Documentation

Mathlib.Data.Real.EReal

The extended reals [-∞, ∞]. #

This file defines EReal, the real numbers together with a top and bottom element, referred to as ⊤ and ⊥. It is implemented as WithBot (WithTop ℝ)

Addition and multiplication are problematic in the presence of ±∞, but negation has a natural definition and satisfies the usual properties.

An ad hoc addition is defined, for which EReal is an AddCommMonoid, and even an ordered one (if a ≤ a' and b ≤ b' then a + b ≤ a' + b'). Note however that addition is badly behaved at (⊥, ⊤) and (⊤, ⊥) so this can not be upgraded to a group structure. Our choice is that ⊥ + ⊤ = ⊤ + ⊥ = ⊥, to make sure that the exponential and the logarithm between EReal and ℝ≥0∞ respect the operations (notice that the convention 0 * ∞ = 0 on ℝ≥0∞ is enforced by measure theory).

An ad hoc subtraction is then defined by x - y = x + (-y). It does not have nice properties, but it is sometimes convenient to have.

An ad hoc multiplication is defined, for which EReal is a CommMonoidWithZero. We make the choice that 0 * x = x * 0 = 0 for any x (while the other cases are defined non-ambiguously). This does not distribute with addition, as ⊥ = ⊥ + ⊤ = 1*⊥ + (-1)*⊥ ≠ (1 - 1) * ⊥ = 0 * ⊥ = 0.

EReal is a CompleteLinearOrder; this is deduced by type class inference from the fact that WithBot (WithTop L) is a complete linear order if L is a conditionally complete linear order.

Coercions from ℝ and from ℝ≥0∞ are registered, and their basic properties are proved. The main one is the real coercion, and is usually referred to just as coe (lemmas such as EReal.coe_add deal with this coercion). The one from ENNReal is usually called coe_ennreal in the EReal namespace.

We define an absolute value EReal.abs from EReal to ℝ≥0∞. Two elements of EReal coincide if and only if they have the same absolute value and the same sign.

Tags #

real, ereal, complete lattice

def EReal :

ereal : The type [-∞, ∞]

Equations

EReal = WithBot (WithTop ℝ)

Instances For

instance instZeroLEOneClassEReal :

ZeroLEOneClass EReal

Equations

instZeroLEOneClassEReal = ⋯

instance instSupSetEReal :

Equations

instSupSetEReal = inferInstanceAs (SupSet (WithBot (WithTop ℝ)))

instance instInfSetEReal :

Equations

instInfSetEReal = inferInstanceAs (InfSet (WithBot (WithTop ℝ)))

instance instCompleteLinearOrderEReal :

CompleteLinearOrder EReal

Equations

instCompleteLinearOrderEReal = inferInstanceAs (CompleteLinearOrder (WithBot (WithTop ℝ)))

instance instLinearOrderedAddCommMonoidEReal :

LinearOrderedAddCommMonoid EReal

Equations

instLinearOrderedAddCommMonoidEReal = inferInstanceAs (LinearOrderedAddCommMonoid (WithBot (WithTop ℝ)))

instance instAddCommMonoidWithOneEReal :

AddCommMonoidWithOne EReal

Equations

instAddCommMonoidWithOneEReal = inferInstanceAs (AddCommMonoidWithOne (WithBot (WithTop ℝ)))

instance instDenselyOrderedEReal :

DenselyOrdered EReal

Equations

instDenselyOrderedEReal = ⋯

def Real.toEReal :

The canonical inclusion from reals to ereals. Registered as a coercion.

Equations

Real.toEReal = some ∘ some

Instances For

instance EReal.decidableLT :

DecidableRel fun (x x_1 : EReal) => x < x_1

Equations

EReal.decidableLT = WithBot.decidableLT

instance EReal.instTop :

Equations

EReal.instTop = { top := some ⊤ }

instance EReal.instCoeReal :

Equations

EReal.instCoeReal = { coe := Real.toEReal }

theorem EReal.coe_strictMono :

StrictMono Real.toEReal

theorem EReal.coe_injective :

Function.Injective Real.toEReal

@[simp]

theorem EReal.coe_le_coe_iff {x : ℝ} {y : ℝ} :

↑x ≤ ↑y ↔ x ≤ y

@[simp]

theorem EReal.coe_lt_coe_iff {x : ℝ} {y : ℝ} :

↑x < ↑y ↔ x < y

@[simp]

theorem EReal.coe_eq_coe_iff {x : ℝ} {y : ℝ} :

↑x = ↑y ↔ x = y

theorem EReal.coe_ne_coe_iff {x : ℝ} {y : ℝ} :

↑x ≠ ↑y ↔ x ≠ y

def ENNReal.toEReal :

ENNReal → EReal

The canonical map from nonnegative extended reals to extended reals

Equations

↑x = match x with | none => ⊤ | some x => ↑↑x

Instances For

instance EReal.hasCoeENNReal :

Coe ENNReal EReal

Equations

EReal.hasCoeENNReal = { coe := ENNReal.toEReal }

instance EReal.instInhabited :

Inhabited EReal

Equations

EReal.instInhabited = { default := 0 }

@[simp]

theorem EReal.coe_zero :

↑0 = 0

@[simp]

theorem EReal.coe_one :

↑1 = 1

def EReal.rec {C : EReal → Sort u_1} (h_bot : C ⊥) (h_real : (a : ℝ) → C ↑a) (h_top : C ⊤) (a : EReal) :

C a

A recursor for EReal in terms of the coercion.

When working in term mode, note that pattern matching can be used directly.

Equations

EReal.rec h_bot h_real h_top x = match x with | none => h_bot | some (some a) => h_real a | some none => h_top

Instances For

def EReal.mul :

EReal → EReal → EReal

The multiplication on EReal. Our definition satisfies 0 * x = x * 0 = 0 for any x, and picks the only sensible value elsewhere.

