This paper provides a mathematical analysis of the relationship between vernacular and for-malised derivations of programs; in particular, when the vernacular derivations deploy general recursion. We introduce a calculus VPD to represent vernacular derivations and a constructive theory of operations and types, EOT , to represent formal derivations. We provide an intensional semantics for VPD wi...

Surprisingly, skew derivations rather than ordinary derivations are more basic (important) object in study of the Grassmann algebras. Let Λn = K⌊x1, . . . , xn⌋ be the Grassmann algebra over a commutative ring K with 12 ∈ K, and δ be a skew K-derivation of Λn. It is proved that δ is a unique sum δ = δ ev + δ of an even and odd skew derivation. Explicit formulae are given for δ and δ via the ele...

We define a new type of spectrum, called δ-approximate spectrum, of an element a in a complex unital Banach algebra A and show that the δ-approximate spectrum σ_δ (a) of a is compact. The relation between the δ-approximate spectrum and the usual spectrum is investigated. Also an analogue of the classical Gleason-Kahane-Zelazko theorem is established: For each ε>0, there is δ>0 such that if ϕ is...

We study the properties of input-consuming derivations of moded logic programs. Inputconsuming derivations can be used to model the behavior of logic programs using dynamic scheduling and employing constructs such as delay declarations. We consider the class of nicely-moded programs and queries. We show that for these programs a weak version of the well-known switching lemma holds also for inpu...

In a system of sequents for intuitionistic predicate logic a theorem, which corresponds to Prawitz’s Normal Form Theorem for natural deduction, are proved. In sequent derivations a special kind of cuts, maximum cuts, are defined. Maximum cuts from sequent derivations are connected with maximum segments from natural deduction derivations, i.e., sequent derivations without maximum cuts correspond...

Structured derivations were introduced by Back and von Wright as an extension of the calculational proof style originally proposed by E.W. Dijkstra and his colleagues. Structured derivations added nested subderivations and inherited assumptions to the original calculational style. This paper introduces a further extension of the structured derivation format, and gives a precise syntax and seman...

1. Derivations in general 1 1.1. Definitions and conventions . . . . . . . . . . . . . . . . . . . . . . . 1 1.2. Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3. The module of derivations . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4. The k-algebra RA (M) . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5. Compositions and tensor products . ....