----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Fri Nov 21 23:36:22 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). 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("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("dDef.txt"). ------- start included file dDef.txt------- formula_list(usable). all x y (d(x,y)=c(k(c(x),c(y)))). end_of_list. -------> usable clausifies to: list(usable). 0 [] d(x,y)=c(k(c(x),c(y))). end_of_list. ------- end included file dDef.txt------- include("ix.txt"). ------- start included file ix.txt------- formula_list(usable). all x (k(x,I)=x). all x (k(I,x)=x). all x i(x,k(x,U)). all x i(x,k(U,x)). all x i(d(Z,x),x). -(all x i(d(x,Z),x)). end_of_list. -------> usable clausifies to: list(usable). 0 [] k(x,I)=x. 0 [] k(I,x)=x. 0 [] i(x,k(x,U)). 0 [] i(x,k(U,x)). 0 [] i(d(Z,x),x). 0 [] -i(d($c1,Z),$c1). end_of_list. ------- end included file ix.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)|u(x,y)=y. ** KEPT (pick-wt=8): 2 [] i(x,y)|u(x,y)!=y. ** KEPT (pick-wt=5): 3 [] -i(d($c1,Z),$c1). ------------> process sos: ** KEPT (pick-wt=7): 4 [] u(x,y)=u(y,x). ** KEPT (pick-wt=11): 6 [copy,5,flip.1] u(u(x,y),z)=u(x,u(y,z)). ---> New Demodulator: 7 [new_demod,6] u(u(x,y),z)=u(x,u(y,z)). ** KEPT (pick-wt=14): 8 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ---> New Demodulator: 9 [new_demod,8] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ** KEPT (pick-wt=11): 11 [copy,10,flip.1] k(k(x,y),z)=k(x,k(y,z)). ---> New Demodulator: 12 [new_demod,11] k(k(x,y),z)=k(x,k(y,z)). ** KEPT (pick-wt=5): 13 [] r(r(x))=x. ---> New Demodulator: 14 [new_demod,13] r(r(x))=x. ** KEPT (pick-wt=10): 15 [] r(u(x,y))=u(r(x),r(y)). ---> New Demodulator: 16 [new_demod,15] r(u(x,y))=u(r(x),r(y)). ** KEPT (pick-wt=10): 17 [] r(k(x,y))=k(r(y),r(x)). ---> New Demodulator: 18 [new_demod,17] r(k(x,y))=k(r(y),r(x)). ** KEPT (pick-wt=13): 20 [copy,19,flip.1] u(k(x,y),k(z,y))=k(u(x,z),y). ---> New Demodulator: 21 [new_demod,20] u(k(x,y),k(z,y))=k(u(x,z),y). ** KEPT (pick-wt=5): 22 [] k(x,I)=x. ---> New Demodulator: 23 [new_demod,22] k(x,I)=x. ** KEPT (pick-wt=13): 24 [] u(k(r(x),c(k(x,y))),c(y))=c(y). ---> New Demodulator: 25 [new_demod,24] u(k(r(x),c(k(x,y))),c(y))=c(y). ** KEPT (pick-wt=10): 27 [copy,26,flip.1] c(u(c(x),c(y)))=n(x,y). ---> New Demodulator: 28 [new_demod,27] c(u(c(x),c(y)))=n(x,y). ** KEPT (pick-wt=4): 30 [copy,29,flip.1] c(I)=D. ---> New Demodulator: 31 [new_demod,30] c(I)=D. ** KEPT (pick-wt=5): 33 [copy,32,flip.1] u(I,D)=U. ---> New Demodulator: 34 [new_demod,33] u(I,D)=U. ** KEPT (pick-wt=4): 36 [copy,35,flip.1] c(U)=Z. ---> New Demodulator: 37 [new_demod,36] c(U)=Z. ** KEPT (pick-wt=10): 39 [copy,38,flip.1] c(k(c(x),c(y)))=d(x,y). ---> New Demodulator: 40 [new_demod,39] c(k(c(x),c(y)))=d(x,y). ** KEPT (pick-wt=3): 42 [copy,41,demod,23] x=x. ** KEPT (pick-wt=5): 43 [] k(I,x)=x. ---> New Demodulator: 44 [new_demod,43] k(I,x)=x. ** KEPT (pick-wt=5): 45 [] i(x,k(x,U)). ** KEPT (pick-wt=5): 46 [] i(x,k(U,x)). ** KEPT (pick-wt=5): 47 [] i(d(Z,x),x). Following clause subsumed by 4 during input processing: 0 [copy,4,flip.1] u(x,y)=u(y,x). >>>> Starting back demodulation with 7. >>>> Starting back demodulation with 9. >>>> Starting back demodulation with 12. >>>> Starting back demodulation with 14. >>>> Starting back demodulation with 16. >>>> Starting back demodulation with 18. >>>> Starting back demodulation with 21. >>>> Starting back demodulation with 23. >>>> Starting back demodulation with 25. >>>> Starting back demodulation with 28. >> back demodulating 8 with 28. >>>> Starting back demodulation with 31. >>>> Starting back demodulation with 34. >>>> Starting back demodulation with 37. >>>> Starting back demodulation with 40. Following clause subsumed by 42 during input processing: 0 [copy,42,flip.1] x=x. >>>> Starting back demodulation with 44. >>>> Starting back demodulation with 49. ======= end of input processing ======= =========== start of search =========== Resetting weight limit to 9. sos_size=2166 ----> UNIT CONFLICT at 0.67 sec ----> 5376 [binary,5375.1,3.1] $F. Length of proof is 66. Level of proof is 20. ---------------- PROOF ---------------- 1 [] -i(x,y)|u(x,y)=y. 2 [] i(x,y)|u(x,y)!=y. 3 [] -i(d($c1,Z),$c1). 4 [] u(x,y)=u(y,x). 5 [] u(x,u(y,z))=u(u(x,y),z). 6 [copy,5,flip.1] u(u(x,y),z)=u(x,u(y,z)). 8 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. 14,13 [] r(r(x))=x. 17 [] r(k(x,y))=k(r(y),r(x)). 19 [] k(u(x,y),z)=u(k(x,z),k(y,z)). 20 [copy,19,flip.1] u(k(x,y),k(z,y))=k(u(x,z),y). 23,22 [] k(x,I)=x. 24 [] u(k(r(x),c(k(x,y))),c(y))=c(y). 26 [] n(x,y)=c(u(c(x),c(y))). 28,27 [copy,26,flip.1] c(u(c(x),c(y)))=n(x,y). 29 [] D=c(I). 31,30 [copy,29,flip.1] c(I)=D. 32 [] U=u(I,D). 34,33 [copy,32,flip.1] u(I,D)=U. 35 [] Z=c(U). 37,36 [copy,35,flip.1] c(U)=Z. 38 [] d(x,y)=c(k(c(x),c(y))). 39 [copy,38,flip.1] c(k(c(x),c(y)))=d(x,y). 44,43 [] k(I,x)=x. 45 [] i(x,k(x,U)). 46 [] i(x,k(U,x)). 47 [] i(d(Z,x),x). 48 [back_demod,8,demod,28] u(c(u(c(x),y)),n(x,y))=x. 55 [para_from,33.1.1,6.1.1.1] u(U,x)=u(I,u(D,x)). 57 [hyper,45,1] u(x,k(x,U))=k(x,U). 59 [para_into,45.1.2,43.1.1] i(I,U). 62,61 [hyper,59,1] u(I,U)=U. 63 [hyper,46,1] u(x,k(U,x))=k(U,x). 65 [hyper,47,1] u(d(Z,x),x)=x. 77 [para_from,65.1.1,6.1.1.1,flip.1] u(d(Z,x),u(x,y))=u(x,y). 85 [para_into,17.1.1.1,43.1.1,flip.1] k(r(x),r(I))=r(x). 93 [para_into,85.1.1.1,13.1.1,demod,14] k(x,r(I))=x. 95 [para_into,93.1.1,43.1.1] r(I)=I. 116 [para_into,55.1.1,4.1.1] u(x,U)=u(I,u(D,x)). 117 [copy,116,flip.1] u(I,u(D,x))=u(x,U). 128 [para_from,57.1.1,6.1.1.1,flip.1] u(x,u(k(x,U),y))=u(k(x,U),y). 131 [para_into,63.1.1,6.1.1] u(x,u(y,k(U,u(x,y))))=k(U,u(x,y)). 