----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Sun Nov 30 17:43:22 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). assign(max_literals,1). assign(max_mem,64000). assign(max_weight,19). formula_list(usable). all x (x=x). end_of_list. -------> usable clausifies to: list(usable). 0 [] x=x. end_of_list. 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("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("nDef.txt"). ------- start included file nDef.txt------- formula_list(usable). all x y (n(x,y)=c(u(c(x),c(y)))). end_of_list. -------> usable clausifies to: list(usable). 0 [] n(x,y)=c(u(c(x),c(y))). end_of_list. ------- end included file nDef.txt------- include("dDef.txt"). ------- start included file dDef.txt------- formula_list(usable). all x y (d(x,y)=c(k(c(x),c(y)))). end_of_list. -------> usable clausifies to: list(usable). 0 [] d(x,y)=c(k(c(x),c(y))). end_of_list. ------- end included file dDef.txt------- include("ii.txt"). ------- start included file ii.txt------- formula_list(usable). k(U,U)=U. r(U)=U&r(Z)=Z&r(I)=I&r(D)=D. end_of_list. -------> usable clausifies to: list(usable). 0 [] k(U,U)=U. 0 [] r(U)=U. 0 [] r(Z)=Z. 0 [] r(I)=I. 0 [] r(D)=D. end_of_list. ------- end included file ii.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("xxix_ghost.txt"). ------- start included file xxix_ghost.txt------- formula_list(usable). -(all x i(k(x,c(I)),c(n(x,c(k(x,c(I))))))). end_of_list. -------> usable clausifies to: list(usable). 0 [] -i(k($c1,c(I)),c(n($c1,c(k($c1,c(I)))))). end_of_list. ------- end included file xxix_ghost.txt------- SCAN INPUT: prop=0, horn=1, equality=1, symmetry=0, max_lits=2. This is a Horn set with equality. The strategy will be Knuth-Bendix and hyper_res, with positive clauses in sos and nonpositive clauses in usable. dependent: set(knuth_bendix). dependent: set(para_from). dependent: set(para_into). dependent: clear(para_from_right). dependent: clear(para_into_right). dependent: set(para_from_vars). dependent: set(eq_units_both_ways). dependent: set(dynamic_demod_all). dependent: set(dynamic_demod). dependent: set(order_eq). dependent: set(back_demod). dependent: set(lrpo). dependent: set(hyper_res). dependent: clear(order_hyper). ------------> process usable: ** KEPT (pick-wt=8): 1 [] -i(x,y)|u(x,y)=y. ** KEPT (pick-wt=8): 2 [] i(x,y)|u(x,y)!