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/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
module
public import Mathlib.Data.Set.NAry
public import Mathlib.Data.ULift
public import Mathlib.Order.CompleteLattice.Defs
public import Mathlib.Order.Hom.Set
/-!
# Theory of complete lattices
This file contains basic results on complete lattices.
## Naming conventions
In lemma names,
* `sSup` is called `sSup`
* `sInf` is called `sInf`
* `⨆ i, s i` is called `iSup`
* `⨅ i, s i` is called `iInf`
* `⨆ i j, s i j` is called `iSup₂`. This is an `iSup` inside an `iSup`.
* `⨅ i j, s i j` is called `iInf₂`. This is an `iInf` inside an `iInf`.
* `⨆ i ∈ s, t i` is called `biSup` for "bounded `iSup`". This is the special case of `iSup₂`
where `j : i ∈ s`.
* `⨅ i ∈ s, t i` is called `biInf` for "bounded `iInf`". This is the special case of `iInf₂`
where `j : i ∈ s`.
## Notation
* `⨆ i, f i` : `iSup f`, the supremum of the range of `f`;
* `⨅ i, f i` : `iInf f`, the infimum of the range of `f`.
-/
@[expose] public section
open Function OrderDual Set
variable {α β γ : Type*} {ι ι' : Sort*} {κ : ι → Sort*} {κ' : ι' → Sort*}
@[simp] lemma iSup_ulift {ι : Type*} [SupSet α] (f : ULift ι → α) :
⨆ i : ULift ι, f i = ⨆ i, f (.up i) := by simp only [iSup]; congr with x; simp
@[simp] lemma iInf_ulift {ι : Type*} [InfSet α] (f : ULift ι → α) :
⨅ i : ULift ι, f i = ⨅ i, f (.up i) := by simp only [iInf]; congr with x; simp
section
variable [CompleteSemilatticeSup α] {s t : Set α} {a b : α}
theorem sSup_le_sSup_of_isCofinalFor (h : IsCofinalFor s t) : sSup s ≤ sSup t :=
IsLeast.mono (isLUB_sSup t) (isLUB_sSup s) <| upperBounds_mono_of_isCofinalFor h
-- We will generalize this to conditionally complete lattices in `csSup_singleton`.
@[simp]
theorem sSup_singleton {a : α} : sSup {a} = a :=
isLUB_singleton.sSup_eq
end
section
variable [CompleteSemilatticeInf α] {s t : Set α} {a b : α}
theorem sInf_le_sInf_of_isCoinitialFor (h : IsCoinitialFor s t) : sInf t ≤ sInf s :=
IsGreatest.mono (isGLB_sInf t) (isGLB_sInf s) <| lowerBounds_mono_of_isCoinitialFor h
-- We will generalize this to conditionally complete lattices in `csInf_singleton`.
@[simp]
theorem sInf_singleton {a : α} : sInf {a} = a :=
isGLB_singleton.sInf_eq
end
open OrderDual
section
variable [CompleteLattice α] {s t : Set α} {b : α}
theorem sInf_le_sSup (hs : s.Nonempty) : sInf s ≤ sSup s :=
isGLB_le_isLUB (isGLB_sInf s) (isLUB_sSup s) hs
theorem sSup_union {s t : Set α} : sSup (s ∪ t) = sSup s ⊔ sSup t :=
((isLUB_sSup s).union (isLUB_sSup t)).sSup_eq
theorem sInf_union {s t : Set α} : sInf (s ∪ t) = sInf s ⊓ sInf t :=
((isGLB_sInf s).union (isGLB_sInf t)).sInf_eq
theorem sSup_inter_le {s t : Set α} : sSup (s ∩ t) ≤ sSup s ⊓ sSup t :=
sSup_le fun _ hb => le_inf (le_sSup hb.1) (le_sSup hb.2)
theorem le_sInf_inter {s t : Set α} : sInf s ⊔ sInf t ≤ sInf (s ∩ t) :=
@sSup_inter_le αᵒᵈ _ _ _
@[simp]
theorem sSup_empty : sSup ∅ = (⊥ : α) :=
(@isLUB_empty α _ _).sSup_eq
@[simp]
theorem sInf_empty : sInf ∅ = (⊤ : α) :=
(@isGLB_empty α _ _).sInf_eq
@[simp]
theorem sSup_univ : sSup univ = (⊤ : α) :=
(@isLUB_univ α _ _).sSup_eq
@[simp]
theorem sInf_univ : sInf univ = (⊥ : α) :=
(@isGLB_univ α _ _).sInf_eq
-- TODO(Jeremy): get this automatically
@[simp]
theorem sSup_insert {a : α} {s : Set α} : sSup (insert a s) = a ⊔ sSup s :=
((isLUB_sSup s).insert a).sSup_eq
@[simp]
theorem sInf_insert {a : α} {s : Set α} : sInf (insert a s) = a ⊓ sInf s :=
((isGLB_sInf s).insert a).sInf_eq
theorem sSup_le_sSup_of_subset_insert_bot (h : s ⊆ insert ⊥ t) : sSup s ≤ sSup t :=
(sSup_le_sSup h).trans_eq (sSup_insert.trans (bot_sup_eq _))
theorem sInf_le_sInf_of_subset_insert_top (h : s ⊆ insert ⊤ t) : sInf t ≤ sInf s :=
(sInf_le_sInf h).trans_eq' (sInf_insert.trans (top_inf_eq _))
@[simp]
theorem sSup_diff_singleton_bot (s : Set α) : sSup (s \ {⊥}) = sSup s :=
(sSup_le_sSup diff_subset).antisymm <|
sSup_le_sSup_of_subset_insert_bot <| subset_insert_diff_singleton _ _
@[simp]
theorem sInf_diff_singleton_top (s : Set α) : sInf (s \ {⊤}) = sInf s :=
@sSup_diff_singleton_bot αᵒᵈ _ s
theorem sSup_pair {a b : α} : sSup {a, b} = a ⊔ b :=
(@isLUB_pair α _ a b).sSup_eq
theorem sInf_pair {a b : α} : sInf {a, b} = a ⊓ b :=
(@isGLB_pair α _ a b).sInf_eq
@[simp]
theorem sSup_eq_bot : sSup s = ⊥ ↔ ∀ a ∈ s, a = ⊥ :=
⟨fun h _ ha => bot_unique <| h ▸ le_sSup ha, fun h =>
bot_unique <| sSup_le fun a ha => le_bot_iff.2 <| h a ha⟩
@[simp]
theorem sInf_eq_top : sInf s = ⊤ ↔ ∀ a ∈ s, a = ⊤ :=
@sSup_eq_bot αᵒᵈ _ _
lemma sSup_eq_bot' {s : Set α} : sSup s = ⊥ ↔ s = ∅ ∨ s = {⊥} := by
rw [sSup_eq_bot, ← subset_singleton_iff_eq, subset_singleton_iff]
theorem eq_singleton_bot_of_sSup_eq_bot_of_nonempty {s : Set α} (h_sup : sSup s = ⊥)
(hne : s.Nonempty) : s = {⊥} := by
rw [Set.eq_singleton_iff_nonempty_unique_mem]
rw [sSup_eq_bot] at h_sup
exact ⟨hne, h_sup⟩
theorem eq_singleton_top_of_sInf_eq_top_of_nonempty : sInf s = ⊤ → s.Nonempty → s = {⊤} :=
@eq_singleton_bot_of_sSup_eq_bot_of_nonempty αᵒᵈ _ _
/-- Introduction rule to prove that `b` is the supremum of `s`: it suffices to check that `b`
is larger than all elements of `s`, and that this is not the case of any `w < b`.
