edit: I should probably say how I am currently worked around the problem here. I defined a principle for showing equality of permutations,
Lemma permInd : ∀ (U : Type) (A : Ensemble U) (φ ψ : Perm A),
φ ↓ = ψ ↓ → φ ↑ = ψ ↑ → φ = ψ
then applied the lemma in the proof context that’s giving me trouble below and shows that the eta-equivalent terms are equal. The problem therefore seems to be showing eta-equivalence when the terms are nested inside the record. But I’m not good at working with records, so I might be missing something.
original:
I am having trouble proving the equality of eta-equivalent terms nested in record fields. For reference, eta-reduction is independently provable by reflexivity:
Lemma etaEquivalence : ∀ (A B : Type) (f : A → B), (λ x : A, f x) = f.
Proof. reflexivity. Qed.
In my current proof context, I have two records the equality of which I must prove. Fully destructed and unfolded, the proof context and current subgoal looks like this:
U : Type
A : Ensemble U
perm0 : U → U
pinv0 : U → U
permutes0 : IsPerm A perm0 pinv0
============================
{|
perm := λ x : U, perm0 x;
pinv := λ x : U, pinv0 x;
permutes := permutationComp permutes0 (permutationId A) |} =
{| perm := perm0; pinv := pinv0; permutes := permutes0 |}
The equalities that must be established are
perm0 = λ x : U, perm0 x
pinv0 = λ x : U, pinv0 x
Because these equalities can be established by reflexivity, I’m unsure what the problem is. However, I suspect something is awry, because attempting to replace λ x : U, perm0 x with perm0 generates the appropriate subgoal, but doesn’t replace the term inside the record. Furthermore, rewriting using eqa_reduction causes errors regarding abstraction causing ill-typed terms or nested dependent arguments.
I’ve simplified the context as much as possible and pasted it below. Beyond stylistic problems and the fact that I’m still a beginner, I don’t see any problems with the current development.
Require Import Unicode.Utf8 Utf8_core Ensembles Setoid.
Class IsPerm {U : Type} (A : Ensemble U) (φ ψ : U → U) : Prop := {
pinvLeft : ∀ x : U, ψ (φ x) = x;
pinvRight : ∀ x : U, φ (ψ x) = x;
closedPerm : ∀ x : U, In U A x → In U A (φ x);
closedPinv : ∀ x : U, In U A x → In U A (ψ x)
}.
Record Perm {U : Type} (A : Ensemble U) : Type := {
perm : U → U;
pinv : U → U;
permutes :> IsPerm A perm pinv
}.
Notation "f ∘ g" := (λ x, f (g x)) (at level 45).
Notation "P ↓" := (@perm _ _ P) (at level 2, no associativity).
Notation "P ↑" := (@pinv _ _ P) (at level 2, no associativity).
Instance permutationComp
{U : Type} {A : Ensemble U} {f g k h : U → U}
(P : IsPerm A f k) (Q : IsPerm A g h) : IsPerm A (f ∘ g) (h ∘ k).
Proof.
constructor; intros.
setoid_rewrite pinvLeft. apply pinvLeft.
setoid_rewrite pinvRight. apply pinvRight.
apply closedPerm. apply closedPerm. auto.
apply closedPinv. apply closedPinv. auto.
Defined.
Instance permutationId
{U : Type} (A : Ensemble U) :
IsPerm A (λ x : U, x) (λ x : U, x).
Proof. constructor; intros; auto. Defined.
Definition permComp
{U : Type} (A : Ensemble U)
(φ : Perm A) (ψ : Perm A) : Perm A :=
Build_Perm U A (φ↓ ∘ ψ↓) (ψ↑ ∘ φ↑)
(permutationComp (permutes A φ) (permutes A ψ)).
Definition permId {U : Type} (A : Ensemble U) : Perm A :=
Build_Perm U A (λ x : U, x) (λ x : U, x) (permutationId A).
(* problems occur after the application of the tactic simpl, below: *)
Lemma permCompRightIdentity :
∀ {U : Type} (A : Ensemble U) (φ : Perm A), permComp A φ (permId A) = φ.
Proof. intros. unfold permComp. simpl. admit. Qed.
Finally, I want to thank everyone here for helping me out with Coq and being patient.
Proof irrelevance is not built-in in Coq. You can easily prove what you want if you assume the proof irrelevance axiom: