Hint Extern 1 (RExists _ (_ !! snd) _ _) =>
eapply rel_push_snd_rexists : typeclass_instances.
(** ** Relation currying *)
(* An interesting special case of [rel_pull] is [rel_curry], which takes a relation over [A * B -> C] and turns it into a relation over [A -> B -> C].  [rel_curry] is particularly useful when two (curried) arguments to a given function have to be related in a dependent way. For example in Compcert, memory injections relate memory blocks and offsets jointly, but many operations have those as two separate arguments. To state the relational property of such operations, we can uncurry a joint relation on (block, offset) pairs.  [rel_curry] can be obtained by pulling back the original relation along [uncurry]: *)
Definition curry {A B C} (f: A * B -> C): A -> B -> C :=
fun a b => f (a, b).
Definition uncurry {A B C} (f: A -> B -> C): A * B -> C :=
fun ab => match ab with (a, b) => f a b end.
Definition rel_curry {A1 B1 C1 A2 B2 C2} (R: rel (A1*B1->C1) (A2*B2->C2)) :=
(R @@ uncurry)%rel.
Definition rel_uncurry {A1 B1 C1 A2 B2 C2} (R: rel (A1->B1->C1) (A2->B2->C2)) :=
(R @@ curry)%rel.
(** We use the following notation for [rel_curry], which evokes splitting a pair into two separate arguments. Note that we use the same priority as for [++>], [-->], and [==>], because we usually want [rel_curry] to nest in such a way that for instance, [S ++> % R --> % R ++> impl] will be interpreted as [S ++> rel_curry (R --> rel_curry (R ++> impl))]. *)
Notation "% R" := (rel_curry R) (at level 55, right associativity) : rel_scope.
(** In order to provide an [RElim] instance for [rel_curry], we will rely on the fact that:  uncurry f (x1, x2) x3 ... xn = f x1 x2 x3 ... xn,  and that:  rel_curry R f g <-> R (uncurry f) (uncurry g),  so that when the instance of [RElim] associated with [R] allows us to show:  Rcod (uncurry f (x1, x2) x3 ... xn) (uncurry g (y1, y2) y3 ... yn)  we can convert it into an instance for proving:  Rcod (f x1 x2 x3 ... xn) (g y1 y2 y3 ... yn),  which is what will be expected for [monotonicity]. To accomplish this, we need to [unfold uncurry] in the result provided for [R] before we use it to solve the [rel_uncurry] case. This helper class does this. *)
Class UnfoldUncurry {A} (before: A) (after: A) :=
unfold_uncurry_before_after: before = after.
Ltac unfold_uncurry :=
match goal with
| |- context C[uncurry ?f ?p] =>
is_evar p;
let T := type of p in
let Av := fresh in evar (Av: Type);
let A := eval red in Av in clear Av;
let Bv := fresh in evar (Bv: Type);
let B := eval red in Bv in clear Bv;
unify T (prod A B)%type;
let av := fresh in evar (av : A);
let a := eval red in av in clear av;
let bv := fresh in evar (bv : B);
let b := eval red in bv in clear bv;
let G := context C[f a b] in
unify p (a, b);
change G
end.
Hint Extern 1 (UnfoldUncurry ?P ?Q) =>
repeat unfold_uncurry; constructor : typeclass_instances.
(** Now we can provide a somewhat general [RElim] instance for [rel_uncurry]. *)

Lemma rel_curry_relim {A1 B1 C1 A2 B2 C2} R f g P Q Q':
@RElim (A1 * B1 -> C1) (A2 * B2 -> C2) R (uncurry f) (uncurry g) P Q ->
UnfoldUncurry Q Q' ->
@RElim (A1 -> B1 -> C1) (A2 -> B2 -> C2) (% R) f g P Q'.