state
stringlengths 0
159k
| srcUpToTactic
stringlengths 387
167k
| nextTactic
stringlengths 3
9k
| declUpToTactic
stringlengths 22
11.5k
| declId
stringlengths 38
95
| decl
stringlengths 16
1.89k
| file_tag
stringlengths 17
73
|
|---|---|---|---|---|---|---|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
t : α → Set β
s₁ s₂ : Set α
⊢ ⋃ x ∈ s₁, t x ⊆ ⋃ x ∈ s₂ ∪ s₁ \ s₂, t x
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
/-!
# The set lattice
This file provides usual set notation for unions and intersections, a `CompleteLattice` instance
for `Set α`, and some more set constructions.
## Main declarations
* `Set.iUnion`: **i**ndexed **union**. Union of an indexed family of sets.
* `Set.iInter`: **i**ndexed **inter**section. Intersection of an indexed family of sets.
* `Set.sInter`: **s**et **inter**section. Intersection of sets belonging to a set of sets.
* `Set.sUnion`: **s**et **union**. Union of sets belonging to a set of sets.
* `Set.sInter_eq_biInter`, `Set.sUnion_eq_biInter`: Shows that `⋂₀ s = ⋂ x ∈ s, x` and
`⋃₀ s = ⋃ x ∈ s, x`.
* `Set.completeAtomicBooleanAlgebra`: `Set α` is a `CompleteAtomicBooleanAlgebra` with `≤ = ⊆`,
`< = ⊂`, `⊓ = ∩`, `⊔ = ∪`, `⨅ = ⋂`, `⨆ = ⋃` and `\` as the set difference.
See `Set.BooleanAlgebra`.
* `Set.kernImage`: For a function `f : α → β`, `s.kernImage f` is the set of `y` such that
`f ⁻¹ y ⊆ s`.
* `Set.seq`: Union of the image of a set under a **seq**uence of functions. `seq s t` is the union
of `f '' t` over all `f ∈ s`, where `t : Set α` and `s : Set (α → β)`.
* `Set.iUnion_eq_sigma_of_disjoint`: Equivalence between `⋃ i, t i` and `Σ i, t i`, where `t` is an
indexed family of disjoint sets.
## Naming convention
In lemma names,
* `⋃ i, s i` is called `iUnion`
* `⋂ i, s i` is called `iInter`
* `⋃ i j, s i j` is called `iUnion₂`. This is an `iUnion` inside an `iUnion`.
* `⋂ i j, s i j` is called `iInter₂`. This is an `iInter` inside an `iInter`.
* `⋃ i ∈ s, t i` is called `biUnion` for "bounded `iUnion`". This is the special case of `iUnion₂`
where `j : i ∈ s`.
* `⋂ i ∈ s, t i` is called `biInter` for "bounded `iInter`". This is the special case of `iInter₂`
where `j : i ∈ s`.
## Notation
* `⋃`: `Set.iUnion`
* `⋂`: `Set.iInter`
* `⋃₀`: `Set.sUnion`
* `⋂₀`: `Set.sInter`
-/
set_option autoImplicit true
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι₂ : Sort*} {κ κ₁ κ₂ : ι → Sort*} {κ' : ι' → Sort*}
namespace Set
/-! ### Complete lattice and complete Boolean algebra instances -/
instance : InfSet (Set α) :=
⟨fun s => { a | ∀ t ∈ s, a ∈ t }⟩
instance : SupSet (Set α) :=
⟨fun s => { a | ∃ t ∈ s, a ∈ t }⟩
/-- Intersection of a set of sets. -/
def sInter (S : Set (Set α)) : Set α :=
sInf S
#align set.sInter Set.sInter
/-- Notation for `Set.sInter` Intersection of a set of sets. -/
prefix:110 "⋂₀ " => sInter
/-- Union of a set of sets. -/
def sUnion (S : Set (Set α)) : Set α :=
sSup S
#align set.sUnion Set.sUnion
/-- Notation for `Set.sUnion`. Union of a set of sets. -/
prefix:110 "⋃₀ " => sUnion
@[simp]
theorem mem_sInter {x : α} {S : Set (Set α)} : x ∈ ⋂₀ S ↔ ∀ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sInter Set.mem_sInter
@[simp]
theorem mem_sUnion {x : α} {S : Set (Set α)} : x ∈ ⋃₀ S ↔ ∃ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sUnion Set.mem_sUnion
/-- Indexed union of a family of sets -/
def iUnion (s : ι → Set β) : Set β :=
iSup s
#align set.Union Set.iUnion
/-- Indexed intersection of a family of sets -/
def iInter (s : ι → Set β) : Set β :=
iInf s
#align set.Inter Set.iInter
/-- Notation for `Set.iUnion`. Indexed union of a family of sets -/
notation3 "⋃ "(...)", "r:60:(scoped f => iUnion f) => r
/-- Notation for `Set.iInter`. Indexed intersection of a family of sets -/
notation3 "⋂ "(...)", "r:60:(scoped f => iInter f) => r
section delaborators
open Lean Lean.PrettyPrinter.Delaborator
/-- Delaborator for indexed unions. -/
@[delab app.Set.iUnion]
def iUnion_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋃ (_ : $dom), $body)
else if prop || ppTypes then
`(⋃ ($x:ident : $dom), $body)
else
`(⋃ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋃ $x:ident, ⋃ (_ : $y:ident ∈ $s), $body)
| `(⋃ ($x:ident : $_), ⋃ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋃ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
/-- Delaborator for indexed intersections. -/
@[delab app.Set.iInter]
def sInter_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋂ (_ : $dom), $body)
else if prop || ppTypes then
`(⋂ ($x:ident : $dom), $body)
else
`(⋂ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋂ $x:ident, ⋂ (_ : $y:ident ∈ $s), $body)
| `(⋂ ($x:ident : $_), ⋂ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋂ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
end delaborators
@[simp]
theorem sSup_eq_sUnion (S : Set (Set α)) : sSup S = ⋃₀S :=
rfl
#align set.Sup_eq_sUnion Set.sSup_eq_sUnion
@[simp]
theorem sInf_eq_sInter (S : Set (Set α)) : sInf S = ⋂₀ S :=
rfl
#align set.Inf_eq_sInter Set.sInf_eq_sInter
@[simp]
theorem iSup_eq_iUnion (s : ι → Set α) : iSup s = iUnion s :=
rfl
#align set.supr_eq_Union Set.iSup_eq_iUnion
@[simp]
theorem iInf_eq_iInter (s : ι → Set α) : iInf s = iInter s :=
rfl
#align set.infi_eq_Inter Set.iInf_eq_iInter
@[simp]
theorem mem_iUnion {x : α} {s : ι → Set α} : (x ∈ ⋃ i, s i) ↔ ∃ i, x ∈ s i :=
⟨fun ⟨_, ⟨⟨a, (t_eq : s a = _)⟩, (h : x ∈ _)⟩⟩ => ⟨a, t_eq.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨s a, ⟨⟨a, rfl⟩, h⟩⟩⟩
#align set.mem_Union Set.mem_iUnion
@[simp]
theorem mem_iInter {x : α} {s : ι → Set α} : (x ∈ ⋂ i, s i) ↔ ∀ i, x ∈ s i :=
⟨fun (h : ∀ a ∈ { a : Set α | ∃ i, s i = a }, x ∈ a) a => h (s a) ⟨a, rfl⟩,
fun h _ ⟨a, (eq : s a = _)⟩ => eq ▸ h a⟩
#align set.mem_Inter Set.mem_iInter
theorem mem_iUnion₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋃ (i) (j), s i j) ↔ ∃ i j, x ∈ s i j := by
simp_rw [mem_iUnion]
#align set.mem_Union₂ Set.mem_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋂ (i) (j), s i j) ↔ ∀ i j, x ∈ s i j := by
simp_rw [mem_iInter]
#align set.mem_Inter₂ Set.mem_iInter₂
theorem mem_iUnion_of_mem {s : ι → Set α} {a : α} (i : ι) (ha : a ∈ s i) : a ∈ ⋃ i, s i :=
mem_iUnion.2 ⟨i, ha⟩
#align set.mem_Union_of_mem Set.mem_iUnion_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iUnion₂_of_mem {s : ∀ i, κ i → Set α} {a : α} {i : ι} (j : κ i) (ha : a ∈ s i j) :
a ∈ ⋃ (i) (j), s i j :=
mem_iUnion₂.2 ⟨i, j, ha⟩
#align set.mem_Union₂_of_mem Set.mem_iUnion₂_of_mem
theorem mem_iInter_of_mem {s : ι → Set α} {a : α} (h : ∀ i, a ∈ s i) : a ∈ ⋂ i, s i :=
mem_iInter.2 h
#align set.mem_Inter_of_mem Set.mem_iInter_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂_of_mem {s : ∀ i, κ i → Set α} {a : α} (h : ∀ i j, a ∈ s i j) :
a ∈ ⋂ (i) (j), s i j :=
mem_iInter₂.2 h
#align set.mem_Inter₂_of_mem Set.mem_iInter₂_of_mem
instance Set.completeAtomicBooleanAlgebra : CompleteAtomicBooleanAlgebra (Set α) :=
{ instBooleanAlgebraSet with
le_sSup := fun s t t_in a a_in => ⟨t, t_in, a_in⟩
sSup_le := fun s t h a ⟨t', ⟨t'_in, a_in⟩⟩ => h t' t'_in a_in
le_sInf := fun s t h a a_in t' t'_in => h t' t'_in a_in
sInf_le := fun s t t_in a h => h _ t_in
iInf_iSup_eq := by intros; ext; simp [Classical.skolem] }
/-- `kernImage f s` is the set of `y` such that `f ⁻¹ y ⊆ s`. -/
def kernImage (f : α → β) (s : Set α) : Set β :=
{ y | ∀ ⦃x⦄, f x = y → x ∈ s }
#align set.kern_image Set.kernImage
lemma subset_kernImage_iff {f : α → β} : s ⊆ kernImage f t ↔ f ⁻¹' s ⊆ t :=
⟨fun h _ hx ↦ h hx rfl,
fun h _ hx y hy ↦ h (show f y ∈ s from hy.symm ▸ hx)⟩
section GaloisConnection
variable {f : α → β}
protected theorem image_preimage : GaloisConnection (image f) (preimage f) := fun _ _ =>
image_subset_iff
#align set.image_preimage Set.image_preimage
protected theorem preimage_kernImage : GaloisConnection (preimage f) (kernImage f) := fun _ _ =>
subset_kernImage_iff.symm
#align set.preimage_kern_image Set.preimage_kernImage
end GaloisConnection
section kernImage
variable {f : α → β}
lemma kernImage_mono : Monotone (kernImage f) :=
Set.preimage_kernImage.monotone_u
lemma kernImage_eq_compl {s : Set α} : kernImage f s = (f '' sᶜ)ᶜ :=
Set.preimage_kernImage.u_unique (Set.image_preimage.compl)
(fun t ↦ compl_compl (f ⁻¹' t) ▸ Set.preimage_compl)
lemma kernImage_compl {s : Set α} : kernImage f (sᶜ) = (f '' s)ᶜ := by
rw [kernImage_eq_compl, compl_compl]
lemma kernImage_empty : kernImage f ∅ = (range f)ᶜ := by
rw [kernImage_eq_compl, compl_empty, image_univ]
lemma kernImage_preimage_eq_iff {s : Set β} : kernImage f (f ⁻¹' s) = s ↔ (range f)ᶜ ⊆ s := by
rw [kernImage_eq_compl, ← preimage_compl, compl_eq_comm, eq_comm, image_preimage_eq_iff,
compl_subset_comm]
lemma compl_range_subset_kernImage {s : Set α} : (range f)ᶜ ⊆ kernImage f s := by
rw [← kernImage_empty]
exact kernImage_mono (empty_subset _)
lemma kernImage_union_preimage {s : Set α} {t : Set β} :
kernImage f (s ∪ f ⁻¹' t) = kernImage f s ∪ t := by
rw [kernImage_eq_compl, kernImage_eq_compl, compl_union, ← preimage_compl, image_inter_preimage,
compl_inter, compl_compl]
lemma kernImage_preimage_union {s : Set α} {t : Set β} :
kernImage f (f ⁻¹' t ∪ s) = t ∪ kernImage f s := by
rw [union_comm, kernImage_union_preimage, union_comm]
end kernImage
/-! ### Union and intersection over an indexed family of sets -/
instance : OrderTop (Set α) where
top := univ
le_top := by simp
@[congr]
theorem iUnion_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iUnion f₁ = iUnion f₂ :=
iSup_congr_Prop pq f
#align set.Union_congr_Prop Set.iUnion_congr_Prop
@[congr]
theorem iInter_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iInter f₁ = iInter f₂ :=
iInf_congr_Prop pq f
#align set.Inter_congr_Prop Set.iInter_congr_Prop
theorem iUnion_plift_up (f : PLift ι → Set α) : ⋃ i, f (PLift.up i) = ⋃ i, f i :=
iSup_plift_up _
#align set.Union_plift_up Set.iUnion_plift_up
theorem iUnion_plift_down (f : ι → Set α) : ⋃ i, f (PLift.down i) = ⋃ i, f i :=
iSup_plift_down _
#align set.Union_plift_down Set.iUnion_plift_down
theorem iInter_plift_up (f : PLift ι → Set α) : ⋂ i, f (PLift.up i) = ⋂ i, f i :=
iInf_plift_up _
#align set.Inter_plift_up Set.iInter_plift_up
theorem iInter_plift_down (f : ι → Set α) : ⋂ i, f (PLift.down i) = ⋂ i, f i :=
iInf_plift_down _
#align set.Inter_plift_down Set.iInter_plift_down
theorem iUnion_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋃ _ : p, s = if p then s else ∅ :=
iSup_eq_if _
#align set.Union_eq_if Set.iUnion_eq_if
theorem iUnion_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋃ h : p, s h = if h : p then s h else ∅ :=
iSup_eq_dif _
#align set.Union_eq_dif Set.iUnion_eq_dif
theorem iInter_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋂ _ : p, s = if p then s else univ :=
iInf_eq_if _
#align set.Inter_eq_if Set.iInter_eq_if
theorem iInf_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋂ h : p, s h = if h : p then s h else univ :=
_root_.iInf_eq_dif _
#align set.Infi_eq_dif Set.iInf_eq_dif
theorem exists_set_mem_of_union_eq_top {ι : Type*} (t : Set ι) (s : ι → Set β)
(w : ⋃ i ∈ t, s i = ⊤) (x : β) : ∃ i ∈ t, x ∈ s i := by
have p : x ∈ ⊤ := Set.mem_univ x
rw [← w, Set.mem_iUnion] at p
simpa using p
#align set.exists_set_mem_of_union_eq_top Set.exists_set_mem_of_union_eq_top
theorem nonempty_of_union_eq_top_of_nonempty {ι : Type*} (t : Set ι) (s : ι → Set α)
(H : Nonempty α) (w : ⋃ i ∈ t, s i = ⊤) : t.Nonempty := by
obtain ⟨x, m, -⟩ := exists_set_mem_of_union_eq_top t s w H.some
exact ⟨x, m⟩
#align set.nonempty_of_union_eq_top_of_nonempty Set.nonempty_of_union_eq_top_of_nonempty
theorem nonempty_of_nonempty_iUnion
{s : ι → Set α} (h_Union : (⋃ i, s i).Nonempty) : Nonempty ι := by
obtain ⟨x, hx⟩ := h_Union
exact ⟨Classical.choose $ mem_iUnion.mp hx⟩
theorem nonempty_of_nonempty_iUnion_eq_univ
{s : ι → Set α} [Nonempty α] (h_Union : ⋃ i, s i = univ) : Nonempty ι :=
nonempty_of_nonempty_iUnion (s := s) (by simpa only [h_Union] using univ_nonempty)
theorem setOf_exists (p : ι → β → Prop) : { x | ∃ i, p i x } = ⋃ i, { x | p i x } :=
ext fun _ => mem_iUnion.symm
#align set.set_of_exists Set.setOf_exists
theorem setOf_forall (p : ι → β → Prop) : { x | ∀ i, p i x } = ⋂ i, { x | p i x } :=
ext fun _ => mem_iInter.symm
#align set.set_of_forall Set.setOf_forall
theorem iUnion_subset {s : ι → Set α} {t : Set α} (h : ∀ i, s i ⊆ t) : ⋃ i, s i ⊆ t :=
iSup_le h
#align set.Union_subset Set.iUnion_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset {s : ∀ i, κ i → Set α} {t : Set α} (h : ∀ i j, s i j ⊆ t) :
⋃ (i) (j), s i j ⊆ t :=
iUnion_subset fun x => iUnion_subset (h x)
#align set.Union₂_subset Set.iUnion₂_subset
theorem subset_iInter {t : Set β} {s : ι → Set β} (h : ∀ i, t ⊆ s i) : t ⊆ ⋂ i, s i :=
le_iInf h
#align set.subset_Inter Set.subset_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iInter₂ {s : Set α} {t : ∀ i, κ i → Set α} (h : ∀ i j, s ⊆ t i j) :
s ⊆ ⋂ (i) (j), t i j :=
subset_iInter fun x => subset_iInter <| h x
#align set.subset_Inter₂ Set.subset_iInter₂
@[simp]
theorem iUnion_subset_iff {s : ι → Set α} {t : Set α} : ⋃ i, s i ⊆ t ↔ ∀ i, s i ⊆ t :=
⟨fun h _ => Subset.trans (le_iSup s _) h, iUnion_subset⟩
#align set.Union_subset_iff Set.iUnion_subset_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset_iff {s : ∀ i, κ i → Set α} {t : Set α} :
⋃ (i) (j), s i j ⊆ t ↔ ∀ i j, s i j ⊆ t := by simp_rw [iUnion_subset_iff]
#align set.Union₂_subset_iff Set.iUnion₂_subset_iff
@[simp]
theorem subset_iInter_iff {s : Set α} {t : ι → Set α} : (s ⊆ ⋂ i, t i) ↔ ∀ i, s ⊆ t i :=
le_iInf_iff
#align set.subset_Inter_iff Set.subset_iInter_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem subset_iInter₂_iff {s : Set α} {t : ∀ i, κ i → Set α} :
(s ⊆ ⋂ (i) (j), t i j) ↔ ∀ i j, s ⊆ t i j := by simp_rw [subset_iInter_iff]
#align set.subset_Inter₂_iff Set.subset_iInter₂_iff
theorem subset_iUnion : ∀ (s : ι → Set β) (i : ι), s i ⊆ ⋃ i, s i :=
le_iSup
#align set.subset_Union Set.subset_iUnion
theorem iInter_subset : ∀ (s : ι → Set β) (i : ι), ⋂ i, s i ⊆ s i :=
iInf_le
#align set.Inter_subset Set.iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iUnion₂ {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : s i j ⊆ ⋃ (i') (j'), s i' j' :=
le_iSup₂ i j
#align set.subset_Union₂ Set.subset_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_subset {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : ⋂ (i) (j), s i j ⊆ s i j :=
iInf₂_le i j
#align set.Inter₂_subset Set.iInter₂_subset
/-- This rather trivial consequence of `subset_iUnion`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem subset_iUnion_of_subset {s : Set α} {t : ι → Set α} (i : ι) (h : s ⊆ t i) : s ⊆ ⋃ i, t i :=
le_iSup_of_le i h
#align set.subset_Union_of_subset Set.subset_iUnion_of_subset
/-- This rather trivial consequence of `iInter_subset`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem iInter_subset_of_subset {s : ι → Set α} {t : Set α} (i : ι) (h : s i ⊆ t) :
⋂ i, s i ⊆ t :=
iInf_le_of_le i h
#align set.Inter_subset_of_subset Set.iInter_subset_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `subset_iUnion₂` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem subset_iUnion₂_of_subset {s : Set α} {t : ∀ i, κ i → Set α} (i : ι) (j : κ i)
(h : s ⊆ t i j) : s ⊆ ⋃ (i) (j), t i j :=
le_iSup₂_of_le i j h
#align set.subset_Union₂_of_subset Set.subset_iUnion₂_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `iInter₂_subset` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem iInter₂_subset_of_subset {s : ∀ i, κ i → Set α} {t : Set α} (i : ι) (j : κ i)
(h : s i j ⊆ t) : ⋂ (i) (j), s i j ⊆ t :=
iInf₂_le_of_le i j h
#align set.Inter₂_subset_of_subset Set.iInter₂_subset_of_subset
theorem iUnion_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono h
#align set.Union_mono Set.iUnion_mono
@[gcongr]
theorem iUnion_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iUnion s ⊆ iUnion t :=
iSup_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋃ (i) (j), s i j ⊆ ⋃ (i) (j), t i j :=
iSup₂_mono h
#align set.Union₂_mono Set.iUnion₂_mono
theorem iInter_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋂ i, s i ⊆ ⋂ i, t i :=
iInf_mono h
#align set.Inter_mono Set.iInter_mono
@[gcongr]
theorem iInter_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iInter s ⊆ iInter t :=
iInf_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋂ (i) (j), s i j ⊆ ⋂ (i) (j), t i j :=
iInf₂_mono h
#align set.Inter₂_mono Set.iInter₂_mono
theorem iUnion_mono' {s : ι → Set α} {t : ι₂ → Set α} (h : ∀ i, ∃ j, s i ⊆ t j) :
⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono' h
#align set.Union_mono' Set.iUnion_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iUnion₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i j, ∃ i' j', s i j ⊆ t i' j') : ⋃ (i) (j), s i j ⊆ ⋃ (i') (j'), t i' j' :=
iSup₂_mono' h
#align set.Union₂_mono' Set.iUnion₂_mono'
theorem iInter_mono' {s : ι → Set α} {t : ι' → Set α} (h : ∀ j, ∃ i, s i ⊆ t j) :
⋂ i, s i ⊆ ⋂ j, t j :=
Set.subset_iInter fun j =>
let ⟨i, hi⟩ := h j
iInter_subset_of_subset i hi
#align set.Inter_mono' Set.iInter_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iInter₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i' j', ∃ i j, s i j ⊆ t i' j') : ⋂ (i) (j), s i j ⊆ ⋂ (i') (j'), t i' j' :=
subset_iInter₂_iff.2 fun i' j' =>
let ⟨_, _, hst⟩ := h i' j'
(iInter₂_subset _ _).trans hst
#align set.Inter₂_mono' Set.iInter₂_mono'
theorem iUnion₂_subset_iUnion (κ : ι → Sort*) (s : ι → Set α) :
⋃ (i) (_ : κ i), s i ⊆ ⋃ i, s i :=
iUnion_mono fun _ => iUnion_subset fun _ => Subset.rfl
#align set.Union₂_subset_Union Set.iUnion₂_subset_iUnion
theorem iInter_subset_iInter₂ (κ : ι → Sort*) (s : ι → Set α) :
⋂ i, s i ⊆ ⋂ (i) (_ : κ i), s i :=
iInter_mono fun _ => subset_iInter fun _ => Subset.rfl
#align set.Inter_subset_Inter₂ Set.iInter_subset_iInter₂
theorem iUnion_setOf (P : ι → α → Prop) : ⋃ i, { x : α | P i x } = { x : α | ∃ i, P i x } := by
ext
exact mem_iUnion
#align set.Union_set_of Set.iUnion_setOf
theorem iInter_setOf (P : ι → α → Prop) : ⋂ i, { x : α | P i x } = { x : α | ∀ i, P i x } := by
ext
exact mem_iInter
#align set.Inter_set_of Set.iInter_setOf
theorem iUnion_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋃ x, f x = ⋃ y, g y :=
h1.iSup_congr h h2
#align set.Union_congr_of_surjective Set.iUnion_congr_of_surjective
theorem iInter_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋂ x, f x = ⋂ y, g y :=
h1.iInf_congr h h2
#align set.Inter_congr_of_surjective Set.iInter_congr_of_surjective
lemma iUnion_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋃ i, s i = ⋃ i, t i := iSup_congr h
#align set.Union_congr Set.iUnion_congr
lemma iInter_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋂ i, s i = ⋂ i, t i := iInf_congr h
#align set.Inter_congr Set.iInter_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iUnion₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋃ (i) (j), s i j = ⋃ (i) (j), t i j :=
iUnion_congr fun i => iUnion_congr <| h i
#align set.Union₂_congr Set.iUnion₂_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iInter₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋂ (i) (j), s i j = ⋂ (i) (j), t i j :=
iInter_congr fun i => iInter_congr <| h i
#align set.Inter₂_congr Set.iInter₂_congr
section Nonempty
variable [Nonempty ι] {f : ι → Set α} {s : Set α}
lemma iUnion_const (s : Set β) : ⋃ _ : ι, s = s := iSup_const
#align set.Union_const Set.iUnion_const
lemma iInter_const (s : Set β) : ⋂ _ : ι, s = s := iInf_const
#align set.Inter_const Set.iInter_const
lemma iUnion_eq_const (hf : ∀ i, f i = s) : ⋃ i, f i = s :=
(iUnion_congr hf).trans $ iUnion_const _
#align set.Union_eq_const Set.iUnion_eq_const
lemma iInter_eq_const (hf : ∀ i, f i = s) : ⋂ i, f i = s :=
(iInter_congr hf).trans $ iInter_const _
#align set.Inter_eq_const Set.iInter_eq_const
end Nonempty
@[simp]
theorem compl_iUnion (s : ι → Set β) : (⋃ i, s i)ᶜ = ⋂ i, (s i)ᶜ :=
compl_iSup
#align set.compl_Union Set.compl_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iUnion₂ (s : ∀ i, κ i → Set α) : (⋃ (i) (j), s i j)ᶜ = ⋂ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iUnion]
#align set.compl_Union₂ Set.compl_iUnion₂
@[simp]
theorem compl_iInter (s : ι → Set β) : (⋂ i, s i)ᶜ = ⋃ i, (s i)ᶜ :=
compl_iInf
#align set.compl_Inter Set.compl_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iInter₂ (s : ∀ i, κ i → Set α) : (⋂ (i) (j), s i j)ᶜ = ⋃ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iInter]
#align set.compl_Inter₂ Set.compl_iInter₂
-- classical -- complete_boolean_algebra
theorem iUnion_eq_compl_iInter_compl (s : ι → Set β) : ⋃ i, s i = (⋂ i, (s i)ᶜ)ᶜ := by
simp only [compl_iInter, compl_compl]
#align set.Union_eq_compl_Inter_compl Set.iUnion_eq_compl_iInter_compl
-- classical -- complete_boolean_algebra
theorem iInter_eq_compl_iUnion_compl (s : ι → Set β) : ⋂ i, s i = (⋃ i, (s i)ᶜ)ᶜ := by
simp only [compl_iUnion, compl_compl]
#align set.Inter_eq_compl_Union_compl Set.iInter_eq_compl_iUnion_compl
theorem inter_iUnion (s : Set β) (t : ι → Set β) : (s ∩ ⋃ i, t i) = ⋃ i, s ∩ t i :=
inf_iSup_eq _ _
#align set.inter_Union Set.inter_iUnion
theorem iUnion_inter (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∩ s = ⋃ i, t i ∩ s :=
iSup_inf_eq _ _
#align set.Union_inter Set.iUnion_inter
theorem iUnion_union_distrib (s : ι → Set β) (t : ι → Set β) :
⋃ i, s i ∪ t i = (⋃ i, s i) ∪ ⋃ i, t i :=
iSup_sup_eq
#align set.Union_union_distrib Set.iUnion_union_distrib
theorem iInter_inter_distrib (s : ι → Set β) (t : ι → Set β) :
⋂ i, s i ∩ t i = (⋂ i, s i) ∩ ⋂ i, t i :=
iInf_inf_eq
#align set.Inter_inter_distrib Set.iInter_inter_distrib
theorem union_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∪ ⋃ i, t i) = ⋃ i, s ∪ t i :=
sup_iSup
#align set.union_Union Set.union_iUnion
theorem iUnion_union [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∪ s = ⋃ i, t i ∪ s :=
iSup_sup
#align set.Union_union Set.iUnion_union
theorem inter_iInter [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∩ ⋂ i, t i) = ⋂ i, s ∩ t i :=
inf_iInf
#align set.inter_Inter Set.inter_iInter
theorem iInter_inter [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋂ i, t i) ∩ s = ⋂ i, t i ∩ s :=
iInf_inf
#align set.Inter_inter Set.iInter_inter
-- classical
theorem union_iInter (s : Set β) (t : ι → Set β) : (s ∪ ⋂ i, t i) = ⋂ i, s ∪ t i :=
sup_iInf_eq _ _
#align set.union_Inter Set.union_iInter
theorem iInter_union (s : ι → Set β) (t : Set β) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.Inter_union Set.iInter_union
theorem iUnion_diff (s : Set β) (t : ι → Set β) : (⋃ i, t i) \ s = ⋃ i, t i \ s :=
iUnion_inter _ _
#align set.Union_diff Set.iUnion_diff
theorem diff_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s \ ⋃ i, t i) = ⋂ i, s \ t i := by
rw [diff_eq, compl_iUnion, inter_iInter]; rfl
#align set.diff_Union Set.diff_iUnion
theorem diff_iInter (s : Set β) (t : ι → Set β) : (s \ ⋂ i, t i) = ⋃ i, s \ t i := by
rw [diff_eq, compl_iInter, inter_iUnion]; rfl
#align set.diff_Inter Set.diff_iInter
theorem directed_on_iUnion {r} {f : ι → Set α} (hd : Directed (· ⊆ ·) f)
(h : ∀ x, DirectedOn r (f x)) : DirectedOn r (⋃ x, f x) := by
simp only [DirectedOn, exists_prop, mem_iUnion, exists_imp]
exact fun a₁ b₁ fb₁ a₂ b₂ fb₂ =>
let ⟨z, zb₁, zb₂⟩ := hd b₁ b₂
let ⟨x, xf, xa₁, xa₂⟩ := h z a₁ (zb₁ fb₁) a₂ (zb₂ fb₂)
⟨x, ⟨z, xf⟩, xa₁, xa₂⟩
#align set.directed_on_Union Set.directed_on_iUnion
theorem iUnion_inter_subset {ι α} {s t : ι → Set α} : ⋃ i, s i ∩ t i ⊆ (⋃ i, s i) ∩ ⋃ i, t i :=
le_iSup_inf_iSup s t
#align set.Union_inter_subset Set.iUnion_inter_subset
theorem iUnion_inter_of_monotone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_monotone hs ht
#align set.Union_inter_of_monotone Set.iUnion_inter_of_monotone
theorem iUnion_inter_of_antitone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_antitone hs ht
#align set.Union_inter_of_antitone Set.iUnion_inter_of_antitone
theorem iInter_union_of_monotone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_monotone hs ht
#align set.Inter_union_of_monotone Set.iInter_union_of_monotone
theorem iInter_union_of_antitone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_antitone hs ht
#align set.Inter_union_of_antitone Set.iInter_union_of_antitone
/-- An equality version of this lemma is `iUnion_iInter_of_monotone` in `Data.Set.Finite`. -/
theorem iUnion_iInter_subset {s : ι → ι' → Set α} : (⋃ j, ⋂ i, s i j) ⊆ ⋂ i, ⋃ j, s i j :=
iSup_iInf_le_iInf_iSup (flip s)
#align set.Union_Inter_subset Set.iUnion_iInter_subset
theorem iUnion_option {ι} (s : Option ι → Set α) : ⋃ o, s o = s none ∪ ⋃ i, s (some i) :=
iSup_option s
#align set.Union_option Set.iUnion_option
theorem iInter_option {ι} (s : Option ι → Set α) : ⋂ o, s o = s none ∩ ⋂ i, s (some i) :=
iInf_option s
#align set.Inter_option Set.iInter_option
section
variable (p : ι → Prop) [DecidablePred p]
theorem iUnion_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 _ _ _
#align set.Union_dite Set.iUnion_dite
theorem iUnion_ite (f g : ι → Set α) :
⋃ i, (if p i then f i else g i) = (⋃ (i) (_ : p i), f i) ∪ ⋃ (i) (_ : ¬p i), g i :=
iUnion_dite _ _ _
#align set.Union_ite Set.iUnion_ite
theorem iInter_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 :=
iInf_dite _ _ _
#align set.Inter_dite Set.iInter_dite
theorem iInter_ite (f g : ι → Set α) :
⋂ i, (if p i then f i else g i) = (⋂ (i) (_ : p i), f i) ∩ ⋂ (i) (_ : ¬p i), g i :=
iInter_dite _ _ _
#align set.Inter_ite Set.iInter_ite
end
theorem image_projection_prod {ι : Type*} {α : ι → Type*} {v : ∀ i : ι, Set (α i)}
(hv : (pi univ v).Nonempty) (i : ι) :
((fun x : ∀ i : ι, α i => x i) '' ⋂ k, (fun x : ∀ j : ι, α j => x k) ⁻¹' v k) = v i := by
classical
apply Subset.antisymm
· simp [iInter_subset]
· intro y y_in
simp only [mem_image, mem_iInter, mem_preimage]
rcases hv with ⟨z, hz⟩
refine' ⟨Function.update z i y, _, update_same i y z⟩
rw [@forall_update_iff ι α _ z i y fun i t => t ∈ v i]
exact ⟨y_in, fun j _ => by simpa using hz j⟩
#align set.image_projection_prod Set.image_projection_prod
/-! ### Unions and intersections indexed by `Prop` -/
theorem iInter_false {s : False → Set α} : iInter s = univ :=
iInf_false
#align set.Inter_false Set.iInter_false
theorem iUnion_false {s : False → Set α} : iUnion s = ∅ :=
iSup_false
#align set.Union_false Set.iUnion_false
@[simp]
theorem iInter_true {s : True → Set α} : iInter s = s trivial :=
iInf_true
#align set.Inter_true Set.iInter_true
@[simp]
theorem iUnion_true {s : True → Set α} : iUnion s = s trivial :=
iSup_true
#align set.Union_true Set.iUnion_true
@[simp]
theorem iInter_exists {p : ι → Prop} {f : Exists p → Set α} :
⋂ x, f x = ⋂ (i) (h : p i), f ⟨i, h⟩ :=
iInf_exists
#align set.Inter_exists Set.iInter_exists
@[simp]
theorem iUnion_exists {p : ι → Prop} {f : Exists p → Set α} :
⋃ x, f x = ⋃ (i) (h : p i), f ⟨i, h⟩ :=
iSup_exists
#align set.Union_exists Set.iUnion_exists
@[simp]
theorem iUnion_empty : (⋃ _ : ι, ∅ : Set α) = ∅ :=
iSup_bot
#align set.Union_empty Set.iUnion_empty
@[simp]
theorem iInter_univ : (⋂ _ : ι, univ : Set α) = univ :=
iInf_top
#align set.Inter_univ Set.iInter_univ
section
variable {s : ι → Set α}
@[simp]
theorem iUnion_eq_empty : ⋃ i, s i = ∅ ↔ ∀ i, s i = ∅ :=
iSup_eq_bot
#align set.Union_eq_empty Set.iUnion_eq_empty
@[simp]
theorem iInter_eq_univ : ⋂ i, s i = univ ↔ ∀ i, s i = univ :=
iInf_eq_top
#align set.Inter_eq_univ Set.iInter_eq_univ
@[simp]
theorem nonempty_iUnion : (⋃ i, s i).Nonempty ↔ ∃ i, (s i).Nonempty := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_Union Set.nonempty_iUnion
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_biUnion {t : Set α} {s : α → Set β} :
(⋃ i ∈ t, s i).Nonempty ↔ ∃ i ∈ t, (s i).Nonempty := by simp
#align set.nonempty_bUnion Set.nonempty_biUnion
theorem iUnion_nonempty_index (s : Set α) (t : s.Nonempty → Set β) :
⋃ h, t h = ⋃ x ∈ s, t ⟨x, ‹_›⟩ :=
iSup_exists
#align set.Union_nonempty_index Set.iUnion_nonempty_index
end
@[simp]
theorem iInter_iInter_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋂ (x) (h : x = b), s x h = s b rfl :=
iInf_iInf_eq_left
#align set.Inter_Inter_eq_left Set.iInter_iInter_eq_left
@[simp]
theorem iInter_iInter_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋂ (x) (h : b = x), s x h = s b rfl :=
iInf_iInf_eq_right
#align set.Inter_Inter_eq_right Set.iInter_iInter_eq_right
@[simp]
theorem iUnion_iUnion_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋃ (x) (h : x = b), s x h = s b rfl :=
iSup_iSup_eq_left
#align set.Union_Union_eq_left Set.iUnion_iUnion_eq_left
@[simp]
theorem iUnion_iUnion_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋃ (x) (h : b = x), s x h = s b rfl :=
iSup_iSup_eq_right
#align set.Union_Union_eq_right Set.iUnion_iUnion_eq_right
theorem iInter_or {p q : Prop} (s : p ∨ q → Set α) :
⋂ h, s h = (⋂ h : p, s (Or.inl h)) ∩ ⋂ h : q, s (Or.inr h) :=
iInf_or
#align set.Inter_or Set.iInter_or
theorem iUnion_or {p q : Prop} (s : p ∨ q → Set α) :
⋃ h, s h = (⋃ i, s (Or.inl i)) ∪ ⋃ j, s (Or.inr j) :=
iSup_or
#align set.Union_or Set.iUnion_or
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iUnion_and {p q : Prop} (s : p ∧ q → Set α) : ⋃ h, s h = ⋃ (hp) (hq), s ⟨hp, hq⟩ :=
iSup_and
#align set.Union_and Set.iUnion_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iInter_and {p q : Prop} (s : p ∧ q → Set α) : ⋂ h, s h = ⋂ (hp) (hq), s ⟨hp, hq⟩ :=
iInf_and
#align set.Inter_and Set.iInter_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iUnion_comm (s : ι → ι' → Set α) : ⋃ (i) (i'), s i i' = ⋃ (i') (i), s i i' :=
iSup_comm
#align set.Union_comm Set.iUnion_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iInter_comm (s : ι → ι' → Set α) : ⋂ (i) (i'), s i i' = ⋂ (i') (i), s i i' :=
iInf_comm
#align set.Inter_comm Set.iInter_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iUnion₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋃ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋃ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iSup₂_comm _
#align set.Union₂_comm Set.iUnion₂_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iInter₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋂ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋂ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iInf₂_comm _
#align set.Inter₂_comm Set.iInter₂_comm
@[simp]
theorem biUnion_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋃ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι']
#align set.bUnion_and Set.biUnion_and
@[simp]
theorem biUnion_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋃ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι]
#align set.bUnion_and' Set.biUnion_and'
@[simp]
theorem biInter_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋂ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iInter_and, @iInter_comm _ ι']
#align set.bInter_and Set.biInter_and
@[simp]
theorem biInter_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋂ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iInter_and, @iInter_comm _ ι]
#align set.bInter_and' Set.biInter_and'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iUnion_iUnion_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋃ (x) (h), s x h = s b (Or.inl rfl) ∪ ⋃ (x) (h : p x), s x (Or.inr h) := by
simp only [iUnion_or, iUnion_union_distrib, iUnion_iUnion_eq_left]
#align set.Union_Union_eq_or_left Set.iUnion_iUnion_eq_or_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iInter_iInter_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋂ (x) (h), s x h = s b (Or.inl rfl) ∩ ⋂ (x) (h : p x), s x (Or.inr h) := by
simp only [iInter_or, iInter_inter_distrib, iInter_iInter_eq_left]
#align set.Inter_Inter_eq_or_left Set.iInter_iInter_eq_or_left
/-! ### Bounded unions and intersections -/
/-- A specialization of `mem_iUnion₂`. -/
theorem mem_biUnion {s : Set α} {t : α → Set β} {x : α} {y : β} (xs : x ∈ s) (ytx : y ∈ t x) :
y ∈ ⋃ x ∈ s, t x :=
mem_iUnion₂_of_mem xs ytx
#align set.mem_bUnion Set.mem_biUnion
/-- A specialization of `mem_iInter₂`. -/
theorem mem_biInter {s : Set α} {t : α → Set β} {y : β} (h : ∀ x ∈ s, y ∈ t x) :
y ∈ ⋂ x ∈ s, t x :=
mem_iInter₂_of_mem h
#align set.mem_bInter Set.mem_biInter
/-- A specialization of `subset_iUnion₂`. -/
theorem subset_biUnion_of_mem {s : Set α} {u : α → Set β} {x : α} (xs : x ∈ s) :
u x ⊆ ⋃ x ∈ s, u x :=
--Porting note: Why is this not just `subset_iUnion₂ x xs`?
@subset_iUnion₂ β α (· ∈ s) (fun i _ => u i) x xs
#align set.subset_bUnion_of_mem Set.subset_biUnion_of_mem
/-- A specialization of `iInter₂_subset`. -/
theorem biInter_subset_of_mem {s : Set α} {t : α → Set β} {x : α} (xs : x ∈ s) :
⋂ x ∈ s, t x ⊆ t x :=
iInter₂_subset x xs
#align set.bInter_subset_of_mem Set.biInter_subset_of_mem
theorem biUnion_subset_biUnion_left {s s' : Set α} {t : α → Set β} (h : s ⊆ s') :
⋃ x ∈ s, t x ⊆ ⋃ x ∈ s', t x :=
iUnion₂_subset fun _ hx => subset_biUnion_of_mem <| h hx
#align set.bUnion_subset_bUnion_left Set.biUnion_subset_biUnion_left
theorem biInter_subset_biInter_left {s s' : Set α} {t : α → Set β} (h : s' ⊆ s) :
⋂ x ∈ s, t x ⊆ ⋂ x ∈ s', t x :=
subset_iInter₂ fun _ hx => biInter_subset_of_mem <| h hx
#align set.bInter_subset_bInter_left Set.biInter_subset_biInter_left
theorem biUnion_mono {s s' : Set α} {t t' : α → Set β} (hs : s' ⊆ s) (h : ∀ x ∈ s, t x ⊆ t' x) :
⋃ x ∈ s', t x ⊆ ⋃ x ∈ s, t' x :=
(biUnion_subset_biUnion_left hs).trans <| iUnion₂_mono h
#align set.bUnion_mono Set.biUnion_mono
theorem biInter_mono {s s' : Set α} {t t' : α → Set β} (hs : s ⊆ s') (h : ∀ x ∈ s, t x ⊆ t' x) :
⋂ x ∈ s', t x ⊆ ⋂ x ∈ s, t' x :=
(biInter_subset_biInter_left hs).trans <| iInter₂_mono h
#align set.bInter_mono Set.biInter_mono
theorem biUnion_eq_iUnion (s : Set α) (t : ∀ x ∈ s, Set β) :
⋃ x ∈ s, t x ‹_› = ⋃ x : s, t x x.2 :=
iSup_subtype'
#align set.bUnion_eq_Union Set.biUnion_eq_iUnion
theorem biInter_eq_iInter (s : Set α) (t : ∀ x ∈ s, Set β) :
⋂ x ∈ s, t x ‹_› = ⋂ x : s, t x x.2 :=
iInf_subtype'
#align set.bInter_eq_Inter Set.biInter_eq_iInter
theorem iUnion_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋃ x : { x // p x }, s x = ⋃ (x) (hx : p x), s ⟨x, hx⟩ :=
iSup_subtype
#align set.Union_subtype Set.iUnion_subtype
theorem iInter_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋂ x : { x // p x }, s x = ⋂ (x) (hx : p x), s ⟨x, hx⟩ :=
iInf_subtype
#align set.Inter_subtype Set.iInter_subtype
theorem biInter_empty (u : α → Set β) : ⋂ x ∈ (∅ : Set α), u x = univ :=
iInf_emptyset
#align set.bInter_empty Set.biInter_empty
theorem biInter_univ (u : α → Set β) : ⋂ x ∈ @univ α, u x = ⋂ x, u x :=
iInf_univ
#align set.bInter_univ Set.biInter_univ
@[simp]
theorem biUnion_self (s : Set α) : ⋃ x ∈ s, s = s :=
Subset.antisymm (iUnion₂_subset fun _ _ => Subset.refl s) fun _ hx => mem_biUnion hx hx
#align set.bUnion_self Set.biUnion_self
@[simp]
theorem iUnion_nonempty_self (s : Set α) : ⋃ _ : s.Nonempty, s = s := by
rw [iUnion_nonempty_index, biUnion_self]
#align set.Union_nonempty_self Set.iUnion_nonempty_self
theorem biInter_singleton (a : α) (s : α → Set β) : ⋂ x ∈ ({a} : Set α), s x = s a :=
iInf_singleton
#align set.bInter_singleton Set.biInter_singleton
theorem biInter_union (s t : Set α) (u : α → Set β) :
⋂ x ∈ s ∪ t, u x = (⋂ x ∈ s, u x) ∩ ⋂ x ∈ t, u x :=
iInf_union
#align set.bInter_union Set.biInter_union
theorem biInter_insert (a : α) (s : Set α) (t : α → Set β) :
⋂ x ∈ insert a s, t x = t a ∩ ⋂ x ∈ s, t x := by simp
#align set.bInter_insert Set.biInter_insert
theorem biInter_pair (a b : α) (s : α → Set β) : ⋂ x ∈ ({a, b} : Set α), s x = s a ∩ s b := by
rw [biInter_insert, biInter_singleton]
#align set.bInter_pair Set.biInter_pair
theorem biInter_inter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, f i ∩ t = (⋂ i ∈ s, f i) ∩ t := by
haveI : Nonempty s := hs.to_subtype
simp [biInter_eq_iInter, ← iInter_inter]
#align set.bInter_inter Set.biInter_inter
theorem inter_biInter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, t ∩ f i = t ∩ ⋂ i ∈ s, f i := by
rw [inter_comm, ← biInter_inter hs]
simp [inter_comm]
#align set.inter_bInter Set.inter_biInter
theorem biUnion_empty (s : α → Set β) : ⋃ x ∈ (∅ : Set α), s x = ∅ :=
iSup_emptyset
#align set.bUnion_empty Set.biUnion_empty
theorem biUnion_univ (s : α → Set β) : ⋃ x ∈ @univ α, s x = ⋃ x, s x :=
iSup_univ
#align set.bUnion_univ Set.biUnion_univ
theorem biUnion_singleton (a : α) (s : α → Set β) : ⋃ x ∈ ({a} : Set α), s x = s a :=
iSup_singleton
#align set.bUnion_singleton Set.biUnion_singleton
@[simp]
theorem biUnion_of_singleton (s : Set α) : ⋃ x ∈ s, {x} = s :=
ext <| by simp
#align set.bUnion_of_singleton Set.biUnion_of_singleton
theorem biUnion_union (s t : Set α) (u : α → Set β) :
⋃ x ∈ s ∪ t, u x = (⋃ x ∈ s, u x) ∪ ⋃ x ∈ t, u x :=
iSup_union
#align set.bUnion_union Set.biUnion_union
@[simp]
theorem iUnion_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋃ i, f i = ⋃ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iUnion_subtype _ _
#align set.Union_coe_set Set.iUnion_coe_set
@[simp]
theorem iInter_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋂ i, f i = ⋂ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iInter_subtype _ _
#align set.Inter_coe_set Set.iInter_coe_set
theorem biUnion_insert (a : α) (s : Set α) (t : α → Set β) :
⋃ x ∈ insert a s, t x = t a ∪ ⋃ x ∈ s, t x := by simp
#align set.bUnion_insert Set.biUnion_insert
theorem biUnion_pair (a b : α) (s : α → Set β) : ⋃ x ∈ ({a, b} : Set α), s x = s a ∪ s b :=
by simp
#align set.bUnion_pair Set.biUnion_pair
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem inter_iUnion₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∩ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ∩ t i j := by simp only [inter_iUnion]
#align set.inter_Union₂ Set.inter_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_inter (s : ∀ i, κ i → Set α) (t : Set α) :
(⋃ (i) (j), s i j) ∩ t = ⋃ (i) (j), s i j ∩ t := by simp_rw [iUnion_inter]
#align set.Union₂_inter Set.iUnion₂_inter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_iInter₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_iInter]
#align set.union_Inter₂ Set.union_iInter₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_union (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [iInter_union]
#align set.Inter₂_union Set.iInter₂_union
theorem mem_sUnion_of_mem {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∈ t) (ht : t ∈ S) :
x ∈ ⋃₀S :=
⟨t, ht, hx⟩
#align set.mem_sUnion_of_mem Set.mem_sUnion_of_mem
-- is this theorem really necessary?
theorem not_mem_of_not_mem_sUnion {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∉ ⋃₀S)
(ht : t ∈ S) : x ∉ t := fun h => hx ⟨t, ht, h⟩
#align set.not_mem_of_not_mem_sUnion Set.not_mem_of_not_mem_sUnion
theorem sInter_subset_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : ⋂₀ S ⊆ t :=
sInf_le tS
#align set.sInter_subset_of_mem Set.sInter_subset_of_mem
theorem subset_sUnion_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : t ⊆ ⋃₀S :=
le_sSup tS
#align set.subset_sUnion_of_mem Set.subset_sUnion_of_mem
theorem subset_sUnion_of_subset {s : Set α} (t : Set (Set α)) (u : Set α) (h₁ : s ⊆ u)
(h₂ : u ∈ t) : s ⊆ ⋃₀t :=
Subset.trans h₁ (subset_sUnion_of_mem h₂)
#align set.subset_sUnion_of_subset Set.subset_sUnion_of_subset
theorem sUnion_subset {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t' ⊆ t) : ⋃₀S ⊆ t :=
sSup_le h
#align set.sUnion_subset Set.sUnion_subset
@[simp]
theorem sUnion_subset_iff {s : Set (Set α)} {t : Set α} : ⋃₀s ⊆ t ↔ ∀ t' ∈ s, t' ⊆ t :=
sSup_le_iff
#align set.sUnion_subset_iff Set.sUnion_subset_iff
/-- `sUnion` is monotone under taking a subset of each set. -/
lemma sUnion_mono_subsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, t ⊆ f t) :
⋃₀ s ⊆ ⋃₀ (f '' s) :=
fun _ ⟨t, htx, hxt⟩ ↦ ⟨f t, mem_image_of_mem f htx, hf t hxt⟩
/-- `sUnion` is monotone under taking a superset of each set. -/
lemma sUnion_mono_supsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, f t ⊆ t) :
⋃₀ (f '' s) ⊆ ⋃₀ s :=
-- If t ∈ f '' s is arbitrary; t = f u for some u : Set α.
fun _ ⟨_, ⟨u, hus, hut⟩, hxt⟩ ↦ ⟨u, hus, (hut ▸ hf u) hxt⟩
theorem subset_sInter {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t ⊆ t') : t ⊆ ⋂₀ S :=
le_sInf h
#align set.subset_sInter Set.subset_sInter
@[simp]
theorem subset_sInter_iff {S : Set (Set α)} {t : Set α} : t ⊆ ⋂₀ S ↔ ∀ t' ∈ S, t ⊆ t' :=
le_sInf_iff
#align set.subset_sInter_iff Set.subset_sInter_iff
@[gcongr]
theorem sUnion_subset_sUnion {S T : Set (Set α)} (h : S ⊆ T) : ⋃₀S ⊆ ⋃₀T :=
sUnion_subset fun _ hs => subset_sUnion_of_mem (h hs)
#align set.sUnion_subset_sUnion Set.sUnion_subset_sUnion
@[gcongr]
theorem sInter_subset_sInter {S T : Set (Set α)} (h : S ⊆ T) : ⋂₀ T ⊆ ⋂₀ S :=
subset_sInter fun _ hs => sInter_subset_of_mem (h hs)
#align set.sInter_subset_sInter Set.sInter_subset_sInter
@[simp]
theorem sUnion_empty : ⋃₀∅ = (∅ : Set α) :=
sSup_empty
#align set.sUnion_empty Set.sUnion_empty
@[simp]
theorem sInter_empty : ⋂₀ ∅ = (univ : Set α) :=
sInf_empty
#align set.sInter_empty Set.sInter_empty
@[simp]
theorem sUnion_singleton (s : Set α) : ⋃₀{s} = s :=
sSup_singleton
#align set.sUnion_singleton Set.sUnion_singleton
@[simp]
theorem sInter_singleton (s : Set α) : ⋂₀ {s} = s :=
sInf_singleton
#align set.sInter_singleton Set.sInter_singleton
@[simp]
theorem sUnion_eq_empty {S : Set (Set α)} : ⋃₀S = ∅ ↔ ∀ s ∈ S, s = ∅ :=
sSup_eq_bot
#align set.sUnion_eq_empty Set.sUnion_eq_empty
@[simp]
theorem sInter_eq_univ {S : Set (Set α)} : ⋂₀ S = univ ↔ ∀ s ∈ S, s = univ :=
sInf_eq_top
#align set.sInter_eq_univ Set.sInter_eq_univ
theorem subset_powerset_iff {s : Set (Set α)} {t : Set α} : s ⊆ 𝒫 t ↔ ⋃₀ s ⊆ t :=
sUnion_subset_iff.symm
/-- `⋃₀` and `𝒫` form a Galois connection. -/
theorem sUnion_powerset_gc :
GaloisConnection (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gc_sSup_Iic
/-- `⋃₀` and `𝒫` form a Galois insertion. -/
def sUnion_powerset_gi :
GaloisInsertion (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gi_sSup_Iic
/-- If all sets in a collection are either `∅` or `Set.univ`, then so is their union. -/
theorem sUnion_mem_empty_univ {S : Set (Set α)} (h : S ⊆ {∅, univ}) :
⋃₀ S ∈ ({∅, univ} : Set (Set α)) := by
simp only [mem_insert_iff, mem_singleton_iff, or_iff_not_imp_left, sUnion_eq_empty, not_forall]
rintro ⟨s, hs, hne⟩
obtain rfl : s = univ := (h hs).resolve_left hne
exact univ_subset_iff.1 <| subset_sUnion_of_mem hs
@[simp]
theorem nonempty_sUnion {S : Set (Set α)} : (⋃₀S).Nonempty ↔ ∃ s ∈ S, Set.Nonempty s := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_sUnion Set.nonempty_sUnion
theorem Nonempty.of_sUnion {s : Set (Set α)} (h : (⋃₀s).Nonempty) : s.Nonempty :=
let ⟨s, hs, _⟩ := nonempty_sUnion.1 h
⟨s, hs⟩
#align set.nonempty.of_sUnion Set.Nonempty.of_sUnion
theorem Nonempty.of_sUnion_eq_univ [Nonempty α] {s : Set (Set α)} (h : ⋃₀s = univ) : s.Nonempty :=
Nonempty.of_sUnion <| h.symm ▸ univ_nonempty
#align set.nonempty.of_sUnion_eq_univ Set.Nonempty.of_sUnion_eq_univ
theorem sUnion_union (S T : Set (Set α)) : ⋃₀(S ∪ T) = ⋃₀S ∪ ⋃₀T :=
sSup_union
#align set.sUnion_union Set.sUnion_union
theorem sInter_union (S T : Set (Set α)) : ⋂₀ (S ∪ T) = ⋂₀ S ∩ ⋂₀ T :=
sInf_union
#align set.sInter_union Set.sInter_union
@[simp]
theorem sUnion_insert (s : Set α) (T : Set (Set α)) : ⋃₀insert s T = s ∪ ⋃₀T :=
sSup_insert
#align set.sUnion_insert Set.sUnion_insert
@[simp]
theorem sInter_insert (s : Set α) (T : Set (Set α)) : ⋂₀ insert s T = s ∩ ⋂₀ T :=
sInf_insert
#align set.sInter_insert Set.sInter_insert
@[simp]
theorem sUnion_diff_singleton_empty (s : Set (Set α)) : ⋃₀(s \ {∅}) = ⋃₀s :=
sSup_diff_singleton_bot s
#align set.sUnion_diff_singleton_empty Set.sUnion_diff_singleton_empty
@[simp]
theorem sInter_diff_singleton_univ (s : Set (Set α)) : ⋂₀ (s \ {univ}) = ⋂₀ s :=
sInf_diff_singleton_top s
#align set.sInter_diff_singleton_univ Set.sInter_diff_singleton_univ
theorem sUnion_pair (s t : Set α) : ⋃₀{s, t} = s ∪ t :=
sSup_pair
#align set.sUnion_pair Set.sUnion_pair
theorem sInter_pair (s t : Set α) : ⋂₀ {s, t} = s ∩ t :=
sInf_pair
#align set.sInter_pair Set.sInter_pair
@[simp]
theorem sUnion_image (f : α → Set β) (s : Set α) : ⋃₀(f '' s) = ⋃ x ∈ s, f x :=
sSup_image
#align set.sUnion_image Set.sUnion_image
@[simp]
theorem sInter_image (f : α → Set β) (s : Set α) : ⋂₀ (f '' s) = ⋂ x ∈ s, f x :=
sInf_image
#align set.sInter_image Set.sInter_image
@[simp]
theorem sUnion_range (f : ι → Set β) : ⋃₀range f = ⋃ x, f x :=
rfl
#align set.sUnion_range Set.sUnion_range
@[simp]
theorem sInter_range (f : ι → Set β) : ⋂₀ range f = ⋂ x, f x :=
rfl
#align set.sInter_range Set.sInter_range
theorem iUnion_eq_univ_iff {f : ι → Set α} : ⋃ i, f i = univ ↔ ∀ x, ∃ i, x ∈ f i := by
simp only [eq_univ_iff_forall, mem_iUnion]
#align set.Union_eq_univ_iff Set.iUnion_eq_univ_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_eq_univ_iff {s : ∀ i, κ i → Set α} :
⋃ (i) (j), s i j = univ ↔ ∀ a, ∃ i j, a ∈ s i j :=
by simp only [iUnion_eq_univ_iff, mem_iUnion]
#align set.Union₂_eq_univ_iff Set.iUnion₂_eq_univ_iff
theorem sUnion_eq_univ_iff {c : Set (Set α)} : ⋃₀c = univ ↔ ∀ a, ∃ b ∈ c, a ∈ b := by
simp only [eq_univ_iff_forall, mem_sUnion]
#align set.sUnion_eq_univ_iff Set.sUnion_eq_univ_iff
-- classical
theorem iInter_eq_empty_iff {f : ι → Set α} : ⋂ i, f i = ∅ ↔ ∀ x, ∃ i, x ∉ f i := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.Inter_eq_empty_iff Set.iInter_eq_empty_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
theorem iInter₂_eq_empty_iff {s : ∀ i, κ i → Set α} :
⋂ (i) (j), s i j = ∅ ↔ ∀ a, ∃ i j, a ∉ s i j := by
simp only [eq_empty_iff_forall_not_mem, mem_iInter, not_forall]
#align set.Inter₂_eq_empty_iff Set.iInter₂_eq_empty_iff
-- classical
theorem sInter_eq_empty_iff {c : Set (Set α)} : ⋂₀ c = ∅ ↔ ∀ a, ∃ b ∈ c, a ∉ b := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.sInter_eq_empty_iff Set.sInter_eq_empty_iff
-- classical
@[simp]
theorem nonempty_iInter {f : ι → Set α} : (⋂ i, f i).Nonempty ↔ ∃ x, ∀ i, x ∈ f i := by
simp [nonempty_iff_ne_empty, iInter_eq_empty_iff]
#align set.nonempty_Inter Set.nonempty_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_iInter₂ {s : ∀ i, κ i → Set α} :
(⋂ (i) (j), s i j).Nonempty ↔ ∃ a, ∀ i j, a ∈ s i j := by
simp
#align set.nonempty_Inter₂ Set.nonempty_iInter₂
-- classical
@[simp]
theorem nonempty_sInter {c : Set (Set α)} : (⋂₀ c).Nonempty ↔ ∃ a, ∀ b ∈ c, a ∈ b := by
simp [nonempty_iff_ne_empty, sInter_eq_empty_iff]
#align set.nonempty_sInter Set.nonempty_sInter
-- classical
theorem compl_sUnion (S : Set (Set α)) : (⋃₀S)ᶜ = ⋂₀ (compl '' S) :=
ext fun x => by simp
#align set.compl_sUnion Set.compl_sUnion
-- classical
theorem sUnion_eq_compl_sInter_compl (S : Set (Set α)) : ⋃₀S = (⋂₀ (compl '' S))ᶜ := by
rw [← compl_compl (⋃₀S), compl_sUnion]
#align set.sUnion_eq_compl_sInter_compl Set.sUnion_eq_compl_sInter_compl
-- classical
theorem compl_sInter (S : Set (Set α)) : (⋂₀ S)ᶜ = ⋃₀(compl '' S) := by
rw [sUnion_eq_compl_sInter_compl, compl_compl_image]
#align set.compl_sInter Set.compl_sInter
-- classical
theorem sInter_eq_compl_sUnion_compl (S : Set (Set α)) : ⋂₀ S = (⋃₀(compl '' S))ᶜ := by
rw [← compl_compl (⋂₀ S), compl_sInter]
#align set.sInter_eq_compl_sUnion_compl Set.sInter_eq_compl_sUnion_compl
theorem inter_empty_of_inter_sUnion_empty {s t : Set α} {S : Set (Set α)} (hs : t ∈ S)
(h : s ∩ ⋃₀S = ∅) : s ∩ t = ∅ :=
eq_empty_of_subset_empty <| by
rw [← h]; exact inter_subset_inter_right _ (subset_sUnion_of_mem hs)
#align set.inter_empty_of_inter_sUnion_empty Set.inter_empty_of_inter_sUnion_empty
theorem range_sigma_eq_iUnion_range {γ : α → Type*} (f : Sigma γ → β) :
range f = ⋃ a, range fun b => f ⟨a, b⟩ :=
Set.ext <| by simp
#align set.range_sigma_eq_Union_range Set.range_sigma_eq_iUnion_range
theorem iUnion_eq_range_sigma (s : α → Set β) : ⋃ i, s i = range fun a : Σi, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_sigma Set.iUnion_eq_range_sigma
theorem iUnion_eq_range_psigma (s : ι → Set β) : ⋃ i, s i = range fun a : Σ'i, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_psigma Set.iUnion_eq_range_psigma
theorem iUnion_image_preimage_sigma_mk_eq_self {ι : Type*} {σ : ι → Type*} (s : Set (Sigma σ)) :
⋃ i, Sigma.mk i '' (Sigma.mk i ⁻¹' s) = s := by
ext x
simp only [mem_iUnion, mem_image, mem_preimage]
constructor
· rintro ⟨i, a, h, rfl⟩
exact h
· intro h
cases' x with i a
exact ⟨i, a, h, rfl⟩
#align set.Union_image_preimage_sigma_mk_eq_self Set.iUnion_image_preimage_sigma_mk_eq_self
theorem Sigma.univ (X : α → Type*) : (Set.univ : Set (Σa, X a)) = ⋃ a, range (Sigma.mk a) :=
Set.ext fun x =>
iff_of_true trivial ⟨range (Sigma.mk x.1), Set.mem_range_self _, x.2, Sigma.eta x⟩
#align set.sigma.univ Set.Sigma.univ
alias sUnion_mono := sUnion_subset_sUnion
#align set.sUnion_mono Set.sUnion_mono
theorem iUnion_subset_iUnion_const {s : Set α} (h : ι → ι₂) : ⋃ _ : ι, s ⊆ ⋃ _ : ι₂, s :=
@iSup_const_mono (Set α) ι ι₂ _ s h
#align set.Union_subset_Union_const Set.iUnion_subset_iUnion_const
@[simp]
theorem iUnion_singleton_eq_range {α β : Type*} (f : α → β) : ⋃ x : α, {f x} = range f := by
ext x
simp [@eq_comm _ x]
#align set.Union_singleton_eq_range Set.iUnion_singleton_eq_range
theorem iUnion_of_singleton (α : Type*) : (⋃ x, {x} : Set α) = univ := by simp [Set.ext_iff]
#align set.Union_of_singleton Set.iUnion_of_singleton
theorem iUnion_of_singleton_coe (s : Set α) : ⋃ i : s, ({(i : α)} : Set α) = s := by simp
#align set.Union_of_singleton_coe Set.iUnion_of_singleton_coe
theorem sUnion_eq_biUnion {s : Set (Set α)} : ⋃₀s = ⋃ (i : Set α) (_ : i ∈ s), i := by
rw [← sUnion_image, image_id']
#align set.sUnion_eq_bUnion Set.sUnion_eq_biUnion
theorem sInter_eq_biInter {s : Set (Set α)} : ⋂₀ s = ⋂ (i : Set α) (_ : i ∈ s), i := by
rw [← sInter_image, image_id']
#align set.sInter_eq_bInter Set.sInter_eq_biInter
theorem sUnion_eq_iUnion {s : Set (Set α)} : ⋃₀s = ⋃ i : s, i := by
simp only [← sUnion_range, Subtype.range_coe]
#align set.sUnion_eq_Union Set.sUnion_eq_iUnion
theorem sInter_eq_iInter {s : Set (Set α)} : ⋂₀ s = ⋂ i : s, i := by
simp only [← sInter_range, Subtype.range_coe]
#align set.sInter_eq_Inter Set.sInter_eq_iInter
@[simp]
theorem iUnion_of_empty [IsEmpty ι] (s : ι → Set α) : ⋃ i, s i = ∅ :=
iSup_of_empty _
#align set.Union_of_empty Set.iUnion_of_empty
@[simp]
theorem iInter_of_empty [IsEmpty ι] (s : ι → Set α) : ⋂ i, s i = univ :=
iInf_of_empty _
#align set.Inter_of_empty Set.iInter_of_empty
theorem union_eq_iUnion {s₁ s₂ : Set α} : s₁ ∪ s₂ = ⋃ b : Bool, cond b s₁ s₂ :=
sup_eq_iSup s₁ s₂
#align set.union_eq_Union Set.union_eq_iUnion
theorem inter_eq_iInter {s₁ s₂ : Set α} : s₁ ∩ s₂ = ⋂ b : Bool, cond b s₁ s₂ :=
inf_eq_iInf s₁ s₂
#align set.inter_eq_Inter Set.inter_eq_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_union_sInter {S T : Set (Set α)} :
⋂₀ S ∪ ⋂₀ T = ⋂ p ∈ S ×ˢ T, (p : Set α × Set α).1 ∪ p.2 :=
sInf_sup_sInf
#align set.sInter_union_sInter Set.sInter_union_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_inter_sUnion {s t : Set (Set α)} :
⋃₀s ∩ ⋃₀t = ⋃ p ∈ s ×ˢ t, (p : Set α × Set α).1 ∩ p.2 :=
sSup_inf_sSup
#align set.sUnion_inter_sUnion Set.sUnion_inter_sUnion
theorem biUnion_iUnion (s : ι → Set α) (t : α → Set β) :
⋃ x ∈ ⋃ i, s i, t x = ⋃ (i) (x ∈ s i), t x := by simp [@iUnion_comm _ ι]
#align set.bUnion_Union Set.biUnion_iUnion
theorem biInter_iUnion (s : ι → Set α) (t : α → Set β) :
⋂ x ∈ ⋃ i, s i, t x = ⋂ (i) (x ∈ s i), t x := by simp [@iInter_comm _ ι]
#align set.bInter_Union Set.biInter_iUnion
theorem sUnion_iUnion (s : ι → Set (Set α)) : ⋃₀⋃ i, s i = ⋃ i, ⋃₀s i := by
simp only [sUnion_eq_biUnion, biUnion_iUnion]
#align set.sUnion_Union Set.sUnion_iUnion
theorem sInter_iUnion (s : ι → Set (Set α)) : ⋂₀ ⋃ i, s i = ⋂ i, ⋂₀ s i := by
simp only [sInter_eq_biInter, biInter_iUnion]
#align set.sInter_Union Set.sInter_iUnion
theorem iUnion_range_eq_sUnion {α β : Type*} (C : Set (Set α)) {f : ∀ s : C, β → (s : Type _)}
(hf : ∀ s : C, Surjective (f s)) : ⋃ y : β, range (fun s : C => (f s y).val) = ⋃₀C := by
ext x; constructor
· rintro ⟨s, ⟨y, rfl⟩, ⟨s, hs⟩, rfl⟩
refine' ⟨_, hs, _⟩
exact (f ⟨s, hs⟩ y).2
· rintro ⟨s, hs, hx⟩
cases' hf ⟨s, hs⟩ ⟨x, hx⟩ with y hy
refine' ⟨_, ⟨y, rfl⟩, ⟨s, hs⟩, _⟩
exact congr_arg Subtype.val hy
#align set.Union_range_eq_sUnion Set.iUnion_range_eq_sUnion
theorem iUnion_range_eq_iUnion (C : ι → Set α) {f : ∀ x : ι, β → C x}
(hf : ∀ x : ι, Surjective (f x)) : ⋃ y : β, range (fun x : ι => (f x y).val) = ⋃ x, C x := by
ext x; rw [mem_iUnion, mem_iUnion]; constructor
· rintro ⟨y, i, rfl⟩
exact ⟨i, (f i y).2⟩
· rintro ⟨i, hx⟩
cases' hf i ⟨x, hx⟩ with y hy
exact ⟨y, i, congr_arg Subtype.val hy⟩
#align set.Union_range_eq_Union Set.iUnion_range_eq_iUnion
theorem union_distrib_iInter_left (s : ι → Set α) (t : Set α) : (t ∪ ⋂ i, s i) = ⋂ i, t ∪ s i :=
sup_iInf_eq _ _
#align set.union_distrib_Inter_left Set.union_distrib_iInter_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_left (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_distrib_iInter_left]
#align set.union_distrib_Inter₂_left Set.union_distrib_iInter₂_left
theorem union_distrib_iInter_right (s : ι → Set α) (t : Set α) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.union_distrib_Inter_right Set.union_distrib_iInter_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_right (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [union_distrib_iInter_right]
#align set.union_distrib_Inter₂_right Set.union_distrib_iInter₂_right
section Function
/-! ### `mapsTo` -/
theorem mapsTo_sUnion {S : Set (Set α)} {t : Set β} {f : α → β} (H : ∀ s ∈ S, MapsTo f s t) :
MapsTo f (⋃₀S) t := fun _ ⟨s, hs, hx⟩ => H s hs hx
#align set.maps_to_sUnion Set.mapsTo_sUnion
theorem mapsTo_iUnion {s : ι → Set α} {t : Set β} {f : α → β} (H : ∀ i, MapsTo f (s i) t) :
MapsTo f (⋃ i, s i) t :=
mapsTo_sUnion <| forall_range_iff.2 H
#align set.maps_to_Union Set.mapsTo_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂ {s : ∀ i, κ i → Set α} {t : Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) t) : MapsTo f (⋃ (i) (j), s i j) t :=
mapsTo_iUnion fun i => mapsTo_iUnion (H i)
#align set.maps_to_Union₂ Set.mapsTo_iUnion₂
theorem mapsTo_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋃ i, s i) (⋃ i, t i) :=
mapsTo_iUnion fun i => (H i).mono (Subset.refl _) (subset_iUnion t i)
#align set.maps_to_Union_Union Set.mapsTo_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
mapsTo_iUnion_iUnion fun i => mapsTo_iUnion_iUnion (H i)
#align set.maps_to_Union₂_Union₂ Set.mapsTo_iUnion₂_iUnion₂
theorem mapsTo_sInter {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, MapsTo f s t) :
MapsTo f s (⋂₀ T) := fun _ hx t ht => H t ht hx
#align set.maps_to_sInter Set.mapsTo_sInter
theorem mapsTo_iInter {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, MapsTo f s (t i)) :
MapsTo f s (⋂ i, t i) := fun _ hx => mem_iInter.2 fun i => H i hx
#align set.maps_to_Inter Set.mapsTo_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f s (t i j)) : MapsTo f s (⋂ (i) (j), t i j) :=
mapsTo_iInter fun i => mapsTo_iInter (H i)
#align set.maps_to_Inter₂ Set.mapsTo_iInter₂
theorem mapsTo_iInter_iInter {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋂ i, s i) (⋂ i, t i) :=
mapsTo_iInter fun i => (H i).mono (iInter_subset s i) (Subset.refl _)
#align set.maps_to_Inter_Inter Set.mapsTo_iInter_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂_iInter₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋂ (i) (j), s i j) (⋂ (i) (j), t i j) :=
mapsTo_iInter_iInter fun i => mapsTo_iInter_iInter (H i)
#align set.maps_to_Inter₂_Inter₂ Set.mapsTo_iInter₂_iInter₂
theorem image_iInter_subset (s : ι → Set α) (f : α → β) : (f '' ⋂ i, s i) ⊆ ⋂ i, f '' s i :=
(mapsTo_iInter_iInter fun i => mapsTo_image f (s i)).image_subset
#align set.image_Inter_subset Set.image_iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂_subset (s : ∀ i, κ i → Set α) (f : α → β) :
(f '' ⋂ (i) (j), s i j) ⊆ ⋂ (i) (j), f '' s i j :=
(mapsTo_iInter₂_iInter₂ fun i hi => mapsTo_image f (s i hi)).image_subset
#align set.image_Inter₂_subset Set.image_iInter₂_subset
theorem image_sInter_subset (S : Set (Set α)) (f : α → β) : f '' ⋂₀ S ⊆ ⋂ s ∈ S, f '' s := by
rw [sInter_eq_biInter]
apply image_iInter₂_subset
#align set.image_sInter_subset Set.image_sInter_subset
/-! ### `restrictPreimage` -/
section
open Function
variable (s : Set β) {f : α → β} {U : ι → Set β} (hU : iUnion U = univ)
theorem injective_iff_injective_of_iUnion_eq_univ :
Injective f ↔ ∀ i, Injective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_injective H, fun H x y e => _⟩
obtain ⟨i, hi⟩ := Set.mem_iUnion.mp
(show f x ∈ Set.iUnion U by rw [hU]; triv)
injection @H i ⟨x, hi⟩ ⟨y, show f y ∈ U i from e ▸ hi⟩ (Subtype.ext e)
#align set.injective_iff_injective_of_Union_eq_univ Set.injective_iff_injective_of_iUnion_eq_univ
theorem surjective_iff_surjective_of_iUnion_eq_univ :
Surjective f ↔ ∀ i, Surjective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_surjective H, fun H x => _⟩
obtain ⟨i, hi⟩ :=
Set.mem_iUnion.mp
(show x ∈ Set.iUnion U by rw [hU]; triv)
exact ⟨_, congr_arg Subtype.val (H i ⟨x, hi⟩).choose_spec⟩
#align set.surjective_iff_surjective_of_Union_eq_univ Set.surjective_iff_surjective_of_iUnion_eq_univ
theorem bijective_iff_bijective_of_iUnion_eq_univ :
Bijective f ↔ ∀ i, Bijective ((U i).restrictPreimage f) := by
rw [Bijective, injective_iff_injective_of_iUnion_eq_univ hU,
surjective_iff_surjective_of_iUnion_eq_univ hU]
simp [Bijective, forall_and]
#align set.bijective_iff_bijective_of_Union_eq_univ Set.bijective_iff_bijective_of_iUnion_eq_univ
end
/-! ### `InjOn` -/
theorem InjOn.image_iInter_eq [Nonempty ι] {s : ι → Set α} {f : α → β} (h : InjOn f (⋃ i, s i)) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
inhabit ι
refine' Subset.antisymm (image_iInter_subset s f) fun y hy => _
simp only [mem_iInter, mem_image_iff_bex] at hy
choose x hx hy using hy
refine' ⟨x default, mem_iInter.2 fun i => _, hy _⟩
suffices x default = x i by
rw [this]
apply hx
replace hx : ∀ i, x i ∈ ⋃ j, s j := fun i => (subset_iUnion _ _) (hx i)
apply h (hx _) (hx _)
simp only [hy]
#align set.inj_on.image_Inter_eq Set.InjOn.image_iInter_eq
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
theorem InjOn.image_biInter_eq {p : ι → Prop} {s : ∀ i, p i → Set α} (hp : ∃ i, p i)
{f : α → β} (h : InjOn f (⋃ (i) (hi), s i hi)) :
(f '' ⋂ (i) (hi), s i hi) = ⋂ (i) (hi), f '' s i hi := by
simp only [iInter, iInf_subtype']
haveI : Nonempty { i // p i } := nonempty_subtype.2 hp
apply InjOn.image_iInter_eq
simpa only [iUnion, iSup_subtype'] using h
#align set.inj_on.image_bInter_eq Set.InjOn.image_biInter_eq
theorem image_iInter {f : α → β} (hf : Bijective f) (s : ι → Set α) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
cases isEmpty_or_nonempty ι
· simp_rw [iInter_of_empty, image_univ_of_surjective hf.surjective]
· exact (hf.injective.injOn _).image_iInter_eq
#align set.image_Inter Set.image_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂ {f : α → β} (hf : Bijective f) (s : ∀ i, κ i → Set α) :
(f '' ⋂ (i) (j), s i j) = ⋂ (i) (j), f '' s i j := by simp_rw [image_iInter hf]
#align set.image_Inter₂ Set.image_iInter₂
theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
(hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i) := by
intro x hx y hy hxy
rcases mem_iUnion.1 hx with ⟨i, hx⟩
rcases mem_iUnion.1 hy with ⟨j, hy⟩
rcases hs i j with ⟨k, hi, hj⟩
exact hf k (hi hx) (hj hy) hxy
#align set.inj_on_Union_of_directed Set.inj_on_iUnion_of_directed
/-! ### `SurjOn` -/
theorem surjOn_sUnion {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, SurjOn f s t) :
SurjOn f s (⋃₀T) := fun _ ⟨t, ht, hx⟩ => H t ht hx
#align set.surj_on_sUnion Set.surjOn_sUnion
theorem surjOn_iUnion {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, SurjOn f s (t i)) :
SurjOn f s (⋃ i, t i) :=
surjOn_sUnion <| forall_range_iff.2 H
#align set.surj_on_Union Set.surjOn_iUnion
theorem surjOn_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) : SurjOn f (⋃ i, s i) (⋃ i, t i) :=
surjOn_iUnion fun i => (H i).mono (subset_iUnion _ _) (Subset.refl _)
#align set.surj_on_Union_Union Set.surjOn_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f s (t i j)) : SurjOn f s (⋃ (i) (j), t i j) :=
surjOn_iUnion fun i => surjOn_iUnion (H i)
#align set.surj_on_Union₂ Set.surjOn_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f (s i j) (t i j)) : SurjOn f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
surjOn_iUnion_iUnion fun i => surjOn_iUnion_iUnion (H i)
#align set.surj_on_Union₂_Union₂ Set.surjOn_iUnion₂_iUnion₂
theorem surjOn_iInter [Nonempty ι] {s : ι → Set α} {t : Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) t) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) t := by
intro y hy
rw [Hinj.image_iInter_eq, mem_iInter]
exact fun i => H i hy
#align set.surj_on_Inter Set.surjOn_iInter
theorem surjOn_iInter_iInter [Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) (⋂ i, t i) :=
surjOn_iInter (fun i => (H i).mono (Subset.refl _) (iInter_subset _ _)) Hinj
#align set.surj_on_Inter_Inter Set.surjOn_iInter_iInter
/-! ### `BijOn` -/
theorem bijOn_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i))
(Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
⟨mapsTo_iUnion_iUnion fun i => (H i).mapsTo, Hinj, surjOn_iUnion_iUnion fun i => (H i).surjOn⟩
#align set.bij_on_Union Set.bijOn_iUnion
theorem bijOn_iInter [hi : Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, BijOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
⟨mapsTo_iInter_iInter fun i => (H i).mapsTo,
hi.elim fun i => (H i).injOn.mono (iInter_subset _ _),
surjOn_iInter_iInter (fun i => (H i).surjOn) Hinj⟩
#align set.bij_on_Inter Set.bijOn_iInter
theorem bijOn_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {t : ι → Set β}
{f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
bijOn_iUnion H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Union_of_directed Set.bijOn_iUnion_of_directed
theorem bijOn_iInter_of_directed [Nonempty ι] {s : ι → Set α} (hs : Directed (· ⊆ ·) s)
{t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
bijOn_iInter H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Inter_of_directed Set.bijOn_iInter_of_directed
end Function
/-! ### `image`, `preimage` -/
section Image
theorem image_iUnion {f : α → β} {s : ι → Set α} : (f '' ⋃ i, s i) = ⋃ i, f '' s i := by
ext1 x
simp only [mem_image, mem_iUnion, ← exists_and_right, ← exists_and_left]
--Porting note: `exists_swap` causes a `simp` loop in Lean4 so we use `rw` instead.
rw [exists_swap]
#align set.image_Union Set.image_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iUnion₂ (f : α → β) (s : ∀ i, κ i → Set α) :
(f '' ⋃ (i) (j), s i j) = ⋃ (i) (j), f '' s i j := by simp_rw [image_iUnion]
#align set.image_Union₂ Set.image_iUnion₂
theorem univ_subtype {p : α → Prop} : (univ : Set (Subtype p)) = ⋃ (x) (h : p x), {⟨x, h⟩} :=
Set.ext fun ⟨x, h⟩ => by simp [h]
#align set.univ_subtype Set.univ_subtype
theorem range_eq_iUnion {ι} (f : ι → α) : range f = ⋃ i, {f i} :=
Set.ext fun a => by simp [@eq_comm α a]
#align set.range_eq_Union Set.range_eq_iUnion
theorem image_eq_iUnion (f : α → β) (s : Set α) : f '' s = ⋃ i ∈ s, {f i} :=
Set.ext fun b => by simp [@eq_comm β b]
#align set.image_eq_Union Set.image_eq_iUnion
theorem biUnion_range {f : ι → α} {g : α → Set β} : ⋃ x ∈ range f, g x = ⋃ y, g (f y) :=
iSup_range
#align set.bUnion_range Set.biUnion_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iUnion_iUnion_eq' {f : ι → α} {g : α → Set β} :
⋃ (x) (y) (_ : f y = x), g x = ⋃ y, g (f y) := by simpa using biUnion_range
#align set.Union_Union_eq' Set.iUnion_iUnion_eq'
theorem biInter_range {f : ι → α} {g : α → Set β} : ⋂ x ∈ range f, g x = ⋂ y, g (f y) :=
iInf_range
#align set.bInter_range Set.biInter_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iInter_iInter_eq' {f : ι → α} {g : α → Set β} :
⋂ (x) (y) (_ : f y = x), g x = ⋂ y, g (f y) := by simpa using biInter_range
#align set.Inter_Inter_eq' Set.iInter_iInter_eq'
variable {s : Set γ} {f : γ → α} {g : α → Set β}
theorem biUnion_image : ⋃ x ∈ f '' s, g x = ⋃ y ∈ s, g (f y) :=
iSup_image
#align set.bUnion_image Set.biUnion_image
theorem biInter_image : ⋂ x ∈ f '' s, g x = ⋂ y ∈ s, g (f y) :=
iInf_image
#align set.bInter_image Set.biInter_image
end Image
section Preimage
theorem monotone_preimage {f : α → β} : Monotone (preimage f) := fun _ _ h => preimage_mono h
#align set.monotone_preimage Set.monotone_preimage
@[simp]
theorem preimage_iUnion {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋃ i, s i) = ⋃ i, f ⁻¹' s i :=
Set.ext <| by simp [preimage]
#align set.preimage_Union Set.preimage_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iUnion₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋃ (i) (j), s i j) = ⋃ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iUnion]
#align set.preimage_Union₂ Set.preimage_iUnion₂
@[simp]
theorem preimage_sUnion {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋃₀s = ⋃ t ∈ s, f ⁻¹' t := by
rw [sUnion_eq_biUnion, preimage_iUnion₂]
#align set.preimage_sUnion Set.preimage_sUnion
theorem preimage_iInter {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋂ i, s i) = ⋂ i, f ⁻¹' s i := by
ext; simp
#align set.preimage_Inter Set.preimage_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iInter₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋂ (i) (j), s i j) = ⋂ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iInter]
#align set.preimage_Inter₂ Set.preimage_iInter₂
@[simp]
theorem preimage_sInter {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋂₀ s = ⋂ t ∈ s, f ⁻¹' t := by
rw [sInter_eq_biInter, preimage_iInter₂]
#align set.preimage_sInter Set.preimage_sInter
@[simp]
theorem biUnion_preimage_singleton (f : α → β) (s : Set β) : ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s := by
rw [← preimage_iUnion₂, biUnion_of_singleton]
#align set.bUnion_preimage_singleton Set.biUnion_preimage_singleton
theorem biUnion_range_preimage_singleton (f : α → β) : ⋃ y ∈ range f, f ⁻¹' {y} = univ := by
rw [biUnion_preimage_singleton, preimage_range]
#align set.bUnion_range_preimage_singleton Set.biUnion_range_preimage_singleton
end Preimage
section Prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion {s : Set α} {t : ι → Set β} : (s ×ˢ ⋃ i, t i) = ⋃ i, s ×ˢ t i := by
ext
simp
#align set.prod_Union Set.prod_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} :
(s ×ˢ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ×ˢ t i j := by simp_rw [prod_iUnion]
#align set.prod_Union₂ Set.prod_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sUnion {s : Set α} {C : Set (Set β)} : s ×ˢ ⋃₀C = ⋃₀((fun t => s ×ˢ t) '' C) := by
simp_rw [sUnion_eq_biUnion, biUnion_image, prod_iUnion₂]
#align set.prod_sUnion Set.prod_sUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_const {s : ι → Set α} {t : Set β} : (⋃ i, s i) ×ˢ t = ⋃ i, s i ×ˢ t := by
ext
simp
#align set.Union_prod_const Set.iUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion₂_prod_const {s : ∀ i, κ i → Set α} {t : Set β} :
(⋃ (i) (j), s i j) ×ˢ t = ⋃ (i) (j), s i j ×ˢ t := by simp_rw [iUnion_prod_const]
#align set.Union₂_prod_const Set.iUnion₂_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_prod_const {C : Set (Set α)} {t : Set β} :
⋃₀C ×ˢ t = ⋃₀((fun s : Set α => s ×ˢ t) '' C) := by
simp only [sUnion_eq_biUnion, iUnion₂_prod_const, biUnion_image]
#align set.sUnion_prod_const Set.sUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod {ι ι' α β} (s : ι → Set α) (t : ι' → Set β) :
⋃ x : ι × ι', s x.1 ×ˢ t x.2 = (⋃ i : ι, s i) ×ˢ ⋃ i : ι', t i := by
ext
simp
#align set.Union_prod Set.iUnion_prod
/-- Analogue of `iSup_prod` for sets. -/
lemma iUnion_prod' (f : β × γ → Set α) : ⋃ x : β × γ, f x = ⋃ (i : β) (j : γ), f (i, j) :=
iSup_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x := by
ext ⟨z, w⟩; simp only [mem_prod, mem_iUnion, exists_imp, and_imp, iff_def]; constructor
· intro x hz hw
exact ⟨⟨x, hz⟩, x, hw⟩
· intro x hz x' hw
exact ⟨x ⊔ x', hs le_sup_left hz, ht le_sup_right hw⟩
#align set.Union_prod_of_monotone Set.iUnion_prod_of_monotone
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter_subset (S : Set (Set α)) (T : Set (Set β)) :
⋂₀ S ×ˢ ⋂₀ T ⊆ ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 :=
subset_iInter₂ fun x hx _ hy => ⟨hy.1 x.1 hx.1, hy.2 x.2 hx.2⟩
#align set.sInter_prod_sInter_subset Set.sInter_prod_sInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 := by
obtain ⟨s₁, h₁⟩ := hS
obtain ⟨s₂, h₂⟩ := hT
refine' Set.Subset.antisymm (sInter_prod_sInter_subset S T) fun x hx => _
rw [mem_iInter₂] at hx
exact ⟨fun s₀ h₀ => (hx (s₀, s₂) ⟨h₀, h₂⟩).1, fun s₀ h₀ => (hx (s₁, s₀) ⟨h₁, h₀⟩).2⟩
#align set.sInter_prod_sInter Set.sInter_prod_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod {S : Set (Set α)} (hS : S.Nonempty) (t : Set β) :
⋂₀ S ×ˢ t = ⋂ s ∈ S, s ×ˢ t := by
rw [← sInter_singleton t, sInter_prod_sInter hS (singleton_nonempty t), sInter_singleton]
simp_rw [prod_singleton, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.sInter_prod Set.sInter_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sInter {T : Set (Set β)} (hT : T.Nonempty) (s : Set α) :
s ×ˢ ⋂₀ T = ⋂ t ∈ T, s ×ˢ t := by
rw [← sInter_singleton s, sInter_prod_sInter (singleton_nonempty s) hT, sInter_singleton]
simp_rw [singleton_prod, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.prod_sInter Set.prod_sInter
end Prod
section Image2
variable (f : α → β → γ) {s : Set α} {t : Set β}
/-- The `Set.image2` version of `Set.image_eq_iUnion` -/
theorem image2_eq_iUnion (s : Set α) (t : Set β) : image2 f s t = ⋃ (i ∈ s) (j ∈ t), {f i j} := by
ext; simp [eq_comm]
#align set.image2_eq_Union Set.image2_eq_iUnion
theorem iUnion_image_left : ⋃ a ∈ s, f a '' t = image2 f s t := by
simp only [image2_eq_iUnion, image_eq_iUnion]
#align set.Union_image_left Set.iUnion_image_left
theorem iUnion_image_right : ⋃ b ∈ t, (f · b) '' s = image2 f s t := by
rw [image2_swap, iUnion_image_left]
#align set.Union_image_right Set.iUnion_image_right
theorem image2_iUnion_left (s : ι → Set α) (t : Set β) :
image2 f (⋃ i, s i) t = ⋃ i, image2 f (s i) t := by
simp only [← image_prod, iUnion_prod_const, image_iUnion]
#align set.image2_Union_left Set.image2_iUnion_left
theorem image2_iUnion_right (s : Set α) (t : ι → Set β) :
image2 f s (⋃ i, t i) = ⋃ i, image2 f s (t i) := by
simp only [← image_prod, prod_iUnion, image_iUnion]
#align set.image2_Union_right Set.image2_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋃ (i) (j), s i j) t = ⋃ (i) (j), image2 f (s i j) t := by simp_rw [image2_iUnion_left]
#align set.image2_Union₂_left Set.image2_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋃ (i) (j), t i j) = ⋃ (i) (j), image2 f s (t i j) :=
by simp_rw [image2_iUnion_right]
#align set.image2_Union₂_right Set.image2_iUnion₂_right
theorem image2_iInter_subset_left (s : ι → Set α) (t : Set β) :
image2 f (⋂ i, s i) t ⊆ ⋂ i, image2 f (s i) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem (hx _) hy
#align set.image2_Inter_subset_left Set.image2_iInter_subset_left
theorem image2_iInter_subset_right (s : Set α) (t : ι → Set β) :
image2 f s (⋂ i, t i) ⊆ ⋂ i, image2 f s (t i) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem hx (hy _)
#align set.image2_Inter_subset_right Set.image2_iInter_subset_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋂ (i) (j), s i j) t ⊆ ⋂ (i) (j), image2 f (s i j) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem (hx _ _) hy
#align set.image2_Inter₂_subset_left Set.image2_iInter₂_subset_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), image2 f s (t i j) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem hx (hy _ _)
#align set.image2_Inter₂_subset_right Set.image2_iInter₂_subset_right
theorem prod_eq_biUnion_left : s ×ˢ t = ⋃ a ∈ s, (fun b => (a, b)) '' t := by
rw [iUnion_image_left, image2_mk_eq_prod]
#align set.prod_eq_bUnion_left Set.prod_eq_biUnion_left
theorem prod_eq_biUnion_right : s ×ˢ t = ⋃ b ∈ t, (fun a => (a, b)) '' s := by
rw [iUnion_image_right, image2_mk_eq_prod]
#align set.prod_eq_bUnion_right Set.prod_eq_biUnion_right
end Image2
section Seq
/-- Given a set `s` of functions `α → β` and `t : Set α`, `seq s t` is the union of `f '' t` over
all `f ∈ s`. -/
def seq (s : Set (α → β)) (t : Set α) : Set β :=
{ b | ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b }
#align set.seq Set.seq
@[simp]
theorem mem_seq_iff {s : Set (α → β)} {t : Set α} {b : β} :
b ∈ seq s t ↔ ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b :=
Iff.rfl
#align set.mem_seq_iff Set.mem_seq_iff
lemma seq_eq_image2 (s : Set (α → β)) (t : Set α) : seq s t = image2 (fun f a ↦ f a) s t := by
ext; simp
theorem seq_def {s : Set (α → β)} {t : Set α} : seq s t = ⋃ f ∈ s, f '' t := by
rw [seq_eq_image2, iUnion_image_left]
#align set.seq_def Set.seq_def
theorem seq_subset {s : Set (α → β)} {t : Set α} {u : Set β} :
seq s t ⊆ u ↔ ∀ f ∈ s, ∀ a ∈ t, (f : α → β) a ∈ u := by
rw [seq_eq_image2, image2_subset_iff]
#align set.seq_subset Set.seq_subset
@[gcongr]
theorem seq_mono {s₀ s₁ : Set (α → β)} {t₀ t₁ : Set α} (hs : s₀ ⊆ s₁) (ht : t₀ ⊆ t₁) :
seq s₀ t₀ ⊆ seq s₁ t₁ := fun _ ⟨f, hf, a, ha, eq⟩ => ⟨f, hs hf, a, ht ha, eq⟩
#align set.seq_mono Set.seq_mono
theorem singleton_seq {f : α → β} {t : Set α} : Set.seq ({f} : Set (α → β)) t = f '' t := by
rw [seq_eq_image2, image2_singleton_left]
#align set.singleton_seq Set.singleton_seq
theorem seq_singleton {s : Set (α → β)} {a : α} : Set.seq s {a} = (fun f : α → β => f a) '' s := by
rw [seq_eq_image2, image2_singleton_right]
#align set.seq_singleton Set.seq_singleton
theorem seq_seq {s : Set (β → γ)} {t : Set (α → β)} {u : Set α} :
seq s (seq t u) = seq (seq ((· ∘ ·) '' s) t) u := by
simp only [seq_eq_image2, image2_image_left, image2_image2_left, image2_image2_right, comp_apply]
#align set.seq_seq Set.seq_seq
theorem image_seq {f : β → γ} {s : Set (α → β)} {t : Set α} :
f '' seq s t = seq ((f ∘ ·) '' s) t := by
rw [← singleton_seq, ← singleton_seq, seq_seq, image_singleton]
#align set.image_seq Set.image_seq
theorem prod_eq_seq {s : Set α} {t : Set β} : s ×ˢ t = (Prod.mk '' s).seq t := by
rw [seq_eq_image2, image2_image_left, image2_mk_eq_prod]
#align set.prod_eq_seq Set.prod_eq_seq
theorem prod_image_seq_comm (s : Set α) (t : Set β) :
(Prod.mk '' s).seq t = seq ((fun b a => (a, b)) '' t) s := by
rw [← prod_eq_seq, ← image_swap_prod, prod_eq_seq, image_seq, ← image_comp]; rfl
#align set.prod_image_seq_comm Set.prod_image_seq_comm
theorem image2_eq_seq (f : α → β → γ) (s : Set α) (t : Set β) : image2 f s t = seq (f '' s) t := by
rw [seq_eq_image2, image2_image_left]
#align set.image2_eq_seq Set.image2_eq_seq
end Seq
section Pi
variable {π : α → Type*}
theorem pi_def (i : Set α) (s : ∀ a, Set (π a)) : pi i s = ⋂ a ∈ i, eval a ⁻¹' s a := by
ext
simp
#align set.pi_def Set.pi_def
theorem univ_pi_eq_iInter (t : ∀ i, Set (π i)) : pi univ t = ⋂ i, eval i ⁻¹' t i := by
simp only [pi_def, iInter_true, mem_univ]
#align set.univ_pi_eq_Inter Set.univ_pi_eq_iInter
theorem pi_diff_pi_subset (i : Set α) (s t : ∀ a, Set (π a)) :
pi i s \ pi i t ⊆ ⋃ a ∈ i, eval a ⁻¹' (s a \ t a) := by
refine' diff_subset_comm.2 fun x hx a ha => _
simp only [mem_diff, mem_pi, mem_iUnion, not_exists, mem_preimage, not_and, not_not,
eval_apply] at hx
exact hx.2 _ ha (hx.1 _ ha)
#align set.pi_diff_pi_subset Set.pi_diff_pi_subset
theorem iUnion_univ_pi {ι : α → Type*} (t : (a : α) → ι a → Set (π a)) :
⋃ x : (a : α) → ι a, pi univ (fun a => t a (x a)) = pi univ fun a => ⋃ j : ι a, t a j := by
ext
simp [Classical.skolem]
#align set.Union_univ_pi Set.iUnion_univ_pi
end Pi
end Set
namespace Function
namespace Surjective
theorem iUnion_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋃ x, g (f x) = ⋃ y, g y :=
hf.iSup_comp g
#align function.surjective.Union_comp Function.Surjective.iUnion_comp
theorem iInter_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋂ x, g (f x) = ⋂ y, g y :=
hf.iInf_comp g
#align function.surjective.Inter_comp Function.Surjective.iInter_comp
end Surjective
end Function
/-!
### Disjoint sets
-/
section Disjoint
variable {s t u : Set α} {f : α → β}
namespace Set
@[simp]
theorem disjoint_iUnion_left {ι : Sort*} {s : ι → Set α} :
Disjoint (⋃ i, s i) t ↔ ∀ i, Disjoint (s i) t :=
iSup_disjoint_iff
#align set.disjoint_Union_left Set.disjoint_iUnion_left
@[simp]
theorem disjoint_iUnion_right {ι : Sort*} {s : ι → Set α} :
Disjoint t (⋃ i, s i) ↔ ∀ i, Disjoint t (s i) :=
disjoint_iSup_iff
#align set.disjoint_Union_right Set.disjoint_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_left {s : ∀ i, κ i → Set α} {t : Set α} :
Disjoint (⋃ (i) (j), s i j) t ↔ ∀ i j, Disjoint (s i j) t :=
iSup₂_disjoint_iff
#align set.disjoint_Union₂_left Set.disjoint_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_right {s : Set α} {t : ∀ i, κ i → Set α} :
Disjoint s (⋃ (i) (j), t i j) ↔ ∀ i j, Disjoint s (t i j) :=
disjoint_iSup₂_iff
#align set.disjoint_Union₂_right Set.disjoint_iUnion₂_right
@[simp]
theorem disjoint_sUnion_left {S : Set (Set α)} {t : Set α} :
Disjoint (⋃₀S) t ↔ ∀ s ∈ S, Disjoint s t :=
sSup_disjoint_iff
#align set.disjoint_sUnion_left Set.disjoint_sUnion_left
@[simp]
theorem disjoint_sUnion_right {s : Set α} {S : Set (Set α)} :
Disjoint s (⋃₀S) ↔ ∀ t ∈ S, Disjoint s t :=
disjoint_sSup_iff
#align set.disjoint_sUnion_right Set.disjoint_sUnion_right
end Set
end Disjoint
/-! ### Intervals -/
namespace Set
lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :
(⋂ i, Iic (f i)).Nonempty ↔ BddBelow (range f) := by
have : (⋂ (i : ι), Iic (f i)) = lowerBounds (range f) := by
ext c; simp [lowerBounds]
simp [this, BddBelow]
lemma nonempty_iInter_Ici_iff [Preorder α] {f : ι → α} :
(⋂ i, Ici (f i)).Nonempty ↔ BddAbove (range f) :=
nonempty_iInter_Iic_iff (α := αᵒᵈ)
variable [CompleteLattice α]
theorem Ici_iSup (f : ι → α) : Ici (⨆ i, f i) = ⋂ i, Ici (f i) :=
ext fun _ => by simp only [mem_Ici, iSup_le_iff, mem_iInter]
#align set.Ici_supr Set.Ici_iSup
theorem Iic_iInf (f : ι → α) : Iic (⨅ i, f i) = ⋂ i, Iic (f i) :=
ext fun _ => by simp only [mem_Iic, le_iInf_iff, mem_iInter]
#align set.Iic_infi Set.Iic_iInf
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Ici_iSup₂ (f : ∀ i, κ i → α) : Ici (⨆ (i) (j), f i j) = ⋂ (i) (j), Ici (f i j) := by
simp_rw [Ici_iSup]
#align set.Ici_supr₂ Set.Ici_iSup₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Iic_iInf₂ (f : ∀ i, κ i → α) : Iic (⨅ (i) (j), f i j) = ⋂ (i) (j), Iic (f i j) := by
simp_rw [Iic_iInf]
#align set.Iic_infi₂ Set.Iic_iInf₂
theorem Ici_sSup (s : Set α) : Ici (sSup s) = ⋂ a ∈ s, Ici a := by rw [sSup_eq_iSup, Ici_iSup₂]
#align set.Ici_Sup Set.Ici_sSup
theorem Iic_sInf (s : Set α) : Iic (sInf s) = ⋂ a ∈ s, Iic a := by rw [sInf_eq_iInf, Iic_iInf₂]
#align set.Iic_Inf Set.Iic_sInf
end Set
namespace Set
variable (t : α → Set β)
theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x := by
simp only [diff_subset_iff, ← biUnion_union]
|
apply biUnion_subset_biUnion_left
|
theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x := by
simp only [diff_subset_iff, ← biUnion_union]
|
Mathlib.Data.Set.Lattice.2291_0.5mONj49h3SYSDwc
|
theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x
|
Mathlib_Data_Set_Lattice
|
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
t : α → Set β
s₁ s₂ : Set α
⊢ s₁ ⊆ s₂ ∪ s₁ \ s₂
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
/-!
# The set lattice
This file provides usual set notation for unions and intersections, a `CompleteLattice` instance
for `Set α`, and some more set constructions.
## Main declarations
* `Set.iUnion`: **i**ndexed **union**. Union of an indexed family of sets.
* `Set.iInter`: **i**ndexed **inter**section. Intersection of an indexed family of sets.
* `Set.sInter`: **s**et **inter**section. Intersection of sets belonging to a set of sets.
* `Set.sUnion`: **s**et **union**. Union of sets belonging to a set of sets.
* `Set.sInter_eq_biInter`, `Set.sUnion_eq_biInter`: Shows that `⋂₀ s = ⋂ x ∈ s, x` and
`⋃₀ s = ⋃ x ∈ s, x`.
* `Set.completeAtomicBooleanAlgebra`: `Set α` is a `CompleteAtomicBooleanAlgebra` with `≤ = ⊆`,
`< = ⊂`, `⊓ = ∩`, `⊔ = ∪`, `⨅ = ⋂`, `⨆ = ⋃` and `\` as the set difference.
See `Set.BooleanAlgebra`.
* `Set.kernImage`: For a function `f : α → β`, `s.kernImage f` is the set of `y` such that
`f ⁻¹ y ⊆ s`.
* `Set.seq`: Union of the image of a set under a **seq**uence of functions. `seq s t` is the union
of `f '' t` over all `f ∈ s`, where `t : Set α` and `s : Set (α → β)`.
* `Set.iUnion_eq_sigma_of_disjoint`: Equivalence between `⋃ i, t i` and `Σ i, t i`, where `t` is an
indexed family of disjoint sets.
## Naming convention
In lemma names,
* `⋃ i, s i` is called `iUnion`
* `⋂ i, s i` is called `iInter`
* `⋃ i j, s i j` is called `iUnion₂`. This is an `iUnion` inside an `iUnion`.
* `⋂ i j, s i j` is called `iInter₂`. This is an `iInter` inside an `iInter`.
* `⋃ i ∈ s, t i` is called `biUnion` for "bounded `iUnion`". This is the special case of `iUnion₂`
where `j : i ∈ s`.
* `⋂ i ∈ s, t i` is called `biInter` for "bounded `iInter`". This is the special case of `iInter₂`
where `j : i ∈ s`.
## Notation
* `⋃`: `Set.iUnion`
* `⋂`: `Set.iInter`
* `⋃₀`: `Set.sUnion`
* `⋂₀`: `Set.sInter`
-/
set_option autoImplicit true
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι₂ : Sort*} {κ κ₁ κ₂ : ι → Sort*} {κ' : ι' → Sort*}
namespace Set
/-! ### Complete lattice and complete Boolean algebra instances -/
instance : InfSet (Set α) :=
⟨fun s => { a | ∀ t ∈ s, a ∈ t }⟩
instance : SupSet (Set α) :=
⟨fun s => { a | ∃ t ∈ s, a ∈ t }⟩
/-- Intersection of a set of sets. -/
def sInter (S : Set (Set α)) : Set α :=
sInf S
#align set.sInter Set.sInter
/-- Notation for `Set.sInter` Intersection of a set of sets. -/
prefix:110 "⋂₀ " => sInter
/-- Union of a set of sets. -/
def sUnion (S : Set (Set α)) : Set α :=
sSup S
#align set.sUnion Set.sUnion
/-- Notation for `Set.sUnion`. Union of a set of sets. -/
prefix:110 "⋃₀ " => sUnion
@[simp]
theorem mem_sInter {x : α} {S : Set (Set α)} : x ∈ ⋂₀ S ↔ ∀ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sInter Set.mem_sInter
@[simp]
theorem mem_sUnion {x : α} {S : Set (Set α)} : x ∈ ⋃₀ S ↔ ∃ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sUnion Set.mem_sUnion
/-- Indexed union of a family of sets -/
def iUnion (s : ι → Set β) : Set β :=
iSup s
#align set.Union Set.iUnion
/-- Indexed intersection of a family of sets -/
def iInter (s : ι → Set β) : Set β :=
iInf s
#align set.Inter Set.iInter
/-- Notation for `Set.iUnion`. Indexed union of a family of sets -/
notation3 "⋃ "(...)", "r:60:(scoped f => iUnion f) => r
/-- Notation for `Set.iInter`. Indexed intersection of a family of sets -/
notation3 "⋂ "(...)", "r:60:(scoped f => iInter f) => r
section delaborators
open Lean Lean.PrettyPrinter.Delaborator
/-- Delaborator for indexed unions. -/
@[delab app.Set.iUnion]
def iUnion_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋃ (_ : $dom), $body)
else if prop || ppTypes then
`(⋃ ($x:ident : $dom), $body)
else
`(⋃ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋃ $x:ident, ⋃ (_ : $y:ident ∈ $s), $body)
| `(⋃ ($x:ident : $_), ⋃ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋃ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
/-- Delaborator for indexed intersections. -/
@[delab app.Set.iInter]
def sInter_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋂ (_ : $dom), $body)
else if prop || ppTypes then
`(⋂ ($x:ident : $dom), $body)
else
`(⋂ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋂ $x:ident, ⋂ (_ : $y:ident ∈ $s), $body)
| `(⋂ ($x:ident : $_), ⋂ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋂ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
end delaborators
@[simp]
theorem sSup_eq_sUnion (S : Set (Set α)) : sSup S = ⋃₀S :=
rfl
#align set.Sup_eq_sUnion Set.sSup_eq_sUnion
@[simp]
theorem sInf_eq_sInter (S : Set (Set α)) : sInf S = ⋂₀ S :=
rfl
#align set.Inf_eq_sInter Set.sInf_eq_sInter
@[simp]
theorem iSup_eq_iUnion (s : ι → Set α) : iSup s = iUnion s :=
rfl
#align set.supr_eq_Union Set.iSup_eq_iUnion
@[simp]
theorem iInf_eq_iInter (s : ι → Set α) : iInf s = iInter s :=
rfl
#align set.infi_eq_Inter Set.iInf_eq_iInter
@[simp]
theorem mem_iUnion {x : α} {s : ι → Set α} : (x ∈ ⋃ i, s i) ↔ ∃ i, x ∈ s i :=
⟨fun ⟨_, ⟨⟨a, (t_eq : s a = _)⟩, (h : x ∈ _)⟩⟩ => ⟨a, t_eq.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨s a, ⟨⟨a, rfl⟩, h⟩⟩⟩
#align set.mem_Union Set.mem_iUnion
@[simp]
theorem mem_iInter {x : α} {s : ι → Set α} : (x ∈ ⋂ i, s i) ↔ ∀ i, x ∈ s i :=
⟨fun (h : ∀ a ∈ { a : Set α | ∃ i, s i = a }, x ∈ a) a => h (s a) ⟨a, rfl⟩,
fun h _ ⟨a, (eq : s a = _)⟩ => eq ▸ h a⟩
#align set.mem_Inter Set.mem_iInter
theorem mem_iUnion₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋃ (i) (j), s i j) ↔ ∃ i j, x ∈ s i j := by
simp_rw [mem_iUnion]
#align set.mem_Union₂ Set.mem_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋂ (i) (j), s i j) ↔ ∀ i j, x ∈ s i j := by
simp_rw [mem_iInter]
#align set.mem_Inter₂ Set.mem_iInter₂
theorem mem_iUnion_of_mem {s : ι → Set α} {a : α} (i : ι) (ha : a ∈ s i) : a ∈ ⋃ i, s i :=
mem_iUnion.2 ⟨i, ha⟩
#align set.mem_Union_of_mem Set.mem_iUnion_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iUnion₂_of_mem {s : ∀ i, κ i → Set α} {a : α} {i : ι} (j : κ i) (ha : a ∈ s i j) :
a ∈ ⋃ (i) (j), s i j :=
mem_iUnion₂.2 ⟨i, j, ha⟩
#align set.mem_Union₂_of_mem Set.mem_iUnion₂_of_mem
theorem mem_iInter_of_mem {s : ι → Set α} {a : α} (h : ∀ i, a ∈ s i) : a ∈ ⋂ i, s i :=
mem_iInter.2 h
#align set.mem_Inter_of_mem Set.mem_iInter_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂_of_mem {s : ∀ i, κ i → Set α} {a : α} (h : ∀ i j, a ∈ s i j) :
a ∈ ⋂ (i) (j), s i j :=
mem_iInter₂.2 h
#align set.mem_Inter₂_of_mem Set.mem_iInter₂_of_mem
instance Set.completeAtomicBooleanAlgebra : CompleteAtomicBooleanAlgebra (Set α) :=
{ instBooleanAlgebraSet with
le_sSup := fun s t t_in a a_in => ⟨t, t_in, a_in⟩
sSup_le := fun s t h a ⟨t', ⟨t'_in, a_in⟩⟩ => h t' t'_in a_in
le_sInf := fun s t h a a_in t' t'_in => h t' t'_in a_in
sInf_le := fun s t t_in a h => h _ t_in
iInf_iSup_eq := by intros; ext; simp [Classical.skolem] }
/-- `kernImage f s` is the set of `y` such that `f ⁻¹ y ⊆ s`. -/
def kernImage (f : α → β) (s : Set α) : Set β :=
{ y | ∀ ⦃x⦄, f x = y → x ∈ s }
#align set.kern_image Set.kernImage
lemma subset_kernImage_iff {f : α → β} : s ⊆ kernImage f t ↔ f ⁻¹' s ⊆ t :=
⟨fun h _ hx ↦ h hx rfl,
fun h _ hx y hy ↦ h (show f y ∈ s from hy.symm ▸ hx)⟩
section GaloisConnection
variable {f : α → β}
protected theorem image_preimage : GaloisConnection (image f) (preimage f) := fun _ _ =>
image_subset_iff
#align set.image_preimage Set.image_preimage
protected theorem preimage_kernImage : GaloisConnection (preimage f) (kernImage f) := fun _ _ =>
subset_kernImage_iff.symm
#align set.preimage_kern_image Set.preimage_kernImage
end GaloisConnection
section kernImage
variable {f : α → β}
lemma kernImage_mono : Monotone (kernImage f) :=
Set.preimage_kernImage.monotone_u
lemma kernImage_eq_compl {s : Set α} : kernImage f s = (f '' sᶜ)ᶜ :=
Set.preimage_kernImage.u_unique (Set.image_preimage.compl)
(fun t ↦ compl_compl (f ⁻¹' t) ▸ Set.preimage_compl)
lemma kernImage_compl {s : Set α} : kernImage f (sᶜ) = (f '' s)ᶜ := by
rw [kernImage_eq_compl, compl_compl]
lemma kernImage_empty : kernImage f ∅ = (range f)ᶜ := by
rw [kernImage_eq_compl, compl_empty, image_univ]
lemma kernImage_preimage_eq_iff {s : Set β} : kernImage f (f ⁻¹' s) = s ↔ (range f)ᶜ ⊆ s := by
rw [kernImage_eq_compl, ← preimage_compl, compl_eq_comm, eq_comm, image_preimage_eq_iff,
compl_subset_comm]
lemma compl_range_subset_kernImage {s : Set α} : (range f)ᶜ ⊆ kernImage f s := by
rw [← kernImage_empty]
exact kernImage_mono (empty_subset _)
lemma kernImage_union_preimage {s : Set α} {t : Set β} :
kernImage f (s ∪ f ⁻¹' t) = kernImage f s ∪ t := by
rw [kernImage_eq_compl, kernImage_eq_compl, compl_union, ← preimage_compl, image_inter_preimage,
compl_inter, compl_compl]
lemma kernImage_preimage_union {s : Set α} {t : Set β} :
kernImage f (f ⁻¹' t ∪ s) = t ∪ kernImage f s := by
rw [union_comm, kernImage_union_preimage, union_comm]
end kernImage
/-! ### Union and intersection over an indexed family of sets -/
instance : OrderTop (Set α) where
top := univ
le_top := by simp
@[congr]
theorem iUnion_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iUnion f₁ = iUnion f₂ :=
iSup_congr_Prop pq f
#align set.Union_congr_Prop Set.iUnion_congr_Prop
@[congr]
theorem iInter_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iInter f₁ = iInter f₂ :=
iInf_congr_Prop pq f
#align set.Inter_congr_Prop Set.iInter_congr_Prop
theorem iUnion_plift_up (f : PLift ι → Set α) : ⋃ i, f (PLift.up i) = ⋃ i, f i :=
iSup_plift_up _
#align set.Union_plift_up Set.iUnion_plift_up
theorem iUnion_plift_down (f : ι → Set α) : ⋃ i, f (PLift.down i) = ⋃ i, f i :=
iSup_plift_down _
#align set.Union_plift_down Set.iUnion_plift_down
theorem iInter_plift_up (f : PLift ι → Set α) : ⋂ i, f (PLift.up i) = ⋂ i, f i :=
iInf_plift_up _
#align set.Inter_plift_up Set.iInter_plift_up
theorem iInter_plift_down (f : ι → Set α) : ⋂ i, f (PLift.down i) = ⋂ i, f i :=
iInf_plift_down _
#align set.Inter_plift_down Set.iInter_plift_down
theorem iUnion_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋃ _ : p, s = if p then s else ∅ :=
iSup_eq_if _
#align set.Union_eq_if Set.iUnion_eq_if
theorem iUnion_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋃ h : p, s h = if h : p then s h else ∅ :=
iSup_eq_dif _
#align set.Union_eq_dif Set.iUnion_eq_dif
theorem iInter_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋂ _ : p, s = if p then s else univ :=
iInf_eq_if _
#align set.Inter_eq_if Set.iInter_eq_if
theorem iInf_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋂ h : p, s h = if h : p then s h else univ :=
_root_.iInf_eq_dif _
#align set.Infi_eq_dif Set.iInf_eq_dif
theorem exists_set_mem_of_union_eq_top {ι : Type*} (t : Set ι) (s : ι → Set β)
(w : ⋃ i ∈ t, s i = ⊤) (x : β) : ∃ i ∈ t, x ∈ s i := by
have p : x ∈ ⊤ := Set.mem_univ x
rw [← w, Set.mem_iUnion] at p
simpa using p
#align set.exists_set_mem_of_union_eq_top Set.exists_set_mem_of_union_eq_top
theorem nonempty_of_union_eq_top_of_nonempty {ι : Type*} (t : Set ι) (s : ι → Set α)
(H : Nonempty α) (w : ⋃ i ∈ t, s i = ⊤) : t.Nonempty := by
obtain ⟨x, m, -⟩ := exists_set_mem_of_union_eq_top t s w H.some
exact ⟨x, m⟩
#align set.nonempty_of_union_eq_top_of_nonempty Set.nonempty_of_union_eq_top_of_nonempty
theorem nonempty_of_nonempty_iUnion
{s : ι → Set α} (h_Union : (⋃ i, s i).Nonempty) : Nonempty ι := by
obtain ⟨x, hx⟩ := h_Union
exact ⟨Classical.choose $ mem_iUnion.mp hx⟩
theorem nonempty_of_nonempty_iUnion_eq_univ
{s : ι → Set α} [Nonempty α] (h_Union : ⋃ i, s i = univ) : Nonempty ι :=
nonempty_of_nonempty_iUnion (s := s) (by simpa only [h_Union] using univ_nonempty)
theorem setOf_exists (p : ι → β → Prop) : { x | ∃ i, p i x } = ⋃ i, { x | p i x } :=
ext fun _ => mem_iUnion.symm
#align set.set_of_exists Set.setOf_exists
theorem setOf_forall (p : ι → β → Prop) : { x | ∀ i, p i x } = ⋂ i, { x | p i x } :=
ext fun _ => mem_iInter.symm
#align set.set_of_forall Set.setOf_forall
theorem iUnion_subset {s : ι → Set α} {t : Set α} (h : ∀ i, s i ⊆ t) : ⋃ i, s i ⊆ t :=
iSup_le h
#align set.Union_subset Set.iUnion_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset {s : ∀ i, κ i → Set α} {t : Set α} (h : ∀ i j, s i j ⊆ t) :
⋃ (i) (j), s i j ⊆ t :=
iUnion_subset fun x => iUnion_subset (h x)
#align set.Union₂_subset Set.iUnion₂_subset
theorem subset_iInter {t : Set β} {s : ι → Set β} (h : ∀ i, t ⊆ s i) : t ⊆ ⋂ i, s i :=
le_iInf h
#align set.subset_Inter Set.subset_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iInter₂ {s : Set α} {t : ∀ i, κ i → Set α} (h : ∀ i j, s ⊆ t i j) :
s ⊆ ⋂ (i) (j), t i j :=
subset_iInter fun x => subset_iInter <| h x
#align set.subset_Inter₂ Set.subset_iInter₂
@[simp]
theorem iUnion_subset_iff {s : ι → Set α} {t : Set α} : ⋃ i, s i ⊆ t ↔ ∀ i, s i ⊆ t :=
⟨fun h _ => Subset.trans (le_iSup s _) h, iUnion_subset⟩
#align set.Union_subset_iff Set.iUnion_subset_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset_iff {s : ∀ i, κ i → Set α} {t : Set α} :
⋃ (i) (j), s i j ⊆ t ↔ ∀ i j, s i j ⊆ t := by simp_rw [iUnion_subset_iff]
#align set.Union₂_subset_iff Set.iUnion₂_subset_iff
@[simp]
theorem subset_iInter_iff {s : Set α} {t : ι → Set α} : (s ⊆ ⋂ i, t i) ↔ ∀ i, s ⊆ t i :=
le_iInf_iff
#align set.subset_Inter_iff Set.subset_iInter_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem subset_iInter₂_iff {s : Set α} {t : ∀ i, κ i → Set α} :
(s ⊆ ⋂ (i) (j), t i j) ↔ ∀ i j, s ⊆ t i j := by simp_rw [subset_iInter_iff]
#align set.subset_Inter₂_iff Set.subset_iInter₂_iff
theorem subset_iUnion : ∀ (s : ι → Set β) (i : ι), s i ⊆ ⋃ i, s i :=
le_iSup
#align set.subset_Union Set.subset_iUnion
theorem iInter_subset : ∀ (s : ι → Set β) (i : ι), ⋂ i, s i ⊆ s i :=
iInf_le
#align set.Inter_subset Set.iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iUnion₂ {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : s i j ⊆ ⋃ (i') (j'), s i' j' :=
le_iSup₂ i j
#align set.subset_Union₂ Set.subset_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_subset {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : ⋂ (i) (j), s i j ⊆ s i j :=
iInf₂_le i j
#align set.Inter₂_subset Set.iInter₂_subset
/-- This rather trivial consequence of `subset_iUnion`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem subset_iUnion_of_subset {s : Set α} {t : ι → Set α} (i : ι) (h : s ⊆ t i) : s ⊆ ⋃ i, t i :=
le_iSup_of_le i h
#align set.subset_Union_of_subset Set.subset_iUnion_of_subset
/-- This rather trivial consequence of `iInter_subset`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem iInter_subset_of_subset {s : ι → Set α} {t : Set α} (i : ι) (h : s i ⊆ t) :
⋂ i, s i ⊆ t :=
iInf_le_of_le i h
#align set.Inter_subset_of_subset Set.iInter_subset_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `subset_iUnion₂` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem subset_iUnion₂_of_subset {s : Set α} {t : ∀ i, κ i → Set α} (i : ι) (j : κ i)
(h : s ⊆ t i j) : s ⊆ ⋃ (i) (j), t i j :=
le_iSup₂_of_le i j h
#align set.subset_Union₂_of_subset Set.subset_iUnion₂_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `iInter₂_subset` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem iInter₂_subset_of_subset {s : ∀ i, κ i → Set α} {t : Set α} (i : ι) (j : κ i)
(h : s i j ⊆ t) : ⋂ (i) (j), s i j ⊆ t :=
iInf₂_le_of_le i j h
#align set.Inter₂_subset_of_subset Set.iInter₂_subset_of_subset
theorem iUnion_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono h
#align set.Union_mono Set.iUnion_mono
@[gcongr]
theorem iUnion_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iUnion s ⊆ iUnion t :=
iSup_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋃ (i) (j), s i j ⊆ ⋃ (i) (j), t i j :=
iSup₂_mono h
#align set.Union₂_mono Set.iUnion₂_mono
theorem iInter_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋂ i, s i ⊆ ⋂ i, t i :=
iInf_mono h
#align set.Inter_mono Set.iInter_mono
@[gcongr]
theorem iInter_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iInter s ⊆ iInter t :=
iInf_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋂ (i) (j), s i j ⊆ ⋂ (i) (j), t i j :=
iInf₂_mono h
#align set.Inter₂_mono Set.iInter₂_mono
theorem iUnion_mono' {s : ι → Set α} {t : ι₂ → Set α} (h : ∀ i, ∃ j, s i ⊆ t j) :
⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono' h
#align set.Union_mono' Set.iUnion_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iUnion₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i j, ∃ i' j', s i j ⊆ t i' j') : ⋃ (i) (j), s i j ⊆ ⋃ (i') (j'), t i' j' :=
iSup₂_mono' h
#align set.Union₂_mono' Set.iUnion₂_mono'
theorem iInter_mono' {s : ι → Set α} {t : ι' → Set α} (h : ∀ j, ∃ i, s i ⊆ t j) :
⋂ i, s i ⊆ ⋂ j, t j :=
Set.subset_iInter fun j =>
let ⟨i, hi⟩ := h j
iInter_subset_of_subset i hi
#align set.Inter_mono' Set.iInter_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iInter₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i' j', ∃ i j, s i j ⊆ t i' j') : ⋂ (i) (j), s i j ⊆ ⋂ (i') (j'), t i' j' :=
subset_iInter₂_iff.2 fun i' j' =>
let ⟨_, _, hst⟩ := h i' j'
(iInter₂_subset _ _).trans hst
#align set.Inter₂_mono' Set.iInter₂_mono'
theorem iUnion₂_subset_iUnion (κ : ι → Sort*) (s : ι → Set α) :
⋃ (i) (_ : κ i), s i ⊆ ⋃ i, s i :=
iUnion_mono fun _ => iUnion_subset fun _ => Subset.rfl
#align set.Union₂_subset_Union Set.iUnion₂_subset_iUnion
theorem iInter_subset_iInter₂ (κ : ι → Sort*) (s : ι → Set α) :
⋂ i, s i ⊆ ⋂ (i) (_ : κ i), s i :=
iInter_mono fun _ => subset_iInter fun _ => Subset.rfl
#align set.Inter_subset_Inter₂ Set.iInter_subset_iInter₂
theorem iUnion_setOf (P : ι → α → Prop) : ⋃ i, { x : α | P i x } = { x : α | ∃ i, P i x } := by
ext
exact mem_iUnion
#align set.Union_set_of Set.iUnion_setOf
theorem iInter_setOf (P : ι → α → Prop) : ⋂ i, { x : α | P i x } = { x : α | ∀ i, P i x } := by
ext
exact mem_iInter
#align set.Inter_set_of Set.iInter_setOf
theorem iUnion_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋃ x, f x = ⋃ y, g y :=
h1.iSup_congr h h2
#align set.Union_congr_of_surjective Set.iUnion_congr_of_surjective
theorem iInter_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋂ x, f x = ⋂ y, g y :=
h1.iInf_congr h h2
#align set.Inter_congr_of_surjective Set.iInter_congr_of_surjective
lemma iUnion_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋃ i, s i = ⋃ i, t i := iSup_congr h
#align set.Union_congr Set.iUnion_congr
lemma iInter_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋂ i, s i = ⋂ i, t i := iInf_congr h
#align set.Inter_congr Set.iInter_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iUnion₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋃ (i) (j), s i j = ⋃ (i) (j), t i j :=
iUnion_congr fun i => iUnion_congr <| h i
#align set.Union₂_congr Set.iUnion₂_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iInter₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋂ (i) (j), s i j = ⋂ (i) (j), t i j :=
iInter_congr fun i => iInter_congr <| h i
#align set.Inter₂_congr Set.iInter₂_congr
section Nonempty
variable [Nonempty ι] {f : ι → Set α} {s : Set α}
lemma iUnion_const (s : Set β) : ⋃ _ : ι, s = s := iSup_const
#align set.Union_const Set.iUnion_const
lemma iInter_const (s : Set β) : ⋂ _ : ι, s = s := iInf_const
#align set.Inter_const Set.iInter_const
lemma iUnion_eq_const (hf : ∀ i, f i = s) : ⋃ i, f i = s :=
(iUnion_congr hf).trans $ iUnion_const _
#align set.Union_eq_const Set.iUnion_eq_const
lemma iInter_eq_const (hf : ∀ i, f i = s) : ⋂ i, f i = s :=
(iInter_congr hf).trans $ iInter_const _
#align set.Inter_eq_const Set.iInter_eq_const
end Nonempty
@[simp]
theorem compl_iUnion (s : ι → Set β) : (⋃ i, s i)ᶜ = ⋂ i, (s i)ᶜ :=
compl_iSup
#align set.compl_Union Set.compl_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iUnion₂ (s : ∀ i, κ i → Set α) : (⋃ (i) (j), s i j)ᶜ = ⋂ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iUnion]
#align set.compl_Union₂ Set.compl_iUnion₂
@[simp]
theorem compl_iInter (s : ι → Set β) : (⋂ i, s i)ᶜ = ⋃ i, (s i)ᶜ :=
compl_iInf
#align set.compl_Inter Set.compl_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iInter₂ (s : ∀ i, κ i → Set α) : (⋂ (i) (j), s i j)ᶜ = ⋃ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iInter]
#align set.compl_Inter₂ Set.compl_iInter₂
-- classical -- complete_boolean_algebra
theorem iUnion_eq_compl_iInter_compl (s : ι → Set β) : ⋃ i, s i = (⋂ i, (s i)ᶜ)ᶜ := by
simp only [compl_iInter, compl_compl]
#align set.Union_eq_compl_Inter_compl Set.iUnion_eq_compl_iInter_compl
-- classical -- complete_boolean_algebra
theorem iInter_eq_compl_iUnion_compl (s : ι → Set β) : ⋂ i, s i = (⋃ i, (s i)ᶜ)ᶜ := by
simp only [compl_iUnion, compl_compl]
#align set.Inter_eq_compl_Union_compl Set.iInter_eq_compl_iUnion_compl
theorem inter_iUnion (s : Set β) (t : ι → Set β) : (s ∩ ⋃ i, t i) = ⋃ i, s ∩ t i :=
inf_iSup_eq _ _
#align set.inter_Union Set.inter_iUnion
theorem iUnion_inter (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∩ s = ⋃ i, t i ∩ s :=
iSup_inf_eq _ _
#align set.Union_inter Set.iUnion_inter
theorem iUnion_union_distrib (s : ι → Set β) (t : ι → Set β) :
⋃ i, s i ∪ t i = (⋃ i, s i) ∪ ⋃ i, t i :=
iSup_sup_eq
#align set.Union_union_distrib Set.iUnion_union_distrib
theorem iInter_inter_distrib (s : ι → Set β) (t : ι → Set β) :
⋂ i, s i ∩ t i = (⋂ i, s i) ∩ ⋂ i, t i :=
iInf_inf_eq
#align set.Inter_inter_distrib Set.iInter_inter_distrib
theorem union_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∪ ⋃ i, t i) = ⋃ i, s ∪ t i :=
sup_iSup
#align set.union_Union Set.union_iUnion
theorem iUnion_union [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∪ s = ⋃ i, t i ∪ s :=
iSup_sup
#align set.Union_union Set.iUnion_union
theorem inter_iInter [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∩ ⋂ i, t i) = ⋂ i, s ∩ t i :=
inf_iInf
#align set.inter_Inter Set.inter_iInter
theorem iInter_inter [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋂ i, t i) ∩ s = ⋂ i, t i ∩ s :=
iInf_inf
#align set.Inter_inter Set.iInter_inter
-- classical
theorem union_iInter (s : Set β) (t : ι → Set β) : (s ∪ ⋂ i, t i) = ⋂ i, s ∪ t i :=
sup_iInf_eq _ _
#align set.union_Inter Set.union_iInter
theorem iInter_union (s : ι → Set β) (t : Set β) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.Inter_union Set.iInter_union
theorem iUnion_diff (s : Set β) (t : ι → Set β) : (⋃ i, t i) \ s = ⋃ i, t i \ s :=
iUnion_inter _ _
#align set.Union_diff Set.iUnion_diff
theorem diff_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s \ ⋃ i, t i) = ⋂ i, s \ t i := by
rw [diff_eq, compl_iUnion, inter_iInter]; rfl
#align set.diff_Union Set.diff_iUnion
theorem diff_iInter (s : Set β) (t : ι → Set β) : (s \ ⋂ i, t i) = ⋃ i, s \ t i := by
rw [diff_eq, compl_iInter, inter_iUnion]; rfl
#align set.diff_Inter Set.diff_iInter
theorem directed_on_iUnion {r} {f : ι → Set α} (hd : Directed (· ⊆ ·) f)
(h : ∀ x, DirectedOn r (f x)) : DirectedOn r (⋃ x, f x) := by
simp only [DirectedOn, exists_prop, mem_iUnion, exists_imp]
exact fun a₁ b₁ fb₁ a₂ b₂ fb₂ =>
let ⟨z, zb₁, zb₂⟩ := hd b₁ b₂
let ⟨x, xf, xa₁, xa₂⟩ := h z a₁ (zb₁ fb₁) a₂ (zb₂ fb₂)
⟨x, ⟨z, xf⟩, xa₁, xa₂⟩
#align set.directed_on_Union Set.directed_on_iUnion
theorem iUnion_inter_subset {ι α} {s t : ι → Set α} : ⋃ i, s i ∩ t i ⊆ (⋃ i, s i) ∩ ⋃ i, t i :=
le_iSup_inf_iSup s t
#align set.Union_inter_subset Set.iUnion_inter_subset
theorem iUnion_inter_of_monotone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_monotone hs ht
#align set.Union_inter_of_monotone Set.iUnion_inter_of_monotone
theorem iUnion_inter_of_antitone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_antitone hs ht
#align set.Union_inter_of_antitone Set.iUnion_inter_of_antitone
theorem iInter_union_of_monotone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_monotone hs ht
#align set.Inter_union_of_monotone Set.iInter_union_of_monotone
theorem iInter_union_of_antitone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_antitone hs ht
#align set.Inter_union_of_antitone Set.iInter_union_of_antitone
/-- An equality version of this lemma is `iUnion_iInter_of_monotone` in `Data.Set.Finite`. -/
theorem iUnion_iInter_subset {s : ι → ι' → Set α} : (⋃ j, ⋂ i, s i j) ⊆ ⋂ i, ⋃ j, s i j :=
iSup_iInf_le_iInf_iSup (flip s)
#align set.Union_Inter_subset Set.iUnion_iInter_subset
theorem iUnion_option {ι} (s : Option ι → Set α) : ⋃ o, s o = s none ∪ ⋃ i, s (some i) :=
iSup_option s
#align set.Union_option Set.iUnion_option
theorem iInter_option {ι} (s : Option ι → Set α) : ⋂ o, s o = s none ∩ ⋂ i, s (some i) :=
iInf_option s
#align set.Inter_option Set.iInter_option
section
variable (p : ι → Prop) [DecidablePred p]
theorem iUnion_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 _ _ _
#align set.Union_dite Set.iUnion_dite
theorem iUnion_ite (f g : ι → Set α) :
⋃ i, (if p i then f i else g i) = (⋃ (i) (_ : p i), f i) ∪ ⋃ (i) (_ : ¬p i), g i :=
iUnion_dite _ _ _
#align set.Union_ite Set.iUnion_ite
theorem iInter_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 :=
iInf_dite _ _ _
#align set.Inter_dite Set.iInter_dite
theorem iInter_ite (f g : ι → Set α) :
⋂ i, (if p i then f i else g i) = (⋂ (i) (_ : p i), f i) ∩ ⋂ (i) (_ : ¬p i), g i :=
iInter_dite _ _ _
#align set.Inter_ite Set.iInter_ite
end
theorem image_projection_prod {ι : Type*} {α : ι → Type*} {v : ∀ i : ι, Set (α i)}
(hv : (pi univ v).Nonempty) (i : ι) :
((fun x : ∀ i : ι, α i => x i) '' ⋂ k, (fun x : ∀ j : ι, α j => x k) ⁻¹' v k) = v i := by
classical
apply Subset.antisymm
· simp [iInter_subset]
· intro y y_in
simp only [mem_image, mem_iInter, mem_preimage]
rcases hv with ⟨z, hz⟩
refine' ⟨Function.update z i y, _, update_same i y z⟩
rw [@forall_update_iff ι α _ z i y fun i t => t ∈ v i]
exact ⟨y_in, fun j _ => by simpa using hz j⟩
#align set.image_projection_prod Set.image_projection_prod
/-! ### Unions and intersections indexed by `Prop` -/
theorem iInter_false {s : False → Set α} : iInter s = univ :=
iInf_false
#align set.Inter_false Set.iInter_false
theorem iUnion_false {s : False → Set α} : iUnion s = ∅ :=
iSup_false
#align set.Union_false Set.iUnion_false
@[simp]
theorem iInter_true {s : True → Set α} : iInter s = s trivial :=
iInf_true
#align set.Inter_true Set.iInter_true
@[simp]
theorem iUnion_true {s : True → Set α} : iUnion s = s trivial :=
iSup_true
#align set.Union_true Set.iUnion_true
@[simp]
theorem iInter_exists {p : ι → Prop} {f : Exists p → Set α} :
⋂ x, f x = ⋂ (i) (h : p i), f ⟨i, h⟩ :=
iInf_exists
#align set.Inter_exists Set.iInter_exists
@[simp]
theorem iUnion_exists {p : ι → Prop} {f : Exists p → Set α} :
⋃ x, f x = ⋃ (i) (h : p i), f ⟨i, h⟩ :=
iSup_exists
#align set.Union_exists Set.iUnion_exists
@[simp]
theorem iUnion_empty : (⋃ _ : ι, ∅ : Set α) = ∅ :=
iSup_bot
#align set.Union_empty Set.iUnion_empty
@[simp]
theorem iInter_univ : (⋂ _ : ι, univ : Set α) = univ :=
iInf_top
#align set.Inter_univ Set.iInter_univ
section
variable {s : ι → Set α}
@[simp]
theorem iUnion_eq_empty : ⋃ i, s i = ∅ ↔ ∀ i, s i = ∅ :=
iSup_eq_bot
#align set.Union_eq_empty Set.iUnion_eq_empty
@[simp]
theorem iInter_eq_univ : ⋂ i, s i = univ ↔ ∀ i, s i = univ :=
iInf_eq_top
#align set.Inter_eq_univ Set.iInter_eq_univ
@[simp]
theorem nonempty_iUnion : (⋃ i, s i).Nonempty ↔ ∃ i, (s i).Nonempty := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_Union Set.nonempty_iUnion
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_biUnion {t : Set α} {s : α → Set β} :
(⋃ i ∈ t, s i).Nonempty ↔ ∃ i ∈ t, (s i).Nonempty := by simp
#align set.nonempty_bUnion Set.nonempty_biUnion
theorem iUnion_nonempty_index (s : Set α) (t : s.Nonempty → Set β) :
⋃ h, t h = ⋃ x ∈ s, t ⟨x, ‹_›⟩ :=
iSup_exists
#align set.Union_nonempty_index Set.iUnion_nonempty_index
end
@[simp]
theorem iInter_iInter_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋂ (x) (h : x = b), s x h = s b rfl :=
iInf_iInf_eq_left
#align set.Inter_Inter_eq_left Set.iInter_iInter_eq_left
@[simp]
theorem iInter_iInter_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋂ (x) (h : b = x), s x h = s b rfl :=
iInf_iInf_eq_right
#align set.Inter_Inter_eq_right Set.iInter_iInter_eq_right
@[simp]
theorem iUnion_iUnion_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋃ (x) (h : x = b), s x h = s b rfl :=
iSup_iSup_eq_left
#align set.Union_Union_eq_left Set.iUnion_iUnion_eq_left
@[simp]
theorem iUnion_iUnion_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋃ (x) (h : b = x), s x h = s b rfl :=
iSup_iSup_eq_right
#align set.Union_Union_eq_right Set.iUnion_iUnion_eq_right
theorem iInter_or {p q : Prop} (s : p ∨ q → Set α) :
⋂ h, s h = (⋂ h : p, s (Or.inl h)) ∩ ⋂ h : q, s (Or.inr h) :=
iInf_or
#align set.Inter_or Set.iInter_or
theorem iUnion_or {p q : Prop} (s : p ∨ q → Set α) :
⋃ h, s h = (⋃ i, s (Or.inl i)) ∪ ⋃ j, s (Or.inr j) :=
iSup_or
#align set.Union_or Set.iUnion_or
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iUnion_and {p q : Prop} (s : p ∧ q → Set α) : ⋃ h, s h = ⋃ (hp) (hq), s ⟨hp, hq⟩ :=
iSup_and
#align set.Union_and Set.iUnion_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iInter_and {p q : Prop} (s : p ∧ q → Set α) : ⋂ h, s h = ⋂ (hp) (hq), s ⟨hp, hq⟩ :=
iInf_and
#align set.Inter_and Set.iInter_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iUnion_comm (s : ι → ι' → Set α) : ⋃ (i) (i'), s i i' = ⋃ (i') (i), s i i' :=
iSup_comm
#align set.Union_comm Set.iUnion_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iInter_comm (s : ι → ι' → Set α) : ⋂ (i) (i'), s i i' = ⋂ (i') (i), s i i' :=
iInf_comm
#align set.Inter_comm Set.iInter_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iUnion₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋃ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋃ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iSup₂_comm _
#align set.Union₂_comm Set.iUnion₂_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iInter₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋂ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋂ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iInf₂_comm _
#align set.Inter₂_comm Set.iInter₂_comm
@[simp]
theorem biUnion_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋃ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι']
#align set.bUnion_and Set.biUnion_and
@[simp]
theorem biUnion_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋃ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι]
#align set.bUnion_and' Set.biUnion_and'
@[simp]
theorem biInter_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋂ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iInter_and, @iInter_comm _ ι']
#align set.bInter_and Set.biInter_and
@[simp]
theorem biInter_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋂ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iInter_and, @iInter_comm _ ι]
#align set.bInter_and' Set.biInter_and'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iUnion_iUnion_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋃ (x) (h), s x h = s b (Or.inl rfl) ∪ ⋃ (x) (h : p x), s x (Or.inr h) := by
simp only [iUnion_or, iUnion_union_distrib, iUnion_iUnion_eq_left]
#align set.Union_Union_eq_or_left Set.iUnion_iUnion_eq_or_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iInter_iInter_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋂ (x) (h), s x h = s b (Or.inl rfl) ∩ ⋂ (x) (h : p x), s x (Or.inr h) := by
simp only [iInter_or, iInter_inter_distrib, iInter_iInter_eq_left]
#align set.Inter_Inter_eq_or_left Set.iInter_iInter_eq_or_left
/-! ### Bounded unions and intersections -/
/-- A specialization of `mem_iUnion₂`. -/
theorem mem_biUnion {s : Set α} {t : α → Set β} {x : α} {y : β} (xs : x ∈ s) (ytx : y ∈ t x) :
y ∈ ⋃ x ∈ s, t x :=
mem_iUnion₂_of_mem xs ytx
#align set.mem_bUnion Set.mem_biUnion
/-- A specialization of `mem_iInter₂`. -/
theorem mem_biInter {s : Set α} {t : α → Set β} {y : β} (h : ∀ x ∈ s, y ∈ t x) :
y ∈ ⋂ x ∈ s, t x :=
mem_iInter₂_of_mem h
#align set.mem_bInter Set.mem_biInter
/-- A specialization of `subset_iUnion₂`. -/
theorem subset_biUnion_of_mem {s : Set α} {u : α → Set β} {x : α} (xs : x ∈ s) :
u x ⊆ ⋃ x ∈ s, u x :=
--Porting note: Why is this not just `subset_iUnion₂ x xs`?
@subset_iUnion₂ β α (· ∈ s) (fun i _ => u i) x xs
#align set.subset_bUnion_of_mem Set.subset_biUnion_of_mem
/-- A specialization of `iInter₂_subset`. -/
theorem biInter_subset_of_mem {s : Set α} {t : α → Set β} {x : α} (xs : x ∈ s) :
⋂ x ∈ s, t x ⊆ t x :=
iInter₂_subset x xs
#align set.bInter_subset_of_mem Set.biInter_subset_of_mem
theorem biUnion_subset_biUnion_left {s s' : Set α} {t : α → Set β} (h : s ⊆ s') :
⋃ x ∈ s, t x ⊆ ⋃ x ∈ s', t x :=
iUnion₂_subset fun _ hx => subset_biUnion_of_mem <| h hx
#align set.bUnion_subset_bUnion_left Set.biUnion_subset_biUnion_left
theorem biInter_subset_biInter_left {s s' : Set α} {t : α → Set β} (h : s' ⊆ s) :
⋂ x ∈ s, t x ⊆ ⋂ x ∈ s', t x :=
subset_iInter₂ fun _ hx => biInter_subset_of_mem <| h hx
#align set.bInter_subset_bInter_left Set.biInter_subset_biInter_left
theorem biUnion_mono {s s' : Set α} {t t' : α → Set β} (hs : s' ⊆ s) (h : ∀ x ∈ s, t x ⊆ t' x) :
⋃ x ∈ s', t x ⊆ ⋃ x ∈ s, t' x :=
(biUnion_subset_biUnion_left hs).trans <| iUnion₂_mono h
#align set.bUnion_mono Set.biUnion_mono
theorem biInter_mono {s s' : Set α} {t t' : α → Set β} (hs : s ⊆ s') (h : ∀ x ∈ s, t x ⊆ t' x) :
⋂ x ∈ s', t x ⊆ ⋂ x ∈ s, t' x :=
(biInter_subset_biInter_left hs).trans <| iInter₂_mono h
#align set.bInter_mono Set.biInter_mono
theorem biUnion_eq_iUnion (s : Set α) (t : ∀ x ∈ s, Set β) :
⋃ x ∈ s, t x ‹_› = ⋃ x : s, t x x.2 :=
iSup_subtype'
#align set.bUnion_eq_Union Set.biUnion_eq_iUnion
theorem biInter_eq_iInter (s : Set α) (t : ∀ x ∈ s, Set β) :
⋂ x ∈ s, t x ‹_› = ⋂ x : s, t x x.2 :=
iInf_subtype'
#align set.bInter_eq_Inter Set.biInter_eq_iInter
theorem iUnion_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋃ x : { x // p x }, s x = ⋃ (x) (hx : p x), s ⟨x, hx⟩ :=
iSup_subtype
#align set.Union_subtype Set.iUnion_subtype
theorem iInter_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋂ x : { x // p x }, s x = ⋂ (x) (hx : p x), s ⟨x, hx⟩ :=
iInf_subtype
#align set.Inter_subtype Set.iInter_subtype
theorem biInter_empty (u : α → Set β) : ⋂ x ∈ (∅ : Set α), u x = univ :=
iInf_emptyset
#align set.bInter_empty Set.biInter_empty
theorem biInter_univ (u : α → Set β) : ⋂ x ∈ @univ α, u x = ⋂ x, u x :=
iInf_univ
#align set.bInter_univ Set.biInter_univ
@[simp]
theorem biUnion_self (s : Set α) : ⋃ x ∈ s, s = s :=
Subset.antisymm (iUnion₂_subset fun _ _ => Subset.refl s) fun _ hx => mem_biUnion hx hx
#align set.bUnion_self Set.biUnion_self
@[simp]
theorem iUnion_nonempty_self (s : Set α) : ⋃ _ : s.Nonempty, s = s := by
rw [iUnion_nonempty_index, biUnion_self]
#align set.Union_nonempty_self Set.iUnion_nonempty_self
theorem biInter_singleton (a : α) (s : α → Set β) : ⋂ x ∈ ({a} : Set α), s x = s a :=
iInf_singleton
#align set.bInter_singleton Set.biInter_singleton
theorem biInter_union (s t : Set α) (u : α → Set β) :
⋂ x ∈ s ∪ t, u x = (⋂ x ∈ s, u x) ∩ ⋂ x ∈ t, u x :=
iInf_union
#align set.bInter_union Set.biInter_union
theorem biInter_insert (a : α) (s : Set α) (t : α → Set β) :
⋂ x ∈ insert a s, t x = t a ∩ ⋂ x ∈ s, t x := by simp
#align set.bInter_insert Set.biInter_insert
theorem biInter_pair (a b : α) (s : α → Set β) : ⋂ x ∈ ({a, b} : Set α), s x = s a ∩ s b := by
rw [biInter_insert, biInter_singleton]
#align set.bInter_pair Set.biInter_pair
theorem biInter_inter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, f i ∩ t = (⋂ i ∈ s, f i) ∩ t := by
haveI : Nonempty s := hs.to_subtype
simp [biInter_eq_iInter, ← iInter_inter]
#align set.bInter_inter Set.biInter_inter
theorem inter_biInter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, t ∩ f i = t ∩ ⋂ i ∈ s, f i := by
rw [inter_comm, ← biInter_inter hs]
simp [inter_comm]
#align set.inter_bInter Set.inter_biInter
theorem biUnion_empty (s : α → Set β) : ⋃ x ∈ (∅ : Set α), s x = ∅ :=
iSup_emptyset
#align set.bUnion_empty Set.biUnion_empty
theorem biUnion_univ (s : α → Set β) : ⋃ x ∈ @univ α, s x = ⋃ x, s x :=
iSup_univ
#align set.bUnion_univ Set.biUnion_univ
theorem biUnion_singleton (a : α) (s : α → Set β) : ⋃ x ∈ ({a} : Set α), s x = s a :=
iSup_singleton
#align set.bUnion_singleton Set.biUnion_singleton
@[simp]
theorem biUnion_of_singleton (s : Set α) : ⋃ x ∈ s, {x} = s :=
ext <| by simp
#align set.bUnion_of_singleton Set.biUnion_of_singleton
theorem biUnion_union (s t : Set α) (u : α → Set β) :
⋃ x ∈ s ∪ t, u x = (⋃ x ∈ s, u x) ∪ ⋃ x ∈ t, u x :=
iSup_union
#align set.bUnion_union Set.biUnion_union
@[simp]
theorem iUnion_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋃ i, f i = ⋃ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iUnion_subtype _ _
#align set.Union_coe_set Set.iUnion_coe_set
@[simp]
theorem iInter_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋂ i, f i = ⋂ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iInter_subtype _ _
#align set.Inter_coe_set Set.iInter_coe_set
theorem biUnion_insert (a : α) (s : Set α) (t : α → Set β) :
⋃ x ∈ insert a s, t x = t a ∪ ⋃ x ∈ s, t x := by simp
#align set.bUnion_insert Set.biUnion_insert
theorem biUnion_pair (a b : α) (s : α → Set β) : ⋃ x ∈ ({a, b} : Set α), s x = s a ∪ s b :=
by simp
#align set.bUnion_pair Set.biUnion_pair
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem inter_iUnion₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∩ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ∩ t i j := by simp only [inter_iUnion]
#align set.inter_Union₂ Set.inter_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_inter (s : ∀ i, κ i → Set α) (t : Set α) :
(⋃ (i) (j), s i j) ∩ t = ⋃ (i) (j), s i j ∩ t := by simp_rw [iUnion_inter]
#align set.Union₂_inter Set.iUnion₂_inter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_iInter₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_iInter]
#align set.union_Inter₂ Set.union_iInter₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_union (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [iInter_union]
#align set.Inter₂_union Set.iInter₂_union
theorem mem_sUnion_of_mem {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∈ t) (ht : t ∈ S) :
x ∈ ⋃₀S :=
⟨t, ht, hx⟩
#align set.mem_sUnion_of_mem Set.mem_sUnion_of_mem
-- is this theorem really necessary?
theorem not_mem_of_not_mem_sUnion {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∉ ⋃₀S)
(ht : t ∈ S) : x ∉ t := fun h => hx ⟨t, ht, h⟩
#align set.not_mem_of_not_mem_sUnion Set.not_mem_of_not_mem_sUnion
theorem sInter_subset_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : ⋂₀ S ⊆ t :=
sInf_le tS
#align set.sInter_subset_of_mem Set.sInter_subset_of_mem
theorem subset_sUnion_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : t ⊆ ⋃₀S :=
le_sSup tS
#align set.subset_sUnion_of_mem Set.subset_sUnion_of_mem
theorem subset_sUnion_of_subset {s : Set α} (t : Set (Set α)) (u : Set α) (h₁ : s ⊆ u)
(h₂ : u ∈ t) : s ⊆ ⋃₀t :=
Subset.trans h₁ (subset_sUnion_of_mem h₂)
#align set.subset_sUnion_of_subset Set.subset_sUnion_of_subset
theorem sUnion_subset {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t' ⊆ t) : ⋃₀S ⊆ t :=
sSup_le h
#align set.sUnion_subset Set.sUnion_subset
@[simp]
theorem sUnion_subset_iff {s : Set (Set α)} {t : Set α} : ⋃₀s ⊆ t ↔ ∀ t' ∈ s, t' ⊆ t :=
sSup_le_iff
#align set.sUnion_subset_iff Set.sUnion_subset_iff
/-- `sUnion` is monotone under taking a subset of each set. -/
lemma sUnion_mono_subsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, t ⊆ f t) :
⋃₀ s ⊆ ⋃₀ (f '' s) :=
fun _ ⟨t, htx, hxt⟩ ↦ ⟨f t, mem_image_of_mem f htx, hf t hxt⟩
/-- `sUnion` is monotone under taking a superset of each set. -/
lemma sUnion_mono_supsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, f t ⊆ t) :
⋃₀ (f '' s) ⊆ ⋃₀ s :=
-- If t ∈ f '' s is arbitrary; t = f u for some u : Set α.
fun _ ⟨_, ⟨u, hus, hut⟩, hxt⟩ ↦ ⟨u, hus, (hut ▸ hf u) hxt⟩
theorem subset_sInter {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t ⊆ t') : t ⊆ ⋂₀ S :=
le_sInf h
#align set.subset_sInter Set.subset_sInter
@[simp]
theorem subset_sInter_iff {S : Set (Set α)} {t : Set α} : t ⊆ ⋂₀ S ↔ ∀ t' ∈ S, t ⊆ t' :=
le_sInf_iff
#align set.subset_sInter_iff Set.subset_sInter_iff
@[gcongr]
theorem sUnion_subset_sUnion {S T : Set (Set α)} (h : S ⊆ T) : ⋃₀S ⊆ ⋃₀T :=
sUnion_subset fun _ hs => subset_sUnion_of_mem (h hs)
#align set.sUnion_subset_sUnion Set.sUnion_subset_sUnion
@[gcongr]
theorem sInter_subset_sInter {S T : Set (Set α)} (h : S ⊆ T) : ⋂₀ T ⊆ ⋂₀ S :=
subset_sInter fun _ hs => sInter_subset_of_mem (h hs)
#align set.sInter_subset_sInter Set.sInter_subset_sInter
@[simp]
theorem sUnion_empty : ⋃₀∅ = (∅ : Set α) :=
sSup_empty
#align set.sUnion_empty Set.sUnion_empty
@[simp]
theorem sInter_empty : ⋂₀ ∅ = (univ : Set α) :=
sInf_empty
#align set.sInter_empty Set.sInter_empty
@[simp]
theorem sUnion_singleton (s : Set α) : ⋃₀{s} = s :=
sSup_singleton
#align set.sUnion_singleton Set.sUnion_singleton
@[simp]
theorem sInter_singleton (s : Set α) : ⋂₀ {s} = s :=
sInf_singleton
#align set.sInter_singleton Set.sInter_singleton
@[simp]
theorem sUnion_eq_empty {S : Set (Set α)} : ⋃₀S = ∅ ↔ ∀ s ∈ S, s = ∅ :=
sSup_eq_bot
#align set.sUnion_eq_empty Set.sUnion_eq_empty
@[simp]
theorem sInter_eq_univ {S : Set (Set α)} : ⋂₀ S = univ ↔ ∀ s ∈ S, s = univ :=
sInf_eq_top
#align set.sInter_eq_univ Set.sInter_eq_univ
theorem subset_powerset_iff {s : Set (Set α)} {t : Set α} : s ⊆ 𝒫 t ↔ ⋃₀ s ⊆ t :=
sUnion_subset_iff.symm
/-- `⋃₀` and `𝒫` form a Galois connection. -/
theorem sUnion_powerset_gc :
GaloisConnection (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gc_sSup_Iic
/-- `⋃₀` and `𝒫` form a Galois insertion. -/
def sUnion_powerset_gi :
GaloisInsertion (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gi_sSup_Iic
/-- If all sets in a collection are either `∅` or `Set.univ`, then so is their union. -/
theorem sUnion_mem_empty_univ {S : Set (Set α)} (h : S ⊆ {∅, univ}) :
⋃₀ S ∈ ({∅, univ} : Set (Set α)) := by
simp only [mem_insert_iff, mem_singleton_iff, or_iff_not_imp_left, sUnion_eq_empty, not_forall]
rintro ⟨s, hs, hne⟩
obtain rfl : s = univ := (h hs).resolve_left hne
exact univ_subset_iff.1 <| subset_sUnion_of_mem hs
@[simp]
theorem nonempty_sUnion {S : Set (Set α)} : (⋃₀S).Nonempty ↔ ∃ s ∈ S, Set.Nonempty s := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_sUnion Set.nonempty_sUnion
theorem Nonempty.of_sUnion {s : Set (Set α)} (h : (⋃₀s).Nonempty) : s.Nonempty :=
let ⟨s, hs, _⟩ := nonempty_sUnion.1 h
⟨s, hs⟩
#align set.nonempty.of_sUnion Set.Nonempty.of_sUnion
theorem Nonempty.of_sUnion_eq_univ [Nonempty α] {s : Set (Set α)} (h : ⋃₀s = univ) : s.Nonempty :=
Nonempty.of_sUnion <| h.symm ▸ univ_nonempty
#align set.nonempty.of_sUnion_eq_univ Set.Nonempty.of_sUnion_eq_univ
theorem sUnion_union (S T : Set (Set α)) : ⋃₀(S ∪ T) = ⋃₀S ∪ ⋃₀T :=
sSup_union
#align set.sUnion_union Set.sUnion_union
theorem sInter_union (S T : Set (Set α)) : ⋂₀ (S ∪ T) = ⋂₀ S ∩ ⋂₀ T :=
sInf_union
#align set.sInter_union Set.sInter_union
@[simp]
theorem sUnion_insert (s : Set α) (T : Set (Set α)) : ⋃₀insert s T = s ∪ ⋃₀T :=
sSup_insert
#align set.sUnion_insert Set.sUnion_insert
@[simp]
theorem sInter_insert (s : Set α) (T : Set (Set α)) : ⋂₀ insert s T = s ∩ ⋂₀ T :=
sInf_insert
#align set.sInter_insert Set.sInter_insert
@[simp]
theorem sUnion_diff_singleton_empty (s : Set (Set α)) : ⋃₀(s \ {∅}) = ⋃₀s :=
sSup_diff_singleton_bot s
#align set.sUnion_diff_singleton_empty Set.sUnion_diff_singleton_empty
@[simp]
theorem sInter_diff_singleton_univ (s : Set (Set α)) : ⋂₀ (s \ {univ}) = ⋂₀ s :=
sInf_diff_singleton_top s
#align set.sInter_diff_singleton_univ Set.sInter_diff_singleton_univ
theorem sUnion_pair (s t : Set α) : ⋃₀{s, t} = s ∪ t :=
sSup_pair
#align set.sUnion_pair Set.sUnion_pair
theorem sInter_pair (s t : Set α) : ⋂₀ {s, t} = s ∩ t :=
sInf_pair
#align set.sInter_pair Set.sInter_pair
@[simp]
theorem sUnion_image (f : α → Set β) (s : Set α) : ⋃₀(f '' s) = ⋃ x ∈ s, f x :=
sSup_image
#align set.sUnion_image Set.sUnion_image
@[simp]
theorem sInter_image (f : α → Set β) (s : Set α) : ⋂₀ (f '' s) = ⋂ x ∈ s, f x :=
sInf_image
#align set.sInter_image Set.sInter_image
@[simp]
theorem sUnion_range (f : ι → Set β) : ⋃₀range f = ⋃ x, f x :=
rfl
#align set.sUnion_range Set.sUnion_range
@[simp]
theorem sInter_range (f : ι → Set β) : ⋂₀ range f = ⋂ x, f x :=
rfl
#align set.sInter_range Set.sInter_range
theorem iUnion_eq_univ_iff {f : ι → Set α} : ⋃ i, f i = univ ↔ ∀ x, ∃ i, x ∈ f i := by
simp only [eq_univ_iff_forall, mem_iUnion]
#align set.Union_eq_univ_iff Set.iUnion_eq_univ_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_eq_univ_iff {s : ∀ i, κ i → Set α} :
⋃ (i) (j), s i j = univ ↔ ∀ a, ∃ i j, a ∈ s i j :=
by simp only [iUnion_eq_univ_iff, mem_iUnion]
#align set.Union₂_eq_univ_iff Set.iUnion₂_eq_univ_iff
theorem sUnion_eq_univ_iff {c : Set (Set α)} : ⋃₀c = univ ↔ ∀ a, ∃ b ∈ c, a ∈ b := by
simp only [eq_univ_iff_forall, mem_sUnion]
#align set.sUnion_eq_univ_iff Set.sUnion_eq_univ_iff
-- classical
theorem iInter_eq_empty_iff {f : ι → Set α} : ⋂ i, f i = ∅ ↔ ∀ x, ∃ i, x ∉ f i := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.Inter_eq_empty_iff Set.iInter_eq_empty_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
theorem iInter₂_eq_empty_iff {s : ∀ i, κ i → Set α} :
⋂ (i) (j), s i j = ∅ ↔ ∀ a, ∃ i j, a ∉ s i j := by
simp only [eq_empty_iff_forall_not_mem, mem_iInter, not_forall]
#align set.Inter₂_eq_empty_iff Set.iInter₂_eq_empty_iff
-- classical
theorem sInter_eq_empty_iff {c : Set (Set α)} : ⋂₀ c = ∅ ↔ ∀ a, ∃ b ∈ c, a ∉ b := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.sInter_eq_empty_iff Set.sInter_eq_empty_iff
-- classical
@[simp]
theorem nonempty_iInter {f : ι → Set α} : (⋂ i, f i).Nonempty ↔ ∃ x, ∀ i, x ∈ f i := by
simp [nonempty_iff_ne_empty, iInter_eq_empty_iff]
#align set.nonempty_Inter Set.nonempty_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_iInter₂ {s : ∀ i, κ i → Set α} :
(⋂ (i) (j), s i j).Nonempty ↔ ∃ a, ∀ i j, a ∈ s i j := by
simp
#align set.nonempty_Inter₂ Set.nonempty_iInter₂
-- classical
@[simp]
theorem nonempty_sInter {c : Set (Set α)} : (⋂₀ c).Nonempty ↔ ∃ a, ∀ b ∈ c, a ∈ b := by
simp [nonempty_iff_ne_empty, sInter_eq_empty_iff]
#align set.nonempty_sInter Set.nonempty_sInter
-- classical
theorem compl_sUnion (S : Set (Set α)) : (⋃₀S)ᶜ = ⋂₀ (compl '' S) :=
ext fun x => by simp
#align set.compl_sUnion Set.compl_sUnion
-- classical
theorem sUnion_eq_compl_sInter_compl (S : Set (Set α)) : ⋃₀S = (⋂₀ (compl '' S))ᶜ := by
rw [← compl_compl (⋃₀S), compl_sUnion]
#align set.sUnion_eq_compl_sInter_compl Set.sUnion_eq_compl_sInter_compl
-- classical
theorem compl_sInter (S : Set (Set α)) : (⋂₀ S)ᶜ = ⋃₀(compl '' S) := by
rw [sUnion_eq_compl_sInter_compl, compl_compl_image]
#align set.compl_sInter Set.compl_sInter
-- classical
theorem sInter_eq_compl_sUnion_compl (S : Set (Set α)) : ⋂₀ S = (⋃₀(compl '' S))ᶜ := by
rw [← compl_compl (⋂₀ S), compl_sInter]
#align set.sInter_eq_compl_sUnion_compl Set.sInter_eq_compl_sUnion_compl
theorem inter_empty_of_inter_sUnion_empty {s t : Set α} {S : Set (Set α)} (hs : t ∈ S)
(h : s ∩ ⋃₀S = ∅) : s ∩ t = ∅ :=
eq_empty_of_subset_empty <| by
rw [← h]; exact inter_subset_inter_right _ (subset_sUnion_of_mem hs)
#align set.inter_empty_of_inter_sUnion_empty Set.inter_empty_of_inter_sUnion_empty
theorem range_sigma_eq_iUnion_range {γ : α → Type*} (f : Sigma γ → β) :
range f = ⋃ a, range fun b => f ⟨a, b⟩ :=
Set.ext <| by simp
#align set.range_sigma_eq_Union_range Set.range_sigma_eq_iUnion_range
theorem iUnion_eq_range_sigma (s : α → Set β) : ⋃ i, s i = range fun a : Σi, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_sigma Set.iUnion_eq_range_sigma
theorem iUnion_eq_range_psigma (s : ι → Set β) : ⋃ i, s i = range fun a : Σ'i, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_psigma Set.iUnion_eq_range_psigma
theorem iUnion_image_preimage_sigma_mk_eq_self {ι : Type*} {σ : ι → Type*} (s : Set (Sigma σ)) :
⋃ i, Sigma.mk i '' (Sigma.mk i ⁻¹' s) = s := by
ext x
simp only [mem_iUnion, mem_image, mem_preimage]
constructor
· rintro ⟨i, a, h, rfl⟩
exact h
· intro h
cases' x with i a
exact ⟨i, a, h, rfl⟩
#align set.Union_image_preimage_sigma_mk_eq_self Set.iUnion_image_preimage_sigma_mk_eq_self
theorem Sigma.univ (X : α → Type*) : (Set.univ : Set (Σa, X a)) = ⋃ a, range (Sigma.mk a) :=
Set.ext fun x =>
iff_of_true trivial ⟨range (Sigma.mk x.1), Set.mem_range_self _, x.2, Sigma.eta x⟩
#align set.sigma.univ Set.Sigma.univ
alias sUnion_mono := sUnion_subset_sUnion
#align set.sUnion_mono Set.sUnion_mono
theorem iUnion_subset_iUnion_const {s : Set α} (h : ι → ι₂) : ⋃ _ : ι, s ⊆ ⋃ _ : ι₂, s :=
@iSup_const_mono (Set α) ι ι₂ _ s h
#align set.Union_subset_Union_const Set.iUnion_subset_iUnion_const
@[simp]
theorem iUnion_singleton_eq_range {α β : Type*} (f : α → β) : ⋃ x : α, {f x} = range f := by
ext x
simp [@eq_comm _ x]
#align set.Union_singleton_eq_range Set.iUnion_singleton_eq_range
theorem iUnion_of_singleton (α : Type*) : (⋃ x, {x} : Set α) = univ := by simp [Set.ext_iff]
#align set.Union_of_singleton Set.iUnion_of_singleton
theorem iUnion_of_singleton_coe (s : Set α) : ⋃ i : s, ({(i : α)} : Set α) = s := by simp
#align set.Union_of_singleton_coe Set.iUnion_of_singleton_coe
theorem sUnion_eq_biUnion {s : Set (Set α)} : ⋃₀s = ⋃ (i : Set α) (_ : i ∈ s), i := by
rw [← sUnion_image, image_id']
#align set.sUnion_eq_bUnion Set.sUnion_eq_biUnion
theorem sInter_eq_biInter {s : Set (Set α)} : ⋂₀ s = ⋂ (i : Set α) (_ : i ∈ s), i := by
rw [← sInter_image, image_id']
#align set.sInter_eq_bInter Set.sInter_eq_biInter
theorem sUnion_eq_iUnion {s : Set (Set α)} : ⋃₀s = ⋃ i : s, i := by
simp only [← sUnion_range, Subtype.range_coe]
#align set.sUnion_eq_Union Set.sUnion_eq_iUnion
theorem sInter_eq_iInter {s : Set (Set α)} : ⋂₀ s = ⋂ i : s, i := by
simp only [← sInter_range, Subtype.range_coe]
#align set.sInter_eq_Inter Set.sInter_eq_iInter
@[simp]
theorem iUnion_of_empty [IsEmpty ι] (s : ι → Set α) : ⋃ i, s i = ∅ :=
iSup_of_empty _
#align set.Union_of_empty Set.iUnion_of_empty
@[simp]
theorem iInter_of_empty [IsEmpty ι] (s : ι → Set α) : ⋂ i, s i = univ :=
iInf_of_empty _
#align set.Inter_of_empty Set.iInter_of_empty
theorem union_eq_iUnion {s₁ s₂ : Set α} : s₁ ∪ s₂ = ⋃ b : Bool, cond b s₁ s₂ :=
sup_eq_iSup s₁ s₂
#align set.union_eq_Union Set.union_eq_iUnion
theorem inter_eq_iInter {s₁ s₂ : Set α} : s₁ ∩ s₂ = ⋂ b : Bool, cond b s₁ s₂ :=
inf_eq_iInf s₁ s₂
#align set.inter_eq_Inter Set.inter_eq_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_union_sInter {S T : Set (Set α)} :
⋂₀ S ∪ ⋂₀ T = ⋂ p ∈ S ×ˢ T, (p : Set α × Set α).1 ∪ p.2 :=
sInf_sup_sInf
#align set.sInter_union_sInter Set.sInter_union_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_inter_sUnion {s t : Set (Set α)} :
⋃₀s ∩ ⋃₀t = ⋃ p ∈ s ×ˢ t, (p : Set α × Set α).1 ∩ p.2 :=
sSup_inf_sSup
#align set.sUnion_inter_sUnion Set.sUnion_inter_sUnion
theorem biUnion_iUnion (s : ι → Set α) (t : α → Set β) :
⋃ x ∈ ⋃ i, s i, t x = ⋃ (i) (x ∈ s i), t x := by simp [@iUnion_comm _ ι]
#align set.bUnion_Union Set.biUnion_iUnion
theorem biInter_iUnion (s : ι → Set α) (t : α → Set β) :
⋂ x ∈ ⋃ i, s i, t x = ⋂ (i) (x ∈ s i), t x := by simp [@iInter_comm _ ι]
#align set.bInter_Union Set.biInter_iUnion
theorem sUnion_iUnion (s : ι → Set (Set α)) : ⋃₀⋃ i, s i = ⋃ i, ⋃₀s i := by
simp only [sUnion_eq_biUnion, biUnion_iUnion]
#align set.sUnion_Union Set.sUnion_iUnion
theorem sInter_iUnion (s : ι → Set (Set α)) : ⋂₀ ⋃ i, s i = ⋂ i, ⋂₀ s i := by
simp only [sInter_eq_biInter, biInter_iUnion]
#align set.sInter_Union Set.sInter_iUnion
theorem iUnion_range_eq_sUnion {α β : Type*} (C : Set (Set α)) {f : ∀ s : C, β → (s : Type _)}
(hf : ∀ s : C, Surjective (f s)) : ⋃ y : β, range (fun s : C => (f s y).val) = ⋃₀C := by
ext x; constructor
· rintro ⟨s, ⟨y, rfl⟩, ⟨s, hs⟩, rfl⟩
refine' ⟨_, hs, _⟩
exact (f ⟨s, hs⟩ y).2
· rintro ⟨s, hs, hx⟩
cases' hf ⟨s, hs⟩ ⟨x, hx⟩ with y hy
refine' ⟨_, ⟨y, rfl⟩, ⟨s, hs⟩, _⟩
exact congr_arg Subtype.val hy
#align set.Union_range_eq_sUnion Set.iUnion_range_eq_sUnion
theorem iUnion_range_eq_iUnion (C : ι → Set α) {f : ∀ x : ι, β → C x}
(hf : ∀ x : ι, Surjective (f x)) : ⋃ y : β, range (fun x : ι => (f x y).val) = ⋃ x, C x := by
ext x; rw [mem_iUnion, mem_iUnion]; constructor
· rintro ⟨y, i, rfl⟩
exact ⟨i, (f i y).2⟩
· rintro ⟨i, hx⟩
cases' hf i ⟨x, hx⟩ with y hy
exact ⟨y, i, congr_arg Subtype.val hy⟩
#align set.Union_range_eq_Union Set.iUnion_range_eq_iUnion
theorem union_distrib_iInter_left (s : ι → Set α) (t : Set α) : (t ∪ ⋂ i, s i) = ⋂ i, t ∪ s i :=
sup_iInf_eq _ _
#align set.union_distrib_Inter_left Set.union_distrib_iInter_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_left (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_distrib_iInter_left]
#align set.union_distrib_Inter₂_left Set.union_distrib_iInter₂_left
theorem union_distrib_iInter_right (s : ι → Set α) (t : Set α) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.union_distrib_Inter_right Set.union_distrib_iInter_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_right (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [union_distrib_iInter_right]
#align set.union_distrib_Inter₂_right Set.union_distrib_iInter₂_right
section Function
/-! ### `mapsTo` -/
theorem mapsTo_sUnion {S : Set (Set α)} {t : Set β} {f : α → β} (H : ∀ s ∈ S, MapsTo f s t) :
MapsTo f (⋃₀S) t := fun _ ⟨s, hs, hx⟩ => H s hs hx
#align set.maps_to_sUnion Set.mapsTo_sUnion
theorem mapsTo_iUnion {s : ι → Set α} {t : Set β} {f : α → β} (H : ∀ i, MapsTo f (s i) t) :
MapsTo f (⋃ i, s i) t :=
mapsTo_sUnion <| forall_range_iff.2 H
#align set.maps_to_Union Set.mapsTo_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂ {s : ∀ i, κ i → Set α} {t : Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) t) : MapsTo f (⋃ (i) (j), s i j) t :=
mapsTo_iUnion fun i => mapsTo_iUnion (H i)
#align set.maps_to_Union₂ Set.mapsTo_iUnion₂
theorem mapsTo_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋃ i, s i) (⋃ i, t i) :=
mapsTo_iUnion fun i => (H i).mono (Subset.refl _) (subset_iUnion t i)
#align set.maps_to_Union_Union Set.mapsTo_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
mapsTo_iUnion_iUnion fun i => mapsTo_iUnion_iUnion (H i)
#align set.maps_to_Union₂_Union₂ Set.mapsTo_iUnion₂_iUnion₂
theorem mapsTo_sInter {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, MapsTo f s t) :
MapsTo f s (⋂₀ T) := fun _ hx t ht => H t ht hx
#align set.maps_to_sInter Set.mapsTo_sInter
theorem mapsTo_iInter {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, MapsTo f s (t i)) :
MapsTo f s (⋂ i, t i) := fun _ hx => mem_iInter.2 fun i => H i hx
#align set.maps_to_Inter Set.mapsTo_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f s (t i j)) : MapsTo f s (⋂ (i) (j), t i j) :=
mapsTo_iInter fun i => mapsTo_iInter (H i)
#align set.maps_to_Inter₂ Set.mapsTo_iInter₂
theorem mapsTo_iInter_iInter {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋂ i, s i) (⋂ i, t i) :=
mapsTo_iInter fun i => (H i).mono (iInter_subset s i) (Subset.refl _)
#align set.maps_to_Inter_Inter Set.mapsTo_iInter_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂_iInter₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋂ (i) (j), s i j) (⋂ (i) (j), t i j) :=
mapsTo_iInter_iInter fun i => mapsTo_iInter_iInter (H i)
#align set.maps_to_Inter₂_Inter₂ Set.mapsTo_iInter₂_iInter₂
theorem image_iInter_subset (s : ι → Set α) (f : α → β) : (f '' ⋂ i, s i) ⊆ ⋂ i, f '' s i :=
(mapsTo_iInter_iInter fun i => mapsTo_image f (s i)).image_subset
#align set.image_Inter_subset Set.image_iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂_subset (s : ∀ i, κ i → Set α) (f : α → β) :
(f '' ⋂ (i) (j), s i j) ⊆ ⋂ (i) (j), f '' s i j :=
(mapsTo_iInter₂_iInter₂ fun i hi => mapsTo_image f (s i hi)).image_subset
#align set.image_Inter₂_subset Set.image_iInter₂_subset
theorem image_sInter_subset (S : Set (Set α)) (f : α → β) : f '' ⋂₀ S ⊆ ⋂ s ∈ S, f '' s := by
rw [sInter_eq_biInter]
apply image_iInter₂_subset
#align set.image_sInter_subset Set.image_sInter_subset
/-! ### `restrictPreimage` -/
section
open Function
variable (s : Set β) {f : α → β} {U : ι → Set β} (hU : iUnion U = univ)
theorem injective_iff_injective_of_iUnion_eq_univ :
Injective f ↔ ∀ i, Injective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_injective H, fun H x y e => _⟩
obtain ⟨i, hi⟩ := Set.mem_iUnion.mp
(show f x ∈ Set.iUnion U by rw [hU]; triv)
injection @H i ⟨x, hi⟩ ⟨y, show f y ∈ U i from e ▸ hi⟩ (Subtype.ext e)
#align set.injective_iff_injective_of_Union_eq_univ Set.injective_iff_injective_of_iUnion_eq_univ
theorem surjective_iff_surjective_of_iUnion_eq_univ :
Surjective f ↔ ∀ i, Surjective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_surjective H, fun H x => _⟩
obtain ⟨i, hi⟩ :=
Set.mem_iUnion.mp
(show x ∈ Set.iUnion U by rw [hU]; triv)
exact ⟨_, congr_arg Subtype.val (H i ⟨x, hi⟩).choose_spec⟩
#align set.surjective_iff_surjective_of_Union_eq_univ Set.surjective_iff_surjective_of_iUnion_eq_univ
theorem bijective_iff_bijective_of_iUnion_eq_univ :
Bijective f ↔ ∀ i, Bijective ((U i).restrictPreimage f) := by
rw [Bijective, injective_iff_injective_of_iUnion_eq_univ hU,
surjective_iff_surjective_of_iUnion_eq_univ hU]
simp [Bijective, forall_and]
#align set.bijective_iff_bijective_of_Union_eq_univ Set.bijective_iff_bijective_of_iUnion_eq_univ
end
/-! ### `InjOn` -/
theorem InjOn.image_iInter_eq [Nonempty ι] {s : ι → Set α} {f : α → β} (h : InjOn f (⋃ i, s i)) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
inhabit ι
refine' Subset.antisymm (image_iInter_subset s f) fun y hy => _
simp only [mem_iInter, mem_image_iff_bex] at hy
choose x hx hy using hy
refine' ⟨x default, mem_iInter.2 fun i => _, hy _⟩
suffices x default = x i by
rw [this]
apply hx
replace hx : ∀ i, x i ∈ ⋃ j, s j := fun i => (subset_iUnion _ _) (hx i)
apply h (hx _) (hx _)
simp only [hy]
#align set.inj_on.image_Inter_eq Set.InjOn.image_iInter_eq
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
theorem InjOn.image_biInter_eq {p : ι → Prop} {s : ∀ i, p i → Set α} (hp : ∃ i, p i)
{f : α → β} (h : InjOn f (⋃ (i) (hi), s i hi)) :
(f '' ⋂ (i) (hi), s i hi) = ⋂ (i) (hi), f '' s i hi := by
simp only [iInter, iInf_subtype']
haveI : Nonempty { i // p i } := nonempty_subtype.2 hp
apply InjOn.image_iInter_eq
simpa only [iUnion, iSup_subtype'] using h
#align set.inj_on.image_bInter_eq Set.InjOn.image_biInter_eq
theorem image_iInter {f : α → β} (hf : Bijective f) (s : ι → Set α) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
cases isEmpty_or_nonempty ι
· simp_rw [iInter_of_empty, image_univ_of_surjective hf.surjective]
· exact (hf.injective.injOn _).image_iInter_eq
#align set.image_Inter Set.image_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂ {f : α → β} (hf : Bijective f) (s : ∀ i, κ i → Set α) :
(f '' ⋂ (i) (j), s i j) = ⋂ (i) (j), f '' s i j := by simp_rw [image_iInter hf]
#align set.image_Inter₂ Set.image_iInter₂
theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
(hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i) := by
intro x hx y hy hxy
rcases mem_iUnion.1 hx with ⟨i, hx⟩
rcases mem_iUnion.1 hy with ⟨j, hy⟩
rcases hs i j with ⟨k, hi, hj⟩
exact hf k (hi hx) (hj hy) hxy
#align set.inj_on_Union_of_directed Set.inj_on_iUnion_of_directed
/-! ### `SurjOn` -/
theorem surjOn_sUnion {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, SurjOn f s t) :
SurjOn f s (⋃₀T) := fun _ ⟨t, ht, hx⟩ => H t ht hx
#align set.surj_on_sUnion Set.surjOn_sUnion
theorem surjOn_iUnion {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, SurjOn f s (t i)) :
SurjOn f s (⋃ i, t i) :=
surjOn_sUnion <| forall_range_iff.2 H
#align set.surj_on_Union Set.surjOn_iUnion
theorem surjOn_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) : SurjOn f (⋃ i, s i) (⋃ i, t i) :=
surjOn_iUnion fun i => (H i).mono (subset_iUnion _ _) (Subset.refl _)
#align set.surj_on_Union_Union Set.surjOn_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f s (t i j)) : SurjOn f s (⋃ (i) (j), t i j) :=
surjOn_iUnion fun i => surjOn_iUnion (H i)
#align set.surj_on_Union₂ Set.surjOn_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f (s i j) (t i j)) : SurjOn f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
surjOn_iUnion_iUnion fun i => surjOn_iUnion_iUnion (H i)
#align set.surj_on_Union₂_Union₂ Set.surjOn_iUnion₂_iUnion₂
theorem surjOn_iInter [Nonempty ι] {s : ι → Set α} {t : Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) t) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) t := by
intro y hy
rw [Hinj.image_iInter_eq, mem_iInter]
exact fun i => H i hy
#align set.surj_on_Inter Set.surjOn_iInter
theorem surjOn_iInter_iInter [Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) (⋂ i, t i) :=
surjOn_iInter (fun i => (H i).mono (Subset.refl _) (iInter_subset _ _)) Hinj
#align set.surj_on_Inter_Inter Set.surjOn_iInter_iInter
/-! ### `BijOn` -/
theorem bijOn_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i))
(Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
⟨mapsTo_iUnion_iUnion fun i => (H i).mapsTo, Hinj, surjOn_iUnion_iUnion fun i => (H i).surjOn⟩
#align set.bij_on_Union Set.bijOn_iUnion
theorem bijOn_iInter [hi : Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, BijOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
⟨mapsTo_iInter_iInter fun i => (H i).mapsTo,
hi.elim fun i => (H i).injOn.mono (iInter_subset _ _),
surjOn_iInter_iInter (fun i => (H i).surjOn) Hinj⟩
#align set.bij_on_Inter Set.bijOn_iInter
theorem bijOn_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {t : ι → Set β}
{f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
bijOn_iUnion H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Union_of_directed Set.bijOn_iUnion_of_directed
theorem bijOn_iInter_of_directed [Nonempty ι] {s : ι → Set α} (hs : Directed (· ⊆ ·) s)
{t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
bijOn_iInter H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Inter_of_directed Set.bijOn_iInter_of_directed
end Function
/-! ### `image`, `preimage` -/
section Image
theorem image_iUnion {f : α → β} {s : ι → Set α} : (f '' ⋃ i, s i) = ⋃ i, f '' s i := by
ext1 x
simp only [mem_image, mem_iUnion, ← exists_and_right, ← exists_and_left]
--Porting note: `exists_swap` causes a `simp` loop in Lean4 so we use `rw` instead.
rw [exists_swap]
#align set.image_Union Set.image_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iUnion₂ (f : α → β) (s : ∀ i, κ i → Set α) :
(f '' ⋃ (i) (j), s i j) = ⋃ (i) (j), f '' s i j := by simp_rw [image_iUnion]
#align set.image_Union₂ Set.image_iUnion₂
theorem univ_subtype {p : α → Prop} : (univ : Set (Subtype p)) = ⋃ (x) (h : p x), {⟨x, h⟩} :=
Set.ext fun ⟨x, h⟩ => by simp [h]
#align set.univ_subtype Set.univ_subtype
theorem range_eq_iUnion {ι} (f : ι → α) : range f = ⋃ i, {f i} :=
Set.ext fun a => by simp [@eq_comm α a]
#align set.range_eq_Union Set.range_eq_iUnion
theorem image_eq_iUnion (f : α → β) (s : Set α) : f '' s = ⋃ i ∈ s, {f i} :=
Set.ext fun b => by simp [@eq_comm β b]
#align set.image_eq_Union Set.image_eq_iUnion
theorem biUnion_range {f : ι → α} {g : α → Set β} : ⋃ x ∈ range f, g x = ⋃ y, g (f y) :=
iSup_range
#align set.bUnion_range Set.biUnion_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iUnion_iUnion_eq' {f : ι → α} {g : α → Set β} :
⋃ (x) (y) (_ : f y = x), g x = ⋃ y, g (f y) := by simpa using biUnion_range
#align set.Union_Union_eq' Set.iUnion_iUnion_eq'
theorem biInter_range {f : ι → α} {g : α → Set β} : ⋂ x ∈ range f, g x = ⋂ y, g (f y) :=
iInf_range
#align set.bInter_range Set.biInter_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iInter_iInter_eq' {f : ι → α} {g : α → Set β} :
⋂ (x) (y) (_ : f y = x), g x = ⋂ y, g (f y) := by simpa using biInter_range
#align set.Inter_Inter_eq' Set.iInter_iInter_eq'
variable {s : Set γ} {f : γ → α} {g : α → Set β}
theorem biUnion_image : ⋃ x ∈ f '' s, g x = ⋃ y ∈ s, g (f y) :=
iSup_image
#align set.bUnion_image Set.biUnion_image
theorem biInter_image : ⋂ x ∈ f '' s, g x = ⋂ y ∈ s, g (f y) :=
iInf_image
#align set.bInter_image Set.biInter_image
end Image
section Preimage
theorem monotone_preimage {f : α → β} : Monotone (preimage f) := fun _ _ h => preimage_mono h
#align set.monotone_preimage Set.monotone_preimage
@[simp]
theorem preimage_iUnion {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋃ i, s i) = ⋃ i, f ⁻¹' s i :=
Set.ext <| by simp [preimage]
#align set.preimage_Union Set.preimage_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iUnion₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋃ (i) (j), s i j) = ⋃ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iUnion]
#align set.preimage_Union₂ Set.preimage_iUnion₂
@[simp]
theorem preimage_sUnion {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋃₀s = ⋃ t ∈ s, f ⁻¹' t := by
rw [sUnion_eq_biUnion, preimage_iUnion₂]
#align set.preimage_sUnion Set.preimage_sUnion
theorem preimage_iInter {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋂ i, s i) = ⋂ i, f ⁻¹' s i := by
ext; simp
#align set.preimage_Inter Set.preimage_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iInter₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋂ (i) (j), s i j) = ⋂ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iInter]
#align set.preimage_Inter₂ Set.preimage_iInter₂
@[simp]
theorem preimage_sInter {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋂₀ s = ⋂ t ∈ s, f ⁻¹' t := by
rw [sInter_eq_biInter, preimage_iInter₂]
#align set.preimage_sInter Set.preimage_sInter
@[simp]
theorem biUnion_preimage_singleton (f : α → β) (s : Set β) : ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s := by
rw [← preimage_iUnion₂, biUnion_of_singleton]
#align set.bUnion_preimage_singleton Set.biUnion_preimage_singleton
theorem biUnion_range_preimage_singleton (f : α → β) : ⋃ y ∈ range f, f ⁻¹' {y} = univ := by
rw [biUnion_preimage_singleton, preimage_range]
#align set.bUnion_range_preimage_singleton Set.biUnion_range_preimage_singleton
end Preimage
section Prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion {s : Set α} {t : ι → Set β} : (s ×ˢ ⋃ i, t i) = ⋃ i, s ×ˢ t i := by
ext
simp
#align set.prod_Union Set.prod_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} :
(s ×ˢ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ×ˢ t i j := by simp_rw [prod_iUnion]
#align set.prod_Union₂ Set.prod_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sUnion {s : Set α} {C : Set (Set β)} : s ×ˢ ⋃₀C = ⋃₀((fun t => s ×ˢ t) '' C) := by
simp_rw [sUnion_eq_biUnion, biUnion_image, prod_iUnion₂]
#align set.prod_sUnion Set.prod_sUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_const {s : ι → Set α} {t : Set β} : (⋃ i, s i) ×ˢ t = ⋃ i, s i ×ˢ t := by
ext
simp
#align set.Union_prod_const Set.iUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion₂_prod_const {s : ∀ i, κ i → Set α} {t : Set β} :
(⋃ (i) (j), s i j) ×ˢ t = ⋃ (i) (j), s i j ×ˢ t := by simp_rw [iUnion_prod_const]
#align set.Union₂_prod_const Set.iUnion₂_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_prod_const {C : Set (Set α)} {t : Set β} :
⋃₀C ×ˢ t = ⋃₀((fun s : Set α => s ×ˢ t) '' C) := by
simp only [sUnion_eq_biUnion, iUnion₂_prod_const, biUnion_image]
#align set.sUnion_prod_const Set.sUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod {ι ι' α β} (s : ι → Set α) (t : ι' → Set β) :
⋃ x : ι × ι', s x.1 ×ˢ t x.2 = (⋃ i : ι, s i) ×ˢ ⋃ i : ι', t i := by
ext
simp
#align set.Union_prod Set.iUnion_prod
/-- Analogue of `iSup_prod` for sets. -/
lemma iUnion_prod' (f : β × γ → Set α) : ⋃ x : β × γ, f x = ⋃ (i : β) (j : γ), f (i, j) :=
iSup_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x := by
ext ⟨z, w⟩; simp only [mem_prod, mem_iUnion, exists_imp, and_imp, iff_def]; constructor
· intro x hz hw
exact ⟨⟨x, hz⟩, x, hw⟩
· intro x hz x' hw
exact ⟨x ⊔ x', hs le_sup_left hz, ht le_sup_right hw⟩
#align set.Union_prod_of_monotone Set.iUnion_prod_of_monotone
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter_subset (S : Set (Set α)) (T : Set (Set β)) :
⋂₀ S ×ˢ ⋂₀ T ⊆ ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 :=
subset_iInter₂ fun x hx _ hy => ⟨hy.1 x.1 hx.1, hy.2 x.2 hx.2⟩
#align set.sInter_prod_sInter_subset Set.sInter_prod_sInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 := by
obtain ⟨s₁, h₁⟩ := hS
obtain ⟨s₂, h₂⟩ := hT
refine' Set.Subset.antisymm (sInter_prod_sInter_subset S T) fun x hx => _
rw [mem_iInter₂] at hx
exact ⟨fun s₀ h₀ => (hx (s₀, s₂) ⟨h₀, h₂⟩).1, fun s₀ h₀ => (hx (s₁, s₀) ⟨h₁, h₀⟩).2⟩
#align set.sInter_prod_sInter Set.sInter_prod_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod {S : Set (Set α)} (hS : S.Nonempty) (t : Set β) :
⋂₀ S ×ˢ t = ⋂ s ∈ S, s ×ˢ t := by
rw [← sInter_singleton t, sInter_prod_sInter hS (singleton_nonempty t), sInter_singleton]
simp_rw [prod_singleton, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.sInter_prod Set.sInter_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sInter {T : Set (Set β)} (hT : T.Nonempty) (s : Set α) :
s ×ˢ ⋂₀ T = ⋂ t ∈ T, s ×ˢ t := by
rw [← sInter_singleton s, sInter_prod_sInter (singleton_nonempty s) hT, sInter_singleton]
simp_rw [singleton_prod, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.prod_sInter Set.prod_sInter
end Prod
section Image2
variable (f : α → β → γ) {s : Set α} {t : Set β}
/-- The `Set.image2` version of `Set.image_eq_iUnion` -/
theorem image2_eq_iUnion (s : Set α) (t : Set β) : image2 f s t = ⋃ (i ∈ s) (j ∈ t), {f i j} := by
ext; simp [eq_comm]
#align set.image2_eq_Union Set.image2_eq_iUnion
theorem iUnion_image_left : ⋃ a ∈ s, f a '' t = image2 f s t := by
simp only [image2_eq_iUnion, image_eq_iUnion]
#align set.Union_image_left Set.iUnion_image_left
theorem iUnion_image_right : ⋃ b ∈ t, (f · b) '' s = image2 f s t := by
rw [image2_swap, iUnion_image_left]
#align set.Union_image_right Set.iUnion_image_right
theorem image2_iUnion_left (s : ι → Set α) (t : Set β) :
image2 f (⋃ i, s i) t = ⋃ i, image2 f (s i) t := by
simp only [← image_prod, iUnion_prod_const, image_iUnion]
#align set.image2_Union_left Set.image2_iUnion_left
theorem image2_iUnion_right (s : Set α) (t : ι → Set β) :
image2 f s (⋃ i, t i) = ⋃ i, image2 f s (t i) := by
simp only [← image_prod, prod_iUnion, image_iUnion]
#align set.image2_Union_right Set.image2_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋃ (i) (j), s i j) t = ⋃ (i) (j), image2 f (s i j) t := by simp_rw [image2_iUnion_left]
#align set.image2_Union₂_left Set.image2_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋃ (i) (j), t i j) = ⋃ (i) (j), image2 f s (t i j) :=
by simp_rw [image2_iUnion_right]
#align set.image2_Union₂_right Set.image2_iUnion₂_right
theorem image2_iInter_subset_left (s : ι → Set α) (t : Set β) :
image2 f (⋂ i, s i) t ⊆ ⋂ i, image2 f (s i) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem (hx _) hy
#align set.image2_Inter_subset_left Set.image2_iInter_subset_left
theorem image2_iInter_subset_right (s : Set α) (t : ι → Set β) :
image2 f s (⋂ i, t i) ⊆ ⋂ i, image2 f s (t i) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem hx (hy _)
#align set.image2_Inter_subset_right Set.image2_iInter_subset_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋂ (i) (j), s i j) t ⊆ ⋂ (i) (j), image2 f (s i j) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem (hx _ _) hy
#align set.image2_Inter₂_subset_left Set.image2_iInter₂_subset_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), image2 f s (t i j) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem hx (hy _ _)
#align set.image2_Inter₂_subset_right Set.image2_iInter₂_subset_right
theorem prod_eq_biUnion_left : s ×ˢ t = ⋃ a ∈ s, (fun b => (a, b)) '' t := by
rw [iUnion_image_left, image2_mk_eq_prod]
#align set.prod_eq_bUnion_left Set.prod_eq_biUnion_left
theorem prod_eq_biUnion_right : s ×ˢ t = ⋃ b ∈ t, (fun a => (a, b)) '' s := by
rw [iUnion_image_right, image2_mk_eq_prod]
#align set.prod_eq_bUnion_right Set.prod_eq_biUnion_right
end Image2
section Seq
/-- Given a set `s` of functions `α → β` and `t : Set α`, `seq s t` is the union of `f '' t` over
all `f ∈ s`. -/
def seq (s : Set (α → β)) (t : Set α) : Set β :=
{ b | ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b }
#align set.seq Set.seq
@[simp]
theorem mem_seq_iff {s : Set (α → β)} {t : Set α} {b : β} :
b ∈ seq s t ↔ ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b :=
Iff.rfl
#align set.mem_seq_iff Set.mem_seq_iff
lemma seq_eq_image2 (s : Set (α → β)) (t : Set α) : seq s t = image2 (fun f a ↦ f a) s t := by
ext; simp
theorem seq_def {s : Set (α → β)} {t : Set α} : seq s t = ⋃ f ∈ s, f '' t := by
rw [seq_eq_image2, iUnion_image_left]
#align set.seq_def Set.seq_def
theorem seq_subset {s : Set (α → β)} {t : Set α} {u : Set β} :
seq s t ⊆ u ↔ ∀ f ∈ s, ∀ a ∈ t, (f : α → β) a ∈ u := by
rw [seq_eq_image2, image2_subset_iff]
#align set.seq_subset Set.seq_subset
@[gcongr]
theorem seq_mono {s₀ s₁ : Set (α → β)} {t₀ t₁ : Set α} (hs : s₀ ⊆ s₁) (ht : t₀ ⊆ t₁) :
seq s₀ t₀ ⊆ seq s₁ t₁ := fun _ ⟨f, hf, a, ha, eq⟩ => ⟨f, hs hf, a, ht ha, eq⟩
#align set.seq_mono Set.seq_mono
theorem singleton_seq {f : α → β} {t : Set α} : Set.seq ({f} : Set (α → β)) t = f '' t := by
rw [seq_eq_image2, image2_singleton_left]
#align set.singleton_seq Set.singleton_seq
theorem seq_singleton {s : Set (α → β)} {a : α} : Set.seq s {a} = (fun f : α → β => f a) '' s := by
rw [seq_eq_image2, image2_singleton_right]
#align set.seq_singleton Set.seq_singleton
theorem seq_seq {s : Set (β → γ)} {t : Set (α → β)} {u : Set α} :
seq s (seq t u) = seq (seq ((· ∘ ·) '' s) t) u := by
simp only [seq_eq_image2, image2_image_left, image2_image2_left, image2_image2_right, comp_apply]
#align set.seq_seq Set.seq_seq
theorem image_seq {f : β → γ} {s : Set (α → β)} {t : Set α} :
f '' seq s t = seq ((f ∘ ·) '' s) t := by
rw [← singleton_seq, ← singleton_seq, seq_seq, image_singleton]
#align set.image_seq Set.image_seq
theorem prod_eq_seq {s : Set α} {t : Set β} : s ×ˢ t = (Prod.mk '' s).seq t := by
rw [seq_eq_image2, image2_image_left, image2_mk_eq_prod]
#align set.prod_eq_seq Set.prod_eq_seq
theorem prod_image_seq_comm (s : Set α) (t : Set β) :
(Prod.mk '' s).seq t = seq ((fun b a => (a, b)) '' t) s := by
rw [← prod_eq_seq, ← image_swap_prod, prod_eq_seq, image_seq, ← image_comp]; rfl
#align set.prod_image_seq_comm Set.prod_image_seq_comm
theorem image2_eq_seq (f : α → β → γ) (s : Set α) (t : Set β) : image2 f s t = seq (f '' s) t := by
rw [seq_eq_image2, image2_image_left]
#align set.image2_eq_seq Set.image2_eq_seq
end Seq
section Pi
variable {π : α → Type*}
theorem pi_def (i : Set α) (s : ∀ a, Set (π a)) : pi i s = ⋂ a ∈ i, eval a ⁻¹' s a := by
ext
simp
#align set.pi_def Set.pi_def
theorem univ_pi_eq_iInter (t : ∀ i, Set (π i)) : pi univ t = ⋂ i, eval i ⁻¹' t i := by
simp only [pi_def, iInter_true, mem_univ]
#align set.univ_pi_eq_Inter Set.univ_pi_eq_iInter
theorem pi_diff_pi_subset (i : Set α) (s t : ∀ a, Set (π a)) :
pi i s \ pi i t ⊆ ⋃ a ∈ i, eval a ⁻¹' (s a \ t a) := by
refine' diff_subset_comm.2 fun x hx a ha => _
simp only [mem_diff, mem_pi, mem_iUnion, not_exists, mem_preimage, not_and, not_not,
eval_apply] at hx
exact hx.2 _ ha (hx.1 _ ha)
#align set.pi_diff_pi_subset Set.pi_diff_pi_subset
theorem iUnion_univ_pi {ι : α → Type*} (t : (a : α) → ι a → Set (π a)) :
⋃ x : (a : α) → ι a, pi univ (fun a => t a (x a)) = pi univ fun a => ⋃ j : ι a, t a j := by
ext
simp [Classical.skolem]
#align set.Union_univ_pi Set.iUnion_univ_pi
end Pi
end Set
namespace Function
namespace Surjective
theorem iUnion_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋃ x, g (f x) = ⋃ y, g y :=
hf.iSup_comp g
#align function.surjective.Union_comp Function.Surjective.iUnion_comp
theorem iInter_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋂ x, g (f x) = ⋂ y, g y :=
hf.iInf_comp g
#align function.surjective.Inter_comp Function.Surjective.iInter_comp
end Surjective
end Function
/-!
### Disjoint sets
-/
section Disjoint
variable {s t u : Set α} {f : α → β}
namespace Set
@[simp]
theorem disjoint_iUnion_left {ι : Sort*} {s : ι → Set α} :
Disjoint (⋃ i, s i) t ↔ ∀ i, Disjoint (s i) t :=
iSup_disjoint_iff
#align set.disjoint_Union_left Set.disjoint_iUnion_left
@[simp]
theorem disjoint_iUnion_right {ι : Sort*} {s : ι → Set α} :
Disjoint t (⋃ i, s i) ↔ ∀ i, Disjoint t (s i) :=
disjoint_iSup_iff
#align set.disjoint_Union_right Set.disjoint_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_left {s : ∀ i, κ i → Set α} {t : Set α} :
Disjoint (⋃ (i) (j), s i j) t ↔ ∀ i j, Disjoint (s i j) t :=
iSup₂_disjoint_iff
#align set.disjoint_Union₂_left Set.disjoint_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_right {s : Set α} {t : ∀ i, κ i → Set α} :
Disjoint s (⋃ (i) (j), t i j) ↔ ∀ i j, Disjoint s (t i j) :=
disjoint_iSup₂_iff
#align set.disjoint_Union₂_right Set.disjoint_iUnion₂_right
@[simp]
theorem disjoint_sUnion_left {S : Set (Set α)} {t : Set α} :
Disjoint (⋃₀S) t ↔ ∀ s ∈ S, Disjoint s t :=
sSup_disjoint_iff
#align set.disjoint_sUnion_left Set.disjoint_sUnion_left
@[simp]
theorem disjoint_sUnion_right {s : Set α} {S : Set (Set α)} :
Disjoint s (⋃₀S) ↔ ∀ t ∈ S, Disjoint s t :=
disjoint_sSup_iff
#align set.disjoint_sUnion_right Set.disjoint_sUnion_right
end Set
end Disjoint
/-! ### Intervals -/
namespace Set
lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :
(⋂ i, Iic (f i)).Nonempty ↔ BddBelow (range f) := by
have : (⋂ (i : ι), Iic (f i)) = lowerBounds (range f) := by
ext c; simp [lowerBounds]
simp [this, BddBelow]
lemma nonempty_iInter_Ici_iff [Preorder α] {f : ι → α} :
(⋂ i, Ici (f i)).Nonempty ↔ BddAbove (range f) :=
nonempty_iInter_Iic_iff (α := αᵒᵈ)
variable [CompleteLattice α]
theorem Ici_iSup (f : ι → α) : Ici (⨆ i, f i) = ⋂ i, Ici (f i) :=
ext fun _ => by simp only [mem_Ici, iSup_le_iff, mem_iInter]
#align set.Ici_supr Set.Ici_iSup
theorem Iic_iInf (f : ι → α) : Iic (⨅ i, f i) = ⋂ i, Iic (f i) :=
ext fun _ => by simp only [mem_Iic, le_iInf_iff, mem_iInter]
#align set.Iic_infi Set.Iic_iInf
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Ici_iSup₂ (f : ∀ i, κ i → α) : Ici (⨆ (i) (j), f i j) = ⋂ (i) (j), Ici (f i j) := by
simp_rw [Ici_iSup]
#align set.Ici_supr₂ Set.Ici_iSup₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Iic_iInf₂ (f : ∀ i, κ i → α) : Iic (⨅ (i) (j), f i j) = ⋂ (i) (j), Iic (f i j) := by
simp_rw [Iic_iInf]
#align set.Iic_infi₂ Set.Iic_iInf₂
theorem Ici_sSup (s : Set α) : Ici (sSup s) = ⋂ a ∈ s, Ici a := by rw [sSup_eq_iSup, Ici_iSup₂]
#align set.Ici_Sup Set.Ici_sSup
theorem Iic_sInf (s : Set α) : Iic (sInf s) = ⋂ a ∈ s, Iic a := by rw [sInf_eq_iInf, Iic_iInf₂]
#align set.Iic_Inf Set.Iic_sInf
end Set
namespace Set
variable (t : α → Set β)
theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x := by
simp only [diff_subset_iff, ← biUnion_union]
apply biUnion_subset_biUnion_left
|
rw [union_diff_self]
|
theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x := by
simp only [diff_subset_iff, ← biUnion_union]
apply biUnion_subset_biUnion_left
|
Mathlib.Data.Set.Lattice.2291_0.5mONj49h3SYSDwc
|
theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x
|
Mathlib_Data_Set_Lattice
|
case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
t : α → Set β
s₁ s₂ : Set α
⊢ s₁ ⊆ s₂ ∪ s₁
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
/-!
