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
Famous quotes containing the words formally, verified, results, ordered and/or rings:
“From a bed in this hotel Seargent S. Prentiss arose in the middle of the night and made a speech in defense of a bedbug that had bitten him. It was heard by a mock jury and judge, and the bedbug was formally acquitted.”
—Federal Writers Project Of The Wor, U.S. public relief program (1935-1943)
“In the course of a life devoted less to living than to reading, I have verified many times that literary intentions and theories are nothing more than stimuli and that the final work usually ignores or even contradicts them.”
—Jorge Luis Borges (18991986)
“It would be easy ... to regard the whole of world 3 as timeless, as Plato suggested of his world of Forms or Ideas.... I propose a different viewone which, I have found, is surprisingly fruitful. I regard world 3 as being essentially the product of the human mind.... More precisely, I regard the world 3 of problems, theories, and critical arguments as one of the results of the evolution of human language, and as acting back on this evolution.”
—Karl Popper (19021994)
“Your mind was wrought in cosmic solitude,
Through which careered an undulous pageantry
Of fiends and suns, darkness and boiling sea,
All held in ordered sway by beautys mood.”
—Allen Tate (18991979)
“It is told that some divorcees, elated by their freedom, pause on leaving the courthouse to kiss a front pillar, or even walk to the Truckee to hurl their wedding rings into the river; but boys who recover the rings declare they are of the dime-store variety, and accuse the throwers of fraudulent practices.”
—Administration in the State of Neva, U.S. public relief program. Nevada: A Guide to the Silver State (The WPA Guide to Nevada)