New PDF release: Interactive Theorem Proving and Program Development:

By Yves Bertot

Coq is an interactive facts assistant for the improvement of mathematical theories and officially qualified software program. it really is in line with a idea known as the calculus of inductive buildings, a version of variety theory.

This ebook offers a realistic advent to the advance of proofs and licensed courses utilizing Coq. With its huge number of examples and workouts it truly is a useful instrument for researchers, scholars, and engineers attracted to formal equipment and the advance of zero-fault software.

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Additional info for Interactive Theorem Proving and Program Development: Coq’Art: The Calculus of Inductive Constructions

Example text

3). Not every order is decidable. For instance, we can consider the type nat--+nat representing the functions from N to N and the following relation: f < g <=}::Ii EN, f(i) < g(i) A (Vj EN. j < i => fU) = gU)) This order relation is undecidable and it is impossible to design a comparison program similar to Z_le_gt_dec for this order. A consequence of this is that we cannot design a program to sort lists of functions. 3 The purpose of function aux is described in the following two lemmas, which are easily proved by induction on l: 3 This kind of problem is not inherent to Coq.

Thanks to the tools provided in this chapter, the proofs described in later chapters can be described more concisely. Chapter 16 describes a design technique for complex tactics that is particularly well-suited for the Calculus of Constructions: namely, the technique of proof by reflection. 8 Lexical Conventions This book provides a wealth of examples that can be input in the Coq system and the answers of the system to these examples. Throughout this book, we use the classical convention of books about computer tools: the typewriter font is used to represent text input by the user, while the italic font is used to represent the text output by the system as answers.

T n " instead of "(f tl) ... t n " thus reducing drastically the number of parentheses used. The Coq system automatically respects these conventions and suppresses extraneous parentheses: Check (((ifb (negb false)) true) false). ifb (negb false) true false: bool However, we should be careful to keep parentheses when they are needed to ensure that the term constructed will be well-formed. The following example shows that removing too many pairs of parentheses leads to a badly formed term: Check (negb negb true).

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