----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Sat Nov 22 00:15:36 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("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("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("xi_a.txt"). ------- start included file xi_a.txt------- formula_list(usable). all x i(k(r(x),c(x)),D). -(all x i(I,d(c(x),r(x)))). end_of_list. -------> usable clausifies to: list(usable). 0 [] i(k(r(x),c(x)),D). 0 [] -i(I,d(c($c1),r($c1))). end_of_list. ------- end included file xi_a.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=7): 3 [] -i(I,d(c($c1),r($c1))). ------------> 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=5): 19 [] k(x,I)=x. ---> New Demodulator: 20 [new_demod,19] k(x,I)=x. ** KEPT (pick-wt=13): 21 [] u(k(r(x),c(k(x,y))),c(y))=c(y). ---> New Demodulator: 22 [new_demod,21] u(k(r(x),c(k(x,y))),c(y))=c(y). ** KEPT (pick-wt=4): 24 [copy,23,flip.1] c(I)=D. ---> New Demodulator: 25 [new_demod,24] c(I)=D. ** KEPT (pick-wt=5): 27 [copy,26,flip.1] u(I,D)=U. ---> New Demodulator: 28 [new_demod,27] u(I,D)=U. ** KEPT (pick-wt=4): 30 [copy,29,flip.1] c(U)=Z. ---> New Demodulator: 31 [new_demod,30] c(U)=Z. ** KEPT (pick-wt=10): 33 [copy,32,flip.1] c(k(c(x),c(y)))=d(x,y). ---> New Demodulator: 34 [new_demod,33] c(k(c(x),c(y)))=d(x,y). ** KEPT (pick-wt=7): 35 [] i(k(r(x),c(x)),D). 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 20. >>>> Starting back demodulation with 22. >>>> Starting back demodulation with 25. >>>> Starting back demodulation with 28. >>>> Starting back demodulation with 31. >>>> Starting back demodulation with 34. ======= end of input processing ======= =========== start of search =========== ----> UNIT CONFLICT at 0.53 sec ----> 5087 [binary,5086.1,3.1] $F. Length of proof is 87. Level of proof is 33. ---------------- PROOF ---------------- 1 [] -i(x,y)|u(x,y)=y. 2 [] i(x,y)|u(x,y)!=y. 3 [] -i(I,d(c($c1),r($c1))). 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)). 9,8 [] 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)). 18,17 [] r(k(x,y))=k(r(y),r(x)). 20,19 [] k(x,I)=x. 21 [] u(k(r(x),c(k(x,y))),c(y))=c(y). 23 [] D=c(I). 25,24 [copy,23,flip.1] c(I)=D. 26 [] U=u(I,D). 28,27 [copy,26,flip.1] u(I,D)=U. 29 [] Z=c(U). 31,30 [copy,29,flip.1] c(U)=Z. 32 [] d(x,y)=c(k(c(x),c(y))). 33 [copy,32,flip.1] c(k(c(x),c(y)))=d(x,y). 35 [] i(k(r(x),c(x)),D). 39 [para_into,6.1.1.1,4.1.1,demod,7] u(x,u(y,z))=u(y,u(x,z)). 40 [para_into,6.1.1,4.1.1] u(x,u(y,z))=u(y,u(z,x)). 41 [copy,40,flip.1] u(x,u(y,z))=u(z,u(x,y)). 