Indexed induction and coinduction, fibrationally.

This paper extends the fibrational approach to induction and coinduction pioneered by Hermida and Jacobs, and developed by the current authors, in two key directions. First, we present a sound coinduction rule for any data type arising as the final coalgebra of a functor, thus relaxing Hermida and Jacobs’ restriction to polynomial data types. For this we introduce the notion of a quotient category with equality (QCE), which both abstracts the standard notion of a fibration of relations constructed from a given fibration, and plays a role in the theory of coinduction dual to that of a comprehension category with unit (CCU) in the theory of induction. Second, we show that indexed inductive and coinductive types also admit sound induction and coinduction rules. Indexed data types often arise as initial algebras and final coalgebras of functors on slice categories, so our key technical results give sufficent conditions under which we can construct, from a CCU (QCE) U : E -> B, a fibration with base B/I that models indexing by I and is also a CCU (QCE)

Refining Inductive Types

Dependently typed programming languages allow sophisticated properties of data to be expressed within the type system. Of particular use in dependently typed programming are indexed types that refine data by computationally useful information. For example, the N-indexed type of vectors refines lists by their lengths. Other data types may be refined in similar ways, but programmers must produce purpose-specific refinements on an ad hoc basis, developers must anticipate which refinements to include in libraries, and implementations must often store redundant information about data and their refinements. In this paper we show how to generically derive inductive characterisations of refinements of inductive types, and argue that these characterisations can alleviate some of the aforementioned difficulties associated with ad hoc refinements. Our characterisations also ensure that standard techniques for programming with and reasoning about inductive types are applicable to refinements, and that refinements can themselves be further refined

Refining Inductive Types

Dependently typed programming languages allow sophisticated properties of data to be expressed within the type system. Of particular use in dependently typed programming are indexed types that refine data by computationally useful information. For example, the N-indexed type of vectors refines lists by their lengths. Other data types may be refined in similar ways, but programmers must produce purpose-specific refinements on an ad hoc basis, developers must anticipate which refinements to include in libraries, and implementations must often store redundant information about data and their refinements. In this paper we show how to generically derive inductive characterizations of refinements of inductive types, and argue that these characterizations can alleviate some of the aforementioned difficulties associated with ad hoc refinements. Our characterizations also ensure that standard techniques for programming with and reasoning about inductive types are applicable to refinements, and that refinements can themselves be further refined

When Is A Type Refinement An Inductive Type?

Dependently typed programming languages allow sophisticated properties of data to be expressed within the type system. Of particular use in dependently typed programming are indexed types that refine data by computationally useful information. For example, the N-indexed type of vectors refines lists by their lengths. Other data types may be refined in similar ways, but programmers must produce purpose-specific refinements on an ad hoc basis, developers must anticipate which refinements to include in libraries, and implementations often store redundant information about data and their refinements. This paper shows how to generically derive inductive characterizations of refinements of inductive types, and argues that these characterizations can alleviate some of the aforementioned difficulties associated with ad hoc refinements. These characterizations also ensure that standard techniques for programming with and reasoning about inductive types are applicable to refinements, and that refinements can themselves be further refined

Abstraction and Invariance for Algebraically Indexed Types

Reynolds’ relational parametricity provides a powerful way to reason about programs in terms of invariance under changes of data representation. A dazzling array of applications of Reynolds’ theory exists, exploiting invariance to yield “free theorems”, non-inhabitation results, and encodings of algebraic datatypes. Outside computer science, invariance is a common theme running through many areas of mathematics and physics. For example, the area of a triangle is unaltered by rotation or flipping. If we scale a triangle, then we scale its area, maintaining an invariant relationship be-tween the two. The transformations under which properties are in-variant are often organised into groups, with the algebraic structure reflecting the composability and invertibility of transformations. In this paper, we investigate programming languages whose types are indexed by algebraic structures such as groups of geometric transformations. Other examples include types indexed by principals–for information flow security–and types indexed by distances–for analysis of analytic uniform continuity properties. Following Reynolds, we prove a general Abstraction Theorem that covers all these instances. Consequences of our Abstraction Theorem include free theorems expressing invariance properties of programs, type isomorphisms based on invariance properties, and non-definability results indicating when certain algebraically indexed types are uninhabited or only inhabited by trivial programs. We have fully formalized our framework and most examples in Coq

Fibrations, logical predicates and indeterminates

Within the framework of categorical logic/type theory, we provide a category-theoretic account of some logical concepts, i.e. first-order logical predicates for simply typed lambda-calculus, structural induction for inductive data types, and indeterminates for polymorphic calculi. The main concept which underlies the issues above is that of fibration, which gives an abstract presentation of the indexing present in all cases: predicates indexed by types/contexts in first-order logic and types indexed by kinds in polymorphic calculi.The characterisation of the logical concepts in terms of fibrations relies on a fundamental property of adjunctions between fibrations, which in particular relates some structure in the total category of a fibration with that of the fibres. Suitable instances of this property reflect the above-mentioned logical concepts in an abstract way, independently of their syntactic presentation, thereby illuminating their main features

A Categorical Semantics for Inductive-Inductive Definitions

Induction-induction is a principle for defining data types in Martin-Löf Type Theory. An inductive-inductive definition consists of a set A, together with an A-indexed family B : A → Set, where both A and B are inductively defined in such a way that the constructors for A can refer to B and vice versa. In addition, the constructors for B can refer to the constructors for A. We extend the usual initial algebra semantics for ordinary inductive data types to the inductive-inductive setting by considering dialgebras instead of ordinary algebras. This gives a new and compact formalisation of inductive-inductive definitions, which we prove is equivalent to the usual formulation with elimination rules