----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Sun Nov 30 19:28:49 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,20). 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("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("ix_a.txt"). ------- start included file ix_a.txt------- formula_list(usable). all x (k(x,I)=x). end_of_list. -------> usable clausifies to: list(usable). 0 [] k(x,I)=x. end_of_list. ------- end included file ix_a.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("cycleLawsB.txt"). ------- start included file cycleLawsB.txt------- formula_list(usable). all x y z (n(y,k(x,n(z,c(k(r(x),y)))))=Z). all x y z v (n(k(x,z),y)=n(u(k(n(x,v),z),k(n(x,c(v)),z)),y)). all x y z (n(y,k(n(x,c(k(y,r(z)))),z))=Z). end_of_list. -------> usable clausifies to: list(usable). 0 [] n(y,k(x,n(z,c(k(r(x),y)))))=Z. 0 [] n(k(x,z),y)=n(u(k(n(x,v),z),k(n(x,c(v)),z)),y). 0 [] n(y,k(n(x,c(k(y,r(z)))),z))=Z. end_of_list. ------- end included file cycleLawsB.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("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("xv_a.txt"). ------- start included file xv_a.txt------- formula_list(usable). all x y z v w i(n(w,k(n(y,z),v)),n(w,u(k(n(y,z),n(v,x)),k(y,n(v,c(x)))))). all y v w i(n(k(y,v),w),k(n(y,k(w,r(v))),n(v,k(r(y),w)))). all y v w i(n(k(y,v),w),k(y,n(v,k(r(y),w)))). end_of_list. -------> usable clausifies to: list(usable). 0 [] i(n(w,k(n(y,z),v)),n(w,u(k(n(y,z),n(v,x)),k(y,n(v,c(x)))))). 0 [] i(n(k(y,v),w),k(n(y,k(w,r(v))),n(v,k(r(y),w)))). 0 [] i(n(k(y,v),w),k(y,n(v,k(r(y),w)))). end_of_list. ------- end included file xv_a.txt------- include("xxi_a_ghost.txt"). ------- start included file xxi_a_ghost.txt------- formula_list(usable). all x (n(x,I)=n(r(x),I)). all x y (n(k(x,n(r(x),y)),I)=n(k(x,y),I)). all x y (n(k(n(x,r(y)),n(r(x),y)),I)=n(k(x,y),I)). end_of_list. -------> usable clausifies to: list(usable). 0 [] n(x,I)=n(r(x),I). 0 [] n(k(x,n(r(x),y)),I)=n(k(x,y),I). 0 [] n(k(n(x,r(y)),n(r(x),y)),I)=n(k(x,y),I). end_of_list. ------- end included file xxi_a_ghost.txt------- include("xxi_b_ghost.txt"). ------- start included file xxi_b_ghost.txt------- formula_list(usable). all x i(k(x,U),k(n(k(x,r(x)),I),U)). all x i(k(n(k(x,r(x)),I),U),k(x,U)). -(all x (k(x,U)=k(n(k(x,r(x)),I),U))). end_of_list. -------> usable clausifies to: list(usable). 0 [] i(k(x,U),k(n(k(x,r(x)),I),U)). 0 [] i(k(n(k(x,r(x)),I),U),k(x,U)). 0 [] k($c1,U)!