----- Otter 3.2, August 2001 -----
The process was started by ??? on ???,
Sun Nov 30 15:29:01 2003
The command was "otter".

set(auto).
   dependent: set(auto1).
   dependent: set(process_input).
   dependent: clear(print_kept).
   dependent: clear(print_new_demod).
   dependent: clear(print_back_demod).
   dependent: clear(print_back_sub).
   dependent: set(control_memory).
   dependent: assign(max_mem, 12000).
   dependent: assign(pick_given_ratio, 4).
   dependent: assign(stats_level, 1).
   dependent: assign(max_seconds, 10800).
clear(print_given).
assign(max_distinct_vars,3).
include("iDef.txt").
------- start included file iDef.txt-------

formula_list(usable).
all x y (i(x,y)<->u(x,y)=y).
end_of_list.

-------> usable clausifies to:

list(usable).
0 [] -i(x,y)|u(x,y)=y.
0 [] i(x,y)|u(x,y)!=y.
end_of_list.
------- end included file iDef.txt-------
include("uzDef.txt").
------- start included file uzDef.txt-------

formula_list(usable).
D=c(I).
U=u(I,D).
Z=c(U).
end_of_list.

-------> usable clausifies to:

list(usable).
0 [] D=c(I).
0 [] U=u(I,D).
0 [] Z=c(U).
end_of_list.
------- end included file uzDef.txt-------
include("booleanAx.txt").
------- start included file booleanAx.txt-------
include("1_2booleanAx.txt").
------- start included file 1_2booleanAx.txt-------

formula_list(usable).
all x y (u(x,y)=u(y,x)).
all x y z (u(x,u(y,z))=u(u(x,y),z)).
end_of_list.

-------> usable clausifies to:

list(usable).
0 [] u(x,y)=u(y,x).
0 [] u(x,u(y,z))=u(u(x,y),z).
end_of_list.
------- end included file 1_2booleanAx.txt-------
include("3booleanAx.txt").
------- start included file 3booleanAx.txt-------

formula_list(usable).
all x y (u(c(u(c(x),y)),c(u(c(x),c(y))))=x).
end_of_list.

-------> usable clausifies to:

list(usable).
0 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x.
end_of_list.
------- end included file 3booleanAx.txt-------
------- end included file booleanAx.txt-------
include("15_16booleanLaws.txt").
------- start included file 15_16booleanLaws.txt-------

formula_list(usable).
all x (c(c(x))=x).
all x y (u(x,y)=y->c(u(y,c(x)))=Z).
all x y (c(u(y,c(x)))=Z->u(x,y)=y).
end_of_list.

-------> usable clausifies to:

list(usable).
0 [] c(c(x))=x.
0 [] u(x,y)!=y|c(u(y,c(x)))=Z.
0 [] c(u(y,c(x)))!=Z|u(x,y)=y.
end_of_list.
------- end included file 15_16booleanLaws.txt-------
include("i.txt").
------- start included file i.txt-------

formula_list(usable).
all x y (i(x,y)->i(r(x),r(y))).
all x y (r(u(x,y))=u(r(x),r(y))).
all x (r(c(x))=c(r(x))).
all x y (r(n(x,y))=n(r(x),r(y))).
all x (r(r(x))=x).
all x y (r(d(x,y))=d(r(y),r(x))).
all x y (r(k(x,y))=k(r(y),r(x))).
r(I)=I.
r(Z)=Z.
r(U)=U.
end_of_list.

-------> usable clausifies to:

list(usable).
0 [] -i(x,y)|i(r(x),r(y)).
0 [] r(u(x,y))=u(r(x),r(y)).
0 [] r(c(x))=c(r(x)).
0 [] r(n(x,y))=n(r(x),r(y)).
0 [] r(r(x))=x.
0 [] r(d(x,y))=d(r(y),r(x)).
0 [] r(k(x,y))=k(r(y),r(x)).
0 [] r(I)=I.
0 [] r(Z)=Z.
0 [] r(U)=U.
end_of_list.
------- end included file i.txt-------
include("xii.txt").
------- start included file xii.txt-------

formula_list(usable).
all x y z (i(k(x,y),c(z))<->i(k(r(x),z),c(y))).
all x y z (i(k(x,y),c(z))<->i(k(z,r(y)),c(x))).
end_of_list.

-------> usable clausifies to:

list(usable).
0 [] -i(k(x,y),c(z))|i(k(r(x),z),c(y)).
0 [] i(k(x,y),c(z))| -i(k(r(x),z),c(y)).
0 [] -i(k(x,y),c(z))|i(k(z,r(y)),c(x)).
0 [] i(k(x,y),c(z))| -i(k(z,r(y)),c(x)).
end_of_list.
------- end included file xii.txt-------
include("xix_a_ghost.txt").
------- start included file xix_a_ghost.txt-------

formula_list(usable).
all x y (k(c(x),c(r(y)))=Z<->i(k(U,c(x)),y)).
all x y (c(k(c(x),c(r(y))))=U<->i(k(U,c(x)),y)).
-(all x y (i(x,y)<->u(c(x),y)=U)).
end_of_list.