Equations

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

Instances For

instance EReal.instMul :

Equations

EReal.instMul = { mul := EReal.mul }

@[simp]

theorem EReal.coe_mul (x : ℝ) (y : ℝ) :

↑(x * y) = ↑x * ↑y

theorem EReal.induction₂ {P : EReal → EReal → Prop} (top_top : P ⊤ ⊤) (top_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x) (top_zero : P ⊤ 0) (top_neg : ∀ x < 0, P ⊤ ↑x) (top_bot : P ⊤ ⊥) (pos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤) (pos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥) (zero_top : P 0 ⊤) (coe_coe : ∀ (x y : ℝ), P ↑x ↑y) (zero_bot : P 0 ⊥) (neg_top : ∀ x < 0, P ↑x ⊤) (neg_bot : ∀ x < 0, P ↑x ⊥) (bot_top : P ⊥ ⊤) (bot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x) (bot_zero : P ⊥ 0) (bot_neg : ∀ x < 0, P ⊥ ↑x) (bot_bot : P ⊥ ⊥) (x : EReal) (y : EReal) :

P x y

Induct on two EReals by performing case splits on the sign of one whenever the other is infinite.

theorem EReal.induction₂_symm {P : EReal → EReal → Prop} (symm : ∀ {x y : EReal}, P x y → P y x) (top_top : P ⊤ ⊤) (top_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x) (top_zero : P ⊤ 0) (top_neg : ∀ x < 0, P ⊤ ↑x) (top_bot : P ⊤ ⊥) (pos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥) (coe_coe : ∀ (x y : ℝ), P ↑x ↑y) (zero_bot : P 0 ⊥) (neg_bot : ∀ x < 0, P ↑x ⊥) (bot_bot : P ⊥ ⊥) (x : EReal) (y : EReal) :

P x y

Induct on two EReals by performing case splits on the sign of one whenever the other is infinite. This version eliminates some cases by assuming that the relation is symmetric.

EReal with its multiplication is a CommMonoidWithZero. However, the proof of associativity by hand is extremely painful (with 125 cases...). Instead, we will deduce it later on from the facts that the absolute value and the sign are multiplicative functions taking value in associative objects, and that they characterize an extended real number. For now, we only record more basic properties of multiplication.

theorem EReal.mul_comm (x : EReal) (y : EReal) :

x * y = y * x

theorem EReal.one_mul (x : EReal) :

1 * x = x

theorem EReal.zero_mul (x : EReal) :

0 * x = 0

instance EReal.instMulZeroOneClass :

MulZeroOneClass EReal

Equations

EReal.instMulZeroOneClass = MulZeroOneClass.mk EReal.zero_mul EReal.instMulZeroOneClass.proof_2

Real coercion #

instance EReal.canLift :

CanLift EReal ℝ Real.toEReal fun (r : EReal) => r ≠ ⊤ ∧ r ≠ ⊥

Equations

EReal.canLift = ⋯

def EReal.toReal :

The map from extended reals to reals sending infinities to zero.

Equations

x.toReal = match x with | none => 0 | some none => 0 | some (some x) => x

Instances For

@[simp]

theorem EReal.toReal_top :

⊤.toReal = 0

@[simp]

theorem EReal.toReal_bot :

⊥.toReal = 0

@[simp]

theorem EReal.toReal_zero :

EReal.toReal 0 = 0

@[simp]

theorem EReal.toReal_one :

EReal.toReal 1 = 1

@[simp]

theorem EReal.toReal_coe (x : ℝ) :

(↑x).toReal = x

@[simp]

theorem EReal.bot_lt_coe (x : ℝ) :

⊥ < ↑x

@[simp]

theorem EReal.coe_ne_bot (x : ℝ) :

↑x ≠ ⊥

@[simp]

theorem EReal.bot_ne_coe (x : ℝ) :

⊥ ≠ ↑x

@[simp]

theorem EReal.coe_lt_top (x : ℝ) :

↑x < ⊤

@[simp]

theorem EReal.coe_ne_top (x : ℝ) :

↑x ≠ ⊤

@[simp]

theorem EReal.top_ne_coe (x : ℝ) :

⊤ ≠ ↑x

@[simp]

theorem EReal.bot_lt_zero :

@[simp]

theorem EReal.bot_ne_zero :

@[simp]

theorem EReal.zero_ne_bot :

@[simp]

theorem EReal.zero_lt_top :

@[simp]

theorem EReal.zero_ne_top :

@[simp]

theorem EReal.top_ne_zero :

theorem EReal.range_coe :

Set.range Real.toEReal = {⊥, ⊤}ᶜ

theorem EReal.range_coe_eq_Ioo :

Set.range Real.toEReal = Set.Ioo ⊥ ⊤

@[simp]

theorem EReal.coe_add (x : ℝ) (y : ℝ) :

↑(x + y) = ↑x + ↑y

theorem EReal.coe_nsmul (n : ℕ) (x : ℝ) :

↑(n • x) = n • ↑x

@[simp]

theorem EReal.coe_eq_zero {x : ℝ} :

↑x = 0 ↔ x = 0

@[simp]

theorem EReal.coe_eq_one {x : ℝ} :

↑x = 1 ↔ x = 1

theorem EReal.coe_ne_zero {x : ℝ} :

↑x ≠ 0 ↔ x ≠ 0

theorem EReal.coe_ne_one {x : ℝ} :

↑x ≠ 1 ↔ x ≠ 1

@[simp]

theorem EReal.coe_nonneg {x : ℝ} :

0 ≤ ↑x ↔ 0 ≤ x

@[simp]

theorem EReal.coe_nonpos {x : ℝ} :

↑x ≤ 0 ↔ x ≤ 0

@[simp]

theorem EReal.coe_pos {x : ℝ} :

0 < ↑x ↔ 0 < x

@[simp]

theorem EReal.coe_neg' {x : ℝ} :

↑x < 0 ↔ x < 0

theorem EReal.toReal_le_toReal {x : EReal} {y : EReal} (h : x ≤ y) (hx : x ≠ ⊥) (hy : y ≠ ⊤) :

x.toReal ≤ y.toReal

theorem EReal.coe_toReal {x : EReal} (hx : x ≠ ⊤) (h'x : x ≠ ⊥) :

