Dependent type theory has been applied to natural language semantics in order to provide a formally precise and computationally adequate account of dynamic aspects of meaning. One of the frameworks of natural language semantics based on dependent type theory is Dependent Type Semantics (DTS), whose focus is on the compositional interpretations of anaphoric expressions. In this paper, we extend the framework of DTS with a mechanism to handle entailment and presupposition associated with factive verbs such as "know". Using the notion of proof objects as first-class objects, we provide a fully compositional account of presuppositional inferences triggered by factive verbs. The proposal also gives a formal reconstruction of the type-distinction between propositions and facts and thereby accounts for the lexical semantic differences between factive and non-factive verbs in a type-theoretical setting.