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 ⌀ |
|---|---|---|---|---|---|---|
piLTLim_symm_apply {f} (k : Iio i) {l : Iio i} (hl : l.1 < k.1) :
(piLTLim (X := X) hi).symm f l = f.1 k ⟨l, hl⟩ := by
conv_rhs => rw [← (piLTLim hi).right_inv f]
rfl | theorem | Order | [
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.UpperLower.Basic"
] | Mathlib/Order/DirectedInverseSystem.lean | piLTLim_symm_apply | null |
piSplitLE : piLT X i × X i ≃ ∀ j : Iic i, X j where
toFun f j := if h : j = i then h.symm ▸ f.2 else f.1 ⟨j, j.2.lt_of_ne h⟩
invFun f := (fun j ↦ f ⟨j, j.2.le⟩, f ⟨i, le_rfl⟩)
left_inv f := by ext j; exacts [dif_neg j.2.ne, dif_pos rfl]
right_inv f := by grind
@[simp] theorem piSplitLE_eq {f : piLT X i × X i} :... | def | Order | [
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.UpperLower.Basic"
] | Mathlib/Order/DirectedInverseSystem.lean | piSplitLE | Splitting off the `X i` factor from the Pi type over `{j | j ≤ i}`. |
piSplitLE_lt {f : piLT X i × X i} {j} (hj : j < i) :
piSplitLE f ⟨j, hj.le⟩ = f.1 ⟨j, hj⟩ := dif_neg hj.ne | theorem | Order | [
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.UpperLower.Basic"
] | Mathlib/Order/DirectedInverseSystem.lean | piSplitLE_lt | null |
piEquivSucc : ∀ j : Iic i⁺, F j ≃ piLT X j :=
piSplitLE (X := fun i ↦ F i ≃ piLT X i)
(fun j ↦ equiv ⟨j, (lt_succ_iff_of_not_isMax hi).mp j.2⟩,
e.trans <| ((equiv ⟨i, le_rfl⟩).prodCongr <| Equiv.refl _).trans <| piSplitLE.trans <|
Equiv.piCongrSet <| Set.ext fun _ ↦ (lt_succ_iff_of_not_isMax hi).symm) | def | Order | [
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.UpperLower.Basic"
] | Mathlib/Order/DirectedInverseSystem.lean | piEquivSucc | Extend a family of bijections to `piLT` by one step. |
piEquivSucc_self {x} :
piEquivSucc equiv e hi ⟨_, le_rfl⟩ x ⟨i, lt_succ_of_not_isMax hi⟩ = (e x).2 := by
simp [piEquivSucc]
variable {equiv e} | theorem | Order | [
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.UpperLower.Basic"
] | Mathlib/Order/DirectedInverseSystem.lean | piEquivSucc_self | null |
isNatEquiv_piEquivSucc [InverseSystem f] (H : ∀ x, (e x).1 = f (le_succ i) x)
(nat : IsNatEquiv f equiv) : IsNatEquiv f (piEquivSucc equiv e hi) := fun j k hj hk h x ↦ by
have lt_succ {j} := (lt_succ_iff_of_not_isMax (b := j) hi).mpr
obtain rfl | hj := le_succ_iff_eq_or_le.mp hj
· obtain rfl | hk := le_succ_i... | theorem | Order | [
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.UpperLower.Basic"
] | Mathlib/Order/DirectedInverseSystem.lean | isNatEquiv_piEquivSucc | null |
@[simps] invLimEquiv : limit f i ≃ limit (piLTProj (X := X)) i where
toFun t := ⟨fun l ↦ equiv l (t.1 l), fun _ _ h ↦ Eq.symm <| by simp_rw [← t.2 h]; apply nat⟩
invFun t := ⟨fun l ↦ (equiv l).symm (t.1 l),
fun _ _ h ↦ (Equiv.eq_symm_apply _).mpr <| by rw [nat, ← t.2 h] <;> simp⟩
left_inv t := by ext; apply E... | def | Order | [
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.UpperLower.Basic"
] | Mathlib/Order/DirectedInverseSystem.lean | invLimEquiv | A natural family of bijections below a limit ordinal
induces a bijection at the limit ordinal. |
