----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Sat Nov 29 22:37:07 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,8). assign(max_distinct_vars,2). assign(max_literals,1). assign(max_mem,44000). 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("peirceanAx1.txt"). ------- start included file peirceanAx1.txt------- formula_list(usable). all x y z (k(u(x,y),z)=u(k(x,z),k(y,z))). end_of_list. -------> usable clausifies to: list(usable). 0 [] k(u(x,y),z)=u(k(x,z),k(y,z)). end_of_list. ------- end included file peirceanAx1.txt------- include("peirceanAx2.txt"). ------- start included file peirceanAx2.txt------- formula_list(usable). all x (k(x,I)=x). all x y (u(k(r(x),c(k(x,y))),c(y))=c(y)). end_of_list. -------> usable clausifies to: list(usable). 0 [] k(x,I)=x. 0 [] u(k(r(x),c(k(x,y))),c(y))=c(y). end_of_list. ------- end included file peirceanAx2.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("xiii.txt"). ------- start included file xiii.txt------- formula_list(usable). all x (k(x,Z)=Z). all x (k(Z,x)=Z). all x (d(x,U)=U). all x (d(U,x)=U). end_of_list. -------> usable clausifies to: list(usable). 0 [] k(x,Z)=Z. 0 [] k(Z,x)=Z. 0 [] d(x,U)=U. 0 [] d(U,x)=U. end_of_list. ------- end included file xiii.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("xvii_b.txt"). ------- start included file xvii_b.txt------- formula_list(usable). -(all x (k(k(U,r(x)),U)=k(k(U,x),U))). end_of_list. -------> usable clausifies to: list(usable). 0 [] k(k(U,r($c1)),U)!=k(k(U,$c1),U). end_of_list. ------- end included file xvii_b.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 [] k(k(U,r($c1)),U)!=k(k(U,$c1),U). ------------> process sos: ** KEPT (pick-wt=7): 3 [] u(x,y)=u(y,x). ** KEPT (pick-wt=11): 5 [copy,4,flip.1] u(u(x,y),z)=u(x,u(y,z)). ---> New Demodulator: 6 [new_demod,5] u(u(x,y),z)=u(x,u(y,z)). ** KEPT (pick-wt=14): 7 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ---> New Demodulator: 8 [new_demod,7] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ** KEPT (pick-wt=11): 10 [copy,9,flip.1] k(k(x,y),z)=k(x,k(y,z)). ---> New Demodulator: 11 [new_demod,10] k(k(x,y),z)=k(x,k(y,z)). ** KEPT (pick-wt=5): 12 [] r(r(x))=x. ---> New Demodulator: 13 [new_demod,12] r(r(x))=x. ** KEPT (pick-wt=10): 14 [] r(u(x,y))=u(r(x),r(y)). ---> New Demodulator: 15 [new_demod,14] r(u(x,y))=u(r(x),r(y)). ** KEPT (pick-wt=10): 16 [] r(k(x,y))=k(r(y),r(x)). ---> New Demodulator: 17 [new_demod,16] r(k(x,y))=k(r(y),r(x)). ** KEPT (pick-wt=13): 19 [copy,18,flip.1] u(k(x,y),k(z,y))=k(u(x,z),y). ---> New Demodulator: 20 [new_demod,19] u(k(x,y),k(z,y))=k(u(x,z),y). ** KEPT (pick-wt=5): 21 [] k(x,I)=x. ---> New Demodulator: 22 [new_demod,21] k(x,I)=x. ** KEPT (pick-wt=13): 23 [] u(k(r(x),c(k(x,y))),c(y))=c(y). ---> New Demodulator: 24 [new_demod,23] u(k(r(x),c(k(x,y))),c(y))=c(y). ** KEPT (pick-wt=4): 26 [copy,25,flip.1] c(I)=D. ---> New Demodulator: 27 [new_demod,26] c(I)=D. ** KEPT (pick-wt=5): 29 [copy,28,flip.1] u(I,D)=U. ---> New Demodulator: 30 [new_demod,29] u(I,D)=U. ** KEPT (pick-wt=4): 32 [copy,31,flip.1] c(U)=Z. ---> New Demodulator: 33 [new_demod,32] c(U)=Z. ** KEPT (pick-wt=5): 34 [] k(x,Z)=Z. ---> New Demodulator: 35 [new_demod,34] k(x,Z)=Z. ** KEPT (pick-wt=5): 36 [] k(Z,x)=Z. ---> New Demodulator: 37 [new_demod,36] k(Z,x)=Z. ** KEPT (pick-wt=5): 38 [] d(x,U)=U. ---> New Demodulator: 39 [new_demod,38] d(x,U)=U. ** KEPT (pick-wt=5): 40 [] d(U,x)=U. ---> New Demodulator: 41 [new_demod,40] d(U,x)=U. ** KEPT (pick-wt=11): 43 [copy,42,demod,15] u(r(x),r(y))=u(r(x),r(y)). ** KEPT (pick-wt=7): 44 [] r(c(x))=c(r(x)). ---> New Demodulator: 45 [new_demod,44] r(c(x))=c(r(x)). ** KEPT (pick-wt=10): 46 [] r(n(x,y))=n(r(x),r(y)). ---> New Demodulator: 47 [new_demod,46] r(n(x,y))=n(r(x),r(y)). ** KEPT (pick-wt=3): 49 [copy,48,demod,13] x=x. ** KEPT (pick-wt=10): 50 [] r(d(x,y))=d(r(y),r(x)). ---> New Demodulator: 51 [new_demod,50] r(d(x,y))=d(r(y),r(x)). Following clause subsumed by 49 during input processing: 0 [demod,17] k(r(y),r(x))=k(r(y),r(x)). ** KEPT (pick-wt=4): 52 [] r(I)=I. ---> New Demodulator: 53 [new_demod,52] r(I)=I. ** KEPT (pick-wt=4): 54 [] r(Z)=Z. ---> New Demodulator: 55 [new_demod,54] r(Z)=Z. ** KEPT (pick-wt=4): 56 [] r(U)=U. ---> New Demodulator: 57 [new_demod,56] r(U)=U. Following clause subsumed by 3 during input processing: 0 [copy,3,flip.1] u(x,y)=u(y,x). >>>> Starting back demodulation with 6. >>>> Starting back demodulation with 8. >>>> Starting back demodulation with 11. >> back demodulating 2 with 11. >>>> Starting back demodulation with 13. >>>> Starting back demodulation with 15. >>>> Starting back demodulation with 17. >>>> Starting back demodulation with 20. >>>> Starting back demodulation with 22. >>>> Starting back demodulation with 24. >>>> Starting back demodulation with 27. >>>> Starting back demodulation with 30. >>>> 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. Following clause subsumed by 49 during input processing: 0 [copy,43,flip.1] u(r(x),r(y))=u(r(x),r(y)). >>>> Starting back demodulation with 45. >>>> Starting back demodulation with 47. Following clause subsumed by 49 during input processing: 0 [copy,49,flip.1] x=x. 49 back subsumes 43. >>>> Starting back demodulation with 51. >>>> Starting back demodulation with 53. >>>> Starting back demodulation with 55. >>>> Starting back demodulation with 57. ======= end of input processing ======= =========== start of search =========== ----> UNIT CONFLICT at 7.69 sec ----> 22200 [binary,22198.1,58.1] $F. Length of proof is 78. Level of proof is 22. ---------------- PROOF ---------------- 2 [] k(k(U,r($c1)),U)!=k(k(U,$c1),U). 3 [] u(x,y)=u(y,x). 4 [] u(x,u(y,z))=u(u(x,y),z). 6,5 [copy,4,flip.1] u(u(x,y),z)=u(x,u(y,z)). 