----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Sat Nov 22 00:40:19 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_literals,1). 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("peirceanAx0.txt"). ------- start included file peirceanAx0.txt------- formula_list(usable). all x y z (k(x,k(y,z))=k(k(x,y),z)). all x (r(r(x))=x). all x y (r(u(x,y))=u(r(x),r(y))). all x y (r(k(x,y))=k(r(y),r(x))). end_of_list. -------> usable clausifies to: list(usable). 0 [] k(x,k(y,z))=k(k(x,y),z). 0 [] r(r(x))=x. 0 [] r(u(x,y))=u(r(x),r(y)). 0 [] r(k(x,y))=k(r(y),r(x)). end_of_list. ------- end included file peirceanAx0.txt------- include("peirceanAx1.txt"). ------- start included file peirceanAx1.txt------- formula_list(usable). all x y z (k(u(x,y),z)=u(k(x,z),k(y,z))). end_of_list. -------> usable clausifies to: list(usable). 0 [] k(u(x,y),z)=u(k(x,z),k(y,z)). end_of_list. ------- end included file peirceanAx1.txt------- include("peirceanAx2.txt"). ------- start included file peirceanAx2.txt------- formula_list(usable). all x (k(x,I)=x). all x y (u(k(r(x),c(k(x,y))),c(y))=c(y)). end_of_list. -------> usable clausifies to: list(usable). 0 [] k(x,I)=x. 0 [] u(k(r(x),c(k(x,y))),c(y))=c(y). end_of_list. ------- end included file peirceanAx2.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("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("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("xiii.txt"). ------- start included file xiii.txt------- formula_list(usable). -(all x (k(x,Z)=Z)). end_of_list. -------> usable clausifies to: list(usable). 0 [] k($c1,Z)!=Z. end_of_list. ------- end included file xiii.txt------- SCAN INPUT: prop=0, horn=1, equality=1, symmetry=0, max_lits=2. This is a Horn set with equality. The strategy will be Knuth-Bendix and hyper_res, with positive clauses in sos and nonpositive clauses in usable. dependent: set(knuth_bendix). dependent: set(para_from). dependent: set(para_into). dependent: clear(para_from_right). dependent: clear(para_into_right). dependent: set(para_from_vars). dependent: set(eq_units_both_ways). dependent: set(dynamic_demod_all). dependent: set(dynamic_demod). dependent: set(order_eq). dependent: set(back_demod). dependent: set(lrpo). dependent: set(hyper_res). dependent: clear(order_hyper). ------------> process usable: ** KEPT (pick-wt=8): 1 [] -i(x,y)|u(x,y)=y. ** KEPT (pick-wt=8): 2 [] i(x,y)|u(x,y)!=y. ** KEPT (pick-wt=5): 3 [] k($c1,Z)!=Z. ------------> process sos: ** KEPT (pick-wt=7): 4 [] u(x,y)=u(y,x). ** KEPT (pick-wt=11): 6 [copy,5,flip.1] u(u(x,y),z)=u(x,u(y,z)). ---> New Demodulator: 7 [new_demod,6] u(u(x,y),z)=u(x,u(y,z)). ** KEPT (pick-wt=14): 8 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ---> New Demodulator: 9 [new_demod,8] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ** KEPT (pick-wt=11): 11 [copy,10,flip.1] k(k(x,y),z)=k(x,k(y,z)). ---> New Demodulator: 12 [new_demod,11] k(k(x,y),z)=k(x,k(y,z)). ** KEPT (pick-wt=5): 13 [] r(r(x))=x. ---> New Demodulator: 14 [new_demod,13] r(r(x))=x. ** KEPT (pick-wt=10): 15 [] r(u(x,y))=u(r(x),r(y)). ---> New Demodulator: 16 [new_demod,15] r(u(x,y))=u(r(x),r(y)). ** KEPT (pick-wt=10): 17 [] r(k(x,y))=k(r(y),r(x)). ---> New Demodulator: 18 [new_demod,17] r(k(x,y))=k(r(y),r(x)). ** KEPT (pick-wt=13): 20 [copy,19,flip.1] u(k(x,y),k(z,y))=k(u(x,z),y). ---> New Demodulator: 21 [new_demod,20] u(k(x,y),k(z,y))=k(u(x,z),y). ** KEPT (pick-wt=5): 22 [] k(x,I)=x. ---> New Demodulator: 23 [new_demod,22] k(x,I)=x. ** KEPT (pick-wt=13): 24 [] u(k(r(x),c(k(x,y))),c(y))=c(y). ---> New Demodulator: 25 [new_demod,24] u(k(r(x),c(k(x,y))),c(y))=c(y). ** KEPT (pick-wt=4): 27 [copy,26,flip.1] c(I)=D. ---> New Demodulator: 28 [new_demod,27] c(I)=D. ** KEPT (pick-wt=5): 30 [copy,29,flip.1] u(I,D)=U. ---> New Demodulator: 31 [new_demod,30] u(I,D)=U. ** KEPT (pick-wt=4): 33 [copy,32,flip.1] c(U)=Z. ---> New Demodulator: 34 [new_demod,33] c(U)=Z. ** KEPT (pick-wt=10): 36 [copy,35,flip.1] c(k(c(x),c(y)))=d(x,y). ---> New Demodulator: 37 [new_demod,36] c(k(c(x),c(y)))=d(x,y). Following clause subsumed by 4 during input processing: 0 [copy,4,flip.1] u(x,y)=u(y,x). >>>> Starting back demodulation with 7. >>>> Starting back demodulation with 9. >>>> Starting back demodulation with 12. >>>> Starting back demodulation with 14. >>>> Starting back demodulation with 16. >>>> Starting back demodulation with 18. >>>> Starting back demodulation with 21. >>>> Starting back demodulation with 23. >>>> Starting back demodulation with 25. >>>> Starting back demodulation with 28. >>>> Starting back demodulation with 31. >>>> Starting back demodulation with 34. >>>> Starting back demodulation with 37. ======= end of input processing ======= =========== start of search =========== ----> UNIT CONFLICT at 0.67 sec ----> 5639 [binary,5637.1,3.1] $F. Length of proof is 98. Level of proof is 28. ---------------- PROOF ---------------- 3 [] k($c1,Z)!=Z. 4 [] u(x,y)=u(y,x). 5 [] u(x,u(y,z))=u(u(x,y),z). 7,6 [copy,5,flip.1] u(u(x,y),z)=u(x,u(y,z)). 9,8 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. 10 [] k(x,k(y,z))=k(k(x,y),z). 12,11 [copy,10,flip.1] k(k(x,y),z)=k(x,k(y,z)). 14,13 [] r(r(x))=x. 15 [] r(u(x,y))=u(r(x),r(y)). 17 [] r(k(x,y))=k(r(y),r(x)). 19 [] k(u(x,y),z)=u(k(x,z),k(y,z)). 20 [copy,19,flip.1] u(k(x,y),k(z,y))=k(u(x,z),y). 23,22 [] k(x,I)=x. 24 [] u(k(r(x),c(k(x,y))),c(y))=c(y). 26 [] D=c(I). 28,27 [copy,26,flip.1] c(I)=D. 29 [] U=u(I,D). 31,30 [copy,29,flip.1] u(I,D)=U. 32 [] Z=c(U). 34,33 [copy,32,flip.1] c(U)=Z. 41 [para_into,6.1.1.1,4.1.1,demod,7] u(x,u(y,z))=u(y,u(x,z)). 42 [para_into,6.1.1,4.1.1] u(x,u(y,z))=u(y,u(z,x)). 43 [copy,42,flip.1] u(x,u(y,z))=u(z,u(x,y)). 47,46 [para_into,30.1.1,4.1.1] u(D,I)=U. 49,48 [para_from,30.1.1,6.1.1.1] u(U,x)=u(I,u(D,x)). 