(* tests S4 *)

(* Specifications: *)
Lemma test18 : forall A B C D, S4 A B C D = S4 B C D A.
Lemma test19 : forall A B C D, S4 A B C D = S4 C D A B.
Lemma test20 : forall A B C D, S4 A B C D = S4 D A B C.
Lemma test21 : forall A B C D, S4 A B C D = - S4 A D C B.
Lemma test22 : forall A B C D, S4 A B C D = - S4  D C B A.
Lemma test23 : forall A B C D, S4 A B C D = - S4 C B A D.
Lemma test24 : forall A B C D, S4 A B C D = - S4 B A D C.
Lemma test25 : forall A B C D, S4 A B C D = - S4 B A D C -> 0=0.
Lemma test26 : forall A B C D, S4 A B C D = - S4 B A D C -> S4 A B C D = - S4 B A D C.
Lemma test27 : forall A C D, S4 A A C D = 0 -> 0=0.
Lemma test28 : forall A C D, S4 C D A A = 0 -> 0=0.
Lemma test29 : forall A C D, S4 A C D A = 0 -> 0=0.
Lemma test30 : forall A C D, S4 C A A D = 0 -> 0=0.
Lemma test31 : forall A C D, S4 C A D A = 0 -> 0=0.
Lemma test32 : forall A C D, S4 A C A D = 0 -> 0=0.
(* tests uniformize_dirseg *)
Lemma test35 : forall A B, A**B=-B**A.
Lemma test36 : forall A B, A**B=-B**A -> 0=0.
(* tests basic_simpl *)
Lemma test40 : forall A C, S A A C + S A C C + S A C A = 0.
Lemma test41 : forall A , A**A = 0.
Lemma test42 : forall A B, 0*A**B + (1*A**B -1*A**B) = 0.
Lemma test43 : forall A B, -A**B + - - A**B + (1*A**B -1*A**B) = 0.
Lemma test44 : forall A B, A**B + (-A**B) = 0.
Lemma test45 : forall A B, - A**B + (A**B) = 0.
Lemma test46 : forall A B, - - A**B + -(A**B) = 0.
Lemma test47 : forall A B C D, A**B + -(C**D) + (C**D) - (A**B)= 0.
Lemma test48 : forall A B C D,  S4 A B A C + A**B + -(C**D) + (C**D) - (A**B)= 0.
Lemma test49 : forall A B C D,  S4 A B A C + S A A C + 1* S A B C - - S B A C +  A**B + -(C**D) + (C**D) - (A**B)= 0.
Lemma test50 : forall A , A<>0 -> A/ A = 1.
Lemma test51 : forall A , A<>0 -> A/ A = 0 -> 0=1.
Lemma test52 : forall A B C D,  S4 A B A C + S A A C + 1* S A B C - - S B A C +  A**B + -(C**D) + (C**D) - (A**B)= 0 -> 0=0.
Lemma test53 : forall A, A<>0 -> (0+A)/A+(-A)/A=0.
Lemma test54 : forall x, -x*-x=x*x.
Lemma test55 : forall x y, -x*y=-(x*y).
Lemma test56 : forall x y, y<>0 -> (-x)/y=-(x/y).
Lemma test57 : forall y, y<>0 -> 0/y=0.
Lemma test58 : forall x y, y<>0 -> (x*y)/y =x.
Lemma test59 : forall x y, x<>0 -> (x*y)/x =y.
Lemma test60 : forall x y, x<>0 -> (x*y)/x =y -> 0=0.
Lemma test61 : forall A B C, False -> S A B C=0.
Lemma test62 : forall A B C, False ->  0 = S A B C.
Lemma test65 : forall A B C D E F:Point, S A C D <> 0 ->
S B A F + S A B F = 0.