Read-back is the process of decoding a -term from its another representation. Previously, [1, Section 7] has presented an embedded read-back mechanism for interaction nets. Here, we will define a novel term rewriting system which resembles that mechanism in order to check its correctness and to study its properties more easily than in the original setting.
We will use notations similar to . is a context with one hole, which will be denoted with . is the result of placing in the hole of the context . is the set of all -terms. is the set of all -terms in -normal form. is the set of all free variables in a -term . is the result of substituting for all free occurrences of variable in . , where . If , then stands for -reduction, and we write when or .
Let . is an atom. is the result of replacing each in with . Additionally, we define set , then extend the definitions of substitution and for according to for any variable .
Note that for any , we have and .
is the minimal set that satisfies the following conditions:
Let us introduce a reduction relation on as follows:
Thanks to , the introduced relation is not to be confused with -reduction.
Mapping is called read-back and defined as follows:
For example, if is a -term, then .
Let . Notice that . Then we have . ∎
A context is normal if and only if .
In particular, contexts and are both normal, however is not. Another example is context : if is a variable and , then is also normal.
will denote the set of all normal contexts.
Let . Then by Definition 5, , and . ∎
Let . Then is in normal form if and only if is an atom.
Notice that in Definition 3 we have if and only if is not an atom. ∎
For any , at most one reduction sequence starts from .
Notice that for any there is at most one such that . ∎
Let . Consider the following four cases:
, where is a variable.
Then and .
, where .
Notice due to and Proposition 3.
, where .
, where is a variable and .
Notice that .
Since , we have for some .
Assume and use induction to conclude .
Notice that any infinite reduction sequence on can only be due to -reduction. Further, the proof of Proposition 6 shows that the leftmost -redex is always contracted, thus we can use [2, Normalization theorem 13.2.2] to conclude ().
This paper introduced a new term rewriting system designed after the embedded read-back mechanism for interaction nets that was presented in our previous work [1, Section 7].
Then we have demonstrated the correctness of our term rewriting system and showed its normalization property in Proposition 7. As a simple corollary, the conjecture in  can thus be proven using Proposition 7 and the preceding work . Proposition 7 can also help investigate the similar conjectures in [1, Section 7] and [5, Section 2].
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