(* Simple let-patterns *)
Variable A B : Type.

Definition l1 (t : A * B * B) : A := let '(x, y, z) := t in x.
Print l1.
Definition l2 (t : (A * B) * B) : A := let '((x, y), z) := t in x.
Definition l3 (t : A * (B * B)) : A := let '(x, (y, z)) := t in x.
Print l3.

Record someT (A : Type) := mkT { a : nat; b: A }.

Definition l4 A (t : someT A) : nat := let 'mkT x y := t in x.
Print l4.
Print sigT.

Definition l5 A (B : A -> Type) (t : sigT B) : B (projT1 t) :=
  let 'existT x y := t return B (projT1 t) in y.

Definition l6 A (B : A -> Type) (t : sigT B) : B (projT1 t) :=
  let 'existT x y as t' := t return B (projT1 t') in y.

Definition l7 A (B : A -> Type) (t : sigT B) : B (projT1 t) :=
  let 'existT x y as t' in sigT _ := t return B (projT1 t') in y.

Definition l8 A (B : A -> Type) (t : sigT B) : B (projT1 t) :=
  match t with
    existT x y => y
  end.

(** An example from algebra, using let' and inference of return clauses
   to deconstruct contexts. *)

Record a_category (A : Type) (hom : A -> A -> Type) := {  }.

Definition category := { A : Type & { hom : A -> A -> Type & a_category A hom } }.

Record a_functor (A : Type) (hom : A -> A -> Type) (C : a_category A hom) := {  }.

Notation " x :& y " := (@existT _ _ x y) (right associativity, at level 55) : core_scope.

Definition functor (c d : category) :=
  let ' A :& homA :& CA := c in
  let ' B :& homB :& CB := d in
    A -> B.

Definition identity_functor (c : category) : functor c c :=
  let 'A :& homA :& CA := c in
    fun x => x.

Definition functor_composition (a b c : category) : functor a b -> functor b c -> functor a c :=
  let 'A :& homA :& CA := a in
  let 'B :& homB :& CB := b in
  let 'C :& homB :& CB := c in
    fun f g =>
      fun x => g (f x).
