----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Fri Nov 21 23:35:27 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))). end_of_list. -------> usable clausifies to: list(usable). 0 [] k(x,I)=x. 0 [] k(I,x)=x. 0 [] -i($c1,k($c1,U)). 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($c1,k($c1,U)). ------------> 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. 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 46. ======= end of input processing ======= =========== start of search =========== Resetting weight limit to 9. sos_size=1289 Resetting weight limit to 8. sos_size=1360 ----> UNIT CONFLICT at 1.45 sec ----> 6442 [binary,6441.1,3.1] $F. Length of proof is 93. Level of proof is 25. ---------------- PROOF ---------------- 1 [] -i(x,y)|u(x,y)=y. 2 [] i(x,y)|u(x,y)!=y. 3 [] -i($c1,k($c1,U)). 4 [] u(x,y)=u(y,x). 5 [] u(x,u(y,z))=u(u(x,y),z). 7,6 [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. 16,15 [] r(u(x,y))=u(r(x),r(y)). 18,17 [] 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. 44,43 [] k(I,x)=x. 45 [back_demod,8,demod,28] u(c(u(c(x),y)),n(x,y))=x. 47 [para_into,6.1.1.1,4.1.1,demod,7] u(x,u(y,z))=u(y,u(x,z)). 48 [para_into,6.1.1,4.1.1] u(x,u(y,z))=u(y,u(z,x)). 51,50 [para_into,33.1.1,4.1.1] u(D,I)=U. 53,52 [para_from,33.1.1,6.1.1.1] u(U,x)=u(I,u(D,x)). 54 [para_into,52.1.1,4.1.1] u(x,U)=u(I,u(D,x)). 55 [copy,54,flip.1] u(I,u(D,x))=u(x,U). 63 [para_into,55.1.1,4.1.1,demod,7] u(D,u(x,I))=u(x,U). 73 [para_into,15.1.1.1,50.1.1] r(U)=u(r(D),r(I)). 78 [para_into,63.1.1.2,4.1.1] u(D,u(I,x))=u(x,U). 80 [copy,78,flip.1] u(x,U)=u(D,u(I,x)). 87 [para_into,17.1.1.1,43.1.1,flip.1] k(r(x),r(I))=r(x). 95 [para_into,87.1.1.1,13.1.1,demod,14] k(x,r(I))=x. 98,97 [para_into,95.1.1,43.1.1] r(I)=I. 101 [back_demod,73,demod,98] r(U)=u(r(D),I). 103 [para_into,20.1.1.1,43.1.1] u(x,k(y,x))=k(u(I,y),x). 111 [copy,103,flip.1] k(u(I,x),y)=u(y,k(x,y)). 118 [para_from,78.1.1,6.1.1.1,demod,7,53,7] u(x,u(I,u(D,y)))=u(D,u(I,u(x,y))). 119 [copy,118,flip.1] u(D,u(I,u(x,y)))=u(x,u(I,u(D,y))). 121 [para_into,80.1.1,6.1.1] u(x,u(y,U))=u(D,u(I,u(x,y))). 124 [para_into,27.1.1.1.1,36.1.1] c(u(Z,c(x)))=n(U,x). 130 [para_into,27.1.1.1.2,36.1.1] c(u(c(x),Z))=n(x,U). 132 [para_into,27.1.1.1.2,30.1.1] c(u(c(x),D))=n(x,I). 136 [para_into,27.1.1.1,4.1.1,demod,28] n(x,y)=n(y,x). 140 [para_into,24.1.1.1.1,97.1.1,demod,44,44] u(c(x),c(x))=c(x). 148 [para_into,24.1.1.1.2.1,22.1.1,demod,31,31] u(k(r(x),c(x)),D)=D. 156 [para_from,24.1.1,15.1.1.1,demod,18,14,flip.1] u(k(r(c(k(x,y))),x),r(c(y)))=r(c(y)). 161 [para_into,140.1.1.1,36.1.1,demod,37,37] u(Z,Z)=Z. 163 [para_into,140.1.1.1,30.1.1,demod,31,31] u(D,D)=D. 168,167 [para_from,140.1.1,27.1.1.1] c(c(x))=n(x,x). 171 [para_from,140.1.1,6.1.1.1,flip.1] u(c(x),u(c(x),y))=u(c(x),y). 205 [para_from,161.1.1,6.1.1.1,flip.1] u(Z,u(Z,x))=u(Z,x). 209 [para_from,163.1.1,15.1.1.1,flip.1] u(r(D),r(D))=r(D). 240 [para_into,45.1.1.1,27.1.1] u(n(x,y),n(x,c(y)))=x. 257,256 [para_into,167.1.1.1,30.1.1] c(D)=n(I,I). 277 [para_into,47.1.1.2,161.1.1,flip.1] u(Z,u(x,Z))=u(x,Z). 302 [para_from,47.1.1,45.1.1.1.1] u(c(u(x,u(c(y),z))),n(y,u(x,z)))=y. 469 [para_from,132.1.1,45.1.1.1] u(n(x,I),n(x,D))=x. 516 [para_into,148.1.1,4.1.1] u(D,k(r(x),c(x)))=D. 798 [para_into,205.1.1.2,48.1.1] u(Z,u(x,u(y,Z)))=u(Z,u(x,y)). 