mathlib docs: data.real.nnreal

data.real.nnreal

nnreal.add_div'

nnreal.add_halves

nnreal.add_sub_cancel

nnreal.add_sub_cancel'

nnreal.archimedean

nnreal.bdd_above_coe

nnreal.bdd_below_coe

nnreal.bot_eq_zero

nnreal.can_lift

nnreal.canonically_ordered_add_monoid

nnreal.canonically_ordered_comm_semiring

nnreal.coe_bit0

nnreal.coe_bit1

nnreal.coe_eq_zero

nnreal.coe_indicator

nnreal.coe_le_coe

nnreal.coe_list_prod

nnreal.coe_list_sum

nnreal.coe_lt_coe

nnreal.coe_mono

nnreal.coe_multiset_prod

nnreal.coe_multiset_sum

nnreal.coe_nat_cast

nnreal.coe_ne_zero

nnreal.coe_nonneg

nnreal.coe_of_real

nnreal.coe_prod

nnreal.coe_to_real_hom

nnreal.coe_zero

nnreal.comm_group_with_zero

nnreal.comm_semiring

nnreal.conditionally_complete_linear_order_bot

nnreal.decidable_linear_order

nnreal.densely_ordered

nnreal.distrib_lattice

nnreal.div_add'

nnreal.div_add_div

nnreal.div_add_div_same

nnreal.div_div_eq_div_mul

nnreal.div_div_eq_mul_div

nnreal.div_eq_div_iff

nnreal.div_eq_iff

nnreal.div_eq_mul_one_div

nnreal.div_le_iff

nnreal.div_lt_iff

nnreal.div_lt_one_of_lt

nnreal.div_mul_eq_mul_div

nnreal.div_self

nnreal.div_self_le

nnreal.eq_div_iff

nnreal.half_lt_self

nnreal.half_pos

nnreal.has_zero

nnreal.inhabited

nnreal.inv_eq_one_div

nnreal.inv_eq_zero

nnreal.inv_le_of_le_mul

nnreal.inv_mul_cancel

nnreal.inv_zero

nnreal.le_coe_of_real

nnreal.le_div_iff_mul_le

nnreal.le_inv_iff_mul_le

nnreal.le_of_forall_epsilon_le

nnreal.le_of_forall_lt_one_mul_lt

nnreal.le_of_real_iff_coe_le

nnreal.linear_ordered_comm_group_with_zero

nnreal.linear_ordered_semiring

nnreal.lt_iff_exists_rat_btwn

nnreal.lt_inv_iff_mul_lt

nnreal.lt_of_real_iff_coe_lt

nnreal.lt_sub_iff_add_lt

nnreal.mul_div_assoc'

nnreal.mul_div_cancel

nnreal.mul_div_cancel'

nnreal.mul_eq_mul_left

nnreal.mul_finset_sup

nnreal.mul_inv_cancel

nnreal.mul_le_iff_le_inv

nnreal.mul_ne_zero'

nnreal.no_top_order

nnreal.nsmul_coe

nnreal.of_real_add

nnreal.of_real_add_le

nnreal.of_real_add_of_real

nnreal.of_real_coe

nnreal.of_real_eq_zero

nnreal.of_real_le_iff_le_coe

nnreal.of_real_le_of_real

nnreal.of_real_le_of_real_iff

nnreal.of_real_lt_iff_lt_coe

nnreal.of_real_lt_of_real_iff

nnreal.of_real_lt_of_real_iff'

nnreal.of_real_lt_of_real_iff_of_nonneg

nnreal.of_real_mono

nnreal.of_real_mul

nnreal.of_real_of_nonpos

nnreal.of_real_one

nnreal.of_real_pos

nnreal.of_real_zero

nnreal.one_div_div

nnreal.order_bot

nnreal.pow_eq_zero

nnreal.pow_ne_zero

nnreal.semilattice_inf_bot

nnreal.semilattice_sup_bot

nnreal.sub_add_cancel_of_le

nnreal.sub_eq_zero

nnreal.sub_le_iff_le_add

nnreal.sub_le_self

nnreal.sub_lt_self

nnreal.sub_self

nnreal.sub_sub_cancel_of_le

nnreal.sub_zero

nnreal.to_real_hom

nnreal.two_inv_lt_one

nnreal.val_eq_coe

nnreal.zero_le_coe

def nnreal :

Type

Nonnegative real numbers.

