I am working through Software Foundations and am a bit stuck. Here is a link for reference: https://softwarefoundations.cis.upenn.edu/vfa-current/Binom.html
I am stuck on the proof "abs_perm," reproduced here.
Theorem abs_perm: forall p al bl,
priq p -> Abs p al -> Abs p bl -> Permutation al bl.
This is a "2 star" question, so it should be pretty easy. The fact that it is proving to be difficult makes me think that the issue is my "Abs" relation. My inductive relation is as follows:
Inductive priqueue_elems: list tree -> list key -> Prop :=
| priqueue_elems_base: priqueue_elems [] nil
| priqueue_elems_next_list: forall l b b' v,
priqueue_elems l b ->
Permutation b' (v::b) ->
priqueue_elems (insert v l) b'
The issue I end up having is that I get something like "insert x1 l1 = insert x2 l2" (via inversions in my proof) and I can't go anywhere from there.
insert in this case is not injective...it seems like insert x [] = insert x [Leaf].
The above relation allows me to directly know that I have "priq l" available, but priq is defined such that if there is more than one element, the first is a Leaf so...yeah.
I think the issue is with my relation (which is modeled after their tree_elems), but I'm having a bit of "proof block." My guess is that I shouldn't have insert in the relation construction, but it's unclear what other structure there is to the priqueue.
Another avenue would just to have it straight up on "list tree," but then it seems like well-formedness could be an issue.
Otherwise I need some sort of theorem on insert. I tried a bunch, but wasn't able to get a proof together.
For example,
Lemma equals_inserts_permute: forall l1 l2 p1 p2 v1 v2,
priqueue_elems l1 p1 ->
priqueue_elems l2 p2 ->
insert v1 l1 = insert v2 l2 -> Permutation (v1::p1) (v2::p2)
Which would give me the ability to relate insert that I need, but...I haven't been able to crack that proof either.
Would appreciate any help!
