----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Mon Dec 29 22:32:45 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). assign(max_literals,2). assign(max_mem,10000). assign(max_weight,20). 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("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("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("1To4booleanLaws.txt"). ------- start included file 1To4booleanLaws.txt------- formula_list(usable). all x y z (i(x,u(y,z))->i(n(x,c(z)),y)). all x y z (i(x,y)->i(u(z,x),u(z,y))). all x y (x=u(n(x,c(y)),n(x,y))). all x y i(n(x,y),y). end_of_list. -------> usable clausifies to: list(usable). 0 [] -i(x,u(y,z))|i(n(x,c(z)),y). 0 [] -i(x,y)|i(u(z,x),u(z,y)). 0 [] x=u(n(x,c(y)),n(x,y)). 0 [] i(n(x,y),y). end_of_list. ------- end included file 1To4booleanLaws.txt------- include("4To8booleanLaws.txt"). ------- start included file 4To8booleanLaws.txt------- formula_list(usable). all x y i(n(x,y),y). all x y z i(n(x,u(y,z)),u(y,n(x,z))). all x y (x=u(n(x,y),n(x,c(y)))). all x y z (i(x,y)->i(u(z,x),u(z,y))). all x y z (i(x,y)->i(n(z,x),n(z,y))). end_of_list. -------> usable clausifies to: list(usable). 0 [] i(n(x,y),y). 0 [] i(n(x,u(y,z)),u(y,n(x,z))). 0 [] x=u(n(x,y),n(x,c(y))). 0 [] -i(x,y)|i(u(z,x),u(z,y)). 0 [] -i(x,y)|i(n(z,x),n(z,y)). end_of_list. ------- end included file 4To8booleanLaws.txt------- include("9_10booleanLaws.txt"). ------- start included file 9_10booleanLaws.txt------- formula_list(usable). all x (u(x,Z)=x). all x y z (i(x,y)&i(y,z)->i(x,z)). end_of_list. -------> usable clausifies to: list(usable). 0 [] u(x,Z)=x. 0 [] -i(x,y)| -i(y,z)|i(x,z). end_of_list. ------- end included file 9_10booleanLaws.txt------- include("13booleanLaws.txt"). ------- start included file 13booleanLaws.txt------- formula_list(usable). all x y (i(x,y)&i(y,x)->x=y). end_of_list. -------> usable clausifies to: list(usable). 0 [] -i(x,y)| -i(y,x)|x=y. end_of_list. ------- end included file 13booleanLaws.txt------- include("26_27booleanLaws.txt"). ------- start included file 26_27booleanLaws.txt------- formula_list(usable). all x y z (i(x,y)&i(x,z)->i(x,n(y,z))). all x y (i(n(x,y),x)&i(n(x,y),y)). end_of_list. -------> usable clausifies to: list(usable). 0 [] -i(x,y)| -i(x,z)|i(x,n(y,z)). 0 [] i(n(x,y),x). 0 [] i(n(x,y),y). end_of_list. ------- end included file 26_27booleanLaws.txt------- include("xxii.txt"). ------- start included file xxii.txt------- formula_list(usable). all x (i(x,I)->k(x,x)=x). all x (i(x,I)->r(x)=x). end_of_list. -------> usable clausifies to: list(usable). 