mathlib docs: order.bounds

A set s is not bounded above if and only if for each x there exists y ∈ s such that x is not greater than or equal to y. This version only assumes preorder structure and uses ¬(y ≤ x). A version for linear orders is called not_bdd_above_iff.

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theorem not_bdd_below_iff' {α : Type u} [preorder α] {s : set α} :

¬bdd_below s ↔ ∀ (x : α), ∃ (y : α) (H : y ∈ s), ¬x ≤ y

A set s is not bounded below if and only if for each x there exists y ∈ s such that x is not less than or equal to y. This version only assumes preorder structure and uses ¬(x ≤ y). A version for linear orders is called not_bdd_below_iff.

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theorem not_bdd_above_iff {α : Type u_1} [linear_order α] {s : set α} :

¬bdd_above s ↔ ∀ (x : α), ∃ (y : α) (H : y ∈ s), x < y

A set s is not bounded above if and only if for each x there exists y ∈ s that is greater than x. A version for preorders is called not_bdd_above_iff'.

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theorem not_bdd_below_iff {α : Type u_1} [linear_order α] {s : set α} :

¬bdd_below s ↔ ∀ (x : α), ∃ (y : α) (H : y ∈ s), y < x

A set s is not bounded below if and only if for each x there exists y ∈ s that is less than x. A version for preorders is called not_bdd_below_iff'.

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theorem upper_bounds_mono_set {α : Type u} [preorder α] ⦃s t : set α⦄ :

s ⊆ t → upper_bounds t ⊆ upper_bounds s

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theorem lower_bounds_mono_set {α : Type u} [preorder α] ⦃s t : set α⦄ :

s ⊆ t → lower_bounds t ⊆ lower_bounds s

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theorem upper_bounds_mono_mem {α : Type u} [preorder α] {s : set α} ⦃a b : α⦄ :

a ≤ b → a ∈ upper_bounds s → b ∈ upper_bounds s

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theorem lower_bounds_mono_mem {α : Type u} [preorder α] {s : set α} ⦃a b : α⦄ :

a ≤ b → b ∈ lower_bounds s → a ∈ lower_bounds s

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theorem upper_bounds_mono {α : Type u} [preorder α] ⦃s t : set α⦄ (hst : s ⊆ t) ⦃a b : α⦄ :

a ≤ b → a ∈ upper_bounds t → b ∈ upper_bounds s

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theorem lower_bounds_mono {α : Type u} [preorder α] ⦃s t : set α⦄ (hst : s ⊆ t) ⦃a b : α⦄ :

a ≤ b → b ∈ lower_bounds t → a ∈ lower_bounds s

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theorem bdd_above.mono {α : Type u} [preorder α] ⦃s t : set α⦄ :

s ⊆ t → bdd_above t → bdd_above s

If s ⊆ t and t is bounded above, then so is s.

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theorem bdd_below.mono {α : Type u} [preorder α] ⦃s t : set α⦄ :

s ⊆ t → bdd_below t → bdd_below s

If s ⊆ t and t is bounded below, then so is s.

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theorem is_lub.of_subset_of_superset {α : Type u} [preorder α] {a : α} {s t p : set α} :

is_lub s a → is_lub p a → s ⊆ t → t ⊆ p → is_lub t a

If a is a least upper bound for sets s and p, then it is a least upper bound for any set t, s ⊆ t ⊆ p.

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theorem is_glb.of_subset_of_superset {α : Type u} [preorder α] {a : α} {s t p : set α} :

is_glb s a → is_glb p a → s ⊆ t → t ⊆ p → is_glb t a

If a is a greatest lower bound for sets s and p, then it is a greater lower bound for any set t, s ⊆ t ⊆ p.