=y. ** KEPT (pick-wt=13): 3 [] -i(k(x,y),c(z))|i(k(r(x),z),c(y)). ** KEPT (pick-wt=13): 4 [] i(k(x,y),c(z))| -i(k(r(x),z),c(y)). ** KEPT (pick-wt=13): 5 [] -i(k(x,y),c(z))|i(k(z,r(y)),c(x)). ** KEPT (pick-wt=13): 6 [] i(k(x,y),c(z))| -i(k(z,r(y)),c(x)). ** KEPT (pick-wt=13): 7 [] -i(k($c1,c(I)),c(n($c1,c(k($c1,c(I)))))). ------------> process sos: ** KEPT (pick-wt=3): 8 [] x=x. ** KEPT (pick-wt=7): 9 [] u(x,y)=u(y,x). ** KEPT (pick-wt=11): 11 [copy,10,flip.1] u(u(x,y),z)=u(x,u(y,z)). ---> New Demodulator: 12 [new_demod,11] u(u(x,y),z)=u(x,u(y,z)). ** KEPT (pick-wt=14): 13 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ---> New Demodulator: 14 [new_demod,13] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ** KEPT (pick-wt=10): 16 [copy,15,flip.1] c(u(c(x),c(y)))=n(x,y). ---> New Demodulator: 17 [new_demod,16] c(u(c(x),c(y)))=n(x,y). ** KEPT (pick-wt=10): 19 [copy,18,flip.1] c(k(c(x),c(y)))=d(x,y). ---> New Demodulator: 20 [new_demod,19] c(k(c(x),c(y)))=d(x,y). ** KEPT (pick-wt=5): 21 [] k(U,U)=U. ---> New Demodulator: 22 [new_demod,21] k(U,U)=U. ** KEPT (pick-wt=4): 23 [] r(U)=U. ---> New Demodulator: 24 [new_demod,23] r(U)=U. ** KEPT (pick-wt=4): 25 [] r(Z)=Z. ---> New Demodulator: 26 [new_demod,25] r(Z)=Z. ** KEPT (pick-wt=4): 27 [] r(I)=I. ---> New Demodulator: 28 [new_demod,27] r(I)=I. ** KEPT (pick-wt=4): 29 [] r(D)=D. ---> New Demodulator: 30 [new_demod,29] r(D)=D. Following clause subsumed by 8 during input processing: 0 [copy,8,flip.1] x=x. Following clause subsumed by 9 during input processing: 0 [copy,9,flip.1] u(x,y)=u(y,x). >>>> Starting back demodulation with 12. >>>> Starting back demodulation with 14. >>>> Starting back demodulation with 17. >> back demodulating 13 with 17. >>>> Starting back demodulation with 20. >>>> Starting back demodulation with 22. >>>> Starting back demodulation with 24. >>>> Starting back demodulation with 26. >>>> Starting back demodulation with 28. >>>> Starting back demodulation with 30. >>>> Starting back demodulation with 32. ======= end of input processing ======= =========== start of search =========== ----> UNIT CONFLICT at 1.24 sec ----> 4404 [binary,4403.1,872.1] $F. Length of proof is 118. Level of proof is 34. ---------------- PROOF ---------------- 1 [] -i(x,y)|u(x,y)=y. 2 [] i(x,y)|u(x,y)!=y. 7 [] -i(k($c1,c(I)),c(n($c1,c(k($c1,c(I)))))). 9 [] u(x,y)=u(y,x). 10 [] u(x,u(y,z))=u(u(x,y),z). 12,11 [copy,10,flip.1] u(u(x,y),z)=u(x,u(y,z)). 13 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. 15 [] n(x,y)=c(u(c(x),c(y))). 17,16 [copy,15,flip.1] c(u(c(x),c(y)))=n(x,y). 31 [back_demod,13,demod,17] u(c(u(c(x),y)),n(x,y))=x. 33 [para_into,16.1.1.1.1,16.1.1] c(u(n(x,y),c(z)))=n(u(c(x),c(y)),z). 35 [para_into,16.1.1.1.2,16.1.1] c(u(c(x),n(y,z)))=n(x,u(c(y),c(z))). 