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 ⌀ |
|---|---|---|---|---|---|---|
instDistribLattice [DistribLattice α] [DistribLattice β] : DistribLattice (α × β) where
le_sup_inf _ _ _ := ⟨le_sup_inf, le_sup_inf⟩ | instance | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | instDistribLattice | null |
protected semilatticeSup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) :
SemilatticeSup { x : α // P x } where
sup x y := ⟨x.1 ⊔ y.1, Psup x.2 y.2⟩
le_sup_left _ _ := le_sup_left
le_sup_right _ _ := le_sup_right
sup_le _ _ _ h1 h2 := sup_le h1 h2 | abbrev | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | semilatticeSup | A subtype forms a `⊔`-semilattice if `⊔` preserves the property.
See note [reducible non-instances]. |
protected semilatticeInf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
SemilatticeInf { x : α // P x } where
inf x y := ⟨x.1 ⊓ y.1, Pinf x.2 y.2⟩
inf_le_left _ _ := inf_le_left
inf_le_right _ _ := inf_le_right
le_inf _ _ _ h1 h2 := le_inf h1 h2 | abbrev | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | semilatticeInf | A subtype forms a `⊓`-semilattice if `⊓` preserves the property.
See note [reducible non-instances]. |
protected lattice [Lattice α] {P : α → Prop} (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y))
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) : Lattice { x : α // P x } where
__ := Subtype.semilatticeInf Pinf
__ := Subtype.semilatticeSup Psup
@[simp, norm_cast] | abbrev | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | lattice | A subtype forms a lattice if `⊔` and `⊓` preserve the property.
See note [reducible non-instances]. |
coe_sup [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeSup Psup; (x ⊔ y : Subtype P) : α) = (x ⊔ y : α) :=
rfl
@[simp, norm_cast] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | coe_sup | null |
coe_inf [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P) :
(haveI := Subtype.semilatticeInf Pinf; (x ⊓ y : Subtype P) : α) = (x ⊓ y : α) :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | coe_inf | null |
mk_sup_mk [SemilatticeSup α] {P : α → Prop}
(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeSup Psup; (⟨x, hx⟩ ⊔ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊔ y, Psup hx hy⟩ :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | mk_sup_mk | null |
mk_inf_mk [SemilatticeInf α] {P : α → Prop}
(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y) :
(haveI := Subtype.semilatticeInf Pinf; (⟨x, hx⟩ ⊓ ⟨y, hy⟩ : Subtype P)) =
⟨x ⊓ y, Pinf hx hy⟩ :=
rfl | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | mk_inf_mk | null |
protected Function.Injective.semilatticeSup [Max α] [SemilatticeSup β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) :
SemilatticeSup α where
__ := PartialOrder.lift f hf_inj
sup a b := max a b
le_sup_left a b := by
change f a ≤ f (a ⊔ b)
rw [map_sup]
exa... | abbrev | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | Function.Injective.semilatticeSup | A type endowed with `⊔` is a `SemilatticeSup`, if it admits an injective map that
preserves `⊔` to a `SemilatticeSup`.
See note [reducible non-instances]. |
protected Function.Injective.semilatticeInf [Min α] [SemilatticeInf β] (f : α → β)
(hf_inj : Function.Injective f) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) :
SemilatticeInf α where
__ := PartialOrder.lift f hf_inj
inf a b := min a b
inf_le_left a b := by
change f (a ⊓ b) ≤ f a
rw [map_inf]
exa... | abbrev | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | Function.Injective.semilatticeInf | A type endowed with `⊓` is a `SemilatticeInf`, if it admits an injective map that
preserves `⊓` to a `SemilatticeInf`.
See note [reducible non-instances]. |
protected Function.Injective.lattice [Max α] [Min α] [Lattice β] (f : α → β)
(hf_inj : Function.Injective f)
(map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) :
Lattice α where
__ := hf_inj.semilatticeSup f map_sup
__ := hf_inj.semilatticeInf f map_inf | abbrev | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | Function.Injective.lattice | A type endowed with `⊔` and `⊓` is a `Lattice`, if it admits an injective map that
preserves `⊔` and `⊓` to a `Lattice`.