↑x.toReal = x

theorem EReal.le_coe_toReal {x : EReal} (h : x ≠ ⊤) :

x ≤ ↑x.toReal

theorem EReal.coe_toReal_le {x : EReal} (h : x ≠ ⊥) :

↑x.toReal ≤ x

theorem EReal.eq_top_iff_forall_lt (x : EReal) :

x = ⊤ ↔ ∀ (y : ℝ), ↑y < x

theorem EReal.eq_bot_iff_forall_lt (x : EReal) :

x = ⊥ ↔ ∀ (y : ℝ), x < ↑y

Intervals and coercion from reals #

theorem EReal.exists_between_coe_real {x : EReal} {z : EReal} (h : x < z) :

∃ (y : ℝ), x < ↑y ∧ ↑y < z

@[simp]

theorem EReal.image_coe_Icc (x : ℝ) (y : ℝ) :

Real.toEReal '' Set.Icc x y = Set.Icc ↑x ↑y

@[simp]

theorem EReal.image_coe_Ico (x : ℝ) (y : ℝ) :

Real.toEReal '' Set.Ico x y = Set.Ico ↑x ↑y

@[simp]

theorem EReal.image_coe_Ici (x : ℝ) :

Real.toEReal '' Set.Ici x = Set.Ico ↑x ⊤

@[simp]

theorem EReal.image_coe_Ioc (x : ℝ) (y : ℝ) :

Real.toEReal '' Set.Ioc x y = Set.Ioc ↑x ↑y

@[simp]

theorem EReal.image_coe_Ioo (x : ℝ) (y : ℝ) :

Real.toEReal '' Set.Ioo x y = Set.Ioo ↑x ↑y

@[simp]

theorem EReal.image_coe_Ioi (x : ℝ) :

Real.toEReal '' Set.Ioi x = Set.Ioo ↑x ⊤

@[simp]

theorem EReal.image_coe_Iic (x : ℝ) :

Real.toEReal '' Set.Iic x = Set.Ioc ⊥ ↑x

@[simp]

theorem EReal.image_coe_Iio (x : ℝ) :

Real.toEReal '' Set.Iio x = Set.Ioo ⊥ ↑x

@[simp]

theorem EReal.preimage_coe_Ici (x : ℝ) :

Real.toEReal ⁻¹' Set.Ici ↑x = Set.Ici x

@[simp]

theorem EReal.preimage_coe_Ioi (x : ℝ) :

Real.toEReal ⁻¹' Set.Ioi ↑x = Set.Ioi x

@[simp]

theorem EReal.preimage_coe_Ioi_bot :

Real.toEReal ⁻¹' Set.Ioi ⊥ = Set.univ

@[simp]

theorem EReal.preimage_coe_Iic (y : ℝ) :

Real.toEReal ⁻¹' Set.Iic ↑y = Set.Iic y

@[simp]

theorem EReal.preimage_coe_Iio (y : ℝ) :

Real.toEReal ⁻¹' Set.Iio ↑y = Set.Iio y

@[simp]

theorem EReal.preimage_coe_Iio_top :

Real.toEReal ⁻¹' Set.Iio ⊤ = Set.univ

@[simp]

theorem EReal.preimage_coe_Icc (x : ℝ) (y : ℝ) :

Real.toEReal ⁻¹' Set.Icc ↑x ↑y = Set.Icc x y

@[simp]

theorem EReal.preimage_coe_Ico (x : ℝ) (y : ℝ) :

Real.toEReal ⁻¹' Set.Ico ↑x ↑y = Set.Ico x y

@[simp]

theorem EReal.preimage_coe_Ioc (x : ℝ) (y : ℝ) :

Real.toEReal ⁻¹' Set.Ioc ↑x ↑y = Set.Ioc x y

@[simp]

theorem EReal.preimage_coe_Ioo (x : ℝ) (y : ℝ) :

Real.toEReal ⁻¹' Set.Ioo ↑x ↑y = Set.Ioo x y

@[simp]

theorem EReal.preimage_coe_Ico_top (x : ℝ) :

Real.toEReal ⁻¹' Set.Ico ↑x ⊤ = Set.Ici x

@[simp]

theorem EReal.preimage_coe_Ioo_top (x : ℝ) :

Real.toEReal ⁻¹' Set.Ioo ↑x ⊤ = Set.Ioi x

@[simp]

theorem EReal.preimage_coe_Ioc_bot (y : ℝ) :

Real.toEReal ⁻¹' Set.Ioc ⊥ ↑y = Set.Iic y

@[simp]

theorem EReal.preimage_coe_Ioo_bot (y : ℝ) :

Real.toEReal ⁻¹' Set.Ioo ⊥ ↑y = Set.Iio y

@[simp]

theorem EReal.preimage_coe_Ioo_bot_top :

Real.toEReal ⁻¹' Set.Ioo ⊥ ⊤ = Set.univ

ennreal coercion #

@[simp]

theorem EReal.toReal_coe_ennreal {x : ENNReal} :

(↑x).toReal = x.toReal

@[simp]

theorem EReal.coe_ennreal_ofReal {x : ℝ} :

↑(ENNReal.ofReal x) = ↑(max x 0)

theorem EReal.coe_nnreal_eq_coe_real (x : NNReal) :

↑↑x = ↑↑x

@[simp]

theorem EReal.coe_ennreal_zero :

↑0 = 0

@[simp]

theorem EReal.coe_ennreal_one :

↑1 = 1

@[simp]

theorem EReal.coe_ennreal_top :

theorem EReal.coe_ennreal_strictMono :

StrictMono ENNReal.toEReal

theorem EReal.coe_ennreal_injective :

Function.Injective ENNReal.toEReal

@[simp]

theorem EReal.coe_ennreal_eq_top_iff {x : ENNReal} :

↑x = ⊤ ↔ x = ⊤

theorem EReal.coe_nnreal_ne_top (x : NNReal) :

↑↑x ≠ ⊤

@[simp]

theorem EReal.coe_nnreal_lt_top (x : NNReal) :

↑↑x < ⊤

@[simp]

theorem EReal.coe_ennreal_le_coe_ennreal_iff {x : ENNReal} {y : ENNReal} :