noncomputable piEquivLim : ∀ j : Iic i, F j ≃ piLT X j :=
piSplitLE (X := fun j ↦ F j ≃ piLT X j)
(equiv, equivLim.trans <| (invLimEquiv nat).trans (piLTLim hi).symm)
variable {equivLim} | def | Order | [
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.UpperLower.Basic"
] | Mathlib/Order/DirectedInverseSystem.lean | piEquivLim | Extend a natural family of bijections to a limit ordinal. |
isNatEquiv_piEquivLim [InverseSystem f] (H : ∀ x l, (equivLim x).1 l = f l.2.le x) :
IsNatEquiv f (piEquivLim nat equivLim hi) := fun j k hj hk h t ↦ by
obtain rfl | hj := hj.eq_or_lt
· obtain rfl | hk := hk.eq_or_lt
· simp [InverseSystem.map_self]
· funext l
simp_rw [piEquivLim, piSplitLE_lt hk, ... | theorem | Order | [
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.UpperLower.Basic"
] | Mathlib/Order/DirectedInverseSystem.lean | isNatEquiv_piEquivLim | null |
@[ext] PEquivOn (s : Set ι) where
/-- A partial family of bijections between `F` and `piLT X` defined on some set in `ι`. -/
equiv (i : s) : F i ≃ piLT X i
/-- It is a natural family of bijections. -/
nat : IsNatEquiv f equiv
/-- It is compatible with a family of bijections relating `F i⁺` to `F i`. -/
comp... | structure | Order | [
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.UpperLower.Basic"
] | Mathlib/Order/DirectedInverseSystem.lean | PEquivOn | A natural partial family of bijections to `piLT` satisfying a compatibility condition. |
@[simps] PEquivOn.restrict (e : PEquivOn f equivSucc t) (h : s ⊆ t) :
PEquivOn f equivSucc s where
equiv i := e.equiv ⟨i, h i.2⟩
nat _ _ _ _ := e.nat _ _
compat _ := e.compat _ | def | Order | [
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.UpperLower.Basic"
] | Mathlib/Order/DirectedInverseSystem.lean | PEquivOn.restrict | Restrict a partial family of bijections to a smaller set. |
unique_pEquivOn (hs : IsLowerSet s) {e₁ e₂ : PEquivOn f equivSucc s} : e₁ = e₂ := by
obtain ⟨e₁, nat₁, compat₁⟩ := e₁
obtain ⟨e₂, nat₂, compat₂⟩ := e₂
ext1; ext1 i; dsimp only
refine SuccOrder.prelimitRecOn i.1 (motive := fun i ↦ ∀ h : i ∈ s, e₁ ⟨i, h⟩ = e₂ ⟨i, h⟩)
(fun i nmax ih hi ↦ ?_) (fun i lim ih hi ↦... | theorem | Order | [
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.UpperLower.Basic"
] | Mathlib/Order/DirectedInverseSystem.lean | unique_pEquivOn | null |
pEquivOn_apply_eq (h : IsLowerSet (s ∩ t))
{e₁ : PEquivOn f equivSucc s} {e₂ : PEquivOn f equivSucc t} {i} {his : i ∈ s} {hit : i ∈ t} :
e₁.equiv ⟨i, his⟩ = e₂.equiv ⟨i, hit⟩ :=
show (e₁.restrict inter_subset_left).equiv ⟨i, his, hit⟩ =
(e₂.restrict inter_subset_right).equiv ⟨i, his, hit⟩ from
congr_... | theorem | Order | [
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.UpperLower.Basic"
] | Mathlib/Order/DirectedInverseSystem.lean | pEquivOn_apply_eq | null |
pEquivOnSucc [InverseSystem f] (hi : ¬IsMax i) (e : PEquivOn f equivSucc (Iic i))
(H : ∀ ⦃i⦄ (hi : ¬ IsMax i) x, (equivSucc hi x).1 = f (le_succ i) x) :
PEquivOn f equivSucc (Iic i⁺) where
equiv := piEquivSucc e.equiv (equivSucc hi) hi
nat := isNatEquiv_piEquivSucc hi (H hi) e.nat
compat hsj hj x := by
... | def | Order | [
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.UpperLower.Basic"
