Three extensional models of type theory

Martin-Löf’s type theory exists in two forms, differing in the formalisation of the identity types. In [15] Per Martin-Löf formulated his type theory with the extensional rules for the identity types, identifying judgmental and propositional equality. This rendered type-checking and well-formedness of formulas undecidable. For this reason (among others), he latter formulated intensional rules for the identity type that do preserve the decidability of type-checking and well-formedness of formulas. For this reason, the intensional version is adopted by most implementations.