----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Fri Nov 21 23:36:47 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). -(all x (d(Z,Z)=Z)). 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. 0 [] d(Z,Z)!=Z. 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 [] d(Z,Z)!=Z. ------------> 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). ** KEPT (pick-wt=5): 49 [] k(U,U)=U. ---> New Demodulator: 50 [new_demod,49] k(U,U)=U. 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. >>>> Starting back demodulation with 52. ======= end of input processing ======= =========== start of search =========== Resetting weight limit to 9. sos_size=2243 ----> UNIT CONFLICT at 0.66 sec ----> 5501 [binary,5499.1,3.1] $F. Length of proof is 89. Level of proof is 26. ---------------- PROOF ---------------- 1 [] -i(x,y)|u(x,y)=y. 2 [] i(x,y)|u(x,y)!=y. 3 [] d(Z,Z)!=Z. 4 [] u(x,y)=u(y,x). 5 [] u(x,u(y,z))=u(u(x,y),z). 7,6 [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. 15 [] r(u(x,y))=u(r(x),r(y)). 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). 22 [] k(x,I)=x. 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). 33 [copy,32,flip.1] u(I,D)=U. 35 [] Z=c(U). 37,36 [copy,35,flip.1] c(U)=Z. 44,43 [] k(I,x)=x. 45 [] i(x,k(x,U)). 46 [] i(x,k(U,x)). 47 [] i(d(Z,x),x). 49 [] k(U,U)=U. 51 [back_demod,8,demod,28] u(c(u(c(x),y)),n(x,y))=x. 54 [para_into,6.1.1,4.1.1] u(x,u(y,z))=u(y,u(z,x)). 55 [copy,54,flip.1] u(x,u(y,z))=u(z,u(x,y)). 57,56 [para_into,33.1.1,4.1.1] u(D,I)=U. 59,58 [para_from,33.1.1,6.1.1.1] u(U,x)=u(I,u(D,x)). 62 [para_into,45.1.2,43.1.1] i(I,U). 65,64 [hyper,62,1] u(I,U)=U. 68 [hyper,47,1] u(d(Z,x),x)=x. 72 [para_into,15.1.1.1,33.1.1] r(U)=u(r(I),r(D)). 74 [para_from,49.1.1,46.1.2] i(U,U). 77 [hyper,74,1,demod,59] u(I,u(D,U))=U. 85 [para_into,17.1.1.1,43.1.1,flip.1] k(r(x),r(I))=r(x). 93 [para_from,68.1.1,6.1.1.1,flip.1] u(d(Z,x),u(x,y))=u(x,y). 104,103 [para_into,77.1.1,4.1.1,demod,7,59,57,65] u(D,U)=U. 118,117 [para_into,20.1.1.2,43.1.1] u(k(x,y),y)=k(u(x,I),y). 145 [hyper,24,2] i(k(r(x),c(k(x,y))),c(y)). 154 [para_into,24.1.1.1.2.1,43.1.1,demod,118] k(u(r(I),I),c(x))=c(x). 156 [para_into,24.1.1.1.2.1,22.1.1,demod,31,31] u(k(r(x),c(x)),D)=D. 159,158 [para_into,24.1.1.2,36.1.1,demod,37] u(k(r(x),c(k(x,U))),Z)=Z. 172 [para_into,85.1.1.1,13.1.1,demod,14] k(x,r(I))=x. 175,174 [para_into,172.1.1,43.1.1] r(I)=I. 178 [back_demod,154,demod,175] k(u(I,I),c(x))=c(x). 191,190 [back_demod,72,demod,175] r(U)=u(I,r(D)). 192 [para_from,174.1.1,24.1.1.1.1,demod,44,44] u(c(x),c(x))=c(x). 194 [para_into,27.1.1.1.1,36.1.1] c(u(Z,c(x)))=n(U,x). 200 [para_into,27.1.1.1.2,36.1.1] c(u(c(x),Z))=n(x,U). 202 [para_into,27.1.1.1.2,30.1.1] c(u(c(x),D))=n(x,I). 206 [para_into,27.1.1.1,4.1.1,demod,28] n(x,y)=n(y,x). 210 [para_into,192.1.1.1,36.1.1,demod,37,37] u(Z,Z)=Z. 213,212 [para_into,192.1.1.1,30.1.1,demod,31,31] u(D,D)=D. 216 [para_from,192.1.1,27.1.1.1] c(c(x))=n(x,x). 220 [para_from,192.1.1,6.1.1.1,flip.1] u(c(x),u(c(x),y))=u(c(x),y). 254 [para_from,210.1.1,6.1.1.1,flip.1] u(Z,u(Z,x))=u(Z,x). 266 [para_into,51.1.1.1.1.1,27.1.1] u(c(u(n(x,y),z)),n(u(c(x),c(y)),z))=u(c(x),c(y)). 