FILE:: orderedGroups_axioms.txt
%-----------------------AXIOMS ON ORDERED GROUPS--------------------
%-------------------------------------------------------------------
formula_list(usable).
% abelian group axioms (think that minuz(X,Y)=pluz(X,rvz(Y)) by def.) 
all x y z ( pluz(pluz(x,y),z) = pluz(x,pluz(y,z))). % associativity
all x ( pluz(x,e) = x). % right unit
all x ( pluz(x,rvz(x)) = e). % right inverse
all x y ( pluz(x,y) = pluz(y,x) ). % commutativity
% ordering axioms (axioms concerning non-negativeness)
all x y ( (nneg(x) & nneg(y)) -> nneg(pluz(x,y)) ).
all x ( nneg(x) | nneg(rvz(x)) ).
all x ( (nneg(x) & nneg(rvz(x))) -> x=e ).
%-------------------------------------------------------------------
end_of_list.

FILE:: orderedGroups_defs.txt
%-----------------------DEFINITIONAL EXTENSIONS---------------------
%-------------------------------------------------------------------
formula_list(usable).
all x ( nneg(x) -> abs(x)=x ). % definition of the absolute value...
all x ( -(nneg(x)) -> abs(x)=rvz(x) ). % ...def.n of the absolute value
all x y ( leq(x,y) <-> nneg(pluz(y,rvz(x))) ). % def.n of comparison
%-------------------------------------------------------------------
end_of_list.

FILE:: orderedGroups_basicLaws.txt
formula_list(usable). % basic laws
( all x y ( pluz(x,y) = pluz(y,x) )). % commutativity (ECHO)
( all x y z ( pluz(x,y)=pluz(x,z) -> y=z )). % A1: cancellation law
( all x y ( rvz(pluz(x,rvz(y)))=pluz(y,rvz(x)) )). % B:
end_of_list.

FILE:: orderedGroups_basic_TESTER.txt
%-------------------------------------------------------------------
set(auto).
assign(max_seconds,1).
clear(print_given).
assign(max_distinct_vars, 3).
assign(max_literals, 1).
assign(max_mem, 20000).
include("orderedGroups_axioms.txt").
include("orderedGroups_basicLaws.txt").

FILE:: orderedGroups_A1.out.txt
3 [] $c2!=$c1.
5,4 [] pluz(pluz(x,y),z)=pluz(x,pluz(y,z)).
6 [] pluz(x,e)=x.
8 [] pluz(x,rvz(x))=e.
10 [] pluz(x,y)=pluz(y,x).
12 [] pluz($c3,$c2)=pluz($c3,$c1).
20 [para_from,8.1.1,4.1.1.1,flip.1] pluz(x,pluz(rvz(x),y))=pluz(e,y).
22 [para_into,10.1.1,8.1.1,flip.1] pluz(rvz(x),x)=e.
25,24 [para_into,10.1.1,6.1.1,flip.1] pluz(e,x)=x.
27 [back_demod,20,demod,25] pluz(x,pluz(rvz(x),y))=y.
34 [para_from,22.1.1,4.1.1.1,demod,25,flip.1] pluz(rvz(x),pluz(x,y))=y.
36 [para_into,12.1.1,10.1.1,flip.1] pluz($c3,$c1)=pluz($c2,$c3).
38 [para_into,36.1.1,10.1.1,flip.1] pluz($c2,$c3)=pluz($c1,$c3).
40 [para_into,38.1.1,10.1.1] pluz($c3,$c2)=pluz($c1,$c3).
53,52 [para_into,27.1.1,10.1.1,demod,5] pluz(rvz(x),pluz(y,x))=y.
66 [para_into,34.1.1.2,40.1.1,demod,53,flip.1] $c2=$c1.
68 [binary,66.1,3.1] $F.

FILE:: orderedGroups_B.out.txt
4 [] rvz(pluz($c2,rvz($c1)))!=pluz($c1,rvz($c2)).
5 [] pluz(pluz(x,y),z)=pluz(x,pluz(y,z)).
8,7 [] pluz(x,e)=x.
9 [] pluz(x,rvz(x))=e.
11 [] pluz(x,y)=pluz(y,x).
17 [para_into,9.1.1,5.1.1] pluz(x,pluz(y,rvz(pluz(x,y))))=e.
19 [para_from,9.1.1,5.1.1.1,flip.1] pluz(x,pluz(rvz(x),y))=pluz(e,y).
21 [para_into,11.1.1,9.1.1,flip.1] pluz(rvz(x),x)=e.
24,23 [para_into,11.1.1,7.1.1,flip.1] pluz(e,x)=x.
26 [back_demod,19,demod,24] pluz(x,pluz(rvz(x),y))=y.
34 [para_from,21.1.1,5.1.1.1,demod,24,flip.1] pluz(rvz(x),pluz(x,y))=y.
56 [para_into,34.1.1.2,17.1.1,demod,8,flip.1] pluz(x,rvz(pluz(y,x)))=rvz(y).
86 [para_from,56.1.1,26.1.1.2,flip.1] rvz(pluz(x,rvz(y)))=pluz(y,rvz(x)).
88 [binary,86.1,4.1] $F.

FILE:: orderedGroups_leqLaws.txt
formula_list(usable). % laws concerning the relation "leq"
( all x y ( leq(x,y) | leq(y,x) )). % C: totality
( all x ( leq(x,x) ) ). % D: reflexivity
( all x y z ( (leq(x,y) & leq(y,z)) -> leq(x,z) )). % E: transitivity
( all x y z ( (leq(x,y) & x != y & leq(y,z)) -> x != z )). % E1: transitivity
( all x y z ( (leq(x,y) & leq(y,z) & y != z) -> x != z )). % E2: transitivity
( all x y z ( leq(x,y) -> leq(pluz(x,z),pluz(y,z)) )). % F: isotony
( all x y z ( pluz(x,z)=pluz(y,z) -> x=y )). % A2: cancellation law
( all x y z ( leq(x,y) & x != y -> pluz(x,z) != pluz(y,z) )). % F1: strictness of isotony
end_of_list.

FILE:: orderedGroups_leq_TESTER.txt
%-------------------------------------------------------------------
set(auto).
assign(max_seconds,1).
clear(print_given).
assign(max_distinct_vars, 3).
assign(max_literals, 2).
assign(max_mem, 20000).
include("orderedGroups_axioms.txt").
include("orderedGroups_defs.txt").
include("orderedGroups_basicLaws.txt").
include("orderedGroups_leqLaws.txt").

FILE:: orderedGroups_C.out.txt
5 [] leq(x,y)| -nneg(pluz(y,rvz(x))).
6 [] pluz(x,y)!=pluz(x,z)|y=z.
7 [] -leq($c2,$c1).
8 [] -leq($c1,$c2).
13,12 [] pluz(x,e)=x.
14 [] pluz(x,rvz(x))=e.
16 [] pluz(x,y)=pluz(y,x).
17 [] nneg(x)|nneg(rvz(x)).
21,20 [] rvz(pluz(x,rvz(y)))=pluz(y,rvz(x)).
28 [para_into,12.1.1,6.2.1,demod,13] x=y|pluz(z,y)!=pluz(z,x).
33 [hyper,17,5,demod,21] nneg(pluz(x,rvz(y)))|leq(x,y).
49,48 [para_into,16.1.1,12.1.1,flip.1] pluz(e,x)=x.
61,60 [para_into,48.1.1,14.1.1,flip.1] rvz(e)=e.
87,86 [para_into,20.1.1.1,48.1.1,demod,61,13] rvz(rvz(x))=x.
213 [para_into,28.2.1,48.1.1,demod,49] x=y|y!=x.
233 [para_from,213.1.1,7.1.1] -leq(x,$c1)|x!=$c2.
242 [hyper,33,5] leq(x,y)|leq(y,x).
247 [hyper,242,233,86,demod,87] leq($c1,$c2).
248 [binary,247.1,8.1] $F.

