Documentation

class FloorSemiring (α : Type u_1) [inst : OrderedSemiring α] :

Type u_1

FloorSemiring.floor a computes the greatest natural n such that (n : α) ≤ a≤ a.
floor : α → ℕ
FloorSemiring.ceil a computes the least natural n such that a ≤ (n : α)≤ (n : α).
ceil : α → ℕ
FloorSemiring.floor of a negative element is zero.
floor_of_neg : ∀ {a : α}, a < 0 → floor a = 0
A natural number n is smaller than FloorSemiring.floor a iff its coercion to α is smaller than a.
gc_floor : ∀ {a : α} {n : ℕ}, 0 ≤ a → (n ≤ floor a ↔ ↑n ≤ a)
FloorSemiring.ceil is the lower adjoint of the coercion ↑ : ℕ → α↑ : ℕ → α→ α.
gc_ceil : GaloisConnection ceil Nat.cast

A FloorSemiring is an ordered semiring over α with a function floor : α → ℕ→ ℕ satisfying ∀ (n : ℕ) (x : α), n ≤ ⌊x⌋ ↔ (n : α) ≤ x)∀ (n : ℕ) (x : α), n ≤ ⌊x⌋ ↔ (n : α) ≤ x)≤ ⌊x⌋ ↔ (n : α) ≤ x)↔ (n : α) ≤ x)≤ x). Note that many lemmas require a LinearOrder. Please see the above TODO.

Instances

instance instFloorSemiringNatOrderedSemiring :

FloorSemiring ℕ

Equations

One or more equations did not get rendered due to their size.

def Nat.floor {α : Type u_1} [inst : OrderedSemiring α] [inst : FloorSemiring α] :

α → ℕ

⌊a⌋₊₊ is the greatest natural n such that n ≤ a≤ a. If a is negative, then ⌊a⌋₊ = 0₊ = 0.

Equations

Nat.floor = FloorSemiring.floor

def Nat.ceil {α : Type u_1} [inst : OrderedSemiring α] [inst : FloorSemiring α] :

α → ℕ

⌈a⌉₊₊ is the least natural n such that a ≤ n≤ n

Equations

Nat.ceil = FloorSemiring.ceil

@[simp]

theorem Nat.floor_nat :

Nat.floor = id

@[simp]

theorem Nat.ceil_nat :

Nat.ceil = id

def Nat.«term⌊_⌋₊» :

⌊a⌋₊₊ is the greatest natural n such that n ≤ a≤ a. If a is negative, then ⌊a⌋₊ = 0₊ = 0.

Equations

One or more equations did not get rendered due to their size.

def Nat.«term⌈_⌉₊» :

⌈a⌉₊₊ is the least natural n such that a ≤ n≤ n

Equations

One or more equations did not get rendered due to their size.

theorem Nat.le_floor_iff {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} {n : ℕ} (ha : 0 ≤ a) :

n ≤ ⌊a⌋₊ ↔ ↑n ≤ a

theorem Nat.le_floor {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} {n : ℕ} (h : ↑n ≤ a) :

n ≤ ⌊a⌋₊

theorem Nat.floor_lt {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} {n : ℕ} (ha : 0 ≤ a) :

⌊a⌋₊ < n ↔ a < ↑n

theorem Nat.floor_lt_one {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} (ha : 0 ≤ a) :

⌊a⌋₊ < 1 ↔ a < 1

theorem Nat.lt_of_floor_lt {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} {n : ℕ} (h : ⌊a⌋₊ < n) :

a < ↑n

theorem Nat.lt_one_of_floor_lt_one {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} (h : ⌊a⌋₊ < 1) :

a < 1

theorem Nat.floor_le {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} (ha : 0 ≤ a) :

↑⌊a⌋₊ ≤ a

theorem Nat.lt_succ_floor {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] (a : α) :

a < ↑(Nat.succ ⌊a⌋₊)

theorem Nat.lt_floor_add_one {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] (a : α) :

a < ↑⌊a⌋₊ + 1

@[simp]

theorem Nat.floor_coe {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] (n : ℕ) :

⌊↑n⌋₊ = n

@[simp]

theorem Nat.floor_zero {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] :

⌊0⌋₊ = 0

@[simp]

theorem Nat.floor_one {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] :

⌊1⌋₊ = 1

theorem Nat.floor_of_nonpos {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} (ha : a ≤ 0) :

⌊a⌋₊ = 0

theorem Nat.floor_mono {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] :

Monotone Nat.floor

theorem Nat.le_floor_iff' {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} {n : ℕ} (hn : n ≠ 0) :

n ≤ ⌊a⌋₊ ↔ ↑n ≤ a

@[simp]

theorem Nat.one_le_floor_iff {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] (x : α) :

1 ≤ ⌊x⌋₊ ↔ 1 ≤ x

theorem Nat.floor_lt' {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} {n : ℕ} (hn : n ≠ 0) :

