Documentation

Mathlib.Order.Filter.FilterProduct

Ultraproducts #

If φ is an ultrafilter, then the space of germs of functions f : α → β at φ is called the ultraproduct. In this file we prove properties of ultraproducts that rely on φ being an ultrafilter. Definitions and properties that work for any filter should go to Order.Filter.Germ.

Tags #

ultrafilter, ultraproduct

instance Filter.Germ.groupWithZero {α : Type u} {β : Type v} {φ : Ultrafilter α} [GroupWithZero β] :
instance Filter.Germ.divisionRing {α : Type u} {β : Type v} {φ : Ultrafilter α} [DivisionRing β] :
instance Filter.Germ.semifield {α : Type u} {β : Type v} {φ : Ultrafilter α} [Semifield β] :
Semifield (Filter.Germ (φ) β)
instance Filter.Germ.field {α : Type u} {β : Type v} {φ : Ultrafilter α} [Field β] :
Field (Filter.Germ (φ) β)
theorem Filter.Germ.coe_lt {α : Type u} {β : Type v} {φ : Ultrafilter α} [Preorder β] {f : αβ} {g : αβ} :
f < g ∀ᶠ (x : α) in φ, f x < g x
theorem Filter.Germ.coe_pos {α : Type u} {β : Type v} {φ : Ultrafilter α} [Preorder β] [Zero β] {f : αβ} :
0 < f ∀ᶠ (x : α) in φ, 0 < f x
theorem Filter.Germ.const_lt {α : Type u} {β : Type v} {φ : Ultrafilter α} [Preorder β] {x : β} {y : β} :
x < yx < y
@[simp]
theorem Filter.Germ.const_lt_iff {α : Type u} {β : Type v} {φ : Ultrafilter α} [Preorder β] {x : β} {y : β} :
x < y x < y
theorem Filter.Germ.lt_def {α : Type u} {β : Type v} {φ : Ultrafilter α} [Preorder β] :
(fun x x_1 => x < x_1) = Filter.Germ.LiftRel fun x x_1 => x < x_1
instance Filter.Germ.isTotal {α : Type u} {β : Type v} {φ : Ultrafilter α} [LE β] [IsTotal β fun x x_1 => x x_1] :
IsTotal (Filter.Germ (φ) β) fun x x_1 => x x_1
noncomputable instance Filter.Germ.linearOrder {α : Type u} {β : Type v} {φ : Ultrafilter α} [LinearOrder β] :

If φ is an ultrafilter then the ultraproduct is a linear order.

theorem Filter.Germ.linearOrderedAddCommGroup.proof_2 {α : Type u_2} {β : Type u_1} {φ : Ultrafilter α} [LinearOrderedAddCommGroup β] (a : Filter.Germ (φ) β) (b : Filter.Germ (φ) β) :
a b b a
theorem Filter.Germ.linearOrderedAddCommGroup.proof_3 {α : Type u_2} {β : Type u_1} {φ : Ultrafilter α} [LinearOrderedAddCommGroup β] (a : Filter.Germ (φ) β) (b : Filter.Germ (φ) β) :
min a b = if a b then a else b
theorem Filter.Germ.linearOrderedAddCommGroup.proof_1 {α : Type u_2} {β : Type u_1} {φ : Ultrafilter α} [LinearOrderedAddCommGroup β] (a : Filter.Germ (φ) β) (b : Filter.Germ (φ) β) :
a b∀ (c : Filter.Germ (φ) β), c + a c + b
theorem Filter.Germ.linearOrderedAddCommGroup.proof_4 {α : Type u_2} {β : Type u_1} {φ : Ultrafilter α} [LinearOrderedAddCommGroup β] (a : Filter.Germ (φ) β) (b : Filter.Germ (φ) β) :
max a b = if a b then b else a
theorem Filter.Germ.linearOrderedAddCommGroup.proof_5 {α : Type u_2} {β : Type u_1} {φ : Ultrafilter α} [LinearOrderedAddCommGroup β] (a : Filter.Germ (φ) β) (b : Filter.Germ (φ) β) :
noncomputable instance Filter.Germ.linearOrderedCommGroup {α : Type u} {β : Type v} {φ : Ultrafilter α} [LinearOrderedCommGroup β] :
noncomputable instance Filter.Germ.linearOrderedRing {α : Type u} {β : Type v} {φ : Ultrafilter α} [LinearOrderedRing β] :
noncomputable instance Filter.Germ.linearOrderedField {α : Type u} {β : Type v} {φ : Ultrafilter α} [LinearOrderedField β] :
noncomputable instance Filter.Germ.linearOrderedCommRing {α : Type u} {β : Type v} {φ : Ultrafilter α} [LinearOrderedCommRing β] :
theorem Filter.Germ.max_def {α : Type u} {β : Type v} {φ : Ultrafilter α} [LinearOrder β] (x : Filter.Germ (φ) β) (y : Filter.Germ (φ) β) :
theorem Filter.Germ.min_def {α : Type u} {β : Type v} {φ : Ultrafilter α} [K : LinearOrder β] (x : Filter.Germ (φ) β) (y : Filter.Germ (φ) β) :
theorem Filter.Germ.abs_def {α : Type u} {β : Type v} {φ : Ultrafilter α} [LinearOrderedAddCommGroup β] (x : Filter.Germ (φ) β) :
|x| = Filter.Germ.map abs x
@[simp]
theorem Filter.Germ.const_max {α : Type u} {β : Type v} {φ : Ultrafilter α} [LinearOrder β] (x : β) (y : β) :
↑(max x y) = max x y
@[simp]
theorem Filter.Germ.const_min {α : Type u} {β : Type v} {φ : Ultrafilter α} [LinearOrder β] (x : β) (y : β) :
↑(min x y) = min x y
@[simp]
theorem Filter.Germ.const_abs {α : Type u} {β : Type v} {φ : Ultrafilter α} [LinearOrderedAddCommGroup β] (x : β) :
|x| = |x|