----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Sun Nov 30 15:27:38 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_distinct_vars,3). 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("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("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("i.txt"). ------- start included file i.txt------- formula_list(usable). all x y (i(x,y)->i(r(x),r(y))). all x y (r(u(x,y))=u(r(x),r(y))). all x (r(c(x))=c(r(x))). all x y (r(n(x,y))=n(r(x),r(y))). all x (r(r(x))=x). all x y (r(d(x,y))=d(r(y),r(x))). all x y (r(k(x,y))=k(r(y),r(x))). r(I)=I. r(Z)=Z. r(U)=U. end_of_list. -------> usable clausifies to: list(usable). 0 [] -i(x,y)|i(r(x),r(y)). 0 [] r(u(x,y))=u(r(x),r(y)). 0 [] r(c(x))=c(r(x)). 0 [] r(n(x,y))=n(r(x),r(y)). 0 [] r(r(x))=x. 0 [] r(d(x,y))=d(r(y),r(x)). 0 [] r(k(x,y))=k(r(y),r(x)). 0 [] r(I)=I. 0 [] r(Z)=Z. 0 [] r(U)=U. end_of_list. ------- end included file i.txt------- include("xii.txt"). ------- start included file xii.txt------- formula_list(usable). all x y z (i(k(x,y),c(z))<->i(k(r(x),z),c(y))). all x y z (i(k(x,y),c(z))<->i(k(z,r(y)),c(x))). end_of_list. -------> usable clausifies to: list(usable). 0 [] -i(k(x,y),c(z))|i(k(r(x),z),c(y)). 0 [] i(k(x,y),c(z))| -i(k(r(x),z),c(y)). 0 [] -i(k(x,y),c(z))|i(k(z,r(y)),c(x)). 0 [] i(k(x,y),c(z))| -i(k(z,r(y)),c(x)). end_of_list. ------- end included file xii.txt------- include("xix_a_ghost.txt"). ------- start included file xix_a_ghost.txt------- formula_list(usable). all x y (k(c(x),c(r(y)))=Z<->i(k(U,c(x)),y)). -(all x y (c(k(c(x),c(r(y))))=U<->i(k(U,c(x)),y))). end_of_list. -------> usable clausifies to: list(usable). 0 [] k(c(x),c(r(y)))!=Z|i(k(U,c(x)),y). 0 [] k(c(x),c(r(y)))=Z| -i(k(U,c(x)),y). 0 [] c(k(c($c2),c(r($c1))))=U|i(k(U,c($c2)),$c1). 0 [] c(k(c($c2),c(r($c1))))!=U| -i(k(U,c($c2)),$c1). end_of_list. ------- end included file xix_a_ghost.txt------- SCAN INPUT: prop=0, horn=0, equality=1, symmetry=0, max_lits=2. This ia a non-Horn set with equality. The strategy will be Knuth-Bendix, ordered hyper_res, factoring, and unit deletion, 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: set(unit_deletion). dependent: set(factor). ------------> 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=12): 3 [] u(x,y)!=y|c(u(y,c(x)))=Z. ** KEPT (pick-wt=12): 4 [] c(u(x,c(y)))!=Z|u(y,x)=x. ** KEPT (pick-wt=8): 5 [] -i(x,y)|i(r(x),r(y)). ** KEPT (pick-wt=13): 6 [] -i(k(x,y),c(z))|i(k(r(x),z),c(y)). ** KEPT (pick-wt=13): 7 [] i(k(x,y),c(z))| -i(k(r(x),z),c(y)). ** KEPT (pick-wt=13): 8 [] -i(k(x,y),c(z))|i(k(z,r(y)),c(x)). ** KEPT (pick-wt=13): 9 [] i(k(x,y),c(z))| -i(k(z,r(y)),c(x)). ** KEPT (pick-wt=14): 10 [] k(c(x),c(r(y)))!=Z|i(k(U,c(x)),y). ** KEPT (pick-wt=14): 11 [] k(c(x),c(r(y)))=Z| -i(k(U,c(x)),y). ** KEPT (pick-wt=15): 12 [] c(k(c($c2),c(r($c1))))!