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]
def
filter.germ.division_ring
{α : Type u}
{β : Type v}
{φ : ultrafilter α}
[division_ring β] :
division_ring (↑φ.germ β)
If φ is an ultrafilter then the ultraproduct is a division ring.
Equations
- filter.germ.division_ring = {add := ring.add filter.germ.ring, add_assoc := _, zero := ring.zero filter.germ.ring, zero_add := _, add_zero := _, nsmul := ring.nsmul filter.germ.ring, nsmul_zero' := _, nsmul_succ' := _, neg := ring.neg filter.germ.ring, sub := ring.sub filter.germ.ring, sub_eq_add_neg := _, gsmul := ring.gsmul filter.germ.ring, gsmul_zero' := _, gsmul_succ' := _, gsmul_neg' := _, add_left_neg := _, add_comm := _, mul := ring.mul filter.germ.ring, mul_assoc := _, one := ring.one filter.germ.ring, one_mul := _, mul_one := _, npow := ring.npow filter.germ.ring, npow_zero' := _, npow_succ' := _, left_distrib := _, right_distrib := _, inv := div_inv_monoid.inv filter.germ.div_inv_monoid, div := div_inv_monoid.div filter.germ.div_inv_monoid, div_eq_mul_inv := _, gpow := gpow filter.germ.div_inv_monoid, gpow_zero' := _, gpow_succ' := _, gpow_neg' := _, exists_pair_ne := _, mul_inv_cancel := _, inv_zero := _}
@[instance]
If φ is an ultrafilter then the ultraproduct is a field.
Equations
- filter.germ.field = {add := comm_ring.add filter.germ.comm_ring, add_assoc := _, zero := comm_ring.zero filter.germ.comm_ring, zero_add := _, add_zero := _, nsmul := comm_ring.nsmul filter.germ.comm_ring, nsmul_zero' := _, nsmul_succ' := _, neg := comm_ring.neg filter.germ.comm_ring, sub := comm_ring.sub filter.germ.comm_ring, sub_eq_add_neg := _, gsmul := comm_ring.gsmul filter.germ.comm_ring, gsmul_zero' := _, gsmul_succ' := _, gsmul_neg' := _, add_left_neg := _, add_comm := _, mul := comm_ring.mul filter.germ.comm_ring, mul_assoc := _, one := comm_ring.one filter.germ.comm_ring, one_mul := _, mul_one := _, npow := comm_ring.npow filter.germ.comm_ring, npow_zero' := _, npow_succ' := _, left_distrib := _, right_distrib := _, mul_comm := _, inv := division_ring.inv filter.germ.division_ring, div := division_ring.div filter.germ.division_ring, div_eq_mul_inv := _, gpow := division_ring.gpow filter.germ.division_ring, gpow_zero' := _, gpow_succ' := _, gpow_neg' := _, exists_pair_ne := _, mul_inv_cancel := _, inv_zero := _}
@[instance]
def
filter.germ.linear_order
{α : Type u}
{β : Type v}
{φ : ultrafilter α}
[linear_order β] :
linear_order (↑φ.germ β)
If φ is an ultrafilter then the ultraproduct is a linear order.
Equations
- filter.germ.linear_order = {le := partial_order.le filter.germ.partial_order, lt := partial_order.lt filter.germ.partial_order, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_total := _, decidable_le := λ (a b : ↑φ.germ β), classical.prop_decidable (a ≤ b), decidable_eq := decidable_eq_of_decidable_le (λ (a b : ↑φ.germ β), classical.prop_decidable (a ≤ b)), decidable_lt := decidable_lt_of_decidable_le (λ (a b : ↑φ.germ β), classical.prop_decidable (a ≤ b))}
@[simp]
theorem
filter.germ.const_div
{α : Type u}
{β : Type v}
{φ : ultrafilter α}
[division_ring β]
(x y : β) :
theorem
filter.germ.coe_lt
{α : Type u}
{β : Type v}
{φ : ultrafilter α}
[preorder β]
{f g : α → β} :
theorem
filter.germ.coe_pos
{α : Type u}
{β : Type v}
{φ : ultrafilter α}
[preorder β]
[has_zero β]
{f : α → β} :
@[instance]
def
filter.germ.ordered_ring
{α : Type u}
{β : Type v}
{φ : ultrafilter α}
[ordered_ring β] :
ordered_ring (↑φ.germ β)
If φ is an ultrafilter then the ultraproduct is an ordered ring.
