# rewriting

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The definition of our outermost-needed narrowing strategy does not determine the computation space for a given inductively sequential rewrite system in a unique way.
Let R be an inductively sequential rewrite system, t an operation-rooted term, T a definitional tree of the root of t, and [Psi](t, T) = (p, R).
Let R be an inductively sequential rewrite system extended by the equality rules, e an equation to solve and V = Var(e).
Let R be an inductively sequential rewrite system, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] a narrowing multiderivation and A the reduction sequence canonically associated to [Alpha].
The resulting graph rewrite system classes can be used for program analysis as well as for certain classes of transformations.
A (relational) graph rewrite system G = (S, Z) consists of a set of graph rewrite rules S and a [Sigma]-graph Z (the axiom).
Figure 6 contains an example of a graph rewrite system.
To ensure termination of a rewrite system, it is important that each rule completes such a termination-subgraph, that no other rule adds nodes to the termination-subgraph of another rule, nor deletes items from it.
1998] we used a notion of simulation of one rewrite system by another rewrite system, to prove correctness of transformations of rewrite systems.
The simulation [Phi] is said to be sound if each reduction in the simulating rewrite system from a b [element of] D([Phi]) to a b' can be extended to a reduction to a b" [element of] D([Phi]), such that [Phi](b) can be rewritten to [Phi](b") in the original rewrite system.
Fokkink and van de Pol [1997] prove that if a simulation is sound, complete, and termination preserving, then no information on normal forms in the original rewrite system is lost.
Huet and Levy [1991] consider orthogonal (so, in particular, nonoverlapping) rewrite systems that are sequential.

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