Formally Verified Results For Commutative Ordered Rings
IsarMathLib, a library for the Isabelle theorem prover, has formal verifications of a few fundamental results on commutative ordered rings. The results are proved in the ring1 context.
Suppose is a commutative ordered ring, and . Then:
| by | |
|---|---|
| The additive group of A is an ordered group | OrdRing_ZF_1_L4 |
| iff | OrdRing_ZF_1_L7 |
| and imply and |
OrdRing_ZF_1_L9 |
| ordring_one_is_nonneg | |
| OrdRing_ZF_2_L5 | |
| ord_ring_triangle_ineq | |
| x is either in the positive set, equal to 0, or in minus the positive set. | OrdRing_ZF_3_L2 |
| The set of positive elements of is closed under multiplication iff A has no zero divisors. | OrdRing_ZF_3_L3 |
| If A is non-trivial, then it is infinite. | ord_ring_infinite |
Read more about this topic: Partially Ordered Ring
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