# The set lattice
This file provides usual set notation for unions and intersections, a `CompleteLattice` instance
for `Set α`, and some more set constructions.
## Main declarations
* `Set.iUnion`: **i**ndexed **union**. Union of an indexed family of sets.
* `Set.iInter`: **i**ndexed **inter**section. Intersection of an indexed family of sets.
* `Set.sInter`: **s**et **inter**section. Intersection of sets belonging to a set of sets.
* `Set.sUnion`: **s**et **union**. Union of sets belonging to a set of sets.
* `Set.sInter_eq_biInter`, `Set.sUnion_eq_biInter`: Shows that `⋂₀ s = ⋂ x ∈ s, x` and
`⋃₀ s = ⋃ x ∈ s, x`.
* `Set.completeAtomicBooleanAlgebra`: `Set α` is a `CompleteAtomicBooleanAlgebra` with `≤ = ⊆`,
`< = ⊂`, `⊓ = ∩`, `⊔ = ∪`, `⨅ = ⋂`, `⨆ = ⋃` and `\` as the set difference.
See `Set.BooleanAlgebra`.
* `Set.kernImage`: For a function `f : α → β`, `s.kernImage f` is the set of `y` such that
`f ⁻¹ y ⊆ s`.
* `Set.seq`: Union of the image of a set under a **seq**uence of functions. `seq s t` is the union
of `f '' t` over all `f ∈ s`, where `t : Set α` and `s : Set (α → β)`.
* `Set.iUnion_eq_sigma_of_disjoint`: Equivalence between `⋃ i, t i` and `Σ i, t i`, where `t` is an
indexed family of disjoint sets.
## Naming convention
In lemma names,
* `⋃ i, s i` is called `iUnion`
* `⋂ i, s i` is called `iInter`
* `⋃ i j, s i j` is called `iUnion₂`. This is an `iUnion` inside an `iUnion`.
* `⋂ i j, s i j` is called `iInter₂`. This is an `iInter` inside an `iInter`.
* `⋃ i ∈ s, t i` is called `biUnion` for "bounded `iUnion`". This is the special case of `iUnion₂`
where `j : i ∈ s`.
* `⋂ i ∈ s, t i` is called `biInter` for "bounded `iInter`". This is the special case of `iInter₂`
where `j : i ∈ s`.
## Notation
* `⋃`: `Set.iUnion`
* `⋂`: `Set.iInter`
* `⋃₀`: `Set.sUnion`
* `⋂₀`: `Set.sInter`
-/
set_option autoImplicit true
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι₂ : Sort*} {κ κ₁ κ₂ : ι → Sort*} {κ' : ι' → Sort*}
namespace Set
/-! ### Complete lattice and complete Boolean algebra instances -/
instance : InfSet (Set α) :=
⟨fun s => { a | ∀ t ∈ s, a ∈ t }⟩
instance : SupSet (Set α) :=
⟨fun s => { a | ∃ t ∈ s, a ∈ t }⟩
/-- Intersection of a set of sets. -/
def sInter (S : Set (Set α)) : Set α :=
sInf S
#align set.sInter Set.sInter
/-- Notation for `Set.sInter` Intersection of a set of sets. -/
prefix:110 "⋂₀ " => sInter
/-- Union of a set of sets. -/
def sUnion (S : Set (Set α)) : Set α :=
sSup S
#align set.sUnion Set.sUnion
/-- Notation for `Set.sUnion`. Union of a set of sets. -/
prefix:110 "⋃₀ " => sUnion
@[simp]
theorem mem_sInter {x : α} {S : Set (Set α)} : x ∈ ⋂₀ S ↔ ∀ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sInter Set.mem_sInter
@[simp]
theorem mem_sUnion {x : α} {S : Set (Set α)} : x ∈ ⋃₀ S ↔ ∃ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sUnion Set.mem_sUnion
/-- Indexed union of a family of sets -/
def iUnion (s : ι → Set β) : Set β :=
iSup s
#align set.Union Set.iUnion
/-- Indexed intersection of a family of sets -/
def iInter (s : ι → Set β) : Set β :=
iInf s
#align set.Inter Set.iInter
/-- Notation for `Set.iUnion`. Indexed union of a family of sets -/
notation3 "⋃ "(...)", "r:60:(scoped f => iUnion f) => r
/-- Notation for `Set.iInter`. Indexed intersection of a family of sets -/
notation3 "⋂ "(...)", "r:60:(scoped f => iInter f) => r
section delaborators
open Lean Lean.PrettyPrinter.Delaborator
/-- Delaborator for indexed unions. -/
@[delab app.Set.iUnion]
def iUnion_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋃ (_ : $dom), $body)
else if prop || ppTypes then
`(⋃ ($x:ident : $dom), $body)
else
`(⋃ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋃ $x:ident, ⋃ (_ : $y:ident ∈ $s), $body)
| `(⋃ ($x:ident : $_), ⋃ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋃ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
/-- Delaborator for indexed intersections. -/
@[delab app.Set.iInter]
def sInter_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋂ (_ : $dom), $body)
else if prop || ppTypes then
`(⋂ ($x:ident : $dom), $body)
else
`(⋂ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋂ $x:ident, ⋂ (_ : $y:ident ∈ $s), $body)
| `(⋂ ($x:ident : $_), ⋂ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋂ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
end delaborators
@[simp]
theorem sSup_eq_sUnion (S : Set (Set α)) : sSup S = ⋃₀S :=
rfl
#align set.Sup_eq_sUnion Set.sSup_eq_sUnion
@[simp]
theorem sInf_eq_sInter (S : Set (Set α)) : sInf S = ⋂₀ S :=
rfl
#align set.Inf_eq_sInter Set.sInf_eq_sInter
@[simp]
theorem iSup_eq_iUnion (s : ι → Set α) : iSup s = iUnion s :=
rfl
#align set.supr_eq_Union Set.iSup_eq_iUnion
@[simp]
theorem iInf_eq_iInter (s : ι → Set α) : iInf s = iInter s :=
rfl
#align set.infi_eq_Inter Set.iInf_eq_iInter
@[simp]
theorem mem_iUnion {x : α} {s : ι → Set α} : (x ∈ ⋃ i, s i) ↔ ∃ i, x ∈ s i :=
⟨fun ⟨_, ⟨⟨a, (t_eq : s a = _)⟩, (h : x ∈ _)⟩⟩ => ⟨a, t_eq.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨s a, ⟨⟨a, rfl⟩, h⟩⟩⟩
#align set.mem_Union Set.mem_iUnion
@[simp]
theorem mem_iInter {x : α} {s : ι → Set α} : (x ∈ ⋂ i, s i) ↔ ∀ i, x ∈ s i :=
⟨fun (h : ∀ a ∈ { a : Set α | ∃ i, s i = a }, x ∈ a) a => h (s a) ⟨a, rfl⟩,
fun h _ ⟨a, (eq : s a = _)⟩ => eq ▸ h a⟩
#align set.mem_Inter Set.mem_iInter
theorem mem_iUnion₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋃ (i) (j), s i j) ↔ ∃ i j, x ∈ s i j := by
simp_rw [mem_iUnion]
#align set.mem_Union₂ Set.mem_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋂ (i) (j), s i j) ↔ ∀ i j, x ∈ s i j := by
simp_rw [mem_iInter]
#align set.mem_Inter₂ Set.mem_iInter₂
theorem mem_iUnion_of_mem {s : ι → Set α} {a : α} (i : ι) (ha : a ∈ s i) : a ∈ ⋃ i, s i :=
mem_iUnion.2 ⟨i, ha⟩
#align set.mem_Union_of_mem Set.mem_iUnion_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iUnion₂_of_mem {s : ∀ i, κ i → Set α} {a : α} {i : ι} (j : κ i) (ha : a ∈ s i j) :
a ∈ ⋃ (i) (j), s i j :=
mem_iUnion₂.2 ⟨i, j, ha⟩
#align set.mem_Union₂_of_mem Set.mem_iUnion₂_of_mem
theorem mem_iInter_of_mem {s : ι → Set α} {a : α} (h : ∀ i, a ∈ s i) : a ∈ ⋂ i, s i :=
mem_iInter.2 h
#align set.mem_Inter_of_mem Set.mem_iInter_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂_of_mem {s : ∀ i, κ i → Set α} {a : α} (h : ∀ i j, a ∈ s i j) :
a ∈ ⋂ (i) (j), s i j :=
mem_iInter₂.2 h
#align set.mem_Inter₂_of_mem Set.mem_iInter₂_of_mem
instance Set.completeAtomicBooleanAlgebra : CompleteAtomicBooleanAlgebra (Set α) :=
{ instBooleanAlgebraSet with
le_sSup := fun s t t_in a a_in => ⟨t, t_in, a_in⟩
sSup_le := fun s t h a ⟨t', ⟨t'_in, a_in⟩⟩ => h t' t'_in a_in
le_sInf := fun s t h a a_in t' t'_in => h t' t'_in a_in
sInf_le := fun s t t_in a h => h _ t_in
iInf_iSup_eq := by intros; ext; simp [Classical.skolem] }
/-- `kernImage f s` is the set of `y` such that `f ⁻¹ y ⊆ s`. -/
def kernImage (f : α → β) (s : Set α) : Set β :=
{ y | ∀ ⦃x⦄, f x = y → x ∈ s }
#align set.kern_image Set.kernImage
lemma subset_kernImage_iff {f : α → β} : s ⊆ kernImage f t ↔ f ⁻¹' s ⊆ t :=
⟨fun h _ hx ↦ h hx rfl,
fun h _ hx y hy ↦ h (show f y ∈ s from hy.symm ▸ hx)⟩
section GaloisConnection
variable {f : α → β}
protected theorem image_preimage : GaloisConnection (image f) (preimage f) := fun _ _ =>
image_subset_iff
#align set.image_preimage Set.image_preimage
protected theorem preimage_kernImage : GaloisConnection (preimage f) (kernImage f) := fun _ _ =>
subset_kernImage_iff.symm
#align set.preimage_kern_image Set.preimage_kernImage
end GaloisConnection
section kernImage
variable {f : α → β}
lemma kernImage_mono : Monotone (kernImage f) :=
Set.preimage_kernImage.monotone_u
lemma kernImage_eq_compl {s : Set α} : kernImage f s = (f '' sᶜ)ᶜ :=
Set.preimage_kernImage.u_unique (Set.image_preimage.compl)
(fun t ↦ compl_compl (f ⁻¹' t) ▸ Set.preimage_compl)
lemma kernImage_compl {s : Set α} : kernImage f (sᶜ) = (f '' s)ᶜ := by
rw [kernImage_eq_compl, compl_compl]
lemma kernImage_empty : kernImage f ∅ = (range f)ᶜ := by
rw [kernImage_eq_compl, compl_empty, image_univ]
lemma kernImage_preimage_eq_iff {s : Set β} : kernImage f (f ⁻¹' s) = s ↔ (range f)ᶜ ⊆ s := by
rw [kernImage_eq_compl, ← preimage_compl, compl_eq_comm, eq_comm, image_preimage_eq_iff,
compl_subset_comm]
lemma compl_range_subset_kernImage {s : Set α} : (range f)ᶜ ⊆ kernImage f s := by
rw [← kernImage_empty]
exact kernImage_mono (empty_subset _)
lemma kernImage_union_preimage {s : Set α} {t : Set β} :
kernImage f (s ∪ f ⁻¹' t) = kernImage f s ∪ t := by
rw [kernImage_eq_compl, kernImage_eq_compl, compl_union, ← preimage_compl, image_inter_preimage,
compl_inter, compl_compl]
lemma kernImage_preimage_union {s : Set α} {t : Set β} :
kernImage f (f ⁻¹' t ∪ s) = t ∪ kernImage f s := by
rw [union_comm, kernImage_union_preimage, union_comm]
end kernImage
/-! ### Union and intersection over an indexed family of sets -/
instance : OrderTop (Set α) where
top := univ
le_top := by simp
@[congr]
theorem iUnion_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iUnion f₁ = iUnion f₂ :=
iSup_congr_Prop pq f
#align set.Union_congr_Prop Set.iUnion_congr_Prop
@[congr]
theorem iInter_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iInter f₁ = iInter f₂ :=
iInf_congr_Prop pq f
#align set.Inter_congr_Prop Set.iInter_congr_Prop
theorem iUnion_plift_up (f : PLift ι → Set α) : ⋃ i, f (PLift.up i) = ⋃ i, f i :=
iSup_plift_up _
#align set.Union_plift_up Set.iUnion_plift_up
theorem iUnion_plift_down (f : ι → Set α) : ⋃ i, f (PLift.down i) = ⋃ i, f i :=
iSup_plift_down _
#align set.Union_plift_down Set.iUnion_plift_down
theorem iInter_plift_up (f : PLift ι → Set α) : ⋂ i, f (PLift.up i) = ⋂ i, f i :=
iInf_plift_up _
#align set.Inter_plift_up Set.iInter_plift_up
theorem iInter_plift_down (f : ι → Set α) : ⋂ i, f (PLift.down i) = ⋂ i, f i :=
iInf_plift_down _
#align set.Inter_plift_down Set.iInter_plift_down
theorem iUnion_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋃ _ : p, s = if p then s else ∅ :=
iSup_eq_if _
#align set.Union_eq_if Set.iUnion_eq_if
theorem iUnion_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋃ h : p, s h = if h : p then s h else ∅ :=
iSup_eq_dif _
#align set.Union_eq_dif Set.iUnion_eq_dif
theorem iInter_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋂ _ : p, s = if p then s else univ :=
iInf_eq_if _
#align set.Inter_eq_if Set.iInter_eq_if
theorem iInf_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋂ h : p, s h = if h : p then s h else univ :=
_root_.iInf_eq_dif _
#align set.Infi_eq_dif Set.iInf_eq_dif
theorem exists_set_mem_of_union_eq_top {ι : Type*} (t : Set ι) (s : ι → Set β)
(w : ⋃ i ∈ t, s i = ⊤) (x : β) : ∃ i ∈ t, x ∈ s i := by
have p : x ∈ ⊤ := Set.mem_univ x
rw [← w, Set.mem_iUnion] at p
simpa using p
#align set.exists_set_mem_of_union_eq_top Set.exists_set_mem_of_union_eq_top
theorem nonempty_of_union_eq_top_of_nonempty {ι : Type*} (t : Set ι) (s : ι → Set α)
(H : Nonempty α) (w : ⋃ i ∈ t, s i = ⊤) : t.Nonempty := by
obtain ⟨x, m, -⟩ := exists_set_mem_of_union_eq_top t s w H.some
exact ⟨x, m⟩
#align set.nonempty_of_union_eq_top_of_nonempty Set.nonempty_of_union_eq_top_of_nonempty
theorem nonempty_of_nonempty_iUnion
{s : ι → Set α} (h_Union : (⋃ i, s i).Nonempty) : Nonempty ι := by
obtain ⟨x, hx⟩ := h_Union
exact ⟨Classical.choose $ mem_iUnion.mp hx⟩
theorem nonempty_of_nonempty_iUnion_eq_univ
{s : ι → Set α} [Nonempty α] (h_Union : ⋃ i, s i = univ) : Nonempty ι :=
nonempty_of_nonempty_iUnion (s := s) (by simpa only [h_Union] using univ_nonempty)
theorem setOf_exists (p : ι → β → Prop) : { x | ∃ i, p i x } = ⋃ i, { x | p i x } :=
ext fun _ => mem_iUnion.symm
#align set.set_of_exists Set.setOf_exists
theorem setOf_forall (p : ι → β → Prop) : { x | ∀ i, p i x } = ⋂ i, { x | p i x } :=
ext fun _ => mem_iInter.symm
#align set.set_of_forall Set.setOf_forall
theorem iUnion_subset {s : ι → Set α} {t : Set α} (h : ∀ i, s i ⊆ t) : ⋃ i, s i ⊆ t :=
iSup_le h
#align set.Union_subset Set.iUnion_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset {s : ∀ i, κ i → Set α} {t : Set α} (h : ∀ i j, s i j ⊆ t) :
⋃ (i) (j), s i j ⊆ t :=
iUnion_subset fun x => iUnion_subset (h x)
#align set.Union₂_subset Set.iUnion₂_subset
theorem subset_iInter {t : Set β} {s : ι → Set β} (h : ∀ i, t ⊆ s i) : t ⊆ ⋂ i, s i :=
le_iInf h
#align set.subset_Inter Set.subset_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iInter₂ {s : Set α} {t : ∀ i, κ i → Set α} (h : ∀ i j, s ⊆ t i j) :
s ⊆ ⋂ (i) (j), t i j :=
subset_iInter fun x => subset_iInter <| h x
#align set.subset_Inter₂ Set.subset_iInter₂
@[simp]
theorem iUnion_subset_iff {s : ι → Set α} {t : Set α} : ⋃ i, s i ⊆ t ↔ ∀ i, s i ⊆ t :=
⟨fun h _ => Subset.trans (le_iSup s _) h, iUnion_subset⟩
#align set.Union_subset_iff Set.iUnion_subset_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset_iff {s : ∀ i, κ i → Set α} {t : Set α} :
⋃ (i) (j), s i j ⊆ t ↔ ∀ i j, s i j ⊆ t := by simp_rw [iUnion_subset_iff]
#align set.Union₂_subset_iff Set.iUnion₂_subset_iff
@[simp]
theorem subset_iInter_iff {s : Set α} {t : ι → Set α} : (s ⊆ ⋂ i, t i) ↔ ∀ i, s ⊆ t i :=
le_iInf_iff
#align set.subset_Inter_iff Set.subset_iInter_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem subset_iInter₂_iff {s : Set α} {t : ∀ i, κ i → Set α} :
(s ⊆ ⋂ (i) (j), t i j) ↔ ∀ i j, s ⊆ t i j := by simp_rw [subset_iInter_iff]
#align set.subset_Inter₂_iff Set.subset_iInter₂_iff
theorem subset_iUnion : ∀ (s : ι → Set β) (i : ι), s i ⊆ ⋃ i, s i :=
le_iSup
#align set.subset_Union Set.subset_iUnion
theorem iInter_subset : ∀ (s : ι → Set β) (i : ι), ⋂ i, s i ⊆ s i :=
iInf_le
#align set.Inter_subset Set.iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iUnion₂ {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : s i j ⊆ ⋃ (i') (j'), s i' j' :=
le_iSup₂ i j
#align set.subset_Union₂ Set.subset_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_subset {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : ⋂ (i) (j), s i j ⊆ s i j :=
iInf₂_le i j
#align set.Inter₂_subset Set.iInter₂_subset
/-- This rather trivial consequence of `subset_iUnion`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem subset_iUnion_of_subset {s : Set α} {t : ι → Set α} (i : ι) (h : s ⊆ t i) : s ⊆ ⋃ i, t i :=
le_iSup_of_le i h
#align set.subset_Union_of_subset Set.subset_iUnion_of_subset
/-- This rather trivial consequence of `iInter_subset`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem iInter_subset_of_subset {s : ι → Set α} {t : Set α} (i : ι) (h : s i ⊆ t) :
⋂ i, s i ⊆ t :=
iInf_le_of_le i h
#align set.Inter_subset_of_subset Set.iInter_subset_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `subset_iUnion₂` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem subset_iUnion₂_of_subset {s : Set α} {t : ∀ i, κ i → Set α} (i : ι) (j : κ i)
(h : s ⊆ t i j) : s ⊆ ⋃ (i) (j), t i j :=
le_iSup₂_of_le i j h
#align set.subset_Union₂_of_subset Set.subset_iUnion₂_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `iInter₂_subset` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem iInter₂_subset_of_subset {s : ∀ i, κ i → Set α} {t : Set α} (i : ι) (j : κ i)
(h : s i j ⊆ t) : ⋂ (i) (j), s i j ⊆ t :=
iInf₂_le_of_le i j h
#align set.Inter₂_subset_of_subset Set.iInter₂_subset_of_subset
theorem iUnion_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono h
#align set.Union_mono Set.iUnion_mono
@[gcongr]
theorem iUnion_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iUnion s ⊆ iUnion t :=
iSup_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋃ (i) (j), s i j ⊆ ⋃ (i) (j), t i j :=
iSup₂_mono h
#align set.Union₂_mono Set.iUnion₂_mono
theorem iInter_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋂ i, s i ⊆ ⋂ i, t i :=
iInf_mono h
#align set.Inter_mono Set.iInter_mono
@[gcongr]
theorem iInter_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iInter s ⊆ iInter t :=
iInf_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋂ (i) (j), s i j ⊆ ⋂ (i) (j), t i j :=
iInf₂_mono h
#align set.Inter₂_mono Set.iInter₂_mono
theorem iUnion_mono' {s : ι → Set α} {t : ι₂ → Set α} (h : ∀ i, ∃ j, s i ⊆ t j) :
⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono' h
#align set.Union_mono' Set.iUnion_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iUnion₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i j, ∃ i' j', s i j ⊆ t i' j') : ⋃ (i) (j), s i j ⊆ ⋃ (i') (j'), t i' j' :=
iSup₂_mono' h
#align set.Union₂_mono' Set.iUnion₂_mono'
theorem iInter_mono' {s : ι → Set α} {t : ι' → Set α} (h : ∀ j, ∃ i, s i ⊆ t j) :
⋂ i, s i ⊆ ⋂ j, t j :=
Set.subset_iInter fun j =>
let ⟨i, hi⟩ := h j
iInter_subset_of_subset i hi
#align set.Inter_mono' Set.iInter_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iInter₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i' j', ∃ i j, s i j ⊆ t i' j') : ⋂ (i) (j), s i j ⊆ ⋂ (i') (j'), t i' j' :=
subset_iInter₂_iff.2 fun i' j' =>
let ⟨_, _, hst⟩ := h i' j'
(iInter₂_subset _ _).trans hst
#align set.Inter₂_mono' Set.iInter₂_mono'
theorem iUnion₂_subset_iUnion (κ : ι → Sort*) (s : ι → Set α) :
⋃ (i) (_ : κ i), s i ⊆ ⋃ i, s i :=
iUnion_mono fun _ => iUnion_subset fun _ => Subset.rfl
#align set.Union₂_subset_Union Set.iUnion₂_subset_iUnion
theorem iInter_subset_iInter₂ (κ : ι → Sort*) (s : ι → Set α) :
⋂ i, s i ⊆ ⋂ (i) (_ : κ i), s i :=
iInter_mono fun _ => subset_iInter fun _ => Subset.rfl
#align set.Inter_subset_Inter₂ Set.iInter_subset_iInter₂
theorem iUnion_setOf (P : ι → α → Prop) : ⋃ i, { x : α | P i x } = { x : α | ∃ i, P i x } := by
ext
exact mem_iUnion
#align set.Union_set_of Set.iUnion_setOf
theorem iInter_setOf (P : ι → α → Prop) : ⋂ i, { x : α | P i x } = { x : α | ∀ i, P i x } := by
ext
exact mem_iInter
#align set.Inter_set_of Set.iInter_setOf
theorem iUnion_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋃ x, f x = ⋃ y, g y :=
h1.iSup_congr h h2
#align set.Union_congr_of_surjective Set.iUnion_congr_of_surjective
theorem iInter_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋂ x, f x = ⋂ y, g y :=
h1.iInf_congr h h2
#align set.Inter_congr_of_surjective Set.iInter_congr_of_surjective
lemma iUnion_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋃ i, s i = ⋃ i, t i := iSup_congr h
#align set.Union_congr Set.iUnion_congr
lemma iInter_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋂ i, s i = ⋂ i, t i := iInf_congr h
#align set.Inter_congr Set.iInter_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iUnion₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋃ (i) (j), s i j = ⋃ (i) (j), t i j :=
iUnion_congr fun i => iUnion_congr <| h i
#align set.Union₂_congr Set.iUnion₂_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iInter₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋂ (i) (j), s i j = ⋂ (i) (j), t i j :=
iInter_congr fun i => iInter_congr <| h i
#align set.Inter₂_congr Set.iInter₂_congr
section Nonempty
variable [Nonempty ι] {f : ι → Set α} {s : Set α}
lemma iUnion_const (s : Set β) : ⋃ _ : ι, s = s := iSup_const
#align set.Union_const Set.iUnion_const
lemma iInter_const (s : Set β) : ⋂ _ : ι, s = s := iInf_const
#align set.Inter_const Set.iInter_const
lemma iUnion_eq_const (hf : ∀ i, f i = s) : ⋃ i, f i = s :=
(iUnion_congr hf).trans $ iUnion_const _
#align set.Union_eq_const Set.iUnion_eq_const
lemma iInter_eq_const (hf : ∀ i, f i = s) : ⋂ i, f i = s :=
(iInter_congr hf).trans $ iInter_const _
#align set.Inter_eq_const Set.iInter_eq_const
end Nonempty
@[simp]
theorem compl_iUnion (s : ι → Set β) : (⋃ i, s i)ᶜ = ⋂ i, (s i)ᶜ :=
compl_iSup
#align set.compl_Union Set.compl_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iUnion₂ (s : ∀ i, κ i → Set α) : (⋃ (i) (j), s i j)ᶜ = ⋂ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iUnion]
#align set.compl_Union₂ Set.compl_iUnion₂
@[simp]
theorem compl_iInter (s : ι → Set β) : (⋂ i, s i)ᶜ = ⋃ i, (s i)ᶜ :=
compl_iInf
#align set.compl_Inter Set.compl_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iInter₂ (s : ∀ i, κ i → Set α) : (⋂ (i) (j), s i j)ᶜ = ⋃ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iInter]
#align set.compl_Inter₂ Set.compl_iInter₂
-- classical -- complete_boolean_algebra
theorem iUnion_eq_compl_iInter_compl (s : ι → Set β) : ⋃ i, s i = (⋂ i, (s i)ᶜ)ᶜ := by
simp only [compl_iInter, compl_compl]
#align set.Union_eq_compl_Inter_compl Set.iUnion_eq_compl_iInter_compl
-- classical -- complete_boolean_algebra
theorem iInter_eq_compl_iUnion_compl (s : ι → Set β) : ⋂ i, s i = (⋃ i, (s i)ᶜ)ᶜ := by
simp only [compl_iUnion, compl_compl]
#align set.Inter_eq_compl_Union_compl Set.iInter_eq_compl_iUnion_compl
theorem inter_iUnion (s : Set β) (t : ι → Set β) : (s ∩ ⋃ i, t i) = ⋃ i, s ∩ t i :=
inf_iSup_eq _ _
#align set.inter_Union Set.inter_iUnion
theorem iUnion_inter (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∩ s = ⋃ i, t i ∩ s :=
iSup_inf_eq _ _
#align set.Union_inter Set.iUnion_inter
theorem iUnion_union_distrib (s : ι → Set β) (t : ι → Set β) :
⋃ i, s i ∪ t i = (⋃ i, s i) ∪ ⋃ i, t i :=
iSup_sup_eq
#align set.Union_union_distrib Set.iUnion_union_distrib
theorem iInter_inter_distrib (s : ι → Set β) (t : ι → Set β) :
⋂ i, s i ∩ t i = (⋂ i, s i) ∩ ⋂ i, t i :=
iInf_inf_eq
#align set.Inter_inter_distrib Set.iInter_inter_distrib
theorem union_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∪ ⋃ i, t i) = ⋃ i, s ∪ t i :=
sup_iSup
#align set.union_Union Set.union_iUnion
theorem iUnion_union [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∪ s = ⋃ i, t i ∪ s :=
iSup_sup
#align set.Union_union Set.iUnion_union
theorem inter_iInter [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∩ ⋂ i, t i) = ⋂ i, s ∩ t i :=
inf_iInf
#align set.inter_Inter Set.inter_iInter
theorem iInter_inter [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋂ i, t i) ∩ s = ⋂ i, t i ∩ s :=
iInf_inf
#align set.Inter_inter Set.iInter_inter
-- classical
theorem union_iInter (s : Set β) (t : ι → Set β) : (s ∪ ⋂ i, t i) = ⋂ i, s ∪ t i :=
sup_iInf_eq _ _
#align set.union_Inter Set.union_iInter
theorem iInter_union (s : ι → Set β) (t : Set β) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.Inter_union Set.iInter_union
theorem iUnion_diff (s : Set β) (t : ι → Set β) : (⋃ i, t i) \ s = ⋃ i, t i \ s :=
iUnion_inter _ _
#align set.Union_diff Set.iUnion_diff
theorem diff_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s \ ⋃ i, t i) = ⋂ i, s \ t i := by
rw [diff_eq, compl_iUnion, inter_iInter]; rfl
#align set.diff_Union Set.diff_iUnion
theorem diff_iInter (s : Set β) (t : ι → Set β) : (s \ ⋂ i, t i) = ⋃ i, s \ t i := by
rw [diff_eq, compl_iInter, inter_iUnion]; rfl
#align set.diff_Inter Set.diff_iInter
theorem directed_on_iUnion {r} {f : ι → Set α} (hd : Directed (· ⊆ ·) f)
(h : ∀ x, DirectedOn r (f x)) : DirectedOn r (⋃ x, f x) := by
simp only [DirectedOn, exists_prop, mem_iUnion, exists_imp]
exact fun a₁ b₁ fb₁ a₂ b₂ fb₂ =>
let ⟨z, zb₁, zb₂⟩ := hd b₁ b₂
let ⟨x, xf, xa₁, xa₂⟩ := h z a₁ (zb₁ fb₁) a₂ (zb₂ fb₂)
⟨x, ⟨z, xf⟩, xa₁, xa₂⟩
#align set.directed_on_Union Set.directed_on_iUnion
theorem iUnion_inter_subset {ι α} {s t : ι → Set α} : ⋃ i, s i ∩ t i ⊆ (⋃ i, s i) ∩ ⋃ i, t i :=
le_iSup_inf_iSup s t
#align set.Union_inter_subset Set.iUnion_inter_subset
theorem iUnion_inter_of_monotone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_monotone hs ht
#align set.Union_inter_of_monotone Set.iUnion_inter_of_monotone
theorem iUnion_inter_of_antitone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_antitone hs ht
#align set.Union_inter_of_antitone Set.iUnion_inter_of_antitone
theorem iInter_union_of_monotone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_monotone hs ht
#align set.Inter_union_of_monotone Set.iInter_union_of_monotone
theorem iInter_union_of_antitone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_antitone hs ht
#align set.Inter_union_of_antitone Set.iInter_union_of_antitone
/-- An equality version of this lemma is `iUnion_iInter_of_monotone` in `Data.Set.Finite`. -/
theorem iUnion_iInter_subset {s : ι → ι' → Set α} : (⋃ j, ⋂ i, s i j) ⊆ ⋂ i, ⋃ j, s i j :=
iSup_iInf_le_iInf_iSup (flip s)
#align set.Union_Inter_subset Set.iUnion_iInter_subset
theorem iUnion_option {ι} (s : Option ι → Set α) : ⋃ o, s o = s none ∪ ⋃ i, s (some i) :=
iSup_option s
#align set.Union_option Set.iUnion_option
theorem iInter_option {ι} (s : Option ι → Set α) : ⋂ o, s o = s none ∩ ⋂ i, s (some i) :=
iInf_option s
#align set.Inter_option Set.iInter_option
section
variable (p : ι → Prop) [DecidablePred p]
theorem iUnion_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 _ _ _
#align set.Union_dite Set.iUnion_dite
theorem iUnion_ite (f g : ι → Set α) :
⋃ i, (if p i then f i else g i) = (⋃ (i) (_ : p i), f i) ∪ ⋃ (i) (_ : ¬p i), g i :=
iUnion_dite _ _ _
#align set.Union_ite Set.iUnion_ite
theorem iInter_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 :=
iInf_dite _ _ _
#align set.Inter_dite Set.iInter_dite
theorem iInter_ite (f g : ι → Set α) :
⋂ i, (if p i then f i else g i) = (⋂ (i) (_ : p i), f i) ∩ ⋂ (i) (_ : ¬p i), g i :=
iInter_dite _ _ _
#align set.Inter_ite Set.iInter_ite
end
theorem image_projection_prod {ι : Type*} {α : ι → Type*} {v : ∀ i : ι, Set (α i)}
(hv : (pi univ v).Nonempty) (i : ι) :
((fun x : ∀ i : ι, α i => x i) '' ⋂ k, (fun x : ∀ j : ι, α j => x k) ⁻¹' v k) = v i := by
classical
apply Subset.antisymm
· simp [iInter_subset]
· intro y y_in
simp only [mem_image, mem_iInter, mem_preimage]
rcases hv with ⟨z, hz⟩
refine' ⟨Function.update z i y, _, update_same i y z⟩
rw [@forall_update_iff ι α _ z i y fun i t => t ∈ v i]
exact ⟨y_in, fun j _ => by simpa using hz j⟩
#align set.image_projection_prod Set.image_projection_prod
/-! ### Unions and intersections indexed by `Prop` -/
theorem iInter_false {s : False → Set α} : iInter s = univ :=
iInf_false
#align set.Inter_false Set.iInter_false
theorem iUnion_false {s : False → Set α} : iUnion s = ∅ :=
iSup_false
#align set.Union_false Set.iUnion_false
@[simp]
theorem iInter_true {s : True → Set α} : iInter s = s trivial :=
iInf_true
#align set.Inter_true Set.iInter_true
@[simp]
theorem iUnion_true {s : True → Set α} : iUnion s = s trivial :=
iSup_true
#align set.Union_true Set.iUnion_true
@[simp]
theorem iInter_exists {p : ι → Prop} {f : Exists p → Set α} :
⋂ x, f x = ⋂ (i) (h : p i), f ⟨i, h⟩ :=
iInf_exists
#align set.Inter_exists Set.iInter_exists
@[simp]
theorem iUnion_exists {p : ι → Prop} {f : Exists p → Set α} :
⋃ x, f x = ⋃ (i) (h : p i), f ⟨i, h⟩ :=
iSup_exists
#align set.Union_exists Set.iUnion_exists
@[simp]
theorem iUnion_empty : (⋃ _ : ι, ∅ : Set α) = ∅ :=
iSup_bot
#align set.Union_empty Set.iUnion_empty
@[simp]
theorem iInter_univ : (⋂ _ : ι, univ : Set α) = univ :=
iInf_top
#align set.Inter_univ Set.iInter_univ
section
variable {s : ι → Set α}
@[simp]
theorem iUnion_eq_empty : ⋃ i, s i = ∅ ↔ ∀ i, s i = ∅ :=
iSup_eq_bot
#align set.Union_eq_empty Set.iUnion_eq_empty
@[simp]
theorem iInter_eq_univ : ⋂ i, s i = univ ↔ ∀ i, s i = univ :=
iInf_eq_top
#align set.Inter_eq_univ Set.iInter_eq_univ
@[simp]
theorem nonempty_iUnion : (⋃ i, s i).Nonempty ↔ ∃ i, (s i).Nonempty := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_Union Set.nonempty_iUnion
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_biUnion {t : Set α} {s : α → Set β} :
(⋃ i ∈ t, s i).Nonempty ↔ ∃ i ∈ t, (s i).Nonempty := by simp
#align set.nonempty_bUnion Set.nonempty_biUnion
theorem iUnion_nonempty_index (s : Set α) (t : s.Nonempty → Set β) :
⋃ h, t h = ⋃ x ∈ s, t ⟨x, ‹_›⟩ :=
iSup_exists
#align set.Union_nonempty_index Set.iUnion_nonempty_index
end
@[simp]
theorem iInter_iInter_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋂ (x) (h : x = b), s x h = s b rfl :=
iInf_iInf_eq_left
#align set.Inter_Inter_eq_left Set.iInter_iInter_eq_left
@[simp]
theorem iInter_iInter_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋂ (x) (h : b = x), s x h = s b rfl :=
iInf_iInf_eq_right
#align set.Inter_Inter_eq_right Set.iInter_iInter_eq_right
@[simp]
theorem iUnion_iUnion_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋃ (x) (h : x = b), s x h = s b rfl :=
iSup_iSup_eq_left
#align set.Union_Union_eq_left Set.iUnion_iUnion_eq_left
@[simp]
theorem iUnion_iUnion_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋃ (x) (h : b = x), s x h = s b rfl :=
iSup_iSup_eq_right
#align set.Union_Union_eq_right Set.iUnion_iUnion_eq_right
theorem iInter_or {p q : Prop} (s : p ∨ q → Set α) :
⋂ h, s h = (⋂ h : p, s (Or.inl h)) ∩ ⋂ h : q, s (Or.inr h) :=
iInf_or
#align set.Inter_or Set.iInter_or
theorem iUnion_or {p q : Prop} (s : p ∨ q → Set α) :
⋃ h, s h = (⋃ i, s (Or.inl i)) ∪ ⋃ j, s (Or.inr j) :=
iSup_or
#align set.Union_or Set.iUnion_or
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iUnion_and {p q : Prop} (s : p ∧ q → Set α) : ⋃ h, s h = ⋃ (hp) (hq), s ⟨hp, hq⟩ :=
iSup_and
#align set.Union_and Set.iUnion_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iInter_and {p q : Prop} (s : p ∧ q → Set α) : ⋂ h, s h = ⋂ (hp) (hq), s ⟨hp, hq⟩ :=
iInf_and
#align set.Inter_and Set.iInter_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iUnion_comm (s : ι → ι' → Set α) : ⋃ (i) (i'), s i i' = ⋃ (i') (i), s i i' :=
iSup_comm
#align set.Union_comm Set.iUnion_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iInter_comm (s : ι → ι' → Set α) : ⋂ (i) (i'), s i i' = ⋂ (i') (i), s i i' :=
iInf_comm
#align set.Inter_comm Set.iInter_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iUnion₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋃ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋃ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iSup₂_comm _
#align set.Union₂_comm Set.iUnion₂_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iInter₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋂ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋂ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iInf₂_comm _
#align set.Inter₂_comm Set.iInter₂_comm
@[simp]
theorem biUnion_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋃ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι']
#align set.bUnion_and Set.biUnion_and
@[simp]
theorem biUnion_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋃ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι]
#align set.bUnion_and' Set.biUnion_and'
@[simp]
theorem biInter_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋂ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iInter_and, @iInter_comm _ ι']
#align set.bInter_and Set.biInter_and
@[simp]
theorem biInter_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋂ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iInter_and, @iInter_comm _ ι]
#align set.bInter_and' Set.biInter_and'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iUnion_iUnion_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋃ (x) (h), s x h = s b (Or.inl rfl) ∪ ⋃ (x) (h : p x), s x (Or.inr h) := by
simp only [iUnion_or, iUnion_union_distrib, iUnion_iUnion_eq_left]
#align set.Union_Union_eq_or_left Set.iUnion_iUnion_eq_or_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iInter_iInter_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋂ (x) (h), s x h = s b (Or.inl rfl) ∩ ⋂ (x) (h : p x), s x (Or.inr h) := by
simp only [iInter_or, iInter_inter_distrib, iInter_iInter_eq_left]
#align set.Inter_Inter_eq_or_left Set.iInter_iInter_eq_or_left
/-! ### Bounded unions and intersections -/
/-- A specialization of `mem_iUnion₂`. -/
theorem mem_biUnion {s : Set α} {t : α → Set β} {x : α} {y : β} (xs : x ∈ s) (ytx : y ∈ t x) :
y ∈ ⋃ x ∈ s, t x :=
mem_iUnion₂_of_mem xs ytx
#align set.mem_bUnion Set.mem_biUnion
/-- A specialization of `mem_iInter₂`. -/
theorem mem_biInter {s : Set α} {t : α → Set β} {y : β} (h : ∀ x ∈ s, y ∈ t x) :
y ∈ ⋂ x ∈ s, t x :=
mem_iInter₂_of_mem h
#align set.mem_bInter Set.mem_biInter
/-- A specialization of `subset_iUnion₂`. -/
theorem subset_biUnion_of_mem {s : Set α} {u : α → Set β} {x : α} (xs : x ∈ s) :
u x ⊆ ⋃ x ∈ s, u x :=
--Porting note: Why is this not just `subset_iUnion₂ x xs`?
@subset_iUnion₂ β α (· ∈ s) (fun i _ => u i) x xs
#align set.subset_bUnion_of_mem Set.subset_biUnion_of_mem
/-- A specialization of `iInter₂_subset`. -/
theorem biInter_subset_of_mem {s : Set α} {t : α → Set β} {x : α} (xs : x ∈ s) :
⋂ x ∈ s, t x ⊆ t x :=
iInter₂_subset x xs
#align set.bInter_subset_of_mem Set.biInter_subset_of_mem
theorem biUnion_subset_biUnion_left {s s' : Set α} {t : α → Set β} (h : s ⊆ s') :
⋃ x ∈ s, t x ⊆ ⋃ x ∈ s', t x :=
iUnion₂_subset fun _ hx => subset_biUnion_of_mem <| h hx
#align set.bUnion_subset_bUnion_left Set.biUnion_subset_biUnion_left
theorem biInter_subset_biInter_left {s s' : Set α} {t : α → Set β} (h : s' ⊆ s) :
⋂ x ∈ s, t x ⊆ ⋂ x ∈ s', t x :=
subset_iInter₂ fun _ hx => biInter_subset_of_mem <| h hx
#align set.bInter_subset_bInter_left Set.biInter_subset_biInter_left
theorem biUnion_mono {s s' : Set α} {t t' : α → Set β} (hs : s' ⊆ s) (h : ∀ x ∈ s, t x ⊆ t' x) :
⋃ x ∈ s', t x ⊆ ⋃ x ∈ s, t' x :=
(biUnion_subset_biUnion_left hs).trans <| iUnion₂_mono h
#align set.bUnion_mono Set.biUnion_mono
theorem biInter_mono {s s' : Set α} {t t' : α → Set β} (hs : s ⊆ s') (h : ∀ x ∈ s, t x ⊆ t' x) :
⋂ x ∈ s', t x ⊆ ⋂ x ∈ s, t' x :=
(biInter_subset_biInter_left hs).trans <| iInter₂_mono h
#align set.bInter_mono Set.biInter_mono
theorem biUnion_eq_iUnion (s : Set α) (t : ∀ x ∈ s, Set β) :
⋃ x ∈ s, t x ‹_› = ⋃ x : s, t x x.2 :=
iSup_subtype'
#align set.bUnion_eq_Union Set.biUnion_eq_iUnion
theorem biInter_eq_iInter (s : Set α) (t : ∀ x ∈ s, Set β) :
⋂ x ∈ s, t x ‹_› = ⋂ x : s, t x x.2 :=
iInf_subtype'
#align set.bInter_eq_Inter Set.biInter_eq_iInter
theorem iUnion_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋃ x : { x // p x }, s x = ⋃ (x) (hx : p x), s ⟨x, hx⟩ :=
iSup_subtype
#align set.Union_subtype Set.iUnion_subtype
theorem iInter_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋂ x : { x // p x }, s x = ⋂ (x) (hx : p x), s ⟨x, hx⟩ :=
iInf_subtype
#align set.Inter_subtype Set.iInter_subtype
theorem biInter_empty (u : α → Set β) : ⋂ x ∈ (∅ : Set α), u x = univ :=
iInf_emptyset
#align set.bInter_empty Set.biInter_empty
theorem biInter_univ (u : α → Set β) : ⋂ x ∈ @univ α, u x = ⋂ x, u x :=
iInf_univ
#align set.bInter_univ Set.biInter_univ
@[simp]
theorem biUnion_self (s : Set α) : ⋃ x ∈ s, s = s :=
Subset.antisymm (iUnion₂_subset fun _ _ => Subset.refl s) fun _ hx => mem_biUnion hx hx
#align set.bUnion_self Set.biUnion_self
@[simp]
theorem iUnion_nonempty_self (s : Set α) : ⋃ _ : s.Nonempty, s = s := by
rw [iUnion_nonempty_index, biUnion_self]
#align set.Union_nonempty_self Set.iUnion_nonempty_self
theorem biInter_singleton (a : α) (s : α → Set β) : ⋂ x ∈ ({a} : Set α), s x = s a :=
iInf_singleton
#align set.bInter_singleton Set.biInter_singleton
theorem biInter_union (s t : Set α) (u : α → Set β) :
⋂ x ∈ s ∪ t, u x = (⋂ x ∈ s, u x) ∩ ⋂ x ∈ t, u x :=
iInf_union
#align set.bInter_union Set.biInter_union
theorem biInter_insert (a : α) (s : Set α) (t : α → Set β) :
⋂ x ∈ insert a s, t x = t a ∩ ⋂ x ∈ s, t x := by simp
#align set.bInter_insert Set.biInter_insert
theorem biInter_pair (a b : α) (s : α → Set β) : ⋂ x ∈ ({a, b} : Set α), s x = s a ∩ s b := by
rw [biInter_insert, biInter_singleton]
#align set.bInter_pair Set.biInter_pair
theorem biInter_inter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, f i ∩ t = (⋂ i ∈ s, f i) ∩ t := by
haveI : Nonempty s := hs.to_subtype
simp [biInter_eq_iInter, ← iInter_inter]
#align set.bInter_inter Set.biInter_inter
theorem inter_biInter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, t ∩ f i = t ∩ ⋂ i ∈ s, f i := by
rw [inter_comm, ← biInter_inter hs]
simp [inter_comm]
#align set.inter_bInter Set.inter_biInter
theorem biUnion_empty (s : α → Set β) : ⋃ x ∈ (∅ : Set α), s x = ∅ :=
iSup_emptyset
#align set.bUnion_empty Set.biUnion_empty
theorem biUnion_univ (s : α → Set β) : ⋃ x ∈ @univ α, s x = ⋃ x, s x :=
iSup_univ
#align set.bUnion_univ Set.biUnion_univ
theorem biUnion_singleton (a : α) (s : α → Set β) : ⋃ x ∈ ({a} : Set α), s x = s a :=
iSup_singleton
#align set.bUnion_singleton Set.biUnion_singleton
@[simp]
theorem biUnion_of_singleton (s : Set α) : ⋃ x ∈ s, {x} = s :=
ext <| by simp
#align set.bUnion_of_singleton Set.biUnion_of_singleton
theorem biUnion_union (s t : Set α) (u : α → Set β) :
⋃ x ∈ s ∪ t, u x = (⋃ x ∈ s, u x) ∪ ⋃ x ∈ t, u x :=
iSup_union
#align set.bUnion_union Set.biUnion_union
@[simp]
theorem iUnion_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋃ i, f i = ⋃ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iUnion_subtype _ _
#align set.Union_coe_set Set.iUnion_coe_set
@[simp]
theorem iInter_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋂ i, f i = ⋂ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iInter_subtype _ _
#align set.Inter_coe_set Set.iInter_coe_set
theorem biUnion_insert (a : α) (s : Set α) (t : α → Set β) :
⋃ x ∈ insert a s, t x = t a ∪ ⋃ x ∈ s, t x := by simp
#align set.bUnion_insert Set.biUnion_insert
theorem biUnion_pair (a b : α) (s : α → Set β) : ⋃ x ∈ ({a, b} : Set α), s x = s a ∪ s b :=
by simp
#align set.bUnion_pair Set.biUnion_pair
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem inter_iUnion₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∩ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ∩ t i j := by simp only [inter_iUnion]
#align set.inter_Union₂ Set.inter_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_inter (s : ∀ i, κ i → Set α) (t : Set α) :
(⋃ (i) (j), s i j) ∩ t = ⋃ (i) (j), s i j ∩ t := by simp_rw [iUnion_inter]
#align set.Union₂_inter Set.iUnion₂_inter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_iInter₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_iInter]
#align set.union_Inter₂ Set.union_iInter₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_union (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [iInter_union]
#align set.Inter₂_union Set.iInter₂_union
theorem mem_sUnion_of_mem {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∈ t) (ht : t ∈ S) :
x ∈ ⋃₀S :=
⟨t, ht, hx⟩
#align set.mem_sUnion_of_mem Set.mem_sUnion_of_mem
-- is this theorem really necessary?
theorem not_mem_of_not_mem_sUnion {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∉ ⋃₀S)
(ht : t ∈ S) : x ∉ t := fun h => hx ⟨t, ht, h⟩
#align set.not_mem_of_not_mem_sUnion Set.not_mem_of_not_mem_sUnion
theorem sInter_subset_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : ⋂₀ S ⊆ t :=
sInf_le tS
#align set.sInter_subset_of_mem Set.sInter_subset_of_mem
theorem subset_sUnion_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : t ⊆ ⋃₀S :=
le_sSup tS
#align set.subset_sUnion_of_mem Set.subset_sUnion_of_mem
theorem subset_sUnion_of_subset {s : Set α} (t : Set (Set α)) (u : Set α) (h₁ : s ⊆ u)
(h₂ : u ∈ t) : s ⊆ ⋃₀t :=
Subset.trans h₁ (subset_sUnion_of_mem h₂)
#align set.subset_sUnion_of_subset Set.subset_sUnion_of_subset
theorem sUnion_subset {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t' ⊆ t) : ⋃₀S ⊆ t :=
sSup_le h
#align set.sUnion_subset Set.sUnion_subset
@[simp]
theorem sUnion_subset_iff {s : Set (Set α)} {t : Set α} : ⋃₀s ⊆ t ↔ ∀ t' ∈ s, t' ⊆ t :=
sSup_le_iff
#align set.sUnion_subset_iff Set.sUnion_subset_iff
/-- `sUnion` is monotone under taking a subset of each set. -/
lemma sUnion_mono_subsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, t ⊆ f t) :
⋃₀ s ⊆ ⋃₀ (f '' s) :=
fun _ ⟨t, htx, hxt⟩ ↦ ⟨f t, mem_image_of_mem f htx, hf t hxt⟩
/-- `sUnion` is monotone under taking a superset of each set. -/
lemma sUnion_mono_supsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, f t ⊆ t) :
⋃₀ (f '' s) ⊆ ⋃₀ s :=
-- If t ∈ f '' s is arbitrary; t = f u for some u : Set α.
fun _ ⟨_, ⟨u, hus, hut⟩, hxt⟩ ↦ ⟨u, hus, (hut ▸ hf u) hxt⟩
theorem subset_sInter {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t ⊆ t') : t ⊆ ⋂₀ S :=
le_sInf h
#align set.subset_sInter Set.subset_sInter
@[simp]
theorem subset_sInter_iff {S : Set (Set α)} {t : Set α} : t ⊆ ⋂₀ S ↔ ∀ t' ∈ S, t ⊆ t' :=
le_sInf_iff
#align set.subset_sInter_iff Set.subset_sInter_iff
@[gcongr]
theorem sUnion_subset_sUnion {S T : Set (Set α)} (h : S ⊆ T) : ⋃₀S ⊆ ⋃₀T :=
sUnion_subset fun _ hs => subset_sUnion_of_mem (h hs)
#align set.sUnion_subset_sUnion Set.sUnion_subset_sUnion
@[gcongr]
theorem sInter_subset_sInter {S T : Set (Set α)} (h : S ⊆ T) : ⋂₀ T ⊆ ⋂₀ S :=
subset_sInter fun _ hs => sInter_subset_of_mem (h hs)
#align set.sInter_subset_sInter Set.sInter_subset_sInter
@[simp]
theorem sUnion_empty : ⋃₀∅ = (∅ : Set α) :=
sSup_empty
#align set.sUnion_empty Set.sUnion_empty
@[simp]
theorem sInter_empty : ⋂₀ ∅ = (univ : Set α) :=
sInf_empty
#align set.sInter_empty Set.sInter_empty
@[simp]
theorem sUnion_singleton (s : Set α) : ⋃₀{s} = s :=
sSup_singleton
#align set.sUnion_singleton Set.sUnion_singleton
@[simp]
theorem sInter_singleton (s : Set α) : ⋂₀ {s} = s :=
sInf_singleton
#align set.sInter_singleton Set.sInter_singleton
@[simp]
theorem sUnion_eq_empty {S : Set (Set α)} : ⋃₀S = ∅ ↔ ∀ s ∈ S, s = ∅ :=
sSup_eq_bot
#align set.sUnion_eq_empty Set.sUnion_eq_empty
@[simp]
theorem sInter_eq_univ {S : Set (Set α)} : ⋂₀ S = univ ↔ ∀ s ∈ S, s = univ :=
sInf_eq_top
#align set.sInter_eq_univ Set.sInter_eq_univ
theorem subset_powerset_iff {s : Set (Set α)} {t : Set α} : s ⊆ 𝒫 t ↔ ⋃₀ s ⊆ t :=
sUnion_subset_iff.symm
/-- `⋃₀` and `𝒫` form a Galois connection. -/
theorem sUnion_powerset_gc :
GaloisConnection (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gc_sSup_Iic
/-- `⋃₀` and `𝒫` form a Galois insertion. -/
def sUnion_powerset_gi :
GaloisInsertion (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gi_sSup_Iic
/-- If all sets in a collection are either `∅` or `Set.univ`, then so is their union. -/
theorem sUnion_mem_empty_univ {S : Set (Set α)} (h : S ⊆ {∅, univ}) :
⋃₀ S ∈ ({∅, univ} : Set (Set α)) := by
simp only [mem_insert_iff, mem_singleton_iff, or_iff_not_imp_left, sUnion_eq_empty, not_forall]
rintro ⟨s, hs, hne⟩
obtain rfl : s = univ := (h hs).resolve_left hne
exact univ_subset_iff.1 <| subset_sUnion_of_mem hs
@[simp]
theorem nonempty_sUnion {S : Set (Set α)} : (⋃₀S).Nonempty ↔ ∃ s ∈ S, Set.Nonempty s := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_sUnion Set.nonempty_sUnion
theorem Nonempty.of_sUnion {s : Set (Set α)} (h : (⋃₀s).Nonempty) : s.Nonempty :=
let ⟨s, hs, _⟩ := nonempty_sUnion.1 h
⟨s, hs⟩
#align set.nonempty.of_sUnion Set.Nonempty.of_sUnion
theorem Nonempty.of_sUnion_eq_univ [Nonempty α] {s : Set (Set α)} (h : ⋃₀s = univ) : s.Nonempty :=
Nonempty.of_sUnion <| h.symm ▸ univ_nonempty
#align set.nonempty.of_sUnion_eq_univ Set.Nonempty.of_sUnion_eq_univ
theorem sUnion_union (S T : Set (Set α)) : ⋃₀(S ∪ T) = ⋃₀S ∪ ⋃₀T :=
sSup_union
#align set.sUnion_union Set.sUnion_union
theorem sInter_union (S T : Set (Set α)) : ⋂₀ (S ∪ T) = ⋂₀ S ∩ ⋂₀ T :=
sInf_union
#align set.sInter_union Set.sInter_union
@[simp]
theorem sUnion_insert (s : Set α) (T : Set (Set α)) : ⋃₀insert s T = s ∪ ⋃₀T :=
sSup_insert
#align set.sUnion_insert Set.sUnion_insert
@[simp]
theorem sInter_insert (s : Set α) (T : Set (Set α)) : ⋂₀ insert s T = s ∩ ⋂₀ T :=
sInf_insert
#align set.sInter_insert Set.sInter_insert
@[simp]
theorem sUnion_diff_singleton_empty (s : Set (Set α)) : ⋃₀(s \ {∅}) = ⋃₀s :=
sSup_diff_singleton_bot s
#align set.sUnion_diff_singleton_empty Set.sUnion_diff_singleton_empty
@[simp]
theorem sInter_diff_singleton_univ (s : Set (Set α)) : ⋂₀ (s \ {univ}) = ⋂₀ s :=
sInf_diff_singleton_top s
#align set.sInter_diff_singleton_univ Set.sInter_diff_singleton_univ
theorem sUnion_pair (s t : Set α) : ⋃₀{s, t} = s ∪ t :=
sSup_pair
#align set.sUnion_pair Set.sUnion_pair
theorem sInter_pair (s t : Set α) : ⋂₀ {s, t} = s ∩ t :=
sInf_pair
#align set.sInter_pair Set.sInter_pair
@[simp]
theorem sUnion_image (f : α → Set β) (s : Set α) : ⋃₀(f '' s) = ⋃ x ∈ s, f x :=
sSup_image
#align set.sUnion_image Set.sUnion_image
@[simp]
theorem sInter_image (f : α → Set β) (s : Set α) : ⋂₀ (f '' s) = ⋂ x ∈ s, f x :=
sInf_image
#align set.sInter_image Set.sInter_image
@[simp]
theorem sUnion_range (f : ι → Set β) : ⋃₀range f = ⋃ x, f x :=
rfl
#align set.sUnion_range Set.sUnion_range
@[simp]
theorem sInter_range (f : ι → Set β) : ⋂₀ range f = ⋂ x, f x :=
rfl
#align set.sInter_range Set.sInter_range
theorem iUnion_eq_univ_iff {f : ι → Set α} : ⋃ i, f i = univ ↔ ∀ x, ∃ i, x ∈ f i := by
simp only [eq_univ_iff_forall, mem_iUnion]
#align set.Union_eq_univ_iff Set.iUnion_eq_univ_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_eq_univ_iff {s : ∀ i, κ i → Set α} :
⋃ (i) (j), s i j = univ ↔ ∀ a, ∃ i j, a ∈ s i j :=
by simp only [iUnion_eq_univ_iff, mem_iUnion]
#align set.Union₂_eq_univ_iff Set.iUnion₂_eq_univ_iff
theorem sUnion_eq_univ_iff {c : Set (Set α)} : ⋃₀c = univ ↔ ∀ a, ∃ b ∈ c, a ∈ b := by
simp only [eq_univ_iff_forall, mem_sUnion]
#align set.sUnion_eq_univ_iff Set.sUnion_eq_univ_iff
-- classical
theorem iInter_eq_empty_iff {f : ι → Set α} : ⋂ i, f i = ∅ ↔ ∀ x, ∃ i, x ∉ f i := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.Inter_eq_empty_iff Set.iInter_eq_empty_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
theorem iInter₂_eq_empty_iff {s : ∀ i, κ i → Set α} :
⋂ (i) (j), s i j = ∅ ↔ ∀ a, ∃ i j, a ∉ s i j := by
simp only [eq_empty_iff_forall_not_mem, mem_iInter, not_forall]
#align set.Inter₂_eq_empty_iff Set.iInter₂_eq_empty_iff
-- classical
theorem sInter_eq_empty_iff {c : Set (Set α)} : ⋂₀ c = ∅ ↔ ∀ a, ∃ b ∈ c, a ∉ b := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.sInter_eq_empty_iff Set.sInter_eq_empty_iff
-- classical
@[simp]
theorem nonempty_iInter {f : ι → Set α} : (⋂ i, f i).Nonempty ↔ ∃ x, ∀ i, x ∈ f i := by
simp [nonempty_iff_ne_empty, iInter_eq_empty_iff]
#align set.nonempty_Inter Set.nonempty_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_iInter₂ {s : ∀ i, κ i → Set α} :
(⋂ (i) (j), s i j).Nonempty ↔ ∃ a, ∀ i j, a ∈ s i j := by
simp
#align set.nonempty_Inter₂ Set.nonempty_iInter₂
-- classical
@[simp]
theorem nonempty_sInter {c : Set (Set α)} : (⋂₀ c).Nonempty ↔ ∃ a, ∀ b ∈ c, a ∈ b := by
simp [nonempty_iff_ne_empty, sInter_eq_empty_iff]
#align set.nonempty_sInter Set.nonempty_sInter
-- classical
theorem compl_sUnion (S : Set (Set α)) : (⋃₀S)ᶜ = ⋂₀ (compl '' S) :=
ext fun x => by simp
#align set.compl_sUnion Set.compl_sUnion
-- classical
theorem sUnion_eq_compl_sInter_compl (S : Set (Set α)) : ⋃₀S = (⋂₀ (compl '' S))ᶜ := by
rw [← compl_compl (⋃₀S), compl_sUnion]
#align set.sUnion_eq_compl_sInter_compl Set.sUnion_eq_compl_sInter_compl
-- classical
theorem compl_sInter (S : Set (Set α)) : (⋂₀ S)ᶜ = ⋃₀(compl '' S) := by
rw [sUnion_eq_compl_sInter_compl, compl_compl_image]
#align set.compl_sInter Set.compl_sInter
-- classical
theorem sInter_eq_compl_sUnion_compl (S : Set (Set α)) : ⋂₀ S = (⋃₀(compl '' S))ᶜ := by
rw [← compl_compl (⋂₀ S), compl_sInter]
#align set.sInter_eq_compl_sUnion_compl Set.sInter_eq_compl_sUnion_compl
theorem inter_empty_of_inter_sUnion_empty {s t : Set α} {S : Set (Set α)} (hs : t ∈ S)
(h : s ∩ ⋃₀S = ∅) : s ∩ t = ∅ :=
eq_empty_of_subset_empty <| by
rw [← h]; exact inter_subset_inter_right _ (subset_sUnion_of_mem hs)
#align set.inter_empty_of_inter_sUnion_empty Set.inter_empty_of_inter_sUnion_empty
theorem range_sigma_eq_iUnion_range {γ : α → Type*} (f : Sigma γ → β) :
range f = ⋃ a, range fun b => f ⟨a, b⟩ :=
Set.ext <| by simp
#align set.range_sigma_eq_Union_range Set.range_sigma_eq_iUnion_range
theorem iUnion_eq_range_sigma (s : α → Set β) : ⋃ i, s i = range fun a : Σi, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_sigma Set.iUnion_eq_range_sigma
theorem iUnion_eq_range_psigma (s : ι → Set β) : ⋃ i, s i = range fun a : Σ'i, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_psigma Set.iUnion_eq_range_psigma
theorem iUnion_image_preimage_sigma_mk_eq_self {ι : Type*} {σ : ι → Type*} (s : Set (Sigma σ)) :
⋃ i, Sigma.mk i '' (Sigma.mk i ⁻¹' s) = s := by
ext x
simp only [mem_iUnion, mem_image, mem_preimage]
constructor
· rintro ⟨i, a, h, rfl⟩
exact h
· intro h
cases' x with i a
exact ⟨i, a, h, rfl⟩
#align set.Union_image_preimage_sigma_mk_eq_self Set.iUnion_image_preimage_sigma_mk_eq_self
theorem Sigma.univ (X : α → Type*) : (Set.univ : Set (Σa, X a)) = ⋃ a, range (Sigma.mk a) :=
Set.ext fun x =>
iff_of_true trivial ⟨range (Sigma.mk x.1), Set.mem_range_self _, x.2, Sigma.eta x⟩
#align set.sigma.univ Set.Sigma.univ
alias sUnion_mono := sUnion_subset_sUnion
#align set.sUnion_mono Set.sUnion_mono
theorem iUnion_subset_iUnion_const {s : Set α} (h : ι → ι₂) : ⋃ _ : ι, s ⊆ ⋃ _ : ι₂, s :=
@iSup_const_mono (Set α) ι ι₂ _ s h
#align set.Union_subset_Union_const Set.iUnion_subset_iUnion_const
@[simp]
theorem iUnion_singleton_eq_range {α β : Type*} (f : α → β) : ⋃ x : α, {f x} = range f := by
ext x
simp [@eq_comm _ x]
#align set.Union_singleton_eq_range Set.iUnion_singleton_eq_range
theorem iUnion_of_singleton (α : Type*) : (⋃ x, {x} : Set α) = univ := by simp [Set.ext_iff]
#align set.Union_of_singleton Set.iUnion_of_singleton
theorem iUnion_of_singleton_coe (s : Set α) : ⋃ i : s, ({(i : α)} : Set α) = s := by simp
#align set.Union_of_singleton_coe Set.iUnion_of_singleton_coe
theorem sUnion_eq_biUnion {s : Set (Set α)} : ⋃₀s = ⋃ (i : Set α) (_ : i ∈ s), i := by
rw [← sUnion_image, image_id']
#align set.sUnion_eq_bUnion Set.sUnion_eq_biUnion
theorem sInter_eq_biInter {s : Set (Set α)} : ⋂₀ s = ⋂ (i : Set α) (_ : i ∈ s), i := by
rw [← sInter_image, image_id']
#align set.sInter_eq_bInter Set.sInter_eq_biInter
theorem sUnion_eq_iUnion {s : Set (Set α)} : ⋃₀s = ⋃ i : s, i := by
simp only [← sUnion_range, Subtype.range_coe]
#align set.sUnion_eq_Union Set.sUnion_eq_iUnion
theorem sInter_eq_iInter {s : Set (Set α)} : ⋂₀ s = ⋂ i : s, i := by
simp only [← sInter_range, Subtype.range_coe]
#align set.sInter_eq_Inter Set.sInter_eq_iInter
@[simp]
theorem iUnion_of_empty [IsEmpty ι] (s : ι → Set α) : ⋃ i, s i = ∅ :=
iSup_of_empty _
#align set.Union_of_empty Set.iUnion_of_empty
@[simp]
theorem iInter_of_empty [IsEmpty ι] (s : ι → Set α) : ⋂ i, s i = univ :=
iInf_of_empty _
#align set.Inter_of_empty Set.iInter_of_empty
theorem union_eq_iUnion {s₁ s₂ : Set α} : s₁ ∪ s₂ = ⋃ b : Bool, cond b s₁ s₂ :=
sup_eq_iSup s₁ s₂
#align set.union_eq_Union Set.union_eq_iUnion
theorem inter_eq_iInter {s₁ s₂ : Set α} : s₁ ∩ s₂ = ⋂ b : Bool, cond b s₁ s₂ :=
inf_eq_iInf s₁ s₂
#align set.inter_eq_Inter Set.inter_eq_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_union_sInter {S T : Set (Set α)} :
⋂₀ S ∪ ⋂₀ T = ⋂ p ∈ S ×ˢ T, (p : Set α × Set α).1 ∪ p.2 :=
sInf_sup_sInf
#align set.sInter_union_sInter Set.sInter_union_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_inter_sUnion {s t : Set (Set α)} :
⋃₀s ∩ ⋃₀t = ⋃ p ∈ s ×ˢ t, (p : Set α × Set α).1 ∩ p.2 :=
sSup_inf_sSup
#align set.sUnion_inter_sUnion Set.sUnion_inter_sUnion
theorem biUnion_iUnion (s : ι → Set α) (t : α → Set β) :
⋃ x ∈ ⋃ i, s i, t x = ⋃ (i) (x ∈ s i), t x := by simp [@iUnion_comm _ ι]
#align set.bUnion_Union Set.biUnion_iUnion
theorem biInter_iUnion (s : ι → Set α) (t : α → Set β) :
⋂ x ∈ ⋃ i, s i, t x = ⋂ (i) (x ∈ s i), t x := by simp [@iInter_comm _ ι]
#align set.bInter_Union Set.biInter_iUnion
theorem sUnion_iUnion (s : ι → Set (Set α)) : ⋃₀⋃ i, s i = ⋃ i, ⋃₀s i := by
simp only [sUnion_eq_biUnion, biUnion_iUnion]
#align set.sUnion_Union Set.sUnion_iUnion
theorem sInter_iUnion (s : ι → Set (Set α)) : ⋂₀ ⋃ i, s i = ⋂ i, ⋂₀ s i := by
simp only [sInter_eq_biInter, biInter_iUnion]
#align set.sInter_Union Set.sInter_iUnion
theorem iUnion_range_eq_sUnion {α β : Type*} (C : Set (Set α)) {f : ∀ s : C, β → (s : Type _)}
(hf : ∀ s : C, Surjective (f s)) : ⋃ y : β, range (fun s : C => (f s y).val) = ⋃₀C := by
ext x; constructor
· rintro ⟨s, ⟨y, rfl⟩, ⟨s, hs⟩, rfl⟩
refine' ⟨_, hs, _⟩
exact (f ⟨s, hs⟩ y).2
· rintro ⟨s, hs, hx⟩
cases' hf ⟨s, hs⟩ ⟨x, hx⟩ with y hy
refine' ⟨_, ⟨y, rfl⟩, ⟨s, hs⟩, _⟩
exact congr_arg Subtype.val hy
#align set.Union_range_eq_sUnion Set.iUnion_range_eq_sUnion
theorem iUnion_range_eq_iUnion (C : ι → Set α) {f : ∀ x : ι, β → C x}
(hf : ∀ x : ι, Surjective (f x)) : ⋃ y : β, range (fun x : ι => (f x y).val) = ⋃ x, C x := by
ext x; rw [mem_iUnion, mem_iUnion]; constructor
· rintro ⟨y, i, rfl⟩
exact ⟨i, (f i y).2⟩
· rintro ⟨i, hx⟩
cases' hf i ⟨x, hx⟩ with y hy
exact ⟨y, i, congr_arg Subtype.val hy⟩
#align set.Union_range_eq_Union Set.iUnion_range_eq_iUnion
theorem union_distrib_iInter_left (s : ι → Set α) (t : Set α) : (t ∪ ⋂ i, s i) = ⋂ i, t ∪ s i :=
sup_iInf_eq _ _
#align set.union_distrib_Inter_left Set.union_distrib_iInter_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_left (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_distrib_iInter_left]
#align set.union_distrib_Inter₂_left Set.union_distrib_iInter₂_left
theorem union_distrib_iInter_right (s : ι → Set α) (t : Set α) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.union_distrib_Inter_right Set.union_distrib_iInter_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_right (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [union_distrib_iInter_right]
#align set.union_distrib_Inter₂_right Set.union_distrib_iInter₂_right
section Function
/-! ### `mapsTo` -/
theorem mapsTo_sUnion {S : Set (Set α)} {t : Set β} {f : α → β} (H : ∀ s ∈ S, MapsTo f s t) :
MapsTo f (⋃₀S) t := fun _ ⟨s, hs, hx⟩ => H s hs hx
#align set.maps_to_sUnion Set.mapsTo_sUnion
theorem mapsTo_iUnion {s : ι → Set α} {t : Set β} {f : α → β} (H : ∀ i, MapsTo f (s i) t) :
MapsTo f (⋃ i, s i) t :=
mapsTo_sUnion <| forall_range_iff.2 H
#align set.maps_to_Union Set.mapsTo_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂ {s : ∀ i, κ i → Set α} {t : Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) t) : MapsTo f (⋃ (i) (j), s i j) t :=
mapsTo_iUnion fun i => mapsTo_iUnion (H i)
#align set.maps_to_Union₂ Set.mapsTo_iUnion₂
theorem mapsTo_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋃ i, s i) (⋃ i, t i) :=
mapsTo_iUnion fun i => (H i).mono (Subset.refl _) (subset_iUnion t i)
#align set.maps_to_Union_Union Set.mapsTo_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
mapsTo_iUnion_iUnion fun i => mapsTo_iUnion_iUnion (H i)
#align set.maps_to_Union₂_Union₂ Set.mapsTo_iUnion₂_iUnion₂
theorem mapsTo_sInter {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, MapsTo f s t) :
MapsTo f s (⋂₀ T) := fun _ hx t ht => H t ht hx
#align set.maps_to_sInter Set.mapsTo_sInter
theorem mapsTo_iInter {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, MapsTo f s (t i)) :
MapsTo f s (⋂ i, t i) := fun _ hx => mem_iInter.2 fun i => H i hx
#align set.maps_to_Inter Set.mapsTo_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f s (t i j)) : MapsTo f s (⋂ (i) (j), t i j) :=
mapsTo_iInter fun i => mapsTo_iInter (H i)
#align set.maps_to_Inter₂ Set.mapsTo_iInter₂
theorem mapsTo_iInter_iInter {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋂ i, s i) (⋂ i, t i) :=
mapsTo_iInter fun i => (H i).mono (iInter_subset s i) (Subset.refl _)
#align set.maps_to_Inter_Inter Set.mapsTo_iInter_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂_iInter₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋂ (i) (j), s i j) (⋂ (i) (j), t i j) :=
mapsTo_iInter_iInter fun i => mapsTo_iInter_iInter (H i)
#align set.maps_to_Inter₂_Inter₂ Set.mapsTo_iInter₂_iInter₂
theorem image_iInter_subset (s : ι → Set α) (f : α → β) : (f '' ⋂ i, s i) ⊆ ⋂ i, f '' s i :=
(mapsTo_iInter_iInter fun i => mapsTo_image f (s i)).image_subset
#align set.image_Inter_subset Set.image_iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂_subset (s : ∀ i, κ i → Set α) (f : α → β) :
(f '' ⋂ (i) (j), s i j) ⊆ ⋂ (i) (j), f '' s i j :=
(mapsTo_iInter₂_iInter₂ fun i hi => mapsTo_image f (s i hi)).image_subset
#align set.image_Inter₂_subset Set.image_iInter₂_subset
theorem image_sInter_subset (S : Set (Set α)) (f : α → β) : f '' ⋂₀ S ⊆ ⋂ s ∈ S, f '' s := by
rw [sInter_eq_biInter]
apply image_iInter₂_subset
#align set.image_sInter_subset Set.image_sInter_subset
/-! ### `restrictPreimage` -/
section
open Function
variable (s : Set β) {f : α → β} {U : ι → Set β} (hU : iUnion U = univ)
theorem injective_iff_injective_of_iUnion_eq_univ :
Injective f ↔ ∀ i, Injective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_injective H, fun H x y e => _⟩
obtain ⟨i, hi⟩ := Set.mem_iUnion.mp
(show f x ∈ Set.iUnion U by rw [hU]; triv)
injection @H i ⟨x, hi⟩ ⟨y, show f y ∈ U i from e ▸ hi⟩ (Subtype.ext e)
#align set.injective_iff_injective_of_Union_eq_univ Set.injective_iff_injective_of_iUnion_eq_univ
theorem surjective_iff_surjective_of_iUnion_eq_univ :
Surjective f ↔ ∀ i, Surjective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_surjective H, fun H x => _⟩
obtain ⟨i, hi⟩ :=
Set.mem_iUnion.mp
(show x ∈ Set.iUnion U by rw [hU]; triv)
exact ⟨_, congr_arg Subtype.val (H i ⟨x, hi⟩).choose_spec⟩
#align set.surjective_iff_surjective_of_Union_eq_univ Set.surjective_iff_surjective_of_iUnion_eq_univ
theorem bijective_iff_bijective_of_iUnion_eq_univ :
Bijective f ↔ ∀ i, Bijective ((U i).restrictPreimage f) := by
rw [Bijective, injective_iff_injective_of_iUnion_eq_univ hU,
surjective_iff_surjective_of_iUnion_eq_univ hU]
simp [Bijective, forall_and]
#align set.bijective_iff_bijective_of_Union_eq_univ Set.bijective_iff_bijective_of_iUnion_eq_univ
end
/-! ### `InjOn` -/
theorem InjOn.image_iInter_eq [Nonempty ι] {s : ι → Set α} {f : α → β} (h : InjOn f (⋃ i, s i)) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
inhabit ι
refine' Subset.antisymm (image_iInter_subset s f) fun y hy => _
simp only [mem_iInter, mem_image_iff_bex] at hy
choose x hx hy using hy
refine' ⟨x default, mem_iInter.2 fun i => _, hy _⟩
suffices x default = x i by
rw [this]
apply hx
replace hx : ∀ i, x i ∈ ⋃ j, s j := fun i => (subset_iUnion _ _) (hx i)
apply h (hx _) (hx _)
simp only [hy]
#align set.inj_on.image_Inter_eq Set.InjOn.image_iInter_eq
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
theorem InjOn.image_biInter_eq {p : ι → Prop} {s : ∀ i, p i → Set α} (hp : ∃ i, p i)
{f : α → β} (h : InjOn f (⋃ (i) (hi), s i hi)) :
(f '' ⋂ (i) (hi), s i hi) = ⋂ (i) (hi), f '' s i hi := by
simp only [iInter, iInf_subtype']
haveI : Nonempty { i // p i } := nonempty_subtype.2 hp
apply InjOn.image_iInter_eq
simpa only [iUnion, iSup_subtype'] using h
#align set.inj_on.image_bInter_eq Set.InjOn.image_biInter_eq
theorem image_iInter {f : α → β} (hf : Bijective f) (s : ι → Set α) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
cases isEmpty_or_nonempty ι
· simp_rw [iInter_of_empty, image_univ_of_surjective hf.surjective]
· exact (hf.injective.injOn _).image_iInter_eq
#align set.image_Inter Set.image_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂ {f : α → β} (hf : Bijective f) (s : ∀ i, κ i → Set α) :
(f '' ⋂ (i) (j), s i j) = ⋂ (i) (j), f '' s i j := by simp_rw [image_iInter hf]
#align set.image_Inter₂ Set.image_iInter₂
theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
(hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i) := by
intro x hx y hy hxy
rcases mem_iUnion.1 hx with ⟨i, hx⟩
rcases mem_iUnion.1 hy with ⟨j, hy⟩
rcases hs i j with ⟨k, hi, hj⟩
exact hf k (hi hx) (hj hy) hxy
#align set.inj_on_Union_of_directed Set.inj_on_iUnion_of_directed
/-! ### `SurjOn` -/
theorem surjOn_sUnion {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, SurjOn f s t) :
SurjOn f s (⋃₀T) := fun _ ⟨t, ht, hx⟩ => H t ht hx
#align set.surj_on_sUnion Set.surjOn_sUnion
theorem surjOn_iUnion {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, SurjOn f s (t i)) :
SurjOn f s (⋃ i, t i) :=
surjOn_sUnion <| forall_range_iff.2 H
#align set.surj_on_Union Set.surjOn_iUnion
theorem surjOn_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) : SurjOn f (⋃ i, s i) (⋃ i, t i) :=
surjOn_iUnion fun i => (H i).mono (subset_iUnion _ _) (Subset.refl _)
#align set.surj_on_Union_Union Set.surjOn_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f s (t i j)) : SurjOn f s (⋃ (i) (j), t i j) :=
surjOn_iUnion fun i => surjOn_iUnion (H i)
#align set.surj_on_Union₂ Set.surjOn_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f (s i j) (t i j)) : SurjOn f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
surjOn_iUnion_iUnion fun i => surjOn_iUnion_iUnion (H i)
#align set.surj_on_Union₂_Union₂ Set.surjOn_iUnion₂_iUnion₂
theorem surjOn_iInter [Nonempty ι] {s : ι → Set α} {t : Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) t) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) t := by
intro y hy
rw [Hinj.image_iInter_eq, mem_iInter]
exact fun i => H i hy
#align set.surj_on_Inter Set.surjOn_iInter
theorem surjOn_iInter_iInter [Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) (⋂ i, t i) :=
surjOn_iInter (fun i => (H i).mono (Subset.refl _) (iInter_subset _ _)) Hinj
#align set.surj_on_Inter_Inter Set.surjOn_iInter_iInter
/-! ### `BijOn` -/
theorem bijOn_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i))
(Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
⟨mapsTo_iUnion_iUnion fun i => (H i).mapsTo, Hinj, surjOn_iUnion_iUnion fun i => (H i).surjOn⟩
#align set.bij_on_Union Set.bijOn_iUnion
theorem bijOn_iInter [hi : Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, BijOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
⟨mapsTo_iInter_iInter fun i => (H i).mapsTo,
hi.elim fun i => (H i).injOn.mono (iInter_subset _ _),
surjOn_iInter_iInter (fun i => (H i).surjOn) Hinj⟩
#align set.bij_on_Inter Set.bijOn_iInter
theorem bijOn_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {t : ι → Set β}
{f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
bijOn_iUnion H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Union_of_directed Set.bijOn_iUnion_of_directed
theorem bijOn_iInter_of_directed [Nonempty ι] {s : ι → Set α} (hs : Directed (· ⊆ ·) s)
{t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
bijOn_iInter H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Inter_of_directed Set.bijOn_iInter_of_directed
end Function
/-! ### `image`, `preimage` -/
section Image
theorem image_iUnion {f : α → β} {s : ι → Set α} : (f '' ⋃ i, s i) = ⋃ i, f '' s i := by
ext1 x
simp only [mem_image, mem_iUnion, ← exists_and_right, ← exists_and_left]
--Porting note: `exists_swap` causes a `simp` loop in Lean4 so we use `rw` instead.
rw [exists_swap]
#align set.image_Union Set.image_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iUnion₂ (f : α → β) (s : ∀ i, κ i → Set α) :
(f '' ⋃ (i) (j), s i j) = ⋃ (i) (j), f '' s i j := by simp_rw [image_iUnion]
#align set.image_Union₂ Set.image_iUnion₂
theorem univ_subtype {p : α → Prop} : (univ : Set (Subtype p)) = ⋃ (x) (h : p x), {⟨x, h⟩} :=
Set.ext fun ⟨x, h⟩ => by simp [h]
#align set.univ_subtype Set.univ_subtype
theorem range_eq_iUnion {ι} (f : ι → α) : range f = ⋃ i, {f i} :=
Set.ext fun a => by simp [@eq_comm α a]
#align set.range_eq_Union Set.range_eq_iUnion
theorem image_eq_iUnion (f : α → β) (s : Set α) : f '' s = ⋃ i ∈ s, {f i} :=
Set.ext fun b => by simp [@eq_comm β b]
#align set.image_eq_Union Set.image_eq_iUnion
theorem biUnion_range {f : ι → α} {g : α → Set β} : ⋃ x ∈ range f, g x = ⋃ y, g (f y) :=
iSup_range
#align set.bUnion_range Set.biUnion_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iUnion_iUnion_eq' {f : ι → α} {g : α → Set β} :
⋃ (x) (y) (_ : f y = x), g x = ⋃ y, g (f y) := by simpa using biUnion_range
#align set.Union_Union_eq' Set.iUnion_iUnion_eq'
theorem biInter_range {f : ι → α} {g : α → Set β} : ⋂ x ∈ range f, g x = ⋂ y, g (f y) :=
iInf_range
#align set.bInter_range Set.biInter_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iInter_iInter_eq' {f : ι → α} {g : α → Set β} :
⋂ (x) (y) (_ : f y = x), g x = ⋂ y, g (f y) := by simpa using biInter_range
#align set.Inter_Inter_eq' Set.iInter_iInter_eq'
variable {s : Set γ} {f : γ → α} {g : α → Set β}
theorem biUnion_image : ⋃ x ∈ f '' s, g x = ⋃ y ∈ s, g (f y) :=
iSup_image
#align set.bUnion_image Set.biUnion_image
theorem biInter_image : ⋂ x ∈ f '' s, g x = ⋂ y ∈ s, g (f y) :=
iInf_image
#align set.bInter_image Set.biInter_image
end Image
section Preimage
theorem monotone_preimage {f : α → β} : Monotone (preimage f) := fun _ _ h => preimage_mono h
#align set.monotone_preimage Set.monotone_preimage
@[simp]
theorem preimage_iUnion {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋃ i, s i) = ⋃ i, f ⁻¹' s i :=
Set.ext <| by simp [preimage]
#align set.preimage_Union Set.preimage_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iUnion₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋃ (i) (j), s i j) = ⋃ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iUnion]
#align set.preimage_Union₂ Set.preimage_iUnion₂
@[simp]
theorem preimage_sUnion {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋃₀s = ⋃ t ∈ s, f ⁻¹' t := by
rw [sUnion_eq_biUnion, preimage_iUnion₂]
#align set.preimage_sUnion Set.preimage_sUnion
theorem preimage_iInter {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋂ i, s i) = ⋂ i, f ⁻¹' s i := by
ext; simp
#align set.preimage_Inter Set.preimage_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iInter₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋂ (i) (j), s i j) = ⋂ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iInter]
#align set.preimage_Inter₂ Set.preimage_iInter₂
@[simp]
theorem preimage_sInter {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋂₀ s = ⋂ t ∈ s, f ⁻¹' t := by
rw [sInter_eq_biInter, preimage_iInter₂]
#align set.preimage_sInter Set.preimage_sInter
@[simp]
theorem biUnion_preimage_singleton (f : α → β) (s : Set β) : ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s := by
rw [← preimage_iUnion₂, biUnion_of_singleton]
#align set.bUnion_preimage_singleton Set.biUnion_preimage_singleton
theorem biUnion_range_preimage_singleton (f : α → β) : ⋃ y ∈ range f, f ⁻¹' {y} = univ := by
rw [biUnion_preimage_singleton, preimage_range]
#align set.bUnion_range_preimage_singleton Set.biUnion_range_preimage_singleton
end Preimage
section Prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion {s : Set α} {t : ι → Set β} : (s ×ˢ ⋃ i, t i) = ⋃ i, s ×ˢ t i := by
ext
simp
#align set.prod_Union Set.prod_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} :
(s ×ˢ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ×ˢ t i j := by simp_rw [prod_iUnion]
#align set.prod_Union₂ Set.prod_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sUnion {s : Set α} {C : Set (Set β)} : s ×ˢ ⋃₀C = ⋃₀((fun t => s ×ˢ t) '' C) := by
simp_rw [sUnion_eq_biUnion, biUnion_image, prod_iUnion₂]
#align set.prod_sUnion Set.prod_sUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_const {s : ι → Set α} {t : Set β} : (⋃ i, s i) ×ˢ t = ⋃ i, s i ×ˢ t := by
ext
simp
#align set.Union_prod_const Set.iUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion₂_prod_const {s : ∀ i, κ i → Set α} {t : Set β} :
(⋃ (i) (j), s i j) ×ˢ t = ⋃ (i) (j), s i j ×ˢ t := by simp_rw [iUnion_prod_const]
#align set.Union₂_prod_const Set.iUnion₂_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_prod_const {C : Set (Set α)} {t : Set β} :
⋃₀C ×ˢ t = ⋃₀((fun s : Set α => s ×ˢ t) '' C) := by
simp only [sUnion_eq_biUnion, iUnion₂_prod_const, biUnion_image]
#align set.sUnion_prod_const Set.sUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod {ι ι' α β} (s : ι → Set α) (t : ι' → Set β) :
⋃ x : ι × ι', s x.1 ×ˢ t x.2 = (⋃ i : ι, s i) ×ˢ ⋃ i : ι', t i := by
ext
simp
#align set.Union_prod Set.iUnion_prod
/-- Analogue of `iSup_prod` for sets. -/
lemma iUnion_prod' (f : β × γ → Set α) : ⋃ x : β × γ, f x = ⋃ (i : β) (j : γ), f (i, j) :=
iSup_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x := by
ext ⟨z, w⟩; simp only [mem_prod, mem_iUnion, exists_imp, and_imp, iff_def]; constructor
· intro x hz hw
exact ⟨⟨x, hz⟩, x, hw⟩
· intro x hz x' hw
exact ⟨x ⊔ x', hs le_sup_left hz, ht le_sup_right hw⟩
#align set.Union_prod_of_monotone Set.iUnion_prod_of_monotone
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter_subset (S : Set (Set α)) (T : Set (Set β)) :
⋂₀ S ×ˢ ⋂₀ T ⊆ ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 :=
subset_iInter₂ fun x hx _ hy => ⟨hy.1 x.1 hx.1, hy.2 x.2 hx.2⟩
#align set.sInter_prod_sInter_subset Set.sInter_prod_sInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 := by
obtain ⟨s₁, h₁⟩ := hS
obtain ⟨s₂, h₂⟩ := hT
refine' Set.Subset.antisymm (sInter_prod_sInter_subset S T) fun x hx => _
rw [mem_iInter₂] at hx
exact ⟨fun s₀ h₀ => (hx (s₀, s₂) ⟨h₀, h₂⟩).1, fun s₀ h₀ => (hx (s₁, s₀) ⟨h₁, h₀⟩).2⟩
#align set.sInter_prod_sInter Set.sInter_prod_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod {S : Set (Set α)} (hS : S.Nonempty) (t : Set β) :
⋂₀ S ×ˢ t = ⋂ s ∈ S, s ×ˢ t := by
rw [← sInter_singleton t, sInter_prod_sInter hS (singleton_nonempty t), sInter_singleton]
simp_rw [prod_singleton, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.sInter_prod Set.sInter_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sInter {T : Set (Set β)} (hT : T.Nonempty) (s : Set α) :
s ×ˢ ⋂₀ T = ⋂ t ∈ T, s ×ˢ t := by
rw [← sInter_singleton s, sInter_prod_sInter (singleton_nonempty s) hT, sInter_singleton]
simp_rw [singleton_prod, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.prod_sInter Set.prod_sInter
end Prod
section Image2
variable (f : α → β → γ) {s : Set α} {t : Set β}
/-- The `Set.image2` version of `Set.image_eq_iUnion` -/
theorem image2_eq_iUnion (s : Set α) (t : Set β) : image2 f s t = ⋃ (i ∈ s) (j ∈ t), {f i j} := by
ext; simp [eq_comm]
#align set.image2_eq_Union Set.image2_eq_iUnion
theorem iUnion_image_left : ⋃ a ∈ s, f a '' t = image2 f s t := by
simp only [image2_eq_iUnion, image_eq_iUnion]
#align set.Union_image_left Set.iUnion_image_left
theorem iUnion_image_right : ⋃ b ∈ t, (f · b) '' s = image2 f s t := by
rw [image2_swap, iUnion_image_left]
#align set.Union_image_right Set.iUnion_image_right
theorem image2_iUnion_left (s : ι → Set α) (t : Set β) :
image2 f (⋃ i, s i) t = ⋃ i, image2 f (s i) t := by
simp only [← image_prod, iUnion_prod_const, image_iUnion]
#align set.image2_Union_left Set.image2_iUnion_left
theorem image2_iUnion_right (s : Set α) (t : ι → Set β) :
image2 f s (⋃ i, t i) = ⋃ i, image2 f s (t i) := by
simp only [← image_prod, prod_iUnion, image_iUnion]
#align set.image2_Union_right Set.image2_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋃ (i) (j), s i j) t = ⋃ (i) (j), image2 f (s i j) t := by simp_rw [image2_iUnion_left]
#align set.image2_Union₂_left Set.image2_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋃ (i) (j), t i j) = ⋃ (i) (j), image2 f s (t i j) :=
by simp_rw [image2_iUnion_right]
#align set.image2_Union₂_right Set.image2_iUnion₂_right
theorem image2_iInter_subset_left (s : ι → Set α) (t : Set β) :
image2 f (⋂ i, s i) t ⊆ ⋂ i, image2 f (s i) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem (hx _) hy
#align set.image2_Inter_subset_left Set.image2_iInter_subset_left
theorem image2_iInter_subset_right (s : Set α) (t : ι → Set β) :
image2 f s (⋂ i, t i) ⊆ ⋂ i, image2 f s (t i) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem hx (hy _)
#align set.image2_Inter_subset_right Set.image2_iInter_subset_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋂ (i) (j), s i j) t ⊆ ⋂ (i) (j), image2 f (s i j) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem (hx _ _) hy
#align set.image2_Inter₂_subset_left Set.image2_iInter₂_subset_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), image2 f s (t i j) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem hx (hy _ _)
#align set.image2_Inter₂_subset_right Set.image2_iInter₂_subset_right
theorem prod_eq_biUnion_left : s ×ˢ t = ⋃ a ∈ s, (fun b => (a, b)) '' t := by
rw [iUnion_image_left, image2_mk_eq_prod]
#align set.prod_eq_bUnion_left Set.prod_eq_biUnion_left
theorem prod_eq_biUnion_right : s ×ˢ t = ⋃ b ∈ t, (fun a => (a, b)) '' s := by
rw [iUnion_image_right, image2_mk_eq_prod]
#align set.prod_eq_bUnion_right Set.prod_eq_biUnion_right
end Image2
section Seq
/-- Given a set `s` of functions `α → β` and `t : Set α`, `seq s t` is the union of `f '' t` over
all `f ∈ s`. -/
def seq (s : Set (α → β)) (t : Set α) : Set β :=
{ b | ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b }
#align set.seq Set.seq
@[simp]
theorem mem_seq_iff {s : Set (α → β)} {t : Set α} {b : β} :
b ∈ seq s t ↔ ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b :=
Iff.rfl
#align set.mem_seq_iff Set.mem_seq_iff
lemma seq_eq_image2 (s : Set (α → β)) (t : Set α) : seq s t = image2 (fun f a ↦ f a) s t := by
ext; simp
theorem seq_def {s : Set (α → β)} {t : Set α} : seq s t = ⋃ f ∈ s, f '' t := by
rw [seq_eq_image2, iUnion_image_left]
#align set.seq_def Set.seq_def
theorem seq_subset {s : Set (α → β)} {t : Set α} {u : Set β} :
seq s t ⊆ u ↔ ∀ f ∈ s, ∀ a ∈ t, (f : α → β) a ∈ u := by
rw [seq_eq_image2, image2_subset_iff]
#align set.seq_subset Set.seq_subset
@[gcongr]
theorem seq_mono {s₀ s₁ : Set (α → β)} {t₀ t₁ : Set α} (hs : s₀ ⊆ s₁) (ht : t₀ ⊆ t₁) :
seq s₀ t₀ ⊆ seq s₁ t₁ := fun _ ⟨f, hf, a, ha, eq⟩ => ⟨f, hs hf, a, ht ha, eq⟩
#align set.seq_mono Set.seq_mono
theorem singleton_seq {f : α → β} {t : Set α} : Set.seq ({f} : Set (α → β)) t = f '' t := by
rw [seq_eq_image2, image2_singleton_left]
#align set.singleton_seq Set.singleton_seq
theorem seq_singleton {s : Set (α → β)} {a : α} : Set.seq s {a} = (fun f : α → β => f a) '' s := by
rw [seq_eq_image2, image2_singleton_right]
#align set.seq_singleton Set.seq_singleton
theorem seq_seq {s : Set (β → γ)} {t : Set (α → β)} {u : Set α} :
seq s (seq t u) = seq (seq ((· ∘ ·) '' s) t) u := by
simp only [seq_eq_image2, image2_image_left, image2_image2_left, image2_image2_right, comp_apply]
#align set.seq_seq Set.seq_seq
theorem image_seq {f : β → γ} {s : Set (α → β)} {t : Set α} :
f '' seq s t = seq ((f ∘ ·) '' s) t := by
rw [← singleton_seq, ← singleton_seq, seq_seq, image_singleton]
#align set.image_seq Set.image_seq
theorem prod_eq_seq {s : Set α} {t : Set β} : s ×ˢ t = (Prod.mk '' s).seq t := by
rw [seq_eq_image2, image2_image_left, image2_mk_eq_prod]
#align set.prod_eq_seq Set.prod_eq_seq
theorem prod_image_seq_comm (s : Set α) (t : Set β) :
(Prod.mk '' s).seq t = seq ((fun b a => (a, b)) '' t) s := by
rw [← prod_eq_seq, ← image_swap_prod, prod_eq_seq, image_seq, ← image_comp]; rfl
#align set.prod_image_seq_comm Set.prod_image_seq_comm
theorem image2_eq_seq (f : α → β → γ) (s : Set α) (t : Set β) : image2 f s t = seq (f '' s) t := by
rw [seq_eq_image2, image2_image_left]
#align set.image2_eq_seq Set.image2_eq_seq
end Seq
section Pi
variable {π : α → Type*}
theorem pi_def (i : Set α) (s : ∀ a, Set (π a)) : pi i s = ⋂ a ∈ i, eval a ⁻¹' s a := by
ext
simp
#align set.pi_def Set.pi_def
theorem univ_pi_eq_iInter (t : ∀ i, Set (π i)) : pi univ t = ⋂ i, eval i ⁻¹' t i := by
simp only [pi_def, iInter_true, mem_univ]
#align set.univ_pi_eq_Inter Set.univ_pi_eq_iInter
theorem pi_diff_pi_subset (i : Set α) (s t : ∀ a, Set (π a)) :
pi i s \ pi i t ⊆ ⋃ a ∈ i, eval a ⁻¹' (s a \ t a) := by
refine' diff_subset_comm.2 fun x hx a ha => _
simp only [mem_diff, mem_pi, mem_iUnion, not_exists, mem_preimage, not_and, not_not,
eval_apply] at hx
exact hx.2 _ ha (hx.1 _ ha)
#align set.pi_diff_pi_subset Set.pi_diff_pi_subset
theorem iUnion_univ_pi {ι : α → Type*} (t : (a : α) → ι a → Set (π a)) :
⋃ x : (a : α) → ι a, pi univ (fun a => t a (x a)) = pi univ fun a => ⋃ j : ι a, t a j := by
ext
simp [Classical.skolem]
#align set.Union_univ_pi Set.iUnion_univ_pi
end Pi
end Set
namespace Function
namespace Surjective
theorem iUnion_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋃ x, g (f x) = ⋃ y, g y :=
hf.iSup_comp g
#align function.surjective.Union_comp Function.Surjective.iUnion_comp
theorem iInter_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋂ x, g (f x) = ⋂ y, g y :=
hf.iInf_comp g
#align function.surjective.Inter_comp Function.Surjective.iInter_comp
end Surjective
end Function
/-!
### Disjoint sets
-/
section Disjoint
variable {s t u : Set α} {f : α → β}
namespace Set
@[simp]
theorem disjoint_iUnion_left {ι : Sort*} {s : ι → Set α} :
Disjoint (⋃ i, s i) t ↔ ∀ i, Disjoint (s i) t :=
iSup_disjoint_iff
#align set.disjoint_Union_left Set.disjoint_iUnion_left
@[simp]
theorem disjoint_iUnion_right {ι : Sort*} {s : ι → Set α} :
Disjoint t (⋃ i, s i) ↔ ∀ i, Disjoint t (s i) :=
disjoint_iSup_iff
#align set.disjoint_Union_right Set.disjoint_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_left {s : ∀ i, κ i → Set α} {t : Set α} :
Disjoint (⋃ (i) (j), s i j) t ↔ ∀ i j, Disjoint (s i j) t :=
iSup₂_disjoint_iff
#align set.disjoint_Union₂_left Set.disjoint_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_right {s : Set α} {t : ∀ i, κ i → Set α} :
Disjoint s (⋃ (i) (j), t i j) ↔ ∀ i j, Disjoint s (t i j) :=
disjoint_iSup₂_iff
#align set.disjoint_Union₂_right Set.disjoint_iUnion₂_right
@[simp]
theorem disjoint_sUnion_left {S : Set (Set α)} {t : Set α} :
Disjoint (⋃₀S) t ↔ ∀ s ∈ S, Disjoint s t :=
sSup_disjoint_iff
#align set.disjoint_sUnion_left Set.disjoint_sUnion_left
@[simp]
theorem disjoint_sUnion_right {s : Set α} {S : Set (Set α)} :
Disjoint s (⋃₀S) ↔ ∀ t ∈ S, Disjoint s t :=
disjoint_sSup_iff
#align set.disjoint_sUnion_right Set.disjoint_sUnion_right
end Set
end Disjoint
/-! ### Intervals -/
namespace Set
lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :
(⋂ i, Iic (f i)).Nonempty ↔ BddBelow (range f) := by
have : (⋂ (i : ι), Iic (f i)) = lowerBounds (range f) := by
ext c; simp [lowerBounds]
simp [this, BddBelow]
lemma nonempty_iInter_Ici_iff [Preorder α] {f : ι → α} :
(⋂ i, Ici (f i)).Nonempty ↔ BddAbove (range f) :=
nonempty_iInter_Iic_iff (α := αᵒᵈ)
variable [CompleteLattice α]
theorem Ici_iSup (f : ι → α) : Ici (⨆ i, f i) = ⋂ i, Ici (f i) :=
ext fun _ => by simp only [mem_Ici, iSup_le_iff, mem_iInter]
#align set.Ici_supr Set.Ici_iSup
theorem Iic_iInf (f : ι → α) : Iic (⨅ i, f i) = ⋂ i, Iic (f i) :=
ext fun _ => by simp only [mem_Iic, le_iInf_iff, mem_iInter]
#align set.Iic_infi Set.Iic_iInf
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Ici_iSup₂ (f : ∀ i, κ i → α) : Ici (⨆ (i) (j), f i j) = ⋂ (i) (j), Ici (f i j) := by
simp_rw [Ici_iSup]
#align set.Ici_supr₂ Set.Ici_iSup₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Iic_iInf₂ (f : ∀ i, κ i → α) : Iic (⨅ (i) (j), f i j) = ⋂ (i) (j), Iic (f i j) := by
simp_rw [Iic_iInf]
#align set.Iic_infi₂ Set.Iic_iInf₂
theorem Ici_sSup (s : Set α) : Ici (sSup s) = ⋂ a ∈ s, Ici a := by rw [sSup_eq_iSup, Ici_iSup₂]
#align set.Ici_Sup Set.Ici_sSup
theorem Iic_sInf (s : Set α) : Iic (sInf s) = ⋂ a ∈ s, Iic a := by rw [sInf_eq_iInf, Iic_iInf₂]
#align set.Iic_Inf Set.Iic_sInf
end Set
namespace Set
variable (t : α → Set β)
theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x := by
simp only [diff_subset_iff, ← biUnion_union]
apply biUnion_subset_biUnion_left
rw [union_diff_self]
|
apply subset_union_right
|
theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x := by
simp only [diff_subset_iff, ← biUnion_union]
apply biUnion_subset_biUnion_left
rw [union_diff_self]
|
Mathlib.Data.Set.Lattice.2291_0.5mONj49h3SYSDwc
|
theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x
|
Mathlib_Data_Set_Lattice
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
t : α → Set β
b : β
hb : b ∈ ⋃ i, t i
⊢ ∃ a, b ∈ t a
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
/-!
# The set lattice
This file provides usual set notation for unions and intersections, a `CompleteLattice` instance
for `Set α`, and some more set constructions.
## Main declarations
* `Set.iUnion`: **i**ndexed **union**. Union of an indexed family of sets.
* `Set.iInter`: **i**ndexed **inter**section. Intersection of an indexed family of sets.
* `Set.sInter`: **s**et **inter**section. Intersection of sets belonging to a set of sets.
* `Set.sUnion`: **s**et **union**. Union of sets belonging to a set of sets.
* `Set.sInter_eq_biInter`, `Set.sUnion_eq_biInter`: Shows that `⋂₀ s = ⋂ x ∈ s, x` and
`⋃₀ s = ⋃ x ∈ s, x`.
* `Set.completeAtomicBooleanAlgebra`: `Set α` is a `CompleteAtomicBooleanAlgebra` with `≤ = ⊆`,
`< = ⊂`, `⊓ = ∩`, `⊔ = ∪`, `⨅ = ⋂`, `⨆ = ⋃` and `\` as the set difference.
See `Set.BooleanAlgebra`.
* `Set.kernImage`: For a function `f : α → β`, `s.kernImage f` is the set of `y` such that
`f ⁻¹ y ⊆ s`.
* `Set.seq`: Union of the image of a set under a **seq**uence of functions. `seq s t` is the union
of `f '' t` over all `f ∈ s`, where `t : Set α` and `s : Set (α → β)`.
* `Set.iUnion_eq_sigma_of_disjoint`: Equivalence between `⋃ i, t i` and `Σ i, t i`, where `t` is an
indexed family of disjoint sets.
## Naming convention
In lemma names,
* `⋃ i, s i` is called `iUnion`
* `⋂ i, s i` is called `iInter`
* `⋃ i j, s i j` is called `iUnion₂`. This is an `iUnion` inside an `iUnion`.
* `⋂ i j, s i j` is called `iInter₂`. This is an `iInter` inside an `iInter`.
* `⋃ i ∈ s, t i` is called `biUnion` for "bounded `iUnion`". This is the special case of `iUnion₂`
where `j : i ∈ s`.
* `⋂ i ∈ s, t i` is called `biInter` for "bounded `iInter`". This is the special case of `iInter₂`
where `j : i ∈ s`.
## Notation
* `⋃`: `Set.iUnion`
* `⋂`: `Set.iInter`
* `⋃₀`: `Set.sUnion`
* `⋂₀`: `Set.sInter`
-/
set_option autoImplicit true
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι₂ : Sort*} {κ κ₁ κ₂ : ι → Sort*} {κ' : ι' → Sort*}
namespace Set
/-! ### Complete lattice and complete Boolean algebra instances -/
instance : InfSet (Set α) :=
⟨fun s => { a | ∀ t ∈ s, a ∈ t }⟩
instance : SupSet (Set α) :=
⟨fun s => { a | ∃ t ∈ s, a ∈ t }⟩
/-- Intersection of a set of sets. -/
def sInter (S : Set (Set α)) : Set α :=
sInf S
#align set.sInter Set.sInter
/-- Notation for `Set.sInter` Intersection of a set of sets. -/
prefix:110 "⋂₀ " => sInter
/-- Union of a set of sets. -/
def sUnion (S : Set (Set α)) : Set α :=
sSup S
#align set.sUnion Set.sUnion
/-- Notation for `Set.sUnion`. Union of a set of sets. -/
prefix:110 "⋃₀ " => sUnion
@[simp]
theorem mem_sInter {x : α} {S : Set (Set α)} : x ∈ ⋂₀ S ↔ ∀ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sInter Set.mem_sInter
@[simp]
theorem mem_sUnion {x : α} {S : Set (Set α)} : x ∈ ⋃₀ S ↔ ∃ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sUnion Set.mem_sUnion
/-- Indexed union of a family of sets -/
def iUnion (s : ι → Set β) : Set β :=
iSup s
#align set.Union Set.iUnion
/-- Indexed intersection of a family of sets -/
def iInter (s : ι → Set β) : Set β :=
iInf s
#align set.Inter Set.iInter
/-- Notation for `Set.iUnion`. Indexed union of a family of sets -/
notation3 "⋃ "(...)", "r:60:(scoped f => iUnion f) => r
/-- Notation for `Set.iInter`. Indexed intersection of a family of sets -/
notation3 "⋂ "(...)", "r:60:(scoped f => iInter f) => r
section delaborators
open Lean Lean.PrettyPrinter.Delaborator
/-- Delaborator for indexed unions. -/
@[delab app.Set.iUnion]
def iUnion_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋃ (_ : $dom), $body)
else if prop || ppTypes then
`(⋃ ($x:ident : $dom), $body)
else
`(⋃ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋃ $x:ident, ⋃ (_ : $y:ident ∈ $s), $body)
| `(⋃ ($x:ident : $_), ⋃ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋃ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
/-- Delaborator for indexed intersections. -/
@[delab app.Set.iInter]
def sInter_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋂ (_ : $dom), $body)
else if prop || ppTypes then
`(⋂ ($x:ident : $dom), $body)
else
`(⋂ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋂ $x:ident, ⋂ (_ : $y:ident ∈ $s), $body)
| `(⋂ ($x:ident : $_), ⋂ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋂ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
end delaborators
@[simp]
theorem sSup_eq_sUnion (S : Set (Set α)) : sSup S = ⋃₀S :=
rfl
#align set.Sup_eq_sUnion Set.sSup_eq_sUnion
@[simp]
theorem sInf_eq_sInter (S : Set (Set α)) : sInf S = ⋂₀ S :=
rfl
#align set.Inf_eq_sInter Set.sInf_eq_sInter
@[simp]
theorem iSup_eq_iUnion (s : ι → Set α) : iSup s = iUnion s :=
rfl
#align set.supr_eq_Union Set.iSup_eq_iUnion
@[simp]
theorem iInf_eq_iInter (s : ι → Set α) : iInf s = iInter s :=
rfl
#align set.infi_eq_Inter Set.iInf_eq_iInter
@[simp]
theorem mem_iUnion {x : α} {s : ι → Set α} : (x ∈ ⋃ i, s i) ↔ ∃ i, x ∈ s i :=
⟨fun ⟨_, ⟨⟨a, (t_eq : s a = _)⟩, (h : x ∈ _)⟩⟩ => ⟨a, t_eq.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨s a, ⟨⟨a, rfl⟩, h⟩⟩⟩
#align set.mem_Union Set.mem_iUnion
@[simp]
theorem mem_iInter {x : α} {s : ι → Set α} : (x ∈ ⋂ i, s i) ↔ ∀ i, x ∈ s i :=
⟨fun (h : ∀ a ∈ { a : Set α | ∃ i, s i = a }, x ∈ a) a => h (s a) ⟨a, rfl⟩,
fun h _ ⟨a, (eq : s a = _)⟩ => eq ▸ h a⟩
#align set.mem_Inter Set.mem_iInter
theorem mem_iUnion₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋃ (i) (j), s i j) ↔ ∃ i j, x ∈ s i j := by
simp_rw [mem_iUnion]
#align set.mem_Union₂ Set.mem_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋂ (i) (j), s i j) ↔ ∀ i j, x ∈ s i j := by
simp_rw [mem_iInter]
#align set.mem_Inter₂ Set.mem_iInter₂
theorem mem_iUnion_of_mem {s : ι → Set α} {a : α} (i : ι) (ha : a ∈ s i) : a ∈ ⋃ i, s i :=
mem_iUnion.2 ⟨i, ha⟩
#align set.mem_Union_of_mem Set.mem_iUnion_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iUnion₂_of_mem {s : ∀ i, κ i → Set α} {a : α} {i : ι} (j : κ i) (ha : a ∈ s i j) :
a ∈ ⋃ (i) (j), s i j :=
mem_iUnion₂.2 ⟨i, j, ha⟩
#align set.mem_Union₂_of_mem Set.mem_iUnion₂_of_mem
theorem mem_iInter_of_mem {s : ι → Set α} {a : α} (h : ∀ i, a ∈ s i) : a ∈ ⋂ i, s i :=
mem_iInter.2 h
#align set.mem_Inter_of_mem Set.mem_iInter_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂_of_mem {s : ∀ i, κ i → Set α} {a : α} (h : ∀ i j, a ∈ s i j) :
a ∈ ⋂ (i) (j), s i j :=
mem_iInter₂.2 h
#align set.mem_Inter₂_of_mem Set.mem_iInter₂_of_mem
instance Set.completeAtomicBooleanAlgebra : CompleteAtomicBooleanAlgebra (Set α) :=
{ instBooleanAlgebraSet with
le_sSup := fun s t t_in a a_in => ⟨t, t_in, a_in⟩
sSup_le := fun s t h a ⟨t', ⟨t'_in, a_in⟩⟩ => h t' t'_in a_in
le_sInf := fun s t h a a_in t' t'_in => h t' t'_in a_in
sInf_le := fun s t t_in a h => h _ t_in
iInf_iSup_eq := by intros; ext; simp [Classical.skolem] }
/-- `kernImage f s` is the set of `y` such that `f ⁻¹ y ⊆ s`. -/
def kernImage (f : α → β) (s : Set α) : Set β :=
{ y | ∀ ⦃x⦄, f x = y → x ∈ s }
#align set.kern_image Set.kernImage
lemma subset_kernImage_iff {f : α → β} : s ⊆ kernImage f t ↔ f ⁻¹' s ⊆ t :=
⟨fun h _ hx ↦ h hx rfl,
fun h _ hx y hy ↦ h (show f y ∈ s from hy.symm ▸ hx)⟩
section GaloisConnection
variable {f : α → β}
protected theorem image_preimage : GaloisConnection (image f) (preimage f) := fun _ _ =>
image_subset_iff
#align set.image_preimage Set.image_preimage
protected theorem preimage_kernImage : GaloisConnection (preimage f) (kernImage f) := fun _ _ =>
subset_kernImage_iff.symm
#align set.preimage_kern_image Set.preimage_kernImage
end GaloisConnection
section kernImage
variable {f : α → β}
lemma kernImage_mono : Monotone (kernImage f) :=
Set.preimage_kernImage.monotone_u
lemma kernImage_eq_compl {s : Set α} : kernImage f s = (f '' sᶜ)ᶜ :=
Set.preimage_kernImage.u_unique (Set.image_preimage.compl)
(fun t ↦ compl_compl (f ⁻¹' t) ▸ Set.preimage_compl)
lemma kernImage_compl {s : Set α} : kernImage f (sᶜ) = (f '' s)ᶜ := by
rw [kernImage_eq_compl, compl_compl]
lemma kernImage_empty : kernImage f ∅ = (range f)ᶜ := by
rw [kernImage_eq_compl, compl_empty, image_univ]
lemma kernImage_preimage_eq_iff {s : Set β} : kernImage f (f ⁻¹' s) = s ↔ (range f)ᶜ ⊆ s := by
rw [kernImage_eq_compl, ← preimage_compl, compl_eq_comm, eq_comm, image_preimage_eq_iff,
compl_subset_comm]
lemma compl_range_subset_kernImage {s : Set α} : (range f)ᶜ ⊆ kernImage f s := by
rw [← kernImage_empty]
exact kernImage_mono (empty_subset _)
lemma kernImage_union_preimage {s : Set α} {t : Set β} :
kernImage f (s ∪ f ⁻¹' t) = kernImage f s ∪ t := by
rw [kernImage_eq_compl, kernImage_eq_compl, compl_union, ← preimage_compl, image_inter_preimage,
compl_inter, compl_compl]
lemma kernImage_preimage_union {s : Set α} {t : Set β} :
kernImage f (f ⁻¹' t ∪ s) = t ∪ kernImage f s := by
rw [union_comm, kernImage_union_preimage, union_comm]
end kernImage
/-! ### Union and intersection over an indexed family of sets -/
instance : OrderTop (Set α) where
top := univ
le_top := by simp
@[congr]
theorem iUnion_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iUnion f₁ = iUnion f₂ :=
iSup_congr_Prop pq f
#align set.Union_congr_Prop Set.iUnion_congr_Prop
@[congr]
theorem iInter_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iInter f₁ = iInter f₂ :=
iInf_congr_Prop pq f
#align set.Inter_congr_Prop Set.iInter_congr_Prop
theorem iUnion_plift_up (f : PLift ι → Set α) : ⋃ i, f (PLift.up i) = ⋃ i, f i :=
iSup_plift_up _
#align set.Union_plift_up Set.iUnion_plift_up
theorem iUnion_plift_down (f : ι → Set α) : ⋃ i, f (PLift.down i) = ⋃ i, f i :=
iSup_plift_down _
#align set.Union_plift_down Set.iUnion_plift_down
theorem iInter_plift_up (f : PLift ι → Set α) : ⋂ i, f (PLift.up i) = ⋂ i, f i :=
iInf_plift_up _
#align set.Inter_plift_up Set.iInter_plift_up
theorem iInter_plift_down (f : ι → Set α) : ⋂ i, f (PLift.down i) = ⋂ i, f i :=
iInf_plift_down _
#align set.Inter_plift_down Set.iInter_plift_down
theorem iUnion_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋃ _ : p, s = if p then s else ∅ :=
iSup_eq_if _
#align set.Union_eq_if Set.iUnion_eq_if
theorem iUnion_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋃ h : p, s h = if h : p then s h else ∅ :=
iSup_eq_dif _
#align set.Union_eq_dif Set.iUnion_eq_dif
theorem iInter_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋂ _ : p, s = if p then s else univ :=
iInf_eq_if _
#align set.Inter_eq_if Set.iInter_eq_if
theorem iInf_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋂ h : p, s h = if h : p then s h else univ :=
_root_.iInf_eq_dif _
#align set.Infi_eq_dif Set.iInf_eq_dif
theorem exists_set_mem_of_union_eq_top {ι : Type*} (t : Set ι) (s : ι → Set β)
(w : ⋃ i ∈ t, s i = ⊤) (x : β) : ∃ i ∈ t, x ∈ s i := by
have p : x ∈ ⊤ := Set.mem_univ x
rw [← w, Set.mem_iUnion] at p
simpa using p
#align set.exists_set_mem_of_union_eq_top Set.exists_set_mem_of_union_eq_top
theorem nonempty_of_union_eq_top_of_nonempty {ι : Type*} (t : Set ι) (s : ι → Set α)
(H : Nonempty α) (w : ⋃ i ∈ t, s i = ⊤) : t.Nonempty := by
obtain ⟨x, m, -⟩ := exists_set_mem_of_union_eq_top t s w H.some
exact ⟨x, m⟩
#align set.nonempty_of_union_eq_top_of_nonempty Set.nonempty_of_union_eq_top_of_nonempty
theorem nonempty_of_nonempty_iUnion
{s : ι → Set α} (h_Union : (⋃ i, s i).Nonempty) : Nonempty ι := by
obtain ⟨x, hx⟩ := h_Union
exact ⟨Classical.choose $ mem_iUnion.mp hx⟩
theorem nonempty_of_nonempty_iUnion_eq_univ
{s : ι → Set α} [Nonempty α] (h_Union : ⋃ i, s i = univ) : Nonempty ι :=
nonempty_of_nonempty_iUnion (s := s) (by simpa only [h_Union] using univ_nonempty)
theorem setOf_exists (p : ι → β → Prop) : { x | ∃ i, p i x } = ⋃ i, { x | p i x } :=
ext fun _ => mem_iUnion.symm
#align set.set_of_exists Set.setOf_exists
theorem setOf_forall (p : ι → β → Prop) : { x | ∀ i, p i x } = ⋂ i, { x | p i x } :=
ext fun _ => mem_iInter.symm
#align set.set_of_forall Set.setOf_forall
theorem iUnion_subset {s : ι → Set α} {t : Set α} (h : ∀ i, s i ⊆ t) : ⋃ i, s i ⊆ t :=
iSup_le h
#align set.Union_subset Set.iUnion_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset {s : ∀ i, κ i → Set α} {t : Set α} (h : ∀ i j, s i j ⊆ t) :
⋃ (i) (j), s i j ⊆ t :=
iUnion_subset fun x => iUnion_subset (h x)
#align set.Union₂_subset Set.iUnion₂_subset
theorem subset_iInter {t : Set β} {s : ι → Set β} (h : ∀ i, t ⊆ s i) : t ⊆ ⋂ i, s i :=
le_iInf h
#align set.subset_Inter Set.subset_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iInter₂ {s : Set α} {t : ∀ i, κ i → Set α} (h : ∀ i j, s ⊆ t i j) :
s ⊆ ⋂ (i) (j), t i j :=
subset_iInter fun x => subset_iInter <| h x
#align set.subset_Inter₂ Set.subset_iInter₂
@[simp]
theorem iUnion_subset_iff {s : ι → Set α} {t : Set α} : ⋃ i, s i ⊆ t ↔ ∀ i, s i ⊆ t :=
⟨fun h _ => Subset.trans (le_iSup s _) h, iUnion_subset⟩
#align set.Union_subset_iff Set.iUnion_subset_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset_iff {s : ∀ i, κ i → Set α} {t : Set α} :
⋃ (i) (j), s i j ⊆ t ↔ ∀ i j, s i j ⊆ t := by simp_rw [iUnion_subset_iff]
#align set.Union₂_subset_iff Set.iUnion₂_subset_iff
@[simp]
theorem subset_iInter_iff {s : Set α} {t : ι → Set α} : (s ⊆ ⋂ i, t i) ↔ ∀ i, s ⊆ t i :=
le_iInf_iff
#align set.subset_Inter_iff Set.subset_iInter_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem subset_iInter₂_iff {s : Set α} {t : ∀ i, κ i → Set α} :
(s ⊆ ⋂ (i) (j), t i j) ↔ ∀ i j, s ⊆ t i j := by simp_rw [subset_iInter_iff]
#align set.subset_Inter₂_iff Set.subset_iInter₂_iff
theorem subset_iUnion : ∀ (s : ι → Set β) (i : ι), s i ⊆ ⋃ i, s i :=
le_iSup
#align set.subset_Union Set.subset_iUnion
theorem iInter_subset : ∀ (s : ι → Set β) (i : ι), ⋂ i, s i ⊆ s i :=
iInf_le
#align set.Inter_subset Set.iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iUnion₂ {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : s i j ⊆ ⋃ (i') (j'), s i' j' :=
le_iSup₂ i j
#align set.subset_Union₂ Set.subset_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_subset {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : ⋂ (i) (j), s i j ⊆ s i j :=
iInf₂_le i j
#align set.Inter₂_subset Set.iInter₂_subset
/-- This rather trivial consequence of `subset_iUnion`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem subset_iUnion_of_subset {s : Set α} {t : ι → Set α} (i : ι) (h : s ⊆ t i) : s ⊆ ⋃ i, t i :=
le_iSup_of_le i h
#align set.subset_Union_of_subset Set.subset_iUnion_of_subset
/-- This rather trivial consequence of `iInter_subset`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem iInter_subset_of_subset {s : ι → Set α} {t : Set α} (i : ι) (h : s i ⊆ t) :
⋂ i, s i ⊆ t :=
iInf_le_of_le i h
#align set.Inter_subset_of_subset Set.iInter_subset_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `subset_iUnion₂` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem subset_iUnion₂_of_subset {s : Set α} {t : ∀ i, κ i → Set α} (i : ι) (j : κ i)
(h : s ⊆ t i j) : s ⊆ ⋃ (i) (j), t i j :=
le_iSup₂_of_le i j h
#align set.subset_Union₂_of_subset Set.subset_iUnion₂_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `iInter₂_subset` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem iInter₂_subset_of_subset {s : ∀ i, κ i → Set α} {t : Set α} (i : ι) (j : κ i)
(h : s i j ⊆ t) : ⋂ (i) (j), s i j ⊆ t :=
iInf₂_le_of_le i j h
#align set.Inter₂_subset_of_subset Set.iInter₂_subset_of_subset
theorem iUnion_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono h
#align set.Union_mono Set.iUnion_mono
@[gcongr]
theorem iUnion_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iUnion s ⊆ iUnion t :=
iSup_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋃ (i) (j), s i j ⊆ ⋃ (i) (j), t i j :=
iSup₂_mono h
#align set.Union₂_mono Set.iUnion₂_mono
theorem iInter_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋂ i, s i ⊆ ⋂ i, t i :=
iInf_mono h
#align set.Inter_mono Set.iInter_mono
@[gcongr]
theorem iInter_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iInter s ⊆ iInter t :=
iInf_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋂ (i) (j), s i j ⊆ ⋂ (i) (j), t i j :=
iInf₂_mono h
#align set.Inter₂_mono Set.iInter₂_mono
theorem iUnion_mono' {s : ι → Set α} {t : ι₂ → Set α} (h : ∀ i, ∃ j, s i ⊆ t j) :
⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono' h
#align set.Union_mono' Set.iUnion_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iUnion₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i j, ∃ i' j', s i j ⊆ t i' j') : ⋃ (i) (j), s i j ⊆ ⋃ (i') (j'), t i' j' :=
iSup₂_mono' h
#align set.Union₂_mono' Set.iUnion₂_mono'
theorem iInter_mono' {s : ι → Set α} {t : ι' → Set α} (h : ∀ j, ∃ i, s i ⊆ t j) :
⋂ i, s i ⊆ ⋂ j, t j :=
Set.subset_iInter fun j =>
let ⟨i, hi⟩ := h j
iInter_subset_of_subset i hi
#align set.Inter_mono' Set.iInter_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iInter₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i' j', ∃ i j, s i j ⊆ t i' j') : ⋂ (i) (j), s i j ⊆ ⋂ (i') (j'), t i' j' :=
subset_iInter₂_iff.2 fun i' j' =>
let ⟨_, _, hst⟩ := h i' j'
(iInter₂_subset _ _).trans hst
#align set.Inter₂_mono' Set.iInter₂_mono'
theorem iUnion₂_subset_iUnion (κ : ι → Sort*) (s : ι → Set α) :
⋃ (i) (_ : κ i), s i ⊆ ⋃ i, s i :=
iUnion_mono fun _ => iUnion_subset fun _ => Subset.rfl
#align set.Union₂_subset_Union Set.iUnion₂_subset_iUnion
theorem iInter_subset_iInter₂ (κ : ι → Sort*) (s : ι → Set α) :
⋂ i, s i ⊆ ⋂ (i) (_ : κ i), s i :=
iInter_mono fun _ => subset_iInter fun _ => Subset.rfl
#align set.Inter_subset_Inter₂ Set.iInter_subset_iInter₂
theorem iUnion_setOf (P : ι → α → Prop) : ⋃ i, { x : α | P i x } = { x : α | ∃ i, P i x } := by
ext
exact mem_iUnion
#align set.Union_set_of Set.iUnion_setOf
theorem iInter_setOf (P : ι → α → Prop) : ⋂ i, { x : α | P i x } = { x : α | ∀ i, P i x } := by
ext
exact mem_iInter
#align set.Inter_set_of Set.iInter_setOf
theorem iUnion_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋃ x, f x = ⋃ y, g y :=
h1.iSup_congr h h2
#align set.Union_congr_of_surjective Set.iUnion_congr_of_surjective
theorem iInter_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋂ x, f x = ⋂ y, g y :=
h1.iInf_congr h h2
#align set.Inter_congr_of_surjective Set.iInter_congr_of_surjective
lemma iUnion_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋃ i, s i = ⋃ i, t i := iSup_congr h
#align set.Union_congr Set.iUnion_congr
lemma iInter_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋂ i, s i = ⋂ i, t i := iInf_congr h
#align set.Inter_congr Set.iInter_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iUnion₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋃ (i) (j), s i j = ⋃ (i) (j), t i j :=
iUnion_congr fun i => iUnion_congr <| h i
#align set.Union₂_congr Set.iUnion₂_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iInter₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋂ (i) (j), s i j = ⋂ (i) (j), t i j :=
iInter_congr fun i => iInter_congr <| h i
#align set.Inter₂_congr Set.iInter₂_congr
section Nonempty
variable [Nonempty ι] {f : ι → Set α} {s : Set α}
lemma iUnion_const (s : Set β) : ⋃ _ : ι, s = s := iSup_const
#align set.Union_const Set.iUnion_const
lemma iInter_const (s : Set β) : ⋂ _ : ι, s = s := iInf_const
#align set.Inter_const Set.iInter_const
lemma iUnion_eq_const (hf : ∀ i, f i = s) : ⋃ i, f i = s :=
(iUnion_congr hf).trans $ iUnion_const _
#align set.Union_eq_const Set.iUnion_eq_const
lemma iInter_eq_const (hf : ∀ i, f i = s) : ⋂ i, f i = s :=
(iInter_congr hf).trans $ iInter_const _
#align set.Inter_eq_const Set.iInter_eq_const
end Nonempty
@[simp]
theorem compl_iUnion (s : ι → Set β) : (⋃ i, s i)ᶜ = ⋂ i, (s i)ᶜ :=
compl_iSup
#align set.compl_Union Set.compl_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iUnion₂ (s : ∀ i, κ i → Set α) : (⋃ (i) (j), s i j)ᶜ = ⋂ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iUnion]
#align set.compl_Union₂ Set.compl_iUnion₂
@[simp]
theorem compl_iInter (s : ι → Set β) : (⋂ i, s i)ᶜ = ⋃ i, (s i)ᶜ :=
compl_iInf
#align set.compl_Inter Set.compl_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iInter₂ (s : ∀ i, κ i → Set α) : (⋂ (i) (j), s i j)ᶜ = ⋃ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iInter]
#align set.compl_Inter₂ Set.compl_iInter₂
-- classical -- complete_boolean_algebra
theorem iUnion_eq_compl_iInter_compl (s : ι → Set β) : ⋃ i, s i = (⋂ i, (s i)ᶜ)ᶜ := by
simp only [compl_iInter, compl_compl]
#align set.Union_eq_compl_Inter_compl Set.iUnion_eq_compl_iInter_compl
-- classical -- complete_boolean_algebra
theorem iInter_eq_compl_iUnion_compl (s : ι → Set β) : ⋂ i, s i = (⋃ i, (s i)ᶜ)ᶜ := by
simp only [compl_iUnion, compl_compl]
#align set.Inter_eq_compl_Union_compl Set.iInter_eq_compl_iUnion_compl
theorem inter_iUnion (s : Set β) (t : ι → Set β) : (s ∩ ⋃ i, t i) = ⋃ i, s ∩ t i :=
inf_iSup_eq _ _
#align set.inter_Union Set.inter_iUnion
theorem iUnion_inter (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∩ s = ⋃ i, t i ∩ s :=
iSup_inf_eq _ _
#align set.Union_inter Set.iUnion_inter
theorem iUnion_union_distrib (s : ι → Set β) (t : ι → Set β) :
⋃ i, s i ∪ t i = (⋃ i, s i) ∪ ⋃ i, t i :=
iSup_sup_eq
#align set.Union_union_distrib Set.iUnion_union_distrib
theorem iInter_inter_distrib (s : ι → Set β) (t : ι → Set β) :
⋂ i, s i ∩ t i = (⋂ i, s i) ∩ ⋂ i, t i :=
iInf_inf_eq
#align set.Inter_inter_distrib Set.iInter_inter_distrib
theorem union_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∪ ⋃ i, t i) = ⋃ i, s ∪ t i :=
sup_iSup
#align set.union_Union Set.union_iUnion
theorem iUnion_union [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∪ s = ⋃ i, t i ∪ s :=
iSup_sup
#align set.Union_union Set.iUnion_union
theorem inter_iInter [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∩ ⋂ i, t i) = ⋂ i, s ∩ t i :=
inf_iInf
#align set.inter_Inter Set.inter_iInter
theorem iInter_inter [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋂ i, t i) ∩ s = ⋂ i, t i ∩ s :=
iInf_inf
#align set.Inter_inter Set.iInter_inter
-- classical
theorem union_iInter (s : Set β) (t : ι → Set β) : (s ∪ ⋂ i, t i) = ⋂ i, s ∪ t i :=
sup_iInf_eq _ _
#align set.union_Inter Set.union_iInter
theorem iInter_union (s : ι → Set β) (t : Set β) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.Inter_union Set.iInter_union
theorem iUnion_diff (s : Set β) (t : ι → Set β) : (⋃ i, t i) \ s = ⋃ i, t i \ s :=
iUnion_inter _ _
#align set.Union_diff Set.iUnion_diff
theorem diff_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s \ ⋃ i, t i) = ⋂ i, s \ t i := by
rw [diff_eq, compl_iUnion, inter_iInter]; rfl
#align set.diff_Union Set.diff_iUnion
theorem diff_iInter (s : Set β) (t : ι → Set β) : (s \ ⋂ i, t i) = ⋃ i, s \ t i := by
rw [diff_eq, compl_iInter, inter_iUnion]; rfl
#align set.diff_Inter Set.diff_iInter
theorem directed_on_iUnion {r} {f : ι → Set α} (hd : Directed (· ⊆ ·) f)
(h : ∀ x, DirectedOn r (f x)) : DirectedOn r (⋃ x, f x) := by
simp only [DirectedOn, exists_prop, mem_iUnion, exists_imp]
exact fun a₁ b₁ fb₁ a₂ b₂ fb₂ =>
let ⟨z, zb₁, zb₂⟩ := hd b₁ b₂
let ⟨x, xf, xa₁, xa₂⟩ := h z a₁ (zb₁ fb₁) a₂ (zb₂ fb₂)
⟨x, ⟨z, xf⟩, xa₁, xa₂⟩
#align set.directed_on_Union Set.directed_on_iUnion
theorem iUnion_inter_subset {ι α} {s t : ι → Set α} : ⋃ i, s i ∩ t i ⊆ (⋃ i, s i) ∩ ⋃ i, t i :=
le_iSup_inf_iSup s t
#align set.Union_inter_subset Set.iUnion_inter_subset
theorem iUnion_inter_of_monotone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_monotone hs ht
#align set.Union_inter_of_monotone Set.iUnion_inter_of_monotone
theorem iUnion_inter_of_antitone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_antitone hs ht
#align set.Union_inter_of_antitone Set.iUnion_inter_of_antitone
theorem iInter_union_of_monotone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_monotone hs ht
#align set.Inter_union_of_monotone Set.iInter_union_of_monotone
theorem iInter_union_of_antitone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_antitone hs ht
#align set.Inter_union_of_antitone Set.iInter_union_of_antitone
/-- An equality version of this lemma is `iUnion_iInter_of_monotone` in `Data.Set.Finite`. -/
theorem iUnion_iInter_subset {s : ι → ι' → Set α} : (⋃ j, ⋂ i, s i j) ⊆ ⋂ i, ⋃ j, s i j :=
iSup_iInf_le_iInf_iSup (flip s)
#align set.Union_Inter_subset Set.iUnion_iInter_subset
theorem iUnion_option {ι} (s : Option ι → Set α) : ⋃ o, s o = s none ∪ ⋃ i, s (some i) :=
iSup_option s
#align set.Union_option Set.iUnion_option
theorem iInter_option {ι} (s : Option ι → Set α) : ⋂ o, s o = s none ∩ ⋂ i, s (some i) :=
iInf_option s
#align set.Inter_option Set.iInter_option
section
variable (p : ι → Prop) [DecidablePred p]
theorem iUnion_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 _ _ _
#align set.Union_dite Set.iUnion_dite
theorem iUnion_ite (f g : ι → Set α) :
⋃ i, (if p i then f i else g i) = (⋃ (i) (_ : p i), f i) ∪ ⋃ (i) (_ : ¬p i), g i :=
iUnion_dite _ _ _
#align set.Union_ite Set.iUnion_ite
theorem iInter_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 :=
iInf_dite _ _ _
#align set.Inter_dite Set.iInter_dite
theorem iInter_ite (f g : ι → Set α) :
⋂ i, (if p i then f i else g i) = (⋂ (i) (_ : p i), f i) ∩ ⋂ (i) (_ : ¬p i), g i :=
iInter_dite _ _ _
#align set.Inter_ite Set.iInter_ite
end
theorem image_projection_prod {ι : Type*} {α : ι → Type*} {v : ∀ i : ι, Set (α i)}
(hv : (pi univ v).Nonempty) (i : ι) :
((fun x : ∀ i : ι, α i => x i) '' ⋂ k, (fun x : ∀ j : ι, α j => x k) ⁻¹' v k) = v i := by
classical
apply Subset.antisymm
· simp [iInter_subset]
· intro y y_in
simp only [mem_image, mem_iInter, mem_preimage]
rcases hv with ⟨z, hz⟩
refine' ⟨Function.update z i y, _, update_same i y z⟩
rw [@forall_update_iff ι α _ z i y fun i t => t ∈ v i]
exact ⟨y_in, fun j _ => by simpa using hz j⟩
#align set.image_projection_prod Set.image_projection_prod
/-! ### Unions and intersections indexed by `Prop` -/
theorem iInter_false {s : False → Set α} : iInter s = univ :=
iInf_false
#align set.Inter_false Set.iInter_false
theorem iUnion_false {s : False → Set α} : iUnion s = ∅ :=
iSup_false
#align set.Union_false Set.iUnion_false
@[simp]
theorem iInter_true {s : True → Set α} : iInter s = s trivial :=
iInf_true
#align set.Inter_true Set.iInter_true
@[simp]
theorem iUnion_true {s : True → Set α} : iUnion s = s trivial :=
iSup_true
#align set.Union_true Set.iUnion_true
@[simp]
theorem iInter_exists {p : ι → Prop} {f : Exists p → Set α} :
⋂ x, f x = ⋂ (i) (h : p i), f ⟨i, h⟩ :=
iInf_exists
#align set.Inter_exists Set.iInter_exists
@[simp]
theorem iUnion_exists {p : ι → Prop} {f : Exists p → Set α} :
⋃ x, f x = ⋃ (i) (h : p i), f ⟨i, h⟩ :=
iSup_exists
#align set.Union_exists Set.iUnion_exists
@[simp]
theorem iUnion_empty : (⋃ _ : ι, ∅ : Set α) = ∅ :=
iSup_bot
#align set.Union_empty Set.iUnion_empty
@[simp]
theorem iInter_univ : (⋂ _ : ι, univ : Set α) = univ :=
iInf_top
#align set.Inter_univ Set.iInter_univ
section
variable {s : ι → Set α}
@[simp]
theorem iUnion_eq_empty : ⋃ i, s i = ∅ ↔ ∀ i, s i = ∅ :=
iSup_eq_bot
#align set.Union_eq_empty Set.iUnion_eq_empty
@[simp]
theorem iInter_eq_univ : ⋂ i, s i = univ ↔ ∀ i, s i = univ :=
iInf_eq_top
#align set.Inter_eq_univ Set.iInter_eq_univ
@[simp]
theorem nonempty_iUnion : (⋃ i, s i).Nonempty ↔ ∃ i, (s i).Nonempty := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_Union Set.nonempty_iUnion
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_biUnion {t : Set α} {s : α → Set β} :
(⋃ i ∈ t, s i).Nonempty ↔ ∃ i ∈ t, (s i).Nonempty := by simp
#align set.nonempty_bUnion Set.nonempty_biUnion
theorem iUnion_nonempty_index (s : Set α) (t : s.Nonempty → Set β) :
⋃ h, t h = ⋃ x ∈ s, t ⟨x, ‹_›⟩ :=
iSup_exists
#align set.Union_nonempty_index Set.iUnion_nonempty_index
end
@[simp]
theorem iInter_iInter_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋂ (x) (h : x = b), s x h = s b rfl :=
iInf_iInf_eq_left
#align set.Inter_Inter_eq_left Set.iInter_iInter_eq_left
@[simp]
theorem iInter_iInter_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋂ (x) (h : b = x), s x h = s b rfl :=
iInf_iInf_eq_right
#align set.Inter_Inter_eq_right Set.iInter_iInter_eq_right
@[simp]
theorem iUnion_iUnion_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋃ (x) (h : x = b), s x h = s b rfl :=
iSup_iSup_eq_left
#align set.Union_Union_eq_left Set.iUnion_iUnion_eq_left
@[simp]
theorem iUnion_iUnion_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋃ (x) (h : b = x), s x h = s b rfl :=
iSup_iSup_eq_right
#align set.Union_Union_eq_right Set.iUnion_iUnion_eq_right
theorem iInter_or {p q : Prop} (s : p ∨ q → Set α) :
⋂ h, s h = (⋂ h : p, s (Or.inl h)) ∩ ⋂ h : q, s (Or.inr h) :=
iInf_or
#align set.Inter_or Set.iInter_or
theorem iUnion_or {p q : Prop} (s : p ∨ q → Set α) :
⋃ h, s h = (⋃ i, s (Or.inl i)) ∪ ⋃ j, s (Or.inr j) :=
iSup_or
#align set.Union_or Set.iUnion_or
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iUnion_and {p q : Prop} (s : p ∧ q → Set α) : ⋃ h, s h = ⋃ (hp) (hq), s ⟨hp, hq⟩ :=
iSup_and
#align set.Union_and Set.iUnion_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iInter_and {p q : Prop} (s : p ∧ q → Set α) : ⋂ h, s h = ⋂ (hp) (hq), s ⟨hp, hq⟩ :=
iInf_and
#align set.Inter_and Set.iInter_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iUnion_comm (s : ι → ι' → Set α) : ⋃ (i) (i'), s i i' = ⋃ (i') (i), s i i' :=
iSup_comm
#align set.Union_comm Set.iUnion_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iInter_comm (s : ι → ι' → Set α) : ⋂ (i) (i'), s i i' = ⋂ (i') (i), s i i' :=
iInf_comm
#align set.Inter_comm Set.iInter_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iUnion₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋃ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋃ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iSup₂_comm _
#align set.Union₂_comm Set.iUnion₂_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iInter₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋂ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋂ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iInf₂_comm _
#align set.Inter₂_comm Set.iInter₂_comm
@[simp]
theorem biUnion_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋃ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι']
#align set.bUnion_and Set.biUnion_and
@[simp]
theorem biUnion_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋃ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι]
#align set.bUnion_and' Set.biUnion_and'
@[simp]
theorem biInter_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋂ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iInter_and, @iInter_comm _ ι']
#align set.bInter_and Set.biInter_and
@[simp]
theorem biInter_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋂ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iInter_and, @iInter_comm _ ι]
#align set.bInter_and' Set.biInter_and'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iUnion_iUnion_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋃ (x) (h), s x h = s b (Or.inl rfl) ∪ ⋃ (x) (h : p x), s x (Or.inr h) := by
simp only [iUnion_or, iUnion_union_distrib, iUnion_iUnion_eq_left]
#align set.Union_Union_eq_or_left Set.iUnion_iUnion_eq_or_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iInter_iInter_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋂ (x) (h), s x h = s b (Or.inl rfl) ∩ ⋂ (x) (h : p x), s x (Or.inr h) := by
simp only [iInter_or, iInter_inter_distrib, iInter_iInter_eq_left]
#align set.Inter_Inter_eq_or_left Set.iInter_iInter_eq_or_left
/-! ### Bounded unions and intersections -/
/-- A specialization of `mem_iUnion₂`. -/
theorem mem_biUnion {s : Set α} {t : α → Set β} {x : α} {y : β} (xs : x ∈ s) (ytx : y ∈ t x) :
y ∈ ⋃ x ∈ s, t x :=
mem_iUnion₂_of_mem xs ytx
#align set.mem_bUnion Set.mem_biUnion
/-- A specialization of `mem_iInter₂`. -/
theorem mem_biInter {s : Set α} {t : α → Set β} {y : β} (h : ∀ x ∈ s, y ∈ t x) :
y ∈ ⋂ x ∈ s, t x :=
mem_iInter₂_of_mem h
#align set.mem_bInter Set.mem_biInter
/-- A specialization of `subset_iUnion₂`. -/
theorem subset_biUnion_of_mem {s : Set α} {u : α → Set β} {x : α} (xs : x ∈ s) :
u x ⊆ ⋃ x ∈ s, u x :=
--Porting note: Why is this not just `subset_iUnion₂ x xs`?
@subset_iUnion₂ β α (· ∈ s) (fun i _ => u i) x xs
#align set.subset_bUnion_of_mem Set.subset_biUnion_of_mem
/-- A specialization of `iInter₂_subset`. -/
theorem biInter_subset_of_mem {s : Set α} {t : α → Set β} {x : α} (xs : x ∈ s) :
⋂ x ∈ s, t x ⊆ t x :=
iInter₂_subset x xs
#align set.bInter_subset_of_mem Set.biInter_subset_of_mem
theorem biUnion_subset_biUnion_left {s s' : Set α} {t : α → Set β} (h : s ⊆ s') :
⋃ x ∈ s, t x ⊆ ⋃ x ∈ s', t x :=
iUnion₂_subset fun _ hx => subset_biUnion_of_mem <| h hx
#align set.bUnion_subset_bUnion_left Set.biUnion_subset_biUnion_left
theorem biInter_subset_biInter_left {s s' : Set α} {t : α → Set β} (h : s' ⊆ s) :
⋂ x ∈ s, t x ⊆ ⋂ x ∈ s', t x :=
subset_iInter₂ fun _ hx => biInter_subset_of_mem <| h hx
#align set.bInter_subset_bInter_left Set.biInter_subset_biInter_left
theorem biUnion_mono {s s' : Set α} {t t' : α → Set β} (hs : s' ⊆ s) (h : ∀ x ∈ s, t x ⊆ t' x) :
⋃ x ∈ s', t x ⊆ ⋃ x ∈ s, t' x :=
(biUnion_subset_biUnion_left hs).trans <| iUnion₂_mono h
#align set.bUnion_mono Set.biUnion_mono
theorem biInter_mono {s s' : Set α} {t t' : α → Set β} (hs : s ⊆ s') (h : ∀ x ∈ s, t x ⊆ t' x) :
⋂ x ∈ s', t x ⊆ ⋂ x ∈ s, t' x :=
(biInter_subset_biInter_left hs).trans <| iInter₂_mono h
#align set.bInter_mono Set.biInter_mono
theorem biUnion_eq_iUnion (s : Set α) (t : ∀ x ∈ s, Set β) :
⋃ x ∈ s, t x ‹_› = ⋃ x : s, t x x.2 :=
iSup_subtype'
#align set.bUnion_eq_Union Set.biUnion_eq_iUnion
theorem biInter_eq_iInter (s : Set α) (t : ∀ x ∈ s, Set β) :
⋂ x ∈ s, t x ‹_› = ⋂ x : s, t x x.2 :=
iInf_subtype'
#align set.bInter_eq_Inter Set.biInter_eq_iInter
theorem iUnion_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋃ x : { x // p x }, s x = ⋃ (x) (hx : p x), s ⟨x, hx⟩ :=
iSup_subtype
#align set.Union_subtype Set.iUnion_subtype
theorem iInter_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋂ x : { x // p x }, s x = ⋂ (x) (hx : p x), s ⟨x, hx⟩ :=
iInf_subtype
#align set.Inter_subtype Set.iInter_subtype
theorem biInter_empty (u : α → Set β) : ⋂ x ∈ (∅ : Set α), u x = univ :=
iInf_emptyset
#align set.bInter_empty Set.biInter_empty
theorem biInter_univ (u : α → Set β) : ⋂ x ∈ @univ α, u x = ⋂ x, u x :=
iInf_univ
#align set.bInter_univ Set.biInter_univ
@[simp]
theorem biUnion_self (s : Set α) : ⋃ x ∈ s, s = s :=
Subset.antisymm (iUnion₂_subset fun _ _ => Subset.refl s) fun _ hx => mem_biUnion hx hx
#align set.bUnion_self Set.biUnion_self
@[simp]
theorem iUnion_nonempty_self (s : Set α) : ⋃ _ : s.Nonempty, s = s := by
rw [iUnion_nonempty_index, biUnion_self]
#align set.Union_nonempty_self Set.iUnion_nonempty_self
theorem biInter_singleton (a : α) (s : α → Set β) : ⋂ x ∈ ({a} : Set α), s x = s a :=
iInf_singleton
#align set.bInter_singleton Set.biInter_singleton
theorem biInter_union (s t : Set α) (u : α → Set β) :
⋂ x ∈ s ∪ t, u x = (⋂ x ∈ s, u x) ∩ ⋂ x ∈ t, u x :=
iInf_union
#align set.bInter_union Set.biInter_union
theorem biInter_insert (a : α) (s : Set α) (t : α → Set β) :
⋂ x ∈ insert a s, t x = t a ∩ ⋂ x ∈ s, t x := by simp
#align set.bInter_insert Set.biInter_insert
theorem biInter_pair (a b : α) (s : α → Set β) : ⋂ x ∈ ({a, b} : Set α), s x = s a ∩ s b := by
rw [biInter_insert, biInter_singleton]
#align set.bInter_pair Set.biInter_pair
theorem biInter_inter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, f i ∩ t = (⋂ i ∈ s, f i) ∩ t := by
haveI : Nonempty s := hs.to_subtype
simp [biInter_eq_iInter, ← iInter_inter]
#align set.bInter_inter Set.biInter_inter
theorem inter_biInter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, t ∩ f i = t ∩ ⋂ i ∈ s, f i := by
rw [inter_comm, ← biInter_inter hs]
simp [inter_comm]
#align set.inter_bInter Set.inter_biInter
theorem biUnion_empty (s : α → Set β) : ⋃ x ∈ (∅ : Set α), s x = ∅ :=
iSup_emptyset
#align set.bUnion_empty Set.biUnion_empty
theorem biUnion_univ (s : α → Set β) : ⋃ x ∈ @univ α, s x = ⋃ x, s x :=
iSup_univ
#align set.bUnion_univ Set.biUnion_univ
theorem biUnion_singleton (a : α) (s : α → Set β) : ⋃ x ∈ ({a} : Set α), s x = s a :=
iSup_singleton
#align set.bUnion_singleton Set.biUnion_singleton
@[simp]
theorem biUnion_of_singleton (s : Set α) : ⋃ x ∈ s, {x} = s :=
ext <| by simp
#align set.bUnion_of_singleton Set.biUnion_of_singleton
theorem biUnion_union (s t : Set α) (u : α → Set β) :
⋃ x ∈ s ∪ t, u x = (⋃ x ∈ s, u x) ∪ ⋃ x ∈ t, u x :=
iSup_union
#align set.bUnion_union Set.biUnion_union
@[simp]
theorem iUnion_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋃ i, f i = ⋃ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iUnion_subtype _ _
#align set.Union_coe_set Set.iUnion_coe_set
@[simp]
theorem iInter_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋂ i, f i = ⋂ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iInter_subtype _ _
#align set.Inter_coe_set Set.iInter_coe_set
theorem biUnion_insert (a : α) (s : Set α) (t : α → Set β) :
⋃ x ∈ insert a s, t x = t a ∪ ⋃ x ∈ s, t x := by simp
#align set.bUnion_insert Set.biUnion_insert
theorem biUnion_pair (a b : α) (s : α → Set β) : ⋃ x ∈ ({a, b} : Set α), s x = s a ∪ s b :=
by simp
#align set.bUnion_pair Set.biUnion_pair
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem inter_iUnion₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∩ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ∩ t i j := by simp only [inter_iUnion]
#align set.inter_Union₂ Set.inter_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_inter (s : ∀ i, κ i → Set α) (t : Set α) :
(⋃ (i) (j), s i j) ∩ t = ⋃ (i) (j), s i j ∩ t := by simp_rw [iUnion_inter]
#align set.Union₂_inter Set.iUnion₂_inter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_iInter₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_iInter]
#align set.union_Inter₂ Set.union_iInter₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_union (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [iInter_union]
#align set.Inter₂_union Set.iInter₂_union
theorem mem_sUnion_of_mem {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∈ t) (ht : t ∈ S) :
x ∈ ⋃₀S :=
⟨t, ht, hx⟩
#align set.mem_sUnion_of_mem Set.mem_sUnion_of_mem
-- is this theorem really necessary?
theorem not_mem_of_not_mem_sUnion {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∉ ⋃₀S)
(ht : t ∈ S) : x ∉ t := fun h => hx ⟨t, ht, h⟩
#align set.not_mem_of_not_mem_sUnion Set.not_mem_of_not_mem_sUnion
theorem sInter_subset_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : ⋂₀ S ⊆ t :=
sInf_le tS
#align set.sInter_subset_of_mem Set.sInter_subset_of_mem
theorem subset_sUnion_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : t ⊆ ⋃₀S :=
le_sSup tS
#align set.subset_sUnion_of_mem Set.subset_sUnion_of_mem
theorem subset_sUnion_of_subset {s : Set α} (t : Set (Set α)) (u : Set α) (h₁ : s ⊆ u)
(h₂ : u ∈ t) : s ⊆ ⋃₀t :=
Subset.trans h₁ (subset_sUnion_of_mem h₂)
#align set.subset_sUnion_of_subset Set.subset_sUnion_of_subset
theorem sUnion_subset {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t' ⊆ t) : ⋃₀S ⊆ t :=
sSup_le h
#align set.sUnion_subset Set.sUnion_subset
@[simp]
theorem sUnion_subset_iff {s : Set (Set α)} {t : Set α} : ⋃₀s ⊆ t ↔ ∀ t' ∈ s, t' ⊆ t :=
sSup_le_iff
#align set.sUnion_subset_iff Set.sUnion_subset_iff
/-- `sUnion` is monotone under taking a subset of each set. -/
lemma sUnion_mono_subsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, t ⊆ f t) :
⋃₀ s ⊆ ⋃₀ (f '' s) :=
fun _ ⟨t, htx, hxt⟩ ↦ ⟨f t, mem_image_of_mem f htx, hf t hxt⟩
/-- `sUnion` is monotone under taking a superset of each set. -/
lemma sUnion_mono_supsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, f t ⊆ t) :
⋃₀ (f '' s) ⊆ ⋃₀ s :=
-- If t ∈ f '' s is arbitrary; t = f u for some u : Set α.
fun _ ⟨_, ⟨u, hus, hut⟩, hxt⟩ ↦ ⟨u, hus, (hut ▸ hf u) hxt⟩
theorem subset_sInter {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t ⊆ t') : t ⊆ ⋂₀ S :=
le_sInf h
#align set.subset_sInter Set.subset_sInter
@[simp]
theorem subset_sInter_iff {S : Set (Set α)} {t : Set α} : t ⊆ ⋂₀ S ↔ ∀ t' ∈ S, t ⊆ t' :=
le_sInf_iff
#align set.subset_sInter_iff Set.subset_sInter_iff
@[gcongr]
theorem sUnion_subset_sUnion {S T : Set (Set α)} (h : S ⊆ T) : ⋃₀S ⊆ ⋃₀T :=
sUnion_subset fun _ hs => subset_sUnion_of_mem (h hs)
#align set.sUnion_subset_sUnion Set.sUnion_subset_sUnion
@[gcongr]
theorem sInter_subset_sInter {S T : Set (Set α)} (h : S ⊆ T) : ⋂₀ T ⊆ ⋂₀ S :=
subset_sInter fun _ hs => sInter_subset_of_mem (h hs)
#align set.sInter_subset_sInter Set.sInter_subset_sInter
@[simp]
theorem sUnion_empty : ⋃₀∅ = (∅ : Set α) :=
sSup_empty
#align set.sUnion_empty Set.sUnion_empty
@[simp]
theorem sInter_empty : ⋂₀ ∅ = (univ : Set α) :=
sInf_empty
#align set.sInter_empty Set.sInter_empty
@[simp]
theorem sUnion_singleton (s : Set α) : ⋃₀{s} = s :=
sSup_singleton
#align set.sUnion_singleton Set.sUnion_singleton
@[simp]
theorem sInter_singleton (s : Set α) : ⋂₀ {s} = s :=
sInf_singleton
#align set.sInter_singleton Set.sInter_singleton
@[simp]
theorem sUnion_eq_empty {S : Set (Set α)} : ⋃₀S = ∅ ↔ ∀ s ∈ S, s = ∅ :=
sSup_eq_bot
#align set.sUnion_eq_empty Set.sUnion_eq_empty
@[simp]
theorem sInter_eq_univ {S : Set (Set α)} : ⋂₀ S = univ ↔ ∀ s ∈ S, s = univ :=
sInf_eq_top
#align set.sInter_eq_univ Set.sInter_eq_univ
theorem subset_powerset_iff {s : Set (Set α)} {t : Set α} : s ⊆ 𝒫 t ↔ ⋃₀ s ⊆ t :=
sUnion_subset_iff.symm
/-- `⋃₀` and `𝒫` form a Galois connection. -/
theorem sUnion_powerset_gc :
GaloisConnection (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gc_sSup_Iic
/-- `⋃₀` and `𝒫` form a Galois insertion. -/
def sUnion_powerset_gi :
GaloisInsertion (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gi_sSup_Iic
/-- If all sets in a collection are either `∅` or `Set.univ`, then so is their union. -/
theorem sUnion_mem_empty_univ {S : Set (Set α)} (h : S ⊆ {∅, univ}) :
⋃₀ S ∈ ({∅, univ} : Set (Set α)) := by
simp only [mem_insert_iff, mem_singleton_iff, or_iff_not_imp_left, sUnion_eq_empty, not_forall]
rintro ⟨s, hs, hne⟩
obtain rfl : s = univ := (h hs).resolve_left hne
exact univ_subset_iff.1 <| subset_sUnion_of_mem hs
@[simp]
theorem nonempty_sUnion {S : Set (Set α)} : (⋃₀S).Nonempty ↔ ∃ s ∈ S, Set.Nonempty s := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_sUnion Set.nonempty_sUnion
theorem Nonempty.of_sUnion {s : Set (Set α)} (h : (⋃₀s).Nonempty) : s.Nonempty :=
let ⟨s, hs, _⟩ := nonempty_sUnion.1 h
⟨s, hs⟩
#align set.nonempty.of_sUnion Set.Nonempty.of_sUnion
theorem Nonempty.of_sUnion_eq_univ [Nonempty α] {s : Set (Set α)} (h : ⋃₀s = univ) : s.Nonempty :=
Nonempty.of_sUnion <| h.symm ▸ univ_nonempty
#align set.nonempty.of_sUnion_eq_univ Set.Nonempty.of_sUnion_eq_univ
theorem sUnion_union (S T : Set (Set α)) : ⋃₀(S ∪ T) = ⋃₀S ∪ ⋃₀T :=
sSup_union
#align set.sUnion_union Set.sUnion_union
theorem sInter_union (S T : Set (Set α)) : ⋂₀ (S ∪ T) = ⋂₀ S ∩ ⋂₀ T :=
sInf_union
#align set.sInter_union Set.sInter_union
@[simp]
theorem sUnion_insert (s : Set α) (T : Set (Set α)) : ⋃₀insert s T = s ∪ ⋃₀T :=
sSup_insert
#align set.sUnion_insert Set.sUnion_insert
@[simp]
theorem sInter_insert (s : Set α) (T : Set (Set α)) : ⋂₀ insert s T = s ∩ ⋂₀ T :=
sInf_insert
#align set.sInter_insert Set.sInter_insert
@[simp]
theorem sUnion_diff_singleton_empty (s : Set (Set α)) : ⋃₀(s \ {∅}) = ⋃₀s :=
sSup_diff_singleton_bot s
#align set.sUnion_diff_singleton_empty Set.sUnion_diff_singleton_empty
@[simp]
theorem sInter_diff_singleton_univ (s : Set (Set α)) : ⋂₀ (s \ {univ}) = ⋂₀ s :=
sInf_diff_singleton_top s
#align set.sInter_diff_singleton_univ Set.sInter_diff_singleton_univ
theorem sUnion_pair (s t : Set α) : ⋃₀{s, t} = s ∪ t :=
sSup_pair
#align set.sUnion_pair Set.sUnion_pair
theorem sInter_pair (s t : Set α) : ⋂₀ {s, t} = s ∩ t :=
sInf_pair
#align set.sInter_pair Set.sInter_pair
@[simp]
theorem sUnion_image (f : α → Set β) (s : Set α) : ⋃₀(f '' s) = ⋃ x ∈ s, f x :=
sSup_image
#align set.sUnion_image Set.sUnion_image
@[simp]
theorem sInter_image (f : α → Set β) (s : Set α) : ⋂₀ (f '' s) = ⋂ x ∈ s, f x :=
sInf_image
#align set.sInter_image Set.sInter_image
@[simp]
theorem sUnion_range (f : ι → Set β) : ⋃₀range f = ⋃ x, f x :=
rfl
#align set.sUnion_range Set.sUnion_range
@[simp]
theorem sInter_range (f : ι → Set β) : ⋂₀ range f = ⋂ x, f x :=
rfl
#align set.sInter_range Set.sInter_range
theorem iUnion_eq_univ_iff {f : ι → Set α} : ⋃ i, f i = univ ↔ ∀ x, ∃ i, x ∈ f i := by
simp only [eq_univ_iff_forall, mem_iUnion]
#align set.Union_eq_univ_iff Set.iUnion_eq_univ_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_eq_univ_iff {s : ∀ i, κ i → Set α} :
⋃ (i) (j), s i j = univ ↔ ∀ a, ∃ i j, a ∈ s i j :=
by simp only [iUnion_eq_univ_iff, mem_iUnion]
#align set.Union₂_eq_univ_iff Set.iUnion₂_eq_univ_iff
theorem sUnion_eq_univ_iff {c : Set (Set α)} : ⋃₀c = univ ↔ ∀ a, ∃ b ∈ c, a ∈ b := by
simp only [eq_univ_iff_forall, mem_sUnion]
#align set.sUnion_eq_univ_iff Set.sUnion_eq_univ_iff
-- classical
theorem iInter_eq_empty_iff {f : ι → Set α} : ⋂ i, f i = ∅ ↔ ∀ x, ∃ i, x ∉ f i := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.Inter_eq_empty_iff Set.iInter_eq_empty_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
theorem iInter₂_eq_empty_iff {s : ∀ i, κ i → Set α} :
⋂ (i) (j), s i j = ∅ ↔ ∀ a, ∃ i j, a ∉ s i j := by
simp only [eq_empty_iff_forall_not_mem, mem_iInter, not_forall]
#align set.Inter₂_eq_empty_iff Set.iInter₂_eq_empty_iff
-- classical
theorem sInter_eq_empty_iff {c : Set (Set α)} : ⋂₀ c = ∅ ↔ ∀ a, ∃ b ∈ c, a ∉ b := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.sInter_eq_empty_iff Set.sInter_eq_empty_iff
-- classical
@[simp]
theorem nonempty_iInter {f : ι → Set α} : (⋂ i, f i).Nonempty ↔ ∃ x, ∀ i, x ∈ f i := by
simp [nonempty_iff_ne_empty, iInter_eq_empty_iff]
#align set.nonempty_Inter Set.nonempty_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_iInter₂ {s : ∀ i, κ i → Set α} :
(⋂ (i) (j), s i j).Nonempty ↔ ∃ a, ∀ i j, a ∈ s i j := by
simp
#align set.nonempty_Inter₂ Set.nonempty_iInter₂
-- classical
@[simp]
theorem nonempty_sInter {c : Set (Set α)} : (⋂₀ c).Nonempty ↔ ∃ a, ∀ b ∈ c, a ∈ b := by
simp [nonempty_iff_ne_empty, sInter_eq_empty_iff]
#align set.nonempty_sInter Set.nonempty_sInter
-- classical
theorem compl_sUnion (S : Set (Set α)) : (⋃₀S)ᶜ = ⋂₀ (compl '' S) :=
ext fun x => by simp
#align set.compl_sUnion Set.compl_sUnion
-- classical
theorem sUnion_eq_compl_sInter_compl (S : Set (Set α)) : ⋃₀S = (⋂₀ (compl '' S))ᶜ := by
rw [← compl_compl (⋃₀S), compl_sUnion]
#align set.sUnion_eq_compl_sInter_compl Set.sUnion_eq_compl_sInter_compl
-- classical
theorem compl_sInter (S : Set (Set α)) : (⋂₀ S)ᶜ = ⋃₀(compl '' S) := by
rw [sUnion_eq_compl_sInter_compl, compl_compl_image]
#align set.compl_sInter Set.compl_sInter
-- classical
theorem sInter_eq_compl_sUnion_compl (S : Set (Set α)) : ⋂₀ S = (⋃₀(compl '' S))ᶜ := by
rw [← compl_compl (⋂₀ S), compl_sInter]
#align set.sInter_eq_compl_sUnion_compl Set.sInter_eq_compl_sUnion_compl
theorem inter_empty_of_inter_sUnion_empty {s t : Set α} {S : Set (Set α)} (hs : t ∈ S)
(h : s ∩ ⋃₀S = ∅) : s ∩ t = ∅ :=
eq_empty_of_subset_empty <| by
rw [← h]; exact inter_subset_inter_right _ (subset_sUnion_of_mem hs)
#align set.inter_empty_of_inter_sUnion_empty Set.inter_empty_of_inter_sUnion_empty
theorem range_sigma_eq_iUnion_range {γ : α → Type*} (f : Sigma γ → β) :
range f = ⋃ a, range fun b => f ⟨a, b⟩ :=
Set.ext <| by simp
#align set.range_sigma_eq_Union_range Set.range_sigma_eq_iUnion_range
theorem iUnion_eq_range_sigma (s : α → Set β) : ⋃ i, s i = range fun a : Σi, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_sigma Set.iUnion_eq_range_sigma
theorem iUnion_eq_range_psigma (s : ι → Set β) : ⋃ i, s i = range fun a : Σ'i, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_psigma Set.iUnion_eq_range_psigma
theorem iUnion_image_preimage_sigma_mk_eq_self {ι : Type*} {σ : ι → Type*} (s : Set (Sigma σ)) :
⋃ i, Sigma.mk i '' (Sigma.mk i ⁻¹' s) = s := by
ext x
simp only [mem_iUnion, mem_image, mem_preimage]
constructor
· rintro ⟨i, a, h, rfl⟩
exact h
· intro h
cases' x with i a
exact ⟨i, a, h, rfl⟩
#align set.Union_image_preimage_sigma_mk_eq_self Set.iUnion_image_preimage_sigma_mk_eq_self
theorem Sigma.univ (X : α → Type*) : (Set.univ : Set (Σa, X a)) = ⋃ a, range (Sigma.mk a) :=
Set.ext fun x =>
iff_of_true trivial ⟨range (Sigma.mk x.1), Set.mem_range_self _, x.2, Sigma.eta x⟩
#align set.sigma.univ Set.Sigma.univ
alias sUnion_mono := sUnion_subset_sUnion
#align set.sUnion_mono Set.sUnion_mono
theorem iUnion_subset_iUnion_const {s : Set α} (h : ι → ι₂) : ⋃ _ : ι, s ⊆ ⋃ _ : ι₂, s :=
@iSup_const_mono (Set α) ι ι₂ _ s h
#align set.Union_subset_Union_const Set.iUnion_subset_iUnion_const
@[simp]
theorem iUnion_singleton_eq_range {α β : Type*} (f : α → β) : ⋃ x : α, {f x} = range f := by
ext x
simp [@eq_comm _ x]
#align set.Union_singleton_eq_range Set.iUnion_singleton_eq_range
theorem iUnion_of_singleton (α : Type*) : (⋃ x, {x} : Set α) = univ := by simp [Set.ext_iff]
#align set.Union_of_singleton Set.iUnion_of_singleton
theorem iUnion_of_singleton_coe (s : Set α) : ⋃ i : s, ({(i : α)} : Set α) = s := by simp
#align set.Union_of_singleton_coe Set.iUnion_of_singleton_coe
theorem sUnion_eq_biUnion {s : Set (Set α)} : ⋃₀s = ⋃ (i : Set α) (_ : i ∈ s), i := by
rw [← sUnion_image, image_id']
#align set.sUnion_eq_bUnion Set.sUnion_eq_biUnion
theorem sInter_eq_biInter {s : Set (Set α)} : ⋂₀ s = ⋂ (i : Set α) (_ : i ∈ s), i := by
rw [← sInter_image, image_id']
#align set.sInter_eq_bInter Set.sInter_eq_biInter
theorem sUnion_eq_iUnion {s : Set (Set α)} : ⋃₀s = ⋃ i : s, i := by
simp only [← sUnion_range, Subtype.range_coe]
#align set.sUnion_eq_Union Set.sUnion_eq_iUnion
theorem sInter_eq_iInter {s : Set (Set α)} : ⋂₀ s = ⋂ i : s, i := by
simp only [← sInter_range, Subtype.range_coe]
#align set.sInter_eq_Inter Set.sInter_eq_iInter
@[simp]
theorem iUnion_of_empty [IsEmpty ι] (s : ι → Set α) : ⋃ i, s i = ∅ :=
iSup_of_empty _
#align set.Union_of_empty Set.iUnion_of_empty
@[simp]
theorem iInter_of_empty [IsEmpty ι] (s : ι → Set α) : ⋂ i, s i = univ :=
iInf_of_empty _
#align set.Inter_of_empty Set.iInter_of_empty
theorem union_eq_iUnion {s₁ s₂ : Set α} : s₁ ∪ s₂ = ⋃ b : Bool, cond b s₁ s₂ :=
sup_eq_iSup s₁ s₂
#align set.union_eq_Union Set.union_eq_iUnion
theorem inter_eq_iInter {s₁ s₂ : Set α} : s₁ ∩ s₂ = ⋂ b : Bool, cond b s₁ s₂ :=
inf_eq_iInf s₁ s₂
#align set.inter_eq_Inter Set.inter_eq_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_union_sInter {S T : Set (Set α)} :
⋂₀ S ∪ ⋂₀ T = ⋂ p ∈ S ×ˢ T, (p : Set α × Set α).1 ∪ p.2 :=
sInf_sup_sInf
#align set.sInter_union_sInter Set.sInter_union_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_inter_sUnion {s t : Set (Set α)} :
⋃₀s ∩ ⋃₀t = ⋃ p ∈ s ×ˢ t, (p : Set α × Set α).1 ∩ p.2 :=
sSup_inf_sSup
#align set.sUnion_inter_sUnion Set.sUnion_inter_sUnion
theorem biUnion_iUnion (s : ι → Set α) (t : α → Set β) :
⋃ x ∈ ⋃ i, s i, t x = ⋃ (i) (x ∈ s i), t x := by simp [@iUnion_comm _ ι]
#align set.bUnion_Union Set.biUnion_iUnion
theorem biInter_iUnion (s : ι → Set α) (t : α → Set β) :
⋂ x ∈ ⋃ i, s i, t x = ⋂ (i) (x ∈ s i), t x := by simp [@iInter_comm _ ι]
#align set.bInter_Union Set.biInter_iUnion
theorem sUnion_iUnion (s : ι → Set (Set α)) : ⋃₀⋃ i, s i = ⋃ i, ⋃₀s i := by
simp only [sUnion_eq_biUnion, biUnion_iUnion]
#align set.sUnion_Union Set.sUnion_iUnion
theorem sInter_iUnion (s : ι → Set (Set α)) : ⋂₀ ⋃ i, s i = ⋂ i, ⋂₀ s i := by
simp only [sInter_eq_biInter, biInter_iUnion]
#align set.sInter_Union Set.sInter_iUnion
theorem iUnion_range_eq_sUnion {α β : Type*} (C : Set (Set α)) {f : ∀ s : C, β → (s : Type _)}
(hf : ∀ s : C, Surjective (f s)) : ⋃ y : β, range (fun s : C => (f s y).val) = ⋃₀C := by
ext x; constructor
· rintro ⟨s, ⟨y, rfl⟩, ⟨s, hs⟩, rfl⟩
refine' ⟨_, hs, _⟩
exact (f ⟨s, hs⟩ y).2
· rintro ⟨s, hs, hx⟩
cases' hf ⟨s, hs⟩ ⟨x, hx⟩ with y hy
refine' ⟨_, ⟨y, rfl⟩, ⟨s, hs⟩, _⟩
exact congr_arg Subtype.val hy
#align set.Union_range_eq_sUnion Set.iUnion_range_eq_sUnion
theorem iUnion_range_eq_iUnion (C : ι → Set α) {f : ∀ x : ι, β → C x}
(hf : ∀ x : ι, Surjective (f x)) : ⋃ y : β, range (fun x : ι => (f x y).val) = ⋃ x, C x := by
ext x; rw [mem_iUnion, mem_iUnion]; constructor
· rintro ⟨y, i, rfl⟩
exact ⟨i, (f i y).2⟩
· rintro ⟨i, hx⟩
cases' hf i ⟨x, hx⟩ with y hy
exact ⟨y, i, congr_arg Subtype.val hy⟩
#align set.Union_range_eq_Union Set.iUnion_range_eq_iUnion
theorem union_distrib_iInter_left (s : ι → Set α) (t : Set α) : (t ∪ ⋂ i, s i) = ⋂ i, t ∪ s i :=
sup_iInf_eq _ _
#align set.union_distrib_Inter_left Set.union_distrib_iInter_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_left (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_distrib_iInter_left]
#align set.union_distrib_Inter₂_left Set.union_distrib_iInter₂_left
theorem union_distrib_iInter_right (s : ι → Set α) (t : Set α) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.union_distrib_Inter_right Set.union_distrib_iInter_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_right (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [union_distrib_iInter_right]
#align set.union_distrib_Inter₂_right Set.union_distrib_iInter₂_right
section Function
/-! ### `mapsTo` -/
theorem mapsTo_sUnion {S : Set (Set α)} {t : Set β} {f : α → β} (H : ∀ s ∈ S, MapsTo f s t) :
MapsTo f (⋃₀S) t := fun _ ⟨s, hs, hx⟩ => H s hs hx
#align set.maps_to_sUnion Set.mapsTo_sUnion
theorem mapsTo_iUnion {s : ι → Set α} {t : Set β} {f : α → β} (H : ∀ i, MapsTo f (s i) t) :
MapsTo f (⋃ i, s i) t :=
mapsTo_sUnion <| forall_range_iff.2 H
#align set.maps_to_Union Set.mapsTo_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂ {s : ∀ i, κ i → Set α} {t : Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) t) : MapsTo f (⋃ (i) (j), s i j) t :=
mapsTo_iUnion fun i => mapsTo_iUnion (H i)
#align set.maps_to_Union₂ Set.mapsTo_iUnion₂
theorem mapsTo_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋃ i, s i) (⋃ i, t i) :=
mapsTo_iUnion fun i => (H i).mono (Subset.refl _) (subset_iUnion t i)
#align set.maps_to_Union_Union Set.mapsTo_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
mapsTo_iUnion_iUnion fun i => mapsTo_iUnion_iUnion (H i)
#align set.maps_to_Union₂_Union₂ Set.mapsTo_iUnion₂_iUnion₂
theorem mapsTo_sInter {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, MapsTo f s t) :
MapsTo f s (⋂₀ T) := fun _ hx t ht => H t ht hx
#align set.maps_to_sInter Set.mapsTo_sInter
theorem mapsTo_iInter {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, MapsTo f s (t i)) :
MapsTo f s (⋂ i, t i) := fun _ hx => mem_iInter.2 fun i => H i hx
#align set.maps_to_Inter Set.mapsTo_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f s (t i j)) : MapsTo f s (⋂ (i) (j), t i j) :=
mapsTo_iInter fun i => mapsTo_iInter (H i)
#align set.maps_to_Inter₂ Set.mapsTo_iInter₂
theorem mapsTo_iInter_iInter {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋂ i, s i) (⋂ i, t i) :=
mapsTo_iInter fun i => (H i).mono (iInter_subset s i) (Subset.refl _)
#align set.maps_to_Inter_Inter Set.mapsTo_iInter_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂_iInter₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋂ (i) (j), s i j) (⋂ (i) (j), t i j) :=
mapsTo_iInter_iInter fun i => mapsTo_iInter_iInter (H i)
#align set.maps_to_Inter₂_Inter₂ Set.mapsTo_iInter₂_iInter₂
theorem image_iInter_subset (s : ι → Set α) (f : α → β) : (f '' ⋂ i, s i) ⊆ ⋂ i, f '' s i :=
(mapsTo_iInter_iInter fun i => mapsTo_image f (s i)).image_subset
#align set.image_Inter_subset Set.image_iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂_subset (s : ∀ i, κ i → Set α) (f : α → β) :
(f '' ⋂ (i) (j), s i j) ⊆ ⋂ (i) (j), f '' s i j :=
(mapsTo_iInter₂_iInter₂ fun i hi => mapsTo_image f (s i hi)).image_subset
#align set.image_Inter₂_subset Set.image_iInter₂_subset
theorem image_sInter_subset (S : Set (Set α)) (f : α → β) : f '' ⋂₀ S ⊆ ⋂ s ∈ S, f '' s := by
rw [sInter_eq_biInter]
apply image_iInter₂_subset
#align set.image_sInter_subset Set.image_sInter_subset
/-! ### `restrictPreimage` -/
section
open Function
variable (s : Set β) {f : α → β} {U : ι → Set β} (hU : iUnion U = univ)
theorem injective_iff_injective_of_iUnion_eq_univ :
Injective f ↔ ∀ i, Injective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_injective H, fun H x y e => _⟩
obtain ⟨i, hi⟩ := Set.mem_iUnion.mp
(show f x ∈ Set.iUnion U by rw [hU]; triv)
injection @H i ⟨x, hi⟩ ⟨y, show f y ∈ U i from e ▸ hi⟩ (Subtype.ext e)
#align set.injective_iff_injective_of_Union_eq_univ Set.injective_iff_injective_of_iUnion_eq_univ
theorem surjective_iff_surjective_of_iUnion_eq_univ :
Surjective f ↔ ∀ i, Surjective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_surjective H, fun H x => _⟩
obtain ⟨i, hi⟩ :=
Set.mem_iUnion.mp
(show x ∈ Set.iUnion U by rw [hU]; triv)
exact ⟨_, congr_arg Subtype.val (H i ⟨x, hi⟩).choose_spec⟩
#align set.surjective_iff_surjective_of_Union_eq_univ Set.surjective_iff_surjective_of_iUnion_eq_univ
theorem bijective_iff_bijective_of_iUnion_eq_univ :
Bijective f ↔ ∀ i, Bijective ((U i).restrictPreimage f) := by
rw [Bijective, injective_iff_injective_of_iUnion_eq_univ hU,
surjective_iff_surjective_of_iUnion_eq_univ hU]
simp [Bijective, forall_and]
#align set.bijective_iff_bijective_of_Union_eq_univ Set.bijective_iff_bijective_of_iUnion_eq_univ
end
/-! ### `InjOn` -/
theorem InjOn.image_iInter_eq [Nonempty ι] {s : ι → Set α} {f : α → β} (h : InjOn f (⋃ i, s i)) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
inhabit ι
refine' Subset.antisymm (image_iInter_subset s f) fun y hy => _
simp only [mem_iInter, mem_image_iff_bex] at hy
choose x hx hy using hy
refine' ⟨x default, mem_iInter.2 fun i => _, hy _⟩
suffices x default = x i by
rw [this]
apply hx
replace hx : ∀ i, x i ∈ ⋃ j, s j := fun i => (subset_iUnion _ _) (hx i)
apply h (hx _) (hx _)
simp only [hy]
#align set.inj_on.image_Inter_eq Set.InjOn.image_iInter_eq
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
theorem InjOn.image_biInter_eq {p : ι → Prop} {s : ∀ i, p i → Set α} (hp : ∃ i, p i)
{f : α → β} (h : InjOn f (⋃ (i) (hi), s i hi)) :
(f '' ⋂ (i) (hi), s i hi) = ⋂ (i) (hi), f '' s i hi := by
simp only [iInter, iInf_subtype']
haveI : Nonempty { i // p i } := nonempty_subtype.2 hp
apply InjOn.image_iInter_eq
simpa only [iUnion, iSup_subtype'] using h
#align set.inj_on.image_bInter_eq Set.InjOn.image_biInter_eq
theorem image_iInter {f : α → β} (hf : Bijective f) (s : ι → Set α) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
cases isEmpty_or_nonempty ι
· simp_rw [iInter_of_empty, image_univ_of_surjective hf.surjective]
· exact (hf.injective.injOn _).image_iInter_eq
#align set.image_Inter Set.image_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂ {f : α → β} (hf : Bijective f) (s : ∀ i, κ i → Set α) :
(f '' ⋂ (i) (j), s i j) = ⋂ (i) (j), f '' s i j := by simp_rw [image_iInter hf]
#align set.image_Inter₂ Set.image_iInter₂
theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
(hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i) := by
intro x hx y hy hxy
rcases mem_iUnion.1 hx with ⟨i, hx⟩
rcases mem_iUnion.1 hy with ⟨j, hy⟩
rcases hs i j with ⟨k, hi, hj⟩
exact hf k (hi hx) (hj hy) hxy
#align set.inj_on_Union_of_directed Set.inj_on_iUnion_of_directed
/-! ### `SurjOn` -/
theorem surjOn_sUnion {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, SurjOn f s t) :
SurjOn f s (⋃₀T) := fun _ ⟨t, ht, hx⟩ => H t ht hx
#align set.surj_on_sUnion Set.surjOn_sUnion
theorem surjOn_iUnion {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, SurjOn f s (t i)) :
SurjOn f s (⋃ i, t i) :=
surjOn_sUnion <| forall_range_iff.2 H
#align set.surj_on_Union Set.surjOn_iUnion
theorem surjOn_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) : SurjOn f (⋃ i, s i) (⋃ i, t i) :=
surjOn_iUnion fun i => (H i).mono (subset_iUnion _ _) (Subset.refl _)
#align set.surj_on_Union_Union Set.surjOn_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f s (t i j)) : SurjOn f s (⋃ (i) (j), t i j) :=
surjOn_iUnion fun i => surjOn_iUnion (H i)
#align set.surj_on_Union₂ Set.surjOn_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f (s i j) (t i j)) : SurjOn f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
surjOn_iUnion_iUnion fun i => surjOn_iUnion_iUnion (H i)
#align set.surj_on_Union₂_Union₂ Set.surjOn_iUnion₂_iUnion₂
theorem surjOn_iInter [Nonempty ι] {s : ι → Set α} {t : Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) t) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) t := by
intro y hy
rw [Hinj.image_iInter_eq, mem_iInter]
exact fun i => H i hy
#align set.surj_on_Inter Set.surjOn_iInter
theorem surjOn_iInter_iInter [Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) (⋂ i, t i) :=
surjOn_iInter (fun i => (H i).mono (Subset.refl _) (iInter_subset _ _)) Hinj
#align set.surj_on_Inter_Inter Set.surjOn_iInter_iInter
/-! ### `BijOn` -/
theorem bijOn_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i))
(Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
⟨mapsTo_iUnion_iUnion fun i => (H i).mapsTo, Hinj, surjOn_iUnion_iUnion fun i => (H i).surjOn⟩
#align set.bij_on_Union Set.bijOn_iUnion
theorem bijOn_iInter [hi : Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, BijOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
⟨mapsTo_iInter_iInter fun i => (H i).mapsTo,
hi.elim fun i => (H i).injOn.mono (iInter_subset _ _),
surjOn_iInter_iInter (fun i => (H i).surjOn) Hinj⟩
#align set.bij_on_Inter Set.bijOn_iInter
theorem bijOn_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {t : ι → Set β}
{f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
bijOn_iUnion H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Union_of_directed Set.bijOn_iUnion_of_directed
theorem bijOn_iInter_of_directed [Nonempty ι] {s : ι → Set α} (hs : Directed (· ⊆ ·) s)
{t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
bijOn_iInter H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Inter_of_directed Set.bijOn_iInter_of_directed
end Function
/-! ### `image`, `preimage` -/
section Image
theorem image_iUnion {f : α → β} {s : ι → Set α} : (f '' ⋃ i, s i) = ⋃ i, f '' s i := by
ext1 x
simp only [mem_image, mem_iUnion, ← exists_and_right, ← exists_and_left]
--Porting note: `exists_swap` causes a `simp` loop in Lean4 so we use `rw` instead.
rw [exists_swap]
#align set.image_Union Set.image_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iUnion₂ (f : α → β) (s : ∀ i, κ i → Set α) :
(f '' ⋃ (i) (j), s i j) = ⋃ (i) (j), f '' s i j := by simp_rw [image_iUnion]
#align set.image_Union₂ Set.image_iUnion₂
theorem univ_subtype {p : α → Prop} : (univ : Set (Subtype p)) = ⋃ (x) (h : p x), {⟨x, h⟩} :=
Set.ext fun ⟨x, h⟩ => by simp [h]
#align set.univ_subtype Set.univ_subtype
theorem range_eq_iUnion {ι} (f : ι → α) : range f = ⋃ i, {f i} :=
Set.ext fun a => by simp [@eq_comm α a]
#align set.range_eq_Union Set.range_eq_iUnion
theorem image_eq_iUnion (f : α → β) (s : Set α) : f '' s = ⋃ i ∈ s, {f i} :=
Set.ext fun b => by simp [@eq_comm β b]
#align set.image_eq_Union Set.image_eq_iUnion
theorem biUnion_range {f : ι → α} {g : α → Set β} : ⋃ x ∈ range f, g x = ⋃ y, g (f y) :=
iSup_range
#align set.bUnion_range Set.biUnion_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iUnion_iUnion_eq' {f : ι → α} {g : α → Set β} :
⋃ (x) (y) (_ : f y = x), g x = ⋃ y, g (f y) := by simpa using biUnion_range
#align set.Union_Union_eq' Set.iUnion_iUnion_eq'
theorem biInter_range {f : ι → α} {g : α → Set β} : ⋂ x ∈ range f, g x = ⋂ y, g (f y) :=
iInf_range
#align set.bInter_range Set.biInter_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iInter_iInter_eq' {f : ι → α} {g : α → Set β} :
⋂ (x) (y) (_ : f y = x), g x = ⋂ y, g (f y) := by simpa using biInter_range
#align set.Inter_Inter_eq' Set.iInter_iInter_eq'
variable {s : Set γ} {f : γ → α} {g : α → Set β}
theorem biUnion_image : ⋃ x ∈ f '' s, g x = ⋃ y ∈ s, g (f y) :=
iSup_image
#align set.bUnion_image Set.biUnion_image
theorem biInter_image : ⋂ x ∈ f '' s, g x = ⋂ y ∈ s, g (f y) :=
iInf_image
#align set.bInter_image Set.biInter_image
end Image
section Preimage
theorem monotone_preimage {f : α → β} : Monotone (preimage f) := fun _ _ h => preimage_mono h
#align set.monotone_preimage Set.monotone_preimage
@[simp]
theorem preimage_iUnion {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋃ i, s i) = ⋃ i, f ⁻¹' s i :=
Set.ext <| by simp [preimage]
#align set.preimage_Union Set.preimage_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iUnion₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋃ (i) (j), s i j) = ⋃ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iUnion]
#align set.preimage_Union₂ Set.preimage_iUnion₂
@[simp]
theorem preimage_sUnion {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋃₀s = ⋃ t ∈ s, f ⁻¹' t := by
rw [sUnion_eq_biUnion, preimage_iUnion₂]
#align set.preimage_sUnion Set.preimage_sUnion
theorem preimage_iInter {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋂ i, s i) = ⋂ i, f ⁻¹' s i := by
ext; simp
#align set.preimage_Inter Set.preimage_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iInter₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋂ (i) (j), s i j) = ⋂ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iInter]
#align set.preimage_Inter₂ Set.preimage_iInter₂
@[simp]
theorem preimage_sInter {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋂₀ s = ⋂ t ∈ s, f ⁻¹' t := by
rw [sInter_eq_biInter, preimage_iInter₂]
#align set.preimage_sInter Set.preimage_sInter
@[simp]
theorem biUnion_preimage_singleton (f : α → β) (s : Set β) : ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s := by
rw [← preimage_iUnion₂, biUnion_of_singleton]
#align set.bUnion_preimage_singleton Set.biUnion_preimage_singleton
theorem biUnion_range_preimage_singleton (f : α → β) : ⋃ y ∈ range f, f ⁻¹' {y} = univ := by
rw [biUnion_preimage_singleton, preimage_range]
#align set.bUnion_range_preimage_singleton Set.biUnion_range_preimage_singleton
end Preimage
section Prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion {s : Set α} {t : ι → Set β} : (s ×ˢ ⋃ i, t i) = ⋃ i, s ×ˢ t i := by
ext
simp
#align set.prod_Union Set.prod_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} :
(s ×ˢ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ×ˢ t i j := by simp_rw [prod_iUnion]
#align set.prod_Union₂ Set.prod_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sUnion {s : Set α} {C : Set (Set β)} : s ×ˢ ⋃₀C = ⋃₀((fun t => s ×ˢ t) '' C) := by
simp_rw [sUnion_eq_biUnion, biUnion_image, prod_iUnion₂]
#align set.prod_sUnion Set.prod_sUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_const {s : ι → Set α} {t : Set β} : (⋃ i, s i) ×ˢ t = ⋃ i, s i ×ˢ t := by
ext
simp
#align set.Union_prod_const Set.iUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion₂_prod_const {s : ∀ i, κ i → Set α} {t : Set β} :
(⋃ (i) (j), s i j) ×ˢ t = ⋃ (i) (j), s i j ×ˢ t := by simp_rw [iUnion_prod_const]
#align set.Union₂_prod_const Set.iUnion₂_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_prod_const {C : Set (Set α)} {t : Set β} :
⋃₀C ×ˢ t = ⋃₀((fun s : Set α => s ×ˢ t) '' C) := by
simp only [sUnion_eq_biUnion, iUnion₂_prod_const, biUnion_image]
#align set.sUnion_prod_const Set.sUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod {ι ι' α β} (s : ι → Set α) (t : ι' → Set β) :
⋃ x : ι × ι', s x.1 ×ˢ t x.2 = (⋃ i : ι, s i) ×ˢ ⋃ i : ι', t i := by
ext
simp
#align set.Union_prod Set.iUnion_prod
/-- Analogue of `iSup_prod` for sets. -/
lemma iUnion_prod' (f : β × γ → Set α) : ⋃ x : β × γ, f x = ⋃ (i : β) (j : γ), f (i, j) :=
iSup_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x := by
ext ⟨z, w⟩; simp only [mem_prod, mem_iUnion, exists_imp, and_imp, iff_def]; constructor
· intro x hz hw
exact ⟨⟨x, hz⟩, x, hw⟩
· intro x hz x' hw
exact ⟨x ⊔ x', hs le_sup_left hz, ht le_sup_right hw⟩
#align set.Union_prod_of_monotone Set.iUnion_prod_of_monotone
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter_subset (S : Set (Set α)) (T : Set (Set β)) :
⋂₀ S ×ˢ ⋂₀ T ⊆ ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 :=
subset_iInter₂ fun x hx _ hy => ⟨hy.1 x.1 hx.1, hy.2 x.2 hx.2⟩
#align set.sInter_prod_sInter_subset Set.sInter_prod_sInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 := by
obtain ⟨s₁, h₁⟩ := hS
obtain ⟨s₂, h₂⟩ := hT
refine' Set.Subset.antisymm (sInter_prod_sInter_subset S T) fun x hx => _
rw [mem_iInter₂] at hx
exact ⟨fun s₀ h₀ => (hx (s₀, s₂) ⟨h₀, h₂⟩).1, fun s₀ h₀ => (hx (s₁, s₀) ⟨h₁, h₀⟩).2⟩
#align set.sInter_prod_sInter Set.sInter_prod_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod {S : Set (Set α)} (hS : S.Nonempty) (t : Set β) :
⋂₀ S ×ˢ t = ⋂ s ∈ S, s ×ˢ t := by
rw [← sInter_singleton t, sInter_prod_sInter hS (singleton_nonempty t), sInter_singleton]
simp_rw [prod_singleton, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.sInter_prod Set.sInter_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sInter {T : Set (Set β)} (hT : T.Nonempty) (s : Set α) :
s ×ˢ ⋂₀ T = ⋂ t ∈ T, s ×ˢ t := by
rw [← sInter_singleton s, sInter_prod_sInter (singleton_nonempty s) hT, sInter_singleton]
simp_rw [singleton_prod, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.prod_sInter Set.prod_sInter
end Prod
section Image2
variable (f : α → β → γ) {s : Set α} {t : Set β}
/-- The `Set.image2` version of `Set.image_eq_iUnion` -/
theorem image2_eq_iUnion (s : Set α) (t : Set β) : image2 f s t = ⋃ (i ∈ s) (j ∈ t), {f i j} := by
ext; simp [eq_comm]
#align set.image2_eq_Union Set.image2_eq_iUnion
theorem iUnion_image_left : ⋃ a ∈ s, f a '' t = image2 f s t := by
simp only [image2_eq_iUnion, image_eq_iUnion]
#align set.Union_image_left Set.iUnion_image_left
theorem iUnion_image_right : ⋃ b ∈ t, (f · b) '' s = image2 f s t := by
rw [image2_swap, iUnion_image_left]
#align set.Union_image_right Set.iUnion_image_right
theorem image2_iUnion_left (s : ι → Set α) (t : Set β) :
image2 f (⋃ i, s i) t = ⋃ i, image2 f (s i) t := by
simp only [← image_prod, iUnion_prod_const, image_iUnion]
#align set.image2_Union_left Set.image2_iUnion_left
theorem image2_iUnion_right (s : Set α) (t : ι → Set β) :
image2 f s (⋃ i, t i) = ⋃ i, image2 f s (t i) := by
simp only [← image_prod, prod_iUnion, image_iUnion]
#align set.image2_Union_right Set.image2_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋃ (i) (j), s i j) t = ⋃ (i) (j), image2 f (s i j) t := by simp_rw [image2_iUnion_left]
#align set.image2_Union₂_left Set.image2_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋃ (i) (j), t i j) = ⋃ (i) (j), image2 f s (t i j) :=
by simp_rw [image2_iUnion_right]
#align set.image2_Union₂_right Set.image2_iUnion₂_right
theorem image2_iInter_subset_left (s : ι → Set α) (t : Set β) :
image2 f (⋂ i, s i) t ⊆ ⋂ i, image2 f (s i) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem (hx _) hy
#align set.image2_Inter_subset_left Set.image2_iInter_subset_left
theorem image2_iInter_subset_right (s : Set α) (t : ι → Set β) :
image2 f s (⋂ i, t i) ⊆ ⋂ i, image2 f s (t i) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem hx (hy _)
#align set.image2_Inter_subset_right Set.image2_iInter_subset_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋂ (i) (j), s i j) t ⊆ ⋂ (i) (j), image2 f (s i j) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem (hx _ _) hy
#align set.image2_Inter₂_subset_left Set.image2_iInter₂_subset_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), image2 f s (t i j) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem hx (hy _ _)
#align set.image2_Inter₂_subset_right Set.image2_iInter₂_subset_right
theorem prod_eq_biUnion_left : s ×ˢ t = ⋃ a ∈ s, (fun b => (a, b)) '' t := by
rw [iUnion_image_left, image2_mk_eq_prod]
#align set.prod_eq_bUnion_left Set.prod_eq_biUnion_left
theorem prod_eq_biUnion_right : s ×ˢ t = ⋃ b ∈ t, (fun a => (a, b)) '' s := by
rw [iUnion_image_right, image2_mk_eq_prod]
#align set.prod_eq_bUnion_right Set.prod_eq_biUnion_right
end Image2
section Seq
/-- Given a set `s` of functions `α → β` and `t : Set α`, `seq s t` is the union of `f '' t` over
all `f ∈ s`. -/
def seq (s : Set (α → β)) (t : Set α) : Set β :=
{ b | ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b }
#align set.seq Set.seq
@[simp]
theorem mem_seq_iff {s : Set (α → β)} {t : Set α} {b : β} :
b ∈ seq s t ↔ ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b :=
Iff.rfl
#align set.mem_seq_iff Set.mem_seq_iff
lemma seq_eq_image2 (s : Set (α → β)) (t : Set α) : seq s t = image2 (fun f a ↦ f a) s t := by
ext; simp
theorem seq_def {s : Set (α → β)} {t : Set α} : seq s t = ⋃ f ∈ s, f '' t := by
rw [seq_eq_image2, iUnion_image_left]
#align set.seq_def Set.seq_def
theorem seq_subset {s : Set (α → β)} {t : Set α} {u : Set β} :
seq s t ⊆ u ↔ ∀ f ∈ s, ∀ a ∈ t, (f : α → β) a ∈ u := by
rw [seq_eq_image2, image2_subset_iff]
#align set.seq_subset Set.seq_subset
@[gcongr]
theorem seq_mono {s₀ s₁ : Set (α → β)} {t₀ t₁ : Set α} (hs : s₀ ⊆ s₁) (ht : t₀ ⊆ t₁) :
seq s₀ t₀ ⊆ seq s₁ t₁ := fun _ ⟨f, hf, a, ha, eq⟩ => ⟨f, hs hf, a, ht ha, eq⟩
#align set.seq_mono Set.seq_mono
theorem singleton_seq {f : α → β} {t : Set α} : Set.seq ({f} : Set (α → β)) t = f '' t := by
rw [seq_eq_image2, image2_singleton_left]
#align set.singleton_seq Set.singleton_seq
theorem seq_singleton {s : Set (α → β)} {a : α} : Set.seq s {a} = (fun f : α → β => f a) '' s := by
rw [seq_eq_image2, image2_singleton_right]
#align set.seq_singleton Set.seq_singleton
theorem seq_seq {s : Set (β → γ)} {t : Set (α → β)} {u : Set α} :
seq s (seq t u) = seq (seq ((· ∘ ·) '' s) t) u := by
simp only [seq_eq_image2, image2_image_left, image2_image2_left, image2_image2_right, comp_apply]
#align set.seq_seq Set.seq_seq
theorem image_seq {f : β → γ} {s : Set (α → β)} {t : Set α} :
f '' seq s t = seq ((f ∘ ·) '' s) t := by
rw [← singleton_seq, ← singleton_seq, seq_seq, image_singleton]
#align set.image_seq Set.image_seq
theorem prod_eq_seq {s : Set α} {t : Set β} : s ×ˢ t = (Prod.mk '' s).seq t := by
rw [seq_eq_image2, image2_image_left, image2_mk_eq_prod]
#align set.prod_eq_seq Set.prod_eq_seq
theorem prod_image_seq_comm (s : Set α) (t : Set β) :
(Prod.mk '' s).seq t = seq ((fun b a => (a, b)) '' t) s := by
rw [← prod_eq_seq, ← image_swap_prod, prod_eq_seq, image_seq, ← image_comp]; rfl
#align set.prod_image_seq_comm Set.prod_image_seq_comm
theorem image2_eq_seq (f : α → β → γ) (s : Set α) (t : Set β) : image2 f s t = seq (f '' s) t := by
rw [seq_eq_image2, image2_image_left]
#align set.image2_eq_seq Set.image2_eq_seq
end Seq
section Pi
variable {π : α → Type*}
theorem pi_def (i : Set α) (s : ∀ a, Set (π a)) : pi i s = ⋂ a ∈ i, eval a ⁻¹' s a := by
ext
simp
#align set.pi_def Set.pi_def
theorem univ_pi_eq_iInter (t : ∀ i, Set (π i)) : pi univ t = ⋂ i, eval i ⁻¹' t i := by
simp only [pi_def, iInter_true, mem_univ]
#align set.univ_pi_eq_Inter Set.univ_pi_eq_iInter
theorem pi_diff_pi_subset (i : Set α) (s t : ∀ a, Set (π a)) :
pi i s \ pi i t ⊆ ⋃ a ∈ i, eval a ⁻¹' (s a \ t a) := by
refine' diff_subset_comm.2 fun x hx a ha => _
simp only [mem_diff, mem_pi, mem_iUnion, not_exists, mem_preimage, not_and, not_not,
eval_apply] at hx
exact hx.2 _ ha (hx.1 _ ha)
#align set.pi_diff_pi_subset Set.pi_diff_pi_subset
theorem iUnion_univ_pi {ι : α → Type*} (t : (a : α) → ι a → Set (π a)) :
⋃ x : (a : α) → ι a, pi univ (fun a => t a (x a)) = pi univ fun a => ⋃ j : ι a, t a j := by
ext
simp [Classical.skolem]
#align set.Union_univ_pi Set.iUnion_univ_pi
end Pi
end Set
namespace Function
namespace Surjective
theorem iUnion_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋃ x, g (f x) = ⋃ y, g y :=
hf.iSup_comp g
#align function.surjective.Union_comp Function.Surjective.iUnion_comp
theorem iInter_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋂ x, g (f x) = ⋂ y, g y :=
hf.iInf_comp g
#align function.surjective.Inter_comp Function.Surjective.iInter_comp
end Surjective
end Function
/-!
