Documentation

Mathlib.Order.GaloisConnection

Galois connections, insertions and coinsertions #

Galois connections are order theoretic adjoints, i.e. a pair of functions u and l, such that ∀ a b, l a ≤ b ↔ a ≤ u b∀ a b, l a ≤ b ↔ a ≤ u b≤ b ↔ a ≤ u b↔ a ≤ u b≤ u b.

Main definitions #

GaloisConnection: A Galois connection is a pair of functions l and u satisfying l a ≤ b ↔ a ≤ u b≤ b ↔ a ≤ u b↔ a ≤ u b≤ u b. They are special cases of adjoint functors in category theory, but do not depend on the category theory library in mathlib.
GaloisInsertion: A Galois insertion is a Galois connection where l ∘ u = id∘ u = id
GaloisCoinsertion: A Galois coinsertion is a Galois connection where u ∘ l = id∘ l = id

Implementation details #

Galois insertions can be used to lift order structures from one type to another. For example if α is a complete lattice, and l : α → β→ β, and u : β → α→ α form a Galois insertion, then β is also a complete lattice. l is the lower adjoint and u is the upper adjoint.

An example of a Galois insertion is in group theory. If G is a group, then there is a Galois insertion between the set of subsets of G, Set G, and the set of subgroups of G, Subgroup G. The lower adjoint is Subgroup.closure, taking the Subgroup generated by a Set, and the upper adjoint is the coercion from Subgroup G to Set G, taking the underlying set of a subgroup.

Naively lifting a lattice structure along this Galois insertion would mean that the definition of inf on subgroups would be Subgroup.closure (↑S ∩ ↑T)↑S ∩ ↑T)∩ ↑T)↑T). This is an undesirable definition because the intersection of subgroups is already a subgroup, so there is no need to take the closure. For this reason a choice function is added as a field to the GaloisInsertion structure. It has type Π S : Set G, ↑(closure S) ≤ S → Subgroup G↑(closure S) ≤ S → Subgroup G≤ S → Subgroup G→ Subgroup G. When ↑(closure S) ≤ S↑(closure S) ≤ S≤ S, then S is already a subgroup, so this function can be defined using Subgroup.mk and not closure. This means the infimum of subgroups will be defined to be the intersection of sets, paired with a proof that intersection of subgroups is a subgroup, rather than the closure of the intersection.

def GaloisConnection {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] (l : α → β) (u : β → α) :

A Galois connection is a pair of functions l and u satisfying l a ≤ b ↔ a ≤ u b≤ b ↔ a ≤ u b↔ a ≤ u b≤ u b. They are special cases of adjoint functors in category theory, but do not depend on the category theory library in mathlib.

Equations

GaloisConnection l u = ∀ (a : α) (b : β), l a ≤ b ↔ a ≤ u b

theorem OrderIso.to_galoisConnection {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] (oi : α ≃o β) :

GaloisConnection ↑(RelIso.toRelEmbedding oi).toEmbedding ↑(RelIso.toRelEmbedding (OrderIso.symm oi)).toEmbedding

Makes a Galois connection from an order-preserving bijection.

theorem GaloisConnection.monotone_intro {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (hu : Monotone u) (hl : Monotone l) (hul : ∀ (a : α), a ≤ u (l a)) (hlu : ∀ (a : β), l (u a) ≤ a) :

GaloisConnection l u

theorem GaloisConnection.dual {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) :

GaloisConnection (↑OrderDual.toDual ∘ u ∘ ↑OrderDual.ofDual) (↑OrderDual.toDual ∘ l ∘ ↑OrderDual.ofDual)

theorem GaloisConnection.le_iff_le {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {a : α} {b : β} :

l a ≤ b ↔ a ≤ u b

theorem GaloisConnection.l_le {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {a : α} {b : β} :

a ≤ u b → l a ≤ b

theorem GaloisConnection.le_u {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {a : α} {b : β} :

l a ≤ b → a ≤ u b

theorem GaloisConnection.le_u_l {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) (a : α) :

a ≤ u (l a)

theorem GaloisConnection.l_u_le {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) (a : β) :

l (u a) ≤ a

theorem GaloisConnection.monotone_u {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) :

theorem GaloisConnection.monotone_l {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) :

theorem GaloisConnection.upperBounds_l_image {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) (s : Set α) :

upperBounds (l '' s) = u ⁻¹' upperBounds s

theorem GaloisConnection.lowerBounds_u_image {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) (s : Set β) :

lowerBounds (u '' s) = l ⁻¹' lowerBounds s

theorem GaloisConnection.bddAbove_l_image {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {s : Set α} :

BddAbove (l '' s) ↔ BddAbove s

theorem GaloisConnection.bddBelow_u_image {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {s : Set β} :

BddBelow (u '' s) ↔ BddBelow s

theorem GaloisConnection.isLUB_l_image {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {s : Set α} {a : α} (h : IsLUB s a) :

IsLUB (l '' s) (l a)

theorem GaloisConnection.isGLB_u_image {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {s : Set β} {b : β} (h : IsGLB s b) :

IsGLB (u '' s) (u b)

theorem GaloisConnection.isLeast_l {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {a : α} :

