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Categorical Semantics WG
Coq for UF
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Formalized Homotopy Theory
Higher Inductive Types
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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)
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
A description of the evaluation algorithm using techniques from functional programming
can be found in
this work of Gregoire and Leroy
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