↑x ≤ ↑y ↔ x ≤ y

@[simp]

theorem EReal.coe_ennreal_lt_coe_ennreal_iff {x : ENNReal} {y : ENNReal} :

↑x < ↑y ↔ x < y

@[simp]

theorem EReal.coe_ennreal_eq_coe_ennreal_iff {x : ENNReal} {y : ENNReal} :

↑x = ↑y ↔ x = y

theorem EReal.coe_ennreal_ne_coe_ennreal_iff {x : ENNReal} {y : ENNReal} :

↑x ≠ ↑y ↔ x ≠ y

@[simp]

theorem EReal.coe_ennreal_eq_zero {x : ENNReal} :

↑x = 0 ↔ x = 0

@[simp]

theorem EReal.coe_ennreal_eq_one {x : ENNReal} :

↑x = 1 ↔ x = 1

theorem EReal.coe_ennreal_ne_zero {x : ENNReal} :

↑x ≠ 0 ↔ x ≠ 0

theorem EReal.coe_ennreal_ne_one {x : ENNReal} :

↑x ≠ 1 ↔ x ≠ 1

theorem EReal.coe_ennreal_nonneg (x : ENNReal) :

0 ≤ ↑x

@[simp]

theorem EReal.range_coe_ennreal :

Set.range ENNReal.toEReal = Set.Ici 0

instance EReal.instCanLiftENNRealToERealLeOfNat :

CanLift EReal ENNReal ENNReal.toEReal fun (x : EReal) => 0 ≤ x

Equations

EReal.instCanLiftENNRealToERealLeOfNat = ⋯

@[simp]

theorem EReal.coe_ennreal_pos {x : ENNReal} :

0 < ↑x ↔ 0 < x

@[simp]

theorem EReal.bot_lt_coe_ennreal (x : ENNReal) :

⊥ < ↑x

@[simp]

theorem EReal.coe_ennreal_ne_bot (x : ENNReal) :

↑x ≠ ⊥

@[simp]

theorem EReal.coe_ennreal_add (x : ENNReal) (y : ENNReal) :

↑(x + y) = ↑x + ↑y

@[simp]

theorem EReal.coe_ennreal_mul (x : ENNReal) (y : ENNReal) :

↑(x * y) = ↑x * ↑y

theorem EReal.coe_ennreal_nsmul (n : ℕ) (x : ENNReal) :

↑(n • x) = n • ↑x

nat coercion #

theorem EReal.coe_coe_eq_natCast (n : ℕ) :

↑↑n = ↑n

theorem EReal.natCast_ne_bot (n : ℕ) :

↑n ≠ ⊥

theorem EReal.natCast_ne_top (n : ℕ) :

↑n ≠ ⊤

@[simp]

theorem EReal.natCast_eq_iff {m : ℕ} {n : ℕ} :

↑m = ↑n ↔ m = n

theorem EReal.natCast_ne_iff {m : ℕ} {n : ℕ} :

↑m ≠ ↑n ↔ m ≠ n

@[simp]

theorem EReal.natCast_le_iff {m : ℕ} {n : ℕ} :

↑m ≤ ↑n ↔ m ≤ n

@[simp]

theorem EReal.natCast_lt_iff {m : ℕ} {n : ℕ} :

↑m < ↑n ↔ m < n

@[simp]

theorem EReal.natCast_mul (m : ℕ) (n : ℕ) :

↑(m * n) = ↑m * ↑n

Order #

theorem EReal.exists_rat_btwn_of_lt {a : EReal} {b : EReal} :

a < b → ∃ (x : ℚ), a < ↑↑x ∧ ↑↑x < b

theorem EReal.lt_iff_exists_rat_btwn {a : EReal} {b : EReal} :

a < b ↔ ∃ (x : ℚ), a < ↑↑x ∧ ↑↑x < b

theorem EReal.lt_iff_exists_real_btwn {a : EReal} {b : EReal} :

a < b ↔ ∃ (x : ℝ), a < ↑x ∧ ↑x < b

def EReal.neTopBotEquivReal :

↑{⊥, ⊤}ᶜ ≃ ℝ

The set of numbers in EReal that are not equal to ±∞ is equivalent to ℝ.

Equations

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

Instances For

Addition #

@[simp]

theorem EReal.add_bot (x : EReal) :

@[simp]

theorem EReal.bot_add (x : EReal) :

@[simp]

theorem EReal.add_eq_bot_iff {x : EReal} {y : EReal} :

x + y = ⊥ ↔ x = ⊥ ∨ y = ⊥

@[simp]

theorem EReal.bot_lt_add_iff {x : EReal} {y : EReal} :

⊥ < x + y ↔ ⊥ < x ∧ ⊥ < y

@[simp]

theorem EReal.top_add_top :

⊤ + ⊤ = ⊤

@[simp]

theorem EReal.top_add_coe (x : ℝ) :

⊤ + ↑x = ⊤

@[simp]

theorem EReal.top_add_of_ne_bot {x : EReal} (h : x ≠ ⊥) :

For any extended real number x which is not ⊥, the sum of ⊤ and x is equal to ⊤.

theorem EReal.top_add_iff_ne_bot {x : EReal} :

⊤ + x = ⊤ ↔ x ≠ ⊥

For any extended real number x, the sum of ⊤ and x is equal to ⊤ if and only if x is not ⊥.

@[simp]

theorem EReal.add_top_of_ne_bot {x : EReal} (h : x ≠ ⊥) :

For any extended real number x which is not ⊥, the sum of x and ⊤ is equal to ⊤.

theorem EReal.add_top_iff_ne_bot {x : EReal} :

x + ⊤ = ⊤ ↔ x ≠ ⊥

For any extended real number x, the sum of x and ⊤ is equal to ⊤ if and only if x is not ⊥.

theorem EReal.add_pos {a : EReal} {b : EReal} (ha : 0 < a) (hb : 0 < b) :

0 < a + b

For any two extended real numbers a and b, if both a and b are greater than 0, then their sum is also greater than 0.

