By M. J. C. Gordon, T. F. Melham

HOL is an evidence improvement procedure meant for purposes to either and software program. it really is mostly utilized in methods: for without delay proving theorems, and as theorem-proving aid for application-specific verification structures. HOL is at present being utilized to a large choice of difficulties, together with the specification and verification of serious platforms. creation to HOL presents a coherent and self-contained description of HOL containing either an academic advent and lots of the fabric that's wanted for daily paintings with the method. After a brief assessment that provides a 'hands-on suppose' for a way HOL is used, there follows a close description of the ML language. The common sense that HOL helps and the way this good judgment is embedded in ML are then defined intimately. this is often by means of an evidence of the theorem-proving infrastructure supplied by way of HOL. ultimately appendices comprise a subset of the reference guide, and an summary of the HOL library, together with an instance of an exact library documentation.

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**Introduction to HOL: A Theorem-Proving Environment for Higher-Order Logic**

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**Extra resources for Introduction to HOL: A Theorem-Proving Environment for Higher-Order Logic**

**Sample text**

As an example, consider the permutation α of {0, . . , 8} deﬁned as follows: α(0) = 3, α(1) = 4, α(2) = 2, α(3) = 0, α(4) = 5, α(5) = 1, α(6) = 8, α(7) = 7, and α(8) = 6. If we write α as a union of disjoint cycles, then we have α = (0 3)(1 4 5)(2)(6 8)(7). The cycle type of α, written as a list of nondecreasing integers, is [1, 1, 2, 2, 3]. Note that α has two ﬁxed points, namely 2 and 7. Any automorphism α of a symmetric BIBD, say (X, A), will permute the blocks in the set A. Hence, we can consider the permutation of A induced by α and deﬁne the cycle type of this permutation in the obvious way.

Then any block of the design forms the desired difference set. 4 is one that can be obtained in this way. 4. We now discuss quadratic residues in a ﬁnite ﬁeld F q , where q is an odd prime power. The quadratic residues of F q are the elements in the set QR(q) = {z2 : z ∈ F q , z = 0}. We will also deﬁne QNR(q) = F q \(QR(q) ∪ {0}). The elements of QNR(q) are called the quadratic nonresidues of F q . Using the fact that z2 = (−z)2 , it is not hard to prove that the mapping z → z2 is a two-to-one mapping if z ∈ F q \{0} and q is odd.

A straightforward generalization to higher dimensions is given in the next theorem. 14. Suppose q ≥ 2 is a prime power and d ≥ 2 is an integer. Then there exists a symmetric qd+1 −1 qd −1 qd−1 −1 q−1 , q−1 , q−1 -BIBD. Proof. Let V = (F q )d+1, let V1 consist of all one-dimensional subspaces of V, and let Vd consist of all d-dimensional subspaces of V. Each d-dimensional subspace gives rise to a block, as before. 14. 14 correspond to the points and hyperplanes of the d-dimensional projective geometry, PGd (q).