The undecidability of semi-unification (unification combined with matching) has been proven by Kfoury, Tiuryn, and Urzyczyn in the 1990s.
The original argument is by Turing reduction from Turing machine immortality (existence of a diverging configuration).
There are several aspects of the existing work which can be improved upon.
First, many-one completeness of semi-unification is not established due to the use of Turing reductions.
Second, existing mechanizations do not cover a comprehensive reduction from Turing machine halting to semi-unification.
Third, reliance on principles such as König's lemma or the fan theorem does not support constructivity of the arguments.
Improving upon the above aspects, the present work gives a constructive many-one reduction from the Turing machine halting problem to semi-unification.
This establishes many-one completeness of semi-unification.
Computability of the reduction function, constructivity of the argument, and correctness of the argument is witnessed by an axiom-free mechanization in the Coq proof assistant.
The mechanization is incorporated into the existing Coq library of undecidability proofs.
Notably, the mechanization relies on a technique invented by Hooper in the 1960s for Turing machine immortality.
An immediate consequence of the present work is an alternative approach to the constructive many-one equivalence of System F typability and System F type checking, compared to the argument established in the 1990s by Wells.