IsLeast { b | a ≤ u b } (l a)

theorem GaloisConnection.isGreatest_u {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {b : β} :

IsGreatest { a | l a ≤ b } (u b)

theorem GaloisConnection.isGLB_l {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {a : α} :

IsGLB { b | a ≤ u b } (l a)

theorem GaloisConnection.isLUB_u {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {b : β} :

IsLUB { a | l a ≤ b } (u b)

theorem GaloisConnection.le_u_l_trans {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {x : α} {y : α} {z : α} (hxy : x ≤ u (l y)) (hyz : y ≤ u (l z)) :

x ≤ u (l z)

If (l, u) is a Galois connection, then the relation x ≤ u (l y)≤ u (l y) is a transitive relation. If l is a closure operator (Submodule.span, Subgroup.closure, ...) and u is the coercion to Set, this reads as "if U is in the closure of V and V is in the closure of W then U is in the closure of W".

theorem GaloisConnection.l_u_le_trans {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {x : β} {y : β} {z : β} (hxy : l (u x) ≤ y) (hyz : l (u y) ≤ z) :

l (u x) ≤ z

theorem GaloisConnection.u_l_u_eq_u {α : Type u} {β : Type v} [inst : PartialOrder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) (b : β) :

u (l (u b)) = u b

theorem GaloisConnection.u_l_u_eq_u' {α : Type u} {β : Type v} [inst : PartialOrder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) :

u ∘ l ∘ u = u

theorem GaloisConnection.u_unique {α : Type u} {β : Type v} [inst : PartialOrder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {l' : α → β} {u' : β → α} (gc' : GaloisConnection l' u') (hl : ∀ (a : α), l a = l' a) {b : β} :

u b = u' b

theorem GaloisConnection.exists_eq_u {α : Type u} {β : Type v} [inst : PartialOrder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) (a : α) :

(∃ b, a = u b) ↔ a = u (l a)

If there exists a b such that a = u a, then b = l a is one such element.

theorem GaloisConnection.u_eq {α : Type u} {β : Type v} [inst : PartialOrder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {z : α} {y : β} :

u y = z ↔ ∀ (x : α), x ≤ z ↔ l x ≤ y

theorem GaloisConnection.l_u_l_eq_l {α : Type u} {β : Type v} [inst : Preorder α] [inst : PartialOrder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) (a : α) :

l (u (l a)) = l a

theorem GaloisConnection.l_u_l_eq_l' {α : Type u} {β : Type v} [inst : Preorder α] [inst : PartialOrder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) :

l ∘ u ∘ l = l

theorem GaloisConnection.l_unique {α : Type u} {β : Type v} [inst : Preorder α] [inst : PartialOrder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {l' : α → β} {u' : β → α} (gc' : GaloisConnection l' u') (hu : ∀ (b : β), u b = u' b) {a : α} :

l a = l' a

theorem GaloisConnection.exists_eq_l {α : Type u} {β : Type v} [inst : Preorder α] [inst : PartialOrder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) (b : β) :

(∃ a, b = l a) ↔ b = l (u b)

If there exists an a such that b = l a, then a = u b is one such element.

theorem GaloisConnection.l_eq {α : Type u} {β : Type v} [inst : Preorder α] [inst : PartialOrder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {x : α} {z : β} :

l x = z ↔ ∀ (y : β), z ≤ y ↔ x ≤ u y

theorem GaloisConnection.u_eq_top {α : Type u} {β : Type v} [inst : PartialOrder α] [inst : Preorder β] [inst : OrderTop α] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {x : β} :

u x = ⊤ ↔ l ⊤ ≤ x

theorem GaloisConnection.u_top {α : Type u} {β : Type v} [inst : PartialOrder α] [inst : Preorder β] [inst : OrderTop α] [inst : OrderTop β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) :

theorem GaloisConnection.l_eq_bot {α : Type u} {β : Type v} [inst : Preorder α] [inst : PartialOrder β] [inst : OrderBot β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {x : α} :

l x = ⊥ ↔ x ≤ u ⊥

theorem GaloisConnection.l_bot {α : Type u} {β : Type v} [inst : Preorder α] [inst : PartialOrder β] [inst : OrderBot β] [inst : OrderBot α] {l : α → β} {u : β → α} (gc : GaloisConnection l u) :

theorem GaloisConnection.l_sup {α : Type u} {β : Type v} {a₁ : α} {a₂ : α} [inst : SemilatticeSup α] [inst : SemilatticeSup β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) :

l (a₁ ⊔ a₂) = l a₁ ⊔ l a₂

theorem GaloisConnection.u_inf {α : Type u} {β : Type v} {b₁ : β} {b₂ : β} [inst : SemilatticeInf α] [inst : SemilatticeInf β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) :

u (b₁ ⊓ b₂) = u b₁ ⊓ u b₂

theorem GaloisConnection.l_supᵢ {α : Type u} {β : Type v} {ι : Sort x} [inst : CompleteLattice α] [inst : CompleteLattice β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {f : ι → α} :

l (supᵢ f) = ⨆ i, l (f i)

theorem GaloisConnection.l_supᵢ₂ {α : Type u} {β : Type v} {ι : Sort x} {κ : ι → Sort u_1} [inst : CompleteLattice α] [inst : CompleteLattice β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {f : (i : ι) → κ i → α} :