⌊a⌋₊ < n ↔ a < ↑n

theorem Nat.floor_pos {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} :

0 < ⌊a⌋₊ ↔ 1 ≤ a

theorem Nat.pos_of_floor_pos {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} (h : 0 < ⌊a⌋₊) :

0 < a

theorem Nat.lt_of_lt_floor {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} {n : ℕ} (h : n < ⌊a⌋₊) :

↑n < a

theorem Nat.floor_le_of_le {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} {n : ℕ} (h : a ≤ ↑n) :

⌊a⌋₊ ≤ n

theorem Nat.floor_le_one_of_le_one {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} (h : a ≤ 1) :

⌊a⌋₊ ≤ 1

@[simp]

theorem Nat.floor_eq_zero {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} :

⌊a⌋₊ = 0 ↔ a < 1

theorem Nat.floor_eq_iff {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} {n : ℕ} (ha : 0 ≤ a) :

⌊a⌋₊ = n ↔ ↑n ≤ a ∧ a < ↑n + 1

theorem Nat.floor_eq_iff' {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} {n : ℕ} (hn : n ≠ 0) :

⌊a⌋₊ = n ↔ ↑n ≤ a ∧ a < ↑n + 1

theorem Nat.floor_eq_on_Ico {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] (n : ℕ) (a : α) :

a ∈ Set.Ico (↑n) (↑n + 1) → ⌊a⌋₊ = n

theorem Nat.floor_eq_on_Ico' {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] (n : ℕ) (a : α) :

a ∈ Set.Ico (↑n) (↑n + 1) → ↑⌊a⌋₊ = ↑n

@[simp]

theorem Nat.preimage_floor_zero {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] :

Nat.floor ⁻¹' {0} = Set.Iio 1

theorem Nat.preimage_floor_of_ne_zero {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {n : ℕ} (hn : n ≠ 0) :

Nat.floor ⁻¹' {n} = Set.Ico (↑n) (↑n + 1)

Ceil #

theorem Nat.gc_ceil_coe {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] :

GaloisConnection Nat.ceil Nat.cast

@[simp]

theorem Nat.ceil_le {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} {n : ℕ} :

⌈a⌉₊ ≤ n ↔ a ≤ ↑n

theorem Nat.lt_ceil {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} {n : ℕ} :

n < ⌈a⌉₊ ↔ ↑n < a

theorem Nat.add_one_le_ceil_iff {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} {n : ℕ} :

n + 1 ≤ ⌈a⌉₊ ↔ ↑n < a

@[simp]

theorem Nat.one_le_ceil_iff {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} :

1 ≤ ⌈a⌉₊ ↔ 0 < a

theorem Nat.ceil_le_floor_add_one {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] (a : α) :

⌈a⌉₊ ≤ ⌊a⌋₊ + 1

theorem Nat.le_ceil {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] (a : α) :

a ≤ ↑⌈a⌉₊

@[simp]

theorem Nat.ceil_intCast {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorSemiring α] (z : ℤ) :

⌈↑z⌉₊ = Int.toNat z

@[simp]

theorem Nat.ceil_natCast {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] (n : ℕ) :

⌈↑n⌉₊ = n

theorem Nat.ceil_mono {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] :

Monotone Nat.ceil

@[simp]

theorem Nat.ceil_zero {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] :

⌈0⌉₊ = 0

@[simp]

theorem Nat.ceil_one {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] :

⌈1⌉₊ = 1

@[simp]

theorem Nat.ceil_eq_zero {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} :

⌈a⌉₊ = 0 ↔ a ≤ 0

@[simp]

theorem Nat.ceil_pos {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} :

0 < ⌈a⌉₊ ↔ 0 < a

theorem Nat.lt_of_ceil_lt {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} {n : ℕ} (h : ⌈a⌉₊ < n) :

a < ↑n

theorem Nat.le_of_ceil_le {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} {n : ℕ} (h : ⌈a⌉₊ ≤ n) :

a ≤ ↑n

theorem Nat.floor_le_ceil {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] (a : α) :

⌊a⌋₊ ≤ ⌈a⌉₊

theorem Nat.floor_lt_ceil_of_lt_of_pos {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} {b : α} (h : a < b) (h' : 0 < b) :

⌊a⌋₊ < ⌈b⌉₊

theorem Nat.ceil_eq_iff {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} {n : ℕ} (hn : n ≠ 0) :

⌈a⌉₊ = n ↔ ↑(n - 1) < a ∧ a ≤ ↑n

@[simp]

theorem Nat.preimage_ceil_zero {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] :

Nat.ceil ⁻¹' {0} = Set.Iic 0

theorem Nat.preimage_ceil_of_ne_zero {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {n : ℕ} (hn : n ≠ 0) :

Nat.ceil ⁻¹' {n} = Set.Ioc ↑(n - 1) ↑n

Intervals #

@[simp]

theorem Nat.preimage_Ioo {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} {b : α} (ha : 0 ≤ a) :

