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 ⌀ |
|---|---|---|---|---|---|---|
sup_le_inf : a ⊔ b ≤ a ⊓ b ↔ a = b := by simp [le_antisymm_iff, and_comm]
@[simp] lemma inf_eq_sup : a ⊓ b = a ⊔ b ↔ a = b := by rw [← inf_le_sup.ge_iff_eq, sup_le_inf]
@[simp] lemma sup_eq_inf : a ⊔ b = a ⊓ b ↔ a = b := eq_comm.trans inf_eq_sup
@[simp] lemma inf_lt_sup : a ⊓ b < a ⊔ b ↔ a ≠ b := by rw [inf_le_sup.lt_i... | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_le_inf | null |
inf_eq_and_sup_eq_iff : a ⊓ b = c ∧ a ⊔ b = c ↔ a = c ∧ b = c := by
refine ⟨fun h ↦ ?_, ?_⟩
· obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)
simpa using h
· rintro ⟨rfl, rfl⟩
exact ⟨inf_idem _, sup_idem _⟩
/-! | lemma | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_eq_and_sup_eq_iff | null |
sup_inf_le : a ⊔ b ⊓ c ≤ (a ⊔ b) ⊓ (a ⊔ c) :=
le_inf (sup_le_sup_left inf_le_left _) (sup_le_sup_left inf_le_right _) | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_inf_le | null |
le_inf_sup : a ⊓ b ⊔ a ⊓ c ≤ a ⊓ (b ⊔ c) :=
sup_le (inf_le_inf_left _ le_sup_left) (inf_le_inf_left _ le_sup_right) | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | le_inf_sup | null |
inf_sup_self : a ⊓ (a ⊔ b) = a := by simp | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_sup_self | null |
sup_inf_self : a ⊔ a ⊓ b = a := by simp | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_inf_self | null |
sup_eq_iff_inf_eq : a ⊔ b = b ↔ a ⊓ b = a := by rw [sup_eq_right, ← inf_eq_left] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_eq_iff_inf_eq | null |
Lattice.ext {α} {A B : Lattice α} (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) :
A = B := by
cases A
cases B
cases SemilatticeSup.ext H
cases SemilatticeInf.ext H
congr | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | Lattice.ext | null |
DistribLattice (α) extends Lattice α where
/-- The infimum distributes over the supremum -/
protected le_sup_inf : ∀ x y z : α, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z | class | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | DistribLattice | A distributive lattice is a lattice that satisfies any of four
equivalent distributive properties (of `sup` over `inf` or `inf` over `sup`,
on the left or right).
The definition here chooses `le_sup_inf`: `(x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ (y ⊓ z)`. To prove distributivity
from the dual law, use `DistribLattice.of_inf_sup_le`.
... |
le_sup_inf : ∀ {x y z : α}, (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z :=
fun {x y z} => DistribLattice.le_sup_inf x y z | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | le_sup_inf | null |
sup_inf_left (a b c : α) : a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c) :=
le_antisymm sup_inf_le le_sup_inf | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_inf_left | null |
sup_inf_right (a b c : α) : a ⊓ b ⊔ c = (a ⊔ c) ⊓ (b ⊔ c) := by
simp only [sup_inf_left, sup_comm _ c] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_inf_right | null |
inf_sup_left (a b c : α) : a ⊓ (b ⊔ c) = a ⊓ b ⊔ a ⊓ c :=
calc
a ⊓ (b ⊔ c) = a ⊓ (a ⊔ c) ⊓ (b ⊔ c) := by rw [inf_sup_self]
_ = a ⊓ (a ⊓ b ⊔ c) := by simp only [inf_assoc, sup_inf_right]
_ = (a ⊔ a ⊓ b) ⊓ (a ⊓ b ⊔ c) := by rw [sup_inf_self]
_ = (a ⊓ b ⊔ a) ⊓ (a ⊓ b ⊔ c) := by rw [sup_comm]
_ = a ⊓ ... | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_sup_left | null |
OrderDual.instDistribLattice (α : Type*) [DistribLattice α] : DistribLattice αᵒᵈ where
le_sup_inf _ _ _ := (inf_sup_left _ _ _).le | instance | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | OrderDual.instDistribLattice | null |
inf_sup_right (a b c : α) : (a ⊔ b) ⊓ c = a ⊓ c ⊔ b ⊓ c := by
simp only [inf_sup_left, inf_comm _ c] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_sup_right | null |
