----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Sun Nov 30 15:35:12 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("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("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("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("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("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("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("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("11booleanLaws.txt"). ------- start included file 11booleanLaws.txt------- formula_list(usable). all x y (c(n(x,y))=u(c(x),c(y))). all x y (c(u(x,y))=n(c(x),c(y))). end_of_list. -------> usable clausifies to: list(usable). 0 [] c(n(x,y))=u(c(x),c(y)). 0 [] c(u(x,y))=n(c(x),c(y)). end_of_list. ------- end included file 11booleanLaws.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("xix_b_ghost.txt"). ------- start included file xix_b_ghost.txt------- formula_list(usable). all x y (k(x,r(y))=Z->n(k(U,x),k(U,y))=Z). all x y (n(k(U,x),k(U,y))=Z->k(x,r(y))=Z). -(all x y (u(d(Z,x),d(Z,y))=U->d(x,r(y))=U)). end_of_list. -------> usable clausifies to: list(usable). 0 [] k(x,r(y))!=Z|n(k(U,x),k(U,y))=Z. 0 [] n(k(U,x),k(U,y))!=Z|k(x,r(y))=Z. 0 [] u(d(Z,$c2),d(Z,$c1))=U. 0 [] d($c2,r($c1))!=U. end_of_list. ------- end included file xix_b_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=12): 1 [] u(x,y)!=y|c(u(y,c(x)))=Z. ** KEPT (pick-wt=12): 2 [] c(u(x,c(y)))!=Z|u(y,x)=x. ** KEPT (pick-wt=8): 3 [] -i(x,y)|i(r(x),r(y)). ** KEPT (pick-wt=15): 4 [] k(x,r(y))!=Z|n(k(U,x),k(U,y))=Z. ** KEPT (pick-wt=15): 5 [] n(k(U,x),k(U,y))!=Z|k(x,r(y))=Z. ** KEPT (pick-wt=6): 6 [] d($c2,r($c1))!=U. ------------> process sos: ** KEPT (pick-wt=10): 8 [copy,7,flip.1] c(u(c(x),c(y)))=n(x,y). ---> New Demodulator: 9 [new_demod,8] c(u(c(x),c(y)))=n(x,y). ** KEPT (pick-wt=10): 11 [copy,10,flip.1] c(k(c(x),c(y)))=d(x,y). ---> New Demodulator: 12 [new_demod,11] c(k(c(x),c(y)))=d(x,y). ** KEPT (pick-wt=4): 14 [copy,13,flip.1] c(I)=D. ---> New Demodulator: 15 [new_demod,14] c(I)=D. ** KEPT (pick-wt=5): 17 [copy,16,flip.1] u(I,D)=U. ---> New Demodulator: 18 [new_demod,17] u(I,D)=U. ** KEPT (pick-wt=4): 20 [copy,19,flip.1] c(U)=Z. ---> New Demodulator: 21 [new_demod,20] c(U)=Z. ** KEPT (pick-wt=7): 22 [] u(x,y)=u(y,x). ** KEPT (pick-wt=11): 24 [copy,23,flip.1] u(u(x,y),z)=u(x,u(y,z)). ---> New Demodulator: 25 [new_demod,24] u(u(x,y),z)=u(x,u(y,z)). ** KEPT (pick-wt=11): 27 [copy,26,demod,9] u(c(u(c(x),y)),n(x,y))=x. ---> New Demodulator: 28 [new_demod,27] u(c(u(c(x),y)),n(x,y))=x. ** KEPT (pick-wt=11): 30 [copy,29,flip.1] k(k(x,y),z)=k(x,k(y,z)). ---> New Demodulator: 31 [new_demod,30] k(k(x,y),z)=k(x,k(y,z)). ** KEPT (pick-wt=5): 32 [] r(r(x))=x. ---> New Demodulator: 33 [new_demod,32] r(r(x))=x. ** KEPT (pick-wt=10): 34 [] r(u(x,y))=u(r(x),r(y)). ---> New Demodulator: 35 [new_demod,34] r(u(x,y))=u(r(x),r(y)). ** KEPT (pick-wt=10): 36 [] r(k(x,y))=k(r(y),r(x)). ---> New Demodulator: 37 [new_demod,36] r(k(x,y))=k(r(y),r(x)). ** KEPT (pick-wt=5): 38 [] k(x,I)=x. ---> New Demodulator: 39 [new_demod,38] k(x,I)=x. ** KEPT (pick-wt=13): 40 [] u(k(r(x),c(k(x,y))),c(y))=c(y). ---> New Demodulator: 41 [new_demod,40] u(k(r(x),c(k(x,y))),c(y))=c(y). ** KEPT (pick-wt=5): 42 [] c(c(x))=x. ---> New Demodulator: 43 [new_demod,42] c(c(x))=x. ** KEPT (pick-wt=11): 45 [copy,44,demod,35] u(r(x),r(y))=u(r(x),r(y)). ** KEPT (pick-wt=7): 46 [] r(c(x))=c(r(x)). ---> New Demodulator: 47 [new_demod,46] r(c(x))=c(r(x)). ** KEPT (pick-wt=10): 48 [] r(n(x,y))=n(r(x),r(y)). ---> New Demodulator: 49 [new_demod,48] r(n(x,y))=n(r(x),r(y)). ** KEPT (pick-wt=3): 51 [copy,50,demod,33] x=x. ** KEPT (pick-wt=10): 52 [] r(d(x,y))=d(r(y),r(x)). ---> New Demodulator: 53 [new_demod,52] r(d(x,y))=d(r(y),r(x)). Following clause subsumed by 51 during input processing: 0 [demod,37] k(r(y),r(x))=k(r(y),r(x)). ** KEPT (pick-wt=4): 54 [] r(I)=I. ---> New Demodulator: 55 [new_demod,54] r(I)=I. ** KEPT (pick-wt=4): 56 [] r(Z)=Z. ---> New Demodulator: 57 [new_demod,56] r(Z)=Z. ** KEPT (pick-wt=4): 58 [] r(U)=U. ---> New Demodulator: 59 [new_demod,58] r(U)=U. ** KEPT (pick-wt=10): 60 [] c(n(x,y))=u(c(x),c(y)). ---> New Demodulator: 61 [new_demod,60] c(n(x,y))=u(c(x),c(y)). ** KEPT (pick-wt=10): 62 [] c(u(x,y))=n(c(x),c(y)). ---> New Demodulator: 63 [new_demod,62] c(u(x,y))=n(c(x),c(y)). ** KEPT (pick-wt=16): 64 [] n(x,k(n(y,k(x,r(z))),z))=n(k(y,z),x). ** KEPT (pick-wt=9): 65 [] u(d(Z,$c2),d(Z,$c1))=U. ---> New Demodulator: 66 [new_demod,65] u(d(Z,$c2),d(Z,$c1))=U. >>>> Starting back demodulation with 9. >>>> Starting back demodulation with 12. >>>> Starting back demodulation with 15. >>>> Starting back demodulation with 18. >>>> Starting back demodulation with 21. Following clause subsumed by 22 during input processing: 0 [copy,22,flip.1] u(x,y)=u(y,x). >>>> Starting back demodulation with 25. >>>> Starting back demodulation with 28. >>>> Starting back demodulation with 31. >>>> 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. >>>> Starting back demodulation with 43. Following clause subsumed by 51 during input processing: 0 [copy,45,flip.1] u(r(x),r(y))=u(r(x),r(y)). >>>> Starting back demodulation with 47. >>>> Starting back demodulation with 49. Following clause subsumed by 51 during input processing: 0 [copy,51,flip.1] x=x. 51 back subsumes 45. >>>> Starting back demodulation with 53. >>>> Starting back demodulation with 55. >>>> Starting back demodulation with 57. >>>> Starting back demodulation with 59. >>>> Starting back demodulation with 61. >>>> Starting back demodulation with 63. >> back demodulating 27 with 63. >> back demodulating 8 with 63. >> back demodulating 2 with 63. >> back demodulating 1 with 63. ** KEPT (pick-wt=16): 71 [copy,64,flip.1] n(k(x,y),z)=n(z,k(n(x,k(z,r(y))),y)). >>>> Starting back demodulation with 66. >>>> Starting back demodulation with 68. Following clause subsumed by 64 during input processing: 0 [copy,71,flip.1] n(x,k(n(y,k(x,r(z))),z))=n(k(y,z),x). ======= end of input processing ======= =========== start of search =========== Resetting weight limit to 10. sos_size=5133 Resetting weight limit to 9. sos_size=4659 -----> EMPTY CLAUSE at 10.87 sec ----> 8192 [hyper,8191,51,51] $F. Length of proof is 35. Level of proof is 16. ---------------- PROOF ---------------- 1 [] u(x,y)!