----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Sun Nov 30 22:11:39 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)))). end_of_list. -------> usable clausifies to: list(usable). 0 [] i(n(x,y),y). 0 [] -i(n($c3,u($c2,$c1)),u($c2,n($c3,$c1))). 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)). ** KEPT (pick-wt=11): 5 [] -i(n($c3,u($c2,$c1)),u($c2,n($c3,$c1))). ------------> 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). 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 29. ======= end of input processing ======= =========== start of search =========== Resetting weight limit to 11. sos_size=3739 ----> UNIT CONFLICT at 1.57 sec ----> 8138 [binary,8137.1,1554.1] $F. Length of proof is 64. Level of proof is 20. ---------------- PROOF ---------------- 1 [] -i(x,y)|u(x,y)=y. 3 [] -i(x,u(y,z))|i(n(x,c(z)),y). 4 [] -i(x,y)|i(u(z,x),u(z,y)). 5 [] -i(n($c3,u($c2,$c1)),u($c2,n($c3,$c1))). 6 [] u(x,y)=u(y,x). 7 [] u(x,u(y,z))=u(u(x,y),z). 9,8 [copy,7,flip.1] u(u(x,y),z)=u(x,u(y,z)). 10 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. 12 [] n(x,y)=c(u(c(x),c(y))). 14,13 [copy,12,flip.1] c(u(c(x),c(y)))=n(x,y). 15 [] D=c(I). 16 [copy,15,flip.1] c(I)=D. 18 [] U=u(I,D). 19 [copy,18,flip.1] u(I,D)=U. 21 [] Z=c(U). 23,22 [copy,21,flip.1] c(U)=Z. 24 [] x=u(n(x,c(y)),n(x,y)). 25 [copy,24,flip.1] u(n(x,c(y)),n(x,y))=x. 27 [] i(n(x,y),y). 28 [back_demod,10,demod,14] u(c(u(c(x),y)),n(x,y))=x. 31 [para_from,6.1.1,5.1.2] -i(n($c3,u($c2,$c1)),u(n($c3,$c1),$c2)). 35 [para_into,8.1.1.1,6.1.1,demod,9] u(x,u(y,z))=u(y,u(x,z)). 36 [para_into,8.1.1,6.1.1] u(x,u(y,z))=u(y,u(z,x)). 37 [copy,36,flip.1] u(x,u(y,z))=u(z,u(x,y)). 40,39 [para_into,19.1.1,6.1.1] u(D,I)=U. 41 [para_from,19.1.1,8.1.1.1] u(U,x)=u(I,u(D,x)). 43 [hyper,27,4] i(u(x,n(y,z)),u(x,z)). 45 [hyper,27,1] u(n(x,y),y)=y. 47 [para_into,13.1.1.1.1,22.1.1] c(u(Z,c(x)))=n(U,x). 59 [para_into,13.1.1.1,6.1.1,demod,14] n(x,y)=n(y,x). 62 [para_into,45.1.1,6.1.1] u(x,n(y,x))=x. 64 [para_from,45.1.1,8.1.1.1,flip.1] u(n(x,y),u(y,z))=u(y,z). 71 [para_into,25.1.1.1.2,22.1.1] u(n(x,Z),n(x,U))=x. 79 [para_into,25.1.1.2,59.1.1] u(n(x,c(y)),n(y,x))=x. 87 [para_into,62.1.1.2,59.1.1] u(x,n(x,y))=x. 125 [hyper,43,3] i(n(u(x,n(y,z)),c(z)),x). 128 [para_into,43.1.1,87.1.1] i(x,u(x,y)). 147 [para_into,128.1.2,87.1.1] i(x,x). 149 [para_into,128.1.2,39.1.1] i(D,U). 150 [para_into,128.1.2,28.1.1] i(c(u(c(x),y)),x). 152 [para_into,128.1.2,6.1.1] i(x,u(y,x)). 155,154 [hyper,147,1] u(x,x)=x. 156 [para_into,31.1.1.2,6.1.1] -i(n($c3,u($c1,$c2)),u(n($c3,$c1),$c2)). 160 [hyper,149,1] u(D,U)=U. 166 [hyper,152,3] i(n(x,c(x)),y). 172,171 [para_from,154.1.1,13.1.1.1] c(c(x))=n(x,x). 187 [para_into,166.1.1.2,22.1.1] i(n(U,Z),x). 193 [hyper,187,1] u(n(U,Z),x)=x. 225 [para_from,35.1.1,128.1.2] i(x,u(y,u(x,z))). 274 [para_into,150.1.1.1.1,22.1.1] i(c(u(Z,x)),U). 275 [para_into,150.1.1.1.1,16.1.1] i(c(u(D,x)),I). 301 [para_into,274.1.1.1,6.1.1] i(c(u(x,Z)),U). 308 [para_into,275.1.1.1,160.1.1,demod,23] i(Z,I). 314 [hyper,308,1] u(Z,I)=I. 341,340 [para_from,314.1.1,37.1.1.2,flip.1] u(I,u(x,Z))=u(x,I). 355 [para_into,301.1.1.1,41.1.1,demod,341,40,23] i(Z,U). 358 [para_into,47.1.1.1.2,22.1.1,demod,155] c(Z)=n(U,U). 375,374 [hyper,355,1] u(Z,U)=U. 615,614 [para_into,64.1.1.2,374.1.1,demod,375] u(n(x,Z),U)=U. 642 [para_into,225.1.2.2,374.1.1] i(Z,u(x,U)). 647 [hyper,642,3,demod,23] i(n(Z,Z),x). 652 [hyper,647,1] u(n(Z,Z),x)=x. 706 [para_into,71.1.1.2,59.1.1] u(n(x,Z),n(U,x))=x. 709,708 [para_into,71.1.1,193.1.1] n(U,U)=U. 717 [back_demod,358,demod,709] c(Z)=U. 719 [para_from,71.1.1,43.1.1,demod,615] i(x,U). 731,730 [hyper,719,1] u(x,U)=U. 752,751 [para_from,717.1.1,13.1.1.1.2,demod,731,23,flip.1] n(x,Z)=Z. 769 [back_demod,706,demod,752] u(Z,n(U,x))=x. 772,771 [back_demod,652,demod,752] u(Z,x)=x. 780,779 [back_demod,769,demod,772] n(U,x)=x. 796,795 [back_demod,47,demod,772,172,780] n(x,x)=x. 819,818 [back_demod,171,demod,796] c(c(x))=x. 906 [para_from,79.1.1,8.1.1.1,flip.1] u(n(x,c(y)),u(n(y,x),z))=u(x,z). 1298 [para_into,125.1.1.1,6.1.1] i(n(u(n(x,y),z),c(y)),z). 1554 [para_into,156.1.1,59.1.1] -i(n(u($c1,$c2),$c3),u(n($c3,$c1),$c2)). 8137 [para_into,1298.1.1.1,906.1.1,demod,819] i(n(u(x,y),z),u(n(z,x),y)). 8138 [binary,8137.1,1554.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 667 clauses generated 60819 clauses kept 6163 clauses forward subsumed 48399 clauses back subsumed 447 Kbytes malloced 4023 ----------- times (seconds) ----------- user CPU time 1.87 (0 hr, 0 min, 1 sec) system CPU time 0.00 (0 hr, 0 min, 0 sec) wall-clock time 2 (0 hr, 0 min, 2 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 22:11:41 2003