[Agda] Well-foundedness of list of well-founded elements

[Ulf Norell]

I think there's something wrong with your definition of _≪_. It doesn't
mention _<_, and it's clearly reflexive on non-empty lists:

refl-≪ : ∀ xs → ∃ (λ x → x ∈ xs) → xs ≪ xs
refl-≪ xs non-empty = non-empty , λ _ i → i

Did you mean

_≪_ : Rel (List A) lzero
xs ≪ ys = ∃ (λ x → x ∈ ys) × (∀ x → x ∈ xs → ∃ (λ y → y ∈ ys × x < y))

?

/ Ulf


On Sun, Mar 4, 2018 at 7:37 AM, How Si Yu <howseeu at gmail.com> wrote:

> Hi, all. given a well-founded relation _<_ on A, define _≪_ on List A
> where  where xs ≪ ys = ∃ (λ x → x ∈ ys) × (∀ x → x ∈ xs → x ∈ ys). Is
> it true that _≪_ is well-founded?
>
> This is definitely true in classical logic since we can argue in the
> following way: Assume not. Then there exists an infinite sequence ...
> ≪ xs3 ≪ xs2 ≪ xs1 ≪ xs0. ∀ i, xs(n+1) ≪ xsn so ∃ x ∈ xsn. If n ≠ 0, ∃
> y ∈ xs(n-1) st x < y. Continuiting moving rightwards we find a
> strictly increasing sequence starting at x and ends in xs0. Hence
> there must exist infinitely such sequences ending at xs0. Since xs0 is
> finite, ∃ x0 ∈ xs0 where infinitely many such sequences ending at x0.
> All such sequences besides one pass through xs1. Since xs1 is finite,
> ∃ x1 ∈ xs1 where infinitely many such sequences pass through x1.
> Continuing such construction we get an infinite descending chain in A.
> But _<_ is well-founded. Contradiction.
>
> Is this true constructively as well? Thanks a lot! I need it in doing
> induction in my proof in
> Agda. Below are codes to make precise my question.
>
> open import Data.List.Base using (List; []; _∷_)
> open import Relation.Binary.Core using (Rel)
> open import Agda.Primitive using (lzero)
> open import Induction.WellFounded using (Well-founded)
> open import Data.Product using (∃; _×_; _,_)
>
> postulate A : Set
> postulate _<_ : Rel A lzero
>
> data _∈_ (x : A) : List A → Set where
>   here : ∀ {xs} → x ∈ (x ∷ xs)
>   there : ∀ {y xs} → x ∈ xs → x ∈ (y ∷ xs)
>
> _≪_ : Rel (List A) lzero
> xs ≪ ys = ∃ (λ x → x ∈ ys) × (∀ x → x ∈ xs → x ∈ ys)
>
> _≪_-wf : Well-founded _<_ → Well-founded _≪_
> _≪_-wf = {!!}
>
> Regards,
> Si Yu
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