le_of_inf_le_sup_le (h₁ : x ⊓ z ≤ y ⊓ z) (h₂ : x ⊔ z ≤ y ⊔ z) : x ≤ y :=
calc
x ≤ y ⊓ z ⊔ x := le_sup_right
_ = (y ⊔ x) ⊓ (x ⊔ z) := by rw [sup_inf_right, sup_comm x]
_ ≤ (y ⊔ x) ⊓ (y ⊔ z) := inf_le_inf_left _ h₂
_ = y ⊔ x ⊓ z := by rw [← sup_inf_left]
_ ≤ y ⊔ y ⊓ z := sup_le_sup_left h₁ _
_ ≤... | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | le_of_inf_le_sup_le | null |
eq_of_inf_eq_sup_eq {a b c : α} (h₁ : b ⊓ a = c ⊓ a) (h₂ : b ⊔ a = c ⊔ a) : b = c :=
le_antisymm (le_of_inf_le_sup_le (le_of_eq h₁) (le_of_eq h₂))
(le_of_inf_le_sup_le (le_of_eq h₁.symm) (le_of_eq h₂.symm)) | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | eq_of_inf_eq_sup_eq | null |
DistribLattice.ofInfSupLe
[Lattice α] (inf_sup_le : ∀ a b c : α, a ⊓ (b ⊔ c) ≤ a ⊓ b ⊔ a ⊓ c) : DistribLattice α where
le_sup_inf := (@OrderDual.instDistribLattice αᵒᵈ {inferInstanceAs (Lattice αᵒᵈ) with
le_sup_inf := inf_sup_le}).le_sup_inf
/-! | abbrev | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | DistribLattice.ofInfSupLe | Prove distributivity of an existing lattice from the dual distributive law. |
sup_ind (a b : α) {p : α → Prop} (ha : p a) (hb : p b) : p (a ⊔ b) :=
(IsTotal.total a b).elim (fun h : a ≤ b => by rwa [sup_eq_right.2 h]) fun h => by
rwa [sup_eq_left.2 h]
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_ind | null |
le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | le_sup_iff | null |
lt_sup_iff : a < b ⊔ c ↔ a < b ∨ a < c := by
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | lt_sup_iff | null |
sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => sup_ind (p := (· < a)) b c h.1 h.2⟩ | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_lt_iff | null |
inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
@sup_ind αᵒᵈ _ _ _ _
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_ind | null |
inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a :=
@le_sup_iff αᵒᵈ _ _ _ _
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_le_iff | null |
inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a :=
@lt_sup_iff αᵒᵈ _ _ _ _
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_lt_iff | null |
lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c :=
@sup_lt_iff αᵒᵈ _ _ _ _
variable (a b c d) | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | lt_inf_iff | null |
max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) :=
sup_sup_sup_comm _ _ _ _ | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | max_max_max_comm | null |
min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) :=
inf_inf_inf_comm _ _ _ _ | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | min_min_min_comm | null |
sup_eq_maxDefault [SemilatticeSup α] [DecidableLE α] [IsTotal α (· ≤ ·)] :
(· ⊔ ·) = (maxDefault : α → α → α) := by
ext x y
unfold maxDefault
split_ifs with h'
exacts [sup_of_le_right h', sup_of_le_left <| (total_of (· ≤ ·) x y).resolve_left h'] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_eq_maxDefault | null |
inf_eq_minDefault [SemilatticeInf α] [DecidableLE α] [IsTotal α (· ≤ ·)] :
(· ⊓ ·) = (minDefault : α → α → α) := by
ext x y
unfold minDefault
split_ifs with h'
exacts [inf_of_le_left h', inf_of_le_right <| (total_of (· ≤ ·) x y).resolve_left h'] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_eq_minDefault | null |
Lattice.toLinearOrder (α : Type u) [Lattice α] [DecidableEq α]
[DecidableLE α] [DecidableLT α] [IsTotal α (· ≤ ·)] : LinearOrder α where
toDecidableLE := ‹_›
toDecidableEq := ‹_›
toDecidableLT := ‹_›
le_total := total_of (· ≤ ·)
max_def := by exact congr_fun₂ sup_eq_maxDefault
min_def := by exact congr_... | abbrev | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | Lattice.toLinearOrder | A lattice with total order is a linear order.