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theorem is_least.mono {α : Type u} [preorder α] {s t : set α} {a b : α} :

is_least s a → is_least t b → s ⊆ t → b ≤ a

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theorem is_greatest.mono {α : Type u} [preorder α] {s t : set α} {a b : α} :

is_greatest s a → is_greatest t b → s ⊆ t → a ≤ b

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theorem is_lub.mono {α : Type u} [preorder α] {s t : set α} {a b : α} :

is_lub s a → is_lub t b → s ⊆ t → a ≤ b

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theorem is_glb.mono {α : Type u} [preorder α] {s t : set α} {a b : α} :

is_glb s a → is_glb t b → s ⊆ t → b ≤ a

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theorem is_least.is_glb {α : Type u} [preorder α] {s : set α} {a : α} :

is_least s a → is_glb s a

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theorem is_greatest.is_lub {α : Type u} [preorder α] {s : set α} {a : α} :

is_greatest s a → is_lub s a

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theorem is_lub.upper_bounds_eq {α : Type u} [preorder α] {s : set α} {a : α} :

is_lub s a → upper_bounds s = set.Ici a

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theorem is_glb.lower_bounds_eq {α : Type u} [preorder α] {s : set α} {a : α} :

is_glb s a → lower_bounds s = set.Iic a

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theorem is_least.lower_bounds_eq {α : Type u} [preorder α] {s : set α} {a : α} :

is_least s a → lower_bounds s = set.Iic a

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theorem is_greatest.upper_bounds_eq {α : Type u} [preorder α] {s : set α} {a : α} :

is_greatest s a → upper_bounds s = set.Ici a

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theorem is_lub_le_iff {α : Type u} [preorder α] {s : set α} {a b : α} :

is_lub s a → (a ≤ b ↔ b ∈ upper_bounds s)

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theorem le_is_glb_iff {α : Type u} [preorder α] {s : set α} {a b : α} :

is_glb s a → (b ≤ a ↔ b ∈ lower_bounds s)

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theorem is_lub.bdd_above {α : Type u} [preorder α] {s : set α} {a : α} :

is_lub s a → bdd_above s

If s has a least upper bound, then it is bounded above.

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theorem is_glb.bdd_below {α : Type u} [preorder α] {s : set α} {a : α} :

is_glb s a → bdd_below s

If s has a greatest lower bound, then it is bounded below.

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theorem is_greatest.bdd_above {α : Type u} [preorder α] {s : set α} {a : α} :

is_greatest s a → bdd_above s

If s has a greatest element, then it is bounded above.

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theorem is_least.bdd_below {α : Type u} [preorder α] {s : set α} {a : α} :

is_least s a → bdd_below s

If s has a least element, then it is bounded below.

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theorem is_least.nonempty {α : Type u} [preorder α] {s : set α} {a : α} :

is_least s a → s.nonempty

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theorem is_greatest.nonempty {α : Type u} [preorder α] {s : set α} {a : α} :

is_greatest s a → s.nonempty

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@[simp]

theorem upper_bounds_union {α : Type u} [preorder α] {s t : set α} :

upper_bounds (s ∪ t) = upper_bounds s ∩ upper_bounds t

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theorem lower_bounds_union {α : Type u} [preorder α] {s t : set α} :

lower_bounds (s ∪ t) = lower_bounds s ∩ lower_bounds t

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theorem union_upper_bounds_subset_upper_bounds_inter {α : Type u} [preorder α] {s t : set α} :

upper_bounds s ∪ upper_bounds t ⊆ upper_bounds (s ∩ t)

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theorem union_lower_bounds_subset_lower_bounds_inter {α : Type u} [preorder α] {s t : set α} :

lower_bounds s ∪ lower_bounds t ⊆ lower_bounds (s ∩ t)

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theorem is_least_union_iff {α : Type u} [preorder α] {a : α} {s t : set α} :

is_least (s ∪ t) a ↔ is_least s a ∧ a ∈ lower_bounds t ∨ a ∈ lower_bounds s ∧ is_least t a

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theorem is_greatest_union_iff {α : Type u} [preorder α] {s t : set α} {a : α} :

is_greatest (s ∪ t) a ↔ is_greatest s a ∧ a ∈ upper_bounds t ∨ a ∈ upper_bounds s ∧ is_greatest t a

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theorem bdd_above.inter_of_left {α : Type u} [preorder α] {s t : set α} :

bdd_above s → bdd_above (s ∩ t)

