WellFounded for lexicographic ordering on Vector.

[mechvel]

Probably WellFounded does not hold for <-lex on lists.
Right?
But for each n, it holds for the vectors of length ≤ n.
So far I have implemented it for vectors of length ≡ n.
But
a) my library uses a different representation for Vector (shown below),
b) I expect, the Agda team can implement this itself.
c) I can send the code as a sample, it is clear enough. One can rewrite it to the standard Vector representation.

The approach is as follows.
The vector representation:

module Vector.Base {α} (A : Set α)                                                
  where                                                                           
  data Vector (n : ℕ) : Set α
    where                                              
    vec : (xs : List A) → length xs ≡ n → Vector n      

 _<_ : Rel A α

A general simple lemma suggested in another issue and used here:

subrelation-WellFounded :  ∀ {r} {~ ~' : Rel A r} → (~' ⇒ ~) → WellFounded ~ → WellFounded ~'

Lexicographic ordering on lists:

  _<llex_ = Lex-< _≡_ _<_

Lexicographic ordering on Vector n:

  <lex :  ∀ n → Rel (Vector n) _
  <lex n =  _<llex_ on toList

Theorem:

  wellFounded-<lex :  WellFounded _<_ → (n : ℕ) → WellFounded (<lex n) 

The proof is by induction on n and by using <lex-1+n ⇒ (×-lex _<_ <lexₙ) on f,
where f : Vector 1+n → A × Vector n is a certain evident map.
This is using that a) <lex-1+n on Vector 1+n equals to a certain composition ×-lex on Product with
<lexₙ on Vector n,
b) WellFounded for ×-lex for Product is in lib-1.7.

[MatthewDaggitt]

I don't think we would want to add your representation of Vector to the standard library. But if you would like to make a PR with a proof that the lexicographic order over the current version of Vector is a well-founded relation if the original relation is well-founded, that would be welcome.

[jamesmckinna]

Closed by #1924 . Cf. #2075 / #2077 .