50 [para_into,8.1.1.1.1.1,33.1.1,demod,34] u(c(u(Z,x)),c(u(Z,c(x))))=U. 52 [para_into,8.1.1.1.1.1,27.1.1,demod,28] u(c(u(D,x)),c(u(D,c(x))))=I. 58 [para_into,8.1.1.2.1.2,33.1.1] u(c(u(c(x),U)),c(u(c(x),Z)))=x. 66 [para_into,8.1.1,4.1.1] u(c(u(c(x),c(y))),c(u(c(x),y)))=x. 68 [para_from,8.1.1,6.1.1.1,flip.1] u(c(u(c(x),y)),u(c(u(c(x),c(y))),z))=u(x,z). 70 [para_into,48.1.1,4.1.1] u(x,U)=u(I,u(D,x)). 71 [copy,70,flip.1] u(I,u(D,x))=u(x,U). 72 [para_into,70.1.1,6.1.1] u(x,u(y,U))=u(I,u(D,u(x,y))). 73 [copy,72,flip.1] u(I,u(D,u(x,y)))=u(x,u(y,U)). 74 [para_from,70.1.1,8.1.1.1.1,demod,34] u(c(u(I,u(D,c(x)))),c(u(c(x),Z)))=x. 80 [para_into,71.1.1.2,4.1.1] u(I,u(x,D))=u(x,U). 81 [para_into,71.1.1,4.1.1,demod,7] u(D,u(x,I))=u(x,U). 83 [copy,81,flip.1] u(x,U)=u(D,u(x,I)). 84 [para_into,11.1.1.1,22.1.1,flip.1] k(x,k(I,y))=k(x,y). 89 [para_into,81.1.1.2,4.1.1] u(D,u(I,x))=u(x,U). 91 [copy,89,flip.1] u(x,U)=u(D,u(I,x)). 110 [para_into,15.1.1.1,46.1.1] r(U)=u(r(D),r(I)). 118 [para_from,89.1.1,6.1.1.1,demod,7,49,7] u(x,u(I,u(D,y)))=u(D,u(I,u(x,y))). 121 [para_into,91.1.1,6.1.1] u(x,u(y,U))=u(D,u(I,u(x,y))). 123 [copy,121,flip.1] u(D,u(I,u(x,y)))=u(x,u(y,U)). 124 [para_from,91.1.1,8.1.1.1.1,demod,34] u(c(u(D,u(I,c(x)))),c(u(c(x),Z)))=x. 128 [para_into,17.1.1.1,22.1.1,flip.1] k(r(I),r(x))=r(x). 137,136 [para_into,128.1.1.2,13.1.1,demod,14] k(r(I),x)=x. 139,138 [para_into,136.1.1,84.1.1,demod,137,flip.1] k(I,x)=x. 141,140 [para_into,136.1.1,22.1.1] r(I)=I. 144 [back_demod,110,demod,141] r(U)=u(r(D),I). 149 [para_into,20.1.1.2,138.1.1] u(k(x,y),y)=k(u(x,I),y). 191,190 [para_into,24.1.1.1.1,140.1.1,demod,139,139] u(c(x),c(x))=c(x). 192 [para_into,24.1.1.1.1,17.1.1,demod,12,12] u(k(r(x),k(r(y),c(k(y,k(x,z))))),c(z))=c(z). 198 [para_into,24.1.1.1.2.1,22.1.1,demod,28,28] u(k(r(x),c(x)),D)=D. 212 [para_into,24.1.1.2,33.1.1,demod,34] u(k(r(x),c(k(x,U))),Z)=Z. 236,235 [para_into,190.1.1.1,33.1.1,demod,34,34] u(Z,Z)=Z. 238,237 [para_into,190.1.1.1,27.1.1,demod,28,28] u(D,D)=D. 278 [para_from,235.1.1,41.1.1.2,flip.1] u(Z,u(x,Z))=u(x,Z). 285,284 [para_from,237.1.1,80.1.1.2,demod,31,flip.1] u(D,U)=U. 385 [para_into,50.1.1.1.1,235.1.1] u(c(Z),c(u(Z,c(Z))))=U. 389 [para_into,50.1.1.1.1,83.1.1,demod,34,236] u(c(u(D,u(Z,I))),c(Z))=U. 534 [para_into,52.1.1.1.1,284.1.1,demod,34,34] u(Z,c(u(D,Z)))=I. 542 [para_into,52.1.1.1.1,46.1.1,demod,34,28,238] u(Z,c(D))=I. 588,587 [para_into,542.1.1,4.1.1] u(c(D),Z)=I. 589 [para_from,542.1.1,50.1.1.2.1,demod,28] u(c(u(Z,D)),D)=U. 720 [para_into,534.1.1.2.1,4.1.1] u(Z,c(u(Z,D)))=I. 849,848 [para_into,720.1.1,4.1.1] u(c(u(Z,D)),Z)=I. 1002,1001 [para_into,278.1.1.2,848.1.1,demod,849] u(Z,I)=I. 1008,1007 [back_demod,389,demod,1002,47,34] u(Z,c(Z))=U. 1009 [back_demod,385,demod,1008,34] u(c(Z),Z)=U. 1023,1022 [para_from,1001.1.1,81.1.1.2,demod,47,flip.1] u(Z,U)=U. 1025,1024 [para_from,1001.1.1,43.1.1.2,flip.1] u(I,u(x,Z))=u(x,I). 1372,1371 [para_into,66.1.1.2.1,1009.1.1,demod,191,34] u(c(c(Z)),Z)=Z. 1379 [para_into,66.1.1.2.1,589.1.1,demod,34] u(c(u(c(u(Z,D)),c(D))),Z)=u(Z,D). 