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%  Tarski-Szmielew's axiom system
%  T is Tarski's B,  non-strict betweenness
%  E is equidistance
%  Names for the axioms as in SST.



set(hyper_res).
set(para_into).
set(para_from).
set(binary_res).
set(ur_res).
%  set(unit_deletion).
set(order_history).
assign(report,5400).
%  assign(max_seconds, 6000).
assign(max_mem,840000).
clear(print_kept).
%set(very_verbose).
set(input_sos_first).
%  set(ancestor_subsume).
set(back_sub).
%  set(sos_queue).
assign(bsub_hint_wt,-1).
set(keep_hint_subsumers).




assign(max_weight,11).
assign(max_distinct_vars,4).
assign(pick_given_ratio,4).
assign(max_proofs,2).
assign(heat,0).
assign(bsub_hint_wt,-1).
set(keep_hint_subsumers).

weight_list(pick_and_purge).
weight(FS($0,$0,$0,$0,$0,$0,$0,$0),1).
weight(f811($0,$0,$0,$0,$0),1).
weight(g811($0,$0,$0,$0,$0),1).
 
end_of_list.



list(usable).
x=x.
 E(x,y,y,x).                                   % A1 from page 10 of sst
 -E(x,y,z,v) | -E(x,y,z2,v2) | E(z,v,z2,v2).   % A2
 -E(x,y,z,z) | x=y.                            % A3
 T(x,y,ext(x,y,w,v)).                          % A4, first half
 E(y,ext(x,y,w,v),w,v).                        % A4, second half
 -E(x,y,x1,y1) | -E(y,z,y1,z1) | -E(x,v,x1,v1) | -E(y,v,y1,v1) |
-T(x,y,z) | -T(x1,y1,z1) | x=y | E(z,v,z1,v1).  % A5
 -T(x,y,x) | x=y.                              % A6
% A7, inner Pasch, two clauses.
-T(xa,xp,xc) | -T(xb,xq,xc) | T(xp,ip(xa,xp,xc,xb,xq),xb).
-T(xa,xp,xc) | -T(xb,xq,xc) | T(xq,ip(xa,xp,xc,xb,xq),xa).
-T(alpha,beta,gamma).  %A8,  three lines.
-T(beta,gamma,alpha).
-T(gamma,alpha,beta).
% We don't need more of Tarski's axioms than that here.
E(x,y,x,y).  % Satz2-1
-E(xa,xb,xc,xd) | E(xc,xd,xa,xb).  % Satz2-2
-E(xa,xb,xc,xd) | E(xb,xa,xc,xd).  % Satz2-4
-E(xa,xb,xc,xd) | -E(xc,xd,xe,xf) | E(xa,xb,xe,xf).  %Satz2-3
-E(xa,xb,xc,xd) | E(xa,xb,xd,xc).  % Satz2-5
E(x,x,y,y).  % Satz 2-8
-T(xa,xb,xc) | -T(xa1,xb1,xc1) | -E(xa,xb,xa1,xb1) |
-E(xb,xc,xb1,xc1) | E(xa,xc,xa1,xc1).  % Satz 2.11
xq = xa | -T(xq,xa,u) | -E(xa,u,xc,xd) | ext(xq,xa,xc,xd) = u.    % Satz 2.12
T(x,y,y).    % Satz 3.1
-E(u,v,x,x) | u=v.   % Not one of Szmielew's theorems but we proved it.
-T(xa,xb,xc) | T(xc,xb,xa).  % Satz 3.2.
T(xa,xa,xb).  % Satz 3.3
-T(xa,xb,xc) | -T(xb,xa,xc) | xa = xb.  % Satz 3.4.
-T(xa,xb,xd) | -T(xb,xc,xd) | T(xa,xb,xc). % Satz 3.51.
-T(xa,xb,xd) | -T(xb,xc,xd) | T(xa,xc,xd). % Satz 3.52.
-T(xa,xb,xc) | -T(xa,xc,xd) | T(xb,xc,xd). % Satz 3.61.
-T(xa,xb,xc) | -T(xa,xc,xd) | T(xa,xb,xd). % Satz 3.61.
-T(xa,xb,xc) | -T(xb,xc,xd) | xb = xc | T(xa,xc,xd).  % Satz 3.71
alpha != beta.   %related to Satz 3.14;  easily provable if added to
sst3h.in.
-T(xa,xb,xc) | -T(xa,xc,xd) | T(xa,xb,xd). % Satz 3.62.
  -T(xa,xb,xc) | -T(xb,xc,xd) | xb = xc | T(xa,xc,xd).  % Satz 3.71
  -T(xa,xb,xc) | -T(xb,xc,xd) | xb = xc | T(xa,xb,xd).  % Satz 3.72
-IFS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | E(xb,xd,xb1,xd1).  % Satz 4.2
-T(xa,xb,xc) | -T(xa1,xb1,xc1) | -E(xa,xc,xa1,xc1)
 | -E(xb,xc,xb1,xc1) | E(xa,xb,xa1,xb1).  % Satz 4.3

