----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Fri Nov 21 23:36:35 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). all x i(d(x,Z),x). -(all x (k(U,U)=U)). 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,x),x). 0 [] i(d(x,Z),x). 0 [] k(U,U)!=U. 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 [] k(U,U)!=U. ------------> 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)). ** KEPT (pick-wt=5): 47 [] i(d(Z,x),x). ** KEPT (pick-wt=5): 48 [] i(d(x,Z),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 50. ======= end of input processing ======= =========== start of search =========== Resetting weight limit to 9. sos_size=2177 ----> UNIT CONFLICT at 0.59 sec ----> 5369 [binary,5367.1,3.1] $F. Length of proof is 80. Level of proof is 23. ---------------- PROOF ---------------- 1 [] -i(x,y)|u(x,y)=y. 2 [] i(x,y)|u(x,y)!=y. 3 [] k(U,U)!=U. 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). 31,30 [copy,29,flip.1] c(I)=D. 32 [] U=u(I,D). 34,33 [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)). 48 [] i(d(x,Z),x). 49 [back_demod,8,demod,28] u(c(u(c(x),y)),n(x,y))=x. 52 [para_into,6.1.1,4.1.1] u(x,u(y,z))=u(y,u(z,x)). 53 [copy,52,flip.1] u(x,u(y,z))=u(z,u(x,y)). 55,54 [para_into,33.1.1,4.1.1] u(D,I)=U. 56 [para_from,33.1.1,6.1.1.1] u(U,x)=u(I,u(D,x)). 58 [hyper,45,1] u(x,k(x,U))=k(x,U). 68 [hyper,48,1] u(d(x,Z),x)=x. 86 [para_from,68.1.1,6.1.1.1,flip.1] u(d(x,Z),u(x,y))=u(x,y). 88 [para_into,17.1.1.1,43.1.1,flip.1] k(r(x),r(I))=r(x). 108 [para_into,88.1.1.1,13.1.1,demod,14] k(x,r(I))=x. 123 [para_into,108.1.1,43.1.1] r(I)=I. 133 [para_into,56.1.1,4.1.1] u(x,U)=u(I,u(D,x)). 134 [copy,133,flip.1] u(I,u(D,x))=u(x,U). 138 [para_into,24.1.1.1.1,123.1.1,demod,44,44] u(c(x),c(x))=c(x). 157 [para_into,138.1.1.1,36.1.1,demod,37,37] u(Z,Z)=Z. 159 [para_into,138.1.1.1,30.1.1,demod,31,31] u(D,D)=D. 163 [para_from,138.1.1,6.1.1.1,flip.1] u(c(x),u(c(x),y))=u(c(x),y). 167 [para_into,27.1.1.1.1,36.1.1] c(u(Z,c(x)))=n(U,x). 175 [para_into,27.1.1.1.2,30.1.1] c(u(c(x),D))=n(x,I). 180,179 [para_into,27.1.1.1,138.1.1] c(c(x))=n(x,x). 181 [para_into,27.1.1.1,4.1.1,demod,28] n(x,y)=n(y,x). 189 [para_from,157.1.1,6.1.1.1,flip.1] u(Z,u(Z,x))=u(Z,x). 198,197 [para_into,179.1.1.1,36.1.1] c(Z)=n(U,U). 200,199 [para_into,179.1.1.1,30.1.1] c(D)=n(I,I). 242 [para_from,39.1.1,27.1.1.1.1] c(u(d(x,y),c(z)))=n(k(c(x),c(y)),z). 251 [para_into,49.1.1.1.1.1,179.1.1] u(c(u(n(x,x),y)),n(c(x),y))=c(x). 253 [para_into,49.1.1.1.1.1,39.1.1] u(c(u(d(x,y),z)),n(k(c(x),c(y)),z))=k(c(x),c(y)). 255 [para_into,49.1.1.1.1.1,36.1.1] u(c(u(Z,x)),n(U,x))=U. 273 [para_into,49.1.1.2,181.1.1] u(c(u(c(x),y)),n(y,x))=x. 279 [para_from,49.1.1,6.1.1.1,flip.1] u(c(u(c(x),y)),u(n(x,y),z))=u(x,z). 