0 [] -i(x,I)|k(x,x)=x. 0 [] -i(x,I)|r(x)=x. end_of_list. ------- end included file xxii.txt------- include("vii.txt"). ------- start included file vii.txt------- formula_list(usable). all x y z (i(x,y)->i(k(x,z),k(y,z))). all x y z (i(x,y)->i(k(z,x),k(z,y))). all x y z (i(x,y)->i(d(x,z),d(y,z))). all x y z (i(x,y)->i(d(z,x),d(z,y))). all x y (i(x,y)->i(r(x),r(y))). end_of_list. -------> usable clausifies to: list(usable). 0 [] -i(x,y)|i(k(x,z),k(y,z)). 0 [] -i(x,y)|i(k(z,x),k(z,y)). 0 [] -i(x,y)|i(d(x,z),d(y,z)). 0 [] -i(x,y)|i(d(z,x),d(z,y)). 0 [] -i(x,y)|i(r(x),r(y)). end_of_list. ------- end included file vii.txt------- include("viii.txt"). ------- start included file viii.txt------- formula_list(usable). all x y w v i(k(n(x,w),n(y,v)),n(k(x,y),k(w,v))). end_of_list. -------> usable clausifies to: list(usable). 0 [] i(k(n(x,w),n(y,v)),n(k(x,y),k(w,v))). end_of_list. ------- end included file viii.txt------- include("xv_a.txt"). ------- start included file xv_a.txt------- formula_list(usable). all x y z v w i(n(w,k(n(y,z),v)),n(w,u(k(n(y,z),n(v,x)),k(y,n(v,c(x)))))). all y v w i(n(k(y,v),w),k(n(y,k(w,r(v))),n(v,k(r(y),w)))). all y v w i(n(k(y,v),w),k(y,n(v,k(r(y),w)))). end_of_list. -------> usable clausifies to: list(usable). 0 [] i(n(w,k(n(y,z),v)),n(w,u(k(n(y,z),n(v,x)),k(y,n(v,c(x)))))). 0 [] i(n(k(y,v),w),k(n(y,k(w,r(v))),n(v,k(r(y),w)))). 0 [] i(n(k(y,v),w),k(y,n(v,k(r(y),w)))). end_of_list. ------- end included file xv_a.txt------- include("xxvii_a.txt"). ------- start included file xxvii_a.txt------- formula_list(usable). all x y z (i(k(r(x),x),I)->i(n(k(x,y),k(x,z)),k(x,n(y,z)))). all x y z (i(k(r(x),x),I)->k(x,n(y,z))=n(k(x,y),k(x,z))). end_of_list. -------> usable clausifies to: list(usable). 0 [] -i(k(r(x),x),I)|i(n(k(x,y),k(x,z)),k(x,n(y,z))). 0 [] -i(k(r(x),x),I)|k(x,n(y,z))=n(k(x,y),k(x,z)). end_of_list. ------- end included file xxvii_a.txt------- include("subId_a_ghost.txt"). ------- start included file subId_a_ghost.txt------- formula_list(usable). all x y z i(k(n(x,y),z),n(k(x,z),k(y,z))). all x y z (i(x,I)&i(y,I)->k(n(x,y),z)=n(k(x,z),k(y,z))). all x y z (i(x,I)->k(x,n(y,z))=n(k(x,y),k(x,z))). end_of_list. -------> usable clausifies to: list(usable). 0 [] i(k(n(x,y),z),n(k(x,z),k(y,z))). 0 [] -i(x,I)| -i(y,I)|k(n(x,y),z)=n(k(x,z),k(y,z)). 0 [] -i(x,I)|k(x,n(y,z))=n(k(x,y),k(x,z)). end_of_list. ------- end included file subId_a_ghost.txt------- include("subIdLaws_a.txt"). ------- start included file subIdLaws_a.txt------- formula_list(usable). -(all x y z w (i(x,I)&i(y,I)->k(n(x,y),n(z,w))=n(n(k(x,z),k(y,z)),n(k(x,w),k(y,w))))). end_of_list. -------> usable clausifies to: list(usable). 0 [] i($c4,I). 0 [] i($c3,I). 0 [] k(n($c4,$c3),n($c2,$c1))!