----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Sun Nov 30 22:28:56 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)). 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($c2,$c1)=$c1. 0 [] n($c2,c($c1))!=Z. end_of_list. ------- end included file 20To24booleanLaws.txt------- ------- end included file booleanLaws.txt------- SCAN INPUT: prop=0, horn=1, equality=1, symmetry=0, max_lits=3. This is a Horn set with equality. The strategy will be Knuth-Bendix and hyper_res, with positive clauses in sos and nonpositive clauses in usable. dependent: set(knuth_bendix). dependent: set(para_from). dependent: set(para_into). dependent: clear(para_from_right). dependent: clear(para_into_right). dependent: set(para_from_vars). dependent: set(eq_units_both_ways). dependent: set(dynamic_demod_all). dependent: set(dynamic_demod). dependent: set(order_eq). dependent: set(back_demod). dependent: set(lrpo). dependent: set(hyper_res). dependent: clear(order_hyper). ------------> process usable: ** KEPT (pick-wt=8): 1 [] -i(x,y)|u(x,y)=y. ** KEPT (pick-wt=8): 2 [] i(x,y)|u(x,y)!=y. ** KEPT (pick-wt=11): 3 [] -i(x,u(y,z))|i(n(x,c(z)),y). ** KEPT (pick-wt=10): 4 [] -i(x,y)|i(u(z,x),u(z,y)). Following clause subsumed by 4 during input processing: 0 [] -i(x,y)|i(u(z,x),u(z,y)). ** KEPT (pick-wt=10): 5 [] -i(x,y)|i(n(z,x),n(z,y)). ** KEPT (pick-wt=9): 6 [] -i(x,y)| -i(y,z)|i(x,z). ** KEPT (pick-wt=11): 7 [] -i(x,y)| -i(z,y)|i(u(x,z),y). ** KEPT (pick-wt=9): 8 [] -i(x,y)| -i(y,x)|x=y. ** KEPT (pick-wt=12): 9 [] u(x,y)!=y|c(u(y,c(x)))=Z. ** KEPT (pick-wt=12): 10 [] c(u(x,c(y)))!=Z|u(y,x)=x. ** KEPT (pick-wt=15): 11 [] u(x,y)!=y|u(y,z)!=z|u(x,z)=z. ** KEPT (pick-wt=13): 12 [] u(x,y)!=y|u(c(y),c(x))=c(x). ** KEPT (pick-wt=11): 13 [] n(x,c(y))!=Z|u(x,y)=y. ** KEPT (pick-wt=6): 14 [] n($c2,c($c1))!=Z. ------------> process sos: ** KEPT (pick-wt=7): 15 [] u(x,y)=u(y,x). ** KEPT (pick-wt=11): 17 [copy,16,flip.1] u(u(x,y),z)=u(x,u(y,z)). ---> New Demodulator: 18 [new_demod,17] u(u(x,y),z)=u(x,u(y,z)). ** KEPT (pick-wt=14): 19 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ---> New Demodulator: 20 [new_demod,19] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ** KEPT (pick-wt=10): 22 [copy,21,flip.1] c(u(c(x),c(y)))=n(x,y). ---> New Demodulator: 23 [new_demod,22] c(u(c(x),c(y)))=n(x,y). ** KEPT (pick-wt=4): 25 [copy,24,flip.1] c(I)=D. ---> New Demodulator: 26 [new_demod,25] c(I)=D. ** KEPT (pick-wt=5): 28 [copy,27,flip.1] u(I,D)=U. ---> New Demodulator: 29 [new_demod,28] u(I,D)=U. ** KEPT (pick-wt=4): 31 [copy,30,flip.1] c(U)=Z. ---> New Demodulator: 32 [new_demod,31] c(U)=Z. ** KEPT (pick-wt=10): 34 [copy,33,flip.1] u(n(x,c(y)),n(x,y))=x. ---> New Demodulator: 35 [new_demod,34] u(n(x,c(y)),n(x,y))=x. ** KEPT (pick-wt=5): 36 [] i(n(x,y),y). Following clause subsumed by 36 during input processing: 0 [] i(n(x,y),y). ** KEPT (pick-wt=11): 37 [] i(n(x,u(y,z)),u(y,n(x,z))). ** KEPT (pick-wt=10): 39 [copy,38,flip.1] u(n(x,y),n(x,c(y)))=x. ---> New Demodulator: 