----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Sun Nov 30 15:46:29 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). assign(max_literals,1). assign(max_mem,30000). assign(max_weight,20). 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("peirceanAx0.txt"). ------- start included file peirceanAx0.txt------- formula_list(usable). all x y z (k(x,k(y,z))=k(k(x,y),z)). all x (r(r(x))=x). all x y (r(u(x,y))=u(r(x),r(y))). all x y (r(k(x,y))=k(r(y),r(x))). end_of_list. -------> usable clausifies to: list(usable). 0 [] k(x,k(y,z))=k(k(x,y),z). 0 [] r(r(x))=x. 0 [] r(u(x,y))=u(r(x),r(y)). 0 [] r(k(x,y))=k(r(y),r(x)). end_of_list. ------- end included file peirceanAx0.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("ix_a.txt"). ------- start included file ix_a.txt------- formula_list(usable). all x (k(x,I)=x). end_of_list. -------> usable clausifies to: list(usable). 0 [] k(x,I)=x. end_of_list. ------- end included file ix_a.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("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("cycleLawsB.txt"). ------- start included file cycleLawsB.txt------- formula_list(usable). all x y z (n(y,k(x,n(z,c(k(r(x),y)))))=Z). all x y z v (n(k(x,z),y)=n(u(k(n(x,v),z),k(n(x,c(v)),z)),y)). all x y z (n(y,k(n(x,c(k(y,r(z)))),z))=Z). end_of_list. -------> usable clausifies to: list(usable). 0 [] n(y,k(x,n(z,c(k(r(x),y)))))=Z. 0 [] n(k(x,z),y)=n(u(k(n(x,v),z),k(n(x,c(v)),z)),y). 0 [] n(y,k(n(x,c(k(y,r(z)))),z))=Z. end_of_list. ------- end included file cycleLawsB.txt------- include("cycleLawC.txt"). ------- start included file cycleLawC.txt------- formula_list(usable). all x y z (n(y,k(n(x,k(y,r(z))),z))=n(k(x,z),y)). end_of_list. -------> usable clausifies to: list(usable). 0 [] n(y,k(n(x,k(y,r(z))),z))=n(k(x,z),y). end_of_list. ------- end included file cycleLawC.txt------- include("xxi_a_ghost.txt"). ------- start included file xxi_a_ghost.txt------- formula_list(usable). all x (n(x,I)=n(r(x),I)). -(all x y (n(k(x,n(r(x),y)),I)=n(k(x,y),I))). end_of_list. -------> usable clausifies to: list(usable). 0 [] n(x,I)=n(r(x),I). 0 [] n(k($c2,n(r($c2),$c1)),I)!=n(k($c2,$c1),I). end_of_list. ------- end included file xxi_a_ghost.txt------- SCAN INPUT: prop=0, horn=1, equality=1, symmetry=0, max_lits=2. This is a Horn set with equality. The strategy will be Knuth-Bendix and hyper_res, with positive clauses in sos and nonpositive clauses in usable. dependent: set(knuth_bendix). dependent: set(para_from). dependent: set(para_into). dependent: clear(para_from_right). dependent: clear(para_into_right). dependent: set(para_from_vars). dependent: set(eq_units_both_ways). dependent: set(dynamic_demod_all). dependent: set(dynamic_demod). dependent: set(order_eq). dependent: set(back_demod). dependent: set(lrpo). dependent: set(hyper_res). dependent: clear(order_hyper). ------------> process usable: ** KEPT (pick-wt=8): 1 [] -i(x,y)|i(r(x),r(y)). ** KEPT (pick-wt=12): 2 [] u(x,y)!=y|c(u(y,c(x)))=Z. ** KEPT (pick-wt=12): 3 [] c(u(x,c(y)))!