45,44 [para_into,27.1.1,4.1.1] u(D,I)=U. 47,46 [para_from,27.1.1,6.1.1.1] u(U,x)=u(I,u(D,x)). 48 [para_into,8.1.1.1.1.1,30.1.1,demod,31] u(c(u(Z,x)),c(u(Z,c(x))))=U. 50 [para_into,8.1.1.1.1.1,24.1.1,demod,25] u(c(u(D,x)),c(u(D,c(x))))=I. 56 [para_into,8.1.1.2.1.2,30.1.1] u(c(u(c(x),U)),c(u(c(x),Z)))=x. 62 [para_into,8.1.1.2.1,4.1.1] u(c(u(c(x),y)),c(u(c(y),c(x))))=x. 64 [para_into,8.1.1,4.1.1] u(c(u(c(x),c(y))),c(u(c(x),y)))=x. 66 [para_from,8.1.1,6.1.1.1,flip.1] u(c(u(c(x),y)),u(c(u(c(x),c(y))),z))=u(x,z). 70 [para_into,35.1.1.1,13.1.1] i(k(x,c(r(x))),D). 72 [para_into,35.1.1.2,24.1.1] i(k(r(I),D),D). 75 [hyper,72,1] u(k(r(I),D),D)=D. 79 [hyper,70,1] u(k(x,c(r(x))),D)=D. 94 [para_into,15.1.1.1,75.1.1,demod,18,14,20,flip.1] u(r(D),r(D))=r(D). 117,116 [para_from,94.1.1,15.1.1.1,demod,14,14,14,flip.1] u(D,D)=D. 121 [para_from,116.1.1,6.1.1.1,flip.1] u(D,u(D,x))=u(D,x). 123 [para_into,17.1.1.1,19.1.1,flip.1] k(r(I),r(x))=r(x). 133 [para_into,123.1.1.2,13.1.1,demod,14] k(r(I),x)=x. 136,135 [para_into,133.1.1,19.1.1] r(I)=I. 138,137 [back_demod,133,demod,136] k(I,x)=x. 167,166 [para_from,135.1.1,21.1.1.1.1,demod,138,138] u(c(x),c(x))=c(x). 171 [para_into,33.1.1.1.1,33.1.1] c(k(d(x,y),c(z)))=d(k(c(x),c(y)),z). 196,195 [para_into,166.1.1.1,30.1.1,demod,31,31] u(Z,Z)=Z. 209 [para_from,195.1.1,6.1.1.1,flip.1] u(Z,u(Z,x))=u(Z,x). 212,211 [para_into,39.1.1.2,195.1.1,flip.1] u(Z,u(x,Z))=u(x,Z). 219 [para_into,39.1.1.2,44.1.1] u(x,U)=u(D,u(x,I)). 221 [para_into,39.1.1.2,27.1.1] u(x,U)=u(I,u(x,D)). 228 [copy,219,flip.1] u(D,u(x,I))=u(x,U). 229 [copy,221,flip.1] u(I,u(x,D))=u(x,U). 239 [para_into,40.1.1.2,44.1.1] u(x,U)=u(D,u(I,x)). 284,283 [para_into,121.1.1.2,44.1.1,demod,45] u(D,U)=U. 311 [para_into,48.1.1.1.1,195.1.1] u(c(Z),c(u(Z,c(Z))))=U. 323 [para_into,48.1.1.2.1.2,30.1.1,demod,196] u(c(u(Z,U)),c(Z))=U. 351 [para_into,50.1.1.1.1,283.1.1,demod,31,31] u(Z,c(u(D,Z)))=I. 357 [para_into,50.1.1.1.1,44.1.1,demod,31,25,117] u(Z,c(D))=I. 386 [para_into,357.1.1,4.1.1] u(c(D),Z)=I. 409,408 [para_into,351.1.1.2.1,4.1.1] u(Z,c(u(Z,D)))=I. 412 [para_from,351.1.1,48.1.1.2.1,demod,212,25] u(c(u(D,Z)),D)=U. 580 [para_into,412.1.1,4.1.1] u(D,c(u(D,Z)))=U. 609 [para_from,580.1.1,50.1.1.1.1,demod,31] u(Z,c(u(D,c(c(u(D,Z))))))=I. 984 [hyper,209,2] i(Z,u(Z,x)). 985 [para_into,209.1.1.2,408.1.1,demod,409] u(Z,I)=I. 997 [para_into,984.1.2,4.1.1] i(Z,u(x,Z)). 1024 [para_into,62.1.1.1.1,386.1.1,demod,25] u(D,c(u(c(Z),c(D))))=D. 1110,1109 [para_into,985.1.1,4.1.1] u(I,Z)=I. 1112,1111 [para_from,985.1.1,41.1.1.2,flip.1] u(I,u(x,Z))=u(x,I). 1120 [para_into,997.1.2,46.1.1,demod,1112,45] i(Z,U). 