----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Sun Nov 30 21:05:04 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,30000). assign(max_weight,25). 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("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("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("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("v.txt"). ------- start included file v.txt------- formula_list(usable). all x y z (k(x,u(y,z))=u(k(x,y),k(x,z))). 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(x,u(y,z))=u(k(x,y),k(x,z)). 0 [] k(u(x,y),z)=u(k(x,z),k(y,z)). end_of_list. ------- end included file v.txt------- include("xxvii_a.txt"). ------- start included file xxvii_a.txt------- formula_list(usable). all x y z (i(k(r(x),x),I)->i(n(k(x,y),k(x,z)),k(x,n(y,z)))). all x y z (i(k(r(x),x),I)->k(x,n(y,z))=n(k(x,y),k(x,z))). end_of_list. -------> usable clausifies to: list(usable). 0 [] -i(k(r(x),x),I)|i(n(k(x,y),k(x,z)),k(x,n(y,z))). 0 [] -i(k(r(x),x),I)|k(x,n(y,z))=n(k(x,y),k(x,z)). end_of_list. ------- end included file xxvii_a.txt------- include("20To24booleanLaws.txt"). ------- start included file 20To24booleanLaws.txt------- formula_list(usable). all x y (n(x,y)=n(y,x)). all x y (n(x,c(y))=Z->u(x,y)=y). all x y (u(x,y)=y->n(x,c(y))=Z). all x y (n(x,y)=Z&u(x,y)=U->c(x)=y). all x y (c(x)=y->n(x,y)=Z). all x y (c(x)=y->u(x,y)=U). end_of_list. -------> usable clausifies to: list(usable). 0 [] n(x,y)=n(y,x). 0 [] n(x,c(y))!=Z|u(x,y)=y. 0 [] u(x,y)!=y|n(x,c(y))=Z. 0 [] n(x,y)!=Z|u(x,y)!=U|c(x)=y. 0 [] c(x)!=y|n(x,y)=Z. 0 [] c(x)!=y|u(x,y)=U. end_of_list. ------- end included file 20To24booleanLaws.txt------- include("11booleanLaws.txt"). ------- start included file 11booleanLaws.txt------- formula_list(usable). all x y (c(n(x,y))=u(c(x),c(y))). all x y (c(u(x,y))=n(c(x),c(y))). end_of_list. -------> usable clausifies to: list(usable). 0 [] c(n(x,y))=u(c(x),c(y)). 0 [] c(u(x,y))=n(c(x),c(y)). end_of_list. ------- end included file 11booleanLaws.txt------- include("xxx_a.txt"). ------- start included file xxx_a.txt------- formula_list(usable). -(all x y (i(k(r(x),x),I)&k(x,U)=U->c(k(x,y))=k(x,c(y)))). end_of_formula. end_of_list. -------> usable clausifies to: list(usable). 0 [] i(k(r($c2),$c2),I). 0 [] k($c2,U)=U. 0 [] c(k($c2,$c1))!=k($c2,c($c1)). 0 [] end_of_formula. end_of_list. ------- end included file xxx_a.txt------- SCAN INPUT: prop=0, horn=1, equality=1, symmetry=0, max_lits=3. This is a Horn set with equality. The strategy will be Knuth-Bendix and hyper_res, with positive clauses in sos and nonpositive clauses in usable. dependent: set(knuth_bendix). dependent: set(para_from). dependent: set(para_into). dependent: clear(para_from_right). dependent: clear(para_into_right). dependent: set(para_from_vars). dependent: set(eq_units_both_ways). dependent: set(dynamic_demod_all). dependent: set(dynamic_demod). dependent: set(order_eq). dependent: set(back_demod). dependent: set(lrpo). dependent: set(hyper_res). dependent: clear(order_hyper). ------------> process usable: ** KEPT (pick-wt=19): 1 [] -i(k(r(x),x),I)|i(n(k(x,y),k(x,z)),k(x,n(y,z))). ** KEPT (pick-wt=19): 3 [copy,2,flip.2] -i(k(r(x),x),I)|n(k(x,y),k(x,z))=k(x,n(y,z)). ** KEPT (pick-wt=11): 4 [] n(x,c(y))!