proposition: Lemma induct_word_S: forall (P : forall n : nat, word (S n) -> Prop), (forall b, P 0 (WS b WO)) -> (forall b b0 n (w : word n), P n (WS b0 w) -> P (S n) (WS b (WS b0 w))) -> forall (n : nat) (w : word (S n)), P n w.
proof: Proof. induction n; intros. - destruct (destruct_word_S w) as [v [b E]]. subst w. rewrite (word0 v). apply H. - destruct (destruct_word_S w) as [v [b E]]. subst w. destruct (destruct_word_S v) as [w [b0 E]]. subst v. apply H0. apply IHn. Qed.
imports: Coq.Classes.Morphisms Coq.Classes.Morphisms_Prop Coq.Classes.Equivalence Coq.Arith.PeanoNat Coq.ZArith.BinInt  Coq.Arith.Arith Coq.NArith.NArith Coq.Bool.Bool Coq.ZArith.ZArith Coq.Logic.Eqdep_dec Coq.Logic.EqdepFacts Coq.Program.Tactics Coq.Program.Equality Coq.setoid_ring.Ring Coq.setoid_ring.Ring_polynom bbv.Nomega bbv.ZLib bbv.NatLib bbv.NLib bbv.N_Z_nat_conversions bbv.DepEq bbv.DepEqNat bbv.ReservedNotations Coq.micromega.Lia