] | Mathlib/Order/DirectedInverseSystem.lean | pEquivOnSucc | Extend a partial family of bijections by one step. |
noncomputable pEquivOnGlue : PEquivOn f equivSucc (Iio i) where
equiv := (piLTLim (X := fun j ↦ F j ≃ piLT X j) hi).symm
⟨fun j ↦ ((e j).restrict fun _ h ↦ h.le).equiv, fun _ _ h ↦ funext fun _ ↦
pEquivOn_apply_eq ((isLowerSet_Iio _).inter <| isLowerSet_Iio _)⟩
nat j k hj hk h := by rw [piLTLim_symm_apply... | def | Order | [
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.UpperLower.Basic"
] | Mathlib/Order/DirectedInverseSystem.lean | pEquivOnGlue | Glue partial families of bijections at a limit ordinal,
obtaining a partial family over a right-open interval. |
noncomputable pEquivOnLim [InverseSystem f]
(equivLim : F i ≃ limit f i) (H : ∀ x l, (equivLim x).1 l = f l.2.le x) :
PEquivOn f equivSucc (Iic i) where
equiv := piEquivLim (pEquivOnGlue hi e).nat equivLim hi
nat := isNatEquiv_piEquivLim (pEquivOnGlue hi e).nat hi H
compat hsj hj x := by
rw [piEquivLi... | def | Order | [
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.UpperLower.Basic"
] | Mathlib/Order/DirectedInverseSystem.lean | pEquivOnLim | Extend `pEquivOnGlue` by one step, obtaining a partial family over a right-closed interval. |
private noncomputable globalEquivAux (i : ι) :
PEquivOn f (fun i hi ↦ (equivSucc i hi).1) (Iic i) :=
SuccOrder.prelimitRecOn i
(fun _ hi e ↦ pEquivOnSucc hi e fun i hi ↦ (equivSucc i hi).2)
fun i hi e ↦ pEquivOnLim hi (fun j ↦ e j j.2) (equivLim i hi).1 (equivLim i hi).2 | def | Order | [
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.UpperLower.Basic"
] | Mathlib/Order/DirectedInverseSystem.lean | globalEquivAux | null |
noncomputable globalEquiv (i : ι) : F i ≃ piLT X i :=
(globalEquivAux equivSucc equivLim i).equiv ⟨i, le_rfl⟩ | def | Order | [
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.UpperLower.Basic"
] | Mathlib/Order/DirectedInverseSystem.lean | globalEquiv | Over a well-ordered type, construct a family of bijections by transfinite recursion. |
globalEquiv_naturality ⦃i j⦄ (h : i ≤ j) (x : F j) :
letI e := globalEquiv equivSucc equivLim
e i (f h x) = piLTProj h (e j x) := by
refine (DFunLike.congr_fun ?_ _).trans ((globalEquivAux equivSucc equivLim j).nat le_rfl h h x)
exact pEquivOn_apply_eq ((isLowerSet_Iic _).inter <| isLowerSet_Iic _) | theorem | Order | [
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.UpperLower.Basic"
] | Mathlib/Order/DirectedInverseSystem.lean | globalEquiv_naturality | null |
globalEquiv_compatibility ⦃i⦄ (hi : ¬IsMax i) (x) :
globalEquiv equivSucc equivLim i⁺ x ⟨i, lt_succ_of_not_isMax hi⟩ = ((equivSucc i hi).1 x).2 :=
(globalEquivAux equivSucc equivLim i⁺).compat le_rfl hi x | theorem | Order | [
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.UpperLower.Basic"
] | Mathlib/Order/DirectedInverseSystem.lean | globalEquiv_compatibility | null |
Disjoint (a b : α) : Prop :=
∀ ⦃x⦄, x ≤ a → x ≤ b → x ≤ ⊥
@[simp] | def | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Disjoint | Two elements of a lattice are disjoint if their inf is the bottom element.
