lt_two_mul {α : Type u_1} [LinearOrderedField α] [FloorSemiring α] {a : α} (ha : 2⁻¹ < a) :
↑⌈a⌉₊ < 2 * a
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theorem Nat.ceil_le_two_mul {α : Type u_1} [LinearOrderedField α] [FloorSemiring α] {a : α} (ha : 2⁻¹ ≤ a) :
↑⌈a⌉₊ ≤ 2 * a
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theorem Nat.floor_congr {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} {β : Type u_2} [LinearOrderedSemiring β] [FloorSemiring β] {b : β} (h : ∀ (n : ℕ), ↑n ≤ a ↔ ↑n ≤ b) :
⌊a⌋₊ = ⌊b⌋₊
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theorem Nat.ceil_congr {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {a : α} {β : Type u_2} [LinearOrderedSemiring β] [FloorSemiring β] {b : β} (h : ∀ (n : ℕ), a ≤ ↑n ↔ b ≤ ↑n) :
⌈a⌉₊ = ⌈b⌉₊
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theorem Nat.map_floor {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {β : Type u_2} [LinearOrderedSemiring β] [FloorSemiring β] {F : Type u_3} [FunLike F α β] [RingHomClass F α β] (f : F) (hf : StrictMono ⇑f) (a : α) :
⌊f a⌋₊ = ⌊a⌋₊
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theorem Nat.map_ceil {α : Type u_1} [LinearOrderedSemiring α] [FloorSemiring α] {β : Type u_2} [LinearOrderedSemiring β] [FloorSemiring β] {F : Type u_3} [FunLike F α β] [RingHomClass F α β] (f : F) (hf : StrictMono ⇑f) (a : α) :
⌈f a⌉₊ = ⌈a⌉₊
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theorem subsingleton_floorSemiring {α : Type u_2} [LinearOrderedSemiring α] :
Subsingleton (FloorSemiring α)

There exists at most one FloorSemiring structure on a linear ordered semiring.