----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Sun Nov 30 15:25:44 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_distinct_vars,3). 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("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("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("xii.txt"). ------- start included file xii.txt------- formula_list(usable). all x y z (i(k(x,y),c(z))<->i(k(r(x),z),c(y))). all x y z (i(k(x,y),c(z))<->i(k(z,r(y)),c(x))). end_of_list. -------> usable clausifies to: list(usable). 0 [] -i(k(x,y),c(z))|i(k(r(x),z),c(y)). 0 [] i(k(x,y),c(z))| -i(k(r(x),z),c(y)). 0 [] -i(k(x,y),c(z))|i(k(z,r(y)),c(x)). 0 [] i(k(x,y),c(z))| -i(k(z,r(y)),c(x)). end_of_list. ------- end included file xii.txt------- include("xix_a_ghost.txt"). ------- start included file xix_a_ghost.txt------- formula_list(usable). -(all x y (k(c(x),c(r(y)))=Z<->i(k(U,c(x)),y))). end_of_list. -------> usable clausifies to: list(usable). 0 [] k(c($c2),c(r($c1)))=Z|i(k(U,c($c2)),$c1). 0 [] k(c($c2),c(r($c1)))!=Z| -i(k(U,c($c2)),$c1). end_of_list. ------- end included file xix_a_ghost.txt------- SCAN INPUT: prop=0, horn=0, equality=1, symmetry=0, max_lits=2. This ia a non-Horn set with equality. The strategy will be Knuth-Bendix, ordered hyper_res, factoring, and unit deletion, 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: set(unit_deletion). dependent: set(factor). ------------> 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=12): 3 [] u(x,y)!=y|c(u(y,c(x)))=Z. ** KEPT (pick-wt=12): 4 [] c(u(x,c(y)))!=Z|u(y,x)=x. ** KEPT (pick-wt=8): 5 [] -i(x,y)|i(r(x),r(y)). ** KEPT (pick-wt=13): 6 [] -i(k(x,y),c(z))|i(k(r(x),z),c(y)). ** KEPT (pick-wt=13): 7 [] i(k(x,y),c(z))| -i(k(r(x),z),c(y)). ** KEPT (pick-wt=13): 8 [] -i(k(x,y),c(z))|i(k(z,r(y)),c(x)). ** KEPT (pick-wt=13): 9 [] i(k(x,y),c(z))| -i(k(z,r(y)),c(x)). ** KEPT (pick-wt=14): 10 [] k(c($c2),c(r($c1)))!=Z| -i(k(U,c($c2)),$c1). ------------> process sos: ** KEPT (pick-wt=4): 12 [copy,11,flip.1] c(I)=D. ---> New Demodulator: 13 [new_demod,12] c(I)=D. ** KEPT (pick-wt=5): 15 [copy,14,flip.1] u(I,D)=U. ---> New Demodulator: 16 [new_demod,15] u(I,D)=U. ** KEPT (pick-wt=4): 18 [copy,17,flip.1] c(U)=Z. ---> New Demodulator: 19 [new_demod,18] c(U)=Z. ** KEPT (pick-wt=7): 20 [] u(x,y)=u(y,x). ** KEPT (pick-wt=11): 22 [copy,21,flip.1] u(u(x,y),z)=u(x,u(y,z)). ---> New Demodulator: 23 [new_demod,22] u(u(x,y),z)=u(x,u(y,z)). ** KEPT (pick-wt=14): 24 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ---> New Demodulator: 25 [new_demod,24] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ** KEPT (pick-wt=5): 