(This generalizes disjoint sets, viewed as members of the subset lattice.)
Note that we define this without reference to `⊓`, as this allows us to talk about orders where
the infimum is not unique, or where implementing `Inf` would require ad... |
disjoint_of_subsingleton [Subsingleton α] : Disjoint a b :=
fun x _ _ ↦ le_of_eq (Subsingleton.elim x ⊥) | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | disjoint_of_subsingleton | null |
disjoint_comm : Disjoint a b ↔ Disjoint b a :=
forall_congr' fun _ ↦ forall_swap
@[symm] | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | disjoint_comm | null |
Disjoint.symm ⦃a b : α⦄ : Disjoint a b → Disjoint b a :=
disjoint_comm.1 | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Disjoint.symm | null |
symmetric_disjoint : Symmetric (Disjoint : α → α → Prop) :=
Disjoint.symm
@[simp] | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | symmetric_disjoint | null |
disjoint_bot_left : Disjoint ⊥ a := fun _ hbot _ ↦ hbot
@[simp] | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | disjoint_bot_left | null |
disjoint_bot_right : Disjoint a ⊥ := fun _ _ hbot ↦ hbot
@[gcongr] theorem Disjoint.mono (h₁ : a ≤ b) (h₂ : c ≤ d) : Disjoint b d → Disjoint a c :=
fun h _ ha hc ↦ h (ha.trans h₁) (hc.trans h₂) | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | disjoint_bot_right | null |
Disjoint.mono_left (h : a ≤ b) : Disjoint b c → Disjoint a c :=
Disjoint.mono h le_rfl | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Disjoint.mono_left | null |
Disjoint.mono_right (h : b ≤ c) : Disjoint a c → Disjoint a b :=
Disjoint.mono le_rfl h
@[simp] | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Disjoint.mono_right | null |
disjoint_self : Disjoint a a ↔ a = ⊥ :=
⟨fun hd ↦ bot_unique <| hd le_rfl le_rfl, fun h _ ha _ ↦ ha.trans_eq h⟩
/- TODO: Rename `Disjoint.eq_bot` to `Disjoint.inf_eq` and `Disjoint.eq_bot_of_self` to
`Disjoint.eq_bot` -/
alias ⟨Disjoint.eq_bot_of_self, _⟩ := disjoint_self | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | disjoint_self | null |
Disjoint.ne (ha : a ≠ ⊥) (hab : Disjoint a b) : a ≠ b :=
fun h ↦ ha <| disjoint_self.1 <| by rwa [← h] at hab | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Disjoint.ne | null |
Disjoint.eq_bot_of_le (hab : Disjoint a b) (h : a ≤ b) : a = ⊥ :=
eq_bot_iff.2 <| hab le_rfl h | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Disjoint.eq_bot_of_le | null |
Disjoint.eq_bot_of_ge (hab : Disjoint a b) : b ≤ a → b = ⊥ :=
hab.symm.eq_bot_of_le | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Disjoint.eq_bot_of_ge | null |
Disjoint.eq_iff (hab : Disjoint a b) : a = b ↔ a = ⊥ ∧ b = ⊥ := by aesop | lemma | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Disjoint.eq_iff | null |
Disjoint.ne_iff (hab : Disjoint a b) : a ≠ b ↔ a ≠ ⊥ ∨ b ≠ ⊥ :=
hab.eq_iff.not.trans not_and_or | lemma | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Disjoint.ne_iff | null |
disjoint_of_le_iff_left_eq_bot (h : a ≤ b) :
Disjoint a b ↔ a = ⊥ :=
⟨fun hd ↦ hd.eq_bot_of_le h, fun h ↦ h ▸ disjoint_bot_left⟩ | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | disjoint_of_le_iff_left_eq_bot | null |
@[simp]
disjoint_top : Disjoint a ⊤ ↔ a = ⊥ :=
⟨fun h ↦ bot_unique <| h le_rfl le_top, fun h _ ha _ ↦ ha.trans_eq h⟩
