----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Fri Nov 21 22:57:00 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). 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("ii.txt"). ------- start included file ii.txt------- formula_list(usable). k(U,U)!=U. end_of_list. -------> usable clausifies to: list(usable). 0 [] k(U,U)!=U. end_of_list. ------- end included file ii.txt------- SCAN INPUT: prop=0, horn=1, equality=1, symmetry=0, max_lits=1. All clauses are units, and equality is present; the strategy will be Knuth-Bendix with positive clauses in sos. 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). ------------> process usable: ** KEPT (pick-wt=5): 1 [] k(U,U)!=U. ------------> process sos: ** KEPT (pick-wt=7): 2 [] u(x,y)=u(y,x). ** KEPT (pick-wt=11): 4 [copy,3,flip.1] u(u(x,y),z)=u(x,u(y,z)). ---> New Demodulator: 5 [new_demod,4] u(u(x,y),z)=u(x,u(y,z)). ** KEPT (pick-wt=14): 6 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ---> New Demodulator: 7 [new_demod,6] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ** KEPT (pick-wt=11): 9 [copy,8,flip.1] k(k(x,y),z)=k(x,k(y,z)). ---> New Demodulator: 10 [new_demod,9] k(k(x,y),z)=k(x,k(y,z)). ** KEPT (pick-wt=5): 11 [] r(r(x))=x. ---> New Demodulator: 12 [new_demod,11] r(r(x))=x. ** KEPT (pick-wt=10): 13 [] r(u(x,y))=u(r(x),r(y)). ---> New Demodulator: 14 [new_demod,13] r(u(x,y))=u(r(x),r(y)). ** KEPT (pick-wt=10): 15 [] r(k(x,y))=k(r(y),r(x)). ---> New Demodulator: 16 [new_demod,15] r(k(x,y))=k(r(y),r(x)). ** KEPT (pick-wt=13): 18 [copy,17,flip.1] u(k(x,y),k(z,y))=k(u(x,z),y). ---> New Demodulator: 19 [new_demod,18] u(k(x,y),k(z,y))=k(u(x,z),y). ** KEPT (pick-wt=5): 20 [] k(x,I)=x. ---> New Demodulator: 21 [new_demod,20] k(x,I)=x. ** KEPT (pick-wt=13): 22 [] u(k(r(x),c(k(x,y))),c(y))=c(y). ---> New Demodulator: 23 [new_demod,22] u(k(r(x),c(k(x,y))),c(y))=c(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. Following clause subsumed by 2 during input processing: 0 [copy,2,flip.1] u(x,y)=u(y,x). >>>> Starting back demodulation with 5. >>>> Starting back demodulation with 7. >>>> Starting back demodulation with 10. >>>> Starting back demodulation with 12. >>>> Starting back demodulation with 14. >>>> Starting back demodulation with 16. >>>> Starting back demodulation with 19. >>>> Starting back demodulation with 21. >>>> Starting back demodulation with 23. >>>> Starting back demodulation with 26. >>>> Starting back demodulation with 29. >>>> Starting back demodulation with 32. ======= end of input processing ======= =========== start of search =========== ----> UNIT CONFLICT at 0.20 sec ----> 2331 [binary,2329.1,1.1] $F. Length of proof is 57. Level of proof is 19. ---------------- PROOF ---------------- 1 [] k(U,U)!=U. 2 [] u(x,y)=u(y,x). 3 [] u(x,u(y,z))=u(u(x,y),z). 5,4 [copy,3,flip.1] u(u(x,y),z)=u(x,u(y,z)). 7,6 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. 8 [] k(x,k(y,z))=k(k(x,y),z). 9 [copy,8,flip.1] k(k(x,y),z)=k(x,k(y,z)). 12,11 [] r(r(x))=x. 15 [] r(k(x,y))=k(r(y),r(x)). 17 [] k(u(x,y),z)=u(k(x,z),k(y,z)). 18 [copy,17,flip.1] u(k(x,y),k(z,y))=k(u(x,z),y). 20 [] k(x,I)=x. 22 [] u(k(r(x),c(k(x,y))),c(y))=c(y). 24 [] D=c(I). 26,25 [copy,24,flip.1] c(I)=D. 27 [] U=u(I,D). 28 [copy,27,flip.1] u(I,D)=U. 30 [] Z=c(U). 32,31 [copy,30,flip.1] c(U)=Z. 37 [para_into,4.1.1,2.1.1] u(x,u(y,z))=u(y,u(z,x)). 38 [copy,37,flip.1] u(x,u(y,z))=u(z,u(x,y)). 