proposition: Theorem split1_split2 : forall n1 n2 n3 Heq w, split1 n2 n3 (split2 n1 (n2 + n3) w) = split2 n1 n2 (split1 (n1 + n2) n3 (match Heq in _ = N return word N with | refl_equal => w end)).
proof: Proof. induction n1; simpl; intros. rewrite (UIP_dec eq_nat_dec Heq (refl_equal _)). reflexivity. rewrite (shatter_word w). simpl. assert (n1 + (n2 + n3) = n1 + n2 + n3) as Heq' by lia. rewrite <- wtl_match with (Heq':=Heq'). rewrite <- IHn1. f_equal. 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