----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Fri Nov 21 23:36:10 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("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("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("ix.txt"). ------- start included file ix.txt------- formula_list(usable). all x (k(x,I)=x). all x (k(I,x)=x). all x i(x,k(x,U)). all x i(x,k(U,x)). -(all x i(d(Z,x),x)). end_of_list. -------> usable clausifies to: list(usable). 0 [] k(x,I)=x. 0 [] k(I,x)=x. 0 [] i(x,k(x,U)). 0 [] i(x,k(U,x)). 0 [] -i(d(Z,$c1),$c1). end_of_list. ------- end included file ix.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 [] -i(d(Z,$c1),$c1). ------------> 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=10): 27 [copy,26,flip.1] c(u(c(x),c(y)))=n(x,y). ---> New Demodulator: 28 [new_demod,27] c(u(c(x),c(y)))=n(x,y). ** KEPT (pick-wt=4): 30 [copy,29,flip.1] c(I)=D. ---> New Demodulator: 31 [new_demod,30] c(I)=D. ** KEPT (pick-wt=5): 33 [copy,32,flip.1] u(I,D)=U. ---> New Demodulator: 34 [new_demod,33] u(I,D)=U. ** KEPT (pick-wt=4): 36 [copy,35,flip.1] c(U)=Z. ---> New Demodulator: 37 [new_demod,36] c(U)=Z. ** KEPT (pick-wt=10): 39 [copy,38,flip.1] c(k(c(x),c(y)))=d(x,y). ---> New Demodulator: 40 [new_demod,39] c(k(c(x),c(y)))=d(x,y). ** KEPT (pick-wt=3): 42 [copy,41,demod,23] x=x. ** KEPT (pick-wt=5): 43 [] k(I,x)=x. ---> New Demodulator: 44 [new_demod,43] k(I,x)=x. ** KEPT (pick-wt=5): 45 [] i(x,k(x,U)). ** KEPT (pick-wt=5): 46 [] i(x,k(U,x)). 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. >> back demodulating 8 with 28. >>>> Starting back demodulation with 31. >>>> Starting back demodulation with 34. >>>> Starting back demodulation with 37. >>>> Starting back demodulation with 40. Following clause subsumed by 42 during input processing: 0 [copy,42,flip.1] x=x. >>>> Starting back demodulation with 44. >>>> Starting back demodulation with 48. ======= end of input processing ======= =========== start of search =========== Resetting weight limit to 9. sos_size=2275 ----> UNIT CONFLICT at 0.67 sec ----> 5582 [binary,5581.1,3.1] $F. Length of proof is 50. Level of proof is 17. ---------------- PROOF ---------------- 1 [] -i(x,y)|u(x,y)=y. 2 [] i(x,y)|u(x,y)!=y. 3 [] -i(d(Z,$c1),$c1). 4 [] u(x,y)=u(y,x). 5 [] u(x,u(y,z))=u(u(x,y),z). 6 [copy,5,flip.1] u(u(x,y),z)=u(x,u(y,z)). 8 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. 14,13 [] r(r(x))=x. 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). 24 [] u(k(r(x),c(k(x,y))),c(y))=c(y). 26 [] n(x,y)=c(u(c(x),c(y))). 28,27 [copy,26,flip.1] c(u(c(x),c(y)))=n(x,y). 29 [] D=c(I). 30 [copy,29,flip.1] c(I)=D. 32 [] U=u(I,D). 33 [copy,32,flip.1] u(I,D)=U. 35 [] Z=c(U). 