See note [reducible non-instances]. |
@[simp]
ofDual_inf [Max α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | ofDual_inf | null |
ofDual_sup [Min α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | ofDual_sup | null |
toDual_inf [Min α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | toDual_inf | null |
toDual_sup [Max α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b :=
rfl | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | toDual_sup | null |
@[simp]
ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | ofDual_min | null |
ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | ofDual_max | null |
toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | toDual_min | null |
toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) :=
rfl | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | toDual_max | null |
@[simp]
sup_apply [∀ i, Max (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_apply | null |
sup_def [∀ i, Max (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_def | null |
@[simp]
inf_apply [∀ i, Min (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i :=
rfl | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_apply | null |
inf_def [∀ i, Min (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i :=
rfl | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_def | null |
instSemilatticeSup [∀ i, SemilatticeSup (α' i)] : SemilatticeSup (∀ i, α' i) where
sup x y i := x i ⊔ y i
le_sup_left _ _ _ := le_sup_left
le_sup_right _ _ _ := le_sup_right
sup_le _ _ _ ac bc i := sup_le (ac i) (bc i) | instance | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | instSemilatticeSup | null |
instSemilatticeInf [∀ i, SemilatticeInf (α' i)] : SemilatticeInf (∀ i, α' i) where
inf x y i := x i ⊓ y i
inf_le_left _ _ _ := inf_le_left
inf_le_right _ _ _ := inf_le_right
le_inf _ _ _ ac bc i := le_inf (ac i) (bc i) | instance | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | instSemilatticeInf | null |
instLattice [∀ i, Lattice (α' i)] : Lattice (∀ i, α' i) where | instance | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | instLattice | null |
instDistribLattice [∀ i, DistribLattice (α' i)] : DistribLattice (∀ i, α' i) where
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 |
update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊔ b) = update f i a ⊔ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_of_ne, *] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | update_sup | null |
update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
update f i (a ⊓ b) = update f i a ⊓ update f i b :=
funext fun j => by obtain rfl | hji := eq_or_ne j i <;> simp [update_of_ne, *] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | update_inf | null |
protected sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h) | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup | Pointwise supremum of two monotone functions is a monotone function. |
protected inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h) | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf | Pointwise infimum of two monotone functions is a monotone function. |
protected max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => max (f x) (g x) :=
hf.sup hg | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | max | Pointwise maximum of two monotone functions is a monotone function. |
protected min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone fun x => min (f x) (g x) :=
hf.inf hg | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | min | Pointwise minimum of two monotone functions is a monotone function. |
le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊔ y) :=
sup_le (h le_sup_left) (h le_sup_right) | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | le_map_sup | null |
map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
f (x ⊓ y) ≤ f x ⊓ f y :=
le_inf (h inf_le_left) (h inf_le_right) | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | map_inf_le | null |
of_map_inf_le_left [SemilatticeInf α] [Preorder β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) ≤ f x) : Monotone f := by
intro x y hxy
rw [← inf_eq_right.2 hxy]
apply h | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | of_map_inf_le_left | null |
of_map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) ≤ f x ⊓ f y) : Monotone f :=
of_map_inf_le_left fun x y ↦ (h x y).trans inf_le_left | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | of_map_inf_le | null |
of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
(h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
of_map_inf_le fun x y ↦ (h x y).le | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | of_map_inf | null |
of_left_le_map_sup [SemilatticeSup α] [Preorder β] {f : α → β}
(h : ∀ x y, f x ≤ f (x ⊔ y)) : Monotone f :=
monotone_dual_iff.1 <| of_map_inf_le_left h | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | of_left_le_map_sup | null |
of_le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f x ⊔ f y ≤ f (x ⊔ y)) : Monotone f :=
monotone_dual_iff.mp <| of_map_inf_le h | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | of_le_map_sup | null |
of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
(h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
(@of_map_inf (OrderDual α) (OrderDual β) _ _ _ h).dual
variable [LinearOrder α] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | of_map_sup | null |
map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊔ y) = f x ⊔ f y :=
(IsTotal.total x y).elim (fun h : x ≤ y => by simp only [h, hf h, sup_of_le_right]) fun h => by
simp only [h, hf h, sup_of_le_left] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | map_sup | null |
map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup _ _ | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | map_inf | null |
protected sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h) | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup | Pointwise supremum of two monotone functions is a monotone function. |
protected inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf | Pointwise infimum of two monotone functions is a monotone function. |
protected max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | max | Pointwise maximum of two monotone functions is a monotone function. |
protected min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
(hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | min | Pointwise minimum of two monotone functions is a monotone function. |
of_map_inf [SemilatticeInf α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s := fun x hx y hy hxy =>
inf_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | of_map_inf | null |
of_map_sup [SemilatticeSup α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
variable [LinearOrder α] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | of_map_sup | null |
map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊔ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | map_sup | null |
map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊓ f y :=
hf.dual.map_sup hx hy | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | map_inf | null |
protected sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h) | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup | Pointwise supremum of two monotone functions is a monotone function. |
protected inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone (f ⊓ g) := fun _ _ h => inf_le_inf (hf h) (hg h) | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf | Pointwise infimum of two monotone functions is a monotone function. |
protected max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => max (f x) (g x) :=
hf.sup hg | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | max | Pointwise maximum of two monotone functions is a monotone function. |
protected min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
(hg : Antitone g) :
Antitone fun x => min (f x) (g x) :=
hf.inf hg | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | min | Pointwise minimum of two monotone functions is a monotone function. |
map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
f (x ⊔ y) ≤ f x ⊓ f y :=
h.dual_right.le_map_sup x y | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | map_sup_le | null |
le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
f x ⊔ f y ≤ f (x ⊓ y) :=
h.dual_right.map_inf_le x y
variable [LinearOrder α] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | le_map_inf | null |
map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊔ y) = f x ⊓ f y :=
hf.dual_right.map_sup x y | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | map_sup | null |
map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual_right.map_inf x y | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | map_inf | null |
protected sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s :=
fun _ hx _ hy h => sup_le_sup (hf hx hy h) (hg hx hy h) | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup | Pointwise supremum of two antitone functions is an antitone function. |
protected inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
(hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf | Pointwise infimum of two antitone functions is an antitone function. |
protected max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s :=
hf.sup hg | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | max | Pointwise maximum of two antitone functions is an antitone function. |
protected min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s :=
hf.inf hg | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | min | Pointwise minimum of two antitone functions is an antitone function. |
of_map_inf [SemilatticeInf α] [SemilatticeSup β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s := fun x hx y hy hxy =>
sup_eq_left.1 <| by rw [← h _ hx _ hy, inf_eq_left.2 hxy] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | of_map_inf | null |
of_map_sup [SemilatticeSup α] [SemilatticeInf β]
(h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s :=
(@of_map_inf αᵒᵈ βᵒᵈ _ _ _ _ h).dual
variable [LinearOrder α] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | of_map_sup | null |
map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊔ y) = f x ⊓ f y := by
cases le_total x y <;> have := hf ?_ ?_ ‹_› <;>
first
| assumption
| simp only [*, sup_of_le_left, sup_of_le_right, inf_of_le_left, inf_of_le_right] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | map_sup | null |
map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
f (x ⊓ y) = f x ⊔ f y :=
hf.dual.map_sup hx hy | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | map_inf | null |
@[simp]
mk_sup_mk [Max α] [Max β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂) :=
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 [Min α] [Min β] (a₁ a₂ : α) (b₁ b₂ : β) :
(a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂) :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | mk_inf_mk | null |
fst_sup [Max α] [Max β] (p q : α × β) : (p ⊔ q).fst = p.fst ⊔ q.fst :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | fst_sup | null |
fst_inf [Min α] [Min β] (p q : α × β) : (p ⊓ q).fst = p.fst ⊓ q.fst :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | fst_inf | null |
snd_sup [Max α] [Max β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | snd_sup | null |
snd_inf [Min α] [Min β] (p q : α × β) : (p ⊓ q).snd = p.snd ⊓ q.snd :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | snd_inf | null |
swap_sup [Max α] [Max β] (p q : α × β) : (p ⊔ q).swap = p.swap ⊔ q.swap :=
rfl
@[simp] | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | swap_sup | null |
swap_inf [Min α] [Min β] (p q : α × β) : (p ⊓ q).swap = p.swap ⊓ q.swap :=
rfl | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | swap_inf | null |
sup_def [Max α] [Max β] (p q : α × β) : p ⊔ q = (p.fst ⊔ q.fst, p.snd ⊔ q.snd) :=
rfl | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | sup_def | null |
inf_def [Min α] [Min β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) :=
rfl | theorem | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | inf_def | null |
instSemilatticeSup [SemilatticeSup α] [SemilatticeSup β] : SemilatticeSup (α × β) where
sup a b := ⟨a.1 ⊔ b.1, a.2 ⊔ b.2⟩
sup_le _ _ _ h₁ h₂ := ⟨sup_le h₁.1 h₂.1, sup_le h₁.2 h₂.2⟩
le_sup_left _ _ := ⟨le_sup_left, le_sup_left⟩
le_sup_right _ _ := ⟨le_sup_right, le_sup_right⟩ | instance | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | instSemilatticeSup | null |
instSemilatticeInf [SemilatticeInf α] [SemilatticeInf β] : SemilatticeInf (α × β) where
inf a b := ⟨a.1 ⊓ b.1, a.2 ⊓ b.2⟩
le_inf _ _ _ h₁ h₂ := ⟨le_inf h₁.1 h₂.1, le_inf h₁.2 h₂.2⟩
inf_le_left _ _ := ⟨inf_le_left, inf_le_left⟩
inf_le_right _ _ := ⟨inf_le_right, inf_le_right⟩ | instance | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | instSemilatticeInf | null |
instLattice [Lattice α] [Lattice β] : Lattice (α × β) where | instance | Order | [
"Mathlib.Data.Bool.Basic",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift"
] | Mathlib/Order/Lattice.lean | instLattice | null |
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