FILE:: orderedGroups_D.out.txt
7 [] -leq($c1,$c1).
21 [] leq(x,y)|leq(y,x).
22 [factor,21.1.2] leq(x,x).
23 [binary,22.1,7.1] $F.

FILE:: orderedGroups_E.out.txt
1 [] -nneg(x)| -nneg(y)|nneg(pluz(x,y)).
4 [] -leq(x,y)|nneg(pluz(y,rvz(x))).
5 [] leq(x,y)| -nneg(pluz(y,rvz(x))).
7 [] -leq($c3,$c1).
10,9 [] pluz(pluz(x,y),z)=pluz(x,pluz(y,z)).
11 [] pluz(x,e)=x.
13 [] pluz(x,rvz(x))=e.
15 [] pluz(x,y)=pluz(y,x).
23 [] leq($c3,$c2).
24 [] leq($c2,$c1).
33 [hyper,23,4] nneg(pluz($c2,rvz($c3))).
36 [hyper,24,4] nneg(pluz($c1,rvz($c2))).
70 [para_from,13.1.1,9.1.1.1,flip.1] pluz(x,pluz(rvz(x),y))=pluz(e,y).
110,109 [para_into,15.1.1,11.1.1,flip.1] pluz(e,x)=x.
113,112 [back_demod,70,demod,110] pluz(x,pluz(rvz(x),y))=y.
116 [para_from,15.1.1,36.1.1] nneg(pluz(rvz($c2),$c1)).
117 [para_from,15.1.1,33.1.1] nneg(pluz(rvz($c3),$c2)).
206 [hyper,117,1,116,demod,10,113] nneg(pluz(rvz($c3),$c1)).
383 [para_into,206.1.1,15.1.1] nneg(pluz($c1,rvz($c3))).
534 [hyper,383,5] leq($c3,$c1).
535 [binary,534.1,7.1] $F.

FILE:: orderedGroups_E1.out.txt
2 [] -nneg(x)| -nneg(rvz(x))|x=e.
3 [] -nneg(x)|abs(x)=x.
4 [] -leq(x,y)|nneg(pluz(y,rvz(x))).
6 [] pluz(x,y)!=pluz(x,z)|y=z.
8 [] $c3!=$c2.
13,12 [] pluz(x,e)=x.
15,14 [] pluz(x,rvz(x))=e.
16 [] pluz(x,y)=pluz(y,x).
20 [] rvz(pluz(x,rvz(y)))=pluz(y,rvz(x)).
23 [] leq(x,x).
24 [] leq($c3,$c2).
25 [] leq($c2,$c1).
27,26 [] $c3=$c1.
28 [back_demod,24,demod,27] leq($c1,$c2).
29 [back_demod,8,demod,27,flip.1] $c2!=$c1.
34 [hyper,23,4,demod,15] nneg(e).
36,35 [hyper,34,3] abs(e)=e.
38 [hyper,25,4] nneg(pluz($c1,rvz($c2))).
50 [hyper,28,4] nneg(pluz($c2,rvz($c1))).
61 [para_from,14.1.1,6.1.1,flip.1] pluz(x,y)!=e|rvz(x)=y.
70 [hyper,38,3] abs(pluz($c1,rvz($c2)))=pluz($c1,rvz($c2)).
99 [para_into,16.1.1,12.1.1,flip.1] pluz(e,x)=x.
113,112 [para_into,99.1.1,14.1.1,flip.1] rvz(e)=e.
224,223 [para_into,20.1.1.1,99.1.1,demod,113,13] rvz(rvz(x))=x.
244,243 [para_from,223.1.1,20.1.1.1.2] rvz(pluz(x,y))=pluz(rvz(y),rvz(x)).
1349 [para_into,70.1.1.1,2.3.1,demod,36,244,224,unit_del,38,50,flip.1] pluz($c1,rvz($c2))=e.
1377 [hyper,1349,61,flip.1] rvz($c2)=rvz($c1).
1454 [para_from,1377.1.1,223.1.1.1,demod,224,flip.1] $c2=$c1.
1456 [binary,1454.1,29.1] $F.

FILE:: orderedGroups_E2.out.txt
8 [] -leq(x,y)|x=y| -leq(y,z)|x!=z.
9 [] $c2!=$c1.
13 [] pluz(x,e)=x.
25 [] leq($c3,$c2).
26 [] leq($c2,$c1).
28,27 [] $c3=$c1.
29 [back_demod,25,demod,28] leq($c1,$c2).
45 [para_into,13.1.1,13.1.1] x=x.
53 [hyper,45,8,29,26,flip.1] $c2=$c1.
55 [binary,53.1,9.1] $F.

FILE:: orderedGroups_F.out.txt
4 [] -leq(x,y)|nneg(pluz(y,rvz(x))).
5 [] leq(x,y)| -nneg(pluz(y,rvz(x))).
9 [] -leq(x,y)| -leq(y,z)|y=z|x!=z.
10 [] -leq(pluz($c3,$c1),pluz($c2,$c1)).
12 [] pluz(pluz(x,y),z)=pluz(x,pluz(y,z)).
15,14 [] pluz(x,e)=x.
16 [] pluz(x,rvz(x))=e.
18 [] pluz(x,y)=pluz(y,x).
22 [] rvz(pluz(x,rvz(y)))=pluz(y,rvz(x)).
24 [] leq(x,y)|leq(y,x).
25 [] leq(x,x).
26 [] leq($c3,$c2).
30 [para_from,12.1.1,4.2.1] -leq(x,pluz(y,z))|nneg(pluz(y,pluz(z,rvz(x)))).
44 [para_into,14.1.1,14.1.1] x=x.
71 [para_from,16.1.1,12.1.1.1,flip.1] pluz(x,pluz(rvz(x),y))=pluz(e,y).
77,76 [para_into,18.1.1,14.1.1,flip.1] pluz(e,x)=x.
80,79 [back_demod,71,demod,77] pluz(x,pluz(rvz(x),y))=y.
83 [para_from,18.1.1,10.1.2] -leq(pluz($c3,$c1),pluz($c1,$c2)).
91,90 [para_into,76.1.1,16.1.1,flip.1] rvz(e)=e.
147 [hyper,24,10] leq(pluz($c2,$c1),pluz($c3,$c1)).
161 [para_into,22.1.1.1,76.1.1,demod,91,15] rvz(rvz(x))=x.
181,180 [para_from,161.1.1,22.1.1.1.2] rvz(pluz(x,y))=pluz(rvz(y),rvz(x)).
387 [para_into,83.1.1,18.1.1] -leq(pluz($c1,$c3),pluz($c1,$c2)).
488 [hyper,147,30,demod,181,80] nneg(pluz($c3,rvz($c2))).
510 [hyper,488,5] leq($c2,$c3).
575,574 [hyper,510,9,26,44,flip.1] $c3=$c2.
640 [back_demod,387,demod,575] -leq(pluz($c1,$c2),pluz($c1,$c2)).
641 [binary,640.1,25.1] $F.

FILE:: orderedGroups_A2.out.txt
6 [] pluz(x,y)!=pluz(x,z)|y=z.
11 [] $c3!=$c2.
16,15 [] pluz(x,e)=x.
19 [] pluz(x,y)=pluz(y,x).
27 [] pluz($c3,$c1)=pluz($c2,$c1).
44 [para_into,15.1.1,6.2.1,demod,16] x=y|pluz(z,y)!=pluz(z,x).
158 [para_into,27.1.1,19.1.1,flip.1] pluz($c2,$c1)=pluz($c1,$c3).
307 [para_into,158.1.1,19.1.1,flip.1] pluz($c1,$c3)=pluz($c1,$c2).
405 [hyper,307,44,flip.1] $c3=$c2.
407 [binary,405.1,11.1] $F.