@[simp] | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | disjoint_top | null |
top_disjoint : Disjoint ⊤ a ↔ a = ⊥ :=
⟨fun h ↦ bot_unique <| h le_top le_rfl, fun h _ _ ha ↦ ha.trans_eq h⟩ | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | top_disjoint | null |
disjoint_iff_inf_le : Disjoint a b ↔ a ⊓ b ≤ ⊥ :=
⟨fun hd ↦ hd inf_le_left inf_le_right, fun h _ ha hb ↦ (le_inf ha hb).trans h⟩ | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | disjoint_iff_inf_le | null |
disjoint_iff : Disjoint a b ↔ a ⊓ b = ⊥ :=
disjoint_iff_inf_le.trans le_bot_iff | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | disjoint_iff | null |
Disjoint.le_bot : Disjoint a b → a ⊓ b ≤ ⊥ :=
disjoint_iff_inf_le.mp | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Disjoint.le_bot | null |
Disjoint.eq_bot : Disjoint a b → a ⊓ b = ⊥ :=
bot_unique ∘ Disjoint.le_bot | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Disjoint.eq_bot | null |
disjoint_assoc : Disjoint (a ⊓ b) c ↔ Disjoint a (b ⊓ c) := by
rw [disjoint_iff_inf_le, disjoint_iff_inf_le, inf_assoc] | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | disjoint_assoc | null |
disjoint_left_comm : Disjoint a (b ⊓ c) ↔ Disjoint b (a ⊓ c) := by
simp_rw [disjoint_iff_inf_le, inf_left_comm] | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | disjoint_left_comm | null |
disjoint_right_comm : Disjoint (a ⊓ b) c ↔ Disjoint (a ⊓ c) b := by
simp_rw [disjoint_iff_inf_le, inf_right_comm]
variable (c) | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | disjoint_right_comm | null |
Disjoint.inf_left (h : Disjoint a b) : Disjoint (a ⊓ c) b :=
h.mono_left inf_le_left | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Disjoint.inf_left | null |
Disjoint.inf_left' (h : Disjoint a b) : Disjoint (c ⊓ a) b :=
h.mono_left inf_le_right | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Disjoint.inf_left' | null |
Disjoint.inf_right (h : Disjoint a b) : Disjoint a (b ⊓ c) :=
h.mono_right inf_le_left | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Disjoint.inf_right | null |
Disjoint.inf_right' (h : Disjoint a b) : Disjoint a (c ⊓ b) :=
h.mono_right inf_le_right
variable {c} | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Disjoint.inf_right' | null |
Disjoint.of_disjoint_inf_of_le (h : Disjoint (a ⊓ b) c) (hle : a ≤ c) : Disjoint a b :=
disjoint_iff.2 <| h.eq_bot_of_le <| inf_le_of_left_le hle | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Disjoint.of_disjoint_inf_of_le | null |
Disjoint.of_disjoint_inf_of_le' (h : Disjoint (a ⊓ b) c) (hle : b ≤ c) : Disjoint a b :=
disjoint_iff.2 <| h.eq_bot_of_le <| inf_le_of_right_le hle | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Disjoint.of_disjoint_inf_of_le' | null |
Disjoint.right_lt_sup_of_left_ne_bot [SemilatticeSup α] [OrderBot α] {a b : α}
(h : Disjoint a b) (ha : a ≠ ⊥) : b < a ⊔ b :=
le_sup_right.lt_of_ne fun eq ↦ ha (le_bot_iff.mp <| h le_rfl <| sup_eq_right.mp eq.symm) | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Disjoint.right_lt_sup_of_left_ne_bot | null |
@[simp]
disjoint_sup_left : Disjoint (a ⊔ b) c ↔ Disjoint a c ∧ Disjoint b c := by
simp only [disjoint_iff, inf_sup_right, sup_eq_bot_iff]
