1 Introduction
Readback is the process of decoding a term from its another representation. Previously, [1, Section 7] has presented an embedded readback 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.
Notation.
We will use notations similar to [2]. 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 .
Definition 1.
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 .
Definition 2.
is the minimal set that satisfies the following conditions:
Definition 3.
Let us introduce a reduction relation on as follows:
Thanks to , the introduced relation is not to be confused with reduction.
2 Correctness
Definition 4.
Mapping is called readback and defined as follows:
For example, if is a term, then .
Proposition 1.
.
Proof.
Let . Notice that . Then we have . ∎
Proposition 2.
.
3 Normalization
Definition 5.
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.
Notation.
will denote the set of all normal contexts.
Proposition 3.
.
Proof.
Let . Then by Definition 5, , and . ∎
Proposition 4.
Let . Then is in normal form if and only if is an atom.
Proof.
Notice that in Definition 3 we have if and only if is not an atom. ∎
Proposition 5.
For any , at most one reduction sequence starts from .
Proof.
Notice that for any there is at most one such that . ∎
Proposition 6.
.
Proof.
Let . Consider the following four cases:

, where is a variable.
Then and . 
, where .
Then .
Notice due to and Proposition 3. 
, where .
Then . 
, where is a variable and .
Then .
Notice that .
Since , we have for some .
Assume and use induction to conclude .
Due to Proposition 4 and Proposition 5, the above four cases are exhaustive. ∎
Proposition 7.
.
4 Conclusion
This paper introduced a new term rewriting system designed after the embedded readback 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 [3] can thus be proven using Proposition 7 and the preceding work [4]. Proposition 7 can also help investigate the similar conjectures in [1, Section 7] and [5, Section 2].
In the future, we intend to apply a similar technique to develop the ideas from [5] which we believe to be a very promising direction, especially taking into account the experimental results obtained from their software implementation^{1}^{1}1https://www.npmjs.com/package/@alexo/lambda.
References

[1]
Anton Salikhmetov, 2016.
Tokenpassing Optimal Reduction with Embedded Readback.
TERMGRAPH 2016, pp. 45–54. 
[2]
Henk P. Barendregt, 1984.
The Lambda Calculus: Its Syntax and Semantics.
Studies in Logic and the Foundations of Mathematics, 103. 
[3]
Anton Salikhmetov, 2015.
Macro Lambda Calculus.
arXiv:1304.2290v8 [cs.LO]. 
[4]
FrançoisRégis Sinot, 2006.
TokenPassing Nets: CallbyNeed for Free.
Electronic Notes in Theoretical Computer Science, 135, pp. 129–139. 
[5]
Anton Salikhmetov, 2018.
An impure solution to the problem of matching fans.
arXiv:1710.07516v3 [cs.LO].
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