7 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. 9 [] k(x,k(y,z))=k(k(x,y),z). 11,10 [copy,9,flip.1] k(k(x,y),z)=k(x,k(y,z)). 13,12 [] r(r(x))=x. 15,14 [] r(u(x,y))=u(r(x),r(y)). 17,16 [] r(k(x,y))=k(r(y),r(x)). 18 [] k(u(x,y),z)=u(k(x,z),k(y,z)). 20,19 [copy,18,flip.1] u(k(x,y),k(z,y))=k(u(x,z),y). 21 [] k(x,I)=x. 23 [] u(k(r(x),c(k(x,y))),c(y))=c(y). 25 [] D=c(I). 27,26 [copy,25,flip.1] c(I)=D. 28 [] U=u(I,D). 29 [copy,28,flip.1] u(I,D)=U. 31 [] Z=c(U). 33,32 [copy,31,flip.1] c(U)=Z. 35,34 [] k(x,Z)=Z. 37,36 [] k(Z,x)=Z. 45,44 [] r(c(x))=c(r(x)). 53,52 [] r(I)=I. 55,54 [] r(Z)=Z. 57,56 [] r(U)=U. 58 [back_demod,2,demod,11,11] k(U,k(r($c1),U))!=k(U,k($c1,U)). 59 [para_into,7.1.1.1.1.1,32.1.1,demod,33] u(c(u(Z,x)),c(u(Z,c(x))))=U. 61 [para_into,7.1.1.1.1.1,26.1.1,demod,27] u(c(u(D,x)),c(u(D,c(x))))=I. 64,63 [para_into,7.1.1.1.1,3.1.1] u(c(u(x,c(y))),c(u(c(y),c(x))))=y. 71 [para_into,7.1.1.2.1,3.1.1] u(c(u(c(x),y)),c(u(c(y),c(x))))=x. 76,75 [para_into,29.1.1,3.1.1] u(D,I)=U. 78,77 [para_from,29.1.1,5.1.1.1] u(U,x)=u(I,u(D,x)). 82,81 [para_from,75.1.1,5.1.1.1,demod,78] u(I,u(D,x))=u(D,u(I,x)). 84,83 [back_demod,77,demod,82] u(U,x)=u(D,u(I,x)). 87 [para_into,83.1.1,3.1.1] u(x,U)=u(D,u(I,x)). 88 [copy,87,flip.1] u(D,u(I,x))=u(x,U). 97 [para_into,88.1.1.2,3.1.1] u(D,u(x,I))=u(x,U). 101 [para_into,16.1.1.1,21.1.1,demod,53,flip.1] k(I,r(x))=r(x). 114,113 [para_into,101.1.1.2,12.1.1,demod,13] k(I,x)=x. 125 [para_into,19.1.1.1,113.1.1] u(x,k(y,x))=k(u(I,y),x). 129,128 [para_into,19.1.1.1,34.1.1,demod,35,35] u(Z,Z)=Z. 131,130 [para_into,19.1.1.2,113.1.1] u(k(x,y),y)=k(u(x,I),y). 133,132 [para_into,19.1.1.2,36.1.1] u(k(x,y),Z)=k(u(x,Z),y). 134 [copy,125,flip.1] k(u(I,x),y)=u(y,k(x,y)). 155,154 [para_into,23.1.1.1.1,54.1.1,demod,37,37] u(Z,c(x))=c(x). 157,156 [para_into,23.1.1.1.1,52.1.1,demod,114,114] u(c(x),c(x))=c(x). 162 [para_into,23.1.1.1.2.1,34.1.1,demod,131] k(u(r(x),I),c(Z))=c(Z). 166 [para_into,23.1.1.2,32.1.1,demod,133,33] k(u(r(x),Z),c(k(x,U)))=Z. 170 [back_demod,59,demod,155] u(c(u(Z,x)),c(c(x)))=U. 174 [para_into,154.1.1.2,26.1.1,demod,27] u(Z,D)=D. 177,176 [para_into,154.1.1,3.1.1] u(c(x),Z)=c(x). 187,186 [para_into,174.1.1,3.1.1] u(D,Z)=D. 195,194 [para_from,176.1.1,5.1.1.1,flip.1] u(c(x),u(Z,y))=u(c(x),y). 197,196 [para_into,156.1.1.1,26.1.1,demod,27,27] u(D,D)=D. 204 [para_from,196.1.1,5.1.1.1,flip.1] u(D,u(D,x))=u(D,x). 303,302 [para_into,61.1.1.1.1,75.1.1,demod,33,27,197,155] c(D)=I. 306 [para_into,61.1.1.2.1.2,32.1.1,demod,187,303] u(c(u(D,U)),I)=I. 358 [para_into,63.1.1.1.1.2,32.1.1,demod,33,155] u(c(u(x,Z)),c(c(x)))=U. 388 [para_from,306.1.1,97.1.1.2,demod,76,flip.1] u(c(u(D,U)),U)=U. 404 [para_from,388.1.1,7.1.1.1.1,demod,33,33,177,155] c(c(u(D,U)))=u(D,U). 