-------> usable clausifies to:

list(usable).
0 [] k(c(x),c(r(y)))!=Z|i(k(U,c(x)),y).
0 [] k(c(x),c(r(y)))=Z| -i(k(U,c(x)),y).
0 [] c(k(c(x),c(r(y))))!=U|i(k(U,c(x)),y).
0 [] c(k(c(x),c(r(y))))=U| -i(k(U,c(x)),y).
0 [] i($c2,$c1)|u(c($c2),$c1)=U.
0 [] -i($c2,$c1)|u(c($c2),$c1)!=U.
end_of_list.
------- end included file xix_a_ghost.txt-------

SCAN INPUT: prop=0, horn=0, equality=1, symmetry=0, max_lits=2.

This ia a non-Horn set with equality.  The strategy will be
Knuth-Bendix, ordered hyper_res, factoring, and unit
deletion, with positive clauses in sos and nonpositive
clauses in usable.

   dependent: set(knuth_bendix).
   dependent: set(para_from).
   dependent: set(para_into).
   dependent: clear(para_from_right).
   dependent: clear(para_into_right).
   dependent: set(para_from_vars).
   dependent: set(eq_units_both_ways).
   dependent: set(dynamic_demod_all).
   dependent: set(dynamic_demod).
   dependent: set(order_eq).
   dependent: set(back_demod).
   dependent: set(lrpo).
   dependent: set(hyper_res).
   dependent: set(unit_deletion).
   dependent: set(factor).

------------> process usable:
** KEPT (pick-wt=8): 1 [] -i(x,y)|u(x,y)=y.
** KEPT (pick-wt=8): 2 [] i(x,y)|u(x,y)!=y.
** KEPT (pick-wt=12): 3 [] u(x,y)!=y|c(u(y,c(x)))=Z.
** KEPT (pick-wt=12): 4 [] c(u(x,c(y)))!=Z|u(y,x)=x.
** KEPT (pick-wt=8): 5 [] -i(x,y)|i(r(x),r(y)).
** KEPT (pick-wt=13): 6 [] -i(k(x,y),c(z))|i(k(r(x),z),c(y)).
** KEPT (pick-wt=13): 7 [] i(k(x,y),c(z))| -i(k(r(x),z),c(y)).
** KEPT (pick-wt=13): 8 [] -i(k(x,y),c(z))|i(k(z,r(y)),c(x)).
** KEPT (pick-wt=13): 9 [] i(k(x,y),c(z))| -i(k(z,r(y)),c(x)).
** KEPT (pick-wt=14): 10 [] k(c(x),c(r(y)))!=Z|i(k(U,c(x)),y).
** KEPT (pick-wt=14): 11 [] k(c(x),c(r(y)))=Z| -i(k(U,c(x)),y).
** KEPT (pick-wt=15): 12 [] c(k(c(x),c(r(y))))!=U|i(k(U,c(x)),y).
** KEPT (pick-wt=15): 13 [] c(k(c(x),c(r(y))))=U| -i(k(U,c(x)),y).
** KEPT (pick-wt=9): 14 [] -i($c2,$c1)|u(c($c2),$c1)!=U.

