By Yves Bertot
Coq is an interactive facts assistant for the improvement of mathematical theories and officially qualified software program. it's in response to a idea referred to as the calculus of inductive structures, a variation of style theory.
This e-book presents a practical creation to the advance of proofs and licensed courses utilizing Coq. With its huge choice of examples and routines it truly is a useful software for researchers, scholars, and engineers attracted to formal equipment and the improvement of zero-fault software.
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Extra resources for Interactive Theorem Proving and Program Development: Coq’Art: The Calculus of Inductive Constructions
Van Heijenoort "From Frege to G6del" [841. 7 Contents of This Book The Calculus of Constructions Chapters 2 to 4 describe the Calculus of Constructions. Chapter 2 presents the simply typed A-calculus and its relation with functional programming. Important notions of terms, types, sorts, and reductions are presented in this chapter, together with the syntax used in Coq. Chapter 3 introduces the logical aspects of Coq, mainly with the CurryHoward isomorphism; this introduction uses the restricted framework that combines simply typed A-calculus and minimal propositional logic.
Require Import Bool. 1 Terms, Expressions, Types The notion of term covers a very general syntactic category in the Gallina specification language and corresponds to the intuitive notion of a well-formed expression. We come back to the rules that govern the formation of terms later. In this chapter, we mainly consider two kinds of terms, expressions, which correspond approximately to programs in a functional programming language, and types, which make it possible to determine when terms are well-formed and when they respect their specifications.
5 A Sorting Example 7 equiv _cons: Vz : Z, Vl, l' : list Z, 1 == l' => z:: 1 == z :: l' equiv _perm: Vn,p: Z, Vl, l' : list Z, 1 == l' => n :: p :: 1 == p :: n :: l' These lemmas will be used in the certification of a sorting program. 3 A Specification for a Sorting Program All the elements are now available to specify a sorting function on lists of integers. We have already seen that the Coq type system integrates complex specifications, with which we can constrain the input and output data of programs.