If s is bounded, then so is s ∩ t

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theorem bdd_above.inter_of_right {α : Type u} [preorder α] {s t : set α} :

bdd_above t → bdd_above (s ∩ t)

If t is bounded, then so is s ∩ t

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theorem bdd_below.inter_of_left {α : Type u} [preorder α] {s t : set α} :

bdd_below s → bdd_below (s ∩ t)

If s is bounded, then so is s ∩ t

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theorem bdd_below.inter_of_right {α : Type u} [preorder α] {s t : set α} :

bdd_below t → bdd_below (s ∩ t)

If t is bounded, then so is s ∩ t

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theorem bdd_above.union {γ : Type w} [semilattice_sup γ] {s t : set γ} :

bdd_above s → bdd_above t → bdd_above (s ∪ t)

If s and t are bounded above sets in a semilattice_sup, then so is s ∪ t.

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theorem bdd_above_union {γ : Type w} [semilattice_sup γ] {s t : set γ} :

bdd_above (s ∪ t) ↔ bdd_above s ∧ bdd_above t

The union of two sets is bounded above if and only if each of the sets is.

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theorem bdd_below.union {γ : Type w} [semilattice_inf γ] {s t : set γ} :

bdd_below s → bdd_below t → bdd_below (s ∪ t)

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theorem bdd_below_union {γ : Type w} [semilattice_inf γ] {s t : set γ} :

bdd_below (s ∪ t) ↔ bdd_below s ∧ bdd_below t

The union of two sets is bounded above if and only if each of the sets is.

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theorem is_lub.union {γ : Type w} [semilattice_sup γ] {a b : γ} {s t : set γ} :

is_lub s a → is_lub t b → is_lub (s ∪ t) (a ⊔ b)

If a is the least upper bound of s and b is the least upper bound of t, then a ⊔ b is the least upper bound of s ∪ t.

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theorem is_glb.union {γ : Type w} [semilattice_inf γ] {a₁ a₂ : γ} {s t : set γ} :

is_glb s a₁ → is_glb t a₂ → is_glb (s ∪ t) (a₁ ⊓ a₂)

If a is the greatest lower bound of s and b is the greatest lower bound of t, then a ⊓ b is the greatest lower bound of s ∪ t.

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theorem is_least.union {γ : Type w} [decidable_linear_order γ] {a b : γ} {s t : set γ} :

is_least s a → is_least t b → is_least (s ∪ t) (min a b)

If a is the least element of s and b is the least element of t, then min a b is the least element of s ∪ t.

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theorem is_greatest.union {γ : Type w} [decidable_linear_order γ] {a b : γ} {s t : set γ} :

is_greatest s a → is_greatest t b → is_greatest (s ∪ t) (max a b)

If a is the greatest element of s and b is the greatest element of t, then max a b is the greatest element of s ∪ t.

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theorem is_least_Ici {α : Type u} [preorder α] {a : α} :

is_least (set.Ici a) a

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theorem is_greatest_Iic {α : Type u} [preorder α] {a : α} :

is_greatest (set.Iic a) a

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theorem is_lub_Iic {α : Type u} [preorder α] {a : α} :

is_lub (set.Iic a) a

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theorem is_glb_Ici {α : Type u} [preorder α] {a : α} :

is_glb (set.Ici a) a

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theorem upper_bounds_Iic {α : Type u} [preorder α] {a : α} :

upper_bounds (set.Iic a) = set.Ici a

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theorem lower_bounds_Ici {α : Type u} [preorder α] {a : α} :

lower_bounds (set.Ici a) = set.Iic a

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theorem bdd_above_Iic {α : Type u} [preorder α] {a : α} :

bdd_above (set.Iic a)

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theorem bdd_below_Ici {α : Type u} [preorder α] {a : α} :

bdd_below (set.Ici a)

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theorem bdd_above_Iio {α : Type u} [preorder α] {a : α} :

bdd_above (set.Iio a)