@[simp]

theorem EReal.coe_add_top (x : ℝ) :

↑x + ⊤ = ⊤

theorem EReal.toReal_add {x : EReal} {y : EReal} (hx : x ≠ ⊤) (h'x : x ≠ ⊥) (hy : y ≠ ⊤) (h'y : y ≠ ⊥) :

(x + y).toReal = x.toReal + y.toReal

theorem EReal.addLECancellable_coe (x : ℝ) :

AddLECancellable ↑x

theorem EReal.add_lt_add_right_coe {x : EReal} {y : EReal} (h : x < y) (z : ℝ) :

x + ↑z < y + ↑z

theorem EReal.add_lt_add_left_coe {x : EReal} {y : EReal} (h : x < y) (z : ℝ) :

↑z + x < ↑z + y

theorem EReal.add_lt_add {x : EReal} {y : EReal} {z : EReal} {t : EReal} (h1 : x < y) (h2 : z < t) :

x + z < y + t

theorem EReal.add_lt_add_of_lt_of_le' {x : EReal} {y : EReal} {z : EReal} {t : EReal} (h : x < y) (h' : z ≤ t) (hbot : t ≠ ⊥) (htop : t = ⊤ → z = ⊤ → x = ⊥) :

x + z < y + t

theorem EReal.add_lt_add_of_lt_of_le {x : EReal} {y : EReal} {z : EReal} {t : EReal} (h : x < y) (h' : z ≤ t) (hz : z ≠ ⊥) (ht : t ≠ ⊤) :

x + z < y + t

See also EReal.add_lt_add_of_lt_of_le' for a version with weaker but less convenient assumptions.

theorem EReal.add_lt_top {x : EReal} {y : EReal} (hx : x ≠ ⊤) (hy : y ≠ ⊤) :

x + y < ⊤

instance EReal.instLinearOrderedAddCommMonoidWithTopOrderDual :

LinearOrderedAddCommMonoidWithTop EReal ᵒᵈ

We do not have a notion of LinearOrderedAddCommMonoidWithBot but we can at least make the order dual of the extended reals into a LinearOrderedAddCommMonoidWithTop.

Equations

EReal.instLinearOrderedAddCommMonoidWithTopOrderDual = LinearOrderedAddCommMonoidWithTop.mk EReal.instLinearOrderedAddCommMonoidWithTopOrderDual.proof_2

Negation #

def EReal.neg :

EReal → EReal

negation on EReal

Equations

x.neg = match x with | none => ⊤ | some none => ⊥ | some (some x) => ↑(-x)

Instances For

instance EReal.instNeg :

Equations

EReal.instNeg = { neg := EReal.neg }

instance EReal.instSubNegZeroMonoid :

SubNegZeroMonoid EReal

Equations

EReal.instSubNegZeroMonoid = SubNegZeroMonoid.mk EReal.instSubNegZeroMonoid.proof_5

@[simp]

theorem EReal.neg_top :

@[simp]

theorem EReal.neg_bot :

@[simp]

theorem EReal.coe_neg (x : ℝ) :

↑(-x) = -↑x

@[simp]

theorem EReal.coe_sub (x : ℝ) (y : ℝ) :

↑(x - y) = ↑x - ↑y

theorem EReal.coe_zsmul (n : ℤ) (x : ℝ) :

↑(n • x) = n • ↑x

instance EReal.instInvolutiveNeg :

InvolutiveNeg EReal

Equations

EReal.instInvolutiveNeg = InvolutiveNeg.mk EReal.instInvolutiveNeg.proof_1

@[simp]

theorem EReal.toReal_neg {a : EReal} :

(-a).toReal = -a.toReal

@[simp]

theorem EReal.neg_eq_top_iff {x : EReal} :

-x = ⊤ ↔ x = ⊥

@[simp]

theorem EReal.neg_eq_bot_iff {x : EReal} :

-x = ⊥ ↔ x = ⊤

@[simp]

theorem EReal.neg_eq_zero_iff {x : EReal} :

-x = 0 ↔ x = 0

theorem EReal.neg_strictAnti :

StrictAnti fun (x : EReal) => -x

@[simp]

theorem EReal.neg_le_neg_iff {a : EReal} {b : EReal} :

-a ≤ -b ↔ b ≤ a

@[simp]

theorem EReal.neg_lt_neg_iff {a : EReal} {b : EReal} :

-a < -b ↔ b < a

theorem EReal.neg_le {a : EReal} {b : EReal} :

-a ≤ b ↔ -b ≤ a

-a ≤ b ↔ -b ≤ a on EReal.

theorem EReal.neg_le_of_neg_le {a : EReal} {b : EReal} (h : -a ≤ b) :

-b ≤ a

if -a ≤ b then -b ≤ a on EReal.

theorem EReal.le_neg_of_le_neg {a : EReal} {b : EReal} (h : a ≤ -b) :

b ≤ -a

a ≤ -b → b ≤ -a on ereal

def EReal.negOrderIso :

EReal ≃o EReal ᵒᵈ

Negation as an order reversing isomorphism on EReal.

Equations

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

Instances For

theorem EReal.neg_lt_iff_neg_lt {a : EReal} {b : EReal} :

-a < b ↔ -b < a

theorem EReal.neg_lt_of_neg_lt {a : EReal} {b : EReal} (h : -a < b) :

-b < a

theorem EReal.neg_add {x : EReal} {y : EReal} (h1 : x ≠ ⊥ ∨ y ≠ ⊤) (h2 : x ≠ ⊤ ∨ y ≠ ⊥) :

-(x + y) = -x - y

theorem EReal.neg_sub {x : EReal} {y : EReal} (h1 : x ≠ ⊥ ∨ y ≠ ⊥) (h2 : x ≠ ⊤ ∨ y ≠ ⊤) :

-(x - y) = -x + y

Subtraction #

Subtraction on EReal is defined by x - y = x + (-y). Since addition is badly behaved at some points, so is subtraction. There is no standard algebraic typeclass involving subtraction that is registered on EReal, beyond SubNegZeroMonoid, because of this bad behavior.

