----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Sun Nov 30 22:12:21 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_seconds,20). assign(max_distinct_vars,3). 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("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("booleanLaws.txt"). ------- start included file booleanLaws.txt------- include("1To4booleanLaws.txt"). ------- start included file 1To4booleanLaws.txt------- formula_list(usable). all x y z (i(x,u(y,z))->i(n(x,c(z)),y)). all x y z (i(x,y)->i(u(z,x),u(z,y))). all x y (x=u(n(x,c(y)),n(x,y))). all x y i(n(x,y),y). end_of_list. -------> usable clausifies to: list(usable). 0 [] -i(x,u(y,z))|i(n(x,c(z)),y). 0 [] -i(x,y)|i(u(z,x),u(z,y)). 0 [] x=u(n(x,c(y)),n(x,y)). 0 [] i(n(x,y),y). end_of_list. ------- end included file 1To4booleanLaws.txt------- include("4To8booleanLaws.txt"). ------- start included file 4To8booleanLaws.txt------- formula_list(usable). all x y i(n(x,y),y). all x y z i(n(x,u(y,z)),u(y,n(x,z))). all x y (x=u(n(x,y),n(x,c(y)))). all x y z (i(x,y)->i(u(z,x),u(z,y))). -(all x y z (i(x,y)->i(n(z,x),n(z,y)))). end_of_list. -------> usable clausifies to: list(usable). 0 [] i(n(x,y),y). 0 [] i(n(x,u(y,z)),u(y,n(x,z))). 0 [] x=u(n(x,y),n(x,c(y))). 0 [] -i(x,y)|i(u(z,x),u(z,y)). 0 [] i($c3,$c2). 0 [] -i(n($c1,$c3),n($c1,$c2)). end_of_list. ------- end included file 4To8booleanLaws.txt------- ------- end included file booleanLaws.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=11): 3 [] -i(x,u(y,z))|i(n(x,c(z)),y). ** KEPT (pick-wt=10): 4 [] -i(x,y)|i(u(z,x),u(z,y)). Following clause subsumed by 4 during input processing: 0 [] -i(x,y)|i(u(z,x),u(z,y)). ** KEPT (pick-wt=7): 5 [] -i(n($c1,$c3),n($c1,$c2)). ------------> process sos: ** KEPT (pick-wt=7): 6 [] u(x,y)=u(y,x). ** KEPT (pick-wt=11): 8 [copy,7,flip.1] u(u(x,y),z)=u(x,u(y,z)). ---> New Demodulator: 9 [new_demod,8] u(u(x,y),z)=u(x,u(y,z)). ** KEPT (pick-wt=14): 10 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ---> New Demodulator: 11 [new_demod,10] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ** KEPT (pick-wt=10): 13 [copy,12,flip.1] c(u(c(x),c(y)))=n(x,y). ---> New Demodulator: 14 [new_demod,13] c(u(c(x),c(y)))=n(x,y). ** KEPT (pick-wt=4): 16 [copy,15,flip.1] c(I)=D. ---> New Demodulator: 17 [new_demod,16] c(I)=D. ** KEPT (pick-wt=5): 19 [copy,18,flip.1] u(I,D)=U. ---> New Demodulator: 20 [new_demod,19] u(I,D)=U. ** KEPT (pick-wt=4): 22 [copy,21,flip.1] c(U)=Z. ---> New Demodulator: 23 [new_demod,22] c(U)=Z. ** KEPT (pick-wt=10): 25 [copy,24,flip.1] u(n(x,c(y)),n(x,y))=x. ---> New Demodulator: 26 [new_demod,25] u(n(x,c(y)),n(x,y))=x. ** KEPT (pick-wt=5): 27 [] i(n(x,y),y). Following