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+
+Inductive equivalent P Q := Equivalent (P_to_Q : P -> Q) (Q_to_P : Q -> P).
+
+Inductive equal T (x : T) : T -> Type := Equal : equal T x x.
+
+(* Arithmetic *)
+
+Inductive natural := Zero | Add_1_to (n : natural).
+
+Fixpoint add (m n : natural) : natural :=
+  match m with Zero => n | Add_1_to m_minus_1 => add m_minus_1 (Add_1_to n) end.
+
+Definition double (n : natural) : natural := add n n.
+
+Inductive odd (n : natural) :=
+  Odd (half : natural)
+    (n_odd : equal natural n (Add_1_to (double half))).
+
+Inductive less_than (m n : natural) :=
+  LessThan (diff : natural)
+    (m_lt_n : equal natural n (Add_1_to (add m diff))).
+
+(* Finite subsets *)
+
+Definition injective_in T R (D : T -> Type) (f : T -> R) :=
+  forall x y, D x -> D y -> equal R (f x) (f y) -> equal T x y.
+
+Inductive in_image T R (D : T -> Type) (f : T -> R) (a : R) :=
+  InImage (x : T) (x_in_D : D x) (a_is_fx : equal R a (f x)).
+
+Inductive finite_of_order T (D : T -> Type) (n : natural) :=
+  FiniteOfOrder (rank : T -> natural)
+    (rank_injective : injective_in T natural D rank)
+    (rank_onto :
+       forall i, equivalent (less_than i n) (in_image T natural D rank i)).