-inductive arity (g:G): C \to T \to A \to Prop \def
-| arity_sort: \forall (c: C).\forall (n: nat).arity g c (TSort n) (ASort O n)
-| arity_abbr: \forall (c: C).\forall (d: C).\forall (u: T).\forall (i: nat).(getl i c (CHead d (Bind Abbr) u)) \to \forall (a: A).(arity g d u a) \to arity g c (TLRef i) a
-| arity_abst: \forall (c: C).\forall (d: C).\forall (u: T).\forall (i: nat).(getl i c (CHead d (Bind Abst) u)) \to \forall (a: A).(arity g d u (asucc g a)) \to arity g c (TLRef i) a
-| arity_bind: \forall (b: B).(not (eq B b Abst)) \to \forall (c: C).\forall (u: T).\forall (a1: A).(arity g c u a1) \to \forall (t: T).\forall (a2: A).(arity g (CHead c (Bind b) u) t a2) \to arity g c (THead (Bind b) u t) a2
-| arity_head: \forall (c: C).\forall (u: T).\forall (a1: A).(arity g c u (asucc g a1)) \to \forall (t: T).\forall (a2: A).(arity g (CHead c (Bind Abst) u) t a2) \to arity g c (THead (Bind Abst) u t) (AHead a1 a2)
-| arity_appl: \forall (c: C).\forall (u: T).\forall (a1: A).(arity g c u a1) \to \forall (t: T).\forall (a2: A).(arity g c t (AHead a1 a2)) \to arity g c (THead (Flat Appl) u t) a2
-| arity_cast: \forall (c: C).\forall (u: T).\forall (a: A).(arity g c u (asucc g a)) \to \forall (t: T).(arity g c t a) \to arity g c (THead (Flat Cast) u t) a
-| arity_repl: \forall (c: C).\forall (t: T).\forall (a1: A).(arity g c t a1) \to \forall (a2: A).(leq g a1 a2) \to arity g c t a2.
+inductive arity (g:G): C \to (T \to (A \to Prop)) \def
+| arity_sort: \forall (c: C).(\forall (n: nat).(arity g c (TSort n) (ASort O n)))
+| arity_abbr: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (a: A).((arity g d u a) \to (arity g c (TLRef i) a)))))))
+| arity_abst: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d (Bind Abst) u)) \to (\forall (a: A).((arity g d u (asucc g a)) \to (arity g c (TLRef i) a)))))))
+| arity_bind: \forall (b: B).((not (eq B b Abst)) \to (\forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to (\forall (t: T).(\forall (a2: A).((arity g (CHead c (Bind b) u) t a2) \to (arity g c (THead (Bind b) u t) a2)))))))))
+| arity_head: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u (asucc g a1)) \to (\forall (t: T).(\forall (a2: A).((arity g (CHead c (Bind Abst) u) t a2) \to (arity g c (THead (Bind Abst) u t) (AHead a1 a2))))))))
+| arity_appl: \forall (c: C).(\forall (u: T).(\forall (a1: A).((arity g c u a1) \to (\forall (t: T).(\forall (a2: A).((arity g c t (AHead a1 a2)) \to (arity g c (THead (Flat Appl) u t) a2)))))))
+| arity_cast: \forall (c: C).(\forall (u: T).(\forall (a: A).((arity g c u (asucc g a)) \to (\forall (t: T).((arity g c t a) \to (arity g c (THead (Flat Cast) u t) a))))))
+| arity_repl: \forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c t a1) \to (\forall (a2: A).((leq g a1 a2) \to (arity g c t a2)))))).