fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
Order.Frame (α : Type*) extends CompleteLattice α, HeytingAlgebra α where | class | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | Order.Frame | A frame, aka complete Heyting algebra, is a complete lattice whose `⊓` distributes over `⨆`. |
inf_sSup_eq {α : Type*} [Order.Frame α] {s : Set α} {a : α} :
a ⊓ sSup s = ⨆ b ∈ s, a ⊓ b :=
gc_inf_himp.l_sSup | theorem | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | inf_sSup_eq | `⊓` distributes over `⨆`. |
Order.Coframe (α : Type*) extends CompleteLattice α, CoheytingAlgebra α where | class | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | Order.Coframe | A coframe, aka complete Brouwer algebra or complete co-Heyting algebra, is a complete lattice
whose `⊔` distributes over `⨅`. |
sup_sInf_eq {α : Type*} [Order.Coframe α] {s : Set α} {a : α} :
a ⊔ sInf s = ⨅ b ∈ s, a ⊔ b :=
gc_sdiff_sup.u_sInf
open Order | theorem | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | sup_sInf_eq | `⊔` distributes over `⨅`. |
CompleteDistribLattice.MinimalAxioms (α : Type u)
extends CompleteLattice α,
toFrameMinimalAxioms : Frame.MinimalAxioms α,
toCoframeMinimalAxioms : Coframe.MinimalAxioms α where
attribute [nolint docBlame] CompleteDistribLattice.MinimalAxioms.toFrameMinimalAxioms
CompleteDistribLattice.MinimalAxioms.t... | structure | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | CompleteDistribLattice.MinimalAxioms | Structure containing the minimal axioms required to check that an order is a complete
distributive lattice. Do NOT use, except for implementing `CompleteDistribLattice` via
`CompleteDistribLattice.ofMinimalAxioms`.
This structure omits the `himp`, `compl`, `sdiff`, `hnot` fields, which can be recovered using
`Complete... |
CompleteDistribLattice (α : Type*) extends Frame α, Coframe α, BiheytingAlgebra α | class | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | CompleteDistribLattice | A complete distributive lattice is a complete lattice whose `⊔` and `⊓` respectively
distribute over `⨅` and `⨆`. |
CompletelyDistribLattice.MinimalAxioms (α : Type u) extends CompleteLattice α where
protected iInf_iSup_eq {ι : Type u} {κ : ι → Type u} (f : ∀ a, κ a → α) :
(⨅ a, ⨆ b, f a b) = ⨆ g : ∀ a, κ a, ⨅ a, f a (g a) | structure | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | CompletelyDistribLattice.MinimalAxioms | Structure containing the minimal axioms required to check that an order is a completely
distributive. Do NOT use, except for implementing `CompletelyDistribLattice` via
`CompletelyDistribLattice.ofMinimalAxioms`.
This structure omits the `himp`, `compl`, `sdiff`, `hnot` fields, which can be recovered using
`Completely... |
CompletelyDistribLattice (α : Type u) extends CompleteLattice α, BiheytingAlgebra α where
protected iInf_iSup_eq {ι : Type u} {κ : ι → Type u} (f : ∀ a, κ a → α) :
(⨅ a, ⨆ b, f a b) = ⨆ g : ∀ a, κ a, ⨅ a, f a (g a) | class | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | CompletelyDistribLattice | A completely distributive lattice is a complete lattice whose `⨅` and `⨆`
distribute over each other. |
le_iInf_iSup [CompleteLattice α] {f : ∀ a, κ a → α} :
(⨆ g : ∀ a, κ a, ⨅ a, f a (g a)) ≤ ⨅ a, ⨆ b, f a b :=
iSup_le fun _ => le_iInf fun a => le_trans (iInf_le _ a) (le_iSup _ _) | theorem | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | le_iInf_iSup | null |
iSup_iInf_le [CompleteLattice α] {f : ∀ a, κ a → α} :
⨆ a, ⨅ b, f a b ≤ ⨅ g : ∀ a, κ a, ⨆ a, f a (g a) :=
le_iInf_iSup (α := αᵒᵈ) | lemma | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | iSup_iInf_le | null |
inf_sSup_eq : a ⊓ sSup s = ⨆ b ∈ s, a ⊓ b :=
(minAx.inf_sSup_le_iSup_inf _ _).antisymm iSup_inf_le_inf_sSup | lemma | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | inf_sSup_eq | null |
sSup_inf_eq : sSup s ⊓ b = ⨆ a ∈ s, a ⊓ b := by
simpa only [inf_comm] using @inf_sSup_eq α _ s b | lemma | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | sSup_inf_eq | null |
iSup_inf_eq (f : ι → α) (a : α) : (⨆ i, f i) ⊓ a = ⨆ i, f i ⊓ a := by
rw [iSup, sSup_inf_eq, iSup_range] | lemma | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | iSup_inf_eq | null |
inf_iSup_eq (a : α) (f : ι → α) : (a ⊓ ⨆ i, f i) = ⨆ i, a ⊓ f i := by
simpa only [inf_comm] using minAx.iSup_inf_eq f a | lemma | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | inf_iSup_eq | null |
inf_iSup₂_eq {f : ∀ i, κ i → α} (a : α) : (a ⊓ ⨆ i, ⨆ j, f i j) = ⨆ i, ⨆ j, a ⊓ f i j := by
simp only [inf_iSup_eq] | lemma | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | inf_iSup₂_eq | null |
of [Frame α] : MinimalAxioms α where
__ := ‹Frame α›
inf_sSup_le_iSup_inf a s := _root_.inf_sSup_eq.le | def | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | of | The `Order.Frame.MinimalAxioms` element corresponding to a frame. |
of [Coframe α] : MinimalAxioms α where
__ := ‹Coframe α›
iInf_sup_le_sup_sInf a s := _root_.sup_sInf_eq.ge | def | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | of | Construct a frame instance using the minimal amount of work needed.
