proposition: Theorem Strong_confluence : forall (U:Type) (R:Relation U), Strongly_confluent U R -> Confluent U R.
proof: Proof. intros U R H'; red. intro x; red; intros a b H'0. unfold coherent at 1. generalize b; clear b. elim H'0; clear H'0. - intros x0 b H'1; exists b; auto with sets. - intros x0 y z H'1 H'2 H'3 b H'4. generalize (Lemma1 U R); intro h; lapply h; [ intro H'0; generalize (H'0 x0 b); intro h0; lapply h0; [ intro H'5; generalize (H'5 y); intro h1; lapply h1; [ intro h2; elim h2; intros z0 h3; elim h3; intros H'6 H'7; clear h h0 h1 h2 h3 | clear h h0 h1 ] | clear h h0 ] | clear h ]; auto with sets. generalize (H'3 z0); intro h; lapply h; [ intro h0; elim h0; intros z1 h1; elim h1; intros H'8 H'9; clear h h0 h1 | clear h ]; auto with sets. exists z1; split; auto with sets. apply Rstar_n with z0; auto with sets. Qed.
imports: Relations_1 Relations_1_facts Relations_2 Relations_2_facts Relations_3