Because of the binding variable separation convention, all these substitutions are idempotent. Otherwise, by Lemma 4. P is a sum. Below we assume no P i is an abstraction, i.

The proof proceeds by distinguishing two cases depending on the form of Q. Its proof did not depend on these properties. Now we formulate the main result of this section, stating that this instance of solve satisfies C 1 and C 2 and, thus, makes the calculus confluent :. Proving C 1.

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Proving C 2. The following corollary immediately follows from theorems 4.

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The solving function considered in the previous section does not permit to do reduction by a generic commutative symbol: one needs to know explicitly which function symbol is commutative. In this section, we illustrate how this can be done on the example of commutative matching, obtaining yet another extension of the pure pattern calculus with matchable symbols to finitary matching.

First, we start with identifying a subset of variables and, consequently, matchables that are restricted to be instantiated respectively, bound to single symbols only, instead of arbitrary terms. The letters x , y , z , v and their hatted counterparts are still reserved for arbitrary variables and matchables.

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## Model theory

This distinction between unrestricted variables and single-symbol variables should be reflected in substitutions as well. We denote this theory by PC. We can easily modify the rules for solve C to obtain the ones that define the solving function solve PC for polymorphic commutativity. From the commutative matching algorithm in Section 4. These rules have quite intuitive explanations: ME-CS is almost the same as the ME-C rule from solve C , with the only difference that the head of the right side is a commutative variable or matchable symbol, instead of a commutative function symbol.

The theorem can be proved similarly to Theorem 4. We take into account that although free commutative or syntactic variables may appear in the active position in the right-hand side of matching equations, it does not cause a problem because any instantiation replaces them only by single symbols. Essentially, it says that nested occurrences of the symbol f can be flattened out. The theory is quite similar to associativity.

We denote it by F. For the other cases of Definition 4. Otherwise the proof proceeds as in Lemma 4. Hence, by Definition 4. The rule-based flat matching algorithm we described in this section is inspired by the flat matching procedure from [ 24 ]. The rules are given below.

## Manes, Ernest G. 1943-

An important assumption is that all nested occurrences of flat symbols in matching equations are flattened. No two rules apply to the same equation.

Similarly to Lemma 4. Termination follows from the fact that every rule except ME-Flat-1 strictly decreases the multiset of sizes number of symbols of the left-hand sides of matching equations. The rule ME-Flat-1 does not increase that measure, but strictly decreases the number of occurrences of matchable symbols in the right-hand sides of matching equations. Hence, a pattern calculus with solve F as the matching function is yet another extension of the pure pattern calculus with matchable symbols to finitary matching.

Below we give several examples that illustrate solve F. Let f be flat.

## Practical Foundations of Mathematics

Assume f is flat. The following lemma plays the crucial role in proving that solve F makes the calculus confluent. We only prove the second statement. The proof of the first one is similar. Below we only highlight those points which make this proof different from the proof on the analogous lemma for commutative matching Lemma 4. Similar to the analogous cases in Lemma 4. We distinguish between the cases whether head Q is a flat symbol or not. When it is not flat, the proof is the same as for the non-commutative head in Lemma 4. Therefore, assume the head of Q is a flat symbol f.

Then by Corollary 4. Proof is the same as for the sum case in Lemma 4. By the reasoning similar to the case of Theorem 4. From this theorem and Theorem 3. We have presented a general framework for pattern calculi with finitary matching, where patterns are first-class citizens. They can be used as parameters, instantiated, reduced and returned as a result. Unrestricted reductions for pattern calculi lead to non-confluence even in the unitary case.

Nondeterminism introduced by finitary matching makes the situation even more complicated. Our goal was to study conditions under which confluence would be guaranteed for calculi with finitary matching. The approach we considered in this paper builds on the general approaches to proving confluence for pattern calculi with unitary matching proposed by [ 7 ] and [ 22 ].

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Like those works, we also consider the matching function to be a parameter of the general framework. This function is denoted by solve in this paper. Instantiating it by concrete algorithms, one obtains concrete versions of pattern calculi. We were interested to find conditions for solve , which would guarantee a general confluence result for the framework and, thus, would ensure confluence of its concrete instantiations, provided that the conditions are satisfied.

We formulated two such conditions in this paper, denoted by C 1 and C 2. They ensure that the function solve is stable by substitution and by reduction. These conditions generalize their counterparts from the unitary matching [ 7 , 22 ]. The solve function is defined in such a way that it computes solutions to matching problems and does not free bound variables during reduction.

We illustrated two concrete matching functions that satisfy C 1 and C 2 , one for commutative theories and the other one for flat patterns. We also discussed a version of commutative matching, which is polymorphic for commutative symbols and permits abstracting with respect to them. In another development, an interesting approach to show metatheoretical results confluence, standardization of the pure pattern calculus from [ 22 ] has been proposed in [ 31 ].

There, the calculus has been represented as a higher-order pattern rewriting system as defined by [ 26 ] and [ 25 ]. It was shown that by applying general higher-order rewriting results to this rewriting system, one can obtain properties of the pattern calculus.

It is worth to investigate how this approach can be extended to calculi with finitary matching, which would require reasoning with equational higher-order rewriting, but that is beyond the scope of this paper. Yet another direction for future work is to study normalization strategies for some pattern calculi with finitary matching. Recently, [ 5 ] addressed this problem for pure pattern calculus of [ 22 ]. Matching failure gives this calculus non-sequential behaviour.

Non-sequentiality means that for a given term M containing redexes under a certain redex-free context, there is no redex in M that, first, is needed i. Bonelli et al. By this strategy, a set of redexes reduces simultaneously at each step. It would be interesting to study how this approach can be extended to pattern calculi with finitary matching.