Nat.cast ⁻¹' Set.Ioo a b = Set.Ioo ⌊a⌋₊ ⌈b⌉₊

@[simp]

theorem Nat.preimage_Ico {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} {b : α} :

Nat.cast ⁻¹' Set.Ico a b = Set.Ico ⌈a⌉₊ ⌈b⌉₊

@[simp]

theorem Nat.preimage_Ioc {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} {b : α} (ha : 0 ≤ a) (hb : 0 ≤ b) :

Nat.cast ⁻¹' Set.Ioc a b = Set.Ioc ⌊a⌋₊ ⌊b⌋₊

@[simp]

theorem Nat.preimage_Icc {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} {b : α} (hb : 0 ≤ b) :

Nat.cast ⁻¹' Set.Icc a b = Set.Icc ⌈a⌉₊ ⌊b⌋₊

@[simp]

theorem Nat.preimage_Ioi {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} (ha : 0 ≤ a) :

Nat.cast ⁻¹' Set.Ioi a = Set.Ioi ⌊a⌋₊

@[simp]

theorem Nat.preimage_Ici {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} :

Nat.cast ⁻¹' Set.Ici a = Set.Ici ⌈a⌉₊

@[simp]

theorem Nat.preimage_Iio {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} :

Nat.cast ⁻¹' Set.Iio a = Set.Iio ⌈a⌉₊

@[simp]

theorem Nat.preimage_Iic {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} (ha : 0 ≤ a) :

Nat.cast ⁻¹' Set.Iic a = Set.Iic ⌊a⌋₊

theorem Nat.floor_add_nat {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} (ha : 0 ≤ a) (n : ℕ) :

⌊a + ↑n⌋₊ = ⌊a⌋₊ + n

theorem Nat.floor_add_one {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} (ha : 0 ≤ a) :

⌊a + 1⌋₊ = ⌊a⌋₊ + 1

theorem Nat.floor_sub_nat {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] [inst : Sub α] [inst : OrderedSub α] [inst : ExistsAddOfLE α] (a : α) (n : ℕ) :

⌊a - ↑n⌋₊ = ⌊a⌋₊ - n

theorem Nat.ceil_add_nat {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} (ha : 0 ≤ a) (n : ℕ) :

⌈a + ↑n⌉₊ = ⌈a⌉₊ + n

theorem Nat.ceil_add_one {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} (ha : 0 ≤ a) :

⌈a + 1⌉₊ = ⌈a⌉₊ + 1

theorem Nat.ceil_lt_add_one {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] {a : α} (ha : 0 ≤ a) :

↑⌈a⌉₊ < a + 1

theorem Nat.ceil_add_le {α : Type u_1} [inst : LinearOrderedSemiring α] [inst : FloorSemiring α] (a : α) (b : α) :

⌈a + b⌉₊ ≤ ⌈a⌉₊ + ⌈b⌉₊

theorem Nat.sub_one_lt_floor {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorSemiring α] (a : α) :

a - 1 < ↑⌊a⌋₊

theorem Nat.floor_div_nat {α : Type u_1} [inst : LinearOrderedSemifield α] [inst : FloorSemiring α] (a : α) (n : ℕ) :

⌊a / ↑n⌋₊ = ⌊a⌋₊ / n

theorem Nat.floor_div_eq_div {α : Type u_1} [inst : LinearOrderedSemifield α] [inst : FloorSemiring α] (m : ℕ) (n : ℕ) :

⌊↑m / ↑n⌋₊ = m / n

Natural division is the floor of field division.

theorem subsingleton_floorSemiring {α : Type u_1} [inst : LinearOrderedSemiring α] :

Subsingleton (FloorSemiring α)

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

Floor rings #

class FloorRing (α : Type u_1) [inst : LinearOrderedRing α] :

Type u_1

FloorRing.floor a computes the greatest integer z such that (z : α) ≤ a≤ a.
floor : α → ℤ
FloorRing.ceil a computes the least integer z such that a ≤ (z : α)≤ (z : α).
ceil : α → ℤ
FloorRing.ceil is the upper adjoint of the coercion ↑ : ℤ → α↑ : ℤ → α→ α.
gc_coe_floor : GaloisConnection Int.cast floor
FloorRing.ceil is the lower adjoint of the coercion ↑ : ℤ → α↑ : ℤ → α→ α.
gc_ceil_coe : GaloisConnection ceil Int.cast

A FloorRing is a linear ordered ring over α with a function floor : α → ℤ→ ℤ satisfying ∀ (z : ℤ) (a : α), z ≤ floor a ↔ (z : α) ≤ a)∀ (z : ℤ) (a : α), z ≤ floor a ↔ (z : α) ≤ a)≤ floor a ↔ (z : α) ≤ a)↔ (z : α) ≤ a)≤ a).