l (⨆ i, ⨆ j, f i j) = ⨆ i, ⨆ j, l (f i j)

theorem GaloisConnection.u_infᵢ {α : Type u} {β : Type v} {ι : Sort x} [inst : CompleteLattice α] [inst : CompleteLattice β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {f : ι → β} :

u (infᵢ f) = ⨅ i, u (f i)

theorem GaloisConnection.u_infᵢ₂ {α : Type u} {β : Type v} {ι : Sort x} {κ : ι → Sort u_1} [inst : CompleteLattice α] [inst : CompleteLattice β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {f : (i : ι) → κ i → β} :

u (⨅ i, ⨅ j, f i j) = ⨅ i, ⨅ j, u (f i j)

theorem GaloisConnection.l_supₛ {α : Type u} {β : Type v} [inst : CompleteLattice α] [inst : CompleteLattice β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {s : Set α} :

l (supₛ s) = ⨆ a, ⨆ h, l a

theorem GaloisConnection.u_infₛ {α : Type u} {β : Type v} [inst : CompleteLattice α] [inst : CompleteLattice β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {s : Set β} :

u (infₛ s) = ⨅ a, ⨅ h, u a

theorem GaloisConnection.lt_iff_lt {α : Type u} {β : Type v} [inst : LinearOrder α] [inst : LinearOrder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {a : α} {b : β} :

b < l a ↔ u b < a

theorem GaloisConnection.id {α : Type u} [pα : Preorder α] :

GaloisConnection id id

theorem GaloisConnection.compose {α : Type u} {β : Type v} {γ : Type w} [inst : Preorder α] [inst : Preorder β] [inst : Preorder γ] {l1 : α → β} {u1 : β → α} {l2 : β → γ} {u2 : γ → β} (gc1 : GaloisConnection l1 u1) (gc2 : GaloisConnection l2 u2) :

GaloisConnection (l2 ∘ l1) (u1 ∘ u2)

theorem GaloisConnection.dfun {ι : Type u} {α : ι → Type v} {β : ι → Type w} [inst : (i : ι) → Preorder (α i)] [inst : (i : ι) → Preorder (β i)] (l : (i : ι) → α i → β i) (u : (i : ι) → β i → α i) (gc : ∀ (i : ι), GaloisConnection (l i) (u i)) :

GaloisConnection (fun a i => l i (a i)) fun b i => u i (b i)

theorem GaloisConnection.l_comm_of_u_comm {X : Type u_1} [inst : Preorder X] {Y : Type u_2} [inst : Preorder Y] {Z : Type u_3} [inst : Preorder Z] {W : Type u_4} [inst : PartialOrder W] {lYX : X → Y} {uXY : Y → X} (hXY : GaloisConnection lYX uXY) {lWZ : Z → W} {uZW : W → Z} (hZW : GaloisConnection lWZ uZW) {lWY : Y → W} {uYW : W → Y} (hWY : GaloisConnection lWY uYW) {lZX : X → Z} {uXZ : Z → X} (hXZ : GaloisConnection lZX uXZ) (h : ∀ (w : W), uXZ (uZW w) = uXY (uYW w)) {x : X} :

lWZ (lZX x) = lWY (lYX x)

theorem GaloisConnection.u_comm_of_l_comm {X : Type u_1} [inst : PartialOrder X] {Y : Type u_2} [inst : Preorder Y] {Z : Type u_3} [inst : Preorder Z] {W : Type u_4} [inst : Preorder W] {lYX : X → Y} {uXY : Y → X} (hXY : GaloisConnection lYX uXY) {lWZ : Z → W} {uZW : W → Z} (hZW : GaloisConnection lWZ uZW) {lWY : Y → W} {uYW : W → Y} (hWY : GaloisConnection lWY uYW) {lZX : X → Z} {uXZ : Z → X} (hXZ : GaloisConnection lZX uXZ) (h : ∀ (x : X), lWZ (lZX x) = lWY (lYX x)) {w : W} :

uXZ (uZW w) = uXY (uYW w)

theorem GaloisConnection.l_comm_iff_u_comm {X : Type u_1} [inst : PartialOrder X] {Y : Type u_2} [inst : Preorder Y] {Z : Type u_3} [inst : Preorder Z] {W : Type u_4} [inst : PartialOrder W] {lYX : X → Y} {uXY : Y → X} (hXY : GaloisConnection lYX uXY) {lWZ : Z → W} {uZW : W → Z} (hZW : GaloisConnection lWZ uZW) {lWY : Y → W} {uYW : W → Y} (hWY : GaloisConnection lWY uYW) {lZX : X → Z} {uXZ : Z → X} (hXZ : GaloisConnection lZX uXZ) :