  alpha != beta.  % Satz 3.13
  beta != gamma.
  alpha != gamma.
T(xa,xb,ext(xa,xb,alpha,gamma)).   % Satz 3.14, first half
xb != ext(xa,xb,alpha,gamma).    % Satz 3.14, second half
% following is Satz 3.17
-T(xa,xb,xc) | -T(xa1,xb1,xc) | -T(xa,xp,xa1) | T(xp,f317(xa,xb,xc,xb1,xa1,xp),xc).
-T(xa,xb,xc) | -T(xa1,xb1,xc) | -T(xa,xp,xa1) | T(xb,f317(xa,xb,xc,xb1,xa1,xp),xb1).
  % The following many clauses are Definition 4.1
-IFS(xa,xb,xc,xd,za,zb,zc,zd) | T(xa,xb,xc).
-IFS(xa,xb,xc,xd,za,zb,zc,zd) | T(za,zb,zc).
-IFS(xa,xb,xc,xd,za,zb,zc,zd) | E(xa,xc,za,zc).
-IFS(xa,xb,xc,xd,za,zb,zc,zd) | E(xb,xc,zb,zc).
-IFS(xa,xb,xc,xd,za,zb,zc,zd) | E(xa,xd,za,zd).
-IFS(xa,xb,xc,xd,za,zb,zc,zd) | E(xc,xd,zc,zd).
 -T(xa,xb,xc) | -T(za,zb,zc) | -E(xa,xc,za,zc) | -E(xb,xc,zb,zc)
 | -E(xa,xd,za,zd) | -E(xc,xd,zc,zd) | IFS(xa,xb,xc,xd,za,zb,zc,zd).

% Following 4 are definition 4.4 for n=3
-E3(xa1,xa2,xa3,xb1,xb2,xb3) | E(xa1,xa2,xb1,xb2).
-E3(xa1,xa2,xa3,xb1,xb2,xb3) | E(xa1,xa3,xb1,xb3).
-E3(xa1,xa2,xa3,xb1,xb2,xb3) | E(xa2,xa3,xb2,xb3).
-E(xa1,xa2,xb1,xb2) | -E(xa1,xa3,xb1,xb3) | -E(xa2,xa3,xb2,xb3)
| E3(xa1,xa2,xa3,xb1,xb2,xb3).
% Following three lines are Satz 4.5
-T(xa,xb,xc) | -E(xa,xc,xa1,xc1) | T(xa1,insert(xa,xb,xa1,xc1),xc1).
-T(xa,xb,xc) | -E(xa,xc,xa1,xc1) |
E3(xa,xb,xc,xa1,insert(xa,xb,xa1,xc1),xc1).
insert(xa,xb,xa1,xc1) = ext(ext(xc1,xa1,alpha,gamma),xa1,xa,xb).
-E3(x,y,z,u,v,w) | E3(x,z,y,u,w,v).   % See sst4q.in, not in Szmielew
-T(xa,xb,xc) | -E3(xa,xb,xc,xa1,xb1,xc1) | T(xa1,xb1,xc1).  % Satz 4.6