See note [reducible non-instances]. |
protected Function.Injective.distribLattice [Max α] [Min α] [DistribLattice β] (f : α → β)
(hf_inj : Function.Injective f) (map_sup : ∀ a b, f (a ⊔ b) = f a ⊔ f b)
(map_inf : ∀ a b, f (a ⊓ b) = f a ⊓ f b) :
DistribLattice α where
__ := hf_inj.lattice f map_sup map_inf
le_sup_inf a b c := by
change f... | abbrev | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | Function.Injective.distribLattice | A type endowed with `⊔` and `⊓` is a `DistribLattice`, if it admits an injective map that
preserves `⊔` and `⊓` to a `DistribLattice`.
See note [reducible non-instances]. |
Bool.instPartialOrder : PartialOrder Bool := inferInstance | instance | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | Bool.instPartialOrder | null |
Bool.instDistribLattice : DistribLattice Bool := inferInstance | instance | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | Bool.instDistribLattice | null |
semilatticeInf [SemilatticeInf α] {a b : α} : SemilatticeInf (Ico a b) :=
Subtype.semilatticeInf fun _ _ hx hy => ⟨le_inf hx.1 hy.1, lt_of_le_of_lt inf_le_left hx.2⟩
@[simp, norm_cast] | instance | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | semilatticeInf | null |
protected coe_inf [SemilatticeInf α] {a b : α} {x y : Ico a b} :
↑(x ⊓ y) = (↑x ⊓ ↑y : α) :=
rfl | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | coe_inf | null |
orderBot [PartialOrder α] {a b : α} [Fact (a < b)] : OrderBot (Ico a b) :=
(isLeast_Ico Fact.out).orderBot
@[simp, norm_cast] | instance | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | orderBot | `Ico a b` has a bottom element whenever `a < b`. |
protected coe_bot [PartialOrder α] (a b : α) [Fact (a < b)] : ↑(⊥ : Ico a b) = a := rfl | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | coe_bot | null |
protected disjoint_iff [SemilatticeInf α] {a b : α} [Fact (a < b)] {x y : Ico a b} :
Disjoint x y ↔ ↑x ⊓ ↑y = a := by
simp [_root_.disjoint_iff, Subtype.ext_iff] | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | disjoint_iff | null |
semilatticeInf [SemilatticeInf α] {a : α} : SemilatticeInf (Iio a) :=
Subtype.semilatticeInf fun _ _ hx _ => lt_of_le_of_lt inf_le_left hx
@[simp, norm_cast] | instance | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | semilatticeInf | null |
protected coe_inf [SemilatticeInf α] {a : α} {x y : Iio a} :
↑(x ⊓ y) = (↑x ⊓ ↑y : α) :=
rfl | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | coe_inf | null |
semilatticeSup [SemilatticeSup α] {a b : α} : SemilatticeSup (Ioc a b) :=
Subtype.semilatticeSup fun _ _ hx hy => ⟨lt_of_lt_of_le hx.1 le_sup_left, sup_le hx.2 hy.2⟩
@[simp, norm_cast] | instance | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | semilatticeSup | null |
protected coe_sup [SemilatticeSup α] {a b : α} {x y : Ioc a b} :
↑(x ⊔ y) = (↑x ⊔ ↑y : α) :=
rfl | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | coe_sup | null |
orderTop [PartialOrder α] {a b : α} [Fact (a < b)] : OrderTop (Ioc a b) :=
(isGreatest_Ioc Fact.out).orderTop
@[simp, norm_cast] | instance | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | orderTop | `Ioc a b` has a top element whenever `a < b`. |
protected coe_top [PartialOrder α] (a b : α) [Fact (a < b)] : ↑(⊤ : Ioc a b) = b := rfl | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | coe_top | null |
protected codisjoint_iff [SemilatticeSup α] {a b : α} [Fact (a < b)] {x y : Ioc a b} :
Codisjoint x y ↔ ↑x ⊔ ↑y = b := by
simp [_root_.codisjoint_iff, Subtype.ext_iff] | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | codisjoint_iff | null |