(∀ (w : W), uXZ (uZW w) = uXY (uYW w)) ↔ ∀ (x : X), lWZ (lZX x) = lWY (lYX x)

theorem supₛ_image2_eq_supₛ_supₛ {α : Type u} {β : Type v} {γ : Type w} [inst : CompleteLattice α] [inst : CompleteLattice β] [inst : CompleteLattice γ] {s : Set α} {t : Set β} {l : α → β → γ} {u₁ : β → γ → α} {u₂ : α → γ → β} (h₁ : ∀ (b : β), GaloisConnection (Function.swap l b) (u₁ b)) (h₂ : ∀ (a : α), GaloisConnection (l a) (u₂ a)) :

supₛ (Set.image2 l s t) = l (supₛ s) (supₛ t)

theorem supₛ_image2_eq_supₛ_infₛ {α : Type u} {β : Type v} {γ : Type w} [inst : CompleteLattice α] [inst : CompleteLattice β] [inst : CompleteLattice γ] {s : Set α} {t : Set β} {l : α → β → γ} {u₁ : β → γ → α} {u₂ : α → γ → β} (h₁ : ∀ (b : β), GaloisConnection (Function.swap l b) (u₁ b)) (h₂ : ∀ (a : α), GaloisConnection (l a ∘ ↑OrderDual.ofDual) (↑OrderDual.toDual ∘ u₂ a)) :

supₛ (Set.image2 l s t) = l (supₛ s) (infₛ t)

theorem supₛ_image2_eq_infₛ_supₛ {α : Type u} {β : Type v} {γ : Type w} [inst : CompleteLattice α] [inst : CompleteLattice β] [inst : CompleteLattice γ] {s : Set α} {t : Set β} {l : α → β → γ} {u₁ : β → γ → α} {u₂ : α → γ → β} (h₁ : ∀ (b : β), GaloisConnection (Function.swap l b ∘ ↑OrderDual.ofDual) (↑OrderDual.toDual ∘ u₁ b)) (h₂ : ∀ (a : α), GaloisConnection (l a) (u₂ a)) :

supₛ (Set.image2 l s t) = l (infₛ s) (supₛ t)

theorem supₛ_image2_eq_infₛ_infₛ {α : Type u} {β : Type v} {γ : Type w} [inst : CompleteLattice α] [inst : CompleteLattice β] [inst : CompleteLattice γ] {s : Set α} {t : Set β} {l : α → β → γ} {u₁ : β → γ → α} {u₂ : α → γ → β} (h₁ : ∀ (b : β), GaloisConnection (Function.swap l b ∘ ↑OrderDual.ofDual) (↑OrderDual.toDual ∘ u₁ b)) (h₂ : ∀ (a : α), GaloisConnection (l a ∘ ↑OrderDual.ofDual) (↑OrderDual.toDual ∘ u₂ a)) :

supₛ (Set.image2 l s t) = l (infₛ s) (infₛ t)

theorem infₛ_image2_eq_infₛ_infₛ {α : Type u} {β : Type v} {γ : Type w} [inst : CompleteLattice α] [inst : CompleteLattice β] [inst : CompleteLattice γ] {s : Set α} {t : Set β} {u : α → β → γ} {l₁ : β → γ → α} {l₂ : α → γ → β} (h₁ : ∀ (b : β), GaloisConnection (l₁ b) (Function.swap u b)) (h₂ : ∀ (a : α), GaloisConnection (l₂ a) (u a)) :

infₛ (Set.image2 u s t) = u (infₛ s) (infₛ t)

theorem infₛ_image2_eq_infₛ_supₛ {α : Type u} {β : Type v} {γ : Type w} [inst : CompleteLattice α] [inst : CompleteLattice β] [inst : CompleteLattice γ] {s : Set α} {t : Set β} {u : α → β → γ} {l₁ : β → γ → α} {l₂ : α → γ → β} (h₁ : ∀ (b : β), GaloisConnection (l₁ b) (Function.swap u b)) (h₂ : ∀ (a : α), GaloisConnection (↑OrderDual.toDual ∘ l₂ a) (u a ∘ ↑OrderDual.ofDual)) :

infₛ (Set.image2 u s t) = u (infₛ s) (supₛ t)

theorem infₛ_image2_eq_supₛ_infₛ {α : Type u} {β : Type v} {γ : Type w} [inst : CompleteLattice α] [inst : CompleteLattice β] [inst : CompleteLattice γ] {s : Set α} {t : Set β} {u : α → β → γ} {l₁ : β → γ → α} {l₂ : α → γ → β} (h₁ : ∀ (b : β), GaloisConnection (↑OrderDual.toDual ∘ l₁ b) (Function.swap u b ∘ ↑OrderDual.ofDual)) (h₂ : ∀ (a : α), GaloisConnection (l₂ a) (u a)) :

infₛ (Set.image2 u s t) = u (supₛ s) (infₛ t)

theorem infₛ_image2_eq_supₛ_supₛ {α : Type u} {β : Type v} {γ : Type w} [inst : CompleteLattice α] [inst : CompleteLattice β] [inst : CompleteLattice γ] {s : Set α} {t : Set β} {u : α → β → γ} {l₁ : β → γ → α} {l₂ : α → γ → β} (h₁ : ∀ (b : β), GaloisConnection (↑OrderDual.toDual ∘ l₁ b) (Function.swap u b ∘ ↑OrderDual.ofDual)) (h₂ : ∀ (a : α), GaloisConnection (↑OrderDual.toDual ∘ l₂ a) (u a ∘ ↑OrderDual.ofDual)) :

infₛ (Set.image2 u s t) = u (supₛ s) (supₛ t)