See `csSup_eq_of_forall_le_of_forall_lt_exists_gt` for a version in conditionally complete
lattices. -/
theorem sSup_eq_of_forall_le_of_forall_lt_exists_gt (h₁ : ∀ a ∈ s, a ≤ b)
(h₂ : ∀ w, w < b → ∃ a ∈ s, w < a) : sSup s = b :=
(sSup_le h₁).eq_of_not_lt fun h =>
let ⟨_, ha, ha'⟩ := h₂ _ h
((le_sSup ha).trans_lt ha').false
/-- Introduction rule to prove that `b` is the infimum of `s`: it suffices to check that `b`
is smaller than all elements of `s`, and that this is not the case of any `w > b`.
See `csInf_eq_of_forall_ge_of_forall_gt_exists_lt` for a version in conditionally complete
lattices. -/
theorem sInf_eq_of_forall_ge_of_forall_gt_exists_lt :
(∀ a ∈ s, b ≤ a) → (∀ w, b < w → ∃ a ∈ s, a < w) → sInf s = b :=
@sSup_eq_of_forall_le_of_forall_lt_exists_gt αᵒᵈ _ _ _
end
/-
### iSup & iInf
-/
section SupSet
variable [SupSet α] {f g : ι → α}
theorem sSup_range : sSup (range f) = iSup f :=
rfl
theorem sSup_eq_iSup' (s : Set α) : sSup s = ⨆ a : s, (a : α) := by rw [iSup, Subtype.range_coe]
theorem iSup_congr (h : ∀ i, f i = g i) : ⨆ i, f i = ⨆ i, g i :=
congr_arg _ <| funext h
theorem biSup_congr {p : ι → Prop} (h : ∀ i, p i → f i = g i) :
⨆ (i) (_ : p i), f i = ⨆ (i) (_ : p i), g i :=
iSup_congr fun i ↦ iSup_congr (h i)
theorem biSup_congr' {p : ι → Prop} {f g : (i : ι) → p i → α}
(h : ∀ i (hi : p i), f i hi = g i hi) :
⨆ i, ⨆ (hi : p i), f i hi = ⨆ i, ⨆ (hi : p i), g i hi := by
grind
theorem Function.Surjective.iSup_comp {f : ι → ι'} (hf : Surjective f) (g : ι' → α) :
⨆ x, g (f x) = ⨆ y, g y := by
simp only [iSup.eq_1]
congr
exact hf.range_comp g
theorem Equiv.iSup_comp {g : ι' → α} (e : ι ≃ ι') : ⨆ x, g (e x) = ⨆ y, g y :=
e.surjective.iSup_comp _
protected theorem Function.Surjective.iSup_congr {g : ι' → α} (h : ι → ι') (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⨆ x, f x = ⨆ y, g y := by
convert h1.iSup_comp g
exact (h2 _).symm
protected theorem Equiv.iSup_congr {g : ι' → α} (e : ι ≃ ι') (h : ∀ x, g (e x) = f x) :
⨆ x, f x = ⨆ y, g y :=
e.surjective.iSup_congr _ h
@[to_dual (attr := congr)]
theorem iSup_congr_Prop {p q : Prop} {f₁ : p → α} {f₂ : q → α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iSup f₁ = iSup f₂ := by
obtain rfl := propext pq
congr with x
apply f
theorem iSup_plift_up (f : PLift ι → α) : ⨆ i, f (PLift.up i) = ⨆ i, f i :=
(PLift.up_surjective.iSup_congr _) fun _ => rfl
theorem iSup_plift_down (f : ι → α) : ⨆ i, f (PLift.down i) = ⨆ i, f i :=
(PLift.down_surjective.iSup_congr _) fun _ => rfl
theorem iSup_range' (g : β → α) (f : ι → β) : ⨆ b : range f, g b = ⨆ i, g (f i) := by
rw [iSup, iSup, ← image_eq_range, ← range_comp']
@[to_dual]
theorem sSup_image' {s : Set β} {f : β → α} : sSup (f '' s) = ⨆ a : s, f a := by
rw [iSup, image_eq_range]
end SupSet
section InfSet
variable [InfSet α] {f g : ι → α}
theorem sInf_range : sInf (range f) = iInf f :=
rfl
theorem sInf_eq_iInf' (s : Set α) : sInf s = ⨅ a : s, (a : α) :=
@sSup_eq_iSup' αᵒᵈ _ _
theorem iInf_congr (h : ∀ i, f i = g i) : ⨅ i, f i = ⨅ i, g i :=
congr_arg _ <| funext h
theorem biInf_congr {p : ι → Prop} (h : ∀ i, p i → f i = g i) :
⨅ (i) (_ : p i), f i = ⨅ (i) (_ : p i), g i :=
biSup_congr (α := αᵒᵈ) h
theorem biInf_congr' {p : ι → Prop} {f g : (i : ι) → p i → α}
(h : ∀ i (hi : p i), f i hi = g i hi) :
⨅ i, ⨅ (hi : p i), f i hi = ⨅ i, ⨅ (hi : p i), g i hi := by
grind
theorem Function.Surjective.iInf_comp {f : ι → ι'} (hf : Surjective f) (g : ι' → α) :
⨅ x, g (f x) = ⨅ y, g y :=
@Function.Surjective.iSup_comp αᵒᵈ _ _ _ f hf g
theorem Equiv.iInf_comp {g : ι' → α} (e : ι ≃ ι') : ⨅ x, g (e x) = ⨅ y, g y :=
@Equiv.iSup_comp αᵒᵈ _ _ _ _ e
protected theorem Function.Surjective.iInf_congr {g : ι' → α} (h : ι → ι') (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⨅ x, f x = ⨅ y, g y :=
@Function.Surjective.iSup_congr αᵒᵈ _ _ _ _ _ h h1 h2
protected theorem Equiv.iInf_congr {g : ι' → α} (e : ι ≃ ι') (h : ∀ x, g (e x) = f x) :
⨅ x, f x = ⨅ y, g y :=
@Equiv.iSup_congr αᵒᵈ _ _ _ _ _ e h
theorem iInf_plift_up (f : PLift ι → α) : ⨅ i, f (PLift.up i) = ⨅ i, f i :=
(PLift.up_surjective.iInf_congr _) fun _ => rfl
theorem iInf_plift_down (f : ι → α) : ⨅ i, f (PLift.down i) = ⨅ i, f i :=
(PLift.down_surjective.iInf_congr _) fun _ => rfl
theorem iInf_range' (g : β → α) (f : ι → β) : ⨅ b : range f, g b = ⨅ i, g (f i) :=
@iSup_range' αᵒᵈ _ _ _ _ _
end InfSet
section
variable [CompleteLattice α] {f g s : ι → α} {a b : α}
