----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Sun Nov 30 15:35:28 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). 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("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("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("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("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("cycleLawC.txt"). ------- start included file cycleLawC.txt------- formula_list(usable). all x y z (n(y,k(n(x,k(y,r(z))),z))=n(k(x,z),y)). end_of_list. -------> usable clausifies to: list(usable). 0 [] n(y,k(n(x,k(y,r(z))),z))=n(k(x,z),y). end_of_list. ------- end included file cycleLawC.txt------- include("xix_b_ghost.txt"). ------- start included file xix_b_ghost.txt------- formula_list(usable). all x y (k(x,r(y))=Z->n(k(U,x),k(U,y))=Z). all x y (n(k(U,x),k(U,y))=Z->k(x,r(y))=Z). all x y (u(d(Z,x),d(Z,y))=U->d(x,r(y))=U). -(all x y (d(x,r(y))=U->u(d(Z,x),d(Z,y))=U)). end_of_list. -------> usable clausifies to: list(usable). 0 [] k(x,r(y))!=Z|n(k(U,x),k(U,y))=Z. 0 [] n(k(U,x),k(U,y))!=Z|k(x,r(y))=Z. 0 [] u(d(Z,x),d(Z,y))!=U|d(x,r(y))=U. 0 [] d($c2,r($c1))=U. 0 [] u(d(Z,$c2),d(Z,$c1))!=U. end_of_list. ------- end included file xix_b_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=12): 1 [] u(x,y)!=y|c(u(y,c(x)))=Z. ** KEPT (pick-wt=12): 2 [] c(u(x,c(y)))!=Z|u(y,x)=x. ** KEPT (pick-wt=8): 3 [] -i(x,y)|i(r(x),r(y)). ** KEPT (pick-wt=15): 4 [] k(x,r(y))!=Z|n(k(U,x),k(U,y))=Z. ** KEPT (pick-wt=15): 5 [] n(k(U,x),k(U,y))!=Z|k(x,r(y))=Z. ** KEPT (pick-wt=15): 6 [] u(d(Z,x),d(Z,y))!=U|d(x,r(y))=U. ** KEPT (pick-wt=9): 7 [] u(d(Z,$c2),d(Z,$c1))!=U. ------------> process sos: ** KEPT (pick-wt=10): 9 [copy,8,flip.1] c(u(c(x),c(y)))=n(x,y). ---> New Demodulator: 10 [new_demod,9] c(u(c(x),c(y)))=n(x,y). ** KEPT (pick-wt=10): 12 [copy,11,flip.1] c(k(c(x),c(y)))=d(x,y). ---> New Demodulator: 13 [new_demod,12] c(k(c(x),c(y)))=d(x,y). ** KEPT (pick-wt=4): 15 [copy,14,flip.1] c(I)=D. ---> New Demodulator: 16 [new_demod,15] c(I)=D. ** KEPT (pick-wt=5): 18 [copy,17,flip.1] u(I,D)=U. ---> New Demodulator: 19 [new_demod,18] u(I,D)=U. ** KEPT (pick-wt=4): 21 [copy,20,flip.1] c(U)=Z. ---> New Demodulator: 22 [new_demod,21] c(U)=Z. ** KEPT (pick-wt=7): 23 [] u(x,y)=u(y,x). ** KEPT (pick-wt=11): 25 [copy,24,flip.1] u(u(x,y),z)=u(x,u(y,z)). ---> New Demodulator: 26 [new_demod,25] u(u(x,y),z)=u(x,u(y,z)). ** KEPT (pick-wt=11): 28 [copy,27,demod,10] u(c(u(c(x),y)),n(x,y))=x. ---> New Demodulator: 29 [new_demod,28] u(c(u(c(x),y)),n(x,y))=x. ** KEPT (pick-wt=11): 31 [copy,30,flip.1] k(k(x,y),z)=k(x,k(y,z)). ---> New Demodulator: 32 [new_demod,31] k(k(x,y),z)=k(x,k(y,z)). ** KEPT (pick-wt=5): 33 [] r(r(x))=x. ---> New Demodulator: 34 [new_demod,33] r(r(x))=x. ** KEPT (pick-wt=10): 35 [] r(u(x,y))=u(r(x),r(y)). ---> New Demodulator: 36 [new_demod,35] r(u(x,y))=u(r(x),r(y)). ** KEPT (pick-wt=10): 37 [] r(k(x,y))=k(r(y),r(x)). ---> New Demodulator: 38 [new_demod,37] r(k(x,y))=k(r(y),r(x)). ** KEPT (pick-wt=5): 39 [] k(x,I)=x. ---> New Demodulator: 40 [new_demod,39] k(x,I)=x. ** KEPT (pick-wt=13): 41 [] u(k(r(x),c(k(x,y))),c(y))=c(y). ---> New Demodulator: 42 [new_demod,41] u(k(r(x),c(k(x,y))),c(y))=c(y). ** KEPT (pick-wt=5): 43 [] c(c(x))=x. ---> New Demodulator: 44 [new_demod,43] c(c(x))=x. ** KEPT (pick-wt=11): 46 [copy,45,demod,36] u(r(x),r(y))=u(r(x),r(y)). ** KEPT (pick-wt=7): 47 [] r(c(x))=c(r(x)). ---> New Demodulator: 48 [new_demod,47] r(c(x))=c(r(x)). ** KEPT (pick-wt=10): 49 [] r(n(x,y))=n(r(x),r(y)). ---> New Demodulator: 50 [new_demod,49] r(n(x,y))=n(r(x),r(y)). ** KEPT (pick-wt=3): 52 [copy,51,demod,34] x=x. ** KEPT (pick-wt=10): 53 [] r(d(x,y))=d(r(y),r(x)). ---> New Demodulator: 54 [new_demod,53] r(d(x,y))=d(r(y),r(x)). Following clause subsumed by 52 during input processing: 0 [demod,38] k(r(y),r(x))=k(r(y),r(x)). ** KEPT (pick-wt=4): 55 [] r(I)=I. ---> New Demodulator: 56 [new_demod,55] r(I)=I. ** KEPT (pick-wt=4): 57 [] r(Z)=Z. ---> New Demodulator: 58 [new_demod,57] r(Z)=Z. ** KEPT (pick-wt=4): 59 [] r(U)=U. ---> New Demodulator: 60 [new_demod,59] r(U)=U. ** KEPT (pick-wt=10): 61 [] c(n(x,y))=u(c(x),c(y)). ---> New Demodulator: 62 [new_demod,61] c(n(x,y))=u(c(x),c(y)). ** KEPT (pick-wt=10): 63 [] c(u(x,y))=n(c(x),c(y)). ---> New Demodulator: 64 [new_demod,63] c(u(x,y))=n(c(x),c(y)). ** KEPT (pick-wt=16): 65 [] n(x,k(n(y,k(x,r(z))),z))=n(k(y,z),x). ** KEPT (pick-wt=6): 66 [] d($c2,r($c1))=U. ---> New Demodulator: 67 [new_demod,66] d($c2,r($c1))=U. >>>> Starting back demodulation with 10. >>>> Starting back demodulation with 13. >>>> Starting back demodulation with 16. >>>> Starting back demodulation with 19. >>>> Starting back demodulation with 22. Following clause subsumed by 23 during input processing: 0 [copy,23,flip.1] u(x,y)=u(y,x). >>>> Starting back demodulation with 26. >>>> Starting back demodulation with 29. >>>> Starting back demodulation with 32. >>>> Starting back demodulation with 34. >>>> Starting back demodulation with 36. >>>> Starting back demodulation with 38. >>>> Starting back demodulation with 40. >>>> Starting back demodulation with 42. >>>> Starting back demodulation with 44. Following clause subsumed by 52 during input processing: 0 [copy,46,flip.1] u(r(x),r(y))=u(r(x),r(y)). >>>> Starting back demodulation with 48. >>>> Starting back demodulation with 50. Following clause subsumed by 52 during input processing: 0 [copy,52,flip.1] x=x. 52 back subsumes 46. >>>> Starting back demodulation with 54. >>>> Starting back demodulation with 56. >>>> Starting back demodulation with 58. >>>> Starting back demodulation with 60. >>>> Starting back demodulation with 62. >>>> Starting back demodulation with 64. >> back