By Tobias Nipkow

This quantity is a self-contained advent to interactive evidence in excessive- order common sense (HOL), utilizing the facts assistant Isabelle 2002. in comparison with latest Isabelle documentation, it offers an immediate path into higher-order good judgment, which most folk desire nowadays. It bypasses ?rst-order good judgment and minimizes dialogue of meta-theory. it really is written for strength clients instead of for our colleagues within the learn international. one other departure from prior documentation is that we describe Markus Wenzel’s evidence script notation rather than ML tactic scripts. The l- ter allow you to introduce new strategies at the ?y, yet not often anyone does that. Wenzel’s devoted syntax is sublime, exchanging for instance 8 simpli?cation strategies with a unmarried strategy, specifically simp, with linked - tions. The ebook has 3 elements. – The ?rst half, ordinary ideas, exhibits the right way to version sensible courses in higher-order good judgment. Early examples contain lists and the traditional numbers. such a lot proofs are steps lengthy, which include induction on a selected variable via the car tactic. yet even this simple half covers such complex issues as nested and mutual recursion. – the second one half, good judgment and units, provides a set of lower-level strategies for you to use to use ideas selectively. It additionally describes I- belle/HOL’s therapy of units, capabilities, and family members and explains the way to de?ne units inductively. one of many examples issues the idea of version checking, and one other is drawn from a vintage textbook on formal languages.

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**Extra resources for Isabelle/HOL: A Proof Assistant for Higher-Order Logic**

**Example text**

In the recursive case the subtrie tt associated with the first letter a is extracted, recursively updated, and then placed in front of the association list. The old subtrie associated with a is still in the association list but no longer accessible via assoc. Clearly, there is room here for optimizations! split[split] The reason becomes clear when looking (probably after a failed proof attempt) at the body of update : it contains both let and a case distinction over type option. Our main goal is to prove the correct interaction of update and lookup : theorem " ∀ t v bs.

28 3. More Functional Programming The simplifier proves arithmetic goals as described in Sect. 1 above. Arithmetic expressions are simplified using built-in procedures that go beyond mere rewrite rules. New simplification procedures can be coded and installed, but they are definitely not a matter for this tutorial. 2 Simplification Rules To facilitate simplification, the attribute [simp] declares theorems to be simplification rules, which the simplifier will use automatically. In addition, datatype and primrec declarations (and a few others) implicitly declare some simplification rules.

In the Logic for Computable Functions (LCF), types like datatype lam = C "lam → lam" do indeed make sense [25]. Note the different arrow, → instead of ⇒, expressing the type of continuous functions. There is even a version of LCF on top of HOL, called HOLCF [18]. 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.