=n(n(k($c4,$c2),k($c3,$c2)),n(k($c4,$c1),k($c3,$c1))). end_of_list. ------- end included file subIdLaws_a.txt------- SCAN INPUT: prop=0, horn=1, equality=1, symmetry=0, max_lits=3. 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=11): 3 [] -i(x,u(y,z))|i(n(x,c(z)),y). ** KEPT (pick-wt=10): 4 [] -i(x,y)|i(u(z,x),u(z,y)). Following clause subsumed by 4 during input processing: 0 [] -i(x,y)|i(u(z,x),u(z,y)). ** KEPT (pick-wt=10): 5 [] -i(x,y)|i(n(z,x),n(z,y)). ** KEPT (pick-wt=9): 6 [] -i(x,y)| -i(y,z)|i(x,z). ** KEPT (pick-wt=9): 7 [] -i(x,y)| -i(y,x)|x=y. ** KEPT (pick-wt=11): 8 [] -i(x,y)| -i(x,z)|i(x,n(y,z)). ** KEPT (pick-wt=8): 9 [] -i(x,I)|k(x,x)=x. ** KEPT (pick-wt=7): 10 [] -i(x,I)|r(x)=x. ** KEPT (pick-wt=10): 11 [] -i(x,y)|i(k(x,z),k(y,z)). ** KEPT (pick-wt=10): 12 [] -i(x,y)|i(k(z,x),k(z,y)). ** KEPT (pick-wt=10): 13 [] -i(x,y)|i(d(x,z),d(y,z)). ** KEPT (pick-wt=10): 14 [] -i(x,y)|i(d(z,x),d(z,y)). ** KEPT (pick-wt=8): 15 [] -i(x,y)|i(r(x),r(y)). ** KEPT (pick-wt=19): 16 [] -i(k(r(x),x),I)|i(n(k(x,y),k(x,z)),k(x,n(y,z))). ** KEPT (pick-wt=19): 18 [copy,17,flip.2] -i(k(r(x),x),I)|n(k(x,y),k(x,z))=k(x,n(y,z)). ** KEPT (pick-wt=19): 20 [copy,19,flip.3] -i(x,I)| -i(y,I)|n(k(x,z),k(y,z))=k(n(x,y),z). ** KEPT (pick-wt=16): 22 [copy,21,flip.2] -i(x,I)|n(k(x,y),k(x,z))=k(x,n(y,z)). ** KEPT (pick-wt=23): 24 [copy,23,flip.1] n(n(k($c4,$c2),k($c3,$c2)),n(k($c4,$c1),k($c3,$c1)))!=k(n($c4,$c3),n($c2,$c1)). ------------> process sos: ** KEPT (pick-wt=7): 25 [] u(x,y)=u(y,x). ** KEPT (pick-wt=11): 27 [copy,26,flip.1] u(u(x,y),z)=u(x,u(y,z)). ---> New Demodulator: 28 [new_demod,27] u(u(x,y),z)=u(x,u(y,z)). ** KEPT (pick-wt=14): 29 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ---> New Demodulator: 30 [new_demod,29] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ** KEPT (pick-wt=11): 32 [copy,31,flip.1] k(k(x,y),z)=k(x,k(y,z)). ---> New Demodulator: 33 [new_demod,32] k(k(x,y),z)=k(x,k(y,z)). ** 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(u(x,y))=u(r(x),r(y)). ---> New Demodulator: 37 [new_demod,36] r(u(x,y))=u(r(x),r(y)). ** 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=5): 40 [] k(x,I)=x. ---> New Demodulator: 41 [new_demod,40] k(x,I)=x. ** KEPT (pick-wt=13): 42 [] u(k(r(x),c(k(x,y))),c(y))=c(y). ---> New Demodulator: 43 [new_demod,42] u(k(r(x),c(k(x,y))),c(y))=c(y). ** KEPT (pick-wt=10): 45 [copy,44,flip.1] c(u(c(x),c(y)))=n(x,y). ---> New Demodulator: 46 [new_demod,45] c(u(c(x),c(y)))=n(x,y). ** KEPT (pick-wt=10): 48 [copy,47,flip.1] u(n(x,c(y)),n(x,y))=x. ---> New Demodulator: 49 [new_demod,48] u(n(x,c(y)),n(x,y))=x. ** KEPT (pick-wt=5): 50 [] i(n(x,y),y). Following clause subsumed by 50 during