=U| -i(k(U,c($c2)),$c1). ------------> process sos: ** KEPT (pick-wt=4): 14 [copy,13,flip.1] c(I)=D. ---> New Demodulator: 15 [new_demod,14] c(I)=D. ** KEPT (pick-wt=5): 17 [copy,16,flip.1] u(I,D)=U. ---> New Demodulator: 18 [new_demod,17] u(I,D)=U. ** KEPT (pick-wt=4): 20 [copy,19,flip.1] c(U)=Z. ---> New Demodulator: 21 [new_demod,20] c(U)=Z. ** KEPT (pick-wt=7): 22 [] u(x,y)=u(y,x). ** KEPT (pick-wt=11): 24 [copy,23,flip.1] u(u(x,y),z)=u(x,u(y,z)). ---> New Demodulator: 25 [new_demod,24] u(u(x,y),z)=u(x,u(y,z)). ** KEPT (pick-wt=14): 26 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ---> New Demodulator: 27 [new_demod,26] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ** KEPT (pick-wt=5): 28 [] c(c(x))=x. ---> New Demodulator: 29 [new_demod,28] c(c(x))=x. ** KEPT (pick-wt=10): 30 [] r(u(x,y))=u(r(x),r(y)). ---> New Demodulator: 31 [new_demod,30] r(u(x,y))=u(r(x),r(y)). ** KEPT (pick-wt=7): 32 [] r(c(x))=c(r(x)). ---> New Demodulator: 33 [new_demod,32] r(c(x))=c(r(x)). ** KEPT (pick-wt=10): 34 [] r(n(x,y))=n(r(x),r(y)). ---> New Demodulator: 35 [new_demod,34] r(n(x,y))=n(r(x),r(y)). ** KEPT (pick-wt=5): 36 [] r(r(x))=x. ---> New Demodulator: 37 [new_demod,36] r(r(x))=x. ** KEPT (pick-wt=10): 38 [] r(d(x,y))=d(r(y),r(x)). ---> New Demodulator: 39 [new_demod,38] r(d(x,y))=d(r(y),r(x)). ** KEPT (pick-wt=10): 40 [] r(k(x,y))=k(r(y),r(x)). ---> New Demodulator: 41 [new_demod,40] r(k(x,y))=k(r(y),r(x)). ** KEPT (pick-wt=4): 42 [] r(I)=I. ---> New Demodulator: 43 [new_demod,42] r(I)=I. ** KEPT (pick-wt=4): 44 [] r(Z)=Z. ---> New Demodulator: 45 [new_demod,44] r(Z)=Z. ** KEPT (pick-wt=4): 46 [] r(U)=U. ---> New Demodulator: 47 [new_demod,46] r(U)=U. ** KEPT (pick-wt=15): 48 [] c(k(c($c2),c(r($c1))))=U|i(k(U,c($c2)),$c1). >>>> Starting back demodulation with 15. >>>> Starting back demodulation with 18. >>>> Starting back demodulation with 21. Following clause subsumed by 22 during input processing: 0 [copy,22,flip.1] u(x,y)=u(y,x). >>>> Starting back demodulation with 25. >>>> Starting back demodulation with 27. >>>> Starting back demodulation with 29. >>>> Starting back demodulation with 31. >>>> Starting back demodulation with 33. >>>> Starting back demodulation with 35. >>>> Starting back demodulation with 37. >>>> Starting back demodulation with 39. >>>> Starting back demodulation with 41. >>>> Starting back demodulation with 43. >>>> Starting back demodulation with 45. >>>> Starting back demodulation with 47. ======= end of input processing ======= =========== start of search =========== Resetting weight limit to 8. sos_size=4889 ----> UNIT CONFLICT at 63.84 sec ----> 8758 [binary,8757.1,5345.1] $F. Length of proof is 44. Level of proof is 16. ---------------- PROOF ---------------- 1 [] -i(x,y)|u(x,y)=y. 2 [] i(x,y)|u(x,y)!=y. 3 [] u(x,y)!=y|c(u(y,c(x)))=Z. 4 [] c(u(x,c(y)))!=Z|u(y,x)=x. 9 [] i(k(x,y),c(z))| -i(k(z,r(y)),c(x)). 11 [] k(c(x),c(r(y)))=Z| -i(k(U,c(x)),y). 12 [] c(k(c($c2),c(r($c1))))!=U| -i(k(U,c($c2)),$c1). 13 [] D=c(I). 14 [copy,13,flip.1] c(I)=D. 16 [] U=u(I,D). 17 [copy,16,flip.1] u(I,D)=U. 19 [] Z=c(U). 21,20 [copy,19,flip.1] c(U)=Z. 22 [] u(x,y)=u(y,x). 23 [] u(x,u(y,z))=u(u(x,y),z). 25,24 [copy,23,flip.1] u(u(x,y),z)=u(x,u(y,z)). 26 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. 