We present some references/motivations for the notion of definitional equality
and for the rule that a:B follows from a:A and A = B.

The notion of definitional equality was introduced first in AUTOMATH. The following paper presents
a suggestive explanation of this notion and how proof-checking was designed in this system
(especially section 10)

The extension from AUTOMATH is that one adds the notion of data type (natural number),
of constructors (zero and successor) and primitive recursion as definitional equality.
The motivation is that one can consider the schema of primitive recursion as a definition
of a function.

This was also influenced by natural deduction, where constructors correspond
to introduction rules and the work of Godel on system T.

With this extension, one obtains a programming language with dependent types
and where computations correspond to unfolding of definitions (that can be primitive
recursive definitions). This programming language has the feature that all computations
terminate. This has been also considered in functional programming, see e.g.
the discussion in this paper.
A description of the evaluation algorithm using techniques from functional programming
can be found in this work of Gregoire and Leroy

and for the rule that a:B follows from a:A and A = B.

The notion of definitional equality was introduced first in AUTOMATH. The following paper presents

a suggestive explanation of this notion and how proof-checking was designed in this system

(especially section 10)

On the roles of types in mathematics

The notion of definitional equality was later introduced by Per Martin-Lof, first in the context of normalization proofs

for higher-order logic in the paper Hauptsatz for Intuitionistic Simple Type Theory

and generalized in Type Theory. He discusses this notion in the paper

About Models for Intuitionistic Type Theory and The notion of Definitional Equality

The extension from AUTOMATH is that one adds the notion of data type (natural number),

of constructors (zero and successor) and primitive recursion as definitional equality.

The motivation is that one can consider the schema of primitive recursion as a definition

of a function.

This was also influenced by natural deduction, where constructors correspond

to introduction rules and the work of Godel on system T.

With this extension, one obtains a programming language with dependent types

and where computations correspond to unfolding of definitions (that can be primitive

recursive definitions). This programming language has the feature that all computations

terminate. This has been also considered in functional programming, see e.g.

the discussion in this paper.

A description of the evaluation algorithm using techniques from functional programming

can be found in this work of Gregoire and Leroy