=k(n(k($c1,r($c1)),I),U). end_of_list. ------- end included file xxi_b_ghost.txt------- SCAN INPUT: prop=0, horn=1, equality=1, symmetry=0, max_lits=2. This is a Horn set with equality. The strategy will be Knuth-Bendix and hyper_res, with positive clauses in sos and nonpositive clauses in usable. dependent: set(knuth_bendix). dependent: set(para_from). dependent: set(para_into). dependent: clear(para_from_right). dependent: clear(para_into_right). dependent: set(para_from_vars). dependent: set(eq_units_both_ways). dependent: set(dynamic_demod_all). dependent: set(dynamic_demod). dependent: set(order_eq). dependent: set(back_demod). dependent: set(lrpo). dependent: set(hyper_res). dependent: clear(order_hyper). ------------> process usable: ** KEPT (pick-wt=12): 1 [] u(x,y)!=y|c(u(y,c(x)))=Z. ** KEPT (pick-wt=12): 2 [] c(u(x,c(y)))!=Z|u(y,x)=x. ** KEPT (pick-wt=8): 3 [] -i(x,y)|i(r(x),r(y)). ** KEPT (pick-wt=8): 4 [] -i(x,y)|u(x,y)=y. ** KEPT (pick-wt=8): 5 [] i(x,y)|u(x,y)!=y. ** KEPT (pick-wt=12): 7 [copy,6,flip.1] k(n(k($c1,r($c1)),I),U)!=k($c1,U). ------------> process sos: ** KEPT (pick-wt=7): 8 [] u(x,y)=u(y,x). ** KEPT (pick-wt=11): 10 [copy,9,flip.1] u(u(x,y),z)=u(x,u(y,z)). ---> New Demodulator: 11 [new_demod,10] u(u(x,y),z)=u(x,u(y,z)). ** KEPT (pick-wt=14): 12 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ---> New Demodulator: 13 [new_demod,12] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ** KEPT (pick-wt=11): 15 [copy,14,flip.1] k(k(x,y),z)=k(x,k(y,z)). ---> New Demodulator: 16 [new_demod,15] k(k(x,y),z)=k(x,k(y,z)). ** KEPT (pick-wt=5): 17 [] r(r(x))=x. ---> New Demodulator: 18 [new_demod,17] r(r(x))=x. ** KEPT (pick-wt=10): 19 [] r(u(x,y))=u(r(x),r(y)). ---> New Demodulator: 20 [new_demod,19] r(u(x,y))=u(r(x),r(y)). ** KEPT (pick-wt=10): 21 [] r(k(x,y))=k(r(y),r(x)). ---> New Demodulator: 22 [new_demod,21] r(k(x,y))=k(r(y),r(x)). ** KEPT (pick-wt=10): 24 [copy,23,flip.1] c(u(c(x),c(y)))=n(x,y). ---> New Demodulator: 25 [new_demod,24] c(u(c(x),c(y)))=n(x,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=5): 35 [] k(x,I)=x. ---> New Demodulator: 36 [new_demod,35] k(x,I)=x. ** KEPT (pick-wt=5): 37 [] c(c(x))=x. ---> New Demodulator: 38 [new_demod,37] c(c(x))=x. ** KEPT (pick-wt=11): 40 [copy,39,demod,20] u(r(x),r(y))=u(r(x),r(y)). ** KEPT (pick-wt=7): 41 [] r(c(x))=c(r(x)). ---> New Demodulator: 42 [new_demod,41] r(c(x))=c(r(x)). ** KEPT (pick-wt=10): 43 [] r(n(x,y))=n(r(x),r(y)). ---> New Demodulator: 44 [new_demod,43] r(n(x,y))=n(r(x),r(y)). ** KEPT (pick-wt=3): 46 [copy,45,demod,18] x=x. ** KEPT (pick-wt=10): 47 [] r(d(x,y))=d(r(y),r(x)). ---> New Demodulator: 48 [new_demod,47] r(d(x,y))=d(r(y),r(x)). Following clause subsumed by 46 during input processing: 0 [demod,22] k(r(y),r(x))=k(r(y),r(x)). ** KEPT (pick-wt=4): 49 [] r(I)=I. ---> New Demodulator: 50 [new_demod,49] r(I)=I. ** KEPT (pick-wt=4): 51 [] r(Z)=Z. ---> New Demodulator: 52 [new_demod,51] r(Z)=Z. ** KEPT (pick-wt=4): 53 [] r(U)=U. ---> New Demodulator: 54 [new_demod,53] r(U)=U. ** KEPT (pick-wt=13): 55 [] n(x,k(y,n(z,c(k(r(y),x)))))=Z. ---> New Demodulator: 