Equations
- filter.germ.ordered_ring = {add := ring.add filter.germ.ring, add_assoc := _, zero := ring.zero filter.germ.ring, zero_add := _, add_zero := _, nsmul := ring.nsmul filter.germ.ring, nsmul_zero' := _, nsmul_succ' := _, neg := ring.neg filter.germ.ring, sub := ring.sub filter.germ.ring, sub_eq_add_neg := _, gsmul := ring.gsmul filter.germ.ring, gsmul_zero' := _, gsmul_succ' := _, gsmul_neg' := _, add_left_neg := _, add_comm := _, mul := ring.mul filter.germ.ring, mul_assoc := _, one := ring.one filter.germ.ring, one_mul := _, mul_one := _, npow := ring.npow filter.germ.ring, npow_zero' := _, npow_succ' := _, left_distrib := _, right_distrib := _, le := ordered_add_comm_group.le filter.germ.ordered_add_comm_group, lt := ordered_add_comm_group.lt filter.germ.ordered_add_comm_group, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, add_le_add_left := _, zero_le_one := _, mul_pos := _}
@[instance]
def
filter.germ.linear_ordered_ring
{α : Type u}
{β : Type v}
{φ : ultrafilter α}
[linear_ordered_ring β] :
linear_ordered_ring (↑φ.germ β)
If φ is an ultrafilter then the ultraproduct is a linear ordered ring.
Equations
- filter.germ.linear_ordered_ring = {add := ordered_ring.add filter.germ.ordered_ring, add_assoc := _, zero := ordered_ring.zero filter.germ.ordered_ring, zero_add := _, add_zero := _, nsmul := ordered_ring.nsmul filter.germ.ordered_ring, nsmul_zero' := _, nsmul_succ' := _, neg := ordered_ring.neg filter.germ.ordered_ring, sub := ordered_ring.sub filter.germ.ordered_ring, sub_eq_add_neg := _, gsmul := ordered_ring.gsmul filter.germ.ordered_ring, gsmul_zero' := _, gsmul_succ' := _, gsmul_neg' := _, add_left_neg := _, add_comm := _, mul := ordered_ring.mul filter.germ.ordered_ring, mul_assoc := _, one := ordered_ring.one filter.germ.ordered_ring, one_mul := _, mul_one := _, npow := ordered_ring.npow filter.germ.ordered_ring, npow_zero' := _, npow_succ' := _, left_distrib := _, right_distrib := _, le := ordered_ring.le filter.germ.ordered_ring, lt := ordered_ring.lt filter.germ.ordered_ring, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, add_le_add_left := _, zero_le_one := _, mul_pos := _, le_total := _, decidable_le := linear_order.decidable_le filter.germ.linear_order, decidable_eq := linear_order.decidable_eq filter.germ.linear_order, decidable_lt := linear_order.decidable_lt filter.germ.linear_order, exists_pair_ne := _}
@[instance]
def
filter.germ.linear_ordered_field
{α : Type u}
{β : Type v}
{φ : ultrafilter α}
[linear_ordered_field β] :
linear_ordered_field (↑φ.germ β)
If φ is an ultrafilter then the ultraproduct is a linear ordered field.
Equations
- filter.germ.linear_ordered_field = {add := linear_ordered_ring.add filter.germ.linear_ordered_ring, add_assoc := _, zero := linear_ordered_ring.zero filter.germ.linear_ordered_ring, zero_add := _, add_zero := _, nsmul := linear_ordered_ring.nsmul filter.germ.linear_ordered_ring, nsmul_zero' := _, nsmul_succ' := _, neg := linear_ordered_ring.neg filter.germ.linear_ordered_ring, sub := linear_ordered_ring.sub filter.germ.linear_ordered_ring, sub_eq_add_neg := _, gsmul := linear_ordered_ring.gsmul filter.germ.linear_ordered_ring, gsmul_zero' := _, gsmul_succ' := _, gsmul_neg' := _, add_left_neg := _, add_comm := _, mul := linear_ordered_ring.mul filter.germ.linear_ordered_ring, mul_assoc := _, one := linear_ordered_ring.one filter.germ.linear_ordered_ring, one_mul := _, mul_one := _, npow := linear_ordered_ring.npow filter.germ.linear_ordered_ring, npow_zero' := _, npow_succ' := _, left_distrib := _, right_distrib := _, le := linear_ordered_ring.le filter.germ.linear_ordered_ring, lt := linear_ordered_ring.lt filter.germ.linear_ordered_ring, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, add_le_add_left := _, zero_le_one := _, mul_pos := _, le_total := _, decidable_le := linear_ordered_ring.decidable_le filter.germ.linear_ordered_ring, decidable_eq := linear_ordered_ring.decidable_eq filter.germ.linear_ordered_ring, decidable_lt := linear_ordered_ring.decidable_lt filter.germ.linear_ordered_ring, exists_pair_ne := _, mul_comm := _, inv := field.inv filter.germ.field, div := field.div filter.germ.field, div_eq_mul_inv := _, gpow := field.gpow filter.germ.field, gpow_zero' := _, gpow_succ' := _, gpow_neg' := _, mul_inv_cancel := _, inv_zero := _}
@[instance]
def
filter.germ.linear_ordered_comm_ring
{α : Type u}
{β : Type v}
{φ : ultrafilter α}
[linear_ordered_comm_ring β] :
If φ is an ultrafilter then the ultraproduct is a linear ordered commutative ring.