semilatticeSup [SemilatticeSup α] {a : α} : SemilatticeSup (Ioi a) :=
Subtype.semilatticeSup fun _ _ hx _ => lt_of_lt_of_le hx le_sup_left
@[simp, norm_cast] | instance | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | semilatticeSup | null |
protected coe_sup [SemilatticeSup α] {a : α} {x y : Ioi a} :
↑(x ⊔ y) = (↑x ⊔ ↑y : α) :=
rfl | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | coe_sup | null |
semilatticeInf [SemilatticeInf α] : SemilatticeInf (Iic a) :=
Subtype.semilatticeInf fun _ _ hx _ => le_trans inf_le_left hx
@[simp, norm_cast] | instance | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | semilatticeInf | null |
protected coe_inf [SemilatticeInf α] {x y : Iic a} :
↑(x ⊓ y) = (↑x ⊓ ↑y : α) :=
rfl | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | coe_inf | null |
semilatticeSup [SemilatticeSup α] : SemilatticeSup (Iic a) :=
Subtype.semilatticeSup fun _ _ hx hy => sup_le hx hy
@[simp, norm_cast] | instance | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | semilatticeSup | null |
protected coe_sup [SemilatticeSup α] {x y : Iic a} :
↑(x ⊔ y) = (↑x ⊔ ↑y : α) :=
rfl | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | coe_sup | null |
orderTop [Preorder α] :
OrderTop (Iic a) where
top := ⟨a, le_refl a⟩
le_top x := x.prop
@[simp, norm_cast] | instance | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | orderTop | null |
protected coe_top [Preorder α] (a : α) : (⊤ : Iic a) = a := rfl | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | coe_top | null |
protected eq_top_iff [Preorder α] {x : Iic a} :
x = ⊤ ↔ (x : α) = a := by
simp [Subtype.ext_iff] | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | eq_top_iff | null |
orderBot [Preorder α] [OrderBot α] :
OrderBot (Iic a) where
bot := ⟨⊥, bot_le⟩
bot_le := fun ⟨_, _⟩ => Subtype.mk_le_mk.2 bot_le
@[simp, norm_cast] | instance | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | orderBot | null |
protected coe_bot [Preorder α] [OrderBot α] (a : α) : (⊥ : Iic a) = (⊥ : α) := rfl | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | coe_bot | null |
protected disjoint_iff [SemilatticeInf α] [OrderBot α] {x y : Iic a} :
Disjoint x y ↔ Disjoint (x : α) (y : α) := by
simp [_root_.disjoint_iff, Subtype.ext_iff] | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | disjoint_iff | null |
protected codisjoint_iff [SemilatticeSup α] {x y : Iic a} :
Codisjoint x y ↔ ↑x ⊔ ↑y = a := by
simpa only [_root_.codisjoint_iff] using Iic.eq_top_iff | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | codisjoint_iff | null |
protected isCompl_iff [Lattice α] [OrderBot α] {x y : Iic a} :
IsCompl x y ↔ Disjoint (x : α) (y : α) ∧ ↑x ⊔ ↑y = a := by
rw [_root_.isCompl_iff, Iic.disjoint_iff, Iic.codisjoint_iff] | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | isCompl_iff | null |
protected complementedLattice_iff [Lattice α] [OrderBot α] :
ComplementedLattice (Iic a) ↔ ∀ b, b ≤ a → ∃ c ≤ a, b ⊓ c = ⊥ ∧ b ⊔ c = a := by
simp_rw [complementedLattice_iff, Iic.isCompl_iff, Subtype.forall, Subtype.exists, disjoint_iff,
exists_prop, Set.mem_Iic] | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | complementedLattice_iff | null |
semilatticeInf [SemilatticeInf α] {a : α} : SemilatticeInf (Ici a) :=
Subtype.semilatticeInf fun _ _ hx hy => le_inf hx hy
@[simp, norm_cast] | instance | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | semilatticeInf | null |
protected coe_inf [SemilatticeInf α] {a : α} {x y : Ici a} :
