<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">Gradually typed languages are designed to support both dynamically typed
and statically typed programming styles while preserving the benefits of
each.  While existing gradual type soundness theorems for these
languages aim to show that type-based reasoning is preserved when
moving from the fully static setting to a gradual one, these theorems do
not imply that correctness of type-based refactorings and optimizations
is preserved.  Establishing correctness of program transformations is
technically difficult, because it requires reasoning about program
equivalence, and is often neglected in the metatheory of gradual
languages.  

In this paper, we propose an \emph{axiomatic} account of
program equivalence in a gradual cast calculus, which we formalize in a
logic we call \emph{gradual type theory} (GTT). Based on Levy's
call-by-push-value, GTT gives an axiomatic account of both call-by-value
and call-by-name gradual languages.  Based on our axiomatic account we
prove many theorems that justify optimizations and refactorings in
gradually typed languages. For example, 
\emph{uniqueness principles} for gradual type connectives show that if the
$\beta\eta$ laws hold for a connective, then casts between that
connective must be equivalent to the so-called ``lazy'' cast semantics.
Contrapositively, this shows that ``eager'' cast semantics violates the
extensionality of function types.  As another example, we show that
gradual upcasts are pure functions and, dually, gradual downcasts are
strict functions.  We show the consistency and applicability of our
axiomatic theory by proving that a contract-based implementation using
the lazy cast semantics gives a logical relations model of our type
theory, where equivalence in GTT implies contextual equivalence of the
programs.  Since GTT also axiomatizes the dynamic gradual guarantee, our
model also establishes this central theorem of gradual typing.  The
model is parametrized by the implementation of the dynamic types, and so
gives a family of implementations that validate type-based optimization
and the gradual guarantee.
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