toIsOrderedRing — Mathlib

English

In RingPreordering, the carrier of the support equals the intersection of the carrier with its negation as a set.

Русский

В RingPreordering носитель опоры равен пересечению носителя с его отрицанием как множества.

LaTeX

$$P.{support} = (P:Set R) ∩ -(P:Set R)$$

Lean4

/-- A commutative star-ordered semiring is an ordered semiring. -/
instance toIsOrderedRing (R : Type*) [CommSemiring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] :
    IsOrderedRing R where
  add_le_add_left _ _ := add_le_add_left
  zero_le_one := zero_le_one
  mul_le_mul_of_nonneg_left _ _ _ := mul_le_mul_of_nonneg_left
  mul_le_mul_of_nonneg_right a b c := by simpa only [mul_comm _ c] using mul_le_mul_of_nonneg_left

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