Instances

instance instFloorRingIntToLinearOrderedRingLinearOrderedCommRing :

FloorRing ℤ

Equations

One or more equations did not get rendered due to their size.

def FloorRing.ofFloor (α : Type u_1) [inst : LinearOrderedRing α] (floor : α → ℤ) (gc_coe_floor : GaloisConnection Int.cast floor) :

FloorRing α

A FloorRing constructor from the floor function alone.

Equations

One or more equations did not get rendered due to their size.

def FloorRing.ofCeil (α : Type u_1) [inst : LinearOrderedRing α] (ceil : α → ℤ) (gc_ceil_coe : GaloisConnection ceil Int.cast) :

FloorRing α

A FloorRing constructor from the ceil function alone.

Equations

One or more equations did not get rendered due to their size.

def Int.floor {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] :

α → ℤ

Int.floor a is the greatest integer z such that z ≤ a≤ a. It is denoted with ⌊a⌋.

Equations

Int.floor = FloorRing.floor

def Int.ceil {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] :

α → ℤ

Int.ceil a is the smallest integer z such that a ≤ z≤ z. It is denoted with ⌈a⌉.

Equations

Int.ceil = FloorRing.ceil

def Int.fract {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

Int.fract a, the fractional part of a, is a minus its floor.

Equations

Int.fract a = a - ↑⌊a⌋

@[simp]

theorem Int.floor_int :

Int.floor = id

@[simp]

theorem Int.ceil_int :

Int.ceil = id

@[simp]

theorem Int.fract_int :

Int.fract = 0

def Int.«term⌊_⌋» :

Int.floor a is the greatest integer z such that z ≤ a≤ a. It is denoted with ⌊a⌋.

Equations

One or more equations did not get rendered due to their size.

def Int.«term⌈_⌉» :

Int.ceil a is the smallest integer z such that a ≤ z≤ z. It is denoted with ⌈a⌉.

Equations

One or more equations did not get rendered due to their size.

@FloorRing.floor = @Int.floor

@[simp]

theorem Int.floorRing_floor_eq :

@FloorRing.ceil = @Int.ceil

@[simp]

theorem Int.floorRing_ceil_eq :

Floor #

theorem Int.gc_coe_floor {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] :

GaloisConnection Int.cast Int.floor

theorem Int.le_floor {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {z : ℤ} {a : α} :

z ≤ ⌊a⌋ ↔ ↑z ≤ a

theorem Int.floor_lt {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {z : ℤ} {a : α} :

⌊a⌋ < z ↔ a < ↑z

theorem Int.floor_le {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

↑⌊a⌋ ≤ a

theorem Int.floor_nonneg {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} :

0 ≤ ⌊a⌋ ↔ 0 ≤ a

@[simp]

theorem Int.floor_le_sub_one_iff {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {z : ℤ} {a : α} :

⌊a⌋ ≤ z - 1 ↔ a < ↑z

@[simp]

theorem Int.floor_le_neg_one_iff {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} :

⌊a⌋ ≤ -1 ↔ a < 0

theorem Int.floor_nonpos {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} (ha : a ≤ 0) :

⌊a⌋ ≤ 0

theorem Int.lt_succ_floor {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

a < ↑(Int.succ ⌊a⌋)

@[simp]

theorem Int.lt_floor_add_one {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

a < ↑⌊a⌋ + 1

@[simp]

theorem Int.sub_one_lt_floor {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

a - 1 < ↑⌊a⌋

@[simp]

theorem Int.floor_intCast {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (z : ℤ) :

⌊↑z⌋ = z

@[simp]

theorem Int.floor_natCast {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (n : ℕ) :

⌊↑n⌋ = ↑n

@[simp]

theorem Int.floor_zero {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] :

⌊0⌋ = 0

@[simp]

theorem Int.floor_one {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] :

⌊1⌋ = 1

theorem Int.floor_mono {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] :

Monotone Int.floor

theorem Int.floor_pos {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} :

0 < ⌊a⌋ ↔ 1 ≤ a

@[simp]

theorem Int.floor_add_int {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) (z : ℤ) :

⌊a + ↑z⌋ = ⌊a⌋ + z

theorem Int.floor_add_one {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

⌊a + 1⌋ = ⌊a⌋ + 1

theorem Int.le_floor_add {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) (b : α) :

⌊a⌋ + ⌊b⌋ ≤ ⌊a + b⌋

theorem Int.le_floor_add_floor {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) (b : α) :

⌊a + b⌋ - 1 ≤ ⌊a⌋ + ⌊b⌋

@[simp]

theorem Int.floor_int_add {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (z : ℤ) (a : α) :

⌊↑z + a⌋ = z + ⌊a⌋

@[simp]

theorem Int.floor_add_nat {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) (n : ℕ) :

⌊a + ↑n⌋ = ⌊a⌋ + ↑n

@[simp]

theorem Int.floor_nat_add {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (n : ℕ) (a : α) :

⌊↑n + a⌋ = ↑n + ⌊a⌋

@[simp]

theorem Int.floor_sub_int {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) (z : ℤ) :