theorem le_iSup (f : ι → α) (i : ι) : f i ≤ iSup f :=
le_sSup ⟨i, rfl⟩
theorem iInf_le (f : ι → α) (i : ι) : iInf f ≤ f i :=
sInf_le ⟨i, rfl⟩
lemma iInf_le_iSup [Nonempty ι] : ⨅ i, f i ≤ ⨆ i, f i :=
(iInf_le _ (Classical.arbitrary _)).trans <| le_iSup _ (Classical.arbitrary _)
theorem isLUB_iSup : IsLUB (range f) (⨆ j, f j) :=
isLUB_sSup _
theorem isGLB_iInf : IsGLB (range f) (⨅ j, f j) :=
isGLB_sInf _
theorem IsLUB.iSup_eq (h : IsLUB (range f) a) : ⨆ j, f j = a :=
h.sSup_eq
theorem IsGLB.iInf_eq (h : IsGLB (range f) a) : ⨅ j, f j = a :=
h.sInf_eq
theorem le_iSup_of_le (i : ι) (h : a ≤ f i) : a ≤ iSup f :=
h.trans <| le_iSup _ i
theorem iInf_le_of_le (i : ι) (h : f i ≤ a) : iInf f ≤ a :=
(iInf_le _ i).trans h
theorem le_iSup₂ {f : ∀ i, κ i → α} (i : ι) (j : κ i) : f i j ≤ ⨆ (i) (j), f i j :=
le_iSup_of_le i <| le_iSup (f i) j
theorem iInf₂_le {f : ∀ i, κ i → α} (i : ι) (j : κ i) : ⨅ (i) (j), f i j ≤ f i j :=
iInf_le_of_le i <| iInf_le (f i) j
theorem le_iSup₂_of_le {f : ∀ i, κ i → α} (i : ι) (j : κ i) (h : a ≤ f i j) :
a ≤ ⨆ (i) (j), f i j :=
h.trans <| le_iSup₂ i j
theorem iInf₂_le_of_le {f : ∀ i, κ i → α} (i : ι) (j : κ i) (h : f i j ≤ a) :
⨅ (i) (j), f i j ≤ a :=
(iInf₂_le i j).trans h
theorem iSup_le (h : ∀ i, f i ≤ a) : iSup f ≤ a :=
sSup_le fun _ ⟨i, Eq⟩ => Eq ▸ h i
theorem le_iInf (h : ∀ i, a ≤ f i) : a ≤ iInf f :=
le_sInf fun _ ⟨i, Eq⟩ => Eq ▸ h i
theorem iSup₂_le {f : ∀ i, κ i → α} (h : ∀ i j, f i j ≤ a) : ⨆ (i) (j), f i j ≤ a :=
iSup_le fun i => iSup_le <| h i
theorem le_iInf₂ {f : ∀ i, κ i → α} (h : ∀ i j, a ≤ f i j) : a ≤ ⨅ (i) (j), f i j :=
le_iInf fun i => le_iInf <| h i
theorem iSup₂_le_iSup (κ : ι → Sort*) (f : ι → α) : ⨆ (i) (_ : κ i), f i ≤ ⨆ i, f i :=
iSup₂_le fun i _ => le_iSup f i
theorem iInf_le_iInf₂ (κ : ι → Sort*) (f : ι → α) : ⨅ i, f i ≤ ⨅ (i) (_ : κ i), f i :=
le_iInf₂ fun i _ => iInf_le f i
@[gcongr]
theorem iSup_mono (h : ∀ i, f i ≤ g i) : iSup f ≤ iSup g :=
iSup_le fun i => le_iSup_of_le i <| h i
@[gcongr]
theorem iInf_mono (h : ∀ i, f i ≤ g i) : iInf f ≤ iInf g :=
le_iInf fun i => iInf_le_of_le i <| h i
theorem iSup₂_mono {f g : ∀ i, κ i → α} (h : ∀ i j, f i j ≤ g i j) :
⨆ (i) (j), f i j ≤ ⨆ (i) (j), g i j :=
iSup_mono fun i => iSup_mono <| h i
theorem iInf₂_mono {f g : ∀ i, κ i → α} (h : ∀ i j, f i j ≤ g i j) :
⨅ (i) (j), f i j ≤ ⨅ (i) (j), g i j :=
iInf_mono fun i => iInf_mono <| h i
theorem iSup_mono' {g : ι' → α} (h : ∀ i, ∃ i', f i ≤ g i') : iSup f ≤ iSup g :=
iSup_le fun i => Exists.elim (h i) le_iSup_of_le
theorem iInf_mono' {g : ι' → α} (h : ∀ i', ∃ i, f i ≤ g i') : iInf f ≤ iInf g :=
le_iInf fun i' => Exists.elim (h i') iInf_le_of_le
theorem iSup₂_mono' {f : ∀ i, κ i → α} {g : ∀ i', κ' i' → α} (h : ∀ i j, ∃ i' j', f i j ≤ g i' j') :
⨆ (i) (j), f i j ≤ ⨆ (i) (j), g i j :=
iSup₂_le fun i j =>
let ⟨i', j', h⟩ := h i j
le_iSup₂_of_le i' j' h
theorem iInf₂_mono' {f : ∀ i, κ i → α} {g : ∀ i', κ' i' → α} (h : ∀ i j, ∃ i' j', f i' j' ≤ g i j) :
⨅ (i) (j), f i j ≤ ⨅ (i) (j), g i j :=
le_iInf₂ fun i j =>
let ⟨i', j', h⟩ := h i j
iInf₂_le_of_le i' j' h
theorem iSup_const_mono (h : ι → ι') : ⨆ _ : ι, a ≤ ⨆ _ : ι', a :=
iSup_le <| le_iSup _ ∘ h
theorem iInf_const_mono (h : ι' → ι) : ⨅ _ : ι, a ≤ ⨅ _ : ι', a :=
le_iInf <| iInf_le _ ∘ h
theorem iSup_iInf_le_iInf_iSup (f : ι → ι' → α) : ⨆ i, ⨅ j, f i j ≤ ⨅ j, ⨆ i, f i j :=
iSup_le fun i => iInf_mono fun j => le_iSup (fun i => f i j) i
theorem biSup_mono {p q : ι → Prop} (hpq : ∀ i, p i → q i) :
⨆ (i) (_ : p i), f i ≤ ⨆ (i) (_ : q i), f i :=
iSup_mono fun i => iSup_const_mono (hpq i)
theorem biInf_mono {p q : ι → Prop} (hpq : ∀ i, p i → q i) :
⨅ (i) (_ : q i), f i ≤ ⨅ (i) (_ : p i), f i :=
iInf_mono fun i => iInf_const_mono (hpq i)
@[simp]
theorem iSup_le_iff : iSup f ≤ a ↔ ∀ i, f i ≤ a :=
(isLUB_le_iff isLUB_iSup).trans forall_mem_range
@[simp]
theorem le_iInf_iff : a ≤ iInf f ↔ ∀ i, a ≤ f i :=
(le_isGLB_iff isGLB_iInf).trans forall_mem_range
theorem iSup₂_le_iff {f : ∀ i, κ i → α} : ⨆ (i) (j), f i j ≤ a ↔ ∀ i j, f i j ≤ a := by
simp_rw [iSup_le_iff]
theorem le_iInf₂_iff {f : ∀ i, κ i → α} : (a ≤ ⨅ (i) (j), f i j) ↔ ∀ i j, a ≤ f i j := by
simp_rw [le_iInf_iff]
theorem iSup_lt_iff : iSup f < a ↔ ∃ b, b < a ∧ ∀ i, f i ≤ b :=
⟨fun h => ⟨iSup f, h, le_iSup f⟩, fun ⟨_, h, hb⟩ => (iSup_le hb).trans_lt h⟩
theorem lt_iInf_iff : a < iInf f ↔ ∃ b, a < b ∧ ∀ i, b ≤ f i :=