Equations

nnreal = {r // 0 ≤ r}

@[instance]

def nnreal.has_coe :

has_coe nnreal ℝ

Equations

nnreal.has_coe = {coe := subtype.val (λ (r : ℝ), 0 ≤ r)}

@[simp]

theorem nnreal.val_eq_coe (n : nnreal) :

@[instance]

def nnreal.can_lift :

can_lift ℝ nnreal

Equations

nnreal.can_lift = {coe := coe coe_to_lift, cond := λ (r : ℝ), 0 ≤ r, prf := nnreal.can_lift._proof_1}

theorem nnreal.eq {n m : nnreal} :

↑n = ↑m → n = m

theorem nnreal.eq_iff {n m : nnreal} :

↑n = ↑m ↔ n = m

theorem nnreal.ne_iff {x y : nnreal} :

↑x ≠ ↑y ↔ x ≠ y

def nnreal.of_real :

Equations

nnreal.of_real r = ⟨max r 0, _⟩

theorem nnreal.coe_of_real (r : ℝ) :

0 ≤ r → ↑(nnreal.of_real r) = r

theorem nnreal.le_coe_of_real (r : ℝ) :

r ≤ ↑(nnreal.of_real r)

theorem nnreal.coe_nonneg (r : nnreal) :

theorem nnreal.coe_mk (a : ℝ) (ha : 0 ≤ a) :

↑⟨a, ha⟩ = a

@[instance]

def nnreal.has_zero :

has_zero nnreal

Equations

nnreal.has_zero = {zero := ⟨0, nnreal.has_zero._proof_1⟩}

@[instance]

def nnreal.has_one :

Equations

nnreal.has_one = {one := ⟨1, _⟩}

@[instance]

def nnreal.has_add :

Equations

nnreal.has_add = {add := λ (a b : nnreal), ⟨↑a + ↑b, _⟩}

@[instance]

def nnreal.has_sub :

Equations

nnreal.has_sub = {sub := λ (a b : nnreal), nnreal.of_real (↑a - ↑b)}

@[instance]

def nnreal.has_mul :

Equations

nnreal.has_mul = {mul := λ (a b : nnreal), ⟨↑a * ↑b, _⟩}

@[instance]

def nnreal.has_inv :

Equations

nnreal.has_inv = {inv := λ (a : nnreal), ⟨(a.val)⁻¹, _⟩}

@[instance]

def nnreal.has_le :

Equations

nnreal.has_le = {le := λ (r s : nnreal), ↑r ≤ ↑s}

@[instance]

def nnreal.has_bot :

Equations

nnreal.has_bot = {bot := 0}

@[instance]

def nnreal.inhabited :

inhabited nnreal

Equations

nnreal.inhabited = {default := 0}

@[simp]

theorem nnreal.coe_eq {r₁ r₂ : nnreal} :

↑r₁ = ↑r₂ ↔ r₁ = r₂

@[simp]

theorem nnreal.coe_zero :

↑0 = 0

@[simp]

theorem nnreal.coe_one :

↑1 = 1

@[simp]

theorem nnreal.coe_add (r₁ r₂ : nnreal) :

↑(r₁ + r₂) = ↑r₁ + ↑r₂

@[simp]

theorem nnreal.coe_mul (r₁ r₂ : nnreal) :

↑(r₁ * r₂) = ↑r₁ * ↑r₂

@[simp]

theorem nnreal.coe_inv (r : nnreal) :

↑r⁻¹ = (↑r)⁻¹

@[simp]

theorem nnreal.coe_bit0 (r : nnreal) :

↑(bit0 r) = bit0 ↑r

@[simp]

theorem nnreal.coe_bit1 (r : nnreal) :

↑(bit1 r) = bit1 ↑r

@[simp]

theorem nnreal.coe_sub {r₁ r₂ : nnreal} :

r₂ ≤ r₁ → ↑(r₁ - r₂) = ↑r₁ - ↑r₂

@[simp]

theorem nnreal.coe_eq_zero (r : nnreal) :

↑r = 0 ↔ r = 0

theorem nnreal.coe_ne_zero {r : nnreal} :