⌊a - ↑z⌋ = ⌊a⌋ - z

@[simp]

theorem Int.floor_sub_nat {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) (n : ℕ) :

⌊a - ↑n⌋ = ⌊a⌋ - ↑n

theorem Int.abs_sub_lt_one_of_floor_eq_floor {α : Type u_1} [inst : LinearOrderedCommRing α] [inst : FloorRing α] {a : α} {b : α} (h : ⌊a⌋ = ⌊b⌋) :

abs (a - b) < 1

theorem Int.floor_eq_iff {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {z : ℤ} {a : α} :

⌊a⌋ = z ↔ ↑z ≤ a ∧ a < ↑z + 1

@[simp]

theorem Int.floor_eq_zero_iff {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} :

⌊a⌋ = 0 ↔ a ∈ Set.Ico 0 1

theorem Int.floor_eq_on_Ico {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (n : ℤ) (a : α) :

a ∈ Set.Ico (↑n) (↑n + 1) → ⌊a⌋ = n

theorem Int.floor_eq_on_Ico' {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (n : ℤ) (a : α) :

a ∈ Set.Ico (↑n) (↑n + 1) → ↑⌊a⌋ = ↑n

@[simp]

theorem Int.preimage_floor_singleton {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (m : ℤ) :

Int.floor ⁻¹' {m} = Set.Ico (↑m) (↑m + 1)

Fractional part #

@[simp]

theorem Int.self_sub_floor {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

a - ↑⌊a⌋ = Int.fract a

@[simp]

theorem Int.floor_add_fract {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

↑⌊a⌋ + Int.fract a = a

@[simp]

theorem Int.fract_add_floor {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

Int.fract a + ↑⌊a⌋ = a

@[simp]

theorem Int.fract_add_int {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) (m : ℤ) :

Int.fract (a + ↑m) = Int.fract a

@[simp]

theorem Int.fract_add_nat {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) (m : ℕ) :

Int.fract (a + ↑m) = Int.fract a

@[simp]

theorem Int.fract_sub_int {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) (m : ℤ) :

Int.fract (a - ↑m) = Int.fract a

@[simp]

theorem Int.fract_int_add {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (m : ℤ) (a : α) :

Int.fract (↑m + a) = Int.fract a

@[simp]

theorem Int.fract_sub_nat {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) (n : ℕ) :

Int.fract (a - ↑n) = Int.fract a

@[simp]

theorem Int.fract_int_nat {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (n : ℕ) (a : α) :

Int.fract (↑n + a) = Int.fract a

theorem Int.fract_add_le {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) (b : α) :

Int.fract (a + b) ≤ Int.fract a + Int.fract b

theorem Int.fract_add_fract_le {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) (b : α) :

Int.fract a + Int.fract b ≤ Int.fract (a + b) + 1

@[simp]

theorem Int.self_sub_fract {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

a - Int.fract a = ↑⌊a⌋

@[simp]

theorem Int.fract_sub_self {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

Int.fract a - a = -↑⌊a⌋

@[simp]

theorem Int.fract_nonneg {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

0 ≤ Int.fract a

theorem Int.fract_pos {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} :

0 < Int.fract a ↔ a ≠ ↑⌊a⌋

The fractional part of a is positive if and only if a ≠ ⌊a⌋≠ ⌊a⌋.

theorem Int.fract_lt_one {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

Int.fract a < 1

@[simp]

theorem Int.fract_zero {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] :

Int.fract 0 = 0

@[simp]

theorem Int.fract_one {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] :

Int.fract 1 = 0

theorem Int.abs_fract {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} :

abs (Int.fract a) = Int.fract a

@[simp]

theorem Int.abs_one_sub_fract {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} :

abs (1 - Int.fract a) = 1 - Int.fract a

@[simp]

theorem Int.fract_intCast {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (z : ℤ) :

Int.fract ↑z = 0

@[simp]

theorem Int.fract_natCast {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (n : ℕ) :

Int.fract ↑n = 0

theorem Int.fract_floor {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

Int.fract ↑⌊a⌋ = 0

@[simp]

theorem Int.floor_fract {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

⌊Int.fract a⌋ = 0

theorem Int.fract_eq_iff {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} {b : α} :

Int.fract a = b ↔ 0 ≤ b ∧ b < 1 ∧ ∃ z, a - b = ↑z

theorem Int.fract_eq_fract {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} {b : α} :

Int.fract a = Int.fract b ↔ ∃ z, a - b = ↑z

@[simp]

theorem Int.fract_eq_self {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} :

Int.fract a = a ↔ 0 ≤ a ∧ a < 1

@[simp]

theorem Int.fract_fract {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

Int.fract (Int.fract a) = Int.fract a

theorem Int.fract_add {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) (b : α) :

∃ z, Int.fract (a + b) - Int.fract a - Int.fract b = ↑z

theorem Int.fract_neg {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {x : α} (hx : Int.fract x ≠ 0) :

Int.fract (-x) = 1 - Int.fract x

@[simp]

theorem Int.fract_neg_eq_zero {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {x : α} :