@[simp]

theorem OrderIso.bddAbove_image {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] (e : α ≃o β) {s : Set α} :

BddAbove (↑(RelIso.toRelEmbedding e).toEmbedding '' s) ↔ BddAbove s

@[simp]

theorem OrderIso.bddBelow_image {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] (e : α ≃o β) {s : Set α} :

BddBelow (↑(RelIso.toRelEmbedding e).toEmbedding '' s) ↔ BddBelow s

@[simp]

theorem OrderIso.bddAbove_preimage {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] (e : α ≃o β) {s : Set β} :

BddAbove (↑(RelIso.toRelEmbedding e).toEmbedding ⁻¹' s) ↔ BddAbove s

@[simp]

theorem OrderIso.bddBelow_preimage {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] (e : α ≃o β) {s : Set β} :

BddBelow (↑(RelIso.toRelEmbedding e).toEmbedding ⁻¹' s) ↔ BddBelow s

theorem Nat.galoisConnection_mul_div {k : ℕ} (h : 0 < k) :

GaloisConnection (fun n => n * k) fun n => n / k

structure GaloisInsertion {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst : Preorder β] (l : α → β) (u : β → α) :

Type (maxu_1u_2)

A contructive choice function for images of l.
choice : (x : α) → u (l x) ≤ x → β
The Galois connection associated to a Galois insertion.
gc : GaloisConnection l u
Main property of a Galois insertion.
le_l_u : ∀ (x : β), x ≤ l (u x)
Property of the choice function.
choice_eq : ∀ (a : α) (h : u (l a) ≤ a), choice a h = l a

A Galois insertion is a Galois connection where l ∘ u = id∘ u = id. It also contains a constructive choice function, to give better definitional equalities when lifting order structures. Dual to GaloisCoinsertion

Instances For

def GaloisInsertion.monotoneIntro {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (hu : Monotone u) (hl : Monotone l) (hul : ∀ (a : α), a ≤ u (l a)) (hlu : ∀ (b : β), l (u b) = b) :

GaloisInsertion l u

A constructor for a Galois insertion with the trivial choice function.

Equations

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

def OrderIso.toGaloisInsertion {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] (oi : α ≃o β) :

GaloisInsertion ↑(RelIso.toRelEmbedding oi).toEmbedding ↑(RelIso.toRelEmbedding (OrderIso.symm oi)).toEmbedding

Makes a Galois insertion from an order-preserving bijection.

Equations

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

def GaloisConnection.toGaloisInsertion {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) (h : ∀ (b : β), b ≤ l (u b)) :

GaloisInsertion l u

Make a GaloisInsertion l u from a GaloisConnection l u such that ∀ b, b ≤ l (u b)∀ b, b ≤ l (u b)≤ l (u b)

Equations

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

def GaloisConnection.liftOrderBot {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst : OrderBot α] [inst : PartialOrder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) :

Lift the bottom along a Galois connection

Equations

GaloisConnection.liftOrderBot gc = OrderBot.mk (_ : ∀ (x : β), l ⊥ ≤ x)

theorem GaloisInsertion.l_u_eq {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : Preorder α] [inst : PartialOrder β] (gi : GaloisInsertion l u) (b : β) :

l (u b) = b

theorem GaloisInsertion.leftInverse_l_u {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : Preorder α] [inst : PartialOrder β] (gi : GaloisInsertion l u) :

Function.LeftInverse l u

theorem GaloisInsertion.l_surjective {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : Preorder α] [inst : PartialOrder β] (gi : GaloisInsertion l u) :

Function.Surjective l

theorem GaloisInsertion.u_injective {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : Preorder α] [inst : PartialOrder β] (gi : GaloisInsertion l u) :

Function.Injective u

theorem GaloisInsertion.l_sup_u {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : SemilatticeSup α] [inst : SemilatticeSup β] (gi : GaloisInsertion l u) (a : β) (b : β) :

l (u a ⊔ u b) = a ⊔ b

theorem GaloisInsertion.l_supᵢ_u {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : CompleteLattice α] [inst : CompleteLattice β] (gi : GaloisInsertion l u) {ι : Sort x} (f : ι → β) :

l (⨆ i, u (f i)) = ⨆ i, f i

theorem GaloisInsertion.l_bsupᵢ_u {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : CompleteLattice α] [inst : CompleteLattice β] (gi : GaloisInsertion l u) {ι : Sort x} {p : ι → Prop} (f : (i : ι) → p i → β) :

l (⨆ i, ⨆ hi, u (f i hi)) = ⨆ i, ⨆ hi, f i hi

theorem GaloisInsertion.l_supₛ_u_image {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : CompleteLattice α] [inst : CompleteLattice β] (gi : GaloisInsertion l u) (s : Set β) :

l (supₛ (u '' s)) = supₛ s

theorem GaloisInsertion.l_inf_u {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : SemilatticeInf α] [inst : SemilatticeInf β] (gi : GaloisInsertion l u) (a : β) (b : β) :

l (u a ⊓ u b) = a ⊓ b

theorem GaloisInsertion.l_infᵢ_u {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : CompleteLattice α] [inst : CompleteLattice β] (gi : GaloisInsertion l u) {ι : Sort x} (f : ι → β) :