### Disjoint sets
-/
section Disjoint
variable {s t u : Set α} {f : α → β}
namespace Set
@[simp]
theorem disjoint_iUnion_left {ι : Sort*} {s : ι → Set α} :
Disjoint (⋃ i, s i) t ↔ ∀ i, Disjoint (s i) t :=
iSup_disjoint_iff
#align set.disjoint_Union_left Set.disjoint_iUnion_left
@[simp]
theorem disjoint_iUnion_right {ι : Sort*} {s : ι → Set α} :
Disjoint t (⋃ i, s i) ↔ ∀ i, Disjoint t (s i) :=
disjoint_iSup_iff
#align set.disjoint_Union_right Set.disjoint_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_left {s : ∀ i, κ i → Set α} {t : Set α} :
Disjoint (⋃ (i) (j), s i j) t ↔ ∀ i j, Disjoint (s i j) t :=
iSup₂_disjoint_iff
#align set.disjoint_Union₂_left Set.disjoint_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_right {s : Set α} {t : ∀ i, κ i → Set α} :
Disjoint s (⋃ (i) (j), t i j) ↔ ∀ i j, Disjoint s (t i j) :=
disjoint_iSup₂_iff
#align set.disjoint_Union₂_right Set.disjoint_iUnion₂_right
@[simp]
theorem disjoint_sUnion_left {S : Set (Set α)} {t : Set α} :
Disjoint (⋃₀S) t ↔ ∀ s ∈ S, Disjoint s t :=
sSup_disjoint_iff
#align set.disjoint_sUnion_left Set.disjoint_sUnion_left
@[simp]
theorem disjoint_sUnion_right {s : Set α} {S : Set (Set α)} :
Disjoint s (⋃₀S) ↔ ∀ t ∈ S, Disjoint s t :=
disjoint_sSup_iff
#align set.disjoint_sUnion_right Set.disjoint_sUnion_right
end Set
end Disjoint
/-! ### Intervals -/
namespace Set
lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :
(⋂ i, Iic (f i)).Nonempty ↔ BddBelow (range f) := by
have : (⋂ (i : ι), Iic (f i)) = lowerBounds (range f) := by
ext c; simp [lowerBounds]
simp [this, BddBelow]
lemma nonempty_iInter_Ici_iff [Preorder α] {f : ι → α} :
(⋂ i, Ici (f i)).Nonempty ↔ BddAbove (range f) :=
nonempty_iInter_Iic_iff (α := αᵒᵈ)
variable [CompleteLattice α]
theorem Ici_iSup (f : ι → α) : Ici (⨆ i, f i) = ⋂ i, Ici (f i) :=
ext fun _ => by simp only [mem_Ici, iSup_le_iff, mem_iInter]
#align set.Ici_supr Set.Ici_iSup
theorem Iic_iInf (f : ι → α) : Iic (⨅ i, f i) = ⋂ i, Iic (f i) :=
ext fun _ => by simp only [mem_Iic, le_iInf_iff, mem_iInter]
#align set.Iic_infi Set.Iic_iInf
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Ici_iSup₂ (f : ∀ i, κ i → α) : Ici (⨆ (i) (j), f i j) = ⋂ (i) (j), Ici (f i j) := by
simp_rw [Ici_iSup]
#align set.Ici_supr₂ Set.Ici_iSup₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Iic_iInf₂ (f : ∀ i, κ i → α) : Iic (⨅ (i) (j), f i j) = ⋂ (i) (j), Iic (f i j) := by
simp_rw [Iic_iInf]
#align set.Iic_infi₂ Set.Iic_iInf₂
theorem Ici_sSup (s : Set α) : Ici (sSup s) = ⋂ a ∈ s, Ici a := by rw [sSup_eq_iSup, Ici_iSup₂]
#align set.Ici_Sup Set.Ici_sSup
theorem Iic_sInf (s : Set α) : Iic (sInf s) = ⋂ a ∈ s, Iic a := by rw [sInf_eq_iInf, Iic_iInf₂]
#align set.Iic_Inf Set.Iic_sInf
end Set
namespace Set
variable (t : α → Set β)
theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x := by
simp only [diff_subset_iff, ← biUnion_union]
apply biUnion_subset_biUnion_left
rw [union_diff_self]
apply subset_union_right
#align set.bUnion_diff_bUnion_subset Set.biUnion_diff_biUnion_subset
/-- If `t` is an indexed family of sets, then there is a natural map from `Σ i, t i` to `⋃ i, t i`
sending `⟨i, x⟩` to `x`. -/
def sigmaToiUnion (x : Σi, t i) : ⋃ i, t i :=
⟨x.2, mem_iUnion.2 ⟨x.1, x.2.2⟩⟩
#align set.sigma_to_Union Set.sigmaToiUnion
theorem sigmaToiUnion_surjective : Surjective (sigmaToiUnion t)
| ⟨b, hb⟩ =>
have : ∃ a, b ∈ t a := by
|
simpa using hb
|
theorem sigmaToiUnion_surjective : Surjective (sigmaToiUnion t)
| ⟨b, hb⟩ =>
have : ∃ a, b ∈ t a := by
|
Mathlib.Data.Set.Lattice.2305_0.5mONj49h3SYSDwc
|
theorem sigmaToiUnion_surjective : Surjective (sigmaToiUnion t)
| ⟨b, hb⟩ =>
have : ∃ a, b ∈ t a
|
Mathlib_Data_Set_Lattice
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
t : α → Set β
h : ∀ (i j : α), i ≠ j → Disjoint (t i) (t j)
a₁ : α
b₁ : β
h₁ : b₁ ∈ t a₁
a₂ : α
b₂ : β
h₂ : b₂ ∈ t a₂
eq :
sigmaToiUnion t { fst := a₁, snd := { val := b₁, property := h₁ } } =
sigmaToiUnion t { fst := a₂, snd := { val := b₂, property := h₂ } }
b_eq : b₁ = b₂
a_eq : a₁ = a₂
⊢ ↑(Eq.recOn a_eq { fst := a₁, snd := { val := b₁, property := h₁ } }.snd) =
↑{ fst := a₂, snd := { val := b₂, property := h₂ } }.snd
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
/-!
# The set lattice
This file provides usual set notation for unions and intersections, a `CompleteLattice` instance
for `Set α`, and some more set constructions.
## Main declarations
* `Set.iUnion`: **i**ndexed **union**. Union of an indexed family of sets.
* `Set.iInter`: **i**ndexed **inter**section. Intersection of an indexed family of sets.
* `Set.sInter`: **s**et **inter**section. Intersection of sets belonging to a set of sets.
* `Set.sUnion`: **s**et **union**. Union of sets belonging to a set of sets.
* `Set.sInter_eq_biInter`, `Set.sUnion_eq_biInter`: Shows that `⋂₀ s = ⋂ x ∈ s, x` and
`⋃₀ s = ⋃ x ∈ s, x`.
* `Set.completeAtomicBooleanAlgebra`: `Set α` is a `CompleteAtomicBooleanAlgebra` with `≤ = ⊆`,
`< = ⊂`, `⊓ = ∩`, `⊔ = ∪`, `⨅ = ⋂`, `⨆ = ⋃` and `\` as the set difference.
See `Set.BooleanAlgebra`.
* `Set.kernImage`: For a function `f : α → β`, `s.kernImage f` is the set of `y` such that
`f ⁻¹ y ⊆ s`.
* `Set.seq`: Union of the image of a set under a **seq**uence of functions. `seq s t` is the union
of `f '' t` over all `f ∈ s`, where `t : Set α` and `s : Set (α → β)`.
* `Set.iUnion_eq_sigma_of_disjoint`: Equivalence between `⋃ i, t i` and `Σ i, t i`, where `t` is an
indexed family of disjoint sets.
## Naming convention
In lemma names,
* `⋃ i, s i` is called `iUnion`
* `⋂ i, s i` is called `iInter`
* `⋃ i j, s i j` is called `iUnion₂`. This is an `iUnion` inside an `iUnion`.
* `⋂ i j, s i j` is called `iInter₂`. This is an `iInter` inside an `iInter`.
* `⋃ i ∈ s, t i` is called `biUnion` for "bounded `iUnion`". This is the special case of `iUnion₂`
where `j : i ∈ s`.
* `⋂ i ∈ s, t i` is called `biInter` for "bounded `iInter`". This is the special case of `iInter₂`
where `j : i ∈ s`.
## Notation
* `⋃`: `Set.iUnion`
* `⋂`: `Set.iInter`
* `⋃₀`: `Set.sUnion`
* `⋂₀`: `Set.sInter`
-/
set_option autoImplicit true
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι₂ : Sort*} {κ κ₁ κ₂ : ι → Sort*} {κ' : ι' → Sort*}
namespace Set
/-! ### Complete lattice and complete Boolean algebra instances -/
instance : InfSet (Set α) :=
⟨fun s => { a | ∀ t ∈ s, a ∈ t }⟩
instance : SupSet (Set α) :=
⟨fun s => { a | ∃ t ∈ s, a ∈ t }⟩
/-- Intersection of a set of sets. -/
def sInter (S : Set (Set α)) : Set α :=
sInf S
#align set.sInter Set.sInter
/-- Notation for `Set.sInter` Intersection of a set of sets. -/
prefix:110 "⋂₀ " => sInter
/-- Union of a set of sets. -/
def sUnion (S : Set (Set α)) : Set α :=
sSup S
#align set.sUnion Set.sUnion
/-- Notation for `Set.sUnion`. Union of a set of sets. -/
prefix:110 "⋃₀ " => sUnion
@[simp]
theorem mem_sInter {x : α} {S : Set (Set α)} : x ∈ ⋂₀ S ↔ ∀ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sInter Set.mem_sInter
@[simp]
theorem mem_sUnion {x : α} {S : Set (Set α)} : x ∈ ⋃₀ S ↔ ∃ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sUnion Set.mem_sUnion
/-- Indexed union of a family of sets -/
def iUnion (s : ι → Set β) : Set β :=
iSup s
#align set.Union Set.iUnion
/-- Indexed intersection of a family of sets -/
def iInter (s : ι → Set β) : Set β :=
iInf s
#align set.Inter Set.iInter
/-- Notation for `Set.iUnion`. Indexed union of a family of sets -/
notation3 "⋃ "(...)", "r:60:(scoped f => iUnion f) => r
/-- Notation for `Set.iInter`. Indexed intersection of a family of sets -/
notation3 "⋂ "(...)", "r:60:(scoped f => iInter f) => r
section delaborators
open Lean Lean.PrettyPrinter.Delaborator
/-- Delaborator for indexed unions. -/
@[delab app.Set.iUnion]
def iUnion_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋃ (_ : $dom), $body)
else if prop || ppTypes then
`(⋃ ($x:ident : $dom), $body)
else
`(⋃ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋃ $x:ident, ⋃ (_ : $y:ident ∈ $s), $body)
| `(⋃ ($x:ident : $_), ⋃ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋃ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
/-- Delaborator for indexed intersections. -/
@[delab app.Set.iInter]
def sInter_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋂ (_ : $dom), $body)
else if prop || ppTypes then
`(⋂ ($x:ident : $dom), $body)
else
`(⋂ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋂ $x:ident, ⋂ (_ : $y:ident ∈ $s), $body)
| `(⋂ ($x:ident : $_), ⋂ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋂ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
end delaborators
@[simp]
theorem sSup_eq_sUnion (S : Set (Set α)) : sSup S = ⋃₀S :=
rfl
#align set.Sup_eq_sUnion Set.sSup_eq_sUnion
@[simp]
theorem sInf_eq_sInter (S : Set (Set α)) : sInf S = ⋂₀ S :=
rfl
#align set.Inf_eq_sInter Set.sInf_eq_sInter
@[simp]
theorem iSup_eq_iUnion (s : ι → Set α) : iSup s = iUnion s :=
rfl
#align set.supr_eq_Union Set.iSup_eq_iUnion
@[simp]
theorem iInf_eq_iInter (s : ι → Set α) : iInf s = iInter s :=
rfl
#align set.infi_eq_Inter Set.iInf_eq_iInter
@[simp]
theorem mem_iUnion {x : α} {s : ι → Set α} : (x ∈ ⋃ i, s i) ↔ ∃ i, x ∈ s i :=
⟨fun ⟨_, ⟨⟨a, (t_eq : s a = _)⟩, (h : x ∈ _)⟩⟩ => ⟨a, t_eq.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨s a, ⟨⟨a, rfl⟩, h⟩⟩⟩
#align set.mem_Union Set.mem_iUnion
@[simp]
theorem mem_iInter {x : α} {s : ι → Set α} : (x ∈ ⋂ i, s i) ↔ ∀ i, x ∈ s i :=
⟨fun (h : ∀ a ∈ { a : Set α | ∃ i, s i = a }, x ∈ a) a => h (s a) ⟨a, rfl⟩,
fun h _ ⟨a, (eq : s a = _)⟩ => eq ▸ h a⟩
#align set.mem_Inter Set.mem_iInter
theorem mem_iUnion₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋃ (i) (j), s i j) ↔ ∃ i j, x ∈ s i j := by
simp_rw [mem_iUnion]
#align set.mem_Union₂ Set.mem_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋂ (i) (j), s i j) ↔ ∀ i j, x ∈ s i j := by
simp_rw [mem_iInter]
#align set.mem_Inter₂ Set.mem_iInter₂
theorem mem_iUnion_of_mem {s : ι → Set α} {a : α} (i : ι) (ha : a ∈ s i) : a ∈ ⋃ i, s i :=
mem_iUnion.2 ⟨i, ha⟩
#align set.mem_Union_of_mem Set.mem_iUnion_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iUnion₂_of_mem {s : ∀ i, κ i → Set α} {a : α} {i : ι} (j : κ i) (ha : a ∈ s i j) :
a ∈ ⋃ (i) (j), s i j :=
mem_iUnion₂.2 ⟨i, j, ha⟩
#align set.mem_Union₂_of_mem Set.mem_iUnion₂_of_mem
theorem mem_iInter_of_mem {s : ι → Set α} {a : α} (h : ∀ i, a ∈ s i) : a ∈ ⋂ i, s i :=
mem_iInter.2 h
#align set.mem_Inter_of_mem Set.mem_iInter_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂_of_mem {s : ∀ i, κ i → Set α} {a : α} (h : ∀ i j, a ∈ s i j) :
a ∈ ⋂ (i) (j), s i j :=
mem_iInter₂.2 h
#align set.mem_Inter₂_of_mem Set.mem_iInter₂_of_mem
instance Set.completeAtomicBooleanAlgebra : CompleteAtomicBooleanAlgebra (Set α) :=
{ instBooleanAlgebraSet with
le_sSup := fun s t t_in a a_in => ⟨t, t_in, a_in⟩
sSup_le := fun s t h a ⟨t', ⟨t'_in, a_in⟩⟩ => h t' t'_in a_in
le_sInf := fun s t h a a_in t' t'_in => h t' t'_in a_in
sInf_le := fun s t t_in a h => h _ t_in
iInf_iSup_eq := by intros; ext; simp [Classical.skolem] }
/-- `kernImage f s` is the set of `y` such that `f ⁻¹ y ⊆ s`. -/
def kernImage (f : α → β) (s : Set α) : Set β :=
{ y | ∀ ⦃x⦄, f x = y → x ∈ s }
#align set.kern_image Set.kernImage
lemma subset_kernImage_iff {f : α → β} : s ⊆ kernImage f t ↔ f ⁻¹' s ⊆ t :=
⟨fun h _ hx ↦ h hx rfl,
fun h _ hx y hy ↦ h (show f y ∈ s from hy.symm ▸ hx)⟩
section GaloisConnection
variable {f : α → β}
protected theorem image_preimage : GaloisConnection (image f) (preimage f) := fun _ _ =>
image_subset_iff
#align set.image_preimage Set.image_preimage
protected theorem preimage_kernImage : GaloisConnection (preimage f) (kernImage f) := fun _ _ =>
subset_kernImage_iff.symm
#align set.preimage_kern_image Set.preimage_kernImage
end GaloisConnection
section kernImage
variable {f : α → β}
lemma kernImage_mono : Monotone (kernImage f) :=
Set.preimage_kernImage.monotone_u
lemma kernImage_eq_compl {s : Set α} : kernImage f s = (f '' sᶜ)ᶜ :=
Set.preimage_kernImage.u_unique (Set.image_preimage.compl)
(fun t ↦ compl_compl (f ⁻¹' t) ▸ Set.preimage_compl)
lemma kernImage_compl {s : Set α} : kernImage f (sᶜ) = (f '' s)ᶜ := by
rw [kernImage_eq_compl, compl_compl]
lemma kernImage_empty : kernImage f ∅ = (range f)ᶜ := by
rw [kernImage_eq_compl, compl_empty, image_univ]
lemma kernImage_preimage_eq_iff {s : Set β} : kernImage f (f ⁻¹' s) = s ↔ (range f)ᶜ ⊆ s := by
rw [kernImage_eq_compl, ← preimage_compl, compl_eq_comm, eq_comm, image_preimage_eq_iff,
compl_subset_comm]
lemma compl_range_subset_kernImage {s : Set α} : (range f)ᶜ ⊆ kernImage f s := by
rw [← kernImage_empty]
exact kernImage_mono (empty_subset _)
lemma kernImage_union_preimage {s : Set α} {t : Set β} :
kernImage f (s ∪ f ⁻¹' t) = kernImage f s ∪ t := by
rw [kernImage_eq_compl, kernImage_eq_compl, compl_union, ← preimage_compl, image_inter_preimage,
compl_inter, compl_compl]
lemma kernImage_preimage_union {s : Set α} {t : Set β} :
kernImage f (f ⁻¹' t ∪ s) = t ∪ kernImage f s := by
rw [union_comm, kernImage_union_preimage, union_comm]
end kernImage
/-! ### Union and intersection over an indexed family of sets -/
instance : OrderTop (Set α) where
top := univ
le_top := by simp
@[congr]
theorem iUnion_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iUnion f₁ = iUnion f₂ :=
iSup_congr_Prop pq f
#align set.Union_congr_Prop Set.iUnion_congr_Prop
@[congr]
theorem iInter_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iInter f₁ = iInter f₂ :=
iInf_congr_Prop pq f
#align set.Inter_congr_Prop Set.iInter_congr_Prop
theorem iUnion_plift_up (f : PLift ι → Set α) : ⋃ i, f (PLift.up i) = ⋃ i, f i :=
iSup_plift_up _
#align set.Union_plift_up Set.iUnion_plift_up
theorem iUnion_plift_down (f : ι → Set α) : ⋃ i, f (PLift.down i) = ⋃ i, f i :=
iSup_plift_down _
#align set.Union_plift_down Set.iUnion_plift_down
theorem iInter_plift_up (f : PLift ι → Set α) : ⋂ i, f (PLift.up i) = ⋂ i, f i :=
iInf_plift_up _
#align set.Inter_plift_up Set.iInter_plift_up
theorem iInter_plift_down (f : ι → Set α) : ⋂ i, f (PLift.down i) = ⋂ i, f i :=
iInf_plift_down _
#align set.Inter_plift_down Set.iInter_plift_down
theorem iUnion_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋃ _ : p, s = if p then s else ∅ :=
iSup_eq_if _
#align set.Union_eq_if Set.iUnion_eq_if
theorem iUnion_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋃ h : p, s h = if h : p then s h else ∅ :=
iSup_eq_dif _
#align set.Union_eq_dif Set.iUnion_eq_dif
theorem iInter_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋂ _ : p, s = if p then s else univ :=
iInf_eq_if _
#align set.Inter_eq_if Set.iInter_eq_if
theorem iInf_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋂ h : p, s h = if h : p then s h else univ :=
_root_.iInf_eq_dif _
#align set.Infi_eq_dif Set.iInf_eq_dif
theorem exists_set_mem_of_union_eq_top {ι : Type*} (t : Set ι) (s : ι → Set β)
(w : ⋃ i ∈ t, s i = ⊤) (x : β) : ∃ i ∈ t, x ∈ s i := by
have p : x ∈ ⊤ := Set.mem_univ x
rw [← w, Set.mem_iUnion] at p
simpa using p
#align set.exists_set_mem_of_union_eq_top Set.exists_set_mem_of_union_eq_top
theorem nonempty_of_union_eq_top_of_nonempty {ι : Type*} (t : Set ι) (s : ι → Set α)
(H : Nonempty α) (w : ⋃ i ∈ t, s i = ⊤) : t.Nonempty := by
obtain ⟨x, m, -⟩ := exists_set_mem_of_union_eq_top t s w H.some
exact ⟨x, m⟩
#align set.nonempty_of_union_eq_top_of_nonempty Set.nonempty_of_union_eq_top_of_nonempty
theorem nonempty_of_nonempty_iUnion
{s : ι → Set α} (h_Union : (⋃ i, s i).Nonempty) : Nonempty ι := by
obtain ⟨x, hx⟩ := h_Union
exact ⟨Classical.choose $ mem_iUnion.mp hx⟩
theorem nonempty_of_nonempty_iUnion_eq_univ
{s : ι → Set α} [Nonempty α] (h_Union : ⋃ i, s i = univ) : Nonempty ι :=
nonempty_of_nonempty_iUnion (s := s) (by simpa only [h_Union] using univ_nonempty)
theorem setOf_exists (p : ι → β → Prop) : { x | ∃ i, p i x } = ⋃ i, { x | p i x } :=
ext fun _ => mem_iUnion.symm
#align set.set_of_exists Set.setOf_exists
theorem setOf_forall (p : ι → β → Prop) : { x | ∀ i, p i x } = ⋂ i, { x | p i x } :=
ext fun _ => mem_iInter.symm
#align set.set_of_forall Set.setOf_forall
theorem iUnion_subset {s : ι → Set α} {t : Set α} (h : ∀ i, s i ⊆ t) : ⋃ i, s i ⊆ t :=
iSup_le h
#align set.Union_subset Set.iUnion_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset {s : ∀ i, κ i → Set α} {t : Set α} (h : ∀ i j, s i j ⊆ t) :
⋃ (i) (j), s i j ⊆ t :=
iUnion_subset fun x => iUnion_subset (h x)
#align set.Union₂_subset Set.iUnion₂_subset
theorem subset_iInter {t : Set β} {s : ι → Set β} (h : ∀ i, t ⊆ s i) : t ⊆ ⋂ i, s i :=
le_iInf h
#align set.subset_Inter Set.subset_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iInter₂ {s : Set α} {t : ∀ i, κ i → Set α} (h : ∀ i j, s ⊆ t i j) :
s ⊆ ⋂ (i) (j), t i j :=
subset_iInter fun x => subset_iInter <| h x
#align set.subset_Inter₂ Set.subset_iInter₂
@[simp]
theorem iUnion_subset_iff {s : ι → Set α} {t : Set α} : ⋃ i, s i ⊆ t ↔ ∀ i, s i ⊆ t :=
⟨fun h _ => Subset.trans (le_iSup s _) h, iUnion_subset⟩
#align set.Union_subset_iff Set.iUnion_subset_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset_iff {s : ∀ i, κ i → Set α} {t : Set α} :
⋃ (i) (j), s i j ⊆ t ↔ ∀ i j, s i j ⊆ t := by simp_rw [iUnion_subset_iff]
#align set.Union₂_subset_iff Set.iUnion₂_subset_iff
@[simp]
theorem subset_iInter_iff {s : Set α} {t : ι → Set α} : (s ⊆ ⋂ i, t i) ↔ ∀ i, s ⊆ t i :=
le_iInf_iff
#align set.subset_Inter_iff Set.subset_iInter_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem subset_iInter₂_iff {s : Set α} {t : ∀ i, κ i → Set α} :
(s ⊆ ⋂ (i) (j), t i j) ↔ ∀ i j, s ⊆ t i j := by simp_rw [subset_iInter_iff]
#align set.subset_Inter₂_iff Set.subset_iInter₂_iff
theorem subset_iUnion : ∀ (s : ι → Set β) (i : ι), s i ⊆ ⋃ i, s i :=
le_iSup
#align set.subset_Union Set.subset_iUnion
theorem iInter_subset : ∀ (s : ι → Set β) (i : ι), ⋂ i, s i ⊆ s i :=
iInf_le
#align set.Inter_subset Set.iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iUnion₂ {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : s i j ⊆ ⋃ (i') (j'), s i' j' :=
le_iSup₂ i j
#align set.subset_Union₂ Set.subset_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_subset {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : ⋂ (i) (j), s i j ⊆ s i j :=
iInf₂_le i j
#align set.Inter₂_subset Set.iInter₂_subset
/-- This rather trivial consequence of `subset_iUnion`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem subset_iUnion_of_subset {s : Set α} {t : ι → Set α} (i : ι) (h : s ⊆ t i) : s ⊆ ⋃ i, t i :=
le_iSup_of_le i h
#align set.subset_Union_of_subset Set.subset_iUnion_of_subset
/-- This rather trivial consequence of `iInter_subset`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem iInter_subset_of_subset {s : ι → Set α} {t : Set α} (i : ι) (h : s i ⊆ t) :
⋂ i, s i ⊆ t :=
iInf_le_of_le i h
#align set.Inter_subset_of_subset Set.iInter_subset_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `subset_iUnion₂` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem subset_iUnion₂_of_subset {s : Set α} {t : ∀ i, κ i → Set α} (i : ι) (j : κ i)
(h : s ⊆ t i j) : s ⊆ ⋃ (i) (j), t i j :=
le_iSup₂_of_le i j h
#align set.subset_Union₂_of_subset Set.subset_iUnion₂_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `iInter₂_subset` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem iInter₂_subset_of_subset {s : ∀ i, κ i → Set α} {t : Set α} (i : ι) (j : κ i)
(h : s i j ⊆ t) : ⋂ (i) (j), s i j ⊆ t :=
iInf₂_le_of_le i j h
#align set.Inter₂_subset_of_subset Set.iInter₂_subset_of_subset
theorem iUnion_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono h
#align set.Union_mono Set.iUnion_mono
@[gcongr]
theorem iUnion_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iUnion s ⊆ iUnion t :=
iSup_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋃ (i) (j), s i j ⊆ ⋃ (i) (j), t i j :=
iSup₂_mono h
#align set.Union₂_mono Set.iUnion₂_mono
theorem iInter_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋂ i, s i ⊆ ⋂ i, t i :=
iInf_mono h
#align set.Inter_mono Set.iInter_mono
@[gcongr]
theorem iInter_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iInter s ⊆ iInter t :=
iInf_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋂ (i) (j), s i j ⊆ ⋂ (i) (j), t i j :=
iInf₂_mono h
#align set.Inter₂_mono Set.iInter₂_mono
theorem iUnion_mono' {s : ι → Set α} {t : ι₂ → Set α} (h : ∀ i, ∃ j, s i ⊆ t j) :
⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono' h
#align set.Union_mono' Set.iUnion_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iUnion₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i j, ∃ i' j', s i j ⊆ t i' j') : ⋃ (i) (j), s i j ⊆ ⋃ (i') (j'), t i' j' :=
iSup₂_mono' h
#align set.Union₂_mono' Set.iUnion₂_mono'
theorem iInter_mono' {s : ι → Set α} {t : ι' → Set α} (h : ∀ j, ∃ i, s i ⊆ t j) :
⋂ i, s i ⊆ ⋂ j, t j :=
Set.subset_iInter fun j =>
let ⟨i, hi⟩ := h j
iInter_subset_of_subset i hi
#align set.Inter_mono' Set.iInter_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iInter₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i' j', ∃ i j, s i j ⊆ t i' j') : ⋂ (i) (j), s i j ⊆ ⋂ (i') (j'), t i' j' :=
subset_iInter₂_iff.2 fun i' j' =>
let ⟨_, _, hst⟩ := h i' j'
(iInter₂_subset _ _).trans hst
#align set.Inter₂_mono' Set.iInter₂_mono'
theorem iUnion₂_subset_iUnion (κ : ι → Sort*) (s : ι → Set α) :
⋃ (i) (_ : κ i), s i ⊆ ⋃ i, s i :=
iUnion_mono fun _ => iUnion_subset fun _ => Subset.rfl
#align set.Union₂_subset_Union Set.iUnion₂_subset_iUnion
theorem iInter_subset_iInter₂ (κ : ι → Sort*) (s : ι → Set α) :
⋂ i, s i ⊆ ⋂ (i) (_ : κ i), s i :=
iInter_mono fun _ => subset_iInter fun _ => Subset.rfl
#align set.Inter_subset_Inter₂ Set.iInter_subset_iInter₂
theorem iUnion_setOf (P : ι → α → Prop) : ⋃ i, { x : α | P i x } = { x : α | ∃ i, P i x } := by
ext
exact mem_iUnion
#align set.Union_set_of Set.iUnion_setOf
theorem iInter_setOf (P : ι → α → Prop) : ⋂ i, { x : α | P i x } = { x : α | ∀ i, P i x } := by
ext
exact mem_iInter
#align set.Inter_set_of Set.iInter_setOf
theorem iUnion_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋃ x, f x = ⋃ y, g y :=
h1.iSup_congr h h2
#align set.Union_congr_of_surjective Set.iUnion_congr_of_surjective
theorem iInter_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋂ x, f x = ⋂ y, g y :=
h1.iInf_congr h h2
#align set.Inter_congr_of_surjective Set.iInter_congr_of_surjective
lemma iUnion_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋃ i, s i = ⋃ i, t i := iSup_congr h
#align set.Union_congr Set.iUnion_congr
lemma iInter_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋂ i, s i = ⋂ i, t i := iInf_congr h
#align set.Inter_congr Set.iInter_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iUnion₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋃ (i) (j), s i j = ⋃ (i) (j), t i j :=
iUnion_congr fun i => iUnion_congr <| h i
#align set.Union₂_congr Set.iUnion₂_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iInter₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋂ (i) (j), s i j = ⋂ (i) (j), t i j :=
iInter_congr fun i => iInter_congr <| h i
#align set.Inter₂_congr Set.iInter₂_congr
section Nonempty
variable [Nonempty ι] {f : ι → Set α} {s : Set α}
lemma iUnion_const (s : Set β) : ⋃ _ : ι, s = s := iSup_const
#align set.Union_const Set.iUnion_const
lemma iInter_const (s : Set β) : ⋂ _ : ι, s = s := iInf_const
#align set.Inter_const Set.iInter_const
lemma iUnion_eq_const (hf : ∀ i, f i = s) : ⋃ i, f i = s :=
(iUnion_congr hf).trans $ iUnion_const _
#align set.Union_eq_const Set.iUnion_eq_const
lemma iInter_eq_const (hf : ∀ i, f i = s) : ⋂ i, f i = s :=
(iInter_congr hf).trans $ iInter_const _
#align set.Inter_eq_const Set.iInter_eq_const
end Nonempty
@[simp]
theorem compl_iUnion (s : ι → Set β) : (⋃ i, s i)ᶜ = ⋂ i, (s i)ᶜ :=
compl_iSup
#align set.compl_Union Set.compl_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iUnion₂ (s : ∀ i, κ i → Set α) : (⋃ (i) (j), s i j)ᶜ = ⋂ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iUnion]
#align set.compl_Union₂ Set.compl_iUnion₂
@[simp]
theorem compl_iInter (s : ι → Set β) : (⋂ i, s i)ᶜ = ⋃ i, (s i)ᶜ :=
compl_iInf
#align set.compl_Inter Set.compl_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iInter₂ (s : ∀ i, κ i → Set α) : (⋂ (i) (j), s i j)ᶜ = ⋃ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iInter]
#align set.compl_Inter₂ Set.compl_iInter₂
-- classical -- complete_boolean_algebra
theorem iUnion_eq_compl_iInter_compl (s : ι → Set β) : ⋃ i, s i = (⋂ i, (s i)ᶜ)ᶜ := by
simp only [compl_iInter, compl_compl]
#align set.Union_eq_compl_Inter_compl Set.iUnion_eq_compl_iInter_compl
-- classical -- complete_boolean_algebra
theorem iInter_eq_compl_iUnion_compl (s : ι → Set β) : ⋂ i, s i = (⋃ i, (s i)ᶜ)ᶜ := by
simp only [compl_iUnion, compl_compl]
#align set.Inter_eq_compl_Union_compl Set.iInter_eq_compl_iUnion_compl
theorem inter_iUnion (s : Set β) (t : ι → Set β) : (s ∩ ⋃ i, t i) = ⋃ i, s ∩ t i :=
inf_iSup_eq _ _
#align set.inter_Union Set.inter_iUnion
theorem iUnion_inter (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∩ s = ⋃ i, t i ∩ s :=
iSup_inf_eq _ _
#align set.Union_inter Set.iUnion_inter
theorem iUnion_union_distrib (s : ι → Set β) (t : ι → Set β) :
⋃ i, s i ∪ t i = (⋃ i, s i) ∪ ⋃ i, t i :=
iSup_sup_eq
#align set.Union_union_distrib Set.iUnion_union_distrib
theorem iInter_inter_distrib (s : ι → Set β) (t : ι → Set β) :
⋂ i, s i ∩ t i = (⋂ i, s i) ∩ ⋂ i, t i :=
iInf_inf_eq
#align set.Inter_inter_distrib Set.iInter_inter_distrib
theorem union_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∪ ⋃ i, t i) = ⋃ i, s ∪ t i :=
sup_iSup
#align set.union_Union Set.union_iUnion
theorem iUnion_union [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∪ s = ⋃ i, t i ∪ s :=
iSup_sup
#align set.Union_union Set.iUnion_union
theorem inter_iInter [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∩ ⋂ i, t i) = ⋂ i, s ∩ t i :=
inf_iInf
#align set.inter_Inter Set.inter_iInter
theorem iInter_inter [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋂ i, t i) ∩ s = ⋂ i, t i ∩ s :=
iInf_inf
#align set.Inter_inter Set.iInter_inter
-- classical
theorem union_iInter (s : Set β) (t : ι → Set β) : (s ∪ ⋂ i, t i) = ⋂ i, s ∪ t i :=
sup_iInf_eq _ _
#align set.union_Inter Set.union_iInter
theorem iInter_union (s : ι → Set β) (t : Set β) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.Inter_union Set.iInter_union
theorem iUnion_diff (s : Set β) (t : ι → Set β) : (⋃ i, t i) \ s = ⋃ i, t i \ s :=
iUnion_inter _ _
#align set.Union_diff Set.iUnion_diff
theorem diff_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s \ ⋃ i, t i) = ⋂ i, s \ t i := by
rw [diff_eq, compl_iUnion, inter_iInter]; rfl
#align set.diff_Union Set.diff_iUnion
theorem diff_iInter (s : Set β) (t : ι → Set β) : (s \ ⋂ i, t i) = ⋃ i, s \ t i := by
rw [diff_eq, compl_iInter, inter_iUnion]; rfl
#align set.diff_Inter Set.diff_iInter
theorem directed_on_iUnion {r} {f : ι → Set α} (hd : Directed (· ⊆ ·) f)
(h : ∀ x, DirectedOn r (f x)) : DirectedOn r (⋃ x, f x) := by
simp only [DirectedOn, exists_prop, mem_iUnion, exists_imp]
exact fun a₁ b₁ fb₁ a₂ b₂ fb₂ =>
let ⟨z, zb₁, zb₂⟩ := hd b₁ b₂
let ⟨x, xf, xa₁, xa₂⟩ := h z a₁ (zb₁ fb₁) a₂ (zb₂ fb₂)
⟨x, ⟨z, xf⟩, xa₁, xa₂⟩
#align set.directed_on_Union Set.directed_on_iUnion
theorem iUnion_inter_subset {ι α} {s t : ι → Set α} : ⋃ i, s i ∩ t i ⊆ (⋃ i, s i) ∩ ⋃ i, t i :=
le_iSup_inf_iSup s t
#align set.Union_inter_subset Set.iUnion_inter_subset
theorem iUnion_inter_of_monotone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_monotone hs ht
#align set.Union_inter_of_monotone Set.iUnion_inter_of_monotone
theorem iUnion_inter_of_antitone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_antitone hs ht
#align set.Union_inter_of_antitone Set.iUnion_inter_of_antitone
theorem iInter_union_of_monotone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_monotone hs ht
#align set.Inter_union_of_monotone Set.iInter_union_of_monotone
theorem iInter_union_of_antitone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_antitone hs ht
#align set.Inter_union_of_antitone Set.iInter_union_of_antitone
/-- An equality version of this lemma is `iUnion_iInter_of_monotone` in `Data.Set.Finite`. -/
theorem iUnion_iInter_subset {s : ι → ι' → Set α} : (⋃ j, ⋂ i, s i j) ⊆ ⋂ i, ⋃ j, s i j :=
iSup_iInf_le_iInf_iSup (flip s)
#align set.Union_Inter_subset Set.iUnion_iInter_subset
theorem iUnion_option {ι} (s : Option ι → Set α) : ⋃ o, s o = s none ∪ ⋃ i, s (some i) :=
iSup_option s
#align set.Union_option Set.iUnion_option
theorem iInter_option {ι} (s : Option ι → Set α) : ⋂ o, s o = s none ∩ ⋂ i, s (some i) :=
iInf_option s
#align set.Inter_option Set.iInter_option
section
variable (p : ι → Prop) [DecidablePred p]
theorem iUnion_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 _ _ _
#align set.Union_dite Set.iUnion_dite
theorem iUnion_ite (f g : ι → Set α) :
⋃ i, (if p i then f i else g i) = (⋃ (i) (_ : p i), f i) ∪ ⋃ (i) (_ : ¬p i), g i :=
iUnion_dite _ _ _
#align set.Union_ite Set.iUnion_ite
theorem iInter_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 :=
iInf_dite _ _ _
#align set.Inter_dite Set.iInter_dite
theorem iInter_ite (f g : ι → Set α) :
⋂ i, (if p i then f i else g i) = (⋂ (i) (_ : p i), f i) ∩ ⋂ (i) (_ : ¬p i), g i :=
iInter_dite _ _ _
#align set.Inter_ite Set.iInter_ite
end
theorem image_projection_prod {ι : Type*} {α : ι → Type*} {v : ∀ i : ι, Set (α i)}
(hv : (pi univ v).Nonempty) (i : ι) :
((fun x : ∀ i : ι, α i => x i) '' ⋂ k, (fun x : ∀ j : ι, α j => x k) ⁻¹' v k) = v i := by
classical
apply Subset.antisymm
· simp [iInter_subset]
· intro y y_in
simp only [mem_image, mem_iInter, mem_preimage]
rcases hv with ⟨z, hz⟩
refine' ⟨Function.update z i y, _, update_same i y z⟩
rw [@forall_update_iff ι α _ z i y fun i t => t ∈ v i]
exact ⟨y_in, fun j _ => by simpa using hz j⟩
#align set.image_projection_prod Set.image_projection_prod
/-! ### Unions and intersections indexed by `Prop` -/
theorem iInter_false {s : False → Set α} : iInter s = univ :=
iInf_false
#align set.Inter_false Set.iInter_false
theorem iUnion_false {s : False → Set α} : iUnion s = ∅ :=
iSup_false
#align set.Union_false Set.iUnion_false
@[simp]
theorem iInter_true {s : True → Set α} : iInter s = s trivial :=
iInf_true
#align set.Inter_true Set.iInter_true
@[simp]
theorem iUnion_true {s : True → Set α} : iUnion s = s trivial :=
iSup_true
#align set.Union_true Set.iUnion_true
@[simp]
theorem iInter_exists {p : ι → Prop} {f : Exists p → Set α} :
⋂ x, f x = ⋂ (i) (h : p i), f ⟨i, h⟩ :=
iInf_exists
#align set.Inter_exists Set.iInter_exists
@[simp]
theorem iUnion_exists {p : ι → Prop} {f : Exists p → Set α} :
⋃ x, f x = ⋃ (i) (h : p i), f ⟨i, h⟩ :=
iSup_exists
#align set.Union_exists Set.iUnion_exists
@[simp]
theorem iUnion_empty : (⋃ _ : ι, ∅ : Set α) = ∅ :=
iSup_bot
#align set.Union_empty Set.iUnion_empty
@[simp]
theorem iInter_univ : (⋂ _ : ι, univ : Set α) = univ :=
iInf_top
#align set.Inter_univ Set.iInter_univ
section
variable {s : ι → Set α}
@[simp]
theorem iUnion_eq_empty : ⋃ i, s i = ∅ ↔ ∀ i, s i = ∅ :=
iSup_eq_bot
#align set.Union_eq_empty Set.iUnion_eq_empty
@[simp]
theorem iInter_eq_univ : ⋂ i, s i = univ ↔ ∀ i, s i = univ :=
iInf_eq_top
#align set.Inter_eq_univ Set.iInter_eq_univ
@[simp]
theorem nonempty_iUnion : (⋃ i, s i).Nonempty ↔ ∃ i, (s i).Nonempty := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_Union Set.nonempty_iUnion
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_biUnion {t : Set α} {s : α → Set β} :
(⋃ i ∈ t, s i).Nonempty ↔ ∃ i ∈ t, (s i).Nonempty := by simp
#align set.nonempty_bUnion Set.nonempty_biUnion
theorem iUnion_nonempty_index (s : Set α) (t : s.Nonempty → Set β) :
⋃ h, t h = ⋃ x ∈ s, t ⟨x, ‹_›⟩ :=
iSup_exists
#align set.Union_nonempty_index Set.iUnion_nonempty_index
end
@[simp]
theorem iInter_iInter_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋂ (x) (h : x = b), s x h = s b rfl :=
iInf_iInf_eq_left
#align set.Inter_Inter_eq_left Set.iInter_iInter_eq_left
@[simp]
theorem iInter_iInter_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋂ (x) (h : b = x), s x h = s b rfl :=
iInf_iInf_eq_right
#align set.Inter_Inter_eq_right Set.iInter_iInter_eq_right
@[simp]
theorem iUnion_iUnion_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋃ (x) (h : x = b), s x h = s b rfl :=
iSup_iSup_eq_left
#align set.Union_Union_eq_left Set.iUnion_iUnion_eq_left
@[simp]
theorem iUnion_iUnion_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋃ (x) (h : b = x), s x h = s b rfl :=
iSup_iSup_eq_right
#align set.Union_Union_eq_right Set.iUnion_iUnion_eq_right
theorem iInter_or {p q : Prop} (s : p ∨ q → Set α) :
⋂ h, s h = (⋂ h : p, s (Or.inl h)) ∩ ⋂ h : q, s (Or.inr h) :=
iInf_or
#align set.Inter_or Set.iInter_or
theorem iUnion_or {p q : Prop} (s : p ∨ q → Set α) :
⋃ h, s h = (⋃ i, s (Or.inl i)) ∪ ⋃ j, s (Or.inr j) :=
iSup_or
#align set.Union_or Set.iUnion_or
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iUnion_and {p q : Prop} (s : p ∧ q → Set α) : ⋃ h, s h = ⋃ (hp) (hq), s ⟨hp, hq⟩ :=
iSup_and
#align set.Union_and Set.iUnion_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iInter_and {p q : Prop} (s : p ∧ q → Set α) : ⋂ h, s h = ⋂ (hp) (hq), s ⟨hp, hq⟩ :=
iInf_and
#align set.Inter_and Set.iInter_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iUnion_comm (s : ι → ι' → Set α) : ⋃ (i) (i'), s i i' = ⋃ (i') (i), s i i' :=
iSup_comm
#align set.Union_comm Set.iUnion_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iInter_comm (s : ι → ι' → Set α) : ⋂ (i) (i'), s i i' = ⋂ (i') (i), s i i' :=
iInf_comm
#align set.Inter_comm Set.iInter_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iUnion₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋃ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋃ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iSup₂_comm _
#align set.Union₂_comm Set.iUnion₂_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iInter₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋂ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋂ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iInf₂_comm _
#align set.Inter₂_comm Set.iInter₂_comm
@[simp]
theorem biUnion_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋃ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι']
#align set.bUnion_and Set.biUnion_and
@[simp]
theorem biUnion_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋃ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι]
#align set.bUnion_and' Set.biUnion_and'
@[simp]
theorem biInter_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋂ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iInter_and, @iInter_comm _ ι']
#align set.bInter_and Set.biInter_and
@[simp]
theorem biInter_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋂ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iInter_and, @iInter_comm _ ι]
#align set.bInter_and' Set.biInter_and'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iUnion_iUnion_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋃ (x) (h), s x h = s b (Or.inl rfl) ∪ ⋃ (x) (h : p x), s x (Or.inr h) := by
simp only [iUnion_or, iUnion_union_distrib, iUnion_iUnion_eq_left]
#align set.Union_Union_eq_or_left Set.iUnion_iUnion_eq_or_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iInter_iInter_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋂ (x) (h), s x h = s b (Or.inl rfl) ∩ ⋂ (x) (h : p x), s x (Or.inr h) := by
simp only [iInter_or, iInter_inter_distrib, iInter_iInter_eq_left]
#align set.Inter_Inter_eq_or_left Set.iInter_iInter_eq_or_left
/-! ### Bounded unions and intersections -/
/-- A specialization of `mem_iUnion₂`. -/
theorem mem_biUnion {s : Set α} {t : α → Set β} {x : α} {y : β} (xs : x ∈ s) (ytx : y ∈ t x) :
y ∈ ⋃ x ∈ s, t x :=
mem_iUnion₂_of_mem xs ytx
#align set.mem_bUnion Set.mem_biUnion
/-- A specialization of `mem_iInter₂`. -/
theorem mem_biInter {s : Set α} {t : α → Set β} {y : β} (h : ∀ x ∈ s, y ∈ t x) :
y ∈ ⋂ x ∈ s, t x :=
mem_iInter₂_of_mem h
#align set.mem_bInter Set.mem_biInter
/-- A specialization of `subset_iUnion₂`. -/
theorem subset_biUnion_of_mem {s : Set α} {u : α → Set β} {x : α} (xs : x ∈ s) :
u x ⊆ ⋃ x ∈ s, u x :=
--Porting note: Why is this not just `subset_iUnion₂ x xs`?
@subset_iUnion₂ β α (· ∈ s) (fun i _ => u i) x xs
#align set.subset_bUnion_of_mem Set.subset_biUnion_of_mem
/-- A specialization of `iInter₂_subset`. -/
theorem biInter_subset_of_mem {s : Set α} {t : α → Set β} {x : α} (xs : x ∈ s) :
⋂ x ∈ s, t x ⊆ t x :=
iInter₂_subset x xs
#align set.bInter_subset_of_mem Set.biInter_subset_of_mem
theorem biUnion_subset_biUnion_left {s s' : Set α} {t : α → Set β} (h : s ⊆ s') :
⋃ x ∈ s, t x ⊆ ⋃ x ∈ s', t x :=
iUnion₂_subset fun _ hx => subset_biUnion_of_mem <| h hx
#align set.bUnion_subset_bUnion_left Set.biUnion_subset_biUnion_left
theorem biInter_subset_biInter_left {s s' : Set α} {t : α → Set β} (h : s' ⊆ s) :
⋂ x ∈ s, t x ⊆ ⋂ x ∈ s', t x :=
subset_iInter₂ fun _ hx => biInter_subset_of_mem <| h hx
#align set.bInter_subset_bInter_left Set.biInter_subset_biInter_left
theorem biUnion_mono {s s' : Set α} {t t' : α → Set β} (hs : s' ⊆ s) (h : ∀ x ∈ s, t x ⊆ t' x) :
⋃ x ∈ s', t x ⊆ ⋃ x ∈ s, t' x :=
(biUnion_subset_biUnion_left hs).trans <| iUnion₂_mono h
#align set.bUnion_mono Set.biUnion_mono
theorem biInter_mono {s s' : Set α} {t t' : α → Set β} (hs : s ⊆ s') (h : ∀ x ∈ s, t x ⊆ t' x) :
⋂ x ∈ s', t x ⊆ ⋂ x ∈ s, t' x :=
(biInter_subset_biInter_left hs).trans <| iInter₂_mono h
#align set.bInter_mono Set.biInter_mono
theorem biUnion_eq_iUnion (s : Set α) (t : ∀ x ∈ s, Set β) :
⋃ x ∈ s, t x ‹_› = ⋃ x : s, t x x.2 :=
iSup_subtype'
#align set.bUnion_eq_Union Set.biUnion_eq_iUnion
theorem biInter_eq_iInter (s : Set α) (t : ∀ x ∈ s, Set β) :
⋂ x ∈ s, t x ‹_› = ⋂ x : s, t x x.2 :=
iInf_subtype'
#align set.bInter_eq_Inter Set.biInter_eq_iInter
theorem iUnion_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋃ x : { x // p x }, s x = ⋃ (x) (hx : p x), s ⟨x, hx⟩ :=
iSup_subtype
#align set.Union_subtype Set.iUnion_subtype
theorem iInter_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋂ x : { x // p x }, s x = ⋂ (x) (hx : p x), s ⟨x, hx⟩ :=
iInf_subtype
#align set.Inter_subtype Set.iInter_subtype
theorem biInter_empty (u : α → Set β) : ⋂ x ∈ (∅ : Set α), u x = univ :=
iInf_emptyset
#align set.bInter_empty Set.biInter_empty
theorem biInter_univ (u : α → Set β) : ⋂ x ∈ @univ α, u x = ⋂ x, u x :=
iInf_univ
#align set.bInter_univ Set.biInter_univ
@[simp]
theorem biUnion_self (s : Set α) : ⋃ x ∈ s, s = s :=
Subset.antisymm (iUnion₂_subset fun _ _ => Subset.refl s) fun _ hx => mem_biUnion hx hx
#align set.bUnion_self Set.biUnion_self
@[simp]
theorem iUnion_nonempty_self (s : Set α) : ⋃ _ : s.Nonempty, s = s := by
rw [iUnion_nonempty_index, biUnion_self]
#align set.Union_nonempty_self Set.iUnion_nonempty_self
theorem biInter_singleton (a : α) (s : α → Set β) : ⋂ x ∈ ({a} : Set α), s x = s a :=
iInf_singleton
#align set.bInter_singleton Set.biInter_singleton
theorem biInter_union (s t : Set α) (u : α → Set β) :
⋂ x ∈ s ∪ t, u x = (⋂ x ∈ s, u x) ∩ ⋂ x ∈ t, u x :=
iInf_union
#align set.bInter_union Set.biInter_union
theorem biInter_insert (a : α) (s : Set α) (t : α → Set β) :
⋂ x ∈ insert a s, t x = t a ∩ ⋂ x ∈ s, t x := by simp
#align set.bInter_insert Set.biInter_insert
theorem biInter_pair (a b : α) (s : α → Set β) : ⋂ x ∈ ({a, b} : Set α), s x = s a ∩ s b := by
rw [biInter_insert, biInter_singleton]
#align set.bInter_pair Set.biInter_pair
theorem biInter_inter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, f i ∩ t = (⋂ i ∈ s, f i) ∩ t := by
haveI : Nonempty s := hs.to_subtype
simp [biInter_eq_iInter, ← iInter_inter]
#align set.bInter_inter Set.biInter_inter
theorem inter_biInter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, t ∩ f i = t ∩ ⋂ i ∈ s, f i := by
rw [inter_comm, ← biInter_inter hs]
simp [inter_comm]
#align set.inter_bInter Set.inter_biInter
theorem biUnion_empty (s : α → Set β) : ⋃ x ∈ (∅ : Set α), s x = ∅ :=
iSup_emptyset
#align set.bUnion_empty Set.biUnion_empty
theorem biUnion_univ (s : α → Set β) : ⋃ x ∈ @univ α, s x = ⋃ x, s x :=
iSup_univ
#align set.bUnion_univ Set.biUnion_univ
theorem biUnion_singleton (a : α) (s : α → Set β) : ⋃ x ∈ ({a} : Set α), s x = s a :=
iSup_singleton
#align set.bUnion_singleton Set.biUnion_singleton
@[simp]
theorem biUnion_of_singleton (s : Set α) : ⋃ x ∈ s, {x} = s :=
ext <| by simp
#align set.bUnion_of_singleton Set.biUnion_of_singleton
theorem biUnion_union (s t : Set α) (u : α → Set β) :
⋃ x ∈ s ∪ t, u x = (⋃ x ∈ s, u x) ∪ ⋃ x ∈ t, u x :=
iSup_union
#align set.bUnion_union Set.biUnion_union
@[simp]
theorem iUnion_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋃ i, f i = ⋃ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iUnion_subtype _ _
#align set.Union_coe_set Set.iUnion_coe_set
@[simp]
theorem iInter_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋂ i, f i = ⋂ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iInter_subtype _ _
#align set.Inter_coe_set Set.iInter_coe_set
theorem biUnion_insert (a : α) (s : Set α) (t : α → Set β) :
⋃ x ∈ insert a s, t x = t a ∪ ⋃ x ∈ s, t x := by simp
#align set.bUnion_insert Set.biUnion_insert
theorem biUnion_pair (a b : α) (s : α → Set β) : ⋃ x ∈ ({a, b} : Set α), s x = s a ∪ s b :=
by simp
#align set.bUnion_pair Set.biUnion_pair
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem inter_iUnion₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∩ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ∩ t i j := by simp only [inter_iUnion]
#align set.inter_Union₂ Set.inter_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_inter (s : ∀ i, κ i → Set α) (t : Set α) :
(⋃ (i) (j), s i j) ∩ t = ⋃ (i) (j), s i j ∩ t := by simp_rw [iUnion_inter]
#align set.Union₂_inter Set.iUnion₂_inter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_iInter₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_iInter]
#align set.union_Inter₂ Set.union_iInter₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_union (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [iInter_union]
#align set.Inter₂_union Set.iInter₂_union
theorem mem_sUnion_of_mem {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∈ t) (ht : t ∈ S) :
x ∈ ⋃₀S :=
⟨t, ht, hx⟩
#align set.mem_sUnion_of_mem Set.mem_sUnion_of_mem
-- is this theorem really necessary?
theorem not_mem_of_not_mem_sUnion {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∉ ⋃₀S)
(ht : t ∈ S) : x ∉ t := fun h => hx ⟨t, ht, h⟩
#align set.not_mem_of_not_mem_sUnion Set.not_mem_of_not_mem_sUnion
theorem sInter_subset_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : ⋂₀ S ⊆ t :=
sInf_le tS
#align set.sInter_subset_of_mem Set.sInter_subset_of_mem
theorem subset_sUnion_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : t ⊆ ⋃₀S :=
le_sSup tS
#align set.subset_sUnion_of_mem Set.subset_sUnion_of_mem
theorem subset_sUnion_of_subset {s : Set α} (t : Set (Set α)) (u : Set α) (h₁ : s ⊆ u)
(h₂ : u ∈ t) : s ⊆ ⋃₀t :=
Subset.trans h₁ (subset_sUnion_of_mem h₂)
#align set.subset_sUnion_of_subset Set.subset_sUnion_of_subset
theorem sUnion_subset {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t' ⊆ t) : ⋃₀S ⊆ t :=
sSup_le h
#align set.sUnion_subset Set.sUnion_subset
@[simp]
theorem sUnion_subset_iff {s : Set (Set α)} {t : Set α} : ⋃₀s ⊆ t ↔ ∀ t' ∈ s, t' ⊆ t :=
sSup_le_iff
#align set.sUnion_subset_iff Set.sUnion_subset_iff
/-- `sUnion` is monotone under taking a subset of each set. -/
lemma sUnion_mono_subsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, t ⊆ f t) :
⋃₀ s ⊆ ⋃₀ (f '' s) :=
fun _ ⟨t, htx, hxt⟩ ↦ ⟨f t, mem_image_of_mem f htx, hf t hxt⟩
/-- `sUnion` is monotone under taking a superset of each set. -/
lemma sUnion_mono_supsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, f t ⊆ t) :
⋃₀ (f '' s) ⊆ ⋃₀ s :=
-- If t ∈ f '' s is arbitrary; t = f u for some u : Set α.
fun _ ⟨_, ⟨u, hus, hut⟩, hxt⟩ ↦ ⟨u, hus, (hut ▸ hf u) hxt⟩
theorem subset_sInter {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t ⊆ t') : t ⊆ ⋂₀ S :=
le_sInf h
#align set.subset_sInter Set.subset_sInter
@[simp]
theorem subset_sInter_iff {S : Set (Set α)} {t : Set α} : t ⊆ ⋂₀ S ↔ ∀ t' ∈ S, t ⊆ t' :=
le_sInf_iff
#align set.subset_sInter_iff Set.subset_sInter_iff
@[gcongr]
theorem sUnion_subset_sUnion {S T : Set (Set α)} (h : S ⊆ T) : ⋃₀S ⊆ ⋃₀T :=
sUnion_subset fun _ hs => subset_sUnion_of_mem (h hs)
#align set.sUnion_subset_sUnion Set.sUnion_subset_sUnion
@[gcongr]
theorem sInter_subset_sInter {S T : Set (Set α)} (h : S ⊆ T) : ⋂₀ T ⊆ ⋂₀ S :=
subset_sInter fun _ hs => sInter_subset_of_mem (h hs)
#align set.sInter_subset_sInter Set.sInter_subset_sInter
@[simp]
theorem sUnion_empty : ⋃₀∅ = (∅ : Set α) :=
sSup_empty
#align set.sUnion_empty Set.sUnion_empty
@[simp]
theorem sInter_empty : ⋂₀ ∅ = (univ : Set α) :=
sInf_empty
#align set.sInter_empty Set.sInter_empty
@[simp]
theorem sUnion_singleton (s : Set α) : ⋃₀{s} = s :=
sSup_singleton
#align set.sUnion_singleton Set.sUnion_singleton
@[simp]
theorem sInter_singleton (s : Set α) : ⋂₀ {s} = s :=
sInf_singleton
#align set.sInter_singleton Set.sInter_singleton
@[simp]
theorem sUnion_eq_empty {S : Set (Set α)} : ⋃₀S = ∅ ↔ ∀ s ∈ S, s = ∅ :=
sSup_eq_bot
#align set.sUnion_eq_empty Set.sUnion_eq_empty
@[simp]
theorem sInter_eq_univ {S : Set (Set α)} : ⋂₀ S = univ ↔ ∀ s ∈ S, s = univ :=
sInf_eq_top
#align set.sInter_eq_univ Set.sInter_eq_univ
theorem subset_powerset_iff {s : Set (Set α)} {t : Set α} : s ⊆ 𝒫 t ↔ ⋃₀ s ⊆ t :=
sUnion_subset_iff.symm
/-- `⋃₀` and `𝒫` form a Galois connection. -/
theorem sUnion_powerset_gc :
GaloisConnection (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gc_sSup_Iic
/-- `⋃₀` and `𝒫` form a Galois insertion. -/
def sUnion_powerset_gi :
GaloisInsertion (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gi_sSup_Iic
/-- If all sets in a collection are either `∅` or `Set.univ`, then so is their union. -/
theorem sUnion_mem_empty_univ {S : Set (Set α)} (h : S ⊆ {∅, univ}) :
⋃₀ S ∈ ({∅, univ} : Set (Set α)) := by
simp only [mem_insert_iff, mem_singleton_iff, or_iff_not_imp_left, sUnion_eq_empty, not_forall]
rintro ⟨s, hs, hne⟩
obtain rfl : s = univ := (h hs).resolve_left hne
exact univ_subset_iff.1 <| subset_sUnion_of_mem hs
@[simp]
theorem nonempty_sUnion {S : Set (Set α)} : (⋃₀S).Nonempty ↔ ∃ s ∈ S, Set.Nonempty s := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_sUnion Set.nonempty_sUnion
theorem Nonempty.of_sUnion {s : Set (Set α)} (h : (⋃₀s).Nonempty) : s.Nonempty :=
let ⟨s, hs, _⟩ := nonempty_sUnion.1 h
⟨s, hs⟩
#align set.nonempty.of_sUnion Set.Nonempty.of_sUnion
theorem Nonempty.of_sUnion_eq_univ [Nonempty α] {s : Set (Set α)} (h : ⋃₀s = univ) : s.Nonempty :=
Nonempty.of_sUnion <| h.symm ▸ univ_nonempty
#align set.nonempty.of_sUnion_eq_univ Set.Nonempty.of_sUnion_eq_univ
theorem sUnion_union (S T : Set (Set α)) : ⋃₀(S ∪ T) = ⋃₀S ∪ ⋃₀T :=
sSup_union
#align set.sUnion_union Set.sUnion_union
theorem sInter_union (S T : Set (Set α)) : ⋂₀ (S ∪ T) = ⋂₀ S ∩ ⋂₀ T :=
sInf_union
#align set.sInter_union Set.sInter_union
@[simp]
theorem sUnion_insert (s : Set α) (T : Set (Set α)) : ⋃₀insert s T = s ∪ ⋃₀T :=
sSup_insert
#align set.sUnion_insert Set.sUnion_insert
@[simp]
theorem sInter_insert (s : Set α) (T : Set (Set α)) : ⋂₀ insert s T = s ∩ ⋂₀ T :=
sInf_insert
#align set.sInter_insert Set.sInter_insert
@[simp]
theorem sUnion_diff_singleton_empty (s : Set (Set α)) : ⋃₀(s \ {∅}) = ⋃₀s :=
sSup_diff_singleton_bot s
#align set.sUnion_diff_singleton_empty Set.sUnion_diff_singleton_empty
@[simp]
theorem sInter_diff_singleton_univ (s : Set (Set α)) : ⋂₀ (s \ {univ}) = ⋂₀ s :=
sInf_diff_singleton_top s
#align set.sInter_diff_singleton_univ Set.sInter_diff_singleton_univ
theorem sUnion_pair (s t : Set α) : ⋃₀{s, t} = s ∪ t :=
sSup_pair
#align set.sUnion_pair Set.sUnion_pair
theorem sInter_pair (s t : Set α) : ⋂₀ {s, t} = s ∩ t :=
sInf_pair
#align set.sInter_pair Set.sInter_pair
@[simp]
theorem sUnion_image (f : α → Set β) (s : Set α) : ⋃₀(f '' s) = ⋃ x ∈ s, f x :=
sSup_image
#align set.sUnion_image Set.sUnion_image
@[simp]
theorem sInter_image (f : α → Set β) (s : Set α) : ⋂₀ (f '' s) = ⋂ x ∈ s, f x :=
sInf_image
#align set.sInter_image Set.sInter_image
@[simp]
theorem sUnion_range (f : ι → Set β) : ⋃₀range f = ⋃ x, f x :=
rfl
#align set.sUnion_range Set.sUnion_range
@[simp]
theorem sInter_range (f : ι → Set β) : ⋂₀ range f = ⋂ x, f x :=
rfl
#align set.sInter_range Set.sInter_range
theorem iUnion_eq_univ_iff {f : ι → Set α} : ⋃ i, f i = univ ↔ ∀ x, ∃ i, x ∈ f i := by
simp only [eq_univ_iff_forall, mem_iUnion]
#align set.Union_eq_univ_iff Set.iUnion_eq_univ_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_eq_univ_iff {s : ∀ i, κ i → Set α} :
⋃ (i) (j), s i j = univ ↔ ∀ a, ∃ i j, a ∈ s i j :=
by simp only [iUnion_eq_univ_iff, mem_iUnion]
#align set.Union₂_eq_univ_iff Set.iUnion₂_eq_univ_iff
theorem sUnion_eq_univ_iff {c : Set (Set α)} : ⋃₀c = univ ↔ ∀ a, ∃ b ∈ c, a ∈ b := by
simp only [eq_univ_iff_forall, mem_sUnion]
#align set.sUnion_eq_univ_iff Set.sUnion_eq_univ_iff
-- classical
theorem iInter_eq_empty_iff {f : ι → Set α} : ⋂ i, f i = ∅ ↔ ∀ x, ∃ i, x ∉ f i := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.Inter_eq_empty_iff Set.iInter_eq_empty_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
theorem iInter₂_eq_empty_iff {s : ∀ i, κ i → Set α} :
⋂ (i) (j), s i j = ∅ ↔ ∀ a, ∃ i j, a ∉ s i j := by
simp only [eq_empty_iff_forall_not_mem, mem_iInter, not_forall]
#align set.Inter₂_eq_empty_iff Set.iInter₂_eq_empty_iff
-- classical
theorem sInter_eq_empty_iff {c : Set (Set α)} : ⋂₀ c = ∅ ↔ ∀ a, ∃ b ∈ c, a ∉ b := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.sInter_eq_empty_iff Set.sInter_eq_empty_iff
-- classical
@[simp]
theorem nonempty_iInter {f : ι → Set α} : (⋂ i, f i).Nonempty ↔ ∃ x, ∀ i, x ∈ f i := by
simp [nonempty_iff_ne_empty, iInter_eq_empty_iff]
#align set.nonempty_Inter Set.nonempty_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_iInter₂ {s : ∀ i, κ i → Set α} :
(⋂ (i) (j), s i j).Nonempty ↔ ∃ a, ∀ i j, a ∈ s i j := by
simp
#align set.nonempty_Inter₂ Set.nonempty_iInter₂
-- classical
@[simp]
theorem nonempty_sInter {c : Set (Set α)} : (⋂₀ c).Nonempty ↔ ∃ a, ∀ b ∈ c, a ∈ b := by
simp [nonempty_iff_ne_empty, sInter_eq_empty_iff]
#align set.nonempty_sInter Set.nonempty_sInter
-- classical
theorem compl_sUnion (S : Set (Set α)) : (⋃₀S)ᶜ = ⋂₀ (compl '' S) :=
ext fun x => by simp
#align set.compl_sUnion Set.compl_sUnion
-- classical
theorem sUnion_eq_compl_sInter_compl (S : Set (Set α)) : ⋃₀S = (⋂₀ (compl '' S))ᶜ := by
rw [← compl_compl (⋃₀S), compl_sUnion]
#align set.sUnion_eq_compl_sInter_compl Set.sUnion_eq_compl_sInter_compl
-- classical
theorem compl_sInter (S : Set (Set α)) : (⋂₀ S)ᶜ = ⋃₀(compl '' S) := by
rw [sUnion_eq_compl_sInter_compl, compl_compl_image]
#align set.compl_sInter Set.compl_sInter
-- classical
theorem sInter_eq_compl_sUnion_compl (S : Set (Set α)) : ⋂₀ S = (⋃₀(compl '' S))ᶜ := by
rw [← compl_compl (⋂₀ S), compl_sInter]
#align set.sInter_eq_compl_sUnion_compl Set.sInter_eq_compl_sUnion_compl
theorem inter_empty_of_inter_sUnion_empty {s t : Set α} {S : Set (Set α)} (hs : t ∈ S)
(h : s ∩ ⋃₀S = ∅) : s ∩ t = ∅ :=
eq_empty_of_subset_empty <| by
rw [← h]; exact inter_subset_inter_right _ (subset_sUnion_of_mem hs)
#align set.inter_empty_of_inter_sUnion_empty Set.inter_empty_of_inter_sUnion_empty
theorem range_sigma_eq_iUnion_range {γ : α → Type*} (f : Sigma γ → β) :
range f = ⋃ a, range fun b => f ⟨a, b⟩ :=
Set.ext <| by simp
#align set.range_sigma_eq_Union_range Set.range_sigma_eq_iUnion_range
theorem iUnion_eq_range_sigma (s : α → Set β) : ⋃ i, s i = range fun a : Σi, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_sigma Set.iUnion_eq_range_sigma
theorem iUnion_eq_range_psigma (s : ι → Set β) : ⋃ i, s i = range fun a : Σ'i, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_psigma Set.iUnion_eq_range_psigma
theorem iUnion_image_preimage_sigma_mk_eq_self {ι : Type*} {σ : ι → Type*} (s : Set (Sigma σ)) :
⋃ i, Sigma.mk i '' (Sigma.mk i ⁻¹' s) = s := by
ext x
simp only [mem_iUnion, mem_image, mem_preimage]
constructor
· rintro ⟨i, a, h, rfl⟩
exact h
· intro h
cases' x with i a
exact ⟨i, a, h, rfl⟩
#align set.Union_image_preimage_sigma_mk_eq_self Set.iUnion_image_preimage_sigma_mk_eq_self
theorem Sigma.univ (X : α → Type*) : (Set.univ : Set (Σa, X a)) = ⋃ a, range (Sigma.mk a) :=
Set.ext fun x =>
iff_of_true trivial ⟨range (Sigma.mk x.1), Set.mem_range_self _, x.2, Sigma.eta x⟩
#align set.sigma.univ Set.Sigma.univ
alias sUnion_mono := sUnion_subset_sUnion
#align set.sUnion_mono Set.sUnion_mono
theorem iUnion_subset_iUnion_const {s : Set α} (h : ι → ι₂) : ⋃ _ : ι, s ⊆ ⋃ _ : ι₂, s :=
@iSup_const_mono (Set α) ι ι₂ _ s h
#align set.Union_subset_Union_const Set.iUnion_subset_iUnion_const
@[simp]
theorem iUnion_singleton_eq_range {α β : Type*} (f : α → β) : ⋃ x : α, {f x} = range f := by
ext x
simp [@eq_comm _ x]
#align set.Union_singleton_eq_range Set.iUnion_singleton_eq_range
theorem iUnion_of_singleton (α : Type*) : (⋃ x, {x} : Set α) = univ := by simp [Set.ext_iff]
#align set.Union_of_singleton Set.iUnion_of_singleton
theorem iUnion_of_singleton_coe (s : Set α) : ⋃ i : s, ({(i : α)} : Set α) = s := by simp
#align set.Union_of_singleton_coe Set.iUnion_of_singleton_coe
theorem sUnion_eq_biUnion {s : Set (Set α)} : ⋃₀s = ⋃ (i : Set α) (_ : i ∈ s), i := by
rw [← sUnion_image, image_id']
#align set.sUnion_eq_bUnion Set.sUnion_eq_biUnion
theorem sInter_eq_biInter {s : Set (Set α)} : ⋂₀ s = ⋂ (i : Set α) (_ : i ∈ s), i := by
rw [← sInter_image, image_id']
#align set.sInter_eq_bInter Set.sInter_eq_biInter
theorem sUnion_eq_iUnion {s : Set (Set α)} : ⋃₀s = ⋃ i : s, i := by
simp only [← sUnion_range, Subtype.range_coe]
#align set.sUnion_eq_Union Set.sUnion_eq_iUnion
theorem sInter_eq_iInter {s : Set (Set α)} : ⋂₀ s = ⋂ i : s, i := by
simp only [← sInter_range, Subtype.range_coe]
#align set.sInter_eq_Inter Set.sInter_eq_iInter
@[simp]
theorem iUnion_of_empty [IsEmpty ι] (s : ι → Set α) : ⋃ i, s i = ∅ :=
iSup_of_empty _
#align set.Union_of_empty Set.iUnion_of_empty
@[simp]
theorem iInter_of_empty [IsEmpty ι] (s : ι → Set α) : ⋂ i, s i = univ :=
iInf_of_empty _
#align set.Inter_of_empty Set.iInter_of_empty
theorem union_eq_iUnion {s₁ s₂ : Set α} : s₁ ∪ s₂ = ⋃ b : Bool, cond b s₁ s₂ :=
sup_eq_iSup s₁ s₂
#align set.union_eq_Union Set.union_eq_iUnion
theorem inter_eq_iInter {s₁ s₂ : Set α} : s₁ ∩ s₂ = ⋂ b : Bool, cond b s₁ s₂ :=
inf_eq_iInf s₁ s₂
#align set.inter_eq_Inter Set.inter_eq_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_union_sInter {S T : Set (Set α)} :
⋂₀ S ∪ ⋂₀ T = ⋂ p ∈ S ×ˢ T, (p : Set α × Set α).1 ∪ p.2 :=
sInf_sup_sInf
#align set.sInter_union_sInter Set.sInter_union_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_inter_sUnion {s t : Set (Set α)} :
⋃₀s ∩ ⋃₀t = ⋃ p ∈ s ×ˢ t, (p : Set α × Set α).1 ∩ p.2 :=
sSup_inf_sSup
#align set.sUnion_inter_sUnion Set.sUnion_inter_sUnion
theorem biUnion_iUnion (s : ι → Set α) (t : α → Set β) :
⋃ x ∈ ⋃ i, s i, t x = ⋃ (i) (x ∈ s i), t x := by simp [@iUnion_comm _ ι]
#align set.bUnion_Union Set.biUnion_iUnion
theorem biInter_iUnion (s : ι → Set α) (t : α → Set β) :
⋂ x ∈ ⋃ i, s i, t x = ⋂ (i) (x ∈ s i), t x := by simp [@iInter_comm _ ι]
#align set.bInter_Union Set.biInter_iUnion
theorem sUnion_iUnion (s : ι → Set (Set α)) : ⋃₀⋃ i, s i = ⋃ i, ⋃₀s i := by
simp only [sUnion_eq_biUnion, biUnion_iUnion]
#align set.sUnion_Union Set.sUnion_iUnion
theorem sInter_iUnion (s : ι → Set (Set α)) : ⋂₀ ⋃ i, s i = ⋂ i, ⋂₀ s i := by
simp only [sInter_eq_biInter, biInter_iUnion]
#align set.sInter_Union Set.sInter_iUnion
theorem iUnion_range_eq_sUnion {α β : Type*} (C : Set (Set α)) {f : ∀ s : C, β → (s : Type _)}
(hf : ∀ s : C, Surjective (f s)) : ⋃ y : β, range (fun s : C => (f s y).val) = ⋃₀C := by
ext x; constructor
· rintro ⟨s, ⟨y, rfl⟩, ⟨s, hs⟩, rfl⟩
refine' ⟨_, hs, _⟩
exact (f ⟨s, hs⟩ y).2
· rintro ⟨s, hs, hx⟩
cases' hf ⟨s, hs⟩ ⟨x, hx⟩ with y hy
refine' ⟨_, ⟨y, rfl⟩, ⟨s, hs⟩, _⟩
exact congr_arg Subtype.val hy
#align set.Union_range_eq_sUnion Set.iUnion_range_eq_sUnion
theorem iUnion_range_eq_iUnion (C : ι → Set α) {f : ∀ x : ι, β → C x}
(hf : ∀ x : ι, Surjective (f x)) : ⋃ y : β, range (fun x : ι => (f x y).val) = ⋃ x, C x := by
ext x; rw [mem_iUnion, mem_iUnion]; constructor
· rintro ⟨y, i, rfl⟩
exact ⟨i, (f i y).2⟩
· rintro ⟨i, hx⟩
cases' hf i ⟨x, hx⟩ with y hy
exact ⟨y, i, congr_arg Subtype.val hy⟩
#align set.Union_range_eq_Union Set.iUnion_range_eq_iUnion
theorem union_distrib_iInter_left (s : ι → Set α) (t : Set α) : (t ∪ ⋂ i, s i) = ⋂ i, t ∪ s i :=
sup_iInf_eq _ _
#align set.union_distrib_Inter_left Set.union_distrib_iInter_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_left (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_distrib_iInter_left]
#align set.union_distrib_Inter₂_left Set.union_distrib_iInter₂_left
theorem union_distrib_iInter_right (s : ι → Set α) (t : Set α) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.union_distrib_Inter_right Set.union_distrib_iInter_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_right (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [union_distrib_iInter_right]
#align set.union_distrib_Inter₂_right Set.union_distrib_iInter₂_right
section Function
/-! ### `mapsTo` -/
theorem mapsTo_sUnion {S : Set (Set α)} {t : Set β} {f : α → β} (H : ∀ s ∈ S, MapsTo f s t) :
MapsTo f (⋃₀S) t := fun _ ⟨s, hs, hx⟩ => H s hs hx
#align set.maps_to_sUnion Set.mapsTo_sUnion
theorem mapsTo_iUnion {s : ι → Set α} {t : Set β} {f : α → β} (H : ∀ i, MapsTo f (s i) t) :
MapsTo f (⋃ i, s i) t :=
mapsTo_sUnion <| forall_range_iff.2 H
#align set.maps_to_Union Set.mapsTo_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂ {s : ∀ i, κ i → Set α} {t : Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) t) : MapsTo f (⋃ (i) (j), s i j) t :=
mapsTo_iUnion fun i => mapsTo_iUnion (H i)
#align set.maps_to_Union₂ Set.mapsTo_iUnion₂
theorem mapsTo_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋃ i, s i) (⋃ i, t i) :=
mapsTo_iUnion fun i => (H i).mono (Subset.refl _) (subset_iUnion t i)
#align set.maps_to_Union_Union Set.mapsTo_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
mapsTo_iUnion_iUnion fun i => mapsTo_iUnion_iUnion (H i)
#align set.maps_to_Union₂_Union₂ Set.mapsTo_iUnion₂_iUnion₂
theorem mapsTo_sInter {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, MapsTo f s t) :
MapsTo f s (⋂₀ T) := fun _ hx t ht => H t ht hx
#align set.maps_to_sInter Set.mapsTo_sInter
theorem mapsTo_iInter {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, MapsTo f s (t i)) :
MapsTo f s (⋂ i, t i) := fun _ hx => mem_iInter.2 fun i => H i hx
#align set.maps_to_Inter Set.mapsTo_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f s (t i j)) : MapsTo f s (⋂ (i) (j), t i j) :=
mapsTo_iInter fun i => mapsTo_iInter (H i)
#align set.maps_to_Inter₂ Set.mapsTo_iInter₂
theorem mapsTo_iInter_iInter {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋂ i, s i) (⋂ i, t i) :=
mapsTo_iInter fun i => (H i).mono (iInter_subset s i) (Subset.refl _)
#align set.maps_to_Inter_Inter Set.mapsTo_iInter_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂_iInter₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋂ (i) (j), s i j) (⋂ (i) (j), t i j) :=
mapsTo_iInter_iInter fun i => mapsTo_iInter_iInter (H i)
#align set.maps_to_Inter₂_Inter₂ Set.mapsTo_iInter₂_iInter₂
theorem image_iInter_subset (s : ι → Set α) (f : α → β) : (f '' ⋂ i, s i) ⊆ ⋂ i, f '' s i :=
(mapsTo_iInter_iInter fun i => mapsTo_image f (s i)).image_subset
#align set.image_Inter_subset Set.image_iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂_subset (s : ∀ i, κ i → Set α) (f : α → β) :
(f '' ⋂ (i) (j), s i j) ⊆ ⋂ (i) (j), f '' s i j :=
(mapsTo_iInter₂_iInter₂ fun i hi => mapsTo_image f (s i hi)).image_subset
#align set.image_Inter₂_subset Set.image_iInter₂_subset
theorem image_sInter_subset (S : Set (Set α)) (f : α → β) : f '' ⋂₀ S ⊆ ⋂ s ∈ S, f '' s := by
rw [sInter_eq_biInter]
apply image_iInter₂_subset
#align set.image_sInter_subset Set.image_sInter_subset
/-! ### `restrictPreimage` -/
section
open Function
variable (s : Set β) {f : α → β} {U : ι → Set β} (hU : iUnion U = univ)
theorem injective_iff_injective_of_iUnion_eq_univ :
Injective f ↔ ∀ i, Injective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_injective H, fun H x y e => _⟩
obtain ⟨i, hi⟩ := Set.mem_iUnion.mp
(show f x ∈ Set.iUnion U by rw [hU]; triv)
injection @H i ⟨x, hi⟩ ⟨y, show f y ∈ U i from e ▸ hi⟩ (Subtype.ext e)
#align set.injective_iff_injective_of_Union_eq_univ Set.injective_iff_injective_of_iUnion_eq_univ
theorem surjective_iff_surjective_of_iUnion_eq_univ :
Surjective f ↔ ∀ i, Surjective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_surjective H, fun H x => _⟩
obtain ⟨i, hi⟩ :=
Set.mem_iUnion.mp
(show x ∈ Set.iUnion U by rw [hU]; triv)
exact ⟨_, congr_arg Subtype.val (H i ⟨x, hi⟩).choose_spec⟩
#align set.surjective_iff_surjective_of_Union_eq_univ Set.surjective_iff_surjective_of_iUnion_eq_univ
theorem bijective_iff_bijective_of_iUnion_eq_univ :
Bijective f ↔ ∀ i, Bijective ((U i).restrictPreimage f) := by
rw [Bijective, injective_iff_injective_of_iUnion_eq_univ hU,
surjective_iff_surjective_of_iUnion_eq_univ hU]
simp [Bijective, forall_and]
#align set.bijective_iff_bijective_of_Union_eq_univ Set.bijective_iff_bijective_of_iUnion_eq_univ
end
/-! ### `InjOn` -/
theorem InjOn.image_iInter_eq [Nonempty ι] {s : ι → Set α} {f : α → β} (h : InjOn f (⋃ i, s i)) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
inhabit ι
refine' Subset.antisymm (image_iInter_subset s f) fun y hy => _
simp only [mem_iInter, mem_image_iff_bex] at hy
choose x hx hy using hy
refine' ⟨x default, mem_iInter.2 fun i => _, hy _⟩
suffices x default = x i by
rw [this]
apply hx
replace hx : ∀ i, x i ∈ ⋃ j, s j := fun i => (subset_iUnion _ _) (hx i)
apply h (hx _) (hx _)
simp only [hy]
#align set.inj_on.image_Inter_eq Set.InjOn.image_iInter_eq
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
theorem InjOn.image_biInter_eq {p : ι → Prop} {s : ∀ i, p i → Set α} (hp : ∃ i, p i)
{f : α → β} (h : InjOn f (⋃ (i) (hi), s i hi)) :
(f '' ⋂ (i) (hi), s i hi) = ⋂ (i) (hi), f '' s i hi := by
simp only [iInter, iInf_subtype']
haveI : Nonempty { i // p i } := nonempty_subtype.2 hp
apply InjOn.image_iInter_eq
simpa only [iUnion, iSup_subtype'] using h
#align set.inj_on.image_bInter_eq Set.InjOn.image_biInter_eq
theorem image_iInter {f : α → β} (hf : Bijective f) (s : ι → Set α) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
cases isEmpty_or_nonempty ι
· simp_rw [iInter_of_empty, image_univ_of_surjective hf.surjective]
· exact (hf.injective.injOn _).image_iInter_eq
#align set.image_Inter Set.image_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂ {f : α → β} (hf : Bijective f) (s : ∀ i, κ i → Set α) :
(f '' ⋂ (i) (j), s i j) = ⋂ (i) (j), f '' s i j := by simp_rw [image_iInter hf]
#align set.image_Inter₂ Set.image_iInter₂
theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
(hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i) := by
intro x hx y hy hxy
rcases mem_iUnion.1 hx with ⟨i, hx⟩
rcases mem_iUnion.1 hy with ⟨j, hy⟩
rcases hs i j with ⟨k, hi, hj⟩
exact hf k (hi hx) (hj hy) hxy
#align set.inj_on_Union_of_directed Set.inj_on_iUnion_of_directed
/-! ### `SurjOn` -/
theorem surjOn_sUnion {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, SurjOn f s t) :
SurjOn f s (⋃₀T) := fun _ ⟨t, ht, hx⟩ => H t ht hx
#align set.surj_on_sUnion Set.surjOn_sUnion
theorem surjOn_iUnion {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, SurjOn f s (t i)) :
SurjOn f s (⋃ i, t i) :=
surjOn_sUnion <| forall_range_iff.2 H
#align set.surj_on_Union Set.surjOn_iUnion
theorem surjOn_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) : SurjOn f (⋃ i, s i) (⋃ i, t i) :=
surjOn_iUnion fun i => (H i).mono (subset_iUnion _ _) (Subset.refl _)
#align set.surj_on_Union_Union Set.surjOn_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f s (t i j)) : SurjOn f s (⋃ (i) (j), t i j) :=
surjOn_iUnion fun i => surjOn_iUnion (H i)
#align set.surj_on_Union₂ Set.surjOn_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f (s i j) (t i j)) : SurjOn f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
surjOn_iUnion_iUnion fun i => surjOn_iUnion_iUnion (H i)
#align set.surj_on_Union₂_Union₂ Set.surjOn_iUnion₂_iUnion₂
theorem surjOn_iInter [Nonempty ι] {s : ι → Set α} {t : Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) t) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) t := by
intro y hy
rw [Hinj.image_iInter_eq, mem_iInter]
exact fun i => H i hy
#align set.surj_on_Inter Set.surjOn_iInter
theorem surjOn_iInter_iInter [Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) (⋂ i, t i) :=
surjOn_iInter (fun i => (H i).mono (Subset.refl _) (iInter_subset _ _)) Hinj
#align set.surj_on_Inter_Inter Set.surjOn_iInter_iInter
/-! ### `BijOn` -/
theorem bijOn_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i))
(Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
⟨mapsTo_iUnion_iUnion fun i => (H i).mapsTo, Hinj, surjOn_iUnion_iUnion fun i => (H i).surjOn⟩
#align set.bij_on_Union Set.bijOn_iUnion
theorem bijOn_iInter [hi : Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, BijOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
⟨mapsTo_iInter_iInter fun i => (H i).mapsTo,
hi.elim fun i => (H i).injOn.mono (iInter_subset _ _),
surjOn_iInter_iInter (fun i => (H i).surjOn) Hinj⟩
#align set.bij_on_Inter Set.bijOn_iInter
theorem bijOn_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {t : ι → Set β}
{f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
bijOn_iUnion H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Union_of_directed Set.bijOn_iUnion_of_directed
theorem bijOn_iInter_of_directed [Nonempty ι] {s : ι → Set α} (hs : Directed (· ⊆ ·) s)
{t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
bijOn_iInter H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Inter_of_directed Set.bijOn_iInter_of_directed
end Function
/-! ### `image`, `preimage` -/
section Image
theorem image_iUnion {f : α → β} {s : ι → Set α} : (f '' ⋃ i, s i) = ⋃ i, f '' s i := by
ext1 x
simp only [mem_image, mem_iUnion, ← exists_and_right, ← exists_and_left]
--Porting note: `exists_swap` causes a `simp` loop in Lean4 so we use `rw` instead.
rw [exists_swap]
#align set.image_Union Set.image_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iUnion₂ (f : α → β) (s : ∀ i, κ i → Set α) :
(f '' ⋃ (i) (j), s i j) = ⋃ (i) (j), f '' s i j := by simp_rw [image_iUnion]
#align set.image_Union₂ Set.image_iUnion₂
theorem univ_subtype {p : α → Prop} : (univ : Set (Subtype p)) = ⋃ (x) (h : p x), {⟨x, h⟩} :=
Set.ext fun ⟨x, h⟩ => by simp [h]
#align set.univ_subtype Set.univ_subtype
theorem range_eq_iUnion {ι} (f : ι → α) : range f = ⋃ i, {f i} :=
Set.ext fun a => by simp [@eq_comm α a]
#align set.range_eq_Union Set.range_eq_iUnion
theorem image_eq_iUnion (f : α → β) (s : Set α) : f '' s = ⋃ i ∈ s, {f i} :=
Set.ext fun b => by simp [@eq_comm β b]
#align set.image_eq_Union Set.image_eq_iUnion
theorem biUnion_range {f : ι → α} {g : α → Set β} : ⋃ x ∈ range f, g x = ⋃ y, g (f y) :=
iSup_range
#align set.bUnion_range Set.biUnion_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iUnion_iUnion_eq' {f : ι → α} {g : α → Set β} :
⋃ (x) (y) (_ : f y = x), g x = ⋃ y, g (f y) := by simpa using biUnion_range
#align set.Union_Union_eq' Set.iUnion_iUnion_eq'
theorem biInter_range {f : ι → α} {g : α → Set β} : ⋂ x ∈ range f, g x = ⋂ y, g (f y) :=
iInf_range
#align set.bInter_range Set.biInter_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iInter_iInter_eq' {f : ι → α} {g : α → Set β} :
⋂ (x) (y) (_ : f y = x), g x = ⋂ y, g (f y) := by simpa using biInter_range
#align set.Inter_Inter_eq' Set.iInter_iInter_eq'
variable {s : Set γ} {f : γ → α} {g : α → Set β}
theorem biUnion_image : ⋃ x ∈ f '' s, g x = ⋃ y ∈ s, g (f y) :=
iSup_image
#align set.bUnion_image Set.biUnion_image
theorem biInter_image : ⋂ x ∈ f '' s, g x = ⋂ y ∈ s, g (f y) :=
iInf_image
#align set.bInter_image Set.biInter_image
end Image
section Preimage
theorem monotone_preimage {f : α → β} : Monotone (preimage f) := fun _ _ h => preimage_mono h
#align set.monotone_preimage Set.monotone_preimage
@[simp]
theorem preimage_iUnion {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋃ i, s i) = ⋃ i, f ⁻¹' s i :=
Set.ext <| by simp [preimage]
#align set.preimage_Union Set.preimage_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iUnion₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋃ (i) (j), s i j) = ⋃ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iUnion]
#align set.preimage_Union₂ Set.preimage_iUnion₂
@[simp]
theorem preimage_sUnion {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋃₀s = ⋃ t ∈ s, f ⁻¹' t := by
rw [sUnion_eq_biUnion, preimage_iUnion₂]
#align set.preimage_sUnion Set.preimage_sUnion
theorem preimage_iInter {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋂ i, s i) = ⋂ i, f ⁻¹' s i := by
ext; simp
#align set.preimage_Inter Set.preimage_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iInter₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋂ (i) (j), s i j) = ⋂ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iInter]
#align set.preimage_Inter₂ Set.preimage_iInter₂
@[simp]
theorem preimage_sInter {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋂₀ s = ⋂ t ∈ s, f ⁻¹' t := by
rw [sInter_eq_biInter, preimage_iInter₂]
#align set.preimage_sInter Set.preimage_sInter
@[simp]
theorem biUnion_preimage_singleton (f : α → β) (s : Set β) : ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s := by
rw [← preimage_iUnion₂, biUnion_of_singleton]
#align set.bUnion_preimage_singleton Set.biUnion_preimage_singleton
theorem biUnion_range_preimage_singleton (f : α → β) : ⋃ y ∈ range f, f ⁻¹' {y} = univ := by
rw [biUnion_preimage_singleton, preimage_range]
#align set.bUnion_range_preimage_singleton Set.biUnion_range_preimage_singleton
end Preimage
section Prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion {s : Set α} {t : ι → Set β} : (s ×ˢ ⋃ i, t i) = ⋃ i, s ×ˢ t i := by
ext
simp
#align set.prod_Union Set.prod_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} :
(s ×ˢ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ×ˢ t i j := by simp_rw [prod_iUnion]
#align set.prod_Union₂ Set.prod_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sUnion {s : Set α} {C : Set (Set β)} : s ×ˢ ⋃₀C = ⋃₀((fun t => s ×ˢ t) '' C) := by
simp_rw [sUnion_eq_biUnion, biUnion_image, prod_iUnion₂]
#align set.prod_sUnion Set.prod_sUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_const {s : ι → Set α} {t : Set β} : (⋃ i, s i) ×ˢ t = ⋃ i, s i ×ˢ t := by
ext
simp
#align set.Union_prod_const Set.iUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion₂_prod_const {s : ∀ i, κ i → Set α} {t : Set β} :
(⋃ (i) (j), s i j) ×ˢ t = ⋃ (i) (j), s i j ×ˢ t := by simp_rw [iUnion_prod_const]
#align set.Union₂_prod_const Set.iUnion₂_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_prod_const {C : Set (Set α)} {t : Set β} :
⋃₀C ×ˢ t = ⋃₀((fun s : Set α => s ×ˢ t) '' C) := by
simp only [sUnion_eq_biUnion, iUnion₂_prod_const, biUnion_image]
#align set.sUnion_prod_const Set.sUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod {ι ι' α β} (s : ι → Set α) (t : ι' → Set β) :
⋃ x : ι × ι', s x.1 ×ˢ t x.2 = (⋃ i : ι, s i) ×ˢ ⋃ i : ι', t i := by
ext
simp
#align set.Union_prod Set.iUnion_prod
/-- Analogue of `iSup_prod` for sets. -/
lemma iUnion_prod' (f : β × γ → Set α) : ⋃ x : β × γ, f x = ⋃ (i : β) (j : γ), f (i, j) :=
iSup_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x := by
ext ⟨z, w⟩; simp only [mem_prod, mem_iUnion, exists_imp, and_imp, iff_def]; constructor
· intro x hz hw
exact ⟨⟨x, hz⟩, x, hw⟩
· intro x hz x' hw
exact ⟨x ⊔ x', hs le_sup_left hz, ht le_sup_right hw⟩
#align set.Union_prod_of_monotone Set.iUnion_prod_of_monotone
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter_subset (S : Set (Set α)) (T : Set (Set β)) :
⋂₀ S ×ˢ ⋂₀ T ⊆ ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 :=
subset_iInter₂ fun x hx _ hy => ⟨hy.1 x.1 hx.1, hy.2 x.2 hx.2⟩
#align set.sInter_prod_sInter_subset Set.sInter_prod_sInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 := by
obtain ⟨s₁, h₁⟩ := hS
obtain ⟨s₂, h₂⟩ := hT
refine' Set.Subset.antisymm (sInter_prod_sInter_subset S T) fun x hx => _
rw [mem_iInter₂] at hx
exact ⟨fun s₀ h₀ => (hx (s₀, s₂) ⟨h₀, h₂⟩).1, fun s₀ h₀ => (hx (s₁, s₀) ⟨h₁, h₀⟩).2⟩
#align set.sInter_prod_sInter Set.sInter_prod_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod {S : Set (Set α)} (hS : S.Nonempty) (t : Set β) :
⋂₀ S ×ˢ t = ⋂ s ∈ S, s ×ˢ t := by
rw [← sInter_singleton t, sInter_prod_sInter hS (singleton_nonempty t), sInter_singleton]
simp_rw [prod_singleton, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.sInter_prod Set.sInter_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sInter {T : Set (Set β)} (hT : T.Nonempty) (s : Set α) :
s ×ˢ ⋂₀ T = ⋂ t ∈ T, s ×ˢ t := by
rw [← sInter_singleton s, sInter_prod_sInter (singleton_nonempty s) hT, sInter_singleton]
simp_rw [singleton_prod, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.prod_sInter Set.prod_sInter
end Prod
section Image2
variable (f : α → β → γ) {s : Set α} {t : Set β}
/-- The `Set.image2` version of `Set.image_eq_iUnion` -/
theorem image2_eq_iUnion (s : Set α) (t : Set β) : image2 f s t = ⋃ (i ∈ s) (j ∈ t), {f i j} := by
ext; simp [eq_comm]
#align set.image2_eq_Union Set.image2_eq_iUnion
theorem iUnion_image_left : ⋃ a ∈ s, f a '' t = image2 f s t := by
simp only [image2_eq_iUnion, image_eq_iUnion]
#align set.Union_image_left Set.iUnion_image_left
theorem iUnion_image_right : ⋃ b ∈ t, (f · b) '' s = image2 f s t := by
rw [image2_swap, iUnion_image_left]
#align set.Union_image_right Set.iUnion_image_right
theorem image2_iUnion_left (s : ι → Set α) (t : Set β) :
image2 f (⋃ i, s i) t = ⋃ i, image2 f (s i) t := by
simp only [← image_prod, iUnion_prod_const, image_iUnion]
#align set.image2_Union_left Set.image2_iUnion_left
theorem image2_iUnion_right (s : Set α) (t : ι → Set β) :
image2 f s (⋃ i, t i) = ⋃ i, image2 f s (t i) := by
simp only [← image_prod, prod_iUnion, image_iUnion]
#align set.image2_Union_right Set.image2_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋃ (i) (j), s i j) t = ⋃ (i) (j), image2 f (s i j) t := by simp_rw [image2_iUnion_left]
#align set.image2_Union₂_left Set.image2_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋃ (i) (j), t i j) = ⋃ (i) (j), image2 f s (t i j) :=
by simp_rw [image2_iUnion_right]
#align set.image2_Union₂_right Set.image2_iUnion₂_right
theorem image2_iInter_subset_left (s : ι → Set α) (t : Set β) :
image2 f (⋂ i, s i) t ⊆ ⋂ i, image2 f (s i) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem (hx _) hy
#align set.image2_Inter_subset_left Set.image2_iInter_subset_left
theorem image2_iInter_subset_right (s : Set α) (t : ι → Set β) :
image2 f s (⋂ i, t i) ⊆ ⋂ i, image2 f s (t i) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem hx (hy _)
#align set.image2_Inter_subset_right Set.image2_iInter_subset_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋂ (i) (j), s i j) t ⊆ ⋂ (i) (j), image2 f (s i j) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem (hx _ _) hy
#align set.image2_Inter₂_subset_left Set.image2_iInter₂_subset_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), image2 f s (t i j) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem hx (hy _ _)
#align set.image2_Inter₂_subset_right Set.image2_iInter₂_subset_right
theorem prod_eq_biUnion_left : s ×ˢ t = ⋃ a ∈ s, (fun b => (a, b)) '' t := by
rw [iUnion_image_left, image2_mk_eq_prod]
#align set.prod_eq_bUnion_left Set.prod_eq_biUnion_left
theorem prod_eq_biUnion_right : s ×ˢ t = ⋃ b ∈ t, (fun a => (a, b)) '' s := by
rw [iUnion_image_right, image2_mk_eq_prod]
#align set.prod_eq_bUnion_right Set.prod_eq_biUnion_right
end Image2
section Seq
/-- Given a set `s` of functions `α → β` and `t : Set α`, `seq s t` is the union of `f '' t` over
all `f ∈ s`. -/
def seq (s : Set (α → β)) (t : Set α) : Set β :=
{ b | ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b }
#align set.seq Set.seq
@[simp]
theorem mem_seq_iff {s : Set (α → β)} {t : Set α} {b : β} :
b ∈ seq s t ↔ ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b :=
Iff.rfl
#align set.mem_seq_iff Set.mem_seq_iff
lemma seq_eq_image2 (s : Set (α → β)) (t : Set α) : seq s t = image2 (fun f a ↦ f a) s t := by
ext; simp
theorem seq_def {s : Set (α → β)} {t : Set α} : seq s t = ⋃ f ∈ s, f '' t := by
rw [seq_eq_image2, iUnion_image_left]
#align set.seq_def Set.seq_def
theorem seq_subset {s : Set (α → β)} {t : Set α} {u : Set β} :
seq s t ⊆ u ↔ ∀ f ∈ s, ∀ a ∈ t, (f : α → β) a ∈ u := by
rw [seq_eq_image2, image2_subset_iff]
#align set.seq_subset Set.seq_subset
@[gcongr]
theorem seq_mono {s₀ s₁ : Set (α → β)} {t₀ t₁ : Set α} (hs : s₀ ⊆ s₁) (ht : t₀ ⊆ t₁) :
seq s₀ t₀ ⊆ seq s₁ t₁ := fun _ ⟨f, hf, a, ha, eq⟩ => ⟨f, hs hf, a, ht ha, eq⟩
#align set.seq_mono Set.seq_mono
theorem singleton_seq {f : α → β} {t : Set α} : Set.seq ({f} : Set (α → β)) t = f '' t := by
rw [seq_eq_image2, image2_singleton_left]
#align set.singleton_seq Set.singleton_seq
theorem seq_singleton {s : Set (α → β)} {a : α} : Set.seq s {a} = (fun f : α → β => f a) '' s := by
rw [seq_eq_image2, image2_singleton_right]
#align set.seq_singleton Set.seq_singleton
theorem seq_seq {s : Set (β → γ)} {t : Set (α → β)} {u : Set α} :
seq s (seq t u) = seq (seq ((· ∘ ·) '' s) t) u := by
simp only [seq_eq_image2, image2_image_left, image2_image2_left, image2_image2_right, comp_apply]
#align set.seq_seq Set.seq_seq
theorem image_seq {f : β → γ} {s : Set (α → β)} {t : Set α} :
f '' seq s t = seq ((f ∘ ·) '' s) t := by
rw [← singleton_seq, ← singleton_seq, seq_seq, image_singleton]
#align set.image_seq Set.image_seq
theorem prod_eq_seq {s : Set α} {t : Set β} : s ×ˢ t = (Prod.mk '' s).seq t := by
rw [seq_eq_image2, image2_image_left, image2_mk_eq_prod]
#align set.prod_eq_seq Set.prod_eq_seq
theorem prod_image_seq_comm (s : Set α) (t : Set β) :
(Prod.mk '' s).seq t = seq ((fun b a => (a, b)) '' t) s := by
rw [← prod_eq_seq, ← image_swap_prod, prod_eq_seq, image_seq, ← image_comp]; rfl
#align set.prod_image_seq_comm Set.prod_image_seq_comm
theorem image2_eq_seq (f : α → β → γ) (s : Set α) (t : Set β) : image2 f s t = seq (f '' s) t := by
rw [seq_eq_image2, image2_image_left]
#align set.image2_eq_seq Set.image2_eq_seq
end Seq
section Pi
variable {π : α → Type*}
theorem pi_def (i : Set α) (s : ∀ a, Set (π a)) : pi i s = ⋂ a ∈ i, eval a ⁻¹' s a := by
ext
simp
#align set.pi_def Set.pi_def
theorem univ_pi_eq_iInter (t : ∀ i, Set (π i)) : pi univ t = ⋂ i, eval i ⁻¹' t i := by
simp only [pi_def, iInter_true, mem_univ]
#align set.univ_pi_eq_Inter Set.univ_pi_eq_iInter
theorem pi_diff_pi_subset (i : Set α) (s t : ∀ a, Set (π a)) :
pi i s \ pi i t ⊆ ⋃ a ∈ i, eval a ⁻¹' (s a \ t a) := by
refine' diff_subset_comm.2 fun x hx a ha => _
simp only [mem_diff, mem_pi, mem_iUnion, not_exists, mem_preimage, not_and, not_not,
eval_apply] at hx
exact hx.2 _ ha (hx.1 _ ha)
#align set.pi_diff_pi_subset Set.pi_diff_pi_subset
theorem iUnion_univ_pi {ι : α → Type*} (t : (a : α) → ι a → Set (π a)) :
⋃ x : (a : α) → ι a, pi univ (fun a => t a (x a)) = pi univ fun a => ⋃ j : ι a, t a j := by
ext
simp [Classical.skolem]
#align set.Union_univ_pi Set.iUnion_univ_pi
end Pi
end Set
namespace Function
namespace Surjective
theorem iUnion_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋃ x, g (f x) = ⋃ y, g y :=
hf.iSup_comp g
#align function.surjective.Union_comp Function.Surjective.iUnion_comp
theorem iInter_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋂ x, g (f x) = ⋂ y, g y :=
hf.iInf_comp g
#align function.surjective.Inter_comp Function.Surjective.iInter_comp
end Surjective
end Function
/-!
### Disjoint sets
-/
section Disjoint
variable {s t u : Set α} {f : α → β}
namespace Set
@[simp]
theorem disjoint_iUnion_left {ι : Sort*} {s : ι → Set α} :
Disjoint (⋃ i, s i) t ↔ ∀ i, Disjoint (s i) t :=
iSup_disjoint_iff
#align set.disjoint_Union_left Set.disjoint_iUnion_left
@[simp]
theorem disjoint_iUnion_right {ι : Sort*} {s : ι → Set α} :
Disjoint t (⋃ i, s i) ↔ ∀ i, Disjoint t (s i) :=
disjoint_iSup_iff
#align set.disjoint_Union_right Set.disjoint_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_left {s : ∀ i, κ i → Set α} {t : Set α} :
Disjoint (⋃ (i) (j), s i j) t ↔ ∀ i j, Disjoint (s i j) t :=
iSup₂_disjoint_iff
#align set.disjoint_Union₂_left Set.disjoint_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_right {s : Set α} {t : ∀ i, κ i → Set α} :
Disjoint s (⋃ (i) (j), t i j) ↔ ∀ i j, Disjoint s (t i j) :=
disjoint_iSup₂_iff
#align set.disjoint_Union₂_right Set.disjoint_iUnion₂_right
@[simp]
theorem disjoint_sUnion_left {S : Set (Set α)} {t : Set α} :
Disjoint (⋃₀S) t ↔ ∀ s ∈ S, Disjoint s t :=
sSup_disjoint_iff
#align set.disjoint_sUnion_left Set.disjoint_sUnion_left
@[simp]
theorem disjoint_sUnion_right {s : Set α} {S : Set (Set α)} :
Disjoint s (⋃₀S) ↔ ∀ t ∈ S, Disjoint s t :=
disjoint_sSup_iff
#align set.disjoint_sUnion_right Set.disjoint_sUnion_right
end Set
end Disjoint
/-! ### Intervals -/
namespace Set
lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :
(⋂ i, Iic (f i)).Nonempty ↔ BddBelow (range f) := by
have : (⋂ (i : ι), Iic (f i)) = lowerBounds (range f) := by
ext c; simp [lowerBounds]
simp [this, BddBelow]
lemma nonempty_iInter_Ici_iff [Preorder α] {f : ι → α} :
(⋂ i, Ici (f i)).Nonempty ↔ BddAbove (range f) :=
nonempty_iInter_Iic_iff (α := αᵒᵈ)
variable [CompleteLattice α]
theorem Ici_iSup (f : ι → α) : Ici (⨆ i, f i) = ⋂ i, Ici (f i) :=
ext fun _ => by simp only [mem_Ici, iSup_le_iff, mem_iInter]
#align set.Ici_supr Set.Ici_iSup
theorem Iic_iInf (f : ι → α) : Iic (⨅ i, f i) = ⋂ i, Iic (f i) :=
ext fun _ => by simp only [mem_Iic, le_iInf_iff, mem_iInter]
#align set.Iic_infi Set.Iic_iInf
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Ici_iSup₂ (f : ∀ i, κ i → α) : Ici (⨆ (i) (j), f i j) = ⋂ (i) (j), Ici (f i j) := by
simp_rw [Ici_iSup]
#align set.Ici_supr₂ Set.Ici_iSup₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Iic_iInf₂ (f : ∀ i, κ i → α) : Iic (⨅ (i) (j), f i j) = ⋂ (i) (j), Iic (f i j) := by
simp_rw [Iic_iInf]
#align set.Iic_infi₂ Set.Iic_iInf₂
theorem Ici_sSup (s : Set α) : Ici (sSup s) = ⋂ a ∈ s, Ici a := by rw [sSup_eq_iSup, Ici_iSup₂]
#align set.Ici_Sup Set.Ici_sSup
theorem Iic_sInf (s : Set α) : Iic (sInf s) = ⋂ a ∈ s, Iic a := by rw [sInf_eq_iInf, Iic_iInf₂]
#align set.Iic_Inf Set.Iic_sInf
end Set
namespace Set
variable (t : α → Set β)
theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x := by
simp only [diff_subset_iff, ← biUnion_union]
apply biUnion_subset_biUnion_left
rw [union_diff_self]
apply subset_union_right
#align set.bUnion_diff_bUnion_subset Set.biUnion_diff_biUnion_subset
/-- If `t` is an indexed family of sets, then there is a natural map from `Σ i, t i` to `⋃ i, t i`
sending `⟨i, x⟩` to `x`. -/
def sigmaToiUnion (x : Σi, t i) : ⋃ i, t i :=
⟨x.2, mem_iUnion.2 ⟨x.1, x.2.2⟩⟩
#align set.sigma_to_Union Set.sigmaToiUnion
theorem sigmaToiUnion_surjective : Surjective (sigmaToiUnion t)
| ⟨b, hb⟩ =>
have : ∃ a, b ∈ t a := by simpa using hb
let ⟨a, hb⟩ := this
⟨⟨a, b, hb⟩, rfl⟩
#align set.sigma_to_Union_surjective Set.sigmaToiUnion_surjective
theorem sigmaToiUnion_injective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Injective (sigmaToiUnion t)
| ⟨a₁, b₁, h₁⟩, ⟨a₂, b₂, h₂⟩, eq =>
have b_eq : b₁ = b₂ := congr_arg Subtype.val eq
have a_eq : a₁ = a₂ :=
by_contradiction fun ne =>
have : b₁ ∈ t a₁ ∩ t a₂ := ⟨h₁, b_eq.symm ▸ h₂⟩
(h _ _ ne).le_bot this
Sigma.eq a_eq <| Subtype.eq <| by
|
subst b_eq
|
theorem sigmaToiUnion_injective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Injective (sigmaToiUnion t)
| ⟨a₁, b₁, h₁⟩, ⟨a₂, b₂, h₂⟩, eq =>
have b_eq : b₁ = b₂ := congr_arg Subtype.val eq
have a_eq : a₁ = a₂ :=
by_contradiction fun ne =>
have : b₁ ∈ t a₁ ∩ t a₂ := ⟨h₁, b_eq.symm ▸ h₂⟩
(h _ _ ne).le_bot this
Sigma.eq a_eq <| Subtype.eq <| by
|
Mathlib.Data.Set.Lattice.2312_0.5mONj49h3SYSDwc
|
theorem sigmaToiUnion_injective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Injective (sigmaToiUnion t)
| ⟨a₁, b₁, h₁⟩, ⟨a₂, b₂, h₂⟩, eq =>
have b_eq : b₁ = b₂
|
Mathlib_Data_Set_Lattice
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
t : α → Set β
h : ∀ (i j : α), i ≠ j → Disjoint (t i) (t j)
a₁ : α
b₁ : β
h₁ : b₁ ∈ t a₁
a₂ : α
a_eq : a₁ = a₂
h₂ : b₁ ∈ t a₂
eq :
sigmaToiUnion t { fst := a₁, snd := { val := b₁, property := h₁ } } =
sigmaToiUnion t { fst := a₂, snd := { val := b₁, property := h₂ } }
⊢ ↑(Eq.recOn a_eq { fst := a₁, snd := { val := b₁, property := h₁ } }.snd) =
↑{ fst := a₂, snd := { val := b₁, property := h₂ } }.snd
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
/-!
# The set lattice
This file provides usual set notation for unions and intersections, a `CompleteLattice` instance
for `Set α`, and some more set constructions.
## Main declarations
* `Set.iUnion`: **i**ndexed **union**. Union of an indexed family of sets.
* `Set.iInter`: **i**ndexed **inter**section. Intersection of an indexed family of sets.
* `Set.sInter`: **s**et **inter**section. Intersection of sets belonging to a set of sets.
* `Set.sUnion`: **s**et **union**. Union of sets belonging to a set of sets.
* `Set.sInter_eq_biInter`, `Set.sUnion_eq_biInter`: Shows that `⋂₀ s = ⋂ x ∈ s, x` and
`⋃₀ s = ⋃ x ∈ s, x`.
* `Set.completeAtomicBooleanAlgebra`: `Set α` is a `CompleteAtomicBooleanAlgebra` with `≤ = ⊆`,
`< = ⊂`, `⊓ = ∩`, `⊔ = ∪`, `⨅ = ⋂`, `⨆ = ⋃` and `\` as the set difference.
See `Set.BooleanAlgebra`.
* `Set.kernImage`: For a function `f : α → β`, `s.kernImage f` is the set of `y` such that
`f ⁻¹ y ⊆ s`.
* `Set.seq`: Union of the image of a set under a **seq**uence of functions. `seq s t` is the union
of `f '' t` over all `f ∈ s`, where `t : Set α` and `s : Set (α → β)`.
* `Set.iUnion_eq_sigma_of_disjoint`: Equivalence between `⋃ i, t i` and `Σ i, t i`, where `t` is an
indexed family of disjoint sets.
## Naming convention
In lemma names,
* `⋃ i, s i` is called `iUnion`
* `⋂ i, s i` is called `iInter`
* `⋃ i j, s i j` is called `iUnion₂`. This is an `iUnion` inside an `iUnion`.
* `⋂ i j, s i j` is called `iInter₂`. This is an `iInter` inside an `iInter`.
* `⋃ i ∈ s, t i` is called `biUnion` for "bounded `iUnion`". This is the special case of `iUnion₂`
where `j : i ∈ s`.
* `⋂ i ∈ s, t i` is called `biInter` for "bounded `iInter`". This is the special case of `iInter₂`
where `j : i ∈ s`.
## Notation
* `⋃`: `Set.iUnion`
* `⋂`: `Set.iInter`
* `⋃₀`: `Set.sUnion`
* `⋂₀`: `Set.sInter`
-/
set_option autoImplicit true
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι₂ : Sort*} {κ κ₁ κ₂ : ι → Sort*} {κ' : ι' → Sort*}
namespace Set
/-! ### Complete lattice and complete Boolean algebra instances -/
instance : InfSet (Set α) :=
⟨fun s => { a | ∀ t ∈ s, a ∈ t }⟩
instance : SupSet (Set α) :=
⟨fun s => { a | ∃ t ∈ s, a ∈ t }⟩
/-- Intersection of a set of sets. -/
def sInter (S : Set (Set α)) : Set α :=
sInf S
#align set.sInter Set.sInter
/-- Notation for `Set.sInter` Intersection of a set of sets. -/
prefix:110 "⋂₀ " => sInter
/-- Union of a set of sets. -/
def sUnion (S : Set (Set α)) : Set α :=
sSup S
#align set.sUnion Set.sUnion
/-- Notation for `Set.sUnion`. Union of a set of sets. -/
prefix:110 "⋃₀ " => sUnion
@[simp]
theorem mem_sInter {x : α} {S : Set (Set α)} : x ∈ ⋂₀ S ↔ ∀ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sInter Set.mem_sInter
@[simp]
theorem mem_sUnion {x : α} {S : Set (Set α)} : x ∈ ⋃₀ S ↔ ∃ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sUnion Set.mem_sUnion
/-- Indexed union of a family of sets -/
def iUnion (s : ι → Set β) : Set β :=
iSup s
#align set.Union Set.iUnion
/-- Indexed intersection of a family of sets -/
def iInter (s : ι → Set β) : Set β :=
iInf s
#align set.Inter Set.iInter
/-- Notation for `Set.iUnion`. Indexed union of a family of sets -/
notation3 "⋃ "(...)", "r:60:(scoped f => iUnion f) => r
/-- Notation for `Set.iInter`. Indexed intersection of a family of sets -/
notation3 "⋂ "(...)", "r:60:(scoped f => iInter f) => r
section delaborators
open Lean Lean.PrettyPrinter.Delaborator
/-- Delaborator for indexed unions. -/
@[delab app.Set.iUnion]
def iUnion_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋃ (_ : $dom), $body)
else if prop || ppTypes then
`(⋃ ($x:ident : $dom), $body)
else
`(⋃ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋃ $x:ident, ⋃ (_ : $y:ident ∈ $s), $body)
| `(⋃ ($x:ident : $_), ⋃ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋃ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
/-- Delaborator for indexed intersections. -/
@[delab app.Set.iInter]
def sInter_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋂ (_ : $dom), $body)
else if prop || ppTypes then
`(⋂ ($x:ident : $dom), $body)
else
`(⋂ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋂ $x:ident, ⋂ (_ : $y:ident ∈ $s), $body)
| `(⋂ ($x:ident : $_), ⋂ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋂ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
end delaborators
@[simp]
theorem sSup_eq_sUnion (S : Set (Set α)) : sSup S = ⋃₀S :=
rfl
#align set.Sup_eq_sUnion Set.sSup_eq_sUnion
@[simp]
theorem sInf_eq_sInter (S : Set (Set α)) : sInf S = ⋂₀ S :=
rfl
#align set.Inf_eq_sInter Set.sInf_eq_sInter
@[simp]
theorem iSup_eq_iUnion (s : ι → Set α) : iSup s = iUnion s :=
rfl
#align set.supr_eq_Union Set.iSup_eq_iUnion
@[simp]
theorem iInf_eq_iInter (s : ι → Set α) : iInf s = iInter s :=
rfl
#align set.infi_eq_Inter Set.iInf_eq_iInter
@[simp]
theorem mem_iUnion {x : α} {s : ι → Set α} : (x ∈ ⋃ i, s i) ↔ ∃ i, x ∈ s i :=
⟨fun ⟨_, ⟨⟨a, (t_eq : s a = _)⟩, (h : x ∈ _)⟩⟩ => ⟨a, t_eq.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨s a, ⟨⟨a, rfl⟩, h⟩⟩⟩
#align set.mem_Union Set.mem_iUnion
@[simp]
theorem mem_iInter {x : α} {s : ι → Set α} : (x ∈ ⋂ i, s i) ↔ ∀ i, x ∈ s i :=
⟨fun (h : ∀ a ∈ { a : Set α | ∃ i, s i = a }, x ∈ a) a => h (s a) ⟨a, rfl⟩,
fun h _ ⟨a, (eq : s a = _)⟩ => eq ▸ h a⟩
#align set.mem_Inter Set.mem_iInter
theorem mem_iUnion₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋃ (i) (j), s i j) ↔ ∃ i j, x ∈ s i j := by
simp_rw [mem_iUnion]
#align set.mem_Union₂ Set.mem_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋂ (i) (j), s i j) ↔ ∀ i j, x ∈ s i j := by
simp_rw [mem_iInter]
#align set.mem_Inter₂ Set.mem_iInter₂
theorem mem_iUnion_of_mem {s : ι → Set α} {a : α} (i : ι) (ha : a ∈ s i) : a ∈ ⋃ i, s i :=
mem_iUnion.2 ⟨i, ha⟩
#align set.mem_Union_of_mem Set.mem_iUnion_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iUnion₂_of_mem {s : ∀ i, κ i → Set α} {a : α} {i : ι} (j : κ i) (ha : a ∈ s i j) :
a ∈ ⋃ (i) (j), s i j :=
mem_iUnion₂.2 ⟨i, j, ha⟩
#align set.mem_Union₂_of_mem Set.mem_iUnion₂_of_mem
theorem mem_iInter_of_mem {s : ι → Set α} {a : α} (h : ∀ i, a ∈ s i) : a ∈ ⋂ i, s i :=
mem_iInter.2 h
#align set.mem_Inter_of_mem Set.mem_iInter_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂_of_mem {s : ∀ i, κ i → Set α} {a : α} (h : ∀ i j, a ∈ s i j) :
a ∈ ⋂ (i) (j), s i j :=
mem_iInter₂.2 h
#align set.mem_Inter₂_of_mem Set.mem_iInter₂_of_mem
instance Set.completeAtomicBooleanAlgebra : CompleteAtomicBooleanAlgebra (Set α) :=
{ instBooleanAlgebraSet with
le_sSup := fun s t t_in a a_in => ⟨t, t_in, a_in⟩
sSup_le := fun s t h a ⟨t', ⟨t'_in, a_in⟩⟩ => h t' t'_in a_in
le_sInf := fun s t h a a_in t' t'_in => h t' t'_in a_in
sInf_le := fun s t t_in a h => h _ t_in
iInf_iSup_eq := by intros; ext; simp [Classical.skolem] }
/-- `kernImage f s` is the set of `y` such that `f ⁻¹ y ⊆ s`. -/
def kernImage (f : α → β) (s : Set α) : Set β :=
{ y | ∀ ⦃x⦄, f x = y → x ∈ s }
#align set.kern_image Set.kernImage
lemma subset_kernImage_iff {f : α → β} : s ⊆ kernImage f t ↔ f ⁻¹' s ⊆ t :=
⟨fun h _ hx ↦ h hx rfl,
fun h _ hx y hy ↦ h (show f y ∈ s from hy.symm ▸ hx)⟩
section GaloisConnection
variable {f : α → β}
protected theorem image_preimage : GaloisConnection (image f) (preimage f) := fun _ _ =>
image_subset_iff
#align set.image_preimage Set.image_preimage
protected theorem preimage_kernImage : GaloisConnection (preimage f) (kernImage f) := fun _ _ =>
subset_kernImage_iff.symm
#align set.preimage_kern_image Set.preimage_kernImage
end GaloisConnection
section kernImage
variable {f : α → β}
lemma kernImage_mono : Monotone (kernImage f) :=
Set.preimage_kernImage.monotone_u
lemma kernImage_eq_compl {s : Set α} : kernImage f s = (f '' sᶜ)ᶜ :=
Set.preimage_kernImage.u_unique (Set.image_preimage.compl)
(fun t ↦ compl_compl (f ⁻¹' t) ▸ Set.preimage_compl)
lemma kernImage_compl {s : Set α} : kernImage f (sᶜ) = (f '' s)ᶜ := by
rw [kernImage_eq_compl, compl_compl]
lemma kernImage_empty : kernImage f ∅ = (range f)ᶜ := by
rw [kernImage_eq_compl, compl_empty, image_univ]
lemma kernImage_preimage_eq_iff {s : Set β} : kernImage f (f ⁻¹' s) = s ↔ (range f)ᶜ ⊆ s := by
rw [kernImage_eq_compl, ← preimage_compl, compl_eq_comm, eq_comm, image_preimage_eq_iff,
compl_subset_comm]
lemma compl_range_subset_kernImage {s : Set α} : (range f)ᶜ ⊆ kernImage f s := by
rw [← kernImage_empty]
exact kernImage_mono (empty_subset _)
lemma kernImage_union_preimage {s : Set α} {t : Set β} :
kernImage f (s ∪ f ⁻¹' t) = kernImage f s ∪ t := by
rw [kernImage_eq_compl, kernImage_eq_compl, compl_union, ← preimage_compl, image_inter_preimage,
compl_inter, compl_compl]
lemma kernImage_preimage_union {s : Set α} {t : Set β} :
kernImage f (f ⁻¹' t ∪ s) = t ∪ kernImage f s := by
rw [union_comm, kernImage_union_preimage, union_comm]
end kernImage
/-! ### Union and intersection over an indexed family of sets -/
instance : OrderTop (Set α) where
top := univ
le_top := by simp
@[congr]
theorem iUnion_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iUnion f₁ = iUnion f₂ :=
iSup_congr_Prop pq f
#align set.Union_congr_Prop Set.iUnion_congr_Prop
@[congr]
theorem iInter_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iInter f₁ = iInter f₂ :=
iInf_congr_Prop pq f
#align set.Inter_congr_Prop Set.iInter_congr_Prop
theorem iUnion_plift_up (f : PLift ι → Set α) : ⋃ i, f (PLift.up i) = ⋃ i, f i :=
iSup_plift_up _
#align set.Union_plift_up Set.iUnion_plift_up
theorem iUnion_plift_down (f : ι → Set α) : ⋃ i, f (PLift.down i) = ⋃ i, f i :=
iSup_plift_down _
#align set.Union_plift_down Set.iUnion_plift_down
theorem iInter_plift_up (f : PLift ι → Set α) : ⋂ i, f (PLift.up i) = ⋂ i, f i :=
iInf_plift_up _
#align set.Inter_plift_up Set.iInter_plift_up
theorem iInter_plift_down (f : ι → Set α) : ⋂ i, f (PLift.down i) = ⋂ i, f i :=
iInf_plift_down _
#align set.Inter_plift_down Set.iInter_plift_down
theorem iUnion_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋃ _ : p, s = if p then s else ∅ :=
iSup_eq_if _
#align set.Union_eq_if Set.iUnion_eq_if
theorem iUnion_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋃ h : p, s h = if h : p then s h else ∅ :=
iSup_eq_dif _
#align set.Union_eq_dif Set.iUnion_eq_dif
theorem iInter_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋂ _ : p, s = if p then s else univ :=
iInf_eq_if _
#align set.Inter_eq_if Set.iInter_eq_if
theorem iInf_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋂ h : p, s h = if h : p then s h else univ :=
_root_.iInf_eq_dif _
#align set.Infi_eq_dif Set.iInf_eq_dif
theorem exists_set_mem_of_union_eq_top {ι : Type*} (t : Set ι) (s : ι → Set β)
(w : ⋃ i ∈ t, s i = ⊤) (x : β) : ∃ i ∈ t, x ∈ s i := by
have p : x ∈ ⊤ := Set.mem_univ x
rw [← w, Set.mem_iUnion] at p
simpa using p
#align set.exists_set_mem_of_union_eq_top Set.exists_set_mem_of_union_eq_top
theorem nonempty_of_union_eq_top_of_nonempty {ι : Type*} (t : Set ι) (s : ι → Set α)
(H : Nonempty α) (w : ⋃ i ∈ t, s i = ⊤) : t.Nonempty := by
obtain ⟨x, m, -⟩ := exists_set_mem_of_union_eq_top t s w H.some
exact ⟨x, m⟩
#align set.nonempty_of_union_eq_top_of_nonempty Set.nonempty_of_union_eq_top_of_nonempty
theorem nonempty_of_nonempty_iUnion
{s : ι → Set α} (h_Union : (⋃ i, s i).Nonempty) : Nonempty ι := by
obtain ⟨x, hx⟩ := h_Union
exact ⟨Classical.choose $ mem_iUnion.mp hx⟩
theorem nonempty_of_nonempty_iUnion_eq_univ
{s : ι → Set α} [Nonempty α] (h_Union : ⋃ i, s i = univ) : Nonempty ι :=
nonempty_of_nonempty_iUnion (s := s) (by simpa only [h_Union] using univ_nonempty)
theorem setOf_exists (p : ι → β → Prop) : { x | ∃ i, p i x } = ⋃ i, { x | p i x } :=
ext fun _ => mem_iUnion.symm
#align set.set_of_exists Set.setOf_exists
theorem setOf_forall (p : ι → β → Prop) : { x | ∀ i, p i x } = ⋂ i, { x | p i x } :=
ext fun _ => mem_iInter.symm
#align set.set_of_forall Set.setOf_forall
theorem iUnion_subset {s : ι → Set α} {t : Set α} (h : ∀ i, s i ⊆ t) : ⋃ i, s i ⊆ t :=
iSup_le h
#align set.Union_subset Set.iUnion_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset {s : ∀ i, κ i → Set α} {t : Set α} (h : ∀ i j, s i j ⊆ t) :
⋃ (i) (j), s i j ⊆ t :=
iUnion_subset fun x => iUnion_subset (h x)
#align set.Union₂_subset Set.iUnion₂_subset
theorem subset_iInter {t : Set β} {s : ι → Set β} (h : ∀ i, t ⊆ s i) : t ⊆ ⋂ i, s i :=
le_iInf h
#align set.subset_Inter Set.subset_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iInter₂ {s : Set α} {t : ∀ i, κ i → Set α} (h : ∀ i j, s ⊆ t i j) :
s ⊆ ⋂ (i) (j), t i j :=
subset_iInter fun x => subset_iInter <| h x
#align set.subset_Inter₂ Set.subset_iInter₂
@[simp]
theorem iUnion_subset_iff {s : ι → Set α} {t : Set α} : ⋃ i, s i ⊆ t ↔ ∀ i, s i ⊆ t :=
⟨fun h _ => Subset.trans (le_iSup s _) h, iUnion_subset⟩
#align set.Union_subset_iff Set.iUnion_subset_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset_iff {s : ∀ i, κ i → Set α} {t : Set α} :
⋃ (i) (j), s i j ⊆ t ↔ ∀ i j, s i j ⊆ t := by simp_rw [iUnion_subset_iff]
#align set.Union₂_subset_iff Set.iUnion₂_subset_iff
@[simp]
theorem subset_iInter_iff {s : Set α} {t : ι → Set α} : (s ⊆ ⋂ i, t i) ↔ ∀ i, s ⊆ t i :=
le_iInf_iff
#align set.subset_Inter_iff Set.subset_iInter_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem subset_iInter₂_iff {s : Set α} {t : ∀ i, κ i → Set α} :
(s ⊆ ⋂ (i) (j), t i j) ↔ ∀ i j, s ⊆ t i j := by simp_rw [subset_iInter_iff]
#align set.subset_Inter₂_iff Set.subset_iInter₂_iff
theorem subset_iUnion : ∀ (s : ι → Set β) (i : ι), s i ⊆ ⋃ i, s i :=
le_iSup
#align set.subset_Union Set.subset_iUnion
theorem iInter_subset : ∀ (s : ι → Set β) (i : ι), ⋂ i, s i ⊆ s i :=
iInf_le
#align set.Inter_subset Set.iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iUnion₂ {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : s i j ⊆ ⋃ (i') (j'), s i' j' :=
le_iSup₂ i j
#align set.subset_Union₂ Set.subset_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_subset {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : ⋂ (i) (j), s i j ⊆ s i j :=
iInf₂_le i j
#align set.Inter₂_subset Set.iInter₂_subset
/-- This rather trivial consequence of `subset_iUnion`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem subset_iUnion_of_subset {s : Set α} {t : ι → Set α} (i : ι) (h : s ⊆ t i) : s ⊆ ⋃ i, t i :=
le_iSup_of_le i h
#align set.subset_Union_of_subset Set.subset_iUnion_of_subset
/-- This rather trivial consequence of `iInter_subset`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem iInter_subset_of_subset {s : ι → Set α} {t : Set α} (i : ι) (h : s i ⊆ t) :
⋂ i, s i ⊆ t :=
iInf_le_of_le i h
#align set.Inter_subset_of_subset Set.iInter_subset_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `subset_iUnion₂` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem subset_iUnion₂_of_subset {s : Set α} {t : ∀ i, κ i → Set α} (i : ι) (j : κ i)
(h : s ⊆ t i j) : s ⊆ ⋃ (i) (j), t i j :=
le_iSup₂_of_le i j h
#align set.subset_Union₂_of_subset Set.subset_iUnion₂_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `iInter₂_subset` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem iInter₂_subset_of_subset {s : ∀ i, κ i → Set α} {t : Set α} (i : ι) (j : κ i)
(h : s i j ⊆ t) : ⋂ (i) (j), s i j ⊆ t :=
iInf₂_le_of_le i j h
#align set.Inter₂_subset_of_subset Set.iInter₂_subset_of_subset
theorem iUnion_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono h
#align set.Union_mono Set.iUnion_mono
@[gcongr]
theorem iUnion_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iUnion s ⊆ iUnion t :=
iSup_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋃ (i) (j), s i j ⊆ ⋃ (i) (j), t i j :=
iSup₂_mono h
#align set.Union₂_mono Set.iUnion₂_mono
theorem iInter_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋂ i, s i ⊆ ⋂ i, t i :=
iInf_mono h
#align set.Inter_mono Set.iInter_mono
@[gcongr]
theorem iInter_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iInter s ⊆ iInter t :=
iInf_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋂ (i) (j), s i j ⊆ ⋂ (i) (j), t i j :=
iInf₂_mono h
#align set.Inter₂_mono Set.iInter₂_mono
theorem iUnion_mono' {s : ι → Set α} {t : ι₂ → Set α} (h : ∀ i, ∃ j, s i ⊆ t j) :
⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono' h
#align set.Union_mono' Set.iUnion_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iUnion₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i j, ∃ i' j', s i j ⊆ t i' j') : ⋃ (i) (j), s i j ⊆ ⋃ (i') (j'), t i' j' :=
iSup₂_mono' h
#align set.Union₂_mono' Set.iUnion₂_mono'
theorem iInter_mono' {s : ι → Set α} {t : ι' → Set α} (h : ∀ j, ∃ i, s i ⊆ t j) :
⋂ i, s i ⊆ ⋂ j, t j :=
Set.subset_iInter fun j =>
let ⟨i, hi⟩ := h j
iInter_subset_of_subset i hi
#align set.Inter_mono' Set.iInter_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iInter₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i' j', ∃ i j, s i j ⊆ t i' j') : ⋂ (i) (j), s i j ⊆ ⋂ (i') (j'), t i' j' :=
subset_iInter₂_iff.2 fun i' j' =>
let ⟨_, _, hst⟩ := h i' j'
(iInter₂_subset _ _).trans hst
#align set.Inter₂_mono' Set.iInter₂_mono'
theorem iUnion₂_subset_iUnion (κ : ι → Sort*) (s : ι → Set α) :
⋃ (i) (_ : κ i), s i ⊆ ⋃ i, s i :=
iUnion_mono fun _ => iUnion_subset fun _ => Subset.rfl
#align set.Union₂_subset_Union Set.iUnion₂_subset_iUnion
theorem iInter_subset_iInter₂ (κ : ι → Sort*) (s : ι → Set α) :
⋂ i, s i ⊆ ⋂ (i) (_ : κ i), s i :=
iInter_mono fun _ => subset_iInter fun _ => Subset.rfl
#align set.Inter_subset_Inter₂ Set.iInter_subset_iInter₂
theorem iUnion_setOf (P : ι → α → Prop) : ⋃ i, { x : α | P i x } = { x : α | ∃ i, P i x } := by
ext
exact mem_iUnion
#align set.Union_set_of Set.iUnion_setOf
theorem iInter_setOf (P : ι → α → Prop) : ⋂ i, { x : α | P i x } = { x : α | ∀ i, P i x } := by
ext
exact mem_iInter
#align set.Inter_set_of Set.iInter_setOf
theorem iUnion_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋃ x, f x = ⋃ y, g y :=
h1.iSup_congr h h2
#align set.Union_congr_of_surjective Set.iUnion_congr_of_surjective
theorem iInter_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋂ x, f x = ⋂ y, g y :=
h1.iInf_congr h h2
#align set.Inter_congr_of_surjective Set.iInter_congr_of_surjective
lemma iUnion_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋃ i, s i = ⋃ i, t i := iSup_congr h
#align set.Union_congr Set.iUnion_congr
lemma iInter_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋂ i, s i = ⋂ i, t i := iInf_congr h
#align set.Inter_congr Set.iInter_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iUnion₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋃ (i) (j), s i j = ⋃ (i) (j), t i j :=
iUnion_congr fun i => iUnion_congr <| h i
#align set.Union₂_congr Set.iUnion₂_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iInter₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋂ (i) (j), s i j = ⋂ (i) (j), t i j :=
iInter_congr fun i => iInter_congr <| h i
#align set.Inter₂_congr Set.iInter₂_congr
section Nonempty
variable [Nonempty ι] {f : ι → Set α} {s : Set α}
lemma iUnion_const (s : Set β) : ⋃ _ : ι, s = s := iSup_const
#align set.Union_const Set.iUnion_const
lemma iInter_const (s : Set β) : ⋂ _ : ι, s = s := iInf_const
#align set.Inter_const Set.iInter_const
lemma iUnion_eq_const (hf : ∀ i, f i = s) : ⋃ i, f i = s :=
(iUnion_congr hf).trans $ iUnion_const _
#align set.Union_eq_const Set.iUnion_eq_const
lemma iInter_eq_const (hf : ∀ i, f i = s) : ⋂ i, f i = s :=
(iInter_congr hf).trans $ iInter_const _
#align set.Inter_eq_const Set.iInter_eq_const
end Nonempty
@[simp]
theorem compl_iUnion (s : ι → Set β) : (⋃ i, s i)ᶜ = ⋂ i, (s i)ᶜ :=
compl_iSup
#align set.compl_Union Set.compl_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iUnion₂ (s : ∀ i, κ i → Set α) : (⋃ (i) (j), s i j)ᶜ = ⋂ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iUnion]
#align set.compl_Union₂ Set.compl_iUnion₂
@[simp]
theorem compl_iInter (s : ι → Set β) : (⋂ i, s i)ᶜ = ⋃ i, (s i)ᶜ :=
compl_iInf
#align set.compl_Inter Set.compl_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iInter₂ (s : ∀ i, κ i → Set α) : (⋂ (i) (j), s i j)ᶜ = ⋃ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iInter]
#align set.compl_Inter₂ Set.compl_iInter₂
-- classical -- complete_boolean_algebra
theorem iUnion_eq_compl_iInter_compl (s : ι → Set β) : ⋃ i, s i = (⋂ i, (s i)ᶜ)ᶜ := by
simp only [compl_iInter, compl_compl]
#align set.Union_eq_compl_Inter_compl Set.iUnion_eq_compl_iInter_compl
-- classical -- complete_boolean_algebra
theorem iInter_eq_compl_iUnion_compl (s : ι → Set β) : ⋂ i, s i = (⋃ i, (s i)ᶜ)ᶜ := by
simp only [compl_iUnion, compl_compl]
#align set.Inter_eq_compl_Union_compl Set.iInter_eq_compl_iUnion_compl
theorem inter_iUnion (s : Set β) (t : ι → Set β) : (s ∩ ⋃ i, t i) = ⋃ i, s ∩ t i :=
inf_iSup_eq _ _
#align set.inter_Union Set.inter_iUnion
theorem iUnion_inter (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∩ s = ⋃ i, t i ∩ s :=
iSup_inf_eq _ _
#align set.Union_inter Set.iUnion_inter
theorem iUnion_union_distrib (s : ι → Set β) (t : ι → Set β) :
⋃ i, s i ∪ t i = (⋃ i, s i) ∪ ⋃ i, t i :=
iSup_sup_eq
#align set.Union_union_distrib Set.iUnion_union_distrib
theorem iInter_inter_distrib (s : ι → Set β) (t : ι → Set β) :
⋂ i, s i ∩ t i = (⋂ i, s i) ∩ ⋂ i, t i :=
iInf_inf_eq
#align set.Inter_inter_distrib Set.iInter_inter_distrib
theorem union_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∪ ⋃ i, t i) = ⋃ i, s ∪ t i :=
sup_iSup
#align set.union_Union Set.union_iUnion
theorem iUnion_union [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∪ s = ⋃ i, t i ∪ s :=
iSup_sup
#align set.Union_union Set.iUnion_union
theorem inter_iInter [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∩ ⋂ i, t i) = ⋂ i, s ∩ t i :=
inf_iInf
#align set.inter_Inter Set.inter_iInter
theorem iInter_inter [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋂ i, t i) ∩ s = ⋂ i, t i ∩ s :=
iInf_inf
#align set.Inter_inter Set.iInter_inter
-- classical
theorem union_iInter (s : Set β) (t : ι → Set β) : (s ∪ ⋂ i, t i) = ⋂ i, s ∪ t i :=
sup_iInf_eq _ _
#align set.union_Inter Set.union_iInter
theorem iInter_union (s : ι → Set β) (t : Set β) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.Inter_union Set.iInter_union
theorem iUnion_diff (s : Set β) (t : ι → Set β) : (⋃ i, t i) \ s = ⋃ i, t i \ s :=
iUnion_inter _ _
#align set.Union_diff Set.iUnion_diff
theorem diff_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s \ ⋃ i, t i) = ⋂ i, s \ t i := by
rw [diff_eq, compl_iUnion, inter_iInter]; rfl
#align set.diff_Union Set.diff_iUnion
theorem diff_iInter (s : Set β) (t : ι → Set β) : (s \ ⋂ i, t i) = ⋃ i, s \ t i := by
rw [diff_eq, compl_iInter, inter_iUnion]; rfl
#align set.diff_Inter Set.diff_iInter
theorem directed_on_iUnion {r} {f : ι → Set α} (hd : Directed (· ⊆ ·) f)
(h : ∀ x, DirectedOn r (f x)) : DirectedOn r (⋃ x, f x) := by
simp only [DirectedOn, exists_prop, mem_iUnion, exists_imp]
exact fun a₁ b₁ fb₁ a₂ b₂ fb₂ =>
let ⟨z, zb₁, zb₂⟩ := hd b₁ b₂
let ⟨x, xf, xa₁, xa₂⟩ := h z a₁ (zb₁ fb₁) a₂ (zb₂ fb₂)
⟨x, ⟨z, xf⟩, xa₁, xa₂⟩
#align set.directed_on_Union Set.directed_on_iUnion
theorem iUnion_inter_subset {ι α} {s t : ι → Set α} : ⋃ i, s i ∩ t i ⊆ (⋃ i, s i) ∩ ⋃ i, t i :=
le_iSup_inf_iSup s t
#align set.Union_inter_subset Set.iUnion_inter_subset
theorem iUnion_inter_of_monotone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_monotone hs ht
#align set.Union_inter_of_monotone Set.iUnion_inter_of_monotone
theorem iUnion_inter_of_antitone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_antitone hs ht
#align set.Union_inter_of_antitone Set.iUnion_inter_of_antitone
theorem iInter_union_of_monotone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_monotone hs ht
#align set.Inter_union_of_monotone Set.iInter_union_of_monotone
theorem iInter_union_of_antitone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_antitone hs ht
#align set.Inter_union_of_antitone Set.iInter_union_of_antitone
/-- An equality version of this lemma is `iUnion_iInter_of_monotone` in `Data.Set.Finite`. -/
theorem iUnion_iInter_subset {s : ι → ι' → Set α} : (⋃ j, ⋂ i, s i j) ⊆ ⋂ i, ⋃ j, s i j :=
iSup_iInf_le_iInf_iSup (flip s)
#align set.Union_Inter_subset Set.iUnion_iInter_subset
theorem iUnion_option {ι} (s : Option ι → Set α) : ⋃ o, s o = s none ∪ ⋃ i, s (some i) :=
iSup_option s
#align set.Union_option Set.iUnion_option
theorem iInter_option {ι} (s : Option ι → Set α) : ⋂ o, s o = s none ∩ ⋂ i, s (some i) :=
iInf_option s
#align set.Inter_option Set.iInter_option
section
variable (p : ι → Prop) [DecidablePred p]
theorem iUnion_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 _ _ _
#align set.Union_dite Set.iUnion_dite
theorem iUnion_ite (f g : ι → Set α) :
⋃ i, (if p i then f i else g i) = (⋃ (i) (_ : p i), f i) ∪ ⋃ (i) (_ : ¬p i), g i :=
iUnion_dite _ _ _
#align set.Union_ite Set.iUnion_ite
theorem iInter_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 :=
iInf_dite _ _ _
#align set.Inter_dite Set.iInter_dite
theorem iInter_ite (f g : ι → Set α) :
⋂ i, (if p i then f i else g i) = (⋂ (i) (_ : p i), f i) ∩ ⋂ (i) (_ : ¬p i), g i :=
iInter_dite _ _ _
#align set.Inter_ite Set.iInter_ite
end
theorem image_projection_prod {ι : Type*} {α : ι → Type*} {v : ∀ i : ι, Set (α i)}
(hv : (pi univ v).Nonempty) (i : ι) :
((fun x : ∀ i : ι, α i => x i) '' ⋂ k, (fun x : ∀ j : ι, α j => x k) ⁻¹' v k) = v i := by
classical
apply Subset.antisymm
· simp [iInter_subset]
· intro y y_in
simp only [mem_image, mem_iInter, mem_preimage]
rcases hv with ⟨z, hz⟩
refine' ⟨Function.update z i y, _, update_same i y z⟩
rw [@forall_update_iff ι α _ z i y fun i t => t ∈ v i]
exact ⟨y_in, fun j _ => by simpa using hz j⟩
#align set.image_projection_prod Set.image_projection_prod
/-! ### Unions and intersections indexed by `Prop` -/
theorem iInter_false {s : False → Set α} : iInter s = univ :=
iInf_false
#align set.Inter_false Set.iInter_false
theorem iUnion_false {s : False → Set α} : iUnion s = ∅ :=
iSup_false
#align set.Union_false Set.iUnion_false
@[simp]
theorem iInter_true {s : True → Set α} : iInter s = s trivial :=
iInf_true
#align set.Inter_true Set.iInter_true
@[simp]
theorem iUnion_true {s : True → Set α} : iUnion s = s trivial :=
iSup_true
#align set.Union_true Set.iUnion_true
@[simp]
theorem iInter_exists {p : ι → Prop} {f : Exists p → Set α} :
⋂ x, f x = ⋂ (i) (h : p i), f ⟨i, h⟩ :=
iInf_exists
#align set.Inter_exists Set.iInter_exists
@[simp]
theorem iUnion_exists {p : ι → Prop} {f : Exists p → Set α} :
⋃ x, f x = ⋃ (i) (h : p i), f ⟨i, h⟩ :=
iSup_exists
#align set.Union_exists Set.iUnion_exists
@[simp]
theorem iUnion_empty : (⋃ _ : ι, ∅ : Set α) = ∅ :=
iSup_bot
#align set.Union_empty Set.iUnion_empty
@[simp]
theorem iInter_univ : (⋂ _ : ι, univ : Set α) = univ :=
iInf_top
#align set.Inter_univ Set.iInter_univ
section
variable {s : ι → Set α}
@[simp]
theorem iUnion_eq_empty : ⋃ i, s i = ∅ ↔ ∀ i, s i = ∅ :=
iSup_eq_bot
#align set.Union_eq_empty Set.iUnion_eq_empty
@[simp]
theorem iInter_eq_univ : ⋂ i, s i = univ ↔ ∀ i, s i = univ :=
iInf_eq_top
#align set.Inter_eq_univ Set.iInter_eq_univ
@[simp]
theorem nonempty_iUnion : (⋃ i, s i).Nonempty ↔ ∃ i, (s i).Nonempty := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_Union Set.nonempty_iUnion
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_biUnion {t : Set α} {s : α → Set β} :
(⋃ i ∈ t, s i).Nonempty ↔ ∃ i ∈ t, (s i).Nonempty := by simp
#align set.nonempty_bUnion Set.nonempty_biUnion
theorem iUnion_nonempty_index (s : Set α) (t : s.Nonempty → Set β) :
⋃ h, t h = ⋃ x ∈ s, t ⟨x, ‹_›⟩ :=
iSup_exists
#align set.Union_nonempty_index Set.iUnion_nonempty_index
end
@[simp]
theorem iInter_iInter_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋂ (x) (h : x = b), s x h = s b rfl :=
iInf_iInf_eq_left
#align set.Inter_Inter_eq_left Set.iInter_iInter_eq_left
@[simp]
theorem iInter_iInter_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋂ (x) (h : b = x), s x h = s b rfl :=
iInf_iInf_eq_right
#align set.Inter_Inter_eq_right Set.iInter_iInter_eq_right
@[simp]
theorem iUnion_iUnion_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋃ (x) (h : x = b), s x h = s b rfl :=
iSup_iSup_eq_left
#align set.Union_Union_eq_left Set.iUnion_iUnion_eq_left
@[simp]
theorem iUnion_iUnion_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋃ (x) (h : b = x), s x h = s b rfl :=
iSup_iSup_eq_right
#align set.Union_Union_eq_right Set.iUnion_iUnion_eq_right
theorem iInter_or {p q : Prop} (s : p ∨ q → Set α) :
⋂ h, s h = (⋂ h : p, s (Or.inl h)) ∩ ⋂ h : q, s (Or.inr h) :=
iInf_or
#align set.Inter_or Set.iInter_or
theorem iUnion_or {p q : Prop} (s : p ∨ q → Set α) :
⋃ h, s h = (⋃ i, s (Or.inl i)) ∪ ⋃ j, s (Or.inr j) :=
iSup_or
#align set.Union_or Set.iUnion_or
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iUnion_and {p q : Prop} (s : p ∧ q → Set α) : ⋃ h, s h = ⋃ (hp) (hq), s ⟨hp, hq⟩ :=
iSup_and
#align set.Union_and Set.iUnion_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iInter_and {p q : Prop} (s : p ∧ q → Set α) : ⋂ h, s h = ⋂ (hp) (hq), s ⟨hp, hq⟩ :=
iInf_and
#align set.Inter_and Set.iInter_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iUnion_comm (s : ι → ι' → Set α) : ⋃ (i) (i'), s i i' = ⋃ (i') (i), s i i' :=
iSup_comm
#align set.Union_comm Set.iUnion_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iInter_comm (s : ι → ι' → Set α) : ⋂ (i) (i'), s i i' = ⋂ (i') (i), s i i' :=
iInf_comm
#align set.Inter_comm Set.iInter_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iUnion₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋃ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋃ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iSup₂_comm _
#align set.Union₂_comm Set.iUnion₂_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iInter₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋂ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋂ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iInf₂_comm _
#align set.Inter₂_comm Set.iInter₂_comm
@[simp]
theorem biUnion_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋃ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι']
#align set.bUnion_and Set.biUnion_and
@[simp]
theorem biUnion_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋃ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι]
#align set.bUnion_and' Set.biUnion_and'
@[simp]
theorem biInter_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋂ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iInter_and, @iInter_comm _ ι']
#align set.bInter_and Set.biInter_and
@[simp]
theorem biInter_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋂ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iInter_and, @iInter_comm _ ι]
#align set.bInter_and' Set.biInter_and'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iUnion_iUnion_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋃ (x) (h), s x h = s b (Or.inl rfl) ∪ ⋃ (x) (h : p x), s x (Or.inr h) := by
simp only [iUnion_or, iUnion_union_distrib, iUnion_iUnion_eq_left]
#align set.Union_Union_eq_or_left Set.iUnion_iUnion_eq_or_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iInter_iInter_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋂ (x) (h), s x h = s b (Or.inl rfl) ∩ ⋂ (x) (h : p x), s x (Or.inr h) := by
simp only [iInter_or, iInter_inter_distrib, iInter_iInter_eq_left]
#align set.Inter_Inter_eq_or_left Set.iInter_iInter_eq_or_left
/-! ### Bounded unions and intersections -/
/-- A specialization of `mem_iUnion₂`. -/
theorem mem_biUnion {s : Set α} {t : α → Set β} {x : α} {y : β} (xs : x ∈ s) (ytx : y ∈ t x) :
y ∈ ⋃ x ∈ s, t x :=
mem_iUnion₂_of_mem xs ytx
#align set.mem_bUnion Set.mem_biUnion
/-- A specialization of `mem_iInter₂`. -/
theorem mem_biInter {s : Set α} {t : α → Set β} {y : β} (h : ∀ x ∈ s, y ∈ t x) :
y ∈ ⋂ x ∈ s, t x :=
mem_iInter₂_of_mem h
#align set.mem_bInter Set.mem_biInter
/-- A specialization of `subset_iUnion₂`. -/
theorem subset_biUnion_of_mem {s : Set α} {u : α → Set β} {x : α} (xs : x ∈ s) :
u x ⊆ ⋃ x ∈ s, u x :=
--Porting note: Why is this not just `subset_iUnion₂ x xs`?
@subset_iUnion₂ β α (· ∈ s) (fun i _ => u i) x xs
#align set.subset_bUnion_of_mem Set.subset_biUnion_of_mem
/-- A specialization of `iInter₂_subset`. -/
theorem biInter_subset_of_mem {s : Set α} {t : α → Set β} {x : α} (xs : x ∈ s) :
⋂ x ∈ s, t x ⊆ t x :=
iInter₂_subset x xs
#align set.bInter_subset_of_mem Set.biInter_subset_of_mem
theorem biUnion_subset_biUnion_left {s s' : Set α} {t : α → Set β} (h : s ⊆ s') :
⋃ x ∈ s, t x ⊆ ⋃ x ∈ s', t x :=
iUnion₂_subset fun _ hx => subset_biUnion_of_mem <| h hx
#align set.bUnion_subset_bUnion_left Set.biUnion_subset_biUnion_left
theorem biInter_subset_biInter_left {s s' : Set α} {t : α → Set β} (h : s' ⊆ s) :
⋂ x ∈ s, t x ⊆ ⋂ x ∈ s', t x :=
subset_iInter₂ fun _ hx => biInter_subset_of_mem <| h hx
#align set.bInter_subset_bInter_left Set.biInter_subset_biInter_left
theorem biUnion_mono {s s' : Set α} {t t' : α → Set β} (hs : s' ⊆ s) (h : ∀ x ∈ s, t x ⊆ t' x) :
⋃ x ∈ s', t x ⊆ ⋃ x ∈ s, t' x :=
(biUnion_subset_biUnion_left hs).trans <| iUnion₂_mono h
#align set.bUnion_mono Set.biUnion_mono
theorem biInter_mono {s s' : Set α} {t t' : α → Set β} (hs : s ⊆ s') (h : ∀ x ∈ s, t x ⊆ t' x) :
⋂ x ∈ s', t x ⊆ ⋂ x ∈ s, t' x :=
(biInter_subset_biInter_left hs).trans <| iInter₂_mono h
#align set.bInter_mono Set.biInter_mono
theorem biUnion_eq_iUnion (s : Set α) (t : ∀ x ∈ s, Set β) :
⋃ x ∈ s, t x ‹_› = ⋃ x : s, t x x.2 :=
iSup_subtype'
#align set.bUnion_eq_Union Set.biUnion_eq_iUnion
theorem biInter_eq_iInter (s : Set α) (t : ∀ x ∈ s, Set β) :
⋂ x ∈ s, t x ‹_› = ⋂ x : s, t x x.2 :=
iInf_subtype'
#align set.bInter_eq_Inter Set.biInter_eq_iInter
theorem iUnion_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋃ x : { x // p x }, s x = ⋃ (x) (hx : p x), s ⟨x, hx⟩ :=
iSup_subtype
#align set.Union_subtype Set.iUnion_subtype
theorem iInter_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋂ x : { x // p x }, s x = ⋂ (x) (hx : p x), s ⟨x, hx⟩ :=
iInf_subtype
#align set.Inter_subtype Set.iInter_subtype
theorem biInter_empty (u : α → Set β) : ⋂ x ∈ (∅ : Set α), u x = univ :=
iInf_emptyset
#align set.bInter_empty Set.biInter_empty
theorem biInter_univ (u : α → Set β) : ⋂ x ∈ @univ α, u x = ⋂ x, u x :=
iInf_univ
#align set.bInter_univ Set.biInter_univ
@[simp]
theorem biUnion_self (s : Set α) : ⋃ x ∈ s, s = s :=
Subset.antisymm (iUnion₂_subset fun _ _ => Subset.refl s) fun _ hx => mem_biUnion hx hx
#align set.bUnion_self Set.biUnion_self
@[simp]
theorem iUnion_nonempty_self (s : Set α) : ⋃ _ : s.Nonempty, s = s := by
rw [iUnion_nonempty_index, biUnion_self]
#align set.Union_nonempty_self Set.iUnion_nonempty_self
theorem biInter_singleton (a : α) (s : α → Set β) : ⋂ x ∈ ({a} : Set α), s x = s a :=
iInf_singleton
#align set.bInter_singleton Set.biInter_singleton
theorem biInter_union (s t : Set α) (u : α → Set β) :
⋂ x ∈ s ∪ t, u x = (⋂ x ∈ s, u x) ∩ ⋂ x ∈ t, u x :=
iInf_union
#align set.bInter_union Set.biInter_union
theorem biInter_insert (a : α) (s : Set α) (t : α → Set β) :
⋂ x ∈ insert a s, t x = t a ∩ ⋂ x ∈ s, t x := by simp
#align set.bInter_insert Set.biInter_insert
theorem biInter_pair (a b : α) (s : α → Set β) : ⋂ x ∈ ({a, b} : Set α), s x = s a ∩ s b := by
rw [biInter_insert, biInter_singleton]
#align set.bInter_pair Set.biInter_pair
theorem biInter_inter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, f i ∩ t = (⋂ i ∈ s, f i) ∩ t := by
haveI : Nonempty s := hs.to_subtype
simp [biInter_eq_iInter, ← iInter_inter]
#align set.bInter_inter Set.biInter_inter
theorem inter_biInter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, t ∩ f i = t ∩ ⋂ i ∈ s, f i := by
rw [inter_comm, ← biInter_inter hs]
simp [inter_comm]
#align set.inter_bInter Set.inter_biInter
theorem biUnion_empty (s : α → Set β) : ⋃ x ∈ (∅ : Set α), s x = ∅ :=
iSup_emptyset
#align set.bUnion_empty Set.biUnion_empty
theorem biUnion_univ (s : α → Set β) : ⋃ x ∈ @univ α, s x = ⋃ x, s x :=
iSup_univ
#align set.bUnion_univ Set.biUnion_univ
theorem biUnion_singleton (a : α) (s : α → Set β) : ⋃ x ∈ ({a} : Set α), s x = s a :=
iSup_singleton
#align set.bUnion_singleton Set.biUnion_singleton
@[simp]
theorem biUnion_of_singleton (s : Set α) : ⋃ x ∈ s, {x} = s :=
ext <| by simp
#align set.bUnion_of_singleton Set.biUnion_of_singleton
theorem biUnion_union (s t : Set α) (u : α → Set β) :
⋃ x ∈ s ∪ t, u x = (⋃ x ∈ s, u x) ∪ ⋃ x ∈ t, u x :=
iSup_union
#align set.bUnion_union Set.biUnion_union
@[simp]
theorem iUnion_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋃ i, f i = ⋃ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iUnion_subtype _ _
#align set.Union_coe_set Set.iUnion_coe_set
@[simp]
theorem iInter_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋂ i, f i = ⋂ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iInter_subtype _ _
#align set.Inter_coe_set Set.iInter_coe_set
theorem biUnion_insert (a : α) (s : Set α) (t : α → Set β) :
⋃ x ∈ insert a s, t x = t a ∪ ⋃ x ∈ s, t x := by simp
#align set.bUnion_insert Set.biUnion_insert
theorem biUnion_pair (a b : α) (s : α → Set β) : ⋃ x ∈ ({a, b} : Set α), s x = s a ∪ s b :=
by simp
#align set.bUnion_pair Set.biUnion_pair
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem inter_iUnion₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∩ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ∩ t i j := by simp only [inter_iUnion]
#align set.inter_Union₂ Set.inter_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_inter (s : ∀ i, κ i → Set α) (t : Set α) :
(⋃ (i) (j), s i j) ∩ t = ⋃ (i) (j), s i j ∩ t := by simp_rw [iUnion_inter]
#align set.Union₂_inter Set.iUnion₂_inter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_iInter₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_iInter]
#align set.union_Inter₂ Set.union_iInter₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_union (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [iInter_union]
#align set.Inter₂_union Set.iInter₂_union
theorem mem_sUnion_of_mem {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∈ t) (ht : t ∈ S) :
x ∈ ⋃₀S :=
⟨t, ht, hx⟩
#align set.mem_sUnion_of_mem Set.mem_sUnion_of_mem
-- is this theorem really necessary?
theorem not_mem_of_not_mem_sUnion {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∉ ⋃₀S)
(ht : t ∈ S) : x ∉ t := fun h => hx ⟨t, ht, h⟩
#align set.not_mem_of_not_mem_sUnion Set.not_mem_of_not_mem_sUnion
theorem sInter_subset_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : ⋂₀ S ⊆ t :=
sInf_le tS
#align set.sInter_subset_of_mem Set.sInter_subset_of_mem
theorem subset_sUnion_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : t ⊆ ⋃₀S :=
le_sSup tS
#align set.subset_sUnion_of_mem Set.subset_sUnion_of_mem
theorem subset_sUnion_of_subset {s : Set α} (t : Set (Set α)) (u : Set α) (h₁ : s ⊆ u)
(h₂ : u ∈ t) : s ⊆ ⋃₀t :=
Subset.trans h₁ (subset_sUnion_of_mem h₂)
#align set.subset_sUnion_of_subset Set.subset_sUnion_of_subset
theorem sUnion_subset {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t' ⊆ t) : ⋃₀S ⊆ t :=
sSup_le h
#align set.sUnion_subset Set.sUnion_subset
@[simp]
theorem sUnion_subset_iff {s : Set (Set α)} {t : Set α} : ⋃₀s ⊆ t ↔ ∀ t' ∈ s, t' ⊆ t :=
sSup_le_iff
#align set.sUnion_subset_iff Set.sUnion_subset_iff
/-- `sUnion` is monotone under taking a subset of each set. -/
lemma sUnion_mono_subsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, t ⊆ f t) :
⋃₀ s ⊆ ⋃₀ (f '' s) :=
fun _ ⟨t, htx, hxt⟩ ↦ ⟨f t, mem_image_of_mem f htx, hf t hxt⟩
/-- `sUnion` is monotone under taking a superset of each set. -/
lemma sUnion_mono_supsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, f t ⊆ t) :
⋃₀ (f '' s) ⊆ ⋃₀ s :=
-- If t ∈ f '' s is arbitrary; t = f u for some u : Set α.
fun _ ⟨_, ⟨u, hus, hut⟩, hxt⟩ ↦ ⟨u, hus, (hut ▸ hf u) hxt⟩
theorem subset_sInter {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t ⊆ t') : t ⊆ ⋂₀ S :=
le_sInf h
#align set.subset_sInter Set.subset_sInter
@[simp]
theorem subset_sInter_iff {S : Set (Set α)} {t : Set α} : t ⊆ ⋂₀ S ↔ ∀ t' ∈ S, t ⊆ t' :=
le_sInf_iff
#align set.subset_sInter_iff Set.subset_sInter_iff
@[gcongr]
theorem sUnion_subset_sUnion {S T : Set (Set α)} (h : S ⊆ T) : ⋃₀S ⊆ ⋃₀T :=
sUnion_subset fun _ hs => subset_sUnion_of_mem (h hs)
#align set.sUnion_subset_sUnion Set.sUnion_subset_sUnion
@[gcongr]
theorem sInter_subset_sInter {S T : Set (Set α)} (h : S ⊆ T) : ⋂₀ T ⊆ ⋂₀ S :=
subset_sInter fun _ hs => sInter_subset_of_mem (h hs)
#align set.sInter_subset_sInter Set.sInter_subset_sInter
@[simp]
theorem sUnion_empty : ⋃₀∅ = (∅ : Set α) :=
sSup_empty
#align set.sUnion_empty Set.sUnion_empty
@[simp]
theorem sInter_empty : ⋂₀ ∅ = (univ : Set α) :=
sInf_empty
#align set.sInter_empty Set.sInter_empty
@[simp]
theorem sUnion_singleton (s : Set α) : ⋃₀{s} = s :=
sSup_singleton
#align set.sUnion_singleton Set.sUnion_singleton
@[simp]
theorem sInter_singleton (s : Set α) : ⋂₀ {s} = s :=
sInf_singleton
#align set.sInter_singleton Set.sInter_singleton
@[simp]
theorem sUnion_eq_empty {S : Set (Set α)} : ⋃₀S = ∅ ↔ ∀ s ∈ S, s = ∅ :=
sSup_eq_bot
#align set.sUnion_eq_empty Set.sUnion_eq_empty
@[simp]
theorem sInter_eq_univ {S : Set (Set α)} : ⋂₀ S = univ ↔ ∀ s ∈ S, s = univ :=
sInf_eq_top
#align set.sInter_eq_univ Set.sInter_eq_univ
theorem subset_powerset_iff {s : Set (Set α)} {t : Set α} : s ⊆ 𝒫 t ↔ ⋃₀ s ⊆ t :=
sUnion_subset_iff.symm
/-- `⋃₀` and `𝒫` form a Galois connection. -/
theorem sUnion_powerset_gc :
GaloisConnection (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gc_sSup_Iic
/-- `⋃₀` and `𝒫` form a Galois insertion. -/
def sUnion_powerset_gi :
GaloisInsertion (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gi_sSup_Iic
/-- If all sets in a collection are either `∅` or `Set.univ`, then so is their union. -/
theorem sUnion_mem_empty_univ {S : Set (Set α)} (h : S ⊆ {∅, univ}) :
⋃₀ S ∈ ({∅, univ} : Set (Set α)) := by
simp only [mem_insert_iff, mem_singleton_iff, or_iff_not_imp_left, sUnion_eq_empty, not_forall]
rintro ⟨s, hs, hne⟩
obtain rfl : s = univ := (h hs).resolve_left hne
exact univ_subset_iff.1 <| subset_sUnion_of_mem hs
@[simp]
theorem nonempty_sUnion {S : Set (Set α)} : (⋃₀S).Nonempty ↔ ∃ s ∈ S, Set.Nonempty s := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_sUnion Set.nonempty_sUnion
theorem Nonempty.of_sUnion {s : Set (Set α)} (h : (⋃₀s).Nonempty) : s.Nonempty :=
let ⟨s, hs, _⟩ := nonempty_sUnion.1 h
⟨s, hs⟩
#align set.nonempty.of_sUnion Set.Nonempty.of_sUnion
theorem Nonempty.of_sUnion_eq_univ [Nonempty α] {s : Set (Set α)} (h : ⋃₀s = univ) : s.Nonempty :=
Nonempty.of_sUnion <| h.symm ▸ univ_nonempty
#align set.nonempty.of_sUnion_eq_univ Set.Nonempty.of_sUnion_eq_univ
theorem sUnion_union (S T : Set (Set α)) : ⋃₀(S ∪ T) = ⋃₀S ∪ ⋃₀T :=
sSup_union
#align set.sUnion_union Set.sUnion_union
theorem sInter_union (S T : Set (Set α)) : ⋂₀ (S ∪ T) = ⋂₀ S ∩ ⋂₀ T :=
sInf_union
#align set.sInter_union Set.sInter_union
@[simp]
theorem sUnion_insert (s : Set α) (T : Set (Set α)) : ⋃₀insert s T = s ∪ ⋃₀T :=
sSup_insert
#align set.sUnion_insert Set.sUnion_insert
@[simp]
theorem sInter_insert (s : Set α) (T : Set (Set α)) : ⋂₀ insert s T = s ∩ ⋂₀ T :=
sInf_insert
#align set.sInter_insert Set.sInter_insert
@[simp]
theorem sUnion_diff_singleton_empty (s : Set (Set α)) : ⋃₀(s \ {∅}) = ⋃₀s :=
sSup_diff_singleton_bot s
#align set.sUnion_diff_singleton_empty Set.sUnion_diff_singleton_empty
@[simp]
theorem sInter_diff_singleton_univ (s : Set (Set α)) : ⋂₀ (s \ {univ}) = ⋂₀ s :=
sInf_diff_singleton_top s
#align set.sInter_diff_singleton_univ Set.sInter_diff_singleton_univ
theorem sUnion_pair (s t : Set α) : ⋃₀{s, t} = s ∪ t :=
sSup_pair
#align set.sUnion_pair Set.sUnion_pair
theorem sInter_pair (s t : Set α) : ⋂₀ {s, t} = s ∩ t :=
sInf_pair
#align set.sInter_pair Set.sInter_pair
@[simp]
theorem sUnion_image (f : α → Set β) (s : Set α) : ⋃₀(f '' s) = ⋃ x ∈ s, f x :=
sSup_image
#align set.sUnion_image Set.sUnion_image
@[simp]
theorem sInter_image (f : α → Set β) (s : Set α) : ⋂₀ (f '' s) = ⋂ x ∈ s, f x :=
sInf_image
#align set.sInter_image Set.sInter_image
@[simp]
theorem sUnion_range (f : ι → Set β) : ⋃₀range f = ⋃ x, f x :=
rfl
#align set.sUnion_range Set.sUnion_range
@[simp]
theorem sInter_range (f : ι → Set β) : ⋂₀ range f = ⋂ x, f x :=
rfl
#align set.sInter_range Set.sInter_range
theorem iUnion_eq_univ_iff {f : ι → Set α} : ⋃ i, f i = univ ↔ ∀ x, ∃ i, x ∈ f i := by
simp only [eq_univ_iff_forall, mem_iUnion]
#align set.Union_eq_univ_iff Set.iUnion_eq_univ_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_eq_univ_iff {s : ∀ i, κ i → Set α} :
⋃ (i) (j), s i j = univ ↔ ∀ a, ∃ i j, a ∈ s i j :=
by simp only [iUnion_eq_univ_iff, mem_iUnion]
#align set.Union₂_eq_univ_iff Set.iUnion₂_eq_univ_iff
theorem sUnion_eq_univ_iff {c : Set (Set α)} : ⋃₀c = univ ↔ ∀ a, ∃ b ∈ c, a ∈ b := by
simp only [eq_univ_iff_forall, mem_sUnion]
#align set.sUnion_eq_univ_iff Set.sUnion_eq_univ_iff
-- classical
theorem iInter_eq_empty_iff {f : ι → Set α} : ⋂ i, f i = ∅ ↔ ∀ x, ∃ i, x ∉ f i := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.Inter_eq_empty_iff Set.iInter_eq_empty_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
theorem iInter₂_eq_empty_iff {s : ∀ i, κ i → Set α} :
⋂ (i) (j), s i j = ∅ ↔ ∀ a, ∃ i j, a ∉ s i j := by
simp only [eq_empty_iff_forall_not_mem, mem_iInter, not_forall]
#align set.Inter₂_eq_empty_iff Set.iInter₂_eq_empty_iff
-- classical
theorem sInter_eq_empty_iff {c : Set (Set α)} : ⋂₀ c = ∅ ↔ ∀ a, ∃ b ∈ c, a ∉ b := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.sInter_eq_empty_iff Set.sInter_eq_empty_iff
-- classical
@[simp]
theorem nonempty_iInter {f : ι → Set α} : (⋂ i, f i).Nonempty ↔ ∃ x, ∀ i, x ∈ f i := by
simp [nonempty_iff_ne_empty, iInter_eq_empty_iff]
#align set.nonempty_Inter Set.nonempty_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_iInter₂ {s : ∀ i, κ i → Set α} :
(⋂ (i) (j), s i j).Nonempty ↔ ∃ a, ∀ i j, a ∈ s i j := by
simp
#align set.nonempty_Inter₂ Set.nonempty_iInter₂
-- classical
@[simp]
theorem nonempty_sInter {c : Set (Set α)} : (⋂₀ c).Nonempty ↔ ∃ a, ∀ b ∈ c, a ∈ b := by
simp [nonempty_iff_ne_empty, sInter_eq_empty_iff]
#align set.nonempty_sInter Set.nonempty_sInter
-- classical
theorem compl_sUnion (S : Set (Set α)) : (⋃₀S)ᶜ = ⋂₀ (compl '' S) :=
ext fun x => by simp
#align set.compl_sUnion Set.compl_sUnion
-- classical
theorem sUnion_eq_compl_sInter_compl (S : Set (Set α)) : ⋃₀S = (⋂₀ (compl '' S))ᶜ := by
rw [← compl_compl (⋃₀S), compl_sUnion]
#align set.sUnion_eq_compl_sInter_compl Set.sUnion_eq_compl_sInter_compl
-- classical
theorem compl_sInter (S : Set (Set α)) : (⋂₀ S)ᶜ = ⋃₀(compl '' S) := by
rw [sUnion_eq_compl_sInter_compl, compl_compl_image]
#align set.compl_sInter Set.compl_sInter
-- classical
theorem sInter_eq_compl_sUnion_compl (S : Set (Set α)) : ⋂₀ S = (⋃₀(compl '' S))ᶜ := by
rw [← compl_compl (⋂₀ S), compl_sInter]
#align set.sInter_eq_compl_sUnion_compl Set.sInter_eq_compl_sUnion_compl
theorem inter_empty_of_inter_sUnion_empty {s t : Set α} {S : Set (Set α)} (hs : t ∈ S)
(h : s ∩ ⋃₀S = ∅) : s ∩ t = ∅ :=
eq_empty_of_subset_empty <| by
rw [← h]; exact inter_subset_inter_right _ (subset_sUnion_of_mem hs)
#align set.inter_empty_of_inter_sUnion_empty Set.inter_empty_of_inter_sUnion_empty
theorem range_sigma_eq_iUnion_range {γ : α → Type*} (f : Sigma γ → β) :
range f = ⋃ a, range fun b => f ⟨a, b⟩ :=
Set.ext <| by simp
#align set.range_sigma_eq_Union_range Set.range_sigma_eq_iUnion_range
theorem iUnion_eq_range_sigma (s : α → Set β) : ⋃ i, s i = range fun a : Σi, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_sigma Set.iUnion_eq_range_sigma
theorem iUnion_eq_range_psigma (s : ι → Set β) : ⋃ i, s i = range fun a : Σ'i, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_psigma Set.iUnion_eq_range_psigma
theorem iUnion_image_preimage_sigma_mk_eq_self {ι : Type*} {σ : ι → Type*} (s : Set (Sigma σ)) :
⋃ i, Sigma.mk i '' (Sigma.mk i ⁻¹' s) = s := by
ext x
simp only [mem_iUnion, mem_image, mem_preimage]
constructor
· rintro ⟨i, a, h, rfl⟩
exact h
· intro h
cases' x with i a
exact ⟨i, a, h, rfl⟩
#align set.Union_image_preimage_sigma_mk_eq_self Set.iUnion_image_preimage_sigma_mk_eq_self
theorem Sigma.univ (X : α → Type*) : (Set.univ : Set (Σa, X a)) = ⋃ a, range (Sigma.mk a) :=
Set.ext fun x =>
iff_of_true trivial ⟨range (Sigma.mk x.1), Set.mem_range_self _, x.2, Sigma.eta x⟩
#align set.sigma.univ Set.Sigma.univ
alias sUnion_mono := sUnion_subset_sUnion
#align set.sUnion_mono Set.sUnion_mono
theorem iUnion_subset_iUnion_const {s : Set α} (h : ι → ι₂) : ⋃ _ : ι, s ⊆ ⋃ _ : ι₂, s :=
@iSup_const_mono (Set α) ι ι₂ _ s h
#align set.Union_subset_Union_const Set.iUnion_subset_iUnion_const
@[simp]
theorem iUnion_singleton_eq_range {α β : Type*} (f : α → β) : ⋃ x : α, {f x} = range f := by
ext x
simp [@eq_comm _ x]
#align set.Union_singleton_eq_range Set.iUnion_singleton_eq_range
theorem iUnion_of_singleton (α : Type*) : (⋃ x, {x} : Set α) = univ := by simp [Set.ext_iff]
#align set.Union_of_singleton Set.iUnion_of_singleton
theorem iUnion_of_singleton_coe (s : Set α) : ⋃ i : s, ({(i : α)} : Set α) = s := by simp
#align set.Union_of_singleton_coe Set.iUnion_of_singleton_coe
theorem sUnion_eq_biUnion {s : Set (Set α)} : ⋃₀s = ⋃ (i : Set α) (_ : i ∈ s), i := by
rw [← sUnion_image, image_id']
#align set.sUnion_eq_bUnion Set.sUnion_eq_biUnion
theorem sInter_eq_biInter {s : Set (Set α)} : ⋂₀ s = ⋂ (i : Set α) (_ : i ∈ s), i := by
rw [← sInter_image, image_id']
#align set.sInter_eq_bInter Set.sInter_eq_biInter
theorem sUnion_eq_iUnion {s : Set (Set α)} : ⋃₀s = ⋃ i : s, i := by
simp only [← sUnion_range, Subtype.range_coe]
#align set.sUnion_eq_Union Set.sUnion_eq_iUnion
theorem sInter_eq_iInter {s : Set (Set α)} : ⋂₀ s = ⋂ i : s, i := by
simp only [← sInter_range, Subtype.range_coe]
#align set.sInter_eq_Inter Set.sInter_eq_iInter
@[simp]
theorem iUnion_of_empty [IsEmpty ι] (s : ι → Set α) : ⋃ i, s i = ∅ :=
iSup_of_empty _
#align set.Union_of_empty Set.iUnion_of_empty
@[simp]
theorem iInter_of_empty [IsEmpty ι] (s : ι → Set α) : ⋂ i, s i = univ :=
iInf_of_empty _
#align set.Inter_of_empty Set.iInter_of_empty
theorem union_eq_iUnion {s₁ s₂ : Set α} : s₁ ∪ s₂ = ⋃ b : Bool, cond b s₁ s₂ :=
sup_eq_iSup s₁ s₂
#align set.union_eq_Union Set.union_eq_iUnion
theorem inter_eq_iInter {s₁ s₂ : Set α} : s₁ ∩ s₂ = ⋂ b : Bool, cond b s₁ s₂ :=
inf_eq_iInf s₁ s₂
#align set.inter_eq_Inter Set.inter_eq_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_union_sInter {S T : Set (Set α)} :
⋂₀ S ∪ ⋂₀ T = ⋂ p ∈ S ×ˢ T, (p : Set α × Set α).1 ∪ p.2 :=
sInf_sup_sInf
#align set.sInter_union_sInter Set.sInter_union_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_inter_sUnion {s t : Set (Set α)} :
⋃₀s ∩ ⋃₀t = ⋃ p ∈ s ×ˢ t, (p : Set α × Set α).1 ∩ p.2 :=
sSup_inf_sSup
#align set.sUnion_inter_sUnion Set.sUnion_inter_sUnion
theorem biUnion_iUnion (s : ι → Set α) (t : α → Set β) :
⋃ x ∈ ⋃ i, s i, t x = ⋃ (i) (x ∈ s i), t x := by simp [@iUnion_comm _ ι]
#align set.bUnion_Union Set.biUnion_iUnion
theorem biInter_iUnion (s : ι → Set α) (t : α → Set β) :
⋂ x ∈ ⋃ i, s i, t x = ⋂ (i) (x ∈ s i), t x := by simp [@iInter_comm _ ι]
#align set.bInter_Union Set.biInter_iUnion
theorem sUnion_iUnion (s : ι → Set (Set α)) : ⋃₀⋃ i, s i = ⋃ i, ⋃₀s i := by
simp only [sUnion_eq_biUnion, biUnion_iUnion]
#align set.sUnion_Union Set.sUnion_iUnion
theorem sInter_iUnion (s : ι → Set (Set α)) : ⋂₀ ⋃ i, s i = ⋂ i, ⋂₀ s i := by
simp only [sInter_eq_biInter, biInter_iUnion]
#align set.sInter_Union Set.sInter_iUnion
theorem iUnion_range_eq_sUnion {α β : Type*} (C : Set (Set α)) {f : ∀ s : C, β → (s : Type _)}
(hf : ∀ s : C, Surjective (f s)) : ⋃ y : β, range (fun s : C => (f s y).val) = ⋃₀C := by
ext x; constructor
· rintro ⟨s, ⟨y, rfl⟩, ⟨s, hs⟩, rfl⟩
refine' ⟨_, hs, _⟩
exact (f ⟨s, hs⟩ y).2
· rintro ⟨s, hs, hx⟩
cases' hf ⟨s, hs⟩ ⟨x, hx⟩ with y hy
refine' ⟨_, ⟨y, rfl⟩, ⟨s, hs⟩, _⟩
exact congr_arg Subtype.val hy
#align set.Union_range_eq_sUnion Set.iUnion_range_eq_sUnion
theorem iUnion_range_eq_iUnion (C : ι → Set α) {f : ∀ x : ι, β → C x}
(hf : ∀ x : ι, Surjective (f x)) : ⋃ y : β, range (fun x : ι => (f x y).val) = ⋃ x, C x := by
ext x; rw [mem_iUnion, mem_iUnion]; constructor
· rintro ⟨y, i, rfl⟩
exact ⟨i, (f i y).2⟩
· rintro ⟨i, hx⟩
cases' hf i ⟨x, hx⟩ with y hy
exact ⟨y, i, congr_arg Subtype.val hy⟩
#align set.Union_range_eq_Union Set.iUnion_range_eq_iUnion
theorem union_distrib_iInter_left (s : ι → Set α) (t : Set α) : (t ∪ ⋂ i, s i) = ⋂ i, t ∪ s i :=
sup_iInf_eq _ _
#align set.union_distrib_Inter_left Set.union_distrib_iInter_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_left (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_distrib_iInter_left]
#align set.union_distrib_Inter₂_left Set.union_distrib_iInter₂_left
theorem union_distrib_iInter_right (s : ι → Set α) (t : Set α) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.union_distrib_Inter_right Set.union_distrib_iInter_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_right (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [union_distrib_iInter_right]
#align set.union_distrib_Inter₂_right Set.union_distrib_iInter₂_right
section Function
/-! ### `mapsTo` -/
theorem mapsTo_sUnion {S : Set (Set α)} {t : Set β} {f : α → β} (H : ∀ s ∈ S, MapsTo f s t) :
MapsTo f (⋃₀S) t := fun _ ⟨s, hs, hx⟩ => H s hs hx
#align set.maps_to_sUnion Set.mapsTo_sUnion
theorem mapsTo_iUnion {s : ι → Set α} {t : Set β} {f : α → β} (H : ∀ i, MapsTo f (s i) t) :
MapsTo f (⋃ i, s i) t :=
mapsTo_sUnion <| forall_range_iff.2 H
#align set.maps_to_Union Set.mapsTo_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂ {s : ∀ i, κ i → Set α} {t : Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) t) : MapsTo f (⋃ (i) (j), s i j) t :=
mapsTo_iUnion fun i => mapsTo_iUnion (H i)
#align set.maps_to_Union₂ Set.mapsTo_iUnion₂
theorem mapsTo_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋃ i, s i) (⋃ i, t i) :=
mapsTo_iUnion fun i => (H i).mono (Subset.refl _) (subset_iUnion t i)
#align set.maps_to_Union_Union Set.mapsTo_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
mapsTo_iUnion_iUnion fun i => mapsTo_iUnion_iUnion (H i)
#align set.maps_to_Union₂_Union₂ Set.mapsTo_iUnion₂_iUnion₂
theorem mapsTo_sInter {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, MapsTo f s t) :
MapsTo f s (⋂₀ T) := fun _ hx t ht => H t ht hx
#align set.maps_to_sInter Set.mapsTo_sInter
theorem mapsTo_iInter {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, MapsTo f s (t i)) :
MapsTo f s (⋂ i, t i) := fun _ hx => mem_iInter.2 fun i => H i hx
#align set.maps_to_Inter Set.mapsTo_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f s (t i j)) : MapsTo f s (⋂ (i) (j), t i j) :=
mapsTo_iInter fun i => mapsTo_iInter (H i)
#align set.maps_to_Inter₂ Set.mapsTo_iInter₂
theorem mapsTo_iInter_iInter {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋂ i, s i) (⋂ i, t i) :=
mapsTo_iInter fun i => (H i).mono (iInter_subset s i) (Subset.refl _)
#align set.maps_to_Inter_Inter Set.mapsTo_iInter_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂_iInter₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋂ (i) (j), s i j) (⋂ (i) (j), t i j) :=
mapsTo_iInter_iInter fun i => mapsTo_iInter_iInter (H i)
#align set.maps_to_Inter₂_Inter₂ Set.mapsTo_iInter₂_iInter₂
theorem image_iInter_subset (s : ι → Set α) (f : α → β) : (f '' ⋂ i, s i) ⊆ ⋂ i, f '' s i :=
(mapsTo_iInter_iInter fun i => mapsTo_image f (s i)).image_subset
#align set.image_Inter_subset Set.image_iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂_subset (s : ∀ i, κ i → Set α) (f : α → β) :
(f '' ⋂ (i) (j), s i j) ⊆ ⋂ (i) (j), f '' s i j :=
(mapsTo_iInter₂_iInter₂ fun i hi => mapsTo_image f (s i hi)).image_subset
#align set.image_Inter₂_subset Set.image_iInter₂_subset
theorem image_sInter_subset (S : Set (Set α)) (f : α → β) : f '' ⋂₀ S ⊆ ⋂ s ∈ S, f '' s := by
rw [sInter_eq_biInter]
apply image_iInter₂_subset
#align set.image_sInter_subset Set.image_sInter_subset
/-! ### `restrictPreimage` -/
section
open Function
variable (s : Set β) {f : α → β} {U : ι → Set β} (hU : iUnion U = univ)
theorem injective_iff_injective_of_iUnion_eq_univ :
Injective f ↔ ∀ i, Injective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_injective H, fun H x y e => _⟩
obtain ⟨i, hi⟩ := Set.mem_iUnion.mp
(show f x ∈ Set.iUnion U by rw [hU]; triv)
injection @H i ⟨x, hi⟩ ⟨y, show f y ∈ U i from e ▸ hi⟩ (Subtype.ext e)
#align set.injective_iff_injective_of_Union_eq_univ Set.injective_iff_injective_of_iUnion_eq_univ
theorem surjective_iff_surjective_of_iUnion_eq_univ :
Surjective f ↔ ∀ i, Surjective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_surjective H, fun H x => _⟩
obtain ⟨i, hi⟩ :=
Set.mem_iUnion.mp
(show x ∈ Set.iUnion U by rw [hU]; triv)
exact ⟨_, congr_arg Subtype.val (H i ⟨x, hi⟩).choose_spec⟩
#align set.surjective_iff_surjective_of_Union_eq_univ Set.surjective_iff_surjective_of_iUnion_eq_univ
theorem bijective_iff_bijective_of_iUnion_eq_univ :
Bijective f ↔ ∀ i, Bijective ((U i).restrictPreimage f) := by
rw [Bijective, injective_iff_injective_of_iUnion_eq_univ hU,
surjective_iff_surjective_of_iUnion_eq_univ hU]
simp [Bijective, forall_and]
#align set.bijective_iff_bijective_of_Union_eq_univ Set.bijective_iff_bijective_of_iUnion_eq_univ
end
/-! ### `InjOn` -/
theorem InjOn.image_iInter_eq [Nonempty ι] {s : ι → Set α} {f : α → β} (h : InjOn f (⋃ i, s i)) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
inhabit ι
refine' Subset.antisymm (image_iInter_subset s f) fun y hy => _
simp only [mem_iInter, mem_image_iff_bex] at hy
choose x hx hy using hy
refine' ⟨x default, mem_iInter.2 fun i => _, hy _⟩
suffices x default = x i by
rw [this]
apply hx
replace hx : ∀ i, x i ∈ ⋃ j, s j := fun i => (subset_iUnion _ _) (hx i)
apply h (hx _) (hx _)
simp only [hy]
#align set.inj_on.image_Inter_eq Set.InjOn.image_iInter_eq
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
theorem InjOn.image_biInter_eq {p : ι → Prop} {s : ∀ i, p i → Set α} (hp : ∃ i, p i)
{f : α → β} (h : InjOn f (⋃ (i) (hi), s i hi)) :
(f '' ⋂ (i) (hi), s i hi) = ⋂ (i) (hi), f '' s i hi := by
simp only [iInter, iInf_subtype']
haveI : Nonempty { i // p i } := nonempty_subtype.2 hp
apply InjOn.image_iInter_eq
simpa only [iUnion, iSup_subtype'] using h
#align set.inj_on.image_bInter_eq Set.InjOn.image_biInter_eq
theorem image_iInter {f : α → β} (hf : Bijective f) (s : ι → Set α) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
cases isEmpty_or_nonempty ι
· simp_rw [iInter_of_empty, image_univ_of_surjective hf.surjective]
· exact (hf.injective.injOn _).image_iInter_eq
#align set.image_Inter Set.image_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂ {f : α → β} (hf : Bijective f) (s : ∀ i, κ i → Set α) :
(f '' ⋂ (i) (j), s i j) = ⋂ (i) (j), f '' s i j := by simp_rw [image_iInter hf]
#align set.image_Inter₂ Set.image_iInter₂
theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
(hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i) := by
intro x hx y hy hxy
rcases mem_iUnion.1 hx with ⟨i, hx⟩
rcases mem_iUnion.1 hy with ⟨j, hy⟩
rcases hs i j with ⟨k, hi, hj⟩
exact hf k (hi hx) (hj hy) hxy
#align set.inj_on_Union_of_directed Set.inj_on_iUnion_of_directed
/-! ### `SurjOn` -/
theorem surjOn_sUnion {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, SurjOn f s t) :
SurjOn f s (⋃₀T) := fun _ ⟨t, ht, hx⟩ => H t ht hx
#align set.surj_on_sUnion Set.surjOn_sUnion
theorem surjOn_iUnion {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, SurjOn f s (t i)) :
SurjOn f s (⋃ i, t i) :=
surjOn_sUnion <| forall_range_iff.2 H
#align set.surj_on_Union Set.surjOn_iUnion
theorem surjOn_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) : SurjOn f (⋃ i, s i) (⋃ i, t i) :=
surjOn_iUnion fun i => (H i).mono (subset_iUnion _ _) (Subset.refl _)
#align set.surj_on_Union_Union Set.surjOn_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f s (t i j)) : SurjOn f s (⋃ (i) (j), t i j) :=
surjOn_iUnion fun i => surjOn_iUnion (H i)
#align set.surj_on_Union₂ Set.surjOn_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f (s i j) (t i j)) : SurjOn f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
surjOn_iUnion_iUnion fun i => surjOn_iUnion_iUnion (H i)
#align set.surj_on_Union₂_Union₂ Set.surjOn_iUnion₂_iUnion₂
theorem surjOn_iInter [Nonempty ι] {s : ι → Set α} {t : Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) t) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) t := by
intro y hy
rw [Hinj.image_iInter_eq, mem_iInter]
exact fun i => H i hy
#align set.surj_on_Inter Set.surjOn_iInter
theorem surjOn_iInter_iInter [Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) (⋂ i, t i) :=
surjOn_iInter (fun i => (H i).mono (Subset.refl _) (iInter_subset _ _)) Hinj
#align set.surj_on_Inter_Inter Set.surjOn_iInter_iInter
/-! ### `BijOn` -/
theorem bijOn_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i))
(Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
⟨mapsTo_iUnion_iUnion fun i => (H i).mapsTo, Hinj, surjOn_iUnion_iUnion fun i => (H i).surjOn⟩
#align set.bij_on_Union Set.bijOn_iUnion
theorem bijOn_iInter [hi : Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, BijOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
⟨mapsTo_iInter_iInter fun i => (H i).mapsTo,
hi.elim fun i => (H i).injOn.mono (iInter_subset _ _),
surjOn_iInter_iInter (fun i => (H i).surjOn) Hinj⟩
#align set.bij_on_Inter Set.bijOn_iInter
theorem bijOn_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {t : ι → Set β}
{f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
bijOn_iUnion H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Union_of_directed Set.bijOn_iUnion_of_directed
theorem bijOn_iInter_of_directed [Nonempty ι] {s : ι → Set α} (hs : Directed (· ⊆ ·) s)
{t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
bijOn_iInter H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Inter_of_directed Set.bijOn_iInter_of_directed
end Function
/-! ### `image`, `preimage` -/
section Image
theorem image_iUnion {f : α → β} {s : ι → Set α} : (f '' ⋃ i, s i) = ⋃ i, f '' s i := by
ext1 x
simp only [mem_image, mem_iUnion, ← exists_and_right, ← exists_and_left]
--Porting note: `exists_swap` causes a `simp` loop in Lean4 so we use `rw` instead.
rw [exists_swap]
#align set.image_Union Set.image_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iUnion₂ (f : α → β) (s : ∀ i, κ i → Set α) :
(f '' ⋃ (i) (j), s i j) = ⋃ (i) (j), f '' s i j := by simp_rw [image_iUnion]
#align set.image_Union₂ Set.image_iUnion₂
theorem univ_subtype {p : α → Prop} : (univ : Set (Subtype p)) = ⋃ (x) (h : p x), {⟨x, h⟩} :=
Set.ext fun ⟨x, h⟩ => by simp [h]
#align set.univ_subtype Set.univ_subtype
theorem range_eq_iUnion {ι} (f : ι → α) : range f = ⋃ i, {f i} :=
Set.ext fun a => by simp [@eq_comm α a]
#align set.range_eq_Union Set.range_eq_iUnion
theorem image_eq_iUnion (f : α → β) (s : Set α) : f '' s = ⋃ i ∈ s, {f i} :=
Set.ext fun b => by simp [@eq_comm β b]
#align set.image_eq_Union Set.image_eq_iUnion
theorem biUnion_range {f : ι → α} {g : α → Set β} : ⋃ x ∈ range f, g x = ⋃ y, g (f y) :=
iSup_range
#align set.bUnion_range Set.biUnion_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iUnion_iUnion_eq' {f : ι → α} {g : α → Set β} :
⋃ (x) (y) (_ : f y = x), g x = ⋃ y, g (f y) := by simpa using biUnion_range
#align set.Union_Union_eq' Set.iUnion_iUnion_eq'
theorem biInter_range {f : ι → α} {g : α → Set β} : ⋂ x ∈ range f, g x = ⋂ y, g (f y) :=
iInf_range
#align set.bInter_range Set.biInter_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iInter_iInter_eq' {f : ι → α} {g : α → Set β} :
⋂ (x) (y) (_ : f y = x), g x = ⋂ y, g (f y) := by simpa using biInter_range
#align set.Inter_Inter_eq' Set.iInter_iInter_eq'
variable {s : Set γ} {f : γ → α} {g : α → Set β}
theorem biUnion_image : ⋃ x ∈ f '' s, g x = ⋃ y ∈ s, g (f y) :=
iSup_image
#align set.bUnion_image Set.biUnion_image
theorem biInter_image : ⋂ x ∈ f '' s, g x = ⋂ y ∈ s, g (f y) :=
iInf_image
#align set.bInter_image Set.biInter_image
end Image
section Preimage
theorem monotone_preimage {f : α → β} : Monotone (preimage f) := fun _ _ h => preimage_mono h
#align set.monotone_preimage Set.monotone_preimage
@[simp]
theorem preimage_iUnion {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋃ i, s i) = ⋃ i, f ⁻¹' s i :=
Set.ext <| by simp [preimage]
#align set.preimage_Union Set.preimage_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iUnion₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋃ (i) (j), s i j) = ⋃ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iUnion]
#align set.preimage_Union₂ Set.preimage_iUnion₂
@[simp]
theorem preimage_sUnion {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋃₀s = ⋃ t ∈ s, f ⁻¹' t := by
rw [sUnion_eq_biUnion, preimage_iUnion₂]
#align set.preimage_sUnion Set.preimage_sUnion
theorem preimage_iInter {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋂ i, s i) = ⋂ i, f ⁻¹' s i := by
ext; simp
#align set.preimage_Inter Set.preimage_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iInter₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋂ (i) (j), s i j) = ⋂ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iInter]
#align set.preimage_Inter₂ Set.preimage_iInter₂
@[simp]
theorem preimage_sInter {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋂₀ s = ⋂ t ∈ s, f ⁻¹' t := by
rw [sInter_eq_biInter, preimage_iInter₂]
#align set.preimage_sInter Set.preimage_sInter
@[simp]
theorem biUnion_preimage_singleton (f : α → β) (s : Set β) : ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s := by
rw [← preimage_iUnion₂, biUnion_of_singleton]
#align set.bUnion_preimage_singleton Set.biUnion_preimage_singleton
theorem biUnion_range_preimage_singleton (f : α → β) : ⋃ y ∈ range f, f ⁻¹' {y} = univ := by
rw [biUnion_preimage_singleton, preimage_range]
#align set.bUnion_range_preimage_singleton Set.biUnion_range_preimage_singleton
end Preimage
section Prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion {s : Set α} {t : ι → Set β} : (s ×ˢ ⋃ i, t i) = ⋃ i, s ×ˢ t i := by
ext
simp
#align set.prod_Union Set.prod_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} :
(s ×ˢ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ×ˢ t i j := by simp_rw [prod_iUnion]
#align set.prod_Union₂ Set.prod_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sUnion {s : Set α} {C : Set (Set β)} : s ×ˢ ⋃₀C = ⋃₀((fun t => s ×ˢ t) '' C) := by
simp_rw [sUnion_eq_biUnion, biUnion_image, prod_iUnion₂]
#align set.prod_sUnion Set.prod_sUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_const {s : ι → Set α} {t : Set β} : (⋃ i, s i) ×ˢ t = ⋃ i, s i ×ˢ t := by
ext
simp
#align set.Union_prod_const Set.iUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion₂_prod_const {s : ∀ i, κ i → Set α} {t : Set β} :
(⋃ (i) (j), s i j) ×ˢ t = ⋃ (i) (j), s i j ×ˢ t := by simp_rw [iUnion_prod_const]
#align set.Union₂_prod_const Set.iUnion₂_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_prod_const {C : Set (Set α)} {t : Set β} :
⋃₀C ×ˢ t = ⋃₀((fun s : Set α => s ×ˢ t) '' C) := by
simp only [sUnion_eq_biUnion, iUnion₂_prod_const, biUnion_image]
#align set.sUnion_prod_const Set.sUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod {ι ι' α β} (s : ι → Set α) (t : ι' → Set β) :
⋃ x : ι × ι', s x.1 ×ˢ t x.2 = (⋃ i : ι, s i) ×ˢ ⋃ i : ι', t i := by
ext
simp
#align set.Union_prod Set.iUnion_prod
/-- Analogue of `iSup_prod` for sets. -/
lemma iUnion_prod' (f : β × γ → Set α) : ⋃ x : β × γ, f x = ⋃ (i : β) (j : γ), f (i, j) :=
iSup_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x := by
ext ⟨z, w⟩; simp only [mem_prod, mem_iUnion, exists_imp, and_imp, iff_def]; constructor
· intro x hz hw
exact ⟨⟨x, hz⟩, x, hw⟩
· intro x hz x' hw
exact ⟨x ⊔ x', hs le_sup_left hz, ht le_sup_right hw⟩
#align set.Union_prod_of_monotone Set.iUnion_prod_of_monotone
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter_subset (S : Set (Set α)) (T : Set (Set β)) :
⋂₀ S ×ˢ ⋂₀ T ⊆ ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 :=
subset_iInter₂ fun x hx _ hy => ⟨hy.1 x.1 hx.1, hy.2 x.2 hx.2⟩
#align set.sInter_prod_sInter_subset Set.sInter_prod_sInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 := by
obtain ⟨s₁, h₁⟩ := hS
obtain ⟨s₂, h₂⟩ := hT
refine' Set.Subset.antisymm (sInter_prod_sInter_subset S T) fun x hx => _
rw [mem_iInter₂] at hx
exact ⟨fun s₀ h₀ => (hx (s₀, s₂) ⟨h₀, h₂⟩).1, fun s₀ h₀ => (hx (s₁, s₀) ⟨h₁, h₀⟩).2⟩
#align set.sInter_prod_sInter Set.sInter_prod_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod {S : Set (Set α)} (hS : S.Nonempty) (t : Set β) :
⋂₀ S ×ˢ t = ⋂ s ∈ S, s ×ˢ t := by
rw [← sInter_singleton t, sInter_prod_sInter hS (singleton_nonempty t), sInter_singleton]
simp_rw [prod_singleton, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.sInter_prod Set.sInter_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sInter {T : Set (Set β)} (hT : T.Nonempty) (s : Set α) :
s ×ˢ ⋂₀ T = ⋂ t ∈ T, s ×ˢ t := by
rw [← sInter_singleton s, sInter_prod_sInter (singleton_nonempty s) hT, sInter_singleton]
simp_rw [singleton_prod, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.prod_sInter Set.prod_sInter
end Prod
section Image2
variable (f : α → β → γ) {s : Set α} {t : Set β}
/-- The `Set.image2` version of `Set.image_eq_iUnion` -/
theorem image2_eq_iUnion (s : Set α) (t : Set β) : image2 f s t = ⋃ (i ∈ s) (j ∈ t), {f i j} := by
ext; simp [eq_comm]
#align set.image2_eq_Union Set.image2_eq_iUnion
theorem iUnion_image_left : ⋃ a ∈ s, f a '' t = image2 f s t := by
simp only [image2_eq_iUnion, image_eq_iUnion]
#align set.Union_image_left Set.iUnion_image_left
theorem iUnion_image_right : ⋃ b ∈ t, (f · b) '' s = image2 f s t := by
rw [image2_swap, iUnion_image_left]
#align set.Union_image_right Set.iUnion_image_right
theorem image2_iUnion_left (s : ι → Set α) (t : Set β) :
image2 f (⋃ i, s i) t = ⋃ i, image2 f (s i) t := by
simp only [← image_prod, iUnion_prod_const, image_iUnion]
#align set.image2_Union_left Set.image2_iUnion_left
theorem image2_iUnion_right (s : Set α) (t : ι → Set β) :
image2 f s (⋃ i, t i) = ⋃ i, image2 f s (t i) := by
simp only [← image_prod, prod_iUnion, image_iUnion]
#align set.image2_Union_right Set.image2_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋃ (i) (j), s i j) t = ⋃ (i) (j), image2 f (s i j) t := by simp_rw [image2_iUnion_left]
#align set.image2_Union₂_left Set.image2_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋃ (i) (j), t i j) = ⋃ (i) (j), image2 f s (t i j) :=
by simp_rw [image2_iUnion_right]
#align set.image2_Union₂_right Set.image2_iUnion₂_right
theorem image2_iInter_subset_left (s : ι → Set α) (t : Set β) :
image2 f (⋂ i, s i) t ⊆ ⋂ i, image2 f (s i) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem (hx _) hy
#align set.image2_Inter_subset_left Set.image2_iInter_subset_left
theorem image2_iInter_subset_right (s : Set α) (t : ι → Set β) :
image2 f s (⋂ i, t i) ⊆ ⋂ i, image2 f s (t i) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem hx (hy _)
#align set.image2_Inter_subset_right Set.image2_iInter_subset_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋂ (i) (j), s i j) t ⊆ ⋂ (i) (j), image2 f (s i j) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem (hx _ _) hy
#align set.image2_Inter₂_subset_left Set.image2_iInter₂_subset_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), image2 f s (t i j) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem hx (hy _ _)
#align set.image2_Inter₂_subset_right Set.image2_iInter₂_subset_right
theorem prod_eq_biUnion_left : s ×ˢ t = ⋃ a ∈ s, (fun b => (a, b)) '' t := by
rw [iUnion_image_left, image2_mk_eq_prod]
#align set.prod_eq_bUnion_left Set.prod_eq_biUnion_left
theorem prod_eq_biUnion_right : s ×ˢ t = ⋃ b ∈ t, (fun a => (a, b)) '' s := by
rw [iUnion_image_right, image2_mk_eq_prod]
#align set.prod_eq_bUnion_right Set.prod_eq_biUnion_right
end Image2
section Seq
/-- Given a set `s` of functions `α → β` and `t : Set α`, `seq s t` is the union of `f '' t` over
all `f ∈ s`. -/
def seq (s : Set (α → β)) (t : Set α) : Set β :=
{ b | ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b }
#align set.seq Set.seq
@[simp]
theorem mem_seq_iff {s : Set (α → β)} {t : Set α} {b : β} :
b ∈ seq s t ↔ ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b :=
Iff.rfl
#align set.mem_seq_iff Set.mem_seq_iff
lemma seq_eq_image2 (s : Set (α → β)) (t : Set α) : seq s t = image2 (fun f a ↦ f a) s t := by
ext; simp
theorem seq_def {s : Set (α → β)} {t : Set α} : seq s t = ⋃ f ∈ s, f '' t := by
rw [seq_eq_image2, iUnion_image_left]
#align set.seq_def Set.seq_def
theorem seq_subset {s : Set (α → β)} {t : Set α} {u : Set β} :
seq s t ⊆ u ↔ ∀ f ∈ s, ∀ a ∈ t, (f : α → β) a ∈ u := by
rw [seq_eq_image2, image2_subset_iff]
#align set.seq_subset Set.seq_subset
@[gcongr]
theorem seq_mono {s₀ s₁ : Set (α → β)} {t₀ t₁ : Set α} (hs : s₀ ⊆ s₁) (ht : t₀ ⊆ t₁) :
seq s₀ t₀ ⊆ seq s₁ t₁ := fun _ ⟨f, hf, a, ha, eq⟩ => ⟨f, hs hf, a, ht ha, eq⟩
#align set.seq_mono Set.seq_mono
theorem singleton_seq {f : α → β} {t : Set α} : Set.seq ({f} : Set (α → β)) t = f '' t := by
rw [seq_eq_image2, image2_singleton_left]
#align set.singleton_seq Set.singleton_seq
theorem seq_singleton {s : Set (α → β)} {a : α} : Set.seq s {a} = (fun f : α → β => f a) '' s := by
rw [seq_eq_image2, image2_singleton_right]
#align set.seq_singleton Set.seq_singleton
theorem seq_seq {s : Set (β → γ)} {t : Set (α → β)} {u : Set α} :
seq s (seq t u) = seq (seq ((· ∘ ·) '' s) t) u := by
simp only [seq_eq_image2, image2_image_left, image2_image2_left, image2_image2_right, comp_apply]
#align set.seq_seq Set.seq_seq
theorem image_seq {f : β → γ} {s : Set (α → β)} {t : Set α} :
f '' seq s t = seq ((f ∘ ·) '' s) t := by
rw [← singleton_seq, ← singleton_seq, seq_seq, image_singleton]
#align set.image_seq Set.image_seq
theorem prod_eq_seq {s : Set α} {t : Set β} : s ×ˢ t = (Prod.mk '' s).seq t := by
rw [seq_eq_image2, image2_image_left, image2_mk_eq_prod]
#align set.prod_eq_seq Set.prod_eq_seq
theorem prod_image_seq_comm (s : Set α) (t : Set β) :
(Prod.mk '' s).seq t = seq ((fun b a => (a, b)) '' t) s := by
rw [← prod_eq_seq, ← image_swap_prod, prod_eq_seq, image_seq, ← image_comp]; rfl
#align set.prod_image_seq_comm Set.prod_image_seq_comm
theorem image2_eq_seq (f : α → β → γ) (s : Set α) (t : Set β) : image2 f s t = seq (f '' s) t := by
rw [seq_eq_image2, image2_image_left]
#align set.image2_eq_seq Set.image2_eq_seq
end Seq
section Pi
variable {π : α → Type*}
theorem pi_def (i : Set α) (s : ∀ a, Set (π a)) : pi i s = ⋂ a ∈ i, eval a ⁻¹' s a := by
ext
simp
#align set.pi_def Set.pi_def
theorem univ_pi_eq_iInter (t : ∀ i, Set (π i)) : pi univ t = ⋂ i, eval i ⁻¹' t i := by
simp only [pi_def, iInter_true, mem_univ]
#align set.univ_pi_eq_Inter Set.univ_pi_eq_iInter
theorem pi_diff_pi_subset (i : Set α) (s t : ∀ a, Set (π a)) :
pi i s \ pi i t ⊆ ⋃ a ∈ i, eval a ⁻¹' (s a \ t a) := by
refine' diff_subset_comm.2 fun x hx a ha => _
simp only [mem_diff, mem_pi, mem_iUnion, not_exists, mem_preimage, not_and, not_not,
eval_apply] at hx
exact hx.2 _ ha (hx.1 _ ha)
#align set.pi_diff_pi_subset Set.pi_diff_pi_subset
theorem iUnion_univ_pi {ι : α → Type*} (t : (a : α) → ι a → Set (π a)) :
⋃ x : (a : α) → ι a, pi univ (fun a => t a (x a)) = pi univ fun a => ⋃ j : ι a, t a j := by
ext
simp [Classical.skolem]
#align set.Union_univ_pi Set.iUnion_univ_pi
end Pi
end Set
namespace Function
namespace Surjective
theorem iUnion_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋃ x, g (f x) = ⋃ y, g y :=
hf.iSup_comp g
#align function.surjective.Union_comp Function.Surjective.iUnion_comp
theorem iInter_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋂ x, g (f x) = ⋂ y, g y :=
hf.iInf_comp g
#align function.surjective.Inter_comp Function.Surjective.iInter_comp
end Surjective
end Function
/-!