=Z|u(x,y)=y. ** KEPT (pick-wt=11): 5 [] u(x,y)!=y|n(x,c(y))=Z. ** KEPT (pick-wt=14): 6 [] n(x,y)!=Z|u(x,y)!=U|c(x)=y. ** KEPT (pick-wt=9): 7 [] c(x)!=y|n(x,y)=Z. ** KEPT (pick-wt=9): 8 [] c(x)!=y|u(x,y)=U. ** KEPT (pick-wt=9): 9 [] c(k($c2,$c1))!=k($c2,c($c1)). ------------> process sos: ** KEPT (pick-wt=3): 10 [] x=x. ** KEPT (pick-wt=7): 11 [] u(x,y)=u(y,x). ** KEPT (pick-wt=11): 13 [copy,12,flip.1] u(u(x,y),z)=u(x,u(y,z)). ---> New Demodulator: 14 [new_demod,13] u(u(x,y),z)=u(x,u(y,z)). ** KEPT (pick-wt=14): 15 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ---> New Demodulator: 16 [new_demod,15] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ** KEPT (pick-wt=11): 18 [copy,17,flip.1] k(k(x,y),z)=k(x,k(y,z)). ---> New Demodulator: 19 [new_demod,18] k(k(x,y),z)=k(x,k(y,z)). ** KEPT (pick-wt=5): 20 [] r(r(x))=x. ---> New Demodulator: 21 [new_demod,20] r(r(x))=x. ** KEPT (pick-wt=10): 22 [] r(u(x,y))=u(r(x),r(y)). ---> New Demodulator: 23 [new_demod,22] r(u(x,y))=u(r(x),r(y)). ** KEPT (pick-wt=10): 24 [] r(k(x,y))=k(r(y),r(x)). ---> New Demodulator: 25 [new_demod,24] r(k(x,y))=k(r(y),r(x)). ** KEPT (pick-wt=5): 26 [] k(x,I)=x. ---> New Demodulator: 27 [new_demod,26] k(x,I)=x. ** KEPT (pick-wt=13): 28 [] u(k(r(x),c(k(x,y))),c(y))=c(y). ---> New Demodulator: 29 [new_demod,28] u(k(r(x),c(k(x,y))),c(y))=c(y). ** KEPT (pick-wt=10): 31 [copy,30,flip.1] c(u(c(x),c(y)))=n(x,y). ---> New Demodulator: 32 [new_demod,31] c(u(c(x),c(y)))=n(x,y). ** KEPT (pick-wt=13): 34 [copy,33,flip.1] u(k(x,y),k(x,z))=k(x,u(y,z)). ---> New Demodulator: 35 [new_demod,34] u(k(x,y),k(x,z))=k(x,u(y,z)). ** KEPT (pick-wt=13): 37 [copy,36,flip.1] u(k(x,y),k(z,y))=k(u(x,z),y). ---> New Demodulator: 38 [new_demod,37] u(k(x,y),k(z,y))=k(u(x,z),y). ** KEPT (pick-wt=7): 39 [] n(x,y)=n(y,x). ** KEPT (pick-wt=10): 40 [] c(n(x,y))=u(c(x),c(y)). ---> New Demodulator: 41 [new_demod,40] c(n(x,y))=u(c(x),c(y)). ** KEPT (pick-wt=10): 42 [] c(u(x,y))=n(c(x),c(y)). ---> New Demodulator: 43 [new_demod,42] c(u(x,y))=n(c(x),c(y)). ** KEPT (pick-wt=6): 44 [] i(k(r($c2),$c2),I). ** KEPT (pick-wt=5): 45 [] k($c2,U)=U. ---> New Demodulator: 46 [new_demod,45] k($c2,U)=U. ** KEPT (pick-wt=1): 47 [] end_of_formula. Following clause subsumed by 10 during input processing: 0 [copy,10,flip.1] x=x. Following clause subsumed by 11 during input processing: 0 [copy,11,flip.1] u(x,y)=u(y,x). >>>> Starting back demodulation with 14. >>>> Starting back demodulation with 16. >>>> Starting back demodulation with 19. >>>> Starting back demodulation with 21. >>>> Starting back demodulation with 23. >>>> Starting back demodulation with 25. >>>> Starting back demodulation with 27. >>>> Starting back demodulation with 29. >>>> Starting back demodulation with 32. >> back demodulating 15 with 32. >>>> Starting back demodulation with 35. >>>> Starting back demodulation with 38. Following