↑(x ⊓ y) = (↑x ⊓ ↑y : α) :=
rfl | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | coe_inf | null |
semilatticeSup [SemilatticeSup α] {a : α} : SemilatticeSup (Ici a) :=
Subtype.semilatticeSup fun _ _ hx _ => le_trans hx le_sup_left
@[simp, norm_cast] | instance | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | semilatticeSup | null |
protected coe_sup [SemilatticeSup α] {a : α} {x y : Ici a} :
↑(x ⊔ y) = (↑x ⊔ ↑y : α) :=
rfl | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | coe_sup | null |
lattice [Lattice α] {a : α} : Lattice (Ici a) :=
{ Ici.semilatticeInf, Ici.semilatticeSup with } | instance | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | lattice | null |
distribLattice [DistribLattice α] {a : α} : DistribLattice (Ici a) :=
{ Ici.lattice with le_sup_inf := fun _ _ _ => le_sup_inf } | instance | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | distribLattice | null |
orderBot [Preorder α] {a : α} :
OrderBot (Ici a) where
bot := ⟨a, le_refl a⟩
bot_le x := x.prop
@[simp, norm_cast] | instance | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | orderBot | null |
protected coe_bot [Preorder α] (a : α) : ↑(⊥ : Ici a) = a := rfl | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | coe_bot | null |
orderTop [Preorder α] [OrderTop α] {a : α} :
OrderTop (Ici a) where
top := ⟨⊤, le_top⟩
le_top := fun ⟨_, _⟩ => Subtype.mk_le_mk.2 le_top
@[simp, norm_cast] | instance | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | orderTop | null |
protected coe_top [Preorder α] [OrderTop α] (a : α) : ↑(⊤ : Ici a) = (⊤ : α) := rfl | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | coe_top | null |
boundedOrder [Preorder α] [OrderTop α] {a : α} : BoundedOrder (Ici a) :=
{ Ici.orderTop, Ici.orderBot with } | instance | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | boundedOrder | null |
protected disjoint_iff [SemilatticeInf α] {a : α} {x y : Ici a} :
Disjoint x y ↔ ↑x ⊓ ↑y = a := by
simp [_root_.disjoint_iff, Subtype.ext_iff] | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | disjoint_iff | null |
protected codisjoint_iff [SemilatticeSup α] [OrderTop α] {a : α} {x y : Ici a} :
Codisjoint x y ↔ Codisjoint (x : α) (y : α) := by
simp [_root_.codisjoint_iff, Subtype.ext_iff] | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | codisjoint_iff | null |
protected isCompl_iff [Lattice α] [OrderTop α] {a : α} {x y : Ici a} :
IsCompl x y ↔ ↑x ⊓ ↑y = a ∧ Codisjoint (x : α) (y : α) := by
rw [_root_.isCompl_iff, Ici.disjoint_iff, Ici.codisjoint_iff] | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | isCompl_iff | null |
semilatticeInf [SemilatticeInf α] : SemilatticeInf (Icc a b) :=
Subtype.semilatticeInf fun _ _ hx hy => ⟨le_inf hx.1 hy.1, le_trans inf_le_left hx.2⟩
@[simp, norm_cast] | instance | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | semilatticeInf | null |
protected coe_inf [SemilatticeInf α] {x y : Icc a b} :
↑(x ⊓ y) = (↑x ⊓ ↑y : α) :=
rfl | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | coe_inf | null |
semilatticeSup [SemilatticeSup α] : SemilatticeSup (Icc a b) :=
Subtype.semilatticeSup fun _ _ hx hy => ⟨le_trans hx.1 le_sup_left, sup_le hx.2 hy.2⟩
@[simp, norm_cast] | instance | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | semilatticeSup | null |
protected coe_sup [SemilatticeSup α] {x y : Icc a b} :
↑(x ⊔ y) = (↑x ⊔ ↑y : α) :=
rfl | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | coe_sup | null |
lattice [Lattice α] : Lattice (Icc a b) :=