l (⨅ i, u (f i)) = ⨅ i, f i

theorem GaloisInsertion.l_binfᵢ_u {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : CompleteLattice α] [inst : CompleteLattice β] (gi : GaloisInsertion l u) {ι : Sort x} {p : ι → Prop} (f : (i : ι) → p i → β) :

l (⨅ i, ⨅ hi, u (f i hi)) = ⨅ i, ⨅ hi, f i hi

theorem GaloisInsertion.l_infₛ_u_image {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : CompleteLattice α] [inst : CompleteLattice β] (gi : GaloisInsertion l u) (s : Set β) :

l (infₛ (u '' s)) = infₛ s

theorem GaloisInsertion.l_infᵢ_of_ul_eq_self {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : CompleteLattice α] [inst : CompleteLattice β] (gi : GaloisInsertion l u) {ι : Sort x} (f : ι → α) (hf : ∀ (i : ι), u (l (f i)) = f i) :

l (⨅ i, f i) = ⨅ i, l (f i)

theorem GaloisInsertion.l_binfᵢ_of_ul_eq_self {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : CompleteLattice α] [inst : CompleteLattice β] (gi : GaloisInsertion l u) {ι : Sort x} {p : ι → Prop} (f : (i : ι) → p i → α) (hf : ∀ (i : ι) (hi : p i), u (l (f i hi)) = f i hi) :

l (⨅ i, ⨅ hi, f i hi) = ⨅ i, ⨅ hi, l (f i hi)

theorem GaloisInsertion.u_le_u_iff {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : Preorder α] [inst : Preorder β] (gi : GaloisInsertion l u) {a : β} {b : β} :

u a ≤ u b ↔ a ≤ b

theorem GaloisInsertion.strictMono_u {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : Preorder α] [inst : Preorder β] (gi : GaloisInsertion l u) :

theorem GaloisInsertion.isLUB_of_u_image {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : Preorder α] [inst : Preorder β] (gi : GaloisInsertion l u) {s : Set β} {a : α} (hs : IsLUB (u '' s) a) :

IsLUB s (l a)

theorem GaloisInsertion.isGLB_of_u_image {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : Preorder α] [inst : Preorder β] (gi : GaloisInsertion l u) {s : Set β} {a : α} (hs : IsGLB (u '' s) a) :

IsGLB s (l a)

def GaloisInsertion.liftSemilatticeSup {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : PartialOrder β] [inst : SemilatticeSup α] (gi : GaloisInsertion l u) :

SemilatticeSup β

Lift the suprema along a Galois insertion

Equations

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

def GaloisInsertion.liftSemilatticeInf {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : PartialOrder β] [inst : SemilatticeInf α] (gi : GaloisInsertion l u) :

SemilatticeInf β

Lift the infima along a Galois insertion

Equations

GaloisInsertion.liftSemilatticeInf gi = let src := inst; SemilatticeInf.mk (_ : ∀ (a b : β), a ⊓ b ≤ a) (_ : ∀ (a b : β), a ⊓ b ≤ b) (_ : ∀ (a b c : β), a ≤ b → a ≤ c → a ≤ b ⊓ c)

def GaloisInsertion.liftLattice {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : PartialOrder β] [inst : Lattice α] (gi : GaloisInsertion l u) :

Lift the suprema and infima along a Galois insertion

Equations

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

def GaloisInsertion.liftOrderTop {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : PartialOrder β] [inst : Preorder α] [inst : OrderTop α] (gi : GaloisInsertion l u) :

Lift the top along a Galois insertion

Equations

GaloisInsertion.liftOrderTop gi = OrderTop.mk (_ : ∀ (a : β), a ≤ ⊤)

def GaloisInsertion.liftBoundedOrder {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : PartialOrder β] [inst : Preorder α] [inst : BoundedOrder α] (gi : GaloisInsertion l u) :

BoundedOrder β

Lift the top, bottom, suprema, and infima along a Galois insertion

Equations

GaloisInsertion.liftBoundedOrder gi = let src := GaloisInsertion.liftOrderTop gi; let src_1 := GaloisConnection.liftOrderBot (_ : GaloisConnection l u); BoundedOrder.mk

def GaloisInsertion.liftCompleteLattice {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : PartialOrder β] [inst : CompleteLattice α] (gi : GaloisInsertion l u) :

CompleteLattice β

Lift all suprema and infima along a Galois insertion

Equations

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

structure GaloisCoinsertion {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] (l : α → β) (u : β → α) :

Type (maxuv)

A contructive choice function for images of u.
choice : (x : β) → x ≤ l (u x) → α
The Galois connection associated to a Galois coinsertion.
gc : GaloisConnection l u
Main property of a Galois coinsertion.
u_l_le : ∀ (x : α), u (l x) ≤ x
Property of the choice function.
choice_eq : ∀ (a : β) (h : a ≤ l (u a)), choice a h = u a

A Galois coinsertion is a Galois connection where u ∘ l = id∘ l = id. It also contains a constructive choice function, to give better definitional equalities when lifting order structures. Dual to GaloisInsertion

Instances For

def GaloisCoinsertion.dual {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} :

GaloisCoinsertion l u → GaloisInsertion (↑OrderDual.toDual ∘ u ∘ ↑OrderDual.ofDual) (↑OrderDual.toDual ∘ l ∘ ↑OrderDual.ofDual)

Make a GaloisInsertion between αᵒᵈ and βᵒᵈ from a GaloisCoinsertion between α and β.

