Some preparatory proofs for proving sorting+permutation is equality #2724 #2725
Conversation
| -- Takes a permutation m → n and creates a permutation | ||
| -- suc (suc m) → suc (suc n) by mapping 0 to 1 and 1 to 0 and | ||
| -- then applying the input permutation to everything else | ||
| swap : Permutation m n → Permutation (suc (suc m)) (suc (suc n)) |
There was a problem hiding this comment.
This is a special case of \oplus of two permutations.
There was a problem hiding this comment.
I agree (lift0 should be as well!), but we don't have such an operation in the library. I have added a note that we should refactor it when we add that operator.
There was a problem hiding this comment.
I have it buried in my Rig Categories repo. I need to port the basic operations on permutations over. I've got the full semiring defined there.
There was a problem hiding this comment.
But also: Function.Related.TypeIsomorphisms etc. as per #2489 ?
There was a problem hiding this comment.
Plus, of course: we only need this operation because Permutation.Setoid stipulates swap as a constructor... ;-)
There was a problem hiding this comment.
Yes, absolutely, there is some great stuff in Function.Related.TypeIsomorphism. And a rather apt 'of course'.
src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
Outdated
Show resolved
Hide resolved
|
Having looked at this, and its followup, in particular lemma
If so, then I think this PR and its followup may be simplifiable via the already existing |
| _≰_ : ∀ {n} → Rel (Fin n) 0ℓ | ||
| i ≰ j = ¬ (i ≤ j) | ||
|
|
||
| _≮_ : ∀ {n} → Rel (Fin n) 0ℓ | ||
| i ≮ j = ¬ (i < j) |
| xs↭ys⇒|xs|≡|ys| : ∀ {xs ys} → xs ↭ ys → length xs ≡ length ys | ||
| xs↭ys⇒|xs|≡|ys| (refl eq) = Pointwise.Pointwise-length eq | ||
| xs↭ys⇒|xs|≡|ys| (prep eq xs↭ys) = ≡.cong suc (xs↭ys⇒|xs|≡|ys| xs↭ys) | ||
| xs↭ys⇒|xs|≡|ys| (swap eq₁ eq₂ xs↭ys) = ≡.cong (λ x → suc (suc x)) (xs↭ys⇒|xs|≡|ys| xs↭ys) |
There was a problem hiding this comment.
How about
| xs↭ys⇒|xs|≡|ys| (swap eq₁ eq₂ xs↭ys) = ≡.cong (λ x → suc (suc x)) (xs↭ys⇒|xs|≡|ys| xs↭ys) | |
| xs↭ys⇒|xs|≡|ys| (swap eq₁ eq₂ xs↭ys) = ≡.cong 2+_ (xs↭ys⇒|xs|≡|ys| xs↭ys) |
(possibly at the cost of an additional import from Data.Nat.Base)?
| _≰_ : ∀ {n} → Rel (Fin n) 0ℓ | ||
| _≮_ : ∀ {n} → Rel (Fin n) 0ℓ |
There was a problem hiding this comment.
Can we, by convention, avoid writing the ∀ {n} → prefix here?
I've been sniped by #2723. This is a preparatory PR for the full proof that sorting two lists that are permutations of each other result in equal lists.