FILE:: orderedGroups_F1.out.txt
11 [] pluz(x,y)!=pluz(z,y)|x=z.
12 [] $c3!=$c2.
29 [] pluz($c3,$c1)=pluz($c2,$c1).
233 [hyper,29,11] $c3=$c2.
235 [binary,233.1,12.1] $F.

FILE:: orderedGroups_absLaws.txt
formula_list(usable). % laws concerning the operation "abs"
( all x ( abs(pluz(x,rvz(x)))=e )). %1:
( all x ( leq(x,abs(x)) )). %2:
( all x ( abs(abs(x))=abs(x) )). %3:
( all x ( abs(x)=e <-> x=e )). %4:
( all x ( abs(rvz(x))=abs(x) )). %5:
( all x y ( leq(pluz(x,y),pluz(abs(x),abs(y))) )). %6:
( all x y ( leq(abs(pluz(x,y)),pluz(abs(x),abs(y))) )). %7:
( all x y ( -(nneg(x)) -> leq(x,abs(y)) & x != abs(y) )). %8:
( all x y z (leq(abs(pluz(x,rvz(y))),z) -> leq(y,pluz(x,z)))). %9:
( all x y z ( pluz(x,z)=pluz(pluz(x,rvz(y)),pluz(y,z)) )). %10a:
end_of_list.
%-------------------------------------------------------------------

FILE:: orderedGroups_abs_TESTER.txt
%-------------------------------------------------------------------
set(auto).
assign(max_seconds,9).
clear(print_given).
assign(max_distinct_vars, 3).
assign(max_literals, 2).
assign(max_weight, 12).
assign(max_mem, 20000).
include("orderedGroups_axioms.txt").
include("orderedGroups_defs.txt").
include("orderedGroups_basicLaws.txt").
include("orderedGroups_leqLaws.txt").
include("orderedGroups_absLaws.txt").

FILE:: orderedGroups_1.out.txt
3 [] -nneg(x)|abs(x)=x.
4 [] -leq(x,y)|nneg(pluz(y,rvz(x))).
12 [] abs(pluz($c1,rvz($c1)))!=e.
19,18 [] pluz(x,rvz(x))=e.
27 [] leq(x,x).
28 [back_demod,12,demod,19] abs(e)!=e.
33 [hyper,27,4,demod,19] nneg(e).
34 [hyper,33,3] abs(e)=e.
36 [binary,34.1,28.1] $F.

FILE:: orderedGroups_2.out.txt
1 [] -nneg(x)| -nneg(y)|nneg(pluz(x,y)).
2 [] -nneg(x)| -nneg(rvz(x))|x=e.
3 [] -nneg(x)|abs(x)=x.
5 [] leq(x,y)| -nneg(pluz(y,rvz(x))).
6 [] pluz(x,y)!=pluz(x,z)|y=z.
12 [] -leq($c1,abs($c1)).
13 [factor,1.1.2] -nneg(x)|nneg(pluz(x,x)).
17,16 [] pluz(x,e)=x.
19,18 [] pluz(x,rvz(x))=e.
20 [] pluz(x,y)=pluz(y,x).
21 [] nneg(x)|nneg(rvz(x)).
22 [] nneg(x)|abs(x)=rvz(x).
23 [copy,22,flip.2] nneg(x)|rvz(x)=abs(x).
24 [] rvz(pluz(x,rvz(y)))=pluz(y,rvz(x)).
27 [] leq(x,x).
28 [] abs(pluz(x,rvz(x)))=e.
29 [copy,28,demod,19] abs(e)=e.
40 [para_into,29.1.1.1,6.2.1,demod,17,flip.2] abs(x)=e|pluz(y,x)!=y.
51 [hyper,21,13] nneg(rvz(x))|nneg(pluz(x,x)).
53 [hyper,21,3] nneg(rvz(x))|abs(x)=x.
70 [para_into,20.1.1,16.1.1,flip.1] pluz(e,x)=x.
98,97 [para_into,70.1.1,18.1.1,flip.1] rvz(e)=e.
137,136 [para_into,24.1.1.1,70.1.1,demod,98,17] rvz(rvz(x))=x.
248 [para_from,40.1.1,12.1.2] -leq($c1,e)|pluz(x,$c1)!=x.
254 [hyper,51,5,demod,137] nneg(x)|leq(x,rvz(x)).
267 [para_into,254.2.2,23.2.1,factor_simp] nneg(x)|leq(x,abs(x)).
295 [para_from,53.2.1,12.1.2,unit_del,27] nneg(rvz($c1)).
302,301 [hyper,295,2,267,unit_del,12,flip.1] e=$c1.
354 [back_demod,248,demod,302,unit_del,27] pluz(x,$c1)!=x.
437 [back_demod,70,demod,302] pluz($c1,x)=x.
439 [binary,437.1,354.1] $F.

FILE:: orderedGroups_3.out.txt
3 [] -nneg(x)|abs(x)=x.
4 [] -leq(x,y)|nneg(pluz(y,rvz(x))).
5 [] leq(x,y)| -nneg(pluz(y,rvz(x))).
11 [] pluz(x,y)!=pluz(z,y)|x=z.
12 [] abs(abs($c1))!=abs($c1).
17,16 [] pluz(x,e)=x.
18 [] pluz(x,rvz(x))=e.
20 [] pluz(x,y)=pluz(y,x).
21 [] nneg(x)|nneg(rvz(x)).
22 [] nneg(x)|abs(x)=rvz(x).
23 [copy,22,flip.2] nneg(x)|rvz(x)=abs(x).
24 [] rvz(pluz(x,rvz(y)))=pluz(y,rvz(x)).
44 [hyper,16,11,demod,17] x=x.
59 [hyper,21,3] nneg(rvz(x))|abs(x)=x.
99 [para_into,20.1.1,16.1.1,flip.1] pluz(e,x)=x.
124,123 [para_into,99.1.1,18.1.1,flip.1] rvz(e)=e.
127 [para_from,99.1.1,5.2.1] leq(x,e)| -nneg(rvz(x)).
128 [para_from,99.1.1,4.2.1] -leq(x,e)|nneg(rvz(x)).
151,150 [para_into,24.1.1.1,99.1.1,demod,124,17] rvz(rvz(x))=x.
204 [para_into,127.2.1,150.1.1] leq(rvz(x),e)| -nneg(x).
205 [para_into,128.2.1,150.1.1] -leq(rvz(x),e)|nneg(x).
315 [para_into,205.1.1,23.2.1,factor_simp] -leq(abs(x),e)|nneg(x).
356 [para_from,59.2.1,12.1.1,unit_del,44] nneg(rvz(abs($c1))).
431 [hyper,356,204,demod,151] leq(abs($c1),e).
503 [hyper,431,315] nneg($c1).
517,516 [hyper,503,3] abs($c1)=$c1.
569 [back_demod,12,demod,517,517,517] $c1!=$c1.
570 [binary,569.1,44.1] $F.