clause subsumed by 27 during input processing: 0 [] i(n(x,y),y). ** KEPT (pick-wt=11): 28 [] i(n(x,u(y,z)),u(y,n(x,z))). ** KEPT (pick-wt=10): 30 [copy,29,flip.1] u(n(x,y),n(x,c(y)))=x. ---> New Demodulator: 31 [new_demod,30] u(n(x,y),n(x,c(y)))=x. ** KEPT (pick-wt=3): 32 [] i($c3,$c2). Following clause subsumed by 6 during input processing: 0 [copy,6,flip.1] u(x,y)=u(y,x). >>>> Starting back demodulation with 9. >>>> Starting back demodulation with 11. >>>> Starting back demodulation with 14. >> back demodulating 10 with 14. >>>> Starting back demodulation with 17. >>>> Starting back demodulation with 20. >>>> Starting back demodulation with 23. >>>> Starting back demodulation with 26. >>>> Starting back demodulation with 31. >>>> Starting back demodulation with 34. ======= end of input processing ======= =========== start of search =========== ----> UNIT CONFLICT at 0.19 sec ----> 2336 [binary,2335.1,5.1] $F. Length of proof is 76. Level of proof is 22. ---------------- PROOF ---------------- 1 [] -i(x,y)|u(x,y)=y. 3 [] -i(x,u(y,z))|i(n(x,c(z)),y). 4 [] -i(x,y)|i(u(z,x),u(z,y)). 5 [] -i(n($c1,$c3),n($c1,$c2)). 6 [] u(x,y)=u(y,x). 7 [] u(x,u(y,z))=u(u(x,y),z). 8 [copy,7,flip.1] u(u(x,y),z)=u(x,u(y,z)). 10 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. 12 [] n(x,y)=c(u(c(x),c(y))). 14,13 [copy,12,flip.1] c(u(c(x),c(y)))=n(x,y). 15 [] D=c(I). 16 [copy,15,flip.1] c(I)=D. 18 [] U=u(I,D). 19 [copy,18,flip.1] u(I,D)=U. 21 [] Z=c(U). 23,22 [copy,21,flip.1] c(U)=Z. 24 [] x=u(n(x,c(y)),n(x,y)). 25 [copy,24,flip.1] u(n(x,c(y)),n(x,y))=x. 27 [] i(n(x,y),y). 28 [] i(n(x,u(y,z)),u(y,n(x,z))). 32 [] i($c3,$c2). 33 [back_demod,10,demod,14] u(c(u(c(x),y)),n(x,y))=x. 36 [hyper,32,4] i(u(x,$c3),u(x,$c2)). 42 [para_into,8.1.1,6.1.1] u(x,u(y,z))=u(y,u(z,x)). 43 [copy,42,flip.1] u(x,u(y,z))=u(z,u(x,y)). 46,45 [para_into,19.1.1,6.1.1] u(D,I)=U. 47 [para_from,19.1.1,8.1.1.1] u(U,x)=u(I,u(D,x)). 49 [hyper,27,4] i(u(x,n(y,z)),u(x,z)). 51 [hyper,27,1] u(n(x,y),y)=y. 53 [para_into,13.1.1.1.1,22.1.1] c(u(Z,c(x)))=n(U,x). 59 [para_into,13.1.1.1.2,22.1.1] c(u(c(x),Z))=n(x,U). 65 [para_into,13.1.1.1,6.1.1,demod,14] n(x,y)=n(y,x). 74 [para_into,25.1.1.1.2,22.1.1] u(n(x,Z),n(x,U))=x. 78 [para_into,25.1.1.1.2,13.1.1] u(n(x,n(y,z)),n(x,u(c(y),c(z))))=x. 106 [para_into,28.1.2,6.1.1] i(n(x,u(y,z)),u(n(x,z),y)). 110 [para_into,51.1.1,6.1.1] u(x,n(y,x))=x. 157 [para_into,110.1.1.2,65.1.1] u(x,n(x,y))=x. 167 [para_into,33.1.1.1.1.1,22.1.1] u(c(u(Z,x)),n(U,x))=U. 293 [para_into,47.1.1,110.1.1,flip.1] u(I,u(D,n(x,U)))=U. 338 [para_into,49.1.1,157.1.1] i(x,u(x,y)). 364 [para_into,338.1.2,157.1.1] i(x,x). 366 [para_into,338.1.2,45.1.1] i(D,U). 