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theorem bdd_below_Ioi {α : Type u} [preorder α] {a : α} :

bdd_below (set.Ioi a)

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theorem is_lub_Iio {γ : Type w} [linear_order γ] [densely_ordered γ] {a : γ} :

is_lub (set.Iio a) a

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theorem is_glb_Ioi {γ : Type w} [linear_order γ] [densely_ordered γ] {a : γ} :

is_glb (set.Ioi a) a

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theorem upper_bounds_Iio {γ : Type w} [linear_order γ] [densely_ordered γ] {a : γ} :

upper_bounds (set.Iio a) = set.Ici a

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theorem lower_bounds_Ioi {γ : Type w} [linear_order γ] [densely_ordered γ] {a : γ} :

lower_bounds (set.Ioi a) = set.Iic a

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theorem is_greatest_singleton {α : Type u} [preorder α] {a : α} :

is_greatest {a} a

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theorem is_least_singleton {α : Type u} [preorder α] {a : α} :

is_least {a} a

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theorem is_lub_singleton {α : Type u} [preorder α] {a : α} :

is_lub {a} a

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theorem is_glb_singleton {α : Type u} [preorder α] {a : α} :

is_glb {a} a

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theorem bdd_above_singleton {α : Type u} [preorder α] {a : α} :

bdd_above {a}

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theorem bdd_below_singleton {α : Type u} [preorder α] {a : α} :

bdd_below {a}

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@[simp]

theorem upper_bounds_singleton {α : Type u} [preorder α] {a : α} :

upper_bounds {a} = set.Ici a

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theorem lower_bounds_singleton {α : Type u} [preorder α] {a : α} :

lower_bounds {a} = set.Iic a

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theorem bdd_above_Icc {α : Type u} [preorder α] {a b : α} :

bdd_above (set.Icc a b)

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theorem bdd_above_Ico {α : Type u} [preorder α] {a b : α} :

bdd_above (set.Ico a b)

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theorem bdd_above_Ioc {α : Type u} [preorder α] {a b : α} :

bdd_above (set.Ioc a b)

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theorem bdd_above_Ioo {α : Type u} [preorder α] {a b : α} :

bdd_above (set.Ioo a b)

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theorem is_greatest_Icc {α : Type u} [preorder α] {a b : α} :

a ≤ b → is_greatest (set.Icc a b) b

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theorem is_lub_Icc {α : Type u} [preorder α] {a b : α} :

a ≤ b → is_lub (set.Icc a b) b

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theorem upper_bounds_Icc {α : Type u} [preorder α] {a b : α} :

a ≤ b → upper_bounds (set.Icc a b) = set.Ici b

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theorem is_least_Icc {α : Type u} [preorder α] {a b : α} :

a ≤ b → is_least (set.Icc a b) a

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theorem is_glb_Icc {α : Type u} [preorder α] {a b : α} :

a ≤ b → is_glb (set.Icc a b) a

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theorem lower_bounds_Icc {α : Type u} [preorder α] {a b : α} :

a ≤ b → lower_bounds (set.Icc a b) = set.Iic a

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theorem is_greatest_Ioc {α : Type u} [preorder α] {a b : α} :

a < b → is_greatest (set.Ioc a b) b

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theorem is_lub_Ioc {α : Type u} [preorder α] {a b : α} :

a < b → is_lub (set.Ioc a b) b

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theorem upper_bounds_Ioc {α : Type u} [preorder α] {a b : α} :

a < b → upper_bounds (set.Ioc a b) = set.Ici b

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theorem is_least_Ico {α : Type u} [preorder α] {a b : α} :

a < b → is_least (set.Ico a b) a

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theorem is_glb_Ico {α : Type u} [preorder α] {a b : α} :

a < b → is_glb (set.Ico a b) a

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theorem lower_bounds_Ico {α : Type u} [preorder α] {a b : α} :

a < b → lower_bounds (set.Ico a b) = set.Iic a

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theorem is_glb_Ioo {γ : Type w} [linear_order γ] [densely_ordered γ] {a b : γ} :