40 [new_demod,39] u(n(x,y),n(x,c(y)))=x. ** KEPT (pick-wt=5): 41 [] u(x,Z)=x. ---> New Demodulator: 42 [new_demod,41] u(x,Z)=x. ** KEPT (pick-wt=10): 43 [] c(n(x,y))=u(c(x),c(y)). ---> New Demodulator: 44 [new_demod,43] c(n(x,y))=u(c(x),c(y)). ** KEPT (pick-wt=10): 45 [] c(u(x,y))=n(c(x),c(y)). ---> New Demodulator: 46 [new_demod,45] c(u(x,y))=n(c(x),c(y)). ** KEPT (pick-wt=4): 47 [] c(D)=I. ---> New Demodulator: 48 [new_demod,47] c(D)=I. ** KEPT (pick-wt=4): 49 [] c(Z)=U. ---> New Demodulator: 50 [new_demod,49] c(Z)=U. ** KEPT (pick-wt=5): 51 [] c(c(x))=x. ---> New Demodulator: 52 [new_demod,51] c(c(x))=x. ** KEPT (pick-wt=5): 53 [] u(x,x)=x. ---> New Demodulator: 54 [new_demod,53] u(x,x)=x. ** KEPT (pick-wt=7): 55 [] n(x,y)=n(y,x). ** KEPT (pick-wt=5): 56 [] u($c2,$c1)=$c1. ---> New Demodulator: 57 [new_demod,56] u($c2,$c1)=$c1. Following clause subsumed by 15 during input processing: 0 [copy,15,flip.1] u(x,y)=u(y,x). >>>> Starting back demodulation with 18. >>>> Starting back demodulation with 20. >>>> Starting back demodulation with 23. >> back demodulating 19 with 23. >>>> Starting back demodulation with 26. >>>> Starting back demodulation with 29. >>>> Starting back demodulation with 32. >>>> Starting back demodulation with 35. >>>> Starting back demodulation with 40. >>>> Starting back demodulation with 42. >>>> Starting back demodulation with 44. >>>> Starting back demodulation with 46. >> back demodulating 22 with 46. >> back demodulating 10 with 46. >> back demodulating 9 with 46. >>>> Starting back demodulation with 48. >>>> Starting back demodulation with 50. >>>> Starting back demodulation with 52. >>>> Starting back demodulation with 54. Following clause subsumed by 55 during input processing: 0 [copy,55,flip.1] n(x,y)=n(y,x). >>>> Starting back demodulation with 57. Following clause subsumed by 58 during input processing: 0 [copy,58,flip.1] x=x. ======= end of input processing ======= =========== start of search =========== ----> UNIT CONFLICT at 0.06 sec ----> 349 [binary,347.1,14.1] $F. Length of proof is 8. Level of proof is 7. ---------------- PROOF ---------------- 2 [] i(x,y)|u(x,y)!=y. 3 [] -i(x,u(y,z))|i(n(x,c(z)),y). 4 [] -i(x,y)|i(u(z,x),u(z,y)). 6 [] -i(x,y)| -i(y,z)|i(x,z). 8 [] -i(x,y)| -i(y,x)|x=y. 14 [] n($c2,c($c1))!=Z. 15 [] u(x,y)=u(y,x). 42,41 [] u(x,Z)=x. 56 [] u($c2,$c1)=$c1. 74 [para_into,41.1.1,15.1.1] u(Z,x)=x. 108 [hyper,56,2] i($c2,$c1). 122 [hyper,74,2] i(Z,x). 129 [hyper,122,4,demod,42] i(x,u(x,y)). 159 [hyper,129,6,108] i($c2,u($c1,x)). 222 [para_into,159.1.2,15.1.1] i($c2,u(x,$c1)). 313 [hyper,222,3] i(n($c2,c($c1)),x). 347 [hyper,313,8,122,flip.1] n($c2,c($c1))=Z. 349 [binary,347.1,14.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 52 clauses generated 1289 clauses kept 262 clauses forward subsumed 635 clauses back subsumed 21 Kbytes malloced 287 ----------- times (seconds) ----------- user CPU time 0.26 (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 Sun Nov 30 22:28:56 2003