↑r ≠ 0 ↔ r ≠ 0

@[instance]

def nnreal.comm_semiring :

comm_semiring nnreal

Equations

nnreal.comm_semiring = {add := has_add.add nnreal.has_add, add_assoc := nnreal.comm_semiring._proof_1, zero := 0, zero_add := nnreal.comm_semiring._proof_2, add_zero := nnreal.comm_semiring._proof_3, add_comm := nnreal.comm_semiring._proof_4, mul := has_mul.mul nnreal.has_mul, mul_assoc := nnreal.comm_semiring._proof_5, one := 1, one_mul := nnreal.comm_semiring._proof_6, mul_one := nnreal.comm_semiring._proof_7, zero_mul := nnreal.comm_semiring._proof_8, mul_zero := nnreal.comm_semiring._proof_9, left_distrib := nnreal.comm_semiring._proof_10, right_distrib := nnreal.comm_semiring._proof_11, mul_comm := nnreal.comm_semiring._proof_12}

def nnreal.to_real_hom :

nnreal →+* ℝ

Coercion ℝ≥0 → ℝ as a ring_hom.

Equations

nnreal.to_real_hom = {to_fun := coe coe_to_lift, map_one' := nnreal.coe_one, map_mul' := nnreal.coe_mul, map_zero' := nnreal.coe_zero, map_add' := nnreal.coe_add}

@[simp]

theorem nnreal.coe_to_real_hom :

⇑nnreal.to_real_hom = coe

@[instance]

def nnreal.comm_group_with_zero :

comm_group_with_zero nnreal

Equations

nnreal.comm_group_with_zero = {mul := comm_semiring.mul nnreal.comm_semiring, mul_assoc := _, one := comm_semiring.one nnreal.comm_semiring, one_mul := _, mul_one := _, mul_comm := _, zero := comm_semiring.zero nnreal.comm_semiring, zero_mul := _, mul_zero := _, inv := has_inv.inv nnreal.has_inv, exists_pair_ne := nnreal.comm_group_with_zero._proof_1, inv_zero := nnreal.comm_group_with_zero._proof_2, mul_inv_cancel := nnreal.comm_group_with_zero._proof_3}

@[simp]

theorem nnreal.coe_indicator {α : Type u_1} (s : set α) (f : α → nnreal) (a : α) :

↑(s.indicator f a) = s.indicator (λ (x : α), ↑(f x)) a

@[simp]

theorem nnreal.coe_div (r₁ r₂ : nnreal) :

↑(r₁ / r₂) = ↑r₁ / ↑r₂

theorem nnreal.coe_pow (r : nnreal) (n : ℕ) :

↑(r ^ n) = ↑r ^ n

theorem nnreal.coe_list_sum (l : list nnreal) :

↑(l.sum) = (list.map coe l).sum

theorem nnreal.coe_list_prod (l : list nnreal) :

↑(l.prod) = (list.map coe l).prod

theorem nnreal.coe_multiset_sum (s : multiset nnreal) :

↑(s.sum) = (multiset.map coe s).sum

theorem nnreal.coe_multiset_prod (s : multiset nnreal) :

↑(s.prod) = (multiset.map coe s).prod

theorem nnreal.coe_sum {α : Type u_1} {s : finset α} {f : α → nnreal} :

↑(s.sum (λ (a : α), f a)) = s.sum (λ (a : α), ↑(f a))

theorem nnreal.coe_prod {α : Type u_1} {s : finset α} {f : α → nnreal} :

↑(s.prod (λ (a : α), f a)) = s.prod (λ (a : α), ↑(f a))

theorem nnreal.nsmul_coe (r : nnreal) (n : ℕ) :

↑(n •ℕ r) = n •ℕ ↑r

@[simp]

theorem nnreal.coe_nat_cast (n : ℕ) :

@[instance]

def nnreal.decidable_linear_order :

decidable_linear_order nnreal

Equations

nnreal.decidable_linear_order = decidable_linear_order.lift coe nnreal.decidable_linear_order._proof_1

theorem nnreal.coe_le_coe {r₁ r₂ : nnreal} :

↑r₁ ≤ ↑r₂ ↔ r₁ ≤ r₂

theorem nnreal.coe_lt_coe {r₁ r₂ : nnreal} :

↑r₁ < ↑r₂ ↔ r₁ < r₂

theorem nnreal.coe_pos {r : nnreal} :

0 < ↑r ↔ 0 < r

theorem nnreal.coe_mono :

theorem nnreal.of_real_mono :

monotone nnreal.of_real

@[simp]

theorem nnreal.of_real_coe {r : nnreal} :

nnreal.of_real ↑r = r

def nnreal.gi :

galois_insertion nnreal.of_real coe

nnreal.of_real and coe : ℝ≥0 → ℝ form a Galois insertion.