56 [new_demod,55] n(x,k(y,n(z,c(k(r(y),x)))))=Z. ** KEPT (pick-wt=20): 58 [copy,57,flip.1] n(u(k(n(x,y),z),k(n(x,c(y)),z)),u)=n(k(x,z),u). ---> New Demodulator: 59 [new_demod,58] n(u(k(n(x,y),z),k(n(x,c(y)),z)),u)=n(k(x,z),u). ** KEPT (pick-wt=13): 60 [] n(x,k(n(y,c(k(x,r(z)))),z))=Z. ---> New Demodulator: 61 [new_demod,60] n(x,k(n(y,c(k(x,r(z)))),z))=Z. ** KEPT (pick-wt=16): 62 [] n(x,k(n(y,k(x,r(z))),z))=n(k(y,z),x). ** KEPT (pick-wt=24): 63 [] i(n(x,k(n(y,z),u)),n(x,u(k(n(y,z),n(u,v)),k(y,n(u,c(v)))))). ** KEPT (pick-wt=19): 64 [] i(n(k(x,y),z),k(n(x,k(z,r(y))),n(y,k(r(x),z)))). ** KEPT (pick-wt=14): 65 [] i(n(k(x,y),z),k(x,n(y,k(r(x),z)))). ** KEPT (pick-wt=8): 67 [copy,66,flip.1] n(r(x),I)=n(x,I). ---> New Demodulator: 68 [new_demod,67] n(r(x),I)=n(x,I). ** KEPT (pick-wt=14): 69 [] n(k(x,n(r(x),y)),I)=n(k(x,y),I). ---> New Demodulator: 70 [new_demod,69] n(k(x,n(r(x),y)),I)=n(k(x,y),I). ** KEPT (pick-wt=17): 71 [] n(k(n(x,r(y)),n(r(x),y)),I)=n(k(x,y),I). ---> New Demodulator: 72 [new_demod,71] n(k(n(x,r(y)),n(r(x),y)),I)=n(k(x,y),I). ** KEPT (pick-wt=12): 73 [] i(k(x,U),k(n(k(x,r(x)),I),U)). ** KEPT (pick-wt=12): 74 [] i(k(n(k(x,r(x)),I),U),k(x,U)). Following clause subsumed by 8 during input processing: 0 [copy,8,flip.1] u(x,y)=u(y,x). >>>> Starting back demodulation with 11. >>>> Starting back demodulation with 13. >>>> Starting back demodulation with 16. >>>> Starting back demodulation with 18. >>>> Starting back demodulation with 20. >>>> Starting back demodulation with 22. >>>> Starting back demodulation with 25. >> back demodulating 12 with 25. >>>> Starting back demodulation with 28. >>>> Starting back demodulation with 31. >>>> Starting back demodulation with 34. >>>> Starting back demodulation with 36. >>>> Starting back demodulation with 38. Following clause subsumed by 46 during input processing: 0 [copy,40,flip.1] u(r(x),r(y))=u(r(x),r(y)). >>>> Starting back demodulation with 42. >>>> Starting back demodulation with 44. Following clause subsumed by 46 during input processing: 0 [copy,46,flip.1] x=x. 46 back subsumes 40. >>>> Starting back demodulation with 48. >>>> Starting back demodulation with 50. >>>> Starting back demodulation with 52. >>>> Starting back demodulation with 54. >>>> Starting back demodulation with 56. >>>> Starting back demodulation with 59. >>>> Starting back demodulation with 61. ** KEPT (pick-wt=16): 77 [copy,62,flip.1] n(k(x,y),z)=n(z,k(n(x,k(z,r(y))),y)). >>>> Starting back demodulation with 68. >>>> Starting back demodulation with 70. >>>> Starting back demodulation with 72. >>>> Starting back demodulation with 76. Following clause subsumed by 62 during input processing: 0 [copy,77,flip.1] n(x,k(n(y,k(x,r(z))),z))=n(k(y,z),x). ======= end of input processing ======= =========== start of search =========== ----> UNIT CONFLICT at 3.56 sec ----> 15279 [binary,15277.1,1303.1] $F. Length of proof is 28. Level of proof is 11. ---------------- PROOF ---------------- 4 [] -i(x,y)|u(x,y)=y. 6 [] k($c1,U)!