⟨fun h => ⟨iInf f, h, iInf_le f⟩, fun ⟨_, h, hb⟩ => h.trans_le <| le_iInf hb⟩
theorem sSup_eq_iSup {s : Set α} : sSup s = ⨆ a ∈ s, a :=
le_antisymm (sSup_le le_iSup₂) (iSup₂_le fun _ => le_sSup)
theorem sInf_eq_iInf {s : Set α} : sInf s = ⨅ a ∈ s, a :=
@sSup_eq_iSup αᵒᵈ _ _
lemma sSup_lowerBounds_eq_sInf (s : Set α) : sSup (lowerBounds s) = sInf s :=
(isLUB_sSup _).unique (isGLB_sInf _).isLUB
lemma sInf_upperBounds_eq_sSup (s : Set α) : sInf (upperBounds s) = sSup s :=
(isGLB_sInf _).unique (isLUB_sSup _).isGLB
@[deprecated (since := "2026-02-01")] alias sInf_upperBounds_eq_csSup := sInf_upperBounds_eq_sSup
theorem Monotone.le_map_iSup [CompleteLattice β] {f : α → β} (hf : Monotone f) :
⨆ i, f (s i) ≤ f (iSup s) :=
iSup_le fun _ => hf <| le_iSup _ _
theorem Antitone.le_map_iInf [CompleteLattice β] {f : α → β} (hf : Antitone f) :
⨆ i, f (s i) ≤ f (iInf s) :=
hf.dual_left.le_map_iSup
theorem Monotone.le_map_iSup₂ [CompleteLattice β] {f : α → β} (hf : Monotone f) (s : ∀ i, κ i → α) :
⨆ (i) (j), f (s i j) ≤ f (⨆ (i) (j), s i j) :=
iSup₂_le fun _ _ => hf <| le_iSup₂ _ _
theorem Antitone.le_map_iInf₂ [CompleteLattice β] {f : α → β} (hf : Antitone f) (s : ∀ i, κ i → α) :
⨆ (i) (j), f (s i j) ≤ f (⨅ (i) (j), s i j) :=
hf.dual_left.le_map_iSup₂ _
theorem Monotone.le_map_sSup [CompleteLattice β] {s : Set α} {f : α → β} (hf : Monotone f) :
⨆ a ∈ s, f a ≤ f (sSup s) := by rw [sSup_eq_iSup]; exact hf.le_map_iSup₂ _
theorem Antitone.le_map_sInf [CompleteLattice β] {s : Set α} {f : α → β} (hf : Antitone f) :
⨆ a ∈ s, f a ≤ f (sInf s) :=
hf.dual_left.le_map_sSup
theorem OrderIso.map_iSup [CompleteLattice β] (f : α ≃o β) (x : ι → α) :
f (⨆ i, x i) = ⨆ i, f (x i) :=
eq_of_forall_ge_iff <| f.surjective.forall.2
fun x => by simp only [f.le_iff_le, iSup_le_iff]
lemma OrderIso.map_iSup₂ [CompleteLattice β] (f : α ≃o β) (x : ∀ i, κ i → α) :
f (⨆ i, ⨆ j, x i j) = ⨆ i, ⨆ j, f (x i j) :=
eq_of_forall_ge_iff <| f.surjective.forall.2
fun x => by simp only [f.le_iff_le, iSup_le_iff]
theorem OrderIso.map_iInf [CompleteLattice β] (f : α ≃o β) (x : ι → α) :
f (⨅ i, x i) = ⨅ i, f (x i) :=
OrderIso.map_iSup f.dual _
theorem OrderIso.map_iInf₂ [CompleteLattice β] (f : α ≃o β) (x : ∀ i, κ i → α) :
f (⨅ i, ⨅ j, x i j) = ⨅ i, ⨅ j, f (x i j) :=
OrderIso.map_iSup₂ f.dual _
theorem OrderIso.map_sSup [CompleteLattice β] (f : α ≃o β) (s : Set α) :
f (sSup s) = ⨆ a ∈ s, f a := by
simp only [sSup_eq_iSup, OrderIso.map_iSup]
theorem OrderIso.map_sInf [CompleteLattice β] (f : α ≃o β) (s : Set α) :
f (sInf s) = ⨅ a ∈ s, f a :=
OrderIso.map_sSup f.dual _
theorem iSup_comp_le {ι' : Sort*} (f : ι' → α) (g : ι → ι') : ⨆ x, f (g x) ≤ ⨆ y, f y :=
iSup_mono' fun _ => ⟨_, le_rfl⟩
theorem le_iInf_comp {ι' : Sort*} (f : ι' → α) (g : ι → ι') : ⨅ y, f y ≤ ⨅ x, f (g x) :=
iInf_mono' fun _ => ⟨_, le_rfl⟩
theorem Monotone.iSup_comp_eq [Preorder β] {f : β → α} (hf : Monotone f) {s : ι → β}
(hs : ∀ x, ∃ i, x ≤ s i) : ⨆ x, f (s x) = ⨆ y, f y :=
le_antisymm (iSup_comp_le _ _) (iSup_mono' fun x => (hs x).imp fun _ hi => hf hi)
theorem Monotone.iInf_comp_eq [Preorder β] {f : β → α} (hf : Monotone f) {s : ι → β}
(hs : ∀ x, ∃ i, s i ≤ x) : ⨅ x, f (s x) = ⨅ y, f y :=
le_antisymm (iInf_mono' fun x => (hs x).imp fun _ hi => hf hi) (le_iInf_comp _ _)
theorem Antitone.map_iSup_le [CompleteLattice β] {f : α → β} (hf : Antitone f) :
f (iSup s) ≤ ⨅ i, f (s i) :=
le_iInf fun _ => hf <| le_iSup _ _
theorem Monotone.map_iInf_le [CompleteLattice β] {f : α → β} (hf : Monotone f) :
f (iInf s) ≤ ⨅ i, f (s i) :=
hf.dual_left.map_iSup_le
theorem Antitone.map_iSup₂_le [CompleteLattice β] {f : α → β} (hf : Antitone f) (s : ∀ i, κ i → α) :
f (⨆ (i) (j), s i j) ≤ ⨅ (i) (j), f (s i j) :=
hf.dual.le_map_iInf₂ _
theorem Monotone.map_iInf₂_le [CompleteLattice β] {f : α → β} (hf : Monotone f) (s : ∀ i, κ i → α) :
f (⨅ (i) (j), s i j) ≤ ⨅ (i) (j), f (s i j) :=
hf.dual.le_map_iSup₂ _
theorem Antitone.map_sSup_le [CompleteLattice β] {s : Set α} {f : α → β} (hf : Antitone f) :
f (sSup s) ≤ ⨅ a ∈ s, f a := by
rw [sSup_eq_iSup]
exact hf.map_iSup₂_le _
theorem Monotone.map_sInf_le [CompleteLattice β] {s : Set α} {f : α → β} (hf : Monotone f) :
f (sInf s) ≤ ⨅ a ∈ s, f a :=
hf.dual_left.map_sSup_le
theorem iSup_const_le : ⨆ _ : ι, a ≤ a :=
iSup_le fun _ => le_rfl
theorem le_iInf_const : a ≤ ⨅ _ : ι, a :=
le_iInf fun _ => le_rfl
-- We generalize this to conditionally complete lattices in `ciSup_const` and `ciInf_const`.