Equations

nnreal.gi = galois_insertion.monotone_intro nnreal.coe_mono nnreal.of_real_mono nnreal.le_coe_of_real nnreal.gi._proof_1

@[instance]

def nnreal.order_bot :

order_bot nnreal

Equations

nnreal.order_bot = {bot := ⊥, le := decidable_linear_order.le nnreal.decidable_linear_order, lt := decidable_linear_order.lt nnreal.decidable_linear_order, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, bot_le := nnreal.order_bot._proof_1}

@[instance]

def nnreal.canonically_ordered_add_monoid :

canonically_ordered_add_monoid nnreal

Equations

nnreal.canonically_ordered_add_monoid = {add := comm_semiring.add nnreal.comm_semiring, add_assoc := _, zero := comm_semiring.zero nnreal.comm_semiring, zero_add := _, add_zero := _, add_comm := _, le := order_bot.le nnreal.order_bot, lt := order_bot.lt nnreal.order_bot, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, add_le_add_left := nnreal.canonically_ordered_add_monoid._proof_1, lt_of_add_lt_add_left := nnreal.canonically_ordered_add_monoid._proof_2, bot := order_bot.bot nnreal.order_bot, bot_le := _, le_iff_exists_add := nnreal.canonically_ordered_add_monoid._proof_3}

@[instance]

def nnreal.distrib_lattice :

distrib_lattice nnreal

Equations

nnreal.distrib_lattice = distrib_lattice_of_decidable_linear_order

@[instance]

def nnreal.semilattice_inf_bot :

semilattice_inf_bot nnreal

Equations

nnreal.semilattice_inf_bot = {bot := order_bot.bot nnreal.order_bot, le := order_bot.le nnreal.order_bot, lt := order_bot.lt nnreal.order_bot, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, bot_le := _, inf := distrib_lattice.inf nnreal.distrib_lattice, inf_le_left := _, inf_le_right := _, le_inf := _}

@[instance]

def nnreal.semilattice_sup_bot :

semilattice_sup_bot nnreal

Equations

nnreal.semilattice_sup_bot = {bot := order_bot.bot nnreal.order_bot, le := order_bot.le nnreal.order_bot, lt := order_bot.lt nnreal.order_bot, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, bot_le := _, sup := distrib_lattice.sup nnreal.distrib_lattice, le_sup_left := _, le_sup_right := _, sup_le := _}

@[instance]

def nnreal.linear_ordered_semiring :

linear_ordered_semiring nnreal

Equations

nnreal.linear_ordered_semiring = {add := canonically_ordered_add_monoid.add nnreal.canonically_ordered_add_monoid, add_assoc := _, zero := canonically_ordered_add_monoid.zero nnreal.canonically_ordered_add_monoid, zero_add := _, add_zero := _, add_comm := _, mul := comm_semiring.mul nnreal.comm_semiring, mul_assoc := _, one := comm_semiring.one nnreal.comm_semiring, one_mul := _, mul_one := _, zero_mul := _, mul_zero := _, left_distrib := _, right_distrib := _, add_left_cancel := nnreal.linear_ordered_semiring._proof_1, add_right_cancel := nnreal.linear_ordered_semiring._proof_2, le := decidable_linear_order.le nnreal.decidable_linear_order, lt := decidable_linear_order.lt nnreal.decidable_linear_order, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, add_le_add_left := _, le_of_add_le_add_left := nnreal.linear_ordered_semiring._proof_3, mul_lt_mul_of_pos_left := nnreal.linear_ordered_semiring._proof_4, mul_lt_mul_of_pos_right := nnreal.linear_ordered_semiring._proof_5, le_total := _, zero_lt_one := _}

@[instance]

def nnreal.linear_ordered_comm_group_with_zero :

linear_ordered_comm_group_with_zero nnreal

Equations

nnreal.linear_ordered_comm_group_with_zero = {le := linear_ordered_semiring.le nnreal.linear_ordered_semiring, lt := linear_ordered_semiring.lt nnreal.linear_ordered_semiring, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_total := _, mul := linear_ordered_semiring.mul nnreal.linear_ordered_semiring, mul_assoc := _, one := linear_ordered_semiring.one nnreal.linear_ordered_semiring, one_mul := _, mul_one := _, mul_comm := _, zero := linear_ordered_semiring.zero nnreal.linear_ordered_semiring, zero_mul := _, mul_zero := _, inv := comm_group_with_zero.inv nnreal.comm_group_with_zero, exists_pair_ne := _, inv_zero := _, mul_inv_cancel := _, mul_le_mul_left := nnreal.linear_ordered_comm_group_with_zero._proof_1, zero_le_one := nnreal.linear_ordered_comm_group_with_zero._proof_2}