@[simp] | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | disjoint_sup_left | null |
disjoint_sup_right : Disjoint a (b ⊔ c) ↔ Disjoint a b ∧ Disjoint a c := by
simp only [disjoint_iff, inf_sup_left, sup_eq_bot_iff] | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | disjoint_sup_right | null |
Disjoint.sup_left (ha : Disjoint a c) (hb : Disjoint b c) : Disjoint (a ⊔ b) c :=
disjoint_sup_left.2 ⟨ha, hb⟩ | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Disjoint.sup_left | null |
Disjoint.sup_right (hb : Disjoint a b) (hc : Disjoint a c) : Disjoint a (b ⊔ c) :=
disjoint_sup_right.2 ⟨hb, hc⟩ | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Disjoint.sup_right | null |
Disjoint.left_le_of_le_sup_right (h : a ≤ b ⊔ c) (hd : Disjoint a c) : a ≤ b :=
le_of_inf_le_sup_le (le_trans hd.le_bot bot_le) <| sup_le h le_sup_right | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Disjoint.left_le_of_le_sup_right | null |
Disjoint.left_le_of_le_sup_left (h : a ≤ c ⊔ b) (hd : Disjoint a c) : a ≤ b :=
hd.left_le_of_le_sup_right <| by rwa [sup_comm] | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Disjoint.left_le_of_le_sup_left | null |
Codisjoint (a b : α) : Prop :=
∀ ⦃x⦄, a ≤ x → b ≤ x → ⊤ ≤ x | def | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint | Two elements of a lattice are codisjoint if their sup is the top element.
Note that we define this without reference to `⊔`, as this allows us to talk about orders where
the supremum is not unique, or where implement `Sup` would require additional `Decidable`
arguments. |
codisjoint_comm : Codisjoint a b ↔ Codisjoint b a :=
forall_congr' fun _ ↦ forall_swap
@[symm] | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | codisjoint_comm | null |
Codisjoint.symm ⦃a b : α⦄ : Codisjoint a b → Codisjoint b a :=
codisjoint_comm.1 | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint.symm | null |
symmetric_codisjoint : Symmetric (Codisjoint : α → α → Prop) :=
Codisjoint.symm
@[simp] | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | symmetric_codisjoint | null |
codisjoint_top_left : Codisjoint ⊤ a := fun _ htop _ ↦ htop
@[simp] | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | codisjoint_top_left | null |
codisjoint_top_right : Codisjoint a ⊤ := fun _ _ htop ↦ htop
@[gcongr] theorem Codisjoint.mono (h₁ : a ≤ b) (h₂ : c ≤ d) : Codisjoint a c → Codisjoint b d :=
fun h _ ha hc ↦ h (h₁.trans ha) (h₂.trans hc) | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | codisjoint_top_right | null |
Codisjoint.mono_left (h : a ≤ b) : Codisjoint a c → Codisjoint b c :=
Codisjoint.mono h le_rfl | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint.mono_left | null |
Codisjoint.mono_right : b ≤ c → Codisjoint a b → Codisjoint a c :=
Codisjoint.mono le_rfl
@[simp] | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint.mono_right | null |
codisjoint_self : Codisjoint a a ↔ a = ⊤ :=
⟨fun hd ↦ top_unique <| hd le_rfl le_rfl, fun h _ ha _ ↦ h.symm.trans_le ha⟩
/- TODO: Rename `Codisjoint.eq_top` to `Codisjoint.sup_eq` and `Codisjoint.eq_top_of_self` to
`Codisjoint.eq_top` -/
alias ⟨Codisjoint.eq_top_of_self, _⟩ := codisjoint_self | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | codisjoint_self | null |
Codisjoint.ne (ha : a ≠ ⊤) (hab : Codisjoint a b) : a ≠ b :=
