This paper introduces the theory of a particular kind of computation domains called concrete domains. The purpose of this theory is to find a satisfactory framework for the notions of coroutine computation and sequentiality of evaluation. Diagrams are emphasized because I believe that an important part of learning lattice theory is the acquisition of skill in drawing diagrams. George Gratzer 1 Domains of computation In general, we follow Scott's approach [Sco70]. To every syntactic object one associates a semantic object which is found in an appropriate semantic domain. For technical details, we follow [Mil73] and [Plo78] rather than Scott. Definition 1.1 A partial order is a pair ! D; ? where D is a non-empty set and is a binary relation satisfying: i) 8x 2 D x x (reflexivity) ii) 8x; y 2 D x y; y x ) x = y (antisymmetry) iii) 8x; y; z 2 D x y; y z ) x z (transitivity) One writes x ! y when x y and x 6= y. Two elements x and y are comparable when either x y or y x. W...

Abstract. We present the complete development, in Gallina, of the residual theory of β-reduction in pure λ-calculus. The main result is the Prism Theorem, and its corollary Lévy’s Cube Lemma, a strong form of the parallel-moves lemma, itself a key step towards the confluence theorem and its usual corollaries (Church-Rosser, uniqueness of normal forms). Gallina is the specification language of the Coq Proof Assistant[7, 11]. It is a specific concrete syntax for its abstract framework, the Calculus of Inductive Constructions[15]. It may be thought of as a smooth mixture of higher-order predicate calculus with recursive definitions, inductively defined data-types, and inductive predicate definitions reminiscent of logic programming. The development presented here was fully checked in the current distribution version Coq V5.8. We just state the lemmas in the order in which they are proved, omitting the proof justifications. The full transcript is available as a standard library in the distribution of Coq. 1

Abstract. Residuals have been studied for various forms of rewriting setting. In this article we study residuals in orthogonal Pattern Rewriting Systems (PRSs). First, the rewrite relation is defined by means of a higher-order rewriting logic, and proof terms are defined that witness reductions. Then, we have the formal machinery to define a residual operator for PRSs, and we will prove that an orthogonal PRS together with the residual operator mentioned above, is a residual system. As a side-effect, all results of (abstract) residual theory are inherited by orthogonal PRSs, such as confluence, and the notion of permutation equivalence of reductions. 1

1.1 An overview of the specification language Gallina.................... 5

Two co-initial reductions in a term rewriting system are said to be equivalent if they perform the same steps, albeit maybe in a di#erent order. We present four characterisations of such a notion of equivalence, based on permutation, standardisation, labelling and projection, respectively. We prove that the characterisations all yield the same notion of equivalence, for the class of first-order left-linear term rewriting systems. A crucial role in our development is played by the notion of a proof term. 1

Abstract. Residuals have been studied for various forms of rewriting and residual systems have been defined to capture residuals in an abstract setting. In this article we study residuals in orthogonal Pattern Rewriting Systems (PRSs). First, the rewrite relation is defined by means of a higher-order rewriting logic, and proof terms are defined that witness reductions. Then, we have the formal machinery to define a residual operator for PRSs, and we will prove that an orthogonal PRS together with the residual operator mentioned above, is a residual system. As a side-effect, all results of (abstract) residual theory are inherited by orthogonal PRSs, such as confluence, and the notion of