theorem iSup_const [Nonempty ι] : ⨆ _ : ι, a = a := by rw [iSup, range_const, sSup_singleton]
theorem iInf_const [Nonempty ι] : ⨅ _ : ι, a = a :=
@iSup_const αᵒᵈ _ _ a _
lemma iSup_unique [Unique ι] (f : ι → α) : ⨆ i, f i = f default := by
simp only [congr_arg f (Unique.eq_default _), iSup_const]
lemma iInf_unique [Unique ι] (f : ι → α) : ⨅ i, f i = f default := by
simp only [congr_arg f (Unique.eq_default _), iInf_const]
@[simp]
theorem iSup_bot : (⨆ _ : ι, ⊥ : α) = ⊥ :=
bot_unique iSup_const_le
@[simp]
theorem iInf_top : (⨅ _ : ι, ⊤ : α) = ⊤ :=
top_unique le_iInf_const
@[simp]
theorem iSup_eq_bot : iSup s = ⊥ ↔ ∀ i, s i = ⊥ :=
sSup_eq_bot.trans forall_mem_range
@[simp]
theorem iInf_eq_top : iInf s = ⊤ ↔ ∀ i, s i = ⊤ :=
sInf_eq_top.trans forall_mem_range
@[simp] lemma bot_lt_iSup : ⊥ < ⨆ i, s i ↔ ∃ i, ⊥ < s i := by simp [bot_lt_iff_ne_bot]
@[simp] lemma iInf_lt_top : ⨅ i, s i < ⊤ ↔ ∃ i, s i < ⊤ := by simp [lt_top_iff_ne_top]
theorem iSup₂_eq_bot {f : ∀ i, κ i → α} : ⨆ (i) (j), f i j = ⊥ ↔ ∀ i j, f i j = ⊥ := by
simp
theorem iInf₂_eq_top {f : ∀ i, κ i → α} : ⨅ (i) (j), f i j = ⊤ ↔ ∀ i j, f i j = ⊤ := by
simp
@[simp]
theorem iSup_pos {p : Prop} {f : p → α} (hp : p) : ⨆ h : p, f h = f hp :=
le_antisymm (iSup_le fun _ => le_rfl) (le_iSup _ _)
@[simp]
theorem iInf_pos {p : Prop} {f : p → α} (hp : p) : ⨅ h : p, f h = f hp :=
le_antisymm (iInf_le _ _) (le_iInf fun _ => le_rfl)
@[simp]
theorem iSup_neg {p : Prop} {f : p → α} (hp : ¬p) : ⨆ h : p, f h = ⊥ :=
le_antisymm (iSup_le fun h => (hp h).elim) bot_le
@[simp]
theorem iInf_neg {p : Prop} {f : p → α} (hp : ¬p) : ⨅ h : p, f h = ⊤ :=
le_antisymm le_top <| le_iInf fun h => (hp h).elim
/-- Introduction rule to prove that `b` is the supremum of `f`: it suffices to check that `b`
is larger than `f i` for all `i`, and that this is not the case of any `w<b`.
See `ciSup_eq_of_forall_le_of_forall_lt_exists_gt` for a version in conditionally complete
lattices. -/
theorem iSup_eq_of_forall_le_of_forall_lt_exists_gt {f : ι → α} (h₁ : ∀ i, f i ≤ b)
(h₂ : ∀ w, w < b → ∃ i, w < f i) : ⨆ i : ι, f i = b :=
sSup_eq_of_forall_le_of_forall_lt_exists_gt (forall_mem_range.mpr h₁) fun w hw =>
exists_range_iff.mpr <| h₂ w hw
/-- Introduction rule to prove that `b` is the infimum of `f`: it suffices to check that `b`
is smaller than `f i` for all `i`, and that this is not the case of any `w>b`.
See `ciInf_eq_of_forall_ge_of_forall_gt_exists_lt` for a version in conditionally complete
lattices. -/
theorem iInf_eq_of_forall_ge_of_forall_gt_exists_lt :
(∀ i, b ≤ f i) → (∀ w, b < w → ∃ i, f i < w) → ⨅ i, f i = b :=
@iSup_eq_of_forall_le_of_forall_lt_exists_gt αᵒᵈ _ _ _ _
theorem iSup_eq_dif {p : Prop} [Decidable p] (a : p → α) :
⨆ h : p, a h = if h : p then a h else ⊥ := by by_cases h : p <;> simp [h]
theorem iSup_eq_if {p : Prop} [Decidable p] (a : α) : ⨆ _ : p, a = if p then a else ⊥ :=
iSup_eq_dif fun _ => a
theorem iInf_eq_dif {p : Prop} [Decidable p] (a : p → α) :
⨅ h : p, a h = if h : p then a h else ⊤ :=
@iSup_eq_dif αᵒᵈ _ _ _ _
theorem iInf_eq_if {p : Prop} [Decidable p] (a : α) : ⨅ _ : p, a = if p then a else ⊤ :=
iInf_eq_dif fun _ => a
theorem iSup_comm {f : ι → ι' → α} : ⨆ (i) (j), f i j = ⨆ (j) (i), f i j :=
le_antisymm (iSup_le fun i => iSup_mono fun j => le_iSup (fun i => f i j) i)
(iSup_le fun _ => iSup_mono fun _ => le_iSup _ _)
theorem iInf_comm {f : ι → ι' → α} : ⨅ (i) (j), f i j = ⨅ (j) (i), f i j :=
@iSup_comm αᵒᵈ _ _ _ _
theorem iSup₂_comm {ι₁ ι₂ : Sort*} {κ₁ : ι₁ → Sort*} {κ₂ : ι₂ → Sort*}
(f : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → α) :
⨆ (i₁) (j₁) (i₂) (j₂), f i₁ j₁ i₂ j₂ = ⨆ (i₂) (j₂) (i₁) (j₁), f i₁ j₁ i₂ j₂ := by
simp only [@iSup_comm _ (κ₁ _), @iSup_comm _ ι₁]
theorem iInf₂_comm {ι₁ ι₂ : Sort*} {κ₁ : ι₁ → Sort*} {κ₂ : ι₂ → Sort*}
(f : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → α) :
⨅ (i₁) (j₁) (i₂) (j₂), f i₁ j₁ i₂ j₂ = ⨅ (i₂) (j₂) (i₁) (j₁), f i₁ j₁ i₂ j₂ := by
simp only [@iInf_comm _ (κ₁ _), @iInf_comm _ ι₁]
/- TODO: this is strange. In the proof below, we get exactly the desired among the equalities,
but close does not get it.