FILE:: orderedGroups_4.out.txt
3 [] -nneg(x)|abs(x)=x.
11 [] pluz(x,y)!=pluz(z,y)|x=z.
12 [] abs($c1)!=e|$c1!=e.
13 [copy,12,flip.2] abs($c1)!=e|e!=$c1.
18,17 [] pluz(x,e)=x.
20,19 [] pluz(x,rvz(x))=e.
21 [] pluz(x,y)=pluz(y,x).
23 [] nneg(x)|abs(x)=rvz(x).
24 [copy,23,flip.2] nneg(x)|rvz(x)=abs(x).
25 [] rvz(pluz(x,rvz(y)))=pluz(y,rvz(x)).
29 [] abs(pluz(x,rvz(x)))=e.
30 [copy,29,demod,20] abs(e)=e.
35 [] abs($c1)=e|$c1=e.
36 [copy,35,flip.2] abs($c1)=e|e=$c1.
49 [hyper,17,11,demod,18] x=x.
78 [para_into,21.1.1,17.1.1,flip.1] pluz(e,x)=x.
97,96 [para_into,78.1.1,19.1.1,flip.1] rvz(e)=e.
129 [para_into,25.1.1.1,78.1.1,demod,97,18] rvz(rvz(x))=x.
189,188 [para_from,36.2.1,30.1.1.1,factor_simp] abs($c1)=e.
190 [back_demod,13,demod,189,unit_del,49] e!=$c1.
195 [para_into,188.1.1,3.2.1,unit_del,190] -nneg($c1).
198 [hyper,195,24,demod,189] rvz($c1)=e.
214 [para_from,198.1.1,129.1.1.1,demod,97] e=$c1.
216 [binary,214.1,190.1] $F.

FILE:: orderedGroups_5.out.txt
2 [] -nneg(x)| -nneg(rvz(x))|x=e.
3 [] -nneg(x)|abs(x)=x.
11 [] pluz(x,y)!=pluz(z,y)|x=z.
14 [] abs(rvz($c1))!=abs($c1).
19,18 [] pluz(x,e)=x.
20 [] pluz(x,rvz(x))=e.
22 [] pluz(x,y)=pluz(y,x).
24 [] nneg(x)|abs(x)=rvz(x).
25 [copy,24,flip.2] nneg(x)|rvz(x)=abs(x).
26 [] rvz(pluz(x,rvz(y)))=pluz(y,rvz(x)).
35,34 [] abs(abs(x))=abs(x).
48 [hyper,18,11,demod,19] x=x.
81 [para_into,22.1.1,18.1.1,flip.1] pluz(e,x)=x.
101,100 [para_into,81.1.1,20.1.1,flip.1] rvz(e)=e.
117 [para_from,25.2.1,14.1.1.1,demod,35,unit_del,48] nneg($c1).
123,122 [hyper,117,3] abs($c1)=$c1.
124 [hyper,117,2,25,flip.1] e=$c1|rvz(rvz($c1))=abs(rvz($c1)).
132 [back_demod,14,demod,123] abs(rvz($c1))!=$c1.
164,163 [para_into,26.1.1.1,81.1.1,demod,101,19] rvz(rvz(x))=x.
168,167 [back_demod,124,demod,164,unit_del,132] e=$c1.
187,186 [back_demod,100,demod,168,168] rvz($c1)=$c1.
213 [back_demod,132,demod,187,123] $c1!=$c1.
214 [binary,213.1,48.1] $F.

FILE:: orderedGroups_6.out.txt
3 [] -nneg(x)|abs(x)=x.
5 [] leq(x,y)| -nneg(pluz(y,rvz(x))).
7 [] -leq(x,y)| -leq(y,z)|leq(x,z).
10 [] -leq(x,y)|leq(pluz(x,z),pluz(y,z)).
14 [] -leq(pluz($c2,$c1),pluz(abs($c2),abs($c1))).
19,18 [] pluz(x,e)=x.
20 [] pluz(x,rvz(x))=e.
22 [] pluz(x,y)=pluz(y,x).
23 [] nneg(x)|nneg(rvz(x)).
24 [] nneg(x)|abs(x)=rvz(x).
25 [copy,24,flip.2] nneg(x)|rvz(x)=abs(x).
26 [] rvz(pluz(x,rvz(y)))=pluz(y,rvz(x)).
33 [] leq(x,abs(x)).
34 [] abs(abs(x))=abs(x).
59 [hyper,33,10] leq(pluz(x,y),pluz(abs(x),y)).
67 [hyper,23,3] nneg(rvz(x))|abs(x)=x.
82,81 [para_into,22.1.1,18.1.1,flip.1] pluz(e,x)=x.
92 [para_from,22.1.1,14.1.1] -leq(pluz($c1,$c2),pluz(abs($c2),abs($c1))).
103,102 [para_into,81.1.1,20.1.1,flip.1] rvz(e)=e.
106 [para_from,81.1.1,5.2.1] leq(x,e)| -nneg(rvz(x)).
112 [para_from,102.1.1,5.2.1.2,demod,19] leq(e,x)| -nneg(x).
119 [para_from,25.2.1,23.2.1,factor_simp] nneg(x)|nneg(abs(x)).
135 [para_from,34.1.1,119.2.1,factor_simp] nneg(abs(x)).
139,138 [para_into,26.1.1.1,81.1.1,demod,103,19] rvz(rvz(x))=x.
145 [hyper,135,112] leq(e,abs(x)).
159 [hyper,145,10,demod,82] leq(x,pluz(abs(y),x)).
273 [para_into,106.2.1,138.1.1] leq(rvz(x),e)| -nneg(x).
335 [para_into,159.1.2,22.1.1] leq(x,pluz(x,abs(y))).
733 [para_from,67.2.1,14.1.2.2,unit_del,59] nneg(rvz($c1)).
741 [hyper,733,273,demod,139] leq($c1,e).
747 [hyper,741,10,demod,82] leq(pluz($c1,x),x).
809 [hyper,747,7,33] leq(pluz($c1,x),abs(x)).
1136 [hyper,809,7,335] leq(pluz($c1,x),pluz(abs(x),abs(y))).
1137 [binary,1136.1,92.1] $F.