@[instance]

def nnreal.canonically_ordered_comm_semiring :

canonically_ordered_comm_semiring nnreal

Equations

nnreal.canonically_ordered_comm_semiring = {add := canonically_ordered_add_monoid.add nnreal.canonically_ordered_add_monoid, add_assoc := _, zero := canonically_ordered_add_monoid.zero nnreal.canonically_ordered_add_monoid, zero_add := _, add_zero := _, add_comm := _, le := canonically_ordered_add_monoid.le nnreal.canonically_ordered_add_monoid, lt := canonically_ordered_add_monoid.lt nnreal.canonically_ordered_add_monoid, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, add_le_add_left := _, lt_of_add_lt_add_left := _, bot := canonically_ordered_add_monoid.bot nnreal.canonically_ordered_add_monoid, bot_le := _, le_iff_exists_add := _, mul := comm_semiring.mul nnreal.comm_semiring, mul_assoc := _, one := comm_semiring.one nnreal.comm_semiring, one_mul := _, mul_one := _, zero_mul := _, mul_zero := _, left_distrib := _, right_distrib := _, mul_comm := _, exists_pair_ne := _, eq_zero_or_eq_zero_of_mul_eq_zero := nnreal.canonically_ordered_comm_semiring._proof_1}

@[instance]

def nnreal.densely_ordered :

densely_ordered nnreal

Equations

nnreal.densely_ordered = _
_ = _

@[instance]

def nnreal.no_top_order :

no_top_order nnreal

Equations

nnreal.no_top_order = _
_ = _

theorem nnreal.bdd_above_coe {s : set nnreal} :

bdd_above (coe '' s) ↔ bdd_above s

theorem nnreal.bdd_below_coe (s : set nnreal) :

bdd_below (coe '' s)

@[instance]

def nnreal.has_Sup :

Equations

nnreal.has_Sup = {Sup := λ (s : set nnreal), ⟨has_Sup.Sup (coe '' s), _⟩}

@[instance]

def nnreal.has_Inf :

Equations

nnreal.has_Inf = {Inf := λ (s : set nnreal), ⟨has_Inf.Inf (coe '' s), _⟩}

theorem nnreal.coe_Sup (s : set nnreal) :

↑(has_Sup.Sup s) = has_Sup.Sup (coe '' s)

theorem nnreal.coe_Inf (s : set nnreal) :

↑(has_Inf.Inf s) = has_Inf.Inf (coe '' s)

@[instance]

def nnreal.conditionally_complete_linear_order_bot :

conditionally_complete_linear_order_bot nnreal

Equations

nnreal.conditionally_complete_linear_order_bot = {sup := lattice.sup lattice_of_decidable_linear_order, le := linear_ordered_semiring.le nnreal.linear_ordered_semiring, lt := linear_ordered_semiring.lt nnreal.linear_ordered_semiring, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_sup_left := nnreal.conditionally_complete_linear_order_bot._proof_1, le_sup_right := nnreal.conditionally_complete_linear_order_bot._proof_2, sup_le := nnreal.conditionally_complete_linear_order_bot._proof_3, inf := lattice.inf lattice_of_decidable_linear_order, inf_le_left := nnreal.conditionally_complete_linear_order_bot._proof_4, inf_le_right := nnreal.conditionally_complete_linear_order_bot._proof_5, le_inf := nnreal.conditionally_complete_linear_order_bot._proof_6, Sup := has_Sup.Sup nnreal.has_Sup, Inf := has_Inf.Inf nnreal.has_Inf, le_cSup := nnreal.conditionally_complete_linear_order_bot._proof_7, cSup_le := nnreal.conditionally_complete_linear_order_bot._proof_8, cInf_le := nnreal.conditionally_complete_linear_order_bot._proof_9, le_cInf := nnreal.conditionally_complete_linear_order_bot._proof_10, le_total := _, decidable_le := id (λ (x y : nnreal), classical.dec (x ≤ y)), decidable_eq := decidable_linear_order.decidable_eq._default linear_ordered_semiring.le linear_ordered_semiring.lt linear_ordered_semiring.le_refl linear_ordered_semiring.le_trans linear_ordered_semiring.lt_iff_le_not_le linear_ordered_semiring.le_antisymm linear_ordered_semiring.le_total (id (λ (x y : nnreal), classical.dec (x ≤ y))), decidable_lt := decidable_linear_order.decidable_lt._default linear_ordered_semiring.le linear_ordered_semiring.lt linear_ordered_semiring.le_refl linear_ordered_semiring.le_trans linear_ordered_semiring.lt_iff_le_not_le linear_ordered_semiring.le_antisymm linear_ordered_semiring.le_total (id (λ (x y : nnreal), classical.dec (x ≤ y))), bot := order_bot.bot nnreal.order_bot, bot_le := _, cSup_empty := nnreal.conditionally_complete_linear_order_bot._proof_11}