This sets `a ⇨ b := sSup {c | c ⊓ a ≤ b}` and `aᶜ := a ⇨ ⊥`. -/
-- See note [reducible non-instances]
abbrev ofMinimalAxioms (minAx : MinimalAxioms α) : Frame α where
__ := minAx
compl a := sSup {c | c ⊓ a ≤ ⊥}
himp a b := sSup {c | c ⊓ a ≤ b}
... |
of [CompleteDistribLattice α] : MinimalAxioms α where
__ := ‹CompleteDistribLattice α›
inf_sSup_le_iSup_inf a s:= inf_sSup_eq.le
iInf_sup_le_sup_sInf a s:= sup_sInf_eq.ge | def | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | of | Construct a coframe instance using the minimal amount of work needed.
This sets `a \ b := sInf {c | a ≤ b ⊔ c}` and `￢a := ⊤ \ a`. -/
-- See note [reducible non-instances]
abbrev ofMinimalAxioms (minAx : MinimalAxioms α) : Coframe α where
__ := minAx
hnot a := sInf {c | ⊤ ≤ a ⊔ c}
sdiff a b := sInf {c | a ≤ b ⊔ ... |
toFrame : Frame.MinimalAxioms α := minAx.toFrameMinimalAxioms | abbrev | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | toFrame | Turn minimal axioms for `CompleteDistribLattice` into minimal axioms for `Order.Frame`. |
toCoframe : Coframe.MinimalAxioms α where __ := minAx | abbrev | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | toCoframe | Turn minimal axioms for `CompleteDistribLattice` into minimal axioms for `Order.Coframe`. |
toCompleteDistribLattice : CompleteDistribLattice.MinimalAxioms α where
__ := minAx
inf_sSup_le_iSup_inf a s := by
let _ := minAx.toCompleteLattice
calc
_ = ⨅ i : ULift.{u} Bool, ⨆ j : match i with | .up true => PUnit.{u + 1} | .up false => s,
match i with
| .up true => a
... | abbrev | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | toCompleteDistribLattice | Construct a complete distrib lattice instance using the minimal amount of work needed.
This sets `a ⇨ b := sSup {c | c ⊓ a ≤ b}`, `aᶜ := a ⇨ ⊥`, `a \ b := sInf {c | a ≤ b ⊔ c}` and
`￢a := ⊤ \ a`. -/
-- See note [reducible non-instances]
abbrev ofMinimalAxioms (minAx : MinimalAxioms α) : CompleteDistribLattice α where
... |
of [CompletelyDistribLattice α] : MinimalAxioms α := { ‹CompletelyDistribLattice α› with } | def | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | of | The `CompletelyDistribLattice.MinimalAxioms` element corresponding to a frame. |
iSup_symmDiff_iSup_le {g : ι → α} : (⨆ i, f i) ∆ (⨆ i, g i) ≤ ⨆ i, ((f i) ∆ (g i)) := by
simp_rw [symmDiff_le_iff, ← iSup_sup_eq]
exact ⟨iSup_mono fun i ↦ sup_comm (g i) _ ▸ le_symmDiff_sup_right ..,
iSup_mono fun i ↦ sup_comm (f i) _ ▸ symmDiff_comm (f i) _ ▸ le_symmDiff_sup_right ..⟩
open scoped symmDiff in | theorem | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | iSup_symmDiff_iSup_le | Construct a completely distributive lattice instance using the minimal amount of work needed.
This sets `a ⇨ b := sSup {c | c ⊓ a ≤ b}`, `aᶜ := a ⇨ ⊥`, `a \ b := sInf {c | a ≤ b ⊔ c}` and
`￢a := ⊤ \ a`. -/
-- See note [reducible non-instances]
abbrev ofMinimalAxioms (minAx : MinimalAxioms α) : CompletelyDistribLattice... |
biSup_symmDiff_biSup_le {p : ι → Prop} {f g : (i : ι) → p i → α} :
(⨆ i, ⨆ (h : p i), f i h) ∆ (⨆ i, ⨆ (h : p i), g i h) ≤
⨆ i, ⨆ (h : p i), ((f i h) ∆ (g i h)) :=
le_trans iSup_symmDiff_iSup_le <| iSup_mono fun _ ↦ iSup_symmDiff_iSup_le | theorem | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | biSup_symmDiff_biSup_le | A `biSup` version of `iSup_symmDiff_iSup_le`. |
protected Function.Injective.frameMinimalAxioms [Max α] [Min α] [SupSet α] [InfSet α] [Top α]
[Bot α] (minAx : Frame.MinimalAxioms β) (f : α → β) (hf : Injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b)
(map_sSup : ∀ s, f (sSup s) = ⨆ a ∈ s, f a) (map_sInf : ∀ s, ... | abbrev | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | Function.Injective.frameMinimalAxioms | A complete atomic Boolean algebra is a complete Boolean algebra
that is also completely distributive.