FILE:: orderedGroups_7.out.txt
1 [] -nneg(x)| -nneg(y)|nneg(pluz(x,y)).
2 [] -nneg(x)| -nneg(rvz(x))|x=e.
3 [] -nneg(x)|abs(x)=x.
5 [] leq(x,y)| -nneg(pluz(y,rvz(x))).
6 [] pluz(x,y)!=pluz(x,z)|y=z.
7 [] -leq(x,y)| -leq(y,z)|leq(x,z).
9 [] -leq(x,y)| -leq(y,z)|y=z|x!=z.
10 [] -leq(x,y)|leq(pluz(x,z),pluz(y,z)).
11 [] pluz(x,y)!=pluz(z,y)|x=z.
12 [] abs(x)!=e|x=e.
14 [] -leq(abs(pluz($c2,$c1)),pluz(abs($c2),abs($c1))).
16 [] pluz(pluz(x,y),z)=pluz(x,pluz(y,z)).
19,18 [] pluz(x,e)=x.
21,20 [] pluz(x,rvz(x))=e.
22 [] pluz(x,y)=pluz(y,x).
23 [] nneg(x)|nneg(rvz(x)).
24 [] nneg(x)|abs(x)=rvz(x).
25 [copy,24,flip.2] nneg(x)|rvz(x)=abs(x).
26 [] rvz(pluz(x,rvz(y)))=pluz(y,rvz(x)).
28 [] leq(x,y)|leq(y,x).
29 [] leq(x,x).
30 [] abs(pluz(x,rvz(x)))=e.
32,31 [copy,30,demod,21] abs(e)=e.
33 [] leq(x,abs(x)).
34 [] abs(abs(x))=abs(x).
36 [] abs(rvz(x))=abs(x).
38 [] leq(pluz(x,y),pluz(abs(x),abs(y))).
50 [para_into,31.1.1,6.2.1,demod,32,19,flip.2] x=e|pluz(y,x)!=y.
51 [hyper,18,11,demod,19] x=x.
68 [hyper,23,3] nneg(rvz(x))|abs(x)=x.
74 [para_from,20.1.1,10.2.2] -leq(x,y)|leq(pluz(x,rvz(y)),e).
80 [para_into,22.1.1,20.1.1,flip.1] pluz(rvz(x),x)=e.
83,82 [para_into,22.1.1,18.1.1,flip.1] pluz(e,x)=x.
84 [para_into,22.1.1,16.1.1] pluz(x,pluz(y,z))=pluz(z,pluz(x,y)).
85 [para_into,22.1.1,12.2.1,flip.1] pluz(x,y)=e|abs(pluz(y,x))!=e.
92 [para_from,22.1.1,14.1.2] -leq(abs(pluz($c2,$c1)),pluz(abs($c1),abs($c2))).
94 [para_from,22.1.1,10.2.2] -leq(x,y)|leq(pluz(x,z),pluz(z,y)).
104,103 [para_into,82.1.1,20.1.1,flip.1] rvz(e)=e.
113 [para_from,103.1.1,5.2.1.2,demod,19] leq(e,x)| -nneg(x).
117 [hyper,25,113] rvz(x)=abs(x)|leq(e,x).
120 [para_from,25.2.1,23.2.1,factor_simp] nneg(x)|nneg(abs(x)).
121 [para_from,25.2.1,20.1.1.2] pluz(x,abs(x))=e|nneg(x).
130 [hyper,28,14] leq(pluz(abs($c2),abs($c1)),abs(pluz($c2,$c1))).
136 [para_from,34.1.1,120.2.1,factor_simp] nneg(abs(x)).
139 [para_into,26.1.1.1,82.1.1,demod,104,19] rvz(rvz(x))=x.
146 [hyper,136,113] leq(e,abs(x)).
149 [hyper,136,1,136] nneg(pluz(abs(x),abs(y))).
160 [hyper,146,10,demod,83] leq(x,pluz(abs(y),x)).
166 [para_into,139.1.1.1,25.2.1] rvz(abs(x))=x|nneg(x).
170,169 [para_from,139.1.1,26.1.1.1.2] rvz(pluz(x,y))=pluz(rvz(y),rvz(x)).
176 [para_from,36.1.1,33.1.2] leq(rvz(x),abs(x)).
278,277 [para_from,80.1.1,16.1.1.1,demod,83,flip.1] pluz(rvz(x),pluz(x,y))=y.
311 [hyper,149,113] leq(e,pluz(abs(x),abs(y))).
348 [para_into,160.1.2,22.1.1] leq(x,pluz(x,abs(y))).
457 [para_from,50.1.1,14.1.1.1.2,demod,19,unit_del,348] pluz(x,$c1)!=x.
458 [para_from,50.1.1,14.1.1.1.1,demod,83,unit_del,160] pluz(x,$c2)!=x.
474 [para_into,457.1.1.2,2.3.1,demod,19,unit_del,51] -nneg($c1)| -nneg(rvz($c1)).
475 [para_into,457.1.1,82.1.1,flip.1] e!=$c1.
510 [para_into,458.1.1,82.1.1,flip.1] e!=$c2.
869 [para_from,68.2.1,14.1.1,demod,170,unit_del,38] nneg(pluz(rvz($c1),rvz($c2))).
913 [para_into,869.1.1,22.1.1] nneg(pluz(rvz($c2),rvz($c1))).
1741 [para_from,85.1.1,14.1.1.1,demod,32,unit_del,311] abs(pluz($c1,$c2))!=e.
1764 [para_into,1741.1.1.1,2.3.1,demod,32,170,unit_del,51,913] -nneg(pluz($c1,$c2)).
1772 [hyper,1764,25,demod,170] pluz(rvz($c2),rvz($c1))=abs(pluz($c1,$c2)).
1791 [para_into,1764.1.1,22.1.1] -nneg(pluz($c2,$c1)).
1795,1794 [hyper,1791,166] rvz(abs(pluz($c2,$c1)))=pluz($c2,$c1).
2134 [para_into,92.1.1.1,22.1.1] -leq(abs(pluz($c1,$c2)),pluz(abs($c1),abs($c2))).
2235 [hyper,94,29] leq(pluz(x,y),pluz(y,x)).
3669 [hyper,130,7,348] leq(abs($c2),abs(pluz($c2,$c1))).
3670 [hyper,130,7,160] leq(abs($c1),abs(pluz($c2,$c1))).
4843 [hyper,3669,7,176] leq(rvz($c2),abs(pluz($c2,$c1))).
4862 [hyper,3670,74,demod,1795] leq(pluz(abs($c1),pluz($c2,$c1)),e).
5376 [hyper,4843,74,demod,1795,278] leq($c1,e).
5402,5401 [hyper,5376,9,117,51,unit_del,475] rvz($c1)=abs($c1).
5474 [back_demod,1772,demod,5402] pluz(rvz($c2),abs($c1))=abs(pluz($c1,$c2)).
5504 [back_demod,474,demod,5402,unit_del,136] -nneg($c1).
5521 [hyper,5504,121] pluz($c1,abs($c1))=e.
6056,6055 [para_from,5521.1.1,84.1.1.2,demod,19,flip.1] pluz(abs($c1),pluz(x,$c1))=x.
6058 [back_demod,4862,demod,6056] leq($c2,e).
6067,6066 [hyper,6058,9,117,51,unit_del,510] rvz($c2)=abs($c2).
6090,6089 [back_demod,5474,demod,6067,flip.1] abs(pluz($c1,$c2))=pluz(abs($c2),abs($c1)).
6140 [back_demod,2134,demod,6090] -leq(pluz(abs($c2),abs($c1)),pluz(abs($c1),abs($c2))).
6141 [binary,6140.1,2235.1] $F.

FILE:: orderedGroups_8.out.txt
4 [] -leq(x,y)|nneg(pluz(y,rvz(x))).
5 [] leq(x,y)| -nneg(pluz(y,rvz(x))).
7 [] -leq(x,y)| -leq(y,z)|leq(x,z).
10 [] -leq(x,y)|leq(pluz(x,z),pluz(y,z)).
14 [] -nneg($c2).
15 [] -leq($c2,abs($c1))|$c2=abs($c1).
16 [copy,15,flip.2] -leq($c2,abs($c1))|abs($c1)=$c2.
21,20 [] pluz(x,e)=x.
22 [] pluz(x,rvz(x))=e.
24 [] pluz(x,y)=pluz(y,x).
25 [] nneg(x)|nneg(rvz(x)).
26 [] nneg(x)|abs(x)=rvz(x).
27 [copy,26,flip.2] nneg(x)|rvz(x)=abs(x).
30 [] leq(x,y)|leq(y,x).
36 [] abs(abs(x))=abs(x).
131,130 [para_into,24.1.1,20.1.1,flip.1] pluz(e,x)=x.
150 [para_into,130.1.1,22.1.1,flip.1] rvz(e)=e.
160 [para_from,150.1.1,5.2.1.2,demod,21] leq(e,x)| -nneg(x).
161 [para_from,150.1.1,4.2.1.2,demod,21] -leq(e,x)|nneg(x).
179 [para_from,27.2.1,25.2.1,factor_simp] nneg(x)|nneg(abs(x)).
181 [hyper,161,30] nneg(x)|leq(x,e).
195 [hyper,181,14] leq($c2,e).
201 [hyper,195,10,demod,131] leq(pluz($c2,x),x).
225 [para_into,201.1.1,24.1.1] leq(pluz(x,$c2),x).
261 [para_from,36.1.1,179.2.1,factor_simp] nneg(abs(x)).
262 [hyper,261,160] leq(e,abs(x)).
274 [para_into,261.1.1,16.2.1,unit_del,14] -leq($c2,abs($c1)).
278 [hyper,262,7,225,demod,131] leq($c2,abs(x)).
279 [binary,278.1,274.1] $F.

