The purposes of this note are the following two; we first generalize Okada-Takeutis well quasi ordinal diagram theory, utilizing the recent result of Dershowitz-Tzamerets version of tree embedding theorem with gap conditions. Second, we discuss possible use of such strong ordinal notation systems for the purpose of a typical traditional termination proof method for term rewriting systems, especially for second-order (pattern-matching-based) rewriting systems including a rewrite-theoretic version of Buchholzs hydra game.
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