@[instance]

def nnreal.archimedean :

archimedean nnreal

Equations

nnreal.archimedean = _
_ = _

theorem nnreal.le_of_forall_epsilon_le {a b : nnreal} :

(∀ (ε : nnreal), 0 < ε → a ≤ b + ε) → a ≤ b

theorem nnreal.lt_iff_exists_rat_btwn (a b : nnreal) :

a < b ↔ ∃ (q : ℚ), 0 ≤ q ∧ a < nnreal.of_real ↑q ∧ nnreal.of_real ↑q < b

theorem nnreal.bot_eq_zero :

theorem nnreal.mul_sup (a b c : nnreal) :

a * (b ⊔ c) = a * b ⊔ a * c

theorem nnreal.mul_finset_sup {α : Type u_1} {f : α → nnreal} {s : finset α} (r : nnreal) :

r * s.sup f = s.sup (λ (a : α), r * f a)

@[simp]

theorem nnreal.coe_max (x y : nnreal) :

↑(max x y) = max ↑x ↑y

@[simp]

theorem nnreal.coe_min (x y : nnreal) :

↑(min x y) = min ↑x ↑y

@[simp]

theorem nnreal.zero_le_coe {q : nnreal} :

@[simp]

theorem nnreal.of_real_zero :

nnreal.of_real 0 = 0

@[simp]

theorem nnreal.of_real_one :

nnreal.of_real 1 = 1

@[simp]

theorem nnreal.of_real_pos {r : ℝ} :

0 < nnreal.of_real r ↔ 0 < r

@[simp]

theorem nnreal.of_real_eq_zero {r : ℝ} :

nnreal.of_real r = 0 ↔ r ≤ 0

theorem nnreal.of_real_of_nonpos {r : ℝ} :

r ≤ 0 → nnreal.of_real r = 0

@[simp]

theorem nnreal.of_real_le_of_real_iff {r p : ℝ} :

0 ≤ p → (nnreal.of_real r ≤ nnreal.of_real p ↔ r ≤ p)

@[simp]

theorem nnreal.of_real_lt_of_real_iff' {r p : ℝ} :

nnreal.of_real r < nnreal.of_real p ↔ r < p ∧ 0 < p

theorem nnreal.of_real_lt_of_real_iff {r p : ℝ} :

0 < p → (nnreal.of_real r < nnreal.of_real p ↔ r < p)

theorem nnreal.of_real_lt_of_real_iff_of_nonneg {r p : ℝ} :

0 ≤ r → (nnreal.of_real r < nnreal.of_real p ↔ r < p)

@[simp]

theorem nnreal.of_real_add {r p : ℝ} :

0 ≤ r → 0 ≤ p → nnreal.of_real (r + p) = nnreal.of_real r + nnreal.of_real p

theorem nnreal.of_real_add_of_real {r p : ℝ} :

0 ≤ r → 0 ≤ p → nnreal.of_real r + nnreal.of_real p = nnreal.of_real (r + p)

theorem nnreal.of_real_le_of_real {r p : ℝ} :

r ≤ p → nnreal.of_real r ≤ nnreal.of_real p

theorem nnreal.of_real_add_le {r p : ℝ} :

nnreal.of_real (r + p) ≤ nnreal.of_real r + nnreal.of_real p

theorem nnreal.of_real_le_iff_le_coe {r : ℝ} {p : nnreal} :

nnreal.of_real r ≤ p ↔ r ≤ ↑p

theorem nnreal.le_of_real_iff_coe_le {r : nnreal} {p : ℝ} :

0 ≤ p → (r ≤ nnreal.of_real p ↔ ↑r ≤ p)

theorem nnreal.of_real_lt_iff_lt_coe {r : ℝ} {p : nnreal} :

0 ≤ r → (nnreal.of_real r < p ↔ r < ↑p)

theorem nnreal.lt_of_real_iff_coe_lt {r : nnreal} {p : ℝ} :

r < nnreal.of_real p ↔ ↑r < p

theorem nnreal.mul_eq_mul_left {a b c : nnreal} :

a ≠ 0 → (a * b = a * c ↔ b = c)

theorem nnreal.of_real_mul {p q : ℝ} :

0 ≤ p → nnreal.of_real (p * q) = nnreal.of_real p * nnreal.of_real q

theorem nnreal.mul_ne_zero' {a b : nnreal} :

a ≠ 0 → b ≠ 0 → a * b ≠ 0

theorem nnreal.sub_def {r p : nnreal} :

r - p = nnreal.of_real (↑r - ↑p)

theorem nnreal.sub_eq_zero {r p : nnreal} :

r ≤ p → r - p = 0

@[simp]

theorem nnreal.sub_self {r : nnreal} :

r - r = 0

@[simp]

theorem nnreal.sub_zero {r : nnreal} :

r - 0 = r

theorem nnreal.sub_pos {r p : nnreal} :

0 < r - p ↔ p < r

theorem nnreal.sub_lt_self {r p : nnreal} :

0 < r → 0 < p → r - p < r

@[simp]

theorem nnreal.sub_le_iff_le_add {r p q : nnreal} :

r - p ≤ q ↔ r ≤ q + p

@[simp]

theorem nnreal.sub_le_self {r p : nnreal} :

r - p ≤ r

theorem nnreal.add_sub_cancel {r p : nnreal} :

p + r - r = p

theorem nnreal.add_sub_cancel' {r p : nnreal} :

r + p - r = p

@[simp]

theorem nnreal.sub_add_cancel_of_le {a b : nnreal} :

b ≤ a → a - b + b = a

theorem nnreal.sub_sub_cancel_of_le {r p : nnreal} :

r ≤ p → p - (p - r) = r

theorem nnreal.lt_sub_iff_add_lt {p q r : nnreal} :

p < q - r ↔ p + r < q

theorem nnreal.div_def {r p : nnreal} :

r / p = r * p⁻¹

theorem nnreal.sum_div {ι : Type u_1} (s : finset ι) (f : ι → nnreal) (b : nnreal) :

s.sum (λ (i : ι), f i) / b = s.sum (λ (i : ι), f i / b)

@[simp]

theorem nnreal.inv_zero :

@[simp]

theorem nnreal.inv_eq_zero {r : nnreal} :

r⁻¹ = 0 ↔ r = 0

@[simp]

theorem nnreal.inv_pos {r : nnreal} :

0 < r⁻¹ ↔ 0 < r

theorem nnreal.div_pos {r p : nnreal} :

0 < r → 0 < p → 0 < r / p

@[simp]

theorem nnreal.inv_one :

@[simp]

theorem nnreal.div_one {r : nnreal} :

r / 1 = r

theorem nnreal.mul_inv {r p : nnreal} :

(r * p)⁻¹ = p⁻¹ * r⁻¹

theorem nnreal.inv_pow {r : nnreal} {n : ℕ} :

(r ^ n)⁻¹ = r⁻¹ ^ n

@[simp]

theorem nnreal.inv_mul_cancel {r : nnreal} :

r ≠ 0 → r⁻¹ * r = 1

@[simp]

theorem nnreal.mul_inv_cancel {r : nnreal} :

r ≠ 0 → r * r⁻¹ = 1

@[simp]

theorem nnreal.div_self {r : nnreal} :

r ≠ 0 → r / r = 1

theorem nnreal.div_self_le (r : nnreal) :

r / r ≤ 1

@[simp]

theorem nnreal.mul_div_cancel {r p : nnreal} :

p ≠ 0 → r * p / p = r

@[simp]

theorem nnreal.mul_div_cancel' {r p : nnreal} :

r ≠ 0 → r * (p / r) = p

@[simp]

theorem nnreal.inv_inv {r : nnreal} :

r⁻¹ ⁻¹ = r

@[simp]

theorem nnreal.inv_le {r p : nnreal} :

r ≠ 0 → (r⁻¹ ≤ p ↔ 1 ≤ r * p)

theorem nnreal.inv_le_of_le_mul {r p : nnreal} :

