In the cryptologic jargon, the proof is both complete (proving positive knowledge) and sound (cheating is detectable).
So in order to complete proving that the trefoil is not equivalent to an unknot, we only need prove that our trefoil diagram is colourable.
Canada rushed a military force to the northwest, where the new railway, though not quite completed, proved its worth, as did the company's steamers that operated on the Saskatchewan River.
However, since the problem is NP-complete (proved in the Supplementary Appendix), the running time of the dynamic programming approach increases exponentially with the number of branches in the de Bruijn graph.
The timbers, partly scoured bare by erosion of the river bed, the largest up to a third of a metre in diameter, were discovered in work during exceptionally low tides last February, but carbon dating work – revealed in the new edition of London Archaeologist journal – has only recently been completed, proving that the trees were felled between 4790 BC and 4490 BC.
A feasibility study on single chip assemblies for the ATLAS experiment has been successfully completed, proving the possibility of reworking.
After complete replacement proved impossible due to the age of the pumps.
Validation results for single plates, scale-models, and complete structures proved the model to be an interesting tool for the energy-efficient design of greenhouses.
Mediastinal downstaging and complete resection proved to be very strong positive prognostic factors for survival, and 3-years survival of patients receiving neoadjuvant therapy plus complete resection was 70% (Betticher et al, 2003).
Here, I would write another page restating the claims from the opening paragraph but generally indicating that the research above — though not yet completed — had proved my thesis.
Genetic testing was completed and proved that he is Kennedy's father.