Int.fract (-x) = 0 ↔ Int.fract x = 0

theorem Int.fract_mul_nat {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) (b : ℕ) :

∃ z, Int.fract a * ↑b - Int.fract (a * ↑b) = ↑z

theorem Int.preimage_fract {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (s : Set α) :

Int.fract ⁻¹' s = Set.unionᵢ fun m => (fun x => x - ↑m) ⁻¹' (s ∩ Set.Ico 0 1)

theorem Int.image_fract {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (s : Set α) :

Int.fract '' s = Set.unionᵢ fun m => (fun x => x - ↑m) '' s ∩ Set.Ico 0 1

theorem Int.fract_div_mul_self_mem_Ico {k : Type u_1} [inst : LinearOrderedField k] [inst : FloorRing k] (a : k) (b : k) (ha : 0 < a) :

Int.fract (b / a) * a ∈ Set.Ico 0 a

theorem Int.fract_div_mul_self_add_zsmul_eq {k : Type u_1} [inst : LinearOrderedField k] [inst : FloorRing k] (a : k) (b : k) (ha : a ≠ 0) :

Int.fract (b / a) * a + ⌊b / a⌋ • a = b

theorem Int.sub_floor_div_mul_nonneg {k : Type u_1} [inst : LinearOrderedField k] [inst : FloorRing k] {b : k} (a : k) (hb : 0 < b) :

0 ≤ a - ↑⌊a / b⌋ * b

theorem Int.sub_floor_div_mul_lt {k : Type u_1} [inst : LinearOrderedField k] [inst : FloorRing k] {b : k} (a : k) (hb : 0 < b) :

a - ↑⌊a / b⌋ * b < b

theorem Int.fract_div_natCast_eq_div_natCast_mod {k : Type u_1} [inst : LinearOrderedField k] [inst : FloorRing k] {m : ℕ} {n : ℕ} :

Int.fract (↑m / ↑n) = ↑(m % n) / ↑n

theorem Int.fract_div_intCast_eq_div_intCast_mod {k : Type u_1} [inst : LinearOrderedField k] [inst : FloorRing k] {m : ℤ} {n : ℕ} :

Int.fract (↑m / ↑n) = ↑(m % ↑n) / ↑n

Ceil #

theorem Int.gc_ceil_coe {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] :

GaloisConnection Int.ceil Int.cast

theorem Int.ceil_le {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {z : ℤ} {a : α} :

⌈a⌉ ≤ z ↔ a ≤ ↑z

theorem Int.floor_neg {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} :

⌊-a⌋ = -⌈a⌉

theorem Int.ceil_neg {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} :

⌈-a⌉ = -⌊a⌋

theorem Int.lt_ceil {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {z : ℤ} {a : α} :

z < ⌈a⌉ ↔ ↑z < a

@[simp]

theorem Int.add_one_le_ceil_iff {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {z : ℤ} {a : α} :

z + 1 ≤ ⌈a⌉ ↔ ↑z < a

@[simp]

theorem Int.one_le_ceil_iff {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} :

1 ≤ ⌈a⌉ ↔ 0 < a

theorem Int.ceil_le_floor_add_one {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

⌈a⌉ ≤ ⌊a⌋ + 1

theorem Int.le_ceil {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

a ≤ ↑⌈a⌉

@[simp]

theorem Int.ceil_intCast {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (z : ℤ) :

⌈↑z⌉ = z

@[simp]

theorem Int.ceil_natCast {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (n : ℕ) :

⌈↑n⌉ = ↑n

theorem Int.ceil_mono {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] :

Monotone Int.ceil

@[simp]

theorem Int.ceil_add_int {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) (z : ℤ) :

⌈a + ↑z⌉ = ⌈a⌉ + z

@[simp]

theorem Int.ceil_add_nat {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) (n : ℕ) :

⌈a + ↑n⌉ = ⌈a⌉ + ↑n

@[simp]

theorem Int.ceil_add_one {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

⌈a + 1⌉ = ⌈a⌉ + 1

@[simp]

theorem Int.ceil_sub_int {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) (z : ℤ) :

⌈a - ↑z⌉ = ⌈a⌉ - z

@[simp]

theorem Int.ceil_sub_nat {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) (n : ℕ) :

⌈a - ↑n⌉ = ⌈a⌉ - ↑n

@[simp]

theorem Int.ceil_sub_one {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

⌈a - 1⌉ = ⌈a⌉ - 1

theorem Int.ceil_lt_add_one {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

↑⌈a⌉ < a + 1

theorem Int.ceil_add_le {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) (b : α) :

⌈a + b⌉ ≤ ⌈a⌉ + ⌈b⌉

theorem Int.ceil_add_ceil_le {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) (b : α) :

⌈a⌉ + ⌈b⌉ ≤ ⌈a + b⌉ + 1

@[simp]

theorem Int.ceil_pos {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} :

