Constructing an ordered ring from a ring with a specified positive cone. #
Positive cones #
structure
Ring.PositiveCone
(α : Type u_1)
[inst : Ring α]
extends
AddCommGroup.PositiveCone
:
Type u_1
In a positive cone,
1
isnonneg
one_nonneg : AddCommGroup.PositiveCone.nonneg toPositiveCone 1In a positive cone, if
a
andb
arepos
then so isa * b
mul_pos : ∀ (a b : α), AddCommGroup.PositiveCone.pos toPositiveCone a → AddCommGroup.PositiveCone.pos toPositiveCone b → AddCommGroup.PositiveCone.pos toPositiveCone (a * b)
A positive cone in a ring consists of a positive cone in underlying AddCommGroup
,
which contains 1
and such that the positive elements are closed under multiplication.
Instances For
For any
a
the propositionnonneg a
is decidablenonnegDecidable : DecidablePred toPositiveCone_1.toPositiveCone.nonnegEither
a
or-a
isnonneg
nonneg_total : ∀ (a : α), AddCommGroup.PositiveCone.nonneg toPositiveCone_1.toPositiveCone a ∨ AddCommGroup.PositiveCone.nonneg toPositiveCone_1.toPositiveCone (-a)
A total positive cone in a nontrivial ring induces a linear order.
Instances For
abbrev
Ring.TotalPositiveCone.toTotalPositiveCone
{α : Type u_1}
[inst : Ring α]
(self : Ring.TotalPositiveCone α)
:
Forget that a TotalPositiveCone
in a ring respects the multiplicative structure.
Equations
- One or more equations did not get rendered due to their size.
theorem
Ring.PositiveCone.one_pos
{α : Type u_1}
[inst : Ring α]
[inst : Nontrivial α]
(C : Ring.PositiveCone α)
:
AddCommGroup.PositiveCone.pos C.toPositiveCone 1
def
StrictOrderedRing.mkOfPositiveCone
{α : Type u_1}
[inst : Ring α]
[inst : Nontrivial α]
(C : Ring.PositiveCone α)
:
Construct a StrictOrderedRing
by designating a positive cone in an existing Ring
.
Equations
- One or more equations did not get rendered due to their size.
def
LinearOrderedRing.mkOfPositiveCone
{α : Type u_1}
[inst : Ring α]
[inst : Nontrivial α]
(C : Ring.TotalPositiveCone α)
:
Construct a LinearOrderedRing
by
designating a positive cone in an existing Ring
.
Equations
- One or more equations did not get rendered due to their size.