=k(n(k($c1,r($c1)),I),U). 7 [copy,6,flip.1] k(n(k($c1,r($c1)),I),U)!=k($c1,U). 8 [] u(x,y)=u(y,x). 12 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. 18,17 [] r(r(x))=x. 21 [] r(k(x,y))=k(r(y),r(x)). 23 [] n(x,y)=c(u(c(x),c(y))). 25,24 [copy,23,flip.1] c(u(c(x),c(y)))=n(x,y). 32 [] Z=c(U). 33 [copy,32,flip.1] c(U)=Z. 36,35 [] k(x,I)=x. 38,37 [] c(c(x))=x. 50,49 [] r(I)=I. 53 [] r(U)=U. 55 [] n(x,k(y,n(z,c(k(r(y),x)))))=Z. 62 [] n(x,k(n(y,k(x,r(z))),z))=n(k(y,z),x). 71 [] n(k(n(x,r(y)),n(r(x),y)),I)=n(k(x,y),I). 73 [] i(k(x,U),k(n(k(x,r(x)),I),U)). 74 [] i(k(n(k(x,r(x)),I),U),k(x,U)). 75 [back_demod,12,demod,25] u(c(u(c(x),y)),n(x,y))=x. 103 [para_into,21.1.1.1,35.1.1,demod,50,flip.1] k(I,r(x))=r(x). 118,117 [para_into,103.1.1.2,17.1.1,demod,18] k(I,x)=x. 130,129 [para_into,24.1.1.1.1,37.1.1] c(u(x,c(y)))=n(c(x),y). 138,137 [para_into,24.1.1.1.2,37.1.1] c(u(c(x),y))=n(x,c(y)). 141 [para_into,24.1.1.1,8.1.1,demod,130,38] n(x,y)=n(y,x). 142 [back_demod,75,demod,138] u(n(x,c(y)),n(x,y))=x. 207 [para_into,55.1.1.2.2.2.1.1,49.1.1,demod,118,118] n(x,n(y,c(x)))=Z. 225 [para_into,207.1.1.2.2,37.1.1] n(c(x),n(y,x))=Z. 231 [para_into,207.1.1.2,141.1.1] n(x,n(c(x),y))=Z. 318 [para_into,62.1.1.2.1.2.2,49.1.1,demod,36,36,36] n(x,n(y,x))=n(y,x). 441 [para_into,71.1.1.1.1.2,53.1.1] n(k(n(x,U),n(r(x),U)),I)=n(k(x,U),I). 508 [para_into,225.1.1,141.1.1] n(n(x,y),c(y))=Z. 511,510 [para_into,231.1.1.2,231.1.1] n(x,Z)=Z. 583 [hyper,74,4] u(k(n(k(x,r(x)),I),U),k(x,U))=k(x,U). 686,685 [para_into,318.1.1,141.1.1] n(n(x,y),y)=n(x,y). 1109 [para_into,142.1.1.1.2,33.1.1,demod,511] u(Z,n(x,U))=x. 1128,1127 [para_into,142.1.1.1,508.1.1,demod,686] u(Z,n(x,y))=n(x,y). 1222,1221 [back_demod,1109,demod,1128] n(x,U)=x. 1291,1290 [back_demod,441,demod,1222,1222] n(k(x,r(x)),I)=n(k(x,U),I). 1298 [back_demod,583,demod,1291] u(k(n(k(x,U),I),U),k(x,U))=k(x,U). 1302 [back_demod,73,demod,1291] i(k(x,U),k(n(k(x,U),I),U)). 1303 [back_demod,7,demod,1291] k(n(k($c1,U),I),U)!=k($c1,U). 14990,14989 [para_into,1298.1.1,8.1.1] u(k(x,U),k(n(k(x,U),I),U))=k(x,U). 15277 [hyper,1302,4,demod,14990,flip.1] k(n(k(x,U),I),U)=k(x,U). 15279 [binary,15277.1,1303.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 971 clauses generated 96038 clauses kept 11299 clauses forward subsumed 78295 clauses back subsumed 195 Kbytes malloced 8845 ----------- times (seconds) ----------- user CPU time 3.75 (0 hr, 0 min, 3 sec) system CPU time 0.00 (0 hr, 0 min, 0 sec) wall-clock time 4 (0 hr, 0 min, 4 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 19:28:53 2003