We take iSup_iInf_eq as the definition here,
and prove later on that this implies atomicity.
-/
-- We do not directly extend `CompletelyDistribLattice` to avoid having the `hnot` field
class CompleteAtomicBooleanAlgeb... |
protected Function.Injective.coframeMinimalAxioms [Max α] [Min α] [SupSet α] [InfSet α]
[Top α] [Bot α] (minAx : Coframe.MinimalAxioms β) (f : α → β) (hf : Injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b)
(map_sSup : ∀ s, f (sSup s) = ⨆ a ∈ s, f a) (map_sInf : ∀... | abbrev | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | Function.Injective.coframeMinimalAxioms | Pullback an `Order.Coframe.MinimalAxioms` along an injection. |
protected Function.Injective.frame [Max α] [Min α] [SupSet α] [InfSet α] [Top α] [Bot α]
[HasCompl α] [HImp α] [Frame β] (f : α → β) (hf : Injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b)
(map_sSup : ∀ s, f (sSup s) = ⨆ a ∈ s, f a) (map_sInf : ∀ s, f (sInf s) = ... | abbrev | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | Function.Injective.frame | Pullback an `Order.Frame` along an injection. |
protected Function.Injective.coframe [Max α] [Min α] [SupSet α] [InfSet α] [Top α] [Bot α]
[HNot α] [SDiff α] [Coframe β] (f : α → β) (hf : Injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b)
(map_sSup : ∀ s, f (sSup s) = ⨆ a ∈ s, f a) (map_sInf : ∀ s, f (sInf s) =... | abbrev | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | Function.Injective.coframe | Pullback an `Order.Coframe` along an injection. |
protected Function.Injective.completeDistribLatticeMinimalAxioms [Max α] [Min α] [SupSet α]
[InfSet α] [Top α] [Bot α] (minAx : CompleteDistribLattice.MinimalAxioms β) (f : α → β)
(hf : Injective f) (map_sup : let _ := minAx.toCompleteLattice
∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : let _ := minAx.toCompl... | abbrev | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | Function.Injective.completeDistribLatticeMinimalAxioms | Pullback a `CompleteDistribLattice.MinimalAxioms` along an injection. |
protected Function.Injective.completeDistribLattice [Max α] [Min α] [SupSet α] [InfSet α]
[Top α] [Bot α] [HasCompl α] [HImp α] [HNot α] [SDiff α] [CompleteDistribLattice β] (f : α → β)
(hf : Injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b)
(map_sSup : ∀ s, ... | abbrev | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | Function.Injective.completeDistribLattice | Pullback a `CompleteDistribLattice` along an injection. |
protected Function.Injective.completelyDistribLatticeMinimalAxioms [Max α] [Min α] [SupSet α]
[InfSet α] [Top α] [Bot α] (minAx : CompletelyDistribLattice.MinimalAxioms β) (f : α → β)
(hf : Injective f) (map_sup : let _ := minAx.toCompleteLattice
∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : let _ := minAx.toC... | abbrev | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | Function.Injective.completelyDistribLatticeMinimalAxioms | Pullback a `CompletelyDistribLattice.MinimalAxioms` along an injection. |
protected Function.Injective.completelyDistribLattice [Max α] [Min α] [SupSet α] [InfSet α]
[Top α] [Bot α] [HasCompl α] [HImp α] [HNot α] [SDiff α] [CompletelyDistribLattice β]
(f : α → β) (hf : Injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b)
(map_sSup : ∀... | abbrev | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | Function.Injective.completelyDistribLattice | Pullback a `CompletelyDistribLattice` along an injection. |
protected Function.Injective.completeBooleanAlgebra [Max α] [Min α] [SupSet α] [InfSet α]
[Top α] [Bot α] [HasCompl α] [HImp α] [SDiff α] [CompleteBooleanAlgebra β] (f : α → β)
(hf : Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b)
(map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) (map_sSup : ∀ s, f (sSup s) = ... | abbrev | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | Function.Injective.completeBooleanAlgebra | Pullback a `CompleteBooleanAlgebra` along an injection. |
protected Function.Injective.completeAtomicBooleanAlgebra [Max α] [Min α] [SupSet α]
[InfSet α] [Top α] [Bot α] [HasCompl α] [HImp α] [HNot α] [SDiff α]
[CompleteAtomicBooleanAlgebra β] (f : α → β) (hf : Injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b)
(map_... | abbrev | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | Function.Injective.completeAtomicBooleanAlgebra | Pullback a `CompleteAtomicBooleanAlgebra` along an injection. |
instCompleteAtomicBooleanAlgebra : CompleteAtomicBooleanAlgebra PUnit where
__ := PUnit.instBooleanAlgebra
sSup _ := unit
sInf _ := unit
le_sSup _ _ _ := trivial
sSup_le _ _ _ := trivial
sInf_le _ _ _ := trivial
le_sInf _ _ _ := trivial