clause subsumed by 39 during input processing: 0 [copy,39,flip.1] n(x,y)=n(y,x). >>>> Starting back demodulation with 41. >>>> Starting back demodulation with 43. >> back demodulating 31 with 43. >>>> Starting back demodulation with 46. >>>> Starting back demodulation with 49. >>>> Starting back demodulation with 51. >> back demodulating 48 with 51. >>>> Starting back demodulation with 53. ======= end of input processing ======= =========== start of search =========== Resetting weight limit to 11. sos_size=4846 ----> UNIT CONFLICT at 13.84 sec ----> 17659 [binary,17657.1,9.1] $F. Length of proof is 73. Level of proof is 15. ---------------- PROOF ---------------- 2 [] -i(k(r(x),x),I)|k(x,n(y,z))=n(k(x,y),k(x,z)). 3 [copy,2,flip.2] -i(k(r(x),x),I)|n(k(x,y),k(x,z))=k(x,n(y,z)). 4 [] n(x,c(y))!=Z|u(x,y)=y. 5 [] u(x,y)!=y|n(x,c(y))=Z. 6 [] n(x,y)!=Z|u(x,y)!=U|c(x)=y. 7 [] c(x)!=y|n(x,y)=Z. 8 [] c(x)!=y|u(x,y)=U. 9 [] c(k($c2,$c1))!=k($c2,c($c1)). 10 [] x=x. 11 [] u(x,y)=u(y,x). 12 [] u(x,u(y,z))=u(u(x,y),z). 14,13 [copy,12,flip.1] u(u(x,y),z)=u(x,u(y,z)). 15 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. 20 [] r(r(x))=x. 22 [] r(u(x,y))=u(r(x),r(y)). 28 [] u(k(r(x),c(k(x,y))),c(y))=c(y). 30 [] n(x,y)=c(u(c(x),c(y))). 31 [copy,30,flip.1] c(u(c(x),c(y)))=n(x,y). 33 [] k(x,u(y,z))=u(k(x,y),k(x,z)). 35,34 [copy,33,flip.1] u(k(x,y),k(x,z))=k(x,u(y,z)). 36 [] k(u(x,y),z)=u(k(x,z),k(y,z)). 37 [copy,36,flip.1] u(k(x,y),k(z,y))=k(u(x,z),y). 39 [] n(x,y)=n(y,x). 41,40 [] c(n(x,y))=u(c(x),c(y)). 43,42 [] c(u(x,y))=n(c(x),c(y)). 44 [] i(k(r($c2),$c2),I). 46,45 [] k($c2,U)=U. 48 [back_demod,15,demod,43,43] u(n(c(c(x)),c(y)),n(c(c(x)),c(c(y))))=x. 51,50 [back_demod,31,demod,43] n(c(c(x)),c(c(y)))=n(x,y). 52 [back_demod,48,demod,51] u(n(c(c(x)),c(y)),n(x,y))=x. 55,54 [hyper,10,8] u(x,c(x))=U. 56 [hyper,10,7] n(x,c(x))=Z. 58 [hyper,44,3] n(k($c2,x),k($c2,y))=k($c2,n(x,y)). 61 [para_into,54.1.1,11.1.1] u(c(x),x)=U. 64,63 [hyper,56,4] u(x,x)=x. 65 [para_into,13.1.1.1,63.1.1,flip.1] u(x,u(x,y))=u(x,y). 67 [para_into,13.1.1.1,54.1.1,flip.1] u(x,u(c(x),y))=u(U,y). 69 [para_into,13.1.1.1,11.1.1,demod,14] u(x,u(y,z))=u(y,u(x,z)). 78 [para_from,61.1.1,13.1.1.1,flip.1] u(c(x),u(x,y))=u(U,y). 80 [para_into,39.1.1,56.1.1,flip.1] n(c(x),x)=Z. 83,82 [hyper,80,6,61] c(c(x))=x. 85,84 [back_demod,52,demod,83] u(n(x,c(y)),n(x,y))=x. 95,94 [para_into,65.1.1.2,54.1.1,demod,55] u(x,U)=U. 100 [hyper,94,5] n(x,c(U))=Z. 103,102 [para_into,94.1.1,11.1.1] u(U,x)=U. 104 [back_demod,78,demod,103] u(c(x),u(x,y))=U. 107,106 [back_demod,67,demod,103] u(x,u(c(x),y))=U. 108 [para_into,22.1.1.1,102.1.1,flip.1] u(r(U),r(x))=r(U). 116 [para_into,100.1.1,39.1.1] n(c(U),x)=Z. 118 [hyper,116,4] u(c(U),x)=x. 121,120 [para_into,118.1.1,11.1.1] u(x,c(U))=x. 134 [para_into,104.1.1,11.1.1,demod,14] u(x,u(y,c(x)))=U. 146 [para_into,108.1.1.2,20.1.1] u(r(U),x)=r(U). 149,148 [para_into,146.1.1,106.1.1,flip.1] r(U)=U. 232 [para_into,37.1.1.1,45.1.1,demod,103,flip.1] k(u($c2,x),U)=U. 244 [para_into,232.1.1.1,134.1.1] k(U,U)=U. 253,252 [para_from,244.1.1,28.1.1.1.2.1,demod,149,121] k(U,c(U))=c(U). 