## Stream: maths

### Topic: well-founded recursion & lexicographic ordering ?

#### Jack Crawford (Sep 14 2018 at 09:29):

Hi, I have a pleb question about how Lean infers whether recursion is well-founded or not.
I'm writing a recursive function whose first two arguments are of type fin m and fin n, and my algorithm:
1) always decreases the size of my fin n
2) sometimes decreases the size of my fin m
I would think that Lean should be able to work out that my recursion is well-founded from the first fact alone, but it does not seem able.
However, when I simply change the order that my arguments appear from fin m -> fin n -> ... to fin n -> fin m -> ... and nothing else (nontrivial), suddenly Lean recognises well-foundedness!

It seems that Lean is only trying to show well-foundedness from my first argument and can't work out that there should be well-foundedness on the lexicographic order of my fin m and fin n, or that just my fin n alone should suffice. Everything's sort of "doing what I want it to" except that I have to write the arguments to my function in an unconventional way. How can I tell Lean to just look for well-foundedness on my second argument (the fin n), or on the lexicographic pair of fin m and fin n, and not just purely on my fin m? Thanks!

MWE

#### Kevin Buzzard (Sep 14 2018 at 09:40):

By default Lean uses lexicographic ordering when trying to prove recursive functions are well-defined, so definitely the input order matters. The notes at https://github.com/leanprover/mathlib/blob/master/docs/extras/well_founded_recursion.md probably tell you all you need to know

#### Jack Crawford (Sep 14 2018 at 09:41):

@Kenny Lau
here's an MWE, compare the pair:

def mwe_aux {m n : ℕ} : Π (i : fin m) (j : fin n), bool
| ⟨0, h₁⟩ ⟨k₂, h₂⟩ := ff
| ⟨k₁+1, h₁⟩ ⟨0, h₂⟩ := ff
| ⟨k₁+1, h₁⟩ ⟨k₂+1, h₂⟩ :=
begin
apply mwe_aux,
from if k₂%2 = 0 then ⟨k₁, nat.lt_of_succ_lt h₁⟩ else ⟨k₁+1, h₁⟩,
from ⟨k₂, nat.lt_of_succ_lt h₂⟩
end

def mwe_aux2 {m n : ℕ} : Π (j : fin n) (i : fin m), bool
| ⟨k₂, h₂⟩ ⟨0, h₁⟩ := ff
| ⟨0, h₂⟩ ⟨k₁+1, h₁⟩ := ff
| ⟨k₂+1, h₂⟩ ⟨k₁+1, h₁⟩ :=
begin
apply mwe_aux2,
from ⟨k₂, nat.lt_of_succ_lt h₂⟩,
from if k₂%2 = 0 then ⟨k₁, nat.lt_of_succ_lt h₁⟩ else ⟨k₁+1, h₁⟩,
end


#### Kevin Buzzard (Sep 14 2018 at 09:41):

the linked notes explain how to change the well-order on your input.

#### Jack Crawford (Sep 14 2018 at 09:42):

@Kevin Buzzard ah oops I didn't see that before I responded to Kenny, thanks, I'll have a look!

#### Kenny Lau (Sep 14 2018 at 09:44):

well I mean, what are you trying to do, instead of MWE of the error

Last updated: May 18 2021 at 07:19 UTC