iInf_iSup_eq _ := rfl | instance | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | instCompleteAtomicBooleanAlgebra | null |
instCompleteBooleanAlgebra : CompleteBooleanAlgebra PUnit := inferInstance
@[simp] | instance | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | instCompleteBooleanAlgebra | null |
sSup_eq : sSup s = unit :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | sSup_eq | null |
sInf_eq : sInf s = unit :=
rfl | theorem | Order | [
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.CompleteLattice.Lemmas",
"Mathlib.Order.Directed",
"Mathlib.Order.GaloisConnection.Basic"
] | Mathlib/Order/CompleteBooleanAlgebra.lean | sInf_eq | null |
noncomputable subsetSupSet [Inhabited s] : SupSet s where
sSup t :=
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default
attribute [local instance] subsetSupSet
open Classical in
@[simp] | def | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.Interval.Set.OrdConnected"
] | Mathlib/Order/CompleteLatticeIntervals.lean | subsetSupSet | `SupSet` structure on a nonempty subset `s` of a preorder with `SupSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. |
subset_sSup_def [Inhabited s] :
@sSup s _ = fun t =>
if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
else default :=
rfl | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.Interval.Set.OrdConnected"
] | Mathlib/Order/CompleteLatticeIntervals.lean | subset_sSup_def | null |
subset_sSup_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [h, h', h''] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.Interval.Set.OrdConnected"
] | Mathlib/Order/CompleteLatticeIntervals.lean | subset_sSup_of_within | null |
subset_sSup_emptyset [Inhabited s] :
sSup (∅ : Set s) = default := by
simp [sSup] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.Interval.Set.OrdConnected"
] | Mathlib/Order/CompleteLatticeIntervals.lean | subset_sSup_emptyset | null |
subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
sSup t = default := by
simp [sSup, ht] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.Interval.Set.OrdConnected"
] | Mathlib/Order/CompleteLatticeIntervals.lean | subset_sSup_of_not_bddAbove | null |
noncomputable subsetInfSet [Inhabited s] : InfSet s where
sInf t :=
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩
else default
attribute [local instance] subsetInfSet
open Classical in
@[simp] | def | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.Interval.Set.OrdConnected"
] | Mathlib/Order/CompleteLatticeIntervals.lean | subsetInfSet | `InfSet` structure on a nonempty subset `s` of a preorder with `InfSet`. This definition is
non-canonical (it uses `default s`); it should be used only as here, as an auxiliary instance in the
construction of the `ConditionallyCompleteLinearOrder` structure. |
subset_sInf_def [Inhabited s] :
@sInf s _ = fun t =>
if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
default :=
rfl | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.Interval.Set.OrdConnected"
] | Mathlib/Order/CompleteLatticeIntervals.lean | subset_sInf_def | null |
subset_sInf_of_within [Inhabited s] {t : Set s}
(h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
sInf ((↑) '' t : Set α) = (@sInf s _ t : α) := by simp [h, h', h''] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.Interval.Set.OrdConnected"
] | Mathlib/Order/CompleteLatticeIntervals.lean | subset_sInf_of_within | null |
subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by
simp [sInf] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.Interval.Set.OrdConnected"
] | Mathlib/Order/CompleteLatticeIntervals.lean | subset_sInf_emptyset | null |
subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by
simp [sInf, ht] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.Interval.Set.OrdConnected"
] | Mathlib/Order/CompleteLatticeIntervals.lean | subset_sInf_of_not_bddBelow | null |
noncomputable subsetConditionallyCompleteLinearOrder [Inhabited s]
(h_Sup : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddAbove t), sSup ((↑) '' t : Set α) ∈ s)
(h_Inf : ∀ {t : Set s} (_ : t.Nonempty) (_h_bdd : BddBelow t), sInf ((↑) '' t : Set α) ∈ s) :
ConditionallyCompleteLinearOrder s :=
{ subsetSupSet ... | abbrev | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.Interval.Set.OrdConnected"
] | Mathlib/Order/CompleteLatticeIntervals.lean | subsetConditionallyCompleteLinearOrder | For a nonempty subset of a conditionally complete linear order to be a conditionally complete
linear order, it suffices that it contain the `sSup` of all its nonempty bounded-above subsets, and
the `sInf` of all its nonempty bounded-below subsets.