a < b → is_glb (set.Ioo a b) a

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theorem lower_bounds_Ioo {γ : Type w} [linear_order γ] [densely_ordered γ] {a b : γ} :

a < b → lower_bounds (set.Ioo a b) = set.Iic a

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theorem is_glb_Ioc {γ : Type w} [linear_order γ] [densely_ordered γ] {a b : γ} :

a < b → is_glb (set.Ioc a b) a

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theorem lower_bound_Ioc {γ : Type w} [linear_order γ] [densely_ordered γ] {a b : γ} :

a < b → lower_bounds (set.Ioc a b) = set.Iic a

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theorem is_lub_Ioo {γ : Type w} [linear_order γ] [densely_ordered γ] {a b : γ} :

a < b → is_lub (set.Ioo a b) b

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theorem upper_bounds_Ioo {γ : Type w} [linear_order γ] [densely_ordered γ] {a b : γ} :

a < b → upper_bounds (set.Ioo a b) = set.Ici b

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theorem is_lub_Ico {γ : Type w} [linear_order γ] [densely_ordered γ] {a b : γ} :

a < b → is_lub (set.Ico a b) b

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theorem upper_bounds_Ico {γ : Type w} [linear_order γ] [densely_ordered γ] {a b : γ} :

a < b → upper_bounds (set.Ico a b) = set.Ici b

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theorem bdd_below_iff_subset_Ici {α : Type u} [preorder α] {s : set α} :

bdd_below s ↔ ∃ (a : α), s ⊆ set.Ici a

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theorem bdd_above_iff_subset_Iic {α : Type u} [preorder α] {s : set α} :

bdd_above s ↔ ∃ (a : α), s ⊆ set.Iic a

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theorem bdd_below_bdd_above_iff_subset_Icc {α : Type u} [preorder α] {s : set α} :

bdd_below s ∧ bdd_above s ↔ ∃ (a b : α), s ⊆ set.Icc a b

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theorem order_top.upper_bounds_univ {γ : Type w} [order_top γ] :

upper_bounds set.univ = {⊤}

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theorem is_greatest_univ {γ : Type w} [order_top γ] :

is_greatest set.univ ⊤

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theorem is_lub_univ {γ : Type w} [order_top γ] :

is_lub set.univ ⊤

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theorem order_bot.lower_bounds_univ {γ : Type w} [order_bot γ] :

lower_bounds set.univ = {⊥}

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theorem is_least_univ {γ : Type w} [order_bot γ] :

is_least set.univ ⊥

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theorem is_glb_univ {γ : Type w} [order_bot γ] :

is_glb set.univ ⊥

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theorem no_top_order.upper_bounds_univ {α : Type u} [preorder α] [no_top_order α] :

upper_bounds set.univ = ∅

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theorem no_bot_order.lower_bounds_univ {α : Type u} [preorder α] [no_bot_order α] :

lower_bounds set.univ = ∅

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theorem upper_bounds_empty {α : Type u} [preorder α] :

upper_bounds ∅ = set.univ

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theorem lower_bounds_empty {α : Type u} [preorder α] :

lower_bounds ∅ = set.univ

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theorem bdd_above_empty {α : Type u} [preorder α] [nonempty α] :

bdd_above ∅

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theorem bdd_below_empty {α : Type u} [preorder α] [nonempty α] :

bdd_below ∅

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theorem is_glb_empty {γ : Type w} [order_top γ] :

is_glb ∅ ⊤

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theorem is_lub_empty {γ : Type w} [order_bot γ] :

is_lub ∅ ⊥

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theorem is_lub.nonempty {α : Type u} [preorder α] {s : set α} {a : α} [no_bot_order α] :

is_lub s a → s.nonempty

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theorem is_glb.nonempty {α : Type u} [preorder α] {s : set α} {a : α} [no_top_order α] :

is_glb s a → s.nonempty

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theorem nonempty_of_not_bdd_above {α : Type u} [preorder α] {s : set α} [ha : nonempty α] :

¬bdd_above s → s.nonempty

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theorem nonempty_of_not_bdd_below {α : Type u} [preorder α] {s : set α} [ha : nonempty α] :

¬bdd_below s → s.nonempty

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theorem bdd_above_insert {γ : Type w} [semilattice_sup γ] (a : γ) {s : set γ} :

bdd_above (has_insert.insert a s) ↔ bdd_above s

Adding a point to a set preserves its boundedness above.