42,41 [para_into,28.1.1,2.1.1] u(D,I)=U. 44,43 [para_from,28.1.1,4.1.1.1] u(U,x)=u(I,u(D,x)). 45 [para_into,6.1.1.1.1.1,31.1.1,demod,32] u(c(u(Z,x)),c(u(Z,c(x))))=U. 47 [para_into,6.1.1.1.1.1,25.1.1,demod,26] u(c(u(D,x)),c(u(D,c(x))))=I. 53 [para_into,6.1.1.2.1.2,31.1.1] u(c(u(c(x),U)),c(u(c(x),Z)))=x. 63 [para_from,6.1.1,4.1.1.1,flip.1] u(c(u(c(x),y)),u(c(u(c(x),c(y))),z))=u(x,z). 65 [para_into,43.1.1,2.1.1] u(x,U)=u(I,u(D,x)). 66 [copy,65,flip.1] u(I,u(D,x))=u(x,U). 67 [para_into,65.1.1,4.1.1] u(x,u(y,U))=u(I,u(D,u(x,y))). 75 [para_into,66.1.1.2,2.1.1] u(I,u(x,D))=u(x,U). 76 [para_into,66.1.1,2.1.1,demod,5] u(D,u(x,I))=u(x,U). 78 [copy,76,flip.1] u(x,U)=u(D,u(x,I)). 79 [para_into,9.1.1.1,20.1.1,flip.1] k(x,k(I,y))=k(x,y). 81 [para_into,75.1.1.2,4.1.1,demod,5] u(I,u(x,u(y,D)))=u(x,u(y,U)). 95 [para_from,78.1.1,6.1.1.1.1,demod,32] u(c(u(D,u(c(x),I))),c(u(c(x),Z)))=x. 123 [para_into,15.1.1.1,20.1.1,flip.1] k(r(I),r(x))=r(x). 132,131 [para_into,123.1.1.2,11.1.1,demod,12] k(r(I),x)=x. 134,133 [para_into,131.1.1,79.1.1,demod,132,flip.1] k(I,x)=x. 135 [para_into,131.1.1,20.1.1] r(I)=I. 141 [para_into,18.1.1.1,133.1.1] u(x,k(y,x))=k(u(I,y),x). 224,223 [para_into,22.1.1.1.1,135.1.1,demod,134,134] u(c(x),c(x))=c(x). 246,245 [para_into,223.1.1.1,31.1.1,demod,32,32] u(Z,Z)=Z. 248,247 [para_into,223.1.1.1,25.1.1,demod,26,26] u(D,D)=D. 261 [para_from,245.1.1,4.1.1.1,flip.1] u(Z,u(Z,x))=u(Z,x). 279 [para_into,45.1.1.1.1,245.1.1] u(c(Z),c(u(Z,c(Z))))=U. 283 [para_into,45.1.1.1.1,78.1.1,demod,32,246] u(c(u(D,u(Z,I))),c(Z))=U. 369,368 [para_into,47.1.1.1.1,41.1.1,demod,32,26,248] u(Z,c(D))=I. 399,398 [para_from,368.1.1,261.1.1.2,demod,369] u(Z,I)=I. 413,412 [back_demod,283,demod,399,42,32] u(Z,c(Z))=U. 414 [back_demod,279,demod,413,32] u(c(Z),Z)=U. 417,416 [para_into,398.1.1,2.1.1] u(I,Z)=I. 419,418 [para_from,398.1.1,76.1.1.2,demod,42,flip.1] u(Z,U)=U. 424 [para_from,398.1.1,38.1.1.2,flip.1] u(I,u(x,Z))=u(x,I). 514 [para_from,414.1.1,6.1.1.1.1,demod,32,224] u(Z,c(c(Z)))=Z. 579,578 [para_into,514.1.1,2.1.1] u(c(c(Z)),Z)=Z. 1090 [para_into,424.1.1.2,578.1.1,demod,417,flip.1] u(c(c(Z)),I)=I. 1175,1174 [para_into,63.1.1.2,223.1.1,demod,7,flip.1] u(x,c(u(c(x),c(y))))=x. 1195 [para_into,63.1.1.2,6.1.1,demod,1175] u(c(u(c(x),y)),x)=x. 1216 [para_from,1090.1.1,76.1.1.2,demod,42,flip.1] u(c(c(Z)),U)=U. 1243,1242 [para_from,1216.1.1,53.1.1.1.1,demod,32,579,413,flip.1] c(Z)=U. 2085,2084 [para_from,1195.1.1,424.1.1.2,demod,417,1243,44,flip.1] u(c(u(I,u(D,x))),I)=I. 2089 [para_from,1195.1.1,81.1.1.2,flip.1] u(c(u(c(u(x,D)),y)),u(x,U))=u(I,u(x,D)). 2100 [para_from,1195.1.1,67.1.1.2,demod,32,flip.1] u(I,u(D,u(x,c(u(Z,y)))))=u(x,U). 2111,2110 [para_from,1195.1.1,95.1.1.2.1,demod,1243,44,2085,42,32,1243,419,1243,44,flip.1] u(I,u(D,x))=U. 2133,2132 [back_demod,2100,demod,2111,flip.1] u(x,U)=U. 2167 [back_demod,43,demod,2111] u(U,x)=U. 2171 [back_demod,2089,demod,2133,2133,flip.1] u(I,u(x,D))=U. 2292 [para_into,141.1.1,2167.1.1,flip.1] k(u(I,x),U)=U. 2329 [para_into,2292.1.1.1,2171.1.1] k(U,U)=U. 2331 [binary,2329.1,1.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 148 clauses generated 3183 clauses kept 1278 clauses forward subsumed 3073 clauses back subsumed 27 Kbytes malloced 1500 ----------- times (seconds) ----------- user CPU time 0.40 (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) 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 Fri Nov 21 22:57:00 2003