@[simp]

theorem EReal.bot_sub (x : EReal) :

@[simp]

theorem EReal.sub_top (x : EReal) :

@[simp]

theorem EReal.top_sub_bot :

⊤ - ⊥ = ⊤

@[simp]

theorem EReal.top_sub_coe (x : ℝ) :

⊤ - ↑x = ⊤

@[simp]

theorem EReal.coe_sub_bot (x : ℝ) :

↑x - ⊥ = ⊤

theorem EReal.sub_le_sub {x : EReal} {y : EReal} {z : EReal} {t : EReal} (h : x ≤ y) (h' : t ≤ z) :

x - z ≤ y - t

theorem EReal.sub_lt_sub_of_lt_of_le {x : EReal} {y : EReal} {z : EReal} {t : EReal} (h : x < y) (h' : z ≤ t) (hz : z ≠ ⊥) (ht : t ≠ ⊤) :

x - t < y - z

theorem EReal.coe_real_ereal_eq_coe_toNNReal_sub_coe_toNNReal (x : ℝ) :

↑x = ↑↑x.toNNReal - ↑↑(-x).toNNReal

theorem EReal.toReal_sub {x : EReal} {y : EReal} (hx : x ≠ ⊤) (h'x : x ≠ ⊥) (hy : y ≠ ⊤) (h'y : y ≠ ⊥) :

(x - y).toReal = x.toReal - y.toReal

theorem EReal.add_sub_cancel_right {a : EReal} {b : ℝ} :

a + ↑b - ↑b = a

Multiplication #

@[simp]

theorem EReal.top_mul_top :

⊤ * ⊤ = ⊤

@[simp]

theorem EReal.top_mul_bot :

⊤ * ⊥ = ⊥

@[simp]

theorem EReal.bot_mul_top :

⊥ * ⊤ = ⊥

@[simp]

theorem EReal.bot_mul_bot :

⊥ * ⊥ = ⊤

theorem EReal.coe_mul_top_of_pos {x : ℝ} (h : 0 < x) :

↑x * ⊤ = ⊤

theorem EReal.coe_mul_top_of_neg {x : ℝ} (h : x < 0) :

↑x * ⊤ = ⊥

theorem EReal.top_mul_coe_of_pos {x : ℝ} (h : 0 < x) :

⊤ * ↑x = ⊤

theorem EReal.top_mul_coe_of_neg {x : ℝ} (h : x < 0) :

⊤ * ↑x = ⊥

theorem EReal.mul_top_of_pos {x : EReal} :

0 < x → x * ⊤ = ⊤

theorem EReal.mul_top_of_neg {x : EReal} :

x < 0 → x * ⊤ = ⊥

theorem EReal.top_mul_of_pos {x : EReal} (h : 0 < x) :

theorem EReal.mul_pos {a : EReal} {b : EReal} (ha : 0 < a) (hb : 0 < b) :

0 < a * b

The product of two positive extended real numbers is positive.

theorem EReal.top_mul_of_neg {x : EReal} (h : x < 0) :

theorem EReal.coe_mul_bot_of_pos {x : ℝ} (h : 0 < x) :

↑x * ⊥ = ⊥

theorem EReal.coe_mul_bot_of_neg {x : ℝ} (h : x < 0) :

↑x * ⊥ = ⊤

theorem EReal.bot_mul_coe_of_pos {x : ℝ} (h : 0 < x) :

⊥ * ↑x = ⊥

theorem EReal.bot_mul_coe_of_neg {x : ℝ} (h : x < 0) :

⊥ * ↑x = ⊤

theorem EReal.mul_bot_of_pos {x : EReal} :

0 < x → x * ⊥ = ⊥

theorem EReal.mul_bot_of_neg {x : EReal} :

x < 0 → x * ⊥ = ⊤

theorem EReal.bot_mul_of_pos {x : EReal} (h : 0 < x) :

theorem EReal.bot_mul_of_neg {x : EReal} (h : x < 0) :

theorem EReal.toReal_mul {x : EReal} {y : EReal} :

(x * y).toReal = x.toReal * y.toReal

theorem EReal.induction₂_neg_left {P : EReal → EReal → Prop} (neg_left : ∀ {x y : EReal}, P x y → P (-x) y) (top_top : P ⊤ ⊤) (top_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x) (top_zero : P ⊤ 0) (top_neg : ∀ x < 0, P ⊤ ↑x) (top_bot : P ⊤ ⊥) (zero_top : P 0 ⊤) (zero_bot : P 0 ⊥) (pos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤) (pos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥) (coe_coe : ∀ (x y : ℝ), P ↑x ↑y) (x : EReal) (y : EReal) :

P x y

Induct on two ereals by performing case splits on the sign of one whenever the other is infinite. This version eliminates some cases by assuming that P x y implies P (-x) y for all x, y.

theorem EReal.induction₂_symm_neg {P : EReal → EReal → Prop} (symm : ∀ {x y : EReal}, P x y → P y x) (neg_left : ∀ {x y : EReal}, P x y → P (-x) y) (top_top : P ⊤ ⊤) (top_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x) (top_zero : P ⊤ 0) (coe_coe : ∀ (x y : ℝ), P ↑x ↑y) (x : EReal) (y : EReal) :

P x y

Induct on two ereals by performing case splits on the sign of one whenever the other is infinite. This version eliminates some cases by assuming that P is symmetric and P x y implies P (-x) y for all x, y.

theorem EReal.neg_mul (x : EReal) (y : EReal) :

-x * y = -(x * y)

instance EReal.instHasDistribNeg :

HasDistribNeg EReal

Equations

EReal.instHasDistribNeg = HasDistribNeg.mk EReal.neg_mul EReal.instHasDistribNeg.proof_1

theorem EReal.right_distrib_of_nonneg {a : EReal} {b : EReal} {c : EReal} (ha : 0 ≤ a) (hb : 0 ≤ b) :

(a + b) * c = a * c + b * c

theorem EReal.left_distrib_of_nonneg {a : EReal} {b : EReal} {c : EReal} (ha : 0 ≤ a) (hb : 0 ≤ b) :

c * (a + b) = c * a + c * b

theorem EReal.le_iff_le_forall_real_gt (x : EReal) (y : EReal) :

(∀ (z : ℝ), x < ↑z → y ≤ ↑z) ↔ y ≤ x

theorem EReal.ge_iff_le_forall_real_lt (x : EReal) (y : EReal) :

(∀ (z : ℝ), ↑z < y → ↑z ≤ x) ↔ y ≤ x

Absolute value #

def EReal.abs :

EReal → ENNReal

The absolute value from EReal to ℝ≥0∞, mapping ⊥ and ⊤ to ⊤ and a real x to |x|.