{ Icc.semilatticeInf, Icc.semilatticeSup with } | instance | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | lattice | null |
@[simp, norm_cast]
protected coe_bot [Preorder α] (a b : α) [Fact (a ≤ b)] : ↑(⊥ : Icc a b) = a := rfl | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | coe_bot | null |
@[simp, norm_cast]
protected coe_top [Preorder α] (a b : α) [Fact (a ≤ b)] : ↑(⊤ : Icc a b) = b := rfl | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | coe_top | null |
protected disjoint_iff [SemilatticeInf α] [Fact (a ≤ b)] {x y : Icc a b} :
Disjoint x y ↔ ↑x ⊓ ↑y = a := by
simp [_root_.disjoint_iff, Subtype.ext_iff] | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | disjoint_iff | null |
protected codisjoint_iff [SemilatticeSup α] [Fact (a ≤ b)] {x y : Icc a b} :
Codisjoint x y ↔ ↑x ⊔ (y : α) = b := by
simp [_root_.codisjoint_iff, Subtype.ext_iff] | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | codisjoint_iff | null |
protected isCompl_iff [Lattice α] [Fact (a ≤ b)] {x y : Icc a b} :
IsCompl x y ↔ ↑x ⊓ ↑y = a ∧ ↑x ⊔ ↑y = b := by
rw [_root_.isCompl_iff, Icc.disjoint_iff, Icc.codisjoint_iff] | lemma | Order | [
"Mathlib.Order.Bounds.Basic"
] | Mathlib/Order/LatticeIntervals.lean | isCompl_iff | null |
limsSup (f : Filter α) : α :=
sInf { a | ∀ᶠ n in f, n ≤ a } | def | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsSup | The `limsSup` of a filter `f` is the infimum of the `a` such that the inequality
`x ≤ a` eventually holds for `f`. |
limsInf (f : Filter α) : α :=
sSup { a | ∀ᶠ n in f, a ≤ n } | def | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsInf | The `limsInf` of a filter `f` is the supremum of the `a` such that the inequality
`x ≥ a` eventually holds for `f`. |
limsup (u : β → α) (f : Filter β) : α :=
limsSup (map u f) | def | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsup | The `limsup` of a function `u` along a filter `f` is the infimum of the `a` such that
the inequality `u x ≤ a` eventually holds for `f`. |
liminf (u : β → α) (f : Filter β) : α :=
limsInf (map u f) | def | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | liminf | The `liminf` of a function `u` along a filter `f` is the supremum of the `a` such that
the inequality `u x ≥ a` eventually holds for `f`. |
blimsup (u : β → α) (f : Filter β) (p : β → Prop) :=
sInf { a | ∀ᶠ x in f, p x → u x ≤ a } | def | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | blimsup | The `blimsup` of a function `u` along a filter `f`, bounded by a predicate `p`, is the infimum
of the `a` such that the inequality `u x ≤ a` eventually holds for `f`, whenever `p x` holds. |
bliminf (u : β → α) (f : Filter β) (p : β → Prop) :=
sSup { a | ∀ᶠ x in f, p x → a ≤ u x } | def | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf | The `bliminf` of a function `u` along a filter `f`, bounded by a predicate `p`, is the supremum
of the `a` such that the inequality `a ≤ u x` eventually holds for `f` whenever `p x` holds. |
limsup_eq : limsup u f = sInf { a | ∀ᶠ n in f, u n ≤ a } :=
rfl | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsup_eq | null |
liminf_eq : liminf u f = sSup { a | ∀ᶠ n in f, a ≤ u n } :=
rfl | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | liminf_eq | null |
blimsup_eq : blimsup u f p = sInf { a | ∀ᶠ x in f, p x → u x ≤ a } :=
rfl | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | blimsup_eq | null |
bliminf_eq : bliminf u f p = sSup { a | ∀ᶠ x in f, p x → a ≤ u x } :=
rfl | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_eq | null |
