proposition: Remark RD : dominates_from 1 (F_ (E0_succ alpha)) (F_ alpha).
proof: Proof. generalize RB; intro RB'. rewrite alpha_def . destruct (Halpha beta). rewrite alpha_def ;apply Lt_Succ. intros n Hn. rewrite (F_succ_eqn (E0_succ beta)). apply Nat.lt_le_trans with (F_ (E0_succ beta) (F_ (E0_succ beta) n)). rewrite <- alpha_def. apply RB'. rewrite iterate_S_eqn2. change (F_ (E0_succ beta) (F_ (E0_succ beta) n)) with (iterate (F_ (E0_succ beta)) 1 (F_ (E0_succ beta) n)). apply iterate_le. generalize R1; intro R1'. rewrite <- alpha_def. auto. assumption. Qed.
imports: hydras(Iterates Simple_LexProd Exp2) hydras(E0 Canon Paths primRec Hprime) Coq(ArithRing Lia) Compat815 Equations(Equations) hydras(primRec)