See note [reducible non-instances]. |
sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ upperBounds t := h_bdd
refine hs.out c.2 B.2 ⟨?_, ?_⟩
· exact (Subtype.mono_coe s).le_csSup_image ... | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.Interval.Set.OrdConnected"
] | Mathlib/Order/CompleteLatticeIntervals.lean | sSup_within_of_ordConnected | The `sSup` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-above subsets of `s`. |
sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
(h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s := by
obtain ⟨c, hct⟩ : ∃ c, c ∈ t := ht
obtain ⟨B, hB⟩ : ∃ B, B ∈ lowerBounds t := h_bdd
refine hs.out B.2 c.2 ⟨?_, ?_⟩
· exact (Subtype.mono_coe s).le_csInf_image ... | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.Interval.Set.OrdConnected"
] | Mathlib/Order/CompleteLatticeIntervals.lean | sInf_within_of_ordConnected | The `sInf` function on a nonempty `OrdConnected` set `s` in a conditionally complete linear
order takes values within `s`, for all nonempty bounded-below subsets of `s`. |
noncomputable ordConnectedSubsetConditionallyCompleteLinearOrder [Inhabited s]
[OrdConnected s] : ConditionallyCompleteLinearOrder s :=
subsetConditionallyCompleteLinearOrder s
(fun h => sSup_within_of_ordConnected h)
(fun h => sInf_within_of_ordConnected h) | instance | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.Interval.Set.OrdConnected"
] | Mathlib/Order/CompleteLatticeIntervals.lean | ordConnectedSubsetConditionallyCompleteLinearOrder | A nonempty `OrdConnected` set in a conditionally complete linear order is naturally a
conditionally complete linear order. |
noncomputable Set.Icc.completeLattice [ConditionallyCompleteLattice α]
{a b : α} [Fact (a ≤ b)] : CompleteLattice (Set.Icc a b) where
__ := (inferInstance : BoundedOrder ↑(Icc a b))
sSup S := if hS : S = ∅ then ⟨a, le_rfl, Fact.out⟩ else ⟨sSup ((↑) '' S), by
rw [← Set.not_nonempty_iff_eq_empty, not_not] at ... | instance | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.Interval.Set.OrdConnected"
] | Mathlib/Order/CompleteLatticeIntervals.lean | Set.Icc.completeLattice | Complete lattice structure on `Set.Icc` |
Set.Icc.coe_sSup [ConditionallyCompleteLattice α] {a b : α} (h : a ≤ b)
{S : Set (Set.Icc a b)} (hS : S.Nonempty) : have : Fact (a ≤ b) := ⟨h⟩
↑(sSup S) = sSup ((↑) '' S : Set α) :=
congrArg Subtype.val (dif_neg hS.ne_empty) | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.Interval.Set.OrdConnected"
] | Mathlib/Order/CompleteLatticeIntervals.lean | Set.Icc.coe_sSup | null |
Set.Icc.coe_sInf [ConditionallyCompleteLattice α] {a b : α} (h : a ≤ b)
{S : Set (Set.Icc a b)} (hS : S.Nonempty) : have : Fact (a ≤ b) := ⟨h⟩
↑(sInf S) = sInf ((↑) '' S : Set α) :=
congrArg Subtype.val (dif_neg hS.ne_empty) | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.Interval.Set.OrdConnected"
] | Mathlib/Order/CompleteLatticeIntervals.lean | Set.Icc.coe_sInf | null |
Set.Icc.coe_iSup [ConditionallyCompleteLattice α] {a b : α} (h : a ≤ b)
[Nonempty ι] {S : ι → Set.Icc a b} : have : Fact (a ≤ b) := ⟨h⟩
↑(iSup S) = (⨆ i, S i : α) :=
(Set.Icc.coe_sSup h (range_nonempty S)).trans (congrArg sSup (range_comp Subtype.val S).symm) | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.Interval.Set.OrdConnected"
] | Mathlib/Order/CompleteLatticeIntervals.lean | Set.Icc.coe_iSup | null |
Set.Icc.coe_iInf [ConditionallyCompleteLattice α] {a b : α} (h : a ≤ b)
[Nonempty ι] {S : ι → Set.Icc a b} : have : Fact (a ≤ b) := ⟨h⟩
↑(iInf S) = (⨅ i, S i : α) :=
(Set.Icc.coe_sInf h (range_nonempty S)).trans (congrArg sInf (range_comp Subtype.val S).symm) | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.Interval.Set.OrdConnected"
] | Mathlib/Order/CompleteLatticeIntervals.lean | Set.Icc.coe_iInf | null |
instCompleteLattice : CompleteLattice (Iic a) where
sSup S := ⟨sSup ((↑) '' S), by simpa using fun b hb _ ↦ hb⟩
sInf S := ⟨a ⊓ sInf ((↑) '' S), by simp⟩
le_sSup _ _ hb := le_sSup <| mem_image_of_mem Subtype.val hb
sSup_le _ _ hb := sSup_le <| fun _ ⟨c, hc, hc'⟩ ↦ hc' ▸ hb c hc
sInf_le _ _ hb := inf_le_of_righ... | instance | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.Interval.Set.OrdConnected"
] | Mathlib/Order/CompleteLatticeIntervals.lean | instCompleteLattice | null |