------------> process sos:
** KEPT (pick-wt=4): 16 [copy,15,flip.1] c(I)=D.
---> New Demodulator: 17 [new_demod,16] c(I)=D.
** KEPT (pick-wt=5): 19 [copy,18,flip.1] u(I,D)=U.
---> New Demodulator: 20 [new_demod,19] u(I,D)=U.
** KEPT (pick-wt=4): 22 [copy,21,flip.1] c(U)=Z.
---> New Demodulator: 23 [new_demod,22] c(U)=Z.
** KEPT (pick-wt=7): 24 [] u(x,y)=u(y,x).
** KEPT (pick-wt=11): 26 [copy,25,flip.1] u(u(x,y),z)=u(x,u(y,z)).
---> New Demodulator: 27 [new_demod,26] u(u(x,y),z)=u(x,u(y,z)).
** KEPT (pick-wt=14): 28 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x.
---> New Demodulator: 29 [new_demod,28] u(c(u(c(x),y)),c(u(c(x),c(y))))=x.
** KEPT (pick-wt=5): 30 [] c(c(x))=x.
---> New Demodulator: 31 [new_demod,30] c(c(x))=x.
** KEPT (pick-wt=10): 32 [] r(u(x,y))=u(r(x),r(y)).
---> New Demodulator: 33 [new_demod,32] r(u(x,y))=u(r(x),r(y)).
** KEPT (pick-wt=7): 34 [] r(c(x))=c(r(x)).
---> New Demodulator: 35 [new_demod,34] r(c(x))=c(r(x)).
** KEPT (pick-wt=10): 36 [] r(n(x,y))=n(r(x),r(y)).
---> New Demodulator: 37 [new_demod,36] r(n(x,y))=n(r(x),r(y)).
** KEPT (pick-wt=5): 38 [] r(r(x))=x.
---> New Demodulator: 39 [new_demod,38] r(r(x))=x.
** KEPT (pick-wt=10): 40 [] r(d(x,y))=d(r(y),r(x)).
---> New Demodulator: 41 [new_demod,40] r(d(x,y))=d(r(y),r(x)).
** KEPT (pick-wt=10): 42 [] r(k(x,y))=k(r(y),r(x)).
---> New Demodulator: 43 [new_demod,42] r(k(x,y))=k(r(y),r(x)).
** KEPT (pick-wt=4): 44 [] r(I)=I.
---> New Demodulator: 45 [new_demod,44] r(I)=I.
** KEPT (pick-wt=4): 46 [] r(Z)=Z.
---> New Demodulator: 47 [new_demod,46] r(Z)=Z.
** KEPT (pick-wt=4): 48 [] r(U)=U.
---> New Demodulator: 49 [new_demod,48] r(U)=U.
** KEPT (pick-wt=9): 50 [] i($c2,$c1)|u(c($c2),$c1)=U.
>>>> Starting back demodulation with 17.
>>>> Starting back demodulation with 20.
>>>> Starting back demodulation with 23.
  Following clause subsumed by 24 during input processing: 0 [copy,24,flip.1] u(x,y)=u(y,x).
>>>> Starting back demodulation with 27.
>>>> Starting back demodulation with 29.
>>>> Starting back demodulation with 31.
>>>> Starting back demodulation with 33.
>>>> Starting back demodulation with 35.
>>>> Starting back demodulation with 37.
>>>> Starting back demodulation with 39.
>>>> Starting back demodulation with 41.
>>>> Starting back demodulation with 43.
>>>> Starting back demodulation with 45.
>>>> Starting back demodulation with 47.
>>>> Starting back demodulation with 49.

======= end of input processing =======

=========== start of search ===========

-----> EMPTY CLAUSE at   0.08 sec ----> 966 [back_demod,116,demod,964,unit_del,857,127] $F.

Length of proof is 10.  Level of proof is 5.

---------------- PROOF ----------------

1 [] -i(x,y)|u(x,y)=y.
2 [] i(x,y)|u(x,y)!=y.
3 [] u(x,y)!=y|c(u(y,c(x)))=Z.
4 [] c(u(x,c(y)))!=Z|u(y,x)=x.
14 [] -i($c2,$c1)|u(c($c2),$c1)!=U.
21 [] Z=c(U).
23,22 [copy,21,flip.1] c(U)=Z.
24 [] u(x,y)=u(y,x).
30 [] c(c(x))=x.
50 [] i($c2,$c1)|u(c($c2),$c1)=U.
116 [para_from,24.1.1,14.2.1] -i($c2,$c1)|u($c1,c($c2))!=U.
123,122 [para_into,30.1.1.1,22.1.1] c(Z)=U.
127 [para_into,30.1.1,30.1.1] x=x.
342 [para_into,50.2.1,24.1.1] i($c2,$c1)|u($c1,c($c2))=U.
847 [para_from,342.2.1,4.1.1.1,demod,23,unit_del,127] u($c2,$c1)=$c1|i($c2,$c1).
853 [hyper,847,1,factor_simp] u($c2,$c1)=$c1.
855 [hyper,853,3] c(u($c1,c($c2)))=Z.
857 [hyper,853,2] i($c2,$c1).
964,963 [para_from,855.1.1,30.1.1.1,demod,123,flip.1] u($c1,c($c2))=U.
966 [back_demod,116,demod,964,unit_del,857,127] $F.

------------ end of proof -------------


Search stopped by max_proofs option.

============ end of search ============

-------------- statistics -------------
clauses given                109
clauses generated           1537
clauses kept                 912
clauses forward subsumed     554
clauses back subsumed         50
Kbytes malloced              606

----------- times (seconds) -----------
user CPU time          0.28          (0 hr, 0 min, 0 sec)
system CPU time        0.00          (0 hr, 0 min, 0 sec)
wall-clock time        0             (0 hr, 0 min, 0 sec)
hyper_res time         0.00
para_into time         0.00
para_from time         0.00
for_sub time           0.00
back_sub time          0.00
conflict time          0.00
demod time             0.00

That finishes the proof of the theorem.

Process 0 finished Sun Nov 30 15:29:01 2003