Computation and automata by Arto Salomaa

By Arto Salomaa

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Rewriting” is more honest than “simplification” because the terms do not necessarily become simpler in the process. 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.

9 Automatic Case Splits Goals containing if -expressions are usually proved by case distinction on the boolean condition. Here is an example: lemma " ∀ xs. if xs = [] then rev xs = [] else rev xs = []" The goal can be split by a special method, split : apply(split split_if) 32 3. More Functional Programming 1. ∀ xs. (xs = [] −→ rev xs = []) ∧ (xs = [] −→ rev xs = []) where split_if is a theorem that expresses splitting of if s. Because splitting the if s is usually the right proof strategy, the simplifier does it automatically.

Because splitting the if s is usually the right proof strategy, the simplifier does it automatically. Try apply(simp) on the initial goal above. This splitting idea generalizes from if to case . split) 1. (xs = [] −→ zs = xs @ zs) ∧ ( ∀ a list. xs = a # list −→ a # list @ zs = xs @ zs) The simplifier does not split case -expressions, as it does if -expressions, because with recursive datatypes it could lead to nontermination. Instead, the simplifier has a modifier split for adding splitting rules explicitly.

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