37 [para_into,16.1.1.1,9.1.1,demod,17] n(x,y)=n(y,x). 51 [para_into,11.1.1.1,9.1.1,demod,12] u(x,u(y,z))=u(y,u(x,z)). 52 [para_into,11.1.1,9.1.1] u(x,u(y,z))=u(y,u(z,x)). 53 [copy,52,flip.1] u(x,u(y,z))=u(z,u(x,y)). 56 [para_into,31.1.1.1.1,9.1.1] u(c(u(x,c(y))),n(y,x))=y. 58 [para_into,31.1.1.1,16.1.1] u(n(x,y),n(x,c(y)))=x. 60 [para_into,31.1.1.2,37.1.1] u(c(u(c(x),y)),n(y,x))=x. 63,62 [para_into,31.1.1,9.1.1] u(n(x,y),c(u(c(x),y)))=x. 66 [para_into,58.1.1.1,37.1.1] u(n(x,y),n(y,c(x)))=y. 74 [para_into,58.1.1,9.1.1] u(n(x,c(y)),n(x,y))=x. 76 [para_from,58.1.1,11.1.1.1,flip.1] u(n(x,y),u(n(x,c(y)),z))=u(x,z). 82 [para_into,66.1.1.2,37.1.1] u(n(x,y),n(c(x),y))=y. 86 [para_from,66.1.1,11.1.1.1,flip.1] u(n(x,y),u(n(y,c(x)),z))=u(y,z). 107 [para_into,82.1.1,9.1.1] u(n(c(x),y),n(x,y))=y. 124 [para_into,35.1.1.1,31.1.1,flip.1] n(u(c(x),y),u(c(x),c(y)))=c(x). 147,146 [para_into,51.1.1.2,58.1.1,flip.1] u(n(x,y),u(z,n(x,c(y))))=u(z,x). 150 [para_into,51.1.1,9.1.1,demod,12] u(x,u(y,z))=u(x,u(z,y)). 153 [para_into,52.1.1.2,9.1.1] u(x,u(y,z))=u(z,u(y,x)). 160 [para_into,56.1.1.2,37.1.1] u(c(u(x,c(y))),n(x,y))=y. 162 [para_into,56.1.1,9.1.1] u(n(x,y),c(u(y,c(x))))=x. 164 [para_from,56.1.1,35.1.1.1,flip.1] n(u(x,c(y)),u(c(y),c(x)))=c(y). 187 [para_into,60.1.1,9.1.1] u(n(x,y),c(u(c(y),x)))=y. 189 [para_from,60.1.1,35.1.1.1,flip.1] n(u(c(x),y),u(c(y),c(x)))=c(x). 212 [para_from,62.1.1,33.1.1.1,flip.1] n(u(c(x),c(y)),u(c(x),y))=c(x). 233 [para_into,160.1.1,9.1.1] u(n(x,y),c(u(x,c(y))))=y. 235 [para_from,160.1.1,35.1.1.1,flip.1] n(u(x,c(y)),u(c(x),c(y)))=c(y). 252 [para_from,162.1.1,33.1.1.1,flip.1] n(u(c(x),c(y)),u(y,c(x)))=c(x). 274 [para_from,187.1.1,33.1.1.1,flip.1] n(u(c(x),c(y)),u(c(y),x))=c(y). 279 [para_from,233.1.1,33.1.1.1,flip.1] n(u(c(x),c(y)),u(x,c(y)))=c(y). 425 [para_into,274.1.1.2,160.1.1] n(u(c(n(x,y)),c(u(x,c(y)))),y)=c(u(x,c(y))). 453,452 [para_into,76.1.1.2,233.1.1,flip.1] u(x,c(u(x,c(c(y)))))=u(n(x,y),c(y)). 454 [para_into,76.1.1.2,187.1.1,flip.1] u(x,c(u(c(c(y)),x)))=u(n(x,y),c(y)). 457,456 [para_into,76.1.1.2,162.1.1,demod,17] u(n(x,y),x)=u(x,n(y,x)). 459,458 [para_into,76.1.1.2,107.1.1] u(n(c(x),y),c(y))=u(c(x),n(x,c(y))). 464 [para_into,76.1.1.2,74.1.1,demod,457] u(x,n(y,x))=u(x,n(x,y)). 514 [para_into,464.1.1,9.1.1] u(n(x,y),y)=u(y,n(y,x)). 607 [para_into,86.1.1.2,233.1.1,demod,453] u(n(x,y),c(x))=u(n(y,x),c(x)). 627 [para_from,607.1.1,86.1.1.2,demod,459,147] u(c(x),x)=u(y,c(y)). 645 [copy,627,flip.1] u(x,c(x))=u(c(y),y). 651 [hyper,627,2] i(c(u(x,c(x))),u(x,c(x))). 654 [para_into,627.1.1,627.1.1] u(x,c(x))=u(y,c(y)). 