### Disjoint sets
-/
section Disjoint
variable {s t u : Set α} {f : α → β}
namespace Set
@[simp]
theorem disjoint_iUnion_left {ι : Sort*} {s : ι → Set α} :
Disjoint (⋃ i, s i) t ↔ ∀ i, Disjoint (s i) t :=
iSup_disjoint_iff
#align set.disjoint_Union_left Set.disjoint_iUnion_left
@[simp]
theorem disjoint_iUnion_right {ι : Sort*} {s : ι → Set α} :
Disjoint t (⋃ i, s i) ↔ ∀ i, Disjoint t (s i) :=
disjoint_iSup_iff
#align set.disjoint_Union_right Set.disjoint_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_left {s : ∀ i, κ i → Set α} {t : Set α} :
Disjoint (⋃ (i) (j), s i j) t ↔ ∀ i j, Disjoint (s i j) t :=
iSup₂_disjoint_iff
#align set.disjoint_Union₂_left Set.disjoint_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_right {s : Set α} {t : ∀ i, κ i → Set α} :
Disjoint s (⋃ (i) (j), t i j) ↔ ∀ i j, Disjoint s (t i j) :=
disjoint_iSup₂_iff
#align set.disjoint_Union₂_right Set.disjoint_iUnion₂_right
@[simp]
theorem disjoint_sUnion_left {S : Set (Set α)} {t : Set α} :
Disjoint (⋃₀S) t ↔ ∀ s ∈ S, Disjoint s t :=
sSup_disjoint_iff
#align set.disjoint_sUnion_left Set.disjoint_sUnion_left
@[simp]
theorem disjoint_sUnion_right {s : Set α} {S : Set (Set α)} :
Disjoint s (⋃₀S) ↔ ∀ t ∈ S, Disjoint s t :=
disjoint_sSup_iff
#align set.disjoint_sUnion_right Set.disjoint_sUnion_right
end Set
end Disjoint
/-! ### Intervals -/
namespace Set
lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :
(⋂ i, Iic (f i)).Nonempty ↔ BddBelow (range f) := by
have : (⋂ (i : ι), Iic (f i)) = lowerBounds (range f) := by
ext c; simp [lowerBounds]
simp [this, BddBelow]
lemma nonempty_iInter_Ici_iff [Preorder α] {f : ι → α} :
(⋂ i, Ici (f i)).Nonempty ↔ BddAbove (range f) :=
nonempty_iInter_Iic_iff (α := αᵒᵈ)
variable [CompleteLattice α]
theorem Ici_iSup (f : ι → α) : Ici (⨆ i, f i) = ⋂ i, Ici (f i) :=
ext fun _ => by simp only [mem_Ici, iSup_le_iff, mem_iInter]
#align set.Ici_supr Set.Ici_iSup
theorem Iic_iInf (f : ι → α) : Iic (⨅ i, f i) = ⋂ i, Iic (f i) :=
ext fun _ => by simp only [mem_Iic, le_iInf_iff, mem_iInter]
#align set.Iic_infi Set.Iic_iInf
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Ici_iSup₂ (f : ∀ i, κ i → α) : Ici (⨆ (i) (j), f i j) = ⋂ (i) (j), Ici (f i j) := by
simp_rw [Ici_iSup]
#align set.Ici_supr₂ Set.Ici_iSup₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Iic_iInf₂ (f : ∀ i, κ i → α) : Iic (⨅ (i) (j), f i j) = ⋂ (i) (j), Iic (f i j) := by
simp_rw [Iic_iInf]
#align set.Iic_infi₂ Set.Iic_iInf₂
theorem Ici_sSup (s : Set α) : Ici (sSup s) = ⋂ a ∈ s, Ici a := by rw [sSup_eq_iSup, Ici_iSup₂]
#align set.Ici_Sup Set.Ici_sSup
theorem Iic_sInf (s : Set α) : Iic (sInf s) = ⋂ a ∈ s, Iic a := by rw [sInf_eq_iInf, Iic_iInf₂]
#align set.Iic_Inf Set.Iic_sInf
end Set
namespace Set
variable (t : α → Set β)
theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x := by
simp only [diff_subset_iff, ← biUnion_union]
apply biUnion_subset_biUnion_left
rw [union_diff_self]
apply subset_union_right
#align set.bUnion_diff_bUnion_subset Set.biUnion_diff_biUnion_subset
/-- If `t` is an indexed family of sets, then there is a natural map from `Σ i, t i` to `⋃ i, t i`
sending `⟨i, x⟩` to `x`. -/
def sigmaToiUnion (x : Σi, t i) : ⋃ i, t i :=
⟨x.2, mem_iUnion.2 ⟨x.1, x.2.2⟩⟩
#align set.sigma_to_Union Set.sigmaToiUnion
theorem sigmaToiUnion_surjective : Surjective (sigmaToiUnion t)
| ⟨b, hb⟩ =>
have : ∃ a, b ∈ t a := by simpa using hb
let ⟨a, hb⟩ := this
⟨⟨a, b, hb⟩, rfl⟩
#align set.sigma_to_Union_surjective Set.sigmaToiUnion_surjective
theorem sigmaToiUnion_injective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Injective (sigmaToiUnion t)
| ⟨a₁, b₁, h₁⟩, ⟨a₂, b₂, h₂⟩, eq =>
have b_eq : b₁ = b₂ := congr_arg Subtype.val eq
have a_eq : a₁ = a₂ :=
by_contradiction fun ne =>
have : b₁ ∈ t a₁ ∩ t a₂ := ⟨h₁, b_eq.symm ▸ h₂⟩
(h _ _ ne).le_bot this
Sigma.eq a_eq <| Subtype.eq <| by subst b_eq;
|
subst a_eq
|
theorem sigmaToiUnion_injective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Injective (sigmaToiUnion t)
| ⟨a₁, b₁, h₁⟩, ⟨a₂, b₂, h₂⟩, eq =>
have b_eq : b₁ = b₂ := congr_arg Subtype.val eq
have a_eq : a₁ = a₂ :=
by_contradiction fun ne =>
have : b₁ ∈ t a₁ ∩ t a₂ := ⟨h₁, b_eq.symm ▸ h₂⟩
(h _ _ ne).le_bot this
Sigma.eq a_eq <| Subtype.eq <| by subst b_eq;
|
Mathlib.Data.Set.Lattice.2312_0.5mONj49h3SYSDwc
|
theorem sigmaToiUnion_injective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Injective (sigmaToiUnion t)
| ⟨a₁, b₁, h₁⟩, ⟨a₂, b₂, h₂⟩, eq =>
have b_eq : b₁ = b₂
|
Mathlib_Data_Set_Lattice
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
t : α → Set β
h : ∀ (i j : α), i ≠ j → Disjoint (t i) (t j)
a₁ : α
b₁ : β
h₁ h₂ : b₁ ∈ t a₁
eq :
sigmaToiUnion t { fst := a₁, snd := { val := b₁, property := h₁ } } =
sigmaToiUnion t { fst := a₁, snd := { val := b₁, property := h₂ } }
⊢ ↑(Eq.recOn (_ : a₁ = a₁) { fst := a₁, snd := { val := b₁, property := h₁ } }.snd) =
↑{ fst := a₁, snd := { val := b₁, property := h₂ } }.snd
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
/-!
# The set lattice
This file provides usual set notation for unions and intersections, a `CompleteLattice` instance
for `Set α`, and some more set constructions.
## Main declarations
* `Set.iUnion`: **i**ndexed **union**. Union of an indexed family of sets.
* `Set.iInter`: **i**ndexed **inter**section. Intersection of an indexed family of sets.
* `Set.sInter`: **s**et **inter**section. Intersection of sets belonging to a set of sets.
* `Set.sUnion`: **s**et **union**. Union of sets belonging to a set of sets.
* `Set.sInter_eq_biInter`, `Set.sUnion_eq_biInter`: Shows that `⋂₀ s = ⋂ x ∈ s, x` and
`⋃₀ s = ⋃ x ∈ s, x`.
* `Set.completeAtomicBooleanAlgebra`: `Set α` is a `CompleteAtomicBooleanAlgebra` with `≤ = ⊆`,
`< = ⊂`, `⊓ = ∩`, `⊔ = ∪`, `⨅ = ⋂`, `⨆ = ⋃` and `\` as the set difference.
See `Set.BooleanAlgebra`.
* `Set.kernImage`: For a function `f : α → β`, `s.kernImage f` is the set of `y` such that
`f ⁻¹ y ⊆ s`.
* `Set.seq`: Union of the image of a set under a **seq**uence of functions. `seq s t` is the union
of `f '' t` over all `f ∈ s`, where `t : Set α` and `s : Set (α → β)`.
* `Set.iUnion_eq_sigma_of_disjoint`: Equivalence between `⋃ i, t i` and `Σ i, t i`, where `t` is an
indexed family of disjoint sets.
## Naming convention
In lemma names,
* `⋃ i, s i` is called `iUnion`
* `⋂ i, s i` is called `iInter`
* `⋃ i j, s i j` is called `iUnion₂`. This is an `iUnion` inside an `iUnion`.
* `⋂ i j, s i j` is called `iInter₂`. This is an `iInter` inside an `iInter`.
* `⋃ i ∈ s, t i` is called `biUnion` for "bounded `iUnion`". This is the special case of `iUnion₂`
where `j : i ∈ s`.
* `⋂ i ∈ s, t i` is called `biInter` for "bounded `iInter`". This is the special case of `iInter₂`
where `j : i ∈ s`.
## Notation
* `⋃`: `Set.iUnion`
* `⋂`: `Set.iInter`
* `⋃₀`: `Set.sUnion`
* `⋂₀`: `Set.sInter`
-/
set_option autoImplicit true
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι₂ : Sort*} {κ κ₁ κ₂ : ι → Sort*} {κ' : ι' → Sort*}
namespace Set
/-! ### Complete lattice and complete Boolean algebra instances -/
instance : InfSet (Set α) :=
⟨fun s => { a | ∀ t ∈ s, a ∈ t }⟩
instance : SupSet (Set α) :=
⟨fun s => { a | ∃ t ∈ s, a ∈ t }⟩
/-- Intersection of a set of sets. -/
def sInter (S : Set (Set α)) : Set α :=
sInf S
#align set.sInter Set.sInter
/-- Notation for `Set.sInter` Intersection of a set of sets. -/
prefix:110 "⋂₀ " => sInter
/-- Union of a set of sets. -/
def sUnion (S : Set (Set α)) : Set α :=
sSup S
#align set.sUnion Set.sUnion
/-- Notation for `Set.sUnion`. Union of a set of sets. -/
prefix:110 "⋃₀ " => sUnion
@[simp]
theorem mem_sInter {x : α} {S : Set (Set α)} : x ∈ ⋂₀ S ↔ ∀ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sInter Set.mem_sInter
@[simp]
theorem mem_sUnion {x : α} {S : Set (Set α)} : x ∈ ⋃₀ S ↔ ∃ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sUnion Set.mem_sUnion
/-- Indexed union of a family of sets -/
def iUnion (s : ι → Set β) : Set β :=
iSup s
#align set.Union Set.iUnion
/-- Indexed intersection of a family of sets -/
def iInter (s : ι → Set β) : Set β :=
iInf s
#align set.Inter Set.iInter
/-- Notation for `Set.iUnion`. Indexed union of a family of sets -/
notation3 "⋃ "(...)", "r:60:(scoped f => iUnion f) => r
/-- Notation for `Set.iInter`. Indexed intersection of a family of sets -/
notation3 "⋂ "(...)", "r:60:(scoped f => iInter f) => r
section delaborators
open Lean Lean.PrettyPrinter.Delaborator
/-- Delaborator for indexed unions. -/
@[delab app.Set.iUnion]
def iUnion_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋃ (_ : $dom), $body)
else if prop || ppTypes then
`(⋃ ($x:ident : $dom), $body)
else
`(⋃ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋃ $x:ident, ⋃ (_ : $y:ident ∈ $s), $body)
| `(⋃ ($x:ident : $_), ⋃ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋃ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
/-- Delaborator for indexed intersections. -/
@[delab app.Set.iInter]
def sInter_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋂ (_ : $dom), $body)
else if prop || ppTypes then
`(⋂ ($x:ident : $dom), $body)
else
`(⋂ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋂ $x:ident, ⋂ (_ : $y:ident ∈ $s), $body)
| `(⋂ ($x:ident : $_), ⋂ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋂ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
end delaborators
@[simp]
theorem sSup_eq_sUnion (S : Set (Set α)) : sSup S = ⋃₀S :=
rfl
#align set.Sup_eq_sUnion Set.sSup_eq_sUnion
@[simp]
theorem sInf_eq_sInter (S : Set (Set α)) : sInf S = ⋂₀ S :=
rfl
#align set.Inf_eq_sInter Set.sInf_eq_sInter
@[simp]
theorem iSup_eq_iUnion (s : ι → Set α) : iSup s = iUnion s :=
rfl
#align set.supr_eq_Union Set.iSup_eq_iUnion
@[simp]
theorem iInf_eq_iInter (s : ι → Set α) : iInf s = iInter s :=
rfl
#align set.infi_eq_Inter Set.iInf_eq_iInter
@[simp]
theorem mem_iUnion {x : α} {s : ι → Set α} : (x ∈ ⋃ i, s i) ↔ ∃ i, x ∈ s i :=
⟨fun ⟨_, ⟨⟨a, (t_eq : s a = _)⟩, (h : x ∈ _)⟩⟩ => ⟨a, t_eq.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨s a, ⟨⟨a, rfl⟩, h⟩⟩⟩
#align set.mem_Union Set.mem_iUnion
@[simp]
theorem mem_iInter {x : α} {s : ι → Set α} : (x ∈ ⋂ i, s i) ↔ ∀ i, x ∈ s i :=
⟨fun (h : ∀ a ∈ { a : Set α | ∃ i, s i = a }, x ∈ a) a => h (s a) ⟨a, rfl⟩,
fun h _ ⟨a, (eq : s a = _)⟩ => eq ▸ h a⟩
#align set.mem_Inter Set.mem_iInter
theorem mem_iUnion₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋃ (i) (j), s i j) ↔ ∃ i j, x ∈ s i j := by
simp_rw [mem_iUnion]
#align set.mem_Union₂ Set.mem_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋂ (i) (j), s i j) ↔ ∀ i j, x ∈ s i j := by
simp_rw [mem_iInter]
#align set.mem_Inter₂ Set.mem_iInter₂
theorem mem_iUnion_of_mem {s : ι → Set α} {a : α} (i : ι) (ha : a ∈ s i) : a ∈ ⋃ i, s i :=
mem_iUnion.2 ⟨i, ha⟩
#align set.mem_Union_of_mem Set.mem_iUnion_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iUnion₂_of_mem {s : ∀ i, κ i → Set α} {a : α} {i : ι} (j : κ i) (ha : a ∈ s i j) :
a ∈ ⋃ (i) (j), s i j :=
mem_iUnion₂.2 ⟨i, j, ha⟩
#align set.mem_Union₂_of_mem Set.mem_iUnion₂_of_mem
theorem mem_iInter_of_mem {s : ι → Set α} {a : α} (h : ∀ i, a ∈ s i) : a ∈ ⋂ i, s i :=
mem_iInter.2 h
#align set.mem_Inter_of_mem Set.mem_iInter_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂_of_mem {s : ∀ i, κ i → Set α} {a : α} (h : ∀ i j, a ∈ s i j) :
a ∈ ⋂ (i) (j), s i j :=
mem_iInter₂.2 h
#align set.mem_Inter₂_of_mem Set.mem_iInter₂_of_mem
instance Set.completeAtomicBooleanAlgebra : CompleteAtomicBooleanAlgebra (Set α) :=
{ instBooleanAlgebraSet with
le_sSup := fun s t t_in a a_in => ⟨t, t_in, a_in⟩
sSup_le := fun s t h a ⟨t', ⟨t'_in, a_in⟩⟩ => h t' t'_in a_in
le_sInf := fun s t h a a_in t' t'_in => h t' t'_in a_in
sInf_le := fun s t t_in a h => h _ t_in
iInf_iSup_eq := by intros; ext; simp [Classical.skolem] }
/-- `kernImage f s` is the set of `y` such that `f ⁻¹ y ⊆ s`. -/
def kernImage (f : α → β) (s : Set α) : Set β :=
{ y | ∀ ⦃x⦄, f x = y → x ∈ s }
#align set.kern_image Set.kernImage
lemma subset_kernImage_iff {f : α → β} : s ⊆ kernImage f t ↔ f ⁻¹' s ⊆ t :=
⟨fun h _ hx ↦ h hx rfl,
fun h _ hx y hy ↦ h (show f y ∈ s from hy.symm ▸ hx)⟩
section GaloisConnection
variable {f : α → β}
protected theorem image_preimage : GaloisConnection (image f) (preimage f) := fun _ _ =>
image_subset_iff
#align set.image_preimage Set.image_preimage
protected theorem preimage_kernImage : GaloisConnection (preimage f) (kernImage f) := fun _ _ =>
subset_kernImage_iff.symm
#align set.preimage_kern_image Set.preimage_kernImage
end GaloisConnection
section kernImage
variable {f : α → β}
lemma kernImage_mono : Monotone (kernImage f) :=
Set.preimage_kernImage.monotone_u
lemma kernImage_eq_compl {s : Set α} : kernImage f s = (f '' sᶜ)ᶜ :=
Set.preimage_kernImage.u_unique (Set.image_preimage.compl)
(fun t ↦ compl_compl (f ⁻¹' t) ▸ Set.preimage_compl)
lemma kernImage_compl {s : Set α} : kernImage f (sᶜ) = (f '' s)ᶜ := by
rw [kernImage_eq_compl, compl_compl]
lemma kernImage_empty : kernImage f ∅ = (range f)ᶜ := by
rw [kernImage_eq_compl, compl_empty, image_univ]
lemma kernImage_preimage_eq_iff {s : Set β} : kernImage f (f ⁻¹' s) = s ↔ (range f)ᶜ ⊆ s := by
rw [kernImage_eq_compl, ← preimage_compl, compl_eq_comm, eq_comm, image_preimage_eq_iff,
compl_subset_comm]
lemma compl_range_subset_kernImage {s : Set α} : (range f)ᶜ ⊆ kernImage f s := by
rw [← kernImage_empty]
exact kernImage_mono (empty_subset _)
lemma kernImage_union_preimage {s : Set α} {t : Set β} :
kernImage f (s ∪ f ⁻¹' t) = kernImage f s ∪ t := by
rw [kernImage_eq_compl, kernImage_eq_compl, compl_union, ← preimage_compl, image_inter_preimage,
compl_inter, compl_compl]
lemma kernImage_preimage_union {s : Set α} {t : Set β} :
kernImage f (f ⁻¹' t ∪ s) = t ∪ kernImage f s := by
rw [union_comm, kernImage_union_preimage, union_comm]
end kernImage
/-! ### Union and intersection over an indexed family of sets -/
instance : OrderTop (Set α) where
top := univ
le_top := by simp
@[congr]
theorem iUnion_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iUnion f₁ = iUnion f₂ :=
iSup_congr_Prop pq f
#align set.Union_congr_Prop Set.iUnion_congr_Prop
@[congr]
theorem iInter_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iInter f₁ = iInter f₂ :=
iInf_congr_Prop pq f
#align set.Inter_congr_Prop Set.iInter_congr_Prop
theorem iUnion_plift_up (f : PLift ι → Set α) : ⋃ i, f (PLift.up i) = ⋃ i, f i :=
iSup_plift_up _
#align set.Union_plift_up Set.iUnion_plift_up
theorem iUnion_plift_down (f : ι → Set α) : ⋃ i, f (PLift.down i) = ⋃ i, f i :=
iSup_plift_down _
#align set.Union_plift_down Set.iUnion_plift_down
theorem iInter_plift_up (f : PLift ι → Set α) : ⋂ i, f (PLift.up i) = ⋂ i, f i :=
iInf_plift_up _
#align set.Inter_plift_up Set.iInter_plift_up
theorem iInter_plift_down (f : ι → Set α) : ⋂ i, f (PLift.down i) = ⋂ i, f i :=
iInf_plift_down _
#align set.Inter_plift_down Set.iInter_plift_down
theorem iUnion_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋃ _ : p, s = if p then s else ∅ :=
iSup_eq_if _
#align set.Union_eq_if Set.iUnion_eq_if
theorem iUnion_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋃ h : p, s h = if h : p then s h else ∅ :=
iSup_eq_dif _
#align set.Union_eq_dif Set.iUnion_eq_dif
theorem iInter_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋂ _ : p, s = if p then s else univ :=
iInf_eq_if _
#align set.Inter_eq_if Set.iInter_eq_if
theorem iInf_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋂ h : p, s h = if h : p then s h else univ :=
_root_.iInf_eq_dif _
#align set.Infi_eq_dif Set.iInf_eq_dif
theorem exists_set_mem_of_union_eq_top {ι : Type*} (t : Set ι) (s : ι → Set β)
(w : ⋃ i ∈ t, s i = ⊤) (x : β) : ∃ i ∈ t, x ∈ s i := by
have p : x ∈ ⊤ := Set.mem_univ x
rw [← w, Set.mem_iUnion] at p
simpa using p
#align set.exists_set_mem_of_union_eq_top Set.exists_set_mem_of_union_eq_top
theorem nonempty_of_union_eq_top_of_nonempty {ι : Type*} (t : Set ι) (s : ι → Set α)
(H : Nonempty α) (w : ⋃ i ∈ t, s i = ⊤) : t.Nonempty := by
obtain ⟨x, m, -⟩ := exists_set_mem_of_union_eq_top t s w H.some
exact ⟨x, m⟩
#align set.nonempty_of_union_eq_top_of_nonempty Set.nonempty_of_union_eq_top_of_nonempty
theorem nonempty_of_nonempty_iUnion
{s : ι → Set α} (h_Union : (⋃ i, s i).Nonempty) : Nonempty ι := by
obtain ⟨x, hx⟩ := h_Union
exact ⟨Classical.choose $ mem_iUnion.mp hx⟩
theorem nonempty_of_nonempty_iUnion_eq_univ
{s : ι → Set α} [Nonempty α] (h_Union : ⋃ i, s i = univ) : Nonempty ι :=
nonempty_of_nonempty_iUnion (s := s) (by simpa only [h_Union] using univ_nonempty)
theorem setOf_exists (p : ι → β → Prop) : { x | ∃ i, p i x } = ⋃ i, { x | p i x } :=
ext fun _ => mem_iUnion.symm
#align set.set_of_exists Set.setOf_exists
theorem setOf_forall (p : ι → β → Prop) : { x | ∀ i, p i x } = ⋂ i, { x | p i x } :=
ext fun _ => mem_iInter.symm
#align set.set_of_forall Set.setOf_forall
theorem iUnion_subset {s : ι → Set α} {t : Set α} (h : ∀ i, s i ⊆ t) : ⋃ i, s i ⊆ t :=
iSup_le h
#align set.Union_subset Set.iUnion_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset {s : ∀ i, κ i → Set α} {t : Set α} (h : ∀ i j, s i j ⊆ t) :
⋃ (i) (j), s i j ⊆ t :=
iUnion_subset fun x => iUnion_subset (h x)
#align set.Union₂_subset Set.iUnion₂_subset
theorem subset_iInter {t : Set β} {s : ι → Set β} (h : ∀ i, t ⊆ s i) : t ⊆ ⋂ i, s i :=
le_iInf h
#align set.subset_Inter Set.subset_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iInter₂ {s : Set α} {t : ∀ i, κ i → Set α} (h : ∀ i j, s ⊆ t i j) :
s ⊆ ⋂ (i) (j), t i j :=
subset_iInter fun x => subset_iInter <| h x
#align set.subset_Inter₂ Set.subset_iInter₂
@[simp]
theorem iUnion_subset_iff {s : ι → Set α} {t : Set α} : ⋃ i, s i ⊆ t ↔ ∀ i, s i ⊆ t :=
⟨fun h _ => Subset.trans (le_iSup s _) h, iUnion_subset⟩
#align set.Union_subset_iff Set.iUnion_subset_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset_iff {s : ∀ i, κ i → Set α} {t : Set α} :
⋃ (i) (j), s i j ⊆ t ↔ ∀ i j, s i j ⊆ t := by simp_rw [iUnion_subset_iff]
#align set.Union₂_subset_iff Set.iUnion₂_subset_iff
@[simp]
theorem subset_iInter_iff {s : Set α} {t : ι → Set α} : (s ⊆ ⋂ i, t i) ↔ ∀ i, s ⊆ t i :=
le_iInf_iff
#align set.subset_Inter_iff Set.subset_iInter_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem subset_iInter₂_iff {s : Set α} {t : ∀ i, κ i → Set α} :
(s ⊆ ⋂ (i) (j), t i j) ↔ ∀ i j, s ⊆ t i j := by simp_rw [subset_iInter_iff]
#align set.subset_Inter₂_iff Set.subset_iInter₂_iff
theorem subset_iUnion : ∀ (s : ι → Set β) (i : ι), s i ⊆ ⋃ i, s i :=
le_iSup
#align set.subset_Union Set.subset_iUnion
theorem iInter_subset : ∀ (s : ι → Set β) (i : ι), ⋂ i, s i ⊆ s i :=
iInf_le
#align set.Inter_subset Set.iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iUnion₂ {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : s i j ⊆ ⋃ (i') (j'), s i' j' :=
le_iSup₂ i j
#align set.subset_Union₂ Set.subset_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_subset {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : ⋂ (i) (j), s i j ⊆ s i j :=
iInf₂_le i j
#align set.Inter₂_subset Set.iInter₂_subset
/-- This rather trivial consequence of `subset_iUnion`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem subset_iUnion_of_subset {s : Set α} {t : ι → Set α} (i : ι) (h : s ⊆ t i) : s ⊆ ⋃ i, t i :=
le_iSup_of_le i h
#align set.subset_Union_of_subset Set.subset_iUnion_of_subset
/-- This rather trivial consequence of `iInter_subset`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem iInter_subset_of_subset {s : ι → Set α} {t : Set α} (i : ι) (h : s i ⊆ t) :
⋂ i, s i ⊆ t :=
iInf_le_of_le i h
#align set.Inter_subset_of_subset Set.iInter_subset_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `subset_iUnion₂` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem subset_iUnion₂_of_subset {s : Set α} {t : ∀ i, κ i → Set α} (i : ι) (j : κ i)
(h : s ⊆ t i j) : s ⊆ ⋃ (i) (j), t i j :=
le_iSup₂_of_le i j h
#align set.subset_Union₂_of_subset Set.subset_iUnion₂_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `iInter₂_subset` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem iInter₂_subset_of_subset {s : ∀ i, κ i → Set α} {t : Set α} (i : ι) (j : κ i)
(h : s i j ⊆ t) : ⋂ (i) (j), s i j ⊆ t :=
iInf₂_le_of_le i j h
#align set.Inter₂_subset_of_subset Set.iInter₂_subset_of_subset
theorem iUnion_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono h
#align set.Union_mono Set.iUnion_mono
@[gcongr]
theorem iUnion_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iUnion s ⊆ iUnion t :=
iSup_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋃ (i) (j), s i j ⊆ ⋃ (i) (j), t i j :=
iSup₂_mono h
#align set.Union₂_mono Set.iUnion₂_mono
theorem iInter_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋂ i, s i ⊆ ⋂ i, t i :=
iInf_mono h
#align set.Inter_mono Set.iInter_mono
@[gcongr]
theorem iInter_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iInter s ⊆ iInter t :=
iInf_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋂ (i) (j), s i j ⊆ ⋂ (i) (j), t i j :=
iInf₂_mono h
#align set.Inter₂_mono Set.iInter₂_mono
theorem iUnion_mono' {s : ι → Set α} {t : ι₂ → Set α} (h : ∀ i, ∃ j, s i ⊆ t j) :
⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono' h
#align set.Union_mono' Set.iUnion_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iUnion₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i j, ∃ i' j', s i j ⊆ t i' j') : ⋃ (i) (j), s i j ⊆ ⋃ (i') (j'), t i' j' :=
iSup₂_mono' h
#align set.Union₂_mono' Set.iUnion₂_mono'
theorem iInter_mono' {s : ι → Set α} {t : ι' → Set α} (h : ∀ j, ∃ i, s i ⊆ t j) :
⋂ i, s i ⊆ ⋂ j, t j :=
Set.subset_iInter fun j =>
let ⟨i, hi⟩ := h j
iInter_subset_of_subset i hi
#align set.Inter_mono' Set.iInter_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iInter₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i' j', ∃ i j, s i j ⊆ t i' j') : ⋂ (i) (j), s i j ⊆ ⋂ (i') (j'), t i' j' :=
subset_iInter₂_iff.2 fun i' j' =>
let ⟨_, _, hst⟩ := h i' j'
(iInter₂_subset _ _).trans hst
#align set.Inter₂_mono' Set.iInter₂_mono'
theorem iUnion₂_subset_iUnion (κ : ι → Sort*) (s : ι → Set α) :
⋃ (i) (_ : κ i), s i ⊆ ⋃ i, s i :=
iUnion_mono fun _ => iUnion_subset fun _ => Subset.rfl
#align set.Union₂_subset_Union Set.iUnion₂_subset_iUnion
theorem iInter_subset_iInter₂ (κ : ι → Sort*) (s : ι → Set α) :
⋂ i, s i ⊆ ⋂ (i) (_ : κ i), s i :=
iInter_mono fun _ => subset_iInter fun _ => Subset.rfl
#align set.Inter_subset_Inter₂ Set.iInter_subset_iInter₂
theorem iUnion_setOf (P : ι → α → Prop) : ⋃ i, { x : α | P i x } = { x : α | ∃ i, P i x } := by
ext
exact mem_iUnion
#align set.Union_set_of Set.iUnion_setOf
theorem iInter_setOf (P : ι → α → Prop) : ⋂ i, { x : α | P i x } = { x : α | ∀ i, P i x } := by
ext
exact mem_iInter
#align set.Inter_set_of Set.iInter_setOf
theorem iUnion_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋃ x, f x = ⋃ y, g y :=
h1.iSup_congr h h2
#align set.Union_congr_of_surjective Set.iUnion_congr_of_surjective
theorem iInter_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋂ x, f x = ⋂ y, g y :=
h1.iInf_congr h h2
#align set.Inter_congr_of_surjective Set.iInter_congr_of_surjective
lemma iUnion_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋃ i, s i = ⋃ i, t i := iSup_congr h
#align set.Union_congr Set.iUnion_congr
lemma iInter_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋂ i, s i = ⋂ i, t i := iInf_congr h
#align set.Inter_congr Set.iInter_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iUnion₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋃ (i) (j), s i j = ⋃ (i) (j), t i j :=
iUnion_congr fun i => iUnion_congr <| h i
#align set.Union₂_congr Set.iUnion₂_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iInter₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋂ (i) (j), s i j = ⋂ (i) (j), t i j :=
iInter_congr fun i => iInter_congr <| h i
#align set.Inter₂_congr Set.iInter₂_congr
section Nonempty
variable [Nonempty ι] {f : ι → Set α} {s : Set α}
lemma iUnion_const (s : Set β) : ⋃ _ : ι, s = s := iSup_const
#align set.Union_const Set.iUnion_const
lemma iInter_const (s : Set β) : ⋂ _ : ι, s = s := iInf_const
#align set.Inter_const Set.iInter_const
lemma iUnion_eq_const (hf : ∀ i, f i = s) : ⋃ i, f i = s :=
(iUnion_congr hf).trans $ iUnion_const _
#align set.Union_eq_const Set.iUnion_eq_const
lemma iInter_eq_const (hf : ∀ i, f i = s) : ⋂ i, f i = s :=
(iInter_congr hf).trans $ iInter_const _
#align set.Inter_eq_const Set.iInter_eq_const
end Nonempty
@[simp]
theorem compl_iUnion (s : ι → Set β) : (⋃ i, s i)ᶜ = ⋂ i, (s i)ᶜ :=
compl_iSup
#align set.compl_Union Set.compl_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iUnion₂ (s : ∀ i, κ i → Set α) : (⋃ (i) (j), s i j)ᶜ = ⋂ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iUnion]
#align set.compl_Union₂ Set.compl_iUnion₂
@[simp]
theorem compl_iInter (s : ι → Set β) : (⋂ i, s i)ᶜ = ⋃ i, (s i)ᶜ :=
compl_iInf
#align set.compl_Inter Set.compl_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iInter₂ (s : ∀ i, κ i → Set α) : (⋂ (i) (j), s i j)ᶜ = ⋃ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iInter]
#align set.compl_Inter₂ Set.compl_iInter₂
-- classical -- complete_boolean_algebra
theorem iUnion_eq_compl_iInter_compl (s : ι → Set β) : ⋃ i, s i = (⋂ i, (s i)ᶜ)ᶜ := by
simp only [compl_iInter, compl_compl]
#align set.Union_eq_compl_Inter_compl Set.iUnion_eq_compl_iInter_compl
-- classical -- complete_boolean_algebra
theorem iInter_eq_compl_iUnion_compl (s : ι → Set β) : ⋂ i, s i = (⋃ i, (s i)ᶜ)ᶜ := by
simp only [compl_iUnion, compl_compl]
#align set.Inter_eq_compl_Union_compl Set.iInter_eq_compl_iUnion_compl
theorem inter_iUnion (s : Set β) (t : ι → Set β) : (s ∩ ⋃ i, t i) = ⋃ i, s ∩ t i :=
inf_iSup_eq _ _
#align set.inter_Union Set.inter_iUnion
theorem iUnion_inter (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∩ s = ⋃ i, t i ∩ s :=
iSup_inf_eq _ _
#align set.Union_inter Set.iUnion_inter
theorem iUnion_union_distrib (s : ι → Set β) (t : ι → Set β) :
⋃ i, s i ∪ t i = (⋃ i, s i) ∪ ⋃ i, t i :=
iSup_sup_eq
#align set.Union_union_distrib Set.iUnion_union_distrib
theorem iInter_inter_distrib (s : ι → Set β) (t : ι → Set β) :
⋂ i, s i ∩ t i = (⋂ i, s i) ∩ ⋂ i, t i :=
iInf_inf_eq
#align set.Inter_inter_distrib Set.iInter_inter_distrib
theorem union_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∪ ⋃ i, t i) = ⋃ i, s ∪ t i :=
sup_iSup
#align set.union_Union Set.union_iUnion
theorem iUnion_union [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∪ s = ⋃ i, t i ∪ s :=
iSup_sup
#align set.Union_union Set.iUnion_union
theorem inter_iInter [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∩ ⋂ i, t i) = ⋂ i, s ∩ t i :=
inf_iInf
#align set.inter_Inter Set.inter_iInter
theorem iInter_inter [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋂ i, t i) ∩ s = ⋂ i, t i ∩ s :=
iInf_inf
#align set.Inter_inter Set.iInter_inter
-- classical
theorem union_iInter (s : Set β) (t : ι → Set β) : (s ∪ ⋂ i, t i) = ⋂ i, s ∪ t i :=
sup_iInf_eq _ _
#align set.union_Inter Set.union_iInter
theorem iInter_union (s : ι → Set β) (t : Set β) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.Inter_union Set.iInter_union
theorem iUnion_diff (s : Set β) (t : ι → Set β) : (⋃ i, t i) \ s = ⋃ i, t i \ s :=
iUnion_inter _ _
#align set.Union_diff Set.iUnion_diff
theorem diff_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s \ ⋃ i, t i) = ⋂ i, s \ t i := by
rw [diff_eq, compl_iUnion, inter_iInter]; rfl
#align set.diff_Union Set.diff_iUnion
theorem diff_iInter (s : Set β) (t : ι → Set β) : (s \ ⋂ i, t i) = ⋃ i, s \ t i := by
rw [diff_eq, compl_iInter, inter_iUnion]; rfl
#align set.diff_Inter Set.diff_iInter
theorem directed_on_iUnion {r} {f : ι → Set α} (hd : Directed (· ⊆ ·) f)
(h : ∀ x, DirectedOn r (f x)) : DirectedOn r (⋃ x, f x) := by
simp only [DirectedOn, exists_prop, mem_iUnion, exists_imp]
exact fun a₁ b₁ fb₁ a₂ b₂ fb₂ =>
let ⟨z, zb₁, zb₂⟩ := hd b₁ b₂
let ⟨x, xf, xa₁, xa₂⟩ := h z a₁ (zb₁ fb₁) a₂ (zb₂ fb₂)
⟨x, ⟨z, xf⟩, xa₁, xa₂⟩
#align set.directed_on_Union Set.directed_on_iUnion
theorem iUnion_inter_subset {ι α} {s t : ι → Set α} : ⋃ i, s i ∩ t i ⊆ (⋃ i, s i) ∩ ⋃ i, t i :=
le_iSup_inf_iSup s t
#align set.Union_inter_subset Set.iUnion_inter_subset
theorem iUnion_inter_of_monotone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_monotone hs ht
#align set.Union_inter_of_monotone Set.iUnion_inter_of_monotone
theorem iUnion_inter_of_antitone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_antitone hs ht
#align set.Union_inter_of_antitone Set.iUnion_inter_of_antitone
theorem iInter_union_of_monotone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_monotone hs ht
#align set.Inter_union_of_monotone Set.iInter_union_of_monotone
theorem iInter_union_of_antitone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_antitone hs ht
#align set.Inter_union_of_antitone Set.iInter_union_of_antitone
/-- An equality version of this lemma is `iUnion_iInter_of_monotone` in `Data.Set.Finite`. -/
theorem iUnion_iInter_subset {s : ι → ι' → Set α} : (⋃ j, ⋂ i, s i j) ⊆ ⋂ i, ⋃ j, s i j :=
iSup_iInf_le_iInf_iSup (flip s)
#align set.Union_Inter_subset Set.iUnion_iInter_subset
theorem iUnion_option {ι} (s : Option ι → Set α) : ⋃ o, s o = s none ∪ ⋃ i, s (some i) :=
iSup_option s
#align set.Union_option Set.iUnion_option
theorem iInter_option {ι} (s : Option ι → Set α) : ⋂ o, s o = s none ∩ ⋂ i, s (some i) :=
iInf_option s
#align set.Inter_option Set.iInter_option
section
variable (p : ι → Prop) [DecidablePred p]
theorem iUnion_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 _ _ _
#align set.Union_dite Set.iUnion_dite
theorem iUnion_ite (f g : ι → Set α) :
⋃ i, (if p i then f i else g i) = (⋃ (i) (_ : p i), f i) ∪ ⋃ (i) (_ : ¬p i), g i :=
iUnion_dite _ _ _
#align set.Union_ite Set.iUnion_ite
theorem iInter_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 :=
iInf_dite _ _ _
#align set.Inter_dite Set.iInter_dite
theorem iInter_ite (f g : ι → Set α) :
⋂ i, (if p i then f i else g i) = (⋂ (i) (_ : p i), f i) ∩ ⋂ (i) (_ : ¬p i), g i :=
iInter_dite _ _ _
#align set.Inter_ite Set.iInter_ite
end
theorem image_projection_prod {ι : Type*} {α : ι → Type*} {v : ∀ i : ι, Set (α i)}
(hv : (pi univ v).Nonempty) (i : ι) :
((fun x : ∀ i : ι, α i => x i) '' ⋂ k, (fun x : ∀ j : ι, α j => x k) ⁻¹' v k) = v i := by
classical
apply Subset.antisymm
· simp [iInter_subset]
· intro y y_in
simp only [mem_image, mem_iInter, mem_preimage]
rcases hv with ⟨z, hz⟩
refine' ⟨Function.update z i y, _, update_same i y z⟩
rw [@forall_update_iff ι α _ z i y fun i t => t ∈ v i]
exact ⟨y_in, fun j _ => by simpa using hz j⟩
#align set.image_projection_prod Set.image_projection_prod
/-! ### Unions and intersections indexed by `Prop` -/
theorem iInter_false {s : False → Set α} : iInter s = univ :=
iInf_false
#align set.Inter_false Set.iInter_false
theorem iUnion_false {s : False → Set α} : iUnion s = ∅ :=
iSup_false
#align set.Union_false Set.iUnion_false
@[simp]
theorem iInter_true {s : True → Set α} : iInter s = s trivial :=
iInf_true
#align set.Inter_true Set.iInter_true
@[simp]
theorem iUnion_true {s : True → Set α} : iUnion s = s trivial :=
iSup_true
#align set.Union_true Set.iUnion_true
@[simp]
theorem iInter_exists {p : ι → Prop} {f : Exists p → Set α} :
⋂ x, f x = ⋂ (i) (h : p i), f ⟨i, h⟩ :=
iInf_exists
#align set.Inter_exists Set.iInter_exists
@[simp]
theorem iUnion_exists {p : ι → Prop} {f : Exists p → Set α} :
⋃ x, f x = ⋃ (i) (h : p i), f ⟨i, h⟩ :=
iSup_exists
#align set.Union_exists Set.iUnion_exists
@[simp]
theorem iUnion_empty : (⋃ _ : ι, ∅ : Set α) = ∅ :=
iSup_bot
#align set.Union_empty Set.iUnion_empty
@[simp]
theorem iInter_univ : (⋂ _ : ι, univ : Set α) = univ :=
iInf_top
#align set.Inter_univ Set.iInter_univ
section
variable {s : ι → Set α}
@[simp]
theorem iUnion_eq_empty : ⋃ i, s i = ∅ ↔ ∀ i, s i = ∅ :=
iSup_eq_bot
#align set.Union_eq_empty Set.iUnion_eq_empty
@[simp]
theorem iInter_eq_univ : ⋂ i, s i = univ ↔ ∀ i, s i = univ :=
iInf_eq_top
#align set.Inter_eq_univ Set.iInter_eq_univ
@[simp]
theorem nonempty_iUnion : (⋃ i, s i).Nonempty ↔ ∃ i, (s i).Nonempty := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_Union Set.nonempty_iUnion
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_biUnion {t : Set α} {s : α → Set β} :
(⋃ i ∈ t, s i).Nonempty ↔ ∃ i ∈ t, (s i).Nonempty := by simp
#align set.nonempty_bUnion Set.nonempty_biUnion
theorem iUnion_nonempty_index (s : Set α) (t : s.Nonempty → Set β) :
⋃ h, t h = ⋃ x ∈ s, t ⟨x, ‹_›⟩ :=
iSup_exists
#align set.Union_nonempty_index Set.iUnion_nonempty_index
end
@[simp]
theorem iInter_iInter_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋂ (x) (h : x = b), s x h = s b rfl :=
iInf_iInf_eq_left
#align set.Inter_Inter_eq_left Set.iInter_iInter_eq_left
@[simp]
theorem iInter_iInter_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋂ (x) (h : b = x), s x h = s b rfl :=
iInf_iInf_eq_right
#align set.Inter_Inter_eq_right Set.iInter_iInter_eq_right
@[simp]
theorem iUnion_iUnion_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋃ (x) (h : x = b), s x h = s b rfl :=
iSup_iSup_eq_left
#align set.Union_Union_eq_left Set.iUnion_iUnion_eq_left
@[simp]
theorem iUnion_iUnion_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋃ (x) (h : b = x), s x h = s b rfl :=
iSup_iSup_eq_right
#align set.Union_Union_eq_right Set.iUnion_iUnion_eq_right
theorem iInter_or {p q : Prop} (s : p ∨ q → Set α) :
⋂ h, s h = (⋂ h : p, s (Or.inl h)) ∩ ⋂ h : q, s (Or.inr h) :=
iInf_or
#align set.Inter_or Set.iInter_or
theorem iUnion_or {p q : Prop} (s : p ∨ q → Set α) :
⋃ h, s h = (⋃ i, s (Or.inl i)) ∪ ⋃ j, s (Or.inr j) :=
iSup_or
#align set.Union_or Set.iUnion_or
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iUnion_and {p q : Prop} (s : p ∧ q → Set α) : ⋃ h, s h = ⋃ (hp) (hq), s ⟨hp, hq⟩ :=
iSup_and
#align set.Union_and Set.iUnion_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iInter_and {p q : Prop} (s : p ∧ q → Set α) : ⋂ h, s h = ⋂ (hp) (hq), s ⟨hp, hq⟩ :=
iInf_and
#align set.Inter_and Set.iInter_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iUnion_comm (s : ι → ι' → Set α) : ⋃ (i) (i'), s i i' = ⋃ (i') (i), s i i' :=
iSup_comm
#align set.Union_comm Set.iUnion_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iInter_comm (s : ι → ι' → Set α) : ⋂ (i) (i'), s i i' = ⋂ (i') (i), s i i' :=
iInf_comm
#align set.Inter_comm Set.iInter_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iUnion₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋃ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋃ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iSup₂_comm _
#align set.Union₂_comm Set.iUnion₂_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iInter₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋂ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋂ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iInf₂_comm _
#align set.Inter₂_comm Set.iInter₂_comm
@[simp]
theorem biUnion_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋃ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι']
#align set.bUnion_and Set.biUnion_and
@[simp]
theorem biUnion_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋃ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι]
#align set.bUnion_and' Set.biUnion_and'
@[simp]
theorem biInter_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋂ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iInter_and, @iInter_comm _ ι']
#align set.bInter_and Set.biInter_and
@[simp]
theorem biInter_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋂ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iInter_and, @iInter_comm _ ι]
#align set.bInter_and' Set.biInter_and'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iUnion_iUnion_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋃ (x) (h), s x h = s b (Or.inl rfl) ∪ ⋃ (x) (h : p x), s x (Or.inr h) := by
simp only [iUnion_or, iUnion_union_distrib, iUnion_iUnion_eq_left]
#align set.Union_Union_eq_or_left Set.iUnion_iUnion_eq_or_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iInter_iInter_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋂ (x) (h), s x h = s b (Or.inl rfl) ∩ ⋂ (x) (h : p x), s x (Or.inr h) := by
simp only [iInter_or, iInter_inter_distrib, iInter_iInter_eq_left]
#align set.Inter_Inter_eq_or_left Set.iInter_iInter_eq_or_left
/-! ### Bounded unions and intersections -/
/-- A specialization of `mem_iUnion₂`. -/
theorem mem_biUnion {s : Set α} {t : α → Set β} {x : α} {y : β} (xs : x ∈ s) (ytx : y ∈ t x) :
y ∈ ⋃ x ∈ s, t x :=
mem_iUnion₂_of_mem xs ytx
#align set.mem_bUnion Set.mem_biUnion
/-- A specialization of `mem_iInter₂`. -/
theorem mem_biInter {s : Set α} {t : α → Set β} {y : β} (h : ∀ x ∈ s, y ∈ t x) :
y ∈ ⋂ x ∈ s, t x :=
mem_iInter₂_of_mem h
#align set.mem_bInter Set.mem_biInter
/-- A specialization of `subset_iUnion₂`. -/
theorem subset_biUnion_of_mem {s : Set α} {u : α → Set β} {x : α} (xs : x ∈ s) :
u x ⊆ ⋃ x ∈ s, u x :=
--Porting note: Why is this not just `subset_iUnion₂ x xs`?
@subset_iUnion₂ β α (· ∈ s) (fun i _ => u i) x xs
#align set.subset_bUnion_of_mem Set.subset_biUnion_of_mem
/-- A specialization of `iInter₂_subset`. -/
theorem biInter_subset_of_mem {s : Set α} {t : α → Set β} {x : α} (xs : x ∈ s) :
⋂ x ∈ s, t x ⊆ t x :=
iInter₂_subset x xs
#align set.bInter_subset_of_mem Set.biInter_subset_of_mem
theorem biUnion_subset_biUnion_left {s s' : Set α} {t : α → Set β} (h : s ⊆ s') :
⋃ x ∈ s, t x ⊆ ⋃ x ∈ s', t x :=
iUnion₂_subset fun _ hx => subset_biUnion_of_mem <| h hx
#align set.bUnion_subset_bUnion_left Set.biUnion_subset_biUnion_left
theorem biInter_subset_biInter_left {s s' : Set α} {t : α → Set β} (h : s' ⊆ s) :
⋂ x ∈ s, t x ⊆ ⋂ x ∈ s', t x :=
subset_iInter₂ fun _ hx => biInter_subset_of_mem <| h hx
#align set.bInter_subset_bInter_left Set.biInter_subset_biInter_left
theorem biUnion_mono {s s' : Set α} {t t' : α → Set β} (hs : s' ⊆ s) (h : ∀ x ∈ s, t x ⊆ t' x) :
⋃ x ∈ s', t x ⊆ ⋃ x ∈ s, t' x :=
(biUnion_subset_biUnion_left hs).trans <| iUnion₂_mono h
#align set.bUnion_mono Set.biUnion_mono
theorem biInter_mono {s s' : Set α} {t t' : α → Set β} (hs : s ⊆ s') (h : ∀ x ∈ s, t x ⊆ t' x) :
⋂ x ∈ s', t x ⊆ ⋂ x ∈ s, t' x :=
(biInter_subset_biInter_left hs).trans <| iInter₂_mono h
#align set.bInter_mono Set.biInter_mono
theorem biUnion_eq_iUnion (s : Set α) (t : ∀ x ∈ s, Set β) :
⋃ x ∈ s, t x ‹_› = ⋃ x : s, t x x.2 :=
iSup_subtype'
#align set.bUnion_eq_Union Set.biUnion_eq_iUnion
theorem biInter_eq_iInter (s : Set α) (t : ∀ x ∈ s, Set β) :
⋂ x ∈ s, t x ‹_› = ⋂ x : s, t x x.2 :=
iInf_subtype'
#align set.bInter_eq_Inter Set.biInter_eq_iInter
theorem iUnion_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋃ x : { x // p x }, s x = ⋃ (x) (hx : p x), s ⟨x, hx⟩ :=
iSup_subtype
#align set.Union_subtype Set.iUnion_subtype
theorem iInter_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋂ x : { x // p x }, s x = ⋂ (x) (hx : p x), s ⟨x, hx⟩ :=
iInf_subtype
#align set.Inter_subtype Set.iInter_subtype
theorem biInter_empty (u : α → Set β) : ⋂ x ∈ (∅ : Set α), u x = univ :=
iInf_emptyset
#align set.bInter_empty Set.biInter_empty
theorem biInter_univ (u : α → Set β) : ⋂ x ∈ @univ α, u x = ⋂ x, u x :=
iInf_univ
#align set.bInter_univ Set.biInter_univ
@[simp]
theorem biUnion_self (s : Set α) : ⋃ x ∈ s, s = s :=
Subset.antisymm (iUnion₂_subset fun _ _ => Subset.refl s) fun _ hx => mem_biUnion hx hx
#align set.bUnion_self Set.biUnion_self
@[simp]
theorem iUnion_nonempty_self (s : Set α) : ⋃ _ : s.Nonempty, s = s := by
rw [iUnion_nonempty_index, biUnion_self]
#align set.Union_nonempty_self Set.iUnion_nonempty_self
theorem biInter_singleton (a : α) (s : α → Set β) : ⋂ x ∈ ({a} : Set α), s x = s a :=
iInf_singleton
#align set.bInter_singleton Set.biInter_singleton
theorem biInter_union (s t : Set α) (u : α → Set β) :
⋂ x ∈ s ∪ t, u x = (⋂ x ∈ s, u x) ∩ ⋂ x ∈ t, u x :=
iInf_union
#align set.bInter_union Set.biInter_union
theorem biInter_insert (a : α) (s : Set α) (t : α → Set β) :
⋂ x ∈ insert a s, t x = t a ∩ ⋂ x ∈ s, t x := by simp
#align set.bInter_insert Set.biInter_insert
theorem biInter_pair (a b : α) (s : α → Set β) : ⋂ x ∈ ({a, b} : Set α), s x = s a ∩ s b := by
rw [biInter_insert, biInter_singleton]
#align set.bInter_pair Set.biInter_pair
theorem biInter_inter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, f i ∩ t = (⋂ i ∈ s, f i) ∩ t := by
haveI : Nonempty s := hs.to_subtype
simp [biInter_eq_iInter, ← iInter_inter]
#align set.bInter_inter Set.biInter_inter
theorem inter_biInter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, t ∩ f i = t ∩ ⋂ i ∈ s, f i := by
rw [inter_comm, ← biInter_inter hs]
simp [inter_comm]
#align set.inter_bInter Set.inter_biInter
theorem biUnion_empty (s : α → Set β) : ⋃ x ∈ (∅ : Set α), s x = ∅ :=
iSup_emptyset
#align set.bUnion_empty Set.biUnion_empty
theorem biUnion_univ (s : α → Set β) : ⋃ x ∈ @univ α, s x = ⋃ x, s x :=
iSup_univ
#align set.bUnion_univ Set.biUnion_univ
theorem biUnion_singleton (a : α) (s : α → Set β) : ⋃ x ∈ ({a} : Set α), s x = s a :=
iSup_singleton
#align set.bUnion_singleton Set.biUnion_singleton
@[simp]
theorem biUnion_of_singleton (s : Set α) : ⋃ x ∈ s, {x} = s :=
ext <| by simp
#align set.bUnion_of_singleton Set.biUnion_of_singleton
theorem biUnion_union (s t : Set α) (u : α → Set β) :
⋃ x ∈ s ∪ t, u x = (⋃ x ∈ s, u x) ∪ ⋃ x ∈ t, u x :=
iSup_union
#align set.bUnion_union Set.biUnion_union
@[simp]
theorem iUnion_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋃ i, f i = ⋃ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iUnion_subtype _ _
#align set.Union_coe_set Set.iUnion_coe_set
@[simp]
theorem iInter_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋂ i, f i = ⋂ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iInter_subtype _ _
#align set.Inter_coe_set Set.iInter_coe_set
theorem biUnion_insert (a : α) (s : Set α) (t : α → Set β) :
⋃ x ∈ insert a s, t x = t a ∪ ⋃ x ∈ s, t x := by simp
#align set.bUnion_insert Set.biUnion_insert
theorem biUnion_pair (a b : α) (s : α → Set β) : ⋃ x ∈ ({a, b} : Set α), s x = s a ∪ s b :=
by simp
#align set.bUnion_pair Set.biUnion_pair
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem inter_iUnion₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∩ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ∩ t i j := by simp only [inter_iUnion]
#align set.inter_Union₂ Set.inter_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_inter (s : ∀ i, κ i → Set α) (t : Set α) :
(⋃ (i) (j), s i j) ∩ t = ⋃ (i) (j), s i j ∩ t := by simp_rw [iUnion_inter]
#align set.Union₂_inter Set.iUnion₂_inter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_iInter₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_iInter]
#align set.union_Inter₂ Set.union_iInter₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_union (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [iInter_union]
#align set.Inter₂_union Set.iInter₂_union
theorem mem_sUnion_of_mem {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∈ t) (ht : t ∈ S) :
x ∈ ⋃₀S :=
⟨t, ht, hx⟩
#align set.mem_sUnion_of_mem Set.mem_sUnion_of_mem
-- is this theorem really necessary?
theorem not_mem_of_not_mem_sUnion {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∉ ⋃₀S)
(ht : t ∈ S) : x ∉ t := fun h => hx ⟨t, ht, h⟩
#align set.not_mem_of_not_mem_sUnion Set.not_mem_of_not_mem_sUnion
theorem sInter_subset_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : ⋂₀ S ⊆ t :=
sInf_le tS
#align set.sInter_subset_of_mem Set.sInter_subset_of_mem
theorem subset_sUnion_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : t ⊆ ⋃₀S :=
le_sSup tS
#align set.subset_sUnion_of_mem Set.subset_sUnion_of_mem
theorem subset_sUnion_of_subset {s : Set α} (t : Set (Set α)) (u : Set α) (h₁ : s ⊆ u)
(h₂ : u ∈ t) : s ⊆ ⋃₀t :=
Subset.trans h₁ (subset_sUnion_of_mem h₂)
#align set.subset_sUnion_of_subset Set.subset_sUnion_of_subset
theorem sUnion_subset {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t' ⊆ t) : ⋃₀S ⊆ t :=
sSup_le h
#align set.sUnion_subset Set.sUnion_subset
@[simp]
theorem sUnion_subset_iff {s : Set (Set α)} {t : Set α} : ⋃₀s ⊆ t ↔ ∀ t' ∈ s, t' ⊆ t :=
sSup_le_iff
#align set.sUnion_subset_iff Set.sUnion_subset_iff
/-- `sUnion` is monotone under taking a subset of each set. -/
lemma sUnion_mono_subsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, t ⊆ f t) :
⋃₀ s ⊆ ⋃₀ (f '' s) :=
fun _ ⟨t, htx, hxt⟩ ↦ ⟨f t, mem_image_of_mem f htx, hf t hxt⟩
/-- `sUnion` is monotone under taking a superset of each set. -/
lemma sUnion_mono_supsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, f t ⊆ t) :
⋃₀ (f '' s) ⊆ ⋃₀ s :=
-- If t ∈ f '' s is arbitrary; t = f u for some u : Set α.
fun _ ⟨_, ⟨u, hus, hut⟩, hxt⟩ ↦ ⟨u, hus, (hut ▸ hf u) hxt⟩
theorem subset_sInter {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t ⊆ t') : t ⊆ ⋂₀ S :=
le_sInf h
#align set.subset_sInter Set.subset_sInter
@[simp]
theorem subset_sInter_iff {S : Set (Set α)} {t : Set α} : t ⊆ ⋂₀ S ↔ ∀ t' ∈ S, t ⊆ t' :=
le_sInf_iff
#align set.subset_sInter_iff Set.subset_sInter_iff
@[gcongr]
theorem sUnion_subset_sUnion {S T : Set (Set α)} (h : S ⊆ T) : ⋃₀S ⊆ ⋃₀T :=
sUnion_subset fun _ hs => subset_sUnion_of_mem (h hs)
#align set.sUnion_subset_sUnion Set.sUnion_subset_sUnion
@[gcongr]
theorem sInter_subset_sInter {S T : Set (Set α)} (h : S ⊆ T) : ⋂₀ T ⊆ ⋂₀ S :=
subset_sInter fun _ hs => sInter_subset_of_mem (h hs)
#align set.sInter_subset_sInter Set.sInter_subset_sInter
@[simp]
theorem sUnion_empty : ⋃₀∅ = (∅ : Set α) :=
sSup_empty
#align set.sUnion_empty Set.sUnion_empty
@[simp]
theorem sInter_empty : ⋂₀ ∅ = (univ : Set α) :=
sInf_empty
#align set.sInter_empty Set.sInter_empty
@[simp]
theorem sUnion_singleton (s : Set α) : ⋃₀{s} = s :=
sSup_singleton
#align set.sUnion_singleton Set.sUnion_singleton
@[simp]
theorem sInter_singleton (s : Set α) : ⋂₀ {s} = s :=
sInf_singleton
#align set.sInter_singleton Set.sInter_singleton
@[simp]
theorem sUnion_eq_empty {S : Set (Set α)} : ⋃₀S = ∅ ↔ ∀ s ∈ S, s = ∅ :=
sSup_eq_bot
#align set.sUnion_eq_empty Set.sUnion_eq_empty
@[simp]
theorem sInter_eq_univ {S : Set (Set α)} : ⋂₀ S = univ ↔ ∀ s ∈ S, s = univ :=
sInf_eq_top
#align set.sInter_eq_univ Set.sInter_eq_univ
theorem subset_powerset_iff {s : Set (Set α)} {t : Set α} : s ⊆ 𝒫 t ↔ ⋃₀ s ⊆ t :=
sUnion_subset_iff.symm
/-- `⋃₀` and `𝒫` form a Galois connection. -/
theorem sUnion_powerset_gc :
GaloisConnection (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gc_sSup_Iic
/-- `⋃₀` and `𝒫` form a Galois insertion. -/
def sUnion_powerset_gi :
GaloisInsertion (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gi_sSup_Iic
/-- If all sets in a collection are either `∅` or `Set.univ`, then so is their union. -/
theorem sUnion_mem_empty_univ {S : Set (Set α)} (h : S ⊆ {∅, univ}) :
⋃₀ S ∈ ({∅, univ} : Set (Set α)) := by
simp only [mem_insert_iff, mem_singleton_iff, or_iff_not_imp_left, sUnion_eq_empty, not_forall]
rintro ⟨s, hs, hne⟩
obtain rfl : s = univ := (h hs).resolve_left hne
exact univ_subset_iff.1 <| subset_sUnion_of_mem hs
@[simp]
theorem nonempty_sUnion {S : Set (Set α)} : (⋃₀S).Nonempty ↔ ∃ s ∈ S, Set.Nonempty s := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_sUnion Set.nonempty_sUnion
theorem Nonempty.of_sUnion {s : Set (Set α)} (h : (⋃₀s).Nonempty) : s.Nonempty :=
let ⟨s, hs, _⟩ := nonempty_sUnion.1 h
⟨s, hs⟩
#align set.nonempty.of_sUnion Set.Nonempty.of_sUnion
theorem Nonempty.of_sUnion_eq_univ [Nonempty α] {s : Set (Set α)} (h : ⋃₀s = univ) : s.Nonempty :=
Nonempty.of_sUnion <| h.symm ▸ univ_nonempty
#align set.nonempty.of_sUnion_eq_univ Set.Nonempty.of_sUnion_eq_univ
theorem sUnion_union (S T : Set (Set α)) : ⋃₀(S ∪ T) = ⋃₀S ∪ ⋃₀T :=
sSup_union
#align set.sUnion_union Set.sUnion_union
theorem sInter_union (S T : Set (Set α)) : ⋂₀ (S ∪ T) = ⋂₀ S ∩ ⋂₀ T :=
sInf_union
#align set.sInter_union Set.sInter_union
@[simp]
theorem sUnion_insert (s : Set α) (T : Set (Set α)) : ⋃₀insert s T = s ∪ ⋃₀T :=
sSup_insert
#align set.sUnion_insert Set.sUnion_insert
@[simp]
theorem sInter_insert (s : Set α) (T : Set (Set α)) : ⋂₀ insert s T = s ∩ ⋂₀ T :=
sInf_insert
#align set.sInter_insert Set.sInter_insert
@[simp]
theorem sUnion_diff_singleton_empty (s : Set (Set α)) : ⋃₀(s \ {∅}) = ⋃₀s :=
sSup_diff_singleton_bot s
#align set.sUnion_diff_singleton_empty Set.sUnion_diff_singleton_empty
@[simp]
theorem sInter_diff_singleton_univ (s : Set (Set α)) : ⋂₀ (s \ {univ}) = ⋂₀ s :=
sInf_diff_singleton_top s
#align set.sInter_diff_singleton_univ Set.sInter_diff_singleton_univ
theorem sUnion_pair (s t : Set α) : ⋃₀{s, t} = s ∪ t :=
sSup_pair
#align set.sUnion_pair Set.sUnion_pair
theorem sInter_pair (s t : Set α) : ⋂₀ {s, t} = s ∩ t :=
sInf_pair
#align set.sInter_pair Set.sInter_pair
@[simp]
theorem sUnion_image (f : α → Set β) (s : Set α) : ⋃₀(f '' s) = ⋃ x ∈ s, f x :=
sSup_image
#align set.sUnion_image Set.sUnion_image
@[simp]
theorem sInter_image (f : α → Set β) (s : Set α) : ⋂₀ (f '' s) = ⋂ x ∈ s, f x :=
sInf_image
#align set.sInter_image Set.sInter_image
@[simp]
theorem sUnion_range (f : ι → Set β) : ⋃₀range f = ⋃ x, f x :=
rfl
#align set.sUnion_range Set.sUnion_range
@[simp]
theorem sInter_range (f : ι → Set β) : ⋂₀ range f = ⋂ x, f x :=
rfl
#align set.sInter_range Set.sInter_range
theorem iUnion_eq_univ_iff {f : ι → Set α} : ⋃ i, f i = univ ↔ ∀ x, ∃ i, x ∈ f i := by
simp only [eq_univ_iff_forall, mem_iUnion]
#align set.Union_eq_univ_iff Set.iUnion_eq_univ_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_eq_univ_iff {s : ∀ i, κ i → Set α} :
⋃ (i) (j), s i j = univ ↔ ∀ a, ∃ i j, a ∈ s i j :=
by simp only [iUnion_eq_univ_iff, mem_iUnion]
#align set.Union₂_eq_univ_iff Set.iUnion₂_eq_univ_iff
theorem sUnion_eq_univ_iff {c : Set (Set α)} : ⋃₀c = univ ↔ ∀ a, ∃ b ∈ c, a ∈ b := by
simp only [eq_univ_iff_forall, mem_sUnion]
#align set.sUnion_eq_univ_iff Set.sUnion_eq_univ_iff
-- classical
theorem iInter_eq_empty_iff {f : ι → Set α} : ⋂ i, f i = ∅ ↔ ∀ x, ∃ i, x ∉ f i := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.Inter_eq_empty_iff Set.iInter_eq_empty_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
theorem iInter₂_eq_empty_iff {s : ∀ i, κ i → Set α} :
⋂ (i) (j), s i j = ∅ ↔ ∀ a, ∃ i j, a ∉ s i j := by
simp only [eq_empty_iff_forall_not_mem, mem_iInter, not_forall]
#align set.Inter₂_eq_empty_iff Set.iInter₂_eq_empty_iff
-- classical
theorem sInter_eq_empty_iff {c : Set (Set α)} : ⋂₀ c = ∅ ↔ ∀ a, ∃ b ∈ c, a ∉ b := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.sInter_eq_empty_iff Set.sInter_eq_empty_iff
-- classical
@[simp]
theorem nonempty_iInter {f : ι → Set α} : (⋂ i, f i).Nonempty ↔ ∃ x, ∀ i, x ∈ f i := by
simp [nonempty_iff_ne_empty, iInter_eq_empty_iff]
#align set.nonempty_Inter Set.nonempty_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_iInter₂ {s : ∀ i, κ i → Set α} :
(⋂ (i) (j), s i j).Nonempty ↔ ∃ a, ∀ i j, a ∈ s i j := by
simp
#align set.nonempty_Inter₂ Set.nonempty_iInter₂
-- classical
@[simp]
theorem nonempty_sInter {c : Set (Set α)} : (⋂₀ c).Nonempty ↔ ∃ a, ∀ b ∈ c, a ∈ b := by
simp [nonempty_iff_ne_empty, sInter_eq_empty_iff]
#align set.nonempty_sInter Set.nonempty_sInter
-- classical
theorem compl_sUnion (S : Set (Set α)) : (⋃₀S)ᶜ = ⋂₀ (compl '' S) :=
ext fun x => by simp
#align set.compl_sUnion Set.compl_sUnion
-- classical
theorem sUnion_eq_compl_sInter_compl (S : Set (Set α)) : ⋃₀S = (⋂₀ (compl '' S))ᶜ := by
rw [← compl_compl (⋃₀S), compl_sUnion]
#align set.sUnion_eq_compl_sInter_compl Set.sUnion_eq_compl_sInter_compl
-- classical
theorem compl_sInter (S : Set (Set α)) : (⋂₀ S)ᶜ = ⋃₀(compl '' S) := by
rw [sUnion_eq_compl_sInter_compl, compl_compl_image]
#align set.compl_sInter Set.compl_sInter
-- classical
theorem sInter_eq_compl_sUnion_compl (S : Set (Set α)) : ⋂₀ S = (⋃₀(compl '' S))ᶜ := by
rw [← compl_compl (⋂₀ S), compl_sInter]
#align set.sInter_eq_compl_sUnion_compl Set.sInter_eq_compl_sUnion_compl
theorem inter_empty_of_inter_sUnion_empty {s t : Set α} {S : Set (Set α)} (hs : t ∈ S)
(h : s ∩ ⋃₀S = ∅) : s ∩ t = ∅ :=
eq_empty_of_subset_empty <| by
rw [← h]; exact inter_subset_inter_right _ (subset_sUnion_of_mem hs)
#align set.inter_empty_of_inter_sUnion_empty Set.inter_empty_of_inter_sUnion_empty
theorem range_sigma_eq_iUnion_range {γ : α → Type*} (f : Sigma γ → β) :
range f = ⋃ a, range fun b => f ⟨a, b⟩ :=
Set.ext <| by simp
#align set.range_sigma_eq_Union_range Set.range_sigma_eq_iUnion_range
theorem iUnion_eq_range_sigma (s : α → Set β) : ⋃ i, s i = range fun a : Σi, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_sigma Set.iUnion_eq_range_sigma
theorem iUnion_eq_range_psigma (s : ι → Set β) : ⋃ i, s i = range fun a : Σ'i, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_psigma Set.iUnion_eq_range_psigma
theorem iUnion_image_preimage_sigma_mk_eq_self {ι : Type*} {σ : ι → Type*} (s : Set (Sigma σ)) :
⋃ i, Sigma.mk i '' (Sigma.mk i ⁻¹' s) = s := by
ext x
simp only [mem_iUnion, mem_image, mem_preimage]
constructor
· rintro ⟨i, a, h, rfl⟩
exact h
· intro h
cases' x with i a
exact ⟨i, a, h, rfl⟩
#align set.Union_image_preimage_sigma_mk_eq_self Set.iUnion_image_preimage_sigma_mk_eq_self
theorem Sigma.univ (X : α → Type*) : (Set.univ : Set (Σa, X a)) = ⋃ a, range (Sigma.mk a) :=
Set.ext fun x =>
iff_of_true trivial ⟨range (Sigma.mk x.1), Set.mem_range_self _, x.2, Sigma.eta x⟩
#align set.sigma.univ Set.Sigma.univ
alias sUnion_mono := sUnion_subset_sUnion
#align set.sUnion_mono Set.sUnion_mono
theorem iUnion_subset_iUnion_const {s : Set α} (h : ι → ι₂) : ⋃ _ : ι, s ⊆ ⋃ _ : ι₂, s :=
@iSup_const_mono (Set α) ι ι₂ _ s h
#align set.Union_subset_Union_const Set.iUnion_subset_iUnion_const
@[simp]
theorem iUnion_singleton_eq_range {α β : Type*} (f : α → β) : ⋃ x : α, {f x} = range f := by
ext x
simp [@eq_comm _ x]
#align set.Union_singleton_eq_range Set.iUnion_singleton_eq_range
theorem iUnion_of_singleton (α : Type*) : (⋃ x, {x} : Set α) = univ := by simp [Set.ext_iff]
#align set.Union_of_singleton Set.iUnion_of_singleton
theorem iUnion_of_singleton_coe (s : Set α) : ⋃ i : s, ({(i : α)} : Set α) = s := by simp
#align set.Union_of_singleton_coe Set.iUnion_of_singleton_coe
theorem sUnion_eq_biUnion {s : Set (Set α)} : ⋃₀s = ⋃ (i : Set α) (_ : i ∈ s), i := by
rw [← sUnion_image, image_id']
#align set.sUnion_eq_bUnion Set.sUnion_eq_biUnion
theorem sInter_eq_biInter {s : Set (Set α)} : ⋂₀ s = ⋂ (i : Set α) (_ : i ∈ s), i := by
rw [← sInter_image, image_id']
#align set.sInter_eq_bInter Set.sInter_eq_biInter
theorem sUnion_eq_iUnion {s : Set (Set α)} : ⋃₀s = ⋃ i : s, i := by
simp only [← sUnion_range, Subtype.range_coe]
#align set.sUnion_eq_Union Set.sUnion_eq_iUnion
theorem sInter_eq_iInter {s : Set (Set α)} : ⋂₀ s = ⋂ i : s, i := by
simp only [← sInter_range, Subtype.range_coe]
#align set.sInter_eq_Inter Set.sInter_eq_iInter
@[simp]
theorem iUnion_of_empty [IsEmpty ι] (s : ι → Set α) : ⋃ i, s i = ∅ :=
iSup_of_empty _
#align set.Union_of_empty Set.iUnion_of_empty
@[simp]
theorem iInter_of_empty [IsEmpty ι] (s : ι → Set α) : ⋂ i, s i = univ :=
iInf_of_empty _
#align set.Inter_of_empty Set.iInter_of_empty
theorem union_eq_iUnion {s₁ s₂ : Set α} : s₁ ∪ s₂ = ⋃ b : Bool, cond b s₁ s₂ :=
sup_eq_iSup s₁ s₂
#align set.union_eq_Union Set.union_eq_iUnion
theorem inter_eq_iInter {s₁ s₂ : Set α} : s₁ ∩ s₂ = ⋂ b : Bool, cond b s₁ s₂ :=
inf_eq_iInf s₁ s₂
#align set.inter_eq_Inter Set.inter_eq_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_union_sInter {S T : Set (Set α)} :
⋂₀ S ∪ ⋂₀ T = ⋂ p ∈ S ×ˢ T, (p : Set α × Set α).1 ∪ p.2 :=
sInf_sup_sInf
#align set.sInter_union_sInter Set.sInter_union_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_inter_sUnion {s t : Set (Set α)} :
⋃₀s ∩ ⋃₀t = ⋃ p ∈ s ×ˢ t, (p : Set α × Set α).1 ∩ p.2 :=
sSup_inf_sSup
#align set.sUnion_inter_sUnion Set.sUnion_inter_sUnion
theorem biUnion_iUnion (s : ι → Set α) (t : α → Set β) :
⋃ x ∈ ⋃ i, s i, t x = ⋃ (i) (x ∈ s i), t x := by simp [@iUnion_comm _ ι]
#align set.bUnion_Union Set.biUnion_iUnion
theorem biInter_iUnion (s : ι → Set α) (t : α → Set β) :
⋂ x ∈ ⋃ i, s i, t x = ⋂ (i) (x ∈ s i), t x := by simp [@iInter_comm _ ι]
#align set.bInter_Union Set.biInter_iUnion
theorem sUnion_iUnion (s : ι → Set (Set α)) : ⋃₀⋃ i, s i = ⋃ i, ⋃₀s i := by
simp only [sUnion_eq_biUnion, biUnion_iUnion]
#align set.sUnion_Union Set.sUnion_iUnion
theorem sInter_iUnion (s : ι → Set (Set α)) : ⋂₀ ⋃ i, s i = ⋂ i, ⋂₀ s i := by
simp only [sInter_eq_biInter, biInter_iUnion]
#align set.sInter_Union Set.sInter_iUnion
theorem iUnion_range_eq_sUnion {α β : Type*} (C : Set (Set α)) {f : ∀ s : C, β → (s : Type _)}
(hf : ∀ s : C, Surjective (f s)) : ⋃ y : β, range (fun s : C => (f s y).val) = ⋃₀C := by
ext x; constructor
· rintro ⟨s, ⟨y, rfl⟩, ⟨s, hs⟩, rfl⟩
refine' ⟨_, hs, _⟩
exact (f ⟨s, hs⟩ y).2
· rintro ⟨s, hs, hx⟩
cases' hf ⟨s, hs⟩ ⟨x, hx⟩ with y hy
refine' ⟨_, ⟨y, rfl⟩, ⟨s, hs⟩, _⟩
exact congr_arg Subtype.val hy
#align set.Union_range_eq_sUnion Set.iUnion_range_eq_sUnion
theorem iUnion_range_eq_iUnion (C : ι → Set α) {f : ∀ x : ι, β → C x}
(hf : ∀ x : ι, Surjective (f x)) : ⋃ y : β, range (fun x : ι => (f x y).val) = ⋃ x, C x := by
ext x; rw [mem_iUnion, mem_iUnion]; constructor
· rintro ⟨y, i, rfl⟩
exact ⟨i, (f i y).2⟩
· rintro ⟨i, hx⟩
cases' hf i ⟨x, hx⟩ with y hy
exact ⟨y, i, congr_arg Subtype.val hy⟩
#align set.Union_range_eq_Union Set.iUnion_range_eq_iUnion
theorem union_distrib_iInter_left (s : ι → Set α) (t : Set α) : (t ∪ ⋂ i, s i) = ⋂ i, t ∪ s i :=
sup_iInf_eq _ _
#align set.union_distrib_Inter_left Set.union_distrib_iInter_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_left (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_distrib_iInter_left]
#align set.union_distrib_Inter₂_left Set.union_distrib_iInter₂_left
theorem union_distrib_iInter_right (s : ι → Set α) (t : Set α) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.union_distrib_Inter_right Set.union_distrib_iInter_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_right (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [union_distrib_iInter_right]
#align set.union_distrib_Inter₂_right Set.union_distrib_iInter₂_right
section Function
/-! ### `mapsTo` -/
theorem mapsTo_sUnion {S : Set (Set α)} {t : Set β} {f : α → β} (H : ∀ s ∈ S, MapsTo f s t) :
MapsTo f (⋃₀S) t := fun _ ⟨s, hs, hx⟩ => H s hs hx
#align set.maps_to_sUnion Set.mapsTo_sUnion
theorem mapsTo_iUnion {s : ι → Set α} {t : Set β} {f : α → β} (H : ∀ i, MapsTo f (s i) t) :
MapsTo f (⋃ i, s i) t :=
mapsTo_sUnion <| forall_range_iff.2 H
#align set.maps_to_Union Set.mapsTo_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂ {s : ∀ i, κ i → Set α} {t : Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) t) : MapsTo f (⋃ (i) (j), s i j) t :=
mapsTo_iUnion fun i => mapsTo_iUnion (H i)
#align set.maps_to_Union₂ Set.mapsTo_iUnion₂
theorem mapsTo_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋃ i, s i) (⋃ i, t i) :=
mapsTo_iUnion fun i => (H i).mono (Subset.refl _) (subset_iUnion t i)
#align set.maps_to_Union_Union Set.mapsTo_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
mapsTo_iUnion_iUnion fun i => mapsTo_iUnion_iUnion (H i)
#align set.maps_to_Union₂_Union₂ Set.mapsTo_iUnion₂_iUnion₂
theorem mapsTo_sInter {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, MapsTo f s t) :
MapsTo f s (⋂₀ T) := fun _ hx t ht => H t ht hx
#align set.maps_to_sInter Set.mapsTo_sInter
theorem mapsTo_iInter {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, MapsTo f s (t i)) :
MapsTo f s (⋂ i, t i) := fun _ hx => mem_iInter.2 fun i => H i hx
#align set.maps_to_Inter Set.mapsTo_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f s (t i j)) : MapsTo f s (⋂ (i) (j), t i j) :=
mapsTo_iInter fun i => mapsTo_iInter (H i)
#align set.maps_to_Inter₂ Set.mapsTo_iInter₂
theorem mapsTo_iInter_iInter {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋂ i, s i) (⋂ i, t i) :=
mapsTo_iInter fun i => (H i).mono (iInter_subset s i) (Subset.refl _)
#align set.maps_to_Inter_Inter Set.mapsTo_iInter_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂_iInter₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋂ (i) (j), s i j) (⋂ (i) (j), t i j) :=
mapsTo_iInter_iInter fun i => mapsTo_iInter_iInter (H i)
#align set.maps_to_Inter₂_Inter₂ Set.mapsTo_iInter₂_iInter₂
theorem image_iInter_subset (s : ι → Set α) (f : α → β) : (f '' ⋂ i, s i) ⊆ ⋂ i, f '' s i :=
(mapsTo_iInter_iInter fun i => mapsTo_image f (s i)).image_subset
#align set.image_Inter_subset Set.image_iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂_subset (s : ∀ i, κ i → Set α) (f : α → β) :
(f '' ⋂ (i) (j), s i j) ⊆ ⋂ (i) (j), f '' s i j :=
(mapsTo_iInter₂_iInter₂ fun i hi => mapsTo_image f (s i hi)).image_subset
#align set.image_Inter₂_subset Set.image_iInter₂_subset
theorem image_sInter_subset (S : Set (Set α)) (f : α → β) : f '' ⋂₀ S ⊆ ⋂ s ∈ S, f '' s := by
rw [sInter_eq_biInter]
apply image_iInter₂_subset
#align set.image_sInter_subset Set.image_sInter_subset
/-! ### `restrictPreimage` -/
section
open Function
variable (s : Set β) {f : α → β} {U : ι → Set β} (hU : iUnion U = univ)
theorem injective_iff_injective_of_iUnion_eq_univ :
Injective f ↔ ∀ i, Injective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_injective H, fun H x y e => _⟩
obtain ⟨i, hi⟩ := Set.mem_iUnion.mp
(show f x ∈ Set.iUnion U by rw [hU]; triv)
injection @H i ⟨x, hi⟩ ⟨y, show f y ∈ U i from e ▸ hi⟩ (Subtype.ext e)
#align set.injective_iff_injective_of_Union_eq_univ Set.injective_iff_injective_of_iUnion_eq_univ
theorem surjective_iff_surjective_of_iUnion_eq_univ :
Surjective f ↔ ∀ i, Surjective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_surjective H, fun H x => _⟩
obtain ⟨i, hi⟩ :=
Set.mem_iUnion.mp
(show x ∈ Set.iUnion U by rw [hU]; triv)
exact ⟨_, congr_arg Subtype.val (H i ⟨x, hi⟩).choose_spec⟩
#align set.surjective_iff_surjective_of_Union_eq_univ Set.surjective_iff_surjective_of_iUnion_eq_univ
theorem bijective_iff_bijective_of_iUnion_eq_univ :
Bijective f ↔ ∀ i, Bijective ((U i).restrictPreimage f) := by
rw [Bijective, injective_iff_injective_of_iUnion_eq_univ hU,
surjective_iff_surjective_of_iUnion_eq_univ hU]
simp [Bijective, forall_and]
#align set.bijective_iff_bijective_of_Union_eq_univ Set.bijective_iff_bijective_of_iUnion_eq_univ
end
/-! ### `InjOn` -/
theorem InjOn.image_iInter_eq [Nonempty ι] {s : ι → Set α} {f : α → β} (h : InjOn f (⋃ i, s i)) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
inhabit ι
refine' Subset.antisymm (image_iInter_subset s f) fun y hy => _
simp only [mem_iInter, mem_image_iff_bex] at hy
choose x hx hy using hy
refine' ⟨x default, mem_iInter.2 fun i => _, hy _⟩
suffices x default = x i by
rw [this]
apply hx
replace hx : ∀ i, x i ∈ ⋃ j, s j := fun i => (subset_iUnion _ _) (hx i)
apply h (hx _) (hx _)
simp only [hy]
#align set.inj_on.image_Inter_eq Set.InjOn.image_iInter_eq
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
theorem InjOn.image_biInter_eq {p : ι → Prop} {s : ∀ i, p i → Set α} (hp : ∃ i, p i)
{f : α → β} (h : InjOn f (⋃ (i) (hi), s i hi)) :
(f '' ⋂ (i) (hi), s i hi) = ⋂ (i) (hi), f '' s i hi := by
simp only [iInter, iInf_subtype']
haveI : Nonempty { i // p i } := nonempty_subtype.2 hp
apply InjOn.image_iInter_eq
simpa only [iUnion, iSup_subtype'] using h
#align set.inj_on.image_bInter_eq Set.InjOn.image_biInter_eq
theorem image_iInter {f : α → β} (hf : Bijective f) (s : ι → Set α) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
cases isEmpty_or_nonempty ι
· simp_rw [iInter_of_empty, image_univ_of_surjective hf.surjective]
· exact (hf.injective.injOn _).image_iInter_eq
#align set.image_Inter Set.image_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂ {f : α → β} (hf : Bijective f) (s : ∀ i, κ i → Set α) :
(f '' ⋂ (i) (j), s i j) = ⋂ (i) (j), f '' s i j := by simp_rw [image_iInter hf]
#align set.image_Inter₂ Set.image_iInter₂
theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
(hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i) := by
intro x hx y hy hxy
rcases mem_iUnion.1 hx with ⟨i, hx⟩
rcases mem_iUnion.1 hy with ⟨j, hy⟩
rcases hs i j with ⟨k, hi, hj⟩
exact hf k (hi hx) (hj hy) hxy
#align set.inj_on_Union_of_directed Set.inj_on_iUnion_of_directed
/-! ### `SurjOn` -/
theorem surjOn_sUnion {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, SurjOn f s t) :
SurjOn f s (⋃₀T) := fun _ ⟨t, ht, hx⟩ => H t ht hx
#align set.surj_on_sUnion Set.surjOn_sUnion
theorem surjOn_iUnion {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, SurjOn f s (t i)) :
SurjOn f s (⋃ i, t i) :=
surjOn_sUnion <| forall_range_iff.2 H
#align set.surj_on_Union Set.surjOn_iUnion
theorem surjOn_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) : SurjOn f (⋃ i, s i) (⋃ i, t i) :=
surjOn_iUnion fun i => (H i).mono (subset_iUnion _ _) (Subset.refl _)
#align set.surj_on_Union_Union Set.surjOn_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f s (t i j)) : SurjOn f s (⋃ (i) (j), t i j) :=
surjOn_iUnion fun i => surjOn_iUnion (H i)
#align set.surj_on_Union₂ Set.surjOn_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f (s i j) (t i j)) : SurjOn f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
surjOn_iUnion_iUnion fun i => surjOn_iUnion_iUnion (H i)
#align set.surj_on_Union₂_Union₂ Set.surjOn_iUnion₂_iUnion₂
theorem surjOn_iInter [Nonempty ι] {s : ι → Set α} {t : Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) t) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) t := by
intro y hy
rw [Hinj.image_iInter_eq, mem_iInter]
exact fun i => H i hy
#align set.surj_on_Inter Set.surjOn_iInter
theorem surjOn_iInter_iInter [Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) (⋂ i, t i) :=
surjOn_iInter (fun i => (H i).mono (Subset.refl _) (iInter_subset _ _)) Hinj
#align set.surj_on_Inter_Inter Set.surjOn_iInter_iInter
/-! ### `BijOn` -/
theorem bijOn_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i))
(Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
⟨mapsTo_iUnion_iUnion fun i => (H i).mapsTo, Hinj, surjOn_iUnion_iUnion fun i => (H i).surjOn⟩
#align set.bij_on_Union Set.bijOn_iUnion
theorem bijOn_iInter [hi : Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, BijOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
⟨mapsTo_iInter_iInter fun i => (H i).mapsTo,
hi.elim fun i => (H i).injOn.mono (iInter_subset _ _),
surjOn_iInter_iInter (fun i => (H i).surjOn) Hinj⟩
#align set.bij_on_Inter Set.bijOn_iInter
theorem bijOn_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {t : ι → Set β}
{f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
bijOn_iUnion H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Union_of_directed Set.bijOn_iUnion_of_directed
theorem bijOn_iInter_of_directed [Nonempty ι] {s : ι → Set α} (hs : Directed (· ⊆ ·) s)
{t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
bijOn_iInter H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Inter_of_directed Set.bijOn_iInter_of_directed
end Function
/-! ### `image`, `preimage` -/
section Image
theorem image_iUnion {f : α → β} {s : ι → Set α} : (f '' ⋃ i, s i) = ⋃ i, f '' s i := by
ext1 x
simp only [mem_image, mem_iUnion, ← exists_and_right, ← exists_and_left]
--Porting note: `exists_swap` causes a `simp` loop in Lean4 so we use `rw` instead.
rw [exists_swap]
#align set.image_Union Set.image_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iUnion₂ (f : α → β) (s : ∀ i, κ i → Set α) :
(f '' ⋃ (i) (j), s i j) = ⋃ (i) (j), f '' s i j := by simp_rw [image_iUnion]
#align set.image_Union₂ Set.image_iUnion₂
theorem univ_subtype {p : α → Prop} : (univ : Set (Subtype p)) = ⋃ (x) (h : p x), {⟨x, h⟩} :=
Set.ext fun ⟨x, h⟩ => by simp [h]
#align set.univ_subtype Set.univ_subtype
theorem range_eq_iUnion {ι} (f : ι → α) : range f = ⋃ i, {f i} :=
Set.ext fun a => by simp [@eq_comm α a]
#align set.range_eq_Union Set.range_eq_iUnion
theorem image_eq_iUnion (f : α → β) (s : Set α) : f '' s = ⋃ i ∈ s, {f i} :=
Set.ext fun b => by simp [@eq_comm β b]
#align set.image_eq_Union Set.image_eq_iUnion
theorem biUnion_range {f : ι → α} {g : α → Set β} : ⋃ x ∈ range f, g x = ⋃ y, g (f y) :=
iSup_range
#align set.bUnion_range Set.biUnion_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iUnion_iUnion_eq' {f : ι → α} {g : α → Set β} :
⋃ (x) (y) (_ : f y = x), g x = ⋃ y, g (f y) := by simpa using biUnion_range
#align set.Union_Union_eq' Set.iUnion_iUnion_eq'
theorem biInter_range {f : ι → α} {g : α → Set β} : ⋂ x ∈ range f, g x = ⋂ y, g (f y) :=
iInf_range
#align set.bInter_range Set.biInter_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iInter_iInter_eq' {f : ι → α} {g : α → Set β} :
⋂ (x) (y) (_ : f y = x), g x = ⋂ y, g (f y) := by simpa using biInter_range
#align set.Inter_Inter_eq' Set.iInter_iInter_eq'
variable {s : Set γ} {f : γ → α} {g : α → Set β}
theorem biUnion_image : ⋃ x ∈ f '' s, g x = ⋃ y ∈ s, g (f y) :=
iSup_image
#align set.bUnion_image Set.biUnion_image
theorem biInter_image : ⋂ x ∈ f '' s, g x = ⋂ y ∈ s, g (f y) :=
iInf_image
#align set.bInter_image Set.biInter_image
end Image
section Preimage
theorem monotone_preimage {f : α → β} : Monotone (preimage f) := fun _ _ h => preimage_mono h
#align set.monotone_preimage Set.monotone_preimage
@[simp]
theorem preimage_iUnion {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋃ i, s i) = ⋃ i, f ⁻¹' s i :=
Set.ext <| by simp [preimage]
#align set.preimage_Union Set.preimage_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iUnion₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋃ (i) (j), s i j) = ⋃ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iUnion]
#align set.preimage_Union₂ Set.preimage_iUnion₂
@[simp]
theorem preimage_sUnion {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋃₀s = ⋃ t ∈ s, f ⁻¹' t := by
rw [sUnion_eq_biUnion, preimage_iUnion₂]
#align set.preimage_sUnion Set.preimage_sUnion
theorem preimage_iInter {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋂ i, s i) = ⋂ i, f ⁻¹' s i := by
ext; simp
#align set.preimage_Inter Set.preimage_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iInter₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋂ (i) (j), s i j) = ⋂ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iInter]
#align set.preimage_Inter₂ Set.preimage_iInter₂
@[simp]
theorem preimage_sInter {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋂₀ s = ⋂ t ∈ s, f ⁻¹' t := by
rw [sInter_eq_biInter, preimage_iInter₂]
#align set.preimage_sInter Set.preimage_sInter
@[simp]
theorem biUnion_preimage_singleton (f : α → β) (s : Set β) : ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s := by
rw [← preimage_iUnion₂, biUnion_of_singleton]
#align set.bUnion_preimage_singleton Set.biUnion_preimage_singleton
theorem biUnion_range_preimage_singleton (f : α → β) : ⋃ y ∈ range f, f ⁻¹' {y} = univ := by
rw [biUnion_preimage_singleton, preimage_range]
#align set.bUnion_range_preimage_singleton Set.biUnion_range_preimage_singleton
end Preimage
section Prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion {s : Set α} {t : ι → Set β} : (s ×ˢ ⋃ i, t i) = ⋃ i, s ×ˢ t i := by
ext
simp
#align set.prod_Union Set.prod_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} :
(s ×ˢ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ×ˢ t i j := by simp_rw [prod_iUnion]
#align set.prod_Union₂ Set.prod_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sUnion {s : Set α} {C : Set (Set β)} : s ×ˢ ⋃₀C = ⋃₀((fun t => s ×ˢ t) '' C) := by
simp_rw [sUnion_eq_biUnion, biUnion_image, prod_iUnion₂]
#align set.prod_sUnion Set.prod_sUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_const {s : ι → Set α} {t : Set β} : (⋃ i, s i) ×ˢ t = ⋃ i, s i ×ˢ t := by
ext
simp
#align set.Union_prod_const Set.iUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion₂_prod_const {s : ∀ i, κ i → Set α} {t : Set β} :
(⋃ (i) (j), s i j) ×ˢ t = ⋃ (i) (j), s i j ×ˢ t := by simp_rw [iUnion_prod_const]
#align set.Union₂_prod_const Set.iUnion₂_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_prod_const {C : Set (Set α)} {t : Set β} :
⋃₀C ×ˢ t = ⋃₀((fun s : Set α => s ×ˢ t) '' C) := by
simp only [sUnion_eq_biUnion, iUnion₂_prod_const, biUnion_image]
#align set.sUnion_prod_const Set.sUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod {ι ι' α β} (s : ι → Set α) (t : ι' → Set β) :
⋃ x : ι × ι', s x.1 ×ˢ t x.2 = (⋃ i : ι, s i) ×ˢ ⋃ i : ι', t i := by
ext
simp
#align set.Union_prod Set.iUnion_prod
/-- Analogue of `iSup_prod` for sets. -/
lemma iUnion_prod' (f : β × γ → Set α) : ⋃ x : β × γ, f x = ⋃ (i : β) (j : γ), f (i, j) :=
iSup_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x := by
ext ⟨z, w⟩; simp only [mem_prod, mem_iUnion, exists_imp, and_imp, iff_def]; constructor
· intro x hz hw
exact ⟨⟨x, hz⟩, x, hw⟩
· intro x hz x' hw
exact ⟨x ⊔ x', hs le_sup_left hz, ht le_sup_right hw⟩
#align set.Union_prod_of_monotone Set.iUnion_prod_of_monotone
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter_subset (S : Set (Set α)) (T : Set (Set β)) :
⋂₀ S ×ˢ ⋂₀ T ⊆ ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 :=
subset_iInter₂ fun x hx _ hy => ⟨hy.1 x.1 hx.1, hy.2 x.2 hx.2⟩
#align set.sInter_prod_sInter_subset Set.sInter_prod_sInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 := by
obtain ⟨s₁, h₁⟩ := hS
obtain ⟨s₂, h₂⟩ := hT
refine' Set.Subset.antisymm (sInter_prod_sInter_subset S T) fun x hx => _
rw [mem_iInter₂] at hx
exact ⟨fun s₀ h₀ => (hx (s₀, s₂) ⟨h₀, h₂⟩).1, fun s₀ h₀ => (hx (s₁, s₀) ⟨h₁, h₀⟩).2⟩
#align set.sInter_prod_sInter Set.sInter_prod_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod {S : Set (Set α)} (hS : S.Nonempty) (t : Set β) :
⋂₀ S ×ˢ t = ⋂ s ∈ S, s ×ˢ t := by
rw [← sInter_singleton t, sInter_prod_sInter hS (singleton_nonempty t), sInter_singleton]
simp_rw [prod_singleton, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.sInter_prod Set.sInter_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sInter {T : Set (Set β)} (hT : T.Nonempty) (s : Set α) :
s ×ˢ ⋂₀ T = ⋂ t ∈ T, s ×ˢ t := by
rw [← sInter_singleton s, sInter_prod_sInter (singleton_nonempty s) hT, sInter_singleton]
simp_rw [singleton_prod, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.prod_sInter Set.prod_sInter
end Prod
section Image2
variable (f : α → β → γ) {s : Set α} {t : Set β}
/-- The `Set.image2` version of `Set.image_eq_iUnion` -/
theorem image2_eq_iUnion (s : Set α) (t : Set β) : image2 f s t = ⋃ (i ∈ s) (j ∈ t), {f i j} := by
ext; simp [eq_comm]
#align set.image2_eq_Union Set.image2_eq_iUnion
theorem iUnion_image_left : ⋃ a ∈ s, f a '' t = image2 f s t := by
simp only [image2_eq_iUnion, image_eq_iUnion]
#align set.Union_image_left Set.iUnion_image_left
theorem iUnion_image_right : ⋃ b ∈ t, (f · b) '' s = image2 f s t := by
rw [image2_swap, iUnion_image_left]
#align set.Union_image_right Set.iUnion_image_right
theorem image2_iUnion_left (s : ι → Set α) (t : Set β) :
image2 f (⋃ i, s i) t = ⋃ i, image2 f (s i) t := by
simp only [← image_prod, iUnion_prod_const, image_iUnion]
#align set.image2_Union_left Set.image2_iUnion_left
theorem image2_iUnion_right (s : Set α) (t : ι → Set β) :
image2 f s (⋃ i, t i) = ⋃ i, image2 f s (t i) := by
simp only [← image_prod, prod_iUnion, image_iUnion]
#align set.image2_Union_right Set.image2_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋃ (i) (j), s i j) t = ⋃ (i) (j), image2 f (s i j) t := by simp_rw [image2_iUnion_left]
#align set.image2_Union₂_left Set.image2_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋃ (i) (j), t i j) = ⋃ (i) (j), image2 f s (t i j) :=
by simp_rw [image2_iUnion_right]
#align set.image2_Union₂_right Set.image2_iUnion₂_right
theorem image2_iInter_subset_left (s : ι → Set α) (t : Set β) :
image2 f (⋂ i, s i) t ⊆ ⋂ i, image2 f (s i) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem (hx _) hy
#align set.image2_Inter_subset_left Set.image2_iInter_subset_left
theorem image2_iInter_subset_right (s : Set α) (t : ι → Set β) :
image2 f s (⋂ i, t i) ⊆ ⋂ i, image2 f s (t i) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem hx (hy _)
#align set.image2_Inter_subset_right Set.image2_iInter_subset_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋂ (i) (j), s i j) t ⊆ ⋂ (i) (j), image2 f (s i j) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem (hx _ _) hy
#align set.image2_Inter₂_subset_left Set.image2_iInter₂_subset_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), image2 f s (t i j) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem hx (hy _ _)
#align set.image2_Inter₂_subset_right Set.image2_iInter₂_subset_right
theorem prod_eq_biUnion_left : s ×ˢ t = ⋃ a ∈ s, (fun b => (a, b)) '' t := by
rw [iUnion_image_left, image2_mk_eq_prod]
#align set.prod_eq_bUnion_left Set.prod_eq_biUnion_left
theorem prod_eq_biUnion_right : s ×ˢ t = ⋃ b ∈ t, (fun a => (a, b)) '' s := by
rw [iUnion_image_right, image2_mk_eq_prod]
#align set.prod_eq_bUnion_right Set.prod_eq_biUnion_right
end Image2
section Seq
/-- Given a set `s` of functions `α → β` and `t : Set α`, `seq s t` is the union of `f '' t` over
all `f ∈ s`. -/
def seq (s : Set (α → β)) (t : Set α) : Set β :=
{ b | ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b }
#align set.seq Set.seq
@[simp]
theorem mem_seq_iff {s : Set (α → β)} {t : Set α} {b : β} :
b ∈ seq s t ↔ ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b :=
Iff.rfl
#align set.mem_seq_iff Set.mem_seq_iff
lemma seq_eq_image2 (s : Set (α → β)) (t : Set α) : seq s t = image2 (fun f a ↦ f a) s t := by
ext; simp
theorem seq_def {s : Set (α → β)} {t : Set α} : seq s t = ⋃ f ∈ s, f '' t := by
rw [seq_eq_image2, iUnion_image_left]
#align set.seq_def Set.seq_def
theorem seq_subset {s : Set (α → β)} {t : Set α} {u : Set β} :
seq s t ⊆ u ↔ ∀ f ∈ s, ∀ a ∈ t, (f : α → β) a ∈ u := by
rw [seq_eq_image2, image2_subset_iff]
#align set.seq_subset Set.seq_subset
@[gcongr]
theorem seq_mono {s₀ s₁ : Set (α → β)} {t₀ t₁ : Set α} (hs : s₀ ⊆ s₁) (ht : t₀ ⊆ t₁) :
seq s₀ t₀ ⊆ seq s₁ t₁ := fun _ ⟨f, hf, a, ha, eq⟩ => ⟨f, hs hf, a, ht ha, eq⟩
#align set.seq_mono Set.seq_mono
theorem singleton_seq {f : α → β} {t : Set α} : Set.seq ({f} : Set (α → β)) t = f '' t := by
rw [seq_eq_image2, image2_singleton_left]
#align set.singleton_seq Set.singleton_seq
theorem seq_singleton {s : Set (α → β)} {a : α} : Set.seq s {a} = (fun f : α → β => f a) '' s := by
rw [seq_eq_image2, image2_singleton_right]
#align set.seq_singleton Set.seq_singleton
theorem seq_seq {s : Set (β → γ)} {t : Set (α → β)} {u : Set α} :
seq s (seq t u) = seq (seq ((· ∘ ·) '' s) t) u := by
simp only [seq_eq_image2, image2_image_left, image2_image2_left, image2_image2_right, comp_apply]
#align set.seq_seq Set.seq_seq
theorem image_seq {f : β → γ} {s : Set (α → β)} {t : Set α} :
f '' seq s t = seq ((f ∘ ·) '' s) t := by
rw [← singleton_seq, ← singleton_seq, seq_seq, image_singleton]
#align set.image_seq Set.image_seq
theorem prod_eq_seq {s : Set α} {t : Set β} : s ×ˢ t = (Prod.mk '' s).seq t := by
rw [seq_eq_image2, image2_image_left, image2_mk_eq_prod]
#align set.prod_eq_seq Set.prod_eq_seq
theorem prod_image_seq_comm (s : Set α) (t : Set β) :
(Prod.mk '' s).seq t = seq ((fun b a => (a, b)) '' t) s := by
rw [← prod_eq_seq, ← image_swap_prod, prod_eq_seq, image_seq, ← image_comp]; rfl
#align set.prod_image_seq_comm Set.prod_image_seq_comm
theorem image2_eq_seq (f : α → β → γ) (s : Set α) (t : Set β) : image2 f s t = seq (f '' s) t := by
rw [seq_eq_image2, image2_image_left]
#align set.image2_eq_seq Set.image2_eq_seq
end Seq
section Pi
variable {π : α → Type*}
theorem pi_def (i : Set α) (s : ∀ a, Set (π a)) : pi i s = ⋂ a ∈ i, eval a ⁻¹' s a := by
ext
simp
#align set.pi_def Set.pi_def
theorem univ_pi_eq_iInter (t : ∀ i, Set (π i)) : pi univ t = ⋂ i, eval i ⁻¹' t i := by
simp only [pi_def, iInter_true, mem_univ]
#align set.univ_pi_eq_Inter Set.univ_pi_eq_iInter
theorem pi_diff_pi_subset (i : Set α) (s t : ∀ a, Set (π a)) :
pi i s \ pi i t ⊆ ⋃ a ∈ i, eval a ⁻¹' (s a \ t a) := by
refine' diff_subset_comm.2 fun x hx a ha => _
simp only [mem_diff, mem_pi, mem_iUnion, not_exists, mem_preimage, not_and, not_not,
eval_apply] at hx
exact hx.2 _ ha (hx.1 _ ha)
#align set.pi_diff_pi_subset Set.pi_diff_pi_subset
theorem iUnion_univ_pi {ι : α → Type*} (t : (a : α) → ι a → Set (π a)) :
⋃ x : (a : α) → ι a, pi univ (fun a => t a (x a)) = pi univ fun a => ⋃ j : ι a, t a j := by
ext
simp [Classical.skolem]
#align set.Union_univ_pi Set.iUnion_univ_pi
end Pi
end Set
namespace Function
namespace Surjective
theorem iUnion_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋃ x, g (f x) = ⋃ y, g y :=
hf.iSup_comp g
#align function.surjective.Union_comp Function.Surjective.iUnion_comp
theorem iInter_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋂ x, g (f x) = ⋂ y, g y :=
hf.iInf_comp g
#align function.surjective.Inter_comp Function.Surjective.iInter_comp
end Surjective
end Function
/-!
### Disjoint sets
-/
section Disjoint
variable {s t u : Set α} {f : α → β}
namespace Set
@[simp]
theorem disjoint_iUnion_left {ι : Sort*} {s : ι → Set α} :
Disjoint (⋃ i, s i) t ↔ ∀ i, Disjoint (s i) t :=
iSup_disjoint_iff
#align set.disjoint_Union_left Set.disjoint_iUnion_left
@[simp]
theorem disjoint_iUnion_right {ι : Sort*} {s : ι → Set α} :
Disjoint t (⋃ i, s i) ↔ ∀ i, Disjoint t (s i) :=
disjoint_iSup_iff
#align set.disjoint_Union_right Set.disjoint_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_left {s : ∀ i, κ i → Set α} {t : Set α} :
Disjoint (⋃ (i) (j), s i j) t ↔ ∀ i j, Disjoint (s i j) t :=
iSup₂_disjoint_iff
#align set.disjoint_Union₂_left Set.disjoint_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_right {s : Set α} {t : ∀ i, κ i → Set α} :
Disjoint s (⋃ (i) (j), t i j) ↔ ∀ i j, Disjoint s (t i j) :=
disjoint_iSup₂_iff
#align set.disjoint_Union₂_right Set.disjoint_iUnion₂_right
@[simp]
theorem disjoint_sUnion_left {S : Set (Set α)} {t : Set α} :
Disjoint (⋃₀S) t ↔ ∀ s ∈ S, Disjoint s t :=
sSup_disjoint_iff
#align set.disjoint_sUnion_left Set.disjoint_sUnion_left
@[simp]
theorem disjoint_sUnion_right {s : Set α} {S : Set (Set α)} :
Disjoint s (⋃₀S) ↔ ∀ t ∈ S, Disjoint s t :=
disjoint_sSup_iff
#align set.disjoint_sUnion_right Set.disjoint_sUnion_right
end Set
end Disjoint
/-! ### Intervals -/
namespace Set
lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :
(⋂ i, Iic (f i)).Nonempty ↔ BddBelow (range f) := by
have : (⋂ (i : ι), Iic (f i)) = lowerBounds (range f) := by
ext c; simp [lowerBounds]
simp [this, BddBelow]
lemma nonempty_iInter_Ici_iff [Preorder α] {f : ι → α} :
(⋂ i, Ici (f i)).Nonempty ↔ BddAbove (range f) :=
nonempty_iInter_Iic_iff (α := αᵒᵈ)
variable [CompleteLattice α]
theorem Ici_iSup (f : ι → α) : Ici (⨆ i, f i) = ⋂ i, Ici (f i) :=
ext fun _ => by simp only [mem_Ici, iSup_le_iff, mem_iInter]
#align set.Ici_supr Set.Ici_iSup
theorem Iic_iInf (f : ι → α) : Iic (⨅ i, f i) = ⋂ i, Iic (f i) :=
ext fun _ => by simp only [mem_Iic, le_iInf_iff, mem_iInter]
#align set.Iic_infi Set.Iic_iInf
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Ici_iSup₂ (f : ∀ i, κ i → α) : Ici (⨆ (i) (j), f i j) = ⋂ (i) (j), Ici (f i j) := by
simp_rw [Ici_iSup]
#align set.Ici_supr₂ Set.Ici_iSup₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Iic_iInf₂ (f : ∀ i, κ i → α) : Iic (⨅ (i) (j), f i j) = ⋂ (i) (j), Iic (f i j) := by
simp_rw [Iic_iInf]
#align set.Iic_infi₂ Set.Iic_iInf₂
theorem Ici_sSup (s : Set α) : Ici (sSup s) = ⋂ a ∈ s, Ici a := by rw [sSup_eq_iSup, Ici_iSup₂]
#align set.Ici_Sup Set.Ici_sSup
theorem Iic_sInf (s : Set α) : Iic (sInf s) = ⋂ a ∈ s, Iic a := by rw [sInf_eq_iInf, Iic_iInf₂]
#align set.Iic_Inf Set.Iic_sInf
end Set
namespace Set
variable (t : α → Set β)
theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x := by
simp only [diff_subset_iff, ← biUnion_union]
apply biUnion_subset_biUnion_left
rw [union_diff_self]
apply subset_union_right
#align set.bUnion_diff_bUnion_subset Set.biUnion_diff_biUnion_subset
/-- If `t` is an indexed family of sets, then there is a natural map from `Σ i, t i` to `⋃ i, t i`
sending `⟨i, x⟩` to `x`. -/
def sigmaToiUnion (x : Σi, t i) : ⋃ i, t i :=
⟨x.2, mem_iUnion.2 ⟨x.1, x.2.2⟩⟩
#align set.sigma_to_Union Set.sigmaToiUnion
theorem sigmaToiUnion_surjective : Surjective (sigmaToiUnion t)
| ⟨b, hb⟩ =>
have : ∃ a, b ∈ t a := by simpa using hb
let ⟨a, hb⟩ := this
⟨⟨a, b, hb⟩, rfl⟩
#align set.sigma_to_Union_surjective Set.sigmaToiUnion_surjective
theorem sigmaToiUnion_injective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Injective (sigmaToiUnion t)
| ⟨a₁, b₁, h₁⟩, ⟨a₂, b₂, h₂⟩, eq =>
have b_eq : b₁ = b₂ := congr_arg Subtype.val eq
have a_eq : a₁ = a₂ :=
by_contradiction fun ne =>
have : b₁ ∈ t a₁ ∩ t a₂ := ⟨h₁, b_eq.symm ▸ h₂⟩
(h _ _ ne).le_bot this
Sigma.eq a_eq <| Subtype.eq <| by subst b_eq; subst a_eq;
|
rfl
|
theorem sigmaToiUnion_injective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Injective (sigmaToiUnion t)
| ⟨a₁, b₁, h₁⟩, ⟨a₂, b₂, h₂⟩, eq =>
have b_eq : b₁ = b₂ := congr_arg Subtype.val eq
have a_eq : a₁ = a₂ :=
by_contradiction fun ne =>
have : b₁ ∈ t a₁ ∩ t a₂ := ⟨h₁, b_eq.symm ▸ h₂⟩
(h _ _ ne).le_bot this
Sigma.eq a_eq <| Subtype.eq <| by subst b_eq; subst a_eq;
|
Mathlib.Data.Set.Lattice.2312_0.5mONj49h3SYSDwc
|
theorem sigmaToiUnion_injective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Injective (sigmaToiUnion t)
| ⟨a₁, b₁, h₁⟩, ⟨a₂, b₂, h₂⟩, eq =>
have b_eq : b₁ = b₂
|
Mathlib_Data_Set_Lattice
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : CompleteLattice β
s : ι → Set α
f : α → β
⊢ ⨆ a ∈ ⋃ i, s i, f a = ⨆ i, ⨆ a ∈ s i, f a
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
/-!
# The set lattice
This file provides usual set notation for unions and intersections, a `CompleteLattice` instance
for `Set α`, and some more set constructions.
## Main declarations
* `Set.iUnion`: **i**ndexed **union**. Union of an indexed family of sets.
* `Set.iInter`: **i**ndexed **inter**section. Intersection of an indexed family of sets.
* `Set.sInter`: **s**et **inter**section. Intersection of sets belonging to a set of sets.
* `Set.sUnion`: **s**et **union**. Union of sets belonging to a set of sets.
* `Set.sInter_eq_biInter`, `Set.sUnion_eq_biInter`: Shows that `⋂₀ s = ⋂ x ∈ s, x` and
`⋃₀ s = ⋃ x ∈ s, x`.
* `Set.completeAtomicBooleanAlgebra`: `Set α` is a `CompleteAtomicBooleanAlgebra` with `≤ = ⊆`,
`< = ⊂`, `⊓ = ∩`, `⊔ = ∪`, `⨅ = ⋂`, `⨆ = ⋃` and `\` as the set difference.
See `Set.BooleanAlgebra`.
* `Set.kernImage`: For a function `f : α → β`, `s.kernImage f` is the set of `y` such that
`f ⁻¹ y ⊆ s`.
* `Set.seq`: Union of the image of a set under a **seq**uence of functions. `seq s t` is the union
of `f '' t` over all `f ∈ s`, where `t : Set α` and `s : Set (α → β)`.
* `Set.iUnion_eq_sigma_of_disjoint`: Equivalence between `⋃ i, t i` and `Σ i, t i`, where `t` is an
indexed family of disjoint sets.
## Naming convention
In lemma names,
* `⋃ i, s i` is called `iUnion`
* `⋂ i, s i` is called `iInter`
* `⋃ i j, s i j` is called `iUnion₂`. This is an `iUnion` inside an `iUnion`.
* `⋂ i j, s i j` is called `iInter₂`. This is an `iInter` inside an `iInter`.
* `⋃ i ∈ s, t i` is called `biUnion` for "bounded `iUnion`". This is the special case of `iUnion₂`
where `j : i ∈ s`.
* `⋂ i ∈ s, t i` is called `biInter` for "bounded `iInter`". This is the special case of `iInter₂`
where `j : i ∈ s`.
## Notation
* `⋃`: `Set.iUnion`
* `⋂`: `Set.iInter`
* `⋃₀`: `Set.sUnion`
* `⋂₀`: `Set.sInter`
-/
set_option autoImplicit true
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι₂ : Sort*} {κ κ₁ κ₂ : ι → Sort*} {κ' : ι' → Sort*}
namespace Set
/-! ### Complete lattice and complete Boolean algebra instances -/
instance : InfSet (Set α) :=
⟨fun s => { a | ∀ t ∈ s, a ∈ t }⟩
instance : SupSet (Set α) :=
⟨fun s => { a | ∃ t ∈ s, a ∈ t }⟩
/-- Intersection of a set of sets. -/
def sInter (S : Set (Set α)) : Set α :=
sInf S
#align set.sInter Set.sInter
/-- Notation for `Set.sInter` Intersection of a set of sets. -/
prefix:110 "⋂₀ " => sInter
/-- Union of a set of sets. -/
def sUnion (S : Set (Set α)) : Set α :=
sSup S
#align set.sUnion Set.sUnion
/-- Notation for `Set.sUnion`. Union of a set of sets. -/
prefix:110 "⋃₀ " => sUnion
@[simp]
theorem mem_sInter {x : α} {S : Set (Set α)} : x ∈ ⋂₀ S ↔ ∀ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sInter Set.mem_sInter
@[simp]
theorem mem_sUnion {x : α} {S : Set (Set α)} : x ∈ ⋃₀ S ↔ ∃ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sUnion Set.mem_sUnion
/-- Indexed union of a family of sets -/
def iUnion (s : ι → Set β) : Set β :=
iSup s
#align set.Union Set.iUnion
/-- Indexed intersection of a family of sets -/
def iInter (s : ι → Set β) : Set β :=
iInf s
#align set.Inter Set.iInter
/-- Notation for `Set.iUnion`. Indexed union of a family of sets -/
notation3 "⋃ "(...)", "r:60:(scoped f => iUnion f) => r
/-- Notation for `Set.iInter`. Indexed intersection of a family of sets -/
notation3 "⋂ "(...)", "r:60:(scoped f => iInter f) => r
section delaborators
open Lean Lean.PrettyPrinter.Delaborator
/-- Delaborator for indexed unions. -/
@[delab app.Set.iUnion]
def iUnion_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋃ (_ : $dom), $body)
else if prop || ppTypes then
`(⋃ ($x:ident : $dom), $body)
else
`(⋃ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋃ $x:ident, ⋃ (_ : $y:ident ∈ $s), $body)
| `(⋃ ($x:ident : $_), ⋃ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋃ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
/-- Delaborator for indexed intersections. -/
@[delab app.Set.iInter]
def sInter_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋂ (_ : $dom), $body)
else if prop || ppTypes then
`(⋂ ($x:ident : $dom), $body)
else
`(⋂ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋂ $x:ident, ⋂ (_ : $y:ident ∈ $s), $body)
| `(⋂ ($x:ident : $_), ⋂ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋂ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
end delaborators
@[simp]
theorem sSup_eq_sUnion (S : Set (Set α)) : sSup S = ⋃₀S :=
rfl
#align set.Sup_eq_sUnion Set.sSup_eq_sUnion
@[simp]
theorem sInf_eq_sInter (S : Set (Set α)) : sInf S = ⋂₀ S :=
rfl
#align set.Inf_eq_sInter Set.sInf_eq_sInter
@[simp]
theorem iSup_eq_iUnion (s : ι → Set α) : iSup s = iUnion s :=
rfl
#align set.supr_eq_Union Set.iSup_eq_iUnion
@[simp]
theorem iInf_eq_iInter (s : ι → Set α) : iInf s = iInter s :=
rfl
#align set.infi_eq_Inter Set.iInf_eq_iInter
@[simp]
theorem mem_iUnion {x : α} {s : ι → Set α} : (x ∈ ⋃ i, s i) ↔ ∃ i, x ∈ s i :=
⟨fun ⟨_, ⟨⟨a, (t_eq : s a = _)⟩, (h : x ∈ _)⟩⟩ => ⟨a, t_eq.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨s a, ⟨⟨a, rfl⟩, h⟩⟩⟩
#align set.mem_Union Set.mem_iUnion
@[simp]
theorem mem_iInter {x : α} {s : ι → Set α} : (x ∈ ⋂ i, s i) ↔ ∀ i, x ∈ s i :=
⟨fun (h : ∀ a ∈ { a : Set α | ∃ i, s i = a }, x ∈ a) a => h (s a) ⟨a, rfl⟩,
fun h _ ⟨a, (eq : s a = _)⟩ => eq ▸ h a⟩
#align set.mem_Inter Set.mem_iInter
theorem mem_iUnion₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋃ (i) (j), s i j) ↔ ∃ i j, x ∈ s i j := by
simp_rw [mem_iUnion]
#align set.mem_Union₂ Set.mem_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋂ (i) (j), s i j) ↔ ∀ i j, x ∈ s i j := by
simp_rw [mem_iInter]
#align set.mem_Inter₂ Set.mem_iInter₂
theorem mem_iUnion_of_mem {s : ι → Set α} {a : α} (i : ι) (ha : a ∈ s i) : a ∈ ⋃ i, s i :=
mem_iUnion.2 ⟨i, ha⟩
#align set.mem_Union_of_mem Set.mem_iUnion_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iUnion₂_of_mem {s : ∀ i, κ i → Set α} {a : α} {i : ι} (j : κ i) (ha : a ∈ s i j) :
a ∈ ⋃ (i) (j), s i j :=
mem_iUnion₂.2 ⟨i, j, ha⟩
#align set.mem_Union₂_of_mem Set.mem_iUnion₂_of_mem
theorem mem_iInter_of_mem {s : ι → Set α} {a : α} (h : ∀ i, a ∈ s i) : a ∈ ⋂ i, s i :=
mem_iInter.2 h
#align set.mem_Inter_of_mem Set.mem_iInter_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂_of_mem {s : ∀ i, κ i → Set α} {a : α} (h : ∀ i j, a ∈ s i j) :
a ∈ ⋂ (i) (j), s i j :=
mem_iInter₂.2 h
#align set.mem_Inter₂_of_mem Set.mem_iInter₂_of_mem
instance Set.completeAtomicBooleanAlgebra : CompleteAtomicBooleanAlgebra (Set α) :=
{ instBooleanAlgebraSet with
le_sSup := fun s t t_in a a_in => ⟨t, t_in, a_in⟩
sSup_le := fun s t h a ⟨t', ⟨t'_in, a_in⟩⟩ => h t' t'_in a_in
le_sInf := fun s t h a a_in t' t'_in => h t' t'_in a_in
sInf_le := fun s t t_in a h => h _ t_in
iInf_iSup_eq := by intros; ext; simp [Classical.skolem] }
/-- `kernImage f s` is the set of `y` such that `f ⁻¹ y ⊆ s`. -/
def kernImage (f : α → β) (s : Set α) : Set β :=
{ y | ∀ ⦃x⦄, f x = y → x ∈ s }
#align set.kern_image Set.kernImage
lemma subset_kernImage_iff {f : α → β} : s ⊆ kernImage f t ↔ f ⁻¹' s ⊆ t :=
⟨fun h _ hx ↦ h hx rfl,
fun h _ hx y hy ↦ h (show f y ∈ s from hy.symm ▸ hx)⟩
section GaloisConnection
variable {f : α → β}
protected theorem image_preimage : GaloisConnection (image f) (preimage f) := fun _ _ =>
image_subset_iff
#align set.image_preimage Set.image_preimage
protected theorem preimage_kernImage : GaloisConnection (preimage f) (kernImage f) := fun _ _ =>
subset_kernImage_iff.symm
#align set.preimage_kern_image Set.preimage_kernImage
end GaloisConnection
section kernImage
variable {f : α → β}
lemma kernImage_mono : Monotone (kernImage f) :=
Set.preimage_kernImage.monotone_u
lemma kernImage_eq_compl {s : Set α} : kernImage f s = (f '' sᶜ)ᶜ :=
Set.preimage_kernImage.u_unique (Set.image_preimage.compl)
(fun t ↦ compl_compl (f ⁻¹' t) ▸ Set.preimage_compl)
lemma kernImage_compl {s : Set α} : kernImage f (sᶜ) = (f '' s)ᶜ := by
rw [kernImage_eq_compl, compl_compl]
lemma kernImage_empty : kernImage f ∅ = (range f)ᶜ := by
rw [kernImage_eq_compl, compl_empty, image_univ]
lemma kernImage_preimage_eq_iff {s : Set β} : kernImage f (f ⁻¹' s) = s ↔ (range f)ᶜ ⊆ s := by
rw [kernImage_eq_compl, ← preimage_compl, compl_eq_comm, eq_comm, image_preimage_eq_iff,
compl_subset_comm]
lemma compl_range_subset_kernImage {s : Set α} : (range f)ᶜ ⊆ kernImage f s := by
rw [← kernImage_empty]
exact kernImage_mono (empty_subset _)
lemma kernImage_union_preimage {s : Set α} {t : Set β} :
kernImage f (s ∪ f ⁻¹' t) = kernImage f s ∪ t := by
rw [kernImage_eq_compl, kernImage_eq_compl, compl_union, ← preimage_compl, image_inter_preimage,
compl_inter, compl_compl]
lemma kernImage_preimage_union {s : Set α} {t : Set β} :
kernImage f (f ⁻¹' t ∪ s) = t ∪ kernImage f s := by
rw [union_comm, kernImage_union_preimage, union_comm]
end kernImage
/-! ### Union and intersection over an indexed family of sets -/
instance : OrderTop (Set α) where
top := univ
le_top := by simp
@[congr]
theorem iUnion_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iUnion f₁ = iUnion f₂ :=
iSup_congr_Prop pq f
#align set.Union_congr_Prop Set.iUnion_congr_Prop
@[congr]
theorem iInter_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iInter f₁ = iInter f₂ :=
iInf_congr_Prop pq f
#align set.Inter_congr_Prop Set.iInter_congr_Prop
theorem iUnion_plift_up (f : PLift ι → Set α) : ⋃ i, f (PLift.up i) = ⋃ i, f i :=
iSup_plift_up _
#align set.Union_plift_up Set.iUnion_plift_up
theorem iUnion_plift_down (f : ι → Set α) : ⋃ i, f (PLift.down i) = ⋃ i, f i :=
iSup_plift_down _
#align set.Union_plift_down Set.iUnion_plift_down
theorem iInter_plift_up (f : PLift ι → Set α) : ⋂ i, f (PLift.up i) = ⋂ i, f i :=
iInf_plift_up _
#align set.Inter_plift_up Set.iInter_plift_up
theorem iInter_plift_down (f : ι → Set α) : ⋂ i, f (PLift.down i) = ⋂ i, f i :=
iInf_plift_down _
#align set.Inter_plift_down Set.iInter_plift_down
theorem iUnion_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋃ _ : p, s = if p then s else ∅ :=
iSup_eq_if _
#align set.Union_eq_if Set.iUnion_eq_if
theorem iUnion_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋃ h : p, s h = if h : p then s h else ∅ :=
iSup_eq_dif _
#align set.Union_eq_dif Set.iUnion_eq_dif
theorem iInter_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋂ _ : p, s = if p then s else univ :=
iInf_eq_if _
#align set.Inter_eq_if Set.iInter_eq_if
theorem iInf_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋂ h : p, s h = if h : p then s h else univ :=
_root_.iInf_eq_dif _
#align set.Infi_eq_dif Set.iInf_eq_dif
theorem exists_set_mem_of_union_eq_top {ι : Type*} (t : Set ι) (s : ι → Set β)
(w : ⋃ i ∈ t, s i = ⊤) (x : β) : ∃ i ∈ t, x ∈ s i := by
have p : x ∈ ⊤ := Set.mem_univ x
rw [← w, Set.mem_iUnion] at p
simpa using p
#align set.exists_set_mem_of_union_eq_top Set.exists_set_mem_of_union_eq_top
theorem nonempty_of_union_eq_top_of_nonempty {ι : Type*} (t : Set ι) (s : ι → Set α)
(H : Nonempty α) (w : ⋃ i ∈ t, s i = ⊤) : t.Nonempty := by
obtain ⟨x, m, -⟩ := exists_set_mem_of_union_eq_top t s w H.some
exact ⟨x, m⟩
#align set.nonempty_of_union_eq_top_of_nonempty Set.nonempty_of_union_eq_top_of_nonempty
theorem nonempty_of_nonempty_iUnion
{s : ι → Set α} (h_Union : (⋃ i, s i).Nonempty) : Nonempty ι := by
obtain ⟨x, hx⟩ := h_Union
exact ⟨Classical.choose $ mem_iUnion.mp hx⟩
theorem nonempty_of_nonempty_iUnion_eq_univ
{s : ι → Set α} [Nonempty α] (h_Union : ⋃ i, s i = univ) : Nonempty ι :=
nonempty_of_nonempty_iUnion (s := s) (by simpa only [h_Union] using univ_nonempty)
theorem setOf_exists (p : ι → β → Prop) : { x | ∃ i, p i x } = ⋃ i, { x | p i x } :=
ext fun _ => mem_iUnion.symm
#align set.set_of_exists Set.setOf_exists
theorem setOf_forall (p : ι → β → Prop) : { x | ∀ i, p i x } = ⋂ i, { x | p i x } :=
ext fun _ => mem_iInter.symm
#align set.set_of_forall Set.setOf_forall
theorem iUnion_subset {s : ι → Set α} {t : Set α} (h : ∀ i, s i ⊆ t) : ⋃ i, s i ⊆ t :=
iSup_le h
#align set.Union_subset Set.iUnion_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset {s : ∀ i, κ i → Set α} {t : Set α} (h : ∀ i j, s i j ⊆ t) :
⋃ (i) (j), s i j ⊆ t :=
iUnion_subset fun x => iUnion_subset (h x)
#align set.Union₂_subset Set.iUnion₂_subset
theorem subset_iInter {t : Set β} {s : ι → Set β} (h : ∀ i, t ⊆ s i) : t ⊆ ⋂ i, s i :=
le_iInf h
#align set.subset_Inter Set.subset_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iInter₂ {s : Set α} {t : ∀ i, κ i → Set α} (h : ∀ i j, s ⊆ t i j) :
s ⊆ ⋂ (i) (j), t i j :=
subset_iInter fun x => subset_iInter <| h x
#align set.subset_Inter₂ Set.subset_iInter₂
@[simp]
theorem iUnion_subset_iff {s : ι → Set α} {t : Set α} : ⋃ i, s i ⊆ t ↔ ∀ i, s i ⊆ t :=
⟨fun h _ => Subset.trans (le_iSup s _) h, iUnion_subset⟩
#align set.Union_subset_iff Set.iUnion_subset_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset_iff {s : ∀ i, κ i → Set α} {t : Set α} :
⋃ (i) (j), s i j ⊆ t ↔ ∀ i j, s i j ⊆ t := by simp_rw [iUnion_subset_iff]
#align set.Union₂_subset_iff Set.iUnion₂_subset_iff
@[simp]
theorem subset_iInter_iff {s : Set α} {t : ι → Set α} : (s ⊆ ⋂ i, t i) ↔ ∀ i, s ⊆ t i :=
le_iInf_iff
#align set.subset_Inter_iff Set.subset_iInter_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem subset_iInter₂_iff {s : Set α} {t : ∀ i, κ i → Set α} :
(s ⊆ ⋂ (i) (j), t i j) ↔ ∀ i j, s ⊆ t i j := by simp_rw [subset_iInter_iff]
#align set.subset_Inter₂_iff Set.subset_iInter₂_iff
theorem subset_iUnion : ∀ (s : ι → Set β) (i : ι), s i ⊆ ⋃ i, s i :=
le_iSup
#align set.subset_Union Set.subset_iUnion
theorem iInter_subset : ∀ (s : ι → Set β) (i : ι), ⋂ i, s i ⊆ s i :=
iInf_le
#align set.Inter_subset Set.iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iUnion₂ {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : s i j ⊆ ⋃ (i') (j'), s i' j' :=
le_iSup₂ i j
#align set.subset_Union₂ Set.subset_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_subset {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : ⋂ (i) (j), s i j ⊆ s i j :=
iInf₂_le i j
#align set.Inter₂_subset Set.iInter₂_subset
/-- This rather trivial consequence of `subset_iUnion`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem subset_iUnion_of_subset {s : Set α} {t : ι → Set α} (i : ι) (h : s ⊆ t i) : s ⊆ ⋃ i, t i :=
le_iSup_of_le i h
#align set.subset_Union_of_subset Set.subset_iUnion_of_subset
/-- This rather trivial consequence of `iInter_subset`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem iInter_subset_of_subset {s : ι → Set α} {t : Set α} (i : ι) (h : s i ⊆ t) :
⋂ i, s i ⊆ t :=
iInf_le_of_le i h
#align set.Inter_subset_of_subset Set.iInter_subset_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `subset_iUnion₂` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem subset_iUnion₂_of_subset {s : Set α} {t : ∀ i, κ i → Set α} (i : ι) (j : κ i)
(h : s ⊆ t i j) : s ⊆ ⋃ (i) (j), t i j :=
le_iSup₂_of_le i j h
#align set.subset_Union₂_of_subset Set.subset_iUnion₂_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `iInter₂_subset` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem iInter₂_subset_of_subset {s : ∀ i, κ i → Set α} {t : Set α} (i : ι) (j : κ i)
(h : s i j ⊆ t) : ⋂ (i) (j), s i j ⊆ t :=
iInf₂_le_of_le i j h
#align set.Inter₂_subset_of_subset Set.iInter₂_subset_of_subset
theorem iUnion_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono h
#align set.Union_mono Set.iUnion_mono
@[gcongr]
theorem iUnion_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iUnion s ⊆ iUnion t :=
iSup_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋃ (i) (j), s i j ⊆ ⋃ (i) (j), t i j :=
iSup₂_mono h
#align set.Union₂_mono Set.iUnion₂_mono
theorem iInter_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋂ i, s i ⊆ ⋂ i, t i :=
iInf_mono h
#align set.Inter_mono Set.iInter_mono
@[gcongr]
theorem iInter_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iInter s ⊆ iInter t :=
iInf_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋂ (i) (j), s i j ⊆ ⋂ (i) (j), t i j :=
iInf₂_mono h
#align set.Inter₂_mono Set.iInter₂_mono
theorem iUnion_mono' {s : ι → Set α} {t : ι₂ → Set α} (h : ∀ i, ∃ j, s i ⊆ t j) :
⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono' h
#align set.Union_mono' Set.iUnion_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iUnion₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i j, ∃ i' j', s i j ⊆ t i' j') : ⋃ (i) (j), s i j ⊆ ⋃ (i') (j'), t i' j' :=
iSup₂_mono' h
#align set.Union₂_mono' Set.iUnion₂_mono'
theorem iInter_mono' {s : ι → Set α} {t : ι' → Set α} (h : ∀ j, ∃ i, s i ⊆ t j) :
⋂ i, s i ⊆ ⋂ j, t j :=
Set.subset_iInter fun j =>
let ⟨i, hi⟩ := h j
iInter_subset_of_subset i hi
#align set.Inter_mono' Set.iInter_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iInter₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i' j', ∃ i j, s i j ⊆ t i' j') : ⋂ (i) (j), s i j ⊆ ⋂ (i') (j'), t i' j' :=
subset_iInter₂_iff.2 fun i' j' =>
let ⟨_, _, hst⟩ := h i' j'
(iInter₂_subset _ _).trans hst
#align set.Inter₂_mono' Set.iInter₂_mono'
theorem iUnion₂_subset_iUnion (κ : ι → Sort*) (s : ι → Set α) :
⋃ (i) (_ : κ i), s i ⊆ ⋃ i, s i :=
iUnion_mono fun _ => iUnion_subset fun _ => Subset.rfl
#align set.Union₂_subset_Union Set.iUnion₂_subset_iUnion
theorem iInter_subset_iInter₂ (κ : ι → Sort*) (s : ι → Set α) :
⋂ i, s i ⊆ ⋂ (i) (_ : κ i), s i :=
iInter_mono fun _ => subset_iInter fun _ => Subset.rfl
#align set.Inter_subset_Inter₂ Set.iInter_subset_iInter₂
theorem iUnion_setOf (P : ι → α → Prop) : ⋃ i, { x : α | P i x } = { x : α | ∃ i, P i x } := by
ext
exact mem_iUnion
#align set.Union_set_of Set.iUnion_setOf
theorem iInter_setOf (P : ι → α → Prop) : ⋂ i, { x : α | P i x } = { x : α | ∀ i, P i x } := by
ext
exact mem_iInter
#align set.Inter_set_of Set.iInter_setOf
theorem iUnion_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋃ x, f x = ⋃ y, g y :=
h1.iSup_congr h h2
#align set.Union_congr_of_surjective Set.iUnion_congr_of_surjective
theorem iInter_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋂ x, f x = ⋂ y, g y :=
h1.iInf_congr h h2
#align set.Inter_congr_of_surjective Set.iInter_congr_of_surjective
lemma iUnion_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋃ i, s i = ⋃ i, t i := iSup_congr h
#align set.Union_congr Set.iUnion_congr
lemma iInter_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋂ i, s i = ⋂ i, t i := iInf_congr h
#align set.Inter_congr Set.iInter_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iUnion₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋃ (i) (j), s i j = ⋃ (i) (j), t i j :=
iUnion_congr fun i => iUnion_congr <| h i
#align set.Union₂_congr Set.iUnion₂_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iInter₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋂ (i) (j), s i j = ⋂ (i) (j), t i j :=
iInter_congr fun i => iInter_congr <| h i
#align set.Inter₂_congr Set.iInter₂_congr
section Nonempty
variable [Nonempty ι] {f : ι → Set α} {s : Set α}
lemma iUnion_const (s : Set β) : ⋃ _ : ι, s = s := iSup_const
#align set.Union_const Set.iUnion_const
lemma iInter_const (s : Set β) : ⋂ _ : ι, s = s := iInf_const
#align set.Inter_const Set.iInter_const
lemma iUnion_eq_const (hf : ∀ i, f i = s) : ⋃ i, f i = s :=
(iUnion_congr hf).trans $ iUnion_const _
#align set.Union_eq_const Set.iUnion_eq_const
lemma iInter_eq_const (hf : ∀ i, f i = s) : ⋂ i, f i = s :=
(iInter_congr hf).trans $ iInter_const _
#align set.Inter_eq_const Set.iInter_eq_const
end Nonempty
@[simp]
theorem compl_iUnion (s : ι → Set β) : (⋃ i, s i)ᶜ = ⋂ i, (s i)ᶜ :=
compl_iSup
#align set.compl_Union Set.compl_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iUnion₂ (s : ∀ i, κ i → Set α) : (⋃ (i) (j), s i j)ᶜ = ⋂ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iUnion]
#align set.compl_Union₂ Set.compl_iUnion₂
@[simp]
theorem compl_iInter (s : ι → Set β) : (⋂ i, s i)ᶜ = ⋃ i, (s i)ᶜ :=
compl_iInf
#align set.compl_Inter Set.compl_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iInter₂ (s : ∀ i, κ i → Set α) : (⋂ (i) (j), s i j)ᶜ = ⋃ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iInter]
#align set.compl_Inter₂ Set.compl_iInter₂
-- classical -- complete_boolean_algebra
theorem iUnion_eq_compl_iInter_compl (s : ι → Set β) : ⋃ i, s i = (⋂ i, (s i)ᶜ)ᶜ := by
simp only [compl_iInter, compl_compl]
#align set.Union_eq_compl_Inter_compl Set.iUnion_eq_compl_iInter_compl
-- classical -- complete_boolean_algebra
theorem iInter_eq_compl_iUnion_compl (s : ι → Set β) : ⋂ i, s i = (⋃ i, (s i)ᶜ)ᶜ := by
simp only [compl_iUnion, compl_compl]
#align set.Inter_eq_compl_Union_compl Set.iInter_eq_compl_iUnion_compl
theorem inter_iUnion (s : Set β) (t : ι → Set β) : (s ∩ ⋃ i, t i) = ⋃ i, s ∩ t i :=
inf_iSup_eq _ _
#align set.inter_Union Set.inter_iUnion
theorem iUnion_inter (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∩ s = ⋃ i, t i ∩ s :=
iSup_inf_eq _ _
#align set.Union_inter Set.iUnion_inter
theorem iUnion_union_distrib (s : ι → Set β) (t : ι → Set β) :
⋃ i, s i ∪ t i = (⋃ i, s i) ∪ ⋃ i, t i :=
iSup_sup_eq
#align set.Union_union_distrib Set.iUnion_union_distrib
theorem iInter_inter_distrib (s : ι → Set β) (t : ι → Set β) :
⋂ i, s i ∩ t i = (⋂ i, s i) ∩ ⋂ i, t i :=
iInf_inf_eq
#align set.Inter_inter_distrib Set.iInter_inter_distrib
theorem union_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∪ ⋃ i, t i) = ⋃ i, s ∪ t i :=
sup_iSup
#align set.union_Union Set.union_iUnion
theorem iUnion_union [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∪ s = ⋃ i, t i ∪ s :=
iSup_sup
#align set.Union_union Set.iUnion_union
theorem inter_iInter [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∩ ⋂ i, t i) = ⋂ i, s ∩ t i :=
inf_iInf
#align set.inter_Inter Set.inter_iInter
theorem iInter_inter [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋂ i, t i) ∩ s = ⋂ i, t i ∩ s :=
iInf_inf
#align set.Inter_inter Set.iInter_inter
-- classical
theorem union_iInter (s : Set β) (t : ι → Set β) : (s ∪ ⋂ i, t i) = ⋂ i, s ∪ t i :=
sup_iInf_eq _ _
#align set.union_Inter Set.union_iInter
theorem iInter_union (s : ι → Set β) (t : Set β) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.Inter_union Set.iInter_union
theorem iUnion_diff (s : Set β) (t : ι → Set β) : (⋃ i, t i) \ s = ⋃ i, t i \ s :=
iUnion_inter _ _
#align set.Union_diff Set.iUnion_diff
theorem diff_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s \ ⋃ i, t i) = ⋂ i, s \ t i := by
rw [diff_eq, compl_iUnion, inter_iInter]; rfl
#align set.diff_Union Set.diff_iUnion
theorem diff_iInter (s : Set β) (t : ι → Set β) : (s \ ⋂ i, t i) = ⋃ i, s \ t i := by
rw [diff_eq, compl_iInter, inter_iUnion]; rfl
#align set.diff_Inter Set.diff_iInter
theorem directed_on_iUnion {r} {f : ι → Set α} (hd : Directed (· ⊆ ·) f)
(h : ∀ x, DirectedOn r (f x)) : DirectedOn r (⋃ x, f x) := by
simp only [DirectedOn, exists_prop, mem_iUnion, exists_imp]
exact fun a₁ b₁ fb₁ a₂ b₂ fb₂ =>
let ⟨z, zb₁, zb₂⟩ := hd b₁ b₂
let ⟨x, xf, xa₁, xa₂⟩ := h z a₁ (zb₁ fb₁) a₂ (zb₂ fb₂)
⟨x, ⟨z, xf⟩, xa₁, xa₂⟩
#align set.directed_on_Union Set.directed_on_iUnion
theorem iUnion_inter_subset {ι α} {s t : ι → Set α} : ⋃ i, s i ∩ t i ⊆ (⋃ i, s i) ∩ ⋃ i, t i :=
le_iSup_inf_iSup s t
#align set.Union_inter_subset Set.iUnion_inter_subset
theorem iUnion_inter_of_monotone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_monotone hs ht
#align set.Union_inter_of_monotone Set.iUnion_inter_of_monotone
theorem iUnion_inter_of_antitone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_antitone hs ht
#align set.Union_inter_of_antitone Set.iUnion_inter_of_antitone
theorem iInter_union_of_monotone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_monotone hs ht
#align set.Inter_union_of_monotone Set.iInter_union_of_monotone
theorem iInter_union_of_antitone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_antitone hs ht
#align set.Inter_union_of_antitone Set.iInter_union_of_antitone
/-- An equality version of this lemma is `iUnion_iInter_of_monotone` in `Data.Set.Finite`. -/
theorem iUnion_iInter_subset {s : ι → ι' → Set α} : (⋃ j, ⋂ i, s i j) ⊆ ⋂ i, ⋃ j, s i j :=
iSup_iInf_le_iInf_iSup (flip s)
#align set.Union_Inter_subset Set.iUnion_iInter_subset
theorem iUnion_option {ι} (s : Option ι → Set α) : ⋃ o, s o = s none ∪ ⋃ i, s (some i) :=
iSup_option s
#align set.Union_option Set.iUnion_option
theorem iInter_option {ι} (s : Option ι → Set α) : ⋂ o, s o = s none ∩ ⋂ i, s (some i) :=
iInf_option s
#align set.Inter_option Set.iInter_option
section
variable (p : ι → Prop) [DecidablePred p]
theorem iUnion_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 _ _ _
#align set.Union_dite Set.iUnion_dite
theorem iUnion_ite (f g : ι → Set α) :
⋃ i, (if p i then f i else g i) = (⋃ (i) (_ : p i), f i) ∪ ⋃ (i) (_ : ¬p i), g i :=
iUnion_dite _ _ _
#align set.Union_ite Set.iUnion_ite
theorem iInter_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 :=
iInf_dite _ _ _
#align set.Inter_dite Set.iInter_dite
theorem iInter_ite (f g : ι → Set α) :
⋂ i, (if p i then f i else g i) = (⋂ (i) (_ : p i), f i) ∩ ⋂ (i) (_ : ¬p i), g i :=
iInter_dite _ _ _
#align set.Inter_ite Set.iInter_ite
end
theorem image_projection_prod {ι : Type*} {α : ι → Type*} {v : ∀ i : ι, Set (α i)}
(hv : (pi univ v).Nonempty) (i : ι) :
((fun x : ∀ i : ι, α i => x i) '' ⋂ k, (fun x : ∀ j : ι, α j => x k) ⁻¹' v k) = v i := by
classical
apply Subset.antisymm
· simp [iInter_subset]
· intro y y_in
simp only [mem_image, mem_iInter, mem_preimage]
rcases hv with ⟨z, hz⟩
refine' ⟨Function.update z i y, _, update_same i y z⟩
rw [@forall_update_iff ι α _ z i y fun i t => t ∈ v i]
exact ⟨y_in, fun j _ => by simpa using hz j⟩
#align set.image_projection_prod Set.image_projection_prod
/-! ### Unions and intersections indexed by `Prop` -/
theorem iInter_false {s : False → Set α} : iInter s = univ :=
iInf_false
#align set.Inter_false Set.iInter_false
theorem iUnion_false {s : False → Set α} : iUnion s = ∅ :=
iSup_false
#align set.Union_false Set.iUnion_false
@[simp]
theorem iInter_true {s : True → Set α} : iInter s = s trivial :=
iInf_true
#align set.Inter_true Set.iInter_true
@[simp]
theorem iUnion_true {s : True → Set α} : iUnion s = s trivial :=
iSup_true
#align set.Union_true Set.iUnion_true
@[simp]
theorem iInter_exists {p : ι → Prop} {f : Exists p → Set α} :
⋂ x, f x = ⋂ (i) (h : p i), f ⟨i, h⟩ :=
iInf_exists
#align set.Inter_exists Set.iInter_exists
@[simp]
theorem iUnion_exists {p : ι → Prop} {f : Exists p → Set α} :
⋃ x, f x = ⋃ (i) (h : p i), f ⟨i, h⟩ :=
iSup_exists
#align set.Union_exists Set.iUnion_exists
@[simp]
theorem iUnion_empty : (⋃ _ : ι, ∅ : Set α) = ∅ :=
iSup_bot
#align set.Union_empty Set.iUnion_empty
@[simp]
theorem iInter_univ : (⋂ _ : ι, univ : Set α) = univ :=
iInf_top
#align set.Inter_univ Set.iInter_univ
section
variable {s : ι → Set α}
@[simp]
theorem iUnion_eq_empty : ⋃ i, s i = ∅ ↔ ∀ i, s i = ∅ :=
iSup_eq_bot
#align set.Union_eq_empty Set.iUnion_eq_empty
@[simp]
theorem iInter_eq_univ : ⋂ i, s i = univ ↔ ∀ i, s i = univ :=
iInf_eq_top
#align set.Inter_eq_univ Set.iInter_eq_univ
@[simp]
theorem nonempty_iUnion : (⋃ i, s i).Nonempty ↔ ∃ i, (s i).Nonempty := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_Union Set.nonempty_iUnion
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_biUnion {t : Set α} {s : α → Set β} :
(⋃ i ∈ t, s i).Nonempty ↔ ∃ i ∈ t, (s i).Nonempty := by simp
#align set.nonempty_bUnion Set.nonempty_biUnion
theorem iUnion_nonempty_index (s : Set α) (t : s.Nonempty → Set β) :
⋃ h, t h = ⋃ x ∈ s, t ⟨x, ‹_›⟩ :=
iSup_exists
#align set.Union_nonempty_index Set.iUnion_nonempty_index
end
@[simp]
theorem iInter_iInter_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋂ (x) (h : x = b), s x h = s b rfl :=
iInf_iInf_eq_left
#align set.Inter_Inter_eq_left Set.iInter_iInter_eq_left
@[simp]
theorem iInter_iInter_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋂ (x) (h : b = x), s x h = s b rfl :=
iInf_iInf_eq_right
#align set.Inter_Inter_eq_right Set.iInter_iInter_eq_right
@[simp]
theorem iUnion_iUnion_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋃ (x) (h : x = b), s x h = s b rfl :=
iSup_iSup_eq_left
#align set.Union_Union_eq_left Set.iUnion_iUnion_eq_left
@[simp]
theorem iUnion_iUnion_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋃ (x) (h : b = x), s x h = s b rfl :=
iSup_iSup_eq_right
#align set.Union_Union_eq_right Set.iUnion_iUnion_eq_right
theorem iInter_or {p q : Prop} (s : p ∨ q → Set α) :
⋂ h, s h = (⋂ h : p, s (Or.inl h)) ∩ ⋂ h : q, s (Or.inr h) :=
iInf_or
#align set.Inter_or Set.iInter_or
theorem iUnion_or {p q : Prop} (s : p ∨ q → Set α) :
⋃ h, s h = (⋃ i, s (Or.inl i)) ∪ ⋃ j, s (Or.inr j) :=
iSup_or
#align set.Union_or Set.iUnion_or
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iUnion_and {p q : Prop} (s : p ∧ q → Set α) : ⋃ h, s h = ⋃ (hp) (hq), s ⟨hp, hq⟩ :=
iSup_and
#align set.Union_and Set.iUnion_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iInter_and {p q : Prop} (s : p ∧ q → Set α) : ⋂ h, s h = ⋂ (hp) (hq), s ⟨hp, hq⟩ :=
iInf_and
#align set.Inter_and Set.iInter_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iUnion_comm (s : ι → ι' → Set α) : ⋃ (i) (i'), s i i' = ⋃ (i') (i), s i i' :=
iSup_comm
#align set.Union_comm Set.iUnion_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iInter_comm (s : ι → ι' → Set α) : ⋂ (i) (i'), s i i' = ⋂ (i') (i), s i i' :=
iInf_comm
#align set.Inter_comm Set.iInter_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iUnion₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋃ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋃ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iSup₂_comm _
#align set.Union₂_comm Set.iUnion₂_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iInter₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋂ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋂ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iInf₂_comm _
#align set.Inter₂_comm Set.iInter₂_comm
@[simp]
theorem biUnion_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋃ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι']
#align set.bUnion_and Set.biUnion_and
@[simp]
theorem biUnion_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋃ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι]
#align set.bUnion_and' Set.biUnion_and'
@[simp]
theorem biInter_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋂ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iInter_and, @iInter_comm _ ι']
#align set.bInter_and Set.biInter_and
@[simp]
theorem biInter_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋂ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iInter_and, @iInter_comm _ ι]
#align set.bInter_and' Set.biInter_and'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iUnion_iUnion_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋃ (x) (h), s x h = s b (Or.inl rfl) ∪ ⋃ (x) (h : p x), s x (Or.inr h) := by
simp only [iUnion_or, iUnion_union_distrib, iUnion_iUnion_eq_left]
#align set.Union_Union_eq_or_left Set.iUnion_iUnion_eq_or_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iInter_iInter_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋂ (x) (h), s x h = s b (Or.inl rfl) ∩ ⋂ (x) (h : p x), s x (Or.inr h) := by
simp only [iInter_or, iInter_inter_distrib, iInter_iInter_eq_left]
#align set.Inter_Inter_eq_or_left Set.iInter_iInter_eq_or_left
/-! ### Bounded unions and intersections -/
/-- A specialization of `mem_iUnion₂`. -/
theorem mem_biUnion {s : Set α} {t : α → Set β} {x : α} {y : β} (xs : x ∈ s) (ytx : y ∈ t x) :
y ∈ ⋃ x ∈ s, t x :=
mem_iUnion₂_of_mem xs ytx
#align set.mem_bUnion Set.mem_biUnion
/-- A specialization of `mem_iInter₂`. -/
theorem mem_biInter {s : Set α} {t : α → Set β} {y : β} (h : ∀ x ∈ s, y ∈ t x) :
y ∈ ⋂ x ∈ s, t x :=
mem_iInter₂_of_mem h
#align set.mem_bInter Set.mem_biInter
/-- A specialization of `subset_iUnion₂`. -/
theorem subset_biUnion_of_mem {s : Set α} {u : α → Set β} {x : α} (xs : x ∈ s) :
u x ⊆ ⋃ x ∈ s, u x :=
--Porting note: Why is this not just `subset_iUnion₂ x xs`?
@subset_iUnion₂ β α (· ∈ s) (fun i _ => u i) x xs
#align set.subset_bUnion_of_mem Set.subset_biUnion_of_mem
/-- A specialization of `iInter₂_subset`. -/
theorem biInter_subset_of_mem {s : Set α} {t : α → Set β} {x : α} (xs : x ∈ s) :
⋂ x ∈ s, t x ⊆ t x :=
iInter₂_subset x xs
#align set.bInter_subset_of_mem Set.biInter_subset_of_mem
theorem biUnion_subset_biUnion_left {s s' : Set α} {t : α → Set β} (h : s ⊆ s') :
⋃ x ∈ s, t x ⊆ ⋃ x ∈ s', t x :=
iUnion₂_subset fun _ hx => subset_biUnion_of_mem <| h hx
#align set.bUnion_subset_bUnion_left Set.biUnion_subset_biUnion_left
theorem biInter_subset_biInter_left {s s' : Set α} {t : α → Set β} (h : s' ⊆ s) :
⋂ x ∈ s, t x ⊆ ⋂ x ∈ s', t x :=
subset_iInter₂ fun _ hx => biInter_subset_of_mem <| h hx
#align set.bInter_subset_bInter_left Set.biInter_subset_biInter_left
theorem biUnion_mono {s s' : Set α} {t t' : α → Set β} (hs : s' ⊆ s) (h : ∀ x ∈ s, t x ⊆ t' x) :
⋃ x ∈ s', t x ⊆ ⋃ x ∈ s, t' x :=
(biUnion_subset_biUnion_left hs).trans <| iUnion₂_mono h
#align set.bUnion_mono Set.biUnion_mono
theorem biInter_mono {s s' : Set α} {t t' : α → Set β} (hs : s ⊆ s') (h : ∀ x ∈ s, t x ⊆ t' x) :
⋂ x ∈ s', t x ⊆ ⋂ x ∈ s, t' x :=
(biInter_subset_biInter_left hs).trans <| iInter₂_mono h
#align set.bInter_mono Set.biInter_mono
theorem biUnion_eq_iUnion (s : Set α) (t : ∀ x ∈ s, Set β) :
⋃ x ∈ s, t x ‹_› = ⋃ x : s, t x x.2 :=
iSup_subtype'
#align set.bUnion_eq_Union Set.biUnion_eq_iUnion
theorem biInter_eq_iInter (s : Set α) (t : ∀ x ∈ s, Set β) :
⋂ x ∈ s, t x ‹_› = ⋂ x : s, t x x.2 :=
iInf_subtype'
#align set.bInter_eq_Inter Set.biInter_eq_iInter
theorem iUnion_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋃ x : { x // p x }, s x = ⋃ (x) (hx : p x), s ⟨x, hx⟩ :=
iSup_subtype
#align set.Union_subtype Set.iUnion_subtype
theorem iInter_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋂ x : { x // p x }, s x = ⋂ (x) (hx : p x), s ⟨x, hx⟩ :=
iInf_subtype
#align set.Inter_subtype Set.iInter_subtype
theorem biInter_empty (u : α → Set β) : ⋂ x ∈ (∅ : Set α), u x = univ :=
iInf_emptyset
#align set.bInter_empty Set.biInter_empty
theorem biInter_univ (u : α → Set β) : ⋂ x ∈ @univ α, u x = ⋂ x, u x :=
iInf_univ
#align set.bInter_univ Set.biInter_univ
@[simp]
theorem biUnion_self (s : Set α) : ⋃ x ∈ s, s = s :=
Subset.antisymm (iUnion₂_subset fun _ _ => Subset.refl s) fun _ hx => mem_biUnion hx hx
#align set.bUnion_self Set.biUnion_self
@[simp]
theorem iUnion_nonempty_self (s : Set α) : ⋃ _ : s.Nonempty, s = s := by
rw [iUnion_nonempty_index, biUnion_self]
#align set.Union_nonempty_self Set.iUnion_nonempty_self
theorem biInter_singleton (a : α) (s : α → Set β) : ⋂ x ∈ ({a} : Set α), s x = s a :=
iInf_singleton
#align set.bInter_singleton Set.biInter_singleton
theorem biInter_union (s t : Set α) (u : α → Set β) :
⋂ x ∈ s ∪ t, u x = (⋂ x ∈ s, u x) ∩ ⋂ x ∈ t, u x :=
iInf_union
#align set.bInter_union Set.biInter_union
theorem biInter_insert (a : α) (s : Set α) (t : α → Set β) :
⋂ x ∈ insert a s, t x = t a ∩ ⋂ x ∈ s, t x := by simp
#align set.bInter_insert Set.biInter_insert
theorem biInter_pair (a b : α) (s : α → Set β) : ⋂ x ∈ ({a, b} : Set α), s x = s a ∩ s b := by
rw [biInter_insert, biInter_singleton]
#align set.bInter_pair Set.biInter_pair
theorem biInter_inter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, f i ∩ t = (⋂ i ∈ s, f i) ∩ t := by
haveI : Nonempty s := hs.to_subtype
simp [biInter_eq_iInter, ← iInter_inter]
#align set.bInter_inter Set.biInter_inter
theorem inter_biInter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, t ∩ f i = t ∩ ⋂ i ∈ s, f i := by
rw [inter_comm, ← biInter_inter hs]
simp [inter_comm]
#align set.inter_bInter Set.inter_biInter
theorem biUnion_empty (s : α → Set β) : ⋃ x ∈ (∅ : Set α), s x = ∅ :=
iSup_emptyset
#align set.bUnion_empty Set.biUnion_empty
theorem biUnion_univ (s : α → Set β) : ⋃ x ∈ @univ α, s x = ⋃ x, s x :=
iSup_univ
#align set.bUnion_univ Set.biUnion_univ
theorem biUnion_singleton (a : α) (s : α → Set β) : ⋃ x ∈ ({a} : Set α), s x = s a :=
iSup_singleton
#align set.bUnion_singleton Set.biUnion_singleton
@[simp]
theorem biUnion_of_singleton (s : Set α) : ⋃ x ∈ s, {x} = s :=
ext <| by simp
#align set.bUnion_of_singleton Set.biUnion_of_singleton
theorem biUnion_union (s t : Set α) (u : α → Set β) :
⋃ x ∈ s ∪ t, u x = (⋃ x ∈ s, u x) ∪ ⋃ x ∈ t, u x :=
iSup_union
#align set.bUnion_union Set.biUnion_union
@[simp]
theorem iUnion_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋃ i, f i = ⋃ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iUnion_subtype _ _
#align set.Union_coe_set Set.iUnion_coe_set
@[simp]
theorem iInter_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋂ i, f i = ⋂ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iInter_subtype _ _
#align set.Inter_coe_set Set.iInter_coe_set
theorem biUnion_insert (a : α) (s : Set α) (t : α → Set β) :
⋃ x ∈ insert a s, t x = t a ∪ ⋃ x ∈ s, t x := by simp
#align set.bUnion_insert Set.biUnion_insert
theorem biUnion_pair (a b : α) (s : α → Set β) : ⋃ x ∈ ({a, b} : Set α), s x = s a ∪ s b :=
by simp
#align set.bUnion_pair Set.biUnion_pair
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem inter_iUnion₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∩ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ∩ t i j := by simp only [inter_iUnion]
#align set.inter_Union₂ Set.inter_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_inter (s : ∀ i, κ i → Set α) (t : Set α) :
(⋃ (i) (j), s i j) ∩ t = ⋃ (i) (j), s i j ∩ t := by simp_rw [iUnion_inter]
#align set.Union₂_inter Set.iUnion₂_inter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_iInter₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_iInter]
#align set.union_Inter₂ Set.union_iInter₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_union (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [iInter_union]
#align set.Inter₂_union Set.iInter₂_union
theorem mem_sUnion_of_mem {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∈ t) (ht : t ∈ S) :
x ∈ ⋃₀S :=
⟨t, ht, hx⟩
#align set.mem_sUnion_of_mem Set.mem_sUnion_of_mem
-- is this theorem really necessary?
theorem not_mem_of_not_mem_sUnion {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∉ ⋃₀S)
(ht : t ∈ S) : x ∉ t := fun h => hx ⟨t, ht, h⟩
#align set.not_mem_of_not_mem_sUnion Set.not_mem_of_not_mem_sUnion
theorem sInter_subset_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : ⋂₀ S ⊆ t :=
sInf_le tS
#align set.sInter_subset_of_mem Set.sInter_subset_of_mem
theorem subset_sUnion_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : t ⊆ ⋃₀S :=
le_sSup tS
#align set.subset_sUnion_of_mem Set.subset_sUnion_of_mem
theorem subset_sUnion_of_subset {s : Set α} (t : Set (Set α)) (u : Set α) (h₁ : s ⊆ u)
(h₂ : u ∈ t) : s ⊆ ⋃₀t :=
Subset.trans h₁ (subset_sUnion_of_mem h₂)
#align set.subset_sUnion_of_subset Set.subset_sUnion_of_subset
theorem sUnion_subset {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t' ⊆ t) : ⋃₀S ⊆ t :=
sSup_le h
#align set.sUnion_subset Set.sUnion_subset
@[simp]
theorem sUnion_subset_iff {s : Set (Set α)} {t : Set α} : ⋃₀s ⊆ t ↔ ∀ t' ∈ s, t' ⊆ t :=
sSup_le_iff
#align set.sUnion_subset_iff Set.sUnion_subset_iff
/-- `sUnion` is monotone under taking a subset of each set. -/
lemma sUnion_mono_subsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, t ⊆ f t) :
⋃₀ s ⊆ ⋃₀ (f '' s) :=
fun _ ⟨t, htx, hxt⟩ ↦ ⟨f t, mem_image_of_mem f htx, hf t hxt⟩
/-- `sUnion` is monotone under taking a superset of each set. -/
lemma sUnion_mono_supsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, f t ⊆ t) :
⋃₀ (f '' s) ⊆ ⋃₀ s :=
-- If t ∈ f '' s is arbitrary; t = f u for some u : Set α.
fun _ ⟨_, ⟨u, hus, hut⟩, hxt⟩ ↦ ⟨u, hus, (hut ▸ hf u) hxt⟩
theorem subset_sInter {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t ⊆ t') : t ⊆ ⋂₀ S :=
le_sInf h
#align set.subset_sInter Set.subset_sInter
@[simp]
theorem subset_sInter_iff {S : Set (Set α)} {t : Set α} : t ⊆ ⋂₀ S ↔ ∀ t' ∈ S, t ⊆ t' :=
le_sInf_iff
#align set.subset_sInter_iff Set.subset_sInter_iff
@[gcongr]
theorem sUnion_subset_sUnion {S T : Set (Set α)} (h : S ⊆ T) : ⋃₀S ⊆ ⋃₀T :=
sUnion_subset fun _ hs => subset_sUnion_of_mem (h hs)
#align set.sUnion_subset_sUnion Set.sUnion_subset_sUnion
@[gcongr]
theorem sInter_subset_sInter {S T : Set (Set α)} (h : S ⊆ T) : ⋂₀ T ⊆ ⋂₀ S :=
subset_sInter fun _ hs => sInter_subset_of_mem (h hs)
#align set.sInter_subset_sInter Set.sInter_subset_sInter
@[simp]
theorem sUnion_empty : ⋃₀∅ = (∅ : Set α) :=
sSup_empty
#align set.sUnion_empty Set.sUnion_empty
@[simp]
theorem sInter_empty : ⋂₀ ∅ = (univ : Set α) :=
sInf_empty
#align set.sInter_empty Set.sInter_empty
@[simp]
theorem sUnion_singleton (s : Set α) : ⋃₀{s} = s :=
sSup_singleton
#align set.sUnion_singleton Set.sUnion_singleton
@[simp]
theorem sInter_singleton (s : Set α) : ⋂₀ {s} = s :=
sInf_singleton
#align set.sInter_singleton Set.sInter_singleton
@[simp]
theorem sUnion_eq_empty {S : Set (Set α)} : ⋃₀S = ∅ ↔ ∀ s ∈ S, s = ∅ :=
sSup_eq_bot
#align set.sUnion_eq_empty Set.sUnion_eq_empty
@[simp]
theorem sInter_eq_univ {S : Set (Set α)} : ⋂₀ S = univ ↔ ∀ s ∈ S, s = univ :=
sInf_eq_top
#align set.sInter_eq_univ Set.sInter_eq_univ
theorem subset_powerset_iff {s : Set (Set α)} {t : Set α} : s ⊆ 𝒫 t ↔ ⋃₀ s ⊆ t :=
sUnion_subset_iff.symm
/-- `⋃₀` and `𝒫` form a Galois connection. -/
theorem sUnion_powerset_gc :
GaloisConnection (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gc_sSup_Iic
/-- `⋃₀` and `𝒫` form a Galois insertion. -/
def sUnion_powerset_gi :
GaloisInsertion (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gi_sSup_Iic
/-- If all sets in a collection are either `∅` or `Set.univ`, then so is their union. -/
theorem sUnion_mem_empty_univ {S : Set (Set α)} (h : S ⊆ {∅, univ}) :
⋃₀ S ∈ ({∅, univ} : Set (Set α)) := by
simp only [mem_insert_iff, mem_singleton_iff, or_iff_not_imp_left, sUnion_eq_empty, not_forall]
rintro ⟨s, hs, hne⟩
obtain rfl : s = univ := (h hs).resolve_left hne
exact univ_subset_iff.1 <| subset_sUnion_of_mem hs
@[simp]
theorem nonempty_sUnion {S : Set (Set α)} : (⋃₀S).Nonempty ↔ ∃ s ∈ S, Set.Nonempty s := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_sUnion Set.nonempty_sUnion
theorem Nonempty.of_sUnion {s : Set (Set α)} (h : (⋃₀s).Nonempty) : s.Nonempty :=
let ⟨s, hs, _⟩ := nonempty_sUnion.1 h
⟨s, hs⟩
#align set.nonempty.of_sUnion Set.Nonempty.of_sUnion
theorem Nonempty.of_sUnion_eq_univ [Nonempty α] {s : Set (Set α)} (h : ⋃₀s = univ) : s.Nonempty :=
Nonempty.of_sUnion <| h.symm ▸ univ_nonempty
#align set.nonempty.of_sUnion_eq_univ Set.Nonempty.of_sUnion_eq_univ
theorem sUnion_union (S T : Set (Set α)) : ⋃₀(S ∪ T) = ⋃₀S ∪ ⋃₀T :=
sSup_union
#align set.sUnion_union Set.sUnion_union
theorem sInter_union (S T : Set (Set α)) : ⋂₀ (S ∪ T) = ⋂₀ S ∩ ⋂₀ T :=
sInf_union
#align set.sInter_union Set.sInter_union
@[simp]
theorem sUnion_insert (s : Set α) (T : Set (Set α)) : ⋃₀insert s T = s ∪ ⋃₀T :=
sSup_insert
#align set.sUnion_insert Set.sUnion_insert
@[simp]
theorem sInter_insert (s : Set α) (T : Set (Set α)) : ⋂₀ insert s T = s ∩ ⋂₀ T :=
sInf_insert
#align set.sInter_insert Set.sInter_insert
@[simp]
theorem sUnion_diff_singleton_empty (s : Set (Set α)) : ⋃₀(s \ {∅}) = ⋃₀s :=
sSup_diff_singleton_bot s
#align set.sUnion_diff_singleton_empty Set.sUnion_diff_singleton_empty
@[simp]
theorem sInter_diff_singleton_univ (s : Set (Set α)) : ⋂₀ (s \ {univ}) = ⋂₀ s :=
sInf_diff_singleton_top s
#align set.sInter_diff_singleton_univ Set.sInter_diff_singleton_univ
theorem sUnion_pair (s t : Set α) : ⋃₀{s, t} = s ∪ t :=
sSup_pair
#align set.sUnion_pair Set.sUnion_pair
theorem sInter_pair (s t : Set α) : ⋂₀ {s, t} = s ∩ t :=
sInf_pair
#align set.sInter_pair Set.sInter_pair
@[simp]
theorem sUnion_image (f : α → Set β) (s : Set α) : ⋃₀(f '' s) = ⋃ x ∈ s, f x :=
sSup_image
#align set.sUnion_image Set.sUnion_image
@[simp]
theorem sInter_image (f : α → Set β) (s : Set α) : ⋂₀ (f '' s) = ⋂ x ∈ s, f x :=
sInf_image
#align set.sInter_image Set.sInter_image
@[simp]
theorem sUnion_range (f : ι → Set β) : ⋃₀range f = ⋃ x, f x :=
rfl
#align set.sUnion_range Set.sUnion_range
@[simp]
theorem sInter_range (f : ι → Set β) : ⋂₀ range f = ⋂ x, f x :=
rfl
#align set.sInter_range Set.sInter_range
theorem iUnion_eq_univ_iff {f : ι → Set α} : ⋃ i, f i = univ ↔ ∀ x, ∃ i, x ∈ f i := by
simp only [eq_univ_iff_forall, mem_iUnion]
#align set.Union_eq_univ_iff Set.iUnion_eq_univ_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_eq_univ_iff {s : ∀ i, κ i → Set α} :
⋃ (i) (j), s i j = univ ↔ ∀ a, ∃ i j, a ∈ s i j :=
by simp only [iUnion_eq_univ_iff, mem_iUnion]
#align set.Union₂_eq_univ_iff Set.iUnion₂_eq_univ_iff
theorem sUnion_eq_univ_iff {c : Set (Set α)} : ⋃₀c = univ ↔ ∀ a, ∃ b ∈ c, a ∈ b := by
simp only [eq_univ_iff_forall, mem_sUnion]
#align set.sUnion_eq_univ_iff Set.sUnion_eq_univ_iff
-- classical
theorem iInter_eq_empty_iff {f : ι → Set α} : ⋂ i, f i = ∅ ↔ ∀ x, ∃ i, x ∉ f i := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.Inter_eq_empty_iff Set.iInter_eq_empty_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
theorem iInter₂_eq_empty_iff {s : ∀ i, κ i → Set α} :
⋂ (i) (j), s i j = ∅ ↔ ∀ a, ∃ i j, a ∉ s i j := by
simp only [eq_empty_iff_forall_not_mem, mem_iInter, not_forall]
#align set.Inter₂_eq_empty_iff Set.iInter₂_eq_empty_iff
-- classical
theorem sInter_eq_empty_iff {c : Set (Set α)} : ⋂₀ c = ∅ ↔ ∀ a, ∃ b ∈ c, a ∉ b := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.sInter_eq_empty_iff Set.sInter_eq_empty_iff
-- classical
@[simp]
theorem nonempty_iInter {f : ι → Set α} : (⋂ i, f i).Nonempty ↔ ∃ x, ∀ i, x ∈ f i := by
simp [nonempty_iff_ne_empty, iInter_eq_empty_iff]
#align set.nonempty_Inter Set.nonempty_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_iInter₂ {s : ∀ i, κ i → Set α} :
(⋂ (i) (j), s i j).Nonempty ↔ ∃ a, ∀ i j, a ∈ s i j := by
simp
#align set.nonempty_Inter₂ Set.nonempty_iInter₂
-- classical
@[simp]
theorem nonempty_sInter {c : Set (Set α)} : (⋂₀ c).Nonempty ↔ ∃ a, ∀ b ∈ c, a ∈ b := by
simp [nonempty_iff_ne_empty, sInter_eq_empty_iff]
#align set.nonempty_sInter Set.nonempty_sInter
-- classical
theorem compl_sUnion (S : Set (Set α)) : (⋃₀S)ᶜ = ⋂₀ (compl '' S) :=
ext fun x => by simp
#align set.compl_sUnion Set.compl_sUnion
-- classical
theorem sUnion_eq_compl_sInter_compl (S : Set (Set α)) : ⋃₀S = (⋂₀ (compl '' S))ᶜ := by
rw [← compl_compl (⋃₀S), compl_sUnion]
#align set.sUnion_eq_compl_sInter_compl Set.sUnion_eq_compl_sInter_compl
-- classical
theorem compl_sInter (S : Set (Set α)) : (⋂₀ S)ᶜ = ⋃₀(compl '' S) := by
rw [sUnion_eq_compl_sInter_compl, compl_compl_image]
#align set.compl_sInter Set.compl_sInter
-- classical
theorem sInter_eq_compl_sUnion_compl (S : Set (Set α)) : ⋂₀ S = (⋃₀(compl '' S))ᶜ := by
rw [← compl_compl (⋂₀ S), compl_sInter]
#align set.sInter_eq_compl_sUnion_compl Set.sInter_eq_compl_sUnion_compl
theorem inter_empty_of_inter_sUnion_empty {s t : Set α} {S : Set (Set α)} (hs : t ∈ S)
(h : s ∩ ⋃₀S = ∅) : s ∩ t = ∅ :=
eq_empty_of_subset_empty <| by
rw [← h]; exact inter_subset_inter_right _ (subset_sUnion_of_mem hs)
#align set.inter_empty_of_inter_sUnion_empty Set.inter_empty_of_inter_sUnion_empty
theorem range_sigma_eq_iUnion_range {γ : α → Type*} (f : Sigma γ → β) :
range f = ⋃ a, range fun b => f ⟨a, b⟩ :=
Set.ext <| by simp
#align set.range_sigma_eq_Union_range Set.range_sigma_eq_iUnion_range
theorem iUnion_eq_range_sigma (s : α → Set β) : ⋃ i, s i = range fun a : Σi, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_sigma Set.iUnion_eq_range_sigma
theorem iUnion_eq_range_psigma (s : ι → Set β) : ⋃ i, s i = range fun a : Σ'i, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_psigma Set.iUnion_eq_range_psigma
theorem iUnion_image_preimage_sigma_mk_eq_self {ι : Type*} {σ : ι → Type*} (s : Set (Sigma σ)) :
⋃ i, Sigma.mk i '' (Sigma.mk i ⁻¹' s) = s := by
ext x
simp only [mem_iUnion, mem_image, mem_preimage]
constructor
· rintro ⟨i, a, h, rfl⟩
exact h
· intro h
cases' x with i a
exact ⟨i, a, h, rfl⟩
#align set.Union_image_preimage_sigma_mk_eq_self Set.iUnion_image_preimage_sigma_mk_eq_self
theorem Sigma.univ (X : α → Type*) : (Set.univ : Set (Σa, X a)) = ⋃ a, range (Sigma.mk a) :=
Set.ext fun x =>
iff_of_true trivial ⟨range (Sigma.mk x.1), Set.mem_range_self _, x.2, Sigma.eta x⟩
#align set.sigma.univ Set.Sigma.univ
alias sUnion_mono := sUnion_subset_sUnion
#align set.sUnion_mono Set.sUnion_mono
theorem iUnion_subset_iUnion_const {s : Set α} (h : ι → ι₂) : ⋃ _ : ι, s ⊆ ⋃ _ : ι₂, s :=
@iSup_const_mono (Set α) ι ι₂ _ s h
#align set.Union_subset_Union_const Set.iUnion_subset_iUnion_const
@[simp]
theorem iUnion_singleton_eq_range {α β : Type*} (f : α → β) : ⋃ x : α, {f x} = range f := by
ext x
simp [@eq_comm _ x]
#align set.Union_singleton_eq_range Set.iUnion_singleton_eq_range
theorem iUnion_of_singleton (α : Type*) : (⋃ x, {x} : Set α) = univ := by simp [Set.ext_iff]
#align set.Union_of_singleton Set.iUnion_of_singleton
theorem iUnion_of_singleton_coe (s : Set α) : ⋃ i : s, ({(i : α)} : Set α) = s := by simp
#align set.Union_of_singleton_coe Set.iUnion_of_singleton_coe
theorem sUnion_eq_biUnion {s : Set (Set α)} : ⋃₀s = ⋃ (i : Set α) (_ : i ∈ s), i := by
rw [← sUnion_image, image_id']
#align set.sUnion_eq_bUnion Set.sUnion_eq_biUnion
theorem sInter_eq_biInter {s : Set (Set α)} : ⋂₀ s = ⋂ (i : Set α) (_ : i ∈ s), i := by
rw [← sInter_image, image_id']
#align set.sInter_eq_bInter Set.sInter_eq_biInter
theorem sUnion_eq_iUnion {s : Set (Set α)} : ⋃₀s = ⋃ i : s, i := by
simp only [← sUnion_range, Subtype.range_coe]
#align set.sUnion_eq_Union Set.sUnion_eq_iUnion
theorem sInter_eq_iInter {s : Set (Set α)} : ⋂₀ s = ⋂ i : s, i := by
simp only [← sInter_range, Subtype.range_coe]
#align set.sInter_eq_Inter Set.sInter_eq_iInter
@[simp]
theorem iUnion_of_empty [IsEmpty ι] (s : ι → Set α) : ⋃ i, s i = ∅ :=
iSup_of_empty _
#align set.Union_of_empty Set.iUnion_of_empty
@[simp]
theorem iInter_of_empty [IsEmpty ι] (s : ι → Set α) : ⋂ i, s i = univ :=
iInf_of_empty _
#align set.Inter_of_empty Set.iInter_of_empty
theorem union_eq_iUnion {s₁ s₂ : Set α} : s₁ ∪ s₂ = ⋃ b : Bool, cond b s₁ s₂ :=
sup_eq_iSup s₁ s₂
#align set.union_eq_Union Set.union_eq_iUnion
theorem inter_eq_iInter {s₁ s₂ : Set α} : s₁ ∩ s₂ = ⋂ b : Bool, cond b s₁ s₂ :=
inf_eq_iInf s₁ s₂
#align set.inter_eq_Inter Set.inter_eq_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_union_sInter {S T : Set (Set α)} :
⋂₀ S ∪ ⋂₀ T = ⋂ p ∈ S ×ˢ T, (p : Set α × Set α).1 ∪ p.2 :=
sInf_sup_sInf
#align set.sInter_union_sInter Set.sInter_union_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_inter_sUnion {s t : Set (Set α)} :
⋃₀s ∩ ⋃₀t = ⋃ p ∈ s ×ˢ t, (p : Set α × Set α).1 ∩ p.2 :=
sSup_inf_sSup
#align set.sUnion_inter_sUnion Set.sUnion_inter_sUnion
theorem biUnion_iUnion (s : ι → Set α) (t : α → Set β) :
⋃ x ∈ ⋃ i, s i, t x = ⋃ (i) (x ∈ s i), t x := by simp [@iUnion_comm _ ι]
#align set.bUnion_Union Set.biUnion_iUnion
theorem biInter_iUnion (s : ι → Set α) (t : α → Set β) :
⋂ x ∈ ⋃ i, s i, t x = ⋂ (i) (x ∈ s i), t x := by simp [@iInter_comm _ ι]
#align set.bInter_Union Set.biInter_iUnion
theorem sUnion_iUnion (s : ι → Set (Set α)) : ⋃₀⋃ i, s i = ⋃ i, ⋃₀s i := by
simp only [sUnion_eq_biUnion, biUnion_iUnion]
#align set.sUnion_Union Set.sUnion_iUnion
theorem sInter_iUnion (s : ι → Set (Set α)) : ⋂₀ ⋃ i, s i = ⋂ i, ⋂₀ s i := by
simp only [sInter_eq_biInter, biInter_iUnion]
#align set.sInter_Union Set.sInter_iUnion
theorem iUnion_range_eq_sUnion {α β : Type*} (C : Set (Set α)) {f : ∀ s : C, β → (s : Type _)}
(hf : ∀ s : C, Surjective (f s)) : ⋃ y : β, range (fun s : C => (f s y).val) = ⋃₀C := by
ext x; constructor
· rintro ⟨s, ⟨y, rfl⟩, ⟨s, hs⟩, rfl⟩
refine' ⟨_, hs, _⟩
exact (f ⟨s, hs⟩ y).2
· rintro ⟨s, hs, hx⟩
cases' hf ⟨s, hs⟩ ⟨x, hx⟩ with y hy
refine' ⟨_, ⟨y, rfl⟩, ⟨s, hs⟩, _⟩
exact congr_arg Subtype.val hy
#align set.Union_range_eq_sUnion Set.iUnion_range_eq_sUnion
theorem iUnion_range_eq_iUnion (C : ι → Set α) {f : ∀ x : ι, β → C x}
(hf : ∀ x : ι, Surjective (f x)) : ⋃ y : β, range (fun x : ι => (f x y).val) = ⋃ x, C x := by
ext x; rw [mem_iUnion, mem_iUnion]; constructor
· rintro ⟨y, i, rfl⟩
exact ⟨i, (f i y).2⟩
· rintro ⟨i, hx⟩
cases' hf i ⟨x, hx⟩ with y hy
exact ⟨y, i, congr_arg Subtype.val hy⟩
#align set.Union_range_eq_Union Set.iUnion_range_eq_iUnion
theorem union_distrib_iInter_left (s : ι → Set α) (t : Set α) : (t ∪ ⋂ i, s i) = ⋂ i, t ∪ s i :=
sup_iInf_eq _ _
#align set.union_distrib_Inter_left Set.union_distrib_iInter_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_left (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_distrib_iInter_left]
#align set.union_distrib_Inter₂_left Set.union_distrib_iInter₂_left
theorem union_distrib_iInter_right (s : ι → Set α) (t : Set α) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.union_distrib_Inter_right Set.union_distrib_iInter_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_right (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [union_distrib_iInter_right]
#align set.union_distrib_Inter₂_right Set.union_distrib_iInter₂_right
section Function
/-! ### `mapsTo` -/
theorem mapsTo_sUnion {S : Set (Set α)} {t : Set β} {f : α → β} (H : ∀ s ∈ S, MapsTo f s t) :
MapsTo f (⋃₀S) t := fun _ ⟨s, hs, hx⟩ => H s hs hx
#align set.maps_to_sUnion Set.mapsTo_sUnion
theorem mapsTo_iUnion {s : ι → Set α} {t : Set β} {f : α → β} (H : ∀ i, MapsTo f (s i) t) :
MapsTo f (⋃ i, s i) t :=
mapsTo_sUnion <| forall_range_iff.2 H
#align set.maps_to_Union Set.mapsTo_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂ {s : ∀ i, κ i → Set α} {t : Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) t) : MapsTo f (⋃ (i) (j), s i j) t :=
mapsTo_iUnion fun i => mapsTo_iUnion (H i)
#align set.maps_to_Union₂ Set.mapsTo_iUnion₂
theorem mapsTo_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋃ i, s i) (⋃ i, t i) :=
mapsTo_iUnion fun i => (H i).mono (Subset.refl _) (subset_iUnion t i)
#align set.maps_to_Union_Union Set.mapsTo_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
mapsTo_iUnion_iUnion fun i => mapsTo_iUnion_iUnion (H i)
#align set.maps_to_Union₂_Union₂ Set.mapsTo_iUnion₂_iUnion₂
theorem mapsTo_sInter {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, MapsTo f s t) :
MapsTo f s (⋂₀ T) := fun _ hx t ht => H t ht hx
#align set.maps_to_sInter Set.mapsTo_sInter
theorem mapsTo_iInter {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, MapsTo f s (t i)) :
MapsTo f s (⋂ i, t i) := fun _ hx => mem_iInter.2 fun i => H i hx
#align set.maps_to_Inter Set.mapsTo_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f s (t i j)) : MapsTo f s (⋂ (i) (j), t i j) :=
mapsTo_iInter fun i => mapsTo_iInter (H i)
#align set.maps_to_Inter₂ Set.mapsTo_iInter₂
theorem mapsTo_iInter_iInter {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋂ i, s i) (⋂ i, t i) :=
mapsTo_iInter fun i => (H i).mono (iInter_subset s i) (Subset.refl _)
#align set.maps_to_Inter_Inter Set.mapsTo_iInter_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂_iInter₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋂ (i) (j), s i j) (⋂ (i) (j), t i j) :=
mapsTo_iInter_iInter fun i => mapsTo_iInter_iInter (H i)
#align set.maps_to_Inter₂_Inter₂ Set.mapsTo_iInter₂_iInter₂
theorem image_iInter_subset (s : ι → Set α) (f : α → β) : (f '' ⋂ i, s i) ⊆ ⋂ i, f '' s i :=
(mapsTo_iInter_iInter fun i => mapsTo_image f (s i)).image_subset
#align set.image_Inter_subset Set.image_iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂_subset (s : ∀ i, κ i → Set α) (f : α → β) :
(f '' ⋂ (i) (j), s i j) ⊆ ⋂ (i) (j), f '' s i j :=
(mapsTo_iInter₂_iInter₂ fun i hi => mapsTo_image f (s i hi)).image_subset
#align set.image_Inter₂_subset Set.image_iInter₂_subset
theorem image_sInter_subset (S : Set (Set α)) (f : α → β) : f '' ⋂₀ S ⊆ ⋂ s ∈ S, f '' s := by
rw [sInter_eq_biInter]
apply image_iInter₂_subset
#align set.image_sInter_subset Set.image_sInter_subset
/-! ### `restrictPreimage` -/
section
open Function
variable (s : Set β) {f : α → β} {U : ι → Set β} (hU : iUnion U = univ)
theorem injective_iff_injective_of_iUnion_eq_univ :
Injective f ↔ ∀ i, Injective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_injective H, fun H x y e => _⟩
obtain ⟨i, hi⟩ := Set.mem_iUnion.mp
(show f x ∈ Set.iUnion U by rw [hU]; triv)
injection @H i ⟨x, hi⟩ ⟨y, show f y ∈ U i from e ▸ hi⟩ (Subtype.ext e)
#align set.injective_iff_injective_of_Union_eq_univ Set.injective_iff_injective_of_iUnion_eq_univ
theorem surjective_iff_surjective_of_iUnion_eq_univ :
Surjective f ↔ ∀ i, Surjective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_surjective H, fun H x => _⟩
obtain ⟨i, hi⟩ :=
Set.mem_iUnion.mp
(show x ∈ Set.iUnion U by rw [hU]; triv)
exact ⟨_, congr_arg Subtype.val (H i ⟨x, hi⟩).choose_spec⟩
#align set.surjective_iff_surjective_of_Union_eq_univ Set.surjective_iff_surjective_of_iUnion_eq_univ
theorem bijective_iff_bijective_of_iUnion_eq_univ :
Bijective f ↔ ∀ i, Bijective ((U i).restrictPreimage f) := by
rw [Bijective, injective_iff_injective_of_iUnion_eq_univ hU,
surjective_iff_surjective_of_iUnion_eq_univ hU]
simp [Bijective, forall_and]
#align set.bijective_iff_bijective_of_Union_eq_univ Set.bijective_iff_bijective_of_iUnion_eq_univ
end
/-! ### `InjOn` -/
theorem InjOn.image_iInter_eq [Nonempty ι] {s : ι → Set α} {f : α → β} (h : InjOn f (⋃ i, s i)) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
inhabit ι
refine' Subset.antisymm (image_iInter_subset s f) fun y hy => _
simp only [mem_iInter, mem_image_iff_bex] at hy
choose x hx hy using hy
refine' ⟨x default, mem_iInter.2 fun i => _, hy _⟩
suffices x default = x i by
rw [this]
apply hx
replace hx : ∀ i, x i ∈ ⋃ j, s j := fun i => (subset_iUnion _ _) (hx i)
apply h (hx _) (hx _)
simp only [hy]
#align set.inj_on.image_Inter_eq Set.InjOn.image_iInter_eq
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
theorem InjOn.image_biInter_eq {p : ι → Prop} {s : ∀ i, p i → Set α} (hp : ∃ i, p i)
{f : α → β} (h : InjOn f (⋃ (i) (hi), s i hi)) :
(f '' ⋂ (i) (hi), s i hi) = ⋂ (i) (hi), f '' s i hi := by
simp only [iInter, iInf_subtype']
haveI : Nonempty { i // p i } := nonempty_subtype.2 hp
apply InjOn.image_iInter_eq
simpa only [iUnion, iSup_subtype'] using h
#align set.inj_on.image_bInter_eq Set.InjOn.image_biInter_eq
theorem image_iInter {f : α → β} (hf : Bijective f) (s : ι → Set α) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
cases isEmpty_or_nonempty ι
· simp_rw [iInter_of_empty, image_univ_of_surjective hf.surjective]
· exact (hf.injective.injOn _).image_iInter_eq
#align set.image_Inter Set.image_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂ {f : α → β} (hf : Bijective f) (s : ∀ i, κ i → Set α) :
(f '' ⋂ (i) (j), s i j) = ⋂ (i) (j), f '' s i j := by simp_rw [image_iInter hf]
#align set.image_Inter₂ Set.image_iInter₂
theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
(hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i) := by
intro x hx y hy hxy
rcases mem_iUnion.1 hx with ⟨i, hx⟩
rcases mem_iUnion.1 hy with ⟨j, hy⟩
rcases hs i j with ⟨k, hi, hj⟩
exact hf k (hi hx) (hj hy) hxy
#align set.inj_on_Union_of_directed Set.inj_on_iUnion_of_directed
/-! ### `SurjOn` -/
theorem surjOn_sUnion {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, SurjOn f s t) :
SurjOn f s (⋃₀T) := fun _ ⟨t, ht, hx⟩ => H t ht hx
#align set.surj_on_sUnion Set.surjOn_sUnion
theorem surjOn_iUnion {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, SurjOn f s (t i)) :
SurjOn f s (⋃ i, t i) :=
surjOn_sUnion <| forall_range_iff.2 H
#align set.surj_on_Union Set.surjOn_iUnion
theorem surjOn_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) : SurjOn f (⋃ i, s i) (⋃ i, t i) :=
surjOn_iUnion fun i => (H i).mono (subset_iUnion _ _) (Subset.refl _)
#align set.surj_on_Union_Union Set.surjOn_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f s (t i j)) : SurjOn f s (⋃ (i) (j), t i j) :=
surjOn_iUnion fun i => surjOn_iUnion (H i)
#align set.surj_on_Union₂ Set.surjOn_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f (s i j) (t i j)) : SurjOn f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
surjOn_iUnion_iUnion fun i => surjOn_iUnion_iUnion (H i)
#align set.surj_on_Union₂_Union₂ Set.surjOn_iUnion₂_iUnion₂
theorem surjOn_iInter [Nonempty ι] {s : ι → Set α} {t : Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) t) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) t := by
intro y hy
rw [Hinj.image_iInter_eq, mem_iInter]
exact fun i => H i hy
#align set.surj_on_Inter Set.surjOn_iInter
theorem surjOn_iInter_iInter [Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) (⋂ i, t i) :=
surjOn_iInter (fun i => (H i).mono (Subset.refl _) (iInter_subset _ _)) Hinj
#align set.surj_on_Inter_Inter Set.surjOn_iInter_iInter
/-! ### `BijOn` -/
theorem bijOn_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i))
(Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
⟨mapsTo_iUnion_iUnion fun i => (H i).mapsTo, Hinj, surjOn_iUnion_iUnion fun i => (H i).surjOn⟩
#align set.bij_on_Union Set.bijOn_iUnion
theorem bijOn_iInter [hi : Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, BijOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
⟨mapsTo_iInter_iInter fun i => (H i).mapsTo,
hi.elim fun i => (H i).injOn.mono (iInter_subset _ _),
surjOn_iInter_iInter (fun i => (H i).surjOn) Hinj⟩
#align set.bij_on_Inter Set.bijOn_iInter
theorem bijOn_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {t : ι → Set β}
{f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
bijOn_iUnion H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Union_of_directed Set.bijOn_iUnion_of_directed
theorem bijOn_iInter_of_directed [Nonempty ι] {s : ι → Set α} (hs : Directed (· ⊆ ·) s)
{t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
bijOn_iInter H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Inter_of_directed Set.bijOn_iInter_of_directed
end Function
/-! ### `image`, `preimage` -/
section Image
theorem image_iUnion {f : α → β} {s : ι → Set α} : (f '' ⋃ i, s i) = ⋃ i, f '' s i := by
ext1 x
simp only [mem_image, mem_iUnion, ← exists_and_right, ← exists_and_left]
--Porting note: `exists_swap` causes a `simp` loop in Lean4 so we use `rw` instead.
rw [exists_swap]
#align set.image_Union Set.image_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iUnion₂ (f : α → β) (s : ∀ i, κ i → Set α) :
(f '' ⋃ (i) (j), s i j) = ⋃ (i) (j), f '' s i j := by simp_rw [image_iUnion]
#align set.image_Union₂ Set.image_iUnion₂
theorem univ_subtype {p : α → Prop} : (univ : Set (Subtype p)) = ⋃ (x) (h : p x), {⟨x, h⟩} :=
Set.ext fun ⟨x, h⟩ => by simp [h]
#align set.univ_subtype Set.univ_subtype
theorem range_eq_iUnion {ι} (f : ι → α) : range f = ⋃ i, {f i} :=
Set.ext fun a => by simp [@eq_comm α a]
#align set.range_eq_Union Set.range_eq_iUnion
theorem image_eq_iUnion (f : α → β) (s : Set α) : f '' s = ⋃ i ∈ s, {f i} :=
Set.ext fun b => by simp [@eq_comm β b]
#align set.image_eq_Union Set.image_eq_iUnion
theorem biUnion_range {f : ι → α} {g : α → Set β} : ⋃ x ∈ range f, g x = ⋃ y, g (f y) :=
iSup_range
#align set.bUnion_range Set.biUnion_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iUnion_iUnion_eq' {f : ι → α} {g : α → Set β} :
⋃ (x) (y) (_ : f y = x), g x = ⋃ y, g (f y) := by simpa using biUnion_range
#align set.Union_Union_eq' Set.iUnion_iUnion_eq'
theorem biInter_range {f : ι → α} {g : α → Set β} : ⋂ x ∈ range f, g x = ⋂ y, g (f y) :=
iInf_range
#align set.bInter_range Set.biInter_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iInter_iInter_eq' {f : ι → α} {g : α → Set β} :
⋂ (x) (y) (_ : f y = x), g x = ⋂ y, g (f y) := by simpa using biInter_range
#align set.Inter_Inter_eq' Set.iInter_iInter_eq'
variable {s : Set γ} {f : γ → α} {g : α → Set β}
theorem biUnion_image : ⋃ x ∈ f '' s, g x = ⋃ y ∈ s, g (f y) :=
iSup_image
#align set.bUnion_image Set.biUnion_image
theorem biInter_image : ⋂ x ∈ f '' s, g x = ⋂ y ∈ s, g (f y) :=
iInf_image
#align set.bInter_image Set.biInter_image
end Image
section Preimage
theorem monotone_preimage {f : α → β} : Monotone (preimage f) := fun _ _ h => preimage_mono h
#align set.monotone_preimage Set.monotone_preimage
@[simp]
theorem preimage_iUnion {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋃ i, s i) = ⋃ i, f ⁻¹' s i :=
Set.ext <| by simp [preimage]
#align set.preimage_Union Set.preimage_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iUnion₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋃ (i) (j), s i j) = ⋃ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iUnion]
#align set.preimage_Union₂ Set.preimage_iUnion₂
@[simp]
theorem preimage_sUnion {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋃₀s = ⋃ t ∈ s, f ⁻¹' t := by
rw [sUnion_eq_biUnion, preimage_iUnion₂]
#align set.preimage_sUnion Set.preimage_sUnion
theorem preimage_iInter {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋂ i, s i) = ⋂ i, f ⁻¹' s i := by
ext; simp
#align set.preimage_Inter Set.preimage_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iInter₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋂ (i) (j), s i j) = ⋂ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iInter]
#align set.preimage_Inter₂ Set.preimage_iInter₂
@[simp]
theorem preimage_sInter {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋂₀ s = ⋂ t ∈ s, f ⁻¹' t := by
rw [sInter_eq_biInter, preimage_iInter₂]
#align set.preimage_sInter Set.preimage_sInter
@[simp]
theorem biUnion_preimage_singleton (f : α → β) (s : Set β) : ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s := by
rw [← preimage_iUnion₂, biUnion_of_singleton]
#align set.bUnion_preimage_singleton Set.biUnion_preimage_singleton
theorem biUnion_range_preimage_singleton (f : α → β) : ⋃ y ∈ range f, f ⁻¹' {y} = univ := by
rw [biUnion_preimage_singleton, preimage_range]
#align set.bUnion_range_preimage_singleton Set.biUnion_range_preimage_singleton
end Preimage
section Prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion {s : Set α} {t : ι → Set β} : (s ×ˢ ⋃ i, t i) = ⋃ i, s ×ˢ t i := by
ext
simp
#align set.prod_Union Set.prod_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} :
(s ×ˢ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ×ˢ t i j := by simp_rw [prod_iUnion]
#align set.prod_Union₂ Set.prod_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sUnion {s : Set α} {C : Set (Set β)} : s ×ˢ ⋃₀C = ⋃₀((fun t => s ×ˢ t) '' C) := by
simp_rw [sUnion_eq_biUnion, biUnion_image, prod_iUnion₂]
#align set.prod_sUnion Set.prod_sUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_const {s : ι → Set α} {t : Set β} : (⋃ i, s i) ×ˢ t = ⋃ i, s i ×ˢ t := by
ext
simp
#align set.Union_prod_const Set.iUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion₂_prod_const {s : ∀ i, κ i → Set α} {t : Set β} :
(⋃ (i) (j), s i j) ×ˢ t = ⋃ (i) (j), s i j ×ˢ t := by simp_rw [iUnion_prod_const]
#align set.Union₂_prod_const Set.iUnion₂_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_prod_const {C : Set (Set α)} {t : Set β} :
⋃₀C ×ˢ t = ⋃₀((fun s : Set α => s ×ˢ t) '' C) := by
simp only [sUnion_eq_biUnion, iUnion₂_prod_const, biUnion_image]
#align set.sUnion_prod_const Set.sUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod {ι ι' α β} (s : ι → Set α) (t : ι' → Set β) :
⋃ x : ι × ι', s x.1 ×ˢ t x.2 = (⋃ i : ι, s i) ×ˢ ⋃ i : ι', t i := by
ext
simp
#align set.Union_prod Set.iUnion_prod
/-- Analogue of `iSup_prod` for sets. -/
lemma iUnion_prod' (f : β × γ → Set α) : ⋃ x : β × γ, f x = ⋃ (i : β) (j : γ), f (i, j) :=
iSup_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x := by
ext ⟨z, w⟩; simp only [mem_prod, mem_iUnion, exists_imp, and_imp, iff_def]; constructor
· intro x hz hw
exact ⟨⟨x, hz⟩, x, hw⟩
· intro x hz x' hw
exact ⟨x ⊔ x', hs le_sup_left hz, ht le_sup_right hw⟩
#align set.Union_prod_of_monotone Set.iUnion_prod_of_monotone
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter_subset (S : Set (Set α)) (T : Set (Set β)) :
⋂₀ S ×ˢ ⋂₀ T ⊆ ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 :=
subset_iInter₂ fun x hx _ hy => ⟨hy.1 x.1 hx.1, hy.2 x.2 hx.2⟩
#align set.sInter_prod_sInter_subset Set.sInter_prod_sInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 := by
obtain ⟨s₁, h₁⟩ := hS
obtain ⟨s₂, h₂⟩ := hT
refine' Set.Subset.antisymm (sInter_prod_sInter_subset S T) fun x hx => _
rw [mem_iInter₂] at hx
exact ⟨fun s₀ h₀ => (hx (s₀, s₂) ⟨h₀, h₂⟩).1, fun s₀ h₀ => (hx (s₁, s₀) ⟨h₁, h₀⟩).2⟩
#align set.sInter_prod_sInter Set.sInter_prod_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod {S : Set (Set α)} (hS : S.Nonempty) (t : Set β) :
⋂₀ S ×ˢ t = ⋂ s ∈ S, s ×ˢ t := by
rw [← sInter_singleton t, sInter_prod_sInter hS (singleton_nonempty t), sInter_singleton]
simp_rw [prod_singleton, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.sInter_prod Set.sInter_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sInter {T : Set (Set β)} (hT : T.Nonempty) (s : Set α) :
s ×ˢ ⋂₀ T = ⋂ t ∈ T, s ×ˢ t := by
rw [← sInter_singleton s, sInter_prod_sInter (singleton_nonempty s) hT, sInter_singleton]
simp_rw [singleton_prod, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.prod_sInter Set.prod_sInter
end Prod
section Image2
variable (f : α → β → γ) {s : Set α} {t : Set β}
/-- The `Set.image2` version of `Set.image_eq_iUnion` -/
theorem image2_eq_iUnion (s : Set α) (t : Set β) : image2 f s t = ⋃ (i ∈ s) (j ∈ t), {f i j} := by
ext; simp [eq_comm]
#align set.image2_eq_Union Set.image2_eq_iUnion
theorem iUnion_image_left : ⋃ a ∈ s, f a '' t = image2 f s t := by
simp only [image2_eq_iUnion, image_eq_iUnion]
#align set.Union_image_left Set.iUnion_image_left
theorem iUnion_image_right : ⋃ b ∈ t, (f · b) '' s = image2 f s t := by
rw [image2_swap, iUnion_image_left]
#align set.Union_image_right Set.iUnion_image_right
theorem image2_iUnion_left (s : ι → Set α) (t : Set β) :
image2 f (⋃ i, s i) t = ⋃ i, image2 f (s i) t := by
simp only [← image_prod, iUnion_prod_const, image_iUnion]
#align set.image2_Union_left Set.image2_iUnion_left
theorem image2_iUnion_right (s : Set α) (t : ι → Set β) :
image2 f s (⋃ i, t i) = ⋃ i, image2 f s (t i) := by
simp only [← image_prod, prod_iUnion, image_iUnion]
#align set.image2_Union_right Set.image2_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋃ (i) (j), s i j) t = ⋃ (i) (j), image2 f (s i j) t := by simp_rw [image2_iUnion_left]
#align set.image2_Union₂_left Set.image2_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋃ (i) (j), t i j) = ⋃ (i) (j), image2 f s (t i j) :=
by simp_rw [image2_iUnion_right]
#align set.image2_Union₂_right Set.image2_iUnion₂_right
theorem image2_iInter_subset_left (s : ι → Set α) (t : Set β) :
image2 f (⋂ i, s i) t ⊆ ⋂ i, image2 f (s i) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem (hx _) hy
#align set.image2_Inter_subset_left Set.image2_iInter_subset_left
theorem image2_iInter_subset_right (s : Set α) (t : ι → Set β) :
image2 f s (⋂ i, t i) ⊆ ⋂ i, image2 f s (t i) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem hx (hy _)
#align set.image2_Inter_subset_right Set.image2_iInter_subset_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋂ (i) (j), s i j) t ⊆ ⋂ (i) (j), image2 f (s i j) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem (hx _ _) hy
#align set.image2_Inter₂_subset_left Set.image2_iInter₂_subset_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), image2 f s (t i j) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem hx (hy _ _)
#align set.image2_Inter₂_subset_right Set.image2_iInter₂_subset_right
theorem prod_eq_biUnion_left : s ×ˢ t = ⋃ a ∈ s, (fun b => (a, b)) '' t := by
rw [iUnion_image_left, image2_mk_eq_prod]
#align set.prod_eq_bUnion_left Set.prod_eq_biUnion_left
theorem prod_eq_biUnion_right : s ×ˢ t = ⋃ b ∈ t, (fun a => (a, b)) '' s := by
rw [iUnion_image_right, image2_mk_eq_prod]
#align set.prod_eq_bUnion_right Set.prod_eq_biUnion_right
end Image2
section Seq
/-- Given a set `s` of functions `α → β` and `t : Set α`, `seq s t` is the union of `f '' t` over
all `f ∈ s`. -/
def seq (s : Set (α → β)) (t : Set α) : Set β :=
{ b | ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b }
#align set.seq Set.seq
@[simp]
theorem mem_seq_iff {s : Set (α → β)} {t : Set α} {b : β} :
b ∈ seq s t ↔ ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b :=
Iff.rfl
#align set.mem_seq_iff Set.mem_seq_iff
lemma seq_eq_image2 (s : Set (α → β)) (t : Set α) : seq s t = image2 (fun f a ↦ f a) s t := by
ext; simp
theorem seq_def {s : Set (α → β)} {t : Set α} : seq s t = ⋃ f ∈ s, f '' t := by
rw [seq_eq_image2, iUnion_image_left]
#align set.seq_def Set.seq_def
theorem seq_subset {s : Set (α → β)} {t : Set α} {u : Set β} :
seq s t ⊆ u ↔ ∀ f ∈ s, ∀ a ∈ t, (f : α → β) a ∈ u := by
rw [seq_eq_image2, image2_subset_iff]
#align set.seq_subset Set.seq_subset
@[gcongr]
theorem seq_mono {s₀ s₁ : Set (α → β)} {t₀ t₁ : Set α} (hs : s₀ ⊆ s₁) (ht : t₀ ⊆ t₁) :
seq s₀ t₀ ⊆ seq s₁ t₁ := fun _ ⟨f, hf, a, ha, eq⟩ => ⟨f, hs hf, a, ht ha, eq⟩
#align set.seq_mono Set.seq_mono
theorem singleton_seq {f : α → β} {t : Set α} : Set.seq ({f} : Set (α → β)) t = f '' t := by
rw [seq_eq_image2, image2_singleton_left]
#align set.singleton_seq Set.singleton_seq
theorem seq_singleton {s : Set (α → β)} {a : α} : Set.seq s {a} = (fun f : α → β => f a) '' s := by
rw [seq_eq_image2, image2_singleton_right]
#align set.seq_singleton Set.seq_singleton
theorem seq_seq {s : Set (β → γ)} {t : Set (α → β)} {u : Set α} :
seq s (seq t u) = seq (seq ((· ∘ ·) '' s) t) u := by
simp only [seq_eq_image2, image2_image_left, image2_image2_left, image2_image2_right, comp_apply]
#align set.seq_seq Set.seq_seq
theorem image_seq {f : β → γ} {s : Set (α → β)} {t : Set α} :
f '' seq s t = seq ((f ∘ ·) '' s) t := by
rw [← singleton_seq, ← singleton_seq, seq_seq, image_singleton]
#align set.image_seq Set.image_seq
theorem prod_eq_seq {s : Set α} {t : Set β} : s ×ˢ t = (Prod.mk '' s).seq t := by
rw [seq_eq_image2, image2_image_left, image2_mk_eq_prod]
#align set.prod_eq_seq Set.prod_eq_seq
theorem prod_image_seq_comm (s : Set α) (t : Set β) :
(Prod.mk '' s).seq t = seq ((fun b a => (a, b)) '' t) s := by
rw [← prod_eq_seq, ← image_swap_prod, prod_eq_seq, image_seq, ← image_comp]; rfl
#align set.prod_image_seq_comm Set.prod_image_seq_comm
theorem image2_eq_seq (f : α → β → γ) (s : Set α) (t : Set β) : image2 f s t = seq (f '' s) t := by
rw [seq_eq_image2, image2_image_left]
#align set.image2_eq_seq Set.image2_eq_seq
end Seq
section Pi
variable {π : α → Type*}
theorem pi_def (i : Set α) (s : ∀ a, Set (π a)) : pi i s = ⋂ a ∈ i, eval a ⁻¹' s a := by
ext
simp
#align set.pi_def Set.pi_def
theorem univ_pi_eq_iInter (t : ∀ i, Set (π i)) : pi univ t = ⋂ i, eval i ⁻¹' t i := by
simp only [pi_def, iInter_true, mem_univ]
#align set.univ_pi_eq_Inter Set.univ_pi_eq_iInter
theorem pi_diff_pi_subset (i : Set α) (s t : ∀ a, Set (π a)) :
pi i s \ pi i t ⊆ ⋃ a ∈ i, eval a ⁻¹' (s a \ t a) := by
refine' diff_subset_comm.2 fun x hx a ha => _
simp only [mem_diff, mem_pi, mem_iUnion, not_exists, mem_preimage, not_and, not_not,
eval_apply] at hx
exact hx.2 _ ha (hx.1 _ ha)
#align set.pi_diff_pi_subset Set.pi_diff_pi_subset
theorem iUnion_univ_pi {ι : α → Type*} (t : (a : α) → ι a → Set (π a)) :
⋃ x : (a : α) → ι a, pi univ (fun a => t a (x a)) = pi univ fun a => ⋃ j : ι a, t a j := by
ext
simp [Classical.skolem]
#align set.Union_univ_pi Set.iUnion_univ_pi
end Pi
end Set
namespace Function
namespace Surjective
theorem iUnion_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋃ x, g (f x) = ⋃ y, g y :=
hf.iSup_comp g
#align function.surjective.Union_comp Function.Surjective.iUnion_comp
theorem iInter_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋂ x, g (f x) = ⋂ y, g y :=
hf.iInf_comp g
#align function.surjective.Inter_comp Function.Surjective.iInter_comp
end Surjective
end Function
/-!