262 [para_from,252.1.1,37.1.1.2,demod,121,95,253] k(x,c(U))=c(U). 273,272 [para_into,40.1.1.1,116.1.1,demod,83,103] c(Z)=U. 291,290 [para_from,272.1.1,82.1.1.1] c(U)=Z. 299,298 [back_demod,262,demod,291,291] k(x,Z)=Z. 307,306 [back_demod,120,demod,291] u(x,Z)=x. 320 [para_into,42.1.1.1,306.1.1,demod,273,flip.1] n(c(x),U)=c(x). 371 [para_from,58.1.1,40.1.1.1] c(k($c2,n(x,y)))=u(c(k($c2,x)),c(k($c2,y))). 376,375 [para_into,320.1.1.1,82.1.1,demod,83] n(x,U)=x. 378,377 [para_into,375.1.1,39.1.1] n(U,x)=x. 539 [para_into,84.1.1.1.2,82.1.1] u(n(x,y),n(x,c(y)))=x. 547 [para_into,84.1.1.2,58.1.1] u(n(k($c2,x),c(k($c2,y))),k($c2,n(x,y)))=k($c2,x). 554 [para_from,84.1.1,69.1.1.2,flip.1] u(n(x,c(y)),u(z,n(x,y)))=u(z,x). 556 [para_from,84.1.1,65.1.1.2,demod,85] u(n(x,c(y)),x)=x. 558 [para_from,84.1.1,42.1.1.1,demod,41,83,41,flip.1] n(u(c(x),y),u(c(x),c(y)))=c(x). 566 [para_into,556.1.1.1.2,82.1.1] u(n(x,y),x)=x. 580 [para_into,566.1.1.1,39.1.1] u(n(x,y),y)=y. 589,588 [para_into,566.1.1,11.1.1] u(x,n(x,y))=x. 594,593 [para_from,566.1.1,69.1.1.2,flip.1] u(n(x,y),u(z,x))=u(z,x). 601 [hyper,580,5] n(n(x,y),c(y))=Z. 609 [para_into,580.1.1,11.1.1] u(x,n(y,x))=x. 642 [para_from,609.1.1,13.1.1.1,flip.1] u(x,u(n(y,x),z))=u(x,z). 666 [para_from,601.1.1,84.1.1.2,demod,83,307] n(n(x,y),y)=n(x,y). 3285 [para_into,539.1.1.1,39.1.1] u(n(x,y),n(y,c(x)))=y. 6788 [para_into,371.1.1.1.2,80.1.1,demod,299,273,flip.1] u(c(k($c2,c(x))),c(k($c2,x)))=U. 12117,12116 [para_into,554.1.1.2,609.1.1] u(n(x,c(y)),y)=u(y,x). 12329 [para_into,558.1.1.1.1,82.1.1,demod,83,83] n(u(x,y),u(x,c(y)))=x. 12712 [para_into,12116.1.1.1,666.1.1,demod,12117,flip.1] u(x,n(y,c(x)))=u(x,y). 12970,12969 [para_into,12329.1.1.1,3285.1.1,demod,41,83,594] n(x,u(c(x),y))=n(y,x). 13037 [para_into,12329.1.1.1,11.1.1] n(u(x,y),u(y,c(x)))=y. 13073 [para_into,12329.1.1.2,11.1.1] n(u(x,y),u(c(y),x))=x. 13626,13625 [para_from,12712.1.1,13.1.1.1,demod,14,flip.1] u(x,u(n(y,c(x)),z))=u(x,u(y,z)). 14357 [para_into,12969.1.1.2,65.1.1,demod,12970,flip.1] n(u(c(x),y),x)=n(y,x). 14494 [para_into,13037.1.1.2,11.1.1] n(u(x,y),u(c(x),y))=y. 15856 [para_from,14357.1.1,547.1.1.2.2] u(n(k($c2,u(c(x),y)),c(k($c2,x))),k($c2,n(y,x)))=k($c2,u(c(x),y)). 16761,16760 [para_into,642.1.1.2,547.1.1,flip.1] u(c(k($c2,x)),k($c2,n(y,x)))=u(c(k($c2,x)),k($c2,y)). 16790,16789 [para_from,642.1.1,14494.1.1.2,demod,13626,flip.1] u(n(x,c(y)),z)=n(u(y,u(x,z)),u(c(y),z)). 16811,16810 [back_demod,15856,demod,16790,35,14,589,35,107,46,16761,378] u(c(k($c2,x)),k($c2,y))=k($c2,u(c(x),y)). 17657 [para_from,6788.1.1,13073.1.1.2,demod,16811,64,376,flip.1] c(k($c2,x))=k($c2,c(x)). 17659 [binary,17657.1,9.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 1472 clauses generated 438959 clauses kept 9173 clauses forward subsumed 333047 clauses back subsumed 30 Kbytes malloced 10538 ----------- times (seconds) ----------- user CPU time 14.04 (0 hr, 0 min, 14 sec) system CPU time 0.00 (0 hr, 0 min, 0 sec) wall-clock time 14 (0 hr, 0 min, 14 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 21:05:18 2003