(*****************************************************************************)
 
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.

Theorem simpl_mult_r : forall n m p : nat, 0 < n -> m * n = p * n -> m = p.