### Disjoint sets
-/
section Disjoint
variable {s t u : Set α} {f : α → β}
namespace Set
@[simp]
theorem disjoint_iUnion_left {ι : Sort*} {s : ι → Set α} :
Disjoint (⋃ i, s i) t ↔ ∀ i, Disjoint (s i) t :=
iSup_disjoint_iff
#align set.disjoint_Union_left Set.disjoint_iUnion_left
@[simp]
theorem disjoint_iUnion_right {ι : Sort*} {s : ι → Set α} :
Disjoint t (⋃ i, s i) ↔ ∀ i, Disjoint t (s i) :=
disjoint_iSup_iff
#align set.disjoint_Union_right Set.disjoint_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_left {s : ∀ i, κ i → Set α} {t : Set α} :
Disjoint (⋃ (i) (j), s i j) t ↔ ∀ i j, Disjoint (s i j) t :=
iSup₂_disjoint_iff
#align set.disjoint_Union₂_left Set.disjoint_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_right {s : Set α} {t : ∀ i, κ i → Set α} :
Disjoint s (⋃ (i) (j), t i j) ↔ ∀ i j, Disjoint s (t i j) :=
disjoint_iSup₂_iff
#align set.disjoint_Union₂_right Set.disjoint_iUnion₂_right
@[simp]
theorem disjoint_sUnion_left {S : Set (Set α)} {t : Set α} :
Disjoint (⋃₀S) t ↔ ∀ s ∈ S, Disjoint s t :=
sSup_disjoint_iff
#align set.disjoint_sUnion_left Set.disjoint_sUnion_left
@[simp]
theorem disjoint_sUnion_right {s : Set α} {S : Set (Set α)} :
Disjoint s (⋃₀S) ↔ ∀ t ∈ S, Disjoint s t :=
disjoint_sSup_iff
#align set.disjoint_sUnion_right Set.disjoint_sUnion_right
end Set
end Disjoint
/-! ### Intervals -/
namespace Set
lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :
(⋂ i, Iic (f i)).Nonempty ↔ BddBelow (range f) := by
have : (⋂ (i : ι), Iic (f i)) = lowerBounds (range f) := by
ext c; simp [lowerBounds]
simp [this, BddBelow]
lemma nonempty_iInter_Ici_iff [Preorder α] {f : ι → α} :
(⋂ i, Ici (f i)).Nonempty ↔ BddAbove (range f) :=
nonempty_iInter_Iic_iff (α := αᵒᵈ)
variable [CompleteLattice α]
theorem Ici_iSup (f : ι → α) : Ici (⨆ i, f i) = ⋂ i, Ici (f i) :=
ext fun _ => by simp only [mem_Ici, iSup_le_iff, mem_iInter]
#align set.Ici_supr Set.Ici_iSup
theorem Iic_iInf (f : ι → α) : Iic (⨅ i, f i) = ⋂ i, Iic (f i) :=
ext fun _ => by simp only [mem_Iic, le_iInf_iff, mem_iInter]
#align set.Iic_infi Set.Iic_iInf
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Ici_iSup₂ (f : ∀ i, κ i → α) : Ici (⨆ (i) (j), f i j) = ⋂ (i) (j), Ici (f i j) := by
simp_rw [Ici_iSup]
#align set.Ici_supr₂ Set.Ici_iSup₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Iic_iInf₂ (f : ∀ i, κ i → α) : Iic (⨅ (i) (j), f i j) = ⋂ (i) (j), Iic (f i j) := by
simp_rw [Iic_iInf]
#align set.Iic_infi₂ Set.Iic_iInf₂
theorem Ici_sSup (s : Set α) : Ici (sSup s) = ⋂ a ∈ s, Ici a := by rw [sSup_eq_iSup, Ici_iSup₂]
#align set.Ici_Sup Set.Ici_sSup
theorem Iic_sInf (s : Set α) : Iic (sInf s) = ⋂ a ∈ s, Iic a := by rw [sInf_eq_iInf, Iic_iInf₂]
#align set.Iic_Inf Set.Iic_sInf
end Set
namespace Set
variable (t : α → Set β)
theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x := by
simp only [diff_subset_iff, ← biUnion_union]
apply biUnion_subset_biUnion_left
rw [union_diff_self]
apply subset_union_right
#align set.bUnion_diff_bUnion_subset Set.biUnion_diff_biUnion_subset
/-- If `t` is an indexed family of sets, then there is a natural map from `Σ i, t i` to `⋃ i, t i`
sending `⟨i, x⟩` to `x`. -/
def sigmaToiUnion (x : Σi, t i) : ⋃ i, t i :=
⟨x.2, mem_iUnion.2 ⟨x.1, x.2.2⟩⟩
#align set.sigma_to_Union Set.sigmaToiUnion
theorem sigmaToiUnion_surjective : Surjective (sigmaToiUnion t)
| ⟨b, hb⟩ =>
have : ∃ a, b ∈ t a := by simpa using hb
let ⟨a, hb⟩ := this
⟨⟨a, b, hb⟩, rfl⟩
#align set.sigma_to_Union_surjective Set.sigmaToiUnion_surjective
theorem sigmaToiUnion_injective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Injective (sigmaToiUnion t)
| ⟨a₁, b₁, h₁⟩, ⟨a₂, b₂, h₂⟩, eq =>
have b_eq : b₁ = b₂ := congr_arg Subtype.val eq
have a_eq : a₁ = a₂ :=
by_contradiction fun ne =>
have : b₁ ∈ t a₁ ∩ t a₂ := ⟨h₁, b_eq.symm ▸ h₂⟩
(h _ _ ne).le_bot this
Sigma.eq a_eq <| Subtype.eq <| by subst b_eq; subst a_eq; rfl
#align set.sigma_to_Union_injective Set.sigmaToiUnion_injective
theorem sigmaToiUnion_bijective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Bijective (sigmaToiUnion t) :=
⟨sigmaToiUnion_injective t h, sigmaToiUnion_surjective t⟩
#align set.sigma_to_Union_bijective Set.sigmaToiUnion_bijective
/-- Equivalence between a disjoint union and a dependent sum. -/
noncomputable def unionEqSigmaOfDisjoint {t : α → Set β} (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
(⋃ i, t i) ≃ Σi, t i :=
(Equiv.ofBijective _ <| sigmaToiUnion_bijective t h).symm
#align set.Union_eq_sigma_of_disjoint Set.unionEqSigmaOfDisjoint
theorem iUnion_ge_eq_iUnion_nat_add (u : ℕ → Set α) (n : ℕ) : ⋃ i ≥ n, u i = ⋃ i, u (i + n) :=
iSup_ge_eq_iSup_nat_add u n
#align set.Union_ge_eq_Union_nat_add Set.iUnion_ge_eq_iUnion_nat_add
theorem iInter_ge_eq_iInter_nat_add (u : ℕ → Set α) (n : ℕ) : ⋂ i ≥ n, u i = ⋂ i, u (i + n) :=
iInf_ge_eq_iInf_nat_add u n
#align set.Inter_ge_eq_Inter_nat_add Set.iInter_ge_eq_iInter_nat_add
theorem _root_.Monotone.iUnion_nat_add {f : ℕ → Set α} (hf : Monotone f) (k : ℕ) :
⋃ n, f (n + k) = ⋃ n, f n :=
hf.iSup_nat_add k
#align monotone.Union_nat_add Monotone.iUnion_nat_add
theorem _root_.Antitone.iInter_nat_add {f : ℕ → Set α} (hf : Antitone f) (k : ℕ) :
⋂ n, f (n + k) = ⋂ n, f n :=
hf.iInf_nat_add k
#align antitone.Inter_nat_add Antitone.iInter_nat_add
/-Porting note: removing `simp`. LHS does not simplify. Possible linter bug. Zulip discussion:
https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/complete_lattice.20and.20has_sup/near/316497982
-/
theorem iUnion_iInter_ge_nat_add (f : ℕ → Set α) (k : ℕ) :
⋃ n, ⋂ i ≥ n, f (i + k) = ⋃ n, ⋂ i ≥ n, f i :=
iSup_iInf_ge_nat_add f k
#align set.Union_Inter_ge_nat_add Set.iUnion_iInter_ge_nat_add
theorem union_iUnion_nat_succ (u : ℕ → Set α) : (u 0 ∪ ⋃ i, u (i + 1)) = ⋃ i, u i :=
sup_iSup_nat_succ u
#align set.union_Union_nat_succ Set.union_iUnion_nat_succ
theorem inter_iInter_nat_succ (u : ℕ → Set α) : (u 0 ∩ ⋂ i, u (i + 1)) = ⋂ i, u i :=
inf_iInf_nat_succ u
#align set.inter_Inter_nat_succ Set.inter_iInter_nat_succ
end Set
open Set
variable [CompleteLattice β]
theorem iSup_iUnion (s : ι → Set α) (f : α → β) : ⨆ a ∈ ⋃ i, s i, f a = ⨆ (i) (a ∈ s i), f a := by
|
rw [iSup_comm]
|
theorem iSup_iUnion (s : ι → Set α) (f : α → β) : ⨆ a ∈ ⋃ i, s i, f a = ⨆ (i) (a ∈ s i), f a := by
|
Mathlib.Data.Set.Lattice.2374_0.5mONj49h3SYSDwc
|
theorem iSup_iUnion (s : ι → Set α) (f : α → β) : ⨆ a ∈ ⋃ i, s i, f a = ⨆ (i) (a ∈ s i), f a
|
Mathlib_Data_Set_Lattice
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : CompleteLattice β
s : ι → Set α
f : α → β
⊢ ⨆ a ∈ ⋃ i, s i, f a = ⨆ j, ⨆ i, ⨆ (_ : j ∈ s i), f j
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
/-!
# The set lattice
This file provides usual set notation for unions and intersections, a `CompleteLattice` instance
for `Set α`, and some more set constructions.
## Main declarations
* `Set.iUnion`: **i**ndexed **union**. Union of an indexed family of sets.
* `Set.iInter`: **i**ndexed **inter**section. Intersection of an indexed family of sets.
* `Set.sInter`: **s**et **inter**section. Intersection of sets belonging to a set of sets.
* `Set.sUnion`: **s**et **union**. Union of sets belonging to a set of sets.
* `Set.sInter_eq_biInter`, `Set.sUnion_eq_biInter`: Shows that `⋂₀ s = ⋂ x ∈ s, x` and
`⋃₀ s = ⋃ x ∈ s, x`.
* `Set.completeAtomicBooleanAlgebra`: `Set α` is a `CompleteAtomicBooleanAlgebra` with `≤ = ⊆`,
`< = ⊂`, `⊓ = ∩`, `⊔ = ∪`, `⨅ = ⋂`, `⨆ = ⋃` and `\` as the set difference.
See `Set.BooleanAlgebra`.
* `Set.kernImage`: For a function `f : α → β`, `s.kernImage f` is the set of `y` such that
`f ⁻¹ y ⊆ s`.
* `Set.seq`: Union of the image of a set under a **seq**uence of functions. `seq s t` is the union
of `f '' t` over all `f ∈ s`, where `t : Set α` and `s : Set (α → β)`.
* `Set.iUnion_eq_sigma_of_disjoint`: Equivalence between `⋃ i, t i` and `Σ i, t i`, where `t` is an
indexed family of disjoint sets.
## Naming convention
In lemma names,
* `⋃ i, s i` is called `iUnion`
* `⋂ i, s i` is called `iInter`
* `⋃ i j, s i j` is called `iUnion₂`. This is an `iUnion` inside an `iUnion`.
* `⋂ i j, s i j` is called `iInter₂`. This is an `iInter` inside an `iInter`.
* `⋃ i ∈ s, t i` is called `biUnion` for "bounded `iUnion`". This is the special case of `iUnion₂`
where `j : i ∈ s`.
* `⋂ i ∈ s, t i` is called `biInter` for "bounded `iInter`". This is the special case of `iInter₂`
where `j : i ∈ s`.
## Notation
* `⋃`: `Set.iUnion`
* `⋂`: `Set.iInter`
* `⋃₀`: `Set.sUnion`
* `⋂₀`: `Set.sInter`
-/
set_option autoImplicit true
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι₂ : Sort*} {κ κ₁ κ₂ : ι → Sort*} {κ' : ι' → Sort*}
namespace Set
/-! ### Complete lattice and complete Boolean algebra instances -/
instance : InfSet (Set α) :=
⟨fun s => { a | ∀ t ∈ s, a ∈ t }⟩
instance : SupSet (Set α) :=
⟨fun s => { a | ∃ t ∈ s, a ∈ t }⟩
/-- Intersection of a set of sets. -/
def sInter (S : Set (Set α)) : Set α :=
sInf S
#align set.sInter Set.sInter
/-- Notation for `Set.sInter` Intersection of a set of sets. -/
prefix:110 "⋂₀ " => sInter
/-- Union of a set of sets. -/
def sUnion (S : Set (Set α)) : Set α :=
sSup S
#align set.sUnion Set.sUnion
/-- Notation for `Set.sUnion`. Union of a set of sets. -/
prefix:110 "⋃₀ " => sUnion
@[simp]
theorem mem_sInter {x : α} {S : Set (Set α)} : x ∈ ⋂₀ S ↔ ∀ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sInter Set.mem_sInter
@[simp]
theorem mem_sUnion {x : α} {S : Set (Set α)} : x ∈ ⋃₀ S ↔ ∃ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sUnion Set.mem_sUnion
/-- Indexed union of a family of sets -/
def iUnion (s : ι → Set β) : Set β :=
iSup s
#align set.Union Set.iUnion
/-- Indexed intersection of a family of sets -/
def iInter (s : ι → Set β) : Set β :=
iInf s
#align set.Inter Set.iInter
/-- Notation for `Set.iUnion`. Indexed union of a family of sets -/
notation3 "⋃ "(...)", "r:60:(scoped f => iUnion f) => r
/-- Notation for `Set.iInter`. Indexed intersection of a family of sets -/
notation3 "⋂ "(...)", "r:60:(scoped f => iInter f) => r
section delaborators
open Lean Lean.PrettyPrinter.Delaborator
/-- Delaborator for indexed unions. -/
@[delab app.Set.iUnion]
def iUnion_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋃ (_ : $dom), $body)
else if prop || ppTypes then
`(⋃ ($x:ident : $dom), $body)
else
`(⋃ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋃ $x:ident, ⋃ (_ : $y:ident ∈ $s), $body)
| `(⋃ ($x:ident : $_), ⋃ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋃ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
/-- Delaborator for indexed intersections. -/
@[delab app.Set.iInter]
def sInter_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋂ (_ : $dom), $body)
else if prop || ppTypes then
`(⋂ ($x:ident : $dom), $body)
else
`(⋂ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋂ $x:ident, ⋂ (_ : $y:ident ∈ $s), $body)
| `(⋂ ($x:ident : $_), ⋂ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋂ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
end delaborators
@[simp]
theorem sSup_eq_sUnion (S : Set (Set α)) : sSup S = ⋃₀S :=
rfl
#align set.Sup_eq_sUnion Set.sSup_eq_sUnion
@[simp]
theorem sInf_eq_sInter (S : Set (Set α)) : sInf S = ⋂₀ S :=
rfl
#align set.Inf_eq_sInter Set.sInf_eq_sInter
@[simp]
theorem iSup_eq_iUnion (s : ι → Set α) : iSup s = iUnion s :=
rfl
#align set.supr_eq_Union Set.iSup_eq_iUnion
@[simp]
theorem iInf_eq_iInter (s : ι → Set α) : iInf s = iInter s :=
rfl
#align set.infi_eq_Inter Set.iInf_eq_iInter
@[simp]
theorem mem_iUnion {x : α} {s : ι → Set α} : (x ∈ ⋃ i, s i) ↔ ∃ i, x ∈ s i :=
⟨fun ⟨_, ⟨⟨a, (t_eq : s a = _)⟩, (h : x ∈ _)⟩⟩ => ⟨a, t_eq.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨s a, ⟨⟨a, rfl⟩, h⟩⟩⟩
#align set.mem_Union Set.mem_iUnion
@[simp]
theorem mem_iInter {x : α} {s : ι → Set α} : (x ∈ ⋂ i, s i) ↔ ∀ i, x ∈ s i :=
⟨fun (h : ∀ a ∈ { a : Set α | ∃ i, s i = a }, x ∈ a) a => h (s a) ⟨a, rfl⟩,
fun h _ ⟨a, (eq : s a = _)⟩ => eq ▸ h a⟩
#align set.mem_Inter Set.mem_iInter
theorem mem_iUnion₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋃ (i) (j), s i j) ↔ ∃ i j, x ∈ s i j := by
simp_rw [mem_iUnion]
#align set.mem_Union₂ Set.mem_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋂ (i) (j), s i j) ↔ ∀ i j, x ∈ s i j := by
simp_rw [mem_iInter]
#align set.mem_Inter₂ Set.mem_iInter₂
theorem mem_iUnion_of_mem {s : ι → Set α} {a : α} (i : ι) (ha : a ∈ s i) : a ∈ ⋃ i, s i :=
mem_iUnion.2 ⟨i, ha⟩
#align set.mem_Union_of_mem Set.mem_iUnion_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iUnion₂_of_mem {s : ∀ i, κ i → Set α} {a : α} {i : ι} (j : κ i) (ha : a ∈ s i j) :
a ∈ ⋃ (i) (j), s i j :=
mem_iUnion₂.2 ⟨i, j, ha⟩
#align set.mem_Union₂_of_mem Set.mem_iUnion₂_of_mem
theorem mem_iInter_of_mem {s : ι → Set α} {a : α} (h : ∀ i, a ∈ s i) : a ∈ ⋂ i, s i :=
mem_iInter.2 h
#align set.mem_Inter_of_mem Set.mem_iInter_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂_of_mem {s : ∀ i, κ i → Set α} {a : α} (h : ∀ i j, a ∈ s i j) :
a ∈ ⋂ (i) (j), s i j :=
mem_iInter₂.2 h
#align set.mem_Inter₂_of_mem Set.mem_iInter₂_of_mem
instance Set.completeAtomicBooleanAlgebra : CompleteAtomicBooleanAlgebra (Set α) :=
{ instBooleanAlgebraSet with
le_sSup := fun s t t_in a a_in => ⟨t, t_in, a_in⟩
sSup_le := fun s t h a ⟨t', ⟨t'_in, a_in⟩⟩ => h t' t'_in a_in
le_sInf := fun s t h a a_in t' t'_in => h t' t'_in a_in
sInf_le := fun s t t_in a h => h _ t_in
iInf_iSup_eq := by intros; ext; simp [Classical.skolem] }
/-- `kernImage f s` is the set of `y` such that `f ⁻¹ y ⊆ s`. -/
def kernImage (f : α → β) (s : Set α) : Set β :=
{ y | ∀ ⦃x⦄, f x = y → x ∈ s }
#align set.kern_image Set.kernImage
lemma subset_kernImage_iff {f : α → β} : s ⊆ kernImage f t ↔ f ⁻¹' s ⊆ t :=
⟨fun h _ hx ↦ h hx rfl,
fun h _ hx y hy ↦ h (show f y ∈ s from hy.symm ▸ hx)⟩
section GaloisConnection
variable {f : α → β}
protected theorem image_preimage : GaloisConnection (image f) (preimage f) := fun _ _ =>
image_subset_iff
#align set.image_preimage Set.image_preimage
protected theorem preimage_kernImage : GaloisConnection (preimage f) (kernImage f) := fun _ _ =>
subset_kernImage_iff.symm
#align set.preimage_kern_image Set.preimage_kernImage
end GaloisConnection
section kernImage
variable {f : α → β}
lemma kernImage_mono : Monotone (kernImage f) :=
Set.preimage_kernImage.monotone_u
lemma kernImage_eq_compl {s : Set α} : kernImage f s = (f '' sᶜ)ᶜ :=
Set.preimage_kernImage.u_unique (Set.image_preimage.compl)
(fun t ↦ compl_compl (f ⁻¹' t) ▸ Set.preimage_compl)
lemma kernImage_compl {s : Set α} : kernImage f (sᶜ) = (f '' s)ᶜ := by
rw [kernImage_eq_compl, compl_compl]
lemma kernImage_empty : kernImage f ∅ = (range f)ᶜ := by
rw [kernImage_eq_compl, compl_empty, image_univ]
lemma kernImage_preimage_eq_iff {s : Set β} : kernImage f (f ⁻¹' s) = s ↔ (range f)ᶜ ⊆ s := by
rw [kernImage_eq_compl, ← preimage_compl, compl_eq_comm, eq_comm, image_preimage_eq_iff,
compl_subset_comm]
lemma compl_range_subset_kernImage {s : Set α} : (range f)ᶜ ⊆ kernImage f s := by
rw [← kernImage_empty]
exact kernImage_mono (empty_subset _)
lemma kernImage_union_preimage {s : Set α} {t : Set β} :
kernImage f (s ∪ f ⁻¹' t) = kernImage f s ∪ t := by
rw [kernImage_eq_compl, kernImage_eq_compl, compl_union, ← preimage_compl, image_inter_preimage,
compl_inter, compl_compl]
lemma kernImage_preimage_union {s : Set α} {t : Set β} :
kernImage f (f ⁻¹' t ∪ s) = t ∪ kernImage f s := by
rw [union_comm, kernImage_union_preimage, union_comm]
end kernImage
/-! ### Union and intersection over an indexed family of sets -/
instance : OrderTop (Set α) where
top := univ
le_top := by simp
@[congr]
theorem iUnion_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iUnion f₁ = iUnion f₂ :=
iSup_congr_Prop pq f
#align set.Union_congr_Prop Set.iUnion_congr_Prop
@[congr]
theorem iInter_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iInter f₁ = iInter f₂ :=
iInf_congr_Prop pq f
#align set.Inter_congr_Prop Set.iInter_congr_Prop
theorem iUnion_plift_up (f : PLift ι → Set α) : ⋃ i, f (PLift.up i) = ⋃ i, f i :=
iSup_plift_up _
#align set.Union_plift_up Set.iUnion_plift_up
theorem iUnion_plift_down (f : ι → Set α) : ⋃ i, f (PLift.down i) = ⋃ i, f i :=
iSup_plift_down _
#align set.Union_plift_down Set.iUnion_plift_down
theorem iInter_plift_up (f : PLift ι → Set α) : ⋂ i, f (PLift.up i) = ⋂ i, f i :=
iInf_plift_up _
#align set.Inter_plift_up Set.iInter_plift_up
theorem iInter_plift_down (f : ι → Set α) : ⋂ i, f (PLift.down i) = ⋂ i, f i :=
iInf_plift_down _
#align set.Inter_plift_down Set.iInter_plift_down
theorem iUnion_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋃ _ : p, s = if p then s else ∅ :=
iSup_eq_if _
#align set.Union_eq_if Set.iUnion_eq_if
theorem iUnion_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋃ h : p, s h = if h : p then s h else ∅ :=
iSup_eq_dif _
#align set.Union_eq_dif Set.iUnion_eq_dif
theorem iInter_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋂ _ : p, s = if p then s else univ :=
iInf_eq_if _
#align set.Inter_eq_if Set.iInter_eq_if
theorem iInf_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋂ h : p, s h = if h : p then s h else univ :=
_root_.iInf_eq_dif _
#align set.Infi_eq_dif Set.iInf_eq_dif
theorem exists_set_mem_of_union_eq_top {ι : Type*} (t : Set ι) (s : ι → Set β)
(w : ⋃ i ∈ t, s i = ⊤) (x : β) : ∃ i ∈ t, x ∈ s i := by
have p : x ∈ ⊤ := Set.mem_univ x
rw [← w, Set.mem_iUnion] at p
simpa using p
#align set.exists_set_mem_of_union_eq_top Set.exists_set_mem_of_union_eq_top
theorem nonempty_of_union_eq_top_of_nonempty {ι : Type*} (t : Set ι) (s : ι → Set α)
(H : Nonempty α) (w : ⋃ i ∈ t, s i = ⊤) : t.Nonempty := by
obtain ⟨x, m, -⟩ := exists_set_mem_of_union_eq_top t s w H.some
exact ⟨x, m⟩
#align set.nonempty_of_union_eq_top_of_nonempty Set.nonempty_of_union_eq_top_of_nonempty
theorem nonempty_of_nonempty_iUnion
{s : ι → Set α} (h_Union : (⋃ i, s i).Nonempty) : Nonempty ι := by
obtain ⟨x, hx⟩ := h_Union
exact ⟨Classical.choose $ mem_iUnion.mp hx⟩
theorem nonempty_of_nonempty_iUnion_eq_univ
{s : ι → Set α} [Nonempty α] (h_Union : ⋃ i, s i = univ) : Nonempty ι :=
nonempty_of_nonempty_iUnion (s := s) (by simpa only [h_Union] using univ_nonempty)
theorem setOf_exists (p : ι → β → Prop) : { x | ∃ i, p i x } = ⋃ i, { x | p i x } :=
ext fun _ => mem_iUnion.symm
#align set.set_of_exists Set.setOf_exists
theorem setOf_forall (p : ι → β → Prop) : { x | ∀ i, p i x } = ⋂ i, { x | p i x } :=
ext fun _ => mem_iInter.symm
#align set.set_of_forall Set.setOf_forall
theorem iUnion_subset {s : ι → Set α} {t : Set α} (h : ∀ i, s i ⊆ t) : ⋃ i, s i ⊆ t :=
iSup_le h
#align set.Union_subset Set.iUnion_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset {s : ∀ i, κ i → Set α} {t : Set α} (h : ∀ i j, s i j ⊆ t) :
⋃ (i) (j), s i j ⊆ t :=
iUnion_subset fun x => iUnion_subset (h x)
#align set.Union₂_subset Set.iUnion₂_subset
theorem subset_iInter {t : Set β} {s : ι → Set β} (h : ∀ i, t ⊆ s i) : t ⊆ ⋂ i, s i :=
le_iInf h
#align set.subset_Inter Set.subset_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iInter₂ {s : Set α} {t : ∀ i, κ i → Set α} (h : ∀ i j, s ⊆ t i j) :
s ⊆ ⋂ (i) (j), t i j :=
subset_iInter fun x => subset_iInter <| h x
#align set.subset_Inter₂ Set.subset_iInter₂
@[simp]
theorem iUnion_subset_iff {s : ι → Set α} {t : Set α} : ⋃ i, s i ⊆ t ↔ ∀ i, s i ⊆ t :=
⟨fun h _ => Subset.trans (le_iSup s _) h, iUnion_subset⟩
#align set.Union_subset_iff Set.iUnion_subset_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset_iff {s : ∀ i, κ i → Set α} {t : Set α} :
⋃ (i) (j), s i j ⊆ t ↔ ∀ i j, s i j ⊆ t := by simp_rw [iUnion_subset_iff]
#align set.Union₂_subset_iff Set.iUnion₂_subset_iff
@[simp]
theorem subset_iInter_iff {s : Set α} {t : ι → Set α} : (s ⊆ ⋂ i, t i) ↔ ∀ i, s ⊆ t i :=
le_iInf_iff
#align set.subset_Inter_iff Set.subset_iInter_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem subset_iInter₂_iff {s : Set α} {t : ∀ i, κ i → Set α} :
(s ⊆ ⋂ (i) (j), t i j) ↔ ∀ i j, s ⊆ t i j := by simp_rw [subset_iInter_iff]
#align set.subset_Inter₂_iff Set.subset_iInter₂_iff
theorem subset_iUnion : ∀ (s : ι → Set β) (i : ι), s i ⊆ ⋃ i, s i :=
le_iSup
#align set.subset_Union Set.subset_iUnion
theorem iInter_subset : ∀ (s : ι → Set β) (i : ι), ⋂ i, s i ⊆ s i :=
iInf_le
#align set.Inter_subset Set.iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iUnion₂ {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : s i j ⊆ ⋃ (i') (j'), s i' j' :=
le_iSup₂ i j
#align set.subset_Union₂ Set.subset_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_subset {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : ⋂ (i) (j), s i j ⊆ s i j :=
iInf₂_le i j
#align set.Inter₂_subset Set.iInter₂_subset
/-- This rather trivial consequence of `subset_iUnion`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem subset_iUnion_of_subset {s : Set α} {t : ι → Set α} (i : ι) (h : s ⊆ t i) : s ⊆ ⋃ i, t i :=
le_iSup_of_le i h
#align set.subset_Union_of_subset Set.subset_iUnion_of_subset
/-- This rather trivial consequence of `iInter_subset`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem iInter_subset_of_subset {s : ι → Set α} {t : Set α} (i : ι) (h : s i ⊆ t) :
⋂ i, s i ⊆ t :=
iInf_le_of_le i h
#align set.Inter_subset_of_subset Set.iInter_subset_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `subset_iUnion₂` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem subset_iUnion₂_of_subset {s : Set α} {t : ∀ i, κ i → Set α} (i : ι) (j : κ i)
(h : s ⊆ t i j) : s ⊆ ⋃ (i) (j), t i j :=
le_iSup₂_of_le i j h
#align set.subset_Union₂_of_subset Set.subset_iUnion₂_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `iInter₂_subset` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem iInter₂_subset_of_subset {s : ∀ i, κ i → Set α} {t : Set α} (i : ι) (j : κ i)
(h : s i j ⊆ t) : ⋂ (i) (j), s i j ⊆ t :=
iInf₂_le_of_le i j h
#align set.Inter₂_subset_of_subset Set.iInter₂_subset_of_subset
theorem iUnion_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono h
#align set.Union_mono Set.iUnion_mono
@[gcongr]
theorem iUnion_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iUnion s ⊆ iUnion t :=
iSup_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋃ (i) (j), s i j ⊆ ⋃ (i) (j), t i j :=
iSup₂_mono h
#align set.Union₂_mono Set.iUnion₂_mono
theorem iInter_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋂ i, s i ⊆ ⋂ i, t i :=
iInf_mono h
#align set.Inter_mono Set.iInter_mono
@[gcongr]
theorem iInter_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iInter s ⊆ iInter t :=
iInf_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋂ (i) (j), s i j ⊆ ⋂ (i) (j), t i j :=
iInf₂_mono h
#align set.Inter₂_mono Set.iInter₂_mono
theorem iUnion_mono' {s : ι → Set α} {t : ι₂ → Set α} (h : ∀ i, ∃ j, s i ⊆ t j) :
⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono' h
#align set.Union_mono' Set.iUnion_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iUnion₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i j, ∃ i' j', s i j ⊆ t i' j') : ⋃ (i) (j), s i j ⊆ ⋃ (i') (j'), t i' j' :=
iSup₂_mono' h
#align set.Union₂_mono' Set.iUnion₂_mono'
theorem iInter_mono' {s : ι → Set α} {t : ι' → Set α} (h : ∀ j, ∃ i, s i ⊆ t j) :
⋂ i, s i ⊆ ⋂ j, t j :=
Set.subset_iInter fun j =>
let ⟨i, hi⟩ := h j
iInter_subset_of_subset i hi
#align set.Inter_mono' Set.iInter_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iInter₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i' j', ∃ i j, s i j ⊆ t i' j') : ⋂ (i) (j), s i j ⊆ ⋂ (i') (j'), t i' j' :=
subset_iInter₂_iff.2 fun i' j' =>
let ⟨_, _, hst⟩ := h i' j'
(iInter₂_subset _ _).trans hst
#align set.Inter₂_mono' Set.iInter₂_mono'
theorem iUnion₂_subset_iUnion (κ : ι → Sort*) (s : ι → Set α) :
⋃ (i) (_ : κ i), s i ⊆ ⋃ i, s i :=
iUnion_mono fun _ => iUnion_subset fun _ => Subset.rfl
#align set.Union₂_subset_Union Set.iUnion₂_subset_iUnion
theorem iInter_subset_iInter₂ (κ : ι → Sort*) (s : ι → Set α) :
⋂ i, s i ⊆ ⋂ (i) (_ : κ i), s i :=
iInter_mono fun _ => subset_iInter fun _ => Subset.rfl
#align set.Inter_subset_Inter₂ Set.iInter_subset_iInter₂
theorem iUnion_setOf (P : ι → α → Prop) : ⋃ i, { x : α | P i x } = { x : α | ∃ i, P i x } := by
ext
exact mem_iUnion
#align set.Union_set_of Set.iUnion_setOf
theorem iInter_setOf (P : ι → α → Prop) : ⋂ i, { x : α | P i x } = { x : α | ∀ i, P i x } := by
ext
exact mem_iInter
#align set.Inter_set_of Set.iInter_setOf
theorem iUnion_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋃ x, f x = ⋃ y, g y :=
h1.iSup_congr h h2
#align set.Union_congr_of_surjective Set.iUnion_congr_of_surjective
theorem iInter_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋂ x, f x = ⋂ y, g y :=
h1.iInf_congr h h2
#align set.Inter_congr_of_surjective Set.iInter_congr_of_surjective
lemma iUnion_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋃ i, s i = ⋃ i, t i := iSup_congr h
#align set.Union_congr Set.iUnion_congr
lemma iInter_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋂ i, s i = ⋂ i, t i := iInf_congr h
#align set.Inter_congr Set.iInter_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iUnion₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋃ (i) (j), s i j = ⋃ (i) (j), t i j :=
iUnion_congr fun i => iUnion_congr <| h i
#align set.Union₂_congr Set.iUnion₂_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iInter₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋂ (i) (j), s i j = ⋂ (i) (j), t i j :=
iInter_congr fun i => iInter_congr <| h i
#align set.Inter₂_congr Set.iInter₂_congr
section Nonempty
variable [Nonempty ι] {f : ι → Set α} {s : Set α}
lemma iUnion_const (s : Set β) : ⋃ _ : ι, s = s := iSup_const
#align set.Union_const Set.iUnion_const
lemma iInter_const (s : Set β) : ⋂ _ : ι, s = s := iInf_const
#align set.Inter_const Set.iInter_const
lemma iUnion_eq_const (hf : ∀ i, f i = s) : ⋃ i, f i = s :=
(iUnion_congr hf).trans $ iUnion_const _
#align set.Union_eq_const Set.iUnion_eq_const
lemma iInter_eq_const (hf : ∀ i, f i = s) : ⋂ i, f i = s :=
(iInter_congr hf).trans $ iInter_const _
#align set.Inter_eq_const Set.iInter_eq_const
end Nonempty
@[simp]
theorem compl_iUnion (s : ι → Set β) : (⋃ i, s i)ᶜ = ⋂ i, (s i)ᶜ :=
compl_iSup
#align set.compl_Union Set.compl_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iUnion₂ (s : ∀ i, κ i → Set α) : (⋃ (i) (j), s i j)ᶜ = ⋂ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iUnion]
#align set.compl_Union₂ Set.compl_iUnion₂
@[simp]
theorem compl_iInter (s : ι → Set β) : (⋂ i, s i)ᶜ = ⋃ i, (s i)ᶜ :=
compl_iInf
#align set.compl_Inter Set.compl_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iInter₂ (s : ∀ i, κ i → Set α) : (⋂ (i) (j), s i j)ᶜ = ⋃ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iInter]
#align set.compl_Inter₂ Set.compl_iInter₂
-- classical -- complete_boolean_algebra
theorem iUnion_eq_compl_iInter_compl (s : ι → Set β) : ⋃ i, s i = (⋂ i, (s i)ᶜ)ᶜ := by
simp only [compl_iInter, compl_compl]
#align set.Union_eq_compl_Inter_compl Set.iUnion_eq_compl_iInter_compl
-- classical -- complete_boolean_algebra
theorem iInter_eq_compl_iUnion_compl (s : ι → Set β) : ⋂ i, s i = (⋃ i, (s i)ᶜ)ᶜ := by
simp only [compl_iUnion, compl_compl]
#align set.Inter_eq_compl_Union_compl Set.iInter_eq_compl_iUnion_compl
theorem inter_iUnion (s : Set β) (t : ι → Set β) : (s ∩ ⋃ i, t i) = ⋃ i, s ∩ t i :=
inf_iSup_eq _ _
#align set.inter_Union Set.inter_iUnion
theorem iUnion_inter (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∩ s = ⋃ i, t i ∩ s :=
iSup_inf_eq _ _
#align set.Union_inter Set.iUnion_inter
theorem iUnion_union_distrib (s : ι → Set β) (t : ι → Set β) :
⋃ i, s i ∪ t i = (⋃ i, s i) ∪ ⋃ i, t i :=
iSup_sup_eq
#align set.Union_union_distrib Set.iUnion_union_distrib
theorem iInter_inter_distrib (s : ι → Set β) (t : ι → Set β) :
⋂ i, s i ∩ t i = (⋂ i, s i) ∩ ⋂ i, t i :=
iInf_inf_eq
#align set.Inter_inter_distrib Set.iInter_inter_distrib
theorem union_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∪ ⋃ i, t i) = ⋃ i, s ∪ t i :=
sup_iSup
#align set.union_Union Set.union_iUnion
theorem iUnion_union [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∪ s = ⋃ i, t i ∪ s :=
iSup_sup
#align set.Union_union Set.iUnion_union
theorem inter_iInter [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∩ ⋂ i, t i) = ⋂ i, s ∩ t i :=
inf_iInf
#align set.inter_Inter Set.inter_iInter
theorem iInter_inter [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋂ i, t i) ∩ s = ⋂ i, t i ∩ s :=
iInf_inf
#align set.Inter_inter Set.iInter_inter
-- classical
theorem union_iInter (s : Set β) (t : ι → Set β) : (s ∪ ⋂ i, t i) = ⋂ i, s ∪ t i :=
sup_iInf_eq _ _
#align set.union_Inter Set.union_iInter
theorem iInter_union (s : ι → Set β) (t : Set β) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.Inter_union Set.iInter_union
theorem iUnion_diff (s : Set β) (t : ι → Set β) : (⋃ i, t i) \ s = ⋃ i, t i \ s :=
iUnion_inter _ _
#align set.Union_diff Set.iUnion_diff
theorem diff_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s \ ⋃ i, t i) = ⋂ i, s \ t i := by
rw [diff_eq, compl_iUnion, inter_iInter]; rfl
#align set.diff_Union Set.diff_iUnion
theorem diff_iInter (s : Set β) (t : ι → Set β) : (s \ ⋂ i, t i) = ⋃ i, s \ t i := by
rw [diff_eq, compl_iInter, inter_iUnion]; rfl
#align set.diff_Inter Set.diff_iInter
theorem directed_on_iUnion {r} {f : ι → Set α} (hd : Directed (· ⊆ ·) f)
(h : ∀ x, DirectedOn r (f x)) : DirectedOn r (⋃ x, f x) := by
simp only [DirectedOn, exists_prop, mem_iUnion, exists_imp]
exact fun a₁ b₁ fb₁ a₂ b₂ fb₂ =>
let ⟨z, zb₁, zb₂⟩ := hd b₁ b₂
let ⟨x, xf, xa₁, xa₂⟩ := h z a₁ (zb₁ fb₁) a₂ (zb₂ fb₂)
⟨x, ⟨z, xf⟩, xa₁, xa₂⟩
#align set.directed_on_Union Set.directed_on_iUnion
theorem iUnion_inter_subset {ι α} {s t : ι → Set α} : ⋃ i, s i ∩ t i ⊆ (⋃ i, s i) ∩ ⋃ i, t i :=
le_iSup_inf_iSup s t
#align set.Union_inter_subset Set.iUnion_inter_subset
theorem iUnion_inter_of_monotone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_monotone hs ht
#align set.Union_inter_of_monotone Set.iUnion_inter_of_monotone
theorem iUnion_inter_of_antitone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_antitone hs ht
#align set.Union_inter_of_antitone Set.iUnion_inter_of_antitone
theorem iInter_union_of_monotone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_monotone hs ht
#align set.Inter_union_of_monotone Set.iInter_union_of_monotone
theorem iInter_union_of_antitone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_antitone hs ht
#align set.Inter_union_of_antitone Set.iInter_union_of_antitone
/-- An equality version of this lemma is `iUnion_iInter_of_monotone` in `Data.Set.Finite`. -/
theorem iUnion_iInter_subset {s : ι → ι' → Set α} : (⋃ j, ⋂ i, s i j) ⊆ ⋂ i, ⋃ j, s i j :=
iSup_iInf_le_iInf_iSup (flip s)
#align set.Union_Inter_subset Set.iUnion_iInter_subset
theorem iUnion_option {ι} (s : Option ι → Set α) : ⋃ o, s o = s none ∪ ⋃ i, s (some i) :=
iSup_option s
#align set.Union_option Set.iUnion_option
theorem iInter_option {ι} (s : Option ι → Set α) : ⋂ o, s o = s none ∩ ⋂ i, s (some i) :=
iInf_option s
#align set.Inter_option Set.iInter_option
section
variable (p : ι → Prop) [DecidablePred p]
theorem iUnion_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 _ _ _
#align set.Union_dite Set.iUnion_dite
theorem iUnion_ite (f g : ι → Set α) :
⋃ i, (if p i then f i else g i) = (⋃ (i) (_ : p i), f i) ∪ ⋃ (i) (_ : ¬p i), g i :=
iUnion_dite _ _ _
#align set.Union_ite Set.iUnion_ite
theorem iInter_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 :=
iInf_dite _ _ _
#align set.Inter_dite Set.iInter_dite
theorem iInter_ite (f g : ι → Set α) :
⋂ i, (if p i then f i else g i) = (⋂ (i) (_ : p i), f i) ∩ ⋂ (i) (_ : ¬p i), g i :=
iInter_dite _ _ _
#align set.Inter_ite Set.iInter_ite
end
theorem image_projection_prod {ι : Type*} {α : ι → Type*} {v : ∀ i : ι, Set (α i)}
(hv : (pi univ v).Nonempty) (i : ι) :
((fun x : ∀ i : ι, α i => x i) '' ⋂ k, (fun x : ∀ j : ι, α j => x k) ⁻¹' v k) = v i := by
classical
apply Subset.antisymm
· simp [iInter_subset]
· intro y y_in
simp only [mem_image, mem_iInter, mem_preimage]
rcases hv with ⟨z, hz⟩
refine' ⟨Function.update z i y, _, update_same i y z⟩
rw [@forall_update_iff ι α _ z i y fun i t => t ∈ v i]
exact ⟨y_in, fun j _ => by simpa using hz j⟩
#align set.image_projection_prod Set.image_projection_prod
/-! ### Unions and intersections indexed by `Prop` -/
theorem iInter_false {s : False → Set α} : iInter s = univ :=
iInf_false
#align set.Inter_false Set.iInter_false
theorem iUnion_false {s : False → Set α} : iUnion s = ∅ :=
iSup_false
#align set.Union_false Set.iUnion_false
@[simp]
theorem iInter_true {s : True → Set α} : iInter s = s trivial :=
iInf_true
#align set.Inter_true Set.iInter_true
@[simp]
theorem iUnion_true {s : True → Set α} : iUnion s = s trivial :=
iSup_true
#align set.Union_true Set.iUnion_true
@[simp]
theorem iInter_exists {p : ι → Prop} {f : Exists p → Set α} :
⋂ x, f x = ⋂ (i) (h : p i), f ⟨i, h⟩ :=
iInf_exists
#align set.Inter_exists Set.iInter_exists
@[simp]
theorem iUnion_exists {p : ι → Prop} {f : Exists p → Set α} :
⋃ x, f x = ⋃ (i) (h : p i), f ⟨i, h⟩ :=
iSup_exists
#align set.Union_exists Set.iUnion_exists
@[simp]
theorem iUnion_empty : (⋃ _ : ι, ∅ : Set α) = ∅ :=
iSup_bot
#align set.Union_empty Set.iUnion_empty
@[simp]
theorem iInter_univ : (⋂ _ : ι, univ : Set α) = univ :=
iInf_top
#align set.Inter_univ Set.iInter_univ
section
variable {s : ι → Set α}
@[simp]
theorem iUnion_eq_empty : ⋃ i, s i = ∅ ↔ ∀ i, s i = ∅ :=
iSup_eq_bot
#align set.Union_eq_empty Set.iUnion_eq_empty
@[simp]
theorem iInter_eq_univ : ⋂ i, s i = univ ↔ ∀ i, s i = univ :=
iInf_eq_top
#align set.Inter_eq_univ Set.iInter_eq_univ
@[simp]
theorem nonempty_iUnion : (⋃ i, s i).Nonempty ↔ ∃ i, (s i).Nonempty := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_Union Set.nonempty_iUnion
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_biUnion {t : Set α} {s : α → Set β} :
(⋃ i ∈ t, s i).Nonempty ↔ ∃ i ∈ t, (s i).Nonempty := by simp
#align set.nonempty_bUnion Set.nonempty_biUnion
theorem iUnion_nonempty_index (s : Set α) (t : s.Nonempty → Set β) :
⋃ h, t h = ⋃ x ∈ s, t ⟨x, ‹_›⟩ :=
iSup_exists
#align set.Union_nonempty_index Set.iUnion_nonempty_index
end
@[simp]
theorem iInter_iInter_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋂ (x) (h : x = b), s x h = s b rfl :=
iInf_iInf_eq_left
#align set.Inter_Inter_eq_left Set.iInter_iInter_eq_left
@[simp]
theorem iInter_iInter_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋂ (x) (h : b = x), s x h = s b rfl :=
iInf_iInf_eq_right
#align set.Inter_Inter_eq_right Set.iInter_iInter_eq_right
@[simp]
theorem iUnion_iUnion_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋃ (x) (h : x = b), s x h = s b rfl :=
iSup_iSup_eq_left
#align set.Union_Union_eq_left Set.iUnion_iUnion_eq_left
@[simp]
theorem iUnion_iUnion_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋃ (x) (h : b = x), s x h = s b rfl :=
iSup_iSup_eq_right
#align set.Union_Union_eq_right Set.iUnion_iUnion_eq_right
theorem iInter_or {p q : Prop} (s : p ∨ q → Set α) :
⋂ h, s h = (⋂ h : p, s (Or.inl h)) ∩ ⋂ h : q, s (Or.inr h) :=
iInf_or
#align set.Inter_or Set.iInter_or
theorem iUnion_or {p q : Prop} (s : p ∨ q → Set α) :
⋃ h, s h = (⋃ i, s (Or.inl i)) ∪ ⋃ j, s (Or.inr j) :=
iSup_or
#align set.Union_or Set.iUnion_or
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iUnion_and {p q : Prop} (s : p ∧ q → Set α) : ⋃ h, s h = ⋃ (hp) (hq), s ⟨hp, hq⟩ :=
iSup_and
#align set.Union_and Set.iUnion_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iInter_and {p q : Prop} (s : p ∧ q → Set α) : ⋂ h, s h = ⋂ (hp) (hq), s ⟨hp, hq⟩ :=
iInf_and
#align set.Inter_and Set.iInter_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iUnion_comm (s : ι → ι' → Set α) : ⋃ (i) (i'), s i i' = ⋃ (i') (i), s i i' :=
iSup_comm
#align set.Union_comm Set.iUnion_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iInter_comm (s : ι → ι' → Set α) : ⋂ (i) (i'), s i i' = ⋂ (i') (i), s i i' :=
iInf_comm
#align set.Inter_comm Set.iInter_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iUnion₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋃ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋃ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iSup₂_comm _
#align set.Union₂_comm Set.iUnion₂_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iInter₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋂ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋂ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iInf₂_comm _
#align set.Inter₂_comm Set.iInter₂_comm
@[simp]
theorem biUnion_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋃ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι']
#align set.bUnion_and Set.biUnion_and
@[simp]
theorem biUnion_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋃ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι]
#align set.bUnion_and' Set.biUnion_and'
@[simp]
theorem biInter_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋂ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iInter_and, @iInter_comm _ ι']
#align set.bInter_and Set.biInter_and
@[simp]
theorem biInter_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋂ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iInter_and, @iInter_comm _ ι]
#align set.bInter_and' Set.biInter_and'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iUnion_iUnion_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋃ (x) (h), s x h = s b (Or.inl rfl) ∪ ⋃ (x) (h : p x), s x (Or.inr h) := by
simp only [iUnion_or, iUnion_union_distrib, iUnion_iUnion_eq_left]
#align set.Union_Union_eq_or_left Set.iUnion_iUnion_eq_or_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iInter_iInter_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋂ (x) (h), s x h = s b (Or.inl rfl) ∩ ⋂ (x) (h : p x), s x (Or.inr h) := by
simp only [iInter_or, iInter_inter_distrib, iInter_iInter_eq_left]
#align set.Inter_Inter_eq_or_left Set.iInter_iInter_eq_or_left
/-! ### Bounded unions and intersections -/
/-- A specialization of `mem_iUnion₂`. -/
theorem mem_biUnion {s : Set α} {t : α → Set β} {x : α} {y : β} (xs : x ∈ s) (ytx : y ∈ t x) :
y ∈ ⋃ x ∈ s, t x :=
mem_iUnion₂_of_mem xs ytx
#align set.mem_bUnion Set.mem_biUnion
/-- A specialization of `mem_iInter₂`. -/
theorem mem_biInter {s : Set α} {t : α → Set β} {y : β} (h : ∀ x ∈ s, y ∈ t x) :
y ∈ ⋂ x ∈ s, t x :=
mem_iInter₂_of_mem h
#align set.mem_bInter Set.mem_biInter
/-- A specialization of `subset_iUnion₂`. -/
theorem subset_biUnion_of_mem {s : Set α} {u : α → Set β} {x : α} (xs : x ∈ s) :
u x ⊆ ⋃ x ∈ s, u x :=
--Porting note: Why is this not just `subset_iUnion₂ x xs`?
@subset_iUnion₂ β α (· ∈ s) (fun i _ => u i) x xs
#align set.subset_bUnion_of_mem Set.subset_biUnion_of_mem
/-- A specialization of `iInter₂_subset`. -/
theorem biInter_subset_of_mem {s : Set α} {t : α → Set β} {x : α} (xs : x ∈ s) :
⋂ x ∈ s, t x ⊆ t x :=
iInter₂_subset x xs
#align set.bInter_subset_of_mem Set.biInter_subset_of_mem
theorem biUnion_subset_biUnion_left {s s' : Set α} {t : α → Set β} (h : s ⊆ s') :
⋃ x ∈ s, t x ⊆ ⋃ x ∈ s', t x :=
iUnion₂_subset fun _ hx => subset_biUnion_of_mem <| h hx
#align set.bUnion_subset_bUnion_left Set.biUnion_subset_biUnion_left
theorem biInter_subset_biInter_left {s s' : Set α} {t : α → Set β} (h : s' ⊆ s) :
⋂ x ∈ s, t x ⊆ ⋂ x ∈ s', t x :=
subset_iInter₂ fun _ hx => biInter_subset_of_mem <| h hx
#align set.bInter_subset_bInter_left Set.biInter_subset_biInter_left
theorem biUnion_mono {s s' : Set α} {t t' : α → Set β} (hs : s' ⊆ s) (h : ∀ x ∈ s, t x ⊆ t' x) :
⋃ x ∈ s', t x ⊆ ⋃ x ∈ s, t' x :=
(biUnion_subset_biUnion_left hs).trans <| iUnion₂_mono h
#align set.bUnion_mono Set.biUnion_mono
theorem biInter_mono {s s' : Set α} {t t' : α → Set β} (hs : s ⊆ s') (h : ∀ x ∈ s, t x ⊆ t' x) :
⋂ x ∈ s', t x ⊆ ⋂ x ∈ s, t' x :=
(biInter_subset_biInter_left hs).trans <| iInter₂_mono h
#align set.bInter_mono Set.biInter_mono
theorem biUnion_eq_iUnion (s : Set α) (t : ∀ x ∈ s, Set β) :
⋃ x ∈ s, t x ‹_› = ⋃ x : s, t x x.2 :=
iSup_subtype'
#align set.bUnion_eq_Union Set.biUnion_eq_iUnion
theorem biInter_eq_iInter (s : Set α) (t : ∀ x ∈ s, Set β) :
⋂ x ∈ s, t x ‹_› = ⋂ x : s, t x x.2 :=
iInf_subtype'
#align set.bInter_eq_Inter Set.biInter_eq_iInter
theorem iUnion_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋃ x : { x // p x }, s x = ⋃ (x) (hx : p x), s ⟨x, hx⟩ :=
iSup_subtype
#align set.Union_subtype Set.iUnion_subtype
theorem iInter_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋂ x : { x // p x }, s x = ⋂ (x) (hx : p x), s ⟨x, hx⟩ :=
iInf_subtype
#align set.Inter_subtype Set.iInter_subtype
theorem biInter_empty (u : α → Set β) : ⋂ x ∈ (∅ : Set α), u x = univ :=
iInf_emptyset
#align set.bInter_empty Set.biInter_empty
theorem biInter_univ (u : α → Set β) : ⋂ x ∈ @univ α, u x = ⋂ x, u x :=
iInf_univ
#align set.bInter_univ Set.biInter_univ
@[simp]
theorem biUnion_self (s : Set α) : ⋃ x ∈ s, s = s :=
Subset.antisymm (iUnion₂_subset fun _ _ => Subset.refl s) fun _ hx => mem_biUnion hx hx
#align set.bUnion_self Set.biUnion_self
@[simp]
theorem iUnion_nonempty_self (s : Set α) : ⋃ _ : s.Nonempty, s = s := by
rw [iUnion_nonempty_index, biUnion_self]
#align set.Union_nonempty_self Set.iUnion_nonempty_self
theorem biInter_singleton (a : α) (s : α → Set β) : ⋂ x ∈ ({a} : Set α), s x = s a :=
iInf_singleton
#align set.bInter_singleton Set.biInter_singleton
theorem biInter_union (s t : Set α) (u : α → Set β) :
⋂ x ∈ s ∪ t, u x = (⋂ x ∈ s, u x) ∩ ⋂ x ∈ t, u x :=
iInf_union
#align set.bInter_union Set.biInter_union
theorem biInter_insert (a : α) (s : Set α) (t : α → Set β) :
⋂ x ∈ insert a s, t x = t a ∩ ⋂ x ∈ s, t x := by simp
#align set.bInter_insert Set.biInter_insert
theorem biInter_pair (a b : α) (s : α → Set β) : ⋂ x ∈ ({a, b} : Set α), s x = s a ∩ s b := by
rw [biInter_insert, biInter_singleton]
#align set.bInter_pair Set.biInter_pair
theorem biInter_inter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, f i ∩ t = (⋂ i ∈ s, f i) ∩ t := by
haveI : Nonempty s := hs.to_subtype
simp [biInter_eq_iInter, ← iInter_inter]
#align set.bInter_inter Set.biInter_inter
theorem inter_biInter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, t ∩ f i = t ∩ ⋂ i ∈ s, f i := by
rw [inter_comm, ← biInter_inter hs]
simp [inter_comm]
#align set.inter_bInter Set.inter_biInter
theorem biUnion_empty (s : α → Set β) : ⋃ x ∈ (∅ : Set α), s x = ∅ :=
iSup_emptyset
#align set.bUnion_empty Set.biUnion_empty
theorem biUnion_univ (s : α → Set β) : ⋃ x ∈ @univ α, s x = ⋃ x, s x :=
iSup_univ
#align set.bUnion_univ Set.biUnion_univ
theorem biUnion_singleton (a : α) (s : α → Set β) : ⋃ x ∈ ({a} : Set α), s x = s a :=
iSup_singleton
#align set.bUnion_singleton Set.biUnion_singleton
@[simp]
theorem biUnion_of_singleton (s : Set α) : ⋃ x ∈ s, {x} = s :=
ext <| by simp
#align set.bUnion_of_singleton Set.biUnion_of_singleton
theorem biUnion_union (s t : Set α) (u : α → Set β) :
⋃ x ∈ s ∪ t, u x = (⋃ x ∈ s, u x) ∪ ⋃ x ∈ t, u x :=
iSup_union
#align set.bUnion_union Set.biUnion_union
@[simp]
theorem iUnion_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋃ i, f i = ⋃ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iUnion_subtype _ _
#align set.Union_coe_set Set.iUnion_coe_set
@[simp]
theorem iInter_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋂ i, f i = ⋂ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iInter_subtype _ _
#align set.Inter_coe_set Set.iInter_coe_set
theorem biUnion_insert (a : α) (s : Set α) (t : α → Set β) :
⋃ x ∈ insert a s, t x = t a ∪ ⋃ x ∈ s, t x := by simp
#align set.bUnion_insert Set.biUnion_insert
theorem biUnion_pair (a b : α) (s : α → Set β) : ⋃ x ∈ ({a, b} : Set α), s x = s a ∪ s b :=
by simp
#align set.bUnion_pair Set.biUnion_pair
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem inter_iUnion₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∩ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ∩ t i j := by simp only [inter_iUnion]
#align set.inter_Union₂ Set.inter_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_inter (s : ∀ i, κ i → Set α) (t : Set α) :
(⋃ (i) (j), s i j) ∩ t = ⋃ (i) (j), s i j ∩ t := by simp_rw [iUnion_inter]
#align set.Union₂_inter Set.iUnion₂_inter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_iInter₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_iInter]
#align set.union_Inter₂ Set.union_iInter₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_union (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [iInter_union]
#align set.Inter₂_union Set.iInter₂_union
theorem mem_sUnion_of_mem {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∈ t) (ht : t ∈ S) :
x ∈ ⋃₀S :=
⟨t, ht, hx⟩
#align set.mem_sUnion_of_mem Set.mem_sUnion_of_mem
-- is this theorem really necessary?
theorem not_mem_of_not_mem_sUnion {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∉ ⋃₀S)
(ht : t ∈ S) : x ∉ t := fun h => hx ⟨t, ht, h⟩
#align set.not_mem_of_not_mem_sUnion Set.not_mem_of_not_mem_sUnion
theorem sInter_subset_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : ⋂₀ S ⊆ t :=
sInf_le tS
#align set.sInter_subset_of_mem Set.sInter_subset_of_mem
theorem subset_sUnion_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : t ⊆ ⋃₀S :=
le_sSup tS
#align set.subset_sUnion_of_mem Set.subset_sUnion_of_mem
theorem subset_sUnion_of_subset {s : Set α} (t : Set (Set α)) (u : Set α) (h₁ : s ⊆ u)
(h₂ : u ∈ t) : s ⊆ ⋃₀t :=
Subset.trans h₁ (subset_sUnion_of_mem h₂)
#align set.subset_sUnion_of_subset Set.subset_sUnion_of_subset
theorem sUnion_subset {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t' ⊆ t) : ⋃₀S ⊆ t :=
sSup_le h
#align set.sUnion_subset Set.sUnion_subset
@[simp]
theorem sUnion_subset_iff {s : Set (Set α)} {t : Set α} : ⋃₀s ⊆ t ↔ ∀ t' ∈ s, t' ⊆ t :=
sSup_le_iff
#align set.sUnion_subset_iff Set.sUnion_subset_iff
/-- `sUnion` is monotone under taking a subset of each set. -/
lemma sUnion_mono_subsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, t ⊆ f t) :
⋃₀ s ⊆ ⋃₀ (f '' s) :=
fun _ ⟨t, htx, hxt⟩ ↦ ⟨f t, mem_image_of_mem f htx, hf t hxt⟩
/-- `sUnion` is monotone under taking a superset of each set. -/
lemma sUnion_mono_supsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, f t ⊆ t) :
⋃₀ (f '' s) ⊆ ⋃₀ s :=
-- If t ∈ f '' s is arbitrary; t = f u for some u : Set α.
fun _ ⟨_, ⟨u, hus, hut⟩, hxt⟩ ↦ ⟨u, hus, (hut ▸ hf u) hxt⟩
theorem subset_sInter {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t ⊆ t') : t ⊆ ⋂₀ S :=
le_sInf h
#align set.subset_sInter Set.subset_sInter
@[simp]
theorem subset_sInter_iff {S : Set (Set α)} {t : Set α} : t ⊆ ⋂₀ S ↔ ∀ t' ∈ S, t ⊆ t' :=
le_sInf_iff
#align set.subset_sInter_iff Set.subset_sInter_iff
@[gcongr]
theorem sUnion_subset_sUnion {S T : Set (Set α)} (h : S ⊆ T) : ⋃₀S ⊆ ⋃₀T :=
sUnion_subset fun _ hs => subset_sUnion_of_mem (h hs)
#align set.sUnion_subset_sUnion Set.sUnion_subset_sUnion
@[gcongr]
theorem sInter_subset_sInter {S T : Set (Set α)} (h : S ⊆ T) : ⋂₀ T ⊆ ⋂₀ S :=
subset_sInter fun _ hs => sInter_subset_of_mem (h hs)
#align set.sInter_subset_sInter Set.sInter_subset_sInter
@[simp]
theorem sUnion_empty : ⋃₀∅ = (∅ : Set α) :=
sSup_empty
#align set.sUnion_empty Set.sUnion_empty
@[simp]
theorem sInter_empty : ⋂₀ ∅ = (univ : Set α) :=
sInf_empty
#align set.sInter_empty Set.sInter_empty
@[simp]
theorem sUnion_singleton (s : Set α) : ⋃₀{s} = s :=
sSup_singleton
#align set.sUnion_singleton Set.sUnion_singleton
@[simp]
theorem sInter_singleton (s : Set α) : ⋂₀ {s} = s :=
sInf_singleton
#align set.sInter_singleton Set.sInter_singleton
@[simp]
theorem sUnion_eq_empty {S : Set (Set α)} : ⋃₀S = ∅ ↔ ∀ s ∈ S, s = ∅ :=
sSup_eq_bot
#align set.sUnion_eq_empty Set.sUnion_eq_empty
@[simp]
theorem sInter_eq_univ {S : Set (Set α)} : ⋂₀ S = univ ↔ ∀ s ∈ S, s = univ :=
sInf_eq_top
#align set.sInter_eq_univ Set.sInter_eq_univ
theorem subset_powerset_iff {s : Set (Set α)} {t : Set α} : s ⊆ 𝒫 t ↔ ⋃₀ s ⊆ t :=
sUnion_subset_iff.symm
/-- `⋃₀` and `𝒫` form a Galois connection. -/
theorem sUnion_powerset_gc :
GaloisConnection (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gc_sSup_Iic
/-- `⋃₀` and `𝒫` form a Galois insertion. -/
def sUnion_powerset_gi :
GaloisInsertion (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gi_sSup_Iic
/-- If all sets in a collection are either `∅` or `Set.univ`, then so is their union. -/
theorem sUnion_mem_empty_univ {S : Set (Set α)} (h : S ⊆ {∅, univ}) :
⋃₀ S ∈ ({∅, univ} : Set (Set α)) := by
simp only [mem_insert_iff, mem_singleton_iff, or_iff_not_imp_left, sUnion_eq_empty, not_forall]
rintro ⟨s, hs, hne⟩
obtain rfl : s = univ := (h hs).resolve_left hne
exact univ_subset_iff.1 <| subset_sUnion_of_mem hs
@[simp]
theorem nonempty_sUnion {S : Set (Set α)} : (⋃₀S).Nonempty ↔ ∃ s ∈ S, Set.Nonempty s := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_sUnion Set.nonempty_sUnion
theorem Nonempty.of_sUnion {s : Set (Set α)} (h : (⋃₀s).Nonempty) : s.Nonempty :=
let ⟨s, hs, _⟩ := nonempty_sUnion.1 h
⟨s, hs⟩
#align set.nonempty.of_sUnion Set.Nonempty.of_sUnion
theorem Nonempty.of_sUnion_eq_univ [Nonempty α] {s : Set (Set α)} (h : ⋃₀s = univ) : s.Nonempty :=
Nonempty.of_sUnion <| h.symm ▸ univ_nonempty
#align set.nonempty.of_sUnion_eq_univ Set.Nonempty.of_sUnion_eq_univ
theorem sUnion_union (S T : Set (Set α)) : ⋃₀(S ∪ T) = ⋃₀S ∪ ⋃₀T :=
sSup_union
#align set.sUnion_union Set.sUnion_union
theorem sInter_union (S T : Set (Set α)) : ⋂₀ (S ∪ T) = ⋂₀ S ∩ ⋂₀ T :=
sInf_union
#align set.sInter_union Set.sInter_union
@[simp]
theorem sUnion_insert (s : Set α) (T : Set (Set α)) : ⋃₀insert s T = s ∪ ⋃₀T :=
sSup_insert
#align set.sUnion_insert Set.sUnion_insert
@[simp]
theorem sInter_insert (s : Set α) (T : Set (Set α)) : ⋂₀ insert s T = s ∩ ⋂₀ T :=
sInf_insert
#align set.sInter_insert Set.sInter_insert
@[simp]
theorem sUnion_diff_singleton_empty (s : Set (Set α)) : ⋃₀(s \ {∅}) = ⋃₀s :=
sSup_diff_singleton_bot s
#align set.sUnion_diff_singleton_empty Set.sUnion_diff_singleton_empty
@[simp]
theorem sInter_diff_singleton_univ (s : Set (Set α)) : ⋂₀ (s \ {univ}) = ⋂₀ s :=
sInf_diff_singleton_top s
#align set.sInter_diff_singleton_univ Set.sInter_diff_singleton_univ
theorem sUnion_pair (s t : Set α) : ⋃₀{s, t} = s ∪ t :=
sSup_pair
#align set.sUnion_pair Set.sUnion_pair
theorem sInter_pair (s t : Set α) : ⋂₀ {s, t} = s ∩ t :=
sInf_pair
#align set.sInter_pair Set.sInter_pair
@[simp]
theorem sUnion_image (f : α → Set β) (s : Set α) : ⋃₀(f '' s) = ⋃ x ∈ s, f x :=
sSup_image
#align set.sUnion_image Set.sUnion_image
@[simp]
theorem sInter_image (f : α → Set β) (s : Set α) : ⋂₀ (f '' s) = ⋂ x ∈ s, f x :=
sInf_image
#align set.sInter_image Set.sInter_image
@[simp]
theorem sUnion_range (f : ι → Set β) : ⋃₀range f = ⋃ x, f x :=
rfl
#align set.sUnion_range Set.sUnion_range
@[simp]
theorem sInter_range (f : ι → Set β) : ⋂₀ range f = ⋂ x, f x :=
rfl
#align set.sInter_range Set.sInter_range
theorem iUnion_eq_univ_iff {f : ι → Set α} : ⋃ i, f i = univ ↔ ∀ x, ∃ i, x ∈ f i := by
simp only [eq_univ_iff_forall, mem_iUnion]
#align set.Union_eq_univ_iff Set.iUnion_eq_univ_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_eq_univ_iff {s : ∀ i, κ i → Set α} :
⋃ (i) (j), s i j = univ ↔ ∀ a, ∃ i j, a ∈ s i j :=
by simp only [iUnion_eq_univ_iff, mem_iUnion]
#align set.Union₂_eq_univ_iff Set.iUnion₂_eq_univ_iff
theorem sUnion_eq_univ_iff {c : Set (Set α)} : ⋃₀c = univ ↔ ∀ a, ∃ b ∈ c, a ∈ b := by
simp only [eq_univ_iff_forall, mem_sUnion]
#align set.sUnion_eq_univ_iff Set.sUnion_eq_univ_iff
-- classical
theorem iInter_eq_empty_iff {f : ι → Set α} : ⋂ i, f i = ∅ ↔ ∀ x, ∃ i, x ∉ f i := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.Inter_eq_empty_iff Set.iInter_eq_empty_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
theorem iInter₂_eq_empty_iff {s : ∀ i, κ i → Set α} :
⋂ (i) (j), s i j = ∅ ↔ ∀ a, ∃ i j, a ∉ s i j := by
simp only [eq_empty_iff_forall_not_mem, mem_iInter, not_forall]
#align set.Inter₂_eq_empty_iff Set.iInter₂_eq_empty_iff
-- classical
theorem sInter_eq_empty_iff {c : Set (Set α)} : ⋂₀ c = ∅ ↔ ∀ a, ∃ b ∈ c, a ∉ b := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.sInter_eq_empty_iff Set.sInter_eq_empty_iff
-- classical
@[simp]
theorem nonempty_iInter {f : ι → Set α} : (⋂ i, f i).Nonempty ↔ ∃ x, ∀ i, x ∈ f i := by
simp [nonempty_iff_ne_empty, iInter_eq_empty_iff]
#align set.nonempty_Inter Set.nonempty_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_iInter₂ {s : ∀ i, κ i → Set α} :
(⋂ (i) (j), s i j).Nonempty ↔ ∃ a, ∀ i j, a ∈ s i j := by
simp
#align set.nonempty_Inter₂ Set.nonempty_iInter₂
-- classical
@[simp]
theorem nonempty_sInter {c : Set (Set α)} : (⋂₀ c).Nonempty ↔ ∃ a, ∀ b ∈ c, a ∈ b := by
simp [nonempty_iff_ne_empty, sInter_eq_empty_iff]
#align set.nonempty_sInter Set.nonempty_sInter
-- classical
theorem compl_sUnion (S : Set (Set α)) : (⋃₀S)ᶜ = ⋂₀ (compl '' S) :=
ext fun x => by simp
#align set.compl_sUnion Set.compl_sUnion
-- classical
theorem sUnion_eq_compl_sInter_compl (S : Set (Set α)) : ⋃₀S = (⋂₀ (compl '' S))ᶜ := by
rw [← compl_compl (⋃₀S), compl_sUnion]
#align set.sUnion_eq_compl_sInter_compl Set.sUnion_eq_compl_sInter_compl
-- classical
theorem compl_sInter (S : Set (Set α)) : (⋂₀ S)ᶜ = ⋃₀(compl '' S) := by
rw [sUnion_eq_compl_sInter_compl, compl_compl_image]
#align set.compl_sInter Set.compl_sInter
-- classical
theorem sInter_eq_compl_sUnion_compl (S : Set (Set α)) : ⋂₀ S = (⋃₀(compl '' S))ᶜ := by
rw [← compl_compl (⋂₀ S), compl_sInter]
#align set.sInter_eq_compl_sUnion_compl Set.sInter_eq_compl_sUnion_compl
theorem inter_empty_of_inter_sUnion_empty {s t : Set α} {S : Set (Set α)} (hs : t ∈ S)
(h : s ∩ ⋃₀S = ∅) : s ∩ t = ∅ :=
eq_empty_of_subset_empty <| by
rw [← h]; exact inter_subset_inter_right _ (subset_sUnion_of_mem hs)
#align set.inter_empty_of_inter_sUnion_empty Set.inter_empty_of_inter_sUnion_empty
theorem range_sigma_eq_iUnion_range {γ : α → Type*} (f : Sigma γ → β) :
range f = ⋃ a, range fun b => f ⟨a, b⟩ :=
Set.ext <| by simp
#align set.range_sigma_eq_Union_range Set.range_sigma_eq_iUnion_range
theorem iUnion_eq_range_sigma (s : α → Set β) : ⋃ i, s i = range fun a : Σi, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_sigma Set.iUnion_eq_range_sigma
theorem iUnion_eq_range_psigma (s : ι → Set β) : ⋃ i, s i = range fun a : Σ'i, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_psigma Set.iUnion_eq_range_psigma
theorem iUnion_image_preimage_sigma_mk_eq_self {ι : Type*} {σ : ι → Type*} (s : Set (Sigma σ)) :
⋃ i, Sigma.mk i '' (Sigma.mk i ⁻¹' s) = s := by
ext x
simp only [mem_iUnion, mem_image, mem_preimage]
constructor
· rintro ⟨i, a, h, rfl⟩
exact h
· intro h
cases' x with i a
exact ⟨i, a, h, rfl⟩
#align set.Union_image_preimage_sigma_mk_eq_self Set.iUnion_image_preimage_sigma_mk_eq_self
theorem Sigma.univ (X : α → Type*) : (Set.univ : Set (Σa, X a)) = ⋃ a, range (Sigma.mk a) :=
Set.ext fun x =>
iff_of_true trivial ⟨range (Sigma.mk x.1), Set.mem_range_self _, x.2, Sigma.eta x⟩
#align set.sigma.univ Set.Sigma.univ
alias sUnion_mono := sUnion_subset_sUnion
#align set.sUnion_mono Set.sUnion_mono
theorem iUnion_subset_iUnion_const {s : Set α} (h : ι → ι₂) : ⋃ _ : ι, s ⊆ ⋃ _ : ι₂, s :=
@iSup_const_mono (Set α) ι ι₂ _ s h
#align set.Union_subset_Union_const Set.iUnion_subset_iUnion_const
@[simp]
theorem iUnion_singleton_eq_range {α β : Type*} (f : α → β) : ⋃ x : α, {f x} = range f := by
ext x
simp [@eq_comm _ x]
#align set.Union_singleton_eq_range Set.iUnion_singleton_eq_range
theorem iUnion_of_singleton (α : Type*) : (⋃ x, {x} : Set α) = univ := by simp [Set.ext_iff]
#align set.Union_of_singleton Set.iUnion_of_singleton
theorem iUnion_of_singleton_coe (s : Set α) : ⋃ i : s, ({(i : α)} : Set α) = s := by simp
#align set.Union_of_singleton_coe Set.iUnion_of_singleton_coe
theorem sUnion_eq_biUnion {s : Set (Set α)} : ⋃₀s = ⋃ (i : Set α) (_ : i ∈ s), i := by
rw [← sUnion_image, image_id']
#align set.sUnion_eq_bUnion Set.sUnion_eq_biUnion
theorem sInter_eq_biInter {s : Set (Set α)} : ⋂₀ s = ⋂ (i : Set α) (_ : i ∈ s), i := by
rw [← sInter_image, image_id']
#align set.sInter_eq_bInter Set.sInter_eq_biInter
theorem sUnion_eq_iUnion {s : Set (Set α)} : ⋃₀s = ⋃ i : s, i := by
simp only [← sUnion_range, Subtype.range_coe]
#align set.sUnion_eq_Union Set.sUnion_eq_iUnion
theorem sInter_eq_iInter {s : Set (Set α)} : ⋂₀ s = ⋂ i : s, i := by
simp only [← sInter_range, Subtype.range_coe]
#align set.sInter_eq_Inter Set.sInter_eq_iInter
@[simp]
theorem iUnion_of_empty [IsEmpty ι] (s : ι → Set α) : ⋃ i, s i = ∅ :=
iSup_of_empty _
#align set.Union_of_empty Set.iUnion_of_empty
@[simp]
theorem iInter_of_empty [IsEmpty ι] (s : ι → Set α) : ⋂ i, s i = univ :=
iInf_of_empty _
#align set.Inter_of_empty Set.iInter_of_empty
theorem union_eq_iUnion {s₁ s₂ : Set α} : s₁ ∪ s₂ = ⋃ b : Bool, cond b s₁ s₂ :=
sup_eq_iSup s₁ s₂
#align set.union_eq_Union Set.union_eq_iUnion
theorem inter_eq_iInter {s₁ s₂ : Set α} : s₁ ∩ s₂ = ⋂ b : Bool, cond b s₁ s₂ :=
inf_eq_iInf s₁ s₂
#align set.inter_eq_Inter Set.inter_eq_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_union_sInter {S T : Set (Set α)} :
⋂₀ S ∪ ⋂₀ T = ⋂ p ∈ S ×ˢ T, (p : Set α × Set α).1 ∪ p.2 :=
sInf_sup_sInf
#align set.sInter_union_sInter Set.sInter_union_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_inter_sUnion {s t : Set (Set α)} :
⋃₀s ∩ ⋃₀t = ⋃ p ∈ s ×ˢ t, (p : Set α × Set α).1 ∩ p.2 :=
sSup_inf_sSup
#align set.sUnion_inter_sUnion Set.sUnion_inter_sUnion
theorem biUnion_iUnion (s : ι → Set α) (t : α → Set β) :
⋃ x ∈ ⋃ i, s i, t x = ⋃ (i) (x ∈ s i), t x := by simp [@iUnion_comm _ ι]
#align set.bUnion_Union Set.biUnion_iUnion
theorem biInter_iUnion (s : ι → Set α) (t : α → Set β) :
⋂ x ∈ ⋃ i, s i, t x = ⋂ (i) (x ∈ s i), t x := by simp [@iInter_comm _ ι]
#align set.bInter_Union Set.biInter_iUnion
theorem sUnion_iUnion (s : ι → Set (Set α)) : ⋃₀⋃ i, s i = ⋃ i, ⋃₀s i := by
simp only [sUnion_eq_biUnion, biUnion_iUnion]
#align set.sUnion_Union Set.sUnion_iUnion
theorem sInter_iUnion (s : ι → Set (Set α)) : ⋂₀ ⋃ i, s i = ⋂ i, ⋂₀ s i := by
simp only [sInter_eq_biInter, biInter_iUnion]
#align set.sInter_Union Set.sInter_iUnion
theorem iUnion_range_eq_sUnion {α β : Type*} (C : Set (Set α)) {f : ∀ s : C, β → (s : Type _)}
(hf : ∀ s : C, Surjective (f s)) : ⋃ y : β, range (fun s : C => (f s y).val) = ⋃₀C := by
ext x; constructor
· rintro ⟨s, ⟨y, rfl⟩, ⟨s, hs⟩, rfl⟩
refine' ⟨_, hs, _⟩
exact (f ⟨s, hs⟩ y).2
· rintro ⟨s, hs, hx⟩
cases' hf ⟨s, hs⟩ ⟨x, hx⟩ with y hy
refine' ⟨_, ⟨y, rfl⟩, ⟨s, hs⟩, _⟩
exact congr_arg Subtype.val hy
#align set.Union_range_eq_sUnion Set.iUnion_range_eq_sUnion
theorem iUnion_range_eq_iUnion (C : ι → Set α) {f : ∀ x : ι, β → C x}
(hf : ∀ x : ι, Surjective (f x)) : ⋃ y : β, range (fun x : ι => (f x y).val) = ⋃ x, C x := by
ext x; rw [mem_iUnion, mem_iUnion]; constructor
· rintro ⟨y, i, rfl⟩
exact ⟨i, (f i y).2⟩
· rintro ⟨i, hx⟩
cases' hf i ⟨x, hx⟩ with y hy
exact ⟨y, i, congr_arg Subtype.val hy⟩
#align set.Union_range_eq_Union Set.iUnion_range_eq_iUnion
theorem union_distrib_iInter_left (s : ι → Set α) (t : Set α) : (t ∪ ⋂ i, s i) = ⋂ i, t ∪ s i :=
sup_iInf_eq _ _
#align set.union_distrib_Inter_left Set.union_distrib_iInter_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_left (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_distrib_iInter_left]
#align set.union_distrib_Inter₂_left Set.union_distrib_iInter₂_left
theorem union_distrib_iInter_right (s : ι → Set α) (t : Set α) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.union_distrib_Inter_right Set.union_distrib_iInter_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_right (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [union_distrib_iInter_right]
#align set.union_distrib_Inter₂_right Set.union_distrib_iInter₂_right
section Function
/-! ### `mapsTo` -/
theorem mapsTo_sUnion {S : Set (Set α)} {t : Set β} {f : α → β} (H : ∀ s ∈ S, MapsTo f s t) :
MapsTo f (⋃₀S) t := fun _ ⟨s, hs, hx⟩ => H s hs hx
#align set.maps_to_sUnion Set.mapsTo_sUnion
theorem mapsTo_iUnion {s : ι → Set α} {t : Set β} {f : α → β} (H : ∀ i, MapsTo f (s i) t) :
MapsTo f (⋃ i, s i) t :=
mapsTo_sUnion <| forall_range_iff.2 H
#align set.maps_to_Union Set.mapsTo_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂ {s : ∀ i, κ i → Set α} {t : Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) t) : MapsTo f (⋃ (i) (j), s i j) t :=
mapsTo_iUnion fun i => mapsTo_iUnion (H i)
#align set.maps_to_Union₂ Set.mapsTo_iUnion₂
theorem mapsTo_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋃ i, s i) (⋃ i, t i) :=
mapsTo_iUnion fun i => (H i).mono (Subset.refl _) (subset_iUnion t i)
#align set.maps_to_Union_Union Set.mapsTo_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
mapsTo_iUnion_iUnion fun i => mapsTo_iUnion_iUnion (H i)
#align set.maps_to_Union₂_Union₂ Set.mapsTo_iUnion₂_iUnion₂
theorem mapsTo_sInter {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, MapsTo f s t) :
MapsTo f s (⋂₀ T) := fun _ hx t ht => H t ht hx
#align set.maps_to_sInter Set.mapsTo_sInter
theorem mapsTo_iInter {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, MapsTo f s (t i)) :
MapsTo f s (⋂ i, t i) := fun _ hx => mem_iInter.2 fun i => H i hx
#align set.maps_to_Inter Set.mapsTo_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f s (t i j)) : MapsTo f s (⋂ (i) (j), t i j) :=
mapsTo_iInter fun i => mapsTo_iInter (H i)
#align set.maps_to_Inter₂ Set.mapsTo_iInter₂
theorem mapsTo_iInter_iInter {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋂ i, s i) (⋂ i, t i) :=
mapsTo_iInter fun i => (H i).mono (iInter_subset s i) (Subset.refl _)
#align set.maps_to_Inter_Inter Set.mapsTo_iInter_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂_iInter₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋂ (i) (j), s i j) (⋂ (i) (j), t i j) :=
mapsTo_iInter_iInter fun i => mapsTo_iInter_iInter (H i)
#align set.maps_to_Inter₂_Inter₂ Set.mapsTo_iInter₂_iInter₂
theorem image_iInter_subset (s : ι → Set α) (f : α → β) : (f '' ⋂ i, s i) ⊆ ⋂ i, f '' s i :=
(mapsTo_iInter_iInter fun i => mapsTo_image f (s i)).image_subset
#align set.image_Inter_subset Set.image_iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂_subset (s : ∀ i, κ i → Set α) (f : α → β) :
(f '' ⋂ (i) (j), s i j) ⊆ ⋂ (i) (j), f '' s i j :=
(mapsTo_iInter₂_iInter₂ fun i hi => mapsTo_image f (s i hi)).image_subset
#align set.image_Inter₂_subset Set.image_iInter₂_subset
theorem image_sInter_subset (S : Set (Set α)) (f : α → β) : f '' ⋂₀ S ⊆ ⋂ s ∈ S, f '' s := by
rw [sInter_eq_biInter]
apply image_iInter₂_subset
#align set.image_sInter_subset Set.image_sInter_subset
/-! ### `restrictPreimage` -/
section
open Function
variable (s : Set β) {f : α → β} {U : ι → Set β} (hU : iUnion U = univ)
theorem injective_iff_injective_of_iUnion_eq_univ :
Injective f ↔ ∀ i, Injective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_injective H, fun H x y e => _⟩
obtain ⟨i, hi⟩ := Set.mem_iUnion.mp
(show f x ∈ Set.iUnion U by rw [hU]; triv)
injection @H i ⟨x, hi⟩ ⟨y, show f y ∈ U i from e ▸ hi⟩ (Subtype.ext e)
#align set.injective_iff_injective_of_Union_eq_univ Set.injective_iff_injective_of_iUnion_eq_univ
theorem surjective_iff_surjective_of_iUnion_eq_univ :
Surjective f ↔ ∀ i, Surjective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_surjective H, fun H x => _⟩
obtain ⟨i, hi⟩ :=
Set.mem_iUnion.mp
(show x ∈ Set.iUnion U by rw [hU]; triv)
exact ⟨_, congr_arg Subtype.val (H i ⟨x, hi⟩).choose_spec⟩
#align set.surjective_iff_surjective_of_Union_eq_univ Set.surjective_iff_surjective_of_iUnion_eq_univ
theorem bijective_iff_bijective_of_iUnion_eq_univ :
Bijective f ↔ ∀ i, Bijective ((U i).restrictPreimage f) := by
rw [Bijective, injective_iff_injective_of_iUnion_eq_univ hU,
surjective_iff_surjective_of_iUnion_eq_univ hU]
simp [Bijective, forall_and]
#align set.bijective_iff_bijective_of_Union_eq_univ Set.bijective_iff_bijective_of_iUnion_eq_univ
end
/-! ### `InjOn` -/
theorem InjOn.image_iInter_eq [Nonempty ι] {s : ι → Set α} {f : α → β} (h : InjOn f (⋃ i, s i)) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
inhabit ι
refine' Subset.antisymm (image_iInter_subset s f) fun y hy => _
simp only [mem_iInter, mem_image_iff_bex] at hy
choose x hx hy using hy
refine' ⟨x default, mem_iInter.2 fun i => _, hy _⟩
suffices x default = x i by
rw [this]
apply hx
replace hx : ∀ i, x i ∈ ⋃ j, s j := fun i => (subset_iUnion _ _) (hx i)
apply h (hx _) (hx _)
simp only [hy]
#align set.inj_on.image_Inter_eq Set.InjOn.image_iInter_eq
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
theorem InjOn.image_biInter_eq {p : ι → Prop} {s : ∀ i, p i → Set α} (hp : ∃ i, p i)
{f : α → β} (h : InjOn f (⋃ (i) (hi), s i hi)) :
(f '' ⋂ (i) (hi), s i hi) = ⋂ (i) (hi), f '' s i hi := by
simp only [iInter, iInf_subtype']
haveI : Nonempty { i // p i } := nonempty_subtype.2 hp
apply InjOn.image_iInter_eq
simpa only [iUnion, iSup_subtype'] using h
#align set.inj_on.image_bInter_eq Set.InjOn.image_biInter_eq
theorem image_iInter {f : α → β} (hf : Bijective f) (s : ι → Set α) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
cases isEmpty_or_nonempty ι
· simp_rw [iInter_of_empty, image_univ_of_surjective hf.surjective]
· exact (hf.injective.injOn _).image_iInter_eq
#align set.image_Inter Set.image_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂ {f : α → β} (hf : Bijective f) (s : ∀ i, κ i → Set α) :
(f '' ⋂ (i) (j), s i j) = ⋂ (i) (j), f '' s i j := by simp_rw [image_iInter hf]
#align set.image_Inter₂ Set.image_iInter₂
theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
(hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i) := by
intro x hx y hy hxy
rcases mem_iUnion.1 hx with ⟨i, hx⟩
rcases mem_iUnion.1 hy with ⟨j, hy⟩
rcases hs i j with ⟨k, hi, hj⟩
exact hf k (hi hx) (hj hy) hxy
#align set.inj_on_Union_of_directed Set.inj_on_iUnion_of_directed
/-! ### `SurjOn` -/
theorem surjOn_sUnion {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, SurjOn f s t) :
SurjOn f s (⋃₀T) := fun _ ⟨t, ht, hx⟩ => H t ht hx
#align set.surj_on_sUnion Set.surjOn_sUnion
theorem surjOn_iUnion {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, SurjOn f s (t i)) :
SurjOn f s (⋃ i, t i) :=
surjOn_sUnion <| forall_range_iff.2 H
#align set.surj_on_Union Set.surjOn_iUnion
theorem surjOn_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) : SurjOn f (⋃ i, s i) (⋃ i, t i) :=
surjOn_iUnion fun i => (H i).mono (subset_iUnion _ _) (Subset.refl _)
#align set.surj_on_Union_Union Set.surjOn_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f s (t i j)) : SurjOn f s (⋃ (i) (j), t i j) :=
surjOn_iUnion fun i => surjOn_iUnion (H i)
#align set.surj_on_Union₂ Set.surjOn_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f (s i j) (t i j)) : SurjOn f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
surjOn_iUnion_iUnion fun i => surjOn_iUnion_iUnion (H i)
#align set.surj_on_Union₂_Union₂ Set.surjOn_iUnion₂_iUnion₂
theorem surjOn_iInter [Nonempty ι] {s : ι → Set α} {t : Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) t) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) t := by
intro y hy
rw [Hinj.image_iInter_eq, mem_iInter]
exact fun i => H i hy
#align set.surj_on_Inter Set.surjOn_iInter
theorem surjOn_iInter_iInter [Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) (⋂ i, t i) :=
surjOn_iInter (fun i => (H i).mono (Subset.refl _) (iInter_subset _ _)) Hinj
#align set.surj_on_Inter_Inter Set.surjOn_iInter_iInter
/-! ### `BijOn` -/
theorem bijOn_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i))
(Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
⟨mapsTo_iUnion_iUnion fun i => (H i).mapsTo, Hinj, surjOn_iUnion_iUnion fun i => (H i).surjOn⟩
#align set.bij_on_Union Set.bijOn_iUnion
theorem bijOn_iInter [hi : Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, BijOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
⟨mapsTo_iInter_iInter fun i => (H i).mapsTo,
hi.elim fun i => (H i).injOn.mono (iInter_subset _ _),
surjOn_iInter_iInter (fun i => (H i).surjOn) Hinj⟩
#align set.bij_on_Inter Set.bijOn_iInter
theorem bijOn_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {t : ι → Set β}
{f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
bijOn_iUnion H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Union_of_directed Set.bijOn_iUnion_of_directed
theorem bijOn_iInter_of_directed [Nonempty ι] {s : ι → Set α} (hs : Directed (· ⊆ ·) s)
{t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
bijOn_iInter H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Inter_of_directed Set.bijOn_iInter_of_directed
end Function
/-! ### `image`, `preimage` -/
section Image
theorem image_iUnion {f : α → β} {s : ι → Set α} : (f '' ⋃ i, s i) = ⋃ i, f '' s i := by
ext1 x
simp only [mem_image, mem_iUnion, ← exists_and_right, ← exists_and_left]
--Porting note: `exists_swap` causes a `simp` loop in Lean4 so we use `rw` instead.
rw [exists_swap]
#align set.image_Union Set.image_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iUnion₂ (f : α → β) (s : ∀ i, κ i → Set α) :
(f '' ⋃ (i) (j), s i j) = ⋃ (i) (j), f '' s i j := by simp_rw [image_iUnion]
#align set.image_Union₂ Set.image_iUnion₂
theorem univ_subtype {p : α → Prop} : (univ : Set (Subtype p)) = ⋃ (x) (h : p x), {⟨x, h⟩} :=
Set.ext fun ⟨x, h⟩ => by simp [h]
#align set.univ_subtype Set.univ_subtype
theorem range_eq_iUnion {ι} (f : ι → α) : range f = ⋃ i, {f i} :=
Set.ext fun a => by simp [@eq_comm α a]
#align set.range_eq_Union Set.range_eq_iUnion
theorem image_eq_iUnion (f : α → β) (s : Set α) : f '' s = ⋃ i ∈ s, {f i} :=
Set.ext fun b => by simp [@eq_comm β b]
#align set.image_eq_Union Set.image_eq_iUnion
theorem biUnion_range {f : ι → α} {g : α → Set β} : ⋃ x ∈ range f, g x = ⋃ y, g (f y) :=
iSup_range
#align set.bUnion_range Set.biUnion_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iUnion_iUnion_eq' {f : ι → α} {g : α → Set β} :
⋃ (x) (y) (_ : f y = x), g x = ⋃ y, g (f y) := by simpa using biUnion_range
#align set.Union_Union_eq' Set.iUnion_iUnion_eq'
theorem biInter_range {f : ι → α} {g : α → Set β} : ⋂ x ∈ range f, g x = ⋂ y, g (f y) :=
iInf_range
#align set.bInter_range Set.biInter_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iInter_iInter_eq' {f : ι → α} {g : α → Set β} :
⋂ (x) (y) (_ : f y = x), g x = ⋂ y, g (f y) := by simpa using biInter_range
#align set.Inter_Inter_eq' Set.iInter_iInter_eq'
variable {s : Set γ} {f : γ → α} {g : α → Set β}
theorem biUnion_image : ⋃ x ∈ f '' s, g x = ⋃ y ∈ s, g (f y) :=
iSup_image
#align set.bUnion_image Set.biUnion_image
theorem biInter_image : ⋂ x ∈ f '' s, g x = ⋂ y ∈ s, g (f y) :=
iInf_image
#align set.bInter_image Set.biInter_image
end Image
section Preimage
theorem monotone_preimage {f : α → β} : Monotone (preimage f) := fun _ _ h => preimage_mono h
#align set.monotone_preimage Set.monotone_preimage
@[simp]
theorem preimage_iUnion {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋃ i, s i) = ⋃ i, f ⁻¹' s i :=
Set.ext <| by simp [preimage]
#align set.preimage_Union Set.preimage_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iUnion₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋃ (i) (j), s i j) = ⋃ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iUnion]
#align set.preimage_Union₂ Set.preimage_iUnion₂
@[simp]
theorem preimage_sUnion {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋃₀s = ⋃ t ∈ s, f ⁻¹' t := by
rw [sUnion_eq_biUnion, preimage_iUnion₂]
#align set.preimage_sUnion Set.preimage_sUnion
theorem preimage_iInter {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋂ i, s i) = ⋂ i, f ⁻¹' s i := by
ext; simp
#align set.preimage_Inter Set.preimage_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iInter₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋂ (i) (j), s i j) = ⋂ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iInter]
#align set.preimage_Inter₂ Set.preimage_iInter₂
@[simp]
theorem preimage_sInter {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋂₀ s = ⋂ t ∈ s, f ⁻¹' t := by
rw [sInter_eq_biInter, preimage_iInter₂]
#align set.preimage_sInter Set.preimage_sInter
@[simp]
theorem biUnion_preimage_singleton (f : α → β) (s : Set β) : ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s := by
rw [← preimage_iUnion₂, biUnion_of_singleton]
#align set.bUnion_preimage_singleton Set.biUnion_preimage_singleton
theorem biUnion_range_preimage_singleton (f : α → β) : ⋃ y ∈ range f, f ⁻¹' {y} = univ := by
rw [biUnion_preimage_singleton, preimage_range]
#align set.bUnion_range_preimage_singleton Set.biUnion_range_preimage_singleton
end Preimage
section Prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion {s : Set α} {t : ι → Set β} : (s ×ˢ ⋃ i, t i) = ⋃ i, s ×ˢ t i := by
ext
simp
#align set.prod_Union Set.prod_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} :
(s ×ˢ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ×ˢ t i j := by simp_rw [prod_iUnion]
#align set.prod_Union₂ Set.prod_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sUnion {s : Set α} {C : Set (Set β)} : s ×ˢ ⋃₀C = ⋃₀((fun t => s ×ˢ t) '' C) := by
simp_rw [sUnion_eq_biUnion, biUnion_image, prod_iUnion₂]
#align set.prod_sUnion Set.prod_sUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_const {s : ι → Set α} {t : Set β} : (⋃ i, s i) ×ˢ t = ⋃ i, s i ×ˢ t := by
ext
simp
#align set.Union_prod_const Set.iUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion₂_prod_const {s : ∀ i, κ i → Set α} {t : Set β} :
(⋃ (i) (j), s i j) ×ˢ t = ⋃ (i) (j), s i j ×ˢ t := by simp_rw [iUnion_prod_const]
#align set.Union₂_prod_const Set.iUnion₂_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_prod_const {C : Set (Set α)} {t : Set β} :
⋃₀C ×ˢ t = ⋃₀((fun s : Set α => s ×ˢ t) '' C) := by
simp only [sUnion_eq_biUnion, iUnion₂_prod_const, biUnion_image]
#align set.sUnion_prod_const Set.sUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod {ι ι' α β} (s : ι → Set α) (t : ι' → Set β) :
⋃ x : ι × ι', s x.1 ×ˢ t x.2 = (⋃ i : ι, s i) ×ˢ ⋃ i : ι', t i := by
ext
simp
#align set.Union_prod Set.iUnion_prod
/-- Analogue of `iSup_prod` for sets. -/
lemma iUnion_prod' (f : β × γ → Set α) : ⋃ x : β × γ, f x = ⋃ (i : β) (j : γ), f (i, j) :=
iSup_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x := by
ext ⟨z, w⟩; simp only [mem_prod, mem_iUnion, exists_imp, and_imp, iff_def]; constructor
· intro x hz hw
exact ⟨⟨x, hz⟩, x, hw⟩
· intro x hz x' hw
exact ⟨x ⊔ x', hs le_sup_left hz, ht le_sup_right hw⟩
#align set.Union_prod_of_monotone Set.iUnion_prod_of_monotone
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter_subset (S : Set (Set α)) (T : Set (Set β)) :
⋂₀ S ×ˢ ⋂₀ T ⊆ ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 :=
subset_iInter₂ fun x hx _ hy => ⟨hy.1 x.1 hx.1, hy.2 x.2 hx.2⟩
#align set.sInter_prod_sInter_subset Set.sInter_prod_sInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 := by
obtain ⟨s₁, h₁⟩ := hS
obtain ⟨s₂, h₂⟩ := hT
refine' Set.Subset.antisymm (sInter_prod_sInter_subset S T) fun x hx => _
rw [mem_iInter₂] at hx
exact ⟨fun s₀ h₀ => (hx (s₀, s₂) ⟨h₀, h₂⟩).1, fun s₀ h₀ => (hx (s₁, s₀) ⟨h₁, h₀⟩).2⟩
#align set.sInter_prod_sInter Set.sInter_prod_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod {S : Set (Set α)} (hS : S.Nonempty) (t : Set β) :
⋂₀ S ×ˢ t = ⋂ s ∈ S, s ×ˢ t := by
rw [← sInter_singleton t, sInter_prod_sInter hS (singleton_nonempty t), sInter_singleton]
simp_rw [prod_singleton, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.sInter_prod Set.sInter_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sInter {T : Set (Set β)} (hT : T.Nonempty) (s : Set α) :
s ×ˢ ⋂₀ T = ⋂ t ∈ T, s ×ˢ t := by
rw [← sInter_singleton s, sInter_prod_sInter (singleton_nonempty s) hT, sInter_singleton]
simp_rw [singleton_prod, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.prod_sInter Set.prod_sInter
end Prod
section Image2
variable (f : α → β → γ) {s : Set α} {t : Set β}
/-- The `Set.image2` version of `Set.image_eq_iUnion` -/
theorem image2_eq_iUnion (s : Set α) (t : Set β) : image2 f s t = ⋃ (i ∈ s) (j ∈ t), {f i j} := by
ext; simp [eq_comm]
#align set.image2_eq_Union Set.image2_eq_iUnion
theorem iUnion_image_left : ⋃ a ∈ s, f a '' t = image2 f s t := by
simp only [image2_eq_iUnion, image_eq_iUnion]
#align set.Union_image_left Set.iUnion_image_left
theorem iUnion_image_right : ⋃ b ∈ t, (f · b) '' s = image2 f s t := by
rw [image2_swap, iUnion_image_left]
#align set.Union_image_right Set.iUnion_image_right
theorem image2_iUnion_left (s : ι → Set α) (t : Set β) :
image2 f (⋃ i, s i) t = ⋃ i, image2 f (s i) t := by
simp only [← image_prod, iUnion_prod_const, image_iUnion]
#align set.image2_Union_left Set.image2_iUnion_left
theorem image2_iUnion_right (s : Set α) (t : ι → Set β) :
image2 f s (⋃ i, t i) = ⋃ i, image2 f s (t i) := by
simp only [← image_prod, prod_iUnion, image_iUnion]
#align set.image2_Union_right Set.image2_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋃ (i) (j), s i j) t = ⋃ (i) (j), image2 f (s i j) t := by simp_rw [image2_iUnion_left]
#align set.image2_Union₂_left Set.image2_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋃ (i) (j), t i j) = ⋃ (i) (j), image2 f s (t i j) :=
by simp_rw [image2_iUnion_right]
#align set.image2_Union₂_right Set.image2_iUnion₂_right
theorem image2_iInter_subset_left (s : ι → Set α) (t : Set β) :
image2 f (⋂ i, s i) t ⊆ ⋂ i, image2 f (s i) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem (hx _) hy
#align set.image2_Inter_subset_left Set.image2_iInter_subset_left
theorem image2_iInter_subset_right (s : Set α) (t : ι → Set β) :
image2 f s (⋂ i, t i) ⊆ ⋂ i, image2 f s (t i) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem hx (hy _)
#align set.image2_Inter_subset_right Set.image2_iInter_subset_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋂ (i) (j), s i j) t ⊆ ⋂ (i) (j), image2 f (s i j) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem (hx _ _) hy
#align set.image2_Inter₂_subset_left Set.image2_iInter₂_subset_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), image2 f s (t i j) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem hx (hy _ _)
#align set.image2_Inter₂_subset_right Set.image2_iInter₂_subset_right
theorem prod_eq_biUnion_left : s ×ˢ t = ⋃ a ∈ s, (fun b => (a, b)) '' t := by
rw [iUnion_image_left, image2_mk_eq_prod]
#align set.prod_eq_bUnion_left Set.prod_eq_biUnion_left
theorem prod_eq_biUnion_right : s ×ˢ t = ⋃ b ∈ t, (fun a => (a, b)) '' s := by
rw [iUnion_image_right, image2_mk_eq_prod]
#align set.prod_eq_bUnion_right Set.prod_eq_biUnion_right
end Image2
section Seq
/-- Given a set `s` of functions `α → β` and `t : Set α`, `seq s t` is the union of `f '' t` over
all `f ∈ s`. -/
def seq (s : Set (α → β)) (t : Set α) : Set β :=
{ b | ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b }
#align set.seq Set.seq
@[simp]
theorem mem_seq_iff {s : Set (α → β)} {t : Set α} {b : β} :
b ∈ seq s t ↔ ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b :=
Iff.rfl
#align set.mem_seq_iff Set.mem_seq_iff
lemma seq_eq_image2 (s : Set (α → β)) (t : Set α) : seq s t = image2 (fun f a ↦ f a) s t := by
ext; simp
theorem seq_def {s : Set (α → β)} {t : Set α} : seq s t = ⋃ f ∈ s, f '' t := by
rw [seq_eq_image2, iUnion_image_left]
#align set.seq_def Set.seq_def
theorem seq_subset {s : Set (α → β)} {t : Set α} {u : Set β} :
seq s t ⊆ u ↔ ∀ f ∈ s, ∀ a ∈ t, (f : α → β) a ∈ u := by
rw [seq_eq_image2, image2_subset_iff]
#align set.seq_subset Set.seq_subset
@[gcongr]
theorem seq_mono {s₀ s₁ : Set (α → β)} {t₀ t₁ : Set α} (hs : s₀ ⊆ s₁) (ht : t₀ ⊆ t₁) :
seq s₀ t₀ ⊆ seq s₁ t₁ := fun _ ⟨f, hf, a, ha, eq⟩ => ⟨f, hs hf, a, ht ha, eq⟩
#align set.seq_mono Set.seq_mono
theorem singleton_seq {f : α → β} {t : Set α} : Set.seq ({f} : Set (α → β)) t = f '' t := by
rw [seq_eq_image2, image2_singleton_left]
#align set.singleton_seq Set.singleton_seq
theorem seq_singleton {s : Set (α → β)} {a : α} : Set.seq s {a} = (fun f : α → β => f a) '' s := by
rw [seq_eq_image2, image2_singleton_right]
#align set.seq_singleton Set.seq_singleton
theorem seq_seq {s : Set (β → γ)} {t : Set (α → β)} {u : Set α} :
seq s (seq t u) = seq (seq ((· ∘ ·) '' s) t) u := by
simp only [seq_eq_image2, image2_image_left, image2_image2_left, image2_image2_right, comp_apply]
#align set.seq_seq Set.seq_seq
theorem image_seq {f : β → γ} {s : Set (α → β)} {t : Set α} :
f '' seq s t = seq ((f ∘ ·) '' s) t := by
rw [← singleton_seq, ← singleton_seq, seq_seq, image_singleton]
#align set.image_seq Set.image_seq
theorem prod_eq_seq {s : Set α} {t : Set β} : s ×ˢ t = (Prod.mk '' s).seq t := by
rw [seq_eq_image2, image2_image_left, image2_mk_eq_prod]
#align set.prod_eq_seq Set.prod_eq_seq
theorem prod_image_seq_comm (s : Set α) (t : Set β) :
(Prod.mk '' s).seq t = seq ((fun b a => (a, b)) '' t) s := by
rw [← prod_eq_seq, ← image_swap_prod, prod_eq_seq, image_seq, ← image_comp]; rfl
#align set.prod_image_seq_comm Set.prod_image_seq_comm
theorem image2_eq_seq (f : α → β → γ) (s : Set α) (t : Set β) : image2 f s t = seq (f '' s) t := by
rw [seq_eq_image2, image2_image_left]
#align set.image2_eq_seq Set.image2_eq_seq
end Seq
section Pi
variable {π : α → Type*}
theorem pi_def (i : Set α) (s : ∀ a, Set (π a)) : pi i s = ⋂ a ∈ i, eval a ⁻¹' s a := by
ext
simp
#align set.pi_def Set.pi_def
theorem univ_pi_eq_iInter (t : ∀ i, Set (π i)) : pi univ t = ⋂ i, eval i ⁻¹' t i := by
simp only [pi_def, iInter_true, mem_univ]
#align set.univ_pi_eq_Inter Set.univ_pi_eq_iInter
theorem pi_diff_pi_subset (i : Set α) (s t : ∀ a, Set (π a)) :
pi i s \ pi i t ⊆ ⋃ a ∈ i, eval a ⁻¹' (s a \ t a) := by
refine' diff_subset_comm.2 fun x hx a ha => _
simp only [mem_diff, mem_pi, mem_iUnion, not_exists, mem_preimage, not_and, not_not,
eval_apply] at hx
exact hx.2 _ ha (hx.1 _ ha)
#align set.pi_diff_pi_subset Set.pi_diff_pi_subset
theorem iUnion_univ_pi {ι : α → Type*} (t : (a : α) → ι a → Set (π a)) :
⋃ x : (a : α) → ι a, pi univ (fun a => t a (x a)) = pi univ fun a => ⋃ j : ι a, t a j := by
ext
simp [Classical.skolem]
#align set.Union_univ_pi Set.iUnion_univ_pi
end Pi
end Set
namespace Function
namespace Surjective
theorem iUnion_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋃ x, g (f x) = ⋃ y, g y :=
hf.iSup_comp g
#align function.surjective.Union_comp Function.Surjective.iUnion_comp
theorem iInter_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋂ x, g (f x) = ⋂ y, g y :=
hf.iInf_comp g
#align function.surjective.Inter_comp Function.Surjective.iInter_comp
end Surjective
end Function
/-!
### Disjoint sets
-/
section Disjoint
variable {s t u : Set α} {f : α → β}
namespace Set
@[simp]
theorem disjoint_iUnion_left {ι : Sort*} {s : ι → Set α} :
Disjoint (⋃ i, s i) t ↔ ∀ i, Disjoint (s i) t :=
iSup_disjoint_iff
#align set.disjoint_Union_left Set.disjoint_iUnion_left
@[simp]
theorem disjoint_iUnion_right {ι : Sort*} {s : ι → Set α} :
Disjoint t (⋃ i, s i) ↔ ∀ i, Disjoint t (s i) :=
disjoint_iSup_iff
#align set.disjoint_Union_right Set.disjoint_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_left {s : ∀ i, κ i → Set α} {t : Set α} :
Disjoint (⋃ (i) (j), s i j) t ↔ ∀ i j, Disjoint (s i j) t :=
iSup₂_disjoint_iff
#align set.disjoint_Union₂_left Set.disjoint_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_right {s : Set α} {t : ∀ i, κ i → Set α} :
Disjoint s (⋃ (i) (j), t i j) ↔ ∀ i j, Disjoint s (t i j) :=
disjoint_iSup₂_iff
#align set.disjoint_Union₂_right Set.disjoint_iUnion₂_right
@[simp]
theorem disjoint_sUnion_left {S : Set (Set α)} {t : Set α} :
Disjoint (⋃₀S) t ↔ ∀ s ∈ S, Disjoint s t :=
sSup_disjoint_iff
#align set.disjoint_sUnion_left Set.disjoint_sUnion_left
@[simp]
theorem disjoint_sUnion_right {s : Set α} {S : Set (Set α)} :
Disjoint s (⋃₀S) ↔ ∀ t ∈ S, Disjoint s t :=
disjoint_sSup_iff
#align set.disjoint_sUnion_right Set.disjoint_sUnion_right
end Set
end Disjoint
/-! ### Intervals -/
namespace Set
lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :
(⋂ i, Iic (f i)).Nonempty ↔ BddBelow (range f) := by
have : (⋂ (i : ι), Iic (f i)) = lowerBounds (range f) := by
ext c; simp [lowerBounds]
simp [this, BddBelow]
lemma nonempty_iInter_Ici_iff [Preorder α] {f : ι → α} :
(⋂ i, Ici (f i)).Nonempty ↔ BddAbove (range f) :=
nonempty_iInter_Iic_iff (α := αᵒᵈ)
variable [CompleteLattice α]
theorem Ici_iSup (f : ι → α) : Ici (⨆ i, f i) = ⋂ i, Ici (f i) :=
ext fun _ => by simp only [mem_Ici, iSup_le_iff, mem_iInter]
#align set.Ici_supr Set.Ici_iSup
theorem Iic_iInf (f : ι → α) : Iic (⨅ i, f i) = ⋂ i, Iic (f i) :=
ext fun _ => by simp only [mem_Iic, le_iInf_iff, mem_iInter]
#align set.Iic_infi Set.Iic_iInf
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Ici_iSup₂ (f : ∀ i, κ i → α) : Ici (⨆ (i) (j), f i j) = ⋂ (i) (j), Ici (f i j) := by
simp_rw [Ici_iSup]
#align set.Ici_supr₂ Set.Ici_iSup₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Iic_iInf₂ (f : ∀ i, κ i → α) : Iic (⨅ (i) (j), f i j) = ⋂ (i) (j), Iic (f i j) := by
simp_rw [Iic_iInf]
#align set.Iic_infi₂ Set.Iic_iInf₂
theorem Ici_sSup (s : Set α) : Ici (sSup s) = ⋂ a ∈ s, Ici a := by rw [sSup_eq_iSup, Ici_iSup₂]
#align set.Ici_Sup Set.Ici_sSup
theorem Iic_sInf (s : Set α) : Iic (sInf s) = ⋂ a ∈ s, Iic a := by rw [sInf_eq_iInf, Iic_iInf₂]
#align set.Iic_Inf Set.Iic_sInf
end Set
namespace Set
variable (t : α → Set β)
theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x := by
simp only [diff_subset_iff, ← biUnion_union]
apply biUnion_subset_biUnion_left
rw [union_diff_self]
apply subset_union_right
#align set.bUnion_diff_bUnion_subset Set.biUnion_diff_biUnion_subset
/-- If `t` is an indexed family of sets, then there is a natural map from `Σ i, t i` to `⋃ i, t i`
sending `⟨i, x⟩` to `x`. -/
def sigmaToiUnion (x : Σi, t i) : ⋃ i, t i :=
⟨x.2, mem_iUnion.2 ⟨x.1, x.2.2⟩⟩
#align set.sigma_to_Union Set.sigmaToiUnion
theorem sigmaToiUnion_surjective : Surjective (sigmaToiUnion t)
| ⟨b, hb⟩ =>
have : ∃ a, b ∈ t a := by simpa using hb
let ⟨a, hb⟩ := this
⟨⟨a, b, hb⟩, rfl⟩
#align set.sigma_to_Union_surjective Set.sigmaToiUnion_surjective
theorem sigmaToiUnion_injective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Injective (sigmaToiUnion t)
| ⟨a₁, b₁, h₁⟩, ⟨a₂, b₂, h₂⟩, eq =>
have b_eq : b₁ = b₂ := congr_arg Subtype.val eq
have a_eq : a₁ = a₂ :=
by_contradiction fun ne =>
have : b₁ ∈ t a₁ ∩ t a₂ := ⟨h₁, b_eq.symm ▸ h₂⟩
(h _ _ ne).le_bot this
Sigma.eq a_eq <| Subtype.eq <| by subst b_eq; subst a_eq; rfl
#align set.sigma_to_Union_injective Set.sigmaToiUnion_injective
theorem sigmaToiUnion_bijective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Bijective (sigmaToiUnion t) :=
⟨sigmaToiUnion_injective t h, sigmaToiUnion_surjective t⟩
#align set.sigma_to_Union_bijective Set.sigmaToiUnion_bijective
/-- Equivalence between a disjoint union and a dependent sum. -/
noncomputable def unionEqSigmaOfDisjoint {t : α → Set β} (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
(⋃ i, t i) ≃ Σi, t i :=
(Equiv.ofBijective _ <| sigmaToiUnion_bijective t h).symm
#align set.Union_eq_sigma_of_disjoint Set.unionEqSigmaOfDisjoint
theorem iUnion_ge_eq_iUnion_nat_add (u : ℕ → Set α) (n : ℕ) : ⋃ i ≥ n, u i = ⋃ i, u (i + n) :=
iSup_ge_eq_iSup_nat_add u n
#align set.Union_ge_eq_Union_nat_add Set.iUnion_ge_eq_iUnion_nat_add
theorem iInter_ge_eq_iInter_nat_add (u : ℕ → Set α) (n : ℕ) : ⋂ i ≥ n, u i = ⋂ i, u (i + n) :=
iInf_ge_eq_iInf_nat_add u n
#align set.Inter_ge_eq_Inter_nat_add Set.iInter_ge_eq_iInter_nat_add
theorem _root_.Monotone.iUnion_nat_add {f : ℕ → Set α} (hf : Monotone f) (k : ℕ) :
⋃ n, f (n + k) = ⋃ n, f n :=
hf.iSup_nat_add k
#align monotone.Union_nat_add Monotone.iUnion_nat_add
theorem _root_.Antitone.iInter_nat_add {f : ℕ → Set α} (hf : Antitone f) (k : ℕ) :
⋂ n, f (n + k) = ⋂ n, f n :=
hf.iInf_nat_add k
#align antitone.Inter_nat_add Antitone.iInter_nat_add
/-Porting note: removing `simp`. LHS does not simplify. Possible linter bug. Zulip discussion:
https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/complete_lattice.20and.20has_sup/near/316497982
-/
theorem iUnion_iInter_ge_nat_add (f : ℕ → Set α) (k : ℕ) :
⋃ n, ⋂ i ≥ n, f (i + k) = ⋃ n, ⋂ i ≥ n, f i :=
iSup_iInf_ge_nat_add f k
#align set.Union_Inter_ge_nat_add Set.iUnion_iInter_ge_nat_add
theorem union_iUnion_nat_succ (u : ℕ → Set α) : (u 0 ∪ ⋃ i, u (i + 1)) = ⋃ i, u i :=
sup_iSup_nat_succ u
#align set.union_Union_nat_succ Set.union_iUnion_nat_succ
theorem inter_iInter_nat_succ (u : ℕ → Set α) : (u 0 ∩ ⋂ i, u (i + 1)) = ⋂ i, u i :=
inf_iInf_nat_succ u
#align set.inter_Inter_nat_succ Set.inter_iInter_nat_succ
end Set
open Set
variable [CompleteLattice β]
theorem iSup_iUnion (s : ι → Set α) (f : α → β) : ⨆ a ∈ ⋃ i, s i, f a = ⨆ (i) (a ∈ s i), f a := by
rw [iSup_comm]
|
simp_rw [mem_iUnion, iSup_exists]
|
theorem iSup_iUnion (s : ι → Set α) (f : α → β) : ⨆ a ∈ ⋃ i, s i, f a = ⨆ (i) (a ∈ s i), f a := by
rw [iSup_comm]
|
Mathlib.Data.Set.Lattice.2374_0.5mONj49h3SYSDwc
|
theorem iSup_iUnion (s : ι → Set α) (f : α → β) : ⨆ a ∈ ⋃ i, s i, f a = ⨆ (i) (a ∈ s i), f a
|
Mathlib_Data_Set_Lattice
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : CompleteLattice β
s : Set (Set β)
⊢ sSup (⋃₀ s) = ⨆ t ∈ s, sSup t
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
/-!
# The set lattice
This file provides usual set notation for unions and intersections, a `CompleteLattice` instance
for `Set α`, and some more set constructions.
## Main declarations
* `Set.iUnion`: **i**ndexed **union**. Union of an indexed family of sets.
* `Set.iInter`: **i**ndexed **inter**section. Intersection of an indexed family of sets.
* `Set.sInter`: **s**et **inter**section. Intersection of sets belonging to a set of sets.
* `Set.sUnion`: **s**et **union**. Union of sets belonging to a set of sets.
* `Set.sInter_eq_biInter`, `Set.sUnion_eq_biInter`: Shows that `⋂₀ s = ⋂ x ∈ s, x` and
`⋃₀ s = ⋃ x ∈ s, x`.
* `Set.completeAtomicBooleanAlgebra`: `Set α` is a `CompleteAtomicBooleanAlgebra` with `≤ = ⊆`,
`< = ⊂`, `⊓ = ∩`, `⊔ = ∪`, `⨅ = ⋂`, `⨆ = ⋃` and `\` as the set difference.
See `Set.BooleanAlgebra`.
* `Set.kernImage`: For a function `f : α → β`, `s.kernImage f` is the set of `y` such that
`f ⁻¹ y ⊆ s`.
* `Set.seq`: Union of the image of a set under a **seq**uence of functions. `seq s t` is the union
of `f '' t` over all `f ∈ s`, where `t : Set α` and `s : Set (α → β)`.
* `Set.iUnion_eq_sigma_of_disjoint`: Equivalence between `⋃ i, t i` and `Σ i, t i`, where `t` is an
indexed family of disjoint sets.
## Naming convention
In lemma names,
* `⋃ i, s i` is called `iUnion`
* `⋂ i, s i` is called `iInter`
* `⋃ i j, s i j` is called `iUnion₂`. This is an `iUnion` inside an `iUnion`.
* `⋂ i j, s i j` is called `iInter₂`. This is an `iInter` inside an `iInter`.
* `⋃ i ∈ s, t i` is called `biUnion` for "bounded `iUnion`". This is the special case of `iUnion₂`
where `j : i ∈ s`.
* `⋂ i ∈ s, t i` is called `biInter` for "bounded `iInter`". This is the special case of `iInter₂`
where `j : i ∈ s`.
## Notation
* `⋃`: `Set.iUnion`
* `⋂`: `Set.iInter`
* `⋃₀`: `Set.sUnion`
* `⋂₀`: `Set.sInter`
-/
set_option autoImplicit true
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι₂ : Sort*} {κ κ₁ κ₂ : ι → Sort*} {κ' : ι' → Sort*}
namespace Set
/-! ### Complete lattice and complete Boolean algebra instances -/
instance : InfSet (Set α) :=
⟨fun s => { a | ∀ t ∈ s, a ∈ t }⟩
instance : SupSet (Set α) :=
⟨fun s => { a | ∃ t ∈ s, a ∈ t }⟩
/-- Intersection of a set of sets. -/
def sInter (S : Set (Set α)) : Set α :=
sInf S
#align set.sInter Set.sInter
/-- Notation for `Set.sInter` Intersection of a set of sets. -/
prefix:110 "⋂₀ " => sInter
/-- Union of a set of sets. -/
def sUnion (S : Set (Set α)) : Set α :=
sSup S
#align set.sUnion Set.sUnion
/-- Notation for `Set.sUnion`. Union of a set of sets. -/
prefix:110 "⋃₀ " => sUnion
@[simp]
theorem mem_sInter {x : α} {S : Set (Set α)} : x ∈ ⋂₀ S ↔ ∀ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sInter Set.mem_sInter
@[simp]
theorem mem_sUnion {x : α} {S : Set (Set α)} : x ∈ ⋃₀ S ↔ ∃ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sUnion Set.mem_sUnion
/-- Indexed union of a family of sets -/
def iUnion (s : ι → Set β) : Set β :=
iSup s
#align set.Union Set.iUnion
/-- Indexed intersection of a family of sets -/
def iInter (s : ι → Set β) : Set β :=
iInf s
#align set.Inter Set.iInter
/-- Notation for `Set.iUnion`. Indexed union of a family of sets -/
notation3 "⋃ "(...)", "r:60:(scoped f => iUnion f) => r
/-- Notation for `Set.iInter`. Indexed intersection of a family of sets -/
notation3 "⋂ "(...)", "r:60:(scoped f => iInter f) => r
section delaborators
open Lean Lean.PrettyPrinter.Delaborator
/-- Delaborator for indexed unions. -/
@[delab app.Set.iUnion]
def iUnion_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋃ (_ : $dom), $body)
else if prop || ppTypes then
`(⋃ ($x:ident : $dom), $body)
else
`(⋃ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋃ $x:ident, ⋃ (_ : $y:ident ∈ $s), $body)
| `(⋃ ($x:ident : $_), ⋃ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋃ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
/-- Delaborator for indexed intersections. -/
@[delab app.Set.iInter]
def sInter_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋂ (_ : $dom), $body)
else if prop || ppTypes then
`(⋂ ($x:ident : $dom), $body)
else
`(⋂ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋂ $x:ident, ⋂ (_ : $y:ident ∈ $s), $body)
| `(⋂ ($x:ident : $_), ⋂ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋂ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
end delaborators
@[simp]
theorem sSup_eq_sUnion (S : Set (Set α)) : sSup S = ⋃₀S :=
rfl
#align set.Sup_eq_sUnion Set.sSup_eq_sUnion
@[simp]
theorem sInf_eq_sInter (S : Set (Set α)) : sInf S = ⋂₀ S :=
rfl
#align set.Inf_eq_sInter Set.sInf_eq_sInter
@[simp]
theorem iSup_eq_iUnion (s : ι → Set α) : iSup s = iUnion s :=
rfl
#align set.supr_eq_Union Set.iSup_eq_iUnion
@[simp]
theorem iInf_eq_iInter (s : ι → Set α) : iInf s = iInter s :=
rfl
#align set.infi_eq_Inter Set.iInf_eq_iInter
@[simp]
theorem mem_iUnion {x : α} {s : ι → Set α} : (x ∈ ⋃ i, s i) ↔ ∃ i, x ∈ s i :=
⟨fun ⟨_, ⟨⟨a, (t_eq : s a = _)⟩, (h : x ∈ _)⟩⟩ => ⟨a, t_eq.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨s a, ⟨⟨a, rfl⟩, h⟩⟩⟩
#align set.mem_Union Set.mem_iUnion
@[simp]
theorem mem_iInter {x : α} {s : ι → Set α} : (x ∈ ⋂ i, s i) ↔ ∀ i, x ∈ s i :=
⟨fun (h : ∀ a ∈ { a : Set α | ∃ i, s i = a }, x ∈ a) a => h (s a) ⟨a, rfl⟩,
fun h _ ⟨a, (eq : s a = _)⟩ => eq ▸ h a⟩
#align set.mem_Inter Set.mem_iInter
theorem mem_iUnion₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋃ (i) (j), s i j) ↔ ∃ i j, x ∈ s i j := by
simp_rw [mem_iUnion]
#align set.mem_Union₂ Set.mem_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋂ (i) (j), s i j) ↔ ∀ i j, x ∈ s i j := by
simp_rw [mem_iInter]
#align set.mem_Inter₂ Set.mem_iInter₂
theorem mem_iUnion_of_mem {s : ι → Set α} {a : α} (i : ι) (ha : a ∈ s i) : a ∈ ⋃ i, s i :=
mem_iUnion.2 ⟨i, ha⟩
#align set.mem_Union_of_mem Set.mem_iUnion_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iUnion₂_of_mem {s : ∀ i, κ i → Set α} {a : α} {i : ι} (j : κ i) (ha : a ∈ s i j) :
a ∈ ⋃ (i) (j), s i j :=
mem_iUnion₂.2 ⟨i, j, ha⟩
#align set.mem_Union₂_of_mem Set.mem_iUnion₂_of_mem
theorem mem_iInter_of_mem {s : ι → Set α} {a : α} (h : ∀ i, a ∈ s i) : a ∈ ⋂ i, s i :=
mem_iInter.2 h
#align set.mem_Inter_of_mem Set.mem_iInter_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂_of_mem {s : ∀ i, κ i → Set α} {a : α} (h : ∀ i j, a ∈ s i j) :
a ∈ ⋂ (i) (j), s i j :=
mem_iInter₂.2 h
#align set.mem_Inter₂_of_mem Set.mem_iInter₂_of_mem
instance Set.completeAtomicBooleanAlgebra : CompleteAtomicBooleanAlgebra (Set α) :=
{ instBooleanAlgebraSet with
le_sSup := fun s t t_in a a_in => ⟨t, t_in, a_in⟩
sSup_le := fun s t h a ⟨t', ⟨t'_in, a_in⟩⟩ => h t' t'_in a_in
le_sInf := fun s t h a a_in t' t'_in => h t' t'_in a_in
sInf_le := fun s t t_in a h => h _ t_in
iInf_iSup_eq := by intros; ext; simp [Classical.skolem] }
/-- `kernImage f s` is the set of `y` such that `f ⁻¹ y ⊆ s`. -/
def kernImage (f : α → β) (s : Set α) : Set β :=
{ y | ∀ ⦃x⦄, f x = y → x ∈ s }
#align set.kern_image Set.kernImage
lemma subset_kernImage_iff {f : α → β} : s ⊆ kernImage f t ↔ f ⁻¹' s ⊆ t :=
⟨fun h _ hx ↦ h hx rfl,
fun h _ hx y hy ↦ h (show f y ∈ s from hy.symm ▸ hx)⟩
section GaloisConnection
variable {f : α → β}
protected theorem image_preimage : GaloisConnection (image f) (preimage f) := fun _ _ =>
image_subset_iff
#align set.image_preimage Set.image_preimage
protected theorem preimage_kernImage : GaloisConnection (preimage f) (kernImage f) := fun _ _ =>
subset_kernImage_iff.symm
#align set.preimage_kern_image Set.preimage_kernImage
end GaloisConnection
section kernImage
variable {f : α → β}
lemma kernImage_mono : Monotone (kernImage f) :=
Set.preimage_kernImage.monotone_u
lemma kernImage_eq_compl {s : Set α} : kernImage f s = (f '' sᶜ)ᶜ :=
Set.preimage_kernImage.u_unique (Set.image_preimage.compl)
(fun t ↦ compl_compl (f ⁻¹' t) ▸ Set.preimage_compl)
lemma kernImage_compl {s : Set α} : kernImage f (sᶜ) = (f '' s)ᶜ := by
rw [kernImage_eq_compl, compl_compl]
lemma kernImage_empty : kernImage f ∅ = (range f)ᶜ := by
rw [kernImage_eq_compl, compl_empty, image_univ]
lemma kernImage_preimage_eq_iff {s : Set β} : kernImage f (f ⁻¹' s) = s ↔ (range f)ᶜ ⊆ s := by
rw [kernImage_eq_compl, ← preimage_compl, compl_eq_comm, eq_comm, image_preimage_eq_iff,
compl_subset_comm]
lemma compl_range_subset_kernImage {s : Set α} : (range f)ᶜ ⊆ kernImage f s := by
rw [← kernImage_empty]
exact kernImage_mono (empty_subset _)
lemma kernImage_union_preimage {s : Set α} {t : Set β} :
kernImage f (s ∪ f ⁻¹' t) = kernImage f s ∪ t := by
rw [kernImage_eq_compl, kernImage_eq_compl, compl_union, ← preimage_compl, image_inter_preimage,
compl_inter, compl_compl]
lemma kernImage_preimage_union {s : Set α} {t : Set β} :
kernImage f (f ⁻¹' t ∪ s) = t ∪ kernImage f s := by
rw [union_comm, kernImage_union_preimage, union_comm]
end kernImage
/-! ### Union and intersection over an indexed family of sets -/
instance : OrderTop (Set α) where
top := univ
le_top := by simp
@[congr]
theorem iUnion_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iUnion f₁ = iUnion f₂ :=
iSup_congr_Prop pq f
#align set.Union_congr_Prop Set.iUnion_congr_Prop
@[congr]
theorem iInter_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iInter f₁ = iInter f₂ :=
iInf_congr_Prop pq f
#align set.Inter_congr_Prop Set.iInter_congr_Prop
theorem iUnion_plift_up (f : PLift ι → Set α) : ⋃ i, f (PLift.up i) = ⋃ i, f i :=
iSup_plift_up _
#align set.Union_plift_up Set.iUnion_plift_up
theorem iUnion_plift_down (f : ι → Set α) : ⋃ i, f (PLift.down i) = ⋃ i, f i :=
iSup_plift_down _
#align set.Union_plift_down Set.iUnion_plift_down
theorem iInter_plift_up (f : PLift ι → Set α) : ⋂ i, f (PLift.up i) = ⋂ i, f i :=
iInf_plift_up _
#align set.Inter_plift_up Set.iInter_plift_up
theorem iInter_plift_down (f : ι → Set α) : ⋂ i, f (PLift.down i) = ⋂ i, f i :=
iInf_plift_down _
#align set.Inter_plift_down Set.iInter_plift_down
theorem iUnion_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋃ _ : p, s = if p then s else ∅ :=
iSup_eq_if _
#align set.Union_eq_if Set.iUnion_eq_if
theorem iUnion_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋃ h : p, s h = if h : p then s h else ∅ :=
iSup_eq_dif _
#align set.Union_eq_dif Set.iUnion_eq_dif
theorem iInter_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋂ _ : p, s = if p then s else univ :=
iInf_eq_if _
#align set.Inter_eq_if Set.iInter_eq_if
theorem iInf_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋂ h : p, s h = if h : p then s h else univ :=
_root_.iInf_eq_dif _
#align set.Infi_eq_dif Set.iInf_eq_dif
theorem exists_set_mem_of_union_eq_top {ι : Type*} (t : Set ι) (s : ι → Set β)
(w : ⋃ i ∈ t, s i = ⊤) (x : β) : ∃ i ∈ t, x ∈ s i := by
have p : x ∈ ⊤ := Set.mem_univ x
rw [← w, Set.mem_iUnion] at p
simpa using p
#align set.exists_set_mem_of_union_eq_top Set.exists_set_mem_of_union_eq_top
theorem nonempty_of_union_eq_top_of_nonempty {ι : Type*} (t : Set ι) (s : ι → Set α)
(H : Nonempty α) (w : ⋃ i ∈ t, s i = ⊤) : t.Nonempty := by
obtain ⟨x, m, -⟩ := exists_set_mem_of_union_eq_top t s w H.some
exact ⟨x, m⟩
#align set.nonempty_of_union_eq_top_of_nonempty Set.nonempty_of_union_eq_top_of_nonempty
theorem nonempty_of_nonempty_iUnion
{s : ι → Set α} (h_Union : (⋃ i, s i).Nonempty) : Nonempty ι := by
obtain ⟨x, hx⟩ := h_Union
exact ⟨Classical.choose $ mem_iUnion.mp hx⟩
theorem nonempty_of_nonempty_iUnion_eq_univ
{s : ι → Set α} [Nonempty α] (h_Union : ⋃ i, s i = univ) : Nonempty ι :=
nonempty_of_nonempty_iUnion (s := s) (by simpa only [h_Union] using univ_nonempty)
theorem setOf_exists (p : ι → β → Prop) : { x | ∃ i, p i x } = ⋃ i, { x | p i x } :=
ext fun _ => mem_iUnion.symm
#align set.set_of_exists Set.setOf_exists
theorem setOf_forall (p : ι → β → Prop) : { x | ∀ i, p i x } = ⋂ i, { x | p i x } :=
ext fun _ => mem_iInter.symm
#align set.set_of_forall Set.setOf_forall
theorem iUnion_subset {s : ι → Set α} {t : Set α} (h : ∀ i, s i ⊆ t) : ⋃ i, s i ⊆ t :=
iSup_le h
#align set.Union_subset Set.iUnion_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset {s : ∀ i, κ i → Set α} {t : Set α} (h : ∀ i j, s i j ⊆ t) :
⋃ (i) (j), s i j ⊆ t :=
iUnion_subset fun x => iUnion_subset (h x)
#align set.Union₂_subset Set.iUnion₂_subset
theorem subset_iInter {t : Set β} {s : ι → Set β} (h : ∀ i, t ⊆ s i) : t ⊆ ⋂ i, s i :=
le_iInf h
#align set.subset_Inter Set.subset_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iInter₂ {s : Set α} {t : ∀ i, κ i → Set α} (h : ∀ i j, s ⊆ t i j) :
s ⊆ ⋂ (i) (j), t i j :=
subset_iInter fun x => subset_iInter <| h x
#align set.subset_Inter₂ Set.subset_iInter₂
@[simp]
theorem iUnion_subset_iff {s : ι → Set α} {t : Set α} : ⋃ i, s i ⊆ t ↔ ∀ i, s i ⊆ t :=
⟨fun h _ => Subset.trans (le_iSup s _) h, iUnion_subset⟩
#align set.Union_subset_iff Set.iUnion_subset_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset_iff {s : ∀ i, κ i → Set α} {t : Set α} :
⋃ (i) (j), s i j ⊆ t ↔ ∀ i j, s i j ⊆ t := by simp_rw [iUnion_subset_iff]
#align set.Union₂_subset_iff Set.iUnion₂_subset_iff
@[simp]
theorem subset_iInter_iff {s : Set α} {t : ι → Set α} : (s ⊆ ⋂ i, t i) ↔ ∀ i, s ⊆ t i :=
le_iInf_iff
#align set.subset_Inter_iff Set.subset_iInter_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem subset_iInter₂_iff {s : Set α} {t : ∀ i, κ i → Set α} :
(s ⊆ ⋂ (i) (j), t i j) ↔ ∀ i j, s ⊆ t i j := by simp_rw [subset_iInter_iff]
#align set.subset_Inter₂_iff Set.subset_iInter₂_iff
theorem subset_iUnion : ∀ (s : ι → Set β) (i : ι), s i ⊆ ⋃ i, s i :=
le_iSup
#align set.subset_Union Set.subset_iUnion
theorem iInter_subset : ∀ (s : ι → Set β) (i : ι), ⋂ i, s i ⊆ s i :=
iInf_le
#align set.Inter_subset Set.iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iUnion₂ {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : s i j ⊆ ⋃ (i') (j'), s i' j' :=
le_iSup₂ i j
#align set.subset_Union₂ Set.subset_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_subset {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : ⋂ (i) (j), s i j ⊆ s i j :=
iInf₂_le i j
#align set.Inter₂_subset Set.iInter₂_subset
/-- This rather trivial consequence of `subset_iUnion`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem subset_iUnion_of_subset {s : Set α} {t : ι → Set α} (i : ι) (h : s ⊆ t i) : s ⊆ ⋃ i, t i :=
le_iSup_of_le i h
#align set.subset_Union_of_subset Set.subset_iUnion_of_subset
/-- This rather trivial consequence of `iInter_subset`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem iInter_subset_of_subset {s : ι → Set α} {t : Set α} (i : ι) (h : s i ⊆ t) :
⋂ i, s i ⊆ t :=
iInf_le_of_le i h
#align set.Inter_subset_of_subset Set.iInter_subset_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `subset_iUnion₂` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem subset_iUnion₂_of_subset {s : Set α} {t : ∀ i, κ i → Set α} (i : ι) (j : κ i)
(h : s ⊆ t i j) : s ⊆ ⋃ (i) (j), t i j :=
le_iSup₂_of_le i j h
#align set.subset_Union₂_of_subset Set.subset_iUnion₂_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `iInter₂_subset` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem iInter₂_subset_of_subset {s : ∀ i, κ i → Set α} {t : Set α} (i : ι) (j : κ i)
(h : s i j ⊆ t) : ⋂ (i) (j), s i j ⊆ t :=
iInf₂_le_of_le i j h
#align set.Inter₂_subset_of_subset Set.iInter₂_subset_of_subset
theorem iUnion_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono h
#align set.Union_mono Set.iUnion_mono
@[gcongr]
theorem iUnion_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iUnion s ⊆ iUnion t :=
iSup_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋃ (i) (j), s i j ⊆ ⋃ (i) (j), t i j :=
iSup₂_mono h
#align set.Union₂_mono Set.iUnion₂_mono
theorem iInter_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋂ i, s i ⊆ ⋂ i, t i :=
iInf_mono h
#align set.Inter_mono Set.iInter_mono
@[gcongr]
theorem iInter_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iInter s ⊆ iInter t :=
iInf_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋂ (i) (j), s i j ⊆ ⋂ (i) (j), t i j :=
iInf₂_mono h
#align set.Inter₂_mono Set.iInter₂_mono
theorem iUnion_mono' {s : ι → Set α} {t : ι₂ → Set α} (h : ∀ i, ∃ j, s i ⊆ t j) :
⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono' h
#align set.Union_mono' Set.iUnion_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iUnion₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i j, ∃ i' j', s i j ⊆ t i' j') : ⋃ (i) (j), s i j ⊆ ⋃ (i') (j'), t i' j' :=
iSup₂_mono' h
#align set.Union₂_mono' Set.iUnion₂_mono'
theorem iInter_mono' {s : ι → Set α} {t : ι' → Set α} (h : ∀ j, ∃ i, s i ⊆ t j) :
⋂ i, s i ⊆ ⋂ j, t j :=
Set.subset_iInter fun j =>
let ⟨i, hi⟩ := h j
iInter_subset_of_subset i hi
#align set.Inter_mono' Set.iInter_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iInter₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i' j', ∃ i j, s i j ⊆ t i' j') : ⋂ (i) (j), s i j ⊆ ⋂ (i') (j'), t i' j' :=
subset_iInter₂_iff.2 fun i' j' =>
let ⟨_, _, hst⟩ := h i' j'
(iInter₂_subset _ _).trans hst
#align set.Inter₂_mono' Set.iInter₂_mono'
theorem iUnion₂_subset_iUnion (κ : ι → Sort*) (s : ι → Set α) :
⋃ (i) (_ : κ i), s i ⊆ ⋃ i, s i :=
iUnion_mono fun _ => iUnion_subset fun _ => Subset.rfl
#align set.Union₂_subset_Union Set.iUnion₂_subset_iUnion
theorem iInter_subset_iInter₂ (κ : ι → Sort*) (s : ι → Set α) :
⋂ i, s i ⊆ ⋂ (i) (_ : κ i), s i :=
iInter_mono fun _ => subset_iInter fun _ => Subset.rfl
#align set.Inter_subset_Inter₂ Set.iInter_subset_iInter₂
theorem iUnion_setOf (P : ι → α → Prop) : ⋃ i, { x : α | P i x } = { x : α | ∃ i, P i x } := by
ext
exact mem_iUnion
#align set.Union_set_of Set.iUnion_setOf
theorem iInter_setOf (P : ι → α → Prop) : ⋂ i, { x : α | P i x } = { x : α | ∀ i, P i x } := by
ext
exact mem_iInter
#align set.Inter_set_of Set.iInter_setOf
theorem iUnion_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋃ x, f x = ⋃ y, g y :=
h1.iSup_congr h h2
#align set.Union_congr_of_surjective Set.iUnion_congr_of_surjective
theorem iInter_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋂ x, f x = ⋂ y, g y :=
h1.iInf_congr h h2
#align set.Inter_congr_of_surjective Set.iInter_congr_of_surjective
lemma iUnion_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋃ i, s i = ⋃ i, t i := iSup_congr h
#align set.Union_congr Set.iUnion_congr
lemma iInter_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋂ i, s i = ⋂ i, t i := iInf_congr h
#align set.Inter_congr Set.iInter_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iUnion₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋃ (i) (j), s i j = ⋃ (i) (j), t i j :=
iUnion_congr fun i => iUnion_congr <| h i
#align set.Union₂_congr Set.iUnion₂_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iInter₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋂ (i) (j), s i j = ⋂ (i) (j), t i j :=
iInter_congr fun i => iInter_congr <| h i
#align set.Inter₂_congr Set.iInter₂_congr
section Nonempty
variable [Nonempty ι] {f : ι → Set α} {s : Set α}
lemma iUnion_const (s : Set β) : ⋃ _ : ι, s = s := iSup_const
#align set.Union_const Set.iUnion_const
lemma iInter_const (s : Set β) : ⋂ _ : ι, s = s := iInf_const
#align set.Inter_const Set.iInter_const
lemma iUnion_eq_const (hf : ∀ i, f i = s) : ⋃ i, f i = s :=
(iUnion_congr hf).trans $ iUnion_const _
#align set.Union_eq_const Set.iUnion_eq_const
lemma iInter_eq_const (hf : ∀ i, f i = s) : ⋂ i, f i = s :=
(iInter_congr hf).trans $ iInter_const _
#align set.Inter_eq_const Set.iInter_eq_const
end Nonempty
@[simp]
theorem compl_iUnion (s : ι → Set β) : (⋃ i, s i)ᶜ = ⋂ i, (s i)ᶜ :=
compl_iSup
#align set.compl_Union Set.compl_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iUnion₂ (s : ∀ i, κ i → Set α) : (⋃ (i) (j), s i j)ᶜ = ⋂ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iUnion]
#align set.compl_Union₂ Set.compl_iUnion₂
@[simp]
theorem compl_iInter (s : ι → Set β) : (⋂ i, s i)ᶜ = ⋃ i, (s i)ᶜ :=
compl_iInf
#align set.compl_Inter Set.compl_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iInter₂ (s : ∀ i, κ i → Set α) : (⋂ (i) (j), s i j)ᶜ = ⋃ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iInter]
#align set.compl_Inter₂ Set.compl_iInter₂
-- classical -- complete_boolean_algebra
theorem iUnion_eq_compl_iInter_compl (s : ι → Set β) : ⋃ i, s i = (⋂ i, (s i)ᶜ)ᶜ := by
simp only [compl_iInter, compl_compl]
#align set.Union_eq_compl_Inter_compl Set.iUnion_eq_compl_iInter_compl
-- classical -- complete_boolean_algebra
theorem iInter_eq_compl_iUnion_compl (s : ι → Set β) : ⋂ i, s i = (⋃ i, (s i)ᶜ)ᶜ := by
simp only [compl_iUnion, compl_compl]
#align set.Inter_eq_compl_Union_compl Set.iInter_eq_compl_iUnion_compl
theorem inter_iUnion (s : Set β) (t : ι → Set β) : (s ∩ ⋃ i, t i) = ⋃ i, s ∩ t i :=
inf_iSup_eq _ _
#align set.inter_Union Set.inter_iUnion
theorem iUnion_inter (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∩ s = ⋃ i, t i ∩ s :=
iSup_inf_eq _ _
#align set.Union_inter Set.iUnion_inter
theorem iUnion_union_distrib (s : ι → Set β) (t : ι → Set β) :
⋃ i, s i ∪ t i = (⋃ i, s i) ∪ ⋃ i, t i :=
iSup_sup_eq
#align set.Union_union_distrib Set.iUnion_union_distrib
theorem iInter_inter_distrib (s : ι → Set β) (t : ι → Set β) :
⋂ i, s i ∩ t i = (⋂ i, s i) ∩ ⋂ i, t i :=
iInf_inf_eq
#align set.Inter_inter_distrib Set.iInter_inter_distrib
theorem union_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∪ ⋃ i, t i) = ⋃ i, s ∪ t i :=
sup_iSup
#align set.union_Union Set.union_iUnion
theorem iUnion_union [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∪ s = ⋃ i, t i ∪ s :=
iSup_sup
#align set.Union_union Set.iUnion_union
theorem inter_iInter [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∩ ⋂ i, t i) = ⋂ i, s ∩ t i :=
inf_iInf
#align set.inter_Inter Set.inter_iInter
theorem iInter_inter [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋂ i, t i) ∩ s = ⋂ i, t i ∩ s :=
iInf_inf
#align set.Inter_inter Set.iInter_inter
-- classical
theorem union_iInter (s : Set β) (t : ι → Set β) : (s ∪ ⋂ i, t i) = ⋂ i, s ∪ t i :=
sup_iInf_eq _ _
#align set.union_Inter Set.union_iInter
theorem iInter_union (s : ι → Set β) (t : Set β) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.Inter_union Set.iInter_union
theorem iUnion_diff (s : Set β) (t : ι → Set β) : (⋃ i, t i) \ s = ⋃ i, t i \ s :=
iUnion_inter _ _
#align set.Union_diff Set.iUnion_diff
theorem diff_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s \ ⋃ i, t i) = ⋂ i, s \ t i := by
rw [diff_eq, compl_iUnion, inter_iInter]; rfl
#align set.diff_Union Set.diff_iUnion
theorem diff_iInter (s : Set β) (t : ι → Set β) : (s \ ⋂ i, t i) = ⋃ i, s \ t i := by
rw [diff_eq, compl_iInter, inter_iUnion]; rfl
#align set.diff_Inter Set.diff_iInter
theorem directed_on_iUnion {r} {f : ι → Set α} (hd : Directed (· ⊆ ·) f)
(h : ∀ x, DirectedOn r (f x)) : DirectedOn r (⋃ x, f x) := by
simp only [DirectedOn, exists_prop, mem_iUnion, exists_imp]
exact fun a₁ b₁ fb₁ a₂ b₂ fb₂ =>
let ⟨z, zb₁, zb₂⟩ := hd b₁ b₂
let ⟨x, xf, xa₁, xa₂⟩ := h z a₁ (zb₁ fb₁) a₂ (zb₂ fb₂)
⟨x, ⟨z, xf⟩, xa₁, xa₂⟩
#align set.directed_on_Union Set.directed_on_iUnion
theorem iUnion_inter_subset {ι α} {s t : ι → Set α} : ⋃ i, s i ∩ t i ⊆ (⋃ i, s i) ∩ ⋃ i, t i :=
le_iSup_inf_iSup s t
#align set.Union_inter_subset Set.iUnion_inter_subset
theorem iUnion_inter_of_monotone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_monotone hs ht
#align set.Union_inter_of_monotone Set.iUnion_inter_of_monotone
theorem iUnion_inter_of_antitone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_antitone hs ht
#align set.Union_inter_of_antitone Set.iUnion_inter_of_antitone
theorem iInter_union_of_monotone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_monotone hs ht
#align set.Inter_union_of_monotone Set.iInter_union_of_monotone
theorem iInter_union_of_antitone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_antitone hs ht
#align set.Inter_union_of_antitone Set.iInter_union_of_antitone
/-- An equality version of this lemma is `iUnion_iInter_of_monotone` in `Data.Set.Finite`. -/
theorem iUnion_iInter_subset {s : ι → ι' → Set α} : (⋃ j, ⋂ i, s i j) ⊆ ⋂ i, ⋃ j, s i j :=
iSup_iInf_le_iInf_iSup (flip s)
#align set.Union_Inter_subset Set.iUnion_iInter_subset
theorem iUnion_option {ι} (s : Option ι → Set α) : ⋃ o, s o = s none ∪ ⋃ i, s (some i) :=
iSup_option s
#align set.Union_option Set.iUnion_option
theorem iInter_option {ι} (s : Option ι → Set α) : ⋂ o, s o = s none ∩ ⋂ i, s (some i) :=
iInf_option s
#align set.Inter_option Set.iInter_option
section
variable (p : ι → Prop) [DecidablePred p]
theorem iUnion_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 _ _ _
#align set.Union_dite Set.iUnion_dite
theorem iUnion_ite (f g : ι → Set α) :
⋃ i, (if p i then f i else g i) = (⋃ (i) (_ : p i), f i) ∪ ⋃ (i) (_ : ¬p i), g i :=
iUnion_dite _ _ _
#align set.Union_ite Set.iUnion_ite
theorem iInter_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 :=
iInf_dite _ _ _
#align set.Inter_dite Set.iInter_dite
theorem iInter_ite (f g : ι → Set α) :
⋂ i, (if p i then f i else g i) = (⋂ (i) (_ : p i), f i) ∩ ⋂ (i) (_ : ¬p i), g i :=
iInter_dite _ _ _
#align set.Inter_ite Set.iInter_ite
end
theorem image_projection_prod {ι : Type*} {α : ι → Type*} {v : ∀ i : ι, Set (α i)}
(hv : (pi univ v).Nonempty) (i : ι) :
((fun x : ∀ i : ι, α i => x i) '' ⋂ k, (fun x : ∀ j : ι, α j => x k) ⁻¹' v k) = v i := by
classical
apply Subset.antisymm
· simp [iInter_subset]
· intro y y_in
simp only [mem_image, mem_iInter, mem_preimage]
rcases hv with ⟨z, hz⟩
refine' ⟨Function.update z i y, _, update_same i y z⟩
rw [@forall_update_iff ι α _ z i y fun i t => t ∈ v i]
exact ⟨y_in, fun j _ => by simpa using hz j⟩
#align set.image_projection_prod Set.image_projection_prod
/-! ### Unions and intersections indexed by `Prop` -/
theorem iInter_false {s : False → Set α} : iInter s = univ :=
iInf_false
#align set.Inter_false Set.iInter_false
theorem iUnion_false {s : False → Set α} : iUnion s = ∅ :=
iSup_false
#align set.Union_false Set.iUnion_false
@[simp]
theorem iInter_true {s : True → Set α} : iInter s = s trivial :=
iInf_true
#align set.Inter_true Set.iInter_true
@[simp]
theorem iUnion_true {s : True → Set α} : iUnion s = s trivial :=
iSup_true
#align set.Union_true Set.iUnion_true
@[simp]
theorem iInter_exists {p : ι → Prop} {f : Exists p → Set α} :
⋂ x, f x = ⋂ (i) (h : p i), f ⟨i, h⟩ :=
iInf_exists
#align set.Inter_exists Set.iInter_exists
@[simp]
theorem iUnion_exists {p : ι → Prop} {f : Exists p → Set α} :
⋃ x, f x = ⋃ (i) (h : p i), f ⟨i, h⟩ :=
iSup_exists
#align set.Union_exists Set.iUnion_exists
@[simp]
theorem iUnion_empty : (⋃ _ : ι, ∅ : Set α) = ∅ :=
iSup_bot
#align set.Union_empty Set.iUnion_empty
@[simp]
theorem iInter_univ : (⋂ _ : ι, univ : Set α) = univ :=
iInf_top
#align set.Inter_univ Set.iInter_univ
section
variable {s : ι → Set α}
@[simp]
theorem iUnion_eq_empty : ⋃ i, s i = ∅ ↔ ∀ i, s i = ∅ :=
iSup_eq_bot
#align set.Union_eq_empty Set.iUnion_eq_empty
@[simp]
theorem iInter_eq_univ : ⋂ i, s i = univ ↔ ∀ i, s i = univ :=
iInf_eq_top
#align set.Inter_eq_univ Set.iInter_eq_univ
@[simp]
theorem nonempty_iUnion : (⋃ i, s i).Nonempty ↔ ∃ i, (s i).Nonempty := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_Union Set.nonempty_iUnion
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_biUnion {t : Set α} {s : α → Set β} :
(⋃ i ∈ t, s i).Nonempty ↔ ∃ i ∈ t, (s i).Nonempty := by simp
#align set.nonempty_bUnion Set.nonempty_biUnion
theorem iUnion_nonempty_index (s : Set α) (t : s.Nonempty → Set β) :
⋃ h, t h = ⋃ x ∈ s, t ⟨x, ‹_›⟩ :=
iSup_exists
#align set.Union_nonempty_index Set.iUnion_nonempty_index
end
@[simp]
theorem iInter_iInter_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋂ (x) (h : x = b), s x h = s b rfl :=
iInf_iInf_eq_left
#align set.Inter_Inter_eq_left Set.iInter_iInter_eq_left
@[simp]
theorem iInter_iInter_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋂ (x) (h : b = x), s x h = s b rfl :=
iInf_iInf_eq_right
#align set.Inter_Inter_eq_right Set.iInter_iInter_eq_right
@[simp]
theorem iUnion_iUnion_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋃ (x) (h : x = b), s x h = s b rfl :=
iSup_iSup_eq_left
#align set.Union_Union_eq_left Set.iUnion_iUnion_eq_left
@[simp]
theorem iUnion_iUnion_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋃ (x) (h : b = x), s x h = s b rfl :=
iSup_iSup_eq_right
#align set.Union_Union_eq_right Set.iUnion_iUnion_eq_right
theorem iInter_or {p q : Prop} (s : p ∨ q → Set α) :
⋂ h, s h = (⋂ h : p, s (Or.inl h)) ∩ ⋂ h : q, s (Or.inr h) :=
iInf_or
#align set.Inter_or Set.iInter_or
theorem iUnion_or {p q : Prop} (s : p ∨ q → Set α) :
⋃ h, s h = (⋃ i, s (Or.inl i)) ∪ ⋃ j, s (Or.inr j) :=
iSup_or
#align set.Union_or Set.iUnion_or
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iUnion_and {p q : Prop} (s : p ∧ q → Set α) : ⋃ h, s h = ⋃ (hp) (hq), s ⟨hp, hq⟩ :=
iSup_and
#align set.Union_and Set.iUnion_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iInter_and {p q : Prop} (s : p ∧ q → Set α) : ⋂ h, s h = ⋂ (hp) (hq), s ⟨hp, hq⟩ :=
iInf_and
#align set.Inter_and Set.iInter_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iUnion_comm (s : ι → ι' → Set α) : ⋃ (i) (i'), s i i' = ⋃ (i') (i), s i i' :=
iSup_comm
#align set.Union_comm Set.iUnion_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iInter_comm (s : ι → ι' → Set α) : ⋂ (i) (i'), s i i' = ⋂ (i') (i), s i i' :=
iInf_comm
#align set.Inter_comm Set.iInter_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iUnion₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋃ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋃ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iSup₂_comm _
#align set.Union₂_comm Set.iUnion₂_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iInter₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋂ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋂ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iInf₂_comm _
#align set.Inter₂_comm Set.iInter₂_comm
@[simp]
theorem biUnion_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋃ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι']
#align set.bUnion_and Set.biUnion_and
@[simp]
theorem biUnion_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋃ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι]
#align set.bUnion_and' Set.biUnion_and'
@[simp]
theorem biInter_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋂ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iInter_and, @iInter_comm _ ι']
#align set.bInter_and Set.biInter_and
@[simp]
theorem biInter_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋂ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iInter_and, @iInter_comm _ ι]
#align set.bInter_and' Set.biInter_and'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iUnion_iUnion_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋃ (x) (h), s x h = s b (Or.inl rfl) ∪ ⋃ (x) (h : p x), s x (Or.inr h) := by
simp only [iUnion_or, iUnion_union_distrib, iUnion_iUnion_eq_left]
#align set.Union_Union_eq_or_left Set.iUnion_iUnion_eq_or_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iInter_iInter_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋂ (x) (h), s x h = s b (Or.inl rfl) ∩ ⋂ (x) (h : p x), s x (Or.inr h) := by
simp only [iInter_or, iInter_inter_distrib, iInter_iInter_eq_left]
#align set.Inter_Inter_eq_or_left Set.iInter_iInter_eq_or_left
/-! ### Bounded unions and intersections -/
/-- A specialization of `mem_iUnion₂`. -/
theorem mem_biUnion {s : Set α} {t : α → Set β} {x : α} {y : β} (xs : x ∈ s) (ytx : y ∈ t x) :
y ∈ ⋃ x ∈ s, t x :=
mem_iUnion₂_of_mem xs ytx
#align set.mem_bUnion Set.mem_biUnion
/-- A specialization of `mem_iInter₂`. -/
theorem mem_biInter {s : Set α} {t : α → Set β} {y : β} (h : ∀ x ∈ s, y ∈ t x) :
y ∈ ⋂ x ∈ s, t x :=
mem_iInter₂_of_mem h
#align set.mem_bInter Set.mem_biInter
/-- A specialization of `subset_iUnion₂`. -/
theorem subset_biUnion_of_mem {s : Set α} {u : α → Set β} {x : α} (xs : x ∈ s) :
u x ⊆ ⋃ x ∈ s, u x :=
--Porting note: Why is this not just `subset_iUnion₂ x xs`?
@subset_iUnion₂ β α (· ∈ s) (fun i _ => u i) x xs
#align set.subset_bUnion_of_mem Set.subset_biUnion_of_mem
/-- A specialization of `iInter₂_subset`. -/
theorem biInter_subset_of_mem {s : Set α} {t : α → Set β} {x : α} (xs : x ∈ s) :
⋂ x ∈ s, t x ⊆ t x :=
iInter₂_subset x xs
#align set.bInter_subset_of_mem Set.biInter_subset_of_mem
theorem biUnion_subset_biUnion_left {s s' : Set α} {t : α → Set β} (h : s ⊆ s') :
⋃ x ∈ s, t x ⊆ ⋃ x ∈ s', t x :=
iUnion₂_subset fun _ hx => subset_biUnion_of_mem <| h hx
#align set.bUnion_subset_bUnion_left Set.biUnion_subset_biUnion_left
theorem biInter_subset_biInter_left {s s' : Set α} {t : α → Set β} (h : s' ⊆ s) :
⋂ x ∈ s, t x ⊆ ⋂ x ∈ s', t x :=
subset_iInter₂ fun _ hx => biInter_subset_of_mem <| h hx
#align set.bInter_subset_bInter_left Set.biInter_subset_biInter_left
theorem biUnion_mono {s s' : Set α} {t t' : α → Set β} (hs : s' ⊆ s) (h : ∀ x ∈ s, t x ⊆ t' x) :
⋃ x ∈ s', t x ⊆ ⋃ x ∈ s, t' x :=
(biUnion_subset_biUnion_left hs).trans <| iUnion₂_mono h
#align set.bUnion_mono Set.biUnion_mono
theorem biInter_mono {s s' : Set α} {t t' : α → Set β} (hs : s ⊆ s') (h : ∀ x ∈ s, t x ⊆ t' x) :
⋂ x ∈ s', t x ⊆ ⋂ x ∈ s, t' x :=
(biInter_subset_biInter_left hs).trans <| iInter₂_mono h
#align set.bInter_mono Set.biInter_mono
theorem biUnion_eq_iUnion (s : Set α) (t : ∀ x ∈ s, Set β) :
⋃ x ∈ s, t x ‹_› = ⋃ x : s, t x x.2 :=
iSup_subtype'
#align set.bUnion_eq_Union Set.biUnion_eq_iUnion
theorem biInter_eq_iInter (s : Set α) (t : ∀ x ∈ s, Set β) :
⋂ x ∈ s, t x ‹_› = ⋂ x : s, t x x.2 :=
iInf_subtype'
#align set.bInter_eq_Inter Set.biInter_eq_iInter
theorem iUnion_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋃ x : { x // p x }, s x = ⋃ (x) (hx : p x), s ⟨x, hx⟩ :=
iSup_subtype
#align set.Union_subtype Set.iUnion_subtype
theorem iInter_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋂ x : { x // p x }, s x = ⋂ (x) (hx : p x), s ⟨x, hx⟩ :=
iInf_subtype
#align set.Inter_subtype Set.iInter_subtype
theorem biInter_empty (u : α → Set β) : ⋂ x ∈ (∅ : Set α), u x = univ :=
iInf_emptyset
#align set.bInter_empty Set.biInter_empty
theorem biInter_univ (u : α → Set β) : ⋂ x ∈ @univ α, u x = ⋂ x, u x :=
iInf_univ
#align set.bInter_univ Set.biInter_univ
@[simp]
theorem biUnion_self (s : Set α) : ⋃ x ∈ s, s = s :=
Subset.antisymm (iUnion₂_subset fun _ _ => Subset.refl s) fun _ hx => mem_biUnion hx hx
#align set.bUnion_self Set.biUnion_self
@[simp]
theorem iUnion_nonempty_self (s : Set α) : ⋃ _ : s.Nonempty, s = s := by
rw [iUnion_nonempty_index, biUnion_self]
#align set.Union_nonempty_self Set.iUnion_nonempty_self
theorem biInter_singleton (a : α) (s : α → Set β) : ⋂ x ∈ ({a} : Set α), s x = s a :=
iInf_singleton
#align set.bInter_singleton Set.biInter_singleton
theorem biInter_union (s t : Set α) (u : α → Set β) :
⋂ x ∈ s ∪ t, u x = (⋂ x ∈ s, u x) ∩ ⋂ x ∈ t, u x :=
iInf_union
#align set.bInter_union Set.biInter_union
theorem biInter_insert (a : α) (s : Set α) (t : α → Set β) :
⋂ x ∈ insert a s, t x = t a ∩ ⋂ x ∈ s, t x := by simp
#align set.bInter_insert Set.biInter_insert
theorem biInter_pair (a b : α) (s : α → Set β) : ⋂ x ∈ ({a, b} : Set α), s x = s a ∩ s b := by
rw [biInter_insert, biInter_singleton]
#align set.bInter_pair Set.biInter_pair
theorem biInter_inter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, f i ∩ t = (⋂ i ∈ s, f i) ∩ t := by
haveI : Nonempty s := hs.to_subtype
simp [biInter_eq_iInter, ← iInter_inter]
#align set.bInter_inter Set.biInter_inter
theorem inter_biInter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, t ∩ f i = t ∩ ⋂ i ∈ s, f i := by
rw [inter_comm, ← biInter_inter hs]
simp [inter_comm]
#align set.inter_bInter Set.inter_biInter
theorem biUnion_empty (s : α → Set β) : ⋃ x ∈ (∅ : Set α), s x = ∅ :=
iSup_emptyset
#align set.bUnion_empty Set.biUnion_empty
theorem biUnion_univ (s : α → Set β) : ⋃ x ∈ @univ α, s x = ⋃ x, s x :=
iSup_univ
#align set.bUnion_univ Set.biUnion_univ
theorem biUnion_singleton (a : α) (s : α → Set β) : ⋃ x ∈ ({a} : Set α), s x = s a :=
iSup_singleton
#align set.bUnion_singleton Set.biUnion_singleton
@[simp]
theorem biUnion_of_singleton (s : Set α) : ⋃ x ∈ s, {x} = s :=
ext <| by simp
#align set.bUnion_of_singleton Set.biUnion_of_singleton
theorem biUnion_union (s t : Set α) (u : α → Set β) :
⋃ x ∈ s ∪ t, u x = (⋃ x ∈ s, u x) ∪ ⋃ x ∈ t, u x :=
iSup_union
#align set.bUnion_union Set.biUnion_union
@[simp]
theorem iUnion_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋃ i, f i = ⋃ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iUnion_subtype _ _
#align set.Union_coe_set Set.iUnion_coe_set
@[simp]
theorem iInter_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋂ i, f i = ⋂ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iInter_subtype _ _
#align set.Inter_coe_set Set.iInter_coe_set
theorem biUnion_insert (a : α) (s : Set α) (t : α → Set β) :
⋃ x ∈ insert a s, t x = t a ∪ ⋃ x ∈ s, t x := by simp
#align set.bUnion_insert Set.biUnion_insert
theorem biUnion_pair (a b : α) (s : α → Set β) : ⋃ x ∈ ({a, b} : Set α), s x = s a ∪ s b :=
by simp
#align set.bUnion_pair Set.biUnion_pair
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem inter_iUnion₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∩ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ∩ t i j := by simp only [inter_iUnion]
#align set.inter_Union₂ Set.inter_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_inter (s : ∀ i, κ i → Set α) (t : Set α) :
(⋃ (i) (j), s i j) ∩ t = ⋃ (i) (j), s i j ∩ t := by simp_rw [iUnion_inter]
#align set.Union₂_inter Set.iUnion₂_inter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_iInter₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_iInter]
#align set.union_Inter₂ Set.union_iInter₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_union (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [iInter_union]
#align set.Inter₂_union Set.iInter₂_union
theorem mem_sUnion_of_mem {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∈ t) (ht : t ∈ S) :
x ∈ ⋃₀S :=
⟨t, ht, hx⟩
#align set.mem_sUnion_of_mem Set.mem_sUnion_of_mem
-- is this theorem really necessary?
theorem not_mem_of_not_mem_sUnion {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∉ ⋃₀S)
(ht : t ∈ S) : x ∉ t := fun h => hx ⟨t, ht, h⟩
#align set.not_mem_of_not_mem_sUnion Set.not_mem_of_not_mem_sUnion
theorem sInter_subset_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : ⋂₀ S ⊆ t :=
sInf_le tS
#align set.sInter_subset_of_mem Set.sInter_subset_of_mem
theorem subset_sUnion_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : t ⊆ ⋃₀S :=
le_sSup tS
#align set.subset_sUnion_of_mem Set.subset_sUnion_of_mem
theorem subset_sUnion_of_subset {s : Set α} (t : Set (Set α)) (u : Set α) (h₁ : s ⊆ u)
(h₂ : u ∈ t) : s ⊆ ⋃₀t :=
Subset.trans h₁ (subset_sUnion_of_mem h₂)
#align set.subset_sUnion_of_subset Set.subset_sUnion_of_subset
theorem sUnion_subset {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t' ⊆ t) : ⋃₀S ⊆ t :=
sSup_le h
#align set.sUnion_subset Set.sUnion_subset
@[simp]
theorem sUnion_subset_iff {s : Set (Set α)} {t : Set α} : ⋃₀s ⊆ t ↔ ∀ t' ∈ s, t' ⊆ t :=
sSup_le_iff
#align set.sUnion_subset_iff Set.sUnion_subset_iff
/-- `sUnion` is monotone under taking a subset of each set. -/
lemma sUnion_mono_subsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, t ⊆ f t) :
⋃₀ s ⊆ ⋃₀ (f '' s) :=
fun _ ⟨t, htx, hxt⟩ ↦ ⟨f t, mem_image_of_mem f htx, hf t hxt⟩
/-- `sUnion` is monotone under taking a superset of each set. -/
lemma sUnion_mono_supsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, f t ⊆ t) :
⋃₀ (f '' s) ⊆ ⋃₀ s :=
-- If t ∈ f '' s is arbitrary; t = f u for some u : Set α.
fun _ ⟨_, ⟨u, hus, hut⟩, hxt⟩ ↦ ⟨u, hus, (hut ▸ hf u) hxt⟩
theorem subset_sInter {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t ⊆ t') : t ⊆ ⋂₀ S :=
le_sInf h
#align set.subset_sInter Set.subset_sInter
@[simp]
theorem subset_sInter_iff {S : Set (Set α)} {t : Set α} : t ⊆ ⋂₀ S ↔ ∀ t' ∈ S, t ⊆ t' :=
le_sInf_iff
#align set.subset_sInter_iff Set.subset_sInter_iff
@[gcongr]
theorem sUnion_subset_sUnion {S T : Set (Set α)} (h : S ⊆ T) : ⋃₀S ⊆ ⋃₀T :=
sUnion_subset fun _ hs => subset_sUnion_of_mem (h hs)
#align set.sUnion_subset_sUnion Set.sUnion_subset_sUnion
@[gcongr]
theorem sInter_subset_sInter {S T : Set (Set α)} (h : S ⊆ T) : ⋂₀ T ⊆ ⋂₀ S :=
subset_sInter fun _ hs => sInter_subset_of_mem (h hs)
#align set.sInter_subset_sInter Set.sInter_subset_sInter
@[simp]
theorem sUnion_empty : ⋃₀∅ = (∅ : Set α) :=
sSup_empty
#align set.sUnion_empty Set.sUnion_empty
@[simp]
theorem sInter_empty : ⋂₀ ∅ = (univ : Set α) :=
sInf_empty
#align set.sInter_empty Set.sInter_empty
@[simp]
theorem sUnion_singleton (s : Set α) : ⋃₀{s} = s :=
sSup_singleton
#align set.sUnion_singleton Set.sUnion_singleton
@[simp]
theorem sInter_singleton (s : Set α) : ⋂₀ {s} = s :=
sInf_singleton
#align set.sInter_singleton Set.sInter_singleton
@[simp]
theorem sUnion_eq_empty {S : Set (Set α)} : ⋃₀S = ∅ ↔ ∀ s ∈ S, s = ∅ :=
sSup_eq_bot
#align set.sUnion_eq_empty Set.sUnion_eq_empty
@[simp]
theorem sInter_eq_univ {S : Set (Set α)} : ⋂₀ S = univ ↔ ∀ s ∈ S, s = univ :=
sInf_eq_top
#align set.sInter_eq_univ Set.sInter_eq_univ
theorem subset_powerset_iff {s : Set (Set α)} {t : Set α} : s ⊆ 𝒫 t ↔ ⋃₀ s ⊆ t :=
sUnion_subset_iff.symm
/-- `⋃₀` and `𝒫` form a Galois connection. -/
theorem sUnion_powerset_gc :
GaloisConnection (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gc_sSup_Iic
/-- `⋃₀` and `𝒫` form a Galois insertion. -/
def sUnion_powerset_gi :
GaloisInsertion (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gi_sSup_Iic
/-- If all sets in a collection are either `∅` or `Set.univ`, then so is their union. -/
theorem sUnion_mem_empty_univ {S : Set (Set α)} (h : S ⊆ {∅, univ}) :
⋃₀ S ∈ ({∅, univ} : Set (Set α)) := by
simp only [mem_insert_iff, mem_singleton_iff, or_iff_not_imp_left, sUnion_eq_empty, not_forall]
rintro ⟨s, hs, hne⟩
obtain rfl : s = univ := (h hs).resolve_left hne
exact univ_subset_iff.1 <| subset_sUnion_of_mem hs
@[simp]
theorem nonempty_sUnion {S : Set (Set α)} : (⋃₀S).Nonempty ↔ ∃ s ∈ S, Set.Nonempty s := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_sUnion Set.nonempty_sUnion
theorem Nonempty.of_sUnion {s : Set (Set α)} (h : (⋃₀s).Nonempty) : s.Nonempty :=
let ⟨s, hs, _⟩ := nonempty_sUnion.1 h
⟨s, hs⟩
#align set.nonempty.of_sUnion Set.Nonempty.of_sUnion
theorem Nonempty.of_sUnion_eq_univ [Nonempty α] {s : Set (Set α)} (h : ⋃₀s = univ) : s.Nonempty :=
Nonempty.of_sUnion <| h.symm ▸ univ_nonempty
#align set.nonempty.of_sUnion_eq_univ Set.Nonempty.of_sUnion_eq_univ
theorem sUnion_union (S T : Set (Set α)) : ⋃₀(S ∪ T) = ⋃₀S ∪ ⋃₀T :=
sSup_union
#align set.sUnion_union Set.sUnion_union
theorem sInter_union (S T : Set (Set α)) : ⋂₀ (S ∪ T) = ⋂₀ S ∩ ⋂₀ T :=
sInf_union
#align set.sInter_union Set.sInter_union
@[simp]
theorem sUnion_insert (s : Set α) (T : Set (Set α)) : ⋃₀insert s T = s ∪ ⋃₀T :=
sSup_insert
#align set.sUnion_insert Set.sUnion_insert
@[simp]
theorem sInter_insert (s : Set α) (T : Set (Set α)) : ⋂₀ insert s T = s ∩ ⋂₀ T :=
sInf_insert
#align set.sInter_insert Set.sInter_insert
@[simp]
theorem sUnion_diff_singleton_empty (s : Set (Set α)) : ⋃₀(s \ {∅}) = ⋃₀s :=
sSup_diff_singleton_bot s
#align set.sUnion_diff_singleton_empty Set.sUnion_diff_singleton_empty
@[simp]
theorem sInter_diff_singleton_univ (s : Set (Set α)) : ⋂₀ (s \ {univ}) = ⋂₀ s :=
sInf_diff_singleton_top s
#align set.sInter_diff_singleton_univ Set.sInter_diff_singleton_univ
theorem sUnion_pair (s t : Set α) : ⋃₀{s, t} = s ∪ t :=
sSup_pair
#align set.sUnion_pair Set.sUnion_pair
theorem sInter_pair (s t : Set α) : ⋂₀ {s, t} = s ∩ t :=
sInf_pair
#align set.sInter_pair Set.sInter_pair
@[simp]
theorem sUnion_image (f : α → Set β) (s : Set α) : ⋃₀(f '' s) = ⋃ x ∈ s, f x :=
sSup_image
#align set.sUnion_image Set.sUnion_image
@[simp]
theorem sInter_image (f : α → Set β) (s : Set α) : ⋂₀ (f '' s) = ⋂ x ∈ s, f x :=
sInf_image
#align set.sInter_image Set.sInter_image
@[simp]
theorem sUnion_range (f : ι → Set β) : ⋃₀range f = ⋃ x, f x :=
rfl
#align set.sUnion_range Set.sUnion_range
@[simp]
theorem sInter_range (f : ι → Set β) : ⋂₀ range f = ⋂ x, f x :=
rfl
#align set.sInter_range Set.sInter_range
theorem iUnion_eq_univ_iff {f : ι → Set α} : ⋃ i, f i = univ ↔ ∀ x, ∃ i, x ∈ f i := by
simp only [eq_univ_iff_forall, mem_iUnion]
#align set.Union_eq_univ_iff Set.iUnion_eq_univ_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_eq_univ_iff {s : ∀ i, κ i → Set α} :
⋃ (i) (j), s i j = univ ↔ ∀ a, ∃ i j, a ∈ s i j :=
by simp only [iUnion_eq_univ_iff, mem_iUnion]
#align set.Union₂_eq_univ_iff Set.iUnion₂_eq_univ_iff
theorem sUnion_eq_univ_iff {c : Set (Set α)} : ⋃₀c = univ ↔ ∀ a, ∃ b ∈ c, a ∈ b := by
simp only [eq_univ_iff_forall, mem_sUnion]
#align set.sUnion_eq_univ_iff Set.sUnion_eq_univ_iff
-- classical
theorem iInter_eq_empty_iff {f : ι → Set α} : ⋂ i, f i = ∅ ↔ ∀ x, ∃ i, x ∉ f i := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.Inter_eq_empty_iff Set.iInter_eq_empty_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
theorem iInter₂_eq_empty_iff {s : ∀ i, κ i → Set α} :
⋂ (i) (j), s i j = ∅ ↔ ∀ a, ∃ i j, a ∉ s i j := by
simp only [eq_empty_iff_forall_not_mem, mem_iInter, not_forall]
#align set.Inter₂_eq_empty_iff Set.iInter₂_eq_empty_iff
-- classical
theorem sInter_eq_empty_iff {c : Set (Set α)} : ⋂₀ c = ∅ ↔ ∀ a, ∃ b ∈ c, a ∉ b := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.sInter_eq_empty_iff Set.sInter_eq_empty_iff
-- classical
@[simp]
theorem nonempty_iInter {f : ι → Set α} : (⋂ i, f i).Nonempty ↔ ∃ x, ∀ i, x ∈ f i := by
simp [nonempty_iff_ne_empty, iInter_eq_empty_iff]
#align set.nonempty_Inter Set.nonempty_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_iInter₂ {s : ∀ i, κ i → Set α} :
(⋂ (i) (j), s i j).Nonempty ↔ ∃ a, ∀ i j, a ∈ s i j := by
simp
#align set.nonempty_Inter₂ Set.nonempty_iInter₂
-- classical
@[simp]
theorem nonempty_sInter {c : Set (Set α)} : (⋂₀ c).Nonempty ↔ ∃ a, ∀ b ∈ c, a ∈ b := by
simp [nonempty_iff_ne_empty, sInter_eq_empty_iff]
#align set.nonempty_sInter Set.nonempty_sInter
-- classical
theorem compl_sUnion (S : Set (Set α)) : (⋃₀S)ᶜ = ⋂₀ (compl '' S) :=
ext fun x => by simp
#align set.compl_sUnion Set.compl_sUnion
-- classical
theorem sUnion_eq_compl_sInter_compl (S : Set (Set α)) : ⋃₀S = (⋂₀ (compl '' S))ᶜ := by
rw [← compl_compl (⋃₀S), compl_sUnion]
#align set.sUnion_eq_compl_sInter_compl Set.sUnion_eq_compl_sInter_compl
-- classical
theorem compl_sInter (S : Set (Set α)) : (⋂₀ S)ᶜ = ⋃₀(compl '' S) := by
rw [sUnion_eq_compl_sInter_compl, compl_compl_image]
#align set.compl_sInter Set.compl_sInter
-- classical
theorem sInter_eq_compl_sUnion_compl (S : Set (Set α)) : ⋂₀ S = (⋃₀(compl '' S))ᶜ := by
rw [← compl_compl (⋂₀ S), compl_sInter]
#align set.sInter_eq_compl_sUnion_compl Set.sInter_eq_compl_sUnion_compl
theorem inter_empty_of_inter_sUnion_empty {s t : Set α} {S : Set (Set α)} (hs : t ∈ S)
(h : s ∩ ⋃₀S = ∅) : s ∩ t = ∅ :=
eq_empty_of_subset_empty <| by
rw [← h]; exact inter_subset_inter_right _ (subset_sUnion_of_mem hs)
#align set.inter_empty_of_inter_sUnion_empty Set.inter_empty_of_inter_sUnion_empty
theorem range_sigma_eq_iUnion_range {γ : α → Type*} (f : Sigma γ → β) :
range f = ⋃ a, range fun b => f ⟨a, b⟩ :=
Set.ext <| by simp
#align set.range_sigma_eq_Union_range Set.range_sigma_eq_iUnion_range
theorem iUnion_eq_range_sigma (s : α → Set β) : ⋃ i, s i = range fun a : Σi, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_sigma Set.iUnion_eq_range_sigma
theorem iUnion_eq_range_psigma (s : ι → Set β) : ⋃ i, s i = range fun a : Σ'i, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_psigma Set.iUnion_eq_range_psigma
theorem iUnion_image_preimage_sigma_mk_eq_self {ι : Type*} {σ : ι → Type*} (s : Set (Sigma σ)) :
⋃ i, Sigma.mk i '' (Sigma.mk i ⁻¹' s) = s := by
ext x
simp only [mem_iUnion, mem_image, mem_preimage]
constructor
· rintro ⟨i, a, h, rfl⟩
exact h
· intro h
cases' x with i a
exact ⟨i, a, h, rfl⟩
#align set.Union_image_preimage_sigma_mk_eq_self Set.iUnion_image_preimage_sigma_mk_eq_self
theorem Sigma.univ (X : α → Type*) : (Set.univ : Set (Σa, X a)) = ⋃ a, range (Sigma.mk a) :=
Set.ext fun x =>
iff_of_true trivial ⟨range (Sigma.mk x.1), Set.mem_range_self _, x.2, Sigma.eta x⟩
#align set.sigma.univ Set.Sigma.univ
alias sUnion_mono := sUnion_subset_sUnion
#align set.sUnion_mono Set.sUnion_mono
theorem iUnion_subset_iUnion_const {s : Set α} (h : ι → ι₂) : ⋃ _ : ι, s ⊆ ⋃ _ : ι₂, s :=
@iSup_const_mono (Set α) ι ι₂ _ s h
#align set.Union_subset_Union_const Set.iUnion_subset_iUnion_const
@[simp]
theorem iUnion_singleton_eq_range {α β : Type*} (f : α → β) : ⋃ x : α, {f x} = range f := by
ext x
simp [@eq_comm _ x]
#align set.Union_singleton_eq_range Set.iUnion_singleton_eq_range
theorem iUnion_of_singleton (α : Type*) : (⋃ x, {x} : Set α) = univ := by simp [Set.ext_iff]
#align set.Union_of_singleton Set.iUnion_of_singleton
theorem iUnion_of_singleton_coe (s : Set α) : ⋃ i : s, ({(i : α)} : Set α) = s := by simp
#align set.Union_of_singleton_coe Set.iUnion_of_singleton_coe
theorem sUnion_eq_biUnion {s : Set (Set α)} : ⋃₀s = ⋃ (i : Set α) (_ : i ∈ s), i := by
rw [← sUnion_image, image_id']
#align set.sUnion_eq_bUnion Set.sUnion_eq_biUnion
theorem sInter_eq_biInter {s : Set (Set α)} : ⋂₀ s = ⋂ (i : Set α) (_ : i ∈ s), i := by
rw [← sInter_image, image_id']
#align set.sInter_eq_bInter Set.sInter_eq_biInter
theorem sUnion_eq_iUnion {s : Set (Set α)} : ⋃₀s = ⋃ i : s, i := by
simp only [← sUnion_range, Subtype.range_coe]
#align set.sUnion_eq_Union Set.sUnion_eq_iUnion
theorem sInter_eq_iInter {s : Set (Set α)} : ⋂₀ s = ⋂ i : s, i := by
simp only [← sInter_range, Subtype.range_coe]
#align set.sInter_eq_Inter Set.sInter_eq_iInter
@[simp]
theorem iUnion_of_empty [IsEmpty ι] (s : ι → Set α) : ⋃ i, s i = ∅ :=
iSup_of_empty _
#align set.Union_of_empty Set.iUnion_of_empty
@[simp]
theorem iInter_of_empty [IsEmpty ι] (s : ι → Set α) : ⋂ i, s i = univ :=
iInf_of_empty _
#align set.Inter_of_empty Set.iInter_of_empty
theorem union_eq_iUnion {s₁ s₂ : Set α} : s₁ ∪ s₂ = ⋃ b : Bool, cond b s₁ s₂ :=
sup_eq_iSup s₁ s₂
#align set.union_eq_Union Set.union_eq_iUnion
theorem inter_eq_iInter {s₁ s₂ : Set α} : s₁ ∩ s₂ = ⋂ b : Bool, cond b s₁ s₂ :=
inf_eq_iInf s₁ s₂
#align set.inter_eq_Inter Set.inter_eq_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_union_sInter {S T : Set (Set α)} :
⋂₀ S ∪ ⋂₀ T = ⋂ p ∈ S ×ˢ T, (p : Set α × Set α).1 ∪ p.2 :=
sInf_sup_sInf
#align set.sInter_union_sInter Set.sInter_union_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_inter_sUnion {s t : Set (Set α)} :
⋃₀s ∩ ⋃₀t = ⋃ p ∈ s ×ˢ t, (p : Set α × Set α).1 ∩ p.2 :=
sSup_inf_sSup
#align set.sUnion_inter_sUnion Set.sUnion_inter_sUnion
theorem biUnion_iUnion (s : ι → Set α) (t : α → Set β) :
⋃ x ∈ ⋃ i, s i, t x = ⋃ (i) (x ∈ s i), t x := by simp [@iUnion_comm _ ι]
#align set.bUnion_Union Set.biUnion_iUnion
theorem biInter_iUnion (s : ι → Set α) (t : α → Set β) :
⋂ x ∈ ⋃ i, s i, t x = ⋂ (i) (x ∈ s i), t x := by simp [@iInter_comm _ ι]
#align set.bInter_Union Set.biInter_iUnion
theorem sUnion_iUnion (s : ι → Set (Set α)) : ⋃₀⋃ i, s i = ⋃ i, ⋃₀s i := by
simp only [sUnion_eq_biUnion, biUnion_iUnion]
#align set.sUnion_Union Set.sUnion_iUnion
theorem sInter_iUnion (s : ι → Set (Set α)) : ⋂₀ ⋃ i, s i = ⋂ i, ⋂₀ s i := by
simp only [sInter_eq_biInter, biInter_iUnion]
#align set.sInter_Union Set.sInter_iUnion
theorem iUnion_range_eq_sUnion {α β : Type*} (C : Set (Set α)) {f : ∀ s : C, β → (s : Type _)}
(hf : ∀ s : C, Surjective (f s)) : ⋃ y : β, range (fun s : C => (f s y).val) = ⋃₀C := by
ext x; constructor
· rintro ⟨s, ⟨y, rfl⟩, ⟨s, hs⟩, rfl⟩
refine' ⟨_, hs, _⟩
exact (f ⟨s, hs⟩ y).2
· rintro ⟨s, hs, hx⟩
cases' hf ⟨s, hs⟩ ⟨x, hx⟩ with y hy
refine' ⟨_, ⟨y, rfl⟩, ⟨s, hs⟩, _⟩
exact congr_arg Subtype.val hy
#align set.Union_range_eq_sUnion Set.iUnion_range_eq_sUnion
theorem iUnion_range_eq_iUnion (C : ι → Set α) {f : ∀ x : ι, β → C x}
(hf : ∀ x : ι, Surjective (f x)) : ⋃ y : β, range (fun x : ι => (f x y).val) = ⋃ x, C x := by
ext x; rw [mem_iUnion, mem_iUnion]; constructor
· rintro ⟨y, i, rfl⟩
exact ⟨i, (f i y).2⟩
· rintro ⟨i, hx⟩
cases' hf i ⟨x, hx⟩ with y hy
exact ⟨y, i, congr_arg Subtype.val hy⟩
#align set.Union_range_eq_Union Set.iUnion_range_eq_iUnion
theorem union_distrib_iInter_left (s : ι → Set α) (t : Set α) : (t ∪ ⋂ i, s i) = ⋂ i, t ∪ s i :=
sup_iInf_eq _ _
#align set.union_distrib_Inter_left Set.union_distrib_iInter_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_left (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_distrib_iInter_left]
#align set.union_distrib_Inter₂_left Set.union_distrib_iInter₂_left
theorem union_distrib_iInter_right (s : ι → Set α) (t : Set α) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.union_distrib_Inter_right Set.union_distrib_iInter_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_right (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [union_distrib_iInter_right]
#align set.union_distrib_Inter₂_right Set.union_distrib_iInter₂_right
section Function
/-! ### `mapsTo` -/
theorem mapsTo_sUnion {S : Set (Set α)} {t : Set β} {f : α → β} (H : ∀ s ∈ S, MapsTo f s t) :
MapsTo f (⋃₀S) t := fun _ ⟨s, hs, hx⟩ => H s hs hx
#align set.maps_to_sUnion Set.mapsTo_sUnion
theorem mapsTo_iUnion {s : ι → Set α} {t : Set β} {f : α → β} (H : ∀ i, MapsTo f (s i) t) :
MapsTo f (⋃ i, s i) t :=
mapsTo_sUnion <| forall_range_iff.2 H
#align set.maps_to_Union Set.mapsTo_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂ {s : ∀ i, κ i → Set α} {t : Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) t) : MapsTo f (⋃ (i) (j), s i j) t :=
mapsTo_iUnion fun i => mapsTo_iUnion (H i)
#align set.maps_to_Union₂ Set.mapsTo_iUnion₂
theorem mapsTo_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋃ i, s i) (⋃ i, t i) :=
mapsTo_iUnion fun i => (H i).mono (Subset.refl _) (subset_iUnion t i)
#align set.maps_to_Union_Union Set.mapsTo_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
mapsTo_iUnion_iUnion fun i => mapsTo_iUnion_iUnion (H i)
#align set.maps_to_Union₂_Union₂ Set.mapsTo_iUnion₂_iUnion₂
theorem mapsTo_sInter {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, MapsTo f s t) :
MapsTo f s (⋂₀ T) := fun _ hx t ht => H t ht hx
#align set.maps_to_sInter Set.mapsTo_sInter
theorem mapsTo_iInter {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, MapsTo f s (t i)) :
MapsTo f s (⋂ i, t i) := fun _ hx => mem_iInter.2 fun i => H i hx
#align set.maps_to_Inter Set.mapsTo_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f s (t i j)) : MapsTo f s (⋂ (i) (j), t i j) :=
mapsTo_iInter fun i => mapsTo_iInter (H i)
#align set.maps_to_Inter₂ Set.mapsTo_iInter₂
theorem mapsTo_iInter_iInter {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋂ i, s i) (⋂ i, t i) :=
mapsTo_iInter fun i => (H i).mono (iInter_subset s i) (Subset.refl _)
#align set.maps_to_Inter_Inter Set.mapsTo_iInter_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂_iInter₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋂ (i) (j), s i j) (⋂ (i) (j), t i j) :=
mapsTo_iInter_iInter fun i => mapsTo_iInter_iInter (H i)
#align set.maps_to_Inter₂_Inter₂ Set.mapsTo_iInter₂_iInter₂
theorem image_iInter_subset (s : ι → Set α) (f : α → β) : (f '' ⋂ i, s i) ⊆ ⋂ i, f '' s i :=
(mapsTo_iInter_iInter fun i => mapsTo_image f (s i)).image_subset
#align set.image_Inter_subset Set.image_iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂_subset (s : ∀ i, κ i → Set α) (f : α → β) :
(f '' ⋂ (i) (j), s i j) ⊆ ⋂ (i) (j), f '' s i j :=
(mapsTo_iInter₂_iInter₂ fun i hi => mapsTo_image f (s i hi)).image_subset
#align set.image_Inter₂_subset Set.image_iInter₂_subset
theorem image_sInter_subset (S : Set (Set α)) (f : α → β) : f '' ⋂₀ S ⊆ ⋂ s ∈ S, f '' s := by
rw [sInter_eq_biInter]
apply image_iInter₂_subset
#align set.image_sInter_subset Set.image_sInter_subset
/-! ### `restrictPreimage` -/
section
open Function
variable (s : Set β) {f : α → β} {U : ι → Set β} (hU : iUnion U = univ)
theorem injective_iff_injective_of_iUnion_eq_univ :
Injective f ↔ ∀ i, Injective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_injective H, fun H x y e => _⟩
obtain ⟨i, hi⟩ := Set.mem_iUnion.mp
(show f x ∈ Set.iUnion U by rw [hU]; triv)
injection @H i ⟨x, hi⟩ ⟨y, show f y ∈ U i from e ▸ hi⟩ (Subtype.ext e)
#align set.injective_iff_injective_of_Union_eq_univ Set.injective_iff_injective_of_iUnion_eq_univ
theorem surjective_iff_surjective_of_iUnion_eq_univ :
Surjective f ↔ ∀ i, Surjective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_surjective H, fun H x => _⟩
obtain ⟨i, hi⟩ :=
Set.mem_iUnion.mp
(show x ∈ Set.iUnion U by rw [hU]; triv)
exact ⟨_, congr_arg Subtype.val (H i ⟨x, hi⟩).choose_spec⟩
#align set.surjective_iff_surjective_of_Union_eq_univ Set.surjective_iff_surjective_of_iUnion_eq_univ
theorem bijective_iff_bijective_of_iUnion_eq_univ :
Bijective f ↔ ∀ i, Bijective ((U i).restrictPreimage f) := by
rw [Bijective, injective_iff_injective_of_iUnion_eq_univ hU,
surjective_iff_surjective_of_iUnion_eq_univ hU]
simp [Bijective, forall_and]
#align set.bijective_iff_bijective_of_Union_eq_univ Set.bijective_iff_bijective_of_iUnion_eq_univ
end
/-! ### `InjOn` -/
theorem InjOn.image_iInter_eq [Nonempty ι] {s : ι → Set α} {f : α → β} (h : InjOn f (⋃ i, s i)) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
inhabit ι
refine' Subset.antisymm (image_iInter_subset s f) fun y hy => _
simp only [mem_iInter, mem_image_iff_bex] at hy
choose x hx hy using hy
refine' ⟨x default, mem_iInter.2 fun i => _, hy _⟩
suffices x default = x i by
rw [this]
apply hx
replace hx : ∀ i, x i ∈ ⋃ j, s j := fun i => (subset_iUnion _ _) (hx i)
apply h (hx _) (hx _)
simp only [hy]
#align set.inj_on.image_Inter_eq Set.InjOn.image_iInter_eq
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
theorem InjOn.image_biInter_eq {p : ι → Prop} {s : ∀ i, p i → Set α} (hp : ∃ i, p i)
{f : α → β} (h : InjOn f (⋃ (i) (hi), s i hi)) :
(f '' ⋂ (i) (hi), s i hi) = ⋂ (i) (hi), f '' s i hi := by
simp only [iInter, iInf_subtype']
haveI : Nonempty { i // p i } := nonempty_subtype.2 hp
apply InjOn.image_iInter_eq
simpa only [iUnion, iSup_subtype'] using h
#align set.inj_on.image_bInter_eq Set.InjOn.image_biInter_eq
theorem image_iInter {f : α → β} (hf : Bijective f) (s : ι → Set α) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
cases isEmpty_or_nonempty ι
· simp_rw [iInter_of_empty, image_univ_of_surjective hf.surjective]
· exact (hf.injective.injOn _).image_iInter_eq
#align set.image_Inter Set.image_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂ {f : α → β} (hf : Bijective f) (s : ∀ i, κ i → Set α) :
(f '' ⋂ (i) (j), s i j) = ⋂ (i) (j), f '' s i j := by simp_rw [image_iInter hf]
#align set.image_Inter₂ Set.image_iInter₂
theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
(hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i) := by
intro x hx y hy hxy
rcases mem_iUnion.1 hx with ⟨i, hx⟩
rcases mem_iUnion.1 hy with ⟨j, hy⟩
rcases hs i j with ⟨k, hi, hj⟩
exact hf k (hi hx) (hj hy) hxy
#align set.inj_on_Union_of_directed Set.inj_on_iUnion_of_directed
/-! ### `SurjOn` -/
theorem surjOn_sUnion {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, SurjOn f s t) :
SurjOn f s (⋃₀T) := fun _ ⟨t, ht, hx⟩ => H t ht hx
#align set.surj_on_sUnion Set.surjOn_sUnion
theorem surjOn_iUnion {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, SurjOn f s (t i)) :
SurjOn f s (⋃ i, t i) :=
surjOn_sUnion <| forall_range_iff.2 H
#align set.surj_on_Union Set.surjOn_iUnion
theorem surjOn_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) : SurjOn f (⋃ i, s i) (⋃ i, t i) :=
surjOn_iUnion fun i => (H i).mono (subset_iUnion _ _) (Subset.refl _)
#align set.surj_on_Union_Union Set.surjOn_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f s (t i j)) : SurjOn f s (⋃ (i) (j), t i j) :=
surjOn_iUnion fun i => surjOn_iUnion (H i)
#align set.surj_on_Union₂ Set.surjOn_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f (s i j) (t i j)) : SurjOn f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
surjOn_iUnion_iUnion fun i => surjOn_iUnion_iUnion (H i)
#align set.surj_on_Union₂_Union₂ Set.surjOn_iUnion₂_iUnion₂
theorem surjOn_iInter [Nonempty ι] {s : ι → Set α} {t : Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) t) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) t := by
intro y hy
rw [Hinj.image_iInter_eq, mem_iInter]
exact fun i => H i hy
#align set.surj_on_Inter Set.surjOn_iInter
theorem surjOn_iInter_iInter [Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) (⋂ i, t i) :=
surjOn_iInter (fun i => (H i).mono (Subset.refl _) (iInter_subset _ _)) Hinj
#align set.surj_on_Inter_Inter Set.surjOn_iInter_iInter
/-! ### `BijOn` -/
theorem bijOn_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i))
(Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
⟨mapsTo_iUnion_iUnion fun i => (H i).mapsTo, Hinj, surjOn_iUnion_iUnion fun i => (H i).surjOn⟩
#align set.bij_on_Union Set.bijOn_iUnion
theorem bijOn_iInter [hi : Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, BijOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
⟨mapsTo_iInter_iInter fun i => (H i).mapsTo,
hi.elim fun i => (H i).injOn.mono (iInter_subset _ _),
surjOn_iInter_iInter (fun i => (H i).surjOn) Hinj⟩
#align set.bij_on_Inter Set.bijOn_iInter
theorem bijOn_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {t : ι → Set β}
{f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
bijOn_iUnion H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Union_of_directed Set.bijOn_iUnion_of_directed
theorem bijOn_iInter_of_directed [Nonempty ι] {s : ι → Set α} (hs : Directed (· ⊆ ·) s)
{t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
bijOn_iInter H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Inter_of_directed Set.bijOn_iInter_of_directed
end Function
/-! ### `image`, `preimage` -/
section Image
theorem image_iUnion {f : α → β} {s : ι → Set α} : (f '' ⋃ i, s i) = ⋃ i, f '' s i := by
ext1 x
simp only [mem_image, mem_iUnion, ← exists_and_right, ← exists_and_left]
--Porting note: `exists_swap` causes a `simp` loop in Lean4 so we use `rw` instead.
rw [exists_swap]
#align set.image_Union Set.image_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iUnion₂ (f : α → β) (s : ∀ i, κ i → Set α) :
(f '' ⋃ (i) (j), s i j) = ⋃ (i) (j), f '' s i j := by simp_rw [image_iUnion]
#align set.image_Union₂ Set.image_iUnion₂
theorem univ_subtype {p : α → Prop} : (univ : Set (Subtype p)) = ⋃ (x) (h : p x), {⟨x, h⟩} :=
Set.ext fun ⟨x, h⟩ => by simp [h]
#align set.univ_subtype Set.univ_subtype
theorem range_eq_iUnion {ι} (f : ι → α) : range f = ⋃ i, {f i} :=
Set.ext fun a => by simp [@eq_comm α a]
#align set.range_eq_Union Set.range_eq_iUnion
theorem image_eq_iUnion (f : α → β) (s : Set α) : f '' s = ⋃ i ∈ s, {f i} :=
Set.ext fun b => by simp [@eq_comm β b]
#align set.image_eq_Union Set.image_eq_iUnion
theorem biUnion_range {f : ι → α} {g : α → Set β} : ⋃ x ∈ range f, g x = ⋃ y, g (f y) :=
iSup_range
#align set.bUnion_range Set.biUnion_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iUnion_iUnion_eq' {f : ι → α} {g : α → Set β} :
⋃ (x) (y) (_ : f y = x), g x = ⋃ y, g (f y) := by simpa using biUnion_range
#align set.Union_Union_eq' Set.iUnion_iUnion_eq'
theorem biInter_range {f : ι → α} {g : α → Set β} : ⋂ x ∈ range f, g x = ⋂ y, g (f y) :=
iInf_range
#align set.bInter_range Set.biInter_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iInter_iInter_eq' {f : ι → α} {g : α → Set β} :
⋂ (x) (y) (_ : f y = x), g x = ⋂ y, g (f y) := by simpa using biInter_range
#align set.Inter_Inter_eq' Set.iInter_iInter_eq'
variable {s : Set γ} {f : γ → α} {g : α → Set β}
theorem biUnion_image : ⋃ x ∈ f '' s, g x = ⋃ y ∈ s, g (f y) :=
iSup_image
#align set.bUnion_image Set.biUnion_image
theorem biInter_image : ⋂ x ∈ f '' s, g x = ⋂ y ∈ s, g (f y) :=
iInf_image
#align set.bInter_image Set.biInter_image
end Image
section Preimage
theorem monotone_preimage {f : α → β} : Monotone (preimage f) := fun _ _ h => preimage_mono h
#align set.monotone_preimage Set.monotone_preimage
@[simp]
theorem preimage_iUnion {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋃ i, s i) = ⋃ i, f ⁻¹' s i :=
Set.ext <| by simp [preimage]
#align set.preimage_Union Set.preimage_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iUnion₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋃ (i) (j), s i j) = ⋃ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iUnion]
#align set.preimage_Union₂ Set.preimage_iUnion₂
@[simp]
theorem preimage_sUnion {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋃₀s = ⋃ t ∈ s, f ⁻¹' t := by
rw [sUnion_eq_biUnion, preimage_iUnion₂]
#align set.preimage_sUnion Set.preimage_sUnion
theorem preimage_iInter {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋂ i, s i) = ⋂ i, f ⁻¹' s i := by
ext; simp
#align set.preimage_Inter Set.preimage_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iInter₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋂ (i) (j), s i j) = ⋂ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iInter]
#align set.preimage_Inter₂ Set.preimage_iInter₂
@[simp]
theorem preimage_sInter {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋂₀ s = ⋂ t ∈ s, f ⁻¹' t := by
rw [sInter_eq_biInter, preimage_iInter₂]
#align set.preimage_sInter Set.preimage_sInter
@[simp]
theorem biUnion_preimage_singleton (f : α → β) (s : Set β) : ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s := by
rw [← preimage_iUnion₂, biUnion_of_singleton]
#align set.bUnion_preimage_singleton Set.biUnion_preimage_singleton
theorem biUnion_range_preimage_singleton (f : α → β) : ⋃ y ∈ range f, f ⁻¹' {y} = univ := by
rw [biUnion_preimage_singleton, preimage_range]
#align set.bUnion_range_preimage_singleton Set.biUnion_range_preimage_singleton
end Preimage
section Prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion {s : Set α} {t : ι → Set β} : (s ×ˢ ⋃ i, t i) = ⋃ i, s ×ˢ t i := by
ext
simp
#align set.prod_Union Set.prod_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} :
(s ×ˢ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ×ˢ t i j := by simp_rw [prod_iUnion]
#align set.prod_Union₂ Set.prod_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sUnion {s : Set α} {C : Set (Set β)} : s ×ˢ ⋃₀C = ⋃₀((fun t => s ×ˢ t) '' C) := by
simp_rw [sUnion_eq_biUnion, biUnion_image, prod_iUnion₂]
#align set.prod_sUnion Set.prod_sUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_const {s : ι → Set α} {t : Set β} : (⋃ i, s i) ×ˢ t = ⋃ i, s i ×ˢ t := by
ext
simp
#align set.Union_prod_const Set.iUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion₂_prod_const {s : ∀ i, κ i → Set α} {t : Set β} :
(⋃ (i) (j), s i j) ×ˢ t = ⋃ (i) (j), s i j ×ˢ t := by simp_rw [iUnion_prod_const]
#align set.Union₂_prod_const Set.iUnion₂_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_prod_const {C : Set (Set α)} {t : Set β} :
⋃₀C ×ˢ t = ⋃₀((fun s : Set α => s ×ˢ t) '' C) := by
simp only [sUnion_eq_biUnion, iUnion₂_prod_const, biUnion_image]
#align set.sUnion_prod_const Set.sUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod {ι ι' α β} (s : ι → Set α) (t : ι' → Set β) :
⋃ x : ι × ι', s x.1 ×ˢ t x.2 = (⋃ i : ι, s i) ×ˢ ⋃ i : ι', t i := by
ext
simp
#align set.Union_prod Set.iUnion_prod
/-- Analogue of `iSup_prod` for sets. -/
lemma iUnion_prod' (f : β × γ → Set α) : ⋃ x : β × γ, f x = ⋃ (i : β) (j : γ), f (i, j) :=
iSup_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x := by
ext ⟨z, w⟩; simp only [mem_prod, mem_iUnion, exists_imp, and_imp, iff_def]; constructor
· intro x hz hw
exact ⟨⟨x, hz⟩, x, hw⟩
· intro x hz x' hw
exact ⟨x ⊔ x', hs le_sup_left hz, ht le_sup_right hw⟩
#align set.Union_prod_of_monotone Set.iUnion_prod_of_monotone
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter_subset (S : Set (Set α)) (T : Set (Set β)) :
⋂₀ S ×ˢ ⋂₀ T ⊆ ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 :=
subset_iInter₂ fun x hx _ hy => ⟨hy.1 x.1 hx.1, hy.2 x.2 hx.2⟩
#align set.sInter_prod_sInter_subset Set.sInter_prod_sInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 := by
obtain ⟨s₁, h₁⟩ := hS
obtain ⟨s₂, h₂⟩ := hT
refine' Set.Subset.antisymm (sInter_prod_sInter_subset S T) fun x hx => _
rw [mem_iInter₂] at hx
exact ⟨fun s₀ h₀ => (hx (s₀, s₂) ⟨h₀, h₂⟩).1, fun s₀ h₀ => (hx (s₁, s₀) ⟨h₁, h₀⟩).2⟩
#align set.sInter_prod_sInter Set.sInter_prod_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod {S : Set (Set α)} (hS : S.Nonempty) (t : Set β) :
⋂₀ S ×ˢ t = ⋂ s ∈ S, s ×ˢ t := by
rw [← sInter_singleton t, sInter_prod_sInter hS (singleton_nonempty t), sInter_singleton]
simp_rw [prod_singleton, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.sInter_prod Set.sInter_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sInter {T : Set (Set β)} (hT : T.Nonempty) (s : Set α) :
s ×ˢ ⋂₀ T = ⋂ t ∈ T, s ×ˢ t := by
rw [← sInter_singleton s, sInter_prod_sInter (singleton_nonempty s) hT, sInter_singleton]
simp_rw [singleton_prod, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.prod_sInter Set.prod_sInter
end Prod
section Image2
variable (f : α → β → γ) {s : Set α} {t : Set β}
/-- The `Set.image2` version of `Set.image_eq_iUnion` -/
theorem image2_eq_iUnion (s : Set α) (t : Set β) : image2 f s t = ⋃ (i ∈ s) (j ∈ t), {f i j} := by
ext; simp [eq_comm]
#align set.image2_eq_Union Set.image2_eq_iUnion
theorem iUnion_image_left : ⋃ a ∈ s, f a '' t = image2 f s t := by
simp only [image2_eq_iUnion, image_eq_iUnion]
#align set.Union_image_left Set.iUnion_image_left
theorem iUnion_image_right : ⋃ b ∈ t, (f · b) '' s = image2 f s t := by
rw [image2_swap, iUnion_image_left]
#align set.Union_image_right Set.iUnion_image_right
theorem image2_iUnion_left (s : ι → Set α) (t : Set β) :
image2 f (⋃ i, s i) t = ⋃ i, image2 f (s i) t := by
simp only [← image_prod, iUnion_prod_const, image_iUnion]
#align set.image2_Union_left Set.image2_iUnion_left
theorem image2_iUnion_right (s : Set α) (t : ι → Set β) :
image2 f s (⋃ i, t i) = ⋃ i, image2 f s (t i) := by
simp only [← image_prod, prod_iUnion, image_iUnion]
#align set.image2_Union_right Set.image2_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋃ (i) (j), s i j) t = ⋃ (i) (j), image2 f (s i j) t := by simp_rw [image2_iUnion_left]
#align set.image2_Union₂_left Set.image2_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋃ (i) (j), t i j) = ⋃ (i) (j), image2 f s (t i j) :=
by simp_rw [image2_iUnion_right]
#align set.image2_Union₂_right Set.image2_iUnion₂_right
theorem image2_iInter_subset_left (s : ι → Set α) (t : Set β) :
image2 f (⋂ i, s i) t ⊆ ⋂ i, image2 f (s i) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem (hx _) hy
#align set.image2_Inter_subset_left Set.image2_iInter_subset_left
theorem image2_iInter_subset_right (s : Set α) (t : ι → Set β) :
image2 f s (⋂ i, t i) ⊆ ⋂ i, image2 f s (t i) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem hx (hy _)
#align set.image2_Inter_subset_right Set.image2_iInter_subset_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋂ (i) (j), s i j) t ⊆ ⋂ (i) (j), image2 f (s i j) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem (hx _ _) hy
#align set.image2_Inter₂_subset_left Set.image2_iInter₂_subset_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), image2 f s (t i j) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem hx (hy _ _)
#align set.image2_Inter₂_subset_right Set.image2_iInter₂_subset_right
theorem prod_eq_biUnion_left : s ×ˢ t = ⋃ a ∈ s, (fun b => (a, b)) '' t := by
rw [iUnion_image_left, image2_mk_eq_prod]
#align set.prod_eq_bUnion_left Set.prod_eq_biUnion_left
theorem prod_eq_biUnion_right : s ×ˢ t = ⋃ b ∈ t, (fun a => (a, b)) '' s := by
rw [iUnion_image_right, image2_mk_eq_prod]
#align set.prod_eq_bUnion_right Set.prod_eq_biUnion_right
end Image2
section Seq
/-- Given a set `s` of functions `α → β` and `t : Set α`, `seq s t` is the union of `f '' t` over
all `f ∈ s`. -/
def seq (s : Set (α → β)) (t : Set α) : Set β :=
{ b | ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b }
#align set.seq Set.seq
@[simp]
theorem mem_seq_iff {s : Set (α → β)} {t : Set α} {b : β} :
b ∈ seq s t ↔ ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b :=
Iff.rfl
#align set.mem_seq_iff Set.mem_seq_iff
lemma seq_eq_image2 (s : Set (α → β)) (t : Set α) : seq s t = image2 (fun f a ↦ f a) s t := by
ext; simp
theorem seq_def {s : Set (α → β)} {t : Set α} : seq s t = ⋃ f ∈ s, f '' t := by
rw [seq_eq_image2, iUnion_image_left]
#align set.seq_def Set.seq_def
theorem seq_subset {s : Set (α → β)} {t : Set α} {u : Set β} :
seq s t ⊆ u ↔ ∀ f ∈ s, ∀ a ∈ t, (f : α → β) a ∈ u := by
rw [seq_eq_image2, image2_subset_iff]
#align set.seq_subset Set.seq_subset
@[gcongr]
theorem seq_mono {s₀ s₁ : Set (α → β)} {t₀ t₁ : Set α} (hs : s₀ ⊆ s₁) (ht : t₀ ⊆ t₁) :
seq s₀ t₀ ⊆ seq s₁ t₁ := fun _ ⟨f, hf, a, ha, eq⟩ => ⟨f, hs hf, a, ht ha, eq⟩
#align set.seq_mono Set.seq_mono
theorem singleton_seq {f : α → β} {t : Set α} : Set.seq ({f} : Set (α → β)) t = f '' t := by
rw [seq_eq_image2, image2_singleton_left]
#align set.singleton_seq Set.singleton_seq
theorem seq_singleton {s : Set (α → β)} {a : α} : Set.seq s {a} = (fun f : α → β => f a) '' s := by
rw [seq_eq_image2, image2_singleton_right]
#align set.seq_singleton Set.seq_singleton
theorem seq_seq {s : Set (β → γ)} {t : Set (α → β)} {u : Set α} :
seq s (seq t u) = seq (seq ((· ∘ ·) '' s) t) u := by
simp only [seq_eq_image2, image2_image_left, image2_image2_left, image2_image2_right, comp_apply]
#align set.seq_seq Set.seq_seq
theorem image_seq {f : β → γ} {s : Set (α → β)} {t : Set α} :
f '' seq s t = seq ((f ∘ ·) '' s) t := by
rw [← singleton_seq, ← singleton_seq, seq_seq, image_singleton]
#align set.image_seq Set.image_seq
theorem prod_eq_seq {s : Set α} {t : Set β} : s ×ˢ t = (Prod.mk '' s).seq t := by
rw [seq_eq_image2, image2_image_left, image2_mk_eq_prod]
#align set.prod_eq_seq Set.prod_eq_seq
theorem prod_image_seq_comm (s : Set α) (t : Set β) :
(Prod.mk '' s).seq t = seq ((fun b a => (a, b)) '' t) s := by
rw [← prod_eq_seq, ← image_swap_prod, prod_eq_seq, image_seq, ← image_comp]; rfl
#align set.prod_image_seq_comm Set.prod_image_seq_comm
theorem image2_eq_seq (f : α → β → γ) (s : Set α) (t : Set β) : image2 f s t = seq (f '' s) t := by
rw [seq_eq_image2, image2_image_left]
#align set.image2_eq_seq Set.image2_eq_seq
end Seq
section Pi
variable {π : α → Type*}
theorem pi_def (i : Set α) (s : ∀ a, Set (π a)) : pi i s = ⋂ a ∈ i, eval a ⁻¹' s a := by
ext
simp
#align set.pi_def Set.pi_def
theorem univ_pi_eq_iInter (t : ∀ i, Set (π i)) : pi univ t = ⋂ i, eval i ⁻¹' t i := by
simp only [pi_def, iInter_true, mem_univ]
#align set.univ_pi_eq_Inter Set.univ_pi_eq_iInter
theorem pi_diff_pi_subset (i : Set α) (s t : ∀ a, Set (π a)) :
pi i s \ pi i t ⊆ ⋃ a ∈ i, eval a ⁻¹' (s a \ t a) := by
refine' diff_subset_comm.2 fun x hx a ha => _
simp only [mem_diff, mem_pi, mem_iUnion, not_exists, mem_preimage, not_and, not_not,
eval_apply] at hx
exact hx.2 _ ha (hx.1 _ ha)
#align set.pi_diff_pi_subset Set.pi_diff_pi_subset
theorem iUnion_univ_pi {ι : α → Type*} (t : (a : α) → ι a → Set (π a)) :
⋃ x : (a : α) → ι a, pi univ (fun a => t a (x a)) = pi univ fun a => ⋃ j : ι a, t a j := by
ext
simp [Classical.skolem]
#align set.Union_univ_pi Set.iUnion_univ_pi
end Pi
end Set
namespace Function
namespace Surjective
theorem iUnion_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋃ x, g (f x) = ⋃ y, g y :=
hf.iSup_comp g
#align function.surjective.Union_comp Function.Surjective.iUnion_comp
theorem iInter_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋂ x, g (f x) = ⋂ y, g y :=
hf.iInf_comp g
#align function.surjective.Inter_comp Function.Surjective.iInter_comp
end Surjective
end Function
/-!
