By Tobias Nipkow

This quantity is a self-contained advent to interactive evidence in excessive- order common sense (HOL), utilizing the evidence assistant Isabelle 2002. in comparison with present Isabelle documentation, it offers an immediate path into higher-order good judgment, which most folks favor nowadays. It bypasses ?rst-order good judgment and minimizes dialogue of meta-theory. it truly is written for capability clients instead of for our colleagues within the learn global. one other departure from prior documentation is that we describe Markus Wenzel’s facts script notation rather than ML tactic scripts. The l- ter show you how to introduce new strategies at the ?y, yet rarely anyone does that. Wenzel’s devoted syntax is classy, changing for instance 8 simpli?cation strategies with a unmarried process, particularly simp, with linked - tions. The ebook has 3 components. – The ?rst half, basic recommendations, indicates how you can version practical courses in higher-order good judgment. Early examples contain lists and the average numbers. such a lot proofs are steps lengthy, including induction on a selected variable via the car tactic. yet even this trouble-free half covers such complex themes as nested and mutual recursion. – the second one half, good judgment and units, provides a suite of lower-level strategies so that you can use to use ideas selectively. It additionally describes I- belle/HOL’s remedy of units, features, and kin and explains easy methods to de?ne units inductively. one of many examples matters the speculation of version checking, and one other is drawn from a vintage textbook on formal languages.

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**Example text**

44 3. 4 Case Study: Tries Tries are a classic search tree data structure [15] for fast indexing with strings. 1 gives a graphical example of a trie containing the words “all”, “an”, “ape”, “can”, “car” and “cat”. When searching a string in a trie, the letters of the string are examined sequentially. Each letter determines which subtrie to search next. In this case study we model tries as a datatype, define a lookup and an update function, and prove that they behave as expected. ✑◗ ◗ ◗◗ ✑ ✑ ✑ a ✑ ◗ ◗◗ ✑✑ l n l p e c a ✑ ◗ ◗◗ ✑✑ n r t Fig.

1. 2 in the appendix shows the most important overloaded operations. Constant 1::nat is defined to equal Suc 0. This definition (see Sect. 2) is unfolded automatically by some tactics (like auto, simp and arith ) but not by others (especially the single step tactics in Chapter 5). If you need the full set of numerals, see Sect. 1. Novices are advised to stick to 0 and Suc. Both auto and simp (a method introduced below, Sect. 1) prove simple arithmetic goals automatically: lemma " [[ ¬ m < n; m < n + (1::nat) ]] =⇒ m = n" For efficiency’s sake, this built-in prover ignores quantified formulae, logical connectives, and all arithmetic operations apart from addition.

2. . 3. . =⇒ lookup . . =⇒ lookup . . =⇒ lookup . . bs = lookup t bs bs = lookup t bs bs = lookup t bs Clearly, if we want to make headway we have to instantiate bs as well now. e. [1-3] in our case. g. [2] are also allowed. This proof may look surprisingly straightforward. However, note that this comes at a cost: the proof script is unreadable because the intermediate proof states are invisible, and we rely on the (possibly brittle) magic of auto (simp_all will not do — try it) to split the subgoals of the induction up in such a way that case distinction on bs makes sense and solves the proof.