671 [para_from,627.1.1,252.1.1.2] n(u(c(x),c(c(c(x)))),u(y,c(y)))=c(x). 676,675 [para_from,627.1.1,164.1.1.2] n(u(x,c(c(x))),u(y,c(y)))=c(c(x)). 695 [para_from,627.1.1,212.1.1.1] n(u(x,c(x)),u(c(c(y)),y))=c(c(y)). 697 [para_from,627.1.1,124.1.1.2] n(u(c(c(x)),x),u(y,c(y)))=c(c(x)). 699 [para_from,627.1.1,16.1.1.1] c(u(x,c(x)))=n(c(y),y). 702 [para_from,627.1.1,153.1.1.2] u(x,u(y,c(y)))=u(z,u(c(z),x)). 703 [para_from,627.1.1,150.1.1.2] u(x,u(y,c(y)))=u(x,u(z,c(z))). 704 [para_from,627.1.1,53.1.1.2] u(x,u(y,c(y)))=u(z,u(x,c(z))). 706 [para_from,627.1.1,51.1.1.2] u(x,u(y,c(y)))=u(c(z),u(x,z)). 707 [para_from,627.1.1,11.1.1.1,demod,12] u(x,u(c(x),y))=u(c(z),u(z,y)). 708 [back_demod,671,demod,676] c(c(c(x)))=c(x). 712 [copy,699,flip.1] n(c(x),x)=c(u(y,c(y))). 713 [copy,702,flip.1] u(x,u(c(x),y))=u(y,u(z,c(z))). 714 [copy,704,flip.1] u(x,u(y,c(x)))=u(y,u(z,c(z))). 770,769 [para_from,708.1.1,16.1.1.1.1,demod,17,flip.1] n(c(c(x)),y)=n(x,y). 772,771 [para_from,708.1.1,62.1.1.2.1.1,demod,770,63,flip.1] c(c(x))=x. 781 [back_demod,697,demod,772,772] n(u(x,x),u(y,c(y)))=x. 783 [back_demod,695,demod,772,772] n(u(x,c(x)),u(y,y))=y. 814,813 [back_demod,454,demod,772] u(x,c(u(y,x)))=u(n(x,y),c(y)). 842,841 [para_into,771.1.1.1,16.1.1] c(n(x,y))=u(c(x),c(y)). 870,869 [back_demod,425,demod,842,12,814,459,flip.1] c(u(x,c(y)))=n(u(c(x),u(c(y),n(y,c(x)))),y). 872 [back_demod,7,demod,842,772] -i(k($c1,c(I)),u(c($c1),k($c1,c(I)))). 882,881 [back_demod,16,demod,870,772,772] n(u(x,u(c(y),n(y,x))),y)=n(x,y). 889 [back_demod,712,demod,870,882] n(c(x),x)=n(c(y),y). 898 [back_demod,651,demod,870,882] i(n(c(x),x),u(x,c(x))). 899 [back_demod,869,demod,882] c(u(x,c(y)))=n(c(x),y). 906 [para_from,771.1.1,189.1.1.2.2,demod,772,772] n(u(x,y),u(c(y),x))=x. 920 [para_from,771.1.1,274.1.1.2.1,demod,772,772] n(u(c(x),y),u(y,x))=y. 924 [para_from,771.1.1,279.1.1.2.2,demod,772,772] n(u(c(x),y),u(x,y))=y. 926 [para_from,771.1.1,235.1.1.1.2,demod,772,772] n(u(x,y),u(c(x),y))=y. 935 [para_into,645.1.1.2,771.1.1] u(c(x),x)=u(c(y),y). 942 [para_from,645.1.1,153.1.1.2] u(x,u(c(y),y))=u(c(z),u(z,x)). 944 [para_from,645.1.1,53.1.1.2] u(x,u(c(y),y))=u(c(z),u(x,z)). 950 [copy,944,flip.1] u(c(x),u(y,x))=u(y,u(c(z),z)). 964 [para_into,889.1.1.1,771.1.1] n(x,c(x))=n(c(y),y). 965 [copy,964,flip.1] n(c(x),x)=n(y,c(y)). 984 [para_into,898.1.1.1,771.1.1,demod,772] i(n(x,c(x)),u(c(x),x)). 1017 [para_from,965.1.1,107.1.1.1] u(n(x,c(x)),n(y,y))=y. 1039 [para_into,984.1.2,935.1.1] i(n(x,c(x)),u(c(y),y)). 1065,1064 [hyper,1039,1] u(n(x,c(x)),u(c(y),y))=u(c(y),y). 