1122,1121 [hyper,1120,1] u(Z,U)=U. 1124,1123 [back_demod,323,demod,1122,31] u(Z,c(Z))=U. 1125 [back_demod,311,demod,1124,31] u(c(Z),Z)=U. 1190 [para_into,64.1.1.2.1,412.1.1,demod,31] u(c(u(c(u(D,Z)),c(D))),Z)=u(D,Z). 1361,1360 [para_into,66.1.1.2,166.1.1,demod,9,flip.1] u(x,c(u(c(x),c(y))))=x. 1373 [para_into,66.1.1.2,8.1.1,demod,1361] u(c(u(c(x),y)),x)=x. 1395,1394 [para_from,1125.1.1,64.1.1.2.1,demod,167,31] u(c(c(Z)),Z)=Z. 1802 [para_into,1111.1.1.2,1394.1.1,demod,1110,flip.1] u(c(c(Z)),I)=I. 1810 [para_into,1111.1.1.2,386.1.1,flip.1] u(c(D),I)=u(I,I). 1832 [para_from,1802.1.1,228.1.1.2,demod,45,flip.1] u(c(c(Z)),U)=U. 1868,1867 [para_from,1832.1.1,56.1.1.1.1,demod,31,1395,1124,flip.1] c(Z)=U. 1879 [back_demod,1024,demod,1868,47] u(D,c(u(I,u(D,c(D)))))=D. 1926 [para_from,1867.1.1,166.1.1.2,demod,1868,47,284,1868] u(I,U)=U. 1957,1956 [para_into,1926.1.1,239.1.1] u(D,u(I,I))=U. 1958 [para_from,1926.1.1,41.1.1.2,demod,47] u(x,U)=u(I,u(D,u(x,I))). 1968 [copy,1958,flip.1] u(I,u(D,u(x,I)))=u(x,U). 2013,2012 [para_from,1810.1.1,41.1.1.2,flip.1] u(I,u(x,c(D)))=u(x,u(I,I)). 2027,2026 [back_demod,1879,demod,2013,1957,31] u(D,Z)=D. 2036 [back_demod,1190,demod,2027,167,2027] u(c(c(D)),Z)=D. 2045,2044 [back_demod,609,demod,2027] u(Z,c(u(D,c(c(D)))))=I. 2178 [para_from,2036.1.1,1111.1.1.2,demod,28,flip.1] u(c(c(D)),I)=U. 2301,2300 [para_from,2178.1.1,62.1.1.1.1,demod,31,25,2045,flip.1] c(D)=I. 2335,2334 [para_from,2300.1.1,33.1.1.1.2,demod,20] c(c(x))=d(x,D). 3813 [hyper,1373,2] i(c(u(c(x),y)),x). 4011,4010 [para_from,1373.1.1,229.1.1.2,demod,28,2301,flip.1] u(c(u(I,x)),U)=U. 4025,4024 [para_from,1373.1.1,56.1.1.2.1,demod,1868,47,4011,31,1868,1122,1868,47,flip.1] u(I,u(D,x))=U. 4077 [back_demod,1968,demod,4025,flip.1] u(x,U)=U. 4268,4267 [para_from,4077.1.1,1373.1.1.1.1,demod,31] u(Z,x)=x. 4274,4273 [para_from,4077.1.1,62.1.1.1.1,demod,31,31,4268,2335,4268] d(x,D)=x. 4535 [back_demod,2334,demod,4274] c(c(x))=x. 4711,4710 [para_from,4273.1.1,171.1.1.1.1,demod,2301,20] c(k(x,c(y)))=d(c(x),y). 4821 [para_into,3813.1.1.1.1,4535.1.1] i(c(u(x,y)),c(x)). 5086 [para_into,4821.1.1.1,79.1.1,demod,2301,4711] i(I,d(c(x),r(x))). 5087 [binary,5086.1,3.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 277 clauses generated 7565 clauses kept 2902 clauses forward subsumed 5996 clauses back subsumed 69 Kbytes malloced 3384 ----------- times (seconds) ----------- user CPU time 0.73 (0 hr, 0 min, 0 sec) system CPU time 0.00 (0 hr, 0 min, 0 sec) wall-clock time 0 (0 hr, 0 min, 0 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 Sat Nov 22 00:15:36 2003