% following is Definition 4.10
-Col(xa,xb,xc) | T(xa,xb,xc) | T(xb,xc,xa) | T(xc,xa,xb).
Col(xa,xb,xc) | -T(xa,xb,xc).
Col(xa,xb,xc) | -T(xb,xc,xa).
Col(xa,xb,xc) | -T(xc,xa,xb).
% Following are Satz 4.11
-Col(x,y,z) | Col(y,z,x).
-Col(x,y,z) | Col(z,x,y).
-Col(x,y,z) | Col(z,y,x).
-Col(x,y,z) | Col(y,x,z).
-Col(x,y,z) | Col(x,z,y).
% following is Satz 4.12
Col(x,x,y).
% following is Satz 4.13
-Col(xa,xb,xc) | - E3(xa,xb,xc,xa1,xb1,xc1) | Col(xa1,xb1,xc1).
% following is Satz 4.14
-Col(xa,xb,xc) | -E(xa,xb,xa1,xb1) |
E3(xa,xb,xc,xa1,xb1,insert5(xa,xb,xc,xa1,xb1)).
% following is Definition 4.15
-FS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | Col(xa,xb,xc).
-FS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | E3(xa,xb,xc,xa1,xb1,xc1).
-FS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | E(xa,xd,xa1,xd1).
-FS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | E(xb,xd,xb1,xd1).
-Col(xa,xb,xc) | - E3(xa,xb,xc,xa1,xb1,xc1) | - E(xa,xd,xa1,xd1) |
-E(xb,xd,xb1,xd1) | FS(xa,xb,xc,xd,xa1,xb1,xc1,xd1).
% Following is Satz 4.16
-FS(xa,xb,xc,xd,xa1,xb1,xc1,xd1) | xa = xb | E(xc,xd,xc1,xd1).
% Following is Satz 4.17
xa = xb | -Col(xa,xb,xc) | -E(xa,xp,xa,xq) | -E(xb,xp,xb,xq) |
E(xc,xp,xc,xq).
% Following is Satz 4.18
xa = xb | -Col(xa,xb,xc) | -E(xa,xc,xa,xc1) | -E(xb,xc,xb,xc1) | xc = xc1.
% Following is Satz 4.19
-T(xa,xc,xb) | -E(xa,xc,xa,xc1) | -E(xb,xc,xb,xc1) | xc = xc1.
% Following is Satz 5.1
xa = xb | -T(xa,xb,xc) | -T(xa,xb,xd) | T(xa,xc,xd) | T(xa,xd,xc).
% Two lemmas Narboux needed, proved in sst5c.in and sst5d.in
-T(xa,xb,xc) | -E(xa,xc,xa,xb) | xc = xb.
-T(xa,xd,xb) | -T(xa,xe,xb) | -E(xa,xd,xa,xe) | xd=xe.
% Following is Satz 5.2
xa = xb | -T(xa,xb,xc) | -T(xa,xb,xd)| T(xb,xc,xd) | T(xb,xd,xc).
% Following is Satz 5.3
-T(xa,xb,xd) | -T(xa,xc,xd) | T(xa,xb,xc) | T(xa,xc,xd).
% Following is Definition 5.4
-T(xc,y,xd) | -E(xa,xb,xc,y) | le(xa,xb,xc,xd).
-le(xa,xb,xc,xd) | T(xc,ins( xa,xb,xc,xd),xd). %ab inserted into cd
-le(xa,xb,xc,xd) | E(xa,xb,xc,ins(xa,xb,xc,xd)).
% Following is Satz 5.5a
-le(xa,xb,xc,xd) | T(xa,xb,ins(xc,xd,xa,xb)).
-le(xa,xb,xc,xd) | E(xa,ins(xc,xd,xa,xb),xc,xd).
% Following is Satz 5.5b
-T(xa,xb,x) | -E(xa,x,xc,xd) | le(xa,xb,xc,xd).
% Following is Satz 5.6
-le(xa,xb,xc,xd) | -E(xa,xb,xa1,xb1) | - E(xc,xd,xc1,xd1) |
le(xa1,xb1,xc1,xd1).
% Following is Satz 5.7
le(xa,xb,xa,xb).
% Following is Satz 5.8
-le(xa,xb,xc,xd) | - le(xc,xd,xe,xf) | le(xa,xb,xe,xf).
% Following is Satz 5.9
-le(xa,xb,xc,xd) | -le(xc,xd,xa,xb) | E(xa,xb,xc,xd).
% Following is Satz 5.10
le(xa,xb,xc,xd) | le(xc,xd,xa,xb).
% Following is Satz 5.11
le(xa,xa,xc,xd).
% Following is Satz 5.12a
-Col(xa,xb,xc) | -T(xa,xb,xc) | le(xa,xb,xa,xc).
-Col(xa,xb,xc) | -T(xa,xb,xc) | le(xb,xc,xa,xc).
% Following is Satz 5.12b
-Col(xa,xb,xc) | -le(xa,xb,xa,xc) | -le(xb,xc,xa,xc) | T(xa,xb,xc).
% B is strict betweenness,  "a and c are on opposite sides of b" in SST.
%-B(xa,xb,xc) | T(xa,xb,xc).
%-B(xa,xb,xc) | xa != xb.
%-B(xa,xb,xc) | xb != xc.
%xa = xb | xb = xc | -T(xa,xb,xc) | B(xa,xb,xc).