1 ≤ r * p → r⁻¹ ≤ p

@[simp]

theorem nnreal.le_inv_iff_mul_le {r p : nnreal} :

p ≠ 0 → (r ≤ p⁻¹ ↔ r * p ≤ 1)

@[simp]

theorem nnreal.lt_inv_iff_mul_lt {r p : nnreal} :

p ≠ 0 → (r < p⁻¹ ↔ r * p < 1)

theorem nnreal.mul_le_iff_le_inv {a b r : nnreal} :

r ≠ 0 → (r * a ≤ b ↔ a ≤ r⁻¹ * b)

theorem nnreal.le_div_iff_mul_le {a b r : nnreal} :

r ≠ 0 → (a ≤ b / r ↔ a * r ≤ b)

theorem nnreal.div_le_iff {a b r : nnreal} :

r ≠ 0 → (a / r ≤ b ↔ a ≤ b * r)

theorem nnreal.le_of_forall_lt_one_mul_lt {x y : nnreal} :

(∀ (a : nnreal), a < 1 → a * x ≤ y) → x ≤ y

theorem nnreal.div_add_div_same (a b c : nnreal) :

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

theorem nnreal.half_pos {a : nnreal} :

0 < a → 0 < a / 2

theorem nnreal.add_halves (a : nnreal) :

a / 2 + a / 2 = a

theorem nnreal.half_lt_self {a : nnreal} :

a ≠ 0 → a / 2 < a

theorem nnreal.two_inv_lt_one :

theorem nnreal.div_lt_iff {a b c : nnreal} :

c ≠ 0 → (b / c < a ↔ b < a * c)

theorem nnreal.div_lt_one_of_lt {a b : nnreal} :

a < b → a / b < 1

theorem nnreal.div_pow {a b : nnreal} (n : ℕ) :

(a / b) ^ n = a ^ n / b ^ n

theorem nnreal.mul_div_assoc' (a b c : nnreal) :

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

theorem nnreal.div_add_div (a : nnreal) {b : nnreal} (c : nnreal) {d : nnreal} :

b ≠ 0 → d ≠ 0 → a / b + c / d = (a * d + b * c) / (b * d)

theorem nnreal.inv_eq_one_div (a : nnreal) :

a⁻¹ = 1 / a

theorem nnreal.div_mul_eq_mul_div (a b c : nnreal) :

a / b * c = a * c / b

theorem nnreal.add_div' (a b c : nnreal) :

c ≠ 0 → b + a / c = (b * c + a) / c

theorem nnreal.div_add' (a b c : nnreal) :

c ≠ 0 → a / c + b = (a + b * c) / c

theorem nnreal.one_div (a : nnreal) :

1 / a = a⁻¹

theorem nnreal.one_div_div (a b : nnreal) :

1 / (a / b) = b / a

theorem nnreal.div_eq_mul_one_div (a b : nnreal) :

a / b = a * (1 / b)

theorem nnreal.div_div_eq_mul_div (a b c : nnreal) :

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

theorem nnreal.div_div_eq_div_mul (a b c : nnreal) :

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

theorem nnreal.div_eq_div_iff {a b c d : nnreal} :

b ≠ 0 → d ≠ 0 → (a / b = c / d ↔ a * d = c * b)

theorem nnreal.div_eq_iff {a b c : nnreal} :

b ≠ 0 → (a / b = c ↔ a = c * b)

theorem nnreal.eq_div_iff {a b c : nnreal} :

b ≠ 0 → (c = a / b ↔ c * b = a)

theorem nnreal.pow_eq_zero {a : nnreal} {n : ℕ} :

a ^ n = 0 → a = 0

theorem nnreal.pow_ne_zero {a : nnreal} (n : ℕ) :

a ≠ 0 → a ^ n ≠ 0

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gromov_hausdorff

gromov_hausdorff_realized

hausdorff_distance

isometry

lipschitz

pi_Lp

premetric_space

sheaves

presheaf

presheaf_of_functions

sheaf

sheaf_of_functions

stalks

uniform_space

absolute_value

abstract_completion

basic

cauchy

compact_separated

compare_reals

complete_separated

completion

pi

separation

uniform_convergence

uniform_embedding

bases

basic

bounded_continuous_function

compact_open

compacts

constructions

continuous_map

continuous_on

dense_embedding

extend_from_subset

homeomorph

list

local_extr

local_homeomorph

maps

opens

order

path_connected

separation

sequences

stone_cech

subset_properties

tactic

topological_fiber_bundle