begin
apply @le_antisymm,
simp, intros,
begin [smt]
ematch, ematch, ematch, trace_state, have := le_refl (f i_1 i),
trace_state, close
end
end
-/
@[simp]
theorem iSup_iSup_eq_left {b : β} {f : ∀ x : β, x = b → α} : ⨆ x, ⨆ h : x = b, f x h = f b rfl :=
(@le_iSup₂ _ _ _ _ f b rfl).antisymm'
(iSup_le fun c =>
iSup_le <| by
rintro rfl
rfl)
@[simp]
theorem iInf_iInf_eq_left {b : β} {f : ∀ x : β, x = b → α} : ⨅ x, ⨅ h : x = b, f x h = f b rfl :=
@iSup_iSup_eq_left αᵒᵈ _ _ _ _
@[simp]
theorem iSup_iSup_eq_right {b : β} {f : ∀ x : β, b = x → α} : ⨆ x, ⨆ h : b = x, f x h = f b rfl :=
(le_iSup₂ b rfl).antisymm'
(iSup₂_le fun c => by
rintro rfl
rfl)
@[simp]
theorem iInf_iInf_eq_right {b : β} {f : ∀ x : β, b = x → α} : ⨅ x, ⨅ h : b = x, f x h = f b rfl :=
@iSup_iSup_eq_right αᵒᵈ _ _ _ _
theorem iSup_subtype {p : ι → Prop} {f : Subtype p → α} : iSup f = ⨆ (i) (h : p i), f ⟨i, h⟩ :=
le_antisymm (iSup_le fun ⟨i, h⟩ => @le_iSup₂ _ _ p _ (fun i h => f ⟨i, h⟩) i h)
(iSup₂_le fun _ _ => le_iSup _ _)
theorem iInf_subtype : ∀ {p : ι → Prop} {f : Subtype p → α}, iInf f = ⨅ (i) (h : p i), f ⟨i, h⟩ :=
@iSup_subtype αᵒᵈ _ _
theorem iSup_subtype' {p : ι → Prop} {f : ∀ i, p i → α} :
⨆ (i) (h), f i h = ⨆ x : Subtype p, f x x.property :=
(@iSup_subtype _ _ _ p fun x => f x.val x.property).symm
theorem iInf_subtype' {p : ι → Prop} {f : ∀ i, p i → α} :
⨅ (i) (h : p i), f i h = ⨅ x : Subtype p, f x x.property :=
(@iInf_subtype _ _ _ p fun x => f x.val x.property).symm
theorem iSup_subtype'' {ι} (s : Set ι) (f : ι → α) : ⨆ i : s, f i = ⨆ (t : ι) (_ : t ∈ s), f t :=
iSup_subtype
theorem iInf_subtype'' {ι} (s : Set ι) (f : ι → α) : ⨅ i : s, f i = ⨅ (t : ι) (_ : t ∈ s), f t :=
iInf_subtype
theorem biSup_const {a : α} {s : Set β} (hs : s.Nonempty) : ⨆ i ∈ s, a = a := by
haveI : Nonempty s := Set.nonempty_coe_sort.mpr hs
rw [← iSup_subtype'', iSup_const]
theorem biInf_const {a : α} {s : Set β} (hs : s.Nonempty) : ⨅ i ∈ s, a = a :=
biSup_const (α := αᵒᵈ) hs
theorem iSup_sup_eq : ⨆ x, f x ⊔ g x = (⨆ x, f x) ⊔ ⨆ x, g x :=
le_antisymm (iSup_le fun _ => sup_le_sup (le_iSup _ _) <| le_iSup _ _)
(sup_le (iSup_mono fun _ => le_sup_left) <| iSup_mono fun _ => le_sup_right)
theorem iInf_inf_eq : ⨅ x, f x ⊓ g x = (⨅ x, f x) ⊓ ⨅ x, g x :=
@iSup_sup_eq αᵒᵈ _ _ _ _
lemma Equiv.biSup_comp {ι ι' : Type*} {g : ι' → α} (e : ι ≃ ι') (s : Set ι') :
⨆ i ∈ e.symm '' s, g (e i) = ⨆ i ∈ s, g i := by
simpa only [iSup_subtype'] using (image e.symm s).symm.iSup_comp (g := g ∘ (↑))
lemma Equiv.biInf_comp {ι ι' : Type*} {g : ι' → α} (e : ι ≃ ι') (s : Set ι') :
⨅ i ∈ e.symm '' s, g (e i) = ⨅ i ∈ s, g i :=
e.biSup_comp s (α := αᵒᵈ)
lemma biInf_le {ι : Type*} {s : Set ι} (f : ι → α) {i : ι} (hi : i ∈ s) : ⨅ i ∈ s, f i ≤ f i :=
iInf₂_le i hi
lemma le_biSup {ι : Type*} {s : Set ι} (f : ι → α) {i : ι} (hi : i ∈ s) : f i ≤ ⨆ i ∈ s, f i :=
biInf_le (α := αᵒᵈ) f hi
lemma biInf_le_biSup {ι : Type*} {s : Set ι} (hs : s.Nonempty) {f : ι → α} :
⨅ i ∈ s, f i ≤ ⨆ i ∈ s, f i :=
(biInf_le _ hs.choose_spec).trans <| le_biSup _ hs.choose_spec
/- TODO: here is another example where more flexible pattern matching might help.
begin
apply @le_antisymm,
safe, pose h := f a ⊓ g a, begin [smt] ematch, ematch end
end
-/
theorem iSup_sup [Nonempty ι] {f : ι → α} {a : α} : (⨆ x, f x) ⊔ a = ⨆ x, f x ⊔ a := by
rw [iSup_sup_eq, iSup_const]
theorem iInf_inf [Nonempty ι] {f : ι → α} {a : α} : (⨅ x, f x) ⊓ a = ⨅ x, f x ⊓ a := by
rw [iInf_inf_eq, iInf_const]
theorem sup_iSup [Nonempty ι] {f : ι → α} {a : α} : (a ⊔ ⨆ x, f x) = ⨆ x, a ⊔ f x := by
rw [iSup_sup_eq, iSup_const]
theorem inf_iInf [Nonempty ι] {f : ι → α} {a : α} : (a ⊓ ⨅ x, f x) = ⨅ x, a ⊓ f x := by
rw [iInf_inf_eq, iInf_const]
theorem biSup_sup {p : ι → Prop} {f : ∀ i, p i → α} {a : α} (h : ∃ i, p i) :
(⨆ (i) (h : p i), f i h) ⊔ a = ⨆ (i) (h : p i), f i h ⊔ a := by
haveI : Nonempty { i // p i } :=
let ⟨i, hi⟩ := h
⟨⟨i, hi⟩⟩
rw [iSup_subtype', iSup_subtype', iSup_sup]
theorem sup_biSup {p : ι → Prop} {f : ∀ i, p i → α} {a : α} (h : ∃ i, p i) :
(a ⊔ ⨆ (i) (h : p i), f i h) = ⨆ (i) (h : p i), a ⊔ f i h := by
simpa only [sup_comm] using @biSup_sup α _ _ p _ _ h
theorem biInf_inf {p : ι → Prop} {f : ∀ i, p i → α} {a : α} (h : ∃ i, p i) :
(⨅ (i) (h : p i), f i h) ⊓ a = ⨅ (i) (h : p i), f i h ⊓ a :=
@biSup_sup αᵒᵈ ι _ p f _ h
theorem inf_biInf {p : ι → Prop} {f : ∀ i, p i → α} {a : α} (h : ∃ i, p i) :
(a ⊓ ⨅ (i) (h : p i), f i h) = ⨅ (i) (h : p i), a ⊓ f i h :=
@sup_biSup αᵒᵈ ι _ p f _ h
