----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Mon Dec 29 22:18:36 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,1). assign(max_literals,1). assign(max_mem,20000). assign(max_weight,13). 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("subId_ghost.txt"). ------- start included file subId_ghost.txt------- formula_list(usable). -(all x z (i(x,I)&i(z,I)->k(x,z)=n(x,z))). end_of_list. -------> usable clausifies to: list(usable). 0 [] i($c2,I). 0 [] i($c1,I). 0 [] k($c2,$c1)!=n($c2,$c1). end_of_list. ------- end included file subId_ghost.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=7): 17 [copy,16,flip.1] n($c2,$c1)!=k($c2,$c1). ------------> process sos: ** KEPT (pick-wt=7): 18 [] u(x,y)=u(y,x). ** KEPT (pick-wt=11): 20 [copy,19,flip.1] u(u(x,y),z)=u(x,u(y,z)). ---> New Demodulator: 21 [new_demod,20] u(u(x,y),z)=u(x,u(y,z)). ** KEPT (pick-wt=14): 22 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ---> New Demodulator: 23 [new_demod,22] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ** KEPT (pick-wt=11): 25 [copy,24,flip.1] k(k(x,y),z)=k(x,k(y,z)). ---> New Demodulator: 26 [new_demod,25] k(k(x,y),z)=k(x,k(y,z)). ** KEPT (pick-wt=5): 27 [] r(r(x))=x. ---> New Demodulator: 28 [new_demod,27] r(r(x))=x. ** KEPT (pick-wt=10): 29 [] r(u(x,y))=u(r(x),r(y)). ---> New Demodulator: 30 [new_demod,29] r(u(x,y))=u(r(x),r(y)). ** KEPT (pick-wt=10): 31 [] r(k(x,y))=k(r(y),r(x)). ---> New Demodulator: 32 [new_demod,31] r(k(x,y))=k(r(y),r(x)). ** KEPT (pick-wt=5): 33 [] k(x,I)=x. ---> New Demodulator: 34 [new_demod,33] k(x,I)=x. ** KEPT (pick-wt=13): 35 [] u(k(r(x),c(k(x,y))),c(y))=c(y). ---> New Demodulator: 36 [new_demod,35] u(k(r(x),c(k(x,y))),c(y))=c(y). ** KEPT (pick-wt=10): 38 [copy,37,flip.1] c(u(c(x),c(y)))=n(x,y). ---> New Demodulator: 39 [new_demod,38] c(u(c(x),c(y)))=n(x,y). ** KEPT (pick-wt=10): 41 [copy,40,flip.1] u(n(x,c(y)),n(x,y))=x. ---> New Demodulator: 42 [new_demod,41] u(n(x,c(y)),n(x,y))=x. ** KEPT (pick-wt=5): 43 [] i(n(x,y),y). Following clause subsumed by 43 during input processing: 0 [] i(n(x,y),y). ** KEPT (pick-wt=11): 44 [] i(n(x,u(y,z)),u(y,n(x,z))). ** KEPT (pick-wt=10): 46 [copy,45,flip.1] u(n(x,y),n(x,c(y)))=x. ---> New Demodulator: 47 [new_demod,46] u(n(x,y),n(x,c(y)))=x. ** KEPT (pick-wt=5): 48 [] u(x,Z)=x. ---> New Demodulator: 49 [new_demod,48] u(x,Z)=x. ** KEPT (pick-wt=5): 50 [] i(n(x,y),x). Following clause subsumed by 43 during input processing: 0 [] i(n(x,y),y). ** KEPT (pick-wt=15): 51 [] i(k(n(x,y),n(z,u)),n(k(x,z),k(y,u))). ** KEPT (pick-wt=24): 52 [] 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): 53 [] i(n(k(x,y),z),k(n(x,k(z,r(y))),n(y,k(r(x),z)))). ** KEPT (pick-wt=14): 54 [] i(n(k(x,y),z),k(x,n(y,k(r(x),z)))). ** KEPT (pick-wt=3): 55 [] i($c2,I). ** KEPT (pick-wt=3): 56 [] i($c1,I). Following clause subsumed by 18 during input processing: 0 [copy,18,flip.1] u(x,y)=u(y,x). >>>> Starting back demodulation with 21. >>>> Starting back demodulation with 23. >>>> Starting back demodulation with 26. >>>> Starting back demodulation with 28. >>>> Starting back demodulation with 30. >>>> Starting back demodulation with 32. >>>> Starting back demodulation with 34. >>>> Starting back demodulation with 36. >>>> Starting back demodulation with 39. >> back demodulating 22 with 39. >>>> Starting back demodulation with 42. >>>> Starting back demodulation with 47. >>>> Starting back demodulation with 49. ======= end of input processing ======= =========== start of search =========== ----> UNIT CONFLICT at 0.41 sec ----> 4055 [binary,4053.1,2387.1] $F. Length of proof is 46. Level of proof is 8. ---------------- PROOF ---------------- 2 [] i(x,y)|u(x,y)!