coe_biSup : (↑(⨆ i, ⨆ (_ : p i), f i) : α) = ⨆ i, ⨆ (_ : p i), (f i : α) := by simp
@[simp] theorem coe_sInf : (↑(sInf S) : α) = a ⊓ sInf ((↑) '' S) := rfl
@[simp] theorem coe_iInf : (↑(⨅ i, f i) : α) = a ⊓ ⨅ i, (f i : α) := by
rw [iInf, coe_sInf]; congr; ext; simp | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.Interval.Set.OrdConnected"
] | Mathlib/Order/CompleteLatticeIntervals.lean | coe_biSup | null |
coe_biInf : (↑(⨅ i, ⨅ (_ : p i), f i) : α) = a ⊓ ⨅ i, ⨅ (_ : p i), (f i : α) := by
cases isEmpty_or_nonempty ι
· simp
· simp_rw [coe_iInf, ← inf_iInf, ← inf_assoc, inf_idem] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.Interval.Set.OrdConnected"
] | Mathlib/Order/CompleteLatticeIntervals.lean | coe_biInf | null |
CompletePartialOrder (α : Type*) extends PartialOrder α, SupSet α where
/-- For each directed set `d`, `sSup d` is the least upper bound of `d`. -/
lubOfDirected : ∀ d, DirectedOn (· ≤ ·) d → IsLUB d (sSup d)
variable [CompletePartialOrder α] [Preorder β] {f : ι → α} {d : Set α} {a : α} | class | Order | [
"Mathlib.Order.OmegaCompletePartialOrder"
] | Mathlib/Order/CompletePartialOrder.lean | CompletePartialOrder | Complete partial orders are partial orders where every directed set has a least upper bound. |
protected DirectedOn.isLUB_sSup : DirectedOn (· ≤ ·) d → IsLUB d (sSup d) :=
CompletePartialOrder.lubOfDirected _ | lemma | Order | [
"Mathlib.Order.OmegaCompletePartialOrder"
] | Mathlib/Order/CompletePartialOrder.lean | DirectedOn.isLUB_sSup | null |
protected DirectedOn.le_sSup (hd : DirectedOn (· ≤ ·) d) (ha : a ∈ d) : a ≤ sSup d :=
hd.isLUB_sSup.1 ha | lemma | Order | [
"Mathlib.Order.OmegaCompletePartialOrder"
] | Mathlib/Order/CompletePartialOrder.lean | DirectedOn.le_sSup | null |
protected DirectedOn.sSup_le (hd : DirectedOn (· ≤ ·) d) (ha : ∀ b ∈ d, b ≤ a) : sSup d ≤ a :=
hd.isLUB_sSup.2 ha | lemma | Order | [
"Mathlib.Order.OmegaCompletePartialOrder"
] | Mathlib/Order/CompletePartialOrder.lean | DirectedOn.sSup_le | null |
protected Directed.le_iSup (hf : Directed (· ≤ ·) f) (i : ι) : f i ≤ ⨆ j, f j :=
hf.directedOn_range.le_sSup <| Set.mem_range_self _ | lemma | Order | [
"Mathlib.Order.OmegaCompletePartialOrder"
] | Mathlib/Order/CompletePartialOrder.lean | Directed.le_iSup | null |
protected Directed.iSup_le (hf : Directed (· ≤ ·) f) (ha : ∀ i, f i ≤ a) : ⨆ i, f i ≤ a :=
hf.directedOn_range.sSup_le <| Set.forall_mem_range.2 ha | lemma | Order | [
"Mathlib.Order.OmegaCompletePartialOrder"
] | Mathlib/Order/CompletePartialOrder.lean | Directed.iSup_le | null |
CompletePartialOrder.scottContinuous {f : α → β} :
ScottContinuous f ↔
∀ ⦃d : Set α⦄, d.Nonempty → DirectedOn (· ≤ ·) d → IsLUB (f '' d) (f (sSup d)) := by
refine ⟨fun h d hd₁ hd₂ ↦ h hd₁ hd₂ hd₂.isLUB_sSup, fun h d hne hd a hda ↦ ?_⟩
rw [hda.unique hd.isLUB_sSup]
exact h hne hd
open OmegaCompletePartialO... | lemma | Order | [
"Mathlib.Order.OmegaCompletePartialOrder"
] | Mathlib/Order/CompletePartialOrder.lean | CompletePartialOrder.scottContinuous | Scott-continuity takes on a simpler form in complete partial orders. |
CompletePartialOrder.toOmegaCompletePartialOrder : OmegaCompletePartialOrder α where
ωSup c := ⨆ n, c n
le_ωSup c := c.directed.le_iSup
ωSup_le c _ := c.directed.iSup_le | instance | Order | [
"Mathlib.Order.OmegaCompletePartialOrder"
] | Mathlib/Order/CompletePartialOrder.lean | CompletePartialOrder.toOmegaCompletePartialOrder | A complete partial order is an ω-complete partial order. |
CompleteLattice.toCompletePartialOrder [CompleteLattice α] : CompletePartialOrder α where
sSup := sSup
lubOfDirected _ _ := isLUB_sSup _ | instance | Order | [
"Mathlib.Order.OmegaCompletePartialOrder"
] | Mathlib/Order/CompletePartialOrder.lean | CompleteLattice.toCompletePartialOrder | A complete lattice is a complete partial order. |
CompleteSublattice extends Sublattice α where
sSupClosed' : ∀ ⦃s : Set α⦄, s ⊆ carrier → sSup s ∈ carrier
sInfClosed' : ∀ ⦃s : Set α⦄, s ⊆ carrier → sInf s ∈ carrier
variable {α β} | structure | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | CompleteSublattice | A complete sublattice is a subset of a complete lattice that is closed under arbitrary suprema
and infima. |
@[simps] mk' (carrier : Set α)
