----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Fri Nov 21 18:30:18 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_seconds,1). 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("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("xxv.txt"). ------- start included file xxv.txt------- formula_list(usable). -(all x y (i(I,x)&i(r(x),x)&i(k(x,x),x)->k(x,c(k(x,y)))=c(k(x,y)))). end_of_list. -------> usable clausifies to: list(usable). 0 [] i(I,$c2). 0 [] i(r($c2),$c2). 0 [] i(k($c2,$c2),$c2). 0 [] k($c2,c(k($c2,$c1)))!=c(k($c2,$c1)). end_of_list. ------- end included file xxv.txt------- SCAN INPUT: prop=0, horn=1, equality=1, symmetry=0, max_lits=2. This is a Horn set with equality. The strategy will be Knuth-Bendix and hyper_res, with positive clauses in sos and nonpositive clauses in usable. dependent: set(knuth_bendix). dependent: set(para_from). dependent: set(para_into). dependent: clear(para_from_right). dependent: clear(para_into_right). dependent: set(para_from_vars). dependent: set(eq_units_both_ways). dependent: set(dynamic_demod_all). dependent: set(dynamic_demod). dependent: set(order_eq). dependent: set(back_demod). dependent: set(lrpo). dependent: set(hyper_res). dependent: clear(order_hyper). ------------> process usable: ** KEPT (pick-wt=8): 1 [] -i(x,y)|u(x,y)=y. ** KEPT (pick-wt=8): 2 [] i(x,y)|u(x,y)!=y. ** KEPT (pick-wt=11): 3 [] k($c2,c(k($c2,$c1)))!=c(k($c2,$c1)). ------------> 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=11): 9 [copy,8,flip.1] k(k(x,y),z)=k(x,k(y,z)). ---> New Demodulator: 10 [new_demod,9] k(k(x,y),z)=k(x,k(y,z)). ** KEPT (pick-wt=5): 11 [] r(r(x))=x. ---> New Demodulator: 12 [new_demod,11] r(r(x))=x. ** KEPT (pick-wt=10): 13 [] r(u(x,y))=u(r(x),r(y)). ---> New Demodulator: 14 [new_demod,13] r(u(x,y))=u(r(x),r(y)). ** KEPT (pick-wt=10): 15 [] r(k(x,y))=k(r(y),r(x)). ---> New Demodulator: 16 [new_demod,15] r(k(x,y))=k(r(y),r(x)). ** KEPT (pick-wt=13): 18 [copy,17,flip.1] u(k(x,y),k(z,y))=k(u(x,z),y). ---> New Demodulator: 19 [new_demod,18] u(k(x,y),k(z,y))=k(u(x,z),y). ** KEPT (pick-wt=5): 20 [] k(x,I)=x. ---> New Demodulator: 21 [new_demod,20] k(x,I)=x. ** KEPT (pick-wt=13): 22 [] u(k(r(x),c(k(x,y))),c(y))=c(y). ---> New Demodulator: 23 [new_demod,22] u(k(r(x),c(k(x,y))),c(y))=c(y). ** KEPT (pick-wt=3): 24 [] i(I,$c2). ