Decreasing argument (and what is a Program Fixpoint) - coq

Question

Consider the following fixpoint:
Require Import Coq.Lists.List.
Import ListNotations.
Inductive my_type: Type:=
| Left: my_type
| Right: my_type
.
Fixpoint decrease (which: my_type) (left right: list my_type) : list my_type :=
match which with
| Left =>
match left with
| [] => []
| a::tl => decrease a tl right
end
| Right =>
match right with
| [] => []
| a::tl => decrease a left tl
end
end.
Coq rejects the following fixpoint because it can not guess the decreasing fixpoint (sometimes the left list looses its head, sometimes it is the right one).
This answer shows that one can solve this by using a Program Fixpoint and specifying a {measure ((length left)+(length right))}.
My questions are:
What is the difference between a regular Fixpoint and a Program Fixpoint ?
Is it possible to explicit the decreasing argument in a regular Fixpoint ?
What is the Next Obligation goal ?

Answer 1

The Fixpoint command in Coq constructs a recursive function using Coq's primitive fix, which checks for structural recursion to ensure termination. This is the only recursion in Coq, and all other techniques ultimately desugar to a fix of some sort.
Program Fixpoint is a feature of Program, which allows writing definitions in a slightly extended language that are then compiled into Coq definitions. Program Fixpoint accepts any recursive function, generates an appropriate proof obligation that shows the function terminates (by decreasing some measure of its arguments on each recursive subcall), and then packages up the definition and termination proof into a Coq definition that structurally decreases an argument. If that sounds magical it basically is, but CPDT gives a good explanation of how to do this kind of encoding.
Yes, you can add a {struct arg} annotation to explicitly specify which argument is structurally decreasing: Fixpoint decrease (which: my_type) (left right: list my_type) {struct right} : list my_type. This doesn't help for your example, since your function doesn't in general structurally decrease either argument. All primitive fix constructs have a struct annotation, but Coq can usually infer it automatically when you write a Fixpoint. For example, here's Nat.add:
Nat.add =
fix add (n m : nat) {struct n} : nat :=
match n with
| 0 => m
| S p => S (add p m)
end : nat -> nat -> nat
You get two types of goals from Next Obligation with Program Fixpoint: first that each subcall has a smaller argument (using measure, "smaller" is defined using < on nats), and second, that the "smaller" relation is well-founded; this is, it has no infinitely descending sequences of smaller and smaller objects. I'm not sure why Program Fixpoint doesn't do this automatically for Nat.lt, given that it should know the relevant theorem. Note that Program has more features than just more general recursion, so it can generate other goals related to those features as well, depending on the definition you write.