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] :
Equations
  • Filter.Germ.groupWithZero = let __src := Filter.Germ.divInvMonoid; let __src_1 := Filter.Germ.monoidWithZero; GroupWithZero.mk DivInvMonoid.zpow
instance Filter.Germ.divisionSemiring {α : Type u} {β : Type v} {φ : Ultrafilter α} [DivisionSemiring β] :
Equations
  • Filter.Germ.divisionSemiring = let __spread.0 := Filter.Germ.groupWithZero; DivisionSemiring.mk GroupWithZero.zpow
instance Filter.Germ.divisionRing {α : Type u} {β : Type v} {φ : Ultrafilter α} [DivisionRing β] :
Equations
  • Filter.Germ.divisionRing = let __src := Filter.Germ.ring; let __src_1 := Filter.Germ.divisionSemiring; DivisionRing.mk DivisionSemiring.zpow (qsmulRec fun (a : ) => a)
instance Filter.Germ.semifield {α : Type u} {β : Type v} {φ : Ultrafilter α} [Semifield β] :
Semifield (Filter.Germ (φ) β)
Equations
  • Filter.Germ.semifield = let __src := Filter.Germ.commSemiring; let __src_1 := Filter.Germ.divisionSemiring; Semifield.mk DivisionSemiring.zpow
instance Filter.Germ.field {α : Type u} {β : Type v} {φ : Ultrafilter α} [Field β] :
Field (Filter.Germ (φ) β)
Equations
  • Filter.Germ.field = let __src := Filter.Germ.commRing; let __src_1 := Filter.Germ.divisionRing; Field.mk DivisionRing.zpow DivisionRing.qsmul
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 : Filter.Germ (φ) β) => x < x_1) = Filter.Germ.LiftRel fun (x x_1 : β) => x < x_1
instance Filter.Germ.isTotal {α : Type u} {β : Type v} {φ : Ultrafilter α} [LE β] [IsTotal β fun (x x_1 : β) => x x_1] :
IsTotal (Filter.Germ (φ) β) fun (x x_1 : Filter.Germ (φ) β) => x x_1
Equations
  • =
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.

Equations
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) :
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 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) :
max a b = if a b then b else a
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) :
Equations
  • One or more equations did not get rendered due to their size.
noncomputable instance Filter.Germ.linearOrderedCommGroup {α : Type u} {β : Type v} {φ : Ultrafilter α} [LinearOrderedCommGroup β] :
Equations
  • One or more equations did not get rendered due to their size.
Equations
  • Filter.Germ.strictOrderedSemiring = let __src := Filter.Germ.orderedSemiring; let __src_1 := Filter.Germ.orderedAddCancelCommMonoid; let __src_2 := ; StrictOrderedSemiring.mk
Equations
  • Filter.Germ.strictOrderedCommSemiring = let __src := Filter.Germ.strictOrderedSemiring; let __src_1 := Filter.Germ.orderedCommSemiring; StrictOrderedCommSemiring.mk
instance Filter.Germ.strictOrderedRing {α : Type u} {β : Type v} {φ : Ultrafilter α} [StrictOrderedRing β] :
Equations
  • Filter.Germ.strictOrderedRing = let __src := Filter.Germ.ring; let __src_1 := Filter.Germ.strictOrderedSemiring; StrictOrderedRing.mk
Equations
  • Filter.Germ.strictOrderedCommRing = let __src := Filter.Germ.strictOrderedRing; let __src_1 := Filter.Germ.orderedCommRing; StrictOrderedCommRing.mk
noncomputable instance Filter.Germ.linearOrderedRing {α : Type u} {β : Type v} {φ : Ultrafilter α} [LinearOrderedRing β] :
Equations
  • One or more equations did not get rendered due to their size.
noncomputable instance Filter.Germ.linearOrderedField {α : Type u} {β : Type v} {φ : Ultrafilter α} [LinearOrderedField β] :
Equations
  • Filter.Germ.linearOrderedField = let __src := Filter.Germ.linearOrderedRing; let __src_1 := Filter.Germ.field; LinearOrderedField.mk Field.zpow Field.qsmul
noncomputable instance Filter.Germ.linearOrderedCommRing {α : Type u} {β : Type v} {φ : Ultrafilter α} [LinearOrderedCommRing β] :
Equations
  • Filter.Germ.linearOrderedCommRing = let __src := Filter.Germ.linearOrderedRing; let __src_1 := Filter.Germ.commMonoid; LinearOrderedCommRing.mk
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|