% Following is definition 6.1: a and b are on the same side of p
-sameside(xa,xp,xb) | xa != xp.
-sameside(xa,xp,xb) | xb != xp.
-sameside(xa,xp,xb) | T(xp,xa,xb) | T(xp,xb,xa).
-T(xp,xa,xb) | xb=xp | xp=xa | sameside(xa,xp,xb).
-T(xp,xb,xa) | xb=xp | xp=xa | sameside(xa,xp,xb).

% following is Satz 6.2
xa = xp | xb = xp | xc = xp | -T(xa,xp,xc) | -T(xb,xp,xc) |
sameside(xa,xp,xb).
xa = xp | xb = xp | xc = xp | -T(xa,xp,xc) | T(xb,xp,xc) |
-sameside(xa,xp,xb).
% Following is Satz 6.3
-sameside(xa,xp,xb) | xa != xp.
-sameside(xa,xp,xb) | xb != xp.
-sameside(xa,xp,xb) | c63(xa,xp,xb) != xp.
-sameside(xa,xp,xb) | T(xa,xp,c63(xa,xp,xb)).
-sameside(xa,xp,xb) | T(xb,xp,c63(xa,xp,xb)).
% Following is Satz 6.4
-sameside(xa,xp,xb) | Col(xa,xp,xb).
-sameside(xa,xp,xb) | -T(xa,xp,xb).
-Col(xa,xp,xb) | T(xa,xp,xb) | sameside(xa,xp,xb).
%Following is Satz 6.5
xa = xp | sameside(xa,xp,xa).
% Following is Satz 6.6
-sameside(xa,xp,xb) | sameside(xb,xp,xa).
% Following is Satz 6.7
-sameside(xa,xp,xb) | -sameside(xb,xp,xc) | sameside(xa,xp,xc).
% Following is Satz 6.11
xr = xa | xb = xc | sameside(insert(xb,xc,xa,xr),xa,xr).
xr = xa | xb = xc | E(xa,insert(xb,xc,xa,xr),xb,xc).
xr = xa | xb = xc | -sameside(x,xa,xr) | -sameside(y,xa,xr) |
-E(xa,x,xb,xc) | -E(xa,y,xb,xc) | x=y.
% Following is Satz 6.13
-sameside(xa,xp,xb) | -le(xp,xa,xp,xb) | T(xp,xa,xb).
-sameside(xa,xp,xb) | -T(xp,xa,xb) | le(xp,xa,xp,xb).
% Following is Satz 6.15
xp = xq | xp = xr | -T(xq,xp,xr) | -Col(xa,xp,xq) | xa = xp |
sameside(xa,xp,xq) | sameside(xa,xp,xr).
xp = xq | xp = xr | -T(xq,xp,xr) | -sameside(xa,xp,xq) | Col(xa,xp,xq).
xp = xq | xp = xr | -T(xq,xp,xr) | -sameside(xa,xp,xr) | Col(xa,xp,xq).
xp = xq | xp = xr | -T(xq,xp,xr) | xa != xp | Col(xa,xp,xq).