FILE:: orderedGroups_9.out.txt
4 [] -leq(x,y)|nneg(pluz(y,rvz(x))).
5 [] leq(x,y)| -nneg(pluz(y,rvz(x))).
7 [] -leq(x,y)| -leq(y,z)|leq(x,z).
15 [] -leq($c2,pluz($c3,$c1)).
17 [] pluz(pluz(x,y),z)=pluz(x,pluz(y,z)).
20,19 [] pluz(x,e)=x.
21 [] pluz(x,rvz(x))=e.
23 [] pluz(x,y)=pluz(y,x).
27 [] rvz(pluz(x,rvz(y)))=pluz(y,rvz(x)).
34 [] leq(x,abs(x)).
37 [] abs(rvz(x))=abs(x).
42 [] leq(abs(pluz($c3,rvz($c2))),$c1).
45 [para_from,17.1.1,5.2.1] leq(x,pluz(y,z))| -nneg(pluz(y,pluz(z,rvz(x)))).
132 [para_into,23.1.1,19.1.1,flip.1] pluz(e,x)=x.
145 [para_from,23.1.1,15.1.2] -leq($c2,pluz($c1,$c3)).
156,155 [para_into,132.1.1,21.1.1,flip.1] rvz(e)=e.
219,218 [para_into,27.1.1.1,132.1.1,demod,156,20] rvz(rvz(x))=x.
229,228 [para_from,218.1.1,27.1.1.1.2] rvz(pluz(x,y))=pluz(rvz(y),rvz(x)).
236 [para_from,37.1.1,34.1.2] leq(rvz(x),abs(x)).
392 [hyper,42,7,236,demod,229,219] leq(pluz($c2,rvz($c3)),$c1).
1089 [hyper,392,4,demod,229,219] nneg(pluz($c1,pluz($c3,rvz($c2)))).
5891 [hyper,1089,45] leq($c2,pluz($c1,$c3)).
5892 [binary,5891.1,145.1] $F.

FILE:: orderedGroups_10a.out.txt
11 [] pluz(x,y)!=pluz(z,y)|x=z.
16 [] pluz($c3,$c1)!=pluz(pluz($c3,rvz($c2)),pluz($c2,$c1)).
17 [copy,16,flip.1] pluz(pluz($c3,rvz($c2)),pluz($c2,$c1))!=pluz($c3,$c1).
20,19 [] pluz(pluz(x,y),z)=pluz(x,pluz(y,z)).
22,21 [] pluz(x,e)=x.
23 [] pluz(x,rvz(x))=e.
25 [] pluz(x,y)=pluz(y,x).
44 [back_demod,17,demod,20] pluz($c3,pluz(rvz($c2),pluz($c2,$c1)))!=pluz($c3,$c1).
59 [hyper,21,11,demod,22] x=x.
122 [para_into,25.1.1,23.1.1,flip.1] pluz(rvz(x),x)=e.
125,124 [para_into,25.1.1,21.1.1,flip.1] pluz(e,x)=x.
367,366 [para_from,122.1.1,19.1.1.1,demod,125,flip.1] pluz(rvz(x),pluz(x,y))=y.
370 [back_demod,44,demod,367] pluz($c3,$c1)!=pluz($c3,$c1).
371 [binary,370.1,59.1] $F.

FILE:: orderedGroups_10Laws.txt
formula_list(usable). % one further law concerning the operation "abs"
( all x y z ( leq(abs(pluz(x,rvz(z))), pluz(abs(pluz(x,rvz(y))),abs(pluz(y,rvz(z))))) )). %10: (proved sans axioms)
end_of_list.  
%-------------------------------------------------------------------

FILE:: orderedGroups_10_TESTER.txt
%-------------------------------------------------------------------
set(auto).
assign(max_seconds,1).
clear(print_given).
assign(max_distinct_vars, 3).
assign(max_literals, 1).
assign(max_weight, 20).
assign(max_mem, 20000).
%%include("orderedGroups_axioms.txt").
include("orderedGroups_defs.txt").
include("orderedGroups_basicLaws.txt").
include("orderedGroups_leqLaws.txt").
include("orderedGroups_absLaws.txt").
include("orderedGroups_10Law.txt").

FILE:: orderedGroups_10.out.txt
14 [] -leq(abs(pluz($c3,rvz($c1))),pluz(abs(pluz($c3,rvz($c2))),abs(pluz($c2,rvz($c1))))).
30 [] leq(abs(pluz(x,y)),pluz(abs(x),abs(y))).
32 [] pluz(x,y)=pluz(pluz(x,rvz(z)),pluz(z,y)).
33 [copy,32,flip.1] pluz(pluz(x,rvz(y)),pluz(y,z))=pluz(x,z).
126 [para_from,33.1.1,30.1.1.1] leq(abs(pluz(x,y)),pluz(abs(pluz(x,rvz(z))),abs(pluz(z,y)))).
127 [binary,126.1,14.1] $F.

FILE:: orderedGroups_nnegLaws.txt
formula_list(usable). % further laws concerning the operations "abs" and "nneg"
( all x ( abs(rvz(x))=abs(x) )). %5: (ECHO)
( all x y ( nneg(y) -> leq(pluz(x,rvz(y)),pluz(x,y)) )). %11:
( all x y ( ( nneg(x) & -(nneg(y)) ) -> leq(abs(pluz(abs(x),rvz(abs(y)))),abs(pluz(x,rvz(y)))) )). %12a:
( all x y ( ( nneg(x) & nneg(y) ) -> abs(pluz(abs(x),rvz(abs(y))))=abs(pluz(x,rvz(y))) )). %12b:
( all x y ( ( -(nneg(x)) & -(nneg(y)) ) -> abs(pluz(abs(rvz(x)),rvz(abs(rvz(y)))))=abs(pluz(rvz(x),y)) )). %12c: 
end_of_list.
%-------------------------------------------------------------------

FILE:: orderedGroups_nneg_TESTER.txt
%-------------------------------------------------------------------
set(auto).
assign(max_seconds,1).
clear(print_given).
assign(max_distinct_vars, 2).
assign(max_literals, 2).
assign(max_weight, 20).
assign(max_mem, 25000).
include("orderedGroups_axioms.txt").
include("orderedGroups_defs.txt").
include("orderedGroups_basicLaws.txt").
include("orderedGroups_leqLaws.txt").
include("orderedGroups_absLaws.txt").
include("orderedGroups_10Laws.txt").
include("orderedGroups_nnegLaws.txt").

FILE:: orderedGroups_11.out.txt
5 [] leq(x,y)| -nneg(pluz(y,rvz(x))).
7 [] -leq(x,y)| -leq(y,z)|leq(x,z).
10 [] -leq(x,y)|leq(pluz(x,z),pluz(y,z)).
16 [] -leq(pluz($c2,rvz($c1)),pluz($c2,$c1)).
21,20 [] pluz(x,e)=x.
22 [] pluz(x,rvz(x))=e.
24 [] pluz(x,y)=pluz(y,x).
49 [] nneg($c1).
149,148 [para_into,24.1.1,20.1.1,flip.1] pluz(e,x)=x.
158 [para_from,24.1.1,16.1.1] -leq(pluz(rvz($c1),$c2),pluz($c2,$c1)).
191 [para_into,148.1.1,22.1.1,flip.1] rvz(e)=e.
202 [para_from,191.1.1,5.2.1.2,demod,21] leq(e,x)| -nneg(x).
227 [hyper,202,49] leq(e,$c1).
231 [hyper,227,10,demod,149] leq(x,pluz($c1,x)).
260 [para_into,231.1.2,24.1.1] leq(x,pluz(x,$c1)).
261 [para_into,231.1.2,22.1.1] leq(rvz($c1),e).
263 [hyper,261,10,demod,149] leq(pluz(rvz($c1),x),x).
729 [hyper,263,7,260] leq(pluz(rvz($c1),x),pluz(x,$c1)).
730 [binary,729.1,158.1] $F.