### Disjoint sets
-/
section Disjoint
variable {s t u : Set α} {f : α → β}
namespace Set
@[simp]
theorem disjoint_iUnion_left {ι : Sort*} {s : ι → Set α} :
Disjoint (⋃ i, s i) t ↔ ∀ i, Disjoint (s i) t :=
iSup_disjoint_iff
#align set.disjoint_Union_left Set.disjoint_iUnion_left
@[simp]
theorem disjoint_iUnion_right {ι : Sort*} {s : ι → Set α} :
Disjoint t (⋃ i, s i) ↔ ∀ i, Disjoint t (s i) :=
disjoint_iSup_iff
#align set.disjoint_Union_right Set.disjoint_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_left {s : ∀ i, κ i → Set α} {t : Set α} :
Disjoint (⋃ (i) (j), s i j) t ↔ ∀ i j, Disjoint (s i j) t :=
iSup₂_disjoint_iff
#align set.disjoint_Union₂_left Set.disjoint_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_right {s : Set α} {t : ∀ i, κ i → Set α} :
Disjoint s (⋃ (i) (j), t i j) ↔ ∀ i j, Disjoint s (t i j) :=
disjoint_iSup₂_iff
#align set.disjoint_Union₂_right Set.disjoint_iUnion₂_right
@[simp]
theorem disjoint_sUnion_left {S : Set (Set α)} {t : Set α} :
Disjoint (⋃₀S) t ↔ ∀ s ∈ S, Disjoint s t :=
sSup_disjoint_iff
#align set.disjoint_sUnion_left Set.disjoint_sUnion_left
@[simp]
theorem disjoint_sUnion_right {s : Set α} {S : Set (Set α)} :
Disjoint s (⋃₀S) ↔ ∀ t ∈ S, Disjoint s t :=
disjoint_sSup_iff
#align set.disjoint_sUnion_right Set.disjoint_sUnion_right
end Set
end Disjoint
/-! ### Intervals -/
namespace Set
lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :
(⋂ i, Iic (f i)).Nonempty ↔ BddBelow (range f) := by
have : (⋂ (i : ι), Iic (f i)) = lowerBounds (range f) := by
ext c; simp [lowerBounds]
simp [this, BddBelow]
lemma nonempty_iInter_Ici_iff [Preorder α] {f : ι → α} :
(⋂ i, Ici (f i)).Nonempty ↔ BddAbove (range f) :=
nonempty_iInter_Iic_iff (α := αᵒᵈ)
variable [CompleteLattice α]
theorem Ici_iSup (f : ι → α) : Ici (⨆ i, f i) = ⋂ i, Ici (f i) :=
ext fun _ => by simp only [mem_Ici, iSup_le_iff, mem_iInter]
#align set.Ici_supr Set.Ici_iSup
theorem Iic_iInf (f : ι → α) : Iic (⨅ i, f i) = ⋂ i, Iic (f i) :=
ext fun _ => by simp only [mem_Iic, le_iInf_iff, mem_iInter]
#align set.Iic_infi Set.Iic_iInf
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Ici_iSup₂ (f : ∀ i, κ i → α) : Ici (⨆ (i) (j), f i j) = ⋂ (i) (j), Ici (f i j) := by
simp_rw [Ici_iSup]
#align set.Ici_supr₂ Set.Ici_iSup₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Iic_iInf₂ (f : ∀ i, κ i → α) : Iic (⨅ (i) (j), f i j) = ⋂ (i) (j), Iic (f i j) := by
simp_rw [Iic_iInf]
#align set.Iic_infi₂ Set.Iic_iInf₂
theorem Ici_sSup (s : Set α) : Ici (sSup s) = ⋂ a ∈ s, Ici a := by rw [sSup_eq_iSup, Ici_iSup₂]
#align set.Ici_Sup Set.Ici_sSup
theorem Iic_sInf (s : Set α) : Iic (sInf s) = ⋂ a ∈ s, Iic a := by rw [sInf_eq_iInf, Iic_iInf₂]
#align set.Iic_Inf Set.Iic_sInf
end Set
namespace Set
variable (t : α → Set β)
theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x := by
simp only [diff_subset_iff, ← biUnion_union]
apply biUnion_subset_biUnion_left
rw [union_diff_self]
apply subset_union_right
#align set.bUnion_diff_bUnion_subset Set.biUnion_diff_biUnion_subset
/-- If `t` is an indexed family of sets, then there is a natural map from `Σ i, t i` to `⋃ i, t i`
sending `⟨i, x⟩` to `x`. -/
def sigmaToiUnion (x : Σi, t i) : ⋃ i, t i :=
⟨x.2, mem_iUnion.2 ⟨x.1, x.2.2⟩⟩
#align set.sigma_to_Union Set.sigmaToiUnion
theorem sigmaToiUnion_surjective : Surjective (sigmaToiUnion t)
| ⟨b, hb⟩ =>
have : ∃ a, b ∈ t a := by simpa using hb
let ⟨a, hb⟩ := this
⟨⟨a, b, hb⟩, rfl⟩
#align set.sigma_to_Union_surjective Set.sigmaToiUnion_surjective
theorem sigmaToiUnion_injective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Injective (sigmaToiUnion t)
| ⟨a₁, b₁, h₁⟩, ⟨a₂, b₂, h₂⟩, eq =>
have b_eq : b₁ = b₂ := congr_arg Subtype.val eq
have a_eq : a₁ = a₂ :=
by_contradiction fun ne =>
have : b₁ ∈ t a₁ ∩ t a₂ := ⟨h₁, b_eq.symm ▸ h₂⟩
(h _ _ ne).le_bot this
Sigma.eq a_eq <| Subtype.eq <| by subst b_eq; subst a_eq; rfl
#align set.sigma_to_Union_injective Set.sigmaToiUnion_injective
theorem sigmaToiUnion_bijective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Bijective (sigmaToiUnion t) :=
⟨sigmaToiUnion_injective t h, sigmaToiUnion_surjective t⟩
#align set.sigma_to_Union_bijective Set.sigmaToiUnion_bijective
/-- Equivalence between a disjoint union and a dependent sum. -/
noncomputable def unionEqSigmaOfDisjoint {t : α → Set β} (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
(⋃ i, t i) ≃ Σi, t i :=
(Equiv.ofBijective _ <| sigmaToiUnion_bijective t h).symm
#align set.Union_eq_sigma_of_disjoint Set.unionEqSigmaOfDisjoint
theorem iUnion_ge_eq_iUnion_nat_add (u : ℕ → Set α) (n : ℕ) : ⋃ i ≥ n, u i = ⋃ i, u (i + n) :=
iSup_ge_eq_iSup_nat_add u n
#align set.Union_ge_eq_Union_nat_add Set.iUnion_ge_eq_iUnion_nat_add
theorem iInter_ge_eq_iInter_nat_add (u : ℕ → Set α) (n : ℕ) : ⋂ i ≥ n, u i = ⋂ i, u (i + n) :=
iInf_ge_eq_iInf_nat_add u n
#align set.Inter_ge_eq_Inter_nat_add Set.iInter_ge_eq_iInter_nat_add
theorem _root_.Monotone.iUnion_nat_add {f : ℕ → Set α} (hf : Monotone f) (k : ℕ) :
⋃ n, f (n + k) = ⋃ n, f n :=
hf.iSup_nat_add k
#align monotone.Union_nat_add Monotone.iUnion_nat_add
theorem _root_.Antitone.iInter_nat_add {f : ℕ → Set α} (hf : Antitone f) (k : ℕ) :
⋂ n, f (n + k) = ⋂ n, f n :=
hf.iInf_nat_add k
#align antitone.Inter_nat_add Antitone.iInter_nat_add
/-Porting note: removing `simp`. LHS does not simplify. Possible linter bug. Zulip discussion:
https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/complete_lattice.20and.20has_sup/near/316497982
-/
theorem iUnion_iInter_ge_nat_add (f : ℕ → Set α) (k : ℕ) :
⋃ n, ⋂ i ≥ n, f (i + k) = ⋃ n, ⋂ i ≥ n, f i :=
iSup_iInf_ge_nat_add f k
#align set.Union_Inter_ge_nat_add Set.iUnion_iInter_ge_nat_add
theorem union_iUnion_nat_succ (u : ℕ → Set α) : (u 0 ∪ ⋃ i, u (i + 1)) = ⋃ i, u i :=
sup_iSup_nat_succ u
#align set.union_Union_nat_succ Set.union_iUnion_nat_succ
theorem inter_iInter_nat_succ (u : ℕ → Set α) : (u 0 ∩ ⋂ i, u (i + 1)) = ⋂ i, u i :=
inf_iInf_nat_succ u
#align set.inter_Inter_nat_succ Set.inter_iInter_nat_succ
end Set
open Set
variable [CompleteLattice β]
theorem iSup_iUnion (s : ι → Set α) (f : α → β) : ⨆ a ∈ ⋃ i, s i, f a = ⨆ (i) (a ∈ s i), f a := by
rw [iSup_comm]
simp_rw [mem_iUnion, iSup_exists]
#align supr_Union iSup_iUnion
theorem iInf_iUnion (s : ι → Set α) (f : α → β) : ⨅ a ∈ ⋃ i, s i, f a = ⨅ (i) (a ∈ s i), f a :=
@iSup_iUnion α βᵒᵈ _ _ s f
#align infi_Union iInf_iUnion
theorem sSup_sUnion (s : Set (Set β)) : sSup (⋃₀ s) = ⨆ t ∈ s, sSup t := by
|
simp only [sUnion_eq_biUnion, sSup_eq_iSup, iSup_iUnion]
|
theorem sSup_sUnion (s : Set (Set β)) : sSup (⋃₀ s) = ⨆ t ∈ s, sSup t := by
|
Mathlib.Data.Set.Lattice.2383_0.5mONj49h3SYSDwc
|
theorem sSup_sUnion (s : Set (Set β)) : sSup (⋃₀ s) = ⨆ t ∈ s, sSup t
|
Mathlib_Data_Set_Lattice
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : CompleteLattice β
S : Set (Set α)
f : α → β
⊢ ⨆ x ∈ ⋃₀ S, f x = ⨆ s ∈ S, ⨆ x ∈ s, f x
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
/-!
# The set lattice
This file provides usual set notation for unions and intersections, a `CompleteLattice` instance
for `Set α`, and some more set constructions.
## Main declarations
* `Set.iUnion`: **i**ndexed **union**. Union of an indexed family of sets.
* `Set.iInter`: **i**ndexed **inter**section. Intersection of an indexed family of sets.
* `Set.sInter`: **s**et **inter**section. Intersection of sets belonging to a set of sets.
* `Set.sUnion`: **s**et **union**. Union of sets belonging to a set of sets.
* `Set.sInter_eq_biInter`, `Set.sUnion_eq_biInter`: Shows that `⋂₀ s = ⋂ x ∈ s, x` and
`⋃₀ s = ⋃ x ∈ s, x`.
* `Set.completeAtomicBooleanAlgebra`: `Set α` is a `CompleteAtomicBooleanAlgebra` with `≤ = ⊆`,
`< = ⊂`, `⊓ = ∩`, `⊔ = ∪`, `⨅ = ⋂`, `⨆ = ⋃` and `\` as the set difference.
See `Set.BooleanAlgebra`.
* `Set.kernImage`: For a function `f : α → β`, `s.kernImage f` is the set of `y` such that
`f ⁻¹ y ⊆ s`.
* `Set.seq`: Union of the image of a set under a **seq**uence of functions. `seq s t` is the union
of `f '' t` over all `f ∈ s`, where `t : Set α` and `s : Set (α → β)`.
* `Set.iUnion_eq_sigma_of_disjoint`: Equivalence between `⋃ i, t i` and `Σ i, t i`, where `t` is an
indexed family of disjoint sets.
## Naming convention
In lemma names,
* `⋃ i, s i` is called `iUnion`
* `⋂ i, s i` is called `iInter`
* `⋃ i j, s i j` is called `iUnion₂`. This is an `iUnion` inside an `iUnion`.
* `⋂ i j, s i j` is called `iInter₂`. This is an `iInter` inside an `iInter`.
* `⋃ i ∈ s, t i` is called `biUnion` for "bounded `iUnion`". This is the special case of `iUnion₂`
where `j : i ∈ s`.
* `⋂ i ∈ s, t i` is called `biInter` for "bounded `iInter`". This is the special case of `iInter₂`
where `j : i ∈ s`.
## Notation
* `⋃`: `Set.iUnion`
* `⋂`: `Set.iInter`
* `⋃₀`: `Set.sUnion`
* `⋂₀`: `Set.sInter`
-/
set_option autoImplicit true
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι₂ : Sort*} {κ κ₁ κ₂ : ι → Sort*} {κ' : ι' → Sort*}
namespace Set
/-! ### Complete lattice and complete Boolean algebra instances -/
instance : InfSet (Set α) :=
⟨fun s => { a | ∀ t ∈ s, a ∈ t }⟩
instance : SupSet (Set α) :=
⟨fun s => { a | ∃ t ∈ s, a ∈ t }⟩
/-- Intersection of a set of sets. -/
def sInter (S : Set (Set α)) : Set α :=
sInf S
#align set.sInter Set.sInter
/-- Notation for `Set.sInter` Intersection of a set of sets. -/
prefix:110 "⋂₀ " => sInter
/-- Union of a set of sets. -/
def sUnion (S : Set (Set α)) : Set α :=
sSup S
#align set.sUnion Set.sUnion
/-- Notation for `Set.sUnion`. Union of a set of sets. -/
prefix:110 "⋃₀ " => sUnion
@[simp]
theorem mem_sInter {x : α} {S : Set (Set α)} : x ∈ ⋂₀ S ↔ ∀ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sInter Set.mem_sInter
@[simp]
theorem mem_sUnion {x : α} {S : Set (Set α)} : x ∈ ⋃₀ S ↔ ∃ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sUnion Set.mem_sUnion
/-- Indexed union of a family of sets -/
def iUnion (s : ι → Set β) : Set β :=
iSup s
#align set.Union Set.iUnion
/-- Indexed intersection of a family of sets -/
def iInter (s : ι → Set β) : Set β :=
iInf s
#align set.Inter Set.iInter
/-- Notation for `Set.iUnion`. Indexed union of a family of sets -/
notation3 "⋃ "(...)", "r:60:(scoped f => iUnion f) => r
/-- Notation for `Set.iInter`. Indexed intersection of a family of sets -/
notation3 "⋂ "(...)", "r:60:(scoped f => iInter f) => r
section delaborators
open Lean Lean.PrettyPrinter.Delaborator
/-- Delaborator for indexed unions. -/
@[delab app.Set.iUnion]
def iUnion_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋃ (_ : $dom), $body)
else if prop || ppTypes then
`(⋃ ($x:ident : $dom), $body)
else
`(⋃ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋃ $x:ident, ⋃ (_ : $y:ident ∈ $s), $body)
| `(⋃ ($x:ident : $_), ⋃ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋃ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
/-- Delaborator for indexed intersections. -/
@[delab app.Set.iInter]
def sInter_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋂ (_ : $dom), $body)
else if prop || ppTypes then
`(⋂ ($x:ident : $dom), $body)
else
`(⋂ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋂ $x:ident, ⋂ (_ : $y:ident ∈ $s), $body)
| `(⋂ ($x:ident : $_), ⋂ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋂ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
end delaborators
@[simp]
theorem sSup_eq_sUnion (S : Set (Set α)) : sSup S = ⋃₀S :=
rfl
#align set.Sup_eq_sUnion Set.sSup_eq_sUnion
@[simp]
theorem sInf_eq_sInter (S : Set (Set α)) : sInf S = ⋂₀ S :=
rfl
#align set.Inf_eq_sInter Set.sInf_eq_sInter
@[simp]
theorem iSup_eq_iUnion (s : ι → Set α) : iSup s = iUnion s :=
rfl
#align set.supr_eq_Union Set.iSup_eq_iUnion
@[simp]
theorem iInf_eq_iInter (s : ι → Set α) : iInf s = iInter s :=
rfl
#align set.infi_eq_Inter Set.iInf_eq_iInter
@[simp]
theorem mem_iUnion {x : α} {s : ι → Set α} : (x ∈ ⋃ i, s i) ↔ ∃ i, x ∈ s i :=
⟨fun ⟨_, ⟨⟨a, (t_eq : s a = _)⟩, (h : x ∈ _)⟩⟩ => ⟨a, t_eq.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨s a, ⟨⟨a, rfl⟩, h⟩⟩⟩
#align set.mem_Union Set.mem_iUnion
@[simp]
theorem mem_iInter {x : α} {s : ι → Set α} : (x ∈ ⋂ i, s i) ↔ ∀ i, x ∈ s i :=
⟨fun (h : ∀ a ∈ { a : Set α | ∃ i, s i = a }, x ∈ a) a => h (s a) ⟨a, rfl⟩,
fun h _ ⟨a, (eq : s a = _)⟩ => eq ▸ h a⟩
#align set.mem_Inter Set.mem_iInter
theorem mem_iUnion₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋃ (i) (j), s i j) ↔ ∃ i j, x ∈ s i j := by
simp_rw [mem_iUnion]
#align set.mem_Union₂ Set.mem_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋂ (i) (j), s i j) ↔ ∀ i j, x ∈ s i j := by
simp_rw [mem_iInter]
#align set.mem_Inter₂ Set.mem_iInter₂
theorem mem_iUnion_of_mem {s : ι → Set α} {a : α} (i : ι) (ha : a ∈ s i) : a ∈ ⋃ i, s i :=
mem_iUnion.2 ⟨i, ha⟩
#align set.mem_Union_of_mem Set.mem_iUnion_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iUnion₂_of_mem {s : ∀ i, κ i → Set α} {a : α} {i : ι} (j : κ i) (ha : a ∈ s i j) :
a ∈ ⋃ (i) (j), s i j :=
mem_iUnion₂.2 ⟨i, j, ha⟩
#align set.mem_Union₂_of_mem Set.mem_iUnion₂_of_mem
theorem mem_iInter_of_mem {s : ι → Set α} {a : α} (h : ∀ i, a ∈ s i) : a ∈ ⋂ i, s i :=
mem_iInter.2 h
#align set.mem_Inter_of_mem Set.mem_iInter_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂_of_mem {s : ∀ i, κ i → Set α} {a : α} (h : ∀ i j, a ∈ s i j) :
a ∈ ⋂ (i) (j), s i j :=
mem_iInter₂.2 h
#align set.mem_Inter₂_of_mem Set.mem_iInter₂_of_mem
instance Set.completeAtomicBooleanAlgebra : CompleteAtomicBooleanAlgebra (Set α) :=
{ instBooleanAlgebraSet with
le_sSup := fun s t t_in a a_in => ⟨t, t_in, a_in⟩
sSup_le := fun s t h a ⟨t', ⟨t'_in, a_in⟩⟩ => h t' t'_in a_in
le_sInf := fun s t h a a_in t' t'_in => h t' t'_in a_in
sInf_le := fun s t t_in a h => h _ t_in
iInf_iSup_eq := by intros; ext; simp [Classical.skolem] }
/-- `kernImage f s` is the set of `y` such that `f ⁻¹ y ⊆ s`. -/
def kernImage (f : α → β) (s : Set α) : Set β :=
{ y | ∀ ⦃x⦄, f x = y → x ∈ s }
#align set.kern_image Set.kernImage
lemma subset_kernImage_iff {f : α → β} : s ⊆ kernImage f t ↔ f ⁻¹' s ⊆ t :=
⟨fun h _ hx ↦ h hx rfl,
fun h _ hx y hy ↦ h (show f y ∈ s from hy.symm ▸ hx)⟩
section GaloisConnection
variable {f : α → β}
protected theorem image_preimage : GaloisConnection (image f) (preimage f) := fun _ _ =>
image_subset_iff
#align set.image_preimage Set.image_preimage
protected theorem preimage_kernImage : GaloisConnection (preimage f) (kernImage f) := fun _ _ =>
subset_kernImage_iff.symm
#align set.preimage_kern_image Set.preimage_kernImage
end GaloisConnection
section kernImage
variable {f : α → β}
lemma kernImage_mono : Monotone (kernImage f) :=
Set.preimage_kernImage.monotone_u
lemma kernImage_eq_compl {s : Set α} : kernImage f s = (f '' sᶜ)ᶜ :=
Set.preimage_kernImage.u_unique (Set.image_preimage.compl)
(fun t ↦ compl_compl (f ⁻¹' t) ▸ Set.preimage_compl)
lemma kernImage_compl {s : Set α} : kernImage f (sᶜ) = (f '' s)ᶜ := by
rw [kernImage_eq_compl, compl_compl]
lemma kernImage_empty : kernImage f ∅ = (range f)ᶜ := by
rw [kernImage_eq_compl, compl_empty, image_univ]
lemma kernImage_preimage_eq_iff {s : Set β} : kernImage f (f ⁻¹' s) = s ↔ (range f)ᶜ ⊆ s := by
rw [kernImage_eq_compl, ← preimage_compl, compl_eq_comm, eq_comm, image_preimage_eq_iff,
compl_subset_comm]
lemma compl_range_subset_kernImage {s : Set α} : (range f)ᶜ ⊆ kernImage f s := by
rw [← kernImage_empty]
exact kernImage_mono (empty_subset _)
lemma kernImage_union_preimage {s : Set α} {t : Set β} :
kernImage f (s ∪ f ⁻¹' t) = kernImage f s ∪ t := by
rw [kernImage_eq_compl, kernImage_eq_compl, compl_union, ← preimage_compl, image_inter_preimage,
compl_inter, compl_compl]
lemma kernImage_preimage_union {s : Set α} {t : Set β} :
kernImage f (f ⁻¹' t ∪ s) = t ∪ kernImage f s := by
rw [union_comm, kernImage_union_preimage, union_comm]
end kernImage
/-! ### Union and intersection over an indexed family of sets -/
instance : OrderTop (Set α) where
top := univ
le_top := by simp
@[congr]
theorem iUnion_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iUnion f₁ = iUnion f₂ :=
iSup_congr_Prop pq f
#align set.Union_congr_Prop Set.iUnion_congr_Prop
@[congr]
theorem iInter_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iInter f₁ = iInter f₂ :=
iInf_congr_Prop pq f
#align set.Inter_congr_Prop Set.iInter_congr_Prop
theorem iUnion_plift_up (f : PLift ι → Set α) : ⋃ i, f (PLift.up i) = ⋃ i, f i :=
iSup_plift_up _
#align set.Union_plift_up Set.iUnion_plift_up
theorem iUnion_plift_down (f : ι → Set α) : ⋃ i, f (PLift.down i) = ⋃ i, f i :=
iSup_plift_down _
#align set.Union_plift_down Set.iUnion_plift_down
theorem iInter_plift_up (f : PLift ι → Set α) : ⋂ i, f (PLift.up i) = ⋂ i, f i :=
iInf_plift_up _
#align set.Inter_plift_up Set.iInter_plift_up
theorem iInter_plift_down (f : ι → Set α) : ⋂ i, f (PLift.down i) = ⋂ i, f i :=
iInf_plift_down _
#align set.Inter_plift_down Set.iInter_plift_down
theorem iUnion_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋃ _ : p, s = if p then s else ∅ :=
iSup_eq_if _
#align set.Union_eq_if Set.iUnion_eq_if
theorem iUnion_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋃ h : p, s h = if h : p then s h else ∅ :=
iSup_eq_dif _
#align set.Union_eq_dif Set.iUnion_eq_dif
theorem iInter_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋂ _ : p, s = if p then s else univ :=
iInf_eq_if _
#align set.Inter_eq_if Set.iInter_eq_if
theorem iInf_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋂ h : p, s h = if h : p then s h else univ :=
_root_.iInf_eq_dif _
#align set.Infi_eq_dif Set.iInf_eq_dif
theorem exists_set_mem_of_union_eq_top {ι : Type*} (t : Set ι) (s : ι → Set β)
(w : ⋃ i ∈ t, s i = ⊤) (x : β) : ∃ i ∈ t, x ∈ s i := by
have p : x ∈ ⊤ := Set.mem_univ x
rw [← w, Set.mem_iUnion] at p
simpa using p
#align set.exists_set_mem_of_union_eq_top Set.exists_set_mem_of_union_eq_top
theorem nonempty_of_union_eq_top_of_nonempty {ι : Type*} (t : Set ι) (s : ι → Set α)
(H : Nonempty α) (w : ⋃ i ∈ t, s i = ⊤) : t.Nonempty := by
obtain ⟨x, m, -⟩ := exists_set_mem_of_union_eq_top t s w H.some
exact ⟨x, m⟩
#align set.nonempty_of_union_eq_top_of_nonempty Set.nonempty_of_union_eq_top_of_nonempty
theorem nonempty_of_nonempty_iUnion
{s : ι → Set α} (h_Union : (⋃ i, s i).Nonempty) : Nonempty ι := by
obtain ⟨x, hx⟩ := h_Union
exact ⟨Classical.choose $ mem_iUnion.mp hx⟩
theorem nonempty_of_nonempty_iUnion_eq_univ
{s : ι → Set α} [Nonempty α] (h_Union : ⋃ i, s i = univ) : Nonempty ι :=
nonempty_of_nonempty_iUnion (s := s) (by simpa only [h_Union] using univ_nonempty)
theorem setOf_exists (p : ι → β → Prop) : { x | ∃ i, p i x } = ⋃ i, { x | p i x } :=
ext fun _ => mem_iUnion.symm
#align set.set_of_exists Set.setOf_exists
theorem setOf_forall (p : ι → β → Prop) : { x | ∀ i, p i x } = ⋂ i, { x | p i x } :=
ext fun _ => mem_iInter.symm
#align set.set_of_forall Set.setOf_forall
theorem iUnion_subset {s : ι → Set α} {t : Set α} (h : ∀ i, s i ⊆ t) : ⋃ i, s i ⊆ t :=
iSup_le h
#align set.Union_subset Set.iUnion_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset {s : ∀ i, κ i → Set α} {t : Set α} (h : ∀ i j, s i j ⊆ t) :
⋃ (i) (j), s i j ⊆ t :=
iUnion_subset fun x => iUnion_subset (h x)
#align set.Union₂_subset Set.iUnion₂_subset
theorem subset_iInter {t : Set β} {s : ι → Set β} (h : ∀ i, t ⊆ s i) : t ⊆ ⋂ i, s i :=
le_iInf h
#align set.subset_Inter Set.subset_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iInter₂ {s : Set α} {t : ∀ i, κ i → Set α} (h : ∀ i j, s ⊆ t i j) :
s ⊆ ⋂ (i) (j), t i j :=
subset_iInter fun x => subset_iInter <| h x
#align set.subset_Inter₂ Set.subset_iInter₂
@[simp]
theorem iUnion_subset_iff {s : ι → Set α} {t : Set α} : ⋃ i, s i ⊆ t ↔ ∀ i, s i ⊆ t :=
⟨fun h _ => Subset.trans (le_iSup s _) h, iUnion_subset⟩
#align set.Union_subset_iff Set.iUnion_subset_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset_iff {s : ∀ i, κ i → Set α} {t : Set α} :
⋃ (i) (j), s i j ⊆ t ↔ ∀ i j, s i j ⊆ t := by simp_rw [iUnion_subset_iff]
#align set.Union₂_subset_iff Set.iUnion₂_subset_iff
@[simp]
theorem subset_iInter_iff {s : Set α} {t : ι → Set α} : (s ⊆ ⋂ i, t i) ↔ ∀ i, s ⊆ t i :=
le_iInf_iff
#align set.subset_Inter_iff Set.subset_iInter_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem subset_iInter₂_iff {s : Set α} {t : ∀ i, κ i → Set α} :
(s ⊆ ⋂ (i) (j), t i j) ↔ ∀ i j, s ⊆ t i j := by simp_rw [subset_iInter_iff]
#align set.subset_Inter₂_iff Set.subset_iInter₂_iff
theorem subset_iUnion : ∀ (s : ι → Set β) (i : ι), s i ⊆ ⋃ i, s i :=
le_iSup
#align set.subset_Union Set.subset_iUnion
theorem iInter_subset : ∀ (s : ι → Set β) (i : ι), ⋂ i, s i ⊆ s i :=
iInf_le
#align set.Inter_subset Set.iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iUnion₂ {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : s i j ⊆ ⋃ (i') (j'), s i' j' :=
le_iSup₂ i j
#align set.subset_Union₂ Set.subset_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_subset {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : ⋂ (i) (j), s i j ⊆ s i j :=
iInf₂_le i j
#align set.Inter₂_subset Set.iInter₂_subset
/-- This rather trivial consequence of `subset_iUnion`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem subset_iUnion_of_subset {s : Set α} {t : ι → Set α} (i : ι) (h : s ⊆ t i) : s ⊆ ⋃ i, t i :=
le_iSup_of_le i h
#align set.subset_Union_of_subset Set.subset_iUnion_of_subset
/-- This rather trivial consequence of `iInter_subset`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem iInter_subset_of_subset {s : ι → Set α} {t : Set α} (i : ι) (h : s i ⊆ t) :
⋂ i, s i ⊆ t :=
iInf_le_of_le i h
#align set.Inter_subset_of_subset Set.iInter_subset_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `subset_iUnion₂` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem subset_iUnion₂_of_subset {s : Set α} {t : ∀ i, κ i → Set α} (i : ι) (j : κ i)
(h : s ⊆ t i j) : s ⊆ ⋃ (i) (j), t i j :=
le_iSup₂_of_le i j h
#align set.subset_Union₂_of_subset Set.subset_iUnion₂_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `iInter₂_subset` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem iInter₂_subset_of_subset {s : ∀ i, κ i → Set α} {t : Set α} (i : ι) (j : κ i)
(h : s i j ⊆ t) : ⋂ (i) (j), s i j ⊆ t :=
iInf₂_le_of_le i j h
#align set.Inter₂_subset_of_subset Set.iInter₂_subset_of_subset
theorem iUnion_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono h
#align set.Union_mono Set.iUnion_mono
@[gcongr]
theorem iUnion_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iUnion s ⊆ iUnion t :=
iSup_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋃ (i) (j), s i j ⊆ ⋃ (i) (j), t i j :=
iSup₂_mono h
#align set.Union₂_mono Set.iUnion₂_mono
theorem iInter_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋂ i, s i ⊆ ⋂ i, t i :=
iInf_mono h
#align set.Inter_mono Set.iInter_mono
@[gcongr]
theorem iInter_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iInter s ⊆ iInter t :=
iInf_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋂ (i) (j), s i j ⊆ ⋂ (i) (j), t i j :=
iInf₂_mono h
#align set.Inter₂_mono Set.iInter₂_mono
theorem iUnion_mono' {s : ι → Set α} {t : ι₂ → Set α} (h : ∀ i, ∃ j, s i ⊆ t j) :
⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono' h
#align set.Union_mono' Set.iUnion_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iUnion₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i j, ∃ i' j', s i j ⊆ t i' j') : ⋃ (i) (j), s i j ⊆ ⋃ (i') (j'), t i' j' :=
iSup₂_mono' h
#align set.Union₂_mono' Set.iUnion₂_mono'
theorem iInter_mono' {s : ι → Set α} {t : ι' → Set α} (h : ∀ j, ∃ i, s i ⊆ t j) :
⋂ i, s i ⊆ ⋂ j, t j :=
Set.subset_iInter fun j =>
let ⟨i, hi⟩ := h j
iInter_subset_of_subset i hi
#align set.Inter_mono' Set.iInter_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iInter₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i' j', ∃ i j, s i j ⊆ t i' j') : ⋂ (i) (j), s i j ⊆ ⋂ (i') (j'), t i' j' :=
subset_iInter₂_iff.2 fun i' j' =>
let ⟨_, _, hst⟩ := h i' j'
(iInter₂_subset _ _).trans hst
#align set.Inter₂_mono' Set.iInter₂_mono'
theorem iUnion₂_subset_iUnion (κ : ι → Sort*) (s : ι → Set α) :
⋃ (i) (_ : κ i), s i ⊆ ⋃ i, s i :=
iUnion_mono fun _ => iUnion_subset fun _ => Subset.rfl
#align set.Union₂_subset_Union Set.iUnion₂_subset_iUnion
theorem iInter_subset_iInter₂ (κ : ι → Sort*) (s : ι → Set α) :
⋂ i, s i ⊆ ⋂ (i) (_ : κ i), s i :=
iInter_mono fun _ => subset_iInter fun _ => Subset.rfl
#align set.Inter_subset_Inter₂ Set.iInter_subset_iInter₂
theorem iUnion_setOf (P : ι → α → Prop) : ⋃ i, { x : α | P i x } = { x : α | ∃ i, P i x } := by
ext
exact mem_iUnion
#align set.Union_set_of Set.iUnion_setOf
theorem iInter_setOf (P : ι → α → Prop) : ⋂ i, { x : α | P i x } = { x : α | ∀ i, P i x } := by
ext
exact mem_iInter
#align set.Inter_set_of Set.iInter_setOf
theorem iUnion_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋃ x, f x = ⋃ y, g y :=
h1.iSup_congr h h2
#align set.Union_congr_of_surjective Set.iUnion_congr_of_surjective
theorem iInter_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋂ x, f x = ⋂ y, g y :=
h1.iInf_congr h h2
#align set.Inter_congr_of_surjective Set.iInter_congr_of_surjective
lemma iUnion_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋃ i, s i = ⋃ i, t i := iSup_congr h
#align set.Union_congr Set.iUnion_congr
lemma iInter_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋂ i, s i = ⋂ i, t i := iInf_congr h
#align set.Inter_congr Set.iInter_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iUnion₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋃ (i) (j), s i j = ⋃ (i) (j), t i j :=
iUnion_congr fun i => iUnion_congr <| h i
#align set.Union₂_congr Set.iUnion₂_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iInter₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋂ (i) (j), s i j = ⋂ (i) (j), t i j :=
iInter_congr fun i => iInter_congr <| h i
#align set.Inter₂_congr Set.iInter₂_congr
section Nonempty
variable [Nonempty ι] {f : ι → Set α} {s : Set α}
lemma iUnion_const (s : Set β) : ⋃ _ : ι, s = s := iSup_const
#align set.Union_const Set.iUnion_const
lemma iInter_const (s : Set β) : ⋂ _ : ι, s = s := iInf_const
#align set.Inter_const Set.iInter_const
lemma iUnion_eq_const (hf : ∀ i, f i = s) : ⋃ i, f i = s :=
(iUnion_congr hf).trans $ iUnion_const _
#align set.Union_eq_const Set.iUnion_eq_const
lemma iInter_eq_const (hf : ∀ i, f i = s) : ⋂ i, f i = s :=
(iInter_congr hf).trans $ iInter_const _
#align set.Inter_eq_const Set.iInter_eq_const
end Nonempty
@[simp]
theorem compl_iUnion (s : ι → Set β) : (⋃ i, s i)ᶜ = ⋂ i, (s i)ᶜ :=
compl_iSup
#align set.compl_Union Set.compl_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iUnion₂ (s : ∀ i, κ i → Set α) : (⋃ (i) (j), s i j)ᶜ = ⋂ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iUnion]
#align set.compl_Union₂ Set.compl_iUnion₂
@[simp]
theorem compl_iInter (s : ι → Set β) : (⋂ i, s i)ᶜ = ⋃ i, (s i)ᶜ :=
compl_iInf
#align set.compl_Inter Set.compl_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iInter₂ (s : ∀ i, κ i → Set α) : (⋂ (i) (j), s i j)ᶜ = ⋃ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iInter]
#align set.compl_Inter₂ Set.compl_iInter₂
-- classical -- complete_boolean_algebra
theorem iUnion_eq_compl_iInter_compl (s : ι → Set β) : ⋃ i, s i = (⋂ i, (s i)ᶜ)ᶜ := by
simp only [compl_iInter, compl_compl]
#align set.Union_eq_compl_Inter_compl Set.iUnion_eq_compl_iInter_compl
-- classical -- complete_boolean_algebra
theorem iInter_eq_compl_iUnion_compl (s : ι → Set β) : ⋂ i, s i = (⋃ i, (s i)ᶜ)ᶜ := by
simp only [compl_iUnion, compl_compl]
#align set.Inter_eq_compl_Union_compl Set.iInter_eq_compl_iUnion_compl
theorem inter_iUnion (s : Set β) (t : ι → Set β) : (s ∩ ⋃ i, t i) = ⋃ i, s ∩ t i :=
inf_iSup_eq _ _
#align set.inter_Union Set.inter_iUnion
theorem iUnion_inter (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∩ s = ⋃ i, t i ∩ s :=
iSup_inf_eq _ _
#align set.Union_inter Set.iUnion_inter
theorem iUnion_union_distrib (s : ι → Set β) (t : ι → Set β) :
⋃ i, s i ∪ t i = (⋃ i, s i) ∪ ⋃ i, t i :=
iSup_sup_eq
#align set.Union_union_distrib Set.iUnion_union_distrib
theorem iInter_inter_distrib (s : ι → Set β) (t : ι → Set β) :
⋂ i, s i ∩ t i = (⋂ i, s i) ∩ ⋂ i, t i :=
iInf_inf_eq
#align set.Inter_inter_distrib Set.iInter_inter_distrib
theorem union_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∪ ⋃ i, t i) = ⋃ i, s ∪ t i :=
sup_iSup
#align set.union_Union Set.union_iUnion
theorem iUnion_union [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∪ s = ⋃ i, t i ∪ s :=
iSup_sup
#align set.Union_union Set.iUnion_union
theorem inter_iInter [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∩ ⋂ i, t i) = ⋂ i, s ∩ t i :=
inf_iInf
#align set.inter_Inter Set.inter_iInter
theorem iInter_inter [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋂ i, t i) ∩ s = ⋂ i, t i ∩ s :=
iInf_inf
#align set.Inter_inter Set.iInter_inter
-- classical
theorem union_iInter (s : Set β) (t : ι → Set β) : (s ∪ ⋂ i, t i) = ⋂ i, s ∪ t i :=
sup_iInf_eq _ _
#align set.union_Inter Set.union_iInter
theorem iInter_union (s : ι → Set β) (t : Set β) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.Inter_union Set.iInter_union
theorem iUnion_diff (s : Set β) (t : ι → Set β) : (⋃ i, t i) \ s = ⋃ i, t i \ s :=
iUnion_inter _ _
#align set.Union_diff Set.iUnion_diff
theorem diff_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s \ ⋃ i, t i) = ⋂ i, s \ t i := by
rw [diff_eq, compl_iUnion, inter_iInter]; rfl
#align set.diff_Union Set.diff_iUnion
theorem diff_iInter (s : Set β) (t : ι → Set β) : (s \ ⋂ i, t i) = ⋃ i, s \ t i := by
rw [diff_eq, compl_iInter, inter_iUnion]; rfl
#align set.diff_Inter Set.diff_iInter
theorem directed_on_iUnion {r} {f : ι → Set α} (hd : Directed (· ⊆ ·) f)
(h : ∀ x, DirectedOn r (f x)) : DirectedOn r (⋃ x, f x) := by
simp only [DirectedOn, exists_prop, mem_iUnion, exists_imp]
exact fun a₁ b₁ fb₁ a₂ b₂ fb₂ =>
let ⟨z, zb₁, zb₂⟩ := hd b₁ b₂
let ⟨x, xf, xa₁, xa₂⟩ := h z a₁ (zb₁ fb₁) a₂ (zb₂ fb₂)
⟨x, ⟨z, xf⟩, xa₁, xa₂⟩
#align set.directed_on_Union Set.directed_on_iUnion
theorem iUnion_inter_subset {ι α} {s t : ι → Set α} : ⋃ i, s i ∩ t i ⊆ (⋃ i, s i) ∩ ⋃ i, t i :=
le_iSup_inf_iSup s t
#align set.Union_inter_subset Set.iUnion_inter_subset
theorem iUnion_inter_of_monotone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_monotone hs ht
#align set.Union_inter_of_monotone Set.iUnion_inter_of_monotone
theorem iUnion_inter_of_antitone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_antitone hs ht
#align set.Union_inter_of_antitone Set.iUnion_inter_of_antitone
theorem iInter_union_of_monotone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_monotone hs ht
#align set.Inter_union_of_monotone Set.iInter_union_of_monotone
theorem iInter_union_of_antitone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_antitone hs ht
#align set.Inter_union_of_antitone Set.iInter_union_of_antitone
/-- An equality version of this lemma is `iUnion_iInter_of_monotone` in `Data.Set.Finite`. -/
theorem iUnion_iInter_subset {s : ι → ι' → Set α} : (⋃ j, ⋂ i, s i j) ⊆ ⋂ i, ⋃ j, s i j :=
iSup_iInf_le_iInf_iSup (flip s)
#align set.Union_Inter_subset Set.iUnion_iInter_subset
theorem iUnion_option {ι} (s : Option ι → Set α) : ⋃ o, s o = s none ∪ ⋃ i, s (some i) :=
iSup_option s
#align set.Union_option Set.iUnion_option
theorem iInter_option {ι} (s : Option ι → Set α) : ⋂ o, s o = s none ∩ ⋂ i, s (some i) :=
iInf_option s
#align set.Inter_option Set.iInter_option
section
variable (p : ι → Prop) [DecidablePred p]
theorem iUnion_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 _ _ _
#align set.Union_dite Set.iUnion_dite
theorem iUnion_ite (f g : ι → Set α) :
⋃ i, (if p i then f i else g i) = (⋃ (i) (_ : p i), f i) ∪ ⋃ (i) (_ : ¬p i), g i :=
iUnion_dite _ _ _
#align set.Union_ite Set.iUnion_ite
theorem iInter_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 :=
iInf_dite _ _ _
#align set.Inter_dite Set.iInter_dite
theorem iInter_ite (f g : ι → Set α) :
⋂ i, (if p i then f i else g i) = (⋂ (i) (_ : p i), f i) ∩ ⋂ (i) (_ : ¬p i), g i :=
iInter_dite _ _ _
#align set.Inter_ite Set.iInter_ite
end
theorem image_projection_prod {ι : Type*} {α : ι → Type*} {v : ∀ i : ι, Set (α i)}
(hv : (pi univ v).Nonempty) (i : ι) :
((fun x : ∀ i : ι, α i => x i) '' ⋂ k, (fun x : ∀ j : ι, α j => x k) ⁻¹' v k) = v i := by
classical
apply Subset.antisymm
· simp [iInter_subset]
· intro y y_in
simp only [mem_image, mem_iInter, mem_preimage]
rcases hv with ⟨z, hz⟩
refine' ⟨Function.update z i y, _, update_same i y z⟩
rw [@forall_update_iff ι α _ z i y fun i t => t ∈ v i]
exact ⟨y_in, fun j _ => by simpa using hz j⟩
#align set.image_projection_prod Set.image_projection_prod
/-! ### Unions and intersections indexed by `Prop` -/
theorem iInter_false {s : False → Set α} : iInter s = univ :=
iInf_false
#align set.Inter_false Set.iInter_false
theorem iUnion_false {s : False → Set α} : iUnion s = ∅ :=
iSup_false
#align set.Union_false Set.iUnion_false
@[simp]
theorem iInter_true {s : True → Set α} : iInter s = s trivial :=
iInf_true
#align set.Inter_true Set.iInter_true
@[simp]
theorem iUnion_true {s : True → Set α} : iUnion s = s trivial :=
iSup_true
#align set.Union_true Set.iUnion_true
@[simp]
theorem iInter_exists {p : ι → Prop} {f : Exists p → Set α} :
⋂ x, f x = ⋂ (i) (h : p i), f ⟨i, h⟩ :=
iInf_exists
#align set.Inter_exists Set.iInter_exists
@[simp]
theorem iUnion_exists {p : ι → Prop} {f : Exists p → Set α} :
⋃ x, f x = ⋃ (i) (h : p i), f ⟨i, h⟩ :=
iSup_exists
#align set.Union_exists Set.iUnion_exists
@[simp]
theorem iUnion_empty : (⋃ _ : ι, ∅ : Set α) = ∅ :=
iSup_bot
#align set.Union_empty Set.iUnion_empty
@[simp]
theorem iInter_univ : (⋂ _ : ι, univ : Set α) = univ :=
iInf_top
#align set.Inter_univ Set.iInter_univ
section
variable {s : ι → Set α}
@[simp]
theorem iUnion_eq_empty : ⋃ i, s i = ∅ ↔ ∀ i, s i = ∅ :=
iSup_eq_bot
#align set.Union_eq_empty Set.iUnion_eq_empty
@[simp]
theorem iInter_eq_univ : ⋂ i, s i = univ ↔ ∀ i, s i = univ :=
iInf_eq_top
#align set.Inter_eq_univ Set.iInter_eq_univ
@[simp]
theorem nonempty_iUnion : (⋃ i, s i).Nonempty ↔ ∃ i, (s i).Nonempty := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_Union Set.nonempty_iUnion
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_biUnion {t : Set α} {s : α → Set β} :
(⋃ i ∈ t, s i).Nonempty ↔ ∃ i ∈ t, (s i).Nonempty := by simp
#align set.nonempty_bUnion Set.nonempty_biUnion
theorem iUnion_nonempty_index (s : Set α) (t : s.Nonempty → Set β) :
⋃ h, t h = ⋃ x ∈ s, t ⟨x, ‹_›⟩ :=
iSup_exists
#align set.Union_nonempty_index Set.iUnion_nonempty_index
end
@[simp]
theorem iInter_iInter_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋂ (x) (h : x = b), s x h = s b rfl :=
iInf_iInf_eq_left
#align set.Inter_Inter_eq_left Set.iInter_iInter_eq_left
@[simp]
theorem iInter_iInter_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋂ (x) (h : b = x), s x h = s b rfl :=
iInf_iInf_eq_right
#align set.Inter_Inter_eq_right Set.iInter_iInter_eq_right
@[simp]
theorem iUnion_iUnion_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋃ (x) (h : x = b), s x h = s b rfl :=
iSup_iSup_eq_left
#align set.Union_Union_eq_left Set.iUnion_iUnion_eq_left
@[simp]
theorem iUnion_iUnion_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋃ (x) (h : b = x), s x h = s b rfl :=
iSup_iSup_eq_right
#align set.Union_Union_eq_right Set.iUnion_iUnion_eq_right
theorem iInter_or {p q : Prop} (s : p ∨ q → Set α) :
⋂ h, s h = (⋂ h : p, s (Or.inl h)) ∩ ⋂ h : q, s (Or.inr h) :=
iInf_or
#align set.Inter_or Set.iInter_or
theorem iUnion_or {p q : Prop} (s : p ∨ q → Set α) :
⋃ h, s h = (⋃ i, s (Or.inl i)) ∪ ⋃ j, s (Or.inr j) :=
iSup_or
#align set.Union_or Set.iUnion_or
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iUnion_and {p q : Prop} (s : p ∧ q → Set α) : ⋃ h, s h = ⋃ (hp) (hq), s ⟨hp, hq⟩ :=
iSup_and
#align set.Union_and Set.iUnion_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iInter_and {p q : Prop} (s : p ∧ q → Set α) : ⋂ h, s h = ⋂ (hp) (hq), s ⟨hp, hq⟩ :=
iInf_and
#align set.Inter_and Set.iInter_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iUnion_comm (s : ι → ι' → Set α) : ⋃ (i) (i'), s i i' = ⋃ (i') (i), s i i' :=
iSup_comm
#align set.Union_comm Set.iUnion_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iInter_comm (s : ι → ι' → Set α) : ⋂ (i) (i'), s i i' = ⋂ (i') (i), s i i' :=
iInf_comm
#align set.Inter_comm Set.iInter_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iUnion₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋃ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋃ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iSup₂_comm _
#align set.Union₂_comm Set.iUnion₂_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iInter₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋂ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋂ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iInf₂_comm _
#align set.Inter₂_comm Set.iInter₂_comm
@[simp]
theorem biUnion_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋃ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι']
#align set.bUnion_and Set.biUnion_and
@[simp]
theorem biUnion_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋃ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι]
#align set.bUnion_and' Set.biUnion_and'
@[simp]
theorem biInter_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋂ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iInter_and, @iInter_comm _ ι']
#align set.bInter_and Set.biInter_and
@[simp]
theorem biInter_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋂ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iInter_and, @iInter_comm _ ι]
#align set.bInter_and' Set.biInter_and'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iUnion_iUnion_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋃ (x) (h), s x h = s b (Or.inl rfl) ∪ ⋃ (x) (h : p x), s x (Or.inr h) := by
simp only [iUnion_or, iUnion_union_distrib, iUnion_iUnion_eq_left]
#align set.Union_Union_eq_or_left Set.iUnion_iUnion_eq_or_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iInter_iInter_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋂ (x) (h), s x h = s b (Or.inl rfl) ∩ ⋂ (x) (h : p x), s x (Or.inr h) := by
simp only [iInter_or, iInter_inter_distrib, iInter_iInter_eq_left]
#align set.Inter_Inter_eq_or_left Set.iInter_iInter_eq_or_left
/-! ### Bounded unions and intersections -/
/-- A specialization of `mem_iUnion₂`. -/
theorem mem_biUnion {s : Set α} {t : α → Set β} {x : α} {y : β} (xs : x ∈ s) (ytx : y ∈ t x) :
y ∈ ⋃ x ∈ s, t x :=
mem_iUnion₂_of_mem xs ytx
#align set.mem_bUnion Set.mem_biUnion
/-- A specialization of `mem_iInter₂`. -/
theorem mem_biInter {s : Set α} {t : α → Set β} {y : β} (h : ∀ x ∈ s, y ∈ t x) :
y ∈ ⋂ x ∈ s, t x :=
mem_iInter₂_of_mem h
#align set.mem_bInter Set.mem_biInter
/-- A specialization of `subset_iUnion₂`. -/
theorem subset_biUnion_of_mem {s : Set α} {u : α → Set β} {x : α} (xs : x ∈ s) :
u x ⊆ ⋃ x ∈ s, u x :=
--Porting note: Why is this not just `subset_iUnion₂ x xs`?
@subset_iUnion₂ β α (· ∈ s) (fun i _ => u i) x xs
#align set.subset_bUnion_of_mem Set.subset_biUnion_of_mem
/-- A specialization of `iInter₂_subset`. -/
theorem biInter_subset_of_mem {s : Set α} {t : α → Set β} {x : α} (xs : x ∈ s) :
⋂ x ∈ s, t x ⊆ t x :=
iInter₂_subset x xs
#align set.bInter_subset_of_mem Set.biInter_subset_of_mem
theorem biUnion_subset_biUnion_left {s s' : Set α} {t : α → Set β} (h : s ⊆ s') :
⋃ x ∈ s, t x ⊆ ⋃ x ∈ s', t x :=
iUnion₂_subset fun _ hx => subset_biUnion_of_mem <| h hx
#align set.bUnion_subset_bUnion_left Set.biUnion_subset_biUnion_left
theorem biInter_subset_biInter_left {s s' : Set α} {t : α → Set β} (h : s' ⊆ s) :
⋂ x ∈ s, t x ⊆ ⋂ x ∈ s', t x :=
subset_iInter₂ fun _ hx => biInter_subset_of_mem <| h hx
#align set.bInter_subset_bInter_left Set.biInter_subset_biInter_left
theorem biUnion_mono {s s' : Set α} {t t' : α → Set β} (hs : s' ⊆ s) (h : ∀ x ∈ s, t x ⊆ t' x) :
⋃ x ∈ s', t x ⊆ ⋃ x ∈ s, t' x :=
(biUnion_subset_biUnion_left hs).trans <| iUnion₂_mono h
#align set.bUnion_mono Set.biUnion_mono
theorem biInter_mono {s s' : Set α} {t t' : α → Set β} (hs : s ⊆ s') (h : ∀ x ∈ s, t x ⊆ t' x) :
⋂ x ∈ s', t x ⊆ ⋂ x ∈ s, t' x :=
(biInter_subset_biInter_left hs).trans <| iInter₂_mono h
#align set.bInter_mono Set.biInter_mono
theorem biUnion_eq_iUnion (s : Set α) (t : ∀ x ∈ s, Set β) :
⋃ x ∈ s, t x ‹_› = ⋃ x : s, t x x.2 :=
iSup_subtype'
#align set.bUnion_eq_Union Set.biUnion_eq_iUnion
theorem biInter_eq_iInter (s : Set α) (t : ∀ x ∈ s, Set β) :
⋂ x ∈ s, t x ‹_› = ⋂ x : s, t x x.2 :=
iInf_subtype'
#align set.bInter_eq_Inter Set.biInter_eq_iInter
theorem iUnion_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋃ x : { x // p x }, s x = ⋃ (x) (hx : p x), s ⟨x, hx⟩ :=
iSup_subtype
#align set.Union_subtype Set.iUnion_subtype
theorem iInter_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋂ x : { x // p x }, s x = ⋂ (x) (hx : p x), s ⟨x, hx⟩ :=
iInf_subtype
#align set.Inter_subtype Set.iInter_subtype
theorem biInter_empty (u : α → Set β) : ⋂ x ∈ (∅ : Set α), u x = univ :=
iInf_emptyset
#align set.bInter_empty Set.biInter_empty
theorem biInter_univ (u : α → Set β) : ⋂ x ∈ @univ α, u x = ⋂ x, u x :=
iInf_univ
#align set.bInter_univ Set.biInter_univ
@[simp]
theorem biUnion_self (s : Set α) : ⋃ x ∈ s, s = s :=
Subset.antisymm (iUnion₂_subset fun _ _ => Subset.refl s) fun _ hx => mem_biUnion hx hx
#align set.bUnion_self Set.biUnion_self
@[simp]
theorem iUnion_nonempty_self (s : Set α) : ⋃ _ : s.Nonempty, s = s := by
rw [iUnion_nonempty_index, biUnion_self]
#align set.Union_nonempty_self Set.iUnion_nonempty_self
theorem biInter_singleton (a : α) (s : α → Set β) : ⋂ x ∈ ({a} : Set α), s x = s a :=
iInf_singleton
#align set.bInter_singleton Set.biInter_singleton
theorem biInter_union (s t : Set α) (u : α → Set β) :
⋂ x ∈ s ∪ t, u x = (⋂ x ∈ s, u x) ∩ ⋂ x ∈ t, u x :=
iInf_union
#align set.bInter_union Set.biInter_union
theorem biInter_insert (a : α) (s : Set α) (t : α → Set β) :
⋂ x ∈ insert a s, t x = t a ∩ ⋂ x ∈ s, t x := by simp
#align set.bInter_insert Set.biInter_insert
theorem biInter_pair (a b : α) (s : α → Set β) : ⋂ x ∈ ({a, b} : Set α), s x = s a ∩ s b := by
rw [biInter_insert, biInter_singleton]
#align set.bInter_pair Set.biInter_pair
theorem biInter_inter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, f i ∩ t = (⋂ i ∈ s, f i) ∩ t := by
haveI : Nonempty s := hs.to_subtype
simp [biInter_eq_iInter, ← iInter_inter]
#align set.bInter_inter Set.biInter_inter
theorem inter_biInter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, t ∩ f i = t ∩ ⋂ i ∈ s, f i := by
rw [inter_comm, ← biInter_inter hs]
simp [inter_comm]
#align set.inter_bInter Set.inter_biInter
theorem biUnion_empty (s : α → Set β) : ⋃ x ∈ (∅ : Set α), s x = ∅ :=
iSup_emptyset
#align set.bUnion_empty Set.biUnion_empty
theorem biUnion_univ (s : α → Set β) : ⋃ x ∈ @univ α, s x = ⋃ x, s x :=
iSup_univ
#align set.bUnion_univ Set.biUnion_univ
theorem biUnion_singleton (a : α) (s : α → Set β) : ⋃ x ∈ ({a} : Set α), s x = s a :=
iSup_singleton
#align set.bUnion_singleton Set.biUnion_singleton
@[simp]
theorem biUnion_of_singleton (s : Set α) : ⋃ x ∈ s, {x} = s :=
ext <| by simp
#align set.bUnion_of_singleton Set.biUnion_of_singleton
theorem biUnion_union (s t : Set α) (u : α → Set β) :
⋃ x ∈ s ∪ t, u x = (⋃ x ∈ s, u x) ∪ ⋃ x ∈ t, u x :=
iSup_union
#align set.bUnion_union Set.biUnion_union
@[simp]
theorem iUnion_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋃ i, f i = ⋃ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iUnion_subtype _ _
#align set.Union_coe_set Set.iUnion_coe_set
@[simp]
theorem iInter_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋂ i, f i = ⋂ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iInter_subtype _ _
#align set.Inter_coe_set Set.iInter_coe_set
theorem biUnion_insert (a : α) (s : Set α) (t : α → Set β) :
⋃ x ∈ insert a s, t x = t a ∪ ⋃ x ∈ s, t x := by simp
#align set.bUnion_insert Set.biUnion_insert
theorem biUnion_pair (a b : α) (s : α → Set β) : ⋃ x ∈ ({a, b} : Set α), s x = s a ∪ s b :=
by simp
#align set.bUnion_pair Set.biUnion_pair
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem inter_iUnion₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∩ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ∩ t i j := by simp only [inter_iUnion]
#align set.inter_Union₂ Set.inter_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_inter (s : ∀ i, κ i → Set α) (t : Set α) :
(⋃ (i) (j), s i j) ∩ t = ⋃ (i) (j), s i j ∩ t := by simp_rw [iUnion_inter]
#align set.Union₂_inter Set.iUnion₂_inter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_iInter₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_iInter]
#align set.union_Inter₂ Set.union_iInter₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_union (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [iInter_union]
#align set.Inter₂_union Set.iInter₂_union
theorem mem_sUnion_of_mem {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∈ t) (ht : t ∈ S) :
x ∈ ⋃₀S :=
⟨t, ht, hx⟩
#align set.mem_sUnion_of_mem Set.mem_sUnion_of_mem
-- is this theorem really necessary?
theorem not_mem_of_not_mem_sUnion {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∉ ⋃₀S)
(ht : t ∈ S) : x ∉ t := fun h => hx ⟨t, ht, h⟩
#align set.not_mem_of_not_mem_sUnion Set.not_mem_of_not_mem_sUnion
theorem sInter_subset_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : ⋂₀ S ⊆ t :=
sInf_le tS
#align set.sInter_subset_of_mem Set.sInter_subset_of_mem
theorem subset_sUnion_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : t ⊆ ⋃₀S :=
le_sSup tS
#align set.subset_sUnion_of_mem Set.subset_sUnion_of_mem
theorem subset_sUnion_of_subset {s : Set α} (t : Set (Set α)) (u : Set α) (h₁ : s ⊆ u)
(h₂ : u ∈ t) : s ⊆ ⋃₀t :=
Subset.trans h₁ (subset_sUnion_of_mem h₂)
#align set.subset_sUnion_of_subset Set.subset_sUnion_of_subset
theorem sUnion_subset {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t' ⊆ t) : ⋃₀S ⊆ t :=
sSup_le h
#align set.sUnion_subset Set.sUnion_subset
@[simp]
theorem sUnion_subset_iff {s : Set (Set α)} {t : Set α} : ⋃₀s ⊆ t ↔ ∀ t' ∈ s, t' ⊆ t :=
sSup_le_iff
#align set.sUnion_subset_iff Set.sUnion_subset_iff
/-- `sUnion` is monotone under taking a subset of each set. -/
lemma sUnion_mono_subsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, t ⊆ f t) :
⋃₀ s ⊆ ⋃₀ (f '' s) :=
fun _ ⟨t, htx, hxt⟩ ↦ ⟨f t, mem_image_of_mem f htx, hf t hxt⟩
/-- `sUnion` is monotone under taking a superset of each set. -/
lemma sUnion_mono_supsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, f t ⊆ t) :
⋃₀ (f '' s) ⊆ ⋃₀ s :=
-- If t ∈ f '' s is arbitrary; t = f u for some u : Set α.
fun _ ⟨_, ⟨u, hus, hut⟩, hxt⟩ ↦ ⟨u, hus, (hut ▸ hf u) hxt⟩
theorem subset_sInter {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t ⊆ t') : t ⊆ ⋂₀ S :=
le_sInf h
#align set.subset_sInter Set.subset_sInter
@[simp]
theorem subset_sInter_iff {S : Set (Set α)} {t : Set α} : t ⊆ ⋂₀ S ↔ ∀ t' ∈ S, t ⊆ t' :=
le_sInf_iff
#align set.subset_sInter_iff Set.subset_sInter_iff
@[gcongr]
theorem sUnion_subset_sUnion {S T : Set (Set α)} (h : S ⊆ T) : ⋃₀S ⊆ ⋃₀T :=
sUnion_subset fun _ hs => subset_sUnion_of_mem (h hs)
#align set.sUnion_subset_sUnion Set.sUnion_subset_sUnion
@[gcongr]
theorem sInter_subset_sInter {S T : Set (Set α)} (h : S ⊆ T) : ⋂₀ T ⊆ ⋂₀ S :=
subset_sInter fun _ hs => sInter_subset_of_mem (h hs)
#align set.sInter_subset_sInter Set.sInter_subset_sInter
@[simp]
theorem sUnion_empty : ⋃₀∅ = (∅ : Set α) :=
sSup_empty
#align set.sUnion_empty Set.sUnion_empty
@[simp]
theorem sInter_empty : ⋂₀ ∅ = (univ : Set α) :=
sInf_empty
#align set.sInter_empty Set.sInter_empty
@[simp]
theorem sUnion_singleton (s : Set α) : ⋃₀{s} = s :=
sSup_singleton
#align set.sUnion_singleton Set.sUnion_singleton
@[simp]
theorem sInter_singleton (s : Set α) : ⋂₀ {s} = s :=
sInf_singleton
#align set.sInter_singleton Set.sInter_singleton
@[simp]
theorem sUnion_eq_empty {S : Set (Set α)} : ⋃₀S = ∅ ↔ ∀ s ∈ S, s = ∅ :=
sSup_eq_bot
#align set.sUnion_eq_empty Set.sUnion_eq_empty
@[simp]
theorem sInter_eq_univ {S : Set (Set α)} : ⋂₀ S = univ ↔ ∀ s ∈ S, s = univ :=
sInf_eq_top
#align set.sInter_eq_univ Set.sInter_eq_univ
theorem subset_powerset_iff {s : Set (Set α)} {t : Set α} : s ⊆ 𝒫 t ↔ ⋃₀ s ⊆ t :=
sUnion_subset_iff.symm
/-- `⋃₀` and `𝒫` form a Galois connection. -/
theorem sUnion_powerset_gc :
GaloisConnection (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gc_sSup_Iic
/-- `⋃₀` and `𝒫` form a Galois insertion. -/
def sUnion_powerset_gi :
GaloisInsertion (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gi_sSup_Iic
/-- If all sets in a collection are either `∅` or `Set.univ`, then so is their union. -/
theorem sUnion_mem_empty_univ {S : Set (Set α)} (h : S ⊆ {∅, univ}) :
⋃₀ S ∈ ({∅, univ} : Set (Set α)) := by
simp only [mem_insert_iff, mem_singleton_iff, or_iff_not_imp_left, sUnion_eq_empty, not_forall]
rintro ⟨s, hs, hne⟩
obtain rfl : s = univ := (h hs).resolve_left hne
exact univ_subset_iff.1 <| subset_sUnion_of_mem hs
@[simp]
theorem nonempty_sUnion {S : Set (Set α)} : (⋃₀S).Nonempty ↔ ∃ s ∈ S, Set.Nonempty s := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_sUnion Set.nonempty_sUnion
theorem Nonempty.of_sUnion {s : Set (Set α)} (h : (⋃₀s).Nonempty) : s.Nonempty :=
let ⟨s, hs, _⟩ := nonempty_sUnion.1 h
⟨s, hs⟩
#align set.nonempty.of_sUnion Set.Nonempty.of_sUnion
theorem Nonempty.of_sUnion_eq_univ [Nonempty α] {s : Set (Set α)} (h : ⋃₀s = univ) : s.Nonempty :=
Nonempty.of_sUnion <| h.symm ▸ univ_nonempty
#align set.nonempty.of_sUnion_eq_univ Set.Nonempty.of_sUnion_eq_univ
theorem sUnion_union (S T : Set (Set α)) : ⋃₀(S ∪ T) = ⋃₀S ∪ ⋃₀T :=
sSup_union
#align set.sUnion_union Set.sUnion_union
theorem sInter_union (S T : Set (Set α)) : ⋂₀ (S ∪ T) = ⋂₀ S ∩ ⋂₀ T :=
sInf_union
#align set.sInter_union Set.sInter_union
@[simp]
theorem sUnion_insert (s : Set α) (T : Set (Set α)) : ⋃₀insert s T = s ∪ ⋃₀T :=
sSup_insert
#align set.sUnion_insert Set.sUnion_insert
@[simp]
theorem sInter_insert (s : Set α) (T : Set (Set α)) : ⋂₀ insert s T = s ∩ ⋂₀ T :=
sInf_insert
#align set.sInter_insert Set.sInter_insert
@[simp]
theorem sUnion_diff_singleton_empty (s : Set (Set α)) : ⋃₀(s \ {∅}) = ⋃₀s :=
sSup_diff_singleton_bot s
#align set.sUnion_diff_singleton_empty Set.sUnion_diff_singleton_empty
@[simp]
theorem sInter_diff_singleton_univ (s : Set (Set α)) : ⋂₀ (s \ {univ}) = ⋂₀ s :=
sInf_diff_singleton_top s
#align set.sInter_diff_singleton_univ Set.sInter_diff_singleton_univ
theorem sUnion_pair (s t : Set α) : ⋃₀{s, t} = s ∪ t :=
sSup_pair
#align set.sUnion_pair Set.sUnion_pair
theorem sInter_pair (s t : Set α) : ⋂₀ {s, t} = s ∩ t :=
sInf_pair
#align set.sInter_pair Set.sInter_pair
@[simp]
theorem sUnion_image (f : α → Set β) (s : Set α) : ⋃₀(f '' s) = ⋃ x ∈ s, f x :=
sSup_image
#align set.sUnion_image Set.sUnion_image
@[simp]
theorem sInter_image (f : α → Set β) (s : Set α) : ⋂₀ (f '' s) = ⋂ x ∈ s, f x :=
sInf_image
#align set.sInter_image Set.sInter_image
@[simp]
theorem sUnion_range (f : ι → Set β) : ⋃₀range f = ⋃ x, f x :=
rfl
#align set.sUnion_range Set.sUnion_range
@[simp]
theorem sInter_range (f : ι → Set β) : ⋂₀ range f = ⋂ x, f x :=
rfl
#align set.sInter_range Set.sInter_range
theorem iUnion_eq_univ_iff {f : ι → Set α} : ⋃ i, f i = univ ↔ ∀ x, ∃ i, x ∈ f i := by
simp only [eq_univ_iff_forall, mem_iUnion]
#align set.Union_eq_univ_iff Set.iUnion_eq_univ_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_eq_univ_iff {s : ∀ i, κ i → Set α} :
⋃ (i) (j), s i j = univ ↔ ∀ a, ∃ i j, a ∈ s i j :=
by simp only [iUnion_eq_univ_iff, mem_iUnion]
#align set.Union₂_eq_univ_iff Set.iUnion₂_eq_univ_iff
theorem sUnion_eq_univ_iff {c : Set (Set α)} : ⋃₀c = univ ↔ ∀ a, ∃ b ∈ c, a ∈ b := by
simp only [eq_univ_iff_forall, mem_sUnion]
#align set.sUnion_eq_univ_iff Set.sUnion_eq_univ_iff
-- classical
theorem iInter_eq_empty_iff {f : ι → Set α} : ⋂ i, f i = ∅ ↔ ∀ x, ∃ i, x ∉ f i := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.Inter_eq_empty_iff Set.iInter_eq_empty_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
theorem iInter₂_eq_empty_iff {s : ∀ i, κ i → Set α} :
⋂ (i) (j), s i j = ∅ ↔ ∀ a, ∃ i j, a ∉ s i j := by
simp only [eq_empty_iff_forall_not_mem, mem_iInter, not_forall]
#align set.Inter₂_eq_empty_iff Set.iInter₂_eq_empty_iff
-- classical
theorem sInter_eq_empty_iff {c : Set (Set α)} : ⋂₀ c = ∅ ↔ ∀ a, ∃ b ∈ c, a ∉ b := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.sInter_eq_empty_iff Set.sInter_eq_empty_iff
-- classical
@[simp]
theorem nonempty_iInter {f : ι → Set α} : (⋂ i, f i).Nonempty ↔ ∃ x, ∀ i, x ∈ f i := by
simp [nonempty_iff_ne_empty, iInter_eq_empty_iff]
#align set.nonempty_Inter Set.nonempty_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_iInter₂ {s : ∀ i, κ i → Set α} :
(⋂ (i) (j), s i j).Nonempty ↔ ∃ a, ∀ i j, a ∈ s i j := by
simp
#align set.nonempty_Inter₂ Set.nonempty_iInter₂
-- classical
@[simp]
theorem nonempty_sInter {c : Set (Set α)} : (⋂₀ c).Nonempty ↔ ∃ a, ∀ b ∈ c, a ∈ b := by
simp [nonempty_iff_ne_empty, sInter_eq_empty_iff]
#align set.nonempty_sInter Set.nonempty_sInter
-- classical
theorem compl_sUnion (S : Set (Set α)) : (⋃₀S)ᶜ = ⋂₀ (compl '' S) :=
ext fun x => by simp
#align set.compl_sUnion Set.compl_sUnion
-- classical
theorem sUnion_eq_compl_sInter_compl (S : Set (Set α)) : ⋃₀S = (⋂₀ (compl '' S))ᶜ := by
rw [← compl_compl (⋃₀S), compl_sUnion]
#align set.sUnion_eq_compl_sInter_compl Set.sUnion_eq_compl_sInter_compl
-- classical
theorem compl_sInter (S : Set (Set α)) : (⋂₀ S)ᶜ = ⋃₀(compl '' S) := by
rw [sUnion_eq_compl_sInter_compl, compl_compl_image]
#align set.compl_sInter Set.compl_sInter
-- classical
theorem sInter_eq_compl_sUnion_compl (S : Set (Set α)) : ⋂₀ S = (⋃₀(compl '' S))ᶜ := by
rw [← compl_compl (⋂₀ S), compl_sInter]
#align set.sInter_eq_compl_sUnion_compl Set.sInter_eq_compl_sUnion_compl
theorem inter_empty_of_inter_sUnion_empty {s t : Set α} {S : Set (Set α)} (hs : t ∈ S)
(h : s ∩ ⋃₀S = ∅) : s ∩ t = ∅ :=
eq_empty_of_subset_empty <| by
rw [← h]; exact inter_subset_inter_right _ (subset_sUnion_of_mem hs)
#align set.inter_empty_of_inter_sUnion_empty Set.inter_empty_of_inter_sUnion_empty
theorem range_sigma_eq_iUnion_range {γ : α → Type*} (f : Sigma γ → β) :
range f = ⋃ a, range fun b => f ⟨a, b⟩ :=
Set.ext <| by simp
#align set.range_sigma_eq_Union_range Set.range_sigma_eq_iUnion_range
theorem iUnion_eq_range_sigma (s : α → Set β) : ⋃ i, s i = range fun a : Σi, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_sigma Set.iUnion_eq_range_sigma
theorem iUnion_eq_range_psigma (s : ι → Set β) : ⋃ i, s i = range fun a : Σ'i, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_psigma Set.iUnion_eq_range_psigma
theorem iUnion_image_preimage_sigma_mk_eq_self {ι : Type*} {σ : ι → Type*} (s : Set (Sigma σ)) :
⋃ i, Sigma.mk i '' (Sigma.mk i ⁻¹' s) = s := by
ext x
simp only [mem_iUnion, mem_image, mem_preimage]
constructor
· rintro ⟨i, a, h, rfl⟩
exact h
· intro h
cases' x with i a
exact ⟨i, a, h, rfl⟩
#align set.Union_image_preimage_sigma_mk_eq_self Set.iUnion_image_preimage_sigma_mk_eq_self
theorem Sigma.univ (X : α → Type*) : (Set.univ : Set (Σa, X a)) = ⋃ a, range (Sigma.mk a) :=
Set.ext fun x =>
iff_of_true trivial ⟨range (Sigma.mk x.1), Set.mem_range_self _, x.2, Sigma.eta x⟩
#align set.sigma.univ Set.Sigma.univ
alias sUnion_mono := sUnion_subset_sUnion
#align set.sUnion_mono Set.sUnion_mono
theorem iUnion_subset_iUnion_const {s : Set α} (h : ι → ι₂) : ⋃ _ : ι, s ⊆ ⋃ _ : ι₂, s :=
@iSup_const_mono (Set α) ι ι₂ _ s h
#align set.Union_subset_Union_const Set.iUnion_subset_iUnion_const
@[simp]
theorem iUnion_singleton_eq_range {α β : Type*} (f : α → β) : ⋃ x : α, {f x} = range f := by
ext x
simp [@eq_comm _ x]
#align set.Union_singleton_eq_range Set.iUnion_singleton_eq_range
theorem iUnion_of_singleton (α : Type*) : (⋃ x, {x} : Set α) = univ := by simp [Set.ext_iff]
#align set.Union_of_singleton Set.iUnion_of_singleton
theorem iUnion_of_singleton_coe (s : Set α) : ⋃ i : s, ({(i : α)} : Set α) = s := by simp
#align set.Union_of_singleton_coe Set.iUnion_of_singleton_coe
theorem sUnion_eq_biUnion {s : Set (Set α)} : ⋃₀s = ⋃ (i : Set α) (_ : i ∈ s), i := by
rw [← sUnion_image, image_id']
#align set.sUnion_eq_bUnion Set.sUnion_eq_biUnion
theorem sInter_eq_biInter {s : Set (Set α)} : ⋂₀ s = ⋂ (i : Set α) (_ : i ∈ s), i := by
rw [← sInter_image, image_id']
#align set.sInter_eq_bInter Set.sInter_eq_biInter
theorem sUnion_eq_iUnion {s : Set (Set α)} : ⋃₀s = ⋃ i : s, i := by
simp only [← sUnion_range, Subtype.range_coe]
#align set.sUnion_eq_Union Set.sUnion_eq_iUnion
theorem sInter_eq_iInter {s : Set (Set α)} : ⋂₀ s = ⋂ i : s, i := by
simp only [← sInter_range, Subtype.range_coe]
#align set.sInter_eq_Inter Set.sInter_eq_iInter
@[simp]
theorem iUnion_of_empty [IsEmpty ι] (s : ι → Set α) : ⋃ i, s i = ∅ :=
iSup_of_empty _
#align set.Union_of_empty Set.iUnion_of_empty
@[simp]
theorem iInter_of_empty [IsEmpty ι] (s : ι → Set α) : ⋂ i, s i = univ :=
iInf_of_empty _
#align set.Inter_of_empty Set.iInter_of_empty
theorem union_eq_iUnion {s₁ s₂ : Set α} : s₁ ∪ s₂ = ⋃ b : Bool, cond b s₁ s₂ :=
sup_eq_iSup s₁ s₂
#align set.union_eq_Union Set.union_eq_iUnion
theorem inter_eq_iInter {s₁ s₂ : Set α} : s₁ ∩ s₂ = ⋂ b : Bool, cond b s₁ s₂ :=
inf_eq_iInf s₁ s₂
#align set.inter_eq_Inter Set.inter_eq_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_union_sInter {S T : Set (Set α)} :
⋂₀ S ∪ ⋂₀ T = ⋂ p ∈ S ×ˢ T, (p : Set α × Set α).1 ∪ p.2 :=
sInf_sup_sInf
#align set.sInter_union_sInter Set.sInter_union_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_inter_sUnion {s t : Set (Set α)} :
⋃₀s ∩ ⋃₀t = ⋃ p ∈ s ×ˢ t, (p : Set α × Set α).1 ∩ p.2 :=
sSup_inf_sSup
#align set.sUnion_inter_sUnion Set.sUnion_inter_sUnion
theorem biUnion_iUnion (s : ι → Set α) (t : α → Set β) :
⋃ x ∈ ⋃ i, s i, t x = ⋃ (i) (x ∈ s i), t x := by simp [@iUnion_comm _ ι]
#align set.bUnion_Union Set.biUnion_iUnion
theorem biInter_iUnion (s : ι → Set α) (t : α → Set β) :
⋂ x ∈ ⋃ i, s i, t x = ⋂ (i) (x ∈ s i), t x := by simp [@iInter_comm _ ι]
#align set.bInter_Union Set.biInter_iUnion
theorem sUnion_iUnion (s : ι → Set (Set α)) : ⋃₀⋃ i, s i = ⋃ i, ⋃₀s i := by
simp only [sUnion_eq_biUnion, biUnion_iUnion]
#align set.sUnion_Union Set.sUnion_iUnion
theorem sInter_iUnion (s : ι → Set (Set α)) : ⋂₀ ⋃ i, s i = ⋂ i, ⋂₀ s i := by
simp only [sInter_eq_biInter, biInter_iUnion]
#align set.sInter_Union Set.sInter_iUnion
theorem iUnion_range_eq_sUnion {α β : Type*} (C : Set (Set α)) {f : ∀ s : C, β → (s : Type _)}
(hf : ∀ s : C, Surjective (f s)) : ⋃ y : β, range (fun s : C => (f s y).val) = ⋃₀C := by
ext x; constructor
· rintro ⟨s, ⟨y, rfl⟩, ⟨s, hs⟩, rfl⟩
refine' ⟨_, hs, _⟩
exact (f ⟨s, hs⟩ y).2
· rintro ⟨s, hs, hx⟩
cases' hf ⟨s, hs⟩ ⟨x, hx⟩ with y hy
refine' ⟨_, ⟨y, rfl⟩, ⟨s, hs⟩, _⟩
exact congr_arg Subtype.val hy
#align set.Union_range_eq_sUnion Set.iUnion_range_eq_sUnion
theorem iUnion_range_eq_iUnion (C : ι → Set α) {f : ∀ x : ι, β → C x}
(hf : ∀ x : ι, Surjective (f x)) : ⋃ y : β, range (fun x : ι => (f x y).val) = ⋃ x, C x := by
ext x; rw [mem_iUnion, mem_iUnion]; constructor
· rintro ⟨y, i, rfl⟩
exact ⟨i, (f i y).2⟩
· rintro ⟨i, hx⟩
cases' hf i ⟨x, hx⟩ with y hy
exact ⟨y, i, congr_arg Subtype.val hy⟩
#align set.Union_range_eq_Union Set.iUnion_range_eq_iUnion
theorem union_distrib_iInter_left (s : ι → Set α) (t : Set α) : (t ∪ ⋂ i, s i) = ⋂ i, t ∪ s i :=
sup_iInf_eq _ _
#align set.union_distrib_Inter_left Set.union_distrib_iInter_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_left (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_distrib_iInter_left]
#align set.union_distrib_Inter₂_left Set.union_distrib_iInter₂_left
theorem union_distrib_iInter_right (s : ι → Set α) (t : Set α) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.union_distrib_Inter_right Set.union_distrib_iInter_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_right (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [union_distrib_iInter_right]
#align set.union_distrib_Inter₂_right Set.union_distrib_iInter₂_right
section Function
/-! ### `mapsTo` -/
theorem mapsTo_sUnion {S : Set (Set α)} {t : Set β} {f : α → β} (H : ∀ s ∈ S, MapsTo f s t) :
MapsTo f (⋃₀S) t := fun _ ⟨s, hs, hx⟩ => H s hs hx
#align set.maps_to_sUnion Set.mapsTo_sUnion
theorem mapsTo_iUnion {s : ι → Set α} {t : Set β} {f : α → β} (H : ∀ i, MapsTo f (s i) t) :
MapsTo f (⋃ i, s i) t :=
mapsTo_sUnion <| forall_range_iff.2 H
#align set.maps_to_Union Set.mapsTo_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂ {s : ∀ i, κ i → Set α} {t : Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) t) : MapsTo f (⋃ (i) (j), s i j) t :=
mapsTo_iUnion fun i => mapsTo_iUnion (H i)
#align set.maps_to_Union₂ Set.mapsTo_iUnion₂
theorem mapsTo_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋃ i, s i) (⋃ i, t i) :=
mapsTo_iUnion fun i => (H i).mono (Subset.refl _) (subset_iUnion t i)
#align set.maps_to_Union_Union Set.mapsTo_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
mapsTo_iUnion_iUnion fun i => mapsTo_iUnion_iUnion (H i)
#align set.maps_to_Union₂_Union₂ Set.mapsTo_iUnion₂_iUnion₂
theorem mapsTo_sInter {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, MapsTo f s t) :
MapsTo f s (⋂₀ T) := fun _ hx t ht => H t ht hx
#align set.maps_to_sInter Set.mapsTo_sInter
theorem mapsTo_iInter {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, MapsTo f s (t i)) :
MapsTo f s (⋂ i, t i) := fun _ hx => mem_iInter.2 fun i => H i hx
#align set.maps_to_Inter Set.mapsTo_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f s (t i j)) : MapsTo f s (⋂ (i) (j), t i j) :=
mapsTo_iInter fun i => mapsTo_iInter (H i)
#align set.maps_to_Inter₂ Set.mapsTo_iInter₂
theorem mapsTo_iInter_iInter {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋂ i, s i) (⋂ i, t i) :=
mapsTo_iInter fun i => (H i).mono (iInter_subset s i) (Subset.refl _)
#align set.maps_to_Inter_Inter Set.mapsTo_iInter_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂_iInter₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋂ (i) (j), s i j) (⋂ (i) (j), t i j) :=
mapsTo_iInter_iInter fun i => mapsTo_iInter_iInter (H i)
#align set.maps_to_Inter₂_Inter₂ Set.mapsTo_iInter₂_iInter₂
theorem image_iInter_subset (s : ι → Set α) (f : α → β) : (f '' ⋂ i, s i) ⊆ ⋂ i, f '' s i :=
(mapsTo_iInter_iInter fun i => mapsTo_image f (s i)).image_subset
#align set.image_Inter_subset Set.image_iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂_subset (s : ∀ i, κ i → Set α) (f : α → β) :
(f '' ⋂ (i) (j), s i j) ⊆ ⋂ (i) (j), f '' s i j :=
(mapsTo_iInter₂_iInter₂ fun i hi => mapsTo_image f (s i hi)).image_subset
#align set.image_Inter₂_subset Set.image_iInter₂_subset
theorem image_sInter_subset (S : Set (Set α)) (f : α → β) : f '' ⋂₀ S ⊆ ⋂ s ∈ S, f '' s := by
rw [sInter_eq_biInter]
apply image_iInter₂_subset
#align set.image_sInter_subset Set.image_sInter_subset
/-! ### `restrictPreimage` -/
section
open Function
variable (s : Set β) {f : α → β} {U : ι → Set β} (hU : iUnion U = univ)
theorem injective_iff_injective_of_iUnion_eq_univ :
Injective f ↔ ∀ i, Injective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_injective H, fun H x y e => _⟩
obtain ⟨i, hi⟩ := Set.mem_iUnion.mp
(show f x ∈ Set.iUnion U by rw [hU]; triv)
injection @H i ⟨x, hi⟩ ⟨y, show f y ∈ U i from e ▸ hi⟩ (Subtype.ext e)
#align set.injective_iff_injective_of_Union_eq_univ Set.injective_iff_injective_of_iUnion_eq_univ
theorem surjective_iff_surjective_of_iUnion_eq_univ :
Surjective f ↔ ∀ i, Surjective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_surjective H, fun H x => _⟩
obtain ⟨i, hi⟩ :=
Set.mem_iUnion.mp
(show x ∈ Set.iUnion U by rw [hU]; triv)
exact ⟨_, congr_arg Subtype.val (H i ⟨x, hi⟩).choose_spec⟩
#align set.surjective_iff_surjective_of_Union_eq_univ Set.surjective_iff_surjective_of_iUnion_eq_univ
theorem bijective_iff_bijective_of_iUnion_eq_univ :
Bijective f ↔ ∀ i, Bijective ((U i).restrictPreimage f) := by
rw [Bijective, injective_iff_injective_of_iUnion_eq_univ hU,
surjective_iff_surjective_of_iUnion_eq_univ hU]
simp [Bijective, forall_and]
#align set.bijective_iff_bijective_of_Union_eq_univ Set.bijective_iff_bijective_of_iUnion_eq_univ
end
/-! ### `InjOn` -/
theorem InjOn.image_iInter_eq [Nonempty ι] {s : ι → Set α} {f : α → β} (h : InjOn f (⋃ i, s i)) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
inhabit ι
refine' Subset.antisymm (image_iInter_subset s f) fun y hy => _
simp only [mem_iInter, mem_image_iff_bex] at hy
choose x hx hy using hy
refine' ⟨x default, mem_iInter.2 fun i => _, hy _⟩
suffices x default = x i by
rw [this]
apply hx
replace hx : ∀ i, x i ∈ ⋃ j, s j := fun i => (subset_iUnion _ _) (hx i)
apply h (hx _) (hx _)
simp only [hy]
#align set.inj_on.image_Inter_eq Set.InjOn.image_iInter_eq
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
theorem InjOn.image_biInter_eq {p : ι → Prop} {s : ∀ i, p i → Set α} (hp : ∃ i, p i)
{f : α → β} (h : InjOn f (⋃ (i) (hi), s i hi)) :
(f '' ⋂ (i) (hi), s i hi) = ⋂ (i) (hi), f '' s i hi := by
simp only [iInter, iInf_subtype']
haveI : Nonempty { i // p i } := nonempty_subtype.2 hp
apply InjOn.image_iInter_eq
simpa only [iUnion, iSup_subtype'] using h
#align set.inj_on.image_bInter_eq Set.InjOn.image_biInter_eq
theorem image_iInter {f : α → β} (hf : Bijective f) (s : ι → Set α) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
cases isEmpty_or_nonempty ι
· simp_rw [iInter_of_empty, image_univ_of_surjective hf.surjective]
· exact (hf.injective.injOn _).image_iInter_eq
#align set.image_Inter Set.image_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂ {f : α → β} (hf : Bijective f) (s : ∀ i, κ i → Set α) :
(f '' ⋂ (i) (j), s i j) = ⋂ (i) (j), f '' s i j := by simp_rw [image_iInter hf]
#align set.image_Inter₂ Set.image_iInter₂
theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
(hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i) := by
intro x hx y hy hxy
rcases mem_iUnion.1 hx with ⟨i, hx⟩
rcases mem_iUnion.1 hy with ⟨j, hy⟩
rcases hs i j with ⟨k, hi, hj⟩
exact hf k (hi hx) (hj hy) hxy
#align set.inj_on_Union_of_directed Set.inj_on_iUnion_of_directed
/-! ### `SurjOn` -/
theorem surjOn_sUnion {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, SurjOn f s t) :
SurjOn f s (⋃₀T) := fun _ ⟨t, ht, hx⟩ => H t ht hx
#align set.surj_on_sUnion Set.surjOn_sUnion
theorem surjOn_iUnion {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, SurjOn f s (t i)) :
SurjOn f s (⋃ i, t i) :=
surjOn_sUnion <| forall_range_iff.2 H
#align set.surj_on_Union Set.surjOn_iUnion
theorem surjOn_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) : SurjOn f (⋃ i, s i) (⋃ i, t i) :=
surjOn_iUnion fun i => (H i).mono (subset_iUnion _ _) (Subset.refl _)
#align set.surj_on_Union_Union Set.surjOn_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f s (t i j)) : SurjOn f s (⋃ (i) (j), t i j) :=
surjOn_iUnion fun i => surjOn_iUnion (H i)
#align set.surj_on_Union₂ Set.surjOn_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f (s i j) (t i j)) : SurjOn f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
surjOn_iUnion_iUnion fun i => surjOn_iUnion_iUnion (H i)
#align set.surj_on_Union₂_Union₂ Set.surjOn_iUnion₂_iUnion₂
theorem surjOn_iInter [Nonempty ι] {s : ι → Set α} {t : Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) t) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) t := by
intro y hy
rw [Hinj.image_iInter_eq, mem_iInter]
exact fun i => H i hy
#align set.surj_on_Inter Set.surjOn_iInter
theorem surjOn_iInter_iInter [Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) (⋂ i, t i) :=
surjOn_iInter (fun i => (H i).mono (Subset.refl _) (iInter_subset _ _)) Hinj
#align set.surj_on_Inter_Inter Set.surjOn_iInter_iInter
/-! ### `BijOn` -/
theorem bijOn_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i))
(Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
⟨mapsTo_iUnion_iUnion fun i => (H i).mapsTo, Hinj, surjOn_iUnion_iUnion fun i => (H i).surjOn⟩
#align set.bij_on_Union Set.bijOn_iUnion
theorem bijOn_iInter [hi : Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, BijOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
⟨mapsTo_iInter_iInter fun i => (H i).mapsTo,
hi.elim fun i => (H i).injOn.mono (iInter_subset _ _),
surjOn_iInter_iInter (fun i => (H i).surjOn) Hinj⟩
#align set.bij_on_Inter Set.bijOn_iInter
theorem bijOn_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {t : ι → Set β}
{f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
bijOn_iUnion H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Union_of_directed Set.bijOn_iUnion_of_directed
theorem bijOn_iInter_of_directed [Nonempty ι] {s : ι → Set α} (hs : Directed (· ⊆ ·) s)
{t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
bijOn_iInter H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Inter_of_directed Set.bijOn_iInter_of_directed
end Function
/-! ### `image`, `preimage` -/
section Image
theorem image_iUnion {f : α → β} {s : ι → Set α} : (f '' ⋃ i, s i) = ⋃ i, f '' s i := by
ext1 x
simp only [mem_image, mem_iUnion, ← exists_and_right, ← exists_and_left]
--Porting note: `exists_swap` causes a `simp` loop in Lean4 so we use `rw` instead.
rw [exists_swap]
#align set.image_Union Set.image_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iUnion₂ (f : α → β) (s : ∀ i, κ i → Set α) :
(f '' ⋃ (i) (j), s i j) = ⋃ (i) (j), f '' s i j := by simp_rw [image_iUnion]
#align set.image_Union₂ Set.image_iUnion₂
theorem univ_subtype {p : α → Prop} : (univ : Set (Subtype p)) = ⋃ (x) (h : p x), {⟨x, h⟩} :=
Set.ext fun ⟨x, h⟩ => by simp [h]
#align set.univ_subtype Set.univ_subtype
theorem range_eq_iUnion {ι} (f : ι → α) : range f = ⋃ i, {f i} :=
Set.ext fun a => by simp [@eq_comm α a]
#align set.range_eq_Union Set.range_eq_iUnion
theorem image_eq_iUnion (f : α → β) (s : Set α) : f '' s = ⋃ i ∈ s, {f i} :=
Set.ext fun b => by simp [@eq_comm β b]
#align set.image_eq_Union Set.image_eq_iUnion
theorem biUnion_range {f : ι → α} {g : α → Set β} : ⋃ x ∈ range f, g x = ⋃ y, g (f y) :=
iSup_range
#align set.bUnion_range Set.biUnion_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iUnion_iUnion_eq' {f : ι → α} {g : α → Set β} :
⋃ (x) (y) (_ : f y = x), g x = ⋃ y, g (f y) := by simpa using biUnion_range
#align set.Union_Union_eq' Set.iUnion_iUnion_eq'
theorem biInter_range {f : ι → α} {g : α → Set β} : ⋂ x ∈ range f, g x = ⋂ y, g (f y) :=
iInf_range
#align set.bInter_range Set.biInter_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iInter_iInter_eq' {f : ι → α} {g : α → Set β} :
⋂ (x) (y) (_ : f y = x), g x = ⋂ y, g (f y) := by simpa using biInter_range
#align set.Inter_Inter_eq' Set.iInter_iInter_eq'
variable {s : Set γ} {f : γ → α} {g : α → Set β}
theorem biUnion_image : ⋃ x ∈ f '' s, g x = ⋃ y ∈ s, g (f y) :=
iSup_image
#align set.bUnion_image Set.biUnion_image
theorem biInter_image : ⋂ x ∈ f '' s, g x = ⋂ y ∈ s, g (f y) :=
iInf_image
#align set.bInter_image Set.biInter_image
end Image
section Preimage
theorem monotone_preimage {f : α → β} : Monotone (preimage f) := fun _ _ h => preimage_mono h
#align set.monotone_preimage Set.monotone_preimage
@[simp]
theorem preimage_iUnion {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋃ i, s i) = ⋃ i, f ⁻¹' s i :=
Set.ext <| by simp [preimage]
#align set.preimage_Union Set.preimage_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iUnion₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋃ (i) (j), s i j) = ⋃ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iUnion]
#align set.preimage_Union₂ Set.preimage_iUnion₂
@[simp]
theorem preimage_sUnion {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋃₀s = ⋃ t ∈ s, f ⁻¹' t := by
rw [sUnion_eq_biUnion, preimage_iUnion₂]
#align set.preimage_sUnion Set.preimage_sUnion
theorem preimage_iInter {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋂ i, s i) = ⋂ i, f ⁻¹' s i := by
ext; simp
#align set.preimage_Inter Set.preimage_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iInter₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋂ (i) (j), s i j) = ⋂ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iInter]
#align set.preimage_Inter₂ Set.preimage_iInter₂
@[simp]
theorem preimage_sInter {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋂₀ s = ⋂ t ∈ s, f ⁻¹' t := by
rw [sInter_eq_biInter, preimage_iInter₂]
#align set.preimage_sInter Set.preimage_sInter
@[simp]
theorem biUnion_preimage_singleton (f : α → β) (s : Set β) : ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s := by
rw [← preimage_iUnion₂, biUnion_of_singleton]
#align set.bUnion_preimage_singleton Set.biUnion_preimage_singleton
theorem biUnion_range_preimage_singleton (f : α → β) : ⋃ y ∈ range f, f ⁻¹' {y} = univ := by
rw [biUnion_preimage_singleton, preimage_range]
#align set.bUnion_range_preimage_singleton Set.biUnion_range_preimage_singleton
end Preimage
section Prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion {s : Set α} {t : ι → Set β} : (s ×ˢ ⋃ i, t i) = ⋃ i, s ×ˢ t i := by
ext
simp
#align set.prod_Union Set.prod_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} :
(s ×ˢ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ×ˢ t i j := by simp_rw [prod_iUnion]
#align set.prod_Union₂ Set.prod_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sUnion {s : Set α} {C : Set (Set β)} : s ×ˢ ⋃₀C = ⋃₀((fun t => s ×ˢ t) '' C) := by
simp_rw [sUnion_eq_biUnion, biUnion_image, prod_iUnion₂]
#align set.prod_sUnion Set.prod_sUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_const {s : ι → Set α} {t : Set β} : (⋃ i, s i) ×ˢ t = ⋃ i, s i ×ˢ t := by
ext
simp
#align set.Union_prod_const Set.iUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion₂_prod_const {s : ∀ i, κ i → Set α} {t : Set β} :
(⋃ (i) (j), s i j) ×ˢ t = ⋃ (i) (j), s i j ×ˢ t := by simp_rw [iUnion_prod_const]
#align set.Union₂_prod_const Set.iUnion₂_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_prod_const {C : Set (Set α)} {t : Set β} :
⋃₀C ×ˢ t = ⋃₀((fun s : Set α => s ×ˢ t) '' C) := by
simp only [sUnion_eq_biUnion, iUnion₂_prod_const, biUnion_image]
#align set.sUnion_prod_const Set.sUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod {ι ι' α β} (s : ι → Set α) (t : ι' → Set β) :
⋃ x : ι × ι', s x.1 ×ˢ t x.2 = (⋃ i : ι, s i) ×ˢ ⋃ i : ι', t i := by
ext
simp
#align set.Union_prod Set.iUnion_prod
/-- Analogue of `iSup_prod` for sets. -/
lemma iUnion_prod' (f : β × γ → Set α) : ⋃ x : β × γ, f x = ⋃ (i : β) (j : γ), f (i, j) :=
iSup_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x := by
ext ⟨z, w⟩; simp only [mem_prod, mem_iUnion, exists_imp, and_imp, iff_def]; constructor
· intro x hz hw
exact ⟨⟨x, hz⟩, x, hw⟩
· intro x hz x' hw
exact ⟨x ⊔ x', hs le_sup_left hz, ht le_sup_right hw⟩
#align set.Union_prod_of_monotone Set.iUnion_prod_of_monotone
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter_subset (S : Set (Set α)) (T : Set (Set β)) :
⋂₀ S ×ˢ ⋂₀ T ⊆ ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 :=
subset_iInter₂ fun x hx _ hy => ⟨hy.1 x.1 hx.1, hy.2 x.2 hx.2⟩
#align set.sInter_prod_sInter_subset Set.sInter_prod_sInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 := by
obtain ⟨s₁, h₁⟩ := hS
obtain ⟨s₂, h₂⟩ := hT
refine' Set.Subset.antisymm (sInter_prod_sInter_subset S T) fun x hx => _
rw [mem_iInter₂] at hx
exact ⟨fun s₀ h₀ => (hx (s₀, s₂) ⟨h₀, h₂⟩).1, fun s₀ h₀ => (hx (s₁, s₀) ⟨h₁, h₀⟩).2⟩
#align set.sInter_prod_sInter Set.sInter_prod_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod {S : Set (Set α)} (hS : S.Nonempty) (t : Set β) :
⋂₀ S ×ˢ t = ⋂ s ∈ S, s ×ˢ t := by
rw [← sInter_singleton t, sInter_prod_sInter hS (singleton_nonempty t), sInter_singleton]
simp_rw [prod_singleton, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.sInter_prod Set.sInter_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sInter {T : Set (Set β)} (hT : T.Nonempty) (s : Set α) :
s ×ˢ ⋂₀ T = ⋂ t ∈ T, s ×ˢ t := by
rw [← sInter_singleton s, sInter_prod_sInter (singleton_nonempty s) hT, sInter_singleton]
simp_rw [singleton_prod, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.prod_sInter Set.prod_sInter
end Prod
section Image2
variable (f : α → β → γ) {s : Set α} {t : Set β}
/-- The `Set.image2` version of `Set.image_eq_iUnion` -/
theorem image2_eq_iUnion (s : Set α) (t : Set β) : image2 f s t = ⋃ (i ∈ s) (j ∈ t), {f i j} := by
ext; simp [eq_comm]
#align set.image2_eq_Union Set.image2_eq_iUnion
theorem iUnion_image_left : ⋃ a ∈ s, f a '' t = image2 f s t := by
simp only [image2_eq_iUnion, image_eq_iUnion]
#align set.Union_image_left Set.iUnion_image_left
theorem iUnion_image_right : ⋃ b ∈ t, (f · b) '' s = image2 f s t := by
rw [image2_swap, iUnion_image_left]
#align set.Union_image_right Set.iUnion_image_right
theorem image2_iUnion_left (s : ι → Set α) (t : Set β) :
image2 f (⋃ i, s i) t = ⋃ i, image2 f (s i) t := by
simp only [← image_prod, iUnion_prod_const, image_iUnion]
#align set.image2_Union_left Set.image2_iUnion_left
theorem image2_iUnion_right (s : Set α) (t : ι → Set β) :
image2 f s (⋃ i, t i) = ⋃ i, image2 f s (t i) := by
simp only [← image_prod, prod_iUnion, image_iUnion]
#align set.image2_Union_right Set.image2_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋃ (i) (j), s i j) t = ⋃ (i) (j), image2 f (s i j) t := by simp_rw [image2_iUnion_left]
#align set.image2_Union₂_left Set.image2_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋃ (i) (j), t i j) = ⋃ (i) (j), image2 f s (t i j) :=
by simp_rw [image2_iUnion_right]
#align set.image2_Union₂_right Set.image2_iUnion₂_right
theorem image2_iInter_subset_left (s : ι → Set α) (t : Set β) :
image2 f (⋂ i, s i) t ⊆ ⋂ i, image2 f (s i) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem (hx _) hy
#align set.image2_Inter_subset_left Set.image2_iInter_subset_left
theorem image2_iInter_subset_right (s : Set α) (t : ι → Set β) :
image2 f s (⋂ i, t i) ⊆ ⋂ i, image2 f s (t i) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem hx (hy _)
#align set.image2_Inter_subset_right Set.image2_iInter_subset_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋂ (i) (j), s i j) t ⊆ ⋂ (i) (j), image2 f (s i j) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem (hx _ _) hy
#align set.image2_Inter₂_subset_left Set.image2_iInter₂_subset_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), image2 f s (t i j) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem hx (hy _ _)
#align set.image2_Inter₂_subset_right Set.image2_iInter₂_subset_right
theorem prod_eq_biUnion_left : s ×ˢ t = ⋃ a ∈ s, (fun b => (a, b)) '' t := by
rw [iUnion_image_left, image2_mk_eq_prod]
#align set.prod_eq_bUnion_left Set.prod_eq_biUnion_left
theorem prod_eq_biUnion_right : s ×ˢ t = ⋃ b ∈ t, (fun a => (a, b)) '' s := by
rw [iUnion_image_right, image2_mk_eq_prod]
#align set.prod_eq_bUnion_right Set.prod_eq_biUnion_right
end Image2
section Seq
/-- Given a set `s` of functions `α → β` and `t : Set α`, `seq s t` is the union of `f '' t` over
all `f ∈ s`. -/
def seq (s : Set (α → β)) (t : Set α) : Set β :=
{ b | ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b }
#align set.seq Set.seq
@[simp]
theorem mem_seq_iff {s : Set (α → β)} {t : Set α} {b : β} :
b ∈ seq s t ↔ ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b :=
Iff.rfl
#align set.mem_seq_iff Set.mem_seq_iff
lemma seq_eq_image2 (s : Set (α → β)) (t : Set α) : seq s t = image2 (fun f a ↦ f a) s t := by
ext; simp
theorem seq_def {s : Set (α → β)} {t : Set α} : seq s t = ⋃ f ∈ s, f '' t := by
rw [seq_eq_image2, iUnion_image_left]
#align set.seq_def Set.seq_def
theorem seq_subset {s : Set (α → β)} {t : Set α} {u : Set β} :
seq s t ⊆ u ↔ ∀ f ∈ s, ∀ a ∈ t, (f : α → β) a ∈ u := by
rw [seq_eq_image2, image2_subset_iff]
#align set.seq_subset Set.seq_subset
@[gcongr]
theorem seq_mono {s₀ s₁ : Set (α → β)} {t₀ t₁ : Set α} (hs : s₀ ⊆ s₁) (ht : t₀ ⊆ t₁) :
seq s₀ t₀ ⊆ seq s₁ t₁ := fun _ ⟨f, hf, a, ha, eq⟩ => ⟨f, hs hf, a, ht ha, eq⟩
#align set.seq_mono Set.seq_mono
theorem singleton_seq {f : α → β} {t : Set α} : Set.seq ({f} : Set (α → β)) t = f '' t := by
rw [seq_eq_image2, image2_singleton_left]
#align set.singleton_seq Set.singleton_seq
theorem seq_singleton {s : Set (α → β)} {a : α} : Set.seq s {a} = (fun f : α → β => f a) '' s := by
rw [seq_eq_image2, image2_singleton_right]
#align set.seq_singleton Set.seq_singleton
theorem seq_seq {s : Set (β → γ)} {t : Set (α → β)} {u : Set α} :
seq s (seq t u) = seq (seq ((· ∘ ·) '' s) t) u := by
simp only [seq_eq_image2, image2_image_left, image2_image2_left, image2_image2_right, comp_apply]
#align set.seq_seq Set.seq_seq
theorem image_seq {f : β → γ} {s : Set (α → β)} {t : Set α} :
f '' seq s t = seq ((f ∘ ·) '' s) t := by
rw [← singleton_seq, ← singleton_seq, seq_seq, image_singleton]
#align set.image_seq Set.image_seq
theorem prod_eq_seq {s : Set α} {t : Set β} : s ×ˢ t = (Prod.mk '' s).seq t := by
rw [seq_eq_image2, image2_image_left, image2_mk_eq_prod]
#align set.prod_eq_seq Set.prod_eq_seq
theorem prod_image_seq_comm (s : Set α) (t : Set β) :
(Prod.mk '' s).seq t = seq ((fun b a => (a, b)) '' t) s := by
rw [← prod_eq_seq, ← image_swap_prod, prod_eq_seq, image_seq, ← image_comp]; rfl
#align set.prod_image_seq_comm Set.prod_image_seq_comm
theorem image2_eq_seq (f : α → β → γ) (s : Set α) (t : Set β) : image2 f s t = seq (f '' s) t := by
rw [seq_eq_image2, image2_image_left]
#align set.image2_eq_seq Set.image2_eq_seq
end Seq
section Pi
variable {π : α → Type*}
theorem pi_def (i : Set α) (s : ∀ a, Set (π a)) : pi i s = ⋂ a ∈ i, eval a ⁻¹' s a := by
ext
simp
#align set.pi_def Set.pi_def
theorem univ_pi_eq_iInter (t : ∀ i, Set (π i)) : pi univ t = ⋂ i, eval i ⁻¹' t i := by
simp only [pi_def, iInter_true, mem_univ]
#align set.univ_pi_eq_Inter Set.univ_pi_eq_iInter
theorem pi_diff_pi_subset (i : Set α) (s t : ∀ a, Set (π a)) :
pi i s \ pi i t ⊆ ⋃ a ∈ i, eval a ⁻¹' (s a \ t a) := by
refine' diff_subset_comm.2 fun x hx a ha => _
simp only [mem_diff, mem_pi, mem_iUnion, not_exists, mem_preimage, not_and, not_not,
eval_apply] at hx
exact hx.2 _ ha (hx.1 _ ha)
#align set.pi_diff_pi_subset Set.pi_diff_pi_subset
theorem iUnion_univ_pi {ι : α → Type*} (t : (a : α) → ι a → Set (π a)) :
⋃ x : (a : α) → ι a, pi univ (fun a => t a (x a)) = pi univ fun a => ⋃ j : ι a, t a j := by
ext
simp [Classical.skolem]
#align set.Union_univ_pi Set.iUnion_univ_pi
end Pi
end Set
namespace Function
namespace Surjective
theorem iUnion_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋃ x, g (f x) = ⋃ y, g y :=
hf.iSup_comp g
#align function.surjective.Union_comp Function.Surjective.iUnion_comp
theorem iInter_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋂ x, g (f x) = ⋂ y, g y :=
hf.iInf_comp g
#align function.surjective.Inter_comp Function.Surjective.iInter_comp
end Surjective
end Function
/-!
### Disjoint sets
-/
section Disjoint
variable {s t u : Set α} {f : α → β}
namespace Set
@[simp]
theorem disjoint_iUnion_left {ι : Sort*} {s : ι → Set α} :
Disjoint (⋃ i, s i) t ↔ ∀ i, Disjoint (s i) t :=
iSup_disjoint_iff
#align set.disjoint_Union_left Set.disjoint_iUnion_left
@[simp]
theorem disjoint_iUnion_right {ι : Sort*} {s : ι → Set α} :
Disjoint t (⋃ i, s i) ↔ ∀ i, Disjoint t (s i) :=
disjoint_iSup_iff
#align set.disjoint_Union_right Set.disjoint_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_left {s : ∀ i, κ i → Set α} {t : Set α} :
Disjoint (⋃ (i) (j), s i j) t ↔ ∀ i j, Disjoint (s i j) t :=
iSup₂_disjoint_iff
#align set.disjoint_Union₂_left Set.disjoint_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_right {s : Set α} {t : ∀ i, κ i → Set α} :
Disjoint s (⋃ (i) (j), t i j) ↔ ∀ i j, Disjoint s (t i j) :=
disjoint_iSup₂_iff
#align set.disjoint_Union₂_right Set.disjoint_iUnion₂_right
@[simp]
theorem disjoint_sUnion_left {S : Set (Set α)} {t : Set α} :
Disjoint (⋃₀S) t ↔ ∀ s ∈ S, Disjoint s t :=
sSup_disjoint_iff
#align set.disjoint_sUnion_left Set.disjoint_sUnion_left
@[simp]
theorem disjoint_sUnion_right {s : Set α} {S : Set (Set α)} :
Disjoint s (⋃₀S) ↔ ∀ t ∈ S, Disjoint s t :=
disjoint_sSup_iff
#align set.disjoint_sUnion_right Set.disjoint_sUnion_right
end Set
end Disjoint
/-! ### Intervals -/
namespace Set
lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :
(⋂ i, Iic (f i)).Nonempty ↔ BddBelow (range f) := by
have : (⋂ (i : ι), Iic (f i)) = lowerBounds (range f) := by
ext c; simp [lowerBounds]
simp [this, BddBelow]
lemma nonempty_iInter_Ici_iff [Preorder α] {f : ι → α} :
(⋂ i, Ici (f i)).Nonempty ↔ BddAbove (range f) :=
nonempty_iInter_Iic_iff (α := αᵒᵈ)
variable [CompleteLattice α]
theorem Ici_iSup (f : ι → α) : Ici (⨆ i, f i) = ⋂ i, Ici (f i) :=
ext fun _ => by simp only [mem_Ici, iSup_le_iff, mem_iInter]
#align set.Ici_supr Set.Ici_iSup
theorem Iic_iInf (f : ι → α) : Iic (⨅ i, f i) = ⋂ i, Iic (f i) :=
ext fun _ => by simp only [mem_Iic, le_iInf_iff, mem_iInter]
#align set.Iic_infi Set.Iic_iInf
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Ici_iSup₂ (f : ∀ i, κ i → α) : Ici (⨆ (i) (j), f i j) = ⋂ (i) (j), Ici (f i j) := by
simp_rw [Ici_iSup]
#align set.Ici_supr₂ Set.Ici_iSup₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Iic_iInf₂ (f : ∀ i, κ i → α) : Iic (⨅ (i) (j), f i j) = ⋂ (i) (j), Iic (f i j) := by
simp_rw [Iic_iInf]
#align set.Iic_infi₂ Set.Iic_iInf₂
theorem Ici_sSup (s : Set α) : Ici (sSup s) = ⋂ a ∈ s, Ici a := by rw [sSup_eq_iSup, Ici_iSup₂]
#align set.Ici_Sup Set.Ici_sSup
theorem Iic_sInf (s : Set α) : Iic (sInf s) = ⋂ a ∈ s, Iic a := by rw [sInf_eq_iInf, Iic_iInf₂]
#align set.Iic_Inf Set.Iic_sInf
end Set
namespace Set
variable (t : α → Set β)
theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x := by
simp only [diff_subset_iff, ← biUnion_union]
apply biUnion_subset_biUnion_left
rw [union_diff_self]
apply subset_union_right
#align set.bUnion_diff_bUnion_subset Set.biUnion_diff_biUnion_subset
/-- If `t` is an indexed family of sets, then there is a natural map from `Σ i, t i` to `⋃ i, t i`
sending `⟨i, x⟩` to `x`. -/
def sigmaToiUnion (x : Σi, t i) : ⋃ i, t i :=
⟨x.2, mem_iUnion.2 ⟨x.1, x.2.2⟩⟩
#align set.sigma_to_Union Set.sigmaToiUnion
theorem sigmaToiUnion_surjective : Surjective (sigmaToiUnion t)
| ⟨b, hb⟩ =>
have : ∃ a, b ∈ t a := by simpa using hb
let ⟨a, hb⟩ := this
⟨⟨a, b, hb⟩, rfl⟩
#align set.sigma_to_Union_surjective Set.sigmaToiUnion_surjective
theorem sigmaToiUnion_injective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Injective (sigmaToiUnion t)
| ⟨a₁, b₁, h₁⟩, ⟨a₂, b₂, h₂⟩, eq =>
have b_eq : b₁ = b₂ := congr_arg Subtype.val eq
have a_eq : a₁ = a₂ :=
by_contradiction fun ne =>
have : b₁ ∈ t a₁ ∩ t a₂ := ⟨h₁, b_eq.symm ▸ h₂⟩
(h _ _ ne).le_bot this
Sigma.eq a_eq <| Subtype.eq <| by subst b_eq; subst a_eq; rfl
#align set.sigma_to_Union_injective Set.sigmaToiUnion_injective
theorem sigmaToiUnion_bijective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Bijective (sigmaToiUnion t) :=
⟨sigmaToiUnion_injective t h, sigmaToiUnion_surjective t⟩
#align set.sigma_to_Union_bijective Set.sigmaToiUnion_bijective
/-- Equivalence between a disjoint union and a dependent sum. -/
noncomputable def unionEqSigmaOfDisjoint {t : α → Set β} (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
(⋃ i, t i) ≃ Σi, t i :=
(Equiv.ofBijective _ <| sigmaToiUnion_bijective t h).symm
#align set.Union_eq_sigma_of_disjoint Set.unionEqSigmaOfDisjoint
theorem iUnion_ge_eq_iUnion_nat_add (u : ℕ → Set α) (n : ℕ) : ⋃ i ≥ n, u i = ⋃ i, u (i + n) :=
iSup_ge_eq_iSup_nat_add u n
#align set.Union_ge_eq_Union_nat_add Set.iUnion_ge_eq_iUnion_nat_add
theorem iInter_ge_eq_iInter_nat_add (u : ℕ → Set α) (n : ℕ) : ⋂ i ≥ n, u i = ⋂ i, u (i + n) :=
iInf_ge_eq_iInf_nat_add u n
#align set.Inter_ge_eq_Inter_nat_add Set.iInter_ge_eq_iInter_nat_add
theorem _root_.Monotone.iUnion_nat_add {f : ℕ → Set α} (hf : Monotone f) (k : ℕ) :
⋃ n, f (n + k) = ⋃ n, f n :=
hf.iSup_nat_add k
#align monotone.Union_nat_add Monotone.iUnion_nat_add
theorem _root_.Antitone.iInter_nat_add {f : ℕ → Set α} (hf : Antitone f) (k : ℕ) :
⋂ n, f (n + k) = ⋂ n, f n :=
hf.iInf_nat_add k
#align antitone.Inter_nat_add Antitone.iInter_nat_add
/-Porting note: removing `simp`. LHS does not simplify. Possible linter bug. Zulip discussion:
https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/complete_lattice.20and.20has_sup/near/316497982
-/
theorem iUnion_iInter_ge_nat_add (f : ℕ → Set α) (k : ℕ) :
⋃ n, ⋂ i ≥ n, f (i + k) = ⋃ n, ⋂ i ≥ n, f i :=
iSup_iInf_ge_nat_add f k
#align set.Union_Inter_ge_nat_add Set.iUnion_iInter_ge_nat_add
theorem union_iUnion_nat_succ (u : ℕ → Set α) : (u 0 ∪ ⋃ i, u (i + 1)) = ⋃ i, u i :=
sup_iSup_nat_succ u
#align set.union_Union_nat_succ Set.union_iUnion_nat_succ
theorem inter_iInter_nat_succ (u : ℕ → Set α) : (u 0 ∩ ⋂ i, u (i + 1)) = ⋂ i, u i :=
inf_iInf_nat_succ u
#align set.inter_Inter_nat_succ Set.inter_iInter_nat_succ
end Set
open Set
variable [CompleteLattice β]
theorem iSup_iUnion (s : ι → Set α) (f : α → β) : ⨆ a ∈ ⋃ i, s i, f a = ⨆ (i) (a ∈ s i), f a := by
rw [iSup_comm]
simp_rw [mem_iUnion, iSup_exists]
#align supr_Union iSup_iUnion
theorem iInf_iUnion (s : ι → Set α) (f : α → β) : ⨅ a ∈ ⋃ i, s i, f a = ⨅ (i) (a ∈ s i), f a :=
@iSup_iUnion α βᵒᵈ _ _ s f
#align infi_Union iInf_iUnion
theorem sSup_sUnion (s : Set (Set β)) : sSup (⋃₀ s) = ⨆ t ∈ s, sSup t := by
simp only [sUnion_eq_biUnion, sSup_eq_iSup, iSup_iUnion]
#align Sup_sUnion sSup_sUnion
theorem sInf_sUnion (s : Set (Set β)) : sInf (⋃₀ s) = ⨅ t ∈ s, sInf t :=
@sSup_sUnion βᵒᵈ _ _
#align Inf_sUnion sInf_sUnion
lemma iSup_sUnion (S : Set (Set α)) (f : α → β) :
(⨆ x ∈ ⋃₀ S, f x) = ⨆ (s ∈ S) (x ∈ s), f x := by
|
rw [sUnion_eq_iUnion, iSup_iUnion, ← iSup_subtype'']
|
lemma iSup_sUnion (S : Set (Set α)) (f : α → β) :
(⨆ x ∈ ⋃₀ S, f x) = ⨆ (s ∈ S) (x ∈ s), f x := by
|
Mathlib.Data.Set.Lattice.2391_0.5mONj49h3SYSDwc
|
lemma iSup_sUnion (S : Set (Set α)) (f : α → β) :
(⨆ x ∈ ⋃₀ S, f x) = ⨆ (s ∈ S) (x ∈ s), f x
|
Mathlib_Data_Set_Lattice
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : CompleteLattice β
S : Set (Set α)
f : α → β
⊢ ⨅ x ∈ ⋃₀ S, f x = ⨅ s ∈ S, ⨅ x ∈ s, f x
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
/-!
# The set lattice
This file provides usual set notation for unions and intersections, a `CompleteLattice` instance
for `Set α`, and some more set constructions.
## Main declarations
* `Set.iUnion`: **i**ndexed **union**. Union of an indexed family of sets.
* `Set.iInter`: **i**ndexed **inter**section. Intersection of an indexed family of sets.
* `Set.sInter`: **s**et **inter**section. Intersection of sets belonging to a set of sets.
* `Set.sUnion`: **s**et **union**. Union of sets belonging to a set of sets.
* `Set.sInter_eq_biInter`, `Set.sUnion_eq_biInter`: Shows that `⋂₀ s = ⋂ x ∈ s, x` and
`⋃₀ s = ⋃ x ∈ s, x`.
* `Set.completeAtomicBooleanAlgebra`: `Set α` is a `CompleteAtomicBooleanAlgebra` with `≤ = ⊆`,
`< = ⊂`, `⊓ = ∩`, `⊔ = ∪`, `⨅ = ⋂`, `⨆ = ⋃` and `\` as the set difference.
See `Set.BooleanAlgebra`.
* `Set.kernImage`: For a function `f : α → β`, `s.kernImage f` is the set of `y` such that
`f ⁻¹ y ⊆ s`.
* `Set.seq`: Union of the image of a set under a **seq**uence of functions. `seq s t` is the union
of `f '' t` over all `f ∈ s`, where `t : Set α` and `s : Set (α → β)`.
* `Set.iUnion_eq_sigma_of_disjoint`: Equivalence between `⋃ i, t i` and `Σ i, t i`, where `t` is an
indexed family of disjoint sets.
## Naming convention
In lemma names,
* `⋃ i, s i` is called `iUnion`
* `⋂ i, s i` is called `iInter`
* `⋃ i j, s i j` is called `iUnion₂`. This is an `iUnion` inside an `iUnion`.
* `⋂ i j, s i j` is called `iInter₂`. This is an `iInter` inside an `iInter`.
* `⋃ i ∈ s, t i` is called `biUnion` for "bounded `iUnion`". This is the special case of `iUnion₂`
where `j : i ∈ s`.
* `⋂ i ∈ s, t i` is called `biInter` for "bounded `iInter`". This is the special case of `iInter₂`
where `j : i ∈ s`.
## Notation
* `⋃`: `Set.iUnion`
* `⋂`: `Set.iInter`
* `⋃₀`: `Set.sUnion`
* `⋂₀`: `Set.sInter`
-/
set_option autoImplicit true
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι₂ : Sort*} {κ κ₁ κ₂ : ι → Sort*} {κ' : ι' → Sort*}
namespace Set
/-! ### Complete lattice and complete Boolean algebra instances -/
instance : InfSet (Set α) :=
⟨fun s => { a | ∀ t ∈ s, a ∈ t }⟩
instance : SupSet (Set α) :=
⟨fun s => { a | ∃ t ∈ s, a ∈ t }⟩
/-- Intersection of a set of sets. -/
def sInter (S : Set (Set α)) : Set α :=
sInf S
#align set.sInter Set.sInter
/-- Notation for `Set.sInter` Intersection of a set of sets. -/
prefix:110 "⋂₀ " => sInter
/-- Union of a set of sets. -/
def sUnion (S : Set (Set α)) : Set α :=
sSup S
#align set.sUnion Set.sUnion
/-- Notation for `Set.sUnion`. Union of a set of sets. -/
prefix:110 "⋃₀ " => sUnion
@[simp]
theorem mem_sInter {x : α} {S : Set (Set α)} : x ∈ ⋂₀ S ↔ ∀ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sInter Set.mem_sInter
@[simp]
theorem mem_sUnion {x : α} {S : Set (Set α)} : x ∈ ⋃₀ S ↔ ∃ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sUnion Set.mem_sUnion
/-- Indexed union of a family of sets -/
def iUnion (s : ι → Set β) : Set β :=
iSup s
#align set.Union Set.iUnion
/-- Indexed intersection of a family of sets -/
def iInter (s : ι → Set β) : Set β :=
iInf s
#align set.Inter Set.iInter
/-- Notation for `Set.iUnion`. Indexed union of a family of sets -/
notation3 "⋃ "(...)", "r:60:(scoped f => iUnion f) => r
/-- Notation for `Set.iInter`. Indexed intersection of a family of sets -/
notation3 "⋂ "(...)", "r:60:(scoped f => iInter f) => r
section delaborators
open Lean Lean.PrettyPrinter.Delaborator
/-- Delaborator for indexed unions. -/
@[delab app.Set.iUnion]
def iUnion_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋃ (_ : $dom), $body)
else if prop || ppTypes then
`(⋃ ($x:ident : $dom), $body)
else
`(⋃ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋃ $x:ident, ⋃ (_ : $y:ident ∈ $s), $body)
| `(⋃ ($x:ident : $_), ⋃ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋃ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
/-- Delaborator for indexed intersections. -/
@[delab app.Set.iInter]
def sInter_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋂ (_ : $dom), $body)
else if prop || ppTypes then
`(⋂ ($x:ident : $dom), $body)
else
`(⋂ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋂ $x:ident, ⋂ (_ : $y:ident ∈ $s), $body)
| `(⋂ ($x:ident : $_), ⋂ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋂ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
end delaborators
@[simp]
theorem sSup_eq_sUnion (S : Set (Set α)) : sSup S = ⋃₀S :=
rfl
#align set.Sup_eq_sUnion Set.sSup_eq_sUnion
@[simp]
theorem sInf_eq_sInter (S : Set (Set α)) : sInf S = ⋂₀ S :=
rfl
#align set.Inf_eq_sInter Set.sInf_eq_sInter
@[simp]
theorem iSup_eq_iUnion (s : ι → Set α) : iSup s = iUnion s :=
rfl
#align set.supr_eq_Union Set.iSup_eq_iUnion
@[simp]
theorem iInf_eq_iInter (s : ι → Set α) : iInf s = iInter s :=
rfl
#align set.infi_eq_Inter Set.iInf_eq_iInter
@[simp]
theorem mem_iUnion {x : α} {s : ι → Set α} : (x ∈ ⋃ i, s i) ↔ ∃ i, x ∈ s i :=
⟨fun ⟨_, ⟨⟨a, (t_eq : s a = _)⟩, (h : x ∈ _)⟩⟩ => ⟨a, t_eq.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨s a, ⟨⟨a, rfl⟩, h⟩⟩⟩
#align set.mem_Union Set.mem_iUnion
@[simp]
theorem mem_iInter {x : α} {s : ι → Set α} : (x ∈ ⋂ i, s i) ↔ ∀ i, x ∈ s i :=
⟨fun (h : ∀ a ∈ { a : Set α | ∃ i, s i = a }, x ∈ a) a => h (s a) ⟨a, rfl⟩,
fun h _ ⟨a, (eq : s a = _)⟩ => eq ▸ h a⟩
#align set.mem_Inter Set.mem_iInter
theorem mem_iUnion₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋃ (i) (j), s i j) ↔ ∃ i j, x ∈ s i j := by
simp_rw [mem_iUnion]
#align set.mem_Union₂ Set.mem_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋂ (i) (j), s i j) ↔ ∀ i j, x ∈ s i j := by
simp_rw [mem_iInter]
#align set.mem_Inter₂ Set.mem_iInter₂
theorem mem_iUnion_of_mem {s : ι → Set α} {a : α} (i : ι) (ha : a ∈ s i) : a ∈ ⋃ i, s i :=
mem_iUnion.2 ⟨i, ha⟩
#align set.mem_Union_of_mem Set.mem_iUnion_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iUnion₂_of_mem {s : ∀ i, κ i → Set α} {a : α} {i : ι} (j : κ i) (ha : a ∈ s i j) :
a ∈ ⋃ (i) (j), s i j :=
mem_iUnion₂.2 ⟨i, j, ha⟩
#align set.mem_Union₂_of_mem Set.mem_iUnion₂_of_mem
theorem mem_iInter_of_mem {s : ι → Set α} {a : α} (h : ∀ i, a ∈ s i) : a ∈ ⋂ i, s i :=
mem_iInter.2 h
#align set.mem_Inter_of_mem Set.mem_iInter_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂_of_mem {s : ∀ i, κ i → Set α} {a : α} (h : ∀ i j, a ∈ s i j) :
a ∈ ⋂ (i) (j), s i j :=
mem_iInter₂.2 h
#align set.mem_Inter₂_of_mem Set.mem_iInter₂_of_mem
instance Set.completeAtomicBooleanAlgebra : CompleteAtomicBooleanAlgebra (Set α) :=
{ instBooleanAlgebraSet with
le_sSup := fun s t t_in a a_in => ⟨t, t_in, a_in⟩
sSup_le := fun s t h a ⟨t', ⟨t'_in, a_in⟩⟩ => h t' t'_in a_in
le_sInf := fun s t h a a_in t' t'_in => h t' t'_in a_in
sInf_le := fun s t t_in a h => h _ t_in
iInf_iSup_eq := by intros; ext; simp [Classical.skolem] }
/-- `kernImage f s` is the set of `y` such that `f ⁻¹ y ⊆ s`. -/
def kernImage (f : α → β) (s : Set α) : Set β :=
{ y | ∀ ⦃x⦄, f x = y → x ∈ s }
#align set.kern_image Set.kernImage
lemma subset_kernImage_iff {f : α → β} : s ⊆ kernImage f t ↔ f ⁻¹' s ⊆ t :=
⟨fun h _ hx ↦ h hx rfl,
fun h _ hx y hy ↦ h (show f y ∈ s from hy.symm ▸ hx)⟩
section GaloisConnection
variable {f : α → β}
protected theorem image_preimage : GaloisConnection (image f) (preimage f) := fun _ _ =>
image_subset_iff
#align set.image_preimage Set.image_preimage
protected theorem preimage_kernImage : GaloisConnection (preimage f) (kernImage f) := fun _ _ =>
subset_kernImage_iff.symm
#align set.preimage_kern_image Set.preimage_kernImage
end GaloisConnection
section kernImage
variable {f : α → β}
lemma kernImage_mono : Monotone (kernImage f) :=
Set.preimage_kernImage.monotone_u
lemma kernImage_eq_compl {s : Set α} : kernImage f s = (f '' sᶜ)ᶜ :=
Set.preimage_kernImage.u_unique (Set.image_preimage.compl)
(fun t ↦ compl_compl (f ⁻¹' t) ▸ Set.preimage_compl)
lemma kernImage_compl {s : Set α} : kernImage f (sᶜ) = (f '' s)ᶜ := by
rw [kernImage_eq_compl, compl_compl]
lemma kernImage_empty : kernImage f ∅ = (range f)ᶜ := by
rw [kernImage_eq_compl, compl_empty, image_univ]
lemma kernImage_preimage_eq_iff {s : Set β} : kernImage f (f ⁻¹' s) = s ↔ (range f)ᶜ ⊆ s := by
rw [kernImage_eq_compl, ← preimage_compl, compl_eq_comm, eq_comm, image_preimage_eq_iff,
compl_subset_comm]
lemma compl_range_subset_kernImage {s : Set α} : (range f)ᶜ ⊆ kernImage f s := by
rw [← kernImage_empty]
exact kernImage_mono (empty_subset _)
lemma kernImage_union_preimage {s : Set α} {t : Set β} :
kernImage f (s ∪ f ⁻¹' t) = kernImage f s ∪ t := by
rw [kernImage_eq_compl, kernImage_eq_compl, compl_union, ← preimage_compl, image_inter_preimage,
compl_inter, compl_compl]
lemma kernImage_preimage_union {s : Set α} {t : Set β} :
kernImage f (f ⁻¹' t ∪ s) = t ∪ kernImage f s := by
rw [union_comm, kernImage_union_preimage, union_comm]
end kernImage
/-! ### Union and intersection over an indexed family of sets -/
instance : OrderTop (Set α) where
top := univ
le_top := by simp
@[congr]
theorem iUnion_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iUnion f₁ = iUnion f₂ :=
iSup_congr_Prop pq f
#align set.Union_congr_Prop Set.iUnion_congr_Prop
@[congr]
theorem iInter_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iInter f₁ = iInter f₂ :=
iInf_congr_Prop pq f
#align set.Inter_congr_Prop Set.iInter_congr_Prop
theorem iUnion_plift_up (f : PLift ι → Set α) : ⋃ i, f (PLift.up i) = ⋃ i, f i :=
iSup_plift_up _
#align set.Union_plift_up Set.iUnion_plift_up
theorem iUnion_plift_down (f : ι → Set α) : ⋃ i, f (PLift.down i) = ⋃ i, f i :=
iSup_plift_down _
#align set.Union_plift_down Set.iUnion_plift_down
theorem iInter_plift_up (f : PLift ι → Set α) : ⋂ i, f (PLift.up i) = ⋂ i, f i :=
iInf_plift_up _
#align set.Inter_plift_up Set.iInter_plift_up
theorem iInter_plift_down (f : ι → Set α) : ⋂ i, f (PLift.down i) = ⋂ i, f i :=
iInf_plift_down _
#align set.Inter_plift_down Set.iInter_plift_down
theorem iUnion_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋃ _ : p, s = if p then s else ∅ :=
iSup_eq_if _
#align set.Union_eq_if Set.iUnion_eq_if
theorem iUnion_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋃ h : p, s h = if h : p then s h else ∅ :=
iSup_eq_dif _
#align set.Union_eq_dif Set.iUnion_eq_dif
theorem iInter_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋂ _ : p, s = if p then s else univ :=
iInf_eq_if _
#align set.Inter_eq_if Set.iInter_eq_if
theorem iInf_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋂ h : p, s h = if h : p then s h else univ :=
_root_.iInf_eq_dif _
#align set.Infi_eq_dif Set.iInf_eq_dif
theorem exists_set_mem_of_union_eq_top {ι : Type*} (t : Set ι) (s : ι → Set β)
(w : ⋃ i ∈ t, s i = ⊤) (x : β) : ∃ i ∈ t, x ∈ s i := by
have p : x ∈ ⊤ := Set.mem_univ x
rw [← w, Set.mem_iUnion] at p
simpa using p
#align set.exists_set_mem_of_union_eq_top Set.exists_set_mem_of_union_eq_top
theorem nonempty_of_union_eq_top_of_nonempty {ι : Type*} (t : Set ι) (s : ι → Set α)
(H : Nonempty α) (w : ⋃ i ∈ t, s i = ⊤) : t.Nonempty := by
obtain ⟨x, m, -⟩ := exists_set_mem_of_union_eq_top t s w H.some
exact ⟨x, m⟩
#align set.nonempty_of_union_eq_top_of_nonempty Set.nonempty_of_union_eq_top_of_nonempty
theorem nonempty_of_nonempty_iUnion
{s : ι → Set α} (h_Union : (⋃ i, s i).Nonempty) : Nonempty ι := by
obtain ⟨x, hx⟩ := h_Union
exact ⟨Classical.choose $ mem_iUnion.mp hx⟩
theorem nonempty_of_nonempty_iUnion_eq_univ
{s : ι → Set α} [Nonempty α] (h_Union : ⋃ i, s i = univ) : Nonempty ι :=
nonempty_of_nonempty_iUnion (s := s) (by simpa only [h_Union] using univ_nonempty)
theorem setOf_exists (p : ι → β → Prop) : { x | ∃ i, p i x } = ⋃ i, { x | p i x } :=
ext fun _ => mem_iUnion.symm
#align set.set_of_exists Set.setOf_exists
theorem setOf_forall (p : ι → β → Prop) : { x | ∀ i, p i x } = ⋂ i, { x | p i x } :=
ext fun _ => mem_iInter.symm
#align set.set_of_forall Set.setOf_forall
theorem iUnion_subset {s : ι → Set α} {t : Set α} (h : ∀ i, s i ⊆ t) : ⋃ i, s i ⊆ t :=
iSup_le h
#align set.Union_subset Set.iUnion_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset {s : ∀ i, κ i → Set α} {t : Set α} (h : ∀ i j, s i j ⊆ t) :
⋃ (i) (j), s i j ⊆ t :=
iUnion_subset fun x => iUnion_subset (h x)
#align set.Union₂_subset Set.iUnion₂_subset
theorem subset_iInter {t : Set β} {s : ι → Set β} (h : ∀ i, t ⊆ s i) : t ⊆ ⋂ i, s i :=
le_iInf h
#align set.subset_Inter Set.subset_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iInter₂ {s : Set α} {t : ∀ i, κ i → Set α} (h : ∀ i j, s ⊆ t i j) :
s ⊆ ⋂ (i) (j), t i j :=
subset_iInter fun x => subset_iInter <| h x
#align set.subset_Inter₂ Set.subset_iInter₂
@[simp]
theorem iUnion_subset_iff {s : ι → Set α} {t : Set α} : ⋃ i, s i ⊆ t ↔ ∀ i, s i ⊆ t :=
⟨fun h _ => Subset.trans (le_iSup s _) h, iUnion_subset⟩
#align set.Union_subset_iff Set.iUnion_subset_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset_iff {s : ∀ i, κ i → Set α} {t : Set α} :
⋃ (i) (j), s i j ⊆ t ↔ ∀ i j, s i j ⊆ t := by simp_rw [iUnion_subset_iff]
#align set.Union₂_subset_iff Set.iUnion₂_subset_iff
@[simp]
theorem subset_iInter_iff {s : Set α} {t : ι → Set α} : (s ⊆ ⋂ i, t i) ↔ ∀ i, s ⊆ t i :=
le_iInf_iff
#align set.subset_Inter_iff Set.subset_iInter_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem subset_iInter₂_iff {s : Set α} {t : ∀ i, κ i → Set α} :
(s ⊆ ⋂ (i) (j), t i j) ↔ ∀ i j, s ⊆ t i j := by simp_rw [subset_iInter_iff]
#align set.subset_Inter₂_iff Set.subset_iInter₂_iff
theorem subset_iUnion : ∀ (s : ι → Set β) (i : ι), s i ⊆ ⋃ i, s i :=
le_iSup
#align set.subset_Union Set.subset_iUnion
theorem iInter_subset : ∀ (s : ι → Set β) (i : ι), ⋂ i, s i ⊆ s i :=
iInf_le
#align set.Inter_subset Set.iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iUnion₂ {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : s i j ⊆ ⋃ (i') (j'), s i' j' :=
le_iSup₂ i j
#align set.subset_Union₂ Set.subset_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_subset {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : ⋂ (i) (j), s i j ⊆ s i j :=
iInf₂_le i j
#align set.Inter₂_subset Set.iInter₂_subset
/-- This rather trivial consequence of `subset_iUnion`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem subset_iUnion_of_subset {s : Set α} {t : ι → Set α} (i : ι) (h : s ⊆ t i) : s ⊆ ⋃ i, t i :=
le_iSup_of_le i h
#align set.subset_Union_of_subset Set.subset_iUnion_of_subset
/-- This rather trivial consequence of `iInter_subset`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem iInter_subset_of_subset {s : ι → Set α} {t : Set α} (i : ι) (h : s i ⊆ t) :
⋂ i, s i ⊆ t :=
iInf_le_of_le i h
#align set.Inter_subset_of_subset Set.iInter_subset_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `subset_iUnion₂` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem subset_iUnion₂_of_subset {s : Set α} {t : ∀ i, κ i → Set α} (i : ι) (j : κ i)
(h : s ⊆ t i j) : s ⊆ ⋃ (i) (j), t i j :=
le_iSup₂_of_le i j h
#align set.subset_Union₂_of_subset Set.subset_iUnion₂_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `iInter₂_subset` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem iInter₂_subset_of_subset {s : ∀ i, κ i → Set α} {t : Set α} (i : ι) (j : κ i)
(h : s i j ⊆ t) : ⋂ (i) (j), s i j ⊆ t :=
iInf₂_le_of_le i j h
#align set.Inter₂_subset_of_subset Set.iInter₂_subset_of_subset
theorem iUnion_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono h
#align set.Union_mono Set.iUnion_mono
@[gcongr]
theorem iUnion_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iUnion s ⊆ iUnion t :=
iSup_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋃ (i) (j), s i j ⊆ ⋃ (i) (j), t i j :=
iSup₂_mono h
#align set.Union₂_mono Set.iUnion₂_mono
theorem iInter_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋂ i, s i ⊆ ⋂ i, t i :=
iInf_mono h
#align set.Inter_mono Set.iInter_mono
@[gcongr]
theorem iInter_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iInter s ⊆ iInter t :=
iInf_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋂ (i) (j), s i j ⊆ ⋂ (i) (j), t i j :=
iInf₂_mono h
#align set.Inter₂_mono Set.iInter₂_mono
theorem iUnion_mono' {s : ι → Set α} {t : ι₂ → Set α} (h : ∀ i, ∃ j, s i ⊆ t j) :
⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono' h
#align set.Union_mono' Set.iUnion_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iUnion₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i j, ∃ i' j', s i j ⊆ t i' j') : ⋃ (i) (j), s i j ⊆ ⋃ (i') (j'), t i' j' :=
iSup₂_mono' h
#align set.Union₂_mono' Set.iUnion₂_mono'
theorem iInter_mono' {s : ι → Set α} {t : ι' → Set α} (h : ∀ j, ∃ i, s i ⊆ t j) :
⋂ i, s i ⊆ ⋂ j, t j :=
Set.subset_iInter fun j =>
let ⟨i, hi⟩ := h j
iInter_subset_of_subset i hi
#align set.Inter_mono' Set.iInter_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iInter₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i' j', ∃ i j, s i j ⊆ t i' j') : ⋂ (i) (j), s i j ⊆ ⋂ (i') (j'), t i' j' :=
subset_iInter₂_iff.2 fun i' j' =>
let ⟨_, _, hst⟩ := h i' j'
(iInter₂_subset _ _).trans hst
#align set.Inter₂_mono' Set.iInter₂_mono'
theorem iUnion₂_subset_iUnion (κ : ι → Sort*) (s : ι → Set α) :
⋃ (i) (_ : κ i), s i ⊆ ⋃ i, s i :=
iUnion_mono fun _ => iUnion_subset fun _ => Subset.rfl
#align set.Union₂_subset_Union Set.iUnion₂_subset_iUnion
theorem iInter_subset_iInter₂ (κ : ι → Sort*) (s : ι → Set α) :
⋂ i, s i ⊆ ⋂ (i) (_ : κ i), s i :=
iInter_mono fun _ => subset_iInter fun _ => Subset.rfl
#align set.Inter_subset_Inter₂ Set.iInter_subset_iInter₂
theorem iUnion_setOf (P : ι → α → Prop) : ⋃ i, { x : α | P i x } = { x : α | ∃ i, P i x } := by
ext
exact mem_iUnion
#align set.Union_set_of Set.iUnion_setOf
theorem iInter_setOf (P : ι → α → Prop) : ⋂ i, { x : α | P i x } = { x : α | ∀ i, P i x } := by
ext
exact mem_iInter
#align set.Inter_set_of Set.iInter_setOf
theorem iUnion_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋃ x, f x = ⋃ y, g y :=
h1.iSup_congr h h2
#align set.Union_congr_of_surjective Set.iUnion_congr_of_surjective
theorem iInter_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋂ x, f x = ⋂ y, g y :=
h1.iInf_congr h h2
#align set.Inter_congr_of_surjective Set.iInter_congr_of_surjective
lemma iUnion_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋃ i, s i = ⋃ i, t i := iSup_congr h
#align set.Union_congr Set.iUnion_congr
lemma iInter_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋂ i, s i = ⋂ i, t i := iInf_congr h
#align set.Inter_congr Set.iInter_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iUnion₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋃ (i) (j), s i j = ⋃ (i) (j), t i j :=
iUnion_congr fun i => iUnion_congr <| h i
#align set.Union₂_congr Set.iUnion₂_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iInter₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋂ (i) (j), s i j = ⋂ (i) (j), t i j :=
iInter_congr fun i => iInter_congr <| h i
#align set.Inter₂_congr Set.iInter₂_congr
section Nonempty
variable [Nonempty ι] {f : ι → Set α} {s : Set α}
lemma iUnion_const (s : Set β) : ⋃ _ : ι, s = s := iSup_const
#align set.Union_const Set.iUnion_const
lemma iInter_const (s : Set β) : ⋂ _ : ι, s = s := iInf_const
#align set.Inter_const Set.iInter_const
lemma iUnion_eq_const (hf : ∀ i, f i = s) : ⋃ i, f i = s :=
(iUnion_congr hf).trans $ iUnion_const _
#align set.Union_eq_const Set.iUnion_eq_const
lemma iInter_eq_const (hf : ∀ i, f i = s) : ⋂ i, f i = s :=
(iInter_congr hf).trans $ iInter_const _
#align set.Inter_eq_const Set.iInter_eq_const
end Nonempty
@[simp]
theorem compl_iUnion (s : ι → Set β) : (⋃ i, s i)ᶜ = ⋂ i, (s i)ᶜ :=
compl_iSup
#align set.compl_Union Set.compl_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iUnion₂ (s : ∀ i, κ i → Set α) : (⋃ (i) (j), s i j)ᶜ = ⋂ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iUnion]
#align set.compl_Union₂ Set.compl_iUnion₂
@[simp]
theorem compl_iInter (s : ι → Set β) : (⋂ i, s i)ᶜ = ⋃ i, (s i)ᶜ :=
compl_iInf
#align set.compl_Inter Set.compl_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iInter₂ (s : ∀ i, κ i → Set α) : (⋂ (i) (j), s i j)ᶜ = ⋃ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iInter]
#align set.compl_Inter₂ Set.compl_iInter₂
-- classical -- complete_boolean_algebra
theorem iUnion_eq_compl_iInter_compl (s : ι → Set β) : ⋃ i, s i = (⋂ i, (s i)ᶜ)ᶜ := by
simp only [compl_iInter, compl_compl]
#align set.Union_eq_compl_Inter_compl Set.iUnion_eq_compl_iInter_compl
-- classical -- complete_boolean_algebra
theorem iInter_eq_compl_iUnion_compl (s : ι → Set β) : ⋂ i, s i = (⋃ i, (s i)ᶜ)ᶜ := by
simp only [compl_iUnion, compl_compl]
#align set.Inter_eq_compl_Union_compl Set.iInter_eq_compl_iUnion_compl
theorem inter_iUnion (s : Set β) (t : ι → Set β) : (s ∩ ⋃ i, t i) = ⋃ i, s ∩ t i :=
inf_iSup_eq _ _
#align set.inter_Union Set.inter_iUnion
theorem iUnion_inter (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∩ s = ⋃ i, t i ∩ s :=
iSup_inf_eq _ _
#align set.Union_inter Set.iUnion_inter
theorem iUnion_union_distrib (s : ι → Set β) (t : ι → Set β) :
⋃ i, s i ∪ t i = (⋃ i, s i) ∪ ⋃ i, t i :=
iSup_sup_eq
#align set.Union_union_distrib Set.iUnion_union_distrib
theorem iInter_inter_distrib (s : ι → Set β) (t : ι → Set β) :
⋂ i, s i ∩ t i = (⋂ i, s i) ∩ ⋂ i, t i :=
iInf_inf_eq
#align set.Inter_inter_distrib Set.iInter_inter_distrib
theorem union_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∪ ⋃ i, t i) = ⋃ i, s ∪ t i :=
sup_iSup
#align set.union_Union Set.union_iUnion
theorem iUnion_union [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∪ s = ⋃ i, t i ∪ s :=
iSup_sup
#align set.Union_union Set.iUnion_union
theorem inter_iInter [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∩ ⋂ i, t i) = ⋂ i, s ∩ t i :=
inf_iInf
#align set.inter_Inter Set.inter_iInter
theorem iInter_inter [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋂ i, t i) ∩ s = ⋂ i, t i ∩ s :=
iInf_inf
#align set.Inter_inter Set.iInter_inter
-- classical
theorem union_iInter (s : Set β) (t : ι → Set β) : (s ∪ ⋂ i, t i) = ⋂ i, s ∪ t i :=
sup_iInf_eq _ _
#align set.union_Inter Set.union_iInter
theorem iInter_union (s : ι → Set β) (t : Set β) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.Inter_union Set.iInter_union
theorem iUnion_diff (s : Set β) (t : ι → Set β) : (⋃ i, t i) \ s = ⋃ i, t i \ s :=
iUnion_inter _ _
#align set.Union_diff Set.iUnion_diff
theorem diff_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s \ ⋃ i, t i) = ⋂ i, s \ t i := by
rw [diff_eq, compl_iUnion, inter_iInter]; rfl
#align set.diff_Union Set.diff_iUnion
theorem diff_iInter (s : Set β) (t : ι → Set β) : (s \ ⋂ i, t i) = ⋃ i, s \ t i := by
rw [diff_eq, compl_iInter, inter_iUnion]; rfl
#align set.diff_Inter Set.diff_iInter
theorem directed_on_iUnion {r} {f : ι → Set α} (hd : Directed (· ⊆ ·) f)
(h : ∀ x, DirectedOn r (f x)) : DirectedOn r (⋃ x, f x) := by
simp only [DirectedOn, exists_prop, mem_iUnion, exists_imp]
exact fun a₁ b₁ fb₁ a₂ b₂ fb₂ =>
let ⟨z, zb₁, zb₂⟩ := hd b₁ b₂
let ⟨x, xf, xa₁, xa₂⟩ := h z a₁ (zb₁ fb₁) a₂ (zb₂ fb₂)
⟨x, ⟨z, xf⟩, xa₁, xa₂⟩
#align set.directed_on_Union Set.directed_on_iUnion
theorem iUnion_inter_subset {ι α} {s t : ι → Set α} : ⋃ i, s i ∩ t i ⊆ (⋃ i, s i) ∩ ⋃ i, t i :=
le_iSup_inf_iSup s t
#align set.Union_inter_subset Set.iUnion_inter_subset
theorem iUnion_inter_of_monotone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_monotone hs ht
#align set.Union_inter_of_monotone Set.iUnion_inter_of_monotone
theorem iUnion_inter_of_antitone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_antitone hs ht
#align set.Union_inter_of_antitone Set.iUnion_inter_of_antitone
theorem iInter_union_of_monotone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_monotone hs ht
#align set.Inter_union_of_monotone Set.iInter_union_of_monotone
theorem iInter_union_of_antitone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_antitone hs ht
#align set.Inter_union_of_antitone Set.iInter_union_of_antitone
/-- An equality version of this lemma is `iUnion_iInter_of_monotone` in `Data.Set.Finite`. -/
theorem iUnion_iInter_subset {s : ι → ι' → Set α} : (⋃ j, ⋂ i, s i j) ⊆ ⋂ i, ⋃ j, s i j :=
iSup_iInf_le_iInf_iSup (flip s)
#align set.Union_Inter_subset Set.iUnion_iInter_subset
theorem iUnion_option {ι} (s : Option ι → Set α) : ⋃ o, s o = s none ∪ ⋃ i, s (some i) :=
iSup_option s
#align set.Union_option Set.iUnion_option
theorem iInter_option {ι} (s : Option ι → Set α) : ⋂ o, s o = s none ∩ ⋂ i, s (some i) :=
iInf_option s
#align set.Inter_option Set.iInter_option
section
variable (p : ι → Prop) [DecidablePred p]
theorem iUnion_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 _ _ _
#align set.Union_dite Set.iUnion_dite
theorem iUnion_ite (f g : ι → Set α) :
⋃ i, (if p i then f i else g i) = (⋃ (i) (_ : p i), f i) ∪ ⋃ (i) (_ : ¬p i), g i :=
iUnion_dite _ _ _
#align set.Union_ite Set.iUnion_ite
theorem iInter_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 :=
iInf_dite _ _ _
#align set.Inter_dite Set.iInter_dite
theorem iInter_ite (f g : ι → Set α) :
⋂ i, (if p i then f i else g i) = (⋂ (i) (_ : p i), f i) ∩ ⋂ (i) (_ : ¬p i), g i :=
iInter_dite _ _ _
#align set.Inter_ite Set.iInter_ite
end
theorem image_projection_prod {ι : Type*} {α : ι → Type*} {v : ∀ i : ι, Set (α i)}
(hv : (pi univ v).Nonempty) (i : ι) :
((fun x : ∀ i : ι, α i => x i) '' ⋂ k, (fun x : ∀ j : ι, α j => x k) ⁻¹' v k) = v i := by
classical
apply Subset.antisymm
· simp [iInter_subset]
· intro y y_in
simp only [mem_image, mem_iInter, mem_preimage]
rcases hv with ⟨z, hz⟩
refine' ⟨Function.update z i y, _, update_same i y z⟩
rw [@forall_update_iff ι α _ z i y fun i t => t ∈ v i]
exact ⟨y_in, fun j _ => by simpa using hz j⟩
#align set.image_projection_prod Set.image_projection_prod
/-! ### Unions and intersections indexed by `Prop` -/
theorem iInter_false {s : False → Set α} : iInter s = univ :=
iInf_false
#align set.Inter_false Set.iInter_false
theorem iUnion_false {s : False → Set α} : iUnion s = ∅ :=
iSup_false
#align set.Union_false Set.iUnion_false
@[simp]
theorem iInter_true {s : True → Set α} : iInter s = s trivial :=
iInf_true
#align set.Inter_true Set.iInter_true
@[simp]
theorem iUnion_true {s : True → Set α} : iUnion s = s trivial :=
iSup_true
#align set.Union_true Set.iUnion_true
@[simp]
theorem iInter_exists {p : ι → Prop} {f : Exists p → Set α} :
⋂ x, f x = ⋂ (i) (h : p i), f ⟨i, h⟩ :=
iInf_exists
#align set.Inter_exists Set.iInter_exists
@[simp]
theorem iUnion_exists {p : ι → Prop} {f : Exists p → Set α} :
⋃ x, f x = ⋃ (i) (h : p i), f ⟨i, h⟩ :=
iSup_exists
#align set.Union_exists Set.iUnion_exists
@[simp]
theorem iUnion_empty : (⋃ _ : ι, ∅ : Set α) = ∅ :=
iSup_bot
#align set.Union_empty Set.iUnion_empty
@[simp]
theorem iInter_univ : (⋂ _ : ι, univ : Set α) = univ :=
iInf_top
#align set.Inter_univ Set.iInter_univ
section
variable {s : ι → Set α}
@[simp]
theorem iUnion_eq_empty : ⋃ i, s i = ∅ ↔ ∀ i, s i = ∅ :=
iSup_eq_bot
#align set.Union_eq_empty Set.iUnion_eq_empty
@[simp]
theorem iInter_eq_univ : ⋂ i, s i = univ ↔ ∀ i, s i = univ :=
iInf_eq_top
#align set.Inter_eq_univ Set.iInter_eq_univ
@[simp]
theorem nonempty_iUnion : (⋃ i, s i).Nonempty ↔ ∃ i, (s i).Nonempty := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_Union Set.nonempty_iUnion
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_biUnion {t : Set α} {s : α → Set β} :
(⋃ i ∈ t, s i).Nonempty ↔ ∃ i ∈ t, (s i).Nonempty := by simp
#align set.nonempty_bUnion Set.nonempty_biUnion
theorem iUnion_nonempty_index (s : Set α) (t : s.Nonempty → Set β) :
⋃ h, t h = ⋃ x ∈ s, t ⟨x, ‹_›⟩ :=
iSup_exists
#align set.Union_nonempty_index Set.iUnion_nonempty_index
end
@[simp]
theorem iInter_iInter_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋂ (x) (h : x = b), s x h = s b rfl :=
iInf_iInf_eq_left
#align set.Inter_Inter_eq_left Set.iInter_iInter_eq_left
@[simp]
theorem iInter_iInter_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋂ (x) (h : b = x), s x h = s b rfl :=
iInf_iInf_eq_right
#align set.Inter_Inter_eq_right Set.iInter_iInter_eq_right
@[simp]
theorem iUnion_iUnion_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋃ (x) (h : x = b), s x h = s b rfl :=
iSup_iSup_eq_left
#align set.Union_Union_eq_left Set.iUnion_iUnion_eq_left
@[simp]
theorem iUnion_iUnion_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋃ (x) (h : b = x), s x h = s b rfl :=
iSup_iSup_eq_right
#align set.Union_Union_eq_right Set.iUnion_iUnion_eq_right
theorem iInter_or {p q : Prop} (s : p ∨ q → Set α) :
⋂ h, s h = (⋂ h : p, s (Or.inl h)) ∩ ⋂ h : q, s (Or.inr h) :=
iInf_or
#align set.Inter_or Set.iInter_or
theorem iUnion_or {p q : Prop} (s : p ∨ q → Set α) :
⋃ h, s h = (⋃ i, s (Or.inl i)) ∪ ⋃ j, s (Or.inr j) :=
iSup_or
#align set.Union_or Set.iUnion_or
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iUnion_and {p q : Prop} (s : p ∧ q → Set α) : ⋃ h, s h = ⋃ (hp) (hq), s ⟨hp, hq⟩ :=
iSup_and
#align set.Union_and Set.iUnion_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iInter_and {p q : Prop} (s : p ∧ q → Set α) : ⋂ h, s h = ⋂ (hp) (hq), s ⟨hp, hq⟩ :=
iInf_and
#align set.Inter_and Set.iInter_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iUnion_comm (s : ι → ι' → Set α) : ⋃ (i) (i'), s i i' = ⋃ (i') (i), s i i' :=
iSup_comm
#align set.Union_comm Set.iUnion_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iInter_comm (s : ι → ι' → Set α) : ⋂ (i) (i'), s i i' = ⋂ (i') (i), s i i' :=
iInf_comm
#align set.Inter_comm Set.iInter_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iUnion₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋃ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋃ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iSup₂_comm _
#align set.Union₂_comm Set.iUnion₂_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iInter₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋂ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋂ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iInf₂_comm _
#align set.Inter₂_comm Set.iInter₂_comm
@[simp]
theorem biUnion_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋃ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι']
#align set.bUnion_and Set.biUnion_and
@[simp]
theorem biUnion_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋃ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι]
#align set.bUnion_and' Set.biUnion_and'
@[simp]
theorem biInter_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋂ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iInter_and, @iInter_comm _ ι']
#align set.bInter_and Set.biInter_and
@[simp]
theorem biInter_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋂ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iInter_and, @iInter_comm _ ι]
#align set.bInter_and' Set.biInter_and'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iUnion_iUnion_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋃ (x) (h), s x h = s b (Or.inl rfl) ∪ ⋃ (x) (h : p x), s x (Or.inr h) := by
simp only [iUnion_or, iUnion_union_distrib, iUnion_iUnion_eq_left]
#align set.Union_Union_eq_or_left Set.iUnion_iUnion_eq_or_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iInter_iInter_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋂ (x) (h), s x h = s b (Or.inl rfl) ∩ ⋂ (x) (h : p x), s x (Or.inr h) := by
simp only [iInter_or, iInter_inter_distrib, iInter_iInter_eq_left]
#align set.Inter_Inter_eq_or_left Set.iInter_iInter_eq_or_left
/-! ### Bounded unions and intersections -/
/-- A specialization of `mem_iUnion₂`. -/
theorem mem_biUnion {s : Set α} {t : α → Set β} {x : α} {y : β} (xs : x ∈ s) (ytx : y ∈ t x) :
y ∈ ⋃ x ∈ s, t x :=
mem_iUnion₂_of_mem xs ytx
#align set.mem_bUnion Set.mem_biUnion
/-- A specialization of `mem_iInter₂`. -/
theorem mem_biInter {s : Set α} {t : α → Set β} {y : β} (h : ∀ x ∈ s, y ∈ t x) :
y ∈ ⋂ x ∈ s, t x :=
mem_iInter₂_of_mem h
#align set.mem_bInter Set.mem_biInter
/-- A specialization of `subset_iUnion₂`. -/
theorem subset_biUnion_of_mem {s : Set α} {u : α → Set β} {x : α} (xs : x ∈ s) :
u x ⊆ ⋃ x ∈ s, u x :=
--Porting note: Why is this not just `subset_iUnion₂ x xs`?