37,36 [copy,35,flip.1] c(U)=Z. 38 [] d(x,y)=c(k(c(x),c(y))). 39 [copy,38,flip.1] c(k(c(x),c(y)))=d(x,y). 44,43 [] k(I,x)=x. 45 [] i(x,k(x,U)). 46 [] i(x,k(U,x)). 47 [back_demod,8,demod,28] u(c(u(c(x),y)),n(x,y))=x. 50 [para_into,6.1.1,4.1.1] u(x,u(y,z))=u(y,u(z,x)). 51 [copy,50,flip.1] u(x,u(y,z))=u(z,u(x,y)). 54 [para_from,33.1.1,6.1.1.1] u(U,x)=u(I,u(D,x)). 56 [hyper,45,1] u(x,k(x,U))=k(x,U). 62 [hyper,46,1] u(x,k(U,x))=k(U,x). 72 [para_into,54.1.1,4.1.1] u(x,U)=u(I,u(D,x)). 84 [para_from,56.1.1,6.1.1.1,flip.1] u(x,u(k(x,U),y))=u(k(x,U),y). 87 [para_into,17.1.1.1,43.1.1,flip.1] k(r(x),r(I))=r(x). 95 [para_into,87.1.1.1,13.1.1,demod,14] k(x,r(I))=x. 97 [para_into,95.1.1,43.1.1] r(I)=I. 124 [para_into,72.1.1,6.1.1] u(x,u(y,U))=u(I,u(D,u(x,y))). 125 [copy,124,flip.1] u(I,u(D,u(x,y)))=u(x,u(y,U)). 149 [para_into,24.1.1.1.1,97.1.1,demod,44,44] u(c(x),c(x))=c(x). 176 [para_from,149.1.1,6.1.1.1,flip.1] u(c(x),u(c(x),y))=u(c(x),y). 192 [para_into,27.1.1.1,149.1.1] c(c(x))=n(x,x). 194 [para_into,27.1.1.1,4.1.1,demod,28] n(x,y)=n(y,x). 246 [para_into,192.1.1.1,36.1.1] c(Z)=n(U,U). 274 [para_into,47.1.1.1.1,72.1.1] u(c(u(I,u(D,c(x)))),n(x,U))=x. 578 [hyper,84,2] i(x,u(k(x,U),y)). 622 [para_into,578.1.2,20.1.1] i(x,k(u(x,y),U)). 638 [para_into,622.1.2.1,47.1.1] i(c(u(c(x),y)),k(x,U)). 1341 [para_from,124.1.1,47.1.1.1.1] u(c(u(I,u(D,u(c(x),y)))),n(x,u(y,U)))=x. 2997 [para_into,638.1.1.1.1,30.1.1,demod,44] i(c(u(D,x)),U). 3014 [hyper,2997,1] u(c(u(D,x)),U)=U. 3030 [hyper,176,2] i(c(x),u(c(x),y)). 3293 [para_into,3030.1.2,47.1.1] i(c(u(c(x),y)),x). 3605 [para_into,3293.1.1.1,124.1.1] i(c(u(I,u(D,u(c(x),y)))),x). 3610 [para_into,3293.1.1.1,62.1.1] i(c(k(U,c(x))),x). 3618 [para_into,3293.1.1,27.1.1] i(n(x,y),x). 3621 [para_into,3618.1.1,194.1.1] i(n(x,y),y). 3623,3622 [hyper,3621,1] u(n(x,y),y)=y. 4775,4774 [para_into,3014.1.1.1.1,51.1.1] u(c(u(x,u(D,y))),U)=U. 4786 [para_from,3014.1.1,124.1.1.2,flip.1] u(I,u(D,u(x,c(u(D,y)))))=u(x,U). 5380,5379 [para_from,274.1.1,125.1.1.2.2,demod,3623,4775] u(I,u(D,x))=U. 5387,5386 [back_demod,4786,demod,5380,flip.1] u(x,U)=U. 5388 [back_demod,3605,demod,5380,37] i(Z,x). 5408 [back_demod,1341,demod,5380,37,5387] u(Z,n(x,U))=x. 5461,5460 [hyper,5388,1] u(Z,x)=x. 5463,5462 [back_demod,5408,demod,5461] n(x,U)=x. 5505 [back_demod,246,demod,5463] c(Z)=U. 5571,5570 [para_from,5505.1.1,39.1.1.1.1] c(k(U,c(x)))=d(Z,x). 5581 [back_demod,3610,demod,5571] i(d(Z,x),x). 5582 [binary,5581.1,3.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 349 clauses generated 10164 clauses kept 3532 clauses forward subsumed 7455 clauses back subsumed 381 Kbytes malloced 4119 ----------- times (seconds) ----------- user CPU time 0.90 (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 Fri Nov 21 23:36:11 2003