The third chapter of my dissertation is effectively a longer version of an ESOP 2011 paper, Polymorphic Contracts. We define FH, a polymorphic calculus with manifest contracts. Atsushi Igarashi, with whom I did the original FH work that appeared in ESOP 2011, and his student Taro Sekiyama have been working on continuing some of the FH work. They discovered—after my defense!—a bug in FH’s metatheory.

Short version: FH used parallel reduction as a conversion relation. A key property of this relation is *substitutivity*. We phrased it as “if e1 ⇒ e1′ and e2 ⇒ e2′ then e1{e2/x} ⇒ e1′{e2’/x}”. Unfortunately, this doesn’t hold for FH, due to subtleties in FH’s reduction rules for casts. The cast reduction rules are implicitly performing equality checks on types, and these equality checks can be affected by substitutions to change which reduction rule applies. The (tentative) solution in my thesis is to use a simpler type (and term) conversion relation which we call *common subexpression reduction* (CSR). In CSR, we relate types and terms that are closed by closing substitutions σ_{1} →^{*} σ_{2}. That is, the CSR conversion is the smallest congruence which is substitutive for →^{*}, i.e., where if e →^{*} e’ then T{e/x} ≡ T{e’/x}.

Long version: I’ve excerpted Section 3.5 of my thesis which discusses the System FH type conversion bug.