858 [para_into,103.1.1.2,43.1.1] u(x,x)=k(u(I,I),x). 1082 [para_into,469.1.1.1,136.1.1] u(n(I,x),n(x,D))=x. 1110 [para_into,111.1.1.1,33.1.1,flip.1] u(x,k(D,x))=k(U,x). 1207 [para_into,516.1.1.2.2,256.1.1] u(D,k(r(D),n(I,I)))=D. 1544 [para_into,858.1.1,209.1.1,flip.1] k(u(I,I),r(D))=r(D). 1675 [para_from,121.1.1,45.1.1.1.1] u(c(u(D,u(I,u(c(x),y)))),n(x,u(y,U)))=x. 3007 [para_from,1082.1.1,119.1.1.2.2,flip.1] u(n(I,x),u(I,u(D,n(x,D))))=u(D,u(I,x)). 3254 [para_from,1544.1.1,17.1.1.1,demod,14,14,16,98,98,flip.1] k(D,u(I,I))=D. 3274 [para_from,3254.1.1,156.1.1.1.1.1.1,demod,257] u(k(r(n(I,I)),D),r(c(u(I,I))))=r(c(u(I,I))). 3330 [hyper,171,2] i(c(x),u(c(x),y)). 3413 [para_into,3330.1.2,45.1.1] i(c(u(c(x),y)),x). 3462 [para_into,3413.1.1.1,121.1.1] i(c(u(D,u(I,u(c(x),y)))),x). 3477 [para_into,3413.1.1,27.1.1] i(n(x,y),x). 3563 [hyper,3477,1] u(n(x,y),x)=x. 3565 [para_into,3477.1.1,136.1.1] i(n(x,y),y). 3566 [hyper,3565,1] u(n(x,y),y)=y. 4242 [para_from,3563.1.1,15.1.1.1,flip.1] u(r(n(x,y)),r(x))=r(x). 4245,4244 [para_from,3563.1.1,6.1.1.1,flip.1] u(n(x,y),u(x,z))=u(x,z). 4252 [back_demod,3007,demod,4245] u(I,u(D,n(x,D)))=u(D,u(I,x)). 4256 [para_into,3566.1.1,121.1.1] u(D,u(I,u(n(x,u(y,U)),y)))=u(y,U). 4281,4280 [para_into,3566.1.1,4.1.1] u(x,n(y,x))=x. 4285,4284 [back_demod,4252,demod,4281,34,flip.1] u(D,u(I,x))=U. 4291,4290 [back_demod,4256,demod,4285,flip.1] u(x,U)=U. 4292 [back_demod,3462,demod,4285,37] i(Z,x). 4306 [back_demod,1675,demod,4285,37,4291] u(Z,n(x,U))=x. 4482,4481 [hyper,4292,1] u(Z,x)=x. 4484,4483 [back_demod,4306,demod,4482] n(x,U)=x. 4524,4523 [back_demod,798,demod,4482,4482] u(x,u(y,Z))=u(x,y). 4536,4535 [back_demod,277,demod,4524,4482,flip.1] u(x,Z)=x. 4539 [back_demod,124,demod,4482,168] n(x,x)=n(U,x). 4616,4615 [back_demod,130,demod,4536,168,4484] n(x,x)=x. 4640,4639 [copy,4539,flip.1,demod,4616] n(U,x)=x. 4743 [back_demod,3274,demod,4616,98,44] u(D,r(c(u(I,I))))=r(c(u(I,I))). 4822,4821 [back_demod,1207,demod,4616,23] u(D,r(D))=D. 4838,4837 [back_demod,256,demod,4616] c(D)=I. 4842,4841 [back_demod,167,demod,4616] c(c(x))=x. 5111,5110 [para_from,4837.1.1,140.1.1.2,demod,4838,4838] u(I,I)=I. 5131,5130 [back_demod,4743,demod,5111,31,4822,5111,31,flip.1] r(D)=D. 5283,5282 [back_demod,101,demod,5131,51] r(U)=U. 5634 [para_into,240.1.1.1,4639.1.1,demod,4640] u(x,c(x))=U. 6077 [para_into,302.1.1.1.1.2,5634.1.1,demod,4291,37,4842,4482] n(x,u(y,x))=x. 6163 [para_into,6077.1.1.2,4.1.1] n(x,u(x,y))=x. 6255 [para_into,6163.1.1.2,1110.1.1] n(x,k(U,x))=x. 6270 [para_into,6255.1.1,136.1.1] n(k(U,x),x)=x. 6422 [hyper,4242,2] i(r(n(x,y)),r(x)). 6427 [para_into,6422.1.1.1,6270.1.1,demod,18,5283] i(r(x),k(r(x),U)). 6441 [para_into,6427.1.1,13.1.1,demod,14] i(x,k(x,U)). 6442 [binary,6441.1,3.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 720 clauses generated 45177 clauses kept 4131 clauses forward subsumed 21083 clauses back subsumed 457 Kbytes malloced 4279 ----------- times (seconds) ----------- user CPU time 1.65 (0 hr, 0 min, 1 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 23:35:28 2003