=Z|u(y,x)=x. ** KEPT (pick-wt=14): 4 [] n(k($c2,n(r($c2),$c1)),I)!=n(k($c2,$c1),I). ------------> process sos: ** KEPT (pick-wt=7): 5 [] u(x,y)=u(y,x). ** KEPT (pick-wt=11): 7 [copy,6,flip.1] u(u(x,y),z)=u(x,u(y,z)). ---> New Demodulator: 8 [new_demod,7] u(u(x,y),z)=u(x,u(y,z)). ** KEPT (pick-wt=14): 9 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ---> New Demodulator: 10 [new_demod,9] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ** KEPT (pick-wt=11): 12 [copy,11,flip.1] k(k(x,y),z)=k(x,k(y,z)). ---> New Demodulator: 13 [new_demod,12] k(k(x,y),z)=k(x,k(y,z)). ** KEPT (pick-wt=5): 14 [] r(r(x))=x. ---> New Demodulator: 15 [new_demod,14] r(r(x))=x. ** KEPT (pick-wt=10): 16 [] r(u(x,y))=u(r(x),r(y)). ---> New Demodulator: 17 [new_demod,16] r(u(x,y))=u(r(x),r(y)). ** KEPT (pick-wt=10): 18 [] r(k(x,y))=k(r(y),r(x)). ---> New Demodulator: 19 [new_demod,18] r(k(x,y))=k(r(y),r(x)). ** KEPT (pick-wt=10): 21 [copy,20,flip.1] c(u(c(x),c(y)))=n(x,y). ---> New Demodulator: 22 [new_demod,21] c(u(c(x),c(y)))=n(x,y). ** KEPT (pick-wt=4): 24 [copy,23,flip.1] c(I)=D. ---> New Demodulator: 25 [new_demod,24] c(I)=D. ** KEPT (pick-wt=5): 27 [copy,26,flip.1] u(I,D)=U. ---> New Demodulator: 28 [new_demod,27] u(I,D)=U. ** KEPT (pick-wt=4): 30 [copy,29,flip.1] c(U)=Z. ---> New Demodulator: 31 [new_demod,30] c(U)=Z. ** KEPT (pick-wt=5): 32 [] k(x,I)=x. ---> New Demodulator: 33 [new_demod,32] k(x,I)=x. ** KEPT (pick-wt=11): 35 [copy,34,demod,17] u(r(x),r(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=3): 41 [copy,40,demod,15] 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)). Following clause subsumed by 41 during input processing: 0 [demod,19] k(r(y),r(x))=k(r(y),r(x)). ** KEPT (pick-wt=4): 44 [] r(I)=I. ---> New Demodulator: 45 [new_demod,44] r(I)=I. ** KEPT (pick-wt=4): 46 [] r(Z)=Z. ---> New Demodulator: 47 [new_demod,46] r(Z)=Z. ** KEPT (pick-wt=4): 48 [] r(U)=U. ---> New Demodulator: 49 [new_demod,48] r(U)=U. ** KEPT (pick-wt=5): 50 [] c(c(x))=x. ---> New Demodulator: 51 [new_demod,50] c(c(x))=x. ** KEPT (pick-wt=13): 52 [] n(x,k(y,n(z,c(k(r(y),x)))))=Z. ---> New Demodulator: 53 [new_demod,52] n(x,k(y,n(z,c(k(r(y),x)))))=Z. ** KEPT (pick-wt=20): 55 [copy,54,flip.1] n(u(k(n(x,y),z),k(n(x,c(y)),z)),u)=n(k(x,z),u). ---> New Demodulator: 56 [new_demod,55] n(u(k(n(x,y),z),k(n(x,c(y)),z)),u)=n(k(x,z),u). ** KEPT (pick-wt=13): 57 [] n(x,k(n(y,c(k(x,r(z)))),z))=Z. ---> New Demodulator: 58 [new_demod,57] n(x,k(n(y,c(k(x,r(z)))),z))=Z. ** KEPT (pick-wt=16): 59 [] n(x,k(n(y,k(x,r(z))),z))=n(k(y,z),x). ** KEPT (pick-wt=8): 61 [copy,60,flip.1] n(r(x),I)=n(x,I). ---> New Demodulator: 62 [new_demod,61] n(r(x),I)=n(x,I). Following clause subsumed by 5 during input processing: 0 [copy,5,flip.1] u(x,y)=u(y,x). >>>> Starting back demodulation with 8. >>>> Starting back demodulation with 10. >>>> Starting back demodulation with 13. >>>> Starting back demodulation with 15. >>>> Starting back demodulation with 17. >>>> Starting back