(sSupClosed' : ∀ ⦃s : Set α⦄, s ⊆ carrier → sSup s ∈ carrier)
(sInfClosed' : ∀ ⦃s : Set α⦄, s ⊆ carrier → sInf s ∈ carrier) :
CompleteSublattice α where
carrier := carrier
sSupClosed' := sSupClosed'
sInfClosed' := sInfClosed'
supClosed' := fun x hx y hy ↦ by
suffic... | def | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | mk' | To check that a subset is a complete sublattice, one does not need to check that it is closed
under binary `Sup` since this follows from the stronger `sSup` condition. Likewise for infima. |
instSetLike : SetLike (CompleteSublattice α) α where
coe L := L.carrier
coe_injective' L M h := by cases L; cases M; congr; exact SetLike.coe_injective' h | instance | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | instSetLike | null |
top_mem : ⊤ ∈ L := by simpa using L.sInfClosed' <| empty_subset _ | theorem | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | top_mem | null |
bot_mem : ⊥ ∈ L := by simpa using L.sSupClosed' <| empty_subset _ | theorem | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | bot_mem | null |
instBot : Bot L where
bot := ⟨⊥, bot_mem⟩ | instance | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | instBot | null |
instTop : Top L where
top := ⟨⊤, top_mem⟩ | instance | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | instTop | null |
instSupSet : SupSet L where
sSup s := ⟨sSup <| (↑) '' s, L.sSupClosed' image_val_subset⟩ | instance | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | instSupSet | null |
instInfSet : InfSet L where
sInf s := ⟨sInf <| (↑) '' s, L.sInfClosed' image_val_subset⟩ | instance | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | instInfSet | null |
sSupClosed {s : Set α} (h : s ⊆ L) : sSup s ∈ L := L.sSupClosed' h | theorem | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | sSupClosed | null |
sInfClosed {s : Set α} (h : s ⊆ L) : sInf s ∈ L := L.sInfClosed' h
@[simp] theorem coe_bot : (↑(⊥ : L) : α) = ⊥ := rfl
@[simp] theorem coe_top : (↑(⊤ : L) : α) = ⊤ := rfl
@[simp] theorem coe_sSup (S : Set L) : (↑(sSup S) : α) = sSup {(s : α) | s ∈ S} := rfl | theorem | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | sInfClosed | null |
coe_sSup' (S : Set L) : (↑(sSup S) : α) = ⨆ N ∈ S, (N : α) := by
rw [coe_sSup, ← Set.image, sSup_image]
@[simp] theorem coe_sInf (S : Set L) : (↑(sInf S) : α) = sInf {(s : α) | s ∈ S} := rfl | theorem | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | coe_sSup' | null |
coe_sInf' (S : Set L) : (↑(sInf S) : α) = ⨅ N ∈ S, (N : α) := by
rw [coe_sInf, ← Set.image, sInf_image]
@[simp] theorem coe_iSup {ι} (f : ι → L) : (↑(iSup f) : α) = ⨆ i, (f i : α) := by
rw [iSup, coe_sSup', iSup_range]
@[simp] theorem coe_iInf {ι} (f : ι → L) : (↑(iInf f) : α) = ⨅ i, (f i : α) := by
rw [iInf, coe... | theorem | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | coe_sInf' | null |
instCompleteLattice : CompleteLattice L :=
Subtype.coe_injective.completeLattice _
Sublattice.coe_sup Sublattice.coe_inf coe_sSup' coe_sInf' coe_top coe_bot | instance | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | instCompleteLattice | null |
subtype (L : CompleteSublattice α) : CompleteLatticeHom L α where
toFun := Subtype.val
map_sInf' _ := rfl
map_sSup' _ := rfl
@[simp, norm_cast] lemma coe_subtype (L : CompleteSublattice α) : L.subtype = ((↑) : L → α) := rfl | def | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | subtype | The natural complete lattice hom from a complete sublattice to the original lattice. |
subtype_apply (L : Sublattice α) (a : L) : L.subtype a = a := rfl | lemma | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | subtype_apply | null |
subtype_injective (L : CompleteSublattice α) :
Injective <| subtype L := Subtype.coe_injective | lemma | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | subtype_injective | null |
@[simps] map (L : CompleteSublattice α) : CompleteSublattice β where
carrier := f '' L
supClosed' := L.supClosed.image f
infClosed' := L.infClosed.image f
sSupClosed' := fun s hs ↦ by
obtain ⟨t, ht, rfl⟩ := subset_image_iff.mp hs
rw [← map_sSup]
exact mem_image_of_mem f (sSupClosed ht)
sInfClosed'... | def | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | map | The push forward of a complete sublattice under a complete lattice hom is a complete