276 [para_into,51.1.1.1.1,4.1.1] u(c(u(x,c(y))),n(y,x))=y. 282 [para_into,51.1.1,4.1.1] u(n(x,y),c(u(c(x),y)))=x. 286 [para_from,51.1.1,6.1.1.1,flip.1] u(c(u(c(x),y)),u(n(x,y),z))=u(x,z). 295,294 [para_into,216.1.1.1,30.1.1] c(D)=n(I,I). 1233 [hyper,156,2] i(k(r(x),c(x)),D). 1271 [para_into,1233.1.1.1,190.1.1,demod,37] i(k(u(I,r(D)),Z),D). 1285 [para_into,1271.1.1.1,4.1.1] i(k(u(r(D),I),Z),D). 1357 [para_into,178.1.1.2,36.1.1,demod,37] k(u(I,I),Z)=Z. 1393,1392 [para_from,1357.1.1,20.1.1.2,demod,118,flip.1] k(u(x,u(I,I)),Z)=k(u(x,I),Z). 1491 [para_into,145.1.1.2.1,49.1.1,demod,191,37,37] i(k(u(I,r(D)),Z),Z). 1499 [para_into,1491.1.1.1,4.1.1] i(k(u(r(D),I),Z),Z). 1501,1500 [hyper,1499,1,demod,118,7,1393] k(u(r(D),I),Z)=Z. 1502 [back_demod,1285,demod,1501] i(Z,D). 1504,1503 [hyper,1502,1] u(Z,D)=D. 1898 [para_into,158.1.1,4.1.1] u(Z,k(r(x),c(k(x,U))))=Z. 1913 [para_from,158.1.1,93.1.1.2,demod,159] u(d(Z,k(r(x),c(k(x,U)))),Z)=Z. 2057 [para_into,194.1.1.1.2,30.1.1,demod,1504,295,flip.1] n(U,I)=n(I,I). 2136 [para_into,2057.1.1,206.1.1] n(I,U)=n(I,I). 2143,2142 [para_from,2136.1.1,51.1.1.2,demod,31,104,37] u(Z,n(I,I))=I. 2181,2180 [para_into,2142.1.1,4.1.1] u(n(I,I),Z)=I. 2373 [para_into,200.1.1.1.1,294.1.1,demod,2181,31,flip.1] n(D,U)=D. 2400 [para_into,2373.1.1,206.1.1] n(U,D)=D. 2404 [para_from,2400.1.1,51.1.1.2,demod,37,1504,295] u(n(I,I),D)=U. 2916 [para_into,254.1.1.2,2142.1.1,demod,2143] u(Z,I)=I. 3024 [hyper,220,2] i(c(x),u(c(x),y)). 3455 [para_into,3024.1.2,51.1.1] i(c(u(c(x),y)),x). 3743 [para_into,3455.1.1.1.1,36.1.1] i(c(u(Z,x)),U). 3758 [para_into,3455.1.1,27.1.1] i(n(x,y),x). 3771 [para_into,3758.1.1,206.1.1] i(n(x,y),y). 3774 [hyper,3771,1] u(n(x,y),y)=y. 3781 [para_into,3743.1.1.1,55.1.1] i(c(u(x,u(Z,y))),U). 4163 [para_into,266.1.1.1.1,2404.1.1,demod,37,31,31,213,31,31,213] u(Z,n(D,D))=D. 4743,4742 [para_from,3774.1.1,6.1.1.1,flip.1] u(n(x,y),u(y,z))=u(y,z). 4832 [para_into,276.1.1.1.1,58.1.1] u(c(u(I,u(D,c(x)))),n(x,U))=x. 5336 [para_into,3781.1.1.1.2,4163.1.1] i(c(u(x,D)),U). 5338 [para_into,3781.1.1.1.2,2916.1.1] i(c(u(x,I)),U). 5347 [para_into,5336.1.1,202.1.1] i(n(x,I),U). 5348 [hyper,5347,1] u(n(x,I),U)=U. 5358,5357 [hyper,5338,1] u(c(u(x,I)),U)=U. 5375,5374 [para_from,5348.1.1,286.1.1.2,demod,5358,flip.1] u(x,U)=U. 5377,5376 [para_from,5348.1.1,54.1.1.2,demod,5375,59,4743,flip.1] u(I,u(D,x))=U. 5405 [back_demod,4832,demod,5377,37] u(Z,n(x,U))=x. 5426 [para_from,5374.1.1,3455.1.1.1,demod,37] i(Z,x). 5427 [para_from,5374.1.1,282.1.1.2.1,demod,37] u(n(x,U),Z)=x. 5439,5438 [hyper,5426,1] u(Z,x)=x. 5443,5442 [back_demod,5405,demod,5439] n(x,U)=x. 5466,5465 [back_demod,1898,demod,5439] k(r(x),c(k(x,U)))=Z. 5470,5469 [back_demod,5427,demod,5443] u(x,Z)=x. 5499 [back_demod,1913,demod,5466,5470] d(Z,Z)=Z. 5501 [binary,5499.1,3.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 408 clauses generated 10111 clauses kept 3250 clauses forward subsumed 6892 clauses back subsumed 260 Kbytes malloced 4087 ----------- times (seconds) ----------- user CPU time 0.86 (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 23:36:48 2003