input processing: 0 [] i(n(x,y),y). ** KEPT (pick-wt=11): 51 [] i(n(x,u(y,z)),u(y,n(x,z))). ** KEPT (pick-wt=10): 53 [copy,52,flip.1] u(n(x,y),n(x,c(y)))=x. ---> New Demodulator: 54 [new_demod,53] u(n(x,y),n(x,c(y)))=x. ** KEPT (pick-wt=5): 55 [] u(x,Z)=x. ---> New Demodulator: 56 [new_demod,55] u(x,Z)=x. ** KEPT (pick-wt=5): 57 [] i(n(x,y),x). Following clause subsumed by 50 during input processing: 0 [] i(n(x,y),y). ** KEPT (pick-wt=15): 58 [] i(k(n(x,y),n(z,u)),n(k(x,z),k(y,u))). ** KEPT (pick-wt=24): 59 [] i(n(x,k(n(y,z),u)),n(x,u(k(n(y,z),n(u,v)),k(y,n(u,c(v)))))). ** KEPT (pick-wt=19): 60 [] i(n(k(x,y),z),k(n(x,k(z,r(y))),n(y,k(r(x),z)))). ** KEPT (pick-wt=14): 61 [] i(n(k(x,y),z),k(x,n(y,k(r(x),z)))). ** KEPT (pick-wt=13): 62 [] i(k(n(x,y),z),n(k(x,z),k(y,z))). ** KEPT (pick-wt=3): 63 [] i($c4,I). ** KEPT (pick-wt=3): 64 [] i($c3,I). Following clause subsumed by 25 during input processing: 0 [copy,25,flip.1] u(x,y)=u(y,x). >>>> Starting back demodulation with 28. >>>> Starting back demodulation with 30. >>>> 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 46. >> back demodulating 29 with 46. >>>> Starting back demodulation with 49. >>>> Starting back demodulation with 54. >>>> Starting back demodulation with 56. >>>> Starting back demodulation with 66. ======= end of input processing ======= =========== start of search =========== ----> UNIT CONFLICT at 0.07 sec ----> 497 [binary,495.1,114.1] $F. Length of proof is 7. Level of proof is 3. ---------------- PROOF ---------------- 6 [] -i(x,y)| -i(y,z)|i(x,z). 19 [] -i(x,I)| -i(y,I)|k(n(x,y),z)=n(k(x,z),k(y,z)). 20 [copy,19,flip.3] -i(x,I)| -i(y,I)|n(k(x,z),k(y,z))=k(n(x,y),z). 21 [] -i(x,I)|k(x,n(y,z))=n(k(x,y),k(x,z)). 22 [copy,21,flip.2] -i(x,I)|n(k(x,y),k(x,z))=k(x,n(y,z)). 23 [] k(n($c4,$c3),n($c2,$c1))!=n(n(k($c4,$c2),k($c3,$c2)),n(k($c4,$c1),k($c3,$c1))). 24 [copy,23,flip.1] n(n(k($c4,$c2),k($c3,$c2)),n(k($c4,$c1),k($c3,$c1)))!=k(n($c4,$c3),n($c2,$c1)). 50 [] i(n(x,y),y). 63 [] i($c4,I). 64 [] i($c3,I). 96,95 [hyper,64,20,63] n(k($c4,x),k($c3,x))=k(n($c4,$c3),x). 114 [back_demod,24,demod,96,96] n(k(n($c4,$c3),$c2),k(n($c4,$c3),$c1))!=k(n($c4,$c3),n($c2,$c1)). 197 [hyper,50,6,64] i(n(x,$c3),I). 495 [hyper,197,22] n(k(n(x,$c3),y),k(n(x,$c3),z))=k(n(x,$c3),n(y,z)). 497 [binary,495.1,114.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 33 clauses generated 622 clauses kept 385 clauses forward subsumed 236 clauses back subsumed 17 Kbytes malloced 383 ----------- times (seconds) ----------- user CPU time 0.28 (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) 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 Mon Dec 29 22:32:45 2003