proposition: Theorem combine_assoc : forall n1 (w1 : word n1) n2 n3 (w2 : word n2) (w3 : word n3) Heq, combine (combine w1 w2) w3 = match Heq in _ = N return word N with | refl_equal => combine w1 (combine w2 w3) end.
proof: Proof. induction w1 as [|?? w1 IHw1]; simpl; intros n2 n3 w2 w3 Heq; intuition. rewrite (UIP_dec eq_nat_dec Heq (refl_equal _)); reflexivity. rewrite (IHw1 _ _ _ _ (Nat.add_assoc _ _ _)); clear IHw1. repeat match goal with | [ |- context[match ?pf with refl_equal => _ end] ] => generalize pf end. generalize dependent (combine w1 (combine w2 w3)). rewrite Nat.add_assoc; intros w Heq0 e. rewrite (UIP_dec eq_nat_dec e (refl_equal _)). rewrite (UIP_dec eq_nat_dec Heq0 (refl_equal _)). reflexivity. 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