@subset_iUnion₂ β α (· ∈ s) (fun i _ => u i) x xs
#align set.subset_bUnion_of_mem Set.subset_biUnion_of_mem
/-- A specialization of `iInter₂_subset`. -/
theorem biInter_subset_of_mem {s : Set α} {t : α → Set β} {x : α} (xs : x ∈ s) :
⋂ x ∈ s, t x ⊆ t x :=
iInter₂_subset x xs
#align set.bInter_subset_of_mem Set.biInter_subset_of_mem
theorem biUnion_subset_biUnion_left {s s' : Set α} {t : α → Set β} (h : s ⊆ s') :
⋃ x ∈ s, t x ⊆ ⋃ x ∈ s', t x :=
iUnion₂_subset fun _ hx => subset_biUnion_of_mem <| h hx
#align set.bUnion_subset_bUnion_left Set.biUnion_subset_biUnion_left
theorem biInter_subset_biInter_left {s s' : Set α} {t : α → Set β} (h : s' ⊆ s) :
⋂ x ∈ s, t x ⊆ ⋂ x ∈ s', t x :=
subset_iInter₂ fun _ hx => biInter_subset_of_mem <| h hx
#align set.bInter_subset_bInter_left Set.biInter_subset_biInter_left
theorem biUnion_mono {s s' : Set α} {t t' : α → Set β} (hs : s' ⊆ s) (h : ∀ x ∈ s, t x ⊆ t' x) :
⋃ x ∈ s', t x ⊆ ⋃ x ∈ s, t' x :=
(biUnion_subset_biUnion_left hs).trans <| iUnion₂_mono h
#align set.bUnion_mono Set.biUnion_mono
theorem biInter_mono {s s' : Set α} {t t' : α → Set β} (hs : s ⊆ s') (h : ∀ x ∈ s, t x ⊆ t' x) :
⋂ x ∈ s', t x ⊆ ⋂ x ∈ s, t' x :=
(biInter_subset_biInter_left hs).trans <| iInter₂_mono h
#align set.bInter_mono Set.biInter_mono
theorem biUnion_eq_iUnion (s : Set α) (t : ∀ x ∈ s, Set β) :
⋃ x ∈ s, t x ‹_› = ⋃ x : s, t x x.2 :=
iSup_subtype'
#align set.bUnion_eq_Union Set.biUnion_eq_iUnion
theorem biInter_eq_iInter (s : Set α) (t : ∀ x ∈ s, Set β) :
⋂ x ∈ s, t x ‹_› = ⋂ x : s, t x x.2 :=
iInf_subtype'
#align set.bInter_eq_Inter Set.biInter_eq_iInter
theorem iUnion_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋃ x : { x // p x }, s x = ⋃ (x) (hx : p x), s ⟨x, hx⟩ :=
iSup_subtype
#align set.Union_subtype Set.iUnion_subtype
theorem iInter_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋂ x : { x // p x }, s x = ⋂ (x) (hx : p x), s ⟨x, hx⟩ :=
iInf_subtype
#align set.Inter_subtype Set.iInter_subtype
theorem biInter_empty (u : α → Set β) : ⋂ x ∈ (∅ : Set α), u x = univ :=
iInf_emptyset
#align set.bInter_empty Set.biInter_empty
theorem biInter_univ (u : α → Set β) : ⋂ x ∈ @univ α, u x = ⋂ x, u x :=
iInf_univ
#align set.bInter_univ Set.biInter_univ
@[simp]
theorem biUnion_self (s : Set α) : ⋃ x ∈ s, s = s :=
Subset.antisymm (iUnion₂_subset fun _ _ => Subset.refl s) fun _ hx => mem_biUnion hx hx
#align set.bUnion_self Set.biUnion_self
@[simp]
theorem iUnion_nonempty_self (s : Set α) : ⋃ _ : s.Nonempty, s = s := by
rw [iUnion_nonempty_index, biUnion_self]
#align set.Union_nonempty_self Set.iUnion_nonempty_self
theorem biInter_singleton (a : α) (s : α → Set β) : ⋂ x ∈ ({a} : Set α), s x = s a :=
iInf_singleton
#align set.bInter_singleton Set.biInter_singleton
theorem biInter_union (s t : Set α) (u : α → Set β) :
⋂ x ∈ s ∪ t, u x = (⋂ x ∈ s, u x) ∩ ⋂ x ∈ t, u x :=
iInf_union
#align set.bInter_union Set.biInter_union
theorem biInter_insert (a : α) (s : Set α) (t : α → Set β) :
⋂ x ∈ insert a s, t x = t a ∩ ⋂ x ∈ s, t x := by simp
#align set.bInter_insert Set.biInter_insert
theorem biInter_pair (a b : α) (s : α → Set β) : ⋂ x ∈ ({a, b} : Set α), s x = s a ∩ s b := by
rw [biInter_insert, biInter_singleton]
#align set.bInter_pair Set.biInter_pair
theorem biInter_inter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, f i ∩ t = (⋂ i ∈ s, f i) ∩ t := by
haveI : Nonempty s := hs.to_subtype
simp [biInter_eq_iInter, ← iInter_inter]
#align set.bInter_inter Set.biInter_inter
theorem inter_biInter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, t ∩ f i = t ∩ ⋂ i ∈ s, f i := by
rw [inter_comm, ← biInter_inter hs]
simp [inter_comm]
#align set.inter_bInter Set.inter_biInter
theorem biUnion_empty (s : α → Set β) : ⋃ x ∈ (∅ : Set α), s x = ∅ :=
iSup_emptyset
#align set.bUnion_empty Set.biUnion_empty
theorem biUnion_univ (s : α → Set β) : ⋃ x ∈ @univ α, s x = ⋃ x, s x :=
iSup_univ
#align set.bUnion_univ Set.biUnion_univ
theorem biUnion_singleton (a : α) (s : α → Set β) : ⋃ x ∈ ({a} : Set α), s x = s a :=
iSup_singleton
#align set.bUnion_singleton Set.biUnion_singleton
@[simp]
theorem biUnion_of_singleton (s : Set α) : ⋃ x ∈ s, {x} = s :=
ext <| by simp
#align set.bUnion_of_singleton Set.biUnion_of_singleton
theorem biUnion_union (s t : Set α) (u : α → Set β) :
⋃ x ∈ s ∪ t, u x = (⋃ x ∈ s, u x) ∪ ⋃ x ∈ t, u x :=
iSup_union
#align set.bUnion_union Set.biUnion_union
@[simp]
theorem iUnion_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋃ i, f i = ⋃ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iUnion_subtype _ _
#align set.Union_coe_set Set.iUnion_coe_set
@[simp]
theorem iInter_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋂ i, f i = ⋂ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iInter_subtype _ _
#align set.Inter_coe_set Set.iInter_coe_set
theorem biUnion_insert (a : α) (s : Set α) (t : α → Set β) :
⋃ x ∈ insert a s, t x = t a ∪ ⋃ x ∈ s, t x := by simp
#align set.bUnion_insert Set.biUnion_insert
theorem biUnion_pair (a b : α) (s : α → Set β) : ⋃ x ∈ ({a, b} : Set α), s x = s a ∪ s b :=
by simp
#align set.bUnion_pair Set.biUnion_pair
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem inter_iUnion₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∩ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ∩ t i j := by simp only [inter_iUnion]
#align set.inter_Union₂ Set.inter_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_inter (s : ∀ i, κ i → Set α) (t : Set α) :
(⋃ (i) (j), s i j) ∩ t = ⋃ (i) (j), s i j ∩ t := by simp_rw [iUnion_inter]
#align set.Union₂_inter Set.iUnion₂_inter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_iInter₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_iInter]
#align set.union_Inter₂ Set.union_iInter₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_union (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [iInter_union]
#align set.Inter₂_union Set.iInter₂_union
theorem mem_sUnion_of_mem {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∈ t) (ht : t ∈ S) :
x ∈ ⋃₀S :=
⟨t, ht, hx⟩
#align set.mem_sUnion_of_mem Set.mem_sUnion_of_mem
-- is this theorem really necessary?
theorem not_mem_of_not_mem_sUnion {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∉ ⋃₀S)
(ht : t ∈ S) : x ∉ t := fun h => hx ⟨t, ht, h⟩
#align set.not_mem_of_not_mem_sUnion Set.not_mem_of_not_mem_sUnion
theorem sInter_subset_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : ⋂₀ S ⊆ t :=
sInf_le tS
#align set.sInter_subset_of_mem Set.sInter_subset_of_mem
theorem subset_sUnion_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : t ⊆ ⋃₀S :=
le_sSup tS
#align set.subset_sUnion_of_mem Set.subset_sUnion_of_mem
theorem subset_sUnion_of_subset {s : Set α} (t : Set (Set α)) (u : Set α) (h₁ : s ⊆ u)
(h₂ : u ∈ t) : s ⊆ ⋃₀t :=
Subset.trans h₁ (subset_sUnion_of_mem h₂)
#align set.subset_sUnion_of_subset Set.subset_sUnion_of_subset
theorem sUnion_subset {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t' ⊆ t) : ⋃₀S ⊆ t :=
sSup_le h
#align set.sUnion_subset Set.sUnion_subset
@[simp]
theorem sUnion_subset_iff {s : Set (Set α)} {t : Set α} : ⋃₀s ⊆ t ↔ ∀ t' ∈ s, t' ⊆ t :=
sSup_le_iff
#align set.sUnion_subset_iff Set.sUnion_subset_iff
/-- `sUnion` is monotone under taking a subset of each set. -/
lemma sUnion_mono_subsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, t ⊆ f t) :
⋃₀ s ⊆ ⋃₀ (f '' s) :=
fun _ ⟨t, htx, hxt⟩ ↦ ⟨f t, mem_image_of_mem f htx, hf t hxt⟩
/-- `sUnion` is monotone under taking a superset of each set. -/
lemma sUnion_mono_supsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, f t ⊆ t) :
⋃₀ (f '' s) ⊆ ⋃₀ s :=
-- If t ∈ f '' s is arbitrary; t = f u for some u : Set α.
fun _ ⟨_, ⟨u, hus, hut⟩, hxt⟩ ↦ ⟨u, hus, (hut ▸ hf u) hxt⟩
theorem subset_sInter {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t ⊆ t') : t ⊆ ⋂₀ S :=
le_sInf h
#align set.subset_sInter Set.subset_sInter
@[simp]
theorem subset_sInter_iff {S : Set (Set α)} {t : Set α} : t ⊆ ⋂₀ S ↔ ∀ t' ∈ S, t ⊆ t' :=
le_sInf_iff
#align set.subset_sInter_iff Set.subset_sInter_iff
@[gcongr]
theorem sUnion_subset_sUnion {S T : Set (Set α)} (h : S ⊆ T) : ⋃₀S ⊆ ⋃₀T :=
sUnion_subset fun _ hs => subset_sUnion_of_mem (h hs)
#align set.sUnion_subset_sUnion Set.sUnion_subset_sUnion
@[gcongr]
theorem sInter_subset_sInter {S T : Set (Set α)} (h : S ⊆ T) : ⋂₀ T ⊆ ⋂₀ S :=
subset_sInter fun _ hs => sInter_subset_of_mem (h hs)
#align set.sInter_subset_sInter Set.sInter_subset_sInter
@[simp]
theorem sUnion_empty : ⋃₀∅ = (∅ : Set α) :=
sSup_empty
#align set.sUnion_empty Set.sUnion_empty
@[simp]
theorem sInter_empty : ⋂₀ ∅ = (univ : Set α) :=
sInf_empty
#align set.sInter_empty Set.sInter_empty
@[simp]
theorem sUnion_singleton (s : Set α) : ⋃₀{s} = s :=
sSup_singleton
#align set.sUnion_singleton Set.sUnion_singleton
@[simp]
theorem sInter_singleton (s : Set α) : ⋂₀ {s} = s :=
sInf_singleton
#align set.sInter_singleton Set.sInter_singleton
@[simp]
theorem sUnion_eq_empty {S : Set (Set α)} : ⋃₀S = ∅ ↔ ∀ s ∈ S, s = ∅ :=
sSup_eq_bot
#align set.sUnion_eq_empty Set.sUnion_eq_empty
@[simp]
theorem sInter_eq_univ {S : Set (Set α)} : ⋂₀ S = univ ↔ ∀ s ∈ S, s = univ :=
sInf_eq_top
#align set.sInter_eq_univ Set.sInter_eq_univ
theorem subset_powerset_iff {s : Set (Set α)} {t : Set α} : s ⊆ 𝒫 t ↔ ⋃₀ s ⊆ t :=
sUnion_subset_iff.symm
/-- `⋃₀` and `𝒫` form a Galois connection. -/
theorem sUnion_powerset_gc :
GaloisConnection (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gc_sSup_Iic
/-- `⋃₀` and `𝒫` form a Galois insertion. -/
def sUnion_powerset_gi :
GaloisInsertion (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gi_sSup_Iic
/-- If all sets in a collection are either `∅` or `Set.univ`, then so is their union. -/
theorem sUnion_mem_empty_univ {S : Set (Set α)} (h : S ⊆ {∅, univ}) :
⋃₀ S ∈ ({∅, univ} : Set (Set α)) := by
simp only [mem_insert_iff, mem_singleton_iff, or_iff_not_imp_left, sUnion_eq_empty, not_forall]
rintro ⟨s, hs, hne⟩
obtain rfl : s = univ := (h hs).resolve_left hne
exact univ_subset_iff.1 <| subset_sUnion_of_mem hs
@[simp]
theorem nonempty_sUnion {S : Set (Set α)} : (⋃₀S).Nonempty ↔ ∃ s ∈ S, Set.Nonempty s := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_sUnion Set.nonempty_sUnion
theorem Nonempty.of_sUnion {s : Set (Set α)} (h : (⋃₀s).Nonempty) : s.Nonempty :=
let ⟨s, hs, _⟩ := nonempty_sUnion.1 h
⟨s, hs⟩
#align set.nonempty.of_sUnion Set.Nonempty.of_sUnion
theorem Nonempty.of_sUnion_eq_univ [Nonempty α] {s : Set (Set α)} (h : ⋃₀s = univ) : s.Nonempty :=
Nonempty.of_sUnion <| h.symm ▸ univ_nonempty
#align set.nonempty.of_sUnion_eq_univ Set.Nonempty.of_sUnion_eq_univ
theorem sUnion_union (S T : Set (Set α)) : ⋃₀(S ∪ T) = ⋃₀S ∪ ⋃₀T :=
sSup_union
#align set.sUnion_union Set.sUnion_union
theorem sInter_union (S T : Set (Set α)) : ⋂₀ (S ∪ T) = ⋂₀ S ∩ ⋂₀ T :=
sInf_union
#align set.sInter_union Set.sInter_union
@[simp]
theorem sUnion_insert (s : Set α) (T : Set (Set α)) : ⋃₀insert s T = s ∪ ⋃₀T :=
sSup_insert
#align set.sUnion_insert Set.sUnion_insert
@[simp]
theorem sInter_insert (s : Set α) (T : Set (Set α)) : ⋂₀ insert s T = s ∩ ⋂₀ T :=
sInf_insert
#align set.sInter_insert Set.sInter_insert
@[simp]
theorem sUnion_diff_singleton_empty (s : Set (Set α)) : ⋃₀(s \ {∅}) = ⋃₀s :=
sSup_diff_singleton_bot s
#align set.sUnion_diff_singleton_empty Set.sUnion_diff_singleton_empty
@[simp]
theorem sInter_diff_singleton_univ (s : Set (Set α)) : ⋂₀ (s \ {univ}) = ⋂₀ s :=
sInf_diff_singleton_top s
#align set.sInter_diff_singleton_univ Set.sInter_diff_singleton_univ
theorem sUnion_pair (s t : Set α) : ⋃₀{s, t} = s ∪ t :=
sSup_pair
#align set.sUnion_pair Set.sUnion_pair
theorem sInter_pair (s t : Set α) : ⋂₀ {s, t} = s ∩ t :=
sInf_pair
#align set.sInter_pair Set.sInter_pair
@[simp]
theorem sUnion_image (f : α → Set β) (s : Set α) : ⋃₀(f '' s) = ⋃ x ∈ s, f x :=
sSup_image
#align set.sUnion_image Set.sUnion_image
@[simp]
theorem sInter_image (f : α → Set β) (s : Set α) : ⋂₀ (f '' s) = ⋂ x ∈ s, f x :=
sInf_image
#align set.sInter_image Set.sInter_image
@[simp]
theorem sUnion_range (f : ι → Set β) : ⋃₀range f = ⋃ x, f x :=
rfl
#align set.sUnion_range Set.sUnion_range
@[simp]
theorem sInter_range (f : ι → Set β) : ⋂₀ range f = ⋂ x, f x :=
rfl
#align set.sInter_range Set.sInter_range
theorem iUnion_eq_univ_iff {f : ι → Set α} : ⋃ i, f i = univ ↔ ∀ x, ∃ i, x ∈ f i := by
simp only [eq_univ_iff_forall, mem_iUnion]
#align set.Union_eq_univ_iff Set.iUnion_eq_univ_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_eq_univ_iff {s : ∀ i, κ i → Set α} :
⋃ (i) (j), s i j = univ ↔ ∀ a, ∃ i j, a ∈ s i j :=
by simp only [iUnion_eq_univ_iff, mem_iUnion]
#align set.Union₂_eq_univ_iff Set.iUnion₂_eq_univ_iff
theorem sUnion_eq_univ_iff {c : Set (Set α)} : ⋃₀c = univ ↔ ∀ a, ∃ b ∈ c, a ∈ b := by
simp only [eq_univ_iff_forall, mem_sUnion]
#align set.sUnion_eq_univ_iff Set.sUnion_eq_univ_iff
-- classical
theorem iInter_eq_empty_iff {f : ι → Set α} : ⋂ i, f i = ∅ ↔ ∀ x, ∃ i, x ∉ f i := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.Inter_eq_empty_iff Set.iInter_eq_empty_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
theorem iInter₂_eq_empty_iff {s : ∀ i, κ i → Set α} :
⋂ (i) (j), s i j = ∅ ↔ ∀ a, ∃ i j, a ∉ s i j := by
simp only [eq_empty_iff_forall_not_mem, mem_iInter, not_forall]
#align set.Inter₂_eq_empty_iff Set.iInter₂_eq_empty_iff
-- classical
theorem sInter_eq_empty_iff {c : Set (Set α)} : ⋂₀ c = ∅ ↔ ∀ a, ∃ b ∈ c, a ∉ b := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.sInter_eq_empty_iff Set.sInter_eq_empty_iff
-- classical
@[simp]
theorem nonempty_iInter {f : ι → Set α} : (⋂ i, f i).Nonempty ↔ ∃ x, ∀ i, x ∈ f i := by
simp [nonempty_iff_ne_empty, iInter_eq_empty_iff]
#align set.nonempty_Inter Set.nonempty_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_iInter₂ {s : ∀ i, κ i → Set α} :
(⋂ (i) (j), s i j).Nonempty ↔ ∃ a, ∀ i j, a ∈ s i j := by
simp
#align set.nonempty_Inter₂ Set.nonempty_iInter₂
-- classical
@[simp]
theorem nonempty_sInter {c : Set (Set α)} : (⋂₀ c).Nonempty ↔ ∃ a, ∀ b ∈ c, a ∈ b := by
simp [nonempty_iff_ne_empty, sInter_eq_empty_iff]
#align set.nonempty_sInter Set.nonempty_sInter
-- classical
theorem compl_sUnion (S : Set (Set α)) : (⋃₀S)ᶜ = ⋂₀ (compl '' S) :=
ext fun x => by simp
#align set.compl_sUnion Set.compl_sUnion
-- classical
theorem sUnion_eq_compl_sInter_compl (S : Set (Set α)) : ⋃₀S = (⋂₀ (compl '' S))ᶜ := by
rw [← compl_compl (⋃₀S), compl_sUnion]
#align set.sUnion_eq_compl_sInter_compl Set.sUnion_eq_compl_sInter_compl
-- classical
theorem compl_sInter (S : Set (Set α)) : (⋂₀ S)ᶜ = ⋃₀(compl '' S) := by
rw [sUnion_eq_compl_sInter_compl, compl_compl_image]
#align set.compl_sInter Set.compl_sInter
-- classical
theorem sInter_eq_compl_sUnion_compl (S : Set (Set α)) : ⋂₀ S = (⋃₀(compl '' S))ᶜ := by
rw [← compl_compl (⋂₀ S), compl_sInter]
#align set.sInter_eq_compl_sUnion_compl Set.sInter_eq_compl_sUnion_compl
theorem inter_empty_of_inter_sUnion_empty {s t : Set α} {S : Set (Set α)} (hs : t ∈ S)
(h : s ∩ ⋃₀S = ∅) : s ∩ t = ∅ :=
eq_empty_of_subset_empty <| by
rw [← h]; exact inter_subset_inter_right _ (subset_sUnion_of_mem hs)
#align set.inter_empty_of_inter_sUnion_empty Set.inter_empty_of_inter_sUnion_empty
theorem range_sigma_eq_iUnion_range {γ : α → Type*} (f : Sigma γ → β) :
range f = ⋃ a, range fun b => f ⟨a, b⟩ :=
Set.ext <| by simp
#align set.range_sigma_eq_Union_range Set.range_sigma_eq_iUnion_range
theorem iUnion_eq_range_sigma (s : α → Set β) : ⋃ i, s i = range fun a : Σi, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_sigma Set.iUnion_eq_range_sigma
theorem iUnion_eq_range_psigma (s : ι → Set β) : ⋃ i, s i = range fun a : Σ'i, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_psigma Set.iUnion_eq_range_psigma
theorem iUnion_image_preimage_sigma_mk_eq_self {ι : Type*} {σ : ι → Type*} (s : Set (Sigma σ)) :
⋃ i, Sigma.mk i '' (Sigma.mk i ⁻¹' s) = s := by
ext x
simp only [mem_iUnion, mem_image, mem_preimage]
constructor
· rintro ⟨i, a, h, rfl⟩
exact h
· intro h
cases' x with i a
exact ⟨i, a, h, rfl⟩
#align set.Union_image_preimage_sigma_mk_eq_self Set.iUnion_image_preimage_sigma_mk_eq_self
theorem Sigma.univ (X : α → Type*) : (Set.univ : Set (Σa, X a)) = ⋃ a, range (Sigma.mk a) :=
Set.ext fun x =>
iff_of_true trivial ⟨range (Sigma.mk x.1), Set.mem_range_self _, x.2, Sigma.eta x⟩
#align set.sigma.univ Set.Sigma.univ
alias sUnion_mono := sUnion_subset_sUnion
#align set.sUnion_mono Set.sUnion_mono
theorem iUnion_subset_iUnion_const {s : Set α} (h : ι → ι₂) : ⋃ _ : ι, s ⊆ ⋃ _ : ι₂, s :=
@iSup_const_mono (Set α) ι ι₂ _ s h
#align set.Union_subset_Union_const Set.iUnion_subset_iUnion_const
@[simp]
theorem iUnion_singleton_eq_range {α β : Type*} (f : α → β) : ⋃ x : α, {f x} = range f := by
ext x
simp [@eq_comm _ x]
#align set.Union_singleton_eq_range Set.iUnion_singleton_eq_range
theorem iUnion_of_singleton (α : Type*) : (⋃ x, {x} : Set α) = univ := by simp [Set.ext_iff]
#align set.Union_of_singleton Set.iUnion_of_singleton
theorem iUnion_of_singleton_coe (s : Set α) : ⋃ i : s, ({(i : α)} : Set α) = s := by simp
#align set.Union_of_singleton_coe Set.iUnion_of_singleton_coe
theorem sUnion_eq_biUnion {s : Set (Set α)} : ⋃₀s = ⋃ (i : Set α) (_ : i ∈ s), i := by
rw [← sUnion_image, image_id']
#align set.sUnion_eq_bUnion Set.sUnion_eq_biUnion
theorem sInter_eq_biInter {s : Set (Set α)} : ⋂₀ s = ⋂ (i : Set α) (_ : i ∈ s), i := by
rw [← sInter_image, image_id']
#align set.sInter_eq_bInter Set.sInter_eq_biInter
theorem sUnion_eq_iUnion {s : Set (Set α)} : ⋃₀s = ⋃ i : s, i := by
simp only [← sUnion_range, Subtype.range_coe]
#align set.sUnion_eq_Union Set.sUnion_eq_iUnion
theorem sInter_eq_iInter {s : Set (Set α)} : ⋂₀ s = ⋂ i : s, i := by
simp only [← sInter_range, Subtype.range_coe]
#align set.sInter_eq_Inter Set.sInter_eq_iInter
@[simp]
theorem iUnion_of_empty [IsEmpty ι] (s : ι → Set α) : ⋃ i, s i = ∅ :=
iSup_of_empty _
#align set.Union_of_empty Set.iUnion_of_empty
@[simp]
theorem iInter_of_empty [IsEmpty ι] (s : ι → Set α) : ⋂ i, s i = univ :=
iInf_of_empty _
#align set.Inter_of_empty Set.iInter_of_empty
theorem union_eq_iUnion {s₁ s₂ : Set α} : s₁ ∪ s₂ = ⋃ b : Bool, cond b s₁ s₂ :=
sup_eq_iSup s₁ s₂
#align set.union_eq_Union Set.union_eq_iUnion
theorem inter_eq_iInter {s₁ s₂ : Set α} : s₁ ∩ s₂ = ⋂ b : Bool, cond b s₁ s₂ :=
inf_eq_iInf s₁ s₂
#align set.inter_eq_Inter Set.inter_eq_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_union_sInter {S T : Set (Set α)} :
⋂₀ S ∪ ⋂₀ T = ⋂ p ∈ S ×ˢ T, (p : Set α × Set α).1 ∪ p.2 :=
sInf_sup_sInf
#align set.sInter_union_sInter Set.sInter_union_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_inter_sUnion {s t : Set (Set α)} :
⋃₀s ∩ ⋃₀t = ⋃ p ∈ s ×ˢ t, (p : Set α × Set α).1 ∩ p.2 :=
sSup_inf_sSup
#align set.sUnion_inter_sUnion Set.sUnion_inter_sUnion
theorem biUnion_iUnion (s : ι → Set α) (t : α → Set β) :
⋃ x ∈ ⋃ i, s i, t x = ⋃ (i) (x ∈ s i), t x := by simp [@iUnion_comm _ ι]
#align set.bUnion_Union Set.biUnion_iUnion
theorem biInter_iUnion (s : ι → Set α) (t : α → Set β) :
⋂ x ∈ ⋃ i, s i, t x = ⋂ (i) (x ∈ s i), t x := by simp [@iInter_comm _ ι]
#align set.bInter_Union Set.biInter_iUnion
theorem sUnion_iUnion (s : ι → Set (Set α)) : ⋃₀⋃ i, s i = ⋃ i, ⋃₀s i := by
simp only [sUnion_eq_biUnion, biUnion_iUnion]
#align set.sUnion_Union Set.sUnion_iUnion
theorem sInter_iUnion (s : ι → Set (Set α)) : ⋂₀ ⋃ i, s i = ⋂ i, ⋂₀ s i := by
simp only [sInter_eq_biInter, biInter_iUnion]
#align set.sInter_Union Set.sInter_iUnion
theorem iUnion_range_eq_sUnion {α β : Type*} (C : Set (Set α)) {f : ∀ s : C, β → (s : Type _)}
(hf : ∀ s : C, Surjective (f s)) : ⋃ y : β, range (fun s : C => (f s y).val) = ⋃₀C := by
ext x; constructor
· rintro ⟨s, ⟨y, rfl⟩, ⟨s, hs⟩, rfl⟩
refine' ⟨_, hs, _⟩
exact (f ⟨s, hs⟩ y).2
· rintro ⟨s, hs, hx⟩
cases' hf ⟨s, hs⟩ ⟨x, hx⟩ with y hy
refine' ⟨_, ⟨y, rfl⟩, ⟨s, hs⟩, _⟩
exact congr_arg Subtype.val hy
#align set.Union_range_eq_sUnion Set.iUnion_range_eq_sUnion
theorem iUnion_range_eq_iUnion (C : ι → Set α) {f : ∀ x : ι, β → C x}
(hf : ∀ x : ι, Surjective (f x)) : ⋃ y : β, range (fun x : ι => (f x y).val) = ⋃ x, C x := by
ext x; rw [mem_iUnion, mem_iUnion]; constructor
· rintro ⟨y, i, rfl⟩
exact ⟨i, (f i y).2⟩
· rintro ⟨i, hx⟩
cases' hf i ⟨x, hx⟩ with y hy
exact ⟨y, i, congr_arg Subtype.val hy⟩
#align set.Union_range_eq_Union Set.iUnion_range_eq_iUnion
theorem union_distrib_iInter_left (s : ι → Set α) (t : Set α) : (t ∪ ⋂ i, s i) = ⋂ i, t ∪ s i :=
sup_iInf_eq _ _
#align set.union_distrib_Inter_left Set.union_distrib_iInter_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_left (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_distrib_iInter_left]
#align set.union_distrib_Inter₂_left Set.union_distrib_iInter₂_left
theorem union_distrib_iInter_right (s : ι → Set α) (t : Set α) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.union_distrib_Inter_right Set.union_distrib_iInter_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_right (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [union_distrib_iInter_right]
#align set.union_distrib_Inter₂_right Set.union_distrib_iInter₂_right
section Function
/-! ### `mapsTo` -/
theorem mapsTo_sUnion {S : Set (Set α)} {t : Set β} {f : α → β} (H : ∀ s ∈ S, MapsTo f s t) :
MapsTo f (⋃₀S) t := fun _ ⟨s, hs, hx⟩ => H s hs hx
#align set.maps_to_sUnion Set.mapsTo_sUnion
theorem mapsTo_iUnion {s : ι → Set α} {t : Set β} {f : α → β} (H : ∀ i, MapsTo f (s i) t) :
MapsTo f (⋃ i, s i) t :=
mapsTo_sUnion <| forall_range_iff.2 H
#align set.maps_to_Union Set.mapsTo_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂ {s : ∀ i, κ i → Set α} {t : Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) t) : MapsTo f (⋃ (i) (j), s i j) t :=
mapsTo_iUnion fun i => mapsTo_iUnion (H i)
#align set.maps_to_Union₂ Set.mapsTo_iUnion₂
theorem mapsTo_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋃ i, s i) (⋃ i, t i) :=
mapsTo_iUnion fun i => (H i).mono (Subset.refl _) (subset_iUnion t i)
#align set.maps_to_Union_Union Set.mapsTo_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
mapsTo_iUnion_iUnion fun i => mapsTo_iUnion_iUnion (H i)
#align set.maps_to_Union₂_Union₂ Set.mapsTo_iUnion₂_iUnion₂
theorem mapsTo_sInter {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, MapsTo f s t) :
MapsTo f s (⋂₀ T) := fun _ hx t ht => H t ht hx
#align set.maps_to_sInter Set.mapsTo_sInter
theorem mapsTo_iInter {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, MapsTo f s (t i)) :
MapsTo f s (⋂ i, t i) := fun _ hx => mem_iInter.2 fun i => H i hx
#align set.maps_to_Inter Set.mapsTo_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f s (t i j)) : MapsTo f s (⋂ (i) (j), t i j) :=
mapsTo_iInter fun i => mapsTo_iInter (H i)
#align set.maps_to_Inter₂ Set.mapsTo_iInter₂
theorem mapsTo_iInter_iInter {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋂ i, s i) (⋂ i, t i) :=
mapsTo_iInter fun i => (H i).mono (iInter_subset s i) (Subset.refl _)
#align set.maps_to_Inter_Inter Set.mapsTo_iInter_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂_iInter₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋂ (i) (j), s i j) (⋂ (i) (j), t i j) :=
mapsTo_iInter_iInter fun i => mapsTo_iInter_iInter (H i)
#align set.maps_to_Inter₂_Inter₂ Set.mapsTo_iInter₂_iInter₂
theorem image_iInter_subset (s : ι → Set α) (f : α → β) : (f '' ⋂ i, s i) ⊆ ⋂ i, f '' s i :=
(mapsTo_iInter_iInter fun i => mapsTo_image f (s i)).image_subset
#align set.image_Inter_subset Set.image_iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂_subset (s : ∀ i, κ i → Set α) (f : α → β) :
(f '' ⋂ (i) (j), s i j) ⊆ ⋂ (i) (j), f '' s i j :=
(mapsTo_iInter₂_iInter₂ fun i hi => mapsTo_image f (s i hi)).image_subset
#align set.image_Inter₂_subset Set.image_iInter₂_subset
theorem image_sInter_subset (S : Set (Set α)) (f : α → β) : f '' ⋂₀ S ⊆ ⋂ s ∈ S, f '' s := by
rw [sInter_eq_biInter]
apply image_iInter₂_subset
#align set.image_sInter_subset Set.image_sInter_subset
/-! ### `restrictPreimage` -/
section
open Function
variable (s : Set β) {f : α → β} {U : ι → Set β} (hU : iUnion U = univ)
theorem injective_iff_injective_of_iUnion_eq_univ :
Injective f ↔ ∀ i, Injective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_injective H, fun H x y e => _⟩
obtain ⟨i, hi⟩ := Set.mem_iUnion.mp
(show f x ∈ Set.iUnion U by rw [hU]; triv)
injection @H i ⟨x, hi⟩ ⟨y, show f y ∈ U i from e ▸ hi⟩ (Subtype.ext e)
#align set.injective_iff_injective_of_Union_eq_univ Set.injective_iff_injective_of_iUnion_eq_univ
theorem surjective_iff_surjective_of_iUnion_eq_univ :
Surjective f ↔ ∀ i, Surjective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_surjective H, fun H x => _⟩
obtain ⟨i, hi⟩ :=
Set.mem_iUnion.mp
(show x ∈ Set.iUnion U by rw [hU]; triv)
exact ⟨_, congr_arg Subtype.val (H i ⟨x, hi⟩).choose_spec⟩
#align set.surjective_iff_surjective_of_Union_eq_univ Set.surjective_iff_surjective_of_iUnion_eq_univ
theorem bijective_iff_bijective_of_iUnion_eq_univ :
Bijective f ↔ ∀ i, Bijective ((U i).restrictPreimage f) := by
rw [Bijective, injective_iff_injective_of_iUnion_eq_univ hU,
surjective_iff_surjective_of_iUnion_eq_univ hU]
simp [Bijective, forall_and]
#align set.bijective_iff_bijective_of_Union_eq_univ Set.bijective_iff_bijective_of_iUnion_eq_univ
end
/-! ### `InjOn` -/
theorem InjOn.image_iInter_eq [Nonempty ι] {s : ι → Set α} {f : α → β} (h : InjOn f (⋃ i, s i)) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
inhabit ι
refine' Subset.antisymm (image_iInter_subset s f) fun y hy => _
simp only [mem_iInter, mem_image_iff_bex] at hy
choose x hx hy using hy
refine' ⟨x default, mem_iInter.2 fun i => _, hy _⟩
suffices x default = x i by
rw [this]
apply hx
replace hx : ∀ i, x i ∈ ⋃ j, s j := fun i => (subset_iUnion _ _) (hx i)
apply h (hx _) (hx _)
simp only [hy]
#align set.inj_on.image_Inter_eq Set.InjOn.image_iInter_eq
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
theorem InjOn.image_biInter_eq {p : ι → Prop} {s : ∀ i, p i → Set α} (hp : ∃ i, p i)
{f : α → β} (h : InjOn f (⋃ (i) (hi), s i hi)) :
(f '' ⋂ (i) (hi), s i hi) = ⋂ (i) (hi), f '' s i hi := by
simp only [iInter, iInf_subtype']
haveI : Nonempty { i // p i } := nonempty_subtype.2 hp
apply InjOn.image_iInter_eq
simpa only [iUnion, iSup_subtype'] using h
#align set.inj_on.image_bInter_eq Set.InjOn.image_biInter_eq
theorem image_iInter {f : α → β} (hf : Bijective f) (s : ι → Set α) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
cases isEmpty_or_nonempty ι
· simp_rw [iInter_of_empty, image_univ_of_surjective hf.surjective]
· exact (hf.injective.injOn _).image_iInter_eq
#align set.image_Inter Set.image_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂ {f : α → β} (hf : Bijective f) (s : ∀ i, κ i → Set α) :
(f '' ⋂ (i) (j), s i j) = ⋂ (i) (j), f '' s i j := by simp_rw [image_iInter hf]
#align set.image_Inter₂ Set.image_iInter₂
theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
(hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i) := by
intro x hx y hy hxy
rcases mem_iUnion.1 hx with ⟨i, hx⟩
rcases mem_iUnion.1 hy with ⟨j, hy⟩
rcases hs i j with ⟨k, hi, hj⟩
exact hf k (hi hx) (hj hy) hxy
#align set.inj_on_Union_of_directed Set.inj_on_iUnion_of_directed
/-! ### `SurjOn` -/
theorem surjOn_sUnion {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, SurjOn f s t) :
SurjOn f s (⋃₀T) := fun _ ⟨t, ht, hx⟩ => H t ht hx
#align set.surj_on_sUnion Set.surjOn_sUnion
theorem surjOn_iUnion {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, SurjOn f s (t i)) :
SurjOn f s (⋃ i, t i) :=
surjOn_sUnion <| forall_range_iff.2 H
#align set.surj_on_Union Set.surjOn_iUnion
theorem surjOn_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) : SurjOn f (⋃ i, s i) (⋃ i, t i) :=
surjOn_iUnion fun i => (H i).mono (subset_iUnion _ _) (Subset.refl _)
#align set.surj_on_Union_Union Set.surjOn_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f s (t i j)) : SurjOn f s (⋃ (i) (j), t i j) :=
surjOn_iUnion fun i => surjOn_iUnion (H i)
#align set.surj_on_Union₂ Set.surjOn_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f (s i j) (t i j)) : SurjOn f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
surjOn_iUnion_iUnion fun i => surjOn_iUnion_iUnion (H i)
#align set.surj_on_Union₂_Union₂ Set.surjOn_iUnion₂_iUnion₂
theorem surjOn_iInter [Nonempty ι] {s : ι → Set α} {t : Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) t) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) t := by
intro y hy
rw [Hinj.image_iInter_eq, mem_iInter]
exact fun i => H i hy
#align set.surj_on_Inter Set.surjOn_iInter
theorem surjOn_iInter_iInter [Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) (⋂ i, t i) :=
surjOn_iInter (fun i => (H i).mono (Subset.refl _) (iInter_subset _ _)) Hinj
#align set.surj_on_Inter_Inter Set.surjOn_iInter_iInter
/-! ### `BijOn` -/
theorem bijOn_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i))
(Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
⟨mapsTo_iUnion_iUnion fun i => (H i).mapsTo, Hinj, surjOn_iUnion_iUnion fun i => (H i).surjOn⟩
#align set.bij_on_Union Set.bijOn_iUnion
theorem bijOn_iInter [hi : Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, BijOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
⟨mapsTo_iInter_iInter fun i => (H i).mapsTo,
hi.elim fun i => (H i).injOn.mono (iInter_subset _ _),
surjOn_iInter_iInter (fun i => (H i).surjOn) Hinj⟩
#align set.bij_on_Inter Set.bijOn_iInter
theorem bijOn_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {t : ι → Set β}
{f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
bijOn_iUnion H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Union_of_directed Set.bijOn_iUnion_of_directed
theorem bijOn_iInter_of_directed [Nonempty ι] {s : ι → Set α} (hs : Directed (· ⊆ ·) s)
{t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
bijOn_iInter H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Inter_of_directed Set.bijOn_iInter_of_directed
end Function
/-! ### `image`, `preimage` -/
section Image
theorem image_iUnion {f : α → β} {s : ι → Set α} : (f '' ⋃ i, s i) = ⋃ i, f '' s i := by
ext1 x
simp only [mem_image, mem_iUnion, ← exists_and_right, ← exists_and_left]
--Porting note: `exists_swap` causes a `simp` loop in Lean4 so we use `rw` instead.
rw [exists_swap]
#align set.image_Union Set.image_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iUnion₂ (f : α → β) (s : ∀ i, κ i → Set α) :
(f '' ⋃ (i) (j), s i j) = ⋃ (i) (j), f '' s i j := by simp_rw [image_iUnion]
#align set.image_Union₂ Set.image_iUnion₂
theorem univ_subtype {p : α → Prop} : (univ : Set (Subtype p)) = ⋃ (x) (h : p x), {⟨x, h⟩} :=
Set.ext fun ⟨x, h⟩ => by simp [h]
#align set.univ_subtype Set.univ_subtype
theorem range_eq_iUnion {ι} (f : ι → α) : range f = ⋃ i, {f i} :=
Set.ext fun a => by simp [@eq_comm α a]
#align set.range_eq_Union Set.range_eq_iUnion
theorem image_eq_iUnion (f : α → β) (s : Set α) : f '' s = ⋃ i ∈ s, {f i} :=
Set.ext fun b => by simp [@eq_comm β b]
#align set.image_eq_Union Set.image_eq_iUnion
theorem biUnion_range {f : ι → α} {g : α → Set β} : ⋃ x ∈ range f, g x = ⋃ y, g (f y) :=
iSup_range
#align set.bUnion_range Set.biUnion_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iUnion_iUnion_eq' {f : ι → α} {g : α → Set β} :
⋃ (x) (y) (_ : f y = x), g x = ⋃ y, g (f y) := by simpa using biUnion_range
#align set.Union_Union_eq' Set.iUnion_iUnion_eq'
theorem biInter_range {f : ι → α} {g : α → Set β} : ⋂ x ∈ range f, g x = ⋂ y, g (f y) :=
iInf_range
#align set.bInter_range Set.biInter_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iInter_iInter_eq' {f : ι → α} {g : α → Set β} :
⋂ (x) (y) (_ : f y = x), g x = ⋂ y, g (f y) := by simpa using biInter_range
#align set.Inter_Inter_eq' Set.iInter_iInter_eq'
variable {s : Set γ} {f : γ → α} {g : α → Set β}
theorem biUnion_image : ⋃ x ∈ f '' s, g x = ⋃ y ∈ s, g (f y) :=
iSup_image
#align set.bUnion_image Set.biUnion_image
theorem biInter_image : ⋂ x ∈ f '' s, g x = ⋂ y ∈ s, g (f y) :=
iInf_image
#align set.bInter_image Set.biInter_image
end Image
section Preimage
theorem monotone_preimage {f : α → β} : Monotone (preimage f) := fun _ _ h => preimage_mono h
#align set.monotone_preimage Set.monotone_preimage
@[simp]
theorem preimage_iUnion {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋃ i, s i) = ⋃ i, f ⁻¹' s i :=
Set.ext <| by simp [preimage]
#align set.preimage_Union Set.preimage_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iUnion₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋃ (i) (j), s i j) = ⋃ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iUnion]
#align set.preimage_Union₂ Set.preimage_iUnion₂
@[simp]
theorem preimage_sUnion {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋃₀s = ⋃ t ∈ s, f ⁻¹' t := by
rw [sUnion_eq_biUnion, preimage_iUnion₂]
#align set.preimage_sUnion Set.preimage_sUnion
theorem preimage_iInter {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋂ i, s i) = ⋂ i, f ⁻¹' s i := by
ext; simp
#align set.preimage_Inter Set.preimage_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iInter₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋂ (i) (j), s i j) = ⋂ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iInter]
#align set.preimage_Inter₂ Set.preimage_iInter₂
@[simp]
theorem preimage_sInter {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋂₀ s = ⋂ t ∈ s, f ⁻¹' t := by
rw [sInter_eq_biInter, preimage_iInter₂]
#align set.preimage_sInter Set.preimage_sInter
@[simp]
theorem biUnion_preimage_singleton (f : α → β) (s : Set β) : ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s := by
rw [← preimage_iUnion₂, biUnion_of_singleton]
#align set.bUnion_preimage_singleton Set.biUnion_preimage_singleton
theorem biUnion_range_preimage_singleton (f : α → β) : ⋃ y ∈ range f, f ⁻¹' {y} = univ := by
rw [biUnion_preimage_singleton, preimage_range]
#align set.bUnion_range_preimage_singleton Set.biUnion_range_preimage_singleton
end Preimage
section Prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion {s : Set α} {t : ι → Set β} : (s ×ˢ ⋃ i, t i) = ⋃ i, s ×ˢ t i := by
ext
simp
#align set.prod_Union Set.prod_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} :
(s ×ˢ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ×ˢ t i j := by simp_rw [prod_iUnion]
#align set.prod_Union₂ Set.prod_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sUnion {s : Set α} {C : Set (Set β)} : s ×ˢ ⋃₀C = ⋃₀((fun t => s ×ˢ t) '' C) := by
simp_rw [sUnion_eq_biUnion, biUnion_image, prod_iUnion₂]
#align set.prod_sUnion Set.prod_sUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_const {s : ι → Set α} {t : Set β} : (⋃ i, s i) ×ˢ t = ⋃ i, s i ×ˢ t := by
ext
simp
#align set.Union_prod_const Set.iUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion₂_prod_const {s : ∀ i, κ i → Set α} {t : Set β} :
(⋃ (i) (j), s i j) ×ˢ t = ⋃ (i) (j), s i j ×ˢ t := by simp_rw [iUnion_prod_const]
#align set.Union₂_prod_const Set.iUnion₂_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_prod_const {C : Set (Set α)} {t : Set β} :
⋃₀C ×ˢ t = ⋃₀((fun s : Set α => s ×ˢ t) '' C) := by
simp only [sUnion_eq_biUnion, iUnion₂_prod_const, biUnion_image]
#align set.sUnion_prod_const Set.sUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod {ι ι' α β} (s : ι → Set α) (t : ι' → Set β) :
⋃ x : ι × ι', s x.1 ×ˢ t x.2 = (⋃ i : ι, s i) ×ˢ ⋃ i : ι', t i := by
ext
simp
#align set.Union_prod Set.iUnion_prod
/-- Analogue of `iSup_prod` for sets. -/
lemma iUnion_prod' (f : β × γ → Set α) : ⋃ x : β × γ, f x = ⋃ (i : β) (j : γ), f (i, j) :=
iSup_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x := by
ext ⟨z, w⟩; simp only [mem_prod, mem_iUnion, exists_imp, and_imp, iff_def]; constructor
· intro x hz hw
exact ⟨⟨x, hz⟩, x, hw⟩
· intro x hz x' hw
exact ⟨x ⊔ x', hs le_sup_left hz, ht le_sup_right hw⟩
#align set.Union_prod_of_monotone Set.iUnion_prod_of_monotone
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter_subset (S : Set (Set α)) (T : Set (Set β)) :
⋂₀ S ×ˢ ⋂₀ T ⊆ ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 :=
subset_iInter₂ fun x hx _ hy => ⟨hy.1 x.1 hx.1, hy.2 x.2 hx.2⟩
#align set.sInter_prod_sInter_subset Set.sInter_prod_sInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 := by
obtain ⟨s₁, h₁⟩ := hS
obtain ⟨s₂, h₂⟩ := hT
refine' Set.Subset.antisymm (sInter_prod_sInter_subset S T) fun x hx => _
rw [mem_iInter₂] at hx
exact ⟨fun s₀ h₀ => (hx (s₀, s₂) ⟨h₀, h₂⟩).1, fun s₀ h₀ => (hx (s₁, s₀) ⟨h₁, h₀⟩).2⟩
#align set.sInter_prod_sInter Set.sInter_prod_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod {S : Set (Set α)} (hS : S.Nonempty) (t : Set β) :
⋂₀ S ×ˢ t = ⋂ s ∈ S, s ×ˢ t := by
rw [← sInter_singleton t, sInter_prod_sInter hS (singleton_nonempty t), sInter_singleton]
simp_rw [prod_singleton, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.sInter_prod Set.sInter_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sInter {T : Set (Set β)} (hT : T.Nonempty) (s : Set α) :
s ×ˢ ⋂₀ T = ⋂ t ∈ T, s ×ˢ t := by
rw [← sInter_singleton s, sInter_prod_sInter (singleton_nonempty s) hT, sInter_singleton]
simp_rw [singleton_prod, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.prod_sInter Set.prod_sInter
end Prod
section Image2
variable (f : α → β → γ) {s : Set α} {t : Set β}
/-- The `Set.image2` version of `Set.image_eq_iUnion` -/
theorem image2_eq_iUnion (s : Set α) (t : Set β) : image2 f s t = ⋃ (i ∈ s) (j ∈ t), {f i j} := by
ext; simp [eq_comm]
#align set.image2_eq_Union Set.image2_eq_iUnion
theorem iUnion_image_left : ⋃ a ∈ s, f a '' t = image2 f s t := by
simp only [image2_eq_iUnion, image_eq_iUnion]
#align set.Union_image_left Set.iUnion_image_left
theorem iUnion_image_right : ⋃ b ∈ t, (f · b) '' s = image2 f s t := by
rw [image2_swap, iUnion_image_left]
#align set.Union_image_right Set.iUnion_image_right
theorem image2_iUnion_left (s : ι → Set α) (t : Set β) :
image2 f (⋃ i, s i) t = ⋃ i, image2 f (s i) t := by
simp only [← image_prod, iUnion_prod_const, image_iUnion]
#align set.image2_Union_left Set.image2_iUnion_left
theorem image2_iUnion_right (s : Set α) (t : ι → Set β) :
image2 f s (⋃ i, t i) = ⋃ i, image2 f s (t i) := by
simp only [← image_prod, prod_iUnion, image_iUnion]
#align set.image2_Union_right Set.image2_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋃ (i) (j), s i j) t = ⋃ (i) (j), image2 f (s i j) t := by simp_rw [image2_iUnion_left]
#align set.image2_Union₂_left Set.image2_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋃ (i) (j), t i j) = ⋃ (i) (j), image2 f s (t i j) :=
by simp_rw [image2_iUnion_right]
#align set.image2_Union₂_right Set.image2_iUnion₂_right
theorem image2_iInter_subset_left (s : ι → Set α) (t : Set β) :
image2 f (⋂ i, s i) t ⊆ ⋂ i, image2 f (s i) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem (hx _) hy
#align set.image2_Inter_subset_left Set.image2_iInter_subset_left
theorem image2_iInter_subset_right (s : Set α) (t : ι → Set β) :
image2 f s (⋂ i, t i) ⊆ ⋂ i, image2 f s (t i) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem hx (hy _)
#align set.image2_Inter_subset_right Set.image2_iInter_subset_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋂ (i) (j), s i j) t ⊆ ⋂ (i) (j), image2 f (s i j) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem (hx _ _) hy
#align set.image2_Inter₂_subset_left Set.image2_iInter₂_subset_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), image2 f s (t i j) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem hx (hy _ _)
#align set.image2_Inter₂_subset_right Set.image2_iInter₂_subset_right
theorem prod_eq_biUnion_left : s ×ˢ t = ⋃ a ∈ s, (fun b => (a, b)) '' t := by
rw [iUnion_image_left, image2_mk_eq_prod]
#align set.prod_eq_bUnion_left Set.prod_eq_biUnion_left
theorem prod_eq_biUnion_right : s ×ˢ t = ⋃ b ∈ t, (fun a => (a, b)) '' s := by
rw [iUnion_image_right, image2_mk_eq_prod]
#align set.prod_eq_bUnion_right Set.prod_eq_biUnion_right
end Image2
section Seq
/-- Given a set `s` of functions `α → β` and `t : Set α`, `seq s t` is the union of `f '' t` over
all `f ∈ s`. -/
def seq (s : Set (α → β)) (t : Set α) : Set β :=
{ b | ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b }
#align set.seq Set.seq
@[simp]
theorem mem_seq_iff {s : Set (α → β)} {t : Set α} {b : β} :
b ∈ seq s t ↔ ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b :=
Iff.rfl
#align set.mem_seq_iff Set.mem_seq_iff
lemma seq_eq_image2 (s : Set (α → β)) (t : Set α) : seq s t = image2 (fun f a ↦ f a) s t := by
ext; simp
theorem seq_def {s : Set (α → β)} {t : Set α} : seq s t = ⋃ f ∈ s, f '' t := by
rw [seq_eq_image2, iUnion_image_left]
#align set.seq_def Set.seq_def
theorem seq_subset {s : Set (α → β)} {t : Set α} {u : Set β} :
seq s t ⊆ u ↔ ∀ f ∈ s, ∀ a ∈ t, (f : α → β) a ∈ u := by
rw [seq_eq_image2, image2_subset_iff]
#align set.seq_subset Set.seq_subset
@[gcongr]
theorem seq_mono {s₀ s₁ : Set (α → β)} {t₀ t₁ : Set α} (hs : s₀ ⊆ s₁) (ht : t₀ ⊆ t₁) :
seq s₀ t₀ ⊆ seq s₁ t₁ := fun _ ⟨f, hf, a, ha, eq⟩ => ⟨f, hs hf, a, ht ha, eq⟩
#align set.seq_mono Set.seq_mono
theorem singleton_seq {f : α → β} {t : Set α} : Set.seq ({f} : Set (α → β)) t = f '' t := by
rw [seq_eq_image2, image2_singleton_left]
#align set.singleton_seq Set.singleton_seq
theorem seq_singleton {s : Set (α → β)} {a : α} : Set.seq s {a} = (fun f : α → β => f a) '' s := by
rw [seq_eq_image2, image2_singleton_right]
#align set.seq_singleton Set.seq_singleton
theorem seq_seq {s : Set (β → γ)} {t : Set (α → β)} {u : Set α} :
seq s (seq t u) = seq (seq ((· ∘ ·) '' s) t) u := by
simp only [seq_eq_image2, image2_image_left, image2_image2_left, image2_image2_right, comp_apply]
#align set.seq_seq Set.seq_seq
theorem image_seq {f : β → γ} {s : Set (α → β)} {t : Set α} :
f '' seq s t = seq ((f ∘ ·) '' s) t := by
rw [← singleton_seq, ← singleton_seq, seq_seq, image_singleton]
#align set.image_seq Set.image_seq
theorem prod_eq_seq {s : Set α} {t : Set β} : s ×ˢ t = (Prod.mk '' s).seq t := by
rw [seq_eq_image2, image2_image_left, image2_mk_eq_prod]
#align set.prod_eq_seq Set.prod_eq_seq
theorem prod_image_seq_comm (s : Set α) (t : Set β) :
(Prod.mk '' s).seq t = seq ((fun b a => (a, b)) '' t) s := by
rw [← prod_eq_seq, ← image_swap_prod, prod_eq_seq, image_seq, ← image_comp]; rfl
#align set.prod_image_seq_comm Set.prod_image_seq_comm
theorem image2_eq_seq (f : α → β → γ) (s : Set α) (t : Set β) : image2 f s t = seq (f '' s) t := by
rw [seq_eq_image2, image2_image_left]
#align set.image2_eq_seq Set.image2_eq_seq
end Seq
section Pi
variable {π : α → Type*}
theorem pi_def (i : Set α) (s : ∀ a, Set (π a)) : pi i s = ⋂ a ∈ i, eval a ⁻¹' s a := by
ext
simp
#align set.pi_def Set.pi_def
theorem univ_pi_eq_iInter (t : ∀ i, Set (π i)) : pi univ t = ⋂ i, eval i ⁻¹' t i := by
simp only [pi_def, iInter_true, mem_univ]
#align set.univ_pi_eq_Inter Set.univ_pi_eq_iInter
theorem pi_diff_pi_subset (i : Set α) (s t : ∀ a, Set (π a)) :
pi i s \ pi i t ⊆ ⋃ a ∈ i, eval a ⁻¹' (s a \ t a) := by
refine' diff_subset_comm.2 fun x hx a ha => _
simp only [mem_diff, mem_pi, mem_iUnion, not_exists, mem_preimage, not_and, not_not,
eval_apply] at hx
exact hx.2 _ ha (hx.1 _ ha)
#align set.pi_diff_pi_subset Set.pi_diff_pi_subset
theorem iUnion_univ_pi {ι : α → Type*} (t : (a : α) → ι a → Set (π a)) :
⋃ x : (a : α) → ι a, pi univ (fun a => t a (x a)) = pi univ fun a => ⋃ j : ι a, t a j := by
ext
simp [Classical.skolem]
#align set.Union_univ_pi Set.iUnion_univ_pi
end Pi
end Set
namespace Function
namespace Surjective
theorem iUnion_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋃ x, g (f x) = ⋃ y, g y :=
hf.iSup_comp g
#align function.surjective.Union_comp Function.Surjective.iUnion_comp
theorem iInter_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋂ x, g (f x) = ⋂ y, g y :=
hf.iInf_comp g
#align function.surjective.Inter_comp Function.Surjective.iInter_comp
end Surjective
end Function
/-!
### Disjoint sets
-/
section Disjoint
variable {s t u : Set α} {f : α → β}
namespace Set
@[simp]
theorem disjoint_iUnion_left {ι : Sort*} {s : ι → Set α} :
Disjoint (⋃ i, s i) t ↔ ∀ i, Disjoint (s i) t :=
iSup_disjoint_iff
#align set.disjoint_Union_left Set.disjoint_iUnion_left
@[simp]
theorem disjoint_iUnion_right {ι : Sort*} {s : ι → Set α} :
Disjoint t (⋃ i, s i) ↔ ∀ i, Disjoint t (s i) :=
disjoint_iSup_iff
#align set.disjoint_Union_right Set.disjoint_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_left {s : ∀ i, κ i → Set α} {t : Set α} :
Disjoint (⋃ (i) (j), s i j) t ↔ ∀ i j, Disjoint (s i j) t :=
iSup₂_disjoint_iff
#align set.disjoint_Union₂_left Set.disjoint_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_right {s : Set α} {t : ∀ i, κ i → Set α} :
Disjoint s (⋃ (i) (j), t i j) ↔ ∀ i j, Disjoint s (t i j) :=
disjoint_iSup₂_iff
#align set.disjoint_Union₂_right Set.disjoint_iUnion₂_right
@[simp]
theorem disjoint_sUnion_left {S : Set (Set α)} {t : Set α} :
Disjoint (⋃₀S) t ↔ ∀ s ∈ S, Disjoint s t :=
sSup_disjoint_iff
#align set.disjoint_sUnion_left Set.disjoint_sUnion_left
@[simp]
theorem disjoint_sUnion_right {s : Set α} {S : Set (Set α)} :
Disjoint s (⋃₀S) ↔ ∀ t ∈ S, Disjoint s t :=
disjoint_sSup_iff
#align set.disjoint_sUnion_right Set.disjoint_sUnion_right
end Set
end Disjoint
/-! ### Intervals -/
namespace Set
lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :
(⋂ i, Iic (f i)).Nonempty ↔ BddBelow (range f) := by
have : (⋂ (i : ι), Iic (f i)) = lowerBounds (range f) := by
ext c; simp [lowerBounds]
simp [this, BddBelow]
lemma nonempty_iInter_Ici_iff [Preorder α] {f : ι → α} :
(⋂ i, Ici (f i)).Nonempty ↔ BddAbove (range f) :=
nonempty_iInter_Iic_iff (α := αᵒᵈ)
variable [CompleteLattice α]
theorem Ici_iSup (f : ι → α) : Ici (⨆ i, f i) = ⋂ i, Ici (f i) :=
ext fun _ => by simp only [mem_Ici, iSup_le_iff, mem_iInter]
#align set.Ici_supr Set.Ici_iSup
theorem Iic_iInf (f : ι → α) : Iic (⨅ i, f i) = ⋂ i, Iic (f i) :=
ext fun _ => by simp only [mem_Iic, le_iInf_iff, mem_iInter]
#align set.Iic_infi Set.Iic_iInf
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Ici_iSup₂ (f : ∀ i, κ i → α) : Ici (⨆ (i) (j), f i j) = ⋂ (i) (j), Ici (f i j) := by
simp_rw [Ici_iSup]
#align set.Ici_supr₂ Set.Ici_iSup₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Iic_iInf₂ (f : ∀ i, κ i → α) : Iic (⨅ (i) (j), f i j) = ⋂ (i) (j), Iic (f i j) := by
simp_rw [Iic_iInf]
#align set.Iic_infi₂ Set.Iic_iInf₂
theorem Ici_sSup (s : Set α) : Ici (sSup s) = ⋂ a ∈ s, Ici a := by rw [sSup_eq_iSup, Ici_iSup₂]
#align set.Ici_Sup Set.Ici_sSup
theorem Iic_sInf (s : Set α) : Iic (sInf s) = ⋂ a ∈ s, Iic a := by rw [sInf_eq_iInf, Iic_iInf₂]
#align set.Iic_Inf Set.Iic_sInf
end Set
namespace Set
variable (t : α → Set β)
theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x := by
simp only [diff_subset_iff, ← biUnion_union]
apply biUnion_subset_biUnion_left
rw [union_diff_self]
apply subset_union_right
#align set.bUnion_diff_bUnion_subset Set.biUnion_diff_biUnion_subset
/-- If `t` is an indexed family of sets, then there is a natural map from `Σ i, t i` to `⋃ i, t i`
sending `⟨i, x⟩` to `x`. -/
def sigmaToiUnion (x : Σi, t i) : ⋃ i, t i :=
⟨x.2, mem_iUnion.2 ⟨x.1, x.2.2⟩⟩
#align set.sigma_to_Union Set.sigmaToiUnion
theorem sigmaToiUnion_surjective : Surjective (sigmaToiUnion t)
| ⟨b, hb⟩ =>
have : ∃ a, b ∈ t a := by simpa using hb
let ⟨a, hb⟩ := this
⟨⟨a, b, hb⟩, rfl⟩
#align set.sigma_to_Union_surjective Set.sigmaToiUnion_surjective
theorem sigmaToiUnion_injective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Injective (sigmaToiUnion t)
| ⟨a₁, b₁, h₁⟩, ⟨a₂, b₂, h₂⟩, eq =>
have b_eq : b₁ = b₂ := congr_arg Subtype.val eq
have a_eq : a₁ = a₂ :=
by_contradiction fun ne =>
have : b₁ ∈ t a₁ ∩ t a₂ := ⟨h₁, b_eq.symm ▸ h₂⟩
(h _ _ ne).le_bot this
Sigma.eq a_eq <| Subtype.eq <| by subst b_eq; subst a_eq; rfl
#align set.sigma_to_Union_injective Set.sigmaToiUnion_injective
theorem sigmaToiUnion_bijective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Bijective (sigmaToiUnion t) :=
⟨sigmaToiUnion_injective t h, sigmaToiUnion_surjective t⟩
#align set.sigma_to_Union_bijective Set.sigmaToiUnion_bijective
/-- Equivalence between a disjoint union and a dependent sum. -/
noncomputable def unionEqSigmaOfDisjoint {t : α → Set β} (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
(⋃ i, t i) ≃ Σi, t i :=
(Equiv.ofBijective _ <| sigmaToiUnion_bijective t h).symm
#align set.Union_eq_sigma_of_disjoint Set.unionEqSigmaOfDisjoint
theorem iUnion_ge_eq_iUnion_nat_add (u : ℕ → Set α) (n : ℕ) : ⋃ i ≥ n, u i = ⋃ i, u (i + n) :=
iSup_ge_eq_iSup_nat_add u n
#align set.Union_ge_eq_Union_nat_add Set.iUnion_ge_eq_iUnion_nat_add
theorem iInter_ge_eq_iInter_nat_add (u : ℕ → Set α) (n : ℕ) : ⋂ i ≥ n, u i = ⋂ i, u (i + n) :=
iInf_ge_eq_iInf_nat_add u n
#align set.Inter_ge_eq_Inter_nat_add Set.iInter_ge_eq_iInter_nat_add
theorem _root_.Monotone.iUnion_nat_add {f : ℕ → Set α} (hf : Monotone f) (k : ℕ) :
⋃ n, f (n + k) = ⋃ n, f n :=
hf.iSup_nat_add k
#align monotone.Union_nat_add Monotone.iUnion_nat_add
theorem _root_.Antitone.iInter_nat_add {f : ℕ → Set α} (hf : Antitone f) (k : ℕ) :
⋂ n, f (n + k) = ⋂ n, f n :=
hf.iInf_nat_add k
#align antitone.Inter_nat_add Antitone.iInter_nat_add
/-Porting note: removing `simp`. LHS does not simplify. Possible linter bug. Zulip discussion:
https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/complete_lattice.20and.20has_sup/near/316497982
-/
theorem iUnion_iInter_ge_nat_add (f : ℕ → Set α) (k : ℕ) :
⋃ n, ⋂ i ≥ n, f (i + k) = ⋃ n, ⋂ i ≥ n, f i :=
iSup_iInf_ge_nat_add f k
#align set.Union_Inter_ge_nat_add Set.iUnion_iInter_ge_nat_add
theorem union_iUnion_nat_succ (u : ℕ → Set α) : (u 0 ∪ ⋃ i, u (i + 1)) = ⋃ i, u i :=
sup_iSup_nat_succ u
#align set.union_Union_nat_succ Set.union_iUnion_nat_succ
theorem inter_iInter_nat_succ (u : ℕ → Set α) : (u 0 ∩ ⋂ i, u (i + 1)) = ⋂ i, u i :=
inf_iInf_nat_succ u
#align set.inter_Inter_nat_succ Set.inter_iInter_nat_succ
end Set
open Set
variable [CompleteLattice β]
theorem iSup_iUnion (s : ι → Set α) (f : α → β) : ⨆ a ∈ ⋃ i, s i, f a = ⨆ (i) (a ∈ s i), f a := by
rw [iSup_comm]
simp_rw [mem_iUnion, iSup_exists]
#align supr_Union iSup_iUnion
theorem iInf_iUnion (s : ι → Set α) (f : α → β) : ⨅ a ∈ ⋃ i, s i, f a = ⨅ (i) (a ∈ s i), f a :=
@iSup_iUnion α βᵒᵈ _ _ s f
#align infi_Union iInf_iUnion
theorem sSup_sUnion (s : Set (Set β)) : sSup (⋃₀ s) = ⨆ t ∈ s, sSup t := by
simp only [sUnion_eq_biUnion, sSup_eq_iSup, iSup_iUnion]
#align Sup_sUnion sSup_sUnion
theorem sInf_sUnion (s : Set (Set β)) : sInf (⋃₀ s) = ⨅ t ∈ s, sInf t :=
@sSup_sUnion βᵒᵈ _ _
#align Inf_sUnion sInf_sUnion
lemma iSup_sUnion (S : Set (Set α)) (f : α → β) :
(⨆ x ∈ ⋃₀ S, f x) = ⨆ (s ∈ S) (x ∈ s), f x := by
rw [sUnion_eq_iUnion, iSup_iUnion, ← iSup_subtype'']
lemma iInf_sUnion (S : Set (Set α)) (f : α → β) :
(⨅ x ∈ ⋃₀ S, f x) = ⨅ (s ∈ S) (x ∈ s), f x := by
|
rw [sUnion_eq_iUnion, iInf_iUnion, ← iInf_subtype'']
|
lemma iInf_sUnion (S : Set (Set α)) (f : α → β) :
(⨅ x ∈ ⋃₀ S, f x) = ⨅ (s ∈ S) (x ∈ s), f x := by
|
Mathlib.Data.Set.Lattice.2395_0.5mONj49h3SYSDwc
|
lemma iInf_sUnion (S : Set (Set α)) (f : α → β) :
(⨅ x ∈ ⋃₀ S, f x) = ⨅ (s ∈ S) (x ∈ s), f x
|
Mathlib_Data_Set_Lattice
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : CompleteLattice β
S : Set (Set α)
p : α → Prop
⊢ (∀ x ∈ ⋃₀ S, p x) ↔ ∀ s ∈ S, ∀ x ∈ s, p x
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
/-!
# The set lattice
This file provides usual set notation for unions and intersections, a `CompleteLattice` instance
for `Set α`, and some more set constructions.
## Main declarations
* `Set.iUnion`: **i**ndexed **union**. Union of an indexed family of sets.
* `Set.iInter`: **i**ndexed **inter**section. Intersection of an indexed family of sets.
* `Set.sInter`: **s**et **inter**section. Intersection of sets belonging to a set of sets.
* `Set.sUnion`: **s**et **union**. Union of sets belonging to a set of sets.
* `Set.sInter_eq_biInter`, `Set.sUnion_eq_biInter`: Shows that `⋂₀ s = ⋂ x ∈ s, x` and
`⋃₀ s = ⋃ x ∈ s, x`.
* `Set.completeAtomicBooleanAlgebra`: `Set α` is a `CompleteAtomicBooleanAlgebra` with `≤ = ⊆`,
`< = ⊂`, `⊓ = ∩`, `⊔ = ∪`, `⨅ = ⋂`, `⨆ = ⋃` and `\` as the set difference.
See `Set.BooleanAlgebra`.
* `Set.kernImage`: For a function `f : α → β`, `s.kernImage f` is the set of `y` such that
`f ⁻¹ y ⊆ s`.
* `Set.seq`: Union of the image of a set under a **seq**uence of functions. `seq s t` is the union
of `f '' t` over all `f ∈ s`, where `t : Set α` and `s : Set (α → β)`.
* `Set.iUnion_eq_sigma_of_disjoint`: Equivalence between `⋃ i, t i` and `Σ i, t i`, where `t` is an
indexed family of disjoint sets.
## Naming convention
In lemma names,
* `⋃ i, s i` is called `iUnion`
* `⋂ i, s i` is called `iInter`
* `⋃ i j, s i j` is called `iUnion₂`. This is an `iUnion` inside an `iUnion`.
* `⋂ i j, s i j` is called `iInter₂`. This is an `iInter` inside an `iInter`.
* `⋃ i ∈ s, t i` is called `biUnion` for "bounded `iUnion`". This is the special case of `iUnion₂`
where `j : i ∈ s`.
* `⋂ i ∈ s, t i` is called `biInter` for "bounded `iInter`". This is the special case of `iInter₂`
where `j : i ∈ s`.
## Notation
* `⋃`: `Set.iUnion`
* `⋂`: `Set.iInter`
* `⋃₀`: `Set.sUnion`
* `⋂₀`: `Set.sInter`
-/
set_option autoImplicit true
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι₂ : Sort*} {κ κ₁ κ₂ : ι → Sort*} {κ' : ι' → Sort*}
namespace Set
/-! ### Complete lattice and complete Boolean algebra instances -/
instance : InfSet (Set α) :=
⟨fun s => { a | ∀ t ∈ s, a ∈ t }⟩
instance : SupSet (Set α) :=
⟨fun s => { a | ∃ t ∈ s, a ∈ t }⟩
/-- Intersection of a set of sets. -/
def sInter (S : Set (Set α)) : Set α :=
sInf S
#align set.sInter Set.sInter
/-- Notation for `Set.sInter` Intersection of a set of sets. -/
prefix:110 "⋂₀ " => sInter
/-- Union of a set of sets. -/
def sUnion (S : Set (Set α)) : Set α :=
sSup S
#align set.sUnion Set.sUnion
/-- Notation for `Set.sUnion`. Union of a set of sets. -/
prefix:110 "⋃₀ " => sUnion
@[simp]
theorem mem_sInter {x : α} {S : Set (Set α)} : x ∈ ⋂₀ S ↔ ∀ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sInter Set.mem_sInter
@[simp]
theorem mem_sUnion {x : α} {S : Set (Set α)} : x ∈ ⋃₀ S ↔ ∃ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sUnion Set.mem_sUnion
/-- Indexed union of a family of sets -/
def iUnion (s : ι → Set β) : Set β :=
iSup s
#align set.Union Set.iUnion
/-- Indexed intersection of a family of sets -/
def iInter (s : ι → Set β) : Set β :=
iInf s
#align set.Inter Set.iInter
/-- Notation for `Set.iUnion`. Indexed union of a family of sets -/
notation3 "⋃ "(...)", "r:60:(scoped f => iUnion f) => r
/-- Notation for `Set.iInter`. Indexed intersection of a family of sets -/
notation3 "⋂ "(...)", "r:60:(scoped f => iInter f) => r
section delaborators
open Lean Lean.PrettyPrinter.Delaborator
/-- Delaborator for indexed unions. -/
@[delab app.Set.iUnion]
def iUnion_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋃ (_ : $dom), $body)
else if prop || ppTypes then
`(⋃ ($x:ident : $dom), $body)
else
`(⋃ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋃ $x:ident, ⋃ (_ : $y:ident ∈ $s), $body)
| `(⋃ ($x:ident : $_), ⋃ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋃ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
/-- Delaborator for indexed intersections. -/
@[delab app.Set.iInter]
def sInter_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋂ (_ : $dom), $body)
else if prop || ppTypes then
`(⋂ ($x:ident : $dom), $body)
else
`(⋂ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋂ $x:ident, ⋂ (_ : $y:ident ∈ $s), $body)
| `(⋂ ($x:ident : $_), ⋂ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋂ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
end delaborators
@[simp]
theorem sSup_eq_sUnion (S : Set (Set α)) : sSup S = ⋃₀S :=
rfl
#align set.Sup_eq_sUnion Set.sSup_eq_sUnion
@[simp]
theorem sInf_eq_sInter (S : Set (Set α)) : sInf S = ⋂₀ S :=
rfl
#align set.Inf_eq_sInter Set.sInf_eq_sInter
@[simp]
theorem iSup_eq_iUnion (s : ι → Set α) : iSup s = iUnion s :=
rfl
#align set.supr_eq_Union Set.iSup_eq_iUnion
@[simp]
theorem iInf_eq_iInter (s : ι → Set α) : iInf s = iInter s :=
rfl
#align set.infi_eq_Inter Set.iInf_eq_iInter
@[simp]
theorem mem_iUnion {x : α} {s : ι → Set α} : (x ∈ ⋃ i, s i) ↔ ∃ i, x ∈ s i :=
⟨fun ⟨_, ⟨⟨a, (t_eq : s a = _)⟩, (h : x ∈ _)⟩⟩ => ⟨a, t_eq.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨s a, ⟨⟨a, rfl⟩, h⟩⟩⟩
#align set.mem_Union Set.mem_iUnion
@[simp]
theorem mem_iInter {x : α} {s : ι → Set α} : (x ∈ ⋂ i, s i) ↔ ∀ i, x ∈ s i :=
⟨fun (h : ∀ a ∈ { a : Set α | ∃ i, s i = a }, x ∈ a) a => h (s a) ⟨a, rfl⟩,
fun h _ ⟨a, (eq : s a = _)⟩ => eq ▸ h a⟩
#align set.mem_Inter Set.mem_iInter
theorem mem_iUnion₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋃ (i) (j), s i j) ↔ ∃ i j, x ∈ s i j := by
simp_rw [mem_iUnion]
#align set.mem_Union₂ Set.mem_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋂ (i) (j), s i j) ↔ ∀ i j, x ∈ s i j := by
simp_rw [mem_iInter]
#align set.mem_Inter₂ Set.mem_iInter₂
theorem mem_iUnion_of_mem {s : ι → Set α} {a : α} (i : ι) (ha : a ∈ s i) : a ∈ ⋃ i, s i :=
mem_iUnion.2 ⟨i, ha⟩
#align set.mem_Union_of_mem Set.mem_iUnion_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iUnion₂_of_mem {s : ∀ i, κ i → Set α} {a : α} {i : ι} (j : κ i) (ha : a ∈ s i j) :
a ∈ ⋃ (i) (j), s i j :=
mem_iUnion₂.2 ⟨i, j, ha⟩
#align set.mem_Union₂_of_mem Set.mem_iUnion₂_of_mem
theorem mem_iInter_of_mem {s : ι → Set α} {a : α} (h : ∀ i, a ∈ s i) : a ∈ ⋂ i, s i :=
mem_iInter.2 h
#align set.mem_Inter_of_mem Set.mem_iInter_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂_of_mem {s : ∀ i, κ i → Set α} {a : α} (h : ∀ i j, a ∈ s i j) :
a ∈ ⋂ (i) (j), s i j :=
mem_iInter₂.2 h
#align set.mem_Inter₂_of_mem Set.mem_iInter₂_of_mem
instance Set.completeAtomicBooleanAlgebra : CompleteAtomicBooleanAlgebra (Set α) :=
{ instBooleanAlgebraSet with
le_sSup := fun s t t_in a a_in => ⟨t, t_in, a_in⟩
sSup_le := fun s t h a ⟨t', ⟨t'_in, a_in⟩⟩ => h t' t'_in a_in
le_sInf := fun s t h a a_in t' t'_in => h t' t'_in a_in
sInf_le := fun s t t_in a h => h _ t_in
iInf_iSup_eq := by intros; ext; simp [Classical.skolem] }
/-- `kernImage f s` is the set of `y` such that `f ⁻¹ y ⊆ s`. -/
def kernImage (f : α → β) (s : Set α) : Set β :=
{ y | ∀ ⦃x⦄, f x = y → x ∈ s }
#align set.kern_image Set.kernImage
lemma subset_kernImage_iff {f : α → β} : s ⊆ kernImage f t ↔ f ⁻¹' s ⊆ t :=
⟨fun h _ hx ↦ h hx rfl,
fun h _ hx y hy ↦ h (show f y ∈ s from hy.symm ▸ hx)⟩
section GaloisConnection
variable {f : α → β}
protected theorem image_preimage : GaloisConnection (image f) (preimage f) := fun _ _ =>
image_subset_iff
#align set.image_preimage Set.image_preimage
protected theorem preimage_kernImage : GaloisConnection (preimage f) (kernImage f) := fun _ _ =>
subset_kernImage_iff.symm
#align set.preimage_kern_image Set.preimage_kernImage
end GaloisConnection
section kernImage
variable {f : α → β}
lemma kernImage_mono : Monotone (kernImage f) :=
Set.preimage_kernImage.monotone_u
lemma kernImage_eq_compl {s : Set α} : kernImage f s = (f '' sᶜ)ᶜ :=
Set.preimage_kernImage.u_unique (Set.image_preimage.compl)
(fun t ↦ compl_compl (f ⁻¹' t) ▸ Set.preimage_compl)
lemma kernImage_compl {s : Set α} : kernImage f (sᶜ) = (f '' s)ᶜ := by
rw [kernImage_eq_compl, compl_compl]
lemma kernImage_empty : kernImage f ∅ = (range f)ᶜ := by
rw [kernImage_eq_compl, compl_empty, image_univ]
lemma kernImage_preimage_eq_iff {s : Set β} : kernImage f (f ⁻¹' s) = s ↔ (range f)ᶜ ⊆ s := by
rw [kernImage_eq_compl, ← preimage_compl, compl_eq_comm, eq_comm, image_preimage_eq_iff,
compl_subset_comm]
lemma compl_range_subset_kernImage {s : Set α} : (range f)ᶜ ⊆ kernImage f s := by
rw [← kernImage_empty]
exact kernImage_mono (empty_subset _)
lemma kernImage_union_preimage {s : Set α} {t : Set β} :
kernImage f (s ∪ f ⁻¹' t) = kernImage f s ∪ t := by
rw [kernImage_eq_compl, kernImage_eq_compl, compl_union, ← preimage_compl, image_inter_preimage,
compl_inter, compl_compl]
lemma kernImage_preimage_union {s : Set α} {t : Set β} :
kernImage f (f ⁻¹' t ∪ s) = t ∪ kernImage f s := by
rw [union_comm, kernImage_union_preimage, union_comm]
end kernImage
/-! ### Union and intersection over an indexed family of sets -/
instance : OrderTop (Set α) where
top := univ
le_top := by simp
@[congr]
theorem iUnion_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iUnion f₁ = iUnion f₂ :=
iSup_congr_Prop pq f
#align set.Union_congr_Prop Set.iUnion_congr_Prop
@[congr]
theorem iInter_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iInter f₁ = iInter f₂ :=
iInf_congr_Prop pq f
#align set.Inter_congr_Prop Set.iInter_congr_Prop
theorem iUnion_plift_up (f : PLift ι → Set α) : ⋃ i, f (PLift.up i) = ⋃ i, f i :=
iSup_plift_up _
#align set.Union_plift_up Set.iUnion_plift_up
theorem iUnion_plift_down (f : ι → Set α) : ⋃ i, f (PLift.down i) = ⋃ i, f i :=
iSup_plift_down _
#align set.Union_plift_down Set.iUnion_plift_down
theorem iInter_plift_up (f : PLift ι → Set α) : ⋂ i, f (PLift.up i) = ⋂ i, f i :=
iInf_plift_up _
#align set.Inter_plift_up Set.iInter_plift_up
theorem iInter_plift_down (f : ι → Set α) : ⋂ i, f (PLift.down i) = ⋂ i, f i :=
iInf_plift_down _
#align set.Inter_plift_down Set.iInter_plift_down
theorem iUnion_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋃ _ : p, s = if p then s else ∅ :=
iSup_eq_if _
#align set.Union_eq_if Set.iUnion_eq_if
theorem iUnion_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋃ h : p, s h = if h : p then s h else ∅ :=
iSup_eq_dif _
#align set.Union_eq_dif Set.iUnion_eq_dif
theorem iInter_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋂ _ : p, s = if p then s else univ :=
iInf_eq_if _
#align set.Inter_eq_if Set.iInter_eq_if
theorem iInf_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋂ h : p, s h = if h : p then s h else univ :=
_root_.iInf_eq_dif _
#align set.Infi_eq_dif Set.iInf_eq_dif
theorem exists_set_mem_of_union_eq_top {ι : Type*} (t : Set ι) (s : ι → Set β)
(w : ⋃ i ∈ t, s i = ⊤) (x : β) : ∃ i ∈ t, x ∈ s i := by
have p : x ∈ ⊤ := Set.mem_univ x
rw [← w, Set.mem_iUnion] at p
simpa using p
#align set.exists_set_mem_of_union_eq_top Set.exists_set_mem_of_union_eq_top
theorem nonempty_of_union_eq_top_of_nonempty {ι : Type*} (t : Set ι) (s : ι → Set α)
(H : Nonempty α) (w : ⋃ i ∈ t, s i = ⊤) : t.Nonempty := by
obtain ⟨x, m, -⟩ := exists_set_mem_of_union_eq_top t s w H.some
exact ⟨x, m⟩
#align set.nonempty_of_union_eq_top_of_nonempty Set.nonempty_of_union_eq_top_of_nonempty
theorem nonempty_of_nonempty_iUnion
{s : ι → Set α} (h_Union : (⋃ i, s i).Nonempty) : Nonempty ι := by
obtain ⟨x, hx⟩ := h_Union
exact ⟨Classical.choose $ mem_iUnion.mp hx⟩
theorem nonempty_of_nonempty_iUnion_eq_univ
{s : ι → Set α} [Nonempty α] (h_Union : ⋃ i, s i = univ) : Nonempty ι :=
nonempty_of_nonempty_iUnion (s := s) (by simpa only [h_Union] using univ_nonempty)
theorem setOf_exists (p : ι → β → Prop) : { x | ∃ i, p i x } = ⋃ i, { x | p i x } :=
ext fun _ => mem_iUnion.symm
#align set.set_of_exists Set.setOf_exists
theorem setOf_forall (p : ι → β → Prop) : { x | ∀ i, p i x } = ⋂ i, { x | p i x } :=
ext fun _ => mem_iInter.symm
#align set.set_of_forall Set.setOf_forall
theorem iUnion_subset {s : ι → Set α} {t : Set α} (h : ∀ i, s i ⊆ t) : ⋃ i, s i ⊆ t :=
iSup_le h
#align set.Union_subset Set.iUnion_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset {s : ∀ i, κ i → Set α} {t : Set α} (h : ∀ i j, s i j ⊆ t) :
⋃ (i) (j), s i j ⊆ t :=
iUnion_subset fun x => iUnion_subset (h x)
#align set.Union₂_subset Set.iUnion₂_subset
theorem subset_iInter {t : Set β} {s : ι → Set β} (h : ∀ i, t ⊆ s i) : t ⊆ ⋂ i, s i :=
le_iInf h
#align set.subset_Inter Set.subset_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iInter₂ {s : Set α} {t : ∀ i, κ i → Set α} (h : ∀ i j, s ⊆ t i j) :
s ⊆ ⋂ (i) (j), t i j :=
subset_iInter fun x => subset_iInter <| h x
#align set.subset_Inter₂ Set.subset_iInter₂
@[simp]
theorem iUnion_subset_iff {s : ι → Set α} {t : Set α} : ⋃ i, s i ⊆ t ↔ ∀ i, s i ⊆ t :=
⟨fun h _ => Subset.trans (le_iSup s _) h, iUnion_subset⟩
#align set.Union_subset_iff Set.iUnion_subset_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset_iff {s : ∀ i, κ i → Set α} {t : Set α} :
⋃ (i) (j), s i j ⊆ t ↔ ∀ i j, s i j ⊆ t := by simp_rw [iUnion_subset_iff]
#align set.Union₂_subset_iff Set.iUnion₂_subset_iff
@[simp]
theorem subset_iInter_iff {s : Set α} {t : ι → Set α} : (s ⊆ ⋂ i, t i) ↔ ∀ i, s ⊆ t i :=
le_iInf_iff
#align set.subset_Inter_iff Set.subset_iInter_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem subset_iInter₂_iff {s : Set α} {t : ∀ i, κ i → Set α} :
(s ⊆ ⋂ (i) (j), t i j) ↔ ∀ i j, s ⊆ t i j := by simp_rw [subset_iInter_iff]
#align set.subset_Inter₂_iff Set.subset_iInter₂_iff
theorem subset_iUnion : ∀ (s : ι → Set β) (i : ι), s i ⊆ ⋃ i, s i :=
le_iSup
#align set.subset_Union Set.subset_iUnion
theorem iInter_subset : ∀ (s : ι → Set β) (i : ι), ⋂ i, s i ⊆ s i :=
iInf_le
#align set.Inter_subset Set.iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iUnion₂ {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : s i j ⊆ ⋃ (i') (j'), s i' j' :=
le_iSup₂ i j
#align set.subset_Union₂ Set.subset_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_subset {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : ⋂ (i) (j), s i j ⊆ s i j :=
iInf₂_le i j
#align set.Inter₂_subset Set.iInter₂_subset
/-- This rather trivial consequence of `subset_iUnion`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem subset_iUnion_of_subset {s : Set α} {t : ι → Set α} (i : ι) (h : s ⊆ t i) : s ⊆ ⋃ i, t i :=
le_iSup_of_le i h
#align set.subset_Union_of_subset Set.subset_iUnion_of_subset
/-- This rather trivial consequence of `iInter_subset`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem iInter_subset_of_subset {s : ι → Set α} {t : Set α} (i : ι) (h : s i ⊆ t) :
⋂ i, s i ⊆ t :=
iInf_le_of_le i h
#align set.Inter_subset_of_subset Set.iInter_subset_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `subset_iUnion₂` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem subset_iUnion₂_of_subset {s : Set α} {t : ∀ i, κ i → Set α} (i : ι) (j : κ i)
(h : s ⊆ t i j) : s ⊆ ⋃ (i) (j), t i j :=
le_iSup₂_of_le i j h
#align set.subset_Union₂_of_subset Set.subset_iUnion₂_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `iInter₂_subset` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem iInter₂_subset_of_subset {s : ∀ i, κ i → Set α} {t : Set α} (i : ι) (j : κ i)
(h : s i j ⊆ t) : ⋂ (i) (j), s i j ⊆ t :=
iInf₂_le_of_le i j h
#align set.Inter₂_subset_of_subset Set.iInter₂_subset_of_subset
theorem iUnion_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono h
#align set.Union_mono Set.iUnion_mono
@[gcongr]
theorem iUnion_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iUnion s ⊆ iUnion t :=
iSup_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋃ (i) (j), s i j ⊆ ⋃ (i) (j), t i j :=
iSup₂_mono h
#align set.Union₂_mono Set.iUnion₂_mono
theorem iInter_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋂ i, s i ⊆ ⋂ i, t i :=
iInf_mono h
#align set.Inter_mono Set.iInter_mono
@[gcongr]
theorem iInter_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iInter s ⊆ iInter t :=
iInf_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋂ (i) (j), s i j ⊆ ⋂ (i) (j), t i j :=
iInf₂_mono h
#align set.Inter₂_mono Set.iInter₂_mono
theorem iUnion_mono' {s : ι → Set α} {t : ι₂ → Set α} (h : ∀ i, ∃ j, s i ⊆ t j) :
⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono' h
#align set.Union_mono' Set.iUnion_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iUnion₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i j, ∃ i' j', s i j ⊆ t i' j') : ⋃ (i) (j), s i j ⊆ ⋃ (i') (j'), t i' j' :=
iSup₂_mono' h
#align set.Union₂_mono' Set.iUnion₂_mono'
theorem iInter_mono' {s : ι → Set α} {t : ι' → Set α} (h : ∀ j, ∃ i, s i ⊆ t j) :
⋂ i, s i ⊆ ⋂ j, t j :=
Set.subset_iInter fun j =>
let ⟨i, hi⟩ := h j
iInter_subset_of_subset i hi
#align set.Inter_mono' Set.iInter_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iInter₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i' j', ∃ i j, s i j ⊆ t i' j') : ⋂ (i) (j), s i j ⊆ ⋂ (i') (j'), t i' j' :=
subset_iInter₂_iff.2 fun i' j' =>
let ⟨_, _, hst⟩ := h i' j'
(iInter₂_subset _ _).trans hst
#align set.Inter₂_mono' Set.iInter₂_mono'
theorem iUnion₂_subset_iUnion (κ : ι → Sort*) (s : ι → Set α) :
⋃ (i) (_ : κ i), s i ⊆ ⋃ i, s i :=
iUnion_mono fun _ => iUnion_subset fun _ => Subset.rfl
#align set.Union₂_subset_Union Set.iUnion₂_subset_iUnion
theorem iInter_subset_iInter₂ (κ : ι → Sort*) (s : ι → Set α) :
⋂ i, s i ⊆ ⋂ (i) (_ : κ i), s i :=
iInter_mono fun _ => subset_iInter fun _ => Subset.rfl
#align set.Inter_subset_Inter₂ Set.iInter_subset_iInter₂
theorem iUnion_setOf (P : ι → α → Prop) : ⋃ i, { x : α | P i x } = { x : α | ∃ i, P i x } := by
ext
exact mem_iUnion
#align set.Union_set_of Set.iUnion_setOf
theorem iInter_setOf (P : ι → α → Prop) : ⋂ i, { x : α | P i x } = { x : α | ∀ i, P i x } := by
ext
exact mem_iInter
#align set.Inter_set_of Set.iInter_setOf
theorem iUnion_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋃ x, f x = ⋃ y, g y :=
h1.iSup_congr h h2
#align set.Union_congr_of_surjective Set.iUnion_congr_of_surjective
theorem iInter_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋂ x, f x = ⋂ y, g y :=
h1.iInf_congr h h2
#align set.Inter_congr_of_surjective Set.iInter_congr_of_surjective
lemma iUnion_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋃ i, s i = ⋃ i, t i := iSup_congr h
#align set.Union_congr Set.iUnion_congr
lemma iInter_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋂ i, s i = ⋂ i, t i := iInf_congr h
#align set.Inter_congr Set.iInter_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iUnion₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋃ (i) (j), s i j = ⋃ (i) (j), t i j :=
iUnion_congr fun i => iUnion_congr <| h i
#align set.Union₂_congr Set.iUnion₂_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iInter₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋂ (i) (j), s i j = ⋂ (i) (j), t i j :=
iInter_congr fun i => iInter_congr <| h i
#align set.Inter₂_congr Set.iInter₂_congr
section Nonempty
variable [Nonempty ι] {f : ι → Set α} {s : Set α}
lemma iUnion_const (s : Set β) : ⋃ _ : ι, s = s := iSup_const
#align set.Union_const Set.iUnion_const
lemma iInter_const (s : Set β) : ⋂ _ : ι, s = s := iInf_const
#align set.Inter_const Set.iInter_const
lemma iUnion_eq_const (hf : ∀ i, f i = s) : ⋃ i, f i = s :=
(iUnion_congr hf).trans $ iUnion_const _
#align set.Union_eq_const Set.iUnion_eq_const
lemma iInter_eq_const (hf : ∀ i, f i = s) : ⋂ i, f i = s :=
(iInter_congr hf).trans $ iInter_const _
#align set.Inter_eq_const Set.iInter_eq_const
end Nonempty
@[simp]
theorem compl_iUnion (s : ι → Set β) : (⋃ i, s i)ᶜ = ⋂ i, (s i)ᶜ :=
compl_iSup
#align set.compl_Union Set.compl_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iUnion₂ (s : ∀ i, κ i → Set α) : (⋃ (i) (j), s i j)ᶜ = ⋂ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iUnion]
#align set.compl_Union₂ Set.compl_iUnion₂
@[simp]
theorem compl_iInter (s : ι → Set β) : (⋂ i, s i)ᶜ = ⋃ i, (s i)ᶜ :=
compl_iInf
#align set.compl_Inter Set.compl_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iInter₂ (s : ∀ i, κ i → Set α) : (⋂ (i) (j), s i j)ᶜ = ⋃ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iInter]
#align set.compl_Inter₂ Set.compl_iInter₂
-- classical -- complete_boolean_algebra
theorem iUnion_eq_compl_iInter_compl (s : ι → Set β) : ⋃ i, s i = (⋂ i, (s i)ᶜ)ᶜ := by
simp only [compl_iInter, compl_compl]
#align set.Union_eq_compl_Inter_compl Set.iUnion_eq_compl_iInter_compl
-- classical -- complete_boolean_algebra
theorem iInter_eq_compl_iUnion_compl (s : ι → Set β) : ⋂ i, s i = (⋃ i, (s i)ᶜ)ᶜ := by
simp only [compl_iUnion, compl_compl]
#align set.Inter_eq_compl_Union_compl Set.iInter_eq_compl_iUnion_compl
theorem inter_iUnion (s : Set β) (t : ι → Set β) : (s ∩ ⋃ i, t i) = ⋃ i, s ∩ t i :=
inf_iSup_eq _ _
#align set.inter_Union Set.inter_iUnion
theorem iUnion_inter (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∩ s = ⋃ i, t i ∩ s :=
iSup_inf_eq _ _
#align set.Union_inter Set.iUnion_inter
theorem iUnion_union_distrib (s : ι → Set β) (t : ι → Set β) :
⋃ i, s i ∪ t i = (⋃ i, s i) ∪ ⋃ i, t i :=
iSup_sup_eq
#align set.Union_union_distrib Set.iUnion_union_distrib
theorem iInter_inter_distrib (s : ι → Set β) (t : ι → Set β) :
⋂ i, s i ∩ t i = (⋂ i, s i) ∩ ⋂ i, t i :=
iInf_inf_eq
#align set.Inter_inter_distrib Set.iInter_inter_distrib
theorem union_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∪ ⋃ i, t i) = ⋃ i, s ∪ t i :=
sup_iSup
#align set.union_Union Set.union_iUnion
theorem iUnion_union [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∪ s = ⋃ i, t i ∪ s :=
iSup_sup
#align set.Union_union Set.iUnion_union
theorem inter_iInter [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∩ ⋂ i, t i) = ⋂ i, s ∩ t i :=
inf_iInf
#align set.inter_Inter Set.inter_iInter
theorem iInter_inter [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋂ i, t i) ∩ s = ⋂ i, t i ∩ s :=
iInf_inf
#align set.Inter_inter Set.iInter_inter
-- classical
theorem union_iInter (s : Set β) (t : ι → Set β) : (s ∪ ⋂ i, t i) = ⋂ i, s ∪ t i :=
sup_iInf_eq _ _
#align set.union_Inter Set.union_iInter
theorem iInter_union (s : ι → Set β) (t : Set β) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.Inter_union Set.iInter_union
theorem iUnion_diff (s : Set β) (t : ι → Set β) : (⋃ i, t i) \ s = ⋃ i, t i \ s :=
iUnion_inter _ _
#align set.Union_diff Set.iUnion_diff
theorem diff_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s \ ⋃ i, t i) = ⋂ i, s \ t i := by
rw [diff_eq, compl_iUnion, inter_iInter]; rfl
#align set.diff_Union Set.diff_iUnion
theorem diff_iInter (s : Set β) (t : ι → Set β) : (s \ ⋂ i, t i) = ⋃ i, s \ t i := by
rw [diff_eq, compl_iInter, inter_iUnion]; rfl
#align set.diff_Inter Set.diff_iInter
theorem directed_on_iUnion {r} {f : ι → Set α} (hd : Directed (· ⊆ ·) f)
(h : ∀ x, DirectedOn r (f x)) : DirectedOn r (⋃ x, f x) := by
simp only [DirectedOn, exists_prop, mem_iUnion, exists_imp]
exact fun a₁ b₁ fb₁ a₂ b₂ fb₂ =>
let ⟨z, zb₁, zb₂⟩ := hd b₁ b₂
let ⟨x, xf, xa₁, xa₂⟩ := h z a₁ (zb₁ fb₁) a₂ (zb₂ fb₂)
⟨x, ⟨z, xf⟩, xa₁, xa₂⟩
#align set.directed_on_Union Set.directed_on_iUnion
theorem iUnion_inter_subset {ι α} {s t : ι → Set α} : ⋃ i, s i ∩ t i ⊆ (⋃ i, s i) ∩ ⋃ i, t i :=
le_iSup_inf_iSup s t
#align set.Union_inter_subset Set.iUnion_inter_subset
theorem iUnion_inter_of_monotone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_monotone hs ht
#align set.Union_inter_of_monotone Set.iUnion_inter_of_monotone
theorem iUnion_inter_of_antitone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_antitone hs ht
#align set.Union_inter_of_antitone Set.iUnion_inter_of_antitone
theorem iInter_union_of_monotone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_monotone hs ht
#align set.Inter_union_of_monotone Set.iInter_union_of_monotone
theorem iInter_union_of_antitone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_antitone hs ht
#align set.Inter_union_of_antitone Set.iInter_union_of_antitone
/-- An equality version of this lemma is `iUnion_iInter_of_monotone` in `Data.Set.Finite`. -/
theorem iUnion_iInter_subset {s : ι → ι' → Set α} : (⋃ j, ⋂ i, s i j) ⊆ ⋂ i, ⋃ j, s i j :=
iSup_iInf_le_iInf_iSup (flip s)
#align set.Union_Inter_subset Set.iUnion_iInter_subset
theorem iUnion_option {ι} (s : Option ι → Set α) : ⋃ o, s o = s none ∪ ⋃ i, s (some i) :=
iSup_option s
#align set.Union_option Set.iUnion_option
theorem iInter_option {ι} (s : Option ι → Set α) : ⋂ o, s o = s none ∩ ⋂ i, s (some i) :=
iInf_option s
#align set.Inter_option Set.iInter_option
section
variable (p : ι → Prop) [DecidablePred p]
theorem iUnion_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 _ _ _
#align set.Union_dite Set.iUnion_dite
theorem iUnion_ite (f g : ι → Set α) :
⋃ i, (if p i then f i else g i) = (⋃ (i) (_ : p i), f i) ∪ ⋃ (i) (_ : ¬p i), g i :=
iUnion_dite _ _ _
#align set.Union_ite Set.iUnion_ite
theorem iInter_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 :=
iInf_dite _ _ _
#align set.Inter_dite Set.iInter_dite
theorem iInter_ite (f g : ι → Set α) :
⋂ i, (if p i then f i else g i) = (⋂ (i) (_ : p i), f i) ∩ ⋂ (i) (_ : ¬p i), g i :=
iInter_dite _ _ _
#align set.Inter_ite Set.iInter_ite
end
theorem image_projection_prod {ι : Type*} {α : ι → Type*} {v : ∀ i : ι, Set (α i)}
(hv : (pi univ v).Nonempty) (i : ι) :
((fun x : ∀ i : ι, α i => x i) '' ⋂ k, (fun x : ∀ j : ι, α j => x k) ⁻¹' v k) = v i := by
classical
apply Subset.antisymm
· simp [iInter_subset]
· intro y y_in
simp only [mem_image, mem_iInter, mem_preimage]
rcases hv with ⟨z, hz⟩
refine' ⟨Function.update z i y, _, update_same i y z⟩
rw [@forall_update_iff ι α _ z i y fun i t => t ∈ v i]
exact ⟨y_in, fun j _ => by simpa using hz j⟩
#align set.image_projection_prod Set.image_projection_prod
/-! ### Unions and intersections indexed by `Prop` -/
theorem iInter_false {s : False → Set α} : iInter s = univ :=
iInf_false
#align set.Inter_false Set.iInter_false
theorem iUnion_false {s : False → Set α} : iUnion s = ∅ :=
iSup_false
#align set.Union_false Set.iUnion_false
@[simp]
theorem iInter_true {s : True → Set α} : iInter s = s trivial :=
iInf_true
#align set.Inter_true Set.iInter_true
@[simp]
theorem iUnion_true {s : True → Set α} : iUnion s = s trivial :=
iSup_true
#align set.Union_true Set.iUnion_true
@[simp]
theorem iInter_exists {p : ι → Prop} {f : Exists p → Set α} :
⋂ x, f x = ⋂ (i) (h : p i), f ⟨i, h⟩ :=
iInf_exists
#align set.Inter_exists Set.iInter_exists
@[simp]
theorem iUnion_exists {p : ι → Prop} {f : Exists p → Set α} :
⋃ x, f x = ⋃ (i) (h : p i), f ⟨i, h⟩ :=
iSup_exists
#align set.Union_exists Set.iUnion_exists
@[simp]
theorem iUnion_empty : (⋃ _ : ι, ∅ : Set α) = ∅ :=
iSup_bot
#align set.Union_empty Set.iUnion_empty
@[simp]
theorem iInter_univ : (⋂ _ : ι, univ : Set α) = univ :=
iInf_top
#align set.Inter_univ Set.iInter_univ
section
variable {s : ι → Set α}
@[simp]
theorem iUnion_eq_empty : ⋃ i, s i = ∅ ↔ ∀ i, s i = ∅ :=
iSup_eq_bot
#align set.Union_eq_empty Set.iUnion_eq_empty
@[simp]
theorem iInter_eq_univ : ⋂ i, s i = univ ↔ ∀ i, s i = univ :=
iInf_eq_top
#align set.Inter_eq_univ Set.iInter_eq_univ
@[simp]
theorem nonempty_iUnion : (⋃ i, s i).Nonempty ↔ ∃ i, (s i).Nonempty := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_Union Set.nonempty_iUnion
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_biUnion {t : Set α} {s : α → Set β} :
(⋃ i ∈ t, s i).Nonempty ↔ ∃ i ∈ t, (s i).Nonempty := by simp
#align set.nonempty_bUnion Set.nonempty_biUnion
theorem iUnion_nonempty_index (s : Set α) (t : s.Nonempty → Set β) :
⋃ h, t h = ⋃ x ∈ s, t ⟨x, ‹_›⟩ :=
iSup_exists
#align set.Union_nonempty_index Set.iUnion_nonempty_index
end
@[simp]
theorem iInter_iInter_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋂ (x) (h : x = b), s x h = s b rfl :=
iInf_iInf_eq_left
#align set.Inter_Inter_eq_left Set.iInter_iInter_eq_left
@[simp]
theorem iInter_iInter_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋂ (x) (h : b = x), s x h = s b rfl :=
iInf_iInf_eq_right
#align set.Inter_Inter_eq_right Set.iInter_iInter_eq_right
@[simp]
theorem iUnion_iUnion_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋃ (x) (h : x = b), s x h = s b rfl :=
iSup_iSup_eq_left
#align set.Union_Union_eq_left Set.iUnion_iUnion_eq_left
@[simp]
theorem iUnion_iUnion_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋃ (x) (h : b = x), s x h = s b rfl :=
iSup_iSup_eq_right
#align set.Union_Union_eq_right Set.iUnion_iUnion_eq_right
theorem iInter_or {p q : Prop} (s : p ∨ q → Set α) :
⋂ h, s h = (⋂ h : p, s (Or.inl h)) ∩ ⋂ h : q, s (Or.inr h) :=
iInf_or
#align set.Inter_or Set.iInter_or
theorem iUnion_or {p q : Prop} (s : p ∨ q → Set α) :
⋃ h, s h = (⋃ i, s (Or.inl i)) ∪ ⋃ j, s (Or.inr j) :=
iSup_or
#align set.Union_or Set.iUnion_or
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iUnion_and {p q : Prop} (s : p ∧ q → Set α) : ⋃ h, s h = ⋃ (hp) (hq), s ⟨hp, hq⟩ :=
iSup_and
#align set.Union_and Set.iUnion_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iInter_and {p q : Prop} (s : p ∧ q → Set α) : ⋂ h, s h = ⋂ (hp) (hq), s ⟨hp, hq⟩ :=
iInf_and
#align set.Inter_and Set.iInter_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iUnion_comm (s : ι → ι' → Set α) : ⋃ (i) (i'), s i i' = ⋃ (i') (i), s i i' :=
iSup_comm
#align set.Union_comm Set.iUnion_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iInter_comm (s : ι → ι' → Set α) : ⋂ (i) (i'), s i i' = ⋂ (i') (i), s i i' :=
iInf_comm
#align set.Inter_comm Set.iInter_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iUnion₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋃ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋃ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iSup₂_comm _
#align set.Union₂_comm Set.iUnion₂_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iInter₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋂ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋂ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iInf₂_comm _
#align set.Inter₂_comm Set.iInter₂_comm
@[simp]
theorem biUnion_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋃ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι']
#align set.bUnion_and Set.biUnion_and
@[simp]
theorem biUnion_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋃ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι]
#align set.bUnion_and' Set.biUnion_and'
@[simp]
theorem biInter_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋂ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iInter_and, @iInter_comm _ ι']
#align set.bInter_and Set.biInter_and
@[simp]
theorem biInter_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋂ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iInter_and, @iInter_comm _ ι]
#align set.bInter_and' Set.biInter_and'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iUnion_iUnion_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋃ (x) (h), s x h = s b (Or.inl rfl) ∪ ⋃ (x) (h : p x), s x (Or.inr h) := by
simp only [iUnion_or, iUnion_union_distrib, iUnion_iUnion_eq_left]
#align set.Union_Union_eq_or_left Set.iUnion_iUnion_eq_or_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iInter_iInter_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋂ (x) (h), s x h = s b (Or.inl rfl) ∩ ⋂ (x) (h : p x), s x (Or.inr h) := by
simp only [iInter_or, iInter_inter_distrib, iInter_iInter_eq_left]
#align set.Inter_Inter_eq_or_left Set.iInter_iInter_eq_or_left
/-! ### Bounded unions and intersections -/
/-- A specialization of `mem_iUnion₂`. -/
theorem mem_biUnion {s : Set α} {t : α → Set β} {x : α} {y : β} (xs : x ∈ s) (ytx : y ∈ t x) :
y ∈ ⋃ x ∈ s, t x :=
mem_iUnion₂_of_mem xs ytx
#align set.mem_bUnion Set.mem_biUnion
/-- A specialization of `mem_iInter₂`. -/
theorem mem_biInter {s : Set α} {t : α → Set β} {y : β} (h : ∀ x ∈ s, y ∈ t x) :
y ∈ ⋂ x ∈ s, t x :=
mem_iInter₂_of_mem h
#align set.mem_bInter Set.mem_biInter
/-- A specialization of `subset_iUnion₂`. -/
theorem subset_biUnion_of_mem {s : Set α} {u : α → Set β} {x : α} (xs : x ∈ s) :
u x ⊆ ⋃ x ∈ s, u x :=
--Porting note: Why is this not just `subset_iUnion₂ x xs`?
@subset_iUnion₂ β α (· ∈ s) (fun i _ => u i) x xs
#align set.subset_bUnion_of_mem Set.subset_biUnion_of_mem
/-- A specialization of `iInter₂_subset`. -/
theorem biInter_subset_of_mem {s : Set α} {t : α → Set β} {x : α} (xs : x ∈ s) :
⋂ x ∈ s, t x ⊆ t x :=
iInter₂_subset x xs
#align set.bInter_subset_of_mem Set.biInter_subset_of_mem
theorem biUnion_subset_biUnion_left {s s' : Set α} {t : α → Set β} (h : s ⊆ s') :
⋃ x ∈ s, t x ⊆ ⋃ x ∈ s', t x :=
iUnion₂_subset fun _ hx => subset_biUnion_of_mem <| h hx
#align set.bUnion_subset_bUnion_left Set.biUnion_subset_biUnion_left
theorem biInter_subset_biInter_left {s s' : Set α} {t : α → Set β} (h : s' ⊆ s) :
⋂ x ∈ s, t x ⊆ ⋂ x ∈ s', t x :=
subset_iInter₂ fun _ hx => biInter_subset_of_mem <| h hx
#align set.bInter_subset_bInter_left Set.biInter_subset_biInter_left
theorem biUnion_mono {s s' : Set α} {t t' : α → Set β} (hs : s' ⊆ s) (h : ∀ x ∈ s, t x ⊆ t' x) :
⋃ x ∈ s', t x ⊆ ⋃ x ∈ s, t' x :=
(biUnion_subset_biUnion_left hs).trans <| iUnion₂_mono h
#align set.bUnion_mono Set.biUnion_mono
theorem biInter_mono {s s' : Set α} {t t' : α → Set β} (hs : s ⊆ s') (h : ∀ x ∈ s, t x ⊆ t' x) :
⋂ x ∈ s', t x ⊆ ⋂ x ∈ s, t' x :=
(biInter_subset_biInter_left hs).trans <| iInter₂_mono h
#align set.bInter_mono Set.biInter_mono
theorem biUnion_eq_iUnion (s : Set α) (t : ∀ x ∈ s, Set β) :
⋃ x ∈ s, t x ‹_› = ⋃ x : s, t x x.2 :=
iSup_subtype'
#align set.bUnion_eq_Union Set.biUnion_eq_iUnion
theorem biInter_eq_iInter (s : Set α) (t : ∀ x ∈ s, Set β) :
⋂ x ∈ s, t x ‹_› = ⋂ x : s, t x x.2 :=
iInf_subtype'
#align set.bInter_eq_Inter Set.biInter_eq_iInter
theorem iUnion_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋃ x : { x // p x }, s x = ⋃ (x) (hx : p x), s ⟨x, hx⟩ :=
iSup_subtype
#align set.Union_subtype Set.iUnion_subtype
theorem iInter_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋂ x : { x // p x }, s x = ⋂ (x) (hx : p x), s ⟨x, hx⟩ :=
iInf_subtype
#align set.Inter_subtype Set.iInter_subtype
theorem biInter_empty (u : α → Set β) : ⋂ x ∈ (∅ : Set α), u x = univ :=
iInf_emptyset
#align set.bInter_empty Set.biInter_empty
theorem biInter_univ (u : α → Set β) : ⋂ x ∈ @univ α, u x = ⋂ x, u x :=
iInf_univ
#align set.bInter_univ Set.biInter_univ
@[simp]
theorem biUnion_self (s : Set α) : ⋃ x ∈ s, s = s :=
Subset.antisymm (iUnion₂_subset fun _ _ => Subset.refl s) fun _ hx => mem_biUnion hx hx
#align set.bUnion_self Set.biUnion_self
@[simp]
theorem iUnion_nonempty_self (s : Set α) : ⋃ _ : s.Nonempty, s = s := by
rw [iUnion_nonempty_index, biUnion_self]
#align set.Union_nonempty_self Set.iUnion_nonempty_self
theorem biInter_singleton (a : α) (s : α → Set β) : ⋂ x ∈ ({a} : Set α), s x = s a :=
iInf_singleton
#align set.bInter_singleton Set.biInter_singleton
theorem biInter_union (s t : Set α) (u : α → Set β) :
⋂ x ∈ s ∪ t, u x = (⋂ x ∈ s, u x) ∩ ⋂ x ∈ t, u x :=
iInf_union
#align set.bInter_union Set.biInter_union
theorem biInter_insert (a : α) (s : Set α) (t : α → Set β) :
⋂ x ∈ insert a s, t x = t a ∩ ⋂ x ∈ s, t x := by simp
#align set.bInter_insert Set.biInter_insert
theorem biInter_pair (a b : α) (s : α → Set β) : ⋂ x ∈ ({a, b} : Set α), s x = s a ∩ s b := by
rw [biInter_insert, biInter_singleton]
#align set.bInter_pair Set.biInter_pair
theorem biInter_inter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, f i ∩ t = (⋂ i ∈ s, f i) ∩ t := by
haveI : Nonempty s := hs.to_subtype
simp [biInter_eq_iInter, ← iInter_inter]
#align set.bInter_inter Set.biInter_inter
theorem inter_biInter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, t ∩ f i = t ∩ ⋂ i ∈ s, f i := by
rw [inter_comm, ← biInter_inter hs]
simp [inter_comm]
#align set.inter_bInter Set.inter_biInter
theorem biUnion_empty (s : α → Set β) : ⋃ x ∈ (∅ : Set α), s x = ∅ :=
iSup_emptyset
#align set.bUnion_empty Set.biUnion_empty
theorem biUnion_univ (s : α → Set β) : ⋃ x ∈ @univ α, s x = ⋃ x, s x :=
iSup_univ
#align set.bUnion_univ Set.biUnion_univ
theorem biUnion_singleton (a : α) (s : α → Set β) : ⋃ x ∈ ({a} : Set α), s x = s a :=
iSup_singleton
#align set.bUnion_singleton Set.biUnion_singleton
@[simp]
theorem biUnion_of_singleton (s : Set α) : ⋃ x ∈ s, {x} = s :=
ext <| by simp
#align set.bUnion_of_singleton Set.biUnion_of_singleton
theorem biUnion_union (s t : Set α) (u : α → Set β) :
⋃ x ∈ s ∪ t, u x = (⋃ x ∈ s, u x) ∪ ⋃ x ∈ t, u x :=
iSup_union
#align set.bUnion_union Set.biUnion_union
@[simp]
theorem iUnion_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋃ i, f i = ⋃ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iUnion_subtype _ _
#align set.Union_coe_set Set.iUnion_coe_set
@[simp]
theorem iInter_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋂ i, f i = ⋂ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iInter_subtype _ _
#align set.Inter_coe_set Set.iInter_coe_set
theorem biUnion_insert (a : α) (s : Set α) (t : α → Set β) :
⋃ x ∈ insert a s, t x = t a ∪ ⋃ x ∈ s, t x := by simp
#align set.bUnion_insert Set.biUnion_insert
theorem biUnion_pair (a b : α) (s : α → Set β) : ⋃ x ∈ ({a, b} : Set α), s x = s a ∪ s b :=
by simp
#align set.bUnion_pair Set.biUnion_pair
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem inter_iUnion₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∩ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ∩ t i j := by simp only [inter_iUnion]
#align set.inter_Union₂ Set.inter_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_inter (s : ∀ i, κ i → Set α) (t : Set α) :
(⋃ (i) (j), s i j) ∩ t = ⋃ (i) (j), s i j ∩ t := by simp_rw [iUnion_inter]
#align set.Union₂_inter Set.iUnion₂_inter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_iInter₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_iInter]
#align set.union_Inter₂ Set.union_iInter₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_union (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [iInter_union]
#align set.Inter₂_union Set.iInter₂_union
theorem mem_sUnion_of_mem {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∈ t) (ht : t ∈ S) :
x ∈ ⋃₀S :=
⟨t, ht, hx⟩
#align set.mem_sUnion_of_mem Set.mem_sUnion_of_mem
-- is this theorem really necessary?
theorem not_mem_of_not_mem_sUnion {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∉ ⋃₀S)
(ht : t ∈ S) : x ∉ t := fun h => hx ⟨t, ht, h⟩
#align set.not_mem_of_not_mem_sUnion Set.not_mem_of_not_mem_sUnion
theorem sInter_subset_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : ⋂₀ S ⊆ t :=
sInf_le tS
#align set.sInter_subset_of_mem Set.sInter_subset_of_mem
theorem subset_sUnion_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : t ⊆ ⋃₀S :=
le_sSup tS
#align set.subset_sUnion_of_mem Set.subset_sUnion_of_mem
theorem subset_sUnion_of_subset {s : Set α} (t : Set (Set α)) (u : Set α) (h₁ : s ⊆ u)
(h₂ : u ∈ t) : s ⊆ ⋃₀t :=
Subset.trans h₁ (subset_sUnion_of_mem h₂)
#align set.subset_sUnion_of_subset Set.subset_sUnion_of_subset
theorem sUnion_subset {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t' ⊆ t) : ⋃₀S ⊆ t :=
sSup_le h
#align set.sUnion_subset Set.sUnion_subset
@[simp]
theorem sUnion_subset_iff {s : Set (Set α)} {t : Set α} : ⋃₀s ⊆ t ↔ ∀ t' ∈ s, t' ⊆ t :=
sSup_le_iff
#align set.sUnion_subset_iff Set.sUnion_subset_iff
/-- `sUnion` is monotone under taking a subset of each set. -/
lemma sUnion_mono_subsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, t ⊆ f t) :
⋃₀ s ⊆ ⋃₀ (f '' s) :=
fun _ ⟨t, htx, hxt⟩ ↦ ⟨f t, mem_image_of_mem f htx, hf t hxt⟩
/-- `sUnion` is monotone under taking a superset of each set. -/
lemma sUnion_mono_supsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, f t ⊆ t) :
⋃₀ (f '' s) ⊆ ⋃₀ s :=
-- If t ∈ f '' s is arbitrary; t = f u for some u : Set α.
fun _ ⟨_, ⟨u, hus, hut⟩, hxt⟩ ↦ ⟨u, hus, (hut ▸ hf u) hxt⟩
theorem subset_sInter {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t ⊆ t') : t ⊆ ⋂₀ S :=
le_sInf h
#align set.subset_sInter Set.subset_sInter
@[simp]
theorem subset_sInter_iff {S : Set (Set α)} {t : Set α} : t ⊆ ⋂₀ S ↔ ∀ t' ∈ S, t ⊆ t' :=
le_sInf_iff
#align set.subset_sInter_iff Set.subset_sInter_iff
@[gcongr]
theorem sUnion_subset_sUnion {S T : Set (Set α)} (h : S ⊆ T) : ⋃₀S ⊆ ⋃₀T :=
sUnion_subset fun _ hs => subset_sUnion_of_mem (h hs)
#align set.sUnion_subset_sUnion Set.sUnion_subset_sUnion
@[gcongr]
theorem sInter_subset_sInter {S T : Set (Set α)} (h : S ⊆ T) : ⋂₀ T ⊆ ⋂₀ S :=
subset_sInter fun _ hs => sInter_subset_of_mem (h hs)
#align set.sInter_subset_sInter Set.sInter_subset_sInter
@[simp]
theorem sUnion_empty : ⋃₀∅ = (∅ : Set α) :=
sSup_empty
#align set.sUnion_empty Set.sUnion_empty
@[simp]
theorem sInter_empty : ⋂₀ ∅ = (univ : Set α) :=
sInf_empty
#align set.sInter_empty Set.sInter_empty
@[simp]
theorem sUnion_singleton (s : Set α) : ⋃₀{s} = s :=
sSup_singleton
#align set.sUnion_singleton Set.sUnion_singleton
@[simp]
theorem sInter_singleton (s : Set α) : ⋂₀ {s} = s :=
sInf_singleton
#align set.sInter_singleton Set.sInter_singleton
@[simp]
theorem sUnion_eq_empty {S : Set (Set α)} : ⋃₀S = ∅ ↔ ∀ s ∈ S, s = ∅ :=
sSup_eq_bot
#align set.sUnion_eq_empty Set.sUnion_eq_empty
@[simp]
theorem sInter_eq_univ {S : Set (Set α)} : ⋂₀ S = univ ↔ ∀ s ∈ S, s = univ :=
sInf_eq_top
#align set.sInter_eq_univ Set.sInter_eq_univ
theorem subset_powerset_iff {s : Set (Set α)} {t : Set α} : s ⊆ 𝒫 t ↔ ⋃₀ s ⊆ t :=
sUnion_subset_iff.symm
/-- `⋃₀` and `𝒫` form a Galois connection. -/
theorem sUnion_powerset_gc :
GaloisConnection (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gc_sSup_Iic
/-- `⋃₀` and `𝒫` form a Galois insertion. -/
def sUnion_powerset_gi :
GaloisInsertion (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gi_sSup_Iic
/-- If all sets in a collection are either `∅` or `Set.univ`, then so is their union. -/
theorem sUnion_mem_empty_univ {S : Set (Set α)} (h : S ⊆ {∅, univ}) :
⋃₀ S ∈ ({∅, univ} : Set (Set α)) := by
simp only [mem_insert_iff, mem_singleton_iff, or_iff_not_imp_left, sUnion_eq_empty, not_forall]
rintro ⟨s, hs, hne⟩
obtain rfl : s = univ := (h hs).resolve_left hne
exact univ_subset_iff.1 <| subset_sUnion_of_mem hs
@[simp]
theorem nonempty_sUnion {S : Set (Set α)} : (⋃₀S).Nonempty ↔ ∃ s ∈ S, Set.Nonempty s := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_sUnion Set.nonempty_sUnion
theorem Nonempty.of_sUnion {s : Set (Set α)} (h : (⋃₀s).Nonempty) : s.Nonempty :=
let ⟨s, hs, _⟩ := nonempty_sUnion.1 h
⟨s, hs⟩
#align set.nonempty.of_sUnion Set.Nonempty.of_sUnion
theorem Nonempty.of_sUnion_eq_univ [Nonempty α] {s : Set (Set α)} (h : ⋃₀s = univ) : s.Nonempty :=
Nonempty.of_sUnion <| h.symm ▸ univ_nonempty
#align set.nonempty.of_sUnion_eq_univ Set.Nonempty.of_sUnion_eq_univ
theorem sUnion_union (S T : Set (Set α)) : ⋃₀(S ∪ T) = ⋃₀S ∪ ⋃₀T :=
sSup_union
#align set.sUnion_union Set.sUnion_union
theorem sInter_union (S T : Set (Set α)) : ⋂₀ (S ∪ T) = ⋂₀ S ∩ ⋂₀ T :=
sInf_union
#align set.sInter_union Set.sInter_union
@[simp]
theorem sUnion_insert (s : Set α) (T : Set (Set α)) : ⋃₀insert s T = s ∪ ⋃₀T :=
sSup_insert
#align set.sUnion_insert Set.sUnion_insert
@[simp]
theorem sInter_insert (s : Set α) (T : Set (Set α)) : ⋂₀ insert s T = s ∩ ⋂₀ T :=
sInf_insert
#align set.sInter_insert Set.sInter_insert
@[simp]
theorem sUnion_diff_singleton_empty (s : Set (Set α)) : ⋃₀(s \ {∅}) = ⋃₀s :=
sSup_diff_singleton_bot s
#align set.sUnion_diff_singleton_empty Set.sUnion_diff_singleton_empty
@[simp]
theorem sInter_diff_singleton_univ (s : Set (Set α)) : ⋂₀ (s \ {univ}) = ⋂₀ s :=
sInf_diff_singleton_top s
#align set.sInter_diff_singleton_univ Set.sInter_diff_singleton_univ
theorem sUnion_pair (s t : Set α) : ⋃₀{s, t} = s ∪ t :=
sSup_pair
#align set.sUnion_pair Set.sUnion_pair
theorem sInter_pair (s t : Set α) : ⋂₀ {s, t} = s ∩ t :=
sInf_pair
#align set.sInter_pair Set.sInter_pair
@[simp]
theorem sUnion_image (f : α → Set β) (s : Set α) : ⋃₀(f '' s) = ⋃ x ∈ s, f x :=
sSup_image
#align set.sUnion_image Set.sUnion_image
@[simp]
theorem sInter_image (f : α → Set β) (s : Set α) : ⋂₀ (f '' s) = ⋂ x ∈ s, f x :=
sInf_image
#align set.sInter_image Set.sInter_image
@[simp]
theorem sUnion_range (f : ι → Set β) : ⋃₀range f = ⋃ x, f x :=
rfl
#align set.sUnion_range Set.sUnion_range
@[simp]
theorem sInter_range (f : ι → Set β) : ⋂₀ range f = ⋂ x, f x :=
rfl
#align set.sInter_range Set.sInter_range
theorem iUnion_eq_univ_iff {f : ι → Set α} : ⋃ i, f i = univ ↔ ∀ x, ∃ i, x ∈ f i := by
simp only [eq_univ_iff_forall, mem_iUnion]
#align set.Union_eq_univ_iff Set.iUnion_eq_univ_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_eq_univ_iff {s : ∀ i, κ i → Set α} :
⋃ (i) (j), s i j = univ ↔ ∀ a, ∃ i j, a ∈ s i j :=
by simp only [iUnion_eq_univ_iff, mem_iUnion]
#align set.Union₂_eq_univ_iff Set.iUnion₂_eq_univ_iff
theorem sUnion_eq_univ_iff {c : Set (Set α)} : ⋃₀c = univ ↔ ∀ a, ∃ b ∈ c, a ∈ b := by
simp only [eq_univ_iff_forall, mem_sUnion]
#align set.sUnion_eq_univ_iff Set.sUnion_eq_univ_iff
-- classical
theorem iInter_eq_empty_iff {f : ι → Set α} : ⋂ i, f i = ∅ ↔ ∀ x, ∃ i, x ∉ f i := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.Inter_eq_empty_iff Set.iInter_eq_empty_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
theorem iInter₂_eq_empty_iff {s : ∀ i, κ i → Set α} :
⋂ (i) (j), s i j = ∅ ↔ ∀ a, ∃ i j, a ∉ s i j := by
simp only [eq_empty_iff_forall_not_mem, mem_iInter, not_forall]
#align set.Inter₂_eq_empty_iff Set.iInter₂_eq_empty_iff
-- classical
theorem sInter_eq_empty_iff {c : Set (Set α)} : ⋂₀ c = ∅ ↔ ∀ a, ∃ b ∈ c, a ∉ b := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.sInter_eq_empty_iff Set.sInter_eq_empty_iff
-- classical
@[simp]
theorem nonempty_iInter {f : ι → Set α} : (⋂ i, f i).Nonempty ↔ ∃ x, ∀ i, x ∈ f i := by
simp [nonempty_iff_ne_empty, iInter_eq_empty_iff]
#align set.nonempty_Inter Set.nonempty_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_iInter₂ {s : ∀ i, κ i → Set α} :
(⋂ (i) (j), s i j).Nonempty ↔ ∃ a, ∀ i j, a ∈ s i j := by
simp
#align set.nonempty_Inter₂ Set.nonempty_iInter₂
-- classical
@[simp]
theorem nonempty_sInter {c : Set (Set α)} : (⋂₀ c).Nonempty ↔ ∃ a, ∀ b ∈ c, a ∈ b := by
simp [nonempty_iff_ne_empty, sInter_eq_empty_iff]
#align set.nonempty_sInter Set.nonempty_sInter
-- classical
theorem compl_sUnion (S : Set (Set α)) : (⋃₀S)ᶜ = ⋂₀ (compl '' S) :=
ext fun x => by simp
#align set.compl_sUnion Set.compl_sUnion
-- classical
theorem sUnion_eq_compl_sInter_compl (S : Set (Set α)) : ⋃₀S = (⋂₀ (compl '' S))ᶜ := by
rw [← compl_compl (⋃₀S), compl_sUnion]
#align set.sUnion_eq_compl_sInter_compl Set.sUnion_eq_compl_sInter_compl
-- classical
theorem compl_sInter (S : Set (Set α)) : (⋂₀ S)ᶜ = ⋃₀(compl '' S) := by
rw [sUnion_eq_compl_sInter_compl, compl_compl_image]
#align set.compl_sInter Set.compl_sInter
-- classical
theorem sInter_eq_compl_sUnion_compl (S : Set (Set α)) : ⋂₀ S = (⋃₀(compl '' S))ᶜ := by
rw [← compl_compl (⋂₀ S), compl_sInter]
#align set.sInter_eq_compl_sUnion_compl Set.sInter_eq_compl_sUnion_compl
theorem inter_empty_of_inter_sUnion_empty {s t : Set α} {S : Set (Set α)} (hs : t ∈ S)
(h : s ∩ ⋃₀S = ∅) : s ∩ t = ∅ :=
eq_empty_of_subset_empty <| by
rw [← h]; exact inter_subset_inter_right _ (subset_sUnion_of_mem hs)
#align set.inter_empty_of_inter_sUnion_empty Set.inter_empty_of_inter_sUnion_empty
theorem range_sigma_eq_iUnion_range {γ : α → Type*} (f : Sigma γ → β) :
range f = ⋃ a, range fun b => f ⟨a, b⟩ :=
Set.ext <| by simp
#align set.range_sigma_eq_Union_range Set.range_sigma_eq_iUnion_range
theorem iUnion_eq_range_sigma (s : α → Set β) : ⋃ i, s i = range fun a : Σi, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_sigma Set.iUnion_eq_range_sigma
theorem iUnion_eq_range_psigma (s : ι → Set β) : ⋃ i, s i = range fun a : Σ'i, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_psigma Set.iUnion_eq_range_psigma
theorem iUnion_image_preimage_sigma_mk_eq_self {ι : Type*} {σ : ι → Type*} (s : Set (Sigma σ)) :
⋃ i, Sigma.mk i '' (Sigma.mk i ⁻¹' s) = s := by
ext x
simp only [mem_iUnion, mem_image, mem_preimage]
constructor
· rintro ⟨i, a, h, rfl⟩
exact h
· intro h
cases' x with i a
exact ⟨i, a, h, rfl⟩
#align set.Union_image_preimage_sigma_mk_eq_self Set.iUnion_image_preimage_sigma_mk_eq_self
theorem Sigma.univ (X : α → Type*) : (Set.univ : Set (Σa, X a)) = ⋃ a, range (Sigma.mk a) :=
Set.ext fun x =>
iff_of_true trivial ⟨range (Sigma.mk x.1), Set.mem_range_self _, x.2, Sigma.eta x⟩
#align set.sigma.univ Set.Sigma.univ
alias sUnion_mono := sUnion_subset_sUnion
#align set.sUnion_mono Set.sUnion_mono
theorem iUnion_subset_iUnion_const {s : Set α} (h : ι → ι₂) : ⋃ _ : ι, s ⊆ ⋃ _ : ι₂, s :=
@iSup_const_mono (Set α) ι ι₂ _ s h
#align set.Union_subset_Union_const Set.iUnion_subset_iUnion_const
@[simp]
theorem iUnion_singleton_eq_range {α β : Type*} (f : α → β) : ⋃ x : α, {f x} = range f := by
ext x
simp [@eq_comm _ x]
#align set.Union_singleton_eq_range Set.iUnion_singleton_eq_range
theorem iUnion_of_singleton (α : Type*) : (⋃ x, {x} : Set α) = univ := by simp [Set.ext_iff]
#align set.Union_of_singleton Set.iUnion_of_singleton
theorem iUnion_of_singleton_coe (s : Set α) : ⋃ i : s, ({(i : α)} : Set α) = s := by simp
#align set.Union_of_singleton_coe Set.iUnion_of_singleton_coe
theorem sUnion_eq_biUnion {s : Set (Set α)} : ⋃₀s = ⋃ (i : Set α) (_ : i ∈ s), i := by
rw [← sUnion_image, image_id']
#align set.sUnion_eq_bUnion Set.sUnion_eq_biUnion
theorem sInter_eq_biInter {s : Set (Set α)} : ⋂₀ s = ⋂ (i : Set α) (_ : i ∈ s), i := by
rw [← sInter_image, image_id']
#align set.sInter_eq_bInter Set.sInter_eq_biInter
theorem sUnion_eq_iUnion {s : Set (Set α)} : ⋃₀s = ⋃ i : s, i := by
simp only [← sUnion_range, Subtype.range_coe]
#align set.sUnion_eq_Union Set.sUnion_eq_iUnion
theorem sInter_eq_iInter {s : Set (Set α)} : ⋂₀ s = ⋂ i : s, i := by
simp only [← sInter_range, Subtype.range_coe]
#align set.sInter_eq_Inter Set.sInter_eq_iInter
@[simp]
theorem iUnion_of_empty [IsEmpty ι] (s : ι → Set α) : ⋃ i, s i = ∅ :=
iSup_of_empty _
#align set.Union_of_empty Set.iUnion_of_empty
@[simp]
theorem iInter_of_empty [IsEmpty ι] (s : ι → Set α) : ⋂ i, s i = univ :=
iInf_of_empty _
#align set.Inter_of_empty Set.iInter_of_empty
theorem union_eq_iUnion {s₁ s₂ : Set α} : s₁ ∪ s₂ = ⋃ b : Bool, cond b s₁ s₂ :=
sup_eq_iSup s₁ s₂
#align set.union_eq_Union Set.union_eq_iUnion
theorem inter_eq_iInter {s₁ s₂ : Set α} : s₁ ∩ s₂ = ⋂ b : Bool, cond b s₁ s₂ :=
inf_eq_iInf s₁ s₂
#align set.inter_eq_Inter Set.inter_eq_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_union_sInter {S T : Set (Set α)} :
⋂₀ S ∪ ⋂₀ T = ⋂ p ∈ S ×ˢ T, (p : Set α × Set α).1 ∪ p.2 :=
sInf_sup_sInf
#align set.sInter_union_sInter Set.sInter_union_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_inter_sUnion {s t : Set (Set α)} :
⋃₀s ∩ ⋃₀t = ⋃ p ∈ s ×ˢ t, (p : Set α × Set α).1 ∩ p.2 :=
sSup_inf_sSup
#align set.sUnion_inter_sUnion Set.sUnion_inter_sUnion
theorem biUnion_iUnion (s : ι → Set α) (t : α → Set β) :
⋃ x ∈ ⋃ i, s i, t x = ⋃ (i) (x ∈ s i), t x := by simp [@iUnion_comm _ ι]
#align set.bUnion_Union Set.biUnion_iUnion
theorem biInter_iUnion (s : ι → Set α) (t : α → Set β) :
⋂ x ∈ ⋃ i, s i, t x = ⋂ (i) (x ∈ s i), t x := by simp [@iInter_comm _ ι]
#align set.bInter_Union Set.biInter_iUnion
theorem sUnion_iUnion (s : ι → Set (Set α)) : ⋃₀⋃ i, s i = ⋃ i, ⋃₀s i := by
simp only [sUnion_eq_biUnion, biUnion_iUnion]
#align set.sUnion_Union Set.sUnion_iUnion
theorem sInter_iUnion (s : ι → Set (Set α)) : ⋂₀ ⋃ i, s i = ⋂ i, ⋂₀ s i := by
simp only [sInter_eq_biInter, biInter_iUnion]
#align set.sInter_Union Set.sInter_iUnion
theorem iUnion_range_eq_sUnion {α β : Type*} (C : Set (Set α)) {f : ∀ s : C, β → (s : Type _)}
(hf : ∀ s : C, Surjective (f s)) : ⋃ y : β, range (fun s : C => (f s y).val) = ⋃₀C := by
ext x; constructor
· rintro ⟨s, ⟨y, rfl⟩, ⟨s, hs⟩, rfl⟩
refine' ⟨_, hs, _⟩
exact (f ⟨s, hs⟩ y).2
· rintro ⟨s, hs, hx⟩
cases' hf ⟨s, hs⟩ ⟨x, hx⟩ with y hy
refine' ⟨_, ⟨y, rfl⟩, ⟨s, hs⟩, _⟩
exact congr_arg Subtype.val hy
#align set.Union_range_eq_sUnion Set.iUnion_range_eq_sUnion
theorem iUnion_range_eq_iUnion (C : ι → Set α) {f : ∀ x : ι, β → C x}
(hf : ∀ x : ι, Surjective (f x)) : ⋃ y : β, range (fun x : ι => (f x y).val) = ⋃ x, C x := by
ext x; rw [mem_iUnion, mem_iUnion]; constructor
· rintro ⟨y, i, rfl⟩
exact ⟨i, (f i y).2⟩
· rintro ⟨i, hx⟩
cases' hf i ⟨x, hx⟩ with y hy
exact ⟨y, i, congr_arg Subtype.val hy⟩
#align set.Union_range_eq_Union Set.iUnion_range_eq_iUnion
theorem union_distrib_iInter_left (s : ι → Set α) (t : Set α) : (t ∪ ⋂ i, s i) = ⋂ i, t ∪ s i :=
sup_iInf_eq _ _
#align set.union_distrib_Inter_left Set.union_distrib_iInter_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_left (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_distrib_iInter_left]
#align set.union_distrib_Inter₂_left Set.union_distrib_iInter₂_left
theorem union_distrib_iInter_right (s : ι → Set α) (t : Set α) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.union_distrib_Inter_right Set.union_distrib_iInter_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_right (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [union_distrib_iInter_right]
#align set.union_distrib_Inter₂_right Set.union_distrib_iInter₂_right
section Function
/-! ### `mapsTo` -/
theorem mapsTo_sUnion {S : Set (Set α)} {t : Set β} {f : α → β} (H : ∀ s ∈ S, MapsTo f s t) :
MapsTo f (⋃₀S) t := fun _ ⟨s, hs, hx⟩ => H s hs hx
#align set.maps_to_sUnion Set.mapsTo_sUnion
theorem mapsTo_iUnion {s : ι → Set α} {t : Set β} {f : α → β} (H : ∀ i, MapsTo f (s i) t) :
MapsTo f (⋃ i, s i) t :=
mapsTo_sUnion <| forall_range_iff.2 H
#align set.maps_to_Union Set.mapsTo_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂ {s : ∀ i, κ i → Set α} {t : Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) t) : MapsTo f (⋃ (i) (j), s i j) t :=
mapsTo_iUnion fun i => mapsTo_iUnion (H i)
#align set.maps_to_Union₂ Set.mapsTo_iUnion₂
theorem mapsTo_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋃ i, s i) (⋃ i, t i) :=
mapsTo_iUnion fun i => (H i).mono (Subset.refl _) (subset_iUnion t i)
#align set.maps_to_Union_Union Set.mapsTo_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
mapsTo_iUnion_iUnion fun i => mapsTo_iUnion_iUnion (H i)
#align set.maps_to_Union₂_Union₂ Set.mapsTo_iUnion₂_iUnion₂
theorem mapsTo_sInter {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, MapsTo f s t) :
MapsTo f s (⋂₀ T) := fun _ hx t ht => H t ht hx
#align set.maps_to_sInter Set.mapsTo_sInter
theorem mapsTo_iInter {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, MapsTo f s (t i)) :
MapsTo f s (⋂ i, t i) := fun _ hx => mem_iInter.2 fun i => H i hx
#align set.maps_to_Inter Set.mapsTo_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f s (t i j)) : MapsTo f s (⋂ (i) (j), t i j) :=
mapsTo_iInter fun i => mapsTo_iInter (H i)
#align set.maps_to_Inter₂ Set.mapsTo_iInter₂
theorem mapsTo_iInter_iInter {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋂ i, s i) (⋂ i, t i) :=
mapsTo_iInter fun i => (H i).mono (iInter_subset s i) (Subset.refl _)
#align set.maps_to_Inter_Inter Set.mapsTo_iInter_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂_iInter₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋂ (i) (j), s i j) (⋂ (i) (j), t i j) :=
mapsTo_iInter_iInter fun i => mapsTo_iInter_iInter (H i)
#align set.maps_to_Inter₂_Inter₂ Set.mapsTo_iInter₂_iInter₂
theorem image_iInter_subset (s : ι → Set α) (f : α → β) : (f '' ⋂ i, s i) ⊆ ⋂ i, f '' s i :=
(mapsTo_iInter_iInter fun i => mapsTo_image f (s i)).image_subset
#align set.image_Inter_subset Set.image_iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂_subset (s : ∀ i, κ i → Set α) (f : α → β) :
(f '' ⋂ (i) (j), s i j) ⊆ ⋂ (i) (j), f '' s i j :=
(mapsTo_iInter₂_iInter₂ fun i hi => mapsTo_image f (s i hi)).image_subset
#align set.image_Inter₂_subset Set.image_iInter₂_subset
theorem image_sInter_subset (S : Set (Set α)) (f : α → β) : f '' ⋂₀ S ⊆ ⋂ s ∈ S, f '' s := by
rw [sInter_eq_biInter]
apply image_iInter₂_subset
#align set.image_sInter_subset Set.image_sInter_subset
/-! ### `restrictPreimage` -/
section
open Function
variable (s : Set β) {f : α → β} {U : ι → Set β} (hU : iUnion U = univ)
theorem injective_iff_injective_of_iUnion_eq_univ :
Injective f ↔ ∀ i, Injective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_injective H, fun H x y e => _⟩
obtain ⟨i, hi⟩ := Set.mem_iUnion.mp
(show f x ∈ Set.iUnion U by rw [hU]; triv)
injection @H i ⟨x, hi⟩ ⟨y, show f y ∈ U i from e ▸ hi⟩ (Subtype.ext e)
#align set.injective_iff_injective_of_Union_eq_univ Set.injective_iff_injective_of_iUnion_eq_univ
theorem surjective_iff_surjective_of_iUnion_eq_univ :
Surjective f ↔ ∀ i, Surjective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_surjective H, fun H x => _⟩
obtain ⟨i, hi⟩ :=
Set.mem_iUnion.mp
(show x ∈ Set.iUnion U by rw [hU]; triv)
exact ⟨_, congr_arg Subtype.val (H i ⟨x, hi⟩).choose_spec⟩
#align set.surjective_iff_surjective_of_Union_eq_univ Set.surjective_iff_surjective_of_iUnion_eq_univ
theorem bijective_iff_bijective_of_iUnion_eq_univ :
Bijective f ↔ ∀ i, Bijective ((U i).restrictPreimage f) := by
rw [Bijective, injective_iff_injective_of_iUnion_eq_univ hU,
surjective_iff_surjective_of_iUnion_eq_univ hU]
simp [Bijective, forall_and]
#align set.bijective_iff_bijective_of_Union_eq_univ Set.bijective_iff_bijective_of_iUnion_eq_univ
end
/-! ### `InjOn` -/
theorem InjOn.image_iInter_eq [Nonempty ι] {s : ι → Set α} {f : α → β} (h : InjOn f (⋃ i, s i)) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
inhabit ι
refine' Subset.antisymm (image_iInter_subset s f) fun y hy => _
simp only [mem_iInter, mem_image_iff_bex] at hy
choose x hx hy using hy
refine' ⟨x default, mem_iInter.2 fun i => _, hy _⟩
suffices x default = x i by
rw [this]
apply hx
replace hx : ∀ i, x i ∈ ⋃ j, s j := fun i => (subset_iUnion _ _) (hx i)
apply h (hx _) (hx _)
simp only [hy]
#align set.inj_on.image_Inter_eq Set.InjOn.image_iInter_eq
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
theorem InjOn.image_biInter_eq {p : ι → Prop} {s : ∀ i, p i → Set α} (hp : ∃ i, p i)
{f : α → β} (h : InjOn f (⋃ (i) (hi), s i hi)) :
(f '' ⋂ (i) (hi), s i hi) = ⋂ (i) (hi), f '' s i hi := by
simp only [iInter, iInf_subtype']
haveI : Nonempty { i // p i } := nonempty_subtype.2 hp
apply InjOn.image_iInter_eq
simpa only [iUnion, iSup_subtype'] using h
#align set.inj_on.image_bInter_eq Set.InjOn.image_biInter_eq
theorem image_iInter {f : α → β} (hf : Bijective f) (s : ι → Set α) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
cases isEmpty_or_nonempty ι
· simp_rw [iInter_of_empty, image_univ_of_surjective hf.surjective]
· exact (hf.injective.injOn _).image_iInter_eq
#align set.image_Inter Set.image_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂ {f : α → β} (hf : Bijective f) (s : ∀ i, κ i → Set α) :
(f '' ⋂ (i) (j), s i j) = ⋂ (i) (j), f '' s i j := by simp_rw [image_iInter hf]
#align set.image_Inter₂ Set.image_iInter₂
theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
(hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i) := by
intro x hx y hy hxy
rcases mem_iUnion.1 hx with ⟨i, hx⟩
rcases mem_iUnion.1 hy with ⟨j, hy⟩
rcases hs i j with ⟨k, hi, hj⟩
exact hf k (hi hx) (hj hy) hxy
#align set.inj_on_Union_of_directed Set.inj_on_iUnion_of_directed
/-! ### `SurjOn` -/
theorem surjOn_sUnion {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, SurjOn f s t) :
SurjOn f s (⋃₀T) := fun _ ⟨t, ht, hx⟩ => H t ht hx
#align set.surj_on_sUnion Set.surjOn_sUnion
theorem surjOn_iUnion {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, SurjOn f s (t i)) :
SurjOn f s (⋃ i, t i) :=
surjOn_sUnion <| forall_range_iff.2 H
#align set.surj_on_Union Set.surjOn_iUnion
theorem surjOn_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) : SurjOn f (⋃ i, s i) (⋃ i, t i) :=
surjOn_iUnion fun i => (H i).mono (subset_iUnion _ _) (Subset.refl _)
#align set.surj_on_Union_Union Set.surjOn_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f s (t i j)) : SurjOn f s (⋃ (i) (j), t i j) :=
surjOn_iUnion fun i => surjOn_iUnion (H i)
#align set.surj_on_Union₂ Set.surjOn_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f (s i j) (t i j)) : SurjOn f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
surjOn_iUnion_iUnion fun i => surjOn_iUnion_iUnion (H i)
#align set.surj_on_Union₂_Union₂ Set.surjOn_iUnion₂_iUnion₂
theorem surjOn_iInter [Nonempty ι] {s : ι → Set α} {t : Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) t) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) t := by
intro y hy
rw [Hinj.image_iInter_eq, mem_iInter]
exact fun i => H i hy
#align set.surj_on_Inter Set.surjOn_iInter
theorem surjOn_iInter_iInter [Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) (⋂ i, t i) :=
surjOn_iInter (fun i => (H i).mono (Subset.refl _) (iInter_subset _ _)) Hinj
#align set.surj_on_Inter_Inter Set.surjOn_iInter_iInter
/-! ### `BijOn` -/
theorem bijOn_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i))
(Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
⟨mapsTo_iUnion_iUnion fun i => (H i).mapsTo, Hinj, surjOn_iUnion_iUnion fun i => (H i).surjOn⟩
#align set.bij_on_Union Set.bijOn_iUnion
theorem bijOn_iInter [hi : Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, BijOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
⟨mapsTo_iInter_iInter fun i => (H i).mapsTo,
hi.elim fun i => (H i).injOn.mono (iInter_subset _ _),
surjOn_iInter_iInter (fun i => (H i).surjOn) Hinj⟩
#align set.bij_on_Inter Set.bijOn_iInter
theorem bijOn_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {t : ι → Set β}
{f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
bijOn_iUnion H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Union_of_directed Set.bijOn_iUnion_of_directed
theorem bijOn_iInter_of_directed [Nonempty ι] {s : ι → Set α} (hs : Directed (· ⊆ ·) s)
{t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
bijOn_iInter H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Inter_of_directed Set.bijOn_iInter_of_directed
end Function
/-! ### `image`, `preimage` -/
section Image
theorem image_iUnion {f : α → β} {s : ι → Set α} : (f '' ⋃ i, s i) = ⋃ i, f '' s i := by
ext1 x
simp only [mem_image, mem_iUnion, ← exists_and_right, ← exists_and_left]
--Porting note: `exists_swap` causes a `simp` loop in Lean4 so we use `rw` instead.
rw [exists_swap]
#align set.image_Union Set.image_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iUnion₂ (f : α → β) (s : ∀ i, κ i → Set α) :
(f '' ⋃ (i) (j), s i j) = ⋃ (i) (j), f '' s i j := by simp_rw [image_iUnion]
#align set.image_Union₂ Set.image_iUnion₂
theorem univ_subtype {p : α → Prop} : (univ : Set (Subtype p)) = ⋃ (x) (h : p x), {⟨x, h⟩} :=
Set.ext fun ⟨x, h⟩ => by simp [h]
#align set.univ_subtype Set.univ_subtype
theorem range_eq_iUnion {ι} (f : ι → α) : range f = ⋃ i, {f i} :=
Set.ext fun a => by simp [@eq_comm α a]
#align set.range_eq_Union Set.range_eq_iUnion
theorem image_eq_iUnion (f : α → β) (s : Set α) : f '' s = ⋃ i ∈ s, {f i} :=
Set.ext fun b => by simp [@eq_comm β b]
#align set.image_eq_Union Set.image_eq_iUnion
theorem biUnion_range {f : ι → α} {g : α → Set β} : ⋃ x ∈ range f, g x = ⋃ y, g (f y) :=
iSup_range
#align set.bUnion_range Set.biUnion_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iUnion_iUnion_eq' {f : ι → α} {g : α → Set β} :
⋃ (x) (y) (_ : f y = x), g x = ⋃ y, g (f y) := by simpa using biUnion_range
#align set.Union_Union_eq' Set.iUnion_iUnion_eq'
theorem biInter_range {f : ι → α} {g : α → Set β} : ⋂ x ∈ range f, g x = ⋂ y, g (f y) :=
iInf_range
#align set.bInter_range Set.biInter_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iInter_iInter_eq' {f : ι → α} {g : α → Set β} :
⋂ (x) (y) (_ : f y = x), g x = ⋂ y, g (f y) := by simpa using biInter_range
#align set.Inter_Inter_eq' Set.iInter_iInter_eq'
variable {s : Set γ} {f : γ → α} {g : α → Set β}
theorem biUnion_image : ⋃ x ∈ f '' s, g x = ⋃ y ∈ s, g (f y) :=
iSup_image
#align set.bUnion_image Set.biUnion_image
theorem biInter_image : ⋂ x ∈ f '' s, g x = ⋂ y ∈ s, g (f y) :=
iInf_image
#align set.bInter_image Set.biInter_image
end Image
section Preimage
theorem monotone_preimage {f : α → β} : Monotone (preimage f) := fun _ _ h => preimage_mono h
#align set.monotone_preimage Set.monotone_preimage
@[simp]
theorem preimage_iUnion {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋃ i, s i) = ⋃ i, f ⁻¹' s i :=
Set.ext <| by simp [preimage]
#align set.preimage_Union Set.preimage_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iUnion₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋃ (i) (j), s i j) = ⋃ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iUnion]
#align set.preimage_Union₂ Set.preimage_iUnion₂
@[simp]
theorem preimage_sUnion {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋃₀s = ⋃ t ∈ s, f ⁻¹' t := by
rw [sUnion_eq_biUnion, preimage_iUnion₂]
#align set.preimage_sUnion Set.preimage_sUnion
theorem preimage_iInter {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋂ i, s i) = ⋂ i, f ⁻¹' s i := by
ext; simp
#align set.preimage_Inter Set.preimage_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iInter₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋂ (i) (j), s i j) = ⋂ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iInter]
#align set.preimage_Inter₂ Set.preimage_iInter₂
@[simp]
theorem preimage_sInter {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋂₀ s = ⋂ t ∈ s, f ⁻¹' t := by
rw [sInter_eq_biInter, preimage_iInter₂]
#align set.preimage_sInter Set.preimage_sInter
@[simp]
theorem biUnion_preimage_singleton (f : α → β) (s : Set β) : ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s := by
rw [← preimage_iUnion₂, biUnion_of_singleton]
#align set.bUnion_preimage_singleton Set.biUnion_preimage_singleton
theorem biUnion_range_preimage_singleton (f : α → β) : ⋃ y ∈ range f, f ⁻¹' {y} = univ := by
rw [biUnion_preimage_singleton, preimage_range]
#align set.bUnion_range_preimage_singleton Set.biUnion_range_preimage_singleton
end Preimage
section Prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion {s : Set α} {t : ι → Set β} : (s ×ˢ ⋃ i, t i) = ⋃ i, s ×ˢ t i := by
ext
simp
#align set.prod_Union Set.prod_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} :
(s ×ˢ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ×ˢ t i j := by simp_rw [prod_iUnion]
#align set.prod_Union₂ Set.prod_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sUnion {s : Set α} {C : Set (Set β)} : s ×ˢ ⋃₀C = ⋃₀((fun t => s ×ˢ t) '' C) := by
simp_rw [sUnion_eq_biUnion, biUnion_image, prod_iUnion₂]
#align set.prod_sUnion Set.prod_sUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_const {s : ι → Set α} {t : Set β} : (⋃ i, s i) ×ˢ t = ⋃ i, s i ×ˢ t := by
ext
simp
#align set.Union_prod_const Set.iUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion₂_prod_const {s : ∀ i, κ i → Set α} {t : Set β} :
(⋃ (i) (j), s i j) ×ˢ t = ⋃ (i) (j), s i j ×ˢ t := by simp_rw [iUnion_prod_const]
#align set.Union₂_prod_const Set.iUnion₂_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_prod_const {C : Set (Set α)} {t : Set β} :
⋃₀C ×ˢ t = ⋃₀((fun s : Set α => s ×ˢ t) '' C) := by
simp only [sUnion_eq_biUnion, iUnion₂_prod_const, biUnion_image]
#align set.sUnion_prod_const Set.sUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod {ι ι' α β} (s : ι → Set α) (t : ι' → Set β) :
⋃ x : ι × ι', s x.1 ×ˢ t x.2 = (⋃ i : ι, s i) ×ˢ ⋃ i : ι', t i := by
ext
simp
#align set.Union_prod Set.iUnion_prod
/-- Analogue of `iSup_prod` for sets. -/
lemma iUnion_prod' (f : β × γ → Set α) : ⋃ x : β × γ, f x = ⋃ (i : β) (j : γ), f (i, j) :=
iSup_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x := by
ext ⟨z, w⟩; simp only [mem_prod, mem_iUnion, exists_imp, and_imp, iff_def]; constructor
· intro x hz hw
exact ⟨⟨x, hz⟩, x, hw⟩
· intro x hz x' hw
exact ⟨x ⊔ x', hs le_sup_left hz, ht le_sup_right hw⟩
#align set.Union_prod_of_monotone Set.iUnion_prod_of_monotone
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter_subset (S : Set (Set α)) (T : Set (Set β)) :
⋂₀ S ×ˢ ⋂₀ T ⊆ ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 :=
subset_iInter₂ fun x hx _ hy => ⟨hy.1 x.1 hx.1, hy.2 x.2 hx.2⟩
#align set.sInter_prod_sInter_subset Set.sInter_prod_sInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 := by
obtain ⟨s₁, h₁⟩ := hS
obtain ⟨s₂, h₂⟩ := hT
refine' Set.Subset.antisymm (sInter_prod_sInter_subset S T) fun x hx => _
rw [mem_iInter₂] at hx
exact ⟨fun s₀ h₀ => (hx (s₀, s₂) ⟨h₀, h₂⟩).1, fun s₀ h₀ => (hx (s₁, s₀) ⟨h₁, h₀⟩).2⟩
#align set.sInter_prod_sInter Set.sInter_prod_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod {S : Set (Set α)} (hS : S.Nonempty) (t : Set β) :
⋂₀ S ×ˢ t = ⋂ s ∈ S, s ×ˢ t := by
rw [← sInter_singleton t, sInter_prod_sInter hS (singleton_nonempty t), sInter_singleton]
simp_rw [prod_singleton, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.sInter_prod Set.sInter_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sInter {T : Set (Set β)} (hT : T.Nonempty) (s : Set α) :
s ×ˢ ⋂₀ T = ⋂ t ∈ T, s ×ˢ t := by
rw [← sInter_singleton s, sInter_prod_sInter (singleton_nonempty s) hT, sInter_singleton]
simp_rw [singleton_prod, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.prod_sInter Set.prod_sInter
end Prod
section Image2
variable (f : α → β → γ) {s : Set α} {t : Set β}
/-- The `Set.image2` version of `Set.image_eq_iUnion` -/
theorem image2_eq_iUnion (s : Set α) (t : Set β) : image2 f s t = ⋃ (i ∈ s) (j ∈ t), {f i j} := by
ext; simp [eq_comm]
#align set.image2_eq_Union Set.image2_eq_iUnion
theorem iUnion_image_left : ⋃ a ∈ s, f a '' t = image2 f s t := by
simp only [image2_eq_iUnion, image_eq_iUnion]
#align set.Union_image_left Set.iUnion_image_left
theorem iUnion_image_right : ⋃ b ∈ t, (f · b) '' s = image2 f s t := by
rw [image2_swap, iUnion_image_left]
#align set.Union_image_right Set.iUnion_image_right
theorem image2_iUnion_left (s : ι → Set α) (t : Set β) :
image2 f (⋃ i, s i) t = ⋃ i, image2 f (s i) t := by
simp only [← image_prod, iUnion_prod_const, image_iUnion]
#align set.image2_Union_left Set.image2_iUnion_left
theorem image2_iUnion_right (s : Set α) (t : ι → Set β) :
image2 f s (⋃ i, t i) = ⋃ i, image2 f s (t i) := by
simp only [← image_prod, prod_iUnion, image_iUnion]
#align set.image2_Union_right Set.image2_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋃ (i) (j), s i j) t = ⋃ (i) (j), image2 f (s i j) t := by simp_rw [image2_iUnion_left]
#align set.image2_Union₂_left Set.image2_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋃ (i) (j), t i j) = ⋃ (i) (j), image2 f s (t i j) :=
by simp_rw [image2_iUnion_right]
#align set.image2_Union₂_right Set.image2_iUnion₂_right
theorem image2_iInter_subset_left (s : ι → Set α) (t : Set β) :
image2 f (⋂ i, s i) t ⊆ ⋂ i, image2 f (s i) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem (hx _) hy
#align set.image2_Inter_subset_left Set.image2_iInter_subset_left
theorem image2_iInter_subset_right (s : Set α) (t : ι → Set β) :
image2 f s (⋂ i, t i) ⊆ ⋂ i, image2 f s (t i) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem hx (hy _)
#align set.image2_Inter_subset_right Set.image2_iInter_subset_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋂ (i) (j), s i j) t ⊆ ⋂ (i) (j), image2 f (s i j) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem (hx _ _) hy
#align set.image2_Inter₂_subset_left Set.image2_iInter₂_subset_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), image2 f s (t i j) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem hx (hy _ _)
#align set.image2_Inter₂_subset_right Set.image2_iInter₂_subset_right
theorem prod_eq_biUnion_left : s ×ˢ t = ⋃ a ∈ s, (fun b => (a, b)) '' t := by
rw [iUnion_image_left, image2_mk_eq_prod]
#align set.prod_eq_bUnion_left Set.prod_eq_biUnion_left
theorem prod_eq_biUnion_right : s ×ˢ t = ⋃ b ∈ t, (fun a => (a, b)) '' s := by
rw [iUnion_image_right, image2_mk_eq_prod]
#align set.prod_eq_bUnion_right Set.prod_eq_biUnion_right
end Image2
section Seq
/-- Given a set `s` of functions `α → β` and `t : Set α`, `seq s t` is the union of `f '' t` over
all `f ∈ s`. -/
def seq (s : Set (α → β)) (t : Set α) : Set β :=
{ b | ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b }
#align set.seq Set.seq
@[simp]
theorem mem_seq_iff {s : Set (α → β)} {t : Set α} {b : β} :
b ∈ seq s t ↔ ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b :=
Iff.rfl
#align set.mem_seq_iff Set.mem_seq_iff
lemma seq_eq_image2 (s : Set (α → β)) (t : Set α) : seq s t = image2 (fun f a ↦ f a) s t := by
ext; simp
theorem seq_def {s : Set (α → β)} {t : Set α} : seq s t = ⋃ f ∈ s, f '' t := by
rw [seq_eq_image2, iUnion_image_left]
#align set.seq_def Set.seq_def
theorem seq_subset {s : Set (α → β)} {t : Set α} {u : Set β} :
seq s t ⊆ u ↔ ∀ f ∈ s, ∀ a ∈ t, (f : α → β) a ∈ u := by
rw [seq_eq_image2, image2_subset_iff]
#align set.seq_subset Set.seq_subset
@[gcongr]
theorem seq_mono {s₀ s₁ : Set (α → β)} {t₀ t₁ : Set α} (hs : s₀ ⊆ s₁) (ht : t₀ ⊆ t₁) :
seq s₀ t₀ ⊆ seq s₁ t₁ := fun _ ⟨f, hf, a, ha, eq⟩ => ⟨f, hs hf, a, ht ha, eq⟩
#align set.seq_mono Set.seq_mono
theorem singleton_seq {f : α → β} {t : Set α} : Set.seq ({f} : Set (α → β)) t = f '' t := by
rw [seq_eq_image2, image2_singleton_left]
#align set.singleton_seq Set.singleton_seq
theorem seq_singleton {s : Set (α → β)} {a : α} : Set.seq s {a} = (fun f : α → β => f a) '' s := by
rw [seq_eq_image2, image2_singleton_right]
#align set.seq_singleton Set.seq_singleton
theorem seq_seq {s : Set (β → γ)} {t : Set (α → β)} {u : Set α} :
seq s (seq t u) = seq (seq ((· ∘ ·) '' s) t) u := by
simp only [seq_eq_image2, image2_image_left, image2_image2_left, image2_image2_right, comp_apply]
#align set.seq_seq Set.seq_seq
theorem image_seq {f : β → γ} {s : Set (α → β)} {t : Set α} :
f '' seq s t = seq ((f ∘ ·) '' s) t := by
rw [← singleton_seq, ← singleton_seq, seq_seq, image_singleton]
#align set.image_seq Set.image_seq
theorem prod_eq_seq {s : Set α} {t : Set β} : s ×ˢ t = (Prod.mk '' s).seq t := by
rw [seq_eq_image2, image2_image_left, image2_mk_eq_prod]
#align set.prod_eq_seq Set.prod_eq_seq
theorem prod_image_seq_comm (s : Set α) (t : Set β) :
(Prod.mk '' s).seq t = seq ((fun b a => (a, b)) '' t) s := by
rw [← prod_eq_seq, ← image_swap_prod, prod_eq_seq, image_seq, ← image_comp]; rfl
#align set.prod_image_seq_comm Set.prod_image_seq_comm
theorem image2_eq_seq (f : α → β → γ) (s : Set α) (t : Set β) : image2 f s t = seq (f '' s) t := by
rw [seq_eq_image2, image2_image_left]
#align set.image2_eq_seq Set.image2_eq_seq
end Seq
section Pi
variable {π : α → Type*}
theorem pi_def (i : Set α) (s : ∀ a, Set (π a)) : pi i s = ⋂ a ∈ i, eval a ⁻¹' s a := by
ext
simp
#align set.pi_def Set.pi_def
theorem univ_pi_eq_iInter (t : ∀ i, Set (π i)) : pi univ t = ⋂ i, eval i ⁻¹' t i := by
simp only [pi_def, iInter_true, mem_univ]
#align set.univ_pi_eq_Inter Set.univ_pi_eq_iInter
theorem pi_diff_pi_subset (i : Set α) (s t : ∀ a, Set (π a)) :
pi i s \ pi i t ⊆ ⋃ a ∈ i, eval a ⁻¹' (s a \ t a) := by
refine' diff_subset_comm.2 fun x hx a ha => _
simp only [mem_diff, mem_pi, mem_iUnion, not_exists, mem_preimage, not_and, not_not,
eval_apply] at hx
exact hx.2 _ ha (hx.1 _ ha)
#align set.pi_diff_pi_subset Set.pi_diff_pi_subset
theorem iUnion_univ_pi {ι : α → Type*} (t : (a : α) → ι a → Set (π a)) :
⋃ x : (a : α) → ι a, pi univ (fun a => t a (x a)) = pi univ fun a => ⋃ j : ι a, t a j := by
ext
simp [Classical.skolem]
#align set.Union_univ_pi Set.iUnion_univ_pi
end Pi
end Set
namespace Function
namespace Surjective
theorem iUnion_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋃ x, g (f x) = ⋃ y, g y :=
hf.iSup_comp g
#align function.surjective.Union_comp Function.Surjective.iUnion_comp
theorem iInter_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋂ x, g (f x) = ⋂ y, g y :=
hf.iInf_comp g
#align function.surjective.Inter_comp Function.Surjective.iInter_comp
end Surjective
end Function
/-!
### Disjoint sets
-/
section Disjoint
variable {s t u : Set α} {f : α → β}
namespace Set
@[simp]
theorem disjoint_iUnion_left {ι : Sort*} {s : ι → Set α} :
Disjoint (⋃ i, s i) t ↔ ∀ i, Disjoint (s i) t :=
iSup_disjoint_iff
#align set.disjoint_Union_left Set.disjoint_iUnion_left
@[simp]
theorem disjoint_iUnion_right {ι : Sort*} {s : ι → Set α} :
Disjoint t (⋃ i, s i) ↔ ∀ i, Disjoint t (s i) :=
disjoint_iSup_iff
#align set.disjoint_Union_right Set.disjoint_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_left {s : ∀ i, κ i → Set α} {t : Set α} :
Disjoint (⋃ (i) (j), s i j) t ↔ ∀ i j, Disjoint (s i j) t :=
iSup₂_disjoint_iff
#align set.disjoint_Union₂_left Set.disjoint_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_right {s : Set α} {t : ∀ i, κ i → Set α} :
Disjoint s (⋃ (i) (j), t i j) ↔ ∀ i j, Disjoint s (t i j) :=
disjoint_iSup₂_iff
#align set.disjoint_Union₂_right Set.disjoint_iUnion₂_right
@[simp]
theorem disjoint_sUnion_left {S : Set (Set α)} {t : Set α} :
Disjoint (⋃₀S) t ↔ ∀ s ∈ S, Disjoint s t :=
sSup_disjoint_iff
#align set.disjoint_sUnion_left Set.disjoint_sUnion_left
@[simp]
theorem disjoint_sUnion_right {s : Set α} {S : Set (Set α)} :
Disjoint s (⋃₀S) ↔ ∀ t ∈ S, Disjoint s t :=
disjoint_sSup_iff
#align set.disjoint_sUnion_right Set.disjoint_sUnion_right
end Set
end Disjoint
/-! ### Intervals -/
namespace Set
lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :
(⋂ i, Iic (f i)).Nonempty ↔ BddBelow (range f) := by
have : (⋂ (i : ι), Iic (f i)) = lowerBounds (range f) := by
ext c; simp [lowerBounds]
simp [this, BddBelow]
lemma nonempty_iInter_Ici_iff [Preorder α] {f : ι → α} :
(⋂ i, Ici (f i)).Nonempty ↔ BddAbove (range f) :=
nonempty_iInter_Iic_iff (α := αᵒᵈ)
variable [CompleteLattice α]
theorem Ici_iSup (f : ι → α) : Ici (⨆ i, f i) = ⋂ i, Ici (f i) :=
ext fun _ => by simp only [mem_Ici, iSup_le_iff, mem_iInter]
#align set.Ici_supr Set.Ici_iSup
theorem Iic_iInf (f : ι → α) : Iic (⨅ i, f i) = ⋂ i, Iic (f i) :=
ext fun _ => by simp only [mem_Iic, le_iInf_iff, mem_iInter]
#align set.Iic_infi Set.Iic_iInf
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Ici_iSup₂ (f : ∀ i, κ i → α) : Ici (⨆ (i) (j), f i j) = ⋂ (i) (j), Ici (f i j) := by
simp_rw [Ici_iSup]
#align set.Ici_supr₂ Set.Ici_iSup₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Iic_iInf₂ (f : ∀ i, κ i → α) : Iic (⨅ (i) (j), f i j) = ⋂ (i) (j), Iic (f i j) := by
simp_rw [Iic_iInf]
#align set.Iic_infi₂ Set.Iic_iInf₂
theorem Ici_sSup (s : Set α) : Ici (sSup s) = ⋂ a ∈ s, Ici a := by rw [sSup_eq_iSup, Ici_iSup₂]
#align set.Ici_Sup Set.Ici_sSup
theorem Iic_sInf (s : Set α) : Iic (sInf s) = ⋂ a ∈ s, Iic a := by rw [sInf_eq_iInf, Iic_iInf₂]
#align set.Iic_Inf Set.Iic_sInf
end Set
namespace Set
variable (t : α → Set β)
theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x := by
simp only [diff_subset_iff, ← biUnion_union]
apply biUnion_subset_biUnion_left
rw [union_diff_self]
apply subset_union_right
#align set.bUnion_diff_bUnion_subset Set.biUnion_diff_biUnion_subset
/-- If `t` is an indexed family of sets, then there is a natural map from `Σ i, t i` to `⋃ i, t i`
sending `⟨i, x⟩` to `x`. -/
def sigmaToiUnion (x : Σi, t i) : ⋃ i, t i :=
⟨x.2, mem_iUnion.2 ⟨x.1, x.2.2⟩⟩
#align set.sigma_to_Union Set.sigmaToiUnion
theorem sigmaToiUnion_surjective : Surjective (sigmaToiUnion t)
| ⟨b, hb⟩ =>
have : ∃ a, b ∈ t a := by simpa using hb
let ⟨a, hb⟩ := this
⟨⟨a, b, hb⟩, rfl⟩
#align set.sigma_to_Union_surjective Set.sigmaToiUnion_surjective
theorem sigmaToiUnion_injective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Injective (sigmaToiUnion t)
| ⟨a₁, b₁, h₁⟩, ⟨a₂, b₂, h₂⟩, eq =>
have b_eq : b₁ = b₂ := congr_arg Subtype.val eq
have a_eq : a₁ = a₂ :=
by_contradiction fun ne =>
have : b₁ ∈ t a₁ ∩ t a₂ := ⟨h₁, b_eq.symm ▸ h₂⟩
(h _ _ ne).le_bot this
Sigma.eq a_eq <| Subtype.eq <| by subst b_eq; subst a_eq; rfl
#align set.sigma_to_Union_injective Set.sigmaToiUnion_injective
theorem sigmaToiUnion_bijective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Bijective (sigmaToiUnion t) :=
⟨sigmaToiUnion_injective t h, sigmaToiUnion_surjective t⟩
#align set.sigma_to_Union_bijective Set.sigmaToiUnion_bijective
/-- Equivalence between a disjoint union and a dependent sum. -/
noncomputable def unionEqSigmaOfDisjoint {t : α → Set β} (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
(⋃ i, t i) ≃ Σi, t i :=
(Equiv.ofBijective _ <| sigmaToiUnion_bijective t h).symm
#align set.Union_eq_sigma_of_disjoint Set.unionEqSigmaOfDisjoint
theorem iUnion_ge_eq_iUnion_nat_add (u : ℕ → Set α) (n : ℕ) : ⋃ i ≥ n, u i = ⋃ i, u (i + n) :=
iSup_ge_eq_iSup_nat_add u n
#align set.Union_ge_eq_Union_nat_add Set.iUnion_ge_eq_iUnion_nat_add
theorem iInter_ge_eq_iInter_nat_add (u : ℕ → Set α) (n : ℕ) : ⋂ i ≥ n, u i = ⋂ i, u (i + n) :=
iInf_ge_eq_iInf_nat_add u n
#align set.Inter_ge_eq_Inter_nat_add Set.iInter_ge_eq_iInter_nat_add
theorem _root_.Monotone.iUnion_nat_add {f : ℕ → Set α} (hf : Monotone f) (k : ℕ) :
⋃ n, f (n + k) = ⋃ n, f n :=
hf.iSup_nat_add k
#align monotone.Union_nat_add Monotone.iUnion_nat_add
theorem _root_.Antitone.iInter_nat_add {f : ℕ → Set α} (hf : Antitone f) (k : ℕ) :
⋂ n, f (n + k) = ⋂ n, f n :=
hf.iInf_nat_add k
#align antitone.Inter_nat_add Antitone.iInter_nat_add
/-Porting note: removing `simp`. LHS does not simplify. Possible linter bug. Zulip discussion:
https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/complete_lattice.20and.20has_sup/near/316497982
-/
theorem iUnion_iInter_ge_nat_add (f : ℕ → Set α) (k : ℕ) :
⋃ n, ⋂ i ≥ n, f (i + k) = ⋃ n, ⋂ i ≥ n, f i :=
iSup_iInf_ge_nat_add f k
#align set.Union_Inter_ge_nat_add Set.iUnion_iInter_ge_nat_add
theorem union_iUnion_nat_succ (u : ℕ → Set α) : (u 0 ∪ ⋃ i, u (i + 1)) = ⋃ i, u i :=
sup_iSup_nat_succ u
#align set.union_Union_nat_succ Set.union_iUnion_nat_succ
theorem inter_iInter_nat_succ (u : ℕ → Set α) : (u 0 ∩ ⋂ i, u (i + 1)) = ⋂ i, u i :=
inf_iInf_nat_succ u
#align set.inter_Inter_nat_succ Set.inter_iInter_nat_succ
end Set
open Set
variable [CompleteLattice β]
theorem iSup_iUnion (s : ι → Set α) (f : α → β) : ⨆ a ∈ ⋃ i, s i, f a = ⨆ (i) (a ∈ s i), f a := by
rw [iSup_comm]
simp_rw [mem_iUnion, iSup_exists]
#align supr_Union iSup_iUnion
theorem iInf_iUnion (s : ι → Set α) (f : α → β) : ⨅ a ∈ ⋃ i, s i, f a = ⨅ (i) (a ∈ s i), f a :=
@iSup_iUnion α βᵒᵈ _ _ s f
#align infi_Union iInf_iUnion
theorem sSup_sUnion (s : Set (Set β)) : sSup (⋃₀ s) = ⨆ t ∈ s, sSup t := by
simp only [sUnion_eq_biUnion, sSup_eq_iSup, iSup_iUnion]
#align Sup_sUnion sSup_sUnion
theorem sInf_sUnion (s : Set (Set β)) : sInf (⋃₀ s) = ⨅ t ∈ s, sInf t :=
@sSup_sUnion βᵒᵈ _ _
#align Inf_sUnion sInf_sUnion
lemma iSup_sUnion (S : Set (Set α)) (f : α → β) :
(⨆ x ∈ ⋃₀ S, f x) = ⨆ (s ∈ S) (x ∈ s), f x := by
rw [sUnion_eq_iUnion, iSup_iUnion, ← iSup_subtype'']
lemma iInf_sUnion (S : Set (Set α)) (f : α → β) :
(⨅ x ∈ ⋃₀ S, f x) = ⨅ (s ∈ S) (x ∈ s), f x := by
rw [sUnion_eq_iUnion, iInf_iUnion, ← iInf_subtype'']
lemma forall_sUnion {p : α → Prop} :
(∀ x ∈ ⋃₀ S, p x) ↔ ∀ s ∈ S, ∀ x ∈ s, p x := by
|
simp_rw [← iInf_Prop_eq, iInf_sUnion]
|
lemma forall_sUnion {p : α → Prop} :
(∀ x ∈ ⋃₀ S, p x) ↔ ∀ s ∈ S, ∀ x ∈ s, p x := by
|
Mathlib.Data.Set.Lattice.2399_0.5mONj49h3SYSDwc
|
lemma forall_sUnion {p : α → Prop} :
(∀ x ∈ ⋃₀ S, p x) ↔ ∀ s ∈ S, ∀ x ∈ s, p x
|
Mathlib_Data_Set_Lattice
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
inst✝ : CompleteLattice β
S : Set (Set α)
p : α → Prop
⊢ (∃ x ∈ ⋃₀ S, p x) ↔ ∃ s ∈ S, ∃ x ∈ s, p x
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
/-!
# The set lattice
This file provides usual set notation for unions and intersections, a `CompleteLattice` instance
for `Set α`, and some more set constructions.
## Main declarations
* `Set.iUnion`: **i**ndexed **union**. Union of an indexed family of sets.
* `Set.iInter`: **i**ndexed **inter**section. Intersection of an indexed family of sets.
* `Set.sInter`: **s**et **inter**section. Intersection of sets belonging to a set of sets.
* `Set.sUnion`: **s**et **union**. Union of sets belonging to a set of sets.
* `Set.sInter_eq_biInter`, `Set.sUnion_eq_biInter`: Shows that `⋂₀ s = ⋂ x ∈ s, x` and
`⋃₀ s = ⋃ x ∈ s, x`.
* `Set.completeAtomicBooleanAlgebra`: `Set α` is a `CompleteAtomicBooleanAlgebra` with `≤ = ⊆`,
`< = ⊂`, `⊓ = ∩`, `⊔ = ∪`, `⨅ = ⋂`, `⨆ = ⋃` and `\` as the set difference.
See `Set.BooleanAlgebra`.
* `Set.kernImage`: For a function `f : α → β`, `s.kernImage f` is the set of `y` such that
`f ⁻¹ y ⊆ s`.
* `Set.seq`: Union of the image of a set under a **seq**uence of functions. `seq s t` is the union
of `f '' t` over all `f ∈ s`, where `t : Set α` and `s : Set (α → β)`.
* `Set.iUnion_eq_sigma_of_disjoint`: Equivalence between `⋃ i, t i` and `Σ i, t i`, where `t` is an
indexed family of disjoint sets.
## Naming convention
In lemma names,
* `⋃ i, s i` is called `iUnion`
* `⋂ i, s i` is called `iInter`
* `⋃ i j, s i j` is called `iUnion₂`. This is an `iUnion` inside an `iUnion`.
* `⋂ i j, s i j` is called `iInter₂`. This is an `iInter` inside an `iInter`.
* `⋃ i ∈ s, t i` is called `biUnion` for "bounded `iUnion`". This is the special case of `iUnion₂`
where `j : i ∈ s`.
* `⋂ i ∈ s, t i` is called `biInter` for "bounded `iInter`". This is the special case of `iInter₂`
where `j : i ∈ s`.
## Notation
* `⋃`: `Set.iUnion`
* `⋂`: `Set.iInter`
* `⋃₀`: `Set.sUnion`
* `⋂₀`: `Set.sInter`
-/
set_option autoImplicit true
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι₂ : Sort*} {κ κ₁ κ₂ : ι → Sort*} {κ' : ι' → Sort*}
namespace Set
/-! ### Complete lattice and complete Boolean algebra instances -/
instance : InfSet (Set α) :=
⟨fun s => { a | ∀ t ∈ s, a ∈ t }⟩
instance : SupSet (Set α) :=
⟨fun s => { a | ∃ t ∈ s, a ∈ t }⟩
/-- Intersection of a set of sets. -/
def sInter (S : Set (Set α)) : Set α :=
sInf S
#align set.sInter Set.sInter
/-- Notation for `Set.sInter` Intersection of a set of sets. -/
prefix:110 "⋂₀ " => sInter
/-- Union of a set of sets. -/
def sUnion (S : Set (Set α)) : Set α :=
sSup S
#align set.sUnion Set.sUnion
/-- Notation for `Set.sUnion`. Union of a set of sets. -/
prefix:110 "⋃₀ " => sUnion
@[simp]
theorem mem_sInter {x : α} {S : Set (Set α)} : x ∈ ⋂₀ S ↔ ∀ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sInter Set.mem_sInter
@[simp]
theorem mem_sUnion {x : α} {S : Set (Set α)} : x ∈ ⋃₀ S ↔ ∃ t ∈ S, x ∈ t :=
Iff.rfl
#align set.mem_sUnion Set.mem_sUnion
/-- Indexed union of a family of sets -/
def iUnion (s : ι → Set β) : Set β :=
iSup s
#align set.Union Set.iUnion
/-- Indexed intersection of a family of sets -/
def iInter (s : ι → Set β) : Set β :=
iInf s
#align set.Inter Set.iInter
/-- Notation for `Set.iUnion`. Indexed union of a family of sets -/
notation3 "⋃ "(...)", "r:60:(scoped f => iUnion f) => r
/-- Notation for `Set.iInter`. Indexed intersection of a family of sets -/
notation3 "⋂ "(...)", "r:60:(scoped f => iInter f) => r
section delaborators
open Lean Lean.PrettyPrinter.Delaborator
/-- Delaborator for indexed unions. -/
@[delab app.Set.iUnion]
def iUnion_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋃ (_ : $dom), $body)
else if prop || ppTypes then
`(⋃ ($x:ident : $dom), $body)
else
`(⋃ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋃ $x:ident, ⋃ (_ : $y:ident ∈ $s), $body)
| `(⋃ ($x:ident : $_), ⋃ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋃ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
/-- Delaborator for indexed intersections. -/
@[delab app.Set.iInter]
def sInter_delab : Delab := whenPPOption Lean.getPPNotation do
let #[_, ι, f] := (← SubExpr.getExpr).getAppArgs | failure
unless f.isLambda do failure
let prop ← Meta.isProp ι
let dep := f.bindingBody!.hasLooseBVar 0
let ppTypes ← getPPOption getPPFunBinderTypes
let stx ← SubExpr.withAppArg do
let dom ← SubExpr.withBindingDomain delab
withBindingBodyUnusedName $ fun x => do
let x : TSyntax `ident := .mk x
let body ← delab
if prop && !dep then
`(⋂ (_ : $dom), $body)
else if prop || ppTypes then
`(⋂ ($x:ident : $dom), $body)
else
`(⋂ $x:ident, $body)
-- Cute binders
let stx : Term ←
match stx with
| `(⋂ $x:ident, ⋂ (_ : $y:ident ∈ $s), $body)
| `(⋂ ($x:ident : $_), ⋂ (_ : $y:ident ∈ $s), $body) =>
if x == y then `(⋂ $x:ident ∈ $s, $body) else pure stx
| _ => pure stx
return stx
end delaborators
@[simp]
theorem sSup_eq_sUnion (S : Set (Set α)) : sSup S = ⋃₀S :=
rfl
#align set.Sup_eq_sUnion Set.sSup_eq_sUnion
@[simp]
theorem sInf_eq_sInter (S : Set (Set α)) : sInf S = ⋂₀ S :=
rfl
#align set.Inf_eq_sInter Set.sInf_eq_sInter
@[simp]
theorem iSup_eq_iUnion (s : ι → Set α) : iSup s = iUnion s :=
rfl
#align set.supr_eq_Union Set.iSup_eq_iUnion
@[simp]
theorem iInf_eq_iInter (s : ι → Set α) : iInf s = iInter s :=
rfl
#align set.infi_eq_Inter Set.iInf_eq_iInter
@[simp]
theorem mem_iUnion {x : α} {s : ι → Set α} : (x ∈ ⋃ i, s i) ↔ ∃ i, x ∈ s i :=
⟨fun ⟨_, ⟨⟨a, (t_eq : s a = _)⟩, (h : x ∈ _)⟩⟩ => ⟨a, t_eq.symm ▸ h⟩, fun ⟨a, h⟩ =>
⟨s a, ⟨⟨a, rfl⟩, h⟩⟩⟩
#align set.mem_Union Set.mem_iUnion
@[simp]
theorem mem_iInter {x : α} {s : ι → Set α} : (x ∈ ⋂ i, s i) ↔ ∀ i, x ∈ s i :=
⟨fun (h : ∀ a ∈ { a : Set α | ∃ i, s i = a }, x ∈ a) a => h (s a) ⟨a, rfl⟩,
fun h _ ⟨a, (eq : s a = _)⟩ => eq ▸ h a⟩
#align set.mem_Inter Set.mem_iInter
theorem mem_iUnion₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋃ (i) (j), s i j) ↔ ∃ i j, x ∈ s i j := by
simp_rw [mem_iUnion]
#align set.mem_Union₂ Set.mem_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋂ (i) (j), s i j) ↔ ∀ i j, x ∈ s i j := by
simp_rw [mem_iInter]
#align set.mem_Inter₂ Set.mem_iInter₂
theorem mem_iUnion_of_mem {s : ι → Set α} {a : α} (i : ι) (ha : a ∈ s i) : a ∈ ⋃ i, s i :=
mem_iUnion.2 ⟨i, ha⟩
#align set.mem_Union_of_mem Set.mem_iUnion_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iUnion₂_of_mem {s : ∀ i, κ i → Set α} {a : α} {i : ι} (j : κ i) (ha : a ∈ s i j) :
a ∈ ⋃ (i) (j), s i j :=
mem_iUnion₂.2 ⟨i, j, ha⟩
#align set.mem_Union₂_of_mem Set.mem_iUnion₂_of_mem
theorem mem_iInter_of_mem {s : ι → Set α} {a : α} (h : ∀ i, a ∈ s i) : a ∈ ⋂ i, s i :=
mem_iInter.2 h
#align set.mem_Inter_of_mem Set.mem_iInter_of_mem
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mem_iInter₂_of_mem {s : ∀ i, κ i → Set α} {a : α} (h : ∀ i j, a ∈ s i j) :
a ∈ ⋂ (i) (j), s i j :=
mem_iInter₂.2 h
#align set.mem_Inter₂_of_mem Set.mem_iInter₂_of_mem
instance Set.completeAtomicBooleanAlgebra : CompleteAtomicBooleanAlgebra (Set α) :=
{ instBooleanAlgebraSet with
le_sSup := fun s t t_in a a_in => ⟨t, t_in, a_in⟩
sSup_le := fun s t h a ⟨t', ⟨t'_in, a_in⟩⟩ => h t' t'_in a_in
le_sInf := fun s t h a a_in t' t'_in => h t' t'_in a_in
sInf_le := fun s t t_in a h => h _ t_in
iInf_iSup_eq := by intros; ext; simp [Classical.skolem] }
/-- `kernImage f s` is the set of `y` such that `f ⁻¹ y ⊆ s`. -/
def kernImage (f : α → β) (s : Set α) : Set β :=
{ y | ∀ ⦃x⦄, f x = y → x ∈ s }
#align set.kern_image Set.kernImage
lemma subset_kernImage_iff {f : α → β} : s ⊆ kernImage f t ↔ f ⁻¹' s ⊆ t :=
⟨fun h _ hx ↦ h hx rfl,
fun h _ hx y hy ↦ h (show f y ∈ s from hy.symm ▸ hx)⟩
section GaloisConnection
variable {f : α → β}
protected theorem image_preimage : GaloisConnection (image f) (preimage f) := fun _ _ =>
image_subset_iff
#align set.image_preimage Set.image_preimage
protected theorem preimage_kernImage : GaloisConnection (preimage f) (kernImage f) := fun _ _ =>
subset_kernImage_iff.symm
#align set.preimage_kern_image Set.preimage_kernImage
end GaloisConnection
section kernImage
variable {f : α → β}
lemma kernImage_mono : Monotone (kernImage f) :=
Set.preimage_kernImage.monotone_u
lemma kernImage_eq_compl {s : Set α} : kernImage f s = (f '' sᶜ)ᶜ :=
Set.preimage_kernImage.u_unique (Set.image_preimage.compl)
(fun t ↦ compl_compl (f ⁻¹' t) ▸ Set.preimage_compl)
lemma kernImage_compl {s : Set α} : kernImage f (sᶜ) = (f '' s)ᶜ := by
rw [kernImage_eq_compl, compl_compl]
lemma kernImage_empty : kernImage f ∅ = (range f)ᶜ := by
rw [kernImage_eq_compl, compl_empty, image_univ]
lemma kernImage_preimage_eq_iff {s : Set β} : kernImage f (f ⁻¹' s) = s ↔ (range f)ᶜ ⊆ s := by
rw [kernImage_eq_compl, ← preimage_compl, compl_eq_comm, eq_comm, image_preimage_eq_iff,
compl_subset_comm]
lemma compl_range_subset_kernImage {s : Set α} : (range f)ᶜ ⊆ kernImage f s := by
rw [← kernImage_empty]
exact kernImage_mono (empty_subset _)
lemma kernImage_union_preimage {s : Set α} {t : Set β} :
kernImage f (s ∪ f ⁻¹' t) = kernImage f s ∪ t := by
rw [kernImage_eq_compl, kernImage_eq_compl, compl_union, ← preimage_compl, image_inter_preimage,
compl_inter, compl_compl]
lemma kernImage_preimage_union {s : Set α} {t : Set β} :
kernImage f (f ⁻¹' t ∪ s) = t ∪ kernImage f s := by
rw [union_comm, kernImage_union_preimage, union_comm]
end kernImage
/-! ### Union and intersection over an indexed family of sets -/
instance : OrderTop (Set α) where
top := univ
le_top := by simp
@[congr]
theorem iUnion_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iUnion f₁ = iUnion f₂ :=
iSup_congr_Prop pq f
#align set.Union_congr_Prop Set.iUnion_congr_Prop
@[congr]
theorem iInter_congr_Prop {p q : Prop} {f₁ : p → Set α} {f₂ : q → Set α} (pq : p ↔ q)
(f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iInter f₁ = iInter f₂ :=
iInf_congr_Prop pq f
#align set.Inter_congr_Prop Set.iInter_congr_Prop
theorem iUnion_plift_up (f : PLift ι → Set α) : ⋃ i, f (PLift.up i) = ⋃ i, f i :=
iSup_plift_up _
#align set.Union_plift_up Set.iUnion_plift_up
theorem iUnion_plift_down (f : ι → Set α) : ⋃ i, f (PLift.down i) = ⋃ i, f i :=
iSup_plift_down _
#align set.Union_plift_down Set.iUnion_plift_down
theorem iInter_plift_up (f : PLift ι → Set α) : ⋂ i, f (PLift.up i) = ⋂ i, f i :=
iInf_plift_up _
#align set.Inter_plift_up Set.iInter_plift_up
theorem iInter_plift_down (f : ι → Set α) : ⋂ i, f (PLift.down i) = ⋂ i, f i :=
iInf_plift_down _
#align set.Inter_plift_down Set.iInter_plift_down
theorem iUnion_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋃ _ : p, s = if p then s else ∅ :=
iSup_eq_if _
#align set.Union_eq_if Set.iUnion_eq_if
theorem iUnion_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋃ h : p, s h = if h : p then s h else ∅ :=
iSup_eq_dif _
#align set.Union_eq_dif Set.iUnion_eq_dif
theorem iInter_eq_if {p : Prop} [Decidable p] (s : Set α) : ⋂ _ : p, s = if p then s else univ :=
iInf_eq_if _
#align set.Inter_eq_if Set.iInter_eq_if
theorem iInf_eq_dif {p : Prop} [Decidable p] (s : p → Set α) :
⋂ h : p, s h = if h : p then s h else univ :=
_root_.iInf_eq_dif _
#align set.Infi_eq_dif Set.iInf_eq_dif
theorem exists_set_mem_of_union_eq_top {ι : Type*} (t : Set ι) (s : ι → Set β)
(w : ⋃ i ∈ t, s i = ⊤) (x : β) : ∃ i ∈ t, x ∈ s i := by
have p : x ∈ ⊤ := Set.mem_univ x
rw [← w, Set.mem_iUnion] at p
simpa using p
#align set.exists_set_mem_of_union_eq_top Set.exists_set_mem_of_union_eq_top
theorem nonempty_of_union_eq_top_of_nonempty {ι : Type*} (t : Set ι) (s : ι → Set α)
(H : Nonempty α) (w : ⋃ i ∈ t, s i = ⊤) : t.Nonempty := by
obtain ⟨x, m, -⟩ := exists_set_mem_of_union_eq_top t s w H.some
exact ⟨x, m⟩
#align set.nonempty_of_union_eq_top_of_nonempty Set.nonempty_of_union_eq_top_of_nonempty
theorem nonempty_of_nonempty_iUnion
{s : ι → Set α} (h_Union : (⋃ i, s i).Nonempty) : Nonempty ι := by
obtain ⟨x, hx⟩ := h_Union
exact ⟨Classical.choose $ mem_iUnion.mp hx⟩
theorem nonempty_of_nonempty_iUnion_eq_univ
{s : ι → Set α} [Nonempty α] (h_Union : ⋃ i, s i = univ) : Nonempty ι :=
nonempty_of_nonempty_iUnion (s := s) (by simpa only [h_Union] using univ_nonempty)
theorem setOf_exists (p : ι → β → Prop) : { x | ∃ i, p i x } = ⋃ i, { x | p i x } :=
ext fun _ => mem_iUnion.symm
#align set.set_of_exists Set.setOf_exists
theorem setOf_forall (p : ι → β → Prop) : { x | ∀ i, p i x } = ⋂ i, { x | p i x } :=
ext fun _ => mem_iInter.symm
#align set.set_of_forall Set.setOf_forall
theorem iUnion_subset {s : ι → Set α} {t : Set α} (h : ∀ i, s i ⊆ t) : ⋃ i, s i ⊆ t :=
iSup_le h
#align set.Union_subset Set.iUnion_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset {s : ∀ i, κ i → Set α} {t : Set α} (h : ∀ i j, s i j ⊆ t) :
⋃ (i) (j), s i j ⊆ t :=
iUnion_subset fun x => iUnion_subset (h x)
#align set.Union₂_subset Set.iUnion₂_subset
theorem subset_iInter {t : Set β} {s : ι → Set β} (h : ∀ i, t ⊆ s i) : t ⊆ ⋂ i, s i :=
le_iInf h
#align set.subset_Inter Set.subset_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iInter₂ {s : Set α} {t : ∀ i, κ i → Set α} (h : ∀ i j, s ⊆ t i j) :
s ⊆ ⋂ (i) (j), t i j :=
subset_iInter fun x => subset_iInter <| h x
#align set.subset_Inter₂ Set.subset_iInter₂
@[simp]
theorem iUnion_subset_iff {s : ι → Set α} {t : Set α} : ⋃ i, s i ⊆ t ↔ ∀ i, s i ⊆ t :=
⟨fun h _ => Subset.trans (le_iSup s _) h, iUnion_subset⟩
#align set.Union_subset_iff Set.iUnion_subset_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_subset_iff {s : ∀ i, κ i → Set α} {t : Set α} :
⋃ (i) (j), s i j ⊆ t ↔ ∀ i j, s i j ⊆ t := by simp_rw [iUnion_subset_iff]
#align set.Union₂_subset_iff Set.iUnion₂_subset_iff
@[simp]
theorem subset_iInter_iff {s : Set α} {t : ι → Set α} : (s ⊆ ⋂ i, t i) ↔ ∀ i, s ⊆ t i :=
le_iInf_iff
#align set.subset_Inter_iff Set.subset_iInter_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem subset_iInter₂_iff {s : Set α} {t : ∀ i, κ i → Set α} :
(s ⊆ ⋂ (i) (j), t i j) ↔ ∀ i j, s ⊆ t i j := by simp_rw [subset_iInter_iff]
#align set.subset_Inter₂_iff Set.subset_iInter₂_iff
theorem subset_iUnion : ∀ (s : ι → Set β) (i : ι), s i ⊆ ⋃ i, s i :=
le_iSup
#align set.subset_Union Set.subset_iUnion
theorem iInter_subset : ∀ (s : ι → Set β) (i : ι), ⋂ i, s i ⊆ s i :=
iInf_le
#align set.Inter_subset Set.iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem subset_iUnion₂ {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : s i j ⊆ ⋃ (i') (j'), s i' j' :=
le_iSup₂ i j
#align set.subset_Union₂ Set.subset_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_subset {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : ⋂ (i) (j), s i j ⊆ s i j :=
iInf₂_le i j
#align set.Inter₂_subset Set.iInter₂_subset
/-- This rather trivial consequence of `subset_iUnion`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem subset_iUnion_of_subset {s : Set α} {t : ι → Set α} (i : ι) (h : s ⊆ t i) : s ⊆ ⋃ i, t i :=
le_iSup_of_le i h
#align set.subset_Union_of_subset Set.subset_iUnion_of_subset
/-- This rather trivial consequence of `iInter_subset`is convenient with `apply`, and has `i`
explicit for this purpose. -/
theorem iInter_subset_of_subset {s : ι → Set α} {t : Set α} (i : ι) (h : s i ⊆ t) :
⋂ i, s i ⊆ t :=
iInf_le_of_le i h
#align set.Inter_subset_of_subset Set.iInter_subset_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `subset_iUnion₂` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem subset_iUnion₂_of_subset {s : Set α} {t : ∀ i, κ i → Set α} (i : ι) (j : κ i)
(h : s ⊆ t i j) : s ⊆ ⋃ (i) (j), t i j :=
le_iSup₂_of_le i j h
#align set.subset_Union₂_of_subset Set.subset_iUnion₂_of_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/-- This rather trivial consequence of `iInter₂_subset` is convenient with `apply`, and has `i` and
`j` explicit for this purpose. -/
theorem iInter₂_subset_of_subset {s : ∀ i, κ i → Set α} {t : Set α} (i : ι) (j : κ i)
(h : s i j ⊆ t) : ⋂ (i) (j), s i j ⊆ t :=
iInf₂_le_of_le i j h
#align set.Inter₂_subset_of_subset Set.iInter₂_subset_of_subset
theorem iUnion_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono h
#align set.Union_mono Set.iUnion_mono
@[gcongr]
theorem iUnion_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iUnion s ⊆ iUnion t :=
iSup_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋃ (i) (j), s i j ⊆ ⋃ (i) (j), t i j :=
iSup₂_mono h
#align set.Union₂_mono Set.iUnion₂_mono
theorem iInter_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋂ i, s i ⊆ ⋂ i, t i :=
iInf_mono h
#align set.Inter_mono Set.iInter_mono
@[gcongr]
theorem iInter_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iInter s ⊆ iInter t :=
iInf_mono h
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
⋂ (i) (j), s i j ⊆ ⋂ (i) (j), t i j :=
iInf₂_mono h
#align set.Inter₂_mono Set.iInter₂_mono
theorem iUnion_mono' {s : ι → Set α} {t : ι₂ → Set α} (h : ∀ i, ∃ j, s i ⊆ t j) :
⋃ i, s i ⊆ ⋃ i, t i :=
iSup_mono' h
#align set.Union_mono' Set.iUnion_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iUnion₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i j, ∃ i' j', s i j ⊆ t i' j') : ⋃ (i) (j), s i j ⊆ ⋃ (i') (j'), t i' j' :=
iSup₂_mono' h
#align set.Union₂_mono' Set.iUnion₂_mono'
theorem iInter_mono' {s : ι → Set α} {t : ι' → Set α} (h : ∀ j, ∃ i, s i ⊆ t j) :
⋂ i, s i ⊆ ⋂ j, t j :=
Set.subset_iInter fun j =>
let ⟨i, hi⟩ := h j
iInter_subset_of_subset i hi
#align set.Inter_mono' Set.iInter_mono'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' j') -/
theorem iInter₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
(h : ∀ i' j', ∃ i j, s i j ⊆ t i' j') : ⋂ (i) (j), s i j ⊆ ⋂ (i') (j'), t i' j' :=
subset_iInter₂_iff.2 fun i' j' =>
let ⟨_, _, hst⟩ := h i' j'
(iInter₂_subset _ _).trans hst
#align set.Inter₂_mono' Set.iInter₂_mono'
theorem iUnion₂_subset_iUnion (κ : ι → Sort*) (s : ι → Set α) :
⋃ (i) (_ : κ i), s i ⊆ ⋃ i, s i :=
iUnion_mono fun _ => iUnion_subset fun _ => Subset.rfl
#align set.Union₂_subset_Union Set.iUnion₂_subset_iUnion
theorem iInter_subset_iInter₂ (κ : ι → Sort*) (s : ι → Set α) :
⋂ i, s i ⊆ ⋂ (i) (_ : κ i), s i :=
iInter_mono fun _ => subset_iInter fun _ => Subset.rfl
#align set.Inter_subset_Inter₂ Set.iInter_subset_iInter₂
theorem iUnion_setOf (P : ι → α → Prop) : ⋃ i, { x : α | P i x } = { x : α | ∃ i, P i x } := by
ext
exact mem_iUnion
#align set.Union_set_of Set.iUnion_setOf
theorem iInter_setOf (P : ι → α → Prop) : ⋂ i, { x : α | P i x } = { x : α | ∀ i, P i x } := by
ext
exact mem_iInter
#align set.Inter_set_of Set.iInter_setOf
theorem iUnion_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋃ x, f x = ⋃ y, g y :=
h1.iSup_congr h h2
#align set.Union_congr_of_surjective Set.iUnion_congr_of_surjective
theorem iInter_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
(h2 : ∀ x, g (h x) = f x) : ⋂ x, f x = ⋂ y, g y :=
h1.iInf_congr h h2
#align set.Inter_congr_of_surjective Set.iInter_congr_of_surjective
lemma iUnion_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋃ i, s i = ⋃ i, t i := iSup_congr h
#align set.Union_congr Set.iUnion_congr
lemma iInter_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋂ i, s i = ⋂ i, t i := iInf_congr h
#align set.Inter_congr Set.iInter_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iUnion₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋃ (i) (j), s i j = ⋃ (i) (j), t i j :=
iUnion_congr fun i => iUnion_congr <| h i
#align set.Union₂_congr Set.iUnion₂_congr
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
lemma iInter₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
⋂ (i) (j), s i j = ⋂ (i) (j), t i j :=
iInter_congr fun i => iInter_congr <| h i
#align set.Inter₂_congr Set.iInter₂_congr
section Nonempty
variable [Nonempty ι] {f : ι → Set α} {s : Set α}
lemma iUnion_const (s : Set β) : ⋃ _ : ι, s = s := iSup_const
#align set.Union_const Set.iUnion_const
lemma iInter_const (s : Set β) : ⋂ _ : ι, s = s := iInf_const
#align set.Inter_const Set.iInter_const
lemma iUnion_eq_const (hf : ∀ i, f i = s) : ⋃ i, f i = s :=
(iUnion_congr hf).trans $ iUnion_const _
#align set.Union_eq_const Set.iUnion_eq_const
lemma iInter_eq_const (hf : ∀ i, f i = s) : ⋂ i, f i = s :=
(iInter_congr hf).trans $ iInter_const _
#align set.Inter_eq_const Set.iInter_eq_const
end Nonempty
@[simp]
theorem compl_iUnion (s : ι → Set β) : (⋃ i, s i)ᶜ = ⋂ i, (s i)ᶜ :=
compl_iSup
#align set.compl_Union Set.compl_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iUnion₂ (s : ∀ i, κ i → Set α) : (⋃ (i) (j), s i j)ᶜ = ⋂ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iUnion]
#align set.compl_Union₂ Set.compl_iUnion₂
@[simp]
theorem compl_iInter (s : ι → Set β) : (⋂ i, s i)ᶜ = ⋃ i, (s i)ᶜ :=
compl_iInf
#align set.compl_Inter Set.compl_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem compl_iInter₂ (s : ∀ i, κ i → Set α) : (⋂ (i) (j), s i j)ᶜ = ⋃ (i) (j), (s i j)ᶜ := by
simp_rw [compl_iInter]
#align set.compl_Inter₂ Set.compl_iInter₂
-- classical -- complete_boolean_algebra
theorem iUnion_eq_compl_iInter_compl (s : ι → Set β) : ⋃ i, s i = (⋂ i, (s i)ᶜ)ᶜ := by
simp only [compl_iInter, compl_compl]
#align set.Union_eq_compl_Inter_compl Set.iUnion_eq_compl_iInter_compl
-- classical -- complete_boolean_algebra
theorem iInter_eq_compl_iUnion_compl (s : ι → Set β) : ⋂ i, s i = (⋃ i, (s i)ᶜ)ᶜ := by
simp only [compl_iUnion, compl_compl]
#align set.Inter_eq_compl_Union_compl Set.iInter_eq_compl_iUnion_compl
theorem inter_iUnion (s : Set β) (t : ι → Set β) : (s ∩ ⋃ i, t i) = ⋃ i, s ∩ t i :=
inf_iSup_eq _ _
#align set.inter_Union Set.inter_iUnion
theorem iUnion_inter (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∩ s = ⋃ i, t i ∩ s :=
iSup_inf_eq _ _
#align set.Union_inter Set.iUnion_inter
theorem iUnion_union_distrib (s : ι → Set β) (t : ι → Set β) :
⋃ i, s i ∪ t i = (⋃ i, s i) ∪ ⋃ i, t i :=
iSup_sup_eq
#align set.Union_union_distrib Set.iUnion_union_distrib
theorem iInter_inter_distrib (s : ι → Set β) (t : ι → Set β) :
⋂ i, s i ∩ t i = (⋂ i, s i) ∩ ⋂ i, t i :=
iInf_inf_eq
#align set.Inter_inter_distrib Set.iInter_inter_distrib
theorem union_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∪ ⋃ i, t i) = ⋃ i, s ∪ t i :=
sup_iSup
#align set.union_Union Set.union_iUnion
theorem iUnion_union [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∪ s = ⋃ i, t i ∪ s :=
iSup_sup
#align set.Union_union Set.iUnion_union
theorem inter_iInter [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∩ ⋂ i, t i) = ⋂ i, s ∩ t i :=
inf_iInf
#align set.inter_Inter Set.inter_iInter
theorem iInter_inter [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋂ i, t i) ∩ s = ⋂ i, t i ∩ s :=
iInf_inf
#align set.Inter_inter Set.iInter_inter
-- classical
theorem union_iInter (s : Set β) (t : ι → Set β) : (s ∪ ⋂ i, t i) = ⋂ i, s ∪ t i :=
sup_iInf_eq _ _
#align set.union_Inter Set.union_iInter
theorem iInter_union (s : ι → Set β) (t : Set β) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.Inter_union Set.iInter_union
theorem iUnion_diff (s : Set β) (t : ι → Set β) : (⋃ i, t i) \ s = ⋃ i, t i \ s :=
iUnion_inter _ _
#align set.Union_diff Set.iUnion_diff
theorem diff_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s \ ⋃ i, t i) = ⋂ i, s \ t i := by
rw [diff_eq, compl_iUnion, inter_iInter]; rfl
#align set.diff_Union Set.diff_iUnion
theorem diff_iInter (s : Set β) (t : ι → Set β) : (s \ ⋂ i, t i) = ⋃ i, s \ t i := by
rw [diff_eq, compl_iInter, inter_iUnion]; rfl
#align set.diff_Inter Set.diff_iInter
theorem directed_on_iUnion {r} {f : ι → Set α} (hd : Directed (· ⊆ ·) f)
(h : ∀ x, DirectedOn r (f x)) : DirectedOn r (⋃ x, f x) := by
simp only [DirectedOn, exists_prop, mem_iUnion, exists_imp]
exact fun a₁ b₁ fb₁ a₂ b₂ fb₂ =>
let ⟨z, zb₁, zb₂⟩ := hd b₁ b₂
let ⟨x, xf, xa₁, xa₂⟩ := h z a₁ (zb₁ fb₁) a₂ (zb₂ fb₂)
⟨x, ⟨z, xf⟩, xa₁, xa₂⟩
#align set.directed_on_Union Set.directed_on_iUnion
theorem iUnion_inter_subset {ι α} {s t : ι → Set α} : ⋃ i, s i ∩ t i ⊆ (⋃ i, s i) ∩ ⋃ i, t i :=
le_iSup_inf_iSup s t
#align set.Union_inter_subset Set.iUnion_inter_subset
theorem iUnion_inter_of_monotone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_monotone hs ht
#align set.Union_inter_of_monotone Set.iUnion_inter_of_monotone
theorem iUnion_inter_of_antitone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i :=
iSup_inf_of_antitone hs ht
#align set.Union_inter_of_antitone Set.iUnion_inter_of_antitone
theorem iInter_union_of_monotone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
(hs : Monotone s) (ht : Monotone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_monotone hs ht
#align set.Inter_union_of_monotone Set.iInter_union_of_monotone
theorem iInter_union_of_antitone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
(hs : Antitone s) (ht : Antitone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i :=
iInf_sup_of_antitone hs ht
#align set.Inter_union_of_antitone Set.iInter_union_of_antitone
/-- An equality version of this lemma is `iUnion_iInter_of_monotone` in `Data.Set.Finite`. -/
theorem iUnion_iInter_subset {s : ι → ι' → Set α} : (⋃ j, ⋂ i, s i j) ⊆ ⋂ i, ⋃ j, s i j :=
iSup_iInf_le_iInf_iSup (flip s)
#align set.Union_Inter_subset Set.iUnion_iInter_subset
theorem iUnion_option {ι} (s : Option ι → Set α) : ⋃ o, s o = s none ∪ ⋃ i, s (some i) :=
iSup_option s
#align set.Union_option Set.iUnion_option
theorem iInter_option {ι} (s : Option ι → Set α) : ⋂ o, s o = s none ∩ ⋂ i, s (some i) :=
iInf_option s
#align set.Inter_option Set.iInter_option
section
variable (p : ι → Prop) [DecidablePred p]
theorem iUnion_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 _ _ _
#align set.Union_dite Set.iUnion_dite
theorem iUnion_ite (f g : ι → Set α) :
⋃ i, (if p i then f i else g i) = (⋃ (i) (_ : p i), f i) ∪ ⋃ (i) (_ : ¬p i), g i :=
iUnion_dite _ _ _
#align set.Union_ite Set.iUnion_ite
theorem iInter_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set 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 :=
iInf_dite _ _ _
#align set.Inter_dite Set.iInter_dite
theorem iInter_ite (f g : ι → Set α) :
⋂ i, (if p i then f i else g i) = (⋂ (i) (_ : p i), f i) ∩ ⋂ (i) (_ : ¬p i), g i :=
iInter_dite _ _ _
#align set.Inter_ite Set.iInter_ite
end
theorem image_projection_prod {ι : Type*} {α : ι → Type*} {v : ∀ i : ι, Set (α i)}
(hv : (pi univ v).Nonempty) (i : ι) :
((fun x : ∀ i : ι, α i => x i) '' ⋂ k, (fun x : ∀ j : ι, α j => x k) ⁻¹' v k) = v i := by
classical
apply Subset.antisymm
· simp [iInter_subset]
· intro y y_in
simp only [mem_image, mem_iInter, mem_preimage]
rcases hv with ⟨z, hz⟩
refine' ⟨Function.update z i y, _, update_same i y z⟩
rw [@forall_update_iff ι α _ z i y fun i t => t ∈ v i]
exact ⟨y_in, fun j _ => by simpa using hz j⟩
#align set.image_projection_prod Set.image_projection_prod
/-! ### Unions and intersections indexed by `Prop` -/
theorem iInter_false {s : False → Set α} : iInter s = univ :=
iInf_false
#align set.Inter_false Set.iInter_false
theorem iUnion_false {s : False → Set α} : iUnion s = ∅ :=
iSup_false
#align set.Union_false Set.iUnion_false
@[simp]
theorem iInter_true {s : True → Set α} : iInter s = s trivial :=
iInf_true
#align set.Inter_true Set.iInter_true
@[simp]
theorem iUnion_true {s : True → Set α} : iUnion s = s trivial :=
iSup_true
#align set.Union_true Set.iUnion_true
@[simp]
theorem iInter_exists {p : ι → Prop} {f : Exists p → Set α} :
⋂ x, f x = ⋂ (i) (h : p i), f ⟨i, h⟩ :=
iInf_exists
#align set.Inter_exists Set.iInter_exists
@[simp]
theorem iUnion_exists {p : ι → Prop} {f : Exists p → Set α} :
⋃ x, f x = ⋃ (i) (h : p i), f ⟨i, h⟩ :=
iSup_exists
#align set.Union_exists Set.iUnion_exists
@[simp]
theorem iUnion_empty : (⋃ _ : ι, ∅ : Set α) = ∅ :=
iSup_bot
#align set.Union_empty Set.iUnion_empty
@[simp]
theorem iInter_univ : (⋂ _ : ι, univ : Set α) = univ :=
iInf_top
#align set.Inter_univ Set.iInter_univ
section
variable {s : ι → Set α}
@[simp]
theorem iUnion_eq_empty : ⋃ i, s i = ∅ ↔ ∀ i, s i = ∅ :=
iSup_eq_bot
#align set.Union_eq_empty Set.iUnion_eq_empty
@[simp]
theorem iInter_eq_univ : ⋂ i, s i = univ ↔ ∀ i, s i = univ :=
iInf_eq_top
#align set.Inter_eq_univ Set.iInter_eq_univ
@[simp]
theorem nonempty_iUnion : (⋃ i, s i).Nonempty ↔ ∃ i, (s i).Nonempty := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_Union Set.nonempty_iUnion
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_biUnion {t : Set α} {s : α → Set β} :
(⋃ i ∈ t, s i).Nonempty ↔ ∃ i ∈ t, (s i).Nonempty := by simp
#align set.nonempty_bUnion Set.nonempty_biUnion
theorem iUnion_nonempty_index (s : Set α) (t : s.Nonempty → Set β) :
⋃ h, t h = ⋃ x ∈ s, t ⟨x, ‹_›⟩ :=
iSup_exists
#align set.Union_nonempty_index Set.iUnion_nonempty_index
end
@[simp]
theorem iInter_iInter_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋂ (x) (h : x = b), s x h = s b rfl :=
iInf_iInf_eq_left
#align set.Inter_Inter_eq_left Set.iInter_iInter_eq_left
@[simp]
theorem iInter_iInter_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋂ (x) (h : b = x), s x h = s b rfl :=
iInf_iInf_eq_right
#align set.Inter_Inter_eq_right Set.iInter_iInter_eq_right
@[simp]
theorem iUnion_iUnion_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
⋃ (x) (h : x = b), s x h = s b rfl :=
iSup_iSup_eq_left
#align set.Union_Union_eq_left Set.iUnion_iUnion_eq_left
@[simp]
theorem iUnion_iUnion_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
⋃ (x) (h : b = x), s x h = s b rfl :=
iSup_iSup_eq_right
#align set.Union_Union_eq_right Set.iUnion_iUnion_eq_right
theorem iInter_or {p q : Prop} (s : p ∨ q → Set α) :
⋂ h, s h = (⋂ h : p, s (Or.inl h)) ∩ ⋂ h : q, s (Or.inr h) :=
iInf_or
#align set.Inter_or Set.iInter_or
theorem iUnion_or {p q : Prop} (s : p ∨ q → Set α) :
⋃ h, s h = (⋃ i, s (Or.inl i)) ∪ ⋃ j, s (Or.inr j) :=
iSup_or
#align set.Union_or Set.iUnion_or
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iUnion_and {p q : Prop} (s : p ∧ q → Set α) : ⋃ h, s h = ⋃ (hp) (hq), s ⟨hp, hq⟩ :=
iSup_and
#align set.Union_and Set.iUnion_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (hp hq) -/
theorem iInter_and {p q : Prop} (s : p ∧ q → Set α) : ⋂ h, s h = ⋂ (hp) (hq), s ⟨hp, hq⟩ :=
iInf_and
#align set.Inter_and Set.iInter_and
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iUnion_comm (s : ι → ι' → Set α) : ⋃ (i) (i'), s i i' = ⋃ (i') (i), s i i' :=
iSup_comm
#align set.Union_comm Set.iUnion_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i i') -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i' i) -/
theorem iInter_comm (s : ι → ι' → Set α) : ⋂ (i) (i'), s i i' = ⋂ (i') (i), s i i' :=
iInf_comm
#align set.Inter_comm Set.iInter_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iUnion₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋃ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋃ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iSup₂_comm _
#align set.Union₂_comm Set.iUnion₂_comm
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₁ j₁ i₂ j₂) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i₂ j₂ i₁ j₁) -/
theorem iInter₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
⋂ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋂ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ :=
iInf₂_comm _
#align set.Inter₂_comm Set.iInter₂_comm
@[simp]
theorem biUnion_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋃ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι']
#align set.bUnion_and Set.biUnion_and
@[simp]
theorem biUnion_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋃ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋃ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iUnion_and, @iUnion_comm _ ι]
#align set.bUnion_and' Set.biUnion_and'
@[simp]
theorem biInter_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
⋂ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ :=
by simp only [iInter_and, @iInter_comm _ ι']
#align set.bInter_and Set.biInter_and
@[simp]
theorem biInter_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
⋂ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
⋂ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ :=
by simp only [iInter_and, @iInter_comm _ ι]
#align set.bInter_and' Set.biInter_and'
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iUnion_iUnion_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋃ (x) (h), s x h = s b (Or.inl rfl) ∪ ⋃ (x) (h : p x), s x (Or.inr h) := by
simp only [iUnion_or, iUnion_union_distrib, iUnion_iUnion_eq_left]
#align set.Union_Union_eq_or_left Set.iUnion_iUnion_eq_or_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x h) -/
@[simp]
theorem iInter_iInter_eq_or_left {b : β} {p : β → Prop} {s : ∀ x : β, x = b ∨ p x → Set α} :
⋂ (x) (h), s x h = s b (Or.inl rfl) ∩ ⋂ (x) (h : p x), s x (Or.inr h) := by
simp only [iInter_or, iInter_inter_distrib, iInter_iInter_eq_left]
#align set.Inter_Inter_eq_or_left Set.iInter_iInter_eq_or_left
/-! ### Bounded unions and intersections -/
/-- A specialization of `mem_iUnion₂`. -/
theorem mem_biUnion {s : Set α} {t : α → Set β} {x : α} {y : β} (xs : x ∈ s) (ytx : y ∈ t x) :
y ∈ ⋃ x ∈ s, t x :=
mem_iUnion₂_of_mem xs ytx
#align set.mem_bUnion Set.mem_biUnion
/-- A specialization of `mem_iInter₂`. -/
theorem mem_biInter {s : Set α} {t : α → Set β} {y : β} (h : ∀ x ∈ s, y ∈ t x) :
y ∈ ⋂ x ∈ s, t x :=
mem_iInter₂_of_mem h
#align set.mem_bInter Set.mem_biInter
/-- A specialization of `subset_iUnion₂`. -/
theorem subset_biUnion_of_mem {s : Set α} {u : α → Set β} {x : α} (xs : x ∈ s) :
u x ⊆ ⋃ x ∈ s, u x :=
--Porting note: Why is this not just `subset_iUnion₂ x xs`?