FILE:: orderedGroups_12a.out.txt
1 [] -nneg(x)| -nneg(y)|nneg(pluz(x,y)).
3 [] -nneg(x)|abs(x)=x.
14 [] nneg(x)|x!=abs(y).
17 [] -nneg($c1).
18 [] -leq(abs(pluz(abs($c2),rvz(abs($c1)))),abs(pluz($c2,rvz($c1)))).
23,22 [] pluz(x,e)=x.
24 [] pluz(x,rvz(x))=e.
26 [] pluz(x,y)=pluz(y,x).
27 [] nneg(x)|nneg(rvz(x)).
30 [] rvz(pluz(x,rvz(y)))=pluz(y,rvz(x)).
41,40 [] abs(rvz(x))=abs(x).
43 [] leq(abs(pluz(x,y)),pluz(abs(x),abs(y))).
51 [] nneg($c2).
56,55 [hyper,51,3] abs($c2)=$c2.
57 [back_demod,18,demod,56] -leq(abs(pluz($c2,rvz(abs($c1)))),abs(pluz($c2,rvz($c1)))).
69 [hyper,22,14,demod,23] nneg(abs(x)).
80 [hyper,69,1,51] nneg(pluz($c2,abs(x))).
133 [hyper,27,17] nneg(rvz($c1)).
161 [para_into,26.1.1,22.1.1,flip.1] pluz(e,x)=x.
177,176 [hyper,133,3,demod,41,flip.1] rvz($c1)=abs($c1).
178 [back_demod,57,demod,177] -leq(abs(pluz($c2,rvz(abs($c1)))),abs(pluz($c2,abs($c1)))).
182,181 [hyper,80,3] abs(pluz($c2,abs(x)))=pluz($c2,abs(x)).
194 [back_demod,178,demod,182] -leq(abs(pluz($c2,rvz(abs($c1)))),pluz($c2,abs($c1))).
213,212 [para_into,161.1.1,24.1.1,flip.1] rvz(e)=e.
273 [para_into,30.1.1.1,161.1.1,demod,213,23] rvz(rvz(x))=x.
330,329 [para_into,273.1.1.1,176.1.1] rvz(abs($c1))=$c1.
332 [back_demod,194,demod,330] -leq(abs(pluz($c2,$c1)),pluz($c2,abs($c1))).
515 [para_into,43.1.2.1,55.1.1] leq(abs(pluz($c2,x)),pluz($c2,abs(x))).
516 [binary,515.1,332.1] $F.

FILE:: orderedGroups_12b.out.txt
3 [] -nneg(x)|abs(x)=x.
18 [] abs(pluz(abs($c2),rvz(abs($c1))))!=abs(pluz($c2,rvz($c1))).
41,40 [] abs(rvz(x))=abs(x).
49 [] abs(rvz(x))=abs(x).
50 [copy,49,demod,41] abs(x)=abs(x).
51 [] nneg($c2).
52 [] nneg($c1).
58,57 [hyper,51,3] abs($c2)=$c2.
60 [back_demod,18,demod,58] abs(pluz($c2,rvz(abs($c1))))!=abs(pluz($c2,rvz($c1))).
69,68 [hyper,52,3] abs($c1)=$c1.
73 [back_demod,60,demod,69] abs(pluz($c2,rvz($c1)))!=abs(pluz($c2,rvz($c1))).
74 [binary,73.1,50.1] $F.

FILE:: orderedGroups_12c.out.txt
3 [] -nneg(x)|abs(x)=x.
11 [] pluz(x,y)!=pluz(z,y)|x=z.
19 [] -nneg($c2).
20 [] -nneg($c1).
21 [] abs(pluz(abs(rvz($c2)),rvz(abs(rvz($c1)))))!=abs(pluz(rvz($c2),$c1)).
27,26 [] pluz(x,e)=x.
28 [] pluz(x,rvz(x))=e.
30 [] pluz(x,y)=pluz(y,x).
31 [] nneg(x)|nneg(rvz(x)).
34 [] rvz(pluz(x,rvz(y)))=pluz(y,rvz(x)).
45,44 [] abs(rvz(x))=abs(x).
56 [back_demod,21,demod,45,45] abs(pluz(abs($c2),rvz(abs($c1))))!=abs(pluz(rvz($c2),$c1)).
72 [hyper,26,11,demod,27] x=x.
89 [hyper,31,20] nneg(rvz($c1)).
90 [hyper,31,19] nneg(rvz($c2)).
138 [hyper,89,3,demod,45,flip.1] rvz($c1)=abs($c1).
156,155 [hyper,90,3,demod,45,flip.1] rvz($c2)=abs($c2).
157 [back_demod,56,demod,156] abs(pluz(abs($c2),rvz(abs($c1))))!=abs(pluz(abs($c2),$c1)).
190 [para_into,30.1.1,26.1.1,flip.1] pluz(e,x)=x.
215,214 [para_into,190.1.1,28.1.1,flip.1] rvz(e)=e.
265 [para_into,34.1.1.1,190.1.1,demod,215,27] rvz(rvz(x))=x.
302,301 [para_into,265.1.1.1,138.1.1] rvz(abs($c1))=$c1.
306 [back_demod,157,demod,302] abs(pluz(abs($c2),$c1))!=abs(pluz(abs($c2),$c1)).
307 [binary,306.1,72.1] $F.

FILE:: orderedGroups_12Laws.txt
formula_list(usable).
% to prove the next lemma, it turned out useful to temporarily inhibit:
% * all axioms save commutativity;
% * all definitional extensions;
% * all of the laws concerning "leq" save D (reflexivity) and B;
% * all of the laws concerning "abs" and "nneg" save 5 and 12a--12c
( all x y ( leq(abs(pluz(abs(x),rvz(abs(y)))),abs(pluz(x,rvz(y)))) )). %12:
end_of_list.

FILE:: orderedGroups_12_TESTER.txt
%-------------------------------------------------------------------
set(auto).
assign(max_seconds,1).
clear(print_given).
assign(max_distinct_vars, 2).
assign(max_literals, 3).
assign(max_weight, 17).
include("orderedGroups_basicLaws.txt").
include("orderedGroups_leqLaws.txt").
include("orderedGroups_nnegLaws.txt").
include("orderedGroups_12Laws.txt").