sublattice. |
@[simps] comap (L : CompleteSublattice β) : CompleteSublattice α where
carrier := f ⁻¹' L
supClosed' := L.supClosed.preimage f
infClosed' := L.infClosed.preimage f
sSupClosed' s hs := by
simpa only [mem_preimage, map_sSup, SetLike.mem_coe] using sSupClosed
<| mapsTo_iff_image_subset.mp hs
sInfClosed... | def | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | comap | The pull back of a complete sublattice under a complete lattice hom is a complete sublattice. |
protected disjoint_iff {a b : L} :
Disjoint a b ↔ Disjoint (a : α) (b : α) := by
rw [disjoint_iff, disjoint_iff, ← Sublattice.coe_inf, ← coe_bot (L := L),
Subtype.coe_injective.eq_iff] | lemma | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | disjoint_iff | null |
protected codisjoint_iff {a b : L} :
Codisjoint a b ↔ Codisjoint (a : α) (b : α) := by
rw [codisjoint_iff, codisjoint_iff, ← Sublattice.coe_sup, ← coe_top (L := L),
Subtype.coe_injective.eq_iff] | lemma | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | codisjoint_iff | null |
protected isCompl_iff {a b : L} :
IsCompl a b ↔ IsCompl (a : α) (b : α) := by
rw [isCompl_iff, isCompl_iff, CompleteSublattice.disjoint_iff, CompleteSublattice.codisjoint_iff] | lemma | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | isCompl_iff | null |
isComplemented_iff : ComplementedLattice L ↔ ∀ a ∈ L, ∃ b ∈ L, IsCompl a b := by
refine ⟨fun ⟨h⟩ a ha ↦ ?_, fun h ↦ ⟨fun ⟨a, ha⟩ ↦ ?_⟩⟩
· obtain ⟨b, hb⟩ := h ⟨a, ha⟩
exact ⟨b, b.property, CompleteSublattice.isCompl_iff.mp hb⟩
· obtain ⟨b, hb, hb'⟩ := h a ha
exact ⟨⟨b, hb⟩, CompleteSublattice.isCompl_iff.m... | lemma | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | isComplemented_iff | null |
protected copy (s : Set α) (hs : s = L) : CompleteSublattice α :=
mk' s (hs ▸ L.sSupClosed') (hs ▸ L.sInfClosed')
@[simp, norm_cast] lemma coe_copy (s : Set α) (hs) : L.copy s hs = s := rfl | def | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | copy | Copy of a complete sublattice with a new `carrier` equal to the old one. Useful to fix
definitional equalities. |
copy_eq (s : Set α) (hs) : L.copy s hs = L := SetLike.coe_injective hs | lemma | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | copy_eq | null |
protected range : CompleteSublattice β :=
(CompleteSublattice.map f ⊤).copy (range f) image_univ.symm | def | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | range | The range of a `CompleteLatticeHom` is a `CompleteSublattice`.
See Note [range copy pattern]. |
range_coe : (f.range : Set β) = range f := rfl | theorem | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | range_coe | null |
@[simps! apply] noncomputable toOrderIsoRangeOfInjective (hf : Injective f) : α ≃o f.range :=
(orderEmbeddingOfInjective f hf).orderIso | def | Order | [
"Mathlib.Data.Set.Functor",
"Mathlib.Order.Sublattice",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/CompleteSublattice.lean | toOrderIsoRangeOfInjective | We can regard a complete lattice homomorphism as an order equivalence to its range. |
upperPolar (s : Set α) : Set β :=
{ b | ∀ ⦃a⦄, a ∈ s → r a b }
@[deprecated (since := "2025-07-10")]
alias intentClosure := upperPolar | def | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | upperPolar | The upper polar of `s : Set α` along a relation `r : α → β → Prop` is the set of all elements
which `r` relates to all elements of `s`. |
lowerPolar (t : Set β) : Set α :=
{ a | ∀ ⦃b⦄, b ∈ t → r a b }
@[deprecated (since := "2025-07-10")]
alias extentClosure := lowerPolar
variable {r} | def | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | lowerPolar | The lower polar of `t : Set β` along a relation `r : α → β → Prop` is the set of all elements
which `r` relates to all elements of `t`. |
subset_upperPolar_iff_subset_lowerPolar :
t ⊆ upperPolar r s ↔ s ⊆ lowerPolar r t :=
⟨fun h _ ha _ hb => h hb ha, fun h _ hb _ ha => h ha hb⟩
@[deprecated (since := "2025-07-10")]
alias subset_intentClosure_iff_subset_extentClosure := subset_upperPolar_iff_subset_lowerPolar
variable (r) | theorem | Order | [
"Mathlib.Data.Set.Lattice"
] | Mathlib/Order/Concept.lean | subset_upperPolar_iff_subset_lowerPolar | null |
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