1116 [para_into,783.1.1.1.2,771.1.1] n(u(c(x),x),u(y,y))=y. 1124 [para_into,783.1.1.2,51.1.1,demod,12] n(u(x,c(x)),u(y,u(y,u(z,z))))=u(y,z). 1219,1218 [para_into,899.1.1.1.2,771.1.1] c(u(x,y))=n(c(x),c(y)). 1582 [para_into,920.1.1.2,464.1.1,demod,842,12] n(u(c(x),u(c(y),y)),u(y,n(y,x)))=y. 2142 [para_from,703.1.1,920.1.1.2,demod,1219,772] n(u(n(c(x),x),y),u(y,u(z,c(z))))=y. 2283 [para_from,706.1.1,1116.1.1.1,demod,1219,772] n(u(c(x),u(n(c(y),y),x)),u(z,z))=z. 2344 [para_from,707.1.1,906.1.1.1,demod,1219,772,457] n(u(c(x),u(x,y)),u(z,n(c(y),z)))=z. 2357 [para_from,713.1.1,920.1.1.2,demod,1219,772,457] n(u(x,n(c(y),x)),u(y,u(z,c(z))))=x. 2361 [para_from,713.1.1,924.1.1.1,demod,772,772] n(u(x,u(y,c(y))),u(z,u(z,x)))=u(z,x). 2377 [para_from,714.1.1,906.1.1.2,demod,772,12,772] n(u(x,u(y,y)),u(x,u(z,c(z))))=u(x,y). 3353 [para_from,942.1.1,924.1.1.1] n(u(c(x),u(x,c(y))),u(y,u(c(z),z)))=u(c(z),z). 3586 [para_into,950.1.1.2,1017.1.1,demod,842,12,1065] u(c(x),u(c(x),x))=u(c(y),y). 3604 [hyper,3586,2] i(c(x),u(c(x),x)). 3635 [para_into,3604.1.1,771.1.1,demod,772] i(x,u(x,c(x))). 3653 [para_into,3635.1.2,654.1.1] i(x,u(y,c(y))). 3670,3669 [hyper,3653,1] u(x,u(y,c(y)))=u(y,c(y)). 3725 [back_demod,2377,demod,3670] n(u(x,u(y,y)),u(z,c(z)))=u(x,y). 3736,3735 [back_demod,2361,demod,3670] n(u(x,c(x)),u(y,u(y,z)))=u(y,z). 3739 [back_demod,2357,demod,3670] n(u(x,n(c(y),x)),u(z,c(z)))=x. 3769 [back_demod,2142,demod,3670] n(u(n(c(x),x),y),u(z,c(z)))=y. 3784,3783 [back_demod,1124,demod,3736] u(x,u(y,y))=u(x,y). 3788,3787 [back_demod,3725,demod,3784] n(u(x,y),u(z,c(z)))=u(x,y). 3824,3823 [back_demod,3769,demod,3788] u(n(c(x),x),y)=y. 3830,3829 [back_demod,3739,demod,3788] u(x,n(c(y),x))=x. 3850,3849 [back_demod,781,demod,3788] u(x,x)=x. 3886,3885 [back_demod,2283,demod,3824,3850] n(u(c(x),x),y)=y. 3920,3919 [back_demod,2344,demod,3830] n(u(c(x),u(x,y)),z)=z. 4036,4035 [back_demod,3353,demod,3920] u(x,u(c(y),y))=u(c(y),y). 4046,4045 [back_demod,1582,demod,4036,3886] u(x,n(x,y))=x. 4099 [back_demod,514,demod,4046] u(n(x,y),y)=y. 4375 [hyper,4099,2] i(n(x,y),y). 4403 [para_into,4375.1.1,926.1.1] i(x,u(c(y),x)). 4404 [binary,4403.1,872.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 185 clauses generated 10656 clauses kept 2978 clauses forward subsumed 7524 clauses back subsumed 237 Kbytes malloced 2458 ----------- times (seconds) ----------- user CPU time 1.45 (0 hr, 0 min, 1 sec) system CPU time 0.00 (0 hr, 0 min, 0 sec) wall-clock time 2 (0 hr, 0 min, 2 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 17:43:24 2003