% Following is Definition 7.1, m is the midpoint of ab
-M(xa,xm,xb) | T(xa,xm,xb).
-M(xa,xm,xb) | E(xm,xa,xm,xb).
-T(xa,xm,xb) | -E(xm,xa,xm,xb) | M(xa,xm,xb).

-M(xa,xm,xb) | M(xb,xm,xa).   % Satz 7.2
M(xa,xa,xa).    % Satz 7.3a
-M(xa,xm,xa) | xm = xa.   % Satz 7.3b
-M(x,xa,z) | -M(x,xa,y) | z=y.  %Satz 7.4a
M(x,xa,s(xa,x)).     % Satz 7.4a,  and definition 7.5
-M(x,y,z) | z = s(y,x).   % Satz 7.6
s(x,s(x,y)) = y.    % Satz 7.7
s(xa,xp) != xr | s(xa,xq) != xr | xp = xq.   %Satz 7.8
s(xa,xp) != s(xa,xq) | xp = xq.    % Satz 7.9
s(xa,xp) != xp | xp = xa.   %Satz 7.10a
xp != xa | s(xa,xp)=xp.     % Satz 7.10b
E(xp,xq,s(xa,xp),s(xa,xq)).   % Satz 7.13
-T(xp,xq,xr) | T(s(xa,xp),s(xa,xq),s(xa,xr)).   % Satz 7.15a
T(xp,xq,xr) | -T(s(xa,xp),s(xa,xq),s(xa,xr)).   % Satz 7.15b
-E(xp,xq,xr,xs) | E(s(xa,xp),s(xa,xq),s(xa,xr),s(xa,xs)).  % Satz 7.16a
E(xp,xq,xr,xs) | -E(s(xa,xp),s(xa,xq),s(xa,xr),s(xa,xs)).  % Satz 7.16b
-M(xp,xa,xq) | -M(xp,xb,xq) | xa = xb.   % Satz 7.17
s(xa,xp) != s(xb,xp) | xa = xb.    % Satz 7.18
s(xa,s(xb,xp)) != s(xb,s(xa,xp)) | xa = xb.  % Satz 7.19a
s(xa,s(xb,xp)) = s(xb,s(xa,xp)) | xa != xb.  % Satz 7.19b
-Col(xa,xm,xb) | -E(xm,xa,xm,xb) | xa = xb | M(xa,xm,xb).  % Satz 7.20
% Following 2 lines are Lemma 7.21
Col(xa,xb,xc) | xb = xd | -E(xa,xb,xc,xd) | -E(xb,xc,xd,xa) |
-Col(xa,xp,xc) | -Col(xb,xp,xd) | M(xa,xp,xc).
Col(xa,xb,xc) | xb = xd | -E(xa,xb,xc,xd) | -E(xb,xc,xd,xa) |
-Col(xa,xp,xc) | -Col(xb,xp,xd) | M(xb,xp,xd).
% following is 7.22,  Krippenlemma
-T(xa1,xc,xa2) | -T(xb1,xc,xb2) | -E(xc,xa1,xc,xb1) | -E(xc,xa2,xc,xb2) |
-M(xa1,xm1,xb1) | -M(xa2,xm2,xb2) | T(xm1,xc,xm2).
%following defines a Krippenfigur (definition 7.23)
-T(xa1,xc,xa2) | -T(xb1,xc,xb2) | -E(xc,xa1,xc,xb1) | -E(xc,xa2,xc,xb2) |
-M(xa1,xm1,xb1) | -M(xa2,xm2,xb2) | KF(xa1,xm1,xb1,xc,xb2,xm2,xa2).
-KF(xa1,xm1,xb1,xc,xb2,xm2,xa2) | T(xa1,xc,xa2).
-KF(xa1,xm1,xb1,xc,xb2,xm2,xa2) |  T(xb1,xc,xb2).
-KF(xa1,xm1,xb1,xc,xb2,xm2,xa2) | E(xc,xa1,xc,xb1).
-KF(xa1,xm1,xb1,xc,xb2,xm2,xa2) | E(xc,xa2,xc,xb2).
-KF(xa1,xm1,xb1,xc,xb2,xm2,xa2) | M(xa1,xm1,xb1).
-KF(xa1,xm1,xb1,xc,xb2,xm2,xa2) |  M(xa2,xm2,xb2).
% following is Lemma 7.25
-E(xc,xa,xc,xb) | M(xa,midpoint(xa,xb),xb).
% following is Definition 8.1
-R(xa,xb,xc) | E(xa,xc,xa,s(xb,xc)).
R(xa,xb,xc) | -E(xa,xc,xa,s(xb,xc)).
% following is Satz 8.2
-R(xa,xb,xc) | R(xc,xb,xa).
% following is Satz 8.3
-R(xa,xb,xc) | xa = xb | -Col(xb,xa,xa1) | R(xa1,xb,xc).
% Following is Satz 8.4
-R(xa,xb,xc) | R(xa,xb,s(xb,xc)).
% following is Satz 8.5
R(x,y,y).
% following is Satz 8.6
-R(xa,xb,xc) | -R(xa1,xb,xc) | -T(xa,xc,xa1) | xb = xc.
% following is Satz 8.7
-R(xa,xb,xc) | -R(xa,xc,xb) | xb = xc.
% Following is Satz 8.8
