----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Sun Nov 30 22:30:30 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(x,y)|i(n(z,x),n(z,y)). end_of_list. ------- end included file 4To8booleanLaws.txt------- include("9_10booleanLaws.txt"). ------- start included file 9_10booleanLaws.txt------- formula_list(usable). all x (u(x,Z)=x). all x y z (i(x,y)&i(y,z)->i(x,z)). end_of_list. -------> usable clausifies to: list(usable). 0 [] u(x,Z)=x. 0 [] -i(x,y)| -i(y,z)|i(x,z). end_of_list. ------- end included file 9_10booleanLaws.txt------- include("11booleanLaws.txt"). ------- start included file 11booleanLaws.txt------- formula_list(usable). all x y (c(n(x,y))=u(c(x),c(y))). all x y (c(u(x,y))=n(c(x),c(y))). end_of_list. -------> usable clausifies to: list(usable). 0 [] c(n(x,y))=u(c(x),c(y)). 0 [] c(u(x,y))=n(c(x),c(y)). end_of_list. ------- end included file 11booleanLaws.txt------- include("12booleanLaws.txt"). ------- start included file 12booleanLaws.txt------- formula_list(usable). all x y z (i(x,z)&i(y,z)->i(u(x,y),z)). end_of_list. -------> usable clausifies to: list(usable). 0 [] -i(x,z)| -i(y,z)|i(u(x,y),z). end_of_list. ------- end included file 12booleanLaws.txt------- include("13booleanLaws.txt"). ------- start included file 13booleanLaws.txt------- formula_list(usable). all x y (i(x,y)&i(y,x)->x=y). end_of_list. -------> usable clausifies to: list(usable). 0 [] -i(x,y)| -i(y,x)|x=y. end_of_list. ------- end included file 13booleanLaws.txt------- include("14booleanLaws.txt"). ------- start included file 14booleanLaws.txt------- formula_list(usable). c(D)=I. c(Z)=U. end_of_list. -------> usable clausifies to: list(usable). 0 [] c(D)=I. 0 [] c(Z)=U. end_of_list. ------- end included file 14booleanLaws.txt------- include("15_16booleanLaws.txt"). ------- start included file 15_16booleanLaws.txt------- formula_list(usable). all x (c(c(x))=x). all x y (u(x,y)=y->c(u(y,c(x)))=Z). all x y (c(u(y,c(x)))=Z->u(x,y)=y). end_of_list. -------> usable clausifies to: list(usable). 0 [] c(c(x))=x. 0 [] u(x,y)!=y|c(u(y,c(x)))=Z. 0 [] c(u(y,c(x)))!=Z|u(x,y)=y. end_of_list. ------- end included file 15_16booleanLaws.txt------- include("17To19booleanLaws.txt"). ------- start included file 17To19booleanLaws.txt------- formula_list(usable). all x (u(x,x)=x). all x y z (u(x,y)=y&u(y,z)=z->u(x,z)=z). all x y (u(x,y)=y->u(c(y),c(x))=c(x)). end_of_list. -------> usable clausifies to: list(usable). 0 [] u(x,x)=x. 0 [] u(x,y)!=y|u(y,z)!=z|u(x,z)=z. 0 [] u(x,y)!=y|u(c(y),c(x))=c(x). end_of_list. ------- end included file 17To19booleanLaws.txt------- include("20To24booleanLaws.txt"). ------- start included file 20To24booleanLaws.txt------- formula_list(usable). all x y (n(x,y)=n(y,x)). all x y (n(x,c(y))=Z->u(x,y)=y). all x y (u(x,y)=y->n(x,c(y))=Z). all x y (n(x,y)=Z&u(x,y)=U->c(x)=y). all x y (c(x)=y->n(x,y)=Z). all x y (c(x)=y->u(x,y)=U). end_of_list. -------> usable clausifies to: list(usable). 0 [] n(x,y)=n(y,x). 0 [] n(x,c(y))!=Z|u(x,y)=y. 0 [] u(x,y)!=y|n(x,c(y))=Z. 0 [] n(x,y)!=Z|u(x,y)!=U|c(x)=y. 0 [] c(x)!=y|n(x,y)=Z. 0 [] c(x)!=y|u(x,y)=U. end_of_list. ------- end included file 20To24booleanLaws.txt------- include("25booleanLaws.txt"). ------- start included file 25booleanLaws.txt------- formula_list(usable). -(all x y z (n(z,y)=Z->n(u(x,y),z)=n(z,x))). end_of_list. -------> usable clausifies to: list(usable). 0 [] n($c1,$c2)=Z. 0 [] n(u($c3,$c2),$c1)!