In 2004 Atserias, Kolaitis, and Vardi proposed -based propositional proof systems that prove unsatisfiability of a CNF formula by deduction of an identically false from s representing clauses of the initial formula. All s in such proofs have the same order of variables. We initiate the study of based proof systems that additionally contain a rule that allows changing the order in s. At first we consider a proof system that uses the conjunction (join) rule and the rule that allows changing the order. We exponentially separate this proof system from proof system that uses only the conjunction rule. We prove exponential lower bounds on the size of refutations of Tseitin formulas and the pigeonhole principle. The first lower bound was previously unknown even for proofs and the second one extends the result of Tveretina et al. from to. In 2001 Aguirre and Vardi proposed an approach to the propositional satisfiability problem based on s and symbolic quantifier elimination (we denote algorithms based on this approach as algorithms). We augment these algorithms with the operation of reordering of variables and call the new scheme algorithms. We notice that there exists an algorithm that solves satisfiable and unsatisfiable Tseitin formulas in polynomial time (a standard example of a hard system of linear equations over), but we show that there are formulas representing systems of linear equations over that are hard for algorithms. Our hard instances are satisfiable formulas representing systems of linear equations over that correspond to checksum matrices of error correcting codes.