Abstract This paper introducesdeep induction, and shows that it is the notion of induction most appropriate to nested types and other data types defined over, or mutually recursively with, (other) such types. Standard induction rules induct over only the top-level structure of data, leaving any data internal to the top-level structure untouched. By contrast, deep induction rules induct overallof the structured data present. We give a grammar generating a robust class of nested types (and thus ADTs), and develop a fundamental theory of deep induction for them using their recently defined semantics as fixed points of accessible functors on locally presentable categories. We then use our theory to derive deep induction rules for some common ADTs and nested types, and show how these rules specialize to give the standard structural induction rules for these types. We also show how deep induction specializes to solve the long-standing problem of deriving principled and practically useful structural induction rules for bushes and othertrulynested types. Overall, deep induction opens the way to making induction principles appropriate to richly structured data types available in programming languages and proof assistants. Agda implementations of our development and examples, including two extended case studies, are available.

We present a grammar for a robust class of data types that includes algebraic data types (ADTs), (truly) nested types, generalized algebraic data types (GADTs), and their higher-kinded analogues. All of the data types our grammar defines, as well as their associated type constructors, are shown to have fully functorial initial algebra semantics in locally presentable categories. Since local presentability is a modest hypothesis, needed for such semantics for even the simplest ADTs, our semantic framework is actually quite conservative. Our results thus provide evidence that if a category supports fully functorial initial algebra semantics for standard ADTs, then it does so for advanced higher-kinded data types as well. To give our semantics we introduce a new type former called Lan. that captures on the syntactic level the categorical notion of a left Kan extension. We show how left Kan extensions capture propagation of a data type's syntactic generators across the entire universe of types, via a certain completion procedure, so that the type constructor associated with a data type becomes a bonafide functor with a canonical action on morphisms. A by-product of our semantics is a precise measure of the semantic complexity of data types, given by the least cardinal λ for which the functor underlying a data type is λ-accessible. The proof of our main result allows this cardinal to be read off from a data type definition without much effort. It also gives a sufficient condition for a data type to have semantic complexity ω, thus characterizing those data types whose data elements are effectively enumerable.

The main observational equivalences of the untyped lambda-calculus have been characterized in terms of extensional equalities between B\"ohm trees. It is well known that the lambda-theory H*, arising by taking as observables the head normal forms, equates two lambda-terms whenever their B\"ohm trees are equal up to countably many possibly infinite eta-expansions. Similarly, two lambda-terms are equal in Morris's original observational theory H+, generated by considering as observable the beta-normal forms, whenever their B\"ohm trees are equal up to countably many finite eta-expansions. The lambda-calculus also possesses a strong notion of extensionality called "the omega-rule", which has been the subject of many investigations. It is a longstanding open problem whether the equivalence B-omega obtained by closing the theory of B\"ohm trees under the omega-rule is strictly included in H+, as conjectured by Sall\'e in the seventies. In this paper we demonstrate that the two aforementioned theories actually coincide, thus disproving Sall\'e's conjecture. The proof technique we develop for proving the latter inclusion is general enough to provide as a byproduct a new characterization, based on bounded eta-expansions, of the least extensional equality between B\"ohm trees. Together, these results provide a taxonomy of the different degrees of extensionality in the theory of B\"ohm trees.

Abstract The ${\lambda }$-calculus enjoys the property that each ${\lambda }$-term has at least one fixed point, which is due to the existence of a fixed point combinator. It is unknown whether it enjoys the ‘fixed point property’ stating that each ${\lambda }$-term has either one or infinitely many pairwise distinct fixed points. We show that the fixed point property holds when considering possibly open fixed points. The problem of counting fixed points in the closed setting remains open, but we provide sufficient conditions for a ${\lambda }$-term to have either one or infinitely many fixed points. In the main result of this paper we prove that in every sensible ${\lambda }$-theory there exists a ${\lambda }$-term that violates the fixed point property. We then study the open problem concerning the existence of a double fixed point combinator and propose a proof technique that could lead towards a negative solution. We consider interpretations of the ${\lambda } {\mathtt{Y}}$-calculus into the ${\lambda }$-calculus together with two reduction extension properties, whose validity would entail the non-existence of any double fixed point combinators. We conjecture that both properties hold when typed ${\lambda } {\mathtt{Y}}$-terms are interpreted by arbitrary fixed point combinators. We prove reduction extension property I for a large class of fixed point combinators. Finally, we prove that the ${\lambda }{\mathtt{Y}}$-theory generated by the equation characterizing double fixed point combinators is a conservative extension of the ${\lambda }$-calculus.

We present a coinductive framework for defining and reasoning about the infinitary analogues of equational logic and term rewriting in a uniform, coinductive way. The setup captures rewrite sequences of arbitrary ordinal length, but it has neither the need for ordinals nor for metric convergence. This makes the framework especially suitable for formalizations in theorem provers.

One of the best-known methods for discriminating λ-terms with respect to β-convertibility is due to Corrado Böhm. The idea is to compute the infinitary normal form of a λ-term M, the Böhm Tree (BT) of M. If λ-terms M, N have distinct BTs, then M ≠βN, that is, M and N are not β-convertible. But what if their BTs coincide? For example, all fixed point combinators (FPCs) have the same BT, namely λx.x(x(x(. . .))). We introduce a clocked λ-calculus, an extension of the classical λ-calculus with a unary symbol τ used to witness the β-steps needed in the normalization to the BT. This extension is infinitary strongly normalizing, infinitary confluent and the unique infinitary normal forms constitute enriched BTs, which we call clocked BTs. These are suitable for discriminating a rich class of λ-terms having the same BTs, including the well-known sequence of Böhm's FPCs. We further increase the discrimination power in two directions. First, by a refinement of the calculus: the atomic clocked λ-calculus, where we employ symbols τp that also witness the (relative) positions p of the β-steps. Second, by employing a localized version of the (atomic) clocked BTs that has even more discriminating power.

We present a coinductive framework for defining infinitary analogues of equational reasoning and rewriting in a uniform way. The setup captures rewrite sequences of arbitrary ordinal length, but it has neither the need for ordinals nor for metric convergence. This makes the framework especially suitable for formalizations in theorem provers.

Abstract We work in , the untypedλ-calculus in which all unsolvables are identified. We resolve a conjecture of Barendregt asserting that the range of a definable map is either infinite or a singleton. This is refuted by constructing aλ-term Ξ such that ΞM= ΞI ⇔ ΞM≠ ΞΩ. The construction generalizes to ranges of any finite size, and to some other sensible lambda theories.