413,412 [para_into,204.1.1.2,75.1.1,demod,76] u(D,U)=U. 417,416 [back_demod,404,demod,413,33,413] c(Z)=U. 420 [back_demod,162,demod,417,417] k(u(r(x),I),U)=U. 462 [para_into,420.1.1.1.1,12.1.1] k(u(x,I),U)=U. 477 [para_from,462.1.1,16.1.1.1,demod,57,57,15,53,flip.1] k(U,u(r(x),I))=U. 480,479 [para_from,462.1.1,10.1.1.1,flip.1] k(u(x,I),k(U,y))=k(U,y). 528 [para_into,477.1.1.2.1,12.1.1] k(U,u(x,I))=U. 532 [para_into,528.1.1.2,3.1.1] k(U,u(I,x))=U. 539,538 [para_from,528.1.1,10.1.1.1,flip.1] k(U,k(u(x,I),y))=k(U,y). 566 [para_from,532.1.1,10.1.1.1,flip.1] k(U,k(u(I,x),y))=k(U,y). 947,946 [para_from,170.1.1,71.1.1.2.1,demod,195,33,177] c(u(c(c(x)),x))=c(x). 982 [para_from,946.1.1,63.1.1.2.1.1,demod,947,64,flip.1] u(c(c(x)),x)=x. 1000 [para_into,982.1.1,3.1.1] u(x,c(c(x)))=x. 1009,1008 [para_from,982.1.1,5.1.1.1,flip.1] u(c(c(x)),u(x,y))=u(x,y). 1016 [para_from,1000.1.1,71.1.1.1.1] u(c(c(x)),c(u(c(c(c(c(x)))),c(x))))=x. 1035,1034 [para_from,358.1.1,71.1.1.2.1,demod,1009,33,177] c(u(x,Z))=c(x). 1037,1036 [para_from,358.1.1,63.1.1.2.1,demod,1035,157,33,177] c(c(x))=u(x,Z). 1053,1052 [para_from,358.1.1,3.1.1,demod,1037,1035,6,155,flip.1] u(x,c(x))=U. 1065,1064 [back_demod,1016,demod,1037,1037,1035,1037,6,155,1053,33,6,129] u(x,Z)=x. 1072 [back_demod,1008,demod,1037,1065] u(x,u(x,y))=u(x,y). 1112 [back_demod,166,demod,1065] k(r(x),c(k(x,U)))=Z. 1150 [para_from,1052.1.1,5.1.1.1,demod,84,flip.1] u(x,u(c(x),y))=u(D,u(I,y)). 1165,1164 [para_into,1072.1.1.2,1052.1.1,demod,1053] u(x,U)=U. 1181,1180 [para_into,1072.1.1.2,88.1.1,demod,1165,1165,flip.1] u(D,u(I,x))=U. 1198 [back_demod,1150,demod,1181] u(x,u(c(x),y))=U. 1230 [para_into,1198.1.1,3.1.1,demod,6] u(c(x),u(y,x))=U. 1264 [para_into,1230.1.1.2,130.1.1] u(c(x),k(u(y,I),x))=U. 1290 [para_from,1112.1.1,16.1.1.1,demod,55,45,17,57,13,flip.1] k(c(k(U,r(x))),x)=Z. 1590 [para_from,1290.1.1,19.1.1.2,demod,1065,flip.1] k(u(x,c(k(U,r(y)))),y)=k(x,y). 2004 [para_into,1264.1.1,3.1.1] u(k(u(x,I),y),c(y))=U. 3368 [para_into,566.1.1.2,134.1.1] k(U,u(x,k(y,x)))=k(U,x). 21348 [para_into,1590.1.1.1,2004.1.1,demod,480,11,flip.1] k(U,k(r(x),x))=k(U,x). 21425 [para_into,21348.1.1.2.1,12.1.1] k(U,k(x,r(x)))=k(U,r(x)). 21455 [para_from,21348.1.1,16.1.1.1,demod,17,57,17,13,57,11,flip.1] k(r(x),k(x,U))=k(r(x),U). 22124 [para_from,21425.1.1,16.1.1.1,demod,17,13,57,17,13,57,11,flip.1] k(x,k(r(x),U))=k(x,U). 22173,22172 [para_from,22124.1.1,130.1.1.1,demod,20] k(u(x,r(x)),U)=k(u(x,I),k(r(x),U)). 22198 [para_from,21455.1.1,3368.1.1.2.2,demod,20,22173,539] k(U,k(r(x),U))=k(U,k(x,U)). 22200 [binary,22198.1,58.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 959 clauses generated 164543 clauses kept 11385 clauses forward subsumed 151488 clauses back subsumed 27 Kbytes malloced 12933 ----------- times (seconds) ----------- user CPU time 7.89 (0 hr, 0 min, 7 sec) system CPU time 0.00 (0 hr, 0 min, 0 sec) wall-clock time 8 (0 hr, 0 min, 8 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 Sat Nov 29 22:37:15 2003