29,28 [] c(c(x))=x. 33,32 [] r(c(x))=c(r(x)). 37,36 [] r(r(x))=x. 41,40 [] r(k(x,y))=k(r(y),r(x)). 47,46 [] r(U)=U. 48 [] c(k(c($c2),c(r($c1))))=U|i(k(U,c($c2)),$c1). 56 [para_from,14.1.1,4.1.1.1.2] c(u(x,D))!=Z|u(I,x)=x. 61 [para_from,14.1.1,3.2.1.1.2] u(I,x)!=x|c(u(x,D))=Z. 107 [para_from,22.1.1,2.2.1] i(x,y)|u(y,x)!=y. 110,109 [para_into,28.1.1.1,20.1.1] c(Z)=U. 113 [para_into,28.1.1.1,3.2.1,demod,110,flip.1] u(x,c(y))=U|u(y,x)!=x. 114 [para_into,28.1.1,28.1.1] x=x. 124 [para_from,28.1.1,11.1.1.1,demod,29] k(x,c(r(y)))=Z| -i(k(U,x),y). 135 [para_from,109.1.1,4.1.1.1.2] c(u(x,U))!=Z|u(Z,x)=x. 151,150 [para_into,24.1.1.1,17.1.1] u(U,x)=u(I,u(D,x)). 177 [para_into,26.1.1.1.1.1,28.1.1,demod,29] u(c(u(x,y)),c(u(x,c(y))))=c(x). 190 [para_into,26.1.1.2.1.2,109.1.1] u(c(u(c(x),Z)),c(u(c(x),U)))=x. 194 [para_into,26.1.1.2.1.2,20.1.1] u(c(u(c(x),U)),c(u(c(x),Z)))=x. 239 [para_from,32.1.1,9.2.1.2] i(k(x,c(y)),c(z))| -i(k(z,c(r(y))),c(x)). 336 [para_from,48.1.1,32.1.1.1,demod,47,41,33,37,33,flip.1] c(k(c($c1),c(r($c2))))=U|i(k(U,c($c2)),$c1). 607 [para_into,150.1.1,22.1.1] u(x,U)=u(I,u(D,x)). 610 [para_from,150.1.1,107.2.1] i(x,U)|u(I,u(D,x))!=U. 611 [para_from,150.1.1,4.1.1.1] c(u(I,u(D,c(x))))!=Z|u(x,U)=U. 667 [para_into,56.1.1.1,17.1.1,demod,21,unit_del,114] u(I,I)=I. 682 [para_from,667.1.1,24.1.1.1,flip.1] u(I,u(I,x))=u(I,x). 1657 [para_into,607.1.1,4.2.1,demod,151,flip.1] u(I,u(D,x))=U|c(u(I,u(D,c(x))))!=Z. 2033 [hyper,682,61,demod,25] c(u(I,u(x,D)))=Z. 2477,2476 [para_into,2033.1.1.1.2,22.1.1] c(u(I,u(D,x)))=Z. 2487,2486 [back_demod,1657,demod,2477,unit_del,114] u(I,u(D,x))=U. 2489,2488 [back_demod,611,demod,2487,21,unit_del,114] u(x,U)=U. 2493 [back_demod,194,demod,2489,21] u(Z,c(u(c(x),Z)))=x. 2495 [back_demod,190,demod,2489,21] u(c(u(c(x),Z)),Z)=x. 2498,2497 [back_demod,135,demod,2489,21,unit_del,114] u(Z,x)=x. 2516 [back_demod,610,demod,2487,unit_del,114] i(x,U). 2524,2523 [back_demod,150,demod,2487] u(U,x)=U. 2542,2541 [back_demod,2493,demod,2498] c(u(c(x),Z))=x. 2548,2547 [back_demod,2495,demod,2542] u(x,Z)=x. 2608 [hyper,2497,2] i(Z,x). 3613 [para_into,113.1.1.2,20.1.1,demod,2548,2524] x=U|U!=x. 4035 [para_from,3613.1.1,28.1.1.1,demod,21,flip.2] Z=x|c(x)!=U. 5345 [para_from,124.1.1,12.1.1.1,demod,110,unit_del,114,factor_simp] -i(k(U,c($c2)),$c1). 7830 [para_into,177.1.1.1.1,2497.1.1,demod,2498,29,110] u(c(x),x)=U. 7979 [para_into,7830.1.1,1.2.1] x=U| -i(c(x),x). 8254 [para_from,7979.1.1,4035.2.1,demod,29,unit_del,114] Z=x| -i(x,c(x)). 8412 [para_from,336.1.1,8254.2.2,unit_del,2516,5345,flip.1] k(c($c1),c(r($c2)))=Z. 8757 [para_from,8412.1.1,239.2.1,demod,29,unit_del,2608] i(k(x,c($c2)),$c1). 8758 [binary,8757.1,5345.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 1070 clauses generated 147461 clauses kept 8622 clauses forward subsumed 44444 clauses back subsumed 2409 Kbytes malloced 4279 ----------- times (seconds) ----------- user CPU time 64.04 (0 hr, 1 min, 4 sec) system CPU time 0.00 (0 hr, 0 min, 0 sec) wall-clock time 64 (0 hr, 1 min, 4 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 15:28:42 2003