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In the following discussions, our use of the term "tactic" will always refer to texts based on alt and loop. 2. Elements of Proof Design 31 Remark. It is clear that constructs such as air and l o o p cannot be used arbitrarily, if combinatorial explosion is to be avoided. The above kind of use, however, is quite reasonable since the iteration is unfolded into application sequences with three steps only, one possible sequence is shown below. Moreover, such a use of tactics is quite frequent in the conceptual development of a proof by a human.
The schema variables A and B are represented by the (implicit) parameters of the rules. The structural differences are due to the different handling of assumptions. In the sequent calculus, the assumptions are recorded in a list which is part of a sequent. In the Deva formalization, assumptions are introduced by a (text-) abstraction; in its scope they may be used; they are discharged by a (text-) application. Thus, Deva has already key concepts of a deduction calculus "built-in". A certain overhead with respect to the sequent-based proof results from the fact that Deva enforces the use of explicit pointers (hi, h2, h3) to assumptions, such pointers are not needed in Fig.
The assumptions left implicit in the equational proof have been made explicit by basing the proof on an equational theory of natural numbers. Next, the transformation rules used in the proofs are explicitly stated so that the proof can be checked by a machine. Finally, whereas in the rigorous proof the terms themselves are transformed, in the formalized proof equations are transformed. , given equations are transformed into new equations. This formalization style was chosen because of its clarity and logical adequacy.