demodulation with 19. >>>> Starting back demodulation with 22. >> back demodulating 9 with 22. >>>> Starting back demodulation with 25. >>>> Starting back demodulation with 28. >>>> Starting back demodulation with 31. >>>> Starting back demodulation with 33. Following clause subsumed by 41 during input processing: 0 [copy,35,flip.1] u(r(x),r(y))=u(r(x),r(y)). >>>> Starting back demodulation with 37. >>>> Starting back demodulation with 39. Following clause subsumed by 41 during input processing: 0 [copy,41,flip.1] x=x. 41 back subsumes 35. >>>> 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 53. >>>> Starting back demodulation with 56. >>>> Starting back demodulation with 58. ** KEPT (pick-wt=16): 65 [copy,59,flip.1] n(k(x,y),z)=n(z,k(n(x,k(z,r(y))),y)). >>>> Starting back demodulation with 62. >>>> Starting back demodulation with 64. Following clause subsumed by 59 during input processing: 0 [copy,65,flip.1] n(x,k(n(y,k(x,r(z))),z))=n(k(y,z),x). ======= end of input processing ======= =========== start of search =========== ----> UNIT CONFLICT at 0.43 sec ----> 3969 [binary,3968.1,129.1] $F. Length of proof is 18. Level of proof is 8. ---------------- PROOF ---------------- 4 [] n(k($c2,n(r($c2),$c1)),I)!=n(k($c2,$c1),I). 5 [] u(x,y)=u(y,x). 15,14 [] r(r(x))=x. 18 [] r(k(x,y))=k(r(y),r(x)). 20 [] n(x,y)=c(u(c(x),c(y))). 21 [copy,20,flip.1] c(u(c(x),c(y)))=n(x,y). 32 [] k(x,I)=x. 39,38 [] r(n(x,y))=n(r(x),r(y)). 45,44 [] r(I)=I. 51,50 [] c(c(x))=x. 59 [] n(x,k(n(y,k(x,r(z))),z))=n(k(y,z),x). 60 [] n(x,I)=n(r(x),I). 61 [copy,60,flip.1] n(r(x),I)=n(x,I). 91 [para_into,18.1.1.1,32.1.1,demod,45,flip.1] k(I,r(x))=r(x). 105 [para_into,91.1.1.2,14.1.1,demod,15] k(I,x)=x. 109 [para_into,61.1.1.1,18.1.1] n(k(r(x),r(y)),I)=n(k(y,x),I). 112 [copy,109,flip.1] n(k(x,y),I)=n(k(r(y),r(x)),I). 118,117 [para_into,21.1.1.1.1,50.1.1] c(u(x,c(y)))=n(c(x),y). 129 [para_into,21.1.1.1,5.1.1,demod,118,51] n(x,y)=n(y,x). 132 [para_into,129.1.1,61.1.1,flip.1] n(I,r(x))=n(x,I). 135 [para_from,129.1.1,4.1.1.1.2] n(k($c2,n($c1,r($c2))),I)!=n(k($c2,$c1),I). 138 [para_into,132.1.1.2,18.1.1] n(I,k(r(x),r(y)))=n(k(y,x),I). 141 [copy,138,flip.1] n(k(x,y),I)=n(I,k(r(y),r(x))). 310 [para_into,59.1.1.2.1.2.2,14.1.1] n(x,k(n(y,k(x,z)),r(z)))=n(k(y,r(z)),x). 539 [para_into,112.1.1,129.1.1,flip.1] n(k(r(x),r(y)),I)=n(I,k(y,x)). 1224 [para_from,141.1.1,135.1.1,demod,39,15] n(I,k(n(r($c1),$c2),r($c2)))!=n(k($c2,$c1),I). 3926,3925 [para_into,310.1.1.2.1.2,105.1.1,flip.1] n(k(x,r(y)),I)=n(I,k(n(x,y),r(y))). 3953,3952 [back_demod,539,demod,3926] n(I,k(n(r(x),y),r(y)))=n(I,k(y,x)). 3968 [back_demod,1224,demod,3953,flip.1] n(k($c2,$c1),I)!=n(I,k($c2,$c1)). 3969 [binary,3968.1,129.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 296 clauses generated 13556 clauses kept 2193 clauses forward subsumed 11719 clauses back subsumed 9 Kbytes malloced 2490 ----------- times (seconds) ----------- user CPU time 0.63 (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 Nov 30 15:46:30 2003