demodulating 28 with 64. >> back demodulating 9 with 64. >> back demodulating 2 with 64. >> back demodulating 1 with 64. ** KEPT (pick-wt=16): 72 [copy,65,flip.1] n(k(x,y),z)=n(z,k(n(x,k(z,r(y))),y)). >>>> Starting back demodulation with 67. >>>> Starting back demodulation with 69. Following clause subsumed by 65 during input processing: 0 [copy,72,flip.1] n(x,k(n(y,k(x,r(z))),z))=n(k(y,z),x). ======= end of input processing ======= =========== start of search =========== -----> EMPTY CLAUSE at 1.80 sec ----> 5202 [hyper,5177,52,52] $F. Length of proof is 25. Level of proof is 10. ---------------- PROOF ---------------- 1 [] u(x,y)!=y|c(u(y,c(x)))=Z. 2 [] c(u(x,c(y)))!=Z|u(y,x)=x. 4 [] k(x,r(y))!=Z|n(k(U,x),k(U,y))=Z. 7 [] u(d(Z,$c2),d(Z,$c1))!=U. 11 [] d(x,y)=c(k(c(x),c(y))). 12 [copy,11,flip.1] c(k(c(x),c(y)))=d(x,y). 20 [] Z=c(U). 22,21 [copy,20,flip.1] c(U)=Z. 23 [] u(x,y)=u(y,x). 34,33 [] r(r(x))=x. 37 [] r(k(x,y))=k(r(y),r(x)). 39 [] k(x,I)=x. 41 [] u(k(r(x),c(k(x,y))),c(y))=c(y). 44,43 [] c(c(x))=x. 47 [] r(c(x))=c(r(x)). 51 [] r(r(x))=x. 52 [copy,51,demod,34] x=x. 53 [] r(d(x,y))=d(r(y),r(x)). 56,55 [] r(I)=I. 60,59 [] r(U)=U. 61 [] c(n(x,y))=u(c(x),c(y)). 64,63 [] c(u(x,y))=n(c(x),c(y)). 66 [] d($c2,r($c1))=U. 70 [back_demod,2,demod,64,44] n(c(x),y)!=Z|u(y,x)=x. 71 [back_demod,1,demod,64,44] u(x,y)!=y|n(c(y),x)=Z. 109 [para_from,23.1.1,7.1.1] u(d(Z,$c1),d(Z,$c2))!=U. 112,111 [para_into,43.1.1.1,21.1.1] c(Z)=U. 116,115 [para_into,43.1.1.1,12.1.1] c(d(x,y))=k(c(x),c(y)). 132 [para_from,47.1.1,4.1.1.2] k(x,c(r(y)))!=Z|n(k(U,x),k(U,c(y)))=Z. 176 [para_into,37.1.1.1,39.1.1,demod,56,flip.1] k(I,r(x))=r(x). 213 [para_into,41.1.1.1.1,55.1.1] u(k(I,c(k(I,x))),c(x))=c(x). 262,261 [para_into,176.1.1.2,33.1.1,demod,34] k(I,x)=x. 263 [back_demod,213,demod,262,262] u(c(x),c(x))=c(x). 310 [para_into,263.1.1.1,43.1.1,demod,44,44] u(x,x)=x. 312 [para_into,53.1.1.1,66.1.1,demod,60,34,flip.1] d($c1,r($c2))=U. 371 [para_into,63.1.1.1,310.1.1,flip.1] n(c(x),c(x))=c(x). 428,427 [para_into,371.1.1.1,43.1.1,demod,44,44] n(x,x)=x. 520 [hyper,71,310] n(c(x),x)=Z. 596 [para_into,520.1.1.1,43.1.1] n(x,c(x))=Z. 678 [para_from,596.1.1,61.1.1.1,demod,112,44,flip.1] u(c(x),x)=U. 1014 [para_into,678.1.1,70.2.1,demod,428] x=U|c(x)!=Z. 1838 [para_from,1014.1.1,109.1.1,demod,64,116,112,116,112] U!=U|n(k(U,c($c1)),k(U,c($c2)))!=Z. 2241 [para_into,115.1.1.1,312.1.1,demod,22,flip.1] k(c($c1),c(r($c2)))=Z. 5176,5175 [hyper,2241,132] n(k(U,c($c1)),k(U,c($c2)))=Z. 5177 [back_demod,1838,demod,5176] U!=U|Z!=Z. 5202 [hyper,5177,52,52] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 253 clauses generated 14113 clauses kept 4925 clauses forward subsumed 8942 clauses back subsumed 794 Kbytes malloced 3161 ----------- times (seconds) ----------- user CPU time 2.01 (0 hr, 0 min, 2 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 15:35:30 2003