Section Minus.
Proof using.
intros n; elim n; simpl in |- *; auto.
intros m H1; Contradict H1; auto with arith.
intros n1 Rec m; case m; simpl in |- *; auto.
intros m1 H1; apply Rec; apply lt_S_n; auto.
Qed.
Proof using.
intros a; elim a; simpl in |- *; auto.
intros n H b; case b; simpl in |- *; auto.
Qed.

Theorem minus_minus_simpl4 :
forall a b c : nat, b <= c -> c <= a -> a - b - (a - c) = c - b.