@subset_iUnion₂ β α (· ∈ s) (fun i _ => u i) x xs
#align set.subset_bUnion_of_mem Set.subset_biUnion_of_mem
/-- A specialization of `iInter₂_subset`. -/
theorem biInter_subset_of_mem {s : Set α} {t : α → Set β} {x : α} (xs : x ∈ s) :
⋂ x ∈ s, t x ⊆ t x :=
iInter₂_subset x xs
#align set.bInter_subset_of_mem Set.biInter_subset_of_mem
theorem biUnion_subset_biUnion_left {s s' : Set α} {t : α → Set β} (h : s ⊆ s') :
⋃ x ∈ s, t x ⊆ ⋃ x ∈ s', t x :=
iUnion₂_subset fun _ hx => subset_biUnion_of_mem <| h hx
#align set.bUnion_subset_bUnion_left Set.biUnion_subset_biUnion_left
theorem biInter_subset_biInter_left {s s' : Set α} {t : α → Set β} (h : s' ⊆ s) :
⋂ x ∈ s, t x ⊆ ⋂ x ∈ s', t x :=
subset_iInter₂ fun _ hx => biInter_subset_of_mem <| h hx
#align set.bInter_subset_bInter_left Set.biInter_subset_biInter_left
theorem biUnion_mono {s s' : Set α} {t t' : α → Set β} (hs : s' ⊆ s) (h : ∀ x ∈ s, t x ⊆ t' x) :
⋃ x ∈ s', t x ⊆ ⋃ x ∈ s, t' x :=
(biUnion_subset_biUnion_left hs).trans <| iUnion₂_mono h
#align set.bUnion_mono Set.biUnion_mono
theorem biInter_mono {s s' : Set α} {t t' : α → Set β} (hs : s ⊆ s') (h : ∀ x ∈ s, t x ⊆ t' x) :
⋂ x ∈ s', t x ⊆ ⋂ x ∈ s, t' x :=
(biInter_subset_biInter_left hs).trans <| iInter₂_mono h
#align set.bInter_mono Set.biInter_mono
theorem biUnion_eq_iUnion (s : Set α) (t : ∀ x ∈ s, Set β) :
⋃ x ∈ s, t x ‹_› = ⋃ x : s, t x x.2 :=
iSup_subtype'
#align set.bUnion_eq_Union Set.biUnion_eq_iUnion
theorem biInter_eq_iInter (s : Set α) (t : ∀ x ∈ s, Set β) :
⋂ x ∈ s, t x ‹_› = ⋂ x : s, t x x.2 :=
iInf_subtype'
#align set.bInter_eq_Inter Set.biInter_eq_iInter
theorem iUnion_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋃ x : { x // p x }, s x = ⋃ (x) (hx : p x), s ⟨x, hx⟩ :=
iSup_subtype
#align set.Union_subtype Set.iUnion_subtype
theorem iInter_subtype (p : α → Prop) (s : { x // p x } → Set β) :
⋂ x : { x // p x }, s x = ⋂ (x) (hx : p x), s ⟨x, hx⟩ :=
iInf_subtype
#align set.Inter_subtype Set.iInter_subtype
theorem biInter_empty (u : α → Set β) : ⋂ x ∈ (∅ : Set α), u x = univ :=
iInf_emptyset
#align set.bInter_empty Set.biInter_empty
theorem biInter_univ (u : α → Set β) : ⋂ x ∈ @univ α, u x = ⋂ x, u x :=
iInf_univ
#align set.bInter_univ Set.biInter_univ
@[simp]
theorem biUnion_self (s : Set α) : ⋃ x ∈ s, s = s :=
Subset.antisymm (iUnion₂_subset fun _ _ => Subset.refl s) fun _ hx => mem_biUnion hx hx
#align set.bUnion_self Set.biUnion_self
@[simp]
theorem iUnion_nonempty_self (s : Set α) : ⋃ _ : s.Nonempty, s = s := by
rw [iUnion_nonempty_index, biUnion_self]
#align set.Union_nonempty_self Set.iUnion_nonempty_self
theorem biInter_singleton (a : α) (s : α → Set β) : ⋂ x ∈ ({a} : Set α), s x = s a :=
iInf_singleton
#align set.bInter_singleton Set.biInter_singleton
theorem biInter_union (s t : Set α) (u : α → Set β) :
⋂ x ∈ s ∪ t, u x = (⋂ x ∈ s, u x) ∩ ⋂ x ∈ t, u x :=
iInf_union
#align set.bInter_union Set.biInter_union
theorem biInter_insert (a : α) (s : Set α) (t : α → Set β) :
⋂ x ∈ insert a s, t x = t a ∩ ⋂ x ∈ s, t x := by simp
#align set.bInter_insert Set.biInter_insert
theorem biInter_pair (a b : α) (s : α → Set β) : ⋂ x ∈ ({a, b} : Set α), s x = s a ∩ s b := by
rw [biInter_insert, biInter_singleton]
#align set.bInter_pair Set.biInter_pair
theorem biInter_inter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, f i ∩ t = (⋂ i ∈ s, f i) ∩ t := by
haveI : Nonempty s := hs.to_subtype
simp [biInter_eq_iInter, ← iInter_inter]
#align set.bInter_inter Set.biInter_inter
theorem inter_biInter {ι α : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → Set α) (t : Set α) :
⋂ i ∈ s, t ∩ f i = t ∩ ⋂ i ∈ s, f i := by
rw [inter_comm, ← biInter_inter hs]
simp [inter_comm]
#align set.inter_bInter Set.inter_biInter
theorem biUnion_empty (s : α → Set β) : ⋃ x ∈ (∅ : Set α), s x = ∅ :=
iSup_emptyset
#align set.bUnion_empty Set.biUnion_empty
theorem biUnion_univ (s : α → Set β) : ⋃ x ∈ @univ α, s x = ⋃ x, s x :=
iSup_univ
#align set.bUnion_univ Set.biUnion_univ
theorem biUnion_singleton (a : α) (s : α → Set β) : ⋃ x ∈ ({a} : Set α), s x = s a :=
iSup_singleton
#align set.bUnion_singleton Set.biUnion_singleton
@[simp]
theorem biUnion_of_singleton (s : Set α) : ⋃ x ∈ s, {x} = s :=
ext <| by simp
#align set.bUnion_of_singleton Set.biUnion_of_singleton
theorem biUnion_union (s t : Set α) (u : α → Set β) :
⋃ x ∈ s ∪ t, u x = (⋃ x ∈ s, u x) ∪ ⋃ x ∈ t, u x :=
iSup_union
#align set.bUnion_union Set.biUnion_union
@[simp]
theorem iUnion_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋃ i, f i = ⋃ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iUnion_subtype _ _
#align set.Union_coe_set Set.iUnion_coe_set
@[simp]
theorem iInter_coe_set {α β : Type*} (s : Set α) (f : s → Set β) :
⋂ i, f i = ⋂ i ∈ s, f ⟨i, ‹i ∈ s›⟩ :=
iInter_subtype _ _
#align set.Inter_coe_set Set.iInter_coe_set
theorem biUnion_insert (a : α) (s : Set α) (t : α → Set β) :
⋃ x ∈ insert a s, t x = t a ∪ ⋃ x ∈ s, t x := by simp
#align set.bUnion_insert Set.biUnion_insert
theorem biUnion_pair (a b : α) (s : α → Set β) : ⋃ x ∈ ({a, b} : Set α), s x = s a ∪ s b :=
by simp
#align set.bUnion_pair Set.biUnion_pair
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem inter_iUnion₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∩ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ∩ t i j := by simp only [inter_iUnion]
#align set.inter_Union₂ Set.inter_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_inter (s : ∀ i, κ i → Set α) (t : Set α) :
(⋃ (i) (j), s i j) ∩ t = ⋃ (i) (j), s i j ∩ t := by simp_rw [iUnion_inter]
#align set.Union₂_inter Set.iUnion₂_inter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_iInter₂ (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_iInter]
#align set.union_Inter₂ Set.union_iInter₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iInter₂_union (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [iInter_union]
#align set.Inter₂_union Set.iInter₂_union
theorem mem_sUnion_of_mem {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∈ t) (ht : t ∈ S) :
x ∈ ⋃₀S :=
⟨t, ht, hx⟩
#align set.mem_sUnion_of_mem Set.mem_sUnion_of_mem
-- is this theorem really necessary?
theorem not_mem_of_not_mem_sUnion {x : α} {t : Set α} {S : Set (Set α)} (hx : x ∉ ⋃₀S)
(ht : t ∈ S) : x ∉ t := fun h => hx ⟨t, ht, h⟩
#align set.not_mem_of_not_mem_sUnion Set.not_mem_of_not_mem_sUnion
theorem sInter_subset_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : ⋂₀ S ⊆ t :=
sInf_le tS
#align set.sInter_subset_of_mem Set.sInter_subset_of_mem
theorem subset_sUnion_of_mem {S : Set (Set α)} {t : Set α} (tS : t ∈ S) : t ⊆ ⋃₀S :=
le_sSup tS
#align set.subset_sUnion_of_mem Set.subset_sUnion_of_mem
theorem subset_sUnion_of_subset {s : Set α} (t : Set (Set α)) (u : Set α) (h₁ : s ⊆ u)
(h₂ : u ∈ t) : s ⊆ ⋃₀t :=
Subset.trans h₁ (subset_sUnion_of_mem h₂)
#align set.subset_sUnion_of_subset Set.subset_sUnion_of_subset
theorem sUnion_subset {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t' ⊆ t) : ⋃₀S ⊆ t :=
sSup_le h
#align set.sUnion_subset Set.sUnion_subset
@[simp]
theorem sUnion_subset_iff {s : Set (Set α)} {t : Set α} : ⋃₀s ⊆ t ↔ ∀ t' ∈ s, t' ⊆ t :=
sSup_le_iff
#align set.sUnion_subset_iff Set.sUnion_subset_iff
/-- `sUnion` is monotone under taking a subset of each set. -/
lemma sUnion_mono_subsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, t ⊆ f t) :
⋃₀ s ⊆ ⋃₀ (f '' s) :=
fun _ ⟨t, htx, hxt⟩ ↦ ⟨f t, mem_image_of_mem f htx, hf t hxt⟩
/-- `sUnion` is monotone under taking a superset of each set. -/
lemma sUnion_mono_supsets {s : Set (Set α)} {f : Set α → Set α} (hf : ∀ t : Set α, f t ⊆ t) :
⋃₀ (f '' s) ⊆ ⋃₀ s :=
-- If t ∈ f '' s is arbitrary; t = f u for some u : Set α.
fun _ ⟨_, ⟨u, hus, hut⟩, hxt⟩ ↦ ⟨u, hus, (hut ▸ hf u) hxt⟩
theorem subset_sInter {S : Set (Set α)} {t : Set α} (h : ∀ t' ∈ S, t ⊆ t') : t ⊆ ⋂₀ S :=
le_sInf h
#align set.subset_sInter Set.subset_sInter
@[simp]
theorem subset_sInter_iff {S : Set (Set α)} {t : Set α} : t ⊆ ⋂₀ S ↔ ∀ t' ∈ S, t ⊆ t' :=
le_sInf_iff
#align set.subset_sInter_iff Set.subset_sInter_iff
@[gcongr]
theorem sUnion_subset_sUnion {S T : Set (Set α)} (h : S ⊆ T) : ⋃₀S ⊆ ⋃₀T :=
sUnion_subset fun _ hs => subset_sUnion_of_mem (h hs)
#align set.sUnion_subset_sUnion Set.sUnion_subset_sUnion
@[gcongr]
theorem sInter_subset_sInter {S T : Set (Set α)} (h : S ⊆ T) : ⋂₀ T ⊆ ⋂₀ S :=
subset_sInter fun _ hs => sInter_subset_of_mem (h hs)
#align set.sInter_subset_sInter Set.sInter_subset_sInter
@[simp]
theorem sUnion_empty : ⋃₀∅ = (∅ : Set α) :=
sSup_empty
#align set.sUnion_empty Set.sUnion_empty
@[simp]
theorem sInter_empty : ⋂₀ ∅ = (univ : Set α) :=
sInf_empty
#align set.sInter_empty Set.sInter_empty
@[simp]
theorem sUnion_singleton (s : Set α) : ⋃₀{s} = s :=
sSup_singleton
#align set.sUnion_singleton Set.sUnion_singleton
@[simp]
theorem sInter_singleton (s : Set α) : ⋂₀ {s} = s :=
sInf_singleton
#align set.sInter_singleton Set.sInter_singleton
@[simp]
theorem sUnion_eq_empty {S : Set (Set α)} : ⋃₀S = ∅ ↔ ∀ s ∈ S, s = ∅ :=
sSup_eq_bot
#align set.sUnion_eq_empty Set.sUnion_eq_empty
@[simp]
theorem sInter_eq_univ {S : Set (Set α)} : ⋂₀ S = univ ↔ ∀ s ∈ S, s = univ :=
sInf_eq_top
#align set.sInter_eq_univ Set.sInter_eq_univ
theorem subset_powerset_iff {s : Set (Set α)} {t : Set α} : s ⊆ 𝒫 t ↔ ⋃₀ s ⊆ t :=
sUnion_subset_iff.symm
/-- `⋃₀` and `𝒫` form a Galois connection. -/
theorem sUnion_powerset_gc :
GaloisConnection (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gc_sSup_Iic
/-- `⋃₀` and `𝒫` form a Galois insertion. -/
def sUnion_powerset_gi :
GaloisInsertion (⋃₀ · : Set (Set α) → Set α) (𝒫 · : Set α → Set (Set α)) :=
gi_sSup_Iic
/-- If all sets in a collection are either `∅` or `Set.univ`, then so is their union. -/
theorem sUnion_mem_empty_univ {S : Set (Set α)} (h : S ⊆ {∅, univ}) :
⋃₀ S ∈ ({∅, univ} : Set (Set α)) := by
simp only [mem_insert_iff, mem_singleton_iff, or_iff_not_imp_left, sUnion_eq_empty, not_forall]
rintro ⟨s, hs, hne⟩
obtain rfl : s = univ := (h hs).resolve_left hne
exact univ_subset_iff.1 <| subset_sUnion_of_mem hs
@[simp]
theorem nonempty_sUnion {S : Set (Set α)} : (⋃₀S).Nonempty ↔ ∃ s ∈ S, Set.Nonempty s := by
simp [nonempty_iff_ne_empty]
#align set.nonempty_sUnion Set.nonempty_sUnion
theorem Nonempty.of_sUnion {s : Set (Set α)} (h : (⋃₀s).Nonempty) : s.Nonempty :=
let ⟨s, hs, _⟩ := nonempty_sUnion.1 h
⟨s, hs⟩
#align set.nonempty.of_sUnion Set.Nonempty.of_sUnion
theorem Nonempty.of_sUnion_eq_univ [Nonempty α] {s : Set (Set α)} (h : ⋃₀s = univ) : s.Nonempty :=
Nonempty.of_sUnion <| h.symm ▸ univ_nonempty
#align set.nonempty.of_sUnion_eq_univ Set.Nonempty.of_sUnion_eq_univ
theorem sUnion_union (S T : Set (Set α)) : ⋃₀(S ∪ T) = ⋃₀S ∪ ⋃₀T :=
sSup_union
#align set.sUnion_union Set.sUnion_union
theorem sInter_union (S T : Set (Set α)) : ⋂₀ (S ∪ T) = ⋂₀ S ∩ ⋂₀ T :=
sInf_union
#align set.sInter_union Set.sInter_union
@[simp]
theorem sUnion_insert (s : Set α) (T : Set (Set α)) : ⋃₀insert s T = s ∪ ⋃₀T :=
sSup_insert
#align set.sUnion_insert Set.sUnion_insert
@[simp]
theorem sInter_insert (s : Set α) (T : Set (Set α)) : ⋂₀ insert s T = s ∩ ⋂₀ T :=
sInf_insert
#align set.sInter_insert Set.sInter_insert
@[simp]
theorem sUnion_diff_singleton_empty (s : Set (Set α)) : ⋃₀(s \ {∅}) = ⋃₀s :=
sSup_diff_singleton_bot s
#align set.sUnion_diff_singleton_empty Set.sUnion_diff_singleton_empty
@[simp]
theorem sInter_diff_singleton_univ (s : Set (Set α)) : ⋂₀ (s \ {univ}) = ⋂₀ s :=
sInf_diff_singleton_top s
#align set.sInter_diff_singleton_univ Set.sInter_diff_singleton_univ
theorem sUnion_pair (s t : Set α) : ⋃₀{s, t} = s ∪ t :=
sSup_pair
#align set.sUnion_pair Set.sUnion_pair
theorem sInter_pair (s t : Set α) : ⋂₀ {s, t} = s ∩ t :=
sInf_pair
#align set.sInter_pair Set.sInter_pair
@[simp]
theorem sUnion_image (f : α → Set β) (s : Set α) : ⋃₀(f '' s) = ⋃ x ∈ s, f x :=
sSup_image
#align set.sUnion_image Set.sUnion_image
@[simp]
theorem sInter_image (f : α → Set β) (s : Set α) : ⋂₀ (f '' s) = ⋂ x ∈ s, f x :=
sInf_image
#align set.sInter_image Set.sInter_image
@[simp]
theorem sUnion_range (f : ι → Set β) : ⋃₀range f = ⋃ x, f x :=
rfl
#align set.sUnion_range Set.sUnion_range
@[simp]
theorem sInter_range (f : ι → Set β) : ⋂₀ range f = ⋂ x, f x :=
rfl
#align set.sInter_range Set.sInter_range
theorem iUnion_eq_univ_iff {f : ι → Set α} : ⋃ i, f i = univ ↔ ∀ x, ∃ i, x ∈ f i := by
simp only [eq_univ_iff_forall, mem_iUnion]
#align set.Union_eq_univ_iff Set.iUnion_eq_univ_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem iUnion₂_eq_univ_iff {s : ∀ i, κ i → Set α} :
⋃ (i) (j), s i j = univ ↔ ∀ a, ∃ i j, a ∈ s i j :=
by simp only [iUnion_eq_univ_iff, mem_iUnion]
#align set.Union₂_eq_univ_iff Set.iUnion₂_eq_univ_iff
theorem sUnion_eq_univ_iff {c : Set (Set α)} : ⋃₀c = univ ↔ ∀ a, ∃ b ∈ c, a ∈ b := by
simp only [eq_univ_iff_forall, mem_sUnion]
#align set.sUnion_eq_univ_iff Set.sUnion_eq_univ_iff
-- classical
theorem iInter_eq_empty_iff {f : ι → Set α} : ⋂ i, f i = ∅ ↔ ∀ x, ∃ i, x ∉ f i := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.Inter_eq_empty_iff Set.iInter_eq_empty_iff
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
theorem iInter₂_eq_empty_iff {s : ∀ i, κ i → Set α} :
⋂ (i) (j), s i j = ∅ ↔ ∀ a, ∃ i j, a ∉ s i j := by
simp only [eq_empty_iff_forall_not_mem, mem_iInter, not_forall]
#align set.Inter₂_eq_empty_iff Set.iInter₂_eq_empty_iff
-- classical
theorem sInter_eq_empty_iff {c : Set (Set α)} : ⋂₀ c = ∅ ↔ ∀ a, ∃ b ∈ c, a ∉ b := by
simp [Set.eq_empty_iff_forall_not_mem]
#align set.sInter_eq_empty_iff Set.sInter_eq_empty_iff
-- classical
@[simp]
theorem nonempty_iInter {f : ι → Set α} : (⋂ i, f i).Nonempty ↔ ∃ x, ∀ i, x ∈ f i := by
simp [nonempty_iff_ne_empty, iInter_eq_empty_iff]
#align set.nonempty_Inter Set.nonempty_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
-- classical
--Porting note: removing `simp`. `simp` can prove it
theorem nonempty_iInter₂ {s : ∀ i, κ i → Set α} :
(⋂ (i) (j), s i j).Nonempty ↔ ∃ a, ∀ i j, a ∈ s i j := by
simp
#align set.nonempty_Inter₂ Set.nonempty_iInter₂
-- classical
@[simp]
theorem nonempty_sInter {c : Set (Set α)} : (⋂₀ c).Nonempty ↔ ∃ a, ∀ b ∈ c, a ∈ b := by
simp [nonempty_iff_ne_empty, sInter_eq_empty_iff]
#align set.nonempty_sInter Set.nonempty_sInter
-- classical
theorem compl_sUnion (S : Set (Set α)) : (⋃₀S)ᶜ = ⋂₀ (compl '' S) :=
ext fun x => by simp
#align set.compl_sUnion Set.compl_sUnion
-- classical
theorem sUnion_eq_compl_sInter_compl (S : Set (Set α)) : ⋃₀S = (⋂₀ (compl '' S))ᶜ := by
rw [← compl_compl (⋃₀S), compl_sUnion]
#align set.sUnion_eq_compl_sInter_compl Set.sUnion_eq_compl_sInter_compl
-- classical
theorem compl_sInter (S : Set (Set α)) : (⋂₀ S)ᶜ = ⋃₀(compl '' S) := by
rw [sUnion_eq_compl_sInter_compl, compl_compl_image]
#align set.compl_sInter Set.compl_sInter
-- classical
theorem sInter_eq_compl_sUnion_compl (S : Set (Set α)) : ⋂₀ S = (⋃₀(compl '' S))ᶜ := by
rw [← compl_compl (⋂₀ S), compl_sInter]
#align set.sInter_eq_compl_sUnion_compl Set.sInter_eq_compl_sUnion_compl
theorem inter_empty_of_inter_sUnion_empty {s t : Set α} {S : Set (Set α)} (hs : t ∈ S)
(h : s ∩ ⋃₀S = ∅) : s ∩ t = ∅ :=
eq_empty_of_subset_empty <| by
rw [← h]; exact inter_subset_inter_right _ (subset_sUnion_of_mem hs)
#align set.inter_empty_of_inter_sUnion_empty Set.inter_empty_of_inter_sUnion_empty
theorem range_sigma_eq_iUnion_range {γ : α → Type*} (f : Sigma γ → β) :
range f = ⋃ a, range fun b => f ⟨a, b⟩ :=
Set.ext <| by simp
#align set.range_sigma_eq_Union_range Set.range_sigma_eq_iUnion_range
theorem iUnion_eq_range_sigma (s : α → Set β) : ⋃ i, s i = range fun a : Σi, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_sigma Set.iUnion_eq_range_sigma
theorem iUnion_eq_range_psigma (s : ι → Set β) : ⋃ i, s i = range fun a : Σ'i, s i => a.2 := by
simp [Set.ext_iff]
#align set.Union_eq_range_psigma Set.iUnion_eq_range_psigma
theorem iUnion_image_preimage_sigma_mk_eq_self {ι : Type*} {σ : ι → Type*} (s : Set (Sigma σ)) :
⋃ i, Sigma.mk i '' (Sigma.mk i ⁻¹' s) = s := by
ext x
simp only [mem_iUnion, mem_image, mem_preimage]
constructor
· rintro ⟨i, a, h, rfl⟩
exact h
· intro h
cases' x with i a
exact ⟨i, a, h, rfl⟩
#align set.Union_image_preimage_sigma_mk_eq_self Set.iUnion_image_preimage_sigma_mk_eq_self
theorem Sigma.univ (X : α → Type*) : (Set.univ : Set (Σa, X a)) = ⋃ a, range (Sigma.mk a) :=
Set.ext fun x =>
iff_of_true trivial ⟨range (Sigma.mk x.1), Set.mem_range_self _, x.2, Sigma.eta x⟩
#align set.sigma.univ Set.Sigma.univ
alias sUnion_mono := sUnion_subset_sUnion
#align set.sUnion_mono Set.sUnion_mono
theorem iUnion_subset_iUnion_const {s : Set α} (h : ι → ι₂) : ⋃ _ : ι, s ⊆ ⋃ _ : ι₂, s :=
@iSup_const_mono (Set α) ι ι₂ _ s h
#align set.Union_subset_Union_const Set.iUnion_subset_iUnion_const
@[simp]
theorem iUnion_singleton_eq_range {α β : Type*} (f : α → β) : ⋃ x : α, {f x} = range f := by
ext x
simp [@eq_comm _ x]
#align set.Union_singleton_eq_range Set.iUnion_singleton_eq_range
theorem iUnion_of_singleton (α : Type*) : (⋃ x, {x} : Set α) = univ := by simp [Set.ext_iff]
#align set.Union_of_singleton Set.iUnion_of_singleton
theorem iUnion_of_singleton_coe (s : Set α) : ⋃ i : s, ({(i : α)} : Set α) = s := by simp
#align set.Union_of_singleton_coe Set.iUnion_of_singleton_coe
theorem sUnion_eq_biUnion {s : Set (Set α)} : ⋃₀s = ⋃ (i : Set α) (_ : i ∈ s), i := by
rw [← sUnion_image, image_id']
#align set.sUnion_eq_bUnion Set.sUnion_eq_biUnion
theorem sInter_eq_biInter {s : Set (Set α)} : ⋂₀ s = ⋂ (i : Set α) (_ : i ∈ s), i := by
rw [← sInter_image, image_id']
#align set.sInter_eq_bInter Set.sInter_eq_biInter
theorem sUnion_eq_iUnion {s : Set (Set α)} : ⋃₀s = ⋃ i : s, i := by
simp only [← sUnion_range, Subtype.range_coe]
#align set.sUnion_eq_Union Set.sUnion_eq_iUnion
theorem sInter_eq_iInter {s : Set (Set α)} : ⋂₀ s = ⋂ i : s, i := by
simp only [← sInter_range, Subtype.range_coe]
#align set.sInter_eq_Inter Set.sInter_eq_iInter
@[simp]
theorem iUnion_of_empty [IsEmpty ι] (s : ι → Set α) : ⋃ i, s i = ∅ :=
iSup_of_empty _
#align set.Union_of_empty Set.iUnion_of_empty
@[simp]
theorem iInter_of_empty [IsEmpty ι] (s : ι → Set α) : ⋂ i, s i = univ :=
iInf_of_empty _
#align set.Inter_of_empty Set.iInter_of_empty
theorem union_eq_iUnion {s₁ s₂ : Set α} : s₁ ∪ s₂ = ⋃ b : Bool, cond b s₁ s₂ :=
sup_eq_iSup s₁ s₂
#align set.union_eq_Union Set.union_eq_iUnion
theorem inter_eq_iInter {s₁ s₂ : Set α} : s₁ ∩ s₂ = ⋂ b : Bool, cond b s₁ s₂ :=
inf_eq_iInf s₁ s₂
#align set.inter_eq_Inter Set.inter_eq_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_union_sInter {S T : Set (Set α)} :
⋂₀ S ∪ ⋂₀ T = ⋂ p ∈ S ×ˢ T, (p : Set α × Set α).1 ∪ p.2 :=
sInf_sup_sInf
#align set.sInter_union_sInter Set.sInter_union_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_inter_sUnion {s t : Set (Set α)} :
⋃₀s ∩ ⋃₀t = ⋃ p ∈ s ×ˢ t, (p : Set α × Set α).1 ∩ p.2 :=
sSup_inf_sSup
#align set.sUnion_inter_sUnion Set.sUnion_inter_sUnion
theorem biUnion_iUnion (s : ι → Set α) (t : α → Set β) :
⋃ x ∈ ⋃ i, s i, t x = ⋃ (i) (x ∈ s i), t x := by simp [@iUnion_comm _ ι]
#align set.bUnion_Union Set.biUnion_iUnion
theorem biInter_iUnion (s : ι → Set α) (t : α → Set β) :
⋂ x ∈ ⋃ i, s i, t x = ⋂ (i) (x ∈ s i), t x := by simp [@iInter_comm _ ι]
#align set.bInter_Union Set.biInter_iUnion
theorem sUnion_iUnion (s : ι → Set (Set α)) : ⋃₀⋃ i, s i = ⋃ i, ⋃₀s i := by
simp only [sUnion_eq_biUnion, biUnion_iUnion]
#align set.sUnion_Union Set.sUnion_iUnion
theorem sInter_iUnion (s : ι → Set (Set α)) : ⋂₀ ⋃ i, s i = ⋂ i, ⋂₀ s i := by
simp only [sInter_eq_biInter, biInter_iUnion]
#align set.sInter_Union Set.sInter_iUnion
theorem iUnion_range_eq_sUnion {α β : Type*} (C : Set (Set α)) {f : ∀ s : C, β → (s : Type _)}
(hf : ∀ s : C, Surjective (f s)) : ⋃ y : β, range (fun s : C => (f s y).val) = ⋃₀C := by
ext x; constructor
· rintro ⟨s, ⟨y, rfl⟩, ⟨s, hs⟩, rfl⟩
refine' ⟨_, hs, _⟩
exact (f ⟨s, hs⟩ y).2
· rintro ⟨s, hs, hx⟩
cases' hf ⟨s, hs⟩ ⟨x, hx⟩ with y hy
refine' ⟨_, ⟨y, rfl⟩, ⟨s, hs⟩, _⟩
exact congr_arg Subtype.val hy
#align set.Union_range_eq_sUnion Set.iUnion_range_eq_sUnion
theorem iUnion_range_eq_iUnion (C : ι → Set α) {f : ∀ x : ι, β → C x}
(hf : ∀ x : ι, Surjective (f x)) : ⋃ y : β, range (fun x : ι => (f x y).val) = ⋃ x, C x := by
ext x; rw [mem_iUnion, mem_iUnion]; constructor
· rintro ⟨y, i, rfl⟩
exact ⟨i, (f i y).2⟩
· rintro ⟨i, hx⟩
cases' hf i ⟨x, hx⟩ with y hy
exact ⟨y, i, congr_arg Subtype.val hy⟩
#align set.Union_range_eq_Union Set.iUnion_range_eq_iUnion
theorem union_distrib_iInter_left (s : ι → Set α) (t : Set α) : (t ∪ ⋂ i, s i) = ⋂ i, t ∪ s i :=
sup_iInf_eq _ _
#align set.union_distrib_Inter_left Set.union_distrib_iInter_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_left (s : Set α) (t : ∀ i, κ i → Set α) :
(s ∪ ⋂ (i) (j), t i j) = ⋂ (i) (j), s ∪ t i j := by simp_rw [union_distrib_iInter_left]
#align set.union_distrib_Inter₂_left Set.union_distrib_iInter₂_left
theorem union_distrib_iInter_right (s : ι → Set α) (t : Set α) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t :=
iInf_sup_eq _ _
#align set.union_distrib_Inter_right Set.union_distrib_iInter_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem union_distrib_iInter₂_right (s : ∀ i, κ i → Set α) (t : Set α) :
(⋂ (i) (j), s i j) ∪ t = ⋂ (i) (j), s i j ∪ t := by simp_rw [union_distrib_iInter_right]
#align set.union_distrib_Inter₂_right Set.union_distrib_iInter₂_right
section Function
/-! ### `mapsTo` -/
theorem mapsTo_sUnion {S : Set (Set α)} {t : Set β} {f : α → β} (H : ∀ s ∈ S, MapsTo f s t) :
MapsTo f (⋃₀S) t := fun _ ⟨s, hs, hx⟩ => H s hs hx
#align set.maps_to_sUnion Set.mapsTo_sUnion
theorem mapsTo_iUnion {s : ι → Set α} {t : Set β} {f : α → β} (H : ∀ i, MapsTo f (s i) t) :
MapsTo f (⋃ i, s i) t :=
mapsTo_sUnion <| forall_range_iff.2 H
#align set.maps_to_Union Set.mapsTo_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂ {s : ∀ i, κ i → Set α} {t : Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) t) : MapsTo f (⋃ (i) (j), s i j) t :=
mapsTo_iUnion fun i => mapsTo_iUnion (H i)
#align set.maps_to_Union₂ Set.mapsTo_iUnion₂
theorem mapsTo_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋃ i, s i) (⋃ i, t i) :=
mapsTo_iUnion fun i => (H i).mono (Subset.refl _) (subset_iUnion t i)
#align set.maps_to_Union_Union Set.mapsTo_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
mapsTo_iUnion_iUnion fun i => mapsTo_iUnion_iUnion (H i)
#align set.maps_to_Union₂_Union₂ Set.mapsTo_iUnion₂_iUnion₂
theorem mapsTo_sInter {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, MapsTo f s t) :
MapsTo f s (⋂₀ T) := fun _ hx t ht => H t ht hx
#align set.maps_to_sInter Set.mapsTo_sInter
theorem mapsTo_iInter {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, MapsTo f s (t i)) :
MapsTo f s (⋂ i, t i) := fun _ hx => mem_iInter.2 fun i => H i hx
#align set.maps_to_Inter Set.mapsTo_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f s (t i j)) : MapsTo f s (⋂ (i) (j), t i j) :=
mapsTo_iInter fun i => mapsTo_iInter (H i)
#align set.maps_to_Inter₂ Set.mapsTo_iInter₂
theorem mapsTo_iInter_iInter {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, MapsTo f (s i) (t i)) : MapsTo f (⋂ i, s i) (⋂ i, t i) :=
mapsTo_iInter fun i => (H i).mono (iInter_subset s i) (Subset.refl _)
#align set.maps_to_Inter_Inter Set.mapsTo_iInter_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem mapsTo_iInter₂_iInter₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, MapsTo f (s i j) (t i j)) : MapsTo f (⋂ (i) (j), s i j) (⋂ (i) (j), t i j) :=
mapsTo_iInter_iInter fun i => mapsTo_iInter_iInter (H i)
#align set.maps_to_Inter₂_Inter₂ Set.mapsTo_iInter₂_iInter₂
theorem image_iInter_subset (s : ι → Set α) (f : α → β) : (f '' ⋂ i, s i) ⊆ ⋂ i, f '' s i :=
(mapsTo_iInter_iInter fun i => mapsTo_image f (s i)).image_subset
#align set.image_Inter_subset Set.image_iInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂_subset (s : ∀ i, κ i → Set α) (f : α → β) :
(f '' ⋂ (i) (j), s i j) ⊆ ⋂ (i) (j), f '' s i j :=
(mapsTo_iInter₂_iInter₂ fun i hi => mapsTo_image f (s i hi)).image_subset
#align set.image_Inter₂_subset Set.image_iInter₂_subset
theorem image_sInter_subset (S : Set (Set α)) (f : α → β) : f '' ⋂₀ S ⊆ ⋂ s ∈ S, f '' s := by
rw [sInter_eq_biInter]
apply image_iInter₂_subset
#align set.image_sInter_subset Set.image_sInter_subset
/-! ### `restrictPreimage` -/
section
open Function
variable (s : Set β) {f : α → β} {U : ι → Set β} (hU : iUnion U = univ)
theorem injective_iff_injective_of_iUnion_eq_univ :
Injective f ↔ ∀ i, Injective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_injective H, fun H x y e => _⟩
obtain ⟨i, hi⟩ := Set.mem_iUnion.mp
(show f x ∈ Set.iUnion U by rw [hU]; triv)
injection @H i ⟨x, hi⟩ ⟨y, show f y ∈ U i from e ▸ hi⟩ (Subtype.ext e)
#align set.injective_iff_injective_of_Union_eq_univ Set.injective_iff_injective_of_iUnion_eq_univ
theorem surjective_iff_surjective_of_iUnion_eq_univ :
Surjective f ↔ ∀ i, Surjective ((U i).restrictPreimage f) := by
refine' ⟨fun H i => (U i).restrictPreimage_surjective H, fun H x => _⟩
obtain ⟨i, hi⟩ :=
Set.mem_iUnion.mp
(show x ∈ Set.iUnion U by rw [hU]; triv)
exact ⟨_, congr_arg Subtype.val (H i ⟨x, hi⟩).choose_spec⟩
#align set.surjective_iff_surjective_of_Union_eq_univ Set.surjective_iff_surjective_of_iUnion_eq_univ
theorem bijective_iff_bijective_of_iUnion_eq_univ :
Bijective f ↔ ∀ i, Bijective ((U i).restrictPreimage f) := by
rw [Bijective, injective_iff_injective_of_iUnion_eq_univ hU,
surjective_iff_surjective_of_iUnion_eq_univ hU]
simp [Bijective, forall_and]
#align set.bijective_iff_bijective_of_Union_eq_univ Set.bijective_iff_bijective_of_iUnion_eq_univ
end
/-! ### `InjOn` -/
theorem InjOn.image_iInter_eq [Nonempty ι] {s : ι → Set α} {f : α → β} (h : InjOn f (⋃ i, s i)) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
inhabit ι
refine' Subset.antisymm (image_iInter_subset s f) fun y hy => _
simp only [mem_iInter, mem_image_iff_bex] at hy
choose x hx hy using hy
refine' ⟨x default, mem_iInter.2 fun i => _, hy _⟩
suffices x default = x i by
rw [this]
apply hx
replace hx : ∀ i, x i ∈ ⋃ j, s j := fun i => (subset_iUnion _ _) (hx i)
apply h (hx _) (hx _)
simp only [hy]
#align set.inj_on.image_Inter_eq Set.InjOn.image_iInter_eq
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i hi) -/
theorem InjOn.image_biInter_eq {p : ι → Prop} {s : ∀ i, p i → Set α} (hp : ∃ i, p i)
{f : α → β} (h : InjOn f (⋃ (i) (hi), s i hi)) :
(f '' ⋂ (i) (hi), s i hi) = ⋂ (i) (hi), f '' s i hi := by
simp only [iInter, iInf_subtype']
haveI : Nonempty { i // p i } := nonempty_subtype.2 hp
apply InjOn.image_iInter_eq
simpa only [iUnion, iSup_subtype'] using h
#align set.inj_on.image_bInter_eq Set.InjOn.image_biInter_eq
theorem image_iInter {f : α → β} (hf : Bijective f) (s : ι → Set α) :
(f '' ⋂ i, s i) = ⋂ i, f '' s i := by
cases isEmpty_or_nonempty ι
· simp_rw [iInter_of_empty, image_univ_of_surjective hf.surjective]
· exact (hf.injective.injOn _).image_iInter_eq
#align set.image_Inter Set.image_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iInter₂ {f : α → β} (hf : Bijective f) (s : ∀ i, κ i → Set α) :
(f '' ⋂ (i) (j), s i j) = ⋂ (i) (j), f '' s i j := by simp_rw [image_iInter hf]
#align set.image_Inter₂ Set.image_iInter₂
theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
(hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i) := by
intro x hx y hy hxy
rcases mem_iUnion.1 hx with ⟨i, hx⟩
rcases mem_iUnion.1 hy with ⟨j, hy⟩
rcases hs i j with ⟨k, hi, hj⟩
exact hf k (hi hx) (hj hy) hxy
#align set.inj_on_Union_of_directed Set.inj_on_iUnion_of_directed
/-! ### `SurjOn` -/
theorem surjOn_sUnion {s : Set α} {T : Set (Set β)} {f : α → β} (H : ∀ t ∈ T, SurjOn f s t) :
SurjOn f s (⋃₀T) := fun _ ⟨t, ht, hx⟩ => H t ht hx
#align set.surj_on_sUnion Set.surjOn_sUnion
theorem surjOn_iUnion {s : Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, SurjOn f s (t i)) :
SurjOn f s (⋃ i, t i) :=
surjOn_sUnion <| forall_range_iff.2 H
#align set.surj_on_Union Set.surjOn_iUnion
theorem surjOn_iUnion_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) : SurjOn f (⋃ i, s i) (⋃ i, t i) :=
surjOn_iUnion fun i => (H i).mono (subset_iUnion _ _) (Subset.refl _)
#align set.surj_on_Union_Union Set.surjOn_iUnion_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f s (t i j)) : SurjOn f s (⋃ (i) (j), t i j) :=
surjOn_iUnion fun i => surjOn_iUnion (H i)
#align set.surj_on_Union₂ Set.surjOn_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem surjOn_iUnion₂_iUnion₂ {s : ∀ i, κ i → Set α} {t : ∀ i, κ i → Set β} {f : α → β}
(H : ∀ i j, SurjOn f (s i j) (t i j)) : SurjOn f (⋃ (i) (j), s i j) (⋃ (i) (j), t i j) :=
surjOn_iUnion_iUnion fun i => surjOn_iUnion_iUnion (H i)
#align set.surj_on_Union₂_Union₂ Set.surjOn_iUnion₂_iUnion₂
theorem surjOn_iInter [Nonempty ι] {s : ι → Set α} {t : Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) t) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) t := by
intro y hy
rw [Hinj.image_iInter_eq, mem_iInter]
exact fun i => H i hy
#align set.surj_on_Inter Set.surjOn_iInter
theorem surjOn_iInter_iInter [Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) (⋂ i, t i) :=
surjOn_iInter (fun i => (H i).mono (Subset.refl _) (iInter_subset _ _)) Hinj
#align set.surj_on_Inter_Inter Set.surjOn_iInter_iInter
/-! ### `BijOn` -/
theorem bijOn_iUnion {s : ι → Set α} {t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i))
(Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
⟨mapsTo_iUnion_iUnion fun i => (H i).mapsTo, Hinj, surjOn_iUnion_iUnion fun i => (H i).surjOn⟩
#align set.bij_on_Union Set.bijOn_iUnion
theorem bijOn_iInter [hi : Nonempty ι] {s : ι → Set α} {t : ι → Set β} {f : α → β}
(H : ∀ i, BijOn f (s i) (t i)) (Hinj : InjOn f (⋃ i, s i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
⟨mapsTo_iInter_iInter fun i => (H i).mapsTo,
hi.elim fun i => (H i).injOn.mono (iInter_subset _ _),
surjOn_iInter_iInter (fun i => (H i).surjOn) Hinj⟩
#align set.bij_on_Inter Set.bijOn_iInter
theorem bijOn_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {t : ι → Set β}
{f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋃ i, s i) (⋃ i, t i) :=
bijOn_iUnion H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Union_of_directed Set.bijOn_iUnion_of_directed
theorem bijOn_iInter_of_directed [Nonempty ι] {s : ι → Set α} (hs : Directed (· ⊆ ·) s)
{t : ι → Set β} {f : α → β} (H : ∀ i, BijOn f (s i) (t i)) : BijOn f (⋂ i, s i) (⋂ i, t i) :=
bijOn_iInter H <| inj_on_iUnion_of_directed hs fun i => (H i).injOn
#align set.bij_on_Inter_of_directed Set.bijOn_iInter_of_directed
end Function
/-! ### `image`, `preimage` -/
section Image
theorem image_iUnion {f : α → β} {s : ι → Set α} : (f '' ⋃ i, s i) = ⋃ i, f '' s i := by
ext1 x
simp only [mem_image, mem_iUnion, ← exists_and_right, ← exists_and_left]
--Porting note: `exists_swap` causes a `simp` loop in Lean4 so we use `rw` instead.
rw [exists_swap]
#align set.image_Union Set.image_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image_iUnion₂ (f : α → β) (s : ∀ i, κ i → Set α) :
(f '' ⋃ (i) (j), s i j) = ⋃ (i) (j), f '' s i j := by simp_rw [image_iUnion]
#align set.image_Union₂ Set.image_iUnion₂
theorem univ_subtype {p : α → Prop} : (univ : Set (Subtype p)) = ⋃ (x) (h : p x), {⟨x, h⟩} :=
Set.ext fun ⟨x, h⟩ => by simp [h]
#align set.univ_subtype Set.univ_subtype
theorem range_eq_iUnion {ι} (f : ι → α) : range f = ⋃ i, {f i} :=
Set.ext fun a => by simp [@eq_comm α a]
#align set.range_eq_Union Set.range_eq_iUnion
theorem image_eq_iUnion (f : α → β) (s : Set α) : f '' s = ⋃ i ∈ s, {f i} :=
Set.ext fun b => by simp [@eq_comm β b]
#align set.image_eq_Union Set.image_eq_iUnion
theorem biUnion_range {f : ι → α} {g : α → Set β} : ⋃ x ∈ range f, g x = ⋃ y, g (f y) :=
iSup_range
#align set.bUnion_range Set.biUnion_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iUnion_iUnion_eq' {f : ι → α} {g : α → Set β} :
⋃ (x) (y) (_ : f y = x), g x = ⋃ y, g (f y) := by simpa using biUnion_range
#align set.Union_Union_eq' Set.iUnion_iUnion_eq'
theorem biInter_range {f : ι → α} {g : α → Set β} : ⋂ x ∈ range f, g x = ⋂ y, g (f y) :=
iInf_range
#align set.bInter_range Set.biInter_range
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (x y) -/
@[simp]
theorem iInter_iInter_eq' {f : ι → α} {g : α → Set β} :
⋂ (x) (y) (_ : f y = x), g x = ⋂ y, g (f y) := by simpa using biInter_range
#align set.Inter_Inter_eq' Set.iInter_iInter_eq'
variable {s : Set γ} {f : γ → α} {g : α → Set β}
theorem biUnion_image : ⋃ x ∈ f '' s, g x = ⋃ y ∈ s, g (f y) :=
iSup_image
#align set.bUnion_image Set.biUnion_image
theorem biInter_image : ⋂ x ∈ f '' s, g x = ⋂ y ∈ s, g (f y) :=
iInf_image
#align set.bInter_image Set.biInter_image
end Image
section Preimage
theorem monotone_preimage {f : α → β} : Monotone (preimage f) := fun _ _ h => preimage_mono h
#align set.monotone_preimage Set.monotone_preimage
@[simp]
theorem preimage_iUnion {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋃ i, s i) = ⋃ i, f ⁻¹' s i :=
Set.ext <| by simp [preimage]
#align set.preimage_Union Set.preimage_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iUnion₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋃ (i) (j), s i j) = ⋃ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iUnion]
#align set.preimage_Union₂ Set.preimage_iUnion₂
@[simp]
theorem preimage_sUnion {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋃₀s = ⋃ t ∈ s, f ⁻¹' t := by
rw [sUnion_eq_biUnion, preimage_iUnion₂]
#align set.preimage_sUnion Set.preimage_sUnion
theorem preimage_iInter {f : α → β} {s : ι → Set β} : (f ⁻¹' ⋂ i, s i) = ⋂ i, f ⁻¹' s i := by
ext; simp
#align set.preimage_Inter Set.preimage_iInter
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem preimage_iInter₂ {f : α → β} {s : ∀ i, κ i → Set β} :
(f ⁻¹' ⋂ (i) (j), s i j) = ⋂ (i) (j), f ⁻¹' s i j := by simp_rw [preimage_iInter]
#align set.preimage_Inter₂ Set.preimage_iInter₂
@[simp]
theorem preimage_sInter {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋂₀ s = ⋂ t ∈ s, f ⁻¹' t := by
rw [sInter_eq_biInter, preimage_iInter₂]
#align set.preimage_sInter Set.preimage_sInter
@[simp]
theorem biUnion_preimage_singleton (f : α → β) (s : Set β) : ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s := by
rw [← preimage_iUnion₂, biUnion_of_singleton]
#align set.bUnion_preimage_singleton Set.biUnion_preimage_singleton
theorem biUnion_range_preimage_singleton (f : α → β) : ⋃ y ∈ range f, f ⁻¹' {y} = univ := by
rw [biUnion_preimage_singleton, preimage_range]
#align set.bUnion_range_preimage_singleton Set.biUnion_range_preimage_singleton
end Preimage
section Prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion {s : Set α} {t : ι → Set β} : (s ×ˢ ⋃ i, t i) = ⋃ i, s ×ˢ t i := by
ext
simp
#align set.prod_Union Set.prod_iUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_iUnion₂ {s : Set α} {t : ∀ i, κ i → Set β} :
(s ×ˢ ⋃ (i) (j), t i j) = ⋃ (i) (j), s ×ˢ t i j := by simp_rw [prod_iUnion]
#align set.prod_Union₂ Set.prod_iUnion₂
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sUnion {s : Set α} {C : Set (Set β)} : s ×ˢ ⋃₀C = ⋃₀((fun t => s ×ˢ t) '' C) := by
simp_rw [sUnion_eq_biUnion, biUnion_image, prod_iUnion₂]
#align set.prod_sUnion Set.prod_sUnion
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_const {s : ι → Set α} {t : Set β} : (⋃ i, s i) ×ˢ t = ⋃ i, s i ×ˢ t := by
ext
simp
#align set.Union_prod_const Set.iUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion₂_prod_const {s : ∀ i, κ i → Set α} {t : Set β} :
(⋃ (i) (j), s i j) ×ˢ t = ⋃ (i) (j), s i j ×ˢ t := by simp_rw [iUnion_prod_const]
#align set.Union₂_prod_const Set.iUnion₂_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sUnion_prod_const {C : Set (Set α)} {t : Set β} :
⋃₀C ×ˢ t = ⋃₀((fun s : Set α => s ×ˢ t) '' C) := by
simp only [sUnion_eq_biUnion, iUnion₂_prod_const, biUnion_image]
#align set.sUnion_prod_const Set.sUnion_prod_const
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod {ι ι' α β} (s : ι → Set α) (t : ι' → Set β) :
⋃ x : ι × ι', s x.1 ×ˢ t x.2 = (⋃ i : ι, s i) ×ˢ ⋃ i : ι', t i := by
ext
simp
#align set.Union_prod Set.iUnion_prod
/-- Analogue of `iSup_prod` for sets. -/
lemma iUnion_prod' (f : β × γ → Set α) : ⋃ x : β × γ, f x = ⋃ (i : β) (j : γ), f (i, j) :=
iSup_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem iUnion_prod_of_monotone [SemilatticeSup α] {s : α → Set β} {t : α → Set γ} (hs : Monotone s)
(ht : Monotone t) : ⋃ x, s x ×ˢ t x = (⋃ x, s x) ×ˢ ⋃ x, t x := by
ext ⟨z, w⟩; simp only [mem_prod, mem_iUnion, exists_imp, and_imp, iff_def]; constructor
· intro x hz hw
exact ⟨⟨x, hz⟩, x, hw⟩
· intro x hz x' hw
exact ⟨x ⊔ x', hs le_sup_left hz, ht le_sup_right hw⟩
#align set.Union_prod_of_monotone Set.iUnion_prod_of_monotone
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter_subset (S : Set (Set α)) (T : Set (Set β)) :
⋂₀ S ×ˢ ⋂₀ T ⊆ ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 :=
subset_iInter₂ fun x hx _ hy => ⟨hy.1 x.1 hx.1, hy.2 x.2 hx.2⟩
#align set.sInter_prod_sInter_subset Set.sInter_prod_sInter_subset
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod_sInter {S : Set (Set α)} {T : Set (Set β)} (hS : S.Nonempty) (hT : T.Nonempty) :
⋂₀ S ×ˢ ⋂₀ T = ⋂ r ∈ S ×ˢ T, r.1 ×ˢ r.2 := by
obtain ⟨s₁, h₁⟩ := hS
obtain ⟨s₂, h₂⟩ := hT
refine' Set.Subset.antisymm (sInter_prod_sInter_subset S T) fun x hx => _
rw [mem_iInter₂] at hx
exact ⟨fun s₀ h₀ => (hx (s₀, s₂) ⟨h₀, h₂⟩).1, fun s₀ h₀ => (hx (s₁, s₀) ⟨h₁, h₀⟩).2⟩
#align set.sInter_prod_sInter Set.sInter_prod_sInter
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem sInter_prod {S : Set (Set α)} (hS : S.Nonempty) (t : Set β) :
⋂₀ S ×ˢ t = ⋂ s ∈ S, s ×ˢ t := by
rw [← sInter_singleton t, sInter_prod_sInter hS (singleton_nonempty t), sInter_singleton]
simp_rw [prod_singleton, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.sInter_prod Set.sInter_prod
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
theorem prod_sInter {T : Set (Set β)} (hT : T.Nonempty) (s : Set α) :
s ×ˢ ⋂₀ T = ⋂ t ∈ T, s ×ˢ t := by
rw [← sInter_singleton s, sInter_prod_sInter (singleton_nonempty s) hT, sInter_singleton]
simp_rw [singleton_prod, mem_image, iInter_exists, biInter_and', iInter_iInter_eq_right]
#align set.prod_sInter Set.prod_sInter
end Prod
section Image2
variable (f : α → β → γ) {s : Set α} {t : Set β}
/-- The `Set.image2` version of `Set.image_eq_iUnion` -/
theorem image2_eq_iUnion (s : Set α) (t : Set β) : image2 f s t = ⋃ (i ∈ s) (j ∈ t), {f i j} := by
ext; simp [eq_comm]
#align set.image2_eq_Union Set.image2_eq_iUnion
theorem iUnion_image_left : ⋃ a ∈ s, f a '' t = image2 f s t := by
simp only [image2_eq_iUnion, image_eq_iUnion]
#align set.Union_image_left Set.iUnion_image_left
theorem iUnion_image_right : ⋃ b ∈ t, (f · b) '' s = image2 f s t := by
rw [image2_swap, iUnion_image_left]
#align set.Union_image_right Set.iUnion_image_right
theorem image2_iUnion_left (s : ι → Set α) (t : Set β) :
image2 f (⋃ i, s i) t = ⋃ i, image2 f (s i) t := by
simp only [← image_prod, iUnion_prod_const, image_iUnion]
#align set.image2_Union_left Set.image2_iUnion_left
theorem image2_iUnion_right (s : Set α) (t : ι → Set β) :
image2 f s (⋃ i, t i) = ⋃ i, image2 f s (t i) := by
simp only [← image_prod, prod_iUnion, image_iUnion]
#align set.image2_Union_right Set.image2_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋃ (i) (j), s i j) t = ⋃ (i) (j), image2 f (s i j) t := by simp_rw [image2_iUnion_left]
#align set.image2_Union₂_left Set.image2_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iUnion₂_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋃ (i) (j), t i j) = ⋃ (i) (j), image2 f s (t i j) :=
by simp_rw [image2_iUnion_right]
#align set.image2_Union₂_right Set.image2_iUnion₂_right
theorem image2_iInter_subset_left (s : ι → Set α) (t : Set β) :
image2 f (⋂ i, s i) t ⊆ ⋂ i, image2 f (s i) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem (hx _) hy
#align set.image2_Inter_subset_left Set.image2_iInter_subset_left
theorem image2_iInter_subset_right (s : Set α) (t : ι → Set β) :
image2 f s (⋂ i, t i) ⊆ ⋂ i, image2 f s (t i) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i => mem_image2_of_mem hx (hy _)
#align set.image2_Inter_subset_right Set.image2_iInter_subset_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_left (s : ∀ i, κ i → Set α) (t : Set β) :
image2 f (⋂ (i) (j), s i j) t ⊆ ⋂ (i) (j), image2 f (s i j) t := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem (hx _ _) hy
#align set.image2_Inter₂_subset_left Set.image2_iInter₂_subset_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem image2_iInter₂_subset_right (s : Set α) (t : ∀ i, κ i → Set β) :
image2 f s (⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), image2 f s (t i j) := by
simp_rw [image2_subset_iff, mem_iInter]
exact fun x hx y hy i j => mem_image2_of_mem hx (hy _ _)
#align set.image2_Inter₂_subset_right Set.image2_iInter₂_subset_right
theorem prod_eq_biUnion_left : s ×ˢ t = ⋃ a ∈ s, (fun b => (a, b)) '' t := by
rw [iUnion_image_left, image2_mk_eq_prod]
#align set.prod_eq_bUnion_left Set.prod_eq_biUnion_left
theorem prod_eq_biUnion_right : s ×ˢ t = ⋃ b ∈ t, (fun a => (a, b)) '' s := by
rw [iUnion_image_right, image2_mk_eq_prod]
#align set.prod_eq_bUnion_right Set.prod_eq_biUnion_right
end Image2
section Seq
/-- Given a set `s` of functions `α → β` and `t : Set α`, `seq s t` is the union of `f '' t` over
all `f ∈ s`. -/
def seq (s : Set (α → β)) (t : Set α) : Set β :=
{ b | ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b }
#align set.seq Set.seq
@[simp]
theorem mem_seq_iff {s : Set (α → β)} {t : Set α} {b : β} :
b ∈ seq s t ↔ ∃ f ∈ s, ∃ a ∈ t, (f : α → β) a = b :=
Iff.rfl
#align set.mem_seq_iff Set.mem_seq_iff
lemma seq_eq_image2 (s : Set (α → β)) (t : Set α) : seq s t = image2 (fun f a ↦ f a) s t := by
ext; simp
theorem seq_def {s : Set (α → β)} {t : Set α} : seq s t = ⋃ f ∈ s, f '' t := by
rw [seq_eq_image2, iUnion_image_left]
#align set.seq_def Set.seq_def
theorem seq_subset {s : Set (α → β)} {t : Set α} {u : Set β} :
seq s t ⊆ u ↔ ∀ f ∈ s, ∀ a ∈ t, (f : α → β) a ∈ u := by
rw [seq_eq_image2, image2_subset_iff]
#align set.seq_subset Set.seq_subset
@[gcongr]
theorem seq_mono {s₀ s₁ : Set (α → β)} {t₀ t₁ : Set α} (hs : s₀ ⊆ s₁) (ht : t₀ ⊆ t₁) :
seq s₀ t₀ ⊆ seq s₁ t₁ := fun _ ⟨f, hf, a, ha, eq⟩ => ⟨f, hs hf, a, ht ha, eq⟩
#align set.seq_mono Set.seq_mono
theorem singleton_seq {f : α → β} {t : Set α} : Set.seq ({f} : Set (α → β)) t = f '' t := by
rw [seq_eq_image2, image2_singleton_left]
#align set.singleton_seq Set.singleton_seq
theorem seq_singleton {s : Set (α → β)} {a : α} : Set.seq s {a} = (fun f : α → β => f a) '' s := by
rw [seq_eq_image2, image2_singleton_right]
#align set.seq_singleton Set.seq_singleton
theorem seq_seq {s : Set (β → γ)} {t : Set (α → β)} {u : Set α} :
seq s (seq t u) = seq (seq ((· ∘ ·) '' s) t) u := by
simp only [seq_eq_image2, image2_image_left, image2_image2_left, image2_image2_right, comp_apply]
#align set.seq_seq Set.seq_seq
theorem image_seq {f : β → γ} {s : Set (α → β)} {t : Set α} :
f '' seq s t = seq ((f ∘ ·) '' s) t := by
rw [← singleton_seq, ← singleton_seq, seq_seq, image_singleton]
#align set.image_seq Set.image_seq
theorem prod_eq_seq {s : Set α} {t : Set β} : s ×ˢ t = (Prod.mk '' s).seq t := by
rw [seq_eq_image2, image2_image_left, image2_mk_eq_prod]
#align set.prod_eq_seq Set.prod_eq_seq
theorem prod_image_seq_comm (s : Set α) (t : Set β) :
(Prod.mk '' s).seq t = seq ((fun b a => (a, b)) '' t) s := by
rw [← prod_eq_seq, ← image_swap_prod, prod_eq_seq, image_seq, ← image_comp]; rfl
#align set.prod_image_seq_comm Set.prod_image_seq_comm
theorem image2_eq_seq (f : α → β → γ) (s : Set α) (t : Set β) : image2 f s t = seq (f '' s) t := by
rw [seq_eq_image2, image2_image_left]
#align set.image2_eq_seq Set.image2_eq_seq
end Seq
section Pi
variable {π : α → Type*}
theorem pi_def (i : Set α) (s : ∀ a, Set (π a)) : pi i s = ⋂ a ∈ i, eval a ⁻¹' s a := by
ext
simp
#align set.pi_def Set.pi_def
theorem univ_pi_eq_iInter (t : ∀ i, Set (π i)) : pi univ t = ⋂ i, eval i ⁻¹' t i := by
simp only [pi_def, iInter_true, mem_univ]
#align set.univ_pi_eq_Inter Set.univ_pi_eq_iInter
theorem pi_diff_pi_subset (i : Set α) (s t : ∀ a, Set (π a)) :
pi i s \ pi i t ⊆ ⋃ a ∈ i, eval a ⁻¹' (s a \ t a) := by
refine' diff_subset_comm.2 fun x hx a ha => _
simp only [mem_diff, mem_pi, mem_iUnion, not_exists, mem_preimage, not_and, not_not,
eval_apply] at hx
exact hx.2 _ ha (hx.1 _ ha)
#align set.pi_diff_pi_subset Set.pi_diff_pi_subset
theorem iUnion_univ_pi {ι : α → Type*} (t : (a : α) → ι a → Set (π a)) :
⋃ x : (a : α) → ι a, pi univ (fun a => t a (x a)) = pi univ fun a => ⋃ j : ι a, t a j := by
ext
simp [Classical.skolem]
#align set.Union_univ_pi Set.iUnion_univ_pi
end Pi
end Set
namespace Function
namespace Surjective
theorem iUnion_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋃ x, g (f x) = ⋃ y, g y :=
hf.iSup_comp g
#align function.surjective.Union_comp Function.Surjective.iUnion_comp
theorem iInter_comp {f : ι → ι₂} (hf : Surjective f) (g : ι₂ → Set α) : ⋂ x, g (f x) = ⋂ y, g y :=
hf.iInf_comp g
#align function.surjective.Inter_comp Function.Surjective.iInter_comp
end Surjective
end Function
/-!
### Disjoint sets
-/
section Disjoint
variable {s t u : Set α} {f : α → β}
namespace Set
@[simp]
theorem disjoint_iUnion_left {ι : Sort*} {s : ι → Set α} :
Disjoint (⋃ i, s i) t ↔ ∀ i, Disjoint (s i) t :=
iSup_disjoint_iff
#align set.disjoint_Union_left Set.disjoint_iUnion_left
@[simp]
theorem disjoint_iUnion_right {ι : Sort*} {s : ι → Set α} :
Disjoint t (⋃ i, s i) ↔ ∀ i, Disjoint t (s i) :=
disjoint_iSup_iff
#align set.disjoint_Union_right Set.disjoint_iUnion_right
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_left {s : ∀ i, κ i → Set α} {t : Set α} :
Disjoint (⋃ (i) (j), s i j) t ↔ ∀ i j, Disjoint (s i j) t :=
iSup₂_disjoint_iff
#align set.disjoint_Union₂_left Set.disjoint_iUnion₂_left
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
--Porting note: removing `simp`. `simp` can prove it
theorem disjoint_iUnion₂_right {s : Set α} {t : ∀ i, κ i → Set α} :
Disjoint s (⋃ (i) (j), t i j) ↔ ∀ i j, Disjoint s (t i j) :=
disjoint_iSup₂_iff
#align set.disjoint_Union₂_right Set.disjoint_iUnion₂_right
@[simp]
theorem disjoint_sUnion_left {S : Set (Set α)} {t : Set α} :
Disjoint (⋃₀S) t ↔ ∀ s ∈ S, Disjoint s t :=
sSup_disjoint_iff
#align set.disjoint_sUnion_left Set.disjoint_sUnion_left
@[simp]
theorem disjoint_sUnion_right {s : Set α} {S : Set (Set α)} :
Disjoint s (⋃₀S) ↔ ∀ t ∈ S, Disjoint s t :=
disjoint_sSup_iff
#align set.disjoint_sUnion_right Set.disjoint_sUnion_right
end Set
end Disjoint
/-! ### Intervals -/
namespace Set
lemma nonempty_iInter_Iic_iff [Preorder α] {f : ι → α} :
(⋂ i, Iic (f i)).Nonempty ↔ BddBelow (range f) := by
have : (⋂ (i : ι), Iic (f i)) = lowerBounds (range f) := by
ext c; simp [lowerBounds]
simp [this, BddBelow]
lemma nonempty_iInter_Ici_iff [Preorder α] {f : ι → α} :
(⋂ i, Ici (f i)).Nonempty ↔ BddAbove (range f) :=
nonempty_iInter_Iic_iff (α := αᵒᵈ)
variable [CompleteLattice α]
theorem Ici_iSup (f : ι → α) : Ici (⨆ i, f i) = ⋂ i, Ici (f i) :=
ext fun _ => by simp only [mem_Ici, iSup_le_iff, mem_iInter]
#align set.Ici_supr Set.Ici_iSup
theorem Iic_iInf (f : ι → α) : Iic (⨅ i, f i) = ⋂ i, Iic (f i) :=
ext fun _ => by simp only [mem_Iic, le_iInf_iff, mem_iInter]
#align set.Iic_infi Set.Iic_iInf
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Ici_iSup₂ (f : ∀ i, κ i → α) : Ici (⨆ (i) (j), f i j) = ⋂ (i) (j), Ici (f i j) := by
simp_rw [Ici_iSup]
#align set.Ici_supr₂ Set.Ici_iSup₂
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
/- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
theorem Iic_iInf₂ (f : ∀ i, κ i → α) : Iic (⨅ (i) (j), f i j) = ⋂ (i) (j), Iic (f i j) := by
simp_rw [Iic_iInf]
#align set.Iic_infi₂ Set.Iic_iInf₂
theorem Ici_sSup (s : Set α) : Ici (sSup s) = ⋂ a ∈ s, Ici a := by rw [sSup_eq_iSup, Ici_iSup₂]
#align set.Ici_Sup Set.Ici_sSup
theorem Iic_sInf (s : Set α) : Iic (sInf s) = ⋂ a ∈ s, Iic a := by rw [sInf_eq_iInf, Iic_iInf₂]
#align set.Iic_Inf Set.Iic_sInf
end Set
namespace Set
variable (t : α → Set β)
theorem biUnion_diff_biUnion_subset (s₁ s₂ : Set α) :
((⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x) ⊆ ⋃ x ∈ s₁ \ s₂, t x := by
simp only [diff_subset_iff, ← biUnion_union]
apply biUnion_subset_biUnion_left
rw [union_diff_self]
apply subset_union_right
#align set.bUnion_diff_bUnion_subset Set.biUnion_diff_biUnion_subset
/-- If `t` is an indexed family of sets, then there is a natural map from `Σ i, t i` to `⋃ i, t i`
sending `⟨i, x⟩` to `x`. -/
def sigmaToiUnion (x : Σi, t i) : ⋃ i, t i :=
⟨x.2, mem_iUnion.2 ⟨x.1, x.2.2⟩⟩
#align set.sigma_to_Union Set.sigmaToiUnion
theorem sigmaToiUnion_surjective : Surjective (sigmaToiUnion t)
| ⟨b, hb⟩ =>
have : ∃ a, b ∈ t a := by simpa using hb
let ⟨a, hb⟩ := this
⟨⟨a, b, hb⟩, rfl⟩
#align set.sigma_to_Union_surjective Set.sigmaToiUnion_surjective
theorem sigmaToiUnion_injective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Injective (sigmaToiUnion t)
| ⟨a₁, b₁, h₁⟩, ⟨a₂, b₂, h₂⟩, eq =>
have b_eq : b₁ = b₂ := congr_arg Subtype.val eq
have a_eq : a₁ = a₂ :=
by_contradiction fun ne =>
have : b₁ ∈ t a₁ ∩ t a₂ := ⟨h₁, b_eq.symm ▸ h₂⟩
(h _ _ ne).le_bot this
Sigma.eq a_eq <| Subtype.eq <| by subst b_eq; subst a_eq; rfl
#align set.sigma_to_Union_injective Set.sigmaToiUnion_injective
theorem sigmaToiUnion_bijective (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
Bijective (sigmaToiUnion t) :=
⟨sigmaToiUnion_injective t h, sigmaToiUnion_surjective t⟩
#align set.sigma_to_Union_bijective Set.sigmaToiUnion_bijective
/-- Equivalence between a disjoint union and a dependent sum. -/
noncomputable def unionEqSigmaOfDisjoint {t : α → Set β} (h : ∀ i j, i ≠ j → Disjoint (t i) (t j)) :
(⋃ i, t i) ≃ Σi, t i :=
(Equiv.ofBijective _ <| sigmaToiUnion_bijective t h).symm
#align set.Union_eq_sigma_of_disjoint Set.unionEqSigmaOfDisjoint
theorem iUnion_ge_eq_iUnion_nat_add (u : ℕ → Set α) (n : ℕ) : ⋃ i ≥ n, u i = ⋃ i, u (i + n) :=
iSup_ge_eq_iSup_nat_add u n
#align set.Union_ge_eq_Union_nat_add Set.iUnion_ge_eq_iUnion_nat_add
theorem iInter_ge_eq_iInter_nat_add (u : ℕ → Set α) (n : ℕ) : ⋂ i ≥ n, u i = ⋂ i, u (i + n) :=
iInf_ge_eq_iInf_nat_add u n
#align set.Inter_ge_eq_Inter_nat_add Set.iInter_ge_eq_iInter_nat_add
theorem _root_.Monotone.iUnion_nat_add {f : ℕ → Set α} (hf : Monotone f) (k : ℕ) :
⋃ n, f (n + k) = ⋃ n, f n :=
hf.iSup_nat_add k
#align monotone.Union_nat_add Monotone.iUnion_nat_add
theorem _root_.Antitone.iInter_nat_add {f : ℕ → Set α} (hf : Antitone f) (k : ℕ) :
⋂ n, f (n + k) = ⋂ n, f n :=
hf.iInf_nat_add k
#align antitone.Inter_nat_add Antitone.iInter_nat_add
/-Porting note: removing `simp`. LHS does not simplify. Possible linter bug. Zulip discussion:
https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/complete_lattice.20and.20has_sup/near/316497982
-/
theorem iUnion_iInter_ge_nat_add (f : ℕ → Set α) (k : ℕ) :
⋃ n, ⋂ i ≥ n, f (i + k) = ⋃ n, ⋂ i ≥ n, f i :=
iSup_iInf_ge_nat_add f k
#align set.Union_Inter_ge_nat_add Set.iUnion_iInter_ge_nat_add
theorem union_iUnion_nat_succ (u : ℕ → Set α) : (u 0 ∪ ⋃ i, u (i + 1)) = ⋃ i, u i :=
sup_iSup_nat_succ u
#align set.union_Union_nat_succ Set.union_iUnion_nat_succ
theorem inter_iInter_nat_succ (u : ℕ → Set α) : (u 0 ∩ ⋂ i, u (i + 1)) = ⋂ i, u i :=
inf_iInf_nat_succ u
#align set.inter_Inter_nat_succ Set.inter_iInter_nat_succ
end Set
open Set
variable [CompleteLattice β]
theorem iSup_iUnion (s : ι → Set α) (f : α → β) : ⨆ a ∈ ⋃ i, s i, f a = ⨆ (i) (a ∈ s i), f a := by
rw [iSup_comm]
simp_rw [mem_iUnion, iSup_exists]
#align supr_Union iSup_iUnion
theorem iInf_iUnion (s : ι → Set α) (f : α → β) : ⨅ a ∈ ⋃ i, s i, f a = ⨅ (i) (a ∈ s i), f a :=
@iSup_iUnion α βᵒᵈ _ _ s f
#align infi_Union iInf_iUnion
theorem sSup_sUnion (s : Set (Set β)) : sSup (⋃₀ s) = ⨆ t ∈ s, sSup t := by
simp only [sUnion_eq_biUnion, sSup_eq_iSup, iSup_iUnion]
#align Sup_sUnion sSup_sUnion
theorem sInf_sUnion (s : Set (Set β)) : sInf (⋃₀ s) = ⨅ t ∈ s, sInf t :=
@sSup_sUnion βᵒᵈ _ _
#align Inf_sUnion sInf_sUnion
lemma iSup_sUnion (S : Set (Set α)) (f : α → β) :
(⨆ x ∈ ⋃₀ S, f x) = ⨆ (s ∈ S) (x ∈ s), f x := by
rw [sUnion_eq_iUnion, iSup_iUnion, ← iSup_subtype'']
lemma iInf_sUnion (S : Set (Set α)) (f : α → β) :
(⨅ x ∈ ⋃₀ S, f x) = ⨅ (s ∈ S) (x ∈ s), f x := by
rw [sUnion_eq_iUnion, iInf_iUnion, ← iInf_subtype'']
lemma forall_sUnion {p : α → Prop} :
(∀ x ∈ ⋃₀ S, p x) ↔ ∀ s ∈ S, ∀ x ∈ s, p x := by
simp_rw [← iInf_Prop_eq, iInf_sUnion]
lemma exists_sUnion {p : α → Prop} :
(∃ x ∈ ⋃₀ S, p x) ↔ ∃ s ∈ S, ∃ x ∈ s, p x := by
|
simp_rw [← exists_prop, ← iSup_Prop_eq, iSup_sUnion]
|
lemma exists_sUnion {p : α → Prop} :
(∃ x ∈ ⋃₀ S, p x) ↔ ∃ s ∈ S, ∃ x ∈ s, p x := by
|
Mathlib.Data.Set.Lattice.2403_0.5mONj49h3SYSDwc
|
lemma exists_sUnion {p : α → Prop} :
(∃ x ∈ ⋃₀ S, p x) ↔ ∃ s ∈ S, ∃ x ∈ s, p x
|
Mathlib_Data_Set_Lattice
|
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasShift C ℤ
X : C
X✝ Y✝ : Triangle C
f : X✝ ⟶ Y✝
⊢ (Triangle.rotate X✝).mor₃ ≫ (shiftFunctor C 1).map f.hom₂ = (shiftFunctor C 1).map f.hom₁ ≫ (Triangle.rotate Y✝).mor₃
|
/-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanprover-community/mathlib"@"94d4e70e97c36c896cb70fb42821acfed040de60"
/-!
# Rotate
This file adds the ability to rotate triangles and triangle morphisms.
It also shows that rotation gives an equivalence on the category of triangles.
-/
noncomputable section
open CategoryTheory
open CategoryTheory.Preadditive
open CategoryTheory.Limits
universe v v₀ v₁ v₂ u u₀ u₁ u₂
namespace CategoryTheory.Pretriangulated
open CategoryTheory.Category
variable {C : Type u} [Category.{v} C] [Preadditive C]
variable [HasShift C ℤ]
variable (X : C)
/-- If you rotate a triangle, you get another triangle.
Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `rotate` gives a triangle of the form:
```
g h -f⟦1⟧'
Y ───> Z ───> X⟦1⟧ ───> Y⟦1⟧
```
-/
@[simps!]
def Triangle.rotate (T : Triangle C) : Triangle C :=
Triangle.mk T.mor₂ T.mor₃ (-T.mor₁⟦1⟧')
#align category_theory.pretriangulated.triangle.rotate CategoryTheory.Pretriangulated.Triangle.rotate
section
/-- Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `invRotate` gives a triangle that can be thought of as:
```
-h⟦-1⟧' f g
Z⟦-1⟧ ───> X ───> Y ───> Z
```
(note that this diagram doesn't technically fit the definition of triangle, as `Z⟦-1⟧⟦1⟧` is
not necessarily equal to `Z`, but it is isomorphic, by the `counitIso` of `shiftEquiv C 1`)
-/
@[simps!]
def Triangle.invRotate (T : Triangle C) : Triangle C :=
Triangle.mk (-T.mor₃⟦(-1 : ℤ)⟧' ≫ (shiftEquiv C (1 : ℤ)).unitIso.inv.app _) (T.mor₁)
(T.mor₂ ≫ (shiftEquiv C (1 : ℤ)).counitIso.inv.app _ )
#align category_theory.pretriangulated.triangle.inv_rotate CategoryTheory.Pretriangulated.Triangle.invRotate
end
attribute [local simp] shift_shift_neg' shift_neg_shift'
shift_shiftFunctorCompIsoId_add_neg_self_inv_app
shift_shiftFunctorCompIsoId_add_neg_self_hom_app
variable (C)
/-- Rotating triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def rotate : Triangle C ⥤ Triangle C
where
obj := Triangle.rotate
map f :=
{ hom₁ := f.hom₂
hom₂ := f.hom₃
hom₃ := f.hom₁⟦1⟧'
comm₃ := by
|
dsimp
|
/-- Rotating triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def rotate : Triangle C ⥤ Triangle C
where
obj := Triangle.rotate
map f :=
{ hom₁ := f.hom₂
hom₂ := f.hom₃
hom₃ := f.hom₁⟦1⟧'
comm₃ := by
|
Mathlib.CategoryTheory.Triangulated.Rotate.86_0.sGRpfSsY1fG2rGq
|
/-- Rotating triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def rotate : Triangle C ⥤ Triangle C
where
obj
|
Mathlib_CategoryTheory_Triangulated_Rotate
|
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasShift C ℤ
X : C
X✝ Y✝ : Triangle C
f : X✝ ⟶ Y✝
⊢ (-(shiftFunctor C 1).map X✝.mor₁) ≫ (shiftFunctor C 1).map f.hom₂ =
(shiftFunctor C 1).map f.hom₁ ≫ (-(shiftFunctor C 1).map Y✝.mor₁)
|
/-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanprover-community/mathlib"@"94d4e70e97c36c896cb70fb42821acfed040de60"
/-!
# Rotate
This file adds the ability to rotate triangles and triangle morphisms.
It also shows that rotation gives an equivalence on the category of triangles.
-/
noncomputable section
open CategoryTheory
open CategoryTheory.Preadditive
open CategoryTheory.Limits
universe v v₀ v₁ v₂ u u₀ u₁ u₂
namespace CategoryTheory.Pretriangulated
open CategoryTheory.Category
variable {C : Type u} [Category.{v} C] [Preadditive C]
variable [HasShift C ℤ]
variable (X : C)
/-- If you rotate a triangle, you get another triangle.
Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `rotate` gives a triangle of the form:
```
g h -f⟦1⟧'
Y ───> Z ───> X⟦1⟧ ───> Y⟦1⟧
```
-/
@[simps!]
def Triangle.rotate (T : Triangle C) : Triangle C :=
Triangle.mk T.mor₂ T.mor₃ (-T.mor₁⟦1⟧')
#align category_theory.pretriangulated.triangle.rotate CategoryTheory.Pretriangulated.Triangle.rotate
section
/-- Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `invRotate` gives a triangle that can be thought of as:
```
-h⟦-1⟧' f g
Z⟦-1⟧ ───> X ───> Y ───> Z
```
(note that this diagram doesn't technically fit the definition of triangle, as `Z⟦-1⟧⟦1⟧` is
not necessarily equal to `Z`, but it is isomorphic, by the `counitIso` of `shiftEquiv C 1`)
-/
@[simps!]
def Triangle.invRotate (T : Triangle C) : Triangle C :=
Triangle.mk (-T.mor₃⟦(-1 : ℤ)⟧' ≫ (shiftEquiv C (1 : ℤ)).unitIso.inv.app _) (T.mor₁)
(T.mor₂ ≫ (shiftEquiv C (1 : ℤ)).counitIso.inv.app _ )
#align category_theory.pretriangulated.triangle.inv_rotate CategoryTheory.Pretriangulated.Triangle.invRotate
end
attribute [local simp] shift_shift_neg' shift_neg_shift'
shift_shiftFunctorCompIsoId_add_neg_self_inv_app
shift_shiftFunctorCompIsoId_add_neg_self_hom_app
variable (C)
/-- Rotating triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def rotate : Triangle C ⥤ Triangle C
where
obj := Triangle.rotate
map f :=
{ hom₁ := f.hom₂
hom₂ := f.hom₃
hom₃ := f.hom₁⟦1⟧'
comm₃ := by
dsimp
|
simp only [comp_neg, neg_comp, ← Functor.map_comp, f.comm₁]
|
/-- Rotating triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def rotate : Triangle C ⥤ Triangle C
where
obj := Triangle.rotate
map f :=
{ hom₁ := f.hom₂
hom₂ := f.hom₃
hom₃ := f.hom₁⟦1⟧'
comm₃ := by
dsimp
|
Mathlib.CategoryTheory.Triangulated.Rotate.86_0.sGRpfSsY1fG2rGq
|
/-- Rotating triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def rotate : Triangle C ⥤ Triangle C
where
obj
|
Mathlib_CategoryTheory_Triangulated_Rotate
|
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasShift C ℤ
X : C
X✝ Y✝ : Triangle C
f : X✝ ⟶ Y✝
⊢ (Triangle.invRotate X✝).mor₁ ≫ f.hom₁ = (shiftFunctor C (-1)).map f.hom₃ ≫ (Triangle.invRotate Y✝).mor₁
|
/-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanprover-community/mathlib"@"94d4e70e97c36c896cb70fb42821acfed040de60"
/-!
# Rotate
This file adds the ability to rotate triangles and triangle morphisms.
It also shows that rotation gives an equivalence on the category of triangles.
-/
noncomputable section
open CategoryTheory
open CategoryTheory.Preadditive
open CategoryTheory.Limits
universe v v₀ v₁ v₂ u u₀ u₁ u₂
namespace CategoryTheory.Pretriangulated
open CategoryTheory.Category
variable {C : Type u} [Category.{v} C] [Preadditive C]
variable [HasShift C ℤ]
variable (X : C)
/-- If you rotate a triangle, you get another triangle.
Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `rotate` gives a triangle of the form:
```
g h -f⟦1⟧'
Y ───> Z ───> X⟦1⟧ ───> Y⟦1⟧
```
-/
@[simps!]
def Triangle.rotate (T : Triangle C) : Triangle C :=
Triangle.mk T.mor₂ T.mor₃ (-T.mor₁⟦1⟧')
#align category_theory.pretriangulated.triangle.rotate CategoryTheory.Pretriangulated.Triangle.rotate
section
/-- Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `invRotate` gives a triangle that can be thought of as:
```
-h⟦-1⟧' f g
Z⟦-1⟧ ───> X ───> Y ───> Z
```
(note that this diagram doesn't technically fit the definition of triangle, as `Z⟦-1⟧⟦1⟧` is
not necessarily equal to `Z`, but it is isomorphic, by the `counitIso` of `shiftEquiv C 1`)
-/
@[simps!]
def Triangle.invRotate (T : Triangle C) : Triangle C :=
Triangle.mk (-T.mor₃⟦(-1 : ℤ)⟧' ≫ (shiftEquiv C (1 : ℤ)).unitIso.inv.app _) (T.mor₁)
(T.mor₂ ≫ (shiftEquiv C (1 : ℤ)).counitIso.inv.app _ )
#align category_theory.pretriangulated.triangle.inv_rotate CategoryTheory.Pretriangulated.Triangle.invRotate
end
attribute [local simp] shift_shift_neg' shift_neg_shift'
shift_shiftFunctorCompIsoId_add_neg_self_inv_app
shift_shiftFunctorCompIsoId_add_neg_self_hom_app
variable (C)
/-- Rotating triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def rotate : Triangle C ⥤ Triangle C
where
obj := Triangle.rotate
map f :=
{ hom₁ := f.hom₂
hom₂ := f.hom₃
hom₃ := f.hom₁⟦1⟧'
comm₃ := by
dsimp
simp only [comp_neg, neg_comp, ← Functor.map_comp, f.comm₁] }
#align category_theory.pretriangulated.rotate CategoryTheory.Pretriangulated.rotate
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
|
dsimp
|
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
|
Mathlib.CategoryTheory.Triangulated.Rotate.101_0.sGRpfSsY1fG2rGq
|
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj
|
Mathlib_CategoryTheory_Triangulated_Rotate
|
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasShift C ℤ
X : C
X✝ Y✝ : Triangle C
f : X✝ ⟶ Y✝
⊢ (-(shiftFunctor C (-1)).map X✝.mor₃ ≫ (shiftFunctorCompIsoId C 1 (-1) (_ : 1 + -1 = 0)).hom.app X✝.obj₁) ≫ f.hom₁ =
(shiftFunctor C (-1)).map f.hom₃ ≫
(-(shiftFunctor C (-1)).map Y✝.mor₃ ≫ (shiftFunctorCompIsoId C 1 (-1) (_ : 1 + -1 = 0)).hom.app Y✝.obj₁)
|
/-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanprover-community/mathlib"@"94d4e70e97c36c896cb70fb42821acfed040de60"
/-!
# Rotate
This file adds the ability to rotate triangles and triangle morphisms.
It also shows that rotation gives an equivalence on the category of triangles.
-/
noncomputable section
open CategoryTheory
open CategoryTheory.Preadditive
open CategoryTheory.Limits
universe v v₀ v₁ v₂ u u₀ u₁ u₂
namespace CategoryTheory.Pretriangulated
open CategoryTheory.Category
variable {C : Type u} [Category.{v} C] [Preadditive C]
variable [HasShift C ℤ]
variable (X : C)
/-- If you rotate a triangle, you get another triangle.
Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `rotate` gives a triangle of the form:
```
g h -f⟦1⟧'
Y ───> Z ───> X⟦1⟧ ───> Y⟦1⟧
```
-/
@[simps!]
def Triangle.rotate (T : Triangle C) : Triangle C :=
Triangle.mk T.mor₂ T.mor₃ (-T.mor₁⟦1⟧')
#align category_theory.pretriangulated.triangle.rotate CategoryTheory.Pretriangulated.Triangle.rotate
section
/-- Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `invRotate` gives a triangle that can be thought of as:
```
-h⟦-1⟧' f g
Z⟦-1⟧ ───> X ───> Y ───> Z
```
(note that this diagram doesn't technically fit the definition of triangle, as `Z⟦-1⟧⟦1⟧` is
not necessarily equal to `Z`, but it is isomorphic, by the `counitIso` of `shiftEquiv C 1`)
-/
@[simps!]
def Triangle.invRotate (T : Triangle C) : Triangle C :=
Triangle.mk (-T.mor₃⟦(-1 : ℤ)⟧' ≫ (shiftEquiv C (1 : ℤ)).unitIso.inv.app _) (T.mor₁)
(T.mor₂ ≫ (shiftEquiv C (1 : ℤ)).counitIso.inv.app _ )
#align category_theory.pretriangulated.triangle.inv_rotate CategoryTheory.Pretriangulated.Triangle.invRotate
end
attribute [local simp] shift_shift_neg' shift_neg_shift'
shift_shiftFunctorCompIsoId_add_neg_self_inv_app
shift_shiftFunctorCompIsoId_add_neg_self_hom_app
variable (C)
/-- Rotating triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def rotate : Triangle C ⥤ Triangle C
where
obj := Triangle.rotate
map f :=
{ hom₁ := f.hom₂
hom₂ := f.hom₃
hom₃ := f.hom₁⟦1⟧'
comm₃ := by
dsimp
simp only [comp_neg, neg_comp, ← Functor.map_comp, f.comm₁] }
#align category_theory.pretriangulated.rotate CategoryTheory.Pretriangulated.rotate
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
dsimp
|
simp only [neg_comp, assoc, comp_neg, neg_inj, ← Functor.map_comp_assoc, ← f.comm₃]
|
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
dsimp
|
Mathlib.CategoryTheory.Triangulated.Rotate.101_0.sGRpfSsY1fG2rGq
|
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj
|
Mathlib_CategoryTheory_Triangulated_Rotate
|
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasShift C ℤ
X : C
X✝ Y✝ : Triangle C
f : X✝ ⟶ Y✝
⊢ (shiftFunctor C (-1)).map X✝.mor₃ ≫ (shiftFunctorCompIsoId C 1 (-1) (_ : 1 + -1 = 0)).hom.app X✝.obj₁ ≫ f.hom₁ =
(shiftFunctor C (-1)).map (X✝.mor₃ ≫ (shiftFunctor C 1).map f.hom₁) ≫
(shiftFunctorCompIsoId C 1 (-1) (_ : 1 + -1 = 0)).hom.app Y✝.obj₁
|
/-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanprover-community/mathlib"@"94d4e70e97c36c896cb70fb42821acfed040de60"
/-!
# Rotate
This file adds the ability to rotate triangles and triangle morphisms.
It also shows that rotation gives an equivalence on the category of triangles.
-/
noncomputable section
open CategoryTheory
open CategoryTheory.Preadditive
open CategoryTheory.Limits
universe v v₀ v₁ v₂ u u₀ u₁ u₂
namespace CategoryTheory.Pretriangulated
open CategoryTheory.Category
variable {C : Type u} [Category.{v} C] [Preadditive C]
variable [HasShift C ℤ]
variable (X : C)
/-- If you rotate a triangle, you get another triangle.
Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `rotate` gives a triangle of the form:
```
g h -f⟦1⟧'
Y ───> Z ───> X⟦1⟧ ───> Y⟦1⟧
```
-/
@[simps!]
def Triangle.rotate (T : Triangle C) : Triangle C :=
Triangle.mk T.mor₂ T.mor₃ (-T.mor₁⟦1⟧')
#align category_theory.pretriangulated.triangle.rotate CategoryTheory.Pretriangulated.Triangle.rotate
section
/-- Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `invRotate` gives a triangle that can be thought of as:
```
-h⟦-1⟧' f g
Z⟦-1⟧ ───> X ───> Y ───> Z
```
(note that this diagram doesn't technically fit the definition of triangle, as `Z⟦-1⟧⟦1⟧` is
not necessarily equal to `Z`, but it is isomorphic, by the `counitIso` of `shiftEquiv C 1`)
-/
@[simps!]
def Triangle.invRotate (T : Triangle C) : Triangle C :=
Triangle.mk (-T.mor₃⟦(-1 : ℤ)⟧' ≫ (shiftEquiv C (1 : ℤ)).unitIso.inv.app _) (T.mor₁)
(T.mor₂ ≫ (shiftEquiv C (1 : ℤ)).counitIso.inv.app _ )
#align category_theory.pretriangulated.triangle.inv_rotate CategoryTheory.Pretriangulated.Triangle.invRotate
end
attribute [local simp] shift_shift_neg' shift_neg_shift'
shift_shiftFunctorCompIsoId_add_neg_self_inv_app
shift_shiftFunctorCompIsoId_add_neg_self_hom_app
variable (C)
/-- Rotating triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def rotate : Triangle C ⥤ Triangle C
where
obj := Triangle.rotate
map f :=
{ hom₁ := f.hom₂
hom₂ := f.hom₃
hom₃ := f.hom₁⟦1⟧'
comm₃ := by
dsimp
simp only [comp_neg, neg_comp, ← Functor.map_comp, f.comm₁] }
#align category_theory.pretriangulated.rotate CategoryTheory.Pretriangulated.rotate
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
dsimp
simp only [neg_comp, assoc, comp_neg, neg_inj, ← Functor.map_comp_assoc, ← f.comm₃]
|
rw [Functor.map_comp, assoc]
|
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
dsimp
simp only [neg_comp, assoc, comp_neg, neg_inj, ← Functor.map_comp_assoc, ← f.comm₃]
|
Mathlib.CategoryTheory.Triangulated.Rotate.101_0.sGRpfSsY1fG2rGq
|
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj
|
Mathlib_CategoryTheory_Triangulated_Rotate
|
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasShift C ℤ
X : C
X✝ Y✝ : Triangle C
f : X✝ ⟶ Y✝
⊢ (shiftFunctor C (-1)).map X✝.mor₃ ≫ (shiftFunctorCompIsoId C 1 (-1) (_ : 1 + -1 = 0)).hom.app X✝.obj₁ ≫ f.hom₁ =
(shiftFunctor C (-1)).map X✝.mor₃ ≫
(shiftFunctor C (-1)).map ((shiftFunctor C 1).map f.hom₁) ≫
(shiftFunctorCompIsoId C 1 (-1) (_ : 1 + -1 = 0)).hom.app Y✝.obj₁
|
/-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanprover-community/mathlib"@"94d4e70e97c36c896cb70fb42821acfed040de60"
/-!
# Rotate
This file adds the ability to rotate triangles and triangle morphisms.
It also shows that rotation gives an equivalence on the category of triangles.
-/
noncomputable section
open CategoryTheory
open CategoryTheory.Preadditive
open CategoryTheory.Limits
universe v v₀ v₁ v₂ u u₀ u₁ u₂
namespace CategoryTheory.Pretriangulated
open CategoryTheory.Category
variable {C : Type u} [Category.{v} C] [Preadditive C]
variable [HasShift C ℤ]
variable (X : C)
/-- If you rotate a triangle, you get another triangle.
Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `rotate` gives a triangle of the form:
```
g h -f⟦1⟧'
Y ───> Z ───> X⟦1⟧ ───> Y⟦1⟧
```
-/
@[simps!]
def Triangle.rotate (T : Triangle C) : Triangle C :=
Triangle.mk T.mor₂ T.mor₃ (-T.mor₁⟦1⟧')
#align category_theory.pretriangulated.triangle.rotate CategoryTheory.Pretriangulated.Triangle.rotate
section
/-- Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `invRotate` gives a triangle that can be thought of as:
```
-h⟦-1⟧' f g
Z⟦-1⟧ ───> X ───> Y ───> Z
```
(note that this diagram doesn't technically fit the definition of triangle, as `Z⟦-1⟧⟦1⟧` is
not necessarily equal to `Z`, but it is isomorphic, by the `counitIso` of `shiftEquiv C 1`)
-/
@[simps!]
def Triangle.invRotate (T : Triangle C) : Triangle C :=
Triangle.mk (-T.mor₃⟦(-1 : ℤ)⟧' ≫ (shiftEquiv C (1 : ℤ)).unitIso.inv.app _) (T.mor₁)
(T.mor₂ ≫ (shiftEquiv C (1 : ℤ)).counitIso.inv.app _ )
#align category_theory.pretriangulated.triangle.inv_rotate CategoryTheory.Pretriangulated.Triangle.invRotate
end
attribute [local simp] shift_shift_neg' shift_neg_shift'
shift_shiftFunctorCompIsoId_add_neg_self_inv_app
shift_shiftFunctorCompIsoId_add_neg_self_hom_app
variable (C)
/-- Rotating triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def rotate : Triangle C ⥤ Triangle C
where
obj := Triangle.rotate
map f :=
{ hom₁ := f.hom₂
hom₂ := f.hom₃
hom₃ := f.hom₁⟦1⟧'
comm₃ := by
dsimp
simp only [comp_neg, neg_comp, ← Functor.map_comp, f.comm₁] }
#align category_theory.pretriangulated.rotate CategoryTheory.Pretriangulated.rotate
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
dsimp
simp only [neg_comp, assoc, comp_neg, neg_inj, ← Functor.map_comp_assoc, ← f.comm₃]
rw [Functor.map_comp, assoc]
|
erw [← NatTrans.naturality]
|
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
dsimp
simp only [neg_comp, assoc, comp_neg, neg_inj, ← Functor.map_comp_assoc, ← f.comm₃]
rw [Functor.map_comp, assoc]
|
Mathlib.CategoryTheory.Triangulated.Rotate.101_0.sGRpfSsY1fG2rGq
|
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj
|
Mathlib_CategoryTheory_Triangulated_Rotate
|
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasShift C ℤ
X : C
X✝ Y✝ : Triangle C
f : X✝ ⟶ Y✝
⊢ (shiftFunctor C (-1)).map X✝.mor₃ ≫
(shiftFunctor C 1 ⋙ shiftFunctor C (-1)).map f.1 ≫
(shiftFunctorCompIsoId C 1 (-1) (_ : 1 + -1 = 0)).hom.app Y✝.obj₁ =
(shiftFunctor C (-1)).map X✝.mor₃ ≫
(shiftFunctor C (-1)).map ((shiftFunctor C 1).map f.hom₁) ≫
(shiftFunctorCompIsoId C 1 (-1) (_ : 1 + -1 = 0)).hom.app Y✝.obj₁
|
/-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanprover-community/mathlib"@"94d4e70e97c36c896cb70fb42821acfed040de60"
/-!
# Rotate
This file adds the ability to rotate triangles and triangle morphisms.
It also shows that rotation gives an equivalence on the category of triangles.
-/
noncomputable section
open CategoryTheory
open CategoryTheory.Preadditive
open CategoryTheory.Limits
universe v v₀ v₁ v₂ u u₀ u₁ u₂
namespace CategoryTheory.Pretriangulated
open CategoryTheory.Category
variable {C : Type u} [Category.{v} C] [Preadditive C]
variable [HasShift C ℤ]
variable (X : C)
/-- If you rotate a triangle, you get another triangle.
Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `rotate` gives a triangle of the form:
```
g h -f⟦1⟧'
Y ───> Z ───> X⟦1⟧ ───> Y⟦1⟧
```
-/
@[simps!]
def Triangle.rotate (T : Triangle C) : Triangle C :=
Triangle.mk T.mor₂ T.mor₃ (-T.mor₁⟦1⟧')
#align category_theory.pretriangulated.triangle.rotate CategoryTheory.Pretriangulated.Triangle.rotate
section
/-- Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `invRotate` gives a triangle that can be thought of as:
```
-h⟦-1⟧' f g
Z⟦-1⟧ ───> X ───> Y ───> Z
```
(note that this diagram doesn't technically fit the definition of triangle, as `Z⟦-1⟧⟦1⟧` is
not necessarily equal to `Z`, but it is isomorphic, by the `counitIso` of `shiftEquiv C 1`)
-/
@[simps!]
def Triangle.invRotate (T : Triangle C) : Triangle C :=
Triangle.mk (-T.mor₃⟦(-1 : ℤ)⟧' ≫ (shiftEquiv C (1 : ℤ)).unitIso.inv.app _) (T.mor₁)
(T.mor₂ ≫ (shiftEquiv C (1 : ℤ)).counitIso.inv.app _ )
#align category_theory.pretriangulated.triangle.inv_rotate CategoryTheory.Pretriangulated.Triangle.invRotate
end
attribute [local simp] shift_shift_neg' shift_neg_shift'
shift_shiftFunctorCompIsoId_add_neg_self_inv_app
shift_shiftFunctorCompIsoId_add_neg_self_hom_app
variable (C)
/-- Rotating triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def rotate : Triangle C ⥤ Triangle C
where
obj := Triangle.rotate
map f :=
{ hom₁ := f.hom₂
hom₂ := f.hom₃
hom₃ := f.hom₁⟦1⟧'
comm₃ := by
dsimp
simp only [comp_neg, neg_comp, ← Functor.map_comp, f.comm₁] }
#align category_theory.pretriangulated.rotate CategoryTheory.Pretriangulated.rotate
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
dsimp
simp only [neg_comp, assoc, comp_neg, neg_inj, ← Functor.map_comp_assoc, ← f.comm₃]
rw [Functor.map_comp, assoc]
erw [← NatTrans.naturality]
|
rfl
|
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
dsimp
simp only [neg_comp, assoc, comp_neg, neg_inj, ← Functor.map_comp_assoc, ← f.comm₃]
rw [Functor.map_comp, assoc]
erw [← NatTrans.naturality]
|
Mathlib.CategoryTheory.Triangulated.Rotate.101_0.sGRpfSsY1fG2rGq
|
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj
|
Mathlib_CategoryTheory_Triangulated_Rotate
|
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasShift C ℤ
X : C
X✝ Y✝ : Triangle C
f : X✝ ⟶ Y✝
⊢ (Triangle.invRotate X✝).mor₃ ≫ (shiftFunctor C 1).map ((shiftFunctor C (-1)).map f.hom₃) =
f.hom₂ ≫ (Triangle.invRotate Y✝).mor₃
|
/-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanprover-community/mathlib"@"94d4e70e97c36c896cb70fb42821acfed040de60"
/-!
# Rotate
This file adds the ability to rotate triangles and triangle morphisms.
It also shows that rotation gives an equivalence on the category of triangles.
-/
noncomputable section
open CategoryTheory
open CategoryTheory.Preadditive
open CategoryTheory.Limits
universe v v₀ v₁ v₂ u u₀ u₁ u₂
namespace CategoryTheory.Pretriangulated
open CategoryTheory.Category
variable {C : Type u} [Category.{v} C] [Preadditive C]
variable [HasShift C ℤ]
variable (X : C)
/-- If you rotate a triangle, you get another triangle.
Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `rotate` gives a triangle of the form:
```
g h -f⟦1⟧'
Y ───> Z ───> X⟦1⟧ ───> Y⟦1⟧
```
-/
@[simps!]
def Triangle.rotate (T : Triangle C) : Triangle C :=
Triangle.mk T.mor₂ T.mor₃ (-T.mor₁⟦1⟧')
#align category_theory.pretriangulated.triangle.rotate CategoryTheory.Pretriangulated.Triangle.rotate
section
/-- Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `invRotate` gives a triangle that can be thought of as:
```
-h⟦-1⟧' f g
Z⟦-1⟧ ───> X ───> Y ───> Z
```
(note that this diagram doesn't technically fit the definition of triangle, as `Z⟦-1⟧⟦1⟧` is
not necessarily equal to `Z`, but it is isomorphic, by the `counitIso` of `shiftEquiv C 1`)
-/
@[simps!]
def Triangle.invRotate (T : Triangle C) : Triangle C :=
Triangle.mk (-T.mor₃⟦(-1 : ℤ)⟧' ≫ (shiftEquiv C (1 : ℤ)).unitIso.inv.app _) (T.mor₁)
(T.mor₂ ≫ (shiftEquiv C (1 : ℤ)).counitIso.inv.app _ )
#align category_theory.pretriangulated.triangle.inv_rotate CategoryTheory.Pretriangulated.Triangle.invRotate
end
attribute [local simp] shift_shift_neg' shift_neg_shift'
shift_shiftFunctorCompIsoId_add_neg_self_inv_app
shift_shiftFunctorCompIsoId_add_neg_self_hom_app
variable (C)
/-- Rotating triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def rotate : Triangle C ⥤ Triangle C
where
obj := Triangle.rotate
map f :=
{ hom₁ := f.hom₂
hom₂ := f.hom₃
hom₃ := f.hom₁⟦1⟧'
comm₃ := by
dsimp
simp only [comp_neg, neg_comp, ← Functor.map_comp, f.comm₁] }
#align category_theory.pretriangulated.rotate CategoryTheory.Pretriangulated.rotate
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
dsimp
simp only [neg_comp, assoc, comp_neg, neg_inj, ← Functor.map_comp_assoc, ← f.comm₃]
rw [Functor.map_comp, assoc]
erw [← NatTrans.naturality]
rfl
comm₃ := by
|
erw [← reassoc_of% f.comm₂, Category.assoc, ← NatTrans.naturality]
|
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
dsimp
simp only [neg_comp, assoc, comp_neg, neg_inj, ← Functor.map_comp_assoc, ← f.comm₃]
rw [Functor.map_comp, assoc]
erw [← NatTrans.naturality]
rfl
comm₃ := by
|
Mathlib.CategoryTheory.Triangulated.Rotate.101_0.sGRpfSsY1fG2rGq
|
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj
|
Mathlib_CategoryTheory_Triangulated_Rotate
|
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasShift C ℤ
X : C
X✝ Y✝ : Triangle C
f : X✝ ⟶ Y✝
⊢ X✝.mor₂ ≫ (𝟭 C).map f.hom₃ ≫ (shiftEquiv C 1).counitIso.inv.app Y✝.obj₃ =
X✝.mor₂ ≫ f.hom₃ ≫ (shiftEquiv C 1).counitIso.inv.app Y✝.3
|
/-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanprover-community/mathlib"@"94d4e70e97c36c896cb70fb42821acfed040de60"
/-!
# Rotate
This file adds the ability to rotate triangles and triangle morphisms.
It also shows that rotation gives an equivalence on the category of triangles.
-/
noncomputable section
open CategoryTheory
open CategoryTheory.Preadditive
open CategoryTheory.Limits
universe v v₀ v₁ v₂ u u₀ u₁ u₂
namespace CategoryTheory.Pretriangulated
open CategoryTheory.Category
variable {C : Type u} [Category.{v} C] [Preadditive C]
variable [HasShift C ℤ]
variable (X : C)
/-- If you rotate a triangle, you get another triangle.
Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `rotate` gives a triangle of the form:
```
g h -f⟦1⟧'
Y ───> Z ───> X⟦1⟧ ───> Y⟦1⟧
```
-/
@[simps!]
def Triangle.rotate (T : Triangle C) : Triangle C :=
Triangle.mk T.mor₂ T.mor₃ (-T.mor₁⟦1⟧')
#align category_theory.pretriangulated.triangle.rotate CategoryTheory.Pretriangulated.Triangle.rotate
section
/-- Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `invRotate` gives a triangle that can be thought of as:
```
-h⟦-1⟧' f g
Z⟦-1⟧ ───> X ───> Y ───> Z
```
(note that this diagram doesn't technically fit the definition of triangle, as `Z⟦-1⟧⟦1⟧` is
not necessarily equal to `Z`, but it is isomorphic, by the `counitIso` of `shiftEquiv C 1`)
-/
@[simps!]
def Triangle.invRotate (T : Triangle C) : Triangle C :=
Triangle.mk (-T.mor₃⟦(-1 : ℤ)⟧' ≫ (shiftEquiv C (1 : ℤ)).unitIso.inv.app _) (T.mor₁)
(T.mor₂ ≫ (shiftEquiv C (1 : ℤ)).counitIso.inv.app _ )
#align category_theory.pretriangulated.triangle.inv_rotate CategoryTheory.Pretriangulated.Triangle.invRotate
end
attribute [local simp] shift_shift_neg' shift_neg_shift'
shift_shiftFunctorCompIsoId_add_neg_self_inv_app
shift_shiftFunctorCompIsoId_add_neg_self_hom_app
variable (C)
/-- Rotating triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def rotate : Triangle C ⥤ Triangle C
where
obj := Triangle.rotate
map f :=
{ hom₁ := f.hom₂
hom₂ := f.hom₃
hom₃ := f.hom₁⟦1⟧'
comm₃ := by
dsimp
simp only [comp_neg, neg_comp, ← Functor.map_comp, f.comm₁] }
#align category_theory.pretriangulated.rotate CategoryTheory.Pretriangulated.rotate
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
dsimp
simp only [neg_comp, assoc, comp_neg, neg_inj, ← Functor.map_comp_assoc, ← f.comm₃]
rw [Functor.map_comp, assoc]
erw [← NatTrans.naturality]
rfl
comm₃ := by
erw [← reassoc_of% f.comm₂, Category.assoc, ← NatTrans.naturality]
|
rfl
|
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
dsimp
simp only [neg_comp, assoc, comp_neg, neg_inj, ← Functor.map_comp_assoc, ← f.comm₃]
rw [Functor.map_comp, assoc]
erw [← NatTrans.naturality]
rfl
comm₃ := by
erw [← reassoc_of% f.comm₂, Category.assoc, ← NatTrans.naturality]
|
Mathlib.CategoryTheory.Triangulated.Rotate.101_0.sGRpfSsY1fG2rGq
|
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj
|
Mathlib_CategoryTheory_Triangulated_Rotate
|
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
X : C
inst✝ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)
⊢ IsEquivalence (rotate C)
|
/-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanprover-community/mathlib"@"94d4e70e97c36c896cb70fb42821acfed040de60"
/-!
# Rotate
This file adds the ability to rotate triangles and triangle morphisms.
It also shows that rotation gives an equivalence on the category of triangles.
-/
noncomputable section
open CategoryTheory
open CategoryTheory.Preadditive
open CategoryTheory.Limits
universe v v₀ v₁ v₂ u u₀ u₁ u₂
namespace CategoryTheory.Pretriangulated
open CategoryTheory.Category
variable {C : Type u} [Category.{v} C] [Preadditive C]
variable [HasShift C ℤ]
variable (X : C)
/-- If you rotate a triangle, you get another triangle.
Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `rotate` gives a triangle of the form:
```
g h -f⟦1⟧'
Y ───> Z ───> X⟦1⟧ ───> Y⟦1⟧
```
-/
@[simps!]
def Triangle.rotate (T : Triangle C) : Triangle C :=
Triangle.mk T.mor₂ T.mor₃ (-T.mor₁⟦1⟧')
#align category_theory.pretriangulated.triangle.rotate CategoryTheory.Pretriangulated.Triangle.rotate
section
/-- Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `invRotate` gives a triangle that can be thought of as:
```
-h⟦-1⟧' f g
Z⟦-1⟧ ───> X ───> Y ───> Z
```
(note that this diagram doesn't technically fit the definition of triangle, as `Z⟦-1⟧⟦1⟧` is
not necessarily equal to `Z`, but it is isomorphic, by the `counitIso` of `shiftEquiv C 1`)
-/
@[simps!]
def Triangle.invRotate (T : Triangle C) : Triangle C :=
Triangle.mk (-T.mor₃⟦(-1 : ℤ)⟧' ≫ (shiftEquiv C (1 : ℤ)).unitIso.inv.app _) (T.mor₁)
(T.mor₂ ≫ (shiftEquiv C (1 : ℤ)).counitIso.inv.app _ )
#align category_theory.pretriangulated.triangle.inv_rotate CategoryTheory.Pretriangulated.Triangle.invRotate
end
attribute [local simp] shift_shift_neg' shift_neg_shift'
shift_shiftFunctorCompIsoId_add_neg_self_inv_app
shift_shiftFunctorCompIsoId_add_neg_self_hom_app
variable (C)
/-- Rotating triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def rotate : Triangle C ⥤ Triangle C
where
obj := Triangle.rotate
map f :=
{ hom₁ := f.hom₂
hom₂ := f.hom₃
hom₃ := f.hom₁⟦1⟧'
comm₃ := by
dsimp
simp only [comp_neg, neg_comp, ← Functor.map_comp, f.comm₁] }
#align category_theory.pretriangulated.rotate CategoryTheory.Pretriangulated.rotate
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
dsimp
simp only [neg_comp, assoc, comp_neg, neg_inj, ← Functor.map_comp_assoc, ← f.comm₃]
rw [Functor.map_comp, assoc]
erw [← NatTrans.naturality]
rfl
comm₃ := by
erw [← reassoc_of% f.comm₂, Category.assoc, ← NatTrans.naturality]
rfl }
#align category_theory.pretriangulated.inv_rotate CategoryTheory.Pretriangulated.invRotate
variable {C}
variable [∀ n : ℤ, Functor.Additive (shiftFunctor C n)]
/-- The unit isomorphism of the auto-equivalence of categories `triangleRotation C` of
`Triangle C` given by the rotation of triangles. -/
@[simps!]
def rotCompInvRot : 𝟭 (Triangle C) ≅ rotate C ⋙ invRotate C :=
NatIso.ofComponents fun T => Triangle.isoMk _ _
((shiftEquiv C (1 : ℤ)).unitIso.app T.obj₁) (Iso.refl _) (Iso.refl _)
#align category_theory.pretriangulated.rot_comp_inv_rot CategoryTheory.Pretriangulated.rotCompInvRot
/-- The counit isomorphism of the auto-equivalence of categories `triangleRotation C` of
`Triangle C` given by the rotation of triangles. -/
@[simps!]
def invRotCompRot : invRotate C ⋙ rotate C ≅ 𝟭 (Triangle C) :=
NatIso.ofComponents fun T => Triangle.isoMk _ _ (Iso.refl _) (Iso.refl _)
((shiftEquiv C (1 : ℤ)).counitIso.app T.obj₃)
#align category_theory.pretriangulated.inv_rot_comp_rot CategoryTheory.Pretriangulated.invRotCompRot
variable (C)
/-- Rotating triangles gives an auto-equivalence on the category of triangles in `C`.
-/
@[simps]
def triangleRotation : Equivalence (Triangle C) (Triangle C)
where
functor := rotate C
inverse := invRotate C
unitIso := rotCompInvRot
counitIso := invRotCompRot
#align category_theory.pretriangulated.triangle_rotation CategoryTheory.Pretriangulated.triangleRotation
variable {C}
instance : IsEquivalence (rotate C) := by
|
change IsEquivalence (triangleRotation C).functor
|
instance : IsEquivalence (rotate C) := by
|
Mathlib.CategoryTheory.Triangulated.Rotate.157_0.sGRpfSsY1fG2rGq
|
instance : IsEquivalence (rotate C)
|
Mathlib_CategoryTheory_Triangulated_Rotate
|
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
X : C
inst✝ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)
⊢ IsEquivalence (triangleRotation C).functor
|
/-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanprover-community/mathlib"@"94d4e70e97c36c896cb70fb42821acfed040de60"
/-!
# Rotate
This file adds the ability to rotate triangles and triangle morphisms.
It also shows that rotation gives an equivalence on the category of triangles.
-/
noncomputable section
open CategoryTheory
open CategoryTheory.Preadditive
open CategoryTheory.Limits
universe v v₀ v₁ v₂ u u₀ u₁ u₂
namespace CategoryTheory.Pretriangulated
open CategoryTheory.Category
variable {C : Type u} [Category.{v} C] [Preadditive C]
variable [HasShift C ℤ]
variable (X : C)
/-- If you rotate a triangle, you get another triangle.
Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `rotate` gives a triangle of the form:
```
g h -f⟦1⟧'
Y ───> Z ───> X⟦1⟧ ───> Y⟦1⟧
```
-/
@[simps!]
def Triangle.rotate (T : Triangle C) : Triangle C :=
Triangle.mk T.mor₂ T.mor₃ (-T.mor₁⟦1⟧')
#align category_theory.pretriangulated.triangle.rotate CategoryTheory.Pretriangulated.Triangle.rotate
section
/-- Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `invRotate` gives a triangle that can be thought of as:
```
-h⟦-1⟧' f g
Z⟦-1⟧ ───> X ───> Y ───> Z
```
(note that this diagram doesn't technically fit the definition of triangle, as `Z⟦-1⟧⟦1⟧` is
not necessarily equal to `Z`, but it is isomorphic, by the `counitIso` of `shiftEquiv C 1`)
-/
@[simps!]
def Triangle.invRotate (T : Triangle C) : Triangle C :=
Triangle.mk (-T.mor₃⟦(-1 : ℤ)⟧' ≫ (shiftEquiv C (1 : ℤ)).unitIso.inv.app _) (T.mor₁)
(T.mor₂ ≫ (shiftEquiv C (1 : ℤ)).counitIso.inv.app _ )
#align category_theory.pretriangulated.triangle.inv_rotate CategoryTheory.Pretriangulated.Triangle.invRotate
end
attribute [local simp] shift_shift_neg' shift_neg_shift'
shift_shiftFunctorCompIsoId_add_neg_self_inv_app
shift_shiftFunctorCompIsoId_add_neg_self_hom_app
variable (C)
/-- Rotating triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def rotate : Triangle C ⥤ Triangle C
where
obj := Triangle.rotate
map f :=
{ hom₁ := f.hom₂
hom₂ := f.hom₃
hom₃ := f.hom₁⟦1⟧'
comm₃ := by
dsimp
simp only [comp_neg, neg_comp, ← Functor.map_comp, f.comm₁] }
#align category_theory.pretriangulated.rotate CategoryTheory.Pretriangulated.rotate
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
dsimp
simp only [neg_comp, assoc, comp_neg, neg_inj, ← Functor.map_comp_assoc, ← f.comm₃]
rw [Functor.map_comp, assoc]
erw [← NatTrans.naturality]
rfl
comm₃ := by
erw [← reassoc_of% f.comm₂, Category.assoc, ← NatTrans.naturality]
rfl }
#align category_theory.pretriangulated.inv_rotate CategoryTheory.Pretriangulated.invRotate
variable {C}
variable [∀ n : ℤ, Functor.Additive (shiftFunctor C n)]
/-- The unit isomorphism of the auto-equivalence of categories `triangleRotation C` of
`Triangle C` given by the rotation of triangles. -/
@[simps!]
def rotCompInvRot : 𝟭 (Triangle C) ≅ rotate C ⋙ invRotate C :=
NatIso.ofComponents fun T => Triangle.isoMk _ _
((shiftEquiv C (1 : ℤ)).unitIso.app T.obj₁) (Iso.refl _) (Iso.refl _)
#align category_theory.pretriangulated.rot_comp_inv_rot CategoryTheory.Pretriangulated.rotCompInvRot
/-- The counit isomorphism of the auto-equivalence of categories `triangleRotation C` of
`Triangle C` given by the rotation of triangles. -/
@[simps!]
def invRotCompRot : invRotate C ⋙ rotate C ≅ 𝟭 (Triangle C) :=
NatIso.ofComponents fun T => Triangle.isoMk _ _ (Iso.refl _) (Iso.refl _)
((shiftEquiv C (1 : ℤ)).counitIso.app T.obj₃)
#align category_theory.pretriangulated.inv_rot_comp_rot CategoryTheory.Pretriangulated.invRotCompRot
variable (C)
/-- Rotating triangles gives an auto-equivalence on the category of triangles in `C`.
-/
@[simps]
def triangleRotation : Equivalence (Triangle C) (Triangle C)
where
functor := rotate C
inverse := invRotate C
unitIso := rotCompInvRot
counitIso := invRotCompRot
#align category_theory.pretriangulated.triangle_rotation CategoryTheory.Pretriangulated.triangleRotation
variable {C}
instance : IsEquivalence (rotate C) := by
change IsEquivalence (triangleRotation C).functor
|
infer_instance
|
instance : IsEquivalence (rotate C) := by
change IsEquivalence (triangleRotation C).functor
|
Mathlib.CategoryTheory.Triangulated.Rotate.157_0.sGRpfSsY1fG2rGq
|
instance : IsEquivalence (rotate C)
|
Mathlib_CategoryTheory_Triangulated_Rotate
|
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
X : C
inst✝ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)
⊢ IsEquivalence (invRotate C)
|
/-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanprover-community/mathlib"@"94d4e70e97c36c896cb70fb42821acfed040de60"
/-!
# Rotate
This file adds the ability to rotate triangles and triangle morphisms.
It also shows that rotation gives an equivalence on the category of triangles.
-/
noncomputable section
open CategoryTheory
open CategoryTheory.Preadditive
open CategoryTheory.Limits
universe v v₀ v₁ v₂ u u₀ u₁ u₂
namespace CategoryTheory.Pretriangulated
open CategoryTheory.Category
variable {C : Type u} [Category.{v} C] [Preadditive C]
variable [HasShift C ℤ]
variable (X : C)
/-- If you rotate a triangle, you get another triangle.
Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `rotate` gives a triangle of the form:
```
g h -f⟦1⟧'
Y ───> Z ───> X⟦1⟧ ───> Y⟦1⟧
```
-/
@[simps!]
def Triangle.rotate (T : Triangle C) : Triangle C :=
Triangle.mk T.mor₂ T.mor₃ (-T.mor₁⟦1⟧')
#align category_theory.pretriangulated.triangle.rotate CategoryTheory.Pretriangulated.Triangle.rotate
section
/-- Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `invRotate` gives a triangle that can be thought of as:
```
-h⟦-1⟧' f g
Z⟦-1⟧ ───> X ───> Y ───> Z
```
(note that this diagram doesn't technically fit the definition of triangle, as `Z⟦-1⟧⟦1⟧` is
not necessarily equal to `Z`, but it is isomorphic, by the `counitIso` of `shiftEquiv C 1`)
-/
@[simps!]
def Triangle.invRotate (T : Triangle C) : Triangle C :=
Triangle.mk (-T.mor₃⟦(-1 : ℤ)⟧' ≫ (shiftEquiv C (1 : ℤ)).unitIso.inv.app _) (T.mor₁)
(T.mor₂ ≫ (shiftEquiv C (1 : ℤ)).counitIso.inv.app _ )
#align category_theory.pretriangulated.triangle.inv_rotate CategoryTheory.Pretriangulated.Triangle.invRotate
end
attribute [local simp] shift_shift_neg' shift_neg_shift'
shift_shiftFunctorCompIsoId_add_neg_self_inv_app
shift_shiftFunctorCompIsoId_add_neg_self_hom_app
variable (C)
/-- Rotating triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def rotate : Triangle C ⥤ Triangle C
where
obj := Triangle.rotate
map f :=
{ hom₁ := f.hom₂
hom₂ := f.hom₃
hom₃ := f.hom₁⟦1⟧'
comm₃ := by
dsimp
simp only [comp_neg, neg_comp, ← Functor.map_comp, f.comm₁] }
#align category_theory.pretriangulated.rotate CategoryTheory.Pretriangulated.rotate
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
dsimp
simp only [neg_comp, assoc, comp_neg, neg_inj, ← Functor.map_comp_assoc, ← f.comm₃]
rw [Functor.map_comp, assoc]
erw [← NatTrans.naturality]
rfl
comm₃ := by
erw [← reassoc_of% f.comm₂, Category.assoc, ← NatTrans.naturality]
rfl }
#align category_theory.pretriangulated.inv_rotate CategoryTheory.Pretriangulated.invRotate
variable {C}
variable [∀ n : ℤ, Functor.Additive (shiftFunctor C n)]
/-- The unit isomorphism of the auto-equivalence of categories `triangleRotation C` of
`Triangle C` given by the rotation of triangles. -/
@[simps!]
def rotCompInvRot : 𝟭 (Triangle C) ≅ rotate C ⋙ invRotate C :=
NatIso.ofComponents fun T => Triangle.isoMk _ _
((shiftEquiv C (1 : ℤ)).unitIso.app T.obj₁) (Iso.refl _) (Iso.refl _)
#align category_theory.pretriangulated.rot_comp_inv_rot CategoryTheory.Pretriangulated.rotCompInvRot
/-- The counit isomorphism of the auto-equivalence of categories `triangleRotation C` of
`Triangle C` given by the rotation of triangles. -/
@[simps!]
def invRotCompRot : invRotate C ⋙ rotate C ≅ 𝟭 (Triangle C) :=
NatIso.ofComponents fun T => Triangle.isoMk _ _ (Iso.refl _) (Iso.refl _)
((shiftEquiv C (1 : ℤ)).counitIso.app T.obj₃)
#align category_theory.pretriangulated.inv_rot_comp_rot CategoryTheory.Pretriangulated.invRotCompRot
variable (C)
/-- Rotating triangles gives an auto-equivalence on the category of triangles in `C`.
-/
@[simps]
def triangleRotation : Equivalence (Triangle C) (Triangle C)
where
functor := rotate C
inverse := invRotate C
unitIso := rotCompInvRot
counitIso := invRotCompRot
#align category_theory.pretriangulated.triangle_rotation CategoryTheory.Pretriangulated.triangleRotation
variable {C}
instance : IsEquivalence (rotate C) := by
change IsEquivalence (triangleRotation C).functor
infer_instance
instance : IsEquivalence (invRotate C) := by
|
change IsEquivalence (triangleRotation C).inverse
|
instance : IsEquivalence (invRotate C) := by
|
Mathlib.CategoryTheory.Triangulated.Rotate.161_0.sGRpfSsY1fG2rGq
|
instance : IsEquivalence (invRotate C)
|
Mathlib_CategoryTheory_Triangulated_Rotate
|
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
X : C
inst✝ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)
⊢ IsEquivalence (triangleRotation C).inverse
|
/-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Triangulated.Basic
#align_import category_theory.triangulated.rotate from "leanprover-community/mathlib"@"94d4e70e97c36c896cb70fb42821acfed040de60"
/-!
# Rotate
This file adds the ability to rotate triangles and triangle morphisms.
It also shows that rotation gives an equivalence on the category of triangles.
-/
noncomputable section
open CategoryTheory
open CategoryTheory.Preadditive
open CategoryTheory.Limits
universe v v₀ v₁ v₂ u u₀ u₁ u₂
namespace CategoryTheory.Pretriangulated
open CategoryTheory.Category
variable {C : Type u} [Category.{v} C] [Preadditive C]
variable [HasShift C ℤ]
variable (X : C)
/-- If you rotate a triangle, you get another triangle.
Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `rotate` gives a triangle of the form:
```
g h -f⟦1⟧'
Y ───> Z ───> X⟦1⟧ ───> Y⟦1⟧
```
-/
@[simps!]
def Triangle.rotate (T : Triangle C) : Triangle C :=
Triangle.mk T.mor₂ T.mor₃ (-T.mor₁⟦1⟧')
#align category_theory.pretriangulated.triangle.rotate CategoryTheory.Pretriangulated.Triangle.rotate
section
/-- Given a triangle of the form:
```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
applying `invRotate` gives a triangle that can be thought of as:
```
-h⟦-1⟧' f g
Z⟦-1⟧ ───> X ───> Y ───> Z
```
(note that this diagram doesn't technically fit the definition of triangle, as `Z⟦-1⟧⟦1⟧` is
not necessarily equal to `Z`, but it is isomorphic, by the `counitIso` of `shiftEquiv C 1`)
-/
@[simps!]
def Triangle.invRotate (T : Triangle C) : Triangle C :=
Triangle.mk (-T.mor₃⟦(-1 : ℤ)⟧' ≫ (shiftEquiv C (1 : ℤ)).unitIso.inv.app _) (T.mor₁)
(T.mor₂ ≫ (shiftEquiv C (1 : ℤ)).counitIso.inv.app _ )
#align category_theory.pretriangulated.triangle.inv_rotate CategoryTheory.Pretriangulated.Triangle.invRotate
end
attribute [local simp] shift_shift_neg' shift_neg_shift'
shift_shiftFunctorCompIsoId_add_neg_self_inv_app
shift_shiftFunctorCompIsoId_add_neg_self_hom_app
variable (C)
/-- Rotating triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def rotate : Triangle C ⥤ Triangle C
where
obj := Triangle.rotate
map f :=
{ hom₁ := f.hom₂
hom₂ := f.hom₃
hom₃ := f.hom₁⟦1⟧'
comm₃ := by
dsimp
simp only [comp_neg, neg_comp, ← Functor.map_comp, f.comm₁] }
#align category_theory.pretriangulated.rotate CategoryTheory.Pretriangulated.rotate
/-- The inverse rotation of triangles gives an endofunctor on the category of triangles in `C`.
-/
@[simps]
def invRotate : Triangle C ⥤ Triangle C
where
obj := Triangle.invRotate
map f :=
{ hom₁ := f.hom₃⟦-1⟧'
hom₂ := f.hom₁
hom₃ := f.hom₂
comm₁ := by
dsimp
simp only [neg_comp, assoc, comp_neg, neg_inj, ← Functor.map_comp_assoc, ← f.comm₃]
rw [Functor.map_comp, assoc]
erw [← NatTrans.naturality]
rfl
comm₃ := by
erw [← reassoc_of% f.comm₂, Category.assoc, ← NatTrans.naturality]
rfl }
#align category_theory.pretriangulated.inv_rotate CategoryTheory.Pretriangulated.invRotate
variable {C}
variable [∀ n : ℤ, Functor.Additive (shiftFunctor C n)]
/-- The unit isomorphism of the auto-equivalence of categories `triangleRotation C` of
`Triangle C` given by the rotation of triangles. -/
@[simps!]
def rotCompInvRot : 𝟭 (Triangle C) ≅ rotate C ⋙ invRotate C :=
NatIso.ofComponents fun T => Triangle.isoMk _ _
((shiftEquiv C (1 : ℤ)).unitIso.app T.obj₁) (Iso.refl _) (Iso.refl _)
#align category_theory.pretriangulated.rot_comp_inv_rot CategoryTheory.Pretriangulated.rotCompInvRot
/-- The counit isomorphism of the auto-equivalence of categories `triangleRotation C` of
`Triangle C` given by the rotation of triangles. -/
@[simps!]
def invRotCompRot : invRotate C ⋙ rotate C ≅ 𝟭 (Triangle C) :=
NatIso.ofComponents fun T => Triangle.isoMk _ _ (Iso.refl _) (Iso.refl _)
((shiftEquiv C (1 : ℤ)).counitIso.app T.obj₃)
#align category_theory.pretriangulated.inv_rot_comp_rot CategoryTheory.Pretriangulated.invRotCompRot
variable (C)
/-- Rotating triangles gives an auto-equivalence on the category of triangles in `C`.
-/
@[simps]
def triangleRotation : Equivalence (Triangle C) (Triangle C)
where
functor := rotate C
inverse := invRotate C
unitIso := rotCompInvRot
counitIso := invRotCompRot
#align category_theory.pretriangulated.triangle_rotation CategoryTheory.Pretriangulated.triangleRotation
variable {C}
instance : IsEquivalence (rotate C) := by
change IsEquivalence (triangleRotation C).functor
infer_instance
instance : IsEquivalence (invRotate C) := by
change IsEquivalence (triangleRotation C).inverse
|
infer_instance
|
instance : IsEquivalence (invRotate C) := by
change IsEquivalence (triangleRotation C).inverse
|
Mathlib.CategoryTheory.Triangulated.Rotate.161_0.sGRpfSsY1fG2rGq
|
instance : IsEquivalence (invRotate C)
|
Mathlib_CategoryTheory_Triangulated_Rotate
|
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
⊢ DevosMulRel x y ↔
card (x.1 * x.2) < card (y.1 * y.2) ∨
card (x.1 * x.2) = card (y.1 * y.2) ∧ card y.1 + card y.2 < card x.1 + card x.2 ∨
card (x.1 * x.2) = card (y.1 * y.2) ∧ card x.1 + card x.2 = card y.1 + card y.2 ∧ card x.1 < card y.1
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
|
simp [DevosMulRel, Prod.lex_iff, and_or_left]
|
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.77_0.yGTPJO6UphimMFs
|
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
⊢ Set.WellFoundedOn {x | Finset.Nonempty x.1 ∧ Finset.Nonempty x.2} DevosMulRel
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
|
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
|
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.98_0.yGTPJO6UphimMFs
|
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop)
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
n : ℕ
⊢ Set.WellFoundedOn ({x | Finset.Nonempty x.1 ∧ Finset.Nonempty x.2} ∩ (fun x => card (x.1 * x.2)) ⁻¹' {n})
((fun x x_1 => x > x_1) on fun x => card x.1 + card x.2)
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
|
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
|
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.98_0.yGTPJO6UphimMFs
|
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop)
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
⊢ min (minOrder α) ↑(card s + card t - 1) ≤ ↑(card (s * t))
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
|
set x := (s, t) with hx
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x✝ y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
x : Finset α × Finset α := (s, t)
hx : x = (s, t)
⊢ min (minOrder α) ↑(card s + card t - 1) ≤ ↑(card (s * t))
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
|
clear_value x
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x✝ y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
x : Finset α × Finset α
hx : x = (s, t)
⊢ min (minOrder α) ↑(card s + card t - 1) ≤ ↑(card (s * t))
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
|
simp only [Prod.ext_iff] at hx
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x✝ y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
x : Finset α × Finset α
hx : x.1 = s ∧ x.2 = t
⊢ min (minOrder α) ↑(card s + card t - 1) ≤ ↑(card (s * t))
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
|
obtain ⟨rfl, rfl⟩ := hx
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x✝ y x : Finset α × Finset α
hs : Finset.Nonempty x.1
ht : Finset.Nonempty x.2
⊢ min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ ↑(card (x.1 * x.2))
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
|
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x✝ y x : Finset α × Finset α
hs : Finset.Nonempty x.1
ht : Finset.Nonempty x.2
⊢ ∀ y ∈ {x | Finset.Nonempty x.1 ∧ Finset.Nonempty x.2},
(∀ z ∈ {x | Finset.Nonempty x.1 ∧ Finset.Nonempty x.2},
DevosMulRel z y → (fun x => min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ ↑(card (x.1 * x.2))) z) →
(fun x => min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ ↑(card (x.1 * x.2))) y
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
|
clear! x
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
⊢ ∀ y ∈ {x | Finset.Nonempty x.1 ∧ Finset.Nonempty x.2},
(∀ z ∈ {x | Finset.Nonempty x.1 ∧ Finset.Nonempty x.2},
DevosMulRel z y → (fun x => min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ ↑(card (x.1 * x.2))) z) →
(fun x => min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ ↑(card (x.1 * x.2))) y
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
|
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro.mk.intro
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty (s, t).1
ht : Finset.Nonempty (s, t).2
ih :
∀ z ∈ {x | Finset.Nonempty x.1 ∧ Finset.Nonempty x.2},
DevosMulRel z (s, t) → (fun x => min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ ↑(card (x.1 * x.2))) z
⊢ min (minOrder α) ↑(card (s, t).1 + card (s, t).2 - 1) ≤ ↑(card ((s, t).1 * (s, t).2))
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
|
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro.mk.intro
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
⊢ minOrder α ≤ ↑(card (s * t)) ∨ card s + card t ≤ card (s * t) + 1
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
|
obtain hts | hst := lt_or_le t.card s.card
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro.mk.intro.inl
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hts : card t < card s
⊢ minOrder α ≤ ↑(card (s * t)) ∨ card s + card t ≤ card (s * t) + 1
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
·
|
simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
·
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hts : card t < card s
⊢ card ((t⁻¹, s⁻¹).1 * (t⁻¹, s⁻¹).2) = card ((s, t).1 * (s, t).2) ∧
card (t⁻¹, s⁻¹).1 + card (t⁻¹, s⁻¹).2 = card (s, t).1 + card (s, t).2 ∧ card (t⁻¹, s⁻¹).1 < card (s, t).1
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
|
simpa only [← mul_inv_rev, add_comm, card_inv, true_and]
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro.mk.intro.inr
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
⊢ minOrder α ≤ ↑(card (s * t)) ∨ card s + card t ≤ card (s * t) + 1
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
|
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro.mk.intro.inr.inl.intro
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
t : Finset α
ht : Finset.Nonempty t
a : α
hs : Finset.Nonempty {a}
ih :
∀ (a_1 b : Finset α),
Finset.Nonempty a_1 →
Finset.Nonempty b →
DevosMulRel (a_1, b) ({a}, t) → minOrder α ≤ ↑(card (a_1 * b)) ∨ card a_1 + card b ≤ card (a_1 * b) + 1
hst : card {a} ≤ card t
⊢ minOrder α ≤ ↑(card ({a} * t)) ∨ card {a} + card t ≤ card ({a} * t) + 1
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
·
|
simp [add_comm]
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
·
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro.mk.intro.inr.inr.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
⊢ minOrder α ≤ ↑(card (s * t)) ∨ card s + card t ≤ card (s * t) + 1
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
|
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
⊢ op (b⁻¹ * a) • b = a
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by
|
simp
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
g : α
hg : g ≠ 1
hgs : Finset.Nonempty (s ∩ op g • s)
⊢ minOrder α ≤ ↑(card (s * t)) ∨ card s + card t ≤ card (s * t) + 1
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
|
obtain hsg | hsg := eq_or_ne (op g • s) s
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
g : α
hg : g ≠ 1
hgs : Finset.Nonempty (s ∩ op g • s)
hsg : op g • s = s
⊢ minOrder α ≤ ↑(card (s * t)) ∨ card s + card t ≤ card (s * t) + 1
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
·
|
have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
·
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
g : α
hg : g ≠ 1
hgs : Finset.Nonempty (s ∩ op g • s)
hsg : op g • s = s
⊢ ↑(zpowers g) ⊆ a⁻¹ • ↑s
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
|
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case refine_1
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
g : α
hg : g ≠ 1
hgs : Finset.Nonempty (s ∩ op g • s)
hsg : op g • s = s
c : α
hc : (fun x => x ∈ a⁻¹ • ↑s) c
⊢ (fun x => x ∈ a⁻¹ • ↑s) (c * g)
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
·
|
rw [← hsg, coe_smul_finset, smul_comm]
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
·
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case refine_1
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
g : α
hg : g ≠ 1
hgs : Finset.Nonempty (s ∩ op g • s)
hsg : op g • s = s
c : α
hc : (fun x => x ∈ a⁻¹ • ↑s) c
⊢ (fun x => x ∈ op g • a⁻¹ • ↑s) (c * g)
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
|
exact Set.smul_mem_smul_set hc
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case refine_2
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
g : α
hg : g ≠ 1
hgs : Finset.Nonempty (s ∩ op g • s)
hsg : op g • s = s
c : α
hc : (fun x => x ∈ a⁻¹ • ↑s) c
⊢ (fun x => x ∈ a⁻¹ • ↑s) (c * g⁻¹)
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
·
|
simp only
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
·
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case refine_2
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
g : α
hg : g ≠ 1
hgs : Finset.Nonempty (s ∩ op g • s)
hsg : op g • s = s
c : α
hc : (fun x => x ∈ a⁻¹ • ↑s) c
⊢ c * g⁻¹ ∈ a⁻¹ • ↑s
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
|
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
g : α
hg : g ≠ 1
hgs : Finset.Nonempty (s ∩ op g • s)
hsg : op g • s = s
hS : ↑(zpowers g) ⊆ a⁻¹ • ↑s
⊢ minOrder α ≤ ↑(card (s * t)) ∨ card s + card t ≤ card (s * t) + 1
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
|
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
g : α
hg : g ≠ 1
hgs : Finset.Nonempty (s ∩ op g • s)
hsg : op g • s = s
hS : ↑(zpowers g) ⊆ a⁻¹ • ↑s
⊢ Nat.card ↑(a⁻¹ • ↑s) = card s
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
|
rw [← coe_smul_finset]
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
g : α
hg : g ≠ 1
hgs : Finset.Nonempty (s ∩ op g • s)
hsg : op g • s = s
hS : ↑(zpowers g) ⊆ a⁻¹ • ↑s
⊢ Nat.card ↑↑(a⁻¹ • s) = card s
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
|
simp [-coe_smul_finset]
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
g : α
hg : g ≠ 1
hgs : Finset.Nonempty (s ∩ op g • s)
hsg : op g • s ≠ s
⊢ minOrder α ≤ ↑(card (s * t)) ∨ card s + card t ≤ card (s * t) + 1
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
|
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
g : α
hg : g ≠ 1
hgs : Finset.Nonempty (s ∩ op g • s)
hsg : card (s ∩ op g • s) < card s
⊢ minOrder α ≤ ↑(card (s * t)) ∨ card s + card t ≤ card (s * t) + 1
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
|
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
g : α
hg : g ≠ 1
hgs : Finset.Nonempty (s ∩ op g • s)
hsg : card (s ∩ op g • s) < card s
aux1 : card ((mulETransformLeft g (s, t)).1 * (mulETransformLeft g (s, t)).2) ≤ card ((s, t).1 * (s, t).2)
⊢ minOrder α ≤ ↑(card (s * t)) ∨ card s + card t ≤ card (s * t) + 1
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
|
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
g : α
hg : g ≠ 1
hgs : Finset.Nonempty (s ∩ op g • s)
hsg : card (s ∩ op g • s) < card s
aux1 : card ((mulETransformLeft g (s, t)).1 * (mulETransformLeft g (s, t)).2) ≤ card ((s, t).1 * (s, t).2)
aux2 : card ((mulETransformRight g (s, t)).1 * (mulETransformRight g (s, t)).2) ≤ card ((s, t).1 * (s, t).2)
⊢ minOrder α ≤ ↑(card (s * t)) ∨ card s + card t ≤ card (s * t) + 1
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
|
obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr.inl
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
g : α
hg : g ≠ 1
hgs : Finset.Nonempty (s ∩ op g • s)
hsg : card (s ∩ op g • s) < card s
aux1 : card ((mulETransformLeft g (s, t)).1 * (mulETransformLeft g (s, t)).2) ≤ card ((s, t).1 * (s, t).2)
aux2 : card ((mulETransformRight g (s, t)).1 * (mulETransformRight g (s, t)).2) ≤ card ((s, t).1 * (s, t).2)
hgt : Disjoint t (g⁻¹ • t)
⊢ minOrder α ≤ ↑(card (s * t)) ∨ card s + card t ≤ card (s * t) + 1
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
·
|
rw [← card_smul_finset g⁻¹ t]
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
·
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr.inl
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
g : α
hg : g ≠ 1
hgs : Finset.Nonempty (s ∩ op g • s)
hsg : card (s ∩ op g • s) < card s
aux1 : card ((mulETransformLeft g (s, t)).1 * (mulETransformLeft g (s, t)).2) ≤ card ((s, t).1 * (s, t).2)
aux2 : card ((mulETransformRight g (s, t)).1 * (mulETransformRight g (s, t)).2) ≤ card ((s, t).1 * (s, t).2)
hgt : Disjoint t (g⁻¹ • t)
⊢ minOrder α ≤ ↑(card (s * t)) ∨ card s + card (g⁻¹ • t) ≤ card (s * t) + 1
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
· rw [← card_smul_finset g⁻¹ t]
|
refine' Or.inr ((add_le_add_right hst _).trans _)
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
· rw [← card_smul_finset g⁻¹ t]
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr.inl
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
g : α
hg : g ≠ 1
hgs : Finset.Nonempty (s ∩ op g • s)
hsg : card (s ∩ op g • s) < card s
aux1 : card ((mulETransformLeft g (s, t)).1 * (mulETransformLeft g (s, t)).2) ≤ card ((s, t).1 * (s, t).2)
aux2 : card ((mulETransformRight g (s, t)).1 * (mulETransformRight g (s, t)).2) ≤ card ((s, t).1 * (s, t).2)
hgt : Disjoint t (g⁻¹ • t)
⊢ card t + card (g⁻¹ • t) ≤ card (s * t) + 1
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
· rw [← card_smul_finset g⁻¹ t]
refine' Or.inr ((add_le_add_right hst _).trans _)
|
rw [← card_union_eq hgt]
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
· rw [← card_smul_finset g⁻¹ t]
refine' Or.inr ((add_le_add_right hst _).trans _)
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr.inl
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
g : α
hg : g ≠ 1
hgs : Finset.Nonempty (s ∩ op g • s)
hsg : card (s ∩ op g • s) < card s
aux1 : card ((mulETransformLeft g (s, t)).1 * (mulETransformLeft g (s, t)).2) ≤ card ((s, t).1 * (s, t).2)
aux2 : card ((mulETransformRight g (s, t)).1 * (mulETransformRight g (s, t)).2) ≤ card ((s, t).1 * (s, t).2)
hgt : Disjoint t (g⁻¹ • t)
⊢ card (t ∪ g⁻¹ • t) ≤ card (s * t) + 1
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
· rw [← card_smul_finset g⁻¹ t]
refine' Or.inr ((add_le_add_right hst _).trans _)
rw [← card_union_eq hgt]
|
exact (card_le_card_mul_left _ hgs).trans (le_add_of_le_left aux1)
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
· rw [← card_smul_finset g⁻¹ t]
refine' Or.inr ((add_le_add_right hst _).trans _)
rw [← card_union_eq hgt]
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr.inr
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
g : α
hg : g ≠ 1
hgs : Finset.Nonempty (s ∩ op g • s)
hsg : card (s ∩ op g • s) < card s
aux1 : card ((mulETransformLeft g (s, t)).1 * (mulETransformLeft g (s, t)).2) ≤ card ((s, t).1 * (s, t).2)
aux2 : card ((mulETransformRight g (s, t)).1 * (mulETransformRight g (s, t)).2) ≤ card ((s, t).1 * (s, t).2)
hgt : Finset.Nonempty (t ∩ g⁻¹ • t)
⊢ minOrder α ≤ ↑(card (s * t)) ∨ card s + card t ≤ card (s * t) + 1
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
· rw [← card_smul_finset g⁻¹ t]
refine' Or.inr ((add_le_add_right hst _).trans _)
rw [← card_union_eq hgt]
exact (card_le_card_mul_left _ hgs).trans (le_add_of_le_left aux1)
-- Else, we're done by induction on either `(s', t')` or `(s'', t'')` depending on whether
-- `|s| + |t| ≤ |s'| + |t'|` or `|s| + |t| < |s''| + |t''|`. One of those two inequalities must
-- hold since `2 * (|s| + |t|) = |s'| + |t'| + |s''| + |t''|`.
|
obtain hstg | hstg := le_or_lt_of_add_le_add (MulETransform.card g (s, t)).ge
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
· rw [← card_smul_finset g⁻¹ t]
refine' Or.inr ((add_le_add_right hst _).trans _)
rw [← card_union_eq hgt]
exact (card_le_card_mul_left _ hgs).trans (le_add_of_le_left aux1)
-- Else, we're done by induction on either `(s', t')` or `(s'', t'')` depending on whether
-- `|s| + |t| ≤ |s'| + |t'|` or `|s| + |t| < |s''| + |t''|`. One of those two inequalities must
-- hold since `2 * (|s| + |t|) = |s'| + |t'| + |s''| + |t''|`.
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr.inr.inl
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
g : α
hg : g ≠ 1
hgs : Finset.Nonempty (s ∩ op g • s)
hsg : card (s ∩ op g • s) < card s
aux1 : card ((mulETransformLeft g (s, t)).1 * (mulETransformLeft g (s, t)).2) ≤ card ((s, t).1 * (s, t).2)
aux2 : card ((mulETransformRight g (s, t)).1 * (mulETransformRight g (s, t)).2) ≤ card ((s, t).1 * (s, t).2)
hgt : Finset.Nonempty (t ∩ g⁻¹ • t)
hstg : card (s, t).1 + card (s, t).2 ≤ card (mulETransformLeft g (s, t)).1 + card (mulETransformLeft g (s, t)).2
⊢ minOrder α ≤ ↑(card (s * t)) ∨ card s + card t ≤ card (s * t) + 1
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
· rw [← card_smul_finset g⁻¹ t]
refine' Or.inr ((add_le_add_right hst _).trans _)
rw [← card_union_eq hgt]
exact (card_le_card_mul_left _ hgs).trans (le_add_of_le_left aux1)
-- Else, we're done by induction on either `(s', t')` or `(s'', t'')` depending on whether
-- `|s| + |t| ≤ |s'| + |t'|` or `|s| + |t| < |s''| + |t''|`. One of those two inequalities must
-- hold since `2 * (|s| + |t|) = |s'| + |t'| + |s''| + |t''|`.
obtain hstg | hstg := le_or_lt_of_add_le_add (MulETransform.card g (s, t)).ge
·
|
exact (ih _ _ hgs (hgt.mono inter_subset_union) <| devosMulRel_of_le_of_le aux1 hstg hsg).imp
(WithTop.coe_le_coe.2 aux1).trans' fun h ↦ hstg.trans <| h.trans <| add_le_add_right aux1 _
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
· rw [← card_smul_finset g⁻¹ t]
refine' Or.inr ((add_le_add_right hst _).trans _)
rw [← card_union_eq hgt]
exact (card_le_card_mul_left _ hgs).trans (le_add_of_le_left aux1)
-- Else, we're done by induction on either `(s', t')` or `(s'', t'')` depending on whether
-- `|s| + |t| ≤ |s'| + |t'|` or `|s| + |t| < |s''| + |t''|`. One of those two inequalities must
-- hold since `2 * (|s| + |t|) = |s'| + |t'| + |s''| + |t''|`.
obtain hstg | hstg := le_or_lt_of_add_le_add (MulETransform.card g (s, t)).ge
·
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
case intro.mk.intro.inr.inr.intro.intro.intro.intro.intro.intro.inr.inr.inr
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
ih :
∀ (a b : Finset α),
Finset.Nonempty a →
Finset.Nonempty b → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(card (a * b)) ∨ card a + card b ≤ card (a * b) + 1
hst : card s ≤ card t
a : α
ha : a ∈ ↑s
b : α
hb : b ∈ ↑s
hab : a ≠ b
g : α
hg : g ≠ 1
hgs : Finset.Nonempty (s ∩ op g • s)
hsg : card (s ∩ op g • s) < card s
aux1 : card ((mulETransformLeft g (s, t)).1 * (mulETransformLeft g (s, t)).2) ≤ card ((s, t).1 * (s, t).2)
aux2 : card ((mulETransformRight g (s, t)).1 * (mulETransformRight g (s, t)).2) ≤ card ((s, t).1 * (s, t).2)
hgt : Finset.Nonempty (t ∩ g⁻¹ • t)
hstg : card (s, t).1 + card (s, t).2 < card (mulETransformRight g (s, t)).1 + card (mulETransformRight g (s, t)).2
⊢ minOrder α ≤ ↑(card (s * t)) ∨ card s + card t ≤ card (s * t) + 1
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
· rw [← card_smul_finset g⁻¹ t]
refine' Or.inr ((add_le_add_right hst _).trans _)
rw [← card_union_eq hgt]
exact (card_le_card_mul_left _ hgs).trans (le_add_of_le_left aux1)
-- Else, we're done by induction on either `(s', t')` or `(s'', t'')` depending on whether
-- `|s| + |t| ≤ |s'| + |t'|` or `|s| + |t| < |s''| + |t''|`. One of those two inequalities must
-- hold since `2 * (|s| + |t|) = |s'| + |t'| + |s''| + |t''|`.
obtain hstg | hstg := le_or_lt_of_add_le_add (MulETransform.card g (s, t)).ge
· exact (ih _ _ hgs (hgt.mono inter_subset_union) <| devosMulRel_of_le_of_le aux1 hstg hsg).imp
(WithTop.coe_le_coe.2 aux1).trans' fun h ↦ hstg.trans <| h.trans <| add_le_add_right aux1 _
·
|
exact (ih _ _ (hgs.mono inter_subset_union) hgt <| devosMulRel_of_le aux2 hstg).imp
(WithTop.coe_le_coe.2 aux2).trans' fun h ↦
hstg.le.trans <| h.trans <| add_le_add_right aux2 _
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
· rw [← card_smul_finset g⁻¹ t]
refine' Or.inr ((add_le_add_right hst _).trans _)
rw [← card_union_eq hgt]
exact (card_le_card_mul_left _ hgs).trans (le_add_of_le_left aux1)
-- Else, we're done by induction on either `(s', t')` or `(s'', t'')` depending on whether
-- `|s| + |t| ≤ |s'| + |t'|` or `|s| + |t| < |s''| + |t''|`. One of those two inequalities must
-- hold since `2 * (|s| + |t|) = |s'| + |t'| + |s''| + |t''|`.
obtain hstg | hstg := le_or_lt_of_add_le_add (MulETransform.card g (s, t)).ge
· exact (ih _ _ hgs (hgt.mono inter_subset_union) <| devosMulRel_of_le_of_le aux1 hstg hsg).imp
(WithTop.coe_le_coe.2 aux1).trans' fun h ↦ hstg.trans <| h.trans <| add_le_add_right aux1 _
·
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.109_0.yGTPJO6UphimMFs
|
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
α : Type u_1
inst✝¹ : Group α
inst✝ : DecidableEq α
x y : Finset α × Finset α
s t : Finset α
h : IsTorsionFree α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
⊢ card s + card t - 1 ≤ card (s * t)
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
· rw [← card_smul_finset g⁻¹ t]
refine' Or.inr ((add_le_add_right hst _).trans _)
rw [← card_union_eq hgt]
exact (card_le_card_mul_left _ hgs).trans (le_add_of_le_left aux1)
-- Else, we're done by induction on either `(s', t')` or `(s'', t'')` depending on whether
-- `|s| + |t| ≤ |s'| + |t'|` or `|s| + |t| < |s''| + |t''|`. One of those two inequalities must
-- hold since `2 * (|s| + |t|) = |s'| + |t'| + |s''| + |t''|`.
obtain hstg | hstg := le_or_lt_of_add_le_add (MulETransform.card g (s, t)).ge
· exact (ih _ _ hgs (hgt.mono inter_subset_union) <| devosMulRel_of_le_of_le aux1 hstg hsg).imp
(WithTop.coe_le_coe.2 aux1).trans' fun h ↦ hstg.trans <| h.trans <| add_le_add_right aux1 _
· exact (ih _ _ (hgs.mono inter_subset_union) hgt <| devosMulRel_of_le aux2 hstg).imp
(WithTop.coe_le_coe.2 aux2).trans' fun h ↦
hstg.le.trans <| h.trans <| add_le_add_right aux2 _
/-- The **Cauchy-Davenport Theorem** for torsion-free groups. The size of `s * t` is lower-bounded
by `|s| + |t| - 1`. -/
@[to_additive "The **Cauchy-Davenport theorem** for torsion-free groups. The size of `s + t` is
lower-bounded by `|s| + |t| - 1`."]
lemma Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul (h : IsTorsionFree α)
(hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card := by
|
simpa only [h.minOrder, min_eq_right, le_top, Nat.cast_le] using Finset.min_le_card_mul hs ht
|
/-- The **Cauchy-Davenport Theorem** for torsion-free groups. The size of `s * t` is lower-bounded
by `|s| + |t| - 1`. -/
@[to_additive "The **Cauchy-Davenport theorem** for torsion-free groups. The size of `s + t` is
lower-bounded by `|s| + |t| - 1`."]
lemma Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul (h : IsTorsionFree α)
(hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card := by
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.182_0.yGTPJO6UphimMFs
|
/-- The **Cauchy-Davenport Theorem** for torsion-free groups. The size of `s * t` is lower-bounded
by `|s| + |t| - 1`. -/
@[to_additive "The **Cauchy-Davenport theorem** for torsion-free groups. The size of `s + t` is
lower-bounded by `|s| + |t| - 1`."]
lemma Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul (h : IsTorsionFree α)
(hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
α : Type u_1
p : ℕ
hp : Nat.Prime p
s t : Finset (ZMod p)
hs : Finset.Nonempty s
ht : Finset.Nonempty t
⊢ min p (Finset.card s + Finset.card t - 1) ≤ Finset.card (s + t)
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
· rw [← card_smul_finset g⁻¹ t]
refine' Or.inr ((add_le_add_right hst _).trans _)
rw [← card_union_eq hgt]
exact (card_le_card_mul_left _ hgs).trans (le_add_of_le_left aux1)
-- Else, we're done by induction on either `(s', t')` or `(s'', t'')` depending on whether
-- `|s| + |t| ≤ |s'| + |t'|` or `|s| + |t| < |s''| + |t''|`. One of those two inequalities must
-- hold since `2 * (|s| + |t|) = |s'| + |t'| + |s''| + |t''|`.
obtain hstg | hstg := le_or_lt_of_add_le_add (MulETransform.card g (s, t)).ge
· exact (ih _ _ hgs (hgt.mono inter_subset_union) <| devosMulRel_of_le_of_le aux1 hstg hsg).imp
(WithTop.coe_le_coe.2 aux1).trans' fun h ↦ hstg.trans <| h.trans <| add_le_add_right aux1 _
· exact (ih _ _ (hgs.mono inter_subset_union) hgt <| devosMulRel_of_le aux2 hstg).imp
(WithTop.coe_le_coe.2 aux2).trans' fun h ↦
hstg.le.trans <| h.trans <| add_le_add_right aux2 _
/-- The **Cauchy-Davenport Theorem** for torsion-free groups. The size of `s * t` is lower-bounded
by `|s| + |t| - 1`. -/
@[to_additive "The **Cauchy-Davenport theorem** for torsion-free groups. The size of `s + t` is
lower-bounded by `|s| + |t| - 1`."]
lemma Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul (h : IsTorsionFree α)
(hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card := by
simpa only [h.minOrder, min_eq_right, le_top, Nat.cast_le] using Finset.min_le_card_mul hs ht
end General
/-! ### $$ℤ/nℤ$$ -/
/-- The **Cauchy-Davenport Theorem**. If `s`, `t` are nonempty sets in $$ℤ/pℤ$$, then the size of
`s + t` is lower-bounded by `|s| + |t| - 1`, unless this quantity is greater than `p`. -/
lemma ZMod.min_le_card_add {p : ℕ} (hp : p.Prime) {s t : Finset (ZMod p)} (hs : s.Nonempty)
(ht : t.Nonempty) : min p (s.card + t.card - 1) ≤ (s + t).card := by
|
simpa only [ZMod.minOrder_of_prime hp, min_le_iff, Nat.cast_le] using Finset.min_le_card_add hs ht
|
/-- The **Cauchy-Davenport Theorem**. If `s`, `t` are nonempty sets in $$ℤ/pℤ$$, then the size of
`s + t` is lower-bounded by `|s| + |t| - 1`, unless this quantity is greater than `p`. -/
lemma ZMod.min_le_card_add {p : ℕ} (hp : p.Prime) {s t : Finset (ZMod p)} (hs : s.Nonempty)
(ht : t.Nonempty) : min p (s.card + t.card - 1) ≤ (s + t).card := by
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.194_0.yGTPJO6UphimMFs
|
/-- The **Cauchy-Davenport Theorem**. If `s`, `t` are nonempty sets in $$ℤ/pℤ$$, then the size of
`s + t` is lower-bounded by `|s| + |t| - 1`, unless this quantity is greater than `p`. -/
lemma ZMod.min_le_card_add {p : ℕ} (hp : p.Prime) {s t : Finset (ZMod p)} (hs : s.Nonempty)
(ht : t.Nonempty) : min p (s.card + t.card - 1) ≤ (s + t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
α : Type u_1
inst✝⁴ : LinearOrder α
inst✝³ : Semigroup α
inst✝² : IsCancelMul α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
⊢ card s + card t - 1 ≤ card (s * t)
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
· rw [← card_smul_finset g⁻¹ t]
refine' Or.inr ((add_le_add_right hst _).trans _)
rw [← card_union_eq hgt]
exact (card_le_card_mul_left _ hgs).trans (le_add_of_le_left aux1)
-- Else, we're done by induction on either `(s', t')` or `(s'', t'')` depending on whether
-- `|s| + |t| ≤ |s'| + |t'|` or `|s| + |t| < |s''| + |t''|`. One of those two inequalities must
-- hold since `2 * (|s| + |t|) = |s'| + |t'| + |s''| + |t''|`.
obtain hstg | hstg := le_or_lt_of_add_le_add (MulETransform.card g (s, t)).ge
· exact (ih _ _ hgs (hgt.mono inter_subset_union) <| devosMulRel_of_le_of_le aux1 hstg hsg).imp
(WithTop.coe_le_coe.2 aux1).trans' fun h ↦ hstg.trans <| h.trans <| add_le_add_right aux1 _
· exact (ih _ _ (hgs.mono inter_subset_union) hgt <| devosMulRel_of_le aux2 hstg).imp
(WithTop.coe_le_coe.2 aux2).trans' fun h ↦
hstg.le.trans <| h.trans <| add_le_add_right aux2 _
/-- The **Cauchy-Davenport Theorem** for torsion-free groups. The size of `s * t` is lower-bounded
by `|s| + |t| - 1`. -/
@[to_additive "The **Cauchy-Davenport theorem** for torsion-free groups. The size of `s + t` is
lower-bounded by `|s| + |t| - 1`."]
lemma Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul (h : IsTorsionFree α)
(hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card := by
simpa only [h.minOrder, min_eq_right, le_top, Nat.cast_le] using Finset.min_le_card_mul hs ht
end General
/-! ### $$ℤ/nℤ$$ -/
/-- The **Cauchy-Davenport Theorem**. If `s`, `t` are nonempty sets in $$ℤ/pℤ$$, then the size of
`s + t` is lower-bounded by `|s| + |t| - 1`, unless this quantity is greater than `p`. -/
lemma ZMod.min_le_card_add {p : ℕ} (hp : p.Prime) {s t : Finset (ZMod p)} (hs : s.Nonempty)
(ht : t.Nonempty) : min p (s.card + t.card - 1) ≤ (s + t).card := by
simpa only [ZMod.minOrder_of_prime hp, min_le_iff, Nat.cast_le] using Finset.min_le_card_add hs ht
/-! ### Linearly ordered cancellative semigroups -/
/-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.card_add_card_sub_one_le_card_mul [LinearOrder α] [Semigroup α] [IsCancelMul α]
[CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
{s t : Finset α} (hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card := by
|
suffices s * {t.min' ht} ∩ ({s.max' hs} * t) = {s.max' hs * t.min' ht} by
rw [← card_singleton_mul t (s.max' hs), ← card_mul_singleton s (t.min' ht),
← card_union_add_card_inter, ← card_singleton _, ← this, Nat.add_sub_cancel]
exact card_mono (union_subset (mul_subset_mul_left <| singleton_subset_iff.2 <| min'_mem _ _) <|
mul_subset_mul_right <| singleton_subset_iff.2 <| max'_mem _ _)
|
/-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.card_add_card_sub_one_le_card_mul [LinearOrder α] [Semigroup α] [IsCancelMul α]
[CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
{s t : Finset α} (hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card := by
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.202_0.yGTPJO6UphimMFs
|
/-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.card_add_card_sub_one_le_card_mul [LinearOrder α] [Semigroup α] [IsCancelMul α]
[CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
{s t : Finset α} (hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
α : Type u_1
inst✝⁴ : LinearOrder α
inst✝³ : Semigroup α
inst✝² : IsCancelMul α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
this : s * {min' t ht} ∩ ({max' s hs} * t) = {max' s hs * min' t ht}
⊢ card s + card t - 1 ≤ card (s * t)
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
· rw [← card_smul_finset g⁻¹ t]
refine' Or.inr ((add_le_add_right hst _).trans _)
rw [← card_union_eq hgt]
exact (card_le_card_mul_left _ hgs).trans (le_add_of_le_left aux1)
-- Else, we're done by induction on either `(s', t')` or `(s'', t'')` depending on whether
-- `|s| + |t| ≤ |s'| + |t'|` or `|s| + |t| < |s''| + |t''|`. One of those two inequalities must
-- hold since `2 * (|s| + |t|) = |s'| + |t'| + |s''| + |t''|`.
obtain hstg | hstg := le_or_lt_of_add_le_add (MulETransform.card g (s, t)).ge
· exact (ih _ _ hgs (hgt.mono inter_subset_union) <| devosMulRel_of_le_of_le aux1 hstg hsg).imp
(WithTop.coe_le_coe.2 aux1).trans' fun h ↦ hstg.trans <| h.trans <| add_le_add_right aux1 _
· exact (ih _ _ (hgs.mono inter_subset_union) hgt <| devosMulRel_of_le aux2 hstg).imp
(WithTop.coe_le_coe.2 aux2).trans' fun h ↦
hstg.le.trans <| h.trans <| add_le_add_right aux2 _
/-- The **Cauchy-Davenport Theorem** for torsion-free groups. The size of `s * t` is lower-bounded
by `|s| + |t| - 1`. -/
@[to_additive "The **Cauchy-Davenport theorem** for torsion-free groups. The size of `s + t` is
lower-bounded by `|s| + |t| - 1`."]
lemma Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul (h : IsTorsionFree α)
(hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card := by
simpa only [h.minOrder, min_eq_right, le_top, Nat.cast_le] using Finset.min_le_card_mul hs ht
end General
/-! ### $$ℤ/nℤ$$ -/
/-- The **Cauchy-Davenport Theorem**. If `s`, `t` are nonempty sets in $$ℤ/pℤ$$, then the size of
`s + t` is lower-bounded by `|s| + |t| - 1`, unless this quantity is greater than `p`. -/
lemma ZMod.min_le_card_add {p : ℕ} (hp : p.Prime) {s t : Finset (ZMod p)} (hs : s.Nonempty)
(ht : t.Nonempty) : min p (s.card + t.card - 1) ≤ (s + t).card := by
simpa only [ZMod.minOrder_of_prime hp, min_le_iff, Nat.cast_le] using Finset.min_le_card_add hs ht
/-! ### Linearly ordered cancellative semigroups -/
/-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.card_add_card_sub_one_le_card_mul [LinearOrder α] [Semigroup α] [IsCancelMul α]
[CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
{s t : Finset α} (hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card := by
suffices s * {t.min' ht} ∩ ({s.max' hs} * t) = {s.max' hs * t.min' ht} by
|
rw [← card_singleton_mul t (s.max' hs), ← card_mul_singleton s (t.min' ht),
← card_union_add_card_inter, ← card_singleton _, ← this, Nat.add_sub_cancel]
|
/-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.card_add_card_sub_one_le_card_mul [LinearOrder α] [Semigroup α] [IsCancelMul α]
[CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
{s t : Finset α} (hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card := by
suffices s * {t.min' ht} ∩ ({s.max' hs} * t) = {s.max' hs * t.min' ht} by
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.202_0.yGTPJO6UphimMFs
|
/-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.card_add_card_sub_one_le_card_mul [LinearOrder α] [Semigroup α] [IsCancelMul α]
[CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
{s t : Finset α} (hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
α : Type u_1
inst✝⁴ : LinearOrder α
inst✝³ : Semigroup α
inst✝² : IsCancelMul α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
this : s * {min' t ht} ∩ ({max' s hs} * t) = {max' s hs * min' t ht}
⊢ card (s * {min' t ht} ∪ {max' s hs} * t) ≤ card (s * t)
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
· rw [← card_smul_finset g⁻¹ t]
refine' Or.inr ((add_le_add_right hst _).trans _)
rw [← card_union_eq hgt]
exact (card_le_card_mul_left _ hgs).trans (le_add_of_le_left aux1)
-- Else, we're done by induction on either `(s', t')` or `(s'', t'')` depending on whether
-- `|s| + |t| ≤ |s'| + |t'|` or `|s| + |t| < |s''| + |t''|`. One of those two inequalities must
-- hold since `2 * (|s| + |t|) = |s'| + |t'| + |s''| + |t''|`.
obtain hstg | hstg := le_or_lt_of_add_le_add (MulETransform.card g (s, t)).ge
· exact (ih _ _ hgs (hgt.mono inter_subset_union) <| devosMulRel_of_le_of_le aux1 hstg hsg).imp
(WithTop.coe_le_coe.2 aux1).trans' fun h ↦ hstg.trans <| h.trans <| add_le_add_right aux1 _
· exact (ih _ _ (hgs.mono inter_subset_union) hgt <| devosMulRel_of_le aux2 hstg).imp
(WithTop.coe_le_coe.2 aux2).trans' fun h ↦
hstg.le.trans <| h.trans <| add_le_add_right aux2 _
/-- The **Cauchy-Davenport Theorem** for torsion-free groups. The size of `s * t` is lower-bounded
by `|s| + |t| - 1`. -/
@[to_additive "The **Cauchy-Davenport theorem** for torsion-free groups. The size of `s + t` is
lower-bounded by `|s| + |t| - 1`."]
lemma Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul (h : IsTorsionFree α)
(hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card := by
simpa only [h.minOrder, min_eq_right, le_top, Nat.cast_le] using Finset.min_le_card_mul hs ht
end General
/-! ### $$ℤ/nℤ$$ -/
/-- The **Cauchy-Davenport Theorem**. If `s`, `t` are nonempty sets in $$ℤ/pℤ$$, then the size of
`s + t` is lower-bounded by `|s| + |t| - 1`, unless this quantity is greater than `p`. -/
lemma ZMod.min_le_card_add {p : ℕ} (hp : p.Prime) {s t : Finset (ZMod p)} (hs : s.Nonempty)
(ht : t.Nonempty) : min p (s.card + t.card - 1) ≤ (s + t).card := by
simpa only [ZMod.minOrder_of_prime hp, min_le_iff, Nat.cast_le] using Finset.min_le_card_add hs ht
/-! ### Linearly ordered cancellative semigroups -/
/-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.card_add_card_sub_one_le_card_mul [LinearOrder α] [Semigroup α] [IsCancelMul α]
[CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
{s t : Finset α} (hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card := by
suffices s * {t.min' ht} ∩ ({s.max' hs} * t) = {s.max' hs * t.min' ht} by
rw [← card_singleton_mul t (s.max' hs), ← card_mul_singleton s (t.min' ht),
← card_union_add_card_inter, ← card_singleton _, ← this, Nat.add_sub_cancel]
|
exact card_mono (union_subset (mul_subset_mul_left <| singleton_subset_iff.2 <| min'_mem _ _) <|
mul_subset_mul_right <| singleton_subset_iff.2 <| max'_mem _ _)
|
/-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.card_add_card_sub_one_le_card_mul [LinearOrder α] [Semigroup α] [IsCancelMul α]
[CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
{s t : Finset α} (hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card := by
suffices s * {t.min' ht} ∩ ({s.max' hs} * t) = {s.max' hs * t.min' ht} by
rw [← card_singleton_mul t (s.max' hs), ← card_mul_singleton s (t.min' ht),
← card_union_add_card_inter, ← card_singleton _, ← this, Nat.add_sub_cancel]
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.202_0.yGTPJO6UphimMFs
|
/-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.card_add_card_sub_one_le_card_mul [LinearOrder α] [Semigroup α] [IsCancelMul α]
[CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
{s t : Finset α} (hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
α : Type u_1
inst✝⁴ : LinearOrder α
inst✝³ : Semigroup α
inst✝² : IsCancelMul α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
⊢ s * {min' t ht} ∩ ({max' s hs} * t) = {max' s hs * min' t ht}
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
· rw [← card_smul_finset g⁻¹ t]
refine' Or.inr ((add_le_add_right hst _).trans _)
rw [← card_union_eq hgt]
exact (card_le_card_mul_left _ hgs).trans (le_add_of_le_left aux1)
-- Else, we're done by induction on either `(s', t')` or `(s'', t'')` depending on whether
-- `|s| + |t| ≤ |s'| + |t'|` or `|s| + |t| < |s''| + |t''|`. One of those two inequalities must
-- hold since `2 * (|s| + |t|) = |s'| + |t'| + |s''| + |t''|`.
obtain hstg | hstg := le_or_lt_of_add_le_add (MulETransform.card g (s, t)).ge
· exact (ih _ _ hgs (hgt.mono inter_subset_union) <| devosMulRel_of_le_of_le aux1 hstg hsg).imp
(WithTop.coe_le_coe.2 aux1).trans' fun h ↦ hstg.trans <| h.trans <| add_le_add_right aux1 _
· exact (ih _ _ (hgs.mono inter_subset_union) hgt <| devosMulRel_of_le aux2 hstg).imp
(WithTop.coe_le_coe.2 aux2).trans' fun h ↦
hstg.le.trans <| h.trans <| add_le_add_right aux2 _
/-- The **Cauchy-Davenport Theorem** for torsion-free groups. The size of `s * t` is lower-bounded
by `|s| + |t| - 1`. -/
@[to_additive "The **Cauchy-Davenport theorem** for torsion-free groups. The size of `s + t` is
lower-bounded by `|s| + |t| - 1`."]
lemma Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul (h : IsTorsionFree α)
(hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card := by
simpa only [h.minOrder, min_eq_right, le_top, Nat.cast_le] using Finset.min_le_card_mul hs ht
end General
/-! ### $$ℤ/nℤ$$ -/
/-- The **Cauchy-Davenport Theorem**. If `s`, `t` are nonempty sets in $$ℤ/pℤ$$, then the size of
`s + t` is lower-bounded by `|s| + |t| - 1`, unless this quantity is greater than `p`. -/
lemma ZMod.min_le_card_add {p : ℕ} (hp : p.Prime) {s t : Finset (ZMod p)} (hs : s.Nonempty)
(ht : t.Nonempty) : min p (s.card + t.card - 1) ≤ (s + t).card := by
simpa only [ZMod.minOrder_of_prime hp, min_le_iff, Nat.cast_le] using Finset.min_le_card_add hs ht
/-! ### Linearly ordered cancellative semigroups -/
/-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.card_add_card_sub_one_le_card_mul [LinearOrder α] [Semigroup α] [IsCancelMul α]
[CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
{s t : Finset α} (hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card := by
suffices s * {t.min' ht} ∩ ({s.max' hs} * t) = {s.max' hs * t.min' ht} by
rw [← card_singleton_mul t (s.max' hs), ← card_mul_singleton s (t.min' ht),
← card_union_add_card_inter, ← card_singleton _, ← this, Nat.add_sub_cancel]
exact card_mono (union_subset (mul_subset_mul_left <| singleton_subset_iff.2 <| min'_mem _ _) <|
mul_subset_mul_right <| singleton_subset_iff.2 <| max'_mem _ _)
|
refine' eq_singleton_iff_unique_mem.2 ⟨mem_inter.2 ⟨mul_mem_mul (max'_mem _ _) <|
mem_singleton_self _, mul_mem_mul (mem_singleton_self _) <| min'_mem _ _⟩, _⟩
|
/-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.card_add_card_sub_one_le_card_mul [LinearOrder α] [Semigroup α] [IsCancelMul α]
[CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
{s t : Finset α} (hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card := by
suffices s * {t.min' ht} ∩ ({s.max' hs} * t) = {s.max' hs * t.min' ht} by
rw [← card_singleton_mul t (s.max' hs), ← card_mul_singleton s (t.min' ht),
← card_union_add_card_inter, ← card_singleton _, ← this, Nat.add_sub_cancel]
exact card_mono (union_subset (mul_subset_mul_left <| singleton_subset_iff.2 <| min'_mem _ _) <|
mul_subset_mul_right <| singleton_subset_iff.2 <| max'_mem _ _)
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.202_0.yGTPJO6UphimMFs
|
/-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.card_add_card_sub_one_le_card_mul [LinearOrder α] [Semigroup α] [IsCancelMul α]
[CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
{s t : Finset α} (hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
α : Type u_1
inst✝⁴ : LinearOrder α
inst✝³ : Semigroup α
inst✝² : IsCancelMul α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
⊢ ∀ x ∈ s * {min' t ht} ∩ ({max' s hs} * t), x = max' s hs * min' t ht
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
· rw [← card_smul_finset g⁻¹ t]
refine' Or.inr ((add_le_add_right hst _).trans _)
rw [← card_union_eq hgt]
exact (card_le_card_mul_left _ hgs).trans (le_add_of_le_left aux1)
-- Else, we're done by induction on either `(s', t')` or `(s'', t'')` depending on whether
-- `|s| + |t| ≤ |s'| + |t'|` or `|s| + |t| < |s''| + |t''|`. One of those two inequalities must
-- hold since `2 * (|s| + |t|) = |s'| + |t'| + |s''| + |t''|`.
obtain hstg | hstg := le_or_lt_of_add_le_add (MulETransform.card g (s, t)).ge
· exact (ih _ _ hgs (hgt.mono inter_subset_union) <| devosMulRel_of_le_of_le aux1 hstg hsg).imp
(WithTop.coe_le_coe.2 aux1).trans' fun h ↦ hstg.trans <| h.trans <| add_le_add_right aux1 _
· exact (ih _ _ (hgs.mono inter_subset_union) hgt <| devosMulRel_of_le aux2 hstg).imp
(WithTop.coe_le_coe.2 aux2).trans' fun h ↦
hstg.le.trans <| h.trans <| add_le_add_right aux2 _
/-- The **Cauchy-Davenport Theorem** for torsion-free groups. The size of `s * t` is lower-bounded
by `|s| + |t| - 1`. -/
@[to_additive "The **Cauchy-Davenport theorem** for torsion-free groups. The size of `s + t` is
lower-bounded by `|s| + |t| - 1`."]
lemma Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul (h : IsTorsionFree α)
(hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card := by
simpa only [h.minOrder, min_eq_right, le_top, Nat.cast_le] using Finset.min_le_card_mul hs ht
end General
/-! ### $$ℤ/nℤ$$ -/
/-- The **Cauchy-Davenport Theorem**. If `s`, `t` are nonempty sets in $$ℤ/pℤ$$, then the size of
`s + t` is lower-bounded by `|s| + |t| - 1`, unless this quantity is greater than `p`. -/
lemma ZMod.min_le_card_add {p : ℕ} (hp : p.Prime) {s t : Finset (ZMod p)} (hs : s.Nonempty)
(ht : t.Nonempty) : min p (s.card + t.card - 1) ≤ (s + t).card := by
simpa only [ZMod.minOrder_of_prime hp, min_le_iff, Nat.cast_le] using Finset.min_le_card_add hs ht
/-! ### Linearly ordered cancellative semigroups -/
/-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.card_add_card_sub_one_le_card_mul [LinearOrder α] [Semigroup α] [IsCancelMul α]
[CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
{s t : Finset α} (hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card := by
suffices s * {t.min' ht} ∩ ({s.max' hs} * t) = {s.max' hs * t.min' ht} by
rw [← card_singleton_mul t (s.max' hs), ← card_mul_singleton s (t.min' ht),
← card_union_add_card_inter, ← card_singleton _, ← this, Nat.add_sub_cancel]
exact card_mono (union_subset (mul_subset_mul_left <| singleton_subset_iff.2 <| min'_mem _ _) <|
mul_subset_mul_right <| singleton_subset_iff.2 <| max'_mem _ _)
refine' eq_singleton_iff_unique_mem.2 ⟨mem_inter.2 ⟨mul_mem_mul (max'_mem _ _) <|
mem_singleton_self _, mul_mem_mul (mem_singleton_self _) <| min'_mem _ _⟩, _⟩
|
simp only [mem_inter, and_imp, mem_mul, mem_singleton, exists_and_left, exists_eq_left,
forall_exists_index, and_imp, forall_apply_eq_imp_iff₂, mul_left_inj]
|
/-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.card_add_card_sub_one_le_card_mul [LinearOrder α] [Semigroup α] [IsCancelMul α]
[CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
{s t : Finset α} (hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card := by
suffices s * {t.min' ht} ∩ ({s.max' hs} * t) = {s.max' hs * t.min' ht} by
rw [← card_singleton_mul t (s.max' hs), ← card_mul_singleton s (t.min' ht),
← card_union_add_card_inter, ← card_singleton _, ← this, Nat.add_sub_cancel]
exact card_mono (union_subset (mul_subset_mul_left <| singleton_subset_iff.2 <| min'_mem _ _) <|
mul_subset_mul_right <| singleton_subset_iff.2 <| max'_mem _ _)
refine' eq_singleton_iff_unique_mem.2 ⟨mem_inter.2 ⟨mul_mem_mul (max'_mem _ _) <|
mem_singleton_self _, mul_mem_mul (mem_singleton_self _) <| min'_mem _ _⟩, _⟩
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.202_0.yGTPJO6UphimMFs
|
/-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.card_add_card_sub_one_le_card_mul [LinearOrder α] [Semigroup α] [IsCancelMul α]
[CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
{s t : Finset α} (hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
α : Type u_1
inst✝⁴ : LinearOrder α
inst✝³ : Semigroup α
inst✝² : IsCancelMul α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
s t : Finset α
hs : Finset.Nonempty s
ht : Finset.Nonempty t
⊢ ∀ a ∈ s, ∀ x ∈ t, max' s hs * x = a * min' t ht → a = max' s hs
|
/-
Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.ETransform
import Mathlib.GroupTheory.Order.Min
/-!
# The Cauchy-Davenport theorem
This file proves a generalisation of the Cauchy-Davenport theorem to arbitrary groups.
Cauchy-Davenport provides a lower bound on the size of `s + t` in terms of the sizes of `s` and `t`,
where `s` and `t` are nonempty finite sets in a monoid. Precisely, it says that
`|s + t| ≥ |s| + |t| - 1` unless the RHS is bigger than the size of the smallest nontrivial subgroup
(in which case taking `s` and `t` to be that subgroup would yield a counterexample). The motivating
example is `s = {0, ..., m}`, `t = {0, ..., n}` in the integers, which gives
`s + t = {0, ..., m + n}` and `|s + t| = m + n + 1 = |s| + |t| - 1`.
There are two kinds of proof of Cauchy-Davenport:
* The first one works in linear orders by writing `a₁ < ... < aₖ` the elements of `s`,
`b₁ < ... < bₗ` the elements of `t`, and arguing that `a₁ + b₁ < ... < aₖ + b₁ < ... < aₖ + bₗ`
are distinct elements of `s + t`.
* The second one works in groups by performing an "e-transform". In an abelian group, the
e-transform replaces `s` and `t` by `s ∩ g • s` and `t ∪ g⁻¹ • t`. For a suitably chosen `g`, this
decreases `|s + t|` and keeps `|s| + |t|` the same. In a general group, we use a trickier
e-transform (in fact, a pair of e-transforms), but the idea is the same.
## Main declarations
* `Finset.min_le_card_mul`: A generalisation of the Cauchy-Davenport theorem to arbitrary groups.
* `Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in
torsion-free groups.
* `ZMod.min_le_card_add`: The Cauchy-Davenport theorem.
* `Finset.card_add_card_sub_one_le_card_mul`: The Cauchy-Davenport theorem in linear ordered
cancellative semigroups.
## TODO
Version for `circle`.
## References
* Matt DeVos, *On a generalization of the Cauchy-Davenport theorem*
## Tags
additive combinatorics, number theory, sumset, cauchy-davenport
-/
open Finset Function Monoid MulOpposite Subgroup
open scoped Pointwise
variable {α : Type*}
/-! ### General case -/
section General
variable [Group α] [DecidableEq α] {x y : Finset α × Finset α} {s t : Finset α}
/-- The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ * t₁| < |s₂ * t₂|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ * t₁| = |s₂ * t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`. -/
@[to_additive
"The relation we induct along in the proof by DeVos of the Cauchy-Davenport theorem.
`(s₁, t₁) < (s₂, t₂)` iff
* `|s₁ + t₁| < |s₂ + t₂|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₂| + |t₂| < |s₁| + |t₁|`
* or `|s₁ + t₁| = |s₂ + t₂|` and `|s₁| + |t₁| = |s₂| + |t₂|` and `|s₁| < |s₂|`."]
private def DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop :=
Prod.Lex (· < ·) (Prod.Lex (· > ·) (· < ·)) on fun x ↦
((x.1 * x.2).card, x.1.card + x.2.card, x.1.card)
@[to_additive]
private lemma devosMulRel_iff :
DevosMulRel x y ↔
(x.1 * x.2).card < (y.1 * y.2).card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧ y.1.card + y.2.card < x.1.card + x.2.card ∨
(x.1 * x.2).card = (y.1 * y.2).card ∧
x.1.card + x.2.card = y.1.card + y.2.card ∧ x.1.card < y.1.card := by
simp [DevosMulRel, Prod.lex_iff, and_or_left]
@[to_additive]
private lemma devosMulRel_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card < x.1.card + x.2.card) : DevosMulRel x y :=
devosMulRel_iff.2 <| mul.lt_or_eq.imp_right fun h ↦ Or.inl ⟨h, hadd⟩
@[to_additive]
private lemma devosMulRel_of_le_of_le (mul : (x.1 * x.2).card ≤ (y.1 * y.2).card)
(hadd : y.1.card + y.2.card ≤ x.1.card + x.2.card) (hone : x.1.card < y.1.card) :
DevosMulRel x y :=
devosMulRel_iff.2 <|
mul.lt_or_eq.imp_right fun h ↦ hadd.gt_or_eq.imp (And.intro h) fun h' ↦ ⟨h, h', hone⟩
@[to_additive]
private lemma wellFoundedOn_devosMulRel :
{x : Finset α × Finset α | x.1.Nonempty ∧ x.2.Nonempty}.WellFoundedOn
(DevosMulRel : Finset α × Finset α → Finset α × Finset α → Prop) := by
refine wellFounded_lt.onFun.wellFoundedOn.prod_lex_of_wellFoundedOn_fiber fun n ↦
Set.WellFoundedOn.prod_lex_of_wellFoundedOn_fiber ?_ fun n ↦
wellFounded_lt.onFun.wellFoundedOn
exact wellFounded_lt.onFun.wellFoundedOn.mono' fun x hx y _ ↦ tsub_lt_tsub_left_of_le <|
add_le_add ((card_le_card_mul_right _ hx.1.2).trans_eq hx.2) <|
(card_le_card_mul_left _ hx.1.1).trans_eq hx.2
/-- A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of `s * t` is
lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the smallest
subgroup. -/
@[to_additive "A generalisation of the **Cauchy-Davenport theorem** to arbitrary groups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1` unless this quantity is greater than the size of the
smallest subgroup."]
lemma Finset.min_le_card_mul (hs : s.Nonempty) (ht : t.Nonempty) :
min (minOrder α) ↑(s.card + t.card - 1) ≤ (s * t).card := by
-- Set up the induction on `x := (s, t)` along the `DevosMulRel` relation.
set x := (s, t) with hx
clear_value x
simp only [Prod.ext_iff] at hx
obtain ⟨rfl, rfl⟩ := hx
refine' wellFoundedOn_devosMulRel.induction (P := fun x : Finset α × Finset α ↦
min (minOrder α) ↑(card x.1 + card x.2 - 1) ≤ card (x.1 * x.2)) ⟨hs, ht⟩ _
clear! x
rintro ⟨s, t⟩ ⟨hs, ht⟩ ih
simp only [min_le_iff, tsub_le_iff_right, Prod.forall, Set.mem_setOf_eq, and_imp,
Nat.cast_le] at *
-- If `t.card < s.card`, we're done by the induction hypothesis on `(t⁻¹, s⁻¹)`.
obtain hts | hst := lt_or_le t.card s.card
· simpa only [← mul_inv_rev, add_comm, card_inv] using
ih _ _ ht.inv hs.inv
(devosMulRel_iff.2 <| Or.inr <| Or.inr <| by
simpa only [← mul_inv_rev, add_comm, card_inv, true_and])
-- If `s` is a singleton, then the result is trivial.
obtain ⟨a, rfl⟩ | ⟨a, ha, b, hb, hab⟩ := hs.exists_eq_singleton_or_nontrivial
· simp [add_comm]
-- Else, we have `a, b ∈ s` distinct. So `g := b⁻¹ * a` is a non-identity element such that `s`
-- intersects its right translate by `g`.
obtain ⟨g, hg, hgs⟩ : ∃ g : α, g ≠ 1 ∧ (s ∩ op g • s).Nonempty :=
⟨b⁻¹ * a, inv_mul_eq_one.not.2 hab.symm, _,
mem_inter.2 ⟨ha, mem_smul_finset.2 ⟨_, hb, by simp⟩⟩⟩
-- If `s` is equal to its right translate by `g`, then it contains a nontrivial subgroup, namely
-- the subgroup generated by `g`. So `s * t` has size at least the size of a nontrivial subgroup,
-- as wanted.
obtain hsg | hsg := eq_or_ne (op g • s) s
· have hS : (zpowers g : Set α) ⊆ a⁻¹ • (s : Set α) := by
refine forall_mem_zpowers.2 $ @zpow_induction_right _ _ _ (· ∈ a⁻¹ • (s : Set α))
⟨_, ha, inv_mul_self _⟩ (fun c hc ↦ ?_) fun c hc ↦ ?_
· rw [← hsg, coe_smul_finset, smul_comm]
exact Set.smul_mem_smul_set hc
· simp only
rwa [← op_smul_eq_mul, op_inv, ← Set.mem_smul_set_iff_inv_smul_mem, smul_comm,
← coe_smul_finset, hsg]
refine Or.inl ((minOrder_le_natCard (zpowers_ne_bot.2 hg) <|
s.finite_toSet.smul_set.subset hS).trans <| WithTop.coe_le_coe.2 <|
((Nat.card_mono s.finite_toSet.smul_set hS).trans_eq <| ?_).trans <|
card_le_card_mul_right _ ht)
rw [← coe_smul_finset]
simp [-coe_smul_finset]
-- Else, we can transform `s`, `t` to `s'`, `t'` and `s''`, `t''`, such that one of `(s', t')` and
-- `(s'', t'')` is strictly smaller than `(s, t)` according to `DevosMulRel`.
replace hsg : (s ∩ op g • s).card < s.card := card_lt_card ⟨inter_subset_left _ _, fun h ↦
hsg <| eq_of_superset_of_card_ge (h.trans <| inter_subset_right _ _) (card_smul_finset _ _).le⟩
replace aux1 := card_mono $ mulETransformLeft.fst_mul_snd_subset g (s, t)
replace aux2 := card_mono $ mulETransformRight.fst_mul_snd_subset g (s, t)
-- If the left translate of `t` by `g⁻¹` is disjoint from `t`, then we're easily done.
obtain hgt | hgt := disjoint_or_nonempty_inter t (g⁻¹ • t)
· rw [← card_smul_finset g⁻¹ t]
refine' Or.inr ((add_le_add_right hst _).trans _)
rw [← card_union_eq hgt]
exact (card_le_card_mul_left _ hgs).trans (le_add_of_le_left aux1)
-- Else, we're done by induction on either `(s', t')` or `(s'', t'')` depending on whether
-- `|s| + |t| ≤ |s'| + |t'|` or `|s| + |t| < |s''| + |t''|`. One of those two inequalities must
-- hold since `2 * (|s| + |t|) = |s'| + |t'| + |s''| + |t''|`.
obtain hstg | hstg := le_or_lt_of_add_le_add (MulETransform.card g (s, t)).ge
· exact (ih _ _ hgs (hgt.mono inter_subset_union) <| devosMulRel_of_le_of_le aux1 hstg hsg).imp
(WithTop.coe_le_coe.2 aux1).trans' fun h ↦ hstg.trans <| h.trans <| add_le_add_right aux1 _
· exact (ih _ _ (hgs.mono inter_subset_union) hgt <| devosMulRel_of_le aux2 hstg).imp
(WithTop.coe_le_coe.2 aux2).trans' fun h ↦
hstg.le.trans <| h.trans <| add_le_add_right aux2 _
/-- The **Cauchy-Davenport Theorem** for torsion-free groups. The size of `s * t` is lower-bounded
by `|s| + |t| - 1`. -/
@[to_additive "The **Cauchy-Davenport theorem** for torsion-free groups. The size of `s + t` is
lower-bounded by `|s| + |t| - 1`."]
lemma Monoid.IsTorsionFree.card_add_card_sub_one_le_card_mul (h : IsTorsionFree α)
(hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card := by
simpa only [h.minOrder, min_eq_right, le_top, Nat.cast_le] using Finset.min_le_card_mul hs ht
end General
/-! ### $$ℤ/nℤ$$ -/
/-- The **Cauchy-Davenport Theorem**. If `s`, `t` are nonempty sets in $$ℤ/pℤ$$, then the size of
`s + t` is lower-bounded by `|s| + |t| - 1`, unless this quantity is greater than `p`. -/
lemma ZMod.min_le_card_add {p : ℕ} (hp : p.Prime) {s t : Finset (ZMod p)} (hs : s.Nonempty)
(ht : t.Nonempty) : min p (s.card + t.card - 1) ≤ (s + t).card := by
simpa only [ZMod.minOrder_of_prime hp, min_le_iff, Nat.cast_le] using Finset.min_le_card_add hs ht
/-! ### Linearly ordered cancellative semigroups -/
/-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.card_add_card_sub_one_le_card_mul [LinearOrder α] [Semigroup α] [IsCancelMul α]
[CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
{s t : Finset α} (hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card := by
suffices s * {t.min' ht} ∩ ({s.max' hs} * t) = {s.max' hs * t.min' ht} by
rw [← card_singleton_mul t (s.max' hs), ← card_mul_singleton s (t.min' ht),
← card_union_add_card_inter, ← card_singleton _, ← this, Nat.add_sub_cancel]
exact card_mono (union_subset (mul_subset_mul_left <| singleton_subset_iff.2 <| min'_mem _ _) <|
mul_subset_mul_right <| singleton_subset_iff.2 <| max'_mem _ _)
refine' eq_singleton_iff_unique_mem.2 ⟨mem_inter.2 ⟨mul_mem_mul (max'_mem _ _) <|
mem_singleton_self _, mul_mem_mul (mem_singleton_self _) <| min'_mem _ _⟩, _⟩
simp only [mem_inter, and_imp, mem_mul, mem_singleton, exists_and_left, exists_eq_left,
forall_exists_index, and_imp, forall_apply_eq_imp_iff₂, mul_left_inj]
|
exact fun a' ha' b' hb' h ↦ (le_max' _ _ ha').eq_of_not_lt fun ha ↦
((mul_lt_mul_right' ha _).trans_eq' h).not_le <| mul_le_mul_left' (min'_le _ _ hb') _
|
/-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.card_add_card_sub_one_le_card_mul [LinearOrder α] [Semigroup α] [IsCancelMul α]
[CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
{s t : Finset α} (hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card := by
suffices s * {t.min' ht} ∩ ({s.max' hs} * t) = {s.max' hs * t.min' ht} by
rw [← card_singleton_mul t (s.max' hs), ← card_mul_singleton s (t.min' ht),
← card_union_add_card_inter, ← card_singleton _, ← this, Nat.add_sub_cancel]
exact card_mono (union_subset (mul_subset_mul_left <| singleton_subset_iff.2 <| min'_mem _ _) <|
mul_subset_mul_right <| singleton_subset_iff.2 <| max'_mem _ _)
refine' eq_singleton_iff_unique_mem.2 ⟨mem_inter.2 ⟨mul_mem_mul (max'_mem _ _) <|
mem_singleton_self _, mul_mem_mul (mem_singleton_self _) <| min'_mem _ _⟩, _⟩
simp only [mem_inter, and_imp, mem_mul, mem_singleton, exists_and_left, exists_eq_left,
forall_exists_index, and_imp, forall_apply_eq_imp_iff₂, mul_left_inj]
|
Mathlib.Combinatorics.SetFamily.CauchyDavenport.202_0.yGTPJO6UphimMFs
|
/-- The **Cauchy-Davenport Theorem** for linearly ordered cancellative semigroups. The size of
`s * t` is lower-bounded by `|s| + |t| - 1`. -/
@[to_additive
"The **Cauchy-Davenport theorem** for linearly ordered additive cancellative semigroups. The size of
`s + t` is lower-bounded by `|s| + |t| - 1`."]
lemma Finset.card_add_card_sub_one_le_card_mul [LinearOrder α] [Semigroup α] [IsCancelMul α]
[CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
{s t : Finset α} (hs : s.Nonempty) (ht : t.Nonempty) : s.card + t.card - 1 ≤ (s * t).card
|
Mathlib_Combinatorics_SetFamily_CauchyDavenport
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hm_not : ¬m ≤ m0
⊢ μ[f|m] = 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by
|
rw [condexp, dif_neg hm_not]
|
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.106_0.yd50cWAuCo6hlry
|
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hm : m ≤ m0
hμm_not : ¬SigmaFinite (Measure.trim μ hm)
⊢ μ[f|m] = 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by
|
rw [condexp, dif_pos hm, dif_neg]
|
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.109_0.yd50cWAuCo6hlry
|
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case hnc
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hm : m ≤ m0
hμm_not : ¬SigmaFinite (Measure.trim μ hm)
⊢ ¬(SigmaFinite (Measure.trim μ hm) ∧ Integrable f)
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg];
|
push_neg
|
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg];
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.109_0.yd50cWAuCo6hlry
|
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case hnc
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hm : m ≤ m0
hμm_not : ¬SigmaFinite (Measure.trim μ hm)
⊢ SigmaFinite (Measure.trim μ hm) → ¬Integrable f
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg;
|
exact fun h => absurd h hμm_not
|
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg;
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.109_0.yd50cWAuCo6hlry
|
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
⊢ μ[f|m] =
if Integrable f then
if StronglyMeasurable f then f
else AEStronglyMeasurable'.mk ↑↑(condexpL1 hm μ f) (_ : AEStronglyMeasurable' m (↑↑(condexpL1 hm μ f)) μ)
else 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
|
rw [condexp, dif_pos hm]
|
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.113_0.yd50cWAuCo6hlry
|
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
⊢ (if h : SigmaFinite (Measure.trim μ hm) ∧ Integrable f then
if StronglyMeasurable f then f
else AEStronglyMeasurable'.mk ↑↑(condexpL1 hm μ f) (_ : AEStronglyMeasurable' m (↑↑(condexpL1 hm μ f)) μ)
else 0) =
if Integrable f then
if StronglyMeasurable f then f
else AEStronglyMeasurable'.mk ↑↑(condexpL1 hm μ f) (_ : AEStronglyMeasurable' m (↑↑(condexpL1 hm μ f)) μ)
else 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
|
simp only [hμm, Ne.def, true_and_iff]
|
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.113_0.yd50cWAuCo6hlry
|
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
⊢ (if h : Integrable f then
if StronglyMeasurable f then f
else AEStronglyMeasurable'.mk ↑↑(condexpL1 hm μ f) (_ : AEStronglyMeasurable' m (↑↑(condexpL1 hm μ f)) μ)
else 0) =
if Integrable f then
if StronglyMeasurable f then f
else AEStronglyMeasurable'.mk ↑↑(condexpL1 hm μ f) (_ : AEStronglyMeasurable' m (↑↑(condexpL1 hm μ f)) μ)
else 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
|
by_cases hf : Integrable f μ
|
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.113_0.yd50cWAuCo6hlry
|
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
hf : Integrable f
⊢ (if h : Integrable f then
if StronglyMeasurable f then f
else AEStronglyMeasurable'.mk ↑↑(condexpL1 hm μ f) (_ : AEStronglyMeasurable' m (↑↑(condexpL1 hm μ f)) μ)
else 0) =
if Integrable f then
if StronglyMeasurable f then f
else AEStronglyMeasurable'.mk ↑↑(condexpL1 hm μ f) (_ : AEStronglyMeasurable' m (↑↑(condexpL1 hm μ f)) μ)
else 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
·
|
rw [dif_pos hf, if_pos hf]
|
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
·
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.113_0.yd50cWAuCo6hlry
|
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
hf : ¬Integrable f
⊢ (if h : Integrable f then
if StronglyMeasurable f then f
else AEStronglyMeasurable'.mk ↑↑(condexpL1 hm μ f) (_ : AEStronglyMeasurable' m (↑↑(condexpL1 hm μ f)) μ)
else 0) =
if Integrable f then
if StronglyMeasurable f then f
else AEStronglyMeasurable'.mk ↑↑(condexpL1 hm μ f) (_ : AEStronglyMeasurable' m (↑↑(condexpL1 hm μ f)) μ)
else 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
·
|
rw [dif_neg hf, if_neg hf]
|
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
·
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.113_0.yd50cWAuCo6hlry
|
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
f : α → F'
hf : StronglyMeasurable f
hfi : Integrable f
⊢ μ[f|m] = f
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
|
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
|
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.126_0.yd50cWAuCo6hlry
|
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
f : α → F'
⊢ μ[f|m] =ᵐ[μ] ↑↑(condexpL1 hm μ f)
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
|
rw [condexp_of_sigmaFinite hm]
|
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.136_0.yd50cWAuCo6hlry
|
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
f : α → F'
⊢ (if Integrable f then
if StronglyMeasurable f then f
else AEStronglyMeasurable'.mk ↑↑(condexpL1 hm μ f) (_ : AEStronglyMeasurable' m (↑↑(condexpL1 hm μ f)) μ)
else 0) =ᵐ[μ]
↑↑(condexpL1 hm μ f)
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
|
by_cases hfi : Integrable f μ
|
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.136_0.yd50cWAuCo6hlry
|
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
f : α → F'
hfi : Integrable f
⊢ (if Integrable f then
if StronglyMeasurable f then f
else AEStronglyMeasurable'.mk ↑↑(condexpL1 hm μ f) (_ : AEStronglyMeasurable' m (↑↑(condexpL1 hm μ f)) μ)
else 0) =ᵐ[μ]
↑↑(condexpL1 hm μ f)
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
·
|
rw [if_pos hfi]
|
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
·
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.136_0.yd50cWAuCo6hlry
|
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
f : α → F'
hfi : Integrable f
⊢ (if StronglyMeasurable f then f
else AEStronglyMeasurable'.mk ↑↑(condexpL1 hm μ f) (_ : AEStronglyMeasurable' m (↑↑(condexpL1 hm μ f)) μ)) =ᵐ[μ]
↑↑(condexpL1 hm μ f)
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
|
by_cases hfm : StronglyMeasurable[m] f
|
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.136_0.yd50cWAuCo6hlry
|
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
f : α → F'
hfi : Integrable f
hfm : StronglyMeasurable f
⊢ (if StronglyMeasurable f then f
else AEStronglyMeasurable'.mk ↑↑(condexpL1 hm μ f) (_ : AEStronglyMeasurable' m (↑↑(condexpL1 hm μ f)) μ)) =ᵐ[μ]
↑↑(condexpL1 hm μ f)
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
·
|
rw [if_pos hfm]
|
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
·
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.136_0.yd50cWAuCo6hlry
|
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
f : α → F'
hfi : Integrable f
hfm : StronglyMeasurable f
⊢ f =ᵐ[μ] ↑↑(condexpL1 hm μ f)
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
|
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
|
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.136_0.yd50cWAuCo6hlry
|
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
f : α → F'
hfi : Integrable f
hfm : ¬StronglyMeasurable f
⊢ (if StronglyMeasurable f then f
else AEStronglyMeasurable'.mk ↑↑(condexpL1 hm μ f) (_ : AEStronglyMeasurable' m (↑↑(condexpL1 hm μ f)) μ)) =ᵐ[μ]
↑↑(condexpL1 hm μ f)
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
·
|
rw [if_neg hfm]
|
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
·
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.136_0.yd50cWAuCo6hlry
|
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
f : α → F'
hfi : Integrable f
hfm : ¬StronglyMeasurable f
⊢ AEStronglyMeasurable'.mk ↑↑(condexpL1 hm μ f) (_ : AEStronglyMeasurable' m (↑↑(condexpL1 hm μ f)) μ) =ᵐ[μ]
↑↑(condexpL1 hm μ f)
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
|
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
|
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.136_0.yd50cWAuCo6hlry
|
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
f : α → F'
hfi : ¬Integrable f
⊢ (if Integrable f then
if StronglyMeasurable f then f
else AEStronglyMeasurable'.mk ↑↑(condexpL1 hm μ f) (_ : AEStronglyMeasurable' m (↑↑(condexpL1 hm μ f)) μ)
else 0) =ᵐ[μ]
↑↑(condexpL1 hm μ f)
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
|
rw [if_neg hfi, condexpL1_undef hfi]
|
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.136_0.yd50cWAuCo6hlry
|
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f✝ g : α → F'
s : Set α
hm : m ≤ m0
hμm : SigmaFinite (Measure.trim μ hm)
f : α → F'
hfi : ¬Integrable f
⊢ 0 =ᵐ[μ] ↑↑0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
|
exact (coeFn_zero _ _ _).symm
|
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.136_0.yd50cWAuCo6hlry
|
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace 𝕜 F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hf : Integrable f
⊢ μ[f|m] =ᵐ[μ] ↑↑((condexpL1Clm F' hm μ) (Integrable.toL1 f hf))
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
|
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
|
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.152_0.yd50cWAuCo6hlry
|
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f)
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁷ : IsROrC 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace 𝕜 F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (Measure.trim μ hm)
hf : Integrable f
x : α
⊢ ↑↑(condexpL1 hm μ f) x = ↑↑((condexpL1Clm F' hm μ) (Integrable.toL1 f hf)) x
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
|
rw [condexpL1_eq hf]
|
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.152_0.yd50cWAuCo6hlry
|
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f)
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hf : ¬Integrable f
⊢ μ[f|m] = 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
|
by_cases hm : m ≤ m0
|
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.159_0.yd50cWAuCo6hlry
|
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hf : ¬Integrable f
hm : m ≤ m0
⊢ μ[f|m] = 0
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hf : ¬Integrable f
hm : ¬m ≤ m0
⊢ μ[f|m] = 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
|
swap
|
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.159_0.yd50cWAuCo6hlry
|
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case neg
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hf : ¬Integrable f
hm : ¬m ≤ m0
⊢ μ[f|m] = 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; ·
|
rw [condexp_of_not_le hm]
|
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; ·
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.159_0.yd50cWAuCo6hlry
|
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
case pos
α : Type u_1
F : Type u_2
F' : Type u_3
𝕜 : Type u_4
p : ℝ≥0∞
inst✝⁶ : IsROrC 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace 𝕜 F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
f g : α → F'
s : Set α
hf : ¬Integrable f
hm : m ≤ m0
⊢ μ[f|m] = 0
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` with value in a Banach space
with respect to a measure `μ` (defined on a measurable space structure `m0`) and a measurable space
structure `m` with `hm : m ≤ m0` (a sub-sigma-algebra). This is an `m`-strongly measurable
function `μ[f|hm]` which is integrable and verifies `∫ x in s, μ[f|hm] x ∂μ = ∫ x in s, f x ∂μ`
for all `m`-measurable sets `s`. It is unique as an element of `L¹`.
The construction is done in four steps:
* Define the conditional expectation of an `L²` function, as an element of `L²`. This is the
orthogonal projection on the subspace of almost everywhere `m`-measurable functions.
* Show that the conditional expectation of the indicator of a measurable set with finite measure
is integrable and define a map `Set α → (E →L[ℝ] (α →₁[μ] E))` which to a set associates a linear
map. That linear map sends `x ∈ E` to the conditional expectation of the indicator of the set
with value `x`.
* Extend that map to `condexpL1Clm : (α →₁[μ] E) →L[ℝ] (α →₁[μ] E)`. This is done using the same
construction as the Bochner integral (see the file `MeasureTheory/Integral/SetToL1`).
* Define the conditional expectation of a function `f : α → E`, which is an integrable function
`α → E` equal to 0 if `f` is not integrable, and equal to an `m`-measurable representative of
`condexpL1Clm` applied to `[f]`, the equivalence class of `f` in `L¹`.
The first step is done in `MeasureTheory.Function.ConditionalExpectation.CondexpL2`, the two
next steps in `MeasureTheory.Function.ConditionalExpectation.CondexpL1` and the final step is
performed in this file.
## Main results
The conditional expectation and its properties
* `condexp (m : MeasurableSpace α) (μ : Measure α) (f : α → E)`: conditional expectation of `f`
with respect to `m`.
* `integrable_condexp` : `condexp` is integrable.
* `stronglyMeasurable_condexp` : `condexp` is `m`-strongly-measurable.
* `set_integral_condexp (hf : Integrable f μ) (hs : MeasurableSet[m] s)` : if `m ≤ m0` (the
σ-algebra over which the measure is defined), then the conditional expectation verifies
`∫ x in s, condexp m μ f x ∂μ = ∫ x in s, f x ∂μ` for any `m`-measurable set `s`.
While `condexp` is function-valued, we also define `condexpL1` with value in `L1` and a continuous
linear map `condexpL1Clm` from `L1` to `L1`. `condexp` should be used in most cases.
Uniqueness of the conditional expectation
* `ae_eq_condexp_of_forall_set_integral_eq`: an a.e. `m`-measurable function which verifies the
equality of integrals is a.e. equal to `condexp`.
## Notations
For a measure `μ` defined on a measurable space structure `m0`, another measurable space structure
`m` with `hm : m ≤ m0` (a sub-σ-algebra) and a function `f`, we define the notation
* `μ[f|m] = condexp m μ f`.
## Tags
conditional expectation, conditional expected value
-/
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology BigOperators MeasureTheory
namespace MeasureTheory
variable {α F F' 𝕜 : Type*} {p : ℝ≥0∞} [IsROrC 𝕜]
-- 𝕜 for ℝ or ℂ
-- F for a Lp submodule
[NormedAddCommGroup F]
[NormedSpace 𝕜 F]
-- F' for integrals on a Lp submodule
[NormedAddCommGroup F']
[NormedSpace 𝕜 F'] [NormedSpace ℝ F'] [CompleteSpace F']
open scoped Classical
variable {m m0 : MeasurableSpace α} {μ : Measure α} {f g : α → F'} {s : Set α}
/-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
is true:
- `m` is not a sub-σ-algebra of `m0`,
- `μ` is not σ-finite with respect to `m`,
- `f` is not integrable. -/
noncomputable irreducible_def condexp (m : MeasurableSpace α) {m0 : MeasurableSpace α}
(μ : Measure α) (f : α → F') : α → F' :=
if hm : m ≤ m0 then
if h : SigmaFinite (μ.trim hm) ∧ Integrable f μ then
if StronglyMeasurable[m] f then f
else (@aestronglyMeasurable'_condexpL1 _ _ _ _ _ m m0 μ hm h.1 _).mk
(@condexpL1 _ _ _ _ _ _ _ hm μ h.1 f)
else 0
else 0
#align measure_theory.condexp MeasureTheory.condexp
-- We define notation `μ[f|m]` for the conditional expectation of `f` with respect to `m`.
scoped notation μ "[" f "|" m "]" => MeasureTheory.condexp m μ f
theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by rw [condexp, dif_neg hm_not]
#align measure_theory.condexp_of_not_le MeasureTheory.condexp_of_not_le
theorem condexp_of_not_sigmaFinite (hm : m ≤ m0) (hμm_not : ¬SigmaFinite (μ.trim hm)) :
μ[f|m] = 0 := by rw [condexp, dif_pos hm, dif_neg]; push_neg; exact fun h => absurd h hμm_not
#align measure_theory.condexp_of_not_sigma_finite MeasureTheory.condexp_of_not_sigmaFinite
theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] :
μ[f|m] =
if Integrable f μ then
if StronglyMeasurable[m] f then f
else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
else 0 := by
rw [condexp, dif_pos hm]
simp only [hμm, Ne.def, true_and_iff]
by_cases hf : Integrable f μ
· rw [dif_pos hf, if_pos hf]
· rw [dif_neg hf, if_neg hf]
#align measure_theory.condexp_of_sigma_finite MeasureTheory.condexp_of_sigmaFinite
theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
#align measure_theory.condexp_of_strongly_measurable MeasureTheory.condexp_of_stronglyMeasurable
theorem condexp_const (hm : m ≤ m0) (c : F') [IsFiniteMeasure μ] :
μ[fun _ : α => c|m] = fun _ => c :=
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_const _ _ m _ _) (integrable_const c)
#align measure_theory.condexp_const MeasureTheory.condexp_const
theorem condexp_ae_eq_condexpL1 (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] (f : α → F') :
μ[f|m] =ᵐ[μ] condexpL1 hm μ f := by
rw [condexp_of_sigmaFinite hm]
by_cases hfi : Integrable f μ
· rw [if_pos hfi]
by_cases hfm : StronglyMeasurable[m] f
· rw [if_pos hfm]
exact (condexpL1_of_aestronglyMeasurable' (StronglyMeasurable.aeStronglyMeasurable' hfm)
hfi).symm
· rw [if_neg hfm]
exact (AEStronglyMeasurable'.ae_eq_mk aestronglyMeasurable'_condexpL1).symm
rw [if_neg hfi, condexpL1_undef hfi]
exact (coeFn_zero _ _ _).symm
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1 MeasureTheory.condexp_ae_eq_condexpL1
theorem condexp_ae_eq_condexpL1Clm (hm : m ≤ m0) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ) :
μ[f|m] =ᵐ[μ] condexpL1Clm F' hm μ (hf.toL1 f) := by
refine' (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => _)
rw [condexpL1_eq hf]
set_option linter.uppercaseLean3 false in
#align measure_theory.condexp_ae_eq_condexp_L1_clm MeasureTheory.condexp_ae_eq_condexpL1Clm
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
|
by_cases hμm : SigmaFinite (μ.trim hm)
|
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
|
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic.159_0.yd50cWAuCo6hlry
|
theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0
|
Mathlib_MeasureTheory_Function_ConditionalExpectation_Basic
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.