404,403 [para_into,134.1.1.2,159.1.1,demod,34,flip.1] u(D,U)=U. 1171 [para_into,167.1.1.1.2,30.1.1] c(u(Z,D))=n(U,I). 1427 [hyper,189,2] i(Z,u(Z,x)). 1439 [para_into,1427.1.2,53.1.1] i(Z,u(x,u(Z,y))). 1466 [para_into,1439.1.2.2,58.1.1] i(Z,u(x,k(Z,U))). 1504 [para_into,1466.1.2,20.1.1] i(Z,k(u(x,Z),U)). 1565 [para_into,1504.1.2.1,4.1.1] i(Z,k(u(Z,x),U)). 2478 [hyper,163,2] i(c(x),u(c(x),y)). 2552 [para_into,2478.1.2,49.1.1] i(c(u(c(x),y)),x). 2564 [para_into,2552.1.1.1.1,199.1.1] i(c(u(n(I,I),x)),D). 2572 [para_into,2552.1.1.1.1,36.1.1] i(c(u(Z,x)),U). 2573 [para_into,2552.1.1.1.1,30.1.1] i(c(u(D,x)),I). 2585 [para_into,2552.1.1,27.1.1] i(n(x,y),x). 2586 [hyper,2585,1] u(n(x,y),x)=x. 2657 [para_into,2572.1.1.1,53.1.1] i(c(u(x,u(Z,y))),U). 2662 [para_into,2573.1.1.1,403.1.1,demod,37] i(Z,I). 2670,2669 [hyper,2662,1] u(Z,I)=I. 2674 [para_from,2669.1.1,1565.1.2.1,demod,44] i(Z,U). 2736,2735 [hyper,2674,1] u(Z,U)=U. 2856 [para_from,2735.1.1,86.1.1.2,demod,2736] u(d(Z,Z),U)=U. 3199,3198 [para_from,2586.1.1,53.1.1.2,flip.1] u(x,u(y,n(x,z)))=u(y,x). 3826 [para_into,2564.1.1.1,133.1.1,demod,3199,55,37] i(Z,D). 3831,3830 [hyper,3826,1] u(Z,D)=D. 3861 [back_demod,1171,demod,3831,200,flip.1] n(U,I)=n(I,I). 3912,3911 [para_into,242.1.1.1,68.1.1,demod,180,180,198,flip.1] n(k(n(x,x),n(U,U)),x)=n(x,x). 3954 [para_into,3861.1.1,181.1.1] n(I,U)=n(I,I). 3958 [para_from,3861.1.1,49.1.1.2,demod,37,2670,31] u(D,n(I,I))=U. 4103 [para_from,3954.1.1,49.1.1.2,demod,31,404,37] u(Z,n(I,I))=I. 4172 [para_into,3958.1.1,4.1.1] u(n(I,I),D)=U. 4361,4360 [para_into,253.1.1.1.1,2856.1.1,demod,37,198,198,3912,198,198] u(Z,n(U,U))=k(n(U,U),n(U,U)). 4564 [para_from,4172.1.1,251.1.1.1.1,demod,37,31,31] u(Z,n(D,D))=D. 4622,4621 [para_into,255.1.1.1.1,2735.1.1,demod,37,4361] k(n(U,U),n(U,U))=U. 4651 [back_demod,4360,demod,4622] u(Z,n(U,U))=U. 4811 [para_from,4651.1.1,255.1.1.1.1,demod,37] u(Z,n(U,n(U,U)))=U. 5127 [para_into,2657.1.1.1.2,4564.1.1] i(c(u(x,D)),U). 5128 [para_into,2657.1.1.1.2,4103.1.1] i(c(u(x,I)),U). 5208 [para_into,5127.1.1,175.1.1] i(n(x,I),U). 5209 [hyper,5208,1] u(n(x,I),U)=U. 5215,5214 [hyper,5128,1] u(c(u(x,I)),U)=U. 5295 [para_into,279.1.1.2,5209.1.1,demod,5215,flip.1] u(x,U)=U. 5364,5363 [para_from,5295.1.1,273.1.1.1.1,demod,37] u(Z,n(U,x))=x. 5366,5365 [back_demod,4811,demod,5364] n(U,U)=U. 5367 [back_demod,4621,demod,5366,5366] k(U,U)=U. 5369 [binary,5367.1,3.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 387 clauses generated 10360 clauses kept 3182 clauses forward subsumed 6893 clauses back subsumed 241 Kbytes malloced 4119 ----------- times (seconds) ----------- user CPU time 0.85 (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 23:36:36 2003