fun h ↦ ha <| codisjoint_self.1 <| by rwa [← h] at hab | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint.ne | null |
Codisjoint.eq_top_of_le (hab : Codisjoint a b) (h : b ≤ a) : a = ⊤ :=
eq_top_iff.2 <| hab le_rfl h | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint.eq_top_of_le | null |
Codisjoint.eq_top_of_ge (hab : Codisjoint a b) : a ≤ b → b = ⊤ :=
hab.symm.eq_top_of_le | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint.eq_top_of_ge | null |
Codisjoint.eq_iff (hab : Codisjoint a b) : a = b ↔ a = ⊤ ∧ b = ⊤ := by aesop | lemma | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint.eq_iff | null |
Codisjoint.ne_iff (hab : Codisjoint a b) : a ≠ b ↔ a ≠ ⊤ ∨ b ≠ ⊤ :=
hab.eq_iff.not.trans not_and_or | lemma | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint.ne_iff | null |
@[simp]
codisjoint_bot : Codisjoint a ⊥ ↔ a = ⊤ :=
⟨fun h ↦ top_unique <| h le_rfl bot_le, fun h _ ha _ ↦ h.symm.trans_le ha⟩
@[simp] | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | codisjoint_bot | null |
bot_codisjoint : Codisjoint ⊥ a ↔ a = ⊤ :=
⟨fun h ↦ top_unique <| h bot_le le_rfl, fun h _ _ ha ↦ h.symm.trans_le ha⟩ | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | bot_codisjoint | null |
Codisjoint.ne_bot_of_ne_top (h : Codisjoint a b) (ha : a ≠ ⊤) : b ≠ ⊥ := by
rintro rfl; exact ha <| by simpa using h | lemma | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint.ne_bot_of_ne_top | null |
Codisjoint.ne_bot_of_ne_top' (h : Codisjoint a b) (hb : b ≠ ⊤) : a ≠ ⊥ := by
rintro rfl; exact hb <| by simpa using h | lemma | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint.ne_bot_of_ne_top' | null |
codisjoint_iff_le_sup : Codisjoint a b ↔ ⊤ ≤ a ⊔ b :=
@disjoint_iff_inf_le αᵒᵈ _ _ _ _ | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | codisjoint_iff_le_sup | null |
codisjoint_iff : Codisjoint a b ↔ a ⊔ b = ⊤ :=
@disjoint_iff αᵒᵈ _ _ _ _ | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | codisjoint_iff | null |
Codisjoint.top_le : Codisjoint a b → ⊤ ≤ a ⊔ b :=
@Disjoint.le_bot αᵒᵈ _ _ _ _ | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint.top_le | null |
Codisjoint.eq_top : Codisjoint a b → a ⊔ b = ⊤ :=
@Disjoint.eq_bot αᵒᵈ _ _ _ _ | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint.eq_top | null |
codisjoint_assoc : Codisjoint (a ⊔ b) c ↔ Codisjoint a (b ⊔ c) :=
@disjoint_assoc αᵒᵈ _ _ _ _ _ | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | codisjoint_assoc | null |
codisjoint_left_comm : Codisjoint a (b ⊔ c) ↔ Codisjoint b (a ⊔ c) :=
@disjoint_left_comm αᵒᵈ _ _ _ _ _ | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | codisjoint_left_comm | null |
codisjoint_right_comm : Codisjoint (a ⊔ b) c ↔ Codisjoint (a ⊔ c) b :=
@disjoint_right_comm αᵒᵈ _ _ _ _ _
variable (c) | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | codisjoint_right_comm | null |
Codisjoint.sup_left (h : Codisjoint a b) : Codisjoint (a ⊔ c) b :=
h.mono_left le_sup_left | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint.sup_left | null |
Codisjoint.sup_left' (h : Codisjoint a b) : Codisjoint (c ⊔ a) b :=
h.mono_left le_sup_right | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint.sup_left' | null |
Codisjoint.sup_right (h : Codisjoint a b) : Codisjoint a (b ⊔ c) :=