26 [] c(c(x))=x. ---> New Demodulator: 27 [new_demod,26] c(c(x))=x. ** KEPT (pick-wt=10): 28 [] r(u(x,y))=u(r(x),r(y)). ---> New Demodulator: 29 [new_demod,28] r(u(x,y))=u(r(x),r(y)). ** KEPT (pick-wt=7): 30 [] r(c(x))=c(r(x)). ---> New Demodulator: 31 [new_demod,30] r(c(x))=c(r(x)). ** KEPT (pick-wt=10): 32 [] r(n(x,y))=n(r(x),r(y)). ---> New Demodulator: 33 [new_demod,32] r(n(x,y))=n(r(x),r(y)). ** KEPT (pick-wt=5): 34 [] r(r(x))=x. ---> New Demodulator: 35 [new_demod,34] r(r(x))=x. ** KEPT (pick-wt=10): 36 [] r(d(x,y))=d(r(y),r(x)). ---> New Demodulator: 37 [new_demod,36] r(d(x,y))=d(r(y),r(x)). ** KEPT (pick-wt=10): 38 [] r(k(x,y))=k(r(y),r(x)). ---> New Demodulator: 39 [new_demod,38] r(k(x,y))=k(r(y),r(x)). ** KEPT (pick-wt=4): 40 [] r(I)=I. ---> New Demodulator: 41 [new_demod,40] r(I)=I. ** KEPT (pick-wt=4): 42 [] r(Z)=Z. ---> New Demodulator: 43 [new_demod,42] r(Z)=Z. ** KEPT (pick-wt=4): 44 [] r(U)=U. ---> New Demodulator: 45 [new_demod,44] r(U)=U. ** KEPT (pick-wt=14): 46 [] k(c($c2),c(r($c1)))=Z|i(k(U,c($c2)),$c1). >>>> Starting back demodulation with 13. >>>> Starting back demodulation with 16. >>>> Starting back demodulation with 19. Following clause subsumed by 20 during input processing: 0 [copy,20,flip.1] u(x,y)=u(y,x). >>>> Starting back demodulation with 23. >>>> Starting back demodulation with 25. >>>> Starting back demodulation with 27. >>>> Starting back demodulation with 29. >>>> 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. >>>> Starting back demodulation with 45. ======= end of input processing ======= =========== start of search =========== Resetting weight limit to 9. sos_size=5322 -----> EMPTY CLAUSE at 62.60 sec ----> 8758 [back_demod,5352,demod,8752,98,unit_del,8567,102] $F. Length of proof is 42. Level of proof is 14. ---------------- PROOF ---------------- 1 [] -i(x,y)|u(x,y)=y. 2 [] i(x,y)|u(x,y)!=y. 3 [] u(x,y)!=y|c(u(y,c(x)))=Z. 4 [] c(u(x,c(y)))!=Z|u(y,x)=x. 8 [] -i(k(x,y),c(z))|i(k(z,r(y)),c(x)). 9 [] i(k(x,y),c(z))| -i(k(z,r(y)),c(x)). 10 [] k(c($c2),c(r($c1)))!=Z| -i(k(U,c($c2)),$c1). 11 [] D=c(I). 12 [copy,11,flip.1] c(I)=D. 14 [] U=u(I,D). 15 [copy,14,flip.1] u(I,D)=U. 17 [] Z=c(U). 19,18 [copy,17,flip.1] c(U)=Z. 20 [] u(x,y)=u(y,x). 21 [] u(x,u(y,z))=u(u(x,y),z). 23,22 [copy,21,flip.1] u(u(x,y),z)=u(x,u(y,z)). 24 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. 27,26 [] c(c(x))=x. 31,30 [] r(c(x))=c(r(x)). 35,34 [] r(r(x))=x. 39,38 [] r(k(x,y))=k(r(y),r(x)). 43,42 [] r(Z)=Z. 46 [] k(c($c2),c(r($c1)))=Z|i(k(U,c($c2)),$c1). 54 [para_from,12.1.1,4.1.1.1.2] c(u(x,D))!=Z|u(I,x)=x. 57 [para_from,12.1.1,3.2.1.1.2] u(I,x)!