By Eric C.R. Hehner

ISBN-10: 0387941061

ISBN-13: 9780387941066

Figuring out programming and programming languages calls for wisdom of the underlying theoretical version. This e-book explores elements of programming which are amenable to mathematical evidence. the writer describes a programming idea that's a lot less complicated and extra accomplished than the present theories up to now. within the theoretical version, a specification is simply a boolean expression and refinement is simply a normal implication. the writer develops a pragmatic and wide approach for writing special requisites and designing courses whose executions most likely fulfill the standards. starting with preparatory fabric in common sense, numbers, units, lists, features and kinfolk, the booklet advances additional into application concept, the center of the booklet. next chapters will be chosen or passed over based on direction emphasis. The textual content might be valuable to scholars in classes on programming method or verification on the complex undergraduate or starting graduate point, in addition to for software program engineers within the box. All technical phrases are defined after which tested within the ebook anyplace attainable. No complicated mathematical wisdom or programming language is believed. The booklet comprises various workouts and worked-out ideas for particular workouts. Transparency masters and options for the remainder routines can be found from the writer.

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**Example text**

Function suc was defined earlier as suc = 〈n: nat→n+1〉 . Function nat→nat is an abbreviation of 〈n: nat→nat〉 , which has an unused variable. It is a nondeterministic function whose result, for each element of its domain nat , is the bunch nat . It is also the bunch of all functions whose domain includes nat and whose result is included in nat . suc: nat→nat use Function Inclusion Law = nat: Δsuc ∧ ∀n: nat· suc n: nat definition of suc = nat: nat ∧ ∀n: nat· n+1: nat reflexivity, and nat construction axiom = T We can prove similar inclusions about other functions defined in the first section of this chapter.

If g: A→B , then 〈f: (A→B)→ ... f ... 〉 g applies a higher-order function to a function argument. A parameter stands for an element of the domain, and the Application Law requires the argument to be an element of the domain, but functions are not elements. Therefore we consider a higher-order function applied to an argument, as written above, to be an abbreviation for 〈f: (A→B)→ ... ~f ... 〉 {g} and the set brackets { } just make the parameter and argument into The power operator elements, as required, and the content operator ~ then removes the set structure.

Y′=2×y x>0 ⇒ y′=2x–1 ⇐ x′=x–1. y′=2x y′=2×y ⇐ y:= 2×y. x:= 5 x′=x–1 ⇐ x:= x–1. y:= 7 The first refinement divides the problem into two cases; in the second case x 0 , and since x is natural, x>0 . In the second refinement, since x=0 , we want y′=1 , which we get by the assignment y:= 1 . The other assignment x:= 3 is superfluous, and our solution would be simpler without it; we have included it just to make the point that it is allowed by the specification. The next refinement makes y′=2x in two steps: first y′=2x–1 and then double y .

### A Practical Theory of Programming by Eric C.R. Hehner

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