=y|c(u(y,c(x)))=Z. 2 [] c(u(x,c(y)))!=Z|u(y,x)=x. 5 [] n(k(U,x),k(U,y))!=Z|k(x,r(y))=Z. 6 [] d($c2,r($c1))!=U. 7 [] n(x,y)=c(u(c(x),c(y))). 9,8 [copy,7,flip.1] c(u(c(x),c(y)))=n(x,y). 10 [] d(x,y)=c(k(c(x),c(y))). 11 [copy,10,flip.1] c(k(c(x),c(y)))=d(x,y). 19 [] Z=c(U). 21,20 [copy,19,flip.1] c(U)=Z. 22 [] u(x,y)=u(y,x). 26 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. 27 [copy,26,demod,9] u(c(u(c(x),y)),n(x,y))=x. 33,32 [] r(r(x))=x. 36 [] r(k(x,y))=k(r(y),r(x)). 38 [] k(x,I)=x. 40 [] u(k(r(x),c(k(x,y))),c(y))=c(y). 43,42 [] c(c(x))=x. 47,46 [] r(c(x))=c(r(x)). 50 [] r(r(x))=x. 51 [copy,50,demod,33] x=x. 53,52 [] r(d(x,y))=d(r(y),r(x)). 55,54 [] r(I)=I. 57,56 [] r(Z)=Z. 60 [] c(n(x,y))=u(c(x),c(y)). 63,62 [] c(u(x,y))=n(c(x),c(y)). 65 [] u(d(Z,$c2),d(Z,$c1))=U. 67 [back_demod,27,demod,63,43] u(n(x,c(y)),n(x,y))=x. 69 [back_demod,2,demod,63,43] n(c(x),y)!=Z|u(y,x)=x. 70 [back_demod,1,demod,63,43] u(x,y)!=y|n(c(y),x)=Z. 105,104 [para_into,42.1.1.1,20.1.1] c(Z)=U. 109,108 [para_into,42.1.1.1,11.1.1] c(d(x,y))=k(c(x),c(y)). 168 [para_into,36.1.1.1,38.1.1,demod,55,flip.1] k(I,r(x))=r(x). 170 [para_into,36.1.1.1,5.2.1,demod,57,33,flip.1] k(x,r(y))=Z|n(k(U,y),k(U,x))!=Z. 205 [para_into,40.1.1.1.1,54.1.1] u(k(I,c(k(I,x))),c(x))=c(x). 251,250 [para_into,168.1.1.2,32.1.1,demod,33] k(I,x)=x. 252 [back_demod,205,demod,251,251] u(c(x),c(x))=c(x). 298 [para_into,252.1.1.1,42.1.1,demod,43,43] u(x,x)=x. 369 [para_into,62.1.1.1,298.1.1,flip.1] n(c(x),c(x))=c(x). 375 [para_into,62.1.1.1,65.1.1,demod,21,109,105,109,105,flip.1] n(k(U,c($c2)),k(U,c($c1)))=Z. 427 [para_into,369.1.1.1,42.1.1,demod,43,43] n(x,x)=x. 457 [para_into,67.1.1.2,427.1.1] u(n(x,c(x)),x)=x. 518 [hyper,70,298] n(c(x),x)=Z. 596,595 [para_into,518.1.1.1,42.1.1] n(x,c(x))=Z. 600,599 [back_demod,457,demod,596] u(Z,x)=x. 608 [para_into,599.1.1,22.1.1] u(x,Z)=x. 616 [para_from,599.1.1,62.1.1.1,demod,105,flip.1] n(U,c(x))=c(x). 757 [para_into,608.1.1,69.2.1,demod,105] Z=x|n(U,x)!=Z. 806 [para_from,595.1.1,60.1.1.1,demod,105,43,flip.1] u(c(x),x)=U. 1582,1581 [para_into,616.1.1.2,42.1.1,demod,43] n(U,x)=x. 1596 [back_demod,757,demod,1582] Z=x|x!=Z. 1650 [para_from,1596.1.1,56.1.1.1] r(x)=Z|x!=Z. 2254 [para_into,1650.1.1,32.1.1] x=Z|r(x)!=Z. 2467 [para_from,2254.1.1,806.1.1.1,demod,600,47] x=U|c(r(x))!=Z. 5068 [para_from,2467.1.1,6.1.1,demod,53,33,109] U!=U|k(c($c1),c(r($c2)))!=Z. 8188,8187 [hyper,375,170,demod,47] k(c($c1),c(r($c2)))=Z. 8191 [back_demod,5068,demod,8188] U!=U|Z!=Z. 8192 [hyper,8191,51,51] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 832 clauses generated 195337 clauses kept 7660 clauses forward subsumed 62684 clauses back subsumed 1567 Kbytes malloced 4502 ----------- times (seconds) ----------- user CPU time 11.07 (0 hr, 0 min, 11 sec) system CPU time 0.00 (0 hr, 0 min, 0 sec) wall-clock time 11 (0 hr, 0 min, 11 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 15:35:23 2003