proposition: Theorem combine_inj : forall sz1 sz2 a b c d, @combine sz1 a sz2 b = @combine sz1 c sz2 d -> a = c /\ b = d.
proof: Proof. induction sz1; simpl; intros. - rewrite (word0 a) in *. rewrite (word0 c) in *. simpl in *. intuition. - rewrite (shatter_word a) in *. rewrite (shatter_word c) in *. simpl in *. inversion H. apply (inj_pair2_eq_dec _ eq_nat_dec) in H2. destruct (IHsz1 _ _ _ _ _ H2). intuition. f_equal; auto. 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