----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Sun Dec 14 00:15:18 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_literals,1). assign(max_distinct_vars,4). assign(max_weight,18). 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("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("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("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("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("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("iv.txt"). ------- start included file iv.txt------- formula_list(usable). all x y z (k(x,k(y,z))=k(k(x,y),z)). all x y z (d(x,d(y,z))=d(d(x,y),z)). end_of_list. -------> usable clausifies to: list(usable). 0 [] k(x,k(y,z))=k(k(x,y),z). 0 [] d(x,d(y,z))=d(d(x,y),z). end_of_list. ------- end included file iv.txt------- include("vii.txt"). ------- start included file vii.txt------- formula_list(usable). all x y z (i(x,y)->i(k(x,z),k(y,z))). all x y z (i(x,y)->i(k(z,x),k(z,y))). all x y z (i(x,y)->i(d(x,z),d(y,z))). all x y z (i(x,y)->i(d(z,x),d(z,y))). all x y (i(x,y)->i(r(x),r(y))). end_of_list. -------> usable clausifies to: list(usable). 0 [] -i(x,y)|i(k(x,z),k(y,z)). 0 [] -i(x,y)|i(k(z,x),k(z,y)). 0 [] -i(x,y)|i(d(x,z),d(y,z)). 0 [] -i(x,y)|i(d(z,x),d(z,y)). 0 [] -i(x,y)|i(r(x),r(y)). end_of_list. ------- end included file vii.txt------- include("xv_a.txt"). ------- start included file xv_a.txt------- formula_list(usable). all x y z v w i(n(w,k(n(y,z),v)),n(w,u(k(n(y,z),n(v,x)),k(y,n(v,c(x)))))). all y v w i(n(k(y,v),w),k(n(y,k(w,r(v))),n(v,k(r(y),w)))). all y v w i(n(k(y,v),w),k(y,n(v,k(r(y),w)))). end_of_list. -------> usable clausifies to: list(usable). 0 [] i(n(w,k(n(y,z),v)),n(w,u(k(n(y,z),n(v,x)),k(y,n(v,c(x)))))). 0 [] i(n(k(y,v),w),k(n(y,k(w,r(v))),n(v,k(r(y),w)))). 0 [] i(n(k(y,v),w),k(y,n(v,k(r(y),w)))). end_of_list. ------- end included file xv_a.txt------- include("xvi_a.txt"). ------- start included file xvi_a.txt------- formula_list(usable). -(all y v w i(k(U,n(k(y,v),w)),k(U,n(k(r(y),w),v)))). end_of_list. -------> usable clausifies to: list(usable). 0 [] -i(k(U,n(k($c3,$c2),$c1)),k(U,n(k(r($c3),$c1),$c2))). end_of_list. ------- end included file xvi_a.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=9): 3 [] -i(x,y)| -i(y,z)|i(x,z). ** KEPT (pick-wt=9): 4 [] -i(x,y)| -i(y,x)|x=y. ** KEPT (pick-wt=11): 5 [] n(x,c(y))!=Z|u(x,y)=y. ** KEPT (pick-wt=11): 6 [] u(x,y)!=y|n(x,c(y))=Z. ** KEPT (pick-wt=14): 7 [] n(x,y)!=Z|u(x,y)!=U|c(x)=y. ** KEPT (pick-wt=9): 8 [] c(x)!=y|n(x,y)=Z. ** KEPT (pick-wt=9): 9 [] c(x)!=y|u(x,y)=U. ** KEPT (pick-wt=8): 10 [] -i(x,y)|i(r(x),r(y)). ** KEPT (pick-wt=10): 11 [] -i(x,y)|i(k(x,z),k(y,z)). ** KEPT (pick-wt=10): 12 [] -i(x,y)|i(k(z,x),k(z,y)). ** KEPT (pick-wt=10): 13 [] -i(x,y)|i(d(x,z),d(y,z)). ** KEPT (pick-wt=10): 14 [] -i(x,y)|i(d(z,x),d(z,y)). Following clause subsumed by 10 during input processing: 0 [] -i(x,y)|i(r(x),r(y)). ** KEPT (pick-wt=16): 15 [] -i(k(U,n(k($c3,$c2),$c1)),k(U,n(k(r($c3),$c1),$c2))). ------------> process sos: ** KEPT (pick-wt=7): 16 [] u(x,y)=u(y,x). ** KEPT (pick-wt=11): 18 [copy,17,flip.1] u(u(x,y),z)=u(x,u(y,z)). ---> New Demodulator: 19 [new_demod,18] u(u(x,y),z)=u(x,u(y,z)). ** KEPT (pick-wt=14): 20 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ---> New Demodulator: 21 [new_demod,20] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ** KEPT (pick-wt=4): 23 [copy,22,flip.1] c(I)=D. ---> New Demodulator: 24 [new_demod,23] c(I)=D. ** KEPT (pick-wt=5): 26 [copy,25,flip.1] u(I,D)=U. ---> New Demodulator: 27 [new_demod,26] u(I,D)=U. ** KEPT (pick-wt=4): 29 [copy,28,flip.1] c(U)=Z. ---> New Demodulator: 30 [new_demod,29] c(U)=Z. ** KEPT (pick-wt=5): 31 [] u(x,Z)=x. ---> New Demodulator: 32 [new_demod,31] u(x,Z)=x. ** KEPT (pick-wt=7): 33 [] n(x,y)=n(y,x). ** KEPT (pick-wt=10): 34 [] r(u(x,y))=u(r(x),r(y)). ---> New Demodulator: 35 [new_demod,34] r(u(x,y))=u(r(x),r(y)). ** KEPT (pick-wt=7): 36 [] r(c(x))=c(r(x)). ---> New Demodulator: 37 [new_demod,36] r(c(x))=c(r(x)). ** KEPT (pick-wt=10): 38 [] r(n(x,y))=n(r(x),r(y)). ---> New Demodulator: 39 [new_demod,38] r(n(x,y))=n(r(x),r(y)). ** KEPT (pick-wt=5): 40 [] r(r(x))=x. ---> New Demodulator: 41 [new_demod,40] r(r(x))=x. ** KEPT (pick-wt=10): 42 [] r(d(x,y))=d(r(y),r(x)). ---> New Demodulator: 43 [new_demod,42] r(d(x,y))=d(r(y),r(x)). ** KEPT (pick-wt=10): 44 [] r(k(x,y))=k(r(y),r(x)). ---> New Demodulator: 45 [new_demod,44] r(k(x,y))=k(r(y),r(x)). ** KEPT (pick-wt=4): 46 [] r(I)=I. ---> New Demodulator: 47 [new_demod,46] r(I)=I. ** KEPT (pick-wt=4): 48 [] r(Z)=Z. ---> New Demodulator: 49 [new_demod,48] r(Z)=Z. ** KEPT (pick-wt=4): 50 [] r(U)=U. ---> New Demodulator: 51 [new_demod,50] r(U)=U. ** KEPT (pick-wt=11): 53 [copy,52,flip.1] k(k(x,y),z)=k(x,k(y,z)). ---> New Demodulator: 54 [new_demod,53] k(k(x,y),z)=k(x,k(y,z)). ** KEPT (pick-wt=11): 56 [copy,55,flip.1] d(d(x,y),z)=d(x,d(y,z)). ---> New Demodulator: 57 [new_demod,56] d(d(x,y),z)=d(x,d(y,z)). ** KEPT (pick-wt=24): 58 [] i(n(x,k(n(y,z),u)),n(x,u(k(n(y,z),n(u,v)),k(y,n(u,c(v)))))). ** KEPT (pick-wt=19): 59 [] i(n(k(x,y),z),k(n(x,k(z,r(y))),n(y,k(r(x),z)))). ** KEPT (pick-wt=14): 60 [] i(n(k(x,y),z),k(x,n(y,k(r(x),z)))). Following clause subsumed by 16 during input processing: 0 [copy,16,flip.1] u(x,y)=u(y,x). >>>> Starting back demodulation with 19. >>>> Starting back demodulation with 21. >>>> Starting back demodulation with 24. >>>> Starting back demodulation with 27. >>>> Starting back demodulation with 30. >>>> Starting back demodulation with 32. Following clause subsumed by 33 during input processing: 0 [copy,33,flip.1] n(x,y)=n(y,x). >>>> 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. >>>> Starting back demodulation with 49. >>>> Starting back demodulation with 51. >>>> Starting back demodulation with 54. >>>> Starting back demodulation with 57. ======= end of input processing ======= =========== start of search =========== ----> UNIT CONFLICT at 0.35 sec ----> 3185 [binary,3184.1,103.1] $F. Length of proof is 14. Level of proof is 9. ---------------- PROOF ---------------- 2 [] i(x,y)|u(x,y)!=y. 3 [] -i(x,y)| -i(y,z)|i(x,z). 5 [] n(x,c(y))!=Z|u(x,y)=y. 8 [] c(x)!=y|n(x,y)=Z. 9 [] c(x)!=y|u(x,y)=U. 11 [] -i(x,y)|i(k(x,z),k(y,z)). 12 [] -i(x,y)|i(k(z,x),k(z,y)). 15 [] -i(k(U,n(k($c3,$c2),$c1)),k(U,n(k(r($c3),$c1),$c2))). 17 [] u(x,u(y,z))=u(u(x,y),z). 18 [copy,17,flip.1] u(u(x,y),z)=u(x,u(y,z)). 20 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. 33 [] n(x,y)=n(y,x). 52 [] k(x,k(y,z))=k(k(x,y),z). 53 [copy,52,flip.1] k(k(x,y),z)=k(x,k(y,z)). 60 [] i(n(k(x,y),z),k(x,n(y,k(r(x),z)))). 85 [para_into,20.1.1,20.1.1] x=x. 90 [hyper,85,9] u(x,c(x))=U. 92 [hyper,85,8] n(x,c(x))=Z. 103 [para_from,33.1.1,15.1.2.2] -i(k(U,n(k($c3,$c2),$c1)),k(U,n($c2,k(r($c3),$c1)))). 148 [hyper,92,5] u(x,x)=x. 161 [para_from,148.1.1,18.1.1.1,flip.1] u(x,u(x,y))=u(x,y). 280 [hyper,60,12] i(k(x,n(k(y,z),u)),k(x,k(y,n(z,k(r(y),u))))). 544 [hyper,161,2] i(x,u(x,y)). 612 [para_into,544.1.2,90.1.1] i(x,U). 632 [hyper,612,11] i(k(x,y),k(U,y)). 804 [para_into,632.1.1,53.1.1] i(k(x,k(y,z)),k(U,z)). 3184 [hyper,280,3,804] i(k(x,n(k(y,z),u)),k(U,n(z,k(r(y),u)))). 3185 [binary,3184.1,103.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 311 clauses generated 18633 clauses kept 2729 clauses forward subsumed 11997 clauses back subsumed 237 Kbytes malloced 1692 ----------- times (seconds) ----------- user CPU time 0.55 (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 Dec 14 00:15:19 2003