=x|c(u(x,D))=Z. 68 [para_from,18.1.1,3.2.1.1.2] u(U,x)!=x|c(u(x,Z))=Z. 91 [para_into,20.1.1,1.2.1,flip.1] u(x,y)=x| -i(y,x). 98,97 [para_into,26.1.1.1,18.1.1] c(Z)=U. 102 [para_into,26.1.1,26.1.1] x=x. 104 [para_from,26.1.1,8.1.2] -i(k(x,y),z)|i(k(c(z),r(y)),c(x)). 119 [para_from,97.1.1,4.1.1.1.2] c(u(x,U))!=Z|u(Z,x)=x. 135,134 [para_into,22.1.1.1,15.1.1] u(U,x)=u(I,u(D,x)). 141 [back_demod,68,demod,135] u(I,u(D,x))!=x|c(u(x,Z))=Z. 174 [para_into,24.1.1.2.1.2,97.1.1] u(c(u(c(x),Z)),c(u(c(x),U)))=x. 178 [para_into,24.1.1.2.1.2,18.1.1] u(c(u(c(x),U)),c(u(c(x),Z)))=x. 223 [para_from,30.1.1,9.2.1.2] i(k(x,c(y)),c(z))| -i(k(z,c(r(y))),c(x)). 313 [para_from,46.1.1,38.1.1.1,demod,43,31,35,31,flip.1] k(c($c1),c(r($c2)))=Z|i(k(U,c($c2)),$c1). 582 [para_into,134.1.1,20.1.1] u(x,U)=u(I,u(D,x)). 586 [para_from,134.1.1,4.1.1.1] c(u(I,u(D,c(x))))!=Z|u(x,U)=U. 642 [para_into,54.1.1.1,15.1.1,demod,19,unit_del,102] u(I,I)=I. 657 [para_from,642.1.1,22.1.1.1,flip.1] u(I,u(I,x))=u(I,x). 1612 [para_into,582.1.1,4.2.1,demod,135,flip.1] u(I,u(D,x))=U|c(u(I,u(D,c(x))))!=Z. 1984 [hyper,657,57,demod,23] c(u(I,u(x,D)))=Z. 2435,2434 [para_into,1984.1.1.1.2,20.1.1] c(u(I,u(D,x)))=Z. 2445,2444 [back_demod,1612,demod,2435,unit_del,102] u(I,u(D,x))=U. 2447,2446 [back_demod,586,demod,2445,19,unit_del,102] u(x,U)=U. 2450 [back_demod,178,demod,2447,19] u(Z,c(u(c(x),Z)))=x. 2452 [back_demod,174,demod,2447,19] u(c(u(c(x),Z)),Z)=x. 2455,2454 [back_demod,119,demod,2447,19,unit_del,102] u(Z,x)=x. 2479 [back_demod,141,demod,2445] U!=x|c(u(x,Z))=Z. 2499,2498 [back_demod,2450,demod,2455] c(u(c(x),Z))=x. 2505,2504 [back_demod,2452,demod,2499] u(x,Z)=x. 2507 [back_demod,2479,demod,2505] U!=x|c(x)=Z. 2564 [hyper,2454,2] i(Z,x). 2574 [para_into,2454.1.1,91.1.1] Z=x| -i(x,Z). 3186 [para_from,2574.1.1,42.1.1.1] r(x)=Z| -i(x,Z). 4217 [para_into,2507.2.1,26.1.1,flip.1] c(x)!=U|x=Z. 5352 [para_from,4217.2.1,10.1.1,unit_del,102] -i(k(U,c($c2)),$c1)|c(k(c($c2),c(r($c1))))!=U. 8567 [para_from,313.1.1,223.2.1,demod,27,unit_del,2564,factor_simp] i(k(U,c($c2)),$c1). 8570 [hyper,8567,104,demod,31,19] i(k(c($c1),c(r($c2))),Z). 8752,8751 [hyper,8570,3186,demod,39,31,35,31] k(c($c2),c(r($c1)))=Z. 8758 [back_demod,5352,demod,8752,98,unit_del,8567,102] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 715 clauses generated 113265 clauses kept 8642 clauses forward subsumed 46749 clauses back subsumed 2328 Kbytes malloced 4311 ----------- times (seconds) ----------- user CPU time 62.80 (0 hr, 1 min, 2 sec) system CPU time 0.00 (0 hr, 0 min, 0 sec) wall-clock time 63 (0 hr, 1 min, 3 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 15:26:47 2003