Equations

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

def GaloisInsertion.dual {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} :

GaloisInsertion l u → GaloisCoinsertion (↑OrderDual.toDual ∘ u ∘ ↑OrderDual.ofDual) (↑OrderDual.toDual ∘ l ∘ ↑OrderDual.ofDual)

Make a GaloisCoinsertion between αᵒᵈ and βᵒᵈ from a GaloisInsertion between α and β.

Equations

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

def GaloisCoinsertion.ofDual {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : αᵒᵈ → βᵒᵈ} {u : βᵒᵈ → αᵒᵈ} :

GaloisCoinsertion l u → GaloisInsertion (↑OrderDual.ofDual ∘ u ∘ ↑OrderDual.toDual) (↑OrderDual.ofDual ∘ l ∘ ↑OrderDual.toDual)

Make a GaloisInsertion between α and β from a GaloisCoinsertion between αᵒᵈ and βᵒᵈ.

Equations

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

def GaloisInsertion.ofDual {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : αᵒᵈ → βᵒᵈ} {u : βᵒᵈ → αᵒᵈ} :

GaloisInsertion l u → GaloisCoinsertion (↑OrderDual.ofDual ∘ u ∘ ↑OrderDual.toDual) (↑OrderDual.ofDual ∘ l ∘ ↑OrderDual.toDual)

Make a GaloisCoinsertion between α and β from a GaloisInsertion between αᵒᵈ and βᵒᵈ.

Equations

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

def OrderIso.toGaloisCoinsertion {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] (oi : α ≃o β) :

GaloisCoinsertion ↑(RelIso.toRelEmbedding oi).toEmbedding ↑(RelIso.toRelEmbedding (OrderIso.symm oi)).toEmbedding

Makes a Galois coinsertion from an order-preserving bijection.

Equations

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

def GaloisCoinsertion.monotoneIntro {α : Type u} {β : Type v} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (hu : Monotone u) (hl : Monotone l) (hlu : ∀ (b : β), l (u b) ≤ b) (hul : ∀ (a : α), u (l a) = a) :

GaloisCoinsertion l u

A constructor for a Galois coinsertion with the trivial choice function.

Equations

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

def GaloisConnection.toGaloisCoinsertion {α : Type u_1} {β : Type u_2} [inst : Preorder α] [inst : Preorder β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) (h : ∀ (a : α), u (l a) ≤ a) :

GaloisCoinsertion l u

Make a GaloisCoinsertion l u from a GaloisConnection l u such that ∀ b, b ≤ l (u b)∀ b, b ≤ l (u b)≤ l (u b)

Equations

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

def GaloisConnection.liftOrderTop {α : Type u_1} {β : Type u_2} [inst : PartialOrder α] [inst : Preorder β] [inst : OrderTop β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) :

Lift the top along a Galois connection

Equations

GaloisConnection.liftOrderTop gc = OrderTop.mk (_ : ∀ (x : α), x ≤ u ⊤)

theorem GaloisCoinsertion.u_l_eq {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : PartialOrder α] [inst : Preorder β] (gi : GaloisCoinsertion l u) (a : α) :

u (l a) = a

theorem GaloisCoinsertion.u_l_leftInverse {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : PartialOrder α] [inst : Preorder β] (gi : GaloisCoinsertion l u) :

Function.LeftInverse u l

theorem GaloisCoinsertion.u_surjective {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : PartialOrder α] [inst : Preorder β] (gi : GaloisCoinsertion l u) :

Function.Surjective u

theorem GaloisCoinsertion.l_injective {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : PartialOrder α] [inst : Preorder β] (gi : GaloisCoinsertion l u) :

Function.Injective l

theorem GaloisCoinsertion.u_inf_l {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : SemilatticeInf α] [inst : SemilatticeInf β] (gi : GaloisCoinsertion l u) (a : α) (b : α) :

u (l a ⊓ l b) = a ⊓ b

theorem GaloisCoinsertion.u_infᵢ_l {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : CompleteLattice α] [inst : CompleteLattice β] (gi : GaloisCoinsertion l u) {ι : Sort x} (f : ι → α) :

u (⨅ i, l (f i)) = ⨅ i, f i

theorem GaloisCoinsertion.u_infₛ_l_image {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : CompleteLattice α] [inst : CompleteLattice β] (gi : GaloisCoinsertion l u) (s : Set α) :

u (infₛ (l '' s)) = infₛ s

theorem GaloisCoinsertion.u_sup_l {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : SemilatticeSup α] [inst : SemilatticeSup β] (gi : GaloisCoinsertion l u) (a : α) (b : α) :

u (l a ⊔ l b) = a ⊔ b

theorem GaloisCoinsertion.u_supᵢ_l {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : CompleteLattice α] [inst : CompleteLattice β] (gi : GaloisCoinsertion l u) {ι : Sort x} (f : ι → α) :

