Relating Proof Systems for Recursive Types

Proof Systems The purpose of this appendix is to formally underpin the results concerning derivability and admissibility of inference rules that are reported in Chapter 4. For this purpose, we introduce the notions “abstract pure Hilbert system” (APHS) and “abstract natural deduction system” (ANDS) in Section B.1 and Section B.2 below, respectively. We respectively adapt the notions of derivability, admissibility, and correctness of inference rules to APHS’s and ANDS’s, and subsequently gather basic facts about these notions and their relationship towards each other. Eventually the focus of our interest is, in each kind of proof systems, on statements that clarify the relationship between derivability or admissibility of a rule R in a system S and the possibility to eliminate applications of R from derivations in extensions of S by adding R as a new rule. The concept of APHS defined in Section B.1 is a slight reformulation of the concept of n-AHS (“abstract Hilbert system with names for axioms and rules”) that we have introduced and investigated earlier in [Gra03a] (see also the short overview [Gra04b]). In Section B.1 only a small part of the results developed in the mentioned report is treated, with the statements presented here being generally straightforward adaptations to APHS’s of statements given there. The concept of ANDS and the investigation of rule derivability and admissibility in ANDS’s, the topics treated in Section B.2, are a more recent extension to natural-deduction systems of the study, in the report referred to above, of rule derivability and admissibility in “abstract Hilbert systems”. Only a very small fraction of the statements given here will be proved. As a consequence, this appendix has to be read as of a brief outline of results that we have obtained for concepts that we define here precisely. In the case of APHS’s we may however refer to the mentioned report [Gra03a]: there, for practically all of the results given here, closely corresponding statements can be found, together with detailed proofs. 1We mean statements that correspond to statements given here within the concepts developed there of “abstract Hilbert system” (AHS) and of “abstract Hilbert system with names” (n-AHS). 360 Appendix B: Abstract Proof Systems B.1 Abstract Pure Hilbert Systems For the purpose of collecting general results about the notions or derivability and admissibility of inference rules in pure Hilbert systems, we introduce the framework of “abstract pure Hilbert system”. In such systems, which are defined by analogy with abstract rewrite systems, it is abstracted away from the syntax of the formula language in a pure Hilbert system, and consequently also from the specific ways how rules can be defined syntactically. In an abstract pure Hilbert system a rule is a set of instances that is endowed with a premise and a conclusion function, which respectively assign a finite sequence of premises and a conclusion to every instance. After defining the notion of APHS-rule with respect to a given set (of formulas), abstract pure Hilbert systems are then defined as consisting of a set of formulas, a set of named axioms (which are named formulas), and a set of named rules that ‘operate’ on the set of formulas of the system. The notion of “abstract pure Hilbert system” is analogous to, and in fact has been motivated by, the notion of “abstract rewriting system” in the formulation of van Oostrom and de Vrijer in [vOdV02] and in [Ter03, p. 317]. There, an ARS A is defined as a quadruple 〈A,Φ, src, tgt〉 in which A and Φ are sets whose members are respectively called objects and steps, and src, tgt : Φ→ A are the source and the target functions of A. We start by defining the abstract notion of rule that underlies the notion of “abstract pure Hilbert system”, which will be defined subsequently. Definition B.1. 1. (An abstract notion of rule). Let Fo be a set. An APHS-rule (a rule for an “abstract pure Hilbert system”) on (formulas of) Fo is a triple of the form 〈Insts, prem, concl〉 , where • Insts is a set, the members of which are called the instances or applications of R, • and prem : Insts→ Seqcsf(Fo) and concl : Insts→ Fo are the premise and conclusion functions of R. We will use the symbolic denotations InstsR, premR and conclR, whenever we want to refer directly to the instance set, the premise and conclusion functions of a rule R, respectively. And we will use ι as a syntactical variable for instances. For all sets Fo, we furthermore denote by R(Fo) the class of APHS-rules on Fo. £ In addition to the functions prem and concl associated with a rule, we define for auxiliary purposes the functions arity and prem: for every set Fo of formulas and for every rule R = 〈Insts, prem, concl〉 on Fo , we introduce the function arity : Insts→ ω ι 7→ arity(ι) =def lg(prem(ι)) , 2The concept of “abstract pure Hilbert system” is defined below in Definition B.1. 2. B.1 Abstract Pure Hilbert Systems 361 Figure B.1: Visualization as ‘hypergraph hyperedges’ of two kinds of applications of APHS-rules: of a zero premise application ι of a rule R1 and of an application ι ′ of a rule R2 such that ι ′ has n = arity(ι) ∈ ω\{0} premises.