lemma biSup_lt_eq_iSup {ι : Type*} [LT ι] [NoMaxOrder ι] {f : ι → α} :
⨆ (i) (j < i), f j = ⨆ i, f i := by
apply le_antisymm
· exact iSup_le fun _ ↦ iSup₂_le fun _ _ ↦ le_iSup _ _
· refine iSup_le fun j ↦ ?_
obtain ⟨i, jlt⟩ := exists_gt j
exact le_iSup_of_le i (le_iSup₂_of_le j jlt le_rfl)
lemma biSup_le_eq_iSup {ι : Type*} [Preorder ι] {f : ι → α} :
⨆ (i) (j ≤ i), f j = ⨆ i, f i := by
apply le_antisymm
· exact iSup_le fun _ ↦ iSup₂_le fun _ _ ↦ le_iSup _ _
· exact iSup_le fun j ↦ le_iSup_of_le j (le_iSup₂_of_le j le_rfl le_rfl)
lemma biInf_lt_eq_iInf {ι : Type*} [LT ι] [NoMaxOrder ι] {f : ι → α} :
⨅ (i) (j < i), f j = ⨅ i, f i :=
biSup_lt_eq_iSup (α := αᵒᵈ)
lemma biInf_le_eq_iInf {ι : Type*} [Preorder ι] {f : ι → α} : ⨅ (i) (j ≤ i), f j = ⨅ i, f i :=
biSup_le_eq_iSup (α := αᵒᵈ)
lemma biSup_gt_eq_iSup {ι : Type*} [LT ι] [NoMinOrder ι] {f : ι → α} :
⨆ (i) (j > i), f j = ⨆ i, f i :=
biSup_lt_eq_iSup (ι := ιᵒᵈ)
lemma biSup_ge_eq_iSup {ι : Type*} [Preorder ι] {f : ι → α} : ⨆ (i) (j ≥ i), f j = ⨆ i, f i :=
biSup_le_eq_iSup (ι := ιᵒᵈ)
lemma biInf_gt_eq_iInf {ι : Type*} [LT ι] [NoMinOrder ι] {f : ι → α} :
⨅ (i) (j > i), f j = ⨅ i, f i :=
biInf_lt_eq_iInf (ι := ιᵒᵈ)
lemma biInf_ge_eq_iInf {ι : Type*} [Preorder ι] {f : ι → α} : ⨅ (i) (j ≥ i), f j = ⨅ i, f i :=
biInf_le_eq_iInf (ι := ιᵒᵈ)
lemma biSup_le_eq_of_monotone [Preorder β] {f : β → α} (hf : Monotone f) (b : β) :
⨆ (b' ≤ b), f b' = f b :=
le_antisymm (iSup₂_le_iff.2 (fun _ hji ↦ hf hji))
(le_iSup_of_le b (ge_of_eq (iSup_pos le_rfl)))
lemma biInf_le_eq_of_antitone [Preorder β] {f : β → α} (hf : Antitone f) (b : β) :
⨅ (b' ≤ b), f b' = f b :=
biSup_le_eq_of_monotone (α := αᵒᵈ) hf.dual_right b
lemma biSup_ge_eq_of_antitone [Preorder β] {f : β → α} (hf : Antitone f) (b : β) :
⨆ (b' ≥ b), f b' = f b :=
biSup_le_eq_of_monotone (β := βᵒᵈ) hf.dual_left b
lemma biInf_ge_eq_of_monotone [Preorder β] {f : β → α} (hf : Monotone f) (b : β) :
⨅ (b' ≥ b), f b' = f b :=
biInf_le_eq_of_antitone (β := βᵒᵈ) hf.dual_left b
/-! ### `iSup` and `iInf` under `Prop` -/
theorem iSup_false {s : False → α} : iSup s = ⊥ := by simp
theorem iInf_false {s : False → α} : iInf s = ⊤ := by simp
theorem iSup_true {s : True → α} : iSup s = s trivial :=
iSup_pos trivial
theorem iInf_true {s : True → α} : iInf s = s trivial :=
iInf_pos trivial
@[simp]
theorem iSup_exists {p : ι → Prop} {f : Exists p → α} : ⨆ x, f x = ⨆ (i) (h), f ⟨i, h⟩ :=
le_antisymm (iSup_le fun ⟨i, h⟩ => @le_iSup₂ _ _ _ _ (fun _ _ => _) i h)
(iSup₂_le fun _ _ => le_iSup _ _)
@[simp]
theorem iInf_exists {p : ι → Prop} {f : Exists p → α} : ⨅ x, f x = ⨅ (i) (h), f ⟨i, h⟩ :=
@iSup_exists αᵒᵈ _ _ _ _
theorem iSup_and {p q : Prop} {s : p ∧ q → α} : iSup s = ⨆ (h₁) (h₂), s ⟨h₁, h₂⟩ :=
le_antisymm (iSup_le fun ⟨i, h⟩ => @le_iSup₂ _ _ _ _ (fun _ _ => _) i h)
(iSup₂_le fun _ _ => le_iSup _ _)
theorem iInf_and {p q : Prop} {s : p ∧ q → α} : iInf s = ⨅ (h₁) (h₂), s ⟨h₁, h₂⟩ :=
@iSup_and αᵒᵈ _ _ _ _
/-- The symmetric case of `iSup_and`, useful for rewriting into a supremum over a conjunction -/
theorem iSup_and' {p q : Prop} {s : p → q → α} :
⨆ (h₁ : p) (h₂ : q), s h₁ h₂ = ⨆ h : p ∧ q, s h.1 h.2 :=
Eq.symm iSup_and
/-- The symmetric case of `iInf_and`, useful for rewriting into an infimum over a conjunction -/
theorem iInf_and' {p q : Prop} {s : p → q → α} :
⨅ (h₁ : p) (h₂ : q), s h₁ h₂ = ⨅ h : p ∧ q, s h.1 h.2 :=
Eq.symm iInf_and
theorem iSup_or {p q : Prop} {s : p ∨ q → α} :
⨆ x, s x = (⨆ i, s (Or.inl i)) ⊔ ⨆ j, s (Or.inr j) :=
le_antisymm
(iSup_le fun i =>
match i with
| Or.inl _ => le_sup_of_le_left <| le_iSup (fun _ => s _) _
| Or.inr _ => le_sup_of_le_right <| le_iSup (fun _ => s _) _)
(sup_le (iSup_comp_le _ _) (iSup_comp_le _ _))
theorem iInf_or {p q : Prop} {s : p ∨ q → α} :
⨅ x, s x = (⨅ i, s (Or.inl i)) ⊓ ⨅ j, s (Or.inr j) :=
@iSup_or αᵒᵈ _ _ _ _
section
variable (p : ι → Prop) [DecidablePred p]
theorem iSup_dite (f : ∀ i, p i → α) (g : ∀ i, ¬p i → α) :
⨆ i, (if h : p i then f i h else g i h) = (⨆ (i) (h : p i), f i h) ⊔ ⨆ (i) (h : ¬p i),
g i h := by
rw [← iSup_sup_eq]
congr 1 with i
split_ifs with h <;> simp [h]
theorem iInf_dite (f : ∀ i, p i → α) (g : ∀ i, ¬p i → α) :
⨅ i, (if h : p i then f i h else g i h) = (⨅ (i) (h : p i), f i h) ⊓ ⨅ (i) (h : ¬p i), g i h :=
iSup_dite p (show ∀ i, p i → αᵒᵈ from f) g
theorem iSup_ite (f g : ι → α) :
⨆ i, (if p i then f i else g i) = (⨆ (i) (_ : p i), f i) ⊔ ⨆ (i) (_ : ¬p i), g i :=
iSup_dite _ _ _
theorem iInf_ite (f g : ι → α) :
⨅ i, (if p i then f i else g i) = (⨅ (i) (_ : p i), f i) ⊓ ⨅ (i) (_ : ¬p i), g i :=