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theorem bdd_above.insert {γ : Type w} [semilattice_sup γ] (a : γ) {s : set γ} :

bdd_above s → bdd_above (has_insert.insert a s)

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@[simp]

theorem bdd_below_insert {γ : Type w} [semilattice_inf γ] (a : γ) {s : set γ} :

bdd_below (has_insert.insert a s) ↔ bdd_below s

Adding a point to a set preserves its boundedness below.

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theorem bdd_below.insert {γ : Type w} [semilattice_inf γ] (a : γ) {s : set γ} :

bdd_below s → bdd_below (has_insert.insert a s)

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theorem is_lub.insert {γ : Type w} [semilattice_sup γ] (a : γ) {b : γ} {s : set γ} :

is_lub s b → is_lub (has_insert.insert a s) (a ⊔ b)

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theorem is_glb.insert {γ : Type w} [semilattice_inf γ] (a : γ) {b : γ} {s : set γ} :

is_glb s b → is_glb (has_insert.insert a s) (a ⊓ b)

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theorem is_greatest.insert {γ : Type w} [decidable_linear_order γ] (a : γ) {b : γ} {s : set γ} :

is_greatest s b → is_greatest (has_insert.insert a s) (max a b)

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theorem is_least.insert {γ : Type w} [decidable_linear_order γ] (a : γ) {b : γ} {s : set γ} :

is_least s b → is_least (has_insert.insert a s) (min a b)

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theorem upper_bounds_insert {α : Type u} [preorder α] (a : α) (s : set α) :

upper_bounds (has_insert.insert a s) = set.Ici a ∩ upper_bounds s

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theorem lower_bounds_insert {α : Type u} [preorder α] (a : α) (s : set α) :

lower_bounds (has_insert.insert a s) = set.Iic a ∩ lower_bounds s

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theorem order_top.bdd_above {γ : Type w} [order_top γ] (s : set γ) :

bdd_above s

When there is a global maximum, every set is bounded above.

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@[simp]

theorem order_bot.bdd_below {γ : Type w} [order_bot γ] (s : set γ) :

bdd_below s

When there is a global minimum, every set is bounded below.

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theorem is_lub_pair {γ : Type w} [semilattice_sup γ] {a b : γ} :

is_lub {a, b} (a ⊔ b)

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theorem is_glb_pair {γ : Type w} [semilattice_inf γ] {a b : γ} :

is_glb {a, b} (a ⊓ b)

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theorem is_least_pair {γ : Type w} [decidable_linear_order γ] {a b : γ} :

is_least {a, b} (min a b)

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theorem is_greatest_pair {γ : Type w} [decidable_linear_order γ] {a b : γ} :

is_greatest {a, b} (max a b)

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theorem lower_bounds_le_upper_bounds {α : Type u} [preorder α] {s : set α} {a b : α} :

a ∈ lower_bounds s → b ∈ upper_bounds s → s.nonempty → a ≤ b

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theorem is_glb_le_is_lub {α : Type u} [preorder α] {s : set α} {a b : α} :

is_glb s a → is_lub s b → s.nonempty → a ≤ b

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theorem is_lub_lt_iff {α : Type u} [preorder α] {s : set α} {a b : α} :

is_lub s a → (a < b ↔ ∃ (c : α) (H : c ∈ upper_bounds s), c < b)