-R(xa,xb,xa) | xa = xb.
% following is Satz 8.9
-R(xa,xb,xc) | -Col(xa,xb,xc) | xa=xb | xc=xb.
% following is Satz 8.10
-R(xa,xb,xc) | -E3(xa,xb,xc,xa1,xb1,xc1) | R(xa1,xb1,xc1).
% following is Definition 8.11 (i)
-perpAt(y,z,x,y1,z1) | Col(y,z,x).
-perpAt(y,z,x,y1,z1) | Col(y1,z1,x).
-perpAt(y,z,x,y1,z1) |y != z.
-perpAt(y,z,x,y1,z1) |y1 != z1.
-perpAt(y,z,x,y1,z1) | -Col(y,z,u) | -Col(y1,z1,v) | R(u,x,v).
perpAt(y,z,x,y1,z1) | y = z | y1 = z1 | -Col(y,z,x) | -Col(y1,z1,x) |  Col(y,z,f811(y,z,y1,z1,x)).
perpAt(y,z,x,y1,z1) | y = z | y1 = z1 | -Col(y,z,x) | -Col(y1,z1,x) |  Col(y1,z1,g811(y,z,y1,z1,x)).
perpAt(y,z,x,y1,z1) | y = z | y1 = z1 | -Col(y,z,x) | -Col(y1,z1,x) |  -R(f811(y,z,y1,z1,x),x,g811(y,z,y1,z1,x)).
% following is Definition 8.11 (ii)
-perp(xp,xq,xp1,xq1) | perpAt(xp,xq,il(xp,xq,xp1,xq1),xp1,xq1).
perp(xp,xq,xp1,xq1) | -perpAt(xp,xq,x,xp1,xq1).
% Definition 8.11 (iii) disappears when we reduce to first-order
expressions.
% following is Satz 8.12a
-perpAt(x,y,z,u,v) | perpAt(u,v,z,x,y).
% following is Satz 8.12b
-perp(x,y,u,v) | perp(u,v,x,y).
% following is Satz 8.13a
-perpAt(xa,xb,x,xp,xq) | xa != xb.
-perpAt(xa,xb,x,xp,xq) | xp != xq.
-perpAt(xa,xb,x,xp,xq) | Col(xa,xb,x).
-perpAt(xa,xb,x,xp,xq) | Col(xp,xq,x).
-perpAt(xa,xb,x,xp,xq) | R(xa,x,xp).   % these four lines give the instantiations found in proving Satz 8.13a, so we don't introduce a new skolem symbol here.
-perpAt(xa,xb,x,xp,xq) | R(xb,x,xp).
-perpAt(xa,xb,x,xp,xq) | R(xa,x,xq).
-perpAt(xa,xb,x,xp,xq) | R(xb,x,xq).
% following is Satz 8.13b
-Col(xa,xb,x) | -Col(xp,xq,x) | xa = xb | xp = xq | -Col(xa,xb,u) |
-Col(xp,xq,v) | u = x |  v = x | -R(u,x,v) | perpAt(xa,xb,x,xp,xq).
% following is Satz 8.14a
-perp(xa,xb,xp,xq) | -Col(xa,xb,xp) | -Col(xa,xb,xq).
% following is Satz 8.14b
-perpAt(xa,xb,xc,xp,xq) | Col(xa,xb,xc).
-perpAt(xa,xb,xc,xp,xq) | Col(xp,xq,xc).
% following is Satz 8.14c
-perpAt(xa,xb,xc,xp,xq) | xc = il(xa,xb,xp,xq).
%following is Satz 8.15a
xa=xb | -Col(xa,xb,x) | -perp(xa,xb,xc,x) | perpAt(xa,xb,x,xc,x).
% following is Satz 8.15b
xa=xb | -Col(xa,xb,x) | perp(xa,xb,xc,x) | -perpAt(xa,xb,x,xc,x).
% following two are Satz 8.16a
xa = xb | -Col(xa,xb,xp) | -Col(xa,xb,xq) | xq = xp 
| -perp(xa,xb,xc,xp) | -Col(xa,xb,xc).
xa = xb | -Col(xa,xb,xp) | -Col(xa,xb,xq) | xq = xp 
| -perp(xa,xb,xc,xp) |  R(c,x,q).
% following is Satz8.16b
 xa = xb | -Col(xa,xb,xp) | -Col(xa,xb,xq) | xq = xp 
| perp(xa,xb,xc,xp) | Col(xa,xb,xc) | -R(c,x,q).
% following is Satz 8.18
Col(xa,xb,xc) | Col(xa,xb,foot(xa,xb,xc)).
Col(xa,xb,xc) | perp(xa,xb,xc,foot(xa,xb,xc)).
% following is Satz 8.18a
Col(xa,xb,xc) | -Col(xa,xb,z) | -Col(xa,xb,y) |  -perp(xa,xb,xc,z) | -perp(xa,xb,xc,y) | y=z.