iInf_dite _ _ _
end
theorem iSup_range {g : β → α} {f : ι → β} : ⨆ b ∈ range f, g b = ⨆ i, g (f i) := by
rw [← iSup_subtype'', iSup_range']
theorem iInf_range : ∀ {g : β → α} {f : ι → β}, ⨅ b ∈ range f, g b = ⨅ i, g (f i) :=
@iSup_range αᵒᵈ _ _ _
theorem sSup_image {s : Set β} {f : β → α} : sSup (f '' s) = ⨆ a ∈ s, f a := by
rw [← iSup_subtype'', sSup_image']
theorem sInf_image {s : Set β} {f : β → α} : sInf (f '' s) = ⨅ a ∈ s, f a :=
@sSup_image αᵒᵈ _ _ _ _
theorem OrderIso.map_sSup_eq_sSup_symm_preimage [CompleteLattice β] (f : α ≃o β) (s : Set α) :
f (sSup s) = sSup (f.symm ⁻¹' s) := by
rw [map_sSup, ← sSup_image, f.image_eq_preimage_symm]
theorem OrderIso.map_sInf_eq_sInf_symm_preimage [CompleteLattice β] (f : α ≃o β) (s : Set α) :
f (sInf s) = sInf (f.symm ⁻¹' s) := by
rw [map_sInf, ← sInf_image, f.image_eq_preimage_symm]
/-
### iSup and iInf under set constructions
-/
theorem iSup_emptyset {f : β → α} : ⨆ x ∈ (∅ : Set β), f x = ⊥ := by simp
theorem iInf_emptyset {f : β → α} : ⨅ x ∈ (∅ : Set β), f x = ⊤ := by simp
theorem iSup_univ {f : β → α} : ⨆ x ∈ (univ : Set β), f x = ⨆ x, f x := by simp
theorem iInf_univ {f : β → α} : ⨅ x ∈ (univ : Set β), f x = ⨅ x, f x := by simp
theorem iSup_union {f : β → α} {s t : Set β} :
⨆ x ∈ s ∪ t, f x = (⨆ x ∈ s, f x) ⊔ ⨆ x ∈ t, f x := by
simp_rw [mem_union, iSup_or, iSup_sup_eq]
theorem iInf_union {f : β → α} {s t : Set β} : ⨅ x ∈ s ∪ t, f x = (⨅ x ∈ s, f x) ⊓ ⨅ x ∈ t, f x :=
@iSup_union αᵒᵈ _ _ _ _ _
theorem iSup_split (f : β → α) (p : β → Prop) :
⨆ i, f i = (⨆ (i) (_ : p i), f i) ⊔ ⨆ (i) (_ : ¬p i), f i := by
simpa [Classical.em] using @iSup_union _ _ _ f { i | p i } { i | ¬p i }
theorem iInf_split :
∀ (f : β → α) (p : β → Prop), ⨅ i, f i = (⨅ (i) (_ : p i), f i) ⊓ ⨅ (i) (_ : ¬p i), f i :=
@iSup_split αᵒᵈ _ _
theorem iSup_split_single (f : β → α) (i₀ : β) : ⨆ i, f i = f i₀ ⊔ ⨆ (i) (_ : i ≠ i₀), f i := by
convert iSup_split f (fun i => i = i₀)
simp
theorem iInf_split_single (f : β → α) (i₀ : β) : ⨅ i, f i = f i₀ ⊓ ⨅ (i) (_ : i ≠ i₀), f i :=
@iSup_split_single αᵒᵈ _ _ _ _
theorem iSup_le_iSup_of_subset {f : β → α} {s t : Set β} : s ⊆ t → ⨆ x ∈ s, f x ≤ ⨆ x ∈ t, f x :=
biSup_mono
theorem iInf_le_iInf_of_subset {f : β → α} {s t : Set β} : s ⊆ t → ⨅ x ∈ t, f x ≤ ⨅ x ∈ s, f x :=
biInf_mono
theorem iSup_insert {f : β → α} {s : Set β} {b : β} :
⨆ x ∈ insert b s, f x = f b ⊔ ⨆ x ∈ s, f x :=
Eq.trans iSup_union <| congr_arg (fun x => x ⊔ ⨆ x ∈ s, f x) iSup_iSup_eq_left
theorem iInf_insert {f : β → α} {s : Set β} {b : β} :
⨅ x ∈ insert b s, f x = f b ⊓ ⨅ x ∈ s, f x :=
Eq.trans iInf_union <| congr_arg (fun x => x ⊓ ⨅ x ∈ s, f x) iInf_iInf_eq_left
theorem iSup_singleton {f : β → α} {b : β} : ⨆ x ∈ (singleton b : Set β), f x = f b := by simp
theorem iInf_singleton {f : β → α} {b : β} : ⨅ x ∈ (singleton b : Set β), f x = f b := by simp
theorem iSup_pair {f : β → α} {a b : β} : ⨆ x ∈ ({a, b} : Set β), f x = f a ⊔ f b := by
rw [iSup_insert, iSup_singleton]
theorem iInf_pair {f : β → α} {a b : β} : ⨅ x ∈ ({a, b} : Set β), f x = f a ⊓ f b := by
rw [iInf_insert, iInf_singleton]
theorem iSup_image {γ} {f : β → γ} {g : γ → α} {t : Set β} :
⨆ c ∈ f '' t, g c = ⨆ b ∈ t, g (f b) := by
rw [← sSup_image, ← sSup_image, ← image_comp, comp_def]
theorem iInf_image :
∀ {γ} {f : β → γ} {g : γ → α} {t : Set β}, ⨅ c ∈ f '' t, g c = ⨅ b ∈ t, g (f b) :=
@iSup_image αᵒᵈ _ _
theorem iSup_extend_bot {e : ι → β} (he : Injective e) (f : ι → α) :
⨆ j, extend e f ⊥ j = ⨆ i, f i := by
rw [iSup_split _ fun j => ∃ i, e i = j]
simp +contextual [he.extend_apply, extend_apply', @iSup_comm _ β ι]
theorem iInf_extend_top {e : ι → β} (he : Injective e) (f : ι → α) :
⨅ j, extend e f ⊤ j = iInf f :=
@iSup_extend_bot αᵒᵈ _ _ _ _ he _
theorem Set.BijOn.iSup_comp {s : Set β} {t : Set γ} {f : β → γ} (g : γ → α)
(hf : Set.BijOn f s t) : ⨆ x ∈ s, g (f x) = ⨆ y ∈ t, g y := by
rw [← hf.image_eq, iSup_image]
theorem Set.BijOn.iInf_comp {s : Set β} {t : Set γ} {f : β → γ} (g : γ → α)
(hf : Set.BijOn f s t) : ⨅ x ∈ s, g (f x) = ⨅ y ∈ t, g y := by
rw [← hf.image_eq, iInf_image]
theorem Set.BijOn.iSup_congr {s : Set β} {t : Set γ} (f : β → α) (g : γ → α) {h : β → γ}
(h1 : Set.BijOn h s t) (h2 : ∀ x, g (h x) = f x) : ⨆ x ∈ s, f x = ⨆ y ∈ t, g y := by
simpa only [h2] using h1.iSup_comp g
theorem Set.BijOn.iInf_congr {s : Set β} {t : Set γ} (f : β → α) (g : γ → α) {h : β → γ}
(h1 : Set.BijOn h s t) (h2 : ∀ x, g (h x) = f x) : ⨅ x ∈ s, f x = ⨅ y ∈ t, g y := by