liminf_comp (u : β → α) (v : γ → β) (f : Filter γ) :
liminf (u ∘ v) f = liminf u (map v f) := rfl | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | liminf_comp | null |
limsup_comp (u : β → α) (v : γ → β) (f : Filter γ) :
limsup (u ∘ v) f = limsup u (map v f) := rfl | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsup_comp | null |
@[simp]
blimsup_true (f : Filter β) (u : β → α) : (blimsup u f fun _ => True) = limsup u f := by
simp [blimsup_eq, limsup_eq]
@[simp] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | blimsup_true | null |
bliminf_true (f : Filter β) (u : β → α) : (bliminf u f fun _ => True) = liminf u f := by
simp [bliminf_eq, liminf_eq] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_true | null |
blimsup_eq_limsup {f : Filter β} {u : β → α} {p : β → Prop} :
blimsup u f p = limsup u (f ⊓ 𝓟 {x | p x}) := by
simp only [blimsup_eq, limsup_eq, eventually_inf_principal, mem_setOf_eq] | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | blimsup_eq_limsup | null |
bliminf_eq_liminf {f : Filter β} {u : β → α} {p : β → Prop} :
bliminf u f p = liminf u (f ⊓ 𝓟 {x | p x}) :=
blimsup_eq_limsup (α := αᵒᵈ) | lemma | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_eq_liminf | null |
blimsup_eq_limsup_subtype {f : Filter β} {u : β → α} {p : β → Prop} :
blimsup u f p = limsup (u ∘ ((↑) : { x | p x } → β)) (comap (↑) f) := by
rw [blimsup_eq_limsup, limsup, limsup, ← map_map, map_comap_setCoe_val] | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | blimsup_eq_limsup_subtype | null |
bliminf_eq_liminf_subtype {f : Filter β} {u : β → α} {p : β → Prop} :
bliminf u f p = liminf (u ∘ ((↑) : { x | p x } → β)) (comap (↑) f) :=
blimsup_eq_limsup_subtype (α := αᵒᵈ) | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | bliminf_eq_liminf_subtype | null |
limsSup_le_of_le {f : Filter α} {a}
(hf : f.IsCobounded (· ≤ ·) := by isBoundedDefault)
(h : ∀ᶠ n in f, n ≤ a) : limsSup f ≤ a :=
csInf_le hf h | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsSup_le_of_le | null |
le_limsInf_of_le {f : Filter α} {a}
(hf : f.IsCobounded (· ≥ ·) := by isBoundedDefault)
(h : ∀ᶠ n in f, a ≤ n) : a ≤ limsInf f :=
le_csSup hf h | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | le_limsInf_of_le | null |
limsup_le_of_le {f : Filter β} {u : β → α} {a}
(hf : f.IsCoboundedUnder (· ≤ ·) u := by isBoundedDefault)
(h : ∀ᶠ n in f, u n ≤ a) : limsup u f ≤ a :=
csInf_le hf h | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsup_le_of_le | null |
le_liminf_of_le {f : Filter β} {u : β → α} {a}
(hf : f.IsCoboundedUnder (· ≥ ·) u := by isBoundedDefault)
(h : ∀ᶠ n in f, a ≤ u n) : a ≤ liminf u f :=
le_csSup hf h | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | le_liminf_of_le | null |
le_limsSup_of_le {f : Filter α} {a}
(hf : f.IsBounded (· ≤ ·) := by isBoundedDefault)
(h : ∀ b, (∀ᶠ n in f, n ≤ b) → a ≤ b) : a ≤ limsSup f :=
le_csInf hf h | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | le_limsSup_of_le | null |
limsInf_le_of_le {f : Filter α} {a}
(hf : f.IsBounded (· ≥ ·) := by isBoundedDefault)
(h : ∀ b, (∀ᶠ n in f, b ≤ n) → b ≤ a) : limsInf f ≤ a :=
csSup_le hf h | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsInf_le_of_le | null |
le_limsup_of_le {f : Filter β} {u : β → α} {a}
(hf : f.IsBoundedUnder (· ≤ ·) u := by isBoundedDefault)
(h : ∀ b, (∀ᶠ n in f, u n ≤ b) → a ≤ b) : a ≤ limsup u f :=
le_csInf hf h | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | le_limsup_of_le | null |
