A bridge is presented between term-rewriting theory in computer science and proof theory in logic. It is shown that proof-theoretic tools are very useful for analyzing two basic attributes of term rewriting systems, the termination property and the Church-Rosser property. A counterexample is given to show that Knuth's critical pair lemma does not hold for conditional rewrite systems. Two restrictions on conditional systems under which the critical pair lemma holds are presented. One is considered a generalization of Bergstra-Klop's former result; the other is concerned with a generalization of Kaplan's and Jouannaud-Waldmann's systems.