h.mono_right le_sup_left | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint.sup_right | null |
Codisjoint.sup_right' (h : Codisjoint a b) : Codisjoint a (c ⊔ b) :=
h.mono_right le_sup_right
variable {c} | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint.sup_right' | null |
Codisjoint.of_codisjoint_sup_of_le (h : Codisjoint (a ⊔ b) c) (hle : c ≤ a) :
Codisjoint a b :=
@Disjoint.of_disjoint_inf_of_le αᵒᵈ _ _ _ _ _ h hle | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint.of_codisjoint_sup_of_le | null |
Codisjoint.of_codisjoint_sup_of_le' (h : Codisjoint (a ⊔ b) c) (hle : c ≤ b) :
Codisjoint a b :=
@Disjoint.of_disjoint_inf_of_le' αᵒᵈ _ _ _ _ _ h hle | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint.of_codisjoint_sup_of_le' | null |
@[simp]
codisjoint_inf_left : Codisjoint (a ⊓ b) c ↔ Codisjoint a c ∧ Codisjoint b c := by
simp only [codisjoint_iff, sup_inf_right, inf_eq_top_iff]
@[simp] | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | codisjoint_inf_left | null |
codisjoint_inf_right : Codisjoint a (b ⊓ c) ↔ Codisjoint a b ∧ Codisjoint a c := by
simp only [codisjoint_iff, sup_inf_left, inf_eq_top_iff] | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | codisjoint_inf_right | null |
Codisjoint.inf_left (ha : Codisjoint a c) (hb : Codisjoint b c) : Codisjoint (a ⊓ b) c :=
codisjoint_inf_left.2 ⟨ha, hb⟩ | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint.inf_left | null |
Codisjoint.inf_right (hb : Codisjoint a b) (hc : Codisjoint a c) : Codisjoint a (b ⊓ c) :=
codisjoint_inf_right.2 ⟨hb, hc⟩ | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint.inf_right | null |
Codisjoint.left_le_of_le_inf_right (h : a ⊓ b ≤ c) (hd : Codisjoint b c) : a ≤ c :=
@Disjoint.left_le_of_le_sup_right αᵒᵈ _ _ _ _ _ h hd.symm | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint.left_le_of_le_inf_right | null |
Codisjoint.left_le_of_le_inf_left (h : b ⊓ a ≤ c) (hd : Codisjoint b c) : a ≤ c :=
hd.left_le_of_le_inf_right <| by rwa [inf_comm] | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint.left_le_of_le_inf_left | null |
Disjoint.dual [PartialOrder α] [OrderBot α] {a b : α} :
Disjoint a b → Codisjoint (toDual a) (toDual b) :=
id | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Disjoint.dual | null |
Codisjoint.dual [PartialOrder α] [OrderTop α] {a b : α} :
Codisjoint a b → Disjoint (toDual a) (toDual b) :=
id
@[simp] | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | Codisjoint.dual | null |
disjoint_toDual_iff [PartialOrder α] [OrderTop α] {a b : α} :
Disjoint (toDual a) (toDual b) ↔ Codisjoint a b :=
Iff.rfl
@[simp] | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | disjoint_toDual_iff | null |
disjoint_ofDual_iff [PartialOrder α] [OrderBot α] {a b : αᵒᵈ} :
Disjoint (ofDual a) (ofDual b) ↔ Codisjoint a b :=
Iff.rfl
@[simp] | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | disjoint_ofDual_iff | null |
codisjoint_toDual_iff [PartialOrder α] [OrderBot α] {a b : α} :
Codisjoint (toDual a) (toDual b) ↔ Disjoint a b :=
Iff.rfl
@[simp] | theorem | Order | [
"Aesop",
"Mathlib.Order.BoundedOrder.Lattice"
] | Mathlib/Order/Disjoint.lean | codisjoint_toDual_iff | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.