= Recursive simple envelope by cyclic HNN extensions
A <finitely generated group> with soluble <word problem for a group> embeds in a countably generated <simple group> with a <recursive presentation of a group>. Start with $P_0=G*\langle a\rangle$, retaining decidable membership in $\langle a\rangle$. At stage $n$, add a free infinite cyclic generator $z_n$. For every nonidentity base element $g$, conjugate $[g,z_n]$ to $a$ and conjugate $gz_n$ to $a$; also conjugate $z_n$ to $a$, using distinct <stable letters>. All associated subgroups are infinite cyclic, with decidable membership by <effective cyclic membership in a free product>. <Britton's lemma> preserves embeddings, decidable word problem and decidable membership in the inherited $\langle a\rangle$.
In the union, any nontrivial normal subgroup contains some nonidentity $g$ and hence $a$, by the commutator-conjugation relation. The other relations force $z_n$ and then every old $g$ into the normal closure of $a$. Every stable letter becomes an old element at a later stage, so the normal closure of $a$ is the whole union. The union is nontrivial and simple. Effective word enumeration at each stage makes all relators recursively enumerable. A <two-generator HNN embedding> followed by the <Higman embedding theorem> embeds this simple envelope in a finitely presented group.
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