Equations

x.abs = match x with | none => ⊤ | some none => ⊤ | some (some x) => ENNReal.ofReal |x|

Instances For

@[simp]

theorem EReal.abs_top :

@[simp]

theorem EReal.abs_bot :

theorem EReal.abs_def (x : ℝ) :

(↑x).abs = ENNReal.ofReal |x|

theorem EReal.abs_coe_lt_top (x : ℝ) :

(↑x).abs < ⊤

@[simp]

theorem EReal.abs_eq_zero_iff {x : EReal} :

x.abs = 0 ↔ x = 0

@[simp]

theorem EReal.abs_zero :

EReal.abs 0 = 0

@[simp]

theorem EReal.coe_abs (x : ℝ) :

↑(↑x).abs = ↑|x|

@[simp]

theorem EReal.abs_neg (x : EReal) :

(-x).abs = x.abs

@[simp]

theorem EReal.abs_mul (x : EReal) (y : EReal) :

(x * y).abs = x.abs * y.abs

Sign #

theorem EReal.sign_top :

SignType.sign ⊤ = 1

theorem EReal.sign_bot :

SignType.sign ⊥ = -1

@[simp]

theorem EReal.sign_coe (x : ℝ) :

SignType.sign ↑x = SignType.sign x

@[simp]

theorem EReal.coe_coe_sign (x : SignType) :

↑↑x = ↑x

@[simp]

theorem EReal.sign_neg (x : EReal) :

SignType.sign (-x) = -SignType.sign x

@[simp]

theorem EReal.sign_mul (x : EReal) (y : EReal) :

SignType.sign (x * y) = SignType.sign x * SignType.sign y

@[simp]

theorem EReal.sign_mul_abs (x : EReal) :

↑(SignType.sign x) * ↑x.abs = x

@[simp]

theorem EReal.abs_mul_sign (x : EReal) :

↑x.abs * ↑(SignType.sign x) = x

theorem EReal.sign_eq_and_abs_eq_iff_eq {x : EReal} {y : EReal} :

x.abs = y.abs ∧ SignType.sign x = SignType.sign y ↔ x = y

theorem EReal.le_iff_sign {x : EReal} {y : EReal} :

x ≤ y ↔ SignType.sign x < SignType.sign y ∨ SignType.sign x = SignType.neg ∧ SignType.sign y = SignType.neg ∧ y.abs ≤ x.abs ∨ SignType.sign x = SignType.zero ∧ SignType.sign y = SignType.zero ∨ SignType.sign x = SignType.pos ∧ SignType.sign y = SignType.pos ∧ x.abs ≤ y.abs

instance EReal.instCommMonoidWithZero :

CommMonoidWithZero EReal

Equations

EReal.instCommMonoidWithZero = let __src := inferInstanceAs (MulZeroOneClass EReal); CommMonoidWithZero.mk ⋯ ⋯

instance EReal.instPosMulMono :

PosMulMono EReal

Equations

EReal.instPosMulMono = ⋯

instance EReal.instMulPosMono :

MulPosMono EReal

Equations

EReal.instMulPosMono = ⋯

instance EReal.instPosMulReflectLT :

PosMulReflectLT EReal

Equations

EReal.instPosMulReflectLT = ⋯

instance EReal.instMulPosReflectLT :

MulPosReflectLT EReal

Equations

EReal.instMulPosReflectLT = ⋯

@[simp]

theorem EReal.coe_pow (x : ℝ) (n : ℕ) :

↑(x ^ n) = ↑x ^ n

@[simp]

theorem EReal.coe_ennreal_pow (x : ENNReal) (n : ℕ) :

↑(x ^ n) = ↑x ^ n

Min and Max #

theorem EReal.min_neg_neg (x : EReal) (y : EReal) :

min (-x) (-y) = -max x y

theorem EReal.max_neg_neg (x : EReal) (y : EReal) :

max (-x) (-y) = -min x y

Inverse #

def EReal.inv :

EReal → EReal

Multiplicative inverse of an EReal. We choose 0⁻¹ = 0 to guarantee several good properties, for instance (a * b)⁻¹ = a⁻¹ * b⁻¹.

Equations

x.inv = match x with | none => 0 | some none => 0 | some (some x) => ↑x⁻¹

Instances For

instance EReal.instInv :

Equations

EReal.instInv = { inv := EReal.inv }

noncomputable instance EReal.instDivInvMonoid :

DivInvMonoid EReal

Equations

EReal.instDivInvMonoid = DivInvMonoid.mk EReal.instDivInvMonoid.proof_1 zpowRec EReal.instDivInvMonoid.proof_2 EReal.instDivInvMonoid.proof_3 EReal.instDivInvMonoid.proof_4

@[simp]

theorem EReal.inv_bot :

@[simp]

theorem EReal.inv_top :

theorem EReal.coe_inv (x : ℝ) :

↑x⁻¹ = (↑x)⁻¹

@[simp]

theorem EReal.inv_zero :

noncomputable instance EReal.instDivInvOneMonoid :

DivInvOneMonoid EReal

Equations

EReal.instDivInvOneMonoid = DivInvOneMonoid.mk EReal.instDivInvOneMonoid.proof_1

theorem EReal.inv_neg (a : EReal) :

(-a)⁻¹ = -a⁻¹

theorem EReal.inv_inv {a : EReal} (h : a ≠ ⊥) (h' : a ≠ ⊤) :

a⁻¹⁻¹ = a

theorem EReal.mul_inv (a : EReal) (b : EReal) :

(a * b)⁻¹ = a⁻¹ * b⁻¹

Inversion and Absolute Value #

theorem EReal.sign_mul_inv_abs (a : EReal) :