Equations
- filter.germ.linear_ordered_comm_ring = {add := linear_ordered_ring.add filter.germ.linear_ordered_ring, add_assoc := _, zero := linear_ordered_ring.zero filter.germ.linear_ordered_ring, zero_add := _, add_zero := _, nsmul := linear_ordered_ring.nsmul filter.germ.linear_ordered_ring, nsmul_zero' := _, nsmul_succ' := _, neg := linear_ordered_ring.neg filter.germ.linear_ordered_ring, sub := linear_ordered_ring.sub filter.germ.linear_ordered_ring, sub_eq_add_neg := _, gsmul := linear_ordered_ring.gsmul filter.germ.linear_ordered_ring, gsmul_zero' := _, gsmul_succ' := _, gsmul_neg' := _, add_left_neg := _, add_comm := _, mul := linear_ordered_ring.mul filter.germ.linear_ordered_ring, mul_assoc := _, one := linear_ordered_ring.one filter.germ.linear_ordered_ring, one_mul := _, mul_one := _, npow := linear_ordered_ring.npow filter.germ.linear_ordered_ring, npow_zero' := _, npow_succ' := _, left_distrib := _, right_distrib := _, le := linear_ordered_ring.le filter.germ.linear_ordered_ring, lt := linear_ordered_ring.lt filter.germ.linear_ordered_ring, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, add_le_add_left := _, zero_le_one := _, mul_pos := _, le_total := _, decidable_le := linear_ordered_ring.decidable_le filter.germ.linear_ordered_ring, decidable_eq := linear_ordered_ring.decidable_eq filter.germ.linear_ordered_ring, decidable_lt := linear_ordered_ring.decidable_lt filter.germ.linear_ordered_ring, exists_pair_ne := _, mul_comm := _}
@[instance]
def
filter.germ.linear_ordered_add_comm_group
{α : Type u}
{β : Type v}
{φ : ultrafilter α}
[linear_ordered_add_comm_group β] :
If φ is an ultrafilter then the ultraproduct is a decidable linear ordered commutative
group.
Equations
- filter.germ.linear_ordered_add_comm_group = {add := ordered_add_comm_group.add filter.germ.ordered_add_comm_group, add_assoc := _, zero := ordered_add_comm_group.zero filter.germ.ordered_add_comm_group, zero_add := _, add_zero := _, nsmul := ordered_add_comm_group.nsmul filter.germ.ordered_add_comm_group, nsmul_zero' := _, nsmul_succ' := _, neg := ordered_add_comm_group.neg filter.germ.ordered_add_comm_group, sub := ordered_add_comm_group.sub filter.germ.ordered_add_comm_group, sub_eq_add_neg := _, gsmul := ordered_add_comm_group.gsmul filter.germ.ordered_add_comm_group, gsmul_zero' := _, gsmul_succ' := _, gsmul_neg' := _, add_left_neg := _, add_comm := _, le := ordered_add_comm_group.le filter.germ.ordered_add_comm_group, lt := ordered_add_comm_group.lt filter.germ.ordered_add_comm_group, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, add_le_add_left := _, le_total := _, decidable_le := linear_order.decidable_le filter.germ.linear_order, decidable_eq := linear_order.decidable_eq filter.germ.linear_order, decidable_lt := linear_order.decidable_lt filter.germ.linear_order}
theorem
filter.germ.max_def
{α : Type u}
{β : Type v}
{φ : ultrafilter α}
[linear_order β]
(x 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 : linear_order β]
(x 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}
[linear_ordered_add_comm_group β]
(x : ↑φ.germ β) :
abs x = filter.germ.map abs x
@[simp]
theorem
filter.germ.const_max
{α : Type u}
{β : Type v}
{φ : ultrafilter α}
[linear_order β]
(x y : β) :
@[simp]
theorem
filter.germ.const_min
{α : Type u}
{β : Type v}
{φ : ultrafilter α}
[linear_order β]
(x y : β) :
@[simp]
theorem
filter.germ.const_abs
{α : Type u}
{β : Type v}
{φ : ultrafilter α}
[linear_ordered_add_comm_group β]
(x : β) :