140 [para_into,24.1.1.1.1,95.1.1,demod,44,44] u(c(x),c(x))=c(x). 161 [para_into,140.1.1.1,30.1.1,demod,31,31] u(D,D)=D. 165 [para_from,140.1.1,6.1.1.1,flip.1] u(c(x),u(c(x),y))=u(c(x),y). 178,177 [para_into,27.1.1.1.2,30.1.1] c(u(c(x),D))=n(x,I). 182,181 [para_into,27.1.1.1,140.1.1] c(c(x))=n(x,x). 183 [para_into,27.1.1.1,4.1.1,demod,28] n(x,y)=n(y,x). 200,199 [para_into,181.1.1.1,36.1.1] c(Z)=n(U,U). 244 [para_from,39.1.1,27.1.1.1.1] c(u(d(x,y),c(z)))=n(k(c(x),c(y)),z). 255 [para_into,48.1.1.1.1.1,39.1.1] u(c(u(d(x,y),z)),n(k(c(x),c(y)),z))=k(c(x),c(y)). 257 [para_into,48.1.1.1.1.1,36.1.1] u(c(u(Z,x)),n(U,x))=U. 277 [para_into,48.1.1.2,183.1.1] u(c(u(c(x),y)),n(y,x))=x. 283 [para_from,48.1.1,6.1.1.1,flip.1] u(c(u(c(x),y)),u(n(x,y),z))=u(x,z). 351 [para_into,117.1.1.2,161.1.1,demod,34,flip.1] u(D,U)=U. 1234 [hyper,128,2] i(x,u(k(x,U),y)). 1320 [para_into,1234.1.2,20.1.1] i(x,k(u(x,y),U)). 1533 [para_into,1320.1.2.1,4.1.1] i(x,k(u(y,x),U)). 1791 [para_into,1533.1.2.1,48.1.1] i(n(x,y),k(x,U)). 1848 [para_into,1791.1.2,43.1.1] i(n(I,x),U). 1852 [para_into,1848.1.1,183.1.1] i(n(x,I),U). 1854,1853 [hyper,1852,1] u(n(x,I),U)=U. 2452 [hyper,165,2] i(c(x),u(c(x),y)). 2529 [para_into,2452.1.2,48.1.1] i(c(u(c(x),y)),x). 2550 [para_into,2529.1.1.1.1,30.1.1] i(c(u(D,x)),I). 2558 [para_into,2529.1.1.1,57.1.1] i(c(k(c(x),U)),x). 2564 [para_into,2529.1.1,27.1.1] i(n(x,y),x). 2567 [para_into,2564.1.1,183.1.1] i(n(x,y),y). 2568 [hyper,2567,1] u(n(x,y),y)=y. 2643 [para_into,2550.1.1.1,351.1.1,demod,37] i(Z,I). 2653,2652 [hyper,2643,1] u(Z,I)=I. 2665,2664 [para_from,2652.1.1,131.1.1.2.2.2,demod,23,62,2653,23] u(Z,U)=U. 2841 [para_from,2664.1.1,77.1.1.2,demod,2665] u(d(Z,Z),U)=U. 3307 [para_into,2568.1.1,4.1.1] u(x,n(y,x))=x. 3493 [para_from,3307.1.1,117.1.1.2,demod,34,flip.1] u(n(x,D),U)=U. 3853,3852 [para_into,244.1.1.1,65.1.1,demod,182,200,182,flip.1] n(k(n(U,U),n(x,x)),x)=n(x,x). 4351,4350 [para_into,255.1.1.1.1,2841.1.1,demod,37,200,200,3853,200,200] u(Z,n(U,U))=k(n(U,U),n(U,U)). 4543,4542 [para_into,257.1.1.1.1,2664.1.1,demod,37,4351] k(n(U,U),n(U,U))=U. 4574 [back_demod,4350,demod,4543] u(Z,n(U,U))=U. 5011 [para_from,4574.1.1,257.1.1.1.1,demod,37] u(Z,n(U,n(U,U)))=U. 5191 [para_into,283.1.1.2,3493.1.1,demod,178,1854,flip.1] u(x,U)=U. 5269,5268 [para_from,5191.1.1,277.1.1.1.1,demod,37] u(Z,n(U,x))=x. 5271,5270 [back_demod,5011,demod,5269] n(U,U)=U. 5275 [back_demod,199,demod,5271] c(Z)=U. 5360,5359 [para_from,5275.1.1,39.1.1.1.2] c(k(c(x),U))=d(x,Z). 5375 [back_demod,2558,demod,5360] i(d(x,Z),x). 5376 [binary,5375.1,3.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 390 clauses generated 10367 clauses kept 3242 clauses forward subsumed 6972 clauses back subsumed 279 Kbytes malloced 4119 ----------- times (seconds) ----------- user CPU time 0.87 (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 Fri Nov 21 23:36:23 2003