=n($c1,$c3). end_of_list. ------- end included file 25booleanLaws.txt------- ------- end included file booleanLaws.txt------- SCAN INPUT: prop=0, horn=1, equality=1, symmetry=0, max_lits=3. This is a Horn set with equality. The strategy will be Knuth-Bendix and hyper_res, with positive clauses in sos and nonpositive clauses in usable. dependent: set(knuth_bendix). dependent: set(para_from). dependent: set(para_into). dependent: clear(para_from_right). dependent: clear(para_into_right). dependent: set(para_from_vars). dependent: set(eq_units_both_ways). dependent: set(dynamic_demod_all). dependent: set(dynamic_demod). dependent: set(order_eq). dependent: set(back_demod). dependent: set(lrpo). dependent: set(hyper_res). dependent: clear(order_hyper). ------------> process usable: ** KEPT (pick-wt=8): 1 [] -i(x,y)|u(x,y)=y. ** KEPT (pick-wt=8): 2 [] i(x,y)|u(x,y)!=y. ** KEPT (pick-wt=11): 3 [] -i(x,u(y,z))|i(n(x,c(z)),y). ** KEPT (pick-wt=10): 4 [] -i(x,y)|i(u(z,x),u(z,y)). Following clause subsumed by 4 during input processing: 0 [] -i(x,y)|i(u(z,x),u(z,y)). ** KEPT (pick-wt=10): 5 [] -i(x,y)|i(n(z,x),n(z,y)). ** KEPT (pick-wt=9): 6 [] -i(x,y)| -i(y,z)|i(x,z). ** KEPT (pick-wt=11): 7 [] -i(x,y)| -i(z,y)|i(u(x,z),y). ** KEPT (pick-wt=9): 8 [] -i(x,y)| -i(y,x)|x=y. ** KEPT (pick-wt=12): 9 [] u(x,y)!=y|c(u(y,c(x)))=Z. ** KEPT (pick-wt=12): 10 [] c(u(x,c(y)))!=Z|u(y,x)=x. ** KEPT (pick-wt=15): 11 [] u(x,y)!=y|u(y,z)!=z|u(x,z)=z. ** KEPT (pick-wt=13): 12 [] u(x,y)!=y|u(c(y),c(x))=c(x). ** KEPT (pick-wt=11): 13 [] n(x,c(y))!=Z|u(x,y)=y. ** KEPT (pick-wt=11): 14 [] u(x,y)!=y|n(x,c(y))=Z. ** KEPT (pick-wt=14): 15 [] n(x,y)!=Z|u(x,y)!=U|c(x)=y. ** KEPT (pick-wt=9): 16 [] c(x)!=y|n(x,y)=Z. ** KEPT (pick-wt=9): 17 [] c(x)!=y|u(x,y)=U. ** KEPT (pick-wt=9): 18 [] n(u($c3,$c2),$c1)!=n($c1,$c3). ------------> process sos: ** KEPT (pick-wt=7): 19 [] u(x,y)=u(y,x). ** KEPT (pick-wt=11): 21 [copy,20,flip.1] u(u(x,y),z)=u(x,u(y,z)). ---> New Demodulator: 22 [new_demod,21] u(u(x,y),z)=u(x,u(y,z)). ** KEPT (pick-wt=14): 23 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ---> New Demodulator: 24 [new_demod,23] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ** KEPT (pick-wt=10): 26 [copy,25,flip.1] c(u(c(x),c(y)))=n(x,y). ---> New Demodulator: 27 [new_demod,26] c(u(c(x),c(y)))=n(x,y). ** KEPT (pick-wt=4): 29 [copy,28,flip.1] c(I)=D. ---> New Demodulator: 30 [new_demod,29] c(I)=D. ** KEPT (pick-wt=5): 32 [copy,31,flip.1] u(I,D)=U. ---> New Demodulator: 33 [new_demod,32] u(I,D)=U. ** KEPT (pick-wt=4): 35 [copy,34,flip.1] c(U)=Z. ---> New Demodulator: 36 [new_demod,35] c(U)=Z. ** KEPT (pick-wt=10): 38 [copy,37,flip.1] u(n(x,c(y)),n(x,y))=x. ---> New Demodulator: 39 [new_demod,38] u(n(x,c(y)),n(x,y))=x. ** KEPT (pick-wt=5): 40 [] i(n(x,y),y). Following clause subsumed by 40 during input processing: 0 [] i(n(x,y),y). ** KEPT (pick-wt=11): 41 [] i(n(x,u(y,z)),u(y,n(x,z))). ** KEPT (pick-wt=10): 43 [copy,42,flip.1] u(n(x,y),n(x,c(y)))=x. ---> New Demodulator: 44 [new_demod,43] u(n(x,y),n(x,c(y)))=x. ** KEPT (pick-wt=5): 45 [] u(x,Z)=x. ---> New Demodulator: 46 [new_demod,45] u(x,Z)=x. ** KEPT (pick-wt=10): 47 [] c(n(x,y))=u(c(x),c(y)). ---> New Demodulator: 48 [new_demod,47] c(n(x,y))=u(c(x),c(y)). ** KEPT (pick-wt=10): 49 [] c(u(x,y))=n(c(x),c(y)). ---> New Demodulator: 50 [new_demod,49] c(u(x,y))=n(c(x),c(y)). ** KEPT (pick-wt=4): 51 [] c(D)=I. ---> New Demodulator: 52 [new_demod,51] c(D)=I. ** KEPT (pick-wt=4): 53 [] c(Z)=U. ---> New Demodulator: 54 [new_demod,53] c(Z)=U. ** KEPT (pick-wt=5): 55 [] c(c(x))=x. ---> New Demodulator: 56 [new_demod,55] c(c(x))=x. ** KEPT (pick-wt=5): 57 [] u(x,x)=x. ---> New Demodulator: 58 [new_demod,57] u(x,x)=x. ** KEPT (pick-wt=7): 59 [] n(x,y)=n(y,x). ** KEPT (pick-wt=5): 60 [] n($c1,$c2)=Z. ---> New Demodulator: 61 [new_demod,60] n($c1,$c2)=Z. Following clause subsumed by 19 during input processing: 0 [copy,19,flip.1] u(x,y)=u(y,x). >>>> Starting back demodulation with 22. >>>> Starting back demodulation with 24. >>>> Starting back demodulation with 27. >> back demodulating 23 with 27. >>>> Starting back demodulation with 30. >>>> Starting back demodulation with 33. >>>> Starting back demodulation with 36. >>>> Starting back demodulation with 39. >>>> Starting back demodulation with 44. >>>> Starting back demodulation with 46. >>>> Starting back demodulation with 48. >>>> Starting back demodulation with 50. >> back demodulating 26 with 50. >> back demodulating 10 with 50. >> back demodulating 9 with 50. >>>> Starting back demodulation with 52. >>>> Starting back demodulation with 54. >>>> Starting back demodulation with 56. >>>> Starting back demodulation with 58. Following clause subsumed by 59 during input processing: 0 [copy,59,flip.1] n(x,y)=n(y,x). >>>> Starting back demodulation with 61. Following clause subsumed by 62 during input processing: 0 [copy,62,flip.1] x=x. ======= end of input processing ======= =========== start of search =========== Resetting weight limit to 9. sos_size=6114 ----> UNIT CONFLICT at 17.29 sec ----> 8553 [binary,8551.1,223.1] $F. Length of proof is 20. Level of proof is 10. ---------------- PROOF ---------------- 1 [] -i(x,y)|u(x,y)=y. 2 [] i(x,y)|u(x,y)!=y. 4 [] -i(x,y)|i(u(z,x),u(z,y)). 5 [] -i(x,y)|i(n(z,x),n(z,y)). 8 [] -i(x,y)| -i(y,x)|x=y. 16 [] c(x)!=y|n(x,y)=Z. 18 [] n(u($c3,$c2),$c1)!=n($c1,$c3). 19 [] u(x,y)=u(y,x). 23 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. 37 [] x=u(n(x,c(y)),n(x,y)). 39,38 [copy,37,flip.1] u(n(x,c(y)),n(x,y))=x. 41 [] i(n(x,u(y,z)),u(y,n(x,z))). 42 [] x=u(n(x,y),n(x,c(y))). 43 [copy,42,flip.1] u(n(x,y),n(x,c(y)))=x. 46,45 [] u(x,Z)=x. 50,49 [] c(u(x,y))=n(c(x),c(y)). 56,55 [] c(c(x))=x. 59 [] n(x,y)=n(y,x). 60 [] n($c1,$c2)=Z. 62 [back_demod,23,demod,50,56,50,56,56,39] x=x. 66 [hyper,62,16] n(x,c(x))=Z. 94,93 [para_into,45.1.1,19.1.1] u(Z,x)=x. 129 [para_from,60.1.1,41.1.2.2,demod,46] i(n($c1,u(x,$c2)),x). 178 [hyper,93,2] i(Z,x). 181 [hyper,178,4,demod,46] i(x,u(x,y)). 210 [hyper,181,5] i(n(x,y),n(x,u(y,z))). 214 [para_into,181.1.2,43.1.1] i(n(x,y),x). 223 [para_from,59.1.1,18.1.1] n($c1,u($c3,$c2))!=n($c1,$c3). 273 [hyper,214,1] u(n(x,y),x)=x. 300 [para_from,66.1.1,43.1.1.1,demod,56,94] n(x,x)=x. 614 [hyper,129,5] i(n(x,n($c1,u(y,$c2))),n(x,y)). 7878 [para_into,210.1.1,300.1.1] i(x,n(x,u(x,y))). 7885 [hyper,7878,8,214] n(x,u(x,y))=x. 7895 [para_into,7885.1.1,59.1.1] n(u(x,y),x)=x. 7899 [para_into,7895.1.1.1,273.1.1] n(x,n(x,y))=n(x,y). 8506 [para_into,614.1.1,7899.1.1] i(n($c1,u(x,$c2)),n($c1,x)). 8551 [hyper,8506,8,210,flip.1] n($c1,u(x,$c2))=n($c1,x). 8553 [binary,8551.1,223.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 1207 clauses generated 867398 clauses kept 7924 clauses forward subsumed 202125 clauses back subsumed 187 Kbytes malloced 4311 ----------- times (seconds) ----------- user CPU time 17.50 (0 hr, 0 min, 17 sec) system CPU time 0.00 (0 hr, 0 min, 0 sec) wall-clock time 18 (0 hr, 0 min, 18 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 22:30:48 2003