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Consider the following problem: given a simply typed lambda term of Boolean type and of order r, does it normalize to "true"? A related problem is: given a term M of word type and of order r together with a finite automaton D, does D accept the word represented by the normal form of M ? We prove that these problems are n-EXPTIME complete for r = 2n + 2, and n-EXPSPACE complete for r = 2n + 3. While the hardness part is relatively easy, the membership part is not so obvious; in particular,doi:10.4230/lipics.rta.2012.323 dblp:conf/rta/Terui12 fatcat:gfj6qlibuvbqnh5uo3erfa3rie