A formula is true in a model if it has a realizer in the model, and a formula is valid if it has a realizer in every model.
Steven Cole Kleene, on the interpretation of intuitionistic number theory, JSL, 10:109-124, 1945
Igor Privara, Branislav Rovan, Peter Ruzicka (eds.), lecture notes in computer science, 841, google books, chapter 5 realizability models, p.93
Thomas Streicher, Realizability Models for , unpublished note. In this note WISC is called ( stands for “type-theoretic collection axiom).