1550,1549 [para_into,68.1.1.2,190.1.1,demod,9,flip.1] u(x,c(u(c(x),c(y))))=x. 1570 [para_into,68.1.1.2,8.1.1,demod,1550] u(c(u(c(x),y)),x)=x. 1646 [para_into,73.1.1.2.2,1371.1.1,demod,1025,47,1023,flip.1] u(c(c(Z)),U)=U. 1652 [para_into,73.1.1.2.2,587.1.1,demod,47,1023,flip.1] u(c(D),U)=u(I,U). 1693,1692 [para_from,1646.1.1,58.1.1.1.1,demod,34,1372,1008,flip.1] c(Z)=U. 1747,1746 [para_from,1692.1.1,190.1.1.2,demod,1693,49,285,1693] u(I,U)=U. 1778 [back_demod,1652,demod,1747] u(c(D),U)=U. 1804 [para_into,74.1.1.2.1,587.1.1,demod,28] u(c(u(I,u(D,c(D)))),D)=D. 1866,1865 [para_into,1778.1.1,70.1.1] u(I,u(D,c(D)))=U. 1868,1867 [back_demod,1804,demod,1866,34] u(Z,D)=D. 1873 [back_demod,1379,demod,1868,191,1868] u(c(c(D)),Z)=D. 2044 [para_from,1873.1.1,8.1.1.1.1,demod,1693] u(c(D),c(u(c(c(D)),U)))=c(D). 2052,2051 [para_from,1873.1.1,73.1.1.2.2,demod,238,31,1023,flip.1] u(c(c(D)),U)=U. 2057 [back_demod,2044,demod,2052,34,588,flip.1] c(D)=I. 2085 [para_from,2057.1.1,198.1.1.1.2,demod,23] u(r(D),D)=D. 2463,2462 [para_into,2085.1.1,4.1.1] u(D,r(D))=D. 2473,2472 [para_from,2085.1.1,15.1.1.1,demod,14,2463] r(D)=D. 2509,2508 [back_demod,144,demod,2473,47] r(U)=U. 2686 [para_from,118.1.1,66.1.1.2.1] u(c(u(c(x),c(u(I,u(D,y))))),c(u(D,u(I,u(c(x),y)))))=x. 2692 [para_from,118.1.1,8.1.1.1.1] u(c(u(D,u(I,u(c(x),y)))),c(u(c(x),c(u(I,u(D,y))))))=x. 3097 [para_into,149.1.1,72.1.1] u(I,u(D,u(k(x,u(y,U)),y)))=k(u(x,I),u(y,U)). 4310 [para_into,1570.1.1.1.1,72.1.1] u(c(u(I,u(D,u(c(x),y)))),x)=x. 4525,4524 [para_from,1570.1.1,81.1.1.2,demod,47,28,flip.1] u(c(u(D,x)),U)=U. 4534 [para_from,1570.1.1,72.1.1.2,demod,34,flip.1] u(I,u(D,u(x,c(u(Z,y)))))=u(x,U). 4545,4544 [para_from,1570.1.1,124.1.1.2.1,demod,1693,49,1693,4525,1693,49,flip.1] u(I,u(D,x))=U. 4574,4573 [para_from,1570.1.1,123.1.1.2.2,flip.1] u(c(u(c(x),y)),u(x,U))=u(D,u(I,x)). 4578,4577 [para_from,1570.1.1,73.1.1.2.2,demod,4545,4574,flip.1] u(D,u(I,x))=U. 4584,4583 [back_demod,4534,demod,4545,flip.1] u(x,U)=U. 4596,4595 [back_demod,4310,demod,4545,34] u(Z,x)=x. 4615 [back_demod,3097,demod,4584,4545,4584,flip.1] k(u(x,I),U)=U. 4626,4625 [back_demod,2692,demod,4578,34,4545,34,4596] c(u(c(x),Z))=x. 4632,4631 [back_demod,2686,demod,4545,34,4626,4578,34] u(x,Z)=x. 4649 [back_demod,48,demod,4545] u(U,x)=U. 4920,4919 [back_demod,212,demod,4632] k(r(x),c(k(x,U)))=Z. 5581 [para_into,4615.1.1.1,4649.1.1] k(U,U)=U. 5608,5607 [para_from,5581.1.1,192.1.1.1.2.2.1.2,demod,2509,4920,34,4632,34] k(U,Z)=Z. 5637 [para_from,5607.1.1,20.1.1.2,demod,4632,4584,5608] k(x,Z)=Z. 5639 [binary,5637.1,3.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 279 clauses generated 7698 clauses kept 3208 clauses forward subsumed 6557 clauses back subsumed 135 Kbytes malloced 3704 ----------- times (seconds) ----------- user CPU time 0.87 (0 hr, 0 min, 0 sec) system CPU time 0.00 (0 hr, 0 min, 0 sec) wall-clock time 1 (0 hr, 0 min, 1 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 Sat Nov 22 00:40:20 2003