% following is Lemma 8.20a
-R(xa,xb,xc) | -M(s(xa,xc),xp,s(xb,xc)) | R(xb,xa,xp).
% following is Lemma 8.20b
-R(xa,xb,xc) | -M(s(xa,xc),xp,s(xb,xc)) | -R(xb,xa,xp) | xb = xc | xa != xp. 
 

%Following are not among Szmielew's theorems because she used set theory for lines.  
% These are proved separately in our development. 
 
-perp(x,y,u,v) | perp(y,x,u,v).   % Proved in perp1.in 
-Col(x,y,u) | x=y | -Col(x,y,v) | Col(x,u,v).  % proved in Ext1.in 
end_of_list.

list(passive).
-R(a,cx,c) | $ANS(1).
-R(c,cx,a) | $ANS(1a).
-E(a,c1,a,c) | $ANS(2).
-E(a,c,a,a1) | $ANS(3).
-E(a,c1,a,a1) | $ANS(4).
-M(c1,p,a1) | $ANS(5).
-R(cx,a,p) | $ANS(6).
a = p | $ANS(7).
-T(c,t,p) | $ANS(8).
-T(cx,t,a) | $ANS(9).
-Col(a,b,t) | $ANS(10).
-M(c,a,a1) | $ANS(11).
% following get case 1
s(a,s(b,x))=s(b,s(a,x)) | $ANS(12).
-sameside(a,b,a) | $ANS(13).
M(a,a,b) | $ANS(14).
-perp(a,b,c,cx) | $ANS(15).
-Col(a,b,cx) | $ANS(16).
-M(c,cx,c1) | $ANS(17).
-Col(a,b,a) | $ANS(18).
M(b,a,a) | $ANS(19).
b=a | $ANS(20).
perp(a,b,cx,cx) | $ANS(21).
-M(c1,cx,c) | $ANS(22).
-T(c,cx,c1) | $ANS(23).
-Col(b,a,a) | $ANS(24).
-T(c1,cx,c) | $ANS(25).
-Col(a,b,b) | $ANS(26).
-Col(c,c1,cx) | $ANS(27).
-perpAt(a,b,cx,c,cx) | $ANS(28).
c1=c | $ANS(29).
c=cx | $ANS(30).
M(c1,cx,c1) | $ANS(31).
T(c1,cx,c1) | $ANS(32).
T(c,c1,cx) | $ANS(33).
-sameside(c,c1,cx) | $ANS(34).
cx=c1 | $ANS(35).
c=c1 | $ANS(36).
-sameside(c,c1,c) | $ANS(37).
perp(a,b,c,c1) | $ANS(38).
-Col(c,c1,c) | $ANS(39).
perpAt(a,b,x,c,c1) | $ANS(40).
-R(b,cx,c) | $ANS(41).
-R(a,cx,c) | $ANS(42).
a!=cx | $ANS(43).
b=cx | $ANS(44).
 
