Be based on; of theories and claims, for example(verb)
WordNet. Princeton University. 2010.
Use "repose on" in a sentence
"My situation high and airy, a brisk and cool breeze steadily and incessantly passing over the clear waters of the lake, and fluttering over me through the srrounding groves, wings its way to the moon-light savannas, while I repose on my sweet and healthy couch of the soft Tillandsi ulnea-adscites, and the latter gloomy and still hours of night passed rapidly away as it were in a moment; I arose, strengthened"
"As it was, we were very glad to receive, after a few days of discontented repose on the very ground where we had once been so happy, an order to go out on picket at Port Royal Ferry, with the understanding that we might remain there for some time."
"Persuaded that the middle classes can thrive only when they repose on a basis of religion and practical faith, the Gesellenverein cultivates assiduously the religious and moral sense of its members."