[GregarianChild]

I'm not an expert in this field, but the way I understand it is that Choice Trees extend the ITree signature by adding a choice operator. Some variant of this:

ITrees:

    CoInductive itree (E : Type -> Type) (R : Type) : Type :=
    | Ret (r : R)                                                                                                                                                                                                         
    | Tau (t : itree E R)                                                                                                                                                                                                 
    | Vis {T : Type} (e : E T) (k : T -> itree E R)                                                                                                                                                                       

ChoiceTrees:

    CoInductive ctree (E : Type -> Type) (C : Type -> Type) (R : Type) : Type :=
    | Ret (r : R)                                                                                                                                                                                                         
    | Tau (t : ctree E C R)                                                                                                                                                                                               
    | Vis {T : Type} (e : E T) (k : T -> ctree E C R)                                                                                                                                                                     
    | Choice {T : Type} (c : C T) (k : T -> ctree E C R)                                                                                                                                                                  

One can see "Choice" constructor as modelling internal non-determinism, complementing the external non-determinism that ITrees already allow with "Vis" and that arises from interaction with the environment. (Process calculi like CCS, CSP and Pi, as well as session types and linear logic also make this distinction).

[mzweav]

Ooooh! Those indeed look fun! :)

[GregarianChild]

There are some issues arising from size inconsistencies (AKA Cantor's Paradox) if / when you try to fit the representation of all internal choices (this could be infinite) into a small universe of a theorem prover's inductive types. The ChoiceTree paper solves this with a specific encoding. I'm currently wondering how to port this trick from COq/Rocq to Lean4.