liminf_le_of_le {f : Filter β} {u : β → α} {a}
(hf : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault)
(h : ∀ b, (∀ᶠ n in f, b ≤ u n) → b ≤ a) : liminf u f ≤ a :=
csSup_le hf h | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | liminf_le_of_le | null |
limsInf_le_limsSup {f : Filter α} [NeBot f]
(h₁ : f.IsBounded (· ≤ ·) := by isBoundedDefault)
(h₂ : f.IsBounded (· ≥ ·) := by isBoundedDefault) :
limsInf f ≤ limsSup f :=
liminf_le_of_le h₂ fun a₀ ha₀ =>
le_limsup_of_le h₁ fun a₁ ha₁ =>
show a₀ ≤ a₁ from
let ⟨_, hb₀, hb₁⟩ := (ha₀.and ha₁... | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsInf_le_limsSup | null |
liminf_le_limsup {f : Filter β} [NeBot f] {u : β → α}
(h : f.IsBoundedUnder (· ≤ ·) u := by isBoundedDefault)
(h' : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault) :
liminf u f ≤ limsup u f :=
limsInf_le_limsSup h h' | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | liminf_le_limsup | null |
limsSup_le_limsSup {f g : Filter α}
(hf : f.IsCobounded (· ≤ ·) := by isBoundedDefault)
(hg : g.IsBounded (· ≤ ·) := by isBoundedDefault)
(h : ∀ a, (∀ᶠ n in g, n ≤ a) → ∀ᶠ n in f, n ≤ a) : limsSup f ≤ limsSup g :=
csInf_le_csInf hf hg h | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsSup_le_limsSup | null |
limsInf_le_limsInf {f g : Filter α}
(hf : f.IsBounded (· ≥ ·) := by isBoundedDefault)
(hg : g.IsCobounded (· ≥ ·) := by isBoundedDefault)
(h : ∀ a, (∀ᶠ n in f, a ≤ n) → ∀ᶠ n in g, a ≤ n) : limsInf f ≤ limsInf g :=
csSup_le_csSup hg hf h | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsInf_le_limsInf | null |
limsup_le_limsup {α : Type*} [ConditionallyCompleteLattice β] {f : Filter α} {u v : α → β}
(h : u ≤ᶠ[f] v)
(hu : f.IsCoboundedUnder (· ≤ ·) u := by isBoundedDefault)
(hv : f.IsBoundedUnder (· ≤ ·) v := by isBoundedDefault) :
limsup u f ≤ limsup v f :=
limsSup_le_limsSup hu hv fun _ => h.trans | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsup_le_limsup | null |
liminf_le_liminf {α : Type*} [ConditionallyCompleteLattice β] {f : Filter α} {u v : α → β}
(h : ∀ᶠ a in f, u a ≤ v a)
(hu : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault)
(hv : f.IsCoboundedUnder (· ≥ ·) v := by isBoundedDefault) :
liminf u f ≤ liminf v f :=
limsup_le_limsup (β := βᵒᵈ) h hv hu | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | liminf_le_liminf | null |
limsSup_le_limsSup_of_le {f g : Filter α} (h : f ≤ g)
(hf : f.IsCobounded (· ≤ ·) := by isBoundedDefault)
(hg : g.IsBounded (· ≤ ·) := by isBoundedDefault) :
limsSup f ≤ limsSup g :=
limsSup_le_limsSup hf hg fun _ ha => h ha | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsSup_le_limsSup_of_le | null |
limsInf_le_limsInf_of_le {f g : Filter α} (h : g ≤ f)
(hf : f.IsBounded (· ≥ ·) := by isBoundedDefault)
(hg : g.IsCobounded (· ≥ ·) := by isBoundedDefault) :
limsInf f ≤ limsInf g :=
limsInf_le_limsInf hf hg fun _ ha => h ha | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsInf_le_limsInf_of_le | null |
limsup_le_limsup_of_le {α β} [ConditionallyCompleteLattice β] {f g : Filter α} (h : f ≤ g)
{u : α → β}
(hf : f.IsCoboundedUnder (· ≤ ·) u := by isBoundedDefault)
(hg : g.IsBoundedUnder (· ≤ ·) u := by isBoundedDefault) :
limsup u f ≤ limsup u g :=
limsSup_le_limsSup_of_le (map_mono h) hf hg | theorem | Order | [
"Mathlib.Order.ConditionallyCompleteLattice.Indexed",
"Mathlib.Order.Filter.IsBounded",
"Mathlib.Order.Hom.CompleteLattice"
] | Mathlib/Order/LiminfLimsup.lean | limsup_le_limsup_of_le | null |
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