end_of_list.

list(sos).  % Satz 8.21
a != b.
-perp(a,b,xp,a) | -Col(a,b,xt) | -T(c,xt,xp).
-Col(a,b,c).    % Case 1

cx = foot(a,b,c).
c1 = s(cx,c).
a1 = s(a,c).
p = midpoint(c1,a1).
t = f317(a1,a,c,cx,c1,p).

end_of_list.

list(hints2).


 
 

% proof of Case 1
s(a,s(b,x))!=s(b,s(a,x)).
sameside(a,b,a).
-M(a,a,b).
-Col(a,c,b).
-Col(c,b,a).
perp(a,b,c,cx).
Col(a,b,cx).
-R(x,cx,c)|E(x,c,x,c1).
M(c,cx,c1).
M(c,a,a1).
-E(x,c1,x,a1)|M(c1,p,a1).
b!=a.
Col(a,b,a).
-M(b,a,a).
perp(b,a,c,cx).
perpAt(a,b,cx,c,cx).
Col(cx,a,b).
M(c1,cx,c).
T(c,cx,c1).
s(cx,c)=c1.
E(a,c,a,a1).
T(c,a,a1).
s(a,c)=a1.
c!=b.
Col(b,a,a).
perp(c,cx,b,a).
R(a,cx,c).
c!=cx.
T(c1,cx,c).
-perp(a,b,c1,a).
Col(c,cx,c1).
-T(cx,c,c1).
c1!=c.
T(a1,a,c).
a!=c.
Col(a,b,b).
E(a,c,a,c1).
R(c,cx,a).
Col(cx,c,c1).
Col(c,c1,cx).
-perpAt(a,b,x,c1,a).
sameside(cx,c,c1).
-M(c1,cx,c1).
Col(b,a,b).
R(b,cx,c1).
E(a,c1,a,a1).
cx!=c.
-T(c1,cx,c1).
M(c1,p,a1).
-T(c,c1,cx).
M(a1,p,c1).
sameside(c,c1,cx).
T(a1,p,c1).
M(a1,p,s(cx,c)).
cx!=c1.
T(a,f317(a1,a,c,cx,c1,p),cx).
T(p,f317(a1,a,c,cx,c1,p),c).
M(s(a,c),p,s(cx,c)).
-R(c1,cx,c1).
T(a,t,cx).
T(p,t,c).
R(cx,a,p).
-Col(a,b,c1).
Col(t,cx,a).
Col(cx,a,t).
T(cx,t,a).
Col(t,c,p).
T(c,t,p).
R(p,a,cx).
sameside(a,p,a).
a!=p.
-Col(a,c1,b).
-Col(c1,b,a).
-Col(c1,a,b).
-T(t,cx,a)|cx=t.
Col(a,p,a).
s(a,s(p,x))!=s(p,s(a,x)).
c1!=b.
c1!=a.
Col(p,a,a).
p!=a.
a!=c1.
sameside(c1,a,c1).
-M(c1,c1,a).
sameside(p,a,p).
Col(c1,a,c1).
-M(a,c1,c1).
Col(p,a,p).
Col(a,c1,c1).
Col(c1,a,a).
-R(b,a,c1).
-R(c1,a,b).
-perpAt(c,cx,a,b,a).
-Col(c,cx,a).
-Col(cx,a,c).
cx!=a.
Col(cx,b,t).
perpAt(a,b,a,p,a).
-R(p,cx,a).
Col(t,cx,b).
perp(a,b,p,a).
-Col(cx,c,p).
-Col(a,b,t).
-T(t,cx,a).
sameside(t,cx,a).
t!=cx.
Col(t,b,a).
Col(a,b,t).
end_of_list.

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