↑(SignType.sign a) * (↑a.abs)⁻¹ = a⁻¹

theorem EReal.sign_mul_inv_abs' (a : EReal) :

↑(SignType.sign a) * ↑a.abs⁻¹ = a⁻¹

Inversion and Positivity #

theorem EReal.inv_nonneg_of_nonneg {a : EReal} (h : 0 ≤ a) :

theorem EReal.inv_nonpos_of_nonpos {a : EReal} (h : a ≤ 0) :

theorem EReal.inv_pos_of_pos_ne_top {a : EReal} (h : 0 < a) (h' : a ≠ ⊤) :

theorem EReal.inv_neg_of_neg_ne_bot {a : EReal} (h : a < 0) (h' : a ≠ ⊥) :

Division #

theorem EReal.div_eq_inv_mul (a : EReal) (b : EReal) :

a / b = b⁻¹ * a

theorem EReal.coe_div (a : ℝ) (b : ℝ) :

↑(a / b) = ↑a / ↑b

theorem EReal.natCast_div_le (m : ℕ) (n : ℕ) :

↑(m / n) ≤ ↑m / ↑n

@[simp]

theorem EReal.div_bot {a : EReal} :

a / ⊥ = 0

@[simp]

theorem EReal.div_top {a : EReal} :

a / ⊤ = 0

@[simp]

theorem EReal.div_zero {a : EReal} :

a / 0 = 0

@[simp]

theorem EReal.zero_div {a : EReal} :

0 / a = 0

theorem EReal.top_div_of_pos_ne_top {a : EReal} (h : 0 < a) (h' : a ≠ ⊤) :

theorem EReal.top_div_of_neg_ne_bot {a : EReal} (h : a < 0) (h' : a ≠ ⊥) :

theorem EReal.bot_div_of_pos_ne_top {a : EReal} (h : 0 < a) (h' : a ≠ ⊤) :

theorem EReal.bot_div_of_neg_ne_bot {a : EReal} (h : a < 0) (h' : a ≠ ⊥) :

Division and Multiplication #

theorem EReal.div_self {a : EReal} (h₁ : a ≠ ⊥) (h₂ : a ≠ ⊤) (h₃ : a ≠ 0) :

a / a = 1

theorem EReal.mul_div (a : EReal) (b : EReal) (c : EReal) :

a * (b / c) = a * b / c

theorem EReal.mul_div_right (a : EReal) (b : EReal) (c : EReal) :

a / b * c = a * c / b

theorem EReal.div_div (a : EReal) (b : EReal) (c : EReal) :

a / b / c = a / (b * c)

theorem EReal.div_mul_cancel {a : EReal} {b : EReal} (h₁ : b ≠ ⊥) (h₂ : b ≠ ⊤) (h₃ : b ≠ 0) :

a / b * b = a

theorem EReal.mul_div_cancel {a : EReal} {b : EReal} (h₁ : b ≠ ⊥) (h₂ : b ≠ ⊤) (h₃ : b ≠ 0) :

b * (a / b) = a

theorem EReal.mul_div_mul_cancel {a : EReal} {b : EReal} {c : EReal} (h₁ : c ≠ ⊥) (h₂ : c ≠ ⊤) (h₃ : c ≠ 0) :

a * c / (b * c) = a / b

Division Distributivity #

theorem EReal.div_right_distrib_of_nonneg {a : EReal} {b : EReal} {c : EReal} (h : 0 ≤ a) (h' : 0 ≤ b) :

(a + b) / c = a / c + b / c

Division and Order s #

theorem EReal.monotone_div_right_of_nonneg {b : EReal} (h : 0 ≤ b) :

Monotone fun (a : EReal) => a / b

theorem EReal.div_le_div_right_of_nonneg {a : EReal} {a' : EReal} {b : EReal} (h : 0 ≤ b) (h' : a ≤ a') :

a / b ≤ a' / b

theorem EReal.strictMono_div_right_of_pos {b : EReal} (h : 0 < b) (h' : b ≠ ⊤) :

StrictMono fun (a : EReal) => a / b

theorem EReal.div_lt_div_right_of_pos {a : EReal} {a' : EReal} {b : EReal} (h₁ : 0 < b) (h₂ : b ≠ ⊤) (h₃ : a < a') :

a / b < a' / b

theorem EReal.antitone_div_right_of_nonpos {b : EReal} (h : b ≤ 0) :

Antitone fun (a : EReal) => a / b

theorem EReal.div_le_div_right_of_nonpos {a : EReal} {a' : EReal} {b : EReal} (h : b ≤ 0) (h' : a ≤ a') :

a' / b ≤ a / b

theorem EReal.strictAnti_div_right_of_neg {b : EReal} (h : b < 0) (h' : b ≠ ⊥) :

StrictAnti fun (a : EReal) => a / b

theorem EReal.div_lt_div_right_of_neg {a : EReal} {a' : EReal} {b : EReal} (h₁ : b < 0) (h₂ : b ≠ ⊥) (h₃ : a < a') :

a' / b < a / b

theorem EReal.le_div_iff_mul_le {a : EReal} {b : EReal} {c : EReal} (h : b > 0) (h' : b ≠ ⊤) :

a ≤ c / b ↔ a * b ≤ c

theorem EReal.div_le_iff_le_mul {a : EReal} {b : EReal} {c : EReal} (h : 0 < b) (h' : b ≠ ⊤) :

a / b ≤ c ↔ a ≤ b * c

theorem EReal.div_nonneg {a : EReal} {b : EReal} (h : 0 ≤ a) (h' : 0 ≤ b) :

0 ≤ a / b

theorem EReal.div_nonpos_of_nonpos_of_nonneg {a : EReal} {b : EReal} (h : a ≤ 0) (h' : 0 ≤ b) :

a / b ≤ 0

theorem EReal.div_nonpos_of_nonneg_of_nonpos {a : EReal} {b : EReal} (h : 0 ≤ a) (h' : b ≤ 0) :

a / b ≤ 0

theorem EReal.div_nonneg_of_nonpos_of_nonpos {a : EReal} {b : EReal} (h : a ≤ 0) (h' : b ≤ 0) :

0 ≤ a / b