Editing Openai/693343d7-a38c-8012-a67c-11cbed4c0fd9 (section)

=== Below is a self-contained Coq-like file (uses standard Coq libraries). This is explicit: Tree A is the free-monad-like data type with Leaf and Node where Node carries an operator label (e.g., Merge, SimNode) and a list of children. We define map, eta (ret), join (mu), bind, then prove the monad laws. ===

You can paste into a .v and adapt imports (I write it compactly — proofs are constructive and sketched where appropriate).

<syntaxhighlight lang="coq">('' Obatala_Tree.v  — Coq sketch for Obatala = Tree monad '')

Require Import List.
Import ListNotations.
Require Import Coq.Program.Basics.
Require Import Coq.Program.Tactics.
Require Import Coq.Init.Nat.
Require Import Coq.Logic.FunctionalExtensionality.

Set Implicit Arguments.
Set Asymmetric Patterns.
Generalizable All Variables.

('' --- operator labels for internal nodes (customize as needed) --- '')
Inductive Op : Type :=
| MergeOp
| SimOp
| CustomOp (label : nat).  ('' placeholder '')

('' --- Free-tree type --- '')
Inductive Tree (A : Type) : Type :=
| Leaf : A -> Tree A
| Node : Op -> list (Tree A) -> Tree A.

Arguments Leaf {A} _.
Arguments Node {A} _ _.

('' --- Functor map --- '')
Fixpoint fmap {A B} (f : A -> B) (t : Tree A) : Tree B :=
  match t with
  | Leaf a => Leaf (f a)
  | Node op children => Node op (map (fmap f) children)
  end.

('' --- Monad unit (eta) --- '')
Definition ret {A} (a : A) : Tree A := Leaf a.

('' --- flatten / join --- '')
Fixpoint join {A} (t : Tree (Tree A)) : Tree A :=
  match t with
  | Leaf inner => inner
  | Node op children =>
      ('' graft: flatten any child Leaf/Node by mapping join recursively '')
      let children' := map (fun tt => join tt) children in
      Node op children'
  end.

('' --- bind in terms of fmap + join --- '')
Definition bind {A B} (t : Tree A) (f : A -> Tree B) : Tree B :=
  join (fmap f t).

('' --- Functor & Monad laws proofs (sketches) --- '')

('' Functor identity: fmap id = id '')
Theorem fmap_id {A} (t : Tree A) : fmap (fun x => x) t = t.
Proof.
  induction t.
* reflexivity.
'' simpl. rewrite IHt. ('' IHt is on list, need map lemma *)
    induction l.
    + simpl. reflexivity.
    + simpl. rewrite IHt0. rewrite IHl. reflexivity.
Qed.

('' Functor composition: fmap (g ∘ f) = fmap g ∘ fmap f '')
Theorem fmap_compose {A B C} (f : A -> B) (g : B -> C) (t : Tree A) :
  fmap (fun x => g (f x)) t = fmap g (fmap f t).
Proof.
  induction t; simpl.
* reflexivity.
* f_equal.
    induction l; simpl.
    + reflexivity.
    + rewrite IHt0. rewrite IHl. reflexivity.
Qed.

('' Left identity: bind ret = id '')
Theorem bind_ret_left {A B} (t : Tree A) :
  bind t (@ret A) = t.
Proof.
  unfold bind, ret. induction t; simpl.
* reflexivity.
* f_equal.
    induction l; simpl.
    + reflexivity.
    + rewrite IHt0. rewrite IHl. reflexivity.
Qed.

('' Right identity: bind ret = id for Kleisli? need: bind (ret a) f = f a '')
Theorem bind_ret_right {A B} (a : A) (f : A -> Tree B) :
  bind (ret a) f = f a.
Proof. simpl. reflexivity. Qed.

('' Associativity: bind (bind t f) g = bind t (fun x => bind (f x) g) '')
Theorem bind_assoc {A B C} (t : Tree A) (f : A -> Tree B) (g : B -> Tree C) :
  bind (bind t f) g = bind t (fun x => bind (f x) g).
Proof.
  unfold bind. ('' reduces to join (fmap g (join (fmap f t))) = join (fmap (fun x => join (fmap g (f x))) t) '')
  ('' A full formal proof requires lemmas about join/fmap interactions. '')
Admitted.

('' Note: bind_assoc typically proven by induction on t and list lemmas. '')

('' --- Eilenberg-Moore algebra example: interpret tree into a carrier type --- '')
Section EM_example.
  Variable A : Type.
  Variable carrier : Type.
  Variable interpret_leaf : A -> carrier.
  Variable interpret_node : Op -> list carrier -> carrier.

  Fixpoint algebra (t : Tree A) : carrier :=
    match t with
    | Leaf x => interpret_leaf x
    | Node op ch => interpret_node op (map algebra ch)
    end.

  ('' algebra : Tree A -> carrier is an EM-algebra: algebra ∘ ret = interpret_leaf etc. '')
End EM_example.

('' End of Coq sketch '')

</syntaxhighlight>

Notes & next steps for Coq:
* bind_assoc is left Admitted — it's straightforward but requires standard list-induction lemmas; I can finish it fully if you want the runnable file.
* The above defines a concrete O := Tree with ret, join, bind. Tree is the canonical free-monad-like Obatala instance.
* Extend Op with richer labels for simulation, merge semantics, and provide interpret_node as the EM-algebra.

If you want, I’ll finish the bind_assoc proof and produce a fully checked .v file.