u (⨆ i, l (f i)) = ⨆ i, f i

theorem GaloisCoinsertion.u_bsupᵢ_l {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : CompleteLattice α] [inst : CompleteLattice β] (gi : GaloisCoinsertion l u) {ι : Sort x} {p : ι → Prop} (f : (i : ι) → p i → α) :

u (⨆ i, ⨆ hi, l (f i hi)) = ⨆ i, ⨆ hi, f i hi

theorem GaloisCoinsertion.u_supₛ_l_image {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : CompleteLattice α] [inst : CompleteLattice β] (gi : GaloisCoinsertion l u) (s : Set α) :

u (supₛ (l '' s)) = supₛ s

theorem GaloisCoinsertion.u_supᵢ_of_lu_eq_self {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : CompleteLattice α] [inst : CompleteLattice β] (gi : GaloisCoinsertion l u) {ι : Sort x} (f : ι → β) (hf : ∀ (i : ι), l (u (f i)) = f i) :

u (⨆ i, f i) = ⨆ i, u (f i)

theorem GaloisCoinsertion.u_bsupᵢ_of_lu_eq_self {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : CompleteLattice α] [inst : CompleteLattice β] (gi : GaloisCoinsertion l u) {ι : Sort x} {p : ι → Prop} (f : (i : ι) → p i → β) (hf : ∀ (i : ι) (hi : p i), l (u (f i hi)) = f i hi) :

u (⨆ i, ⨆ hi, f i hi) = ⨆ i, ⨆ hi, u (f i hi)

theorem GaloisCoinsertion.l_le_l_iff {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : Preorder α] [inst : Preorder β] (gi : GaloisCoinsertion l u) {a : α} {b : α} :

l a ≤ l b ↔ a ≤ b

theorem GaloisCoinsertion.strictMono_l {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : Preorder α] [inst : Preorder β] (gi : GaloisCoinsertion l u) :

theorem GaloisCoinsertion.isGLB_of_l_image {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : Preorder α] [inst : Preorder β] (gi : GaloisCoinsertion l u) {s : Set α} {a : β} (hs : IsGLB (l '' s) a) :

IsGLB s (u a)

theorem GaloisCoinsertion.isLUB_of_l_image {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : Preorder α] [inst : Preorder β] (gi : GaloisCoinsertion l u) {s : Set α} {a : β} (hs : IsLUB (l '' s) a) :

IsLUB s (u a)

def GaloisCoinsertion.liftSemilatticeInf {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : PartialOrder α] [inst : SemilatticeInf β] (gi : GaloisCoinsertion l u) :

SemilatticeInf α

Lift the infima along a Galois coinsertion

Equations

GaloisCoinsertion.liftSemilatticeInf gi = let src := inst; SemilatticeInf.mk (_ : ∀ (a b : α), a ⊓ b ≤ a) (_ : ∀ (a b : α), a ⊓ b ≤ b) (_ : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c)

def GaloisCoinsertion.liftSemilatticeSup {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : PartialOrder α] [inst : SemilatticeSup β] (gi : GaloisCoinsertion l u) :

SemilatticeSup α

Lift the suprema along a Galois coinsertion

Equations

GaloisCoinsertion.liftSemilatticeSup gi = let src := inst; SemilatticeSup.mk (_ : ∀ (a b : α), a ≤ a ⊔ b) (_ : ∀ (a b : α), b ≤ a ⊔ b) (_ : ∀ (a b c : α), a ≤ c → b ≤ c → a ⊔ b ≤ c)

def GaloisCoinsertion.liftLattice {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : PartialOrder α] [inst : Lattice β] (gi : GaloisCoinsertion l u) :

Lift the suprema and infima along a Galois coinsertion

Equations

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

def GaloisCoinsertion.liftOrderBot {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : PartialOrder α] [inst : Preorder β] [inst : OrderBot β] (gi : GaloisCoinsertion l u) :

Lift the bot along a Galois coinsertion

Equations

GaloisCoinsertion.liftOrderBot gi = let src := OrderDual.orderBot αᵒᵈ; OrderBot.mk (_ : ∀ (a : αᵒᵈᵒᵈ), ⊥ ≤ a)

def GaloisCoinsertion.liftBoundedOrder {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : PartialOrder α] [inst : Preorder β] [inst : BoundedOrder β] (gi : GaloisCoinsertion l u) :

BoundedOrder α

Lift the top, bottom, suprema, and infima along a Galois coinsertion

Equations

GaloisCoinsertion.liftBoundedOrder gi = let src := GaloisCoinsertion.liftOrderBot gi; let src_1 := GaloisConnection.liftOrderTop (_ : GaloisConnection l u); BoundedOrder.mk

def GaloisCoinsertion.liftCompleteLattice {α : Type u} {β : Type v} {l : α → β} {u : β → α} [inst : PartialOrder α] [inst : CompleteLattice β] (gi : GaloisCoinsertion l u) :

CompleteLattice α

Lift all suprema and infima along a Galois coinsertion

Equations

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

def WithBot.giUnbot'Bot {α : Type u} [inst : Preorder α] [inst : OrderBot α] :

GaloisInsertion (WithBot.unbot' ⊥) WithBot.some

If α is a partial order with bottom element (e.g., ℕ, ℝ≥0≥0), then WithBot.unbot' ⊥⊥ and coercion form a Galois insertion.

Equations

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