=y. 4 [] -i(x,y)|i(u(z,x),u(z,y)). 6 [] -i(x,y)| -i(y,z)|i(x,z). 7 [] -i(x,y)| -i(y,x)|x=y. 8 [] -i(x,y)| -i(x,z)|i(x,n(y,z)). 9 [] -i(x,I)|k(x,x)=x. 10 [] -i(x,I)|r(x)=x. 11 [] -i(x,y)|i(k(x,z),k(y,z)). 12 [] -i(x,y)|i(k(z,x),k(z,y)). 15 [] -i(x,y)|i(r(x),r(y)). 16 [] k($c2,$c1)!=n($c2,$c1). 17 [copy,16,flip.1] n($c2,$c1)!=k($c2,$c1). 24 [] k(x,k(y,z))=k(k(x,y),z). 25 [copy,24,flip.1] k(k(x,y),z)=k(x,k(y,z)). 32,31 [] r(k(x,y))=k(r(y),r(x)). 34,33 [] k(x,I)=x. 43 [] i(n(x,y),y). 49,48 [] u(x,Z)=x. 50 [] i(n(x,y),x). 54 [] i(n(k(x,y),z),k(x,n(y,k(r(x),z)))). 55 [] i($c2,I). 56 [] i($c1,I). 57 [hyper,55,15] i(r($c2),r(I)). 60 [hyper,55,12,demod,34] i(k(x,$c2),x). 61 [hyper,55,11] i(k($c2,x),k(I,x)). 63,62 [hyper,55,10] r($c2)=$c2. 71 [back_demod,57,demod,63] i($c2,r(I)). 72 [hyper,56,15] i(r($c1),r(I)). 75 [hyper,56,12,demod,34] i(k(x,$c1),x). 78,77 [hyper,56,10] r($c1)=$c1. 79 [hyper,56,9] k($c1,$c1)=$c1. 86 [back_demod,72,demod,78] i($c1,r(I)). 119 [hyper,43,6,56] i(n(x,$c1),I). 120 [hyper,43,6,55] i(n(x,$c2),I). 121 [hyper,48,2] i(Z,Z). 133 [hyper,121,4,demod,49,49] i(x,x). 135,134 [hyper,133,10] r(I)=I. 137 [hyper,133,8,86,demod,135] i($c1,n(I,$c1)). 138 [hyper,133,8,71,demod,135] i($c2,n(I,$c2)). 139 [hyper,133,8,86,demod,135] i($c1,n($c1,I)). 140 [hyper,133,8,71,demod,135] i($c2,n($c2,I)). 149 [hyper,50,6,56] i(n($c1,x),I). 150 [hyper,50,6,55] i(n($c2,x),I). 156 [hyper,60,8,133] i(k(x,$c2),n(x,k(x,$c2))). 157 [hyper,60,6,56] i(k($c1,$c2),I). 193,192 [hyper,75,10,demod,32,78,135,34,flip.1] k(I,$c1)=$c1. 206,205 [para_from,79.1.1,25.1.1.1,flip.1] k($c1,k($c1,x))=k($c1,x). 235 [hyper,119,8,50] i(n(x,$c1),n(I,x)). 249 [hyper,120,8,50] i(n(x,$c2),n(I,x)). 304,303 [hyper,137,7,43] n(I,$c1)=$c1. 316,315 [hyper,138,7,43] n(I,$c2)=$c2. 326,325 [hyper,139,7,50] n($c1,I)=$c1. 336,335 [hyper,140,7,50] n($c2,I)=$c2. 345 [hyper,149,8,43] i(n($c1,x),n(x,I)). 358 [hyper,150,8,43] i(n($c2,x),n(x,I)). 373,372 [hyper,157,10,demod,32,63,78] k($c2,$c1)=k($c1,$c2). 384 [back_demod,17,demod,373] n($c2,$c1)!=k($c1,$c2). 440 [para_into,54.1.1.1,79.1.1,demod,78] i(n($c1,x),k($c1,n($c1,k($c1,x)))). 559 [hyper,61,8,75,demod,373,193] i(k($c1,$c2),n($c1,$c2)). 850 [hyper,345,8,249,demod,336,304] i(n($c1,$c2),n($c2,$c1)). 927,926 [hyper,156,7,43] n(x,k(x,$c2))=k(x,$c2). 932 [hyper,358,8,235,demod,326,316] i(n($c2,$c1),n($c1,$c2)). 2386,2385 [hyper,932,7,850,flip.1] n($c2,$c1)=n($c1,$c2). 2387 [back_demod,384,demod,2386] n($c1,$c2)!=k($c1,$c2). 3644 [hyper,440,8,249,demod,304,927,206,927] i(n($c1,$c2),k($c1,$c2)). 4053 [hyper,3644,7,559,flip.1] n($c1,$c2)=k($c1,$c2). 4055 [binary,4053.1,2387.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 373 clauses generated 17965 clauses kept 3644 clauses forward subsumed 9584 clauses back subsumed 133 Kbytes malloced 1660 ----------- times (seconds) ----------- user CPU time 0.61 (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 Mon Dec 29 22:18:37 2003