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.instGroupWithZero {α : Type u} {β : Type v} {φ : Ultrafilter α} [GroupWithZero β] : GroupWithZero ((↑φ).Germ β)

Equations

• Filter.Germ.instGroupWithZero = GroupWithZero.mk ⋯ DivInvMonoid.zpow ⋯ ⋯ ⋯ ⋯ ⋯

instance Filter.Germ.instDivisionSemiring {α : Type u} {β : Type v} {φ : Ultrafilter α} [DivisionSemiring β] : DivisionSemiring ((↑φ).Germ β)

Equations

• Filter.Germ.instDivisionSemiring = DivisionSemiring.mk ⋯ GroupWithZero.zpow ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ (fun (q : ℚ≥0) => HMul.hMul ↑q) ⋯

instance Filter.Germ.instDivisionRing {α : Type u} {β : Type v} {φ : Ultrafilter α} [DivisionRing β] : DivisionRing ((↑φ).Germ β)

Equations

• Filter.Germ.instDivisionRing = DivisionRing.mk ⋯ DivisionSemiring.zpow ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ DivisionSemiring.nnqsmul ⋯ ⋯ (fun (q : ℚ) => HMul.hMul ↑q) ⋯

instance Filter.Germ.instSemifield {α : Type u} {β : Type v} {φ : Ultrafilter α} [Semifield β] : Semifield ((↑φ).Germ β)

Equations

• Filter.Germ.instSemifield = Semifield.mk ⋯ DivisionSemiring.zpow ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ DivisionSemiring.nnqsmul ⋯

instance Filter.Germ.instField {α : Type u} {β : Type v} {φ : Ultrafilter α} [Field β] : Field ((↑φ).Germ β)

Equations

• Filter.Germ.instField = Field.mk ⋯ DivisionRing.zpow ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ DivisionRing.nnqsmul ⋯ ⋯ DivisionRing.qsmul ⋯

theorem Filter.Germ.coe_lt {α : Type u} {β : Type v} {φ : Ultrafilter α} [Preorder β] {f : α → β} {g : α → β} : ↑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 : α → β} : 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 < y → ↑x < ↑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 (x1 x2 : (↑φ).Germ β) => x1 < x2) = Filter.Germ.LiftRel fun (x1 x2 : β) => x1 < x2

instance Filter.Germ.isTotal {α : Type u} {β : Type v} {φ : Ultrafilter α} [LE β] [IsTotal β fun (x1 x2 : β) => x1 ≤ x2] : IsTotal ((↑φ).Germ β) fun (x1 x2 : (↑φ).Germ β) => x1 ≤ x2

Equations

• ⋯ = ⋯

noncomputable instance Filter.Germ.instLinearOrder {α : Type u} {β : Type v} {φ : Ultrafilter α} [LinearOrder β] : LinearOrder ((↑φ).Germ β)

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

Equations

• Filter.Germ.instLinearOrder = Lattice.toLinearOrder ((↑φ).Germ β)

theorem Filter.Germ.linearOrderedAddCommGroup.proof_4 {α : Type u_1} {β : Type u_2} {φ : Ultrafilter α} [LinearOrderedAddCommGroup β] (a : (↑φ).Germ β) (b : (↑φ).Germ β) : compare a b = compareOfLessAndEq a b

theorem Filter.Germ.linearOrderedAddCommGroup.proof_1 {α : Type u_1} {β : Type u_2} {φ : Ultrafilter α} [LinearOrderedAddCommGroup β] (a : (↑φ).Germ β) (b : (↑φ).Germ β) : a ≤ b ∨ b ≤ a

theorem Filter.Germ.linearOrderedAddCommGroup.proof_3 {α : Type u_1} {β : Type u_2} {φ : Ultrafilter α} [LinearOrderedAddCommGroup β] (a : (↑φ).Germ β) (b : (↑φ).Germ β) : max a b = if a ≤ b then b else a

theorem Filter.Germ.linearOrderedAddCommGroup.proof_2 {α : Type u_1} {β : Type u_2} {φ : Ultrafilter α} [LinearOrderedAddCommGroup β] (a : (↑φ).Germ β) (b : (↑φ).Germ β) : min a b = if a ≤ b then a else b

noncomputable instance Filter.Germ.linearOrderedAddCommGroup {α : Type u} {β : Type v} {φ : Ultrafilter α} [LinearOrderedAddCommGroup β] : LinearOrderedAddCommGroup ((↑φ).Germ β)

