Tutorial on Isabelle/HOL

32 3. More Functional ProgrammingThe simplifier does not split case-expressions, as it does if-expressions, becausewith recursive datatypes it could lead to nontermination. Instead, thesimplifier has a modifier split for adding splitting rules explicitly. The lemmaabove can be proved in one step byapply(simp split: list.split)whereas apply(simp) alone will not succeed.Every datatype t comes with a theorem t.split which can be declaredto be a split rule either locally as above, or by giving it the split attributeglobally:declare list.split [split]The split attribute can be removed with the del modifier, either locallyapply(simp split del: split_if)or globally:declare list.split [split del]Polished proofs typically perform splitting within simp rather than invokingthe split method. However, if a goal contains several if and caseexpressions, the split method can be helpful in selectively exploring theeffects of splitting.The split rules shown above are intended to affect only the subgoal’sconclusion. If you want to split an if or case-expression in the assumptions,you have to apply split_if_asm or t.split_asm:lemma "if xs = [] then ys ≠ [] else ys = [] =⇒ xs @ ys ≠ []"apply(split split_if_asm)Unlike splitting the conclusion, this step creates two separate subgoals, whichhere can be solved by simp_all:1. [[xs = []; ys ≠ []]] =⇒ xs @ ys ≠ []2. [[xs ≠ []; ys = []]] =⇒ xs @ ys ≠ []If you need to split both in the assumptions and the conclusion, use t.splitswhich subsumes t.split and t.split_asm. Analogously, there is if_splits.!!The simplifier merely simplifies the condition of an if but not the then orelse parts. The latter are simplified only after the condition reduces to Trueor False, or after splitting. The same is true for case-expressions: only the selectoris simplified at first, until either the expression reduces to one of the cases or it issplit.3.1.10 TracingUsing the simplifier effectively may take a bit of experimentation. Set theProof General flag Isabelle > Settings > Trace Simplifier to get a better ideaof what is going on: