proposition: Theorem eq_rect_combine_dist : forall a b c d (w : word ((a + b) * c)) (H : b * c = d), b * c = d -> let H1 := (eq_rect_combine_dist_helper1 a b c H) in let H2 := (eq_rect_combine_dist_helper2 a b c H) in let w' := eq_rec ((a + b) * c) word w _ H1 in w = eq_rec _ word (combine (split1 (a * c) d w') (split2 (a * c) d w')) _ H2.
proof: Proof. intros. subst d. rewrite combine_split. eq_rect_simpl. generalize dependent w. generalize dependent H2. rewrite H1. intros. eq_rect_simpl; 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