(*****************************************************************************)
 
Inductive is_div : nat -> nat -> nat -> nat -> Prop :=
    is_div_def :
      forall a b q r : nat, r < b -> a = q * b + r -> is_div a b q r.(*****************************************************************************)
 
Inductive is_div : nat -> nat -> nat -> nat -> Prop :=
    is_div_def :
      forall a b q r : nat, r < b -> a = q * b + r -> is_div a b q r.Inductive ex_div : nat -> nat -> Prop :=
    ex_div_def : forall a b q r : nat, is_div a b q r -> ex_div a b.Inductive ex_div : nat -> nat -> Prop :=
    ex_div_def : forall a b q r : nat, is_div a b q r -> ex_div a b.

Lemma div_SxS : forall x y q r : nat, 0 < r -> is_div x y q r -> is_div (pred x) y q (pred r).