Extensional proofs in a propositional logic modulo isomorphisms

System I is a proof language for a fragment of propositional logic where isomorphic propositions, such as $A\wedge B$ and $B\wedge A$, or $A\Rightarrow(B\wedge C)$ and $(A\Rightarrow B)\wedge(A\Rightarrow C)$ are made equal. System I enjoys the strong normalisation property. This is sufficient to prove the existence of empty types, but not to prove the introduction property (every closed term in normal form is an introduction). Moreover, a severe restriction had to be made on the types of the variables in order to obtain the existence of empty types. We show here that adding $\eta$-expansion rules to System I permits to drop this restriction, and yields a strongly normalising calculus with enjoying the full introduction property.Comment: 15 pages plus references and appendi

An Intuitionistic Formula Hierarchy Based on High-School Identities

We revisit the notion of intuitionistic equivalence and formal proof representations by adopting the view of formulas as exponential polynomials. After observing that most of the invertible proof rules of intuitionistic (minimal) propositional sequent calculi are formula (i.e. sequent) isomorphisms corresponding to the high-school identities, we show that one can obtain a more compact variant of a proof system, consisting of non-invertible proof rules only, and where the invertible proof rules have been replaced by a formula normalisation procedure. Moreover, for certain proof systems such as the G4ip sequent calculus of Vorob'ev, Hudelmaier, and Dyckhoff, it is even possible to see all of the non-invertible proof rules as strict inequalities between exponential polynomials; a careful combinatorial treatment is given in order to establish this fact. Finally, we extend the exponential polynomial analogy to the first-order quantifiers, showing that it gives rise to an intuitionistic hierarchy of formulas, resembling the classical arithmetical hierarchy, and the first one that classifies formulas while preserving isomorphism

Automatic and Transparent Transfer of Theorems along Isomorphisms in the Coq Proof Assistant

In mathematics, it is common practice to have several constructions for the same objects. Mathematicians will identify them modulo isomorphism and will not worry later on which construction they use, as theorems proved for one construction will be valid for all. When working with proof assistants, it is also common to see several data-types representing the same objects. This work aims at making the use of several isomorphic constructions as simple and as transparent as it can be done informally in mathematics. This requires inferring automatically the missing proof-steps. We are designing an algorithm which finds and fills these missing proof-steps and we are implementing it as a plugin for Coq

Non-wellfounded trees in Homotopy Type Theory

We prove a conjecture about the constructibility of coinductive types - in the principled form of indexed M-types - in Homotopy Type Theory. The conjecture says that in the presence of inductive types, coinductive types are derivable. Indeed, in this work, we construct coinductive types in a subsystem of Homotopy Type Theory; this subsystem is given by Intensional Martin-L\"of type theory with natural numbers and Voevodsky's Univalence Axiom. Our results are mechanized in the computer proof assistant Agda.Comment: 14 pages, to be published in proceedings of TLCA 2015; ancillary files contain Agda files with formalized proof

Heterogeneous substitution systems revisited

Matthes and Uustalu (TCS 327(1-2):155-174, 2004) presented a categorical description of substitution systems capable of capturing syntax involving binding which is independent of whether the syntax is made up from least or greatest fixed points. We extend this work in two directions: we continue the analysis by creating more categorical structure, in particular by organizing substitution systems into a category and studying its properties, and we develop the proofs of the results of the cited paper and our new ones in UniMath, a recent library of univalent mathematics formalized in the Coq theorem prover.Comment: 24 page

Large and Infinitary Quotient Inductive-Inductive Types

Quotient inductive-inductive types (QIITs) are generalized inductive types which allow sorts to be indexed over previously declared sorts, and allow usage of equality constructors. QIITs are especially useful for algebraic descriptions of type theories and constructive definitions of real, ordinal and surreal numbers. We develop new metatheory for large QIITs, large elimination, recursive equations and infinitary constructors. As in prior work, we describe QIITs using a type theory where each context represents a QIIT signature. However, in our case the theory of signatures can also describe its own signature, modulo universe sizes. We bootstrap the model theory of signatures using self-description and a Church-coded notion of signature, without using complicated raw syntax or assuming an existing internal QIIT of signatures. We give semantics to described QIITs by modeling each signature as a finitely complete CwF (category with families) of algebras. Compared to the case of finitary QIITs, we additionally need to show invariance under algebra isomorphisms in the semantics. We do this by modeling signature types as isofibrations. Finally, we show by a term model construction that every QIIT is constructible from the syntax of the theory of signatures