(** The degenerate segments represent the additive identity *)

(* Specifications: *)
Lemma sums123312 : forall A B C, SumS A B C C A B.
Lemma sums__cong1245 : forall A B C D E, SumS A B C C D E -> Cong A B D E.
Lemma sums__eq34 : forall A B C D, SumS A B C D A B -> C = D.
Lemma sums112323 : forall A B C, SumS A A B C B C.
Lemma sums__cong2345 : forall A B C D E, SumS A A B C D E -> Cong B C D E.
Lemma sums__eq12 : forall A B C D, SumS A B C D C D -> A = B.
(** Some permutation properties *)
Lemma sums_left_comm : forall A B C D E F, SumS A B C D E F -> SumS B A C D E F.
Lemma sums_middle_comm : forall A B C D E F, SumS A B C D E F -> SumS A B D C E F.
Lemma sums_right_comm : forall A B C D E F, SumS A B C D E F -> SumS A B C D F E.
Lemma sums_comm : forall A B C D E F, SumS A B C D E F -> SumS B A D C F E.
(** Basic case of sum *)
Lemma bet__sums : forall A B C, Bet A B C -> SumS A B B C A C.
Lemma sums_assoc_1 : forall A B C D E F G H I J K L,
SumS A B C D G H -> SumS C D E F I J -> SumS G H E F K L ->
SumS A B I J K L.
Lemma sums_assoc_2 : forall A B C D E F G H I J K L,
SumS A B C D G H -> SumS C D E F I J -> SumS A B I J K L ->
SumS G H E F K L.