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theorem lt_is_glb_iff {α : Type u} [preorder α] {s : set α} {a b : α} :

is_glb s a → (b < a ↔ ∃ (c : α) (H : c ∈ lower_bounds s), b < c)

source

theorem is_least.unique {α : Type u} [partial_order α] {s : set α} {a b : α} :

is_least s a → is_least s b → a = b

source

theorem is_least.is_least_iff_eq {α : Type u} [partial_order α] {s : set α} {a b : α} :

is_least s a → (is_least s b ↔ a = b)

source

theorem is_greatest.unique {α : Type u} [partial_order α] {s : set α} {a b : α} :

is_greatest s a → is_greatest s b → a = b

source

theorem is_greatest.is_greatest_iff_eq {α : Type u} [partial_order α] {s : set α} {a b : α} :

is_greatest s a → (is_greatest s b ↔ a = b)

source

theorem is_lub.unique {α : Type u} [partial_order α] {s : set α} {a b : α} :

is_lub s a → is_lub s b → a = b

source

theorem is_glb.unique {α : Type u} [partial_order α] {s : set α} {a b : α} :

is_glb s a → is_glb s b → a = b

source

theorem lt_is_lub_iff {α : Type u} [linear_order α] {s : set α} {a b : α} :

is_lub s a → (b < a ↔ ∃ (c : α) (H : c ∈ s), b < c)

source

theorem is_glb_lt_iff {α : Type u} [linear_order α] {s : set α} {a b : α} :

is_glb s a → (a < b ↔ ∃ (c : α) (H : c ∈ s), c < b)

source

theorem monotone.mem_upper_bounds_image {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (Hf : monotone f) {a : α} {s : set α} :

a ∈ upper_bounds s → f a ∈ upper_bounds (f '' s)

source

theorem monotone.mem_lower_bounds_image {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (Hf : monotone f) {a : α} {s : set α} :

a ∈ lower_bounds s → f a ∈ lower_bounds (f '' s)

source

theorem monotone.map_bdd_above {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} :

monotone f → bdd_above s → bdd_above (f '' s)

The image under a monotone function of a set which is bounded above is bounded above.

source

theorem monotone.map_bdd_below {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} {s : set α} :

monotone f → bdd_below s → bdd_below (f '' s)

The image under a monotone function of a set which is bounded below is bounded below.

source

theorem monotone.map_is_least {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (Hf : monotone f) {a : α} {s : set α} :

is_least s a → is_least (f '' s) (f a)

A monotone map sends a least element of a set to a least element of its image.

source

theorem monotone.map_is_greatest {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (Hf : monotone f) {a : α} {s : set α} :

is_greatest s a → is_greatest (f '' s) (f a)

A monotone map sends a greatest element of a set to a greatest element of its image.

source

theorem monotone.is_lub_image_le {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (Hf : monotone f) {a : α} {s : set α} (Ha : is_lub s a) {b : β} :

is_lub (f '' s) b → b ≤ f a

source

theorem monotone.le_is_glb_image {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (Hf : monotone f) {a : α} {s : set α} (Ha : is_glb s a) {b : β} :

is_glb (f '' s) b → f a ≤ b

source

theorem is_glb.of_image {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (hf : ∀ {x y : α}, f x ≤ f y ↔ x ≤ y) {s : set α} {x : α} :

is_glb (f '' s) (f x) → is_glb s x

source

theorem is_lub.of_image {α : Type u} {β : Type v} [preorder α] [preorder β] {f : α → β} (hf : ∀ {x y : α}, f x ≤ f y ↔ x ≤ y) {s : set α} {x : α} :

is_lub (f '' s) (f x) → is_lub s x

mathlib documentation

order.bounds

Upper / lower bounds

Definitions

Monotonicity

Conversions

Union and intersection

Specific sets

Unbounded intervals

Singleton

Bounded intervals

Univ

Empty set

insert

Pair

(In)equalities with the least upper bound and the greatest lower bound

Images of upper/lower bounds under monotone functions

General documentation

Additional documentation

Library

mathlib documentation

order.​bounds

Upper / lower bounds

Definitions

Monotonicity

Conversions

Union and intersection

Specific sets

Unbounded intervals

Singleton

Bounded intervals

Univ

Empty set

insert

Pair

(In)equalities with the least upper bound and the greatest lower bound

Images of upper/lower bounds under monotone functions

General documentation

Additional documentation

Library

order.bounds