Equations

• Filter.Germ.linearOrderedAddCommGroup = LinearOrderedAddCommGroup.mk ⋯ LinearOrder.decidableLE LinearOrder.decidableEq LinearOrder.decidableLT ⋯ ⋯ ⋯

noncomputable instance Filter.Germ.linearOrderedCommGroup {α : Type u} {β : Type v} {φ : Ultrafilter α} [LinearOrderedCommGroup β] : LinearOrderedCommGroup ((↑φ).Germ β)

Equations

• Filter.Germ.linearOrderedCommGroup = LinearOrderedCommGroup.mk ⋯ LinearOrder.decidableLE LinearOrder.decidableEq LinearOrder.decidableLT ⋯ ⋯ ⋯

instance Filter.Germ.instStrictOrderedSemiring {α : Type u} {β : Type v} {φ : Ultrafilter α} [StrictOrderedSemiring β] : StrictOrderedSemiring ((↑φ).Germ β)

Equations

• Filter.Germ.instStrictOrderedSemiring = StrictOrderedSemiring.mk ⋯ ⋯ ⋯ ⋯ ⋯

instance Filter.Germ.instStrictOrderedCommSemiring {α : Type u} {β : Type v} {φ : Ultrafilter α} [StrictOrderedCommSemiring β] : StrictOrderedCommSemiring ((↑φ).Germ β)

Equations

• Filter.Germ.instStrictOrderedCommSemiring = StrictOrderedCommSemiring.mk ⋯

instance Filter.Germ.instStrictOrderedRing {α : Type u} {β : Type v} {φ : Ultrafilter α} [StrictOrderedRing β] : StrictOrderedRing ((↑φ).Germ β)

Equations

• Filter.Germ.instStrictOrderedRing = StrictOrderedRing.mk ⋯ ⋯ ⋯

instance Filter.Germ.instStrictOrderedCommRing {α : Type u} {β : Type v} {φ : Ultrafilter α} [StrictOrderedCommRing β] : StrictOrderedCommRing ((↑φ).Germ β)

Equations

• Filter.Germ.instStrictOrderedCommRing = StrictOrderedCommRing.mk ⋯

noncomputable instance Filter.Germ.instLinearOrderedRing {α : Type u} {β : Type v} {φ : Ultrafilter α} [LinearOrderedRing β] : LinearOrderedRing ((↑φ).Germ β)

Equations

• Filter.Germ.instLinearOrderedRing = LinearOrderedRing.mk ⋯ LinearOrder.decidableLE LinearOrder.decidableEq LinearOrder.decidableLT ⋯ ⋯ ⋯

noncomputable instance Filter.Germ.instLinearOrderedField {α : Type u} {β : Type v} {φ : Ultrafilter α} [LinearOrderedField β] : LinearOrderedField ((↑φ).Germ β)

Equations

• Filter.Germ.instLinearOrderedField = LinearOrderedField.mk ⋯ Field.zpow ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ Field.nnqsmul ⋯ ⋯ Field.qsmul ⋯

noncomputable instance Filter.Germ.instLinearOrderedCommRing {α : Type u} {β : Type v} {φ : Ultrafilter α} [LinearOrderedCommRing β] : LinearOrderedCommRing ((↑φ).Germ β)

Equations

• Filter.Germ.instLinearOrderedCommRing = LinearOrderedCommRing.mk ⋯

theorem Filter.Germ.max_def {α : Type u} {β : Type v} {φ : Ultrafilter α} [LinearOrder β] (x : (↑φ).Germ β) (y : (↑φ).Germ β) : max x y = Filter.Germ.map₂ max x y

theorem Filter.Germ.min_def {α : Type u} {β : Type v} {φ : Ultrafilter α} [K : LinearOrder β] (x : (↑φ).Germ β) (y : (↑φ).Germ β) : min x y = Filter.Germ.map₂ min x y

theorem Filter.Germ.abs_def {α : Type u} {β : Type v} {φ : Ultrafilter α} [LinearOrderedAddCommGroup β] (x : (↑φ).Germ 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|