0 < ⌈a⌉ ↔ 0 < a

@[simp]

theorem Int.ceil_zero {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] :

⌈0⌉ = 0

@[simp]

theorem Int.ceil_one {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] :

⌈1⌉ = 1

theorem Int.ceil_nonneg {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} (ha : 0 ≤ a) :

0 ≤ ⌈a⌉

theorem Int.ceil_eq_iff {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {z : ℤ} {a : α} :

⌈a⌉ = z ↔ ↑z - 1 < a ∧ a ≤ ↑z

@[simp]

theorem Int.ceil_eq_zero_iff {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} :

⌈a⌉ = 0 ↔ a ∈ Set.Ioc (-1) 0

theorem Int.ceil_eq_on_Ioc {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (z : ℤ) (a : α) :

a ∈ Set.Ioc (↑z - 1) ↑z → ⌈a⌉ = z

theorem Int.ceil_eq_on_Ioc' {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (z : ℤ) (a : α) :

a ∈ Set.Ioc (↑z - 1) ↑z → ↑⌈a⌉ = ↑z

theorem Int.floor_le_ceil {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

⌊a⌋ ≤ ⌈a⌉

theorem Int.floor_lt_ceil_of_lt {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} {b : α} (h : a < b) :

⌊a⌋ < ⌈b⌉

@[simp]

theorem Int.preimage_ceil_singleton {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (m : ℤ) :

Int.ceil ⁻¹' {m} = Set.Ioc (↑m - 1) ↑m

theorem Int.fract_eq_zero_or_add_one_sub_ceil {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

Int.fract a = 0 ∨ Int.fract a = a + 1 - ↑⌈a⌉

theorem Int.ceil_eq_add_one_sub_fract {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} (ha : Int.fract a ≠ 0) :

↑⌈a⌉ = a + 1 - Int.fract a

theorem Int.ceil_sub_self_eq {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} (ha : Int.fract a ≠ 0) :

↑⌈a⌉ - a = 1 - Int.fract a

Intervals #

@[simp]

theorem Int.preimage_Ioo {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} {b : α} :

Int.cast ⁻¹' Set.Ioo a b = Set.Ioo ⌊a⌋ ⌈b⌉

@[simp]

theorem Int.preimage_Ico {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} {b : α} :

Int.cast ⁻¹' Set.Ico a b = Set.Ico ⌈a⌉ ⌈b⌉

@[simp]

theorem Int.preimage_Ioc {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} {b : α} :

Int.cast ⁻¹' Set.Ioc a b = Set.Ioc ⌊a⌋ ⌊b⌋

@[simp]

theorem Int.preimage_Icc {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} {b : α} :

Int.cast ⁻¹' Set.Icc a b = Set.Icc ⌈a⌉ ⌊b⌋

@[simp]

theorem Int.preimage_Ioi {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} :

Int.cast ⁻¹' Set.Ioi a = Set.Ioi ⌊a⌋

@[simp]

theorem Int.preimage_Ici {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} :

Int.cast ⁻¹' Set.Ici a = Set.Ici ⌈a⌉

@[simp]

theorem Int.preimage_Iio {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} :

Int.cast ⁻¹' Set.Iio a = Set.Iio ⌈a⌉

@[simp]

theorem Int.preimage_Iic {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] {a : α} :

Int.cast ⁻¹' Set.Iic a = Set.Iic ⌊a⌋

Round #

def round {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (x : α) :

ℤ

round rounds a number to the nearest integer. round (1 / 2) = 1

Equations

round x = if 2 * Int.fract x < 1 then ⌊x⌋ else ⌈x⌉

@[simp]

theorem round_zero {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] :

round 0 = 0

@[simp]

theorem round_one {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] :

round 1 = 1

@[simp]

theorem round_natCast {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (n : ℕ) :

round ↑n = ↑n

@[simp]

theorem round_intCast {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (n : ℤ) :

round ↑n = n

@[simp]

theorem round_add_int {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (x : α) (y : ℤ) :

round (x + ↑y) = round x + y

@[simp]

theorem round_add_one {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

round (a + 1) = round a + 1

@[simp]

theorem round_sub_int {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (x : α) (y : ℤ) :

round (x - ↑y) = round x - y

@[simp]

theorem round_sub_one {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (a : α) :

round (a - 1) = round a - 1

@[simp]

theorem round_add_nat {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (x : α) (y : ℕ) :

round (x + ↑y) = round x + ↑y

@[simp]

theorem round_sub_nat {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (x : α) (y : ℕ) :

round (x - ↑y) = round x - ↑y

@[simp]

theorem round_int_add {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (x : α) (y : ℤ) :

round (↑y + x) = y + round x

@[simp]

theorem round_nat_add {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (x : α) (y : ℕ) :

round (↑y + x) = ↑y + round x

theorem abs_sub_round_eq_min {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (x : α) :

abs (x - ↑(round x)) = min (Int.fract x) (1 - Int.fract x)