FILE:: orderedGroups_12.out.txt
6 [] pluz(x,y)!=pluz(z,y)|x=z.
8 [] -nneg(x)|nneg(y)|leq(abs(pluz(abs(x),rvz(abs(y)))),abs(pluz(x,rvz(y)))).
9 [] -nneg(x)| -nneg(y)|abs(pluz(abs(x),rvz(abs(y))))=abs(pluz(x,rvz(y))).
10 [] -leq(abs(pluz(abs($c2),rvz(abs($c1)))),abs(pluz($c2,rvz($c1)))).
12 [] pluz(x,y)=pluz(y,x).
13 [] rvz(pluz(x,rvz(y)))=pluz(y,rvz(x)).
16 [] leq(x,x).
18,17 [] abs(rvz(x))=abs(x).
19 [] nneg(x)|nneg(y)|abs(pluz(abs(rvz(x)),rvz(abs(rvz(y)))))=abs(pluz(rvz(x),y)).
20 [copy,19,demod,18,18] nneg(x)|nneg(y)|abs(pluz(abs(x),rvz(abs(y))))=abs(pluz(rvz(x),y)).
25 [para_from,12.1.1,10.1.1.1] -leq(abs(pluz(rvz(abs($c1)),abs($c2))),abs(pluz($c2,rvz($c1)))).
27 [para_from,12.1.1,9.3.1.1] -nneg(x)| -nneg(y)|abs(pluz(rvz(abs(y)),abs(x)))=abs(pluz(x,rvz(y))).
28 [para_from,12.1.1,8.3.1.1] -nneg(x)|nneg(y)|leq(abs(pluz(rvz(abs(y)),abs(x))),abs(pluz(x,rvz(y)))).
32 [para_into,13.1.1.1,12.1.1] rvz(pluz(rvz(x),y))=pluz(x,rvz(y)).
34 [para_from,13.1.1,17.1.1.1] abs(pluz(x,rvz(y)))=abs(pluz(y,rvz(x))).
36 [para_into,32.1.1,13.1.1] pluz(x,rvz(rvz(y)))=pluz(y,rvz(rvz(x))).
37 [para_from,32.1.1,17.1.1.1] abs(pluz(x,rvz(y)))=abs(pluz(rvz(x),y)).
38 [copy,37,flip.1] abs(pluz(rvz(x),y))=abs(pluz(x,rvz(y))).
41 [para_from,20.3.1,10.1.1] -leq(abs(pluz(rvz($c2),$c1)),abs(pluz($c2,rvz($c1))))|nneg($c2)|nneg($c1).
45 [para_from,34.1.1,10.1.2] -leq(abs(pluz(abs($c2),rvz(abs($c1)))),abs(pluz($c1,rvz($c2)))).
50 [para_into,36.1.1,12.1.1] pluz(rvz(rvz(x)),y)=pluz(x,rvz(rvz(y))).
112 [para_into,45.1.1,37.1.1] -leq(abs(pluz(rvz(abs($c2)),abs($c1))),abs(pluz($c1,rvz($c2)))).
123 [para_from,27.3.1,25.1.1,unit_del,16] -nneg($c2)| -nneg($c1).
124 [para_into,123.1.1,6.2.1] -nneg(x)| -nneg($c1)|pluz($c2,y)!=pluz(x,y).
149 [para_into,124.3.1,12.1.1] -nneg(x)| -nneg($c1)|pluz(y,$c2)!=pluz(x,y).
166 [para_into,41.1.1,38.1.1,unit_del,16] nneg($c2)|nneg($c1).
170 [hyper,166,28] nneg($c1)|nneg(x)|leq(abs(pluz(rvz(abs(x)),abs($c2))),abs(pluz($c2,rvz(x)))).
179 [hyper,166,28] nneg($c2)|nneg(x)|leq(abs(pluz(rvz(abs(x)),abs($c1))),abs(pluz($c1,rvz(x)))).
185 [factor,170.1.2,unit_del,25] nneg($c1).
186 [factor,179.1.2,unit_del,112] nneg($c2).
208 [hyper,186,149,185,50] $F.

FILE:: orderedGroups_13Laws.txt
formula_list(usable).
( all x y ( leq(pluz(abs(x),rvz(abs(pluz(abs(y),rvz(abs(x)))))),abs(y)) )). %13:
end_of_list.

FILE:: orderedGroups_13TESTER.txt
%-------------------------------------------------------------------
set(auto).
assign(max_seconds,15).
clear(print_given).
assign(max_distinct_vars, 2).
assign(max_literals, 3).
assign(max_weight, 17).
assign(max_mem, 25000).
include("orderedGroups_axioms.txt").
include("orderedGroups_defs.txt").
include("orderedGroups_basicLaws.txt").
include("orderedGroups_leqLaws.txt").
include("orderedGroups_absLaws.txt").
include("orderedGroups_10Laws.txt").
include("orderedGroups_nnegLaws.txt").
include("orderedGroups_13Laws.txt").

FILE:: orderedGroups_13.out.txt
3 [] -nneg(x)|abs(x)=x.
4 [] -leq(x,y)|nneg(pluz(y,rvz(x))).
5 [] leq(x,y)| -nneg(pluz(y,rvz(x))).
7 [] -leq(x,y)| -leq(y,z)|leq(x,z).
10 [] -leq(x,y)|leq(pluz(x,z),pluz(y,z)).
14 [] nneg(x)|x!=abs(y).
18 [] -nneg(x)| -nneg(y)|abs(pluz(abs(x),rvz(abs(y))))=abs(pluz(x,rvz(y))).
19 [] -leq(pluz(abs($c2),rvz(abs(pluz(abs($c1),rvz(abs($c2)))))),abs($c1)).
22 [] pluz(pluz(x,y),z)=pluz(x,pluz(y,z)).
25,24 [] pluz(x,e)=x.
27,26 [] pluz(x,rvz(x))=e.
28 [] pluz(x,y)=pluz(y,x).
32 [] rvz(pluz(x,rvz(y)))=pluz(y,rvz(x)).
34 [] leq(x,y)|leq(y,x).
35 [] leq(x,x).
36 [] abs(pluz(x,rvz(x)))=e.
38,37 [copy,36,demod,27] abs(e)=e.
41,40 [] abs(abs(x))=abs(x).
43,42 [] abs(rvz(x))=abs(x).
59 [hyper,35,4,demod,27] nneg(e).
71 [hyper,24,14,demod,25] nneg(abs(x)).
127 [para_from,26.1.1,22.1.1.1,flip.1] pluz(x,pluz(rvz(x),y))=pluz(e,y).
140 [hyper,34,19] leq(abs($c1),pluz(abs($c2),rvz(abs(pluz(abs($c1),rvz(abs($c2))))))).
155,154 [para_into,28.1.1,24.1.1,flip.1] pluz(e,x)=x.
159,158 [back_demod,127,demod,155] pluz(x,pluz(rvz(x),y))=y.
164 [para_from,28.1.1,19.1.1.2.1.1] -leq(pluz(abs($c2),rvz(abs(pluz(rvz(abs($c2)),abs($c1))))),abs($c1)).
208,207 [para_into,154.1.1,26.1.1,flip.1] rvz(e)=e.
242 [para_from,207.1.1,5.2.1.2,demod,25] leq(e,x)| -nneg(x).
244 [hyper,242,71] leq(e,abs(x)).
248 [hyper,244,10,demod,155] leq(x,pluz(abs(y),x)).
255,254 [para_into,32.1.1.1,154.1.1,demod,208,25] rvz(rvz(x))=x.
292,291 [para_from,254.1.1,32.1.1.1.2] rvz(pluz(x,y))=pluz(rvz(y),rvz(x)).
530 [para_into,248.1.2,26.1.1] leq(rvz(abs(x)),e).
533 [hyper,530,10,demod,155] leq(pluz(rvz(abs(x)),y),y).
1577 [para_into,533.1.1,28.1.1] leq(pluz(x,rvz(abs(y))),x).
3102 [hyper,140,7,1577] leq(abs($c1),abs($c2)).
3135 [hyper,3102,4] nneg(pluz(abs($c2),rvz(abs($c1)))).
4382 [para_into,3135.1.1,28.1.1] nneg(pluz(rvz(abs($c1)),abs($c2))).
5948,5947 [hyper,4382,18,59,demod,38,155,43,41,292,255,155,flip.1] abs(pluz(rvz(abs($c2)),abs($c1)))=abs(pluz(rvz(abs($c1)),abs($c2))).
5953,5952 [hyper,4382,3] abs(pluz(rvz(abs($c1)),abs($c2)))=pluz(rvz(abs($c1)),abs($c2)).
6073 [back_demod,164,demod,5948,5953,292,255,159] -leq(abs($c1),abs($c1)).
6074 [binary,6073.1,35.1] $F.