367 [para_into,338.1.2,43.1.1] i(x,u(y,u(x,z))). 369 [para_into,338.1.2,33.1.1] i(c(u(c(x),y)),x). 370 [para_into,338.1.2,19.1.1] i(I,U). 371 [para_into,338.1.2,6.1.1] i(x,u(y,x)). 374,373 [hyper,364,1] u(x,x)=x. 376 [hyper,366,1] u(D,U)=U. 403,402 [hyper,370,1] u(I,U)=U. 405 [hyper,371,3] i(n(x,c(x)),y). 412,411 [para_from,373.1.1,13.1.1.1] c(c(x))=n(x,x). 427 [para_into,53.1.1.1.2,22.1.1,demod,374] c(Z)=n(U,U). 451 [para_into,405.1.1.2,22.1.1] i(n(U,Z),x). 457 [hyper,451,1] u(n(U,Z),x)=x. 668 [para_into,369.1.1.1.1,22.1.1] i(c(u(Z,x)),U). 669 [para_into,369.1.1.1.1,16.1.1] i(c(u(D,x)),I). 675 [hyper,668,1] u(c(u(Z,x)),U)=U. 683 [para_into,669.1.1.1,376.1.1,demod,23] i(Z,I). 688 [hyper,683,1] u(Z,I)=I. 701 [para_from,688.1.1,367.1.2.2] i(Z,u(x,I)). 718 [para_into,701.1.2,47.1.1,demod,46,403] i(Z,U). 723,722 [hyper,718,1] u(Z,U)=U. 726 [para_into,74.1.1.2,65.1.1] u(n(x,Z),n(U,x))=x. 742 [para_from,722.1.1,367.1.2.2] i(Z,u(x,U)). 749 [hyper,742,3,demod,23] i(n(Z,Z),x). 753 [hyper,749,1] u(n(Z,Z),x)=x. 816 [para_into,78.1.1.2.2.2,427.1.1] u(n(x,n(y,Z)),n(x,u(c(y),n(U,U))))=x. 938 [para_into,457.1.1,110.1.1,flip.1] n(x,n(U,Z))=n(U,Z). 941,940 [para_into,457.1.1,74.1.1,flip.1] n(U,U)=U. 944 [para_into,457.1.1,6.1.1] u(x,n(U,Z))=x. 948 [back_demod,816,demod,941] u(n(x,n(y,Z)),n(x,u(c(y),U)))=x. 957 [back_demod,427,demod,941] c(Z)=U. 965,964 [para_from,957.1.1,411.1.1.1,demod,23,flip.1] n(Z,Z)=Z. 975,974 [para_from,957.1.1,53.1.1.1.2,demod,723,23,flip.1] n(U,Z)=Z. 981,980 [back_demod,753,demod,965] u(Z,x)=x. 987,986 [back_demod,944,demod,975] u(x,Z)=x. 989,988 [back_demod,938,demod,975,975] n(x,Z)=Z. 1000,999 [back_demod,675,demod,981] u(c(x),U)=U. 1017 [back_demod,167,demod,981] u(c(x),n(U,x))=U. 1045 [back_demod,59,demod,987,412] n(x,x)=n(x,U). 1049,1048 [back_demod,948,demod,989,989,1000,981] n(x,U)=x. 1052,1051 [back_demod,726,demod,989,981] n(U,x)=x. 1071,1070 [copy,1045,flip.1,demod,1049,flip.1] n(x,x)=x. 1087,1086 [back_demod,293,demod,1049] u(I,u(D,x))=U. 1090 [back_demod,1017,demod,1052] u(c(x),x)=U. 1114,1113 [back_demod,411,demod,1071] c(c(x))=x. 1121,1120 [back_demod,47,demod,1087] u(U,x)=U. 1230 [para_from,1090.1.1,36.1.1] i(U,u(c($c3),$c2)). 1265 [hyper,1230,1,demod,1121,flip.1] u(c($c3),$c2)=U. 1284 [para_from,1265.1.1,106.1.1.2,demod,1049] i(x,u(n(x,$c2),c($c3))). 2335 [hyper,1284,3,demod,1114] i(n(x,$c3),n(x,$c2)). 2336 [binary,2335.1,5.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 278 clauses generated 9176 clauses kept 1688 clauses forward subsumed 7001 clauses back subsumed 112 Kbytes malloced 1245 ----------- times (seconds) ----------- user CPU time 0.39 (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 Sun Nov 30 22:12:22 2003