theorem round_le {α : Type u_1} [inst : LinearOrderedRing α] [inst : FloorRing α] (x : α) (z : ℤ) :

abs (x - ↑(round x)) ≤ abs (x - ↑z)

theorem round_eq {α : Type u_1} [inst : LinearOrderedField α] [inst : FloorRing α] (x : α) :

round x = ⌊x + 1 / 2⌋

@[simp]

theorem round_two_inv {α : Type u_1} [inst : LinearOrderedField α] [inst : FloorRing α] :

round 2⁻¹ = 1

@[simp]

theorem round_neg_two_inv {α : Type u_1} [inst : LinearOrderedField α] [inst : FloorRing α] :

round (-2⁻¹) = 0

@[simp]

theorem round_eq_zero_iff {α : Type u_1} [inst : LinearOrderedField α] [inst : FloorRing α] {x : α} :

round x = 0 ↔ x ∈ Set.Ico (-(1 / 2)) (1 / 2)

theorem abs_sub_round {α : Type u_1} [inst : LinearOrderedField α] [inst : FloorRing α] (x : α) :

abs (x - ↑(round x)) ≤ 1 / 2

theorem abs_sub_round_div_natCast_eq {α : Type u_1} [inst : LinearOrderedField α] [inst : FloorRing α] {m : ℕ} {n : ℕ} :

abs (↑m / ↑n - ↑(round (↑m / ↑n))) = ↑(min (m % n) (n - m % n)) / ↑n

theorem Nat.floor_congr {α : Type u_1} {β : Type u_2} [inst : LinearOrderedSemiring α] [inst : LinearOrderedSemiring β] [inst : FloorSemiring α] [inst : FloorSemiring β] {a : α} {b : β} (h : ∀ (n : ℕ), ↑n ≤ a ↔ ↑n ≤ b) :

⌊a⌋₊ = ⌊b⌋₊

theorem Nat.ceil_congr {α : Type u_1} {β : Type u_2} [inst : LinearOrderedSemiring α] [inst : LinearOrderedSemiring β] [inst : FloorSemiring α] [inst : FloorSemiring β] {a : α} {b : β} (h : ∀ (n : ℕ), a ≤ ↑n ↔ b ≤ ↑n) :

⌈a⌉₊ = ⌈b⌉₊

theorem Nat.map_floor {F : Type u_3} {α : Type u_1} {β : Type u_2} [inst : LinearOrderedSemiring α] [inst : LinearOrderedSemiring β] [inst : FloorSemiring α] [inst : FloorSemiring β] [inst : RingHomClass F α β] (f : F) (hf : StrictMono ↑f) (a : α) :

⌊↑f a⌋₊ = ⌊a⌋₊

theorem Nat.map_ceil {F : Type u_3} {α : Type u_1} {β : Type u_2} [inst : LinearOrderedSemiring α] [inst : LinearOrderedSemiring β] [inst : FloorSemiring α] [inst : FloorSemiring β] [inst : RingHomClass F α β] (f : F) (hf : StrictMono ↑f) (a : α) :

⌈↑f a⌉₊ = ⌈a⌉₊

theorem Int.floor_congr {α : Type u_1} {β : Type u_2} [inst : LinearOrderedRing α] [inst : LinearOrderedRing β] [inst : FloorRing α] [inst : FloorRing β] {a : α} {b : β} (h : ∀ (n : ℤ), ↑n ≤ a ↔ ↑n ≤ b) :

⌊a⌋ = ⌊b⌋

theorem Int.ceil_congr {α : Type u_1} {β : Type u_2} [inst : LinearOrderedRing α] [inst : LinearOrderedRing β] [inst : FloorRing α] [inst : FloorRing β] {a : α} {b : β} (h : ∀ (n : ℤ), a ≤ ↑n ↔ b ≤ ↑n) :

⌈a⌉ = ⌈b⌉

theorem Int.map_floor {F : Type u_3} {α : Type u_1} {β : Type u_2} [inst : LinearOrderedRing α] [inst : LinearOrderedRing β] [inst : FloorRing α] [inst : FloorRing β] [inst : RingHomClass F α β] (f : F) (hf : StrictMono ↑f) (a : α) :

⌊↑f a⌋ = ⌊a⌋

theorem Int.map_ceil {F : Type u_3} {α : Type u_1} {β : Type u_2} [inst : LinearOrderedRing α] [inst : LinearOrderedRing β] [inst : FloorRing α] [inst : FloorRing β] [inst : RingHomClass F α β] (f : F) (hf : StrictMono ↑f) (a : α) :

⌈↑f a⌉ = ⌈a⌉

theorem Int.map_fract {F : Type u_3} {α : Type u_1} {β : Type u_2} [inst : LinearOrderedRing α] [inst : LinearOrderedRing β] [inst : FloorRing α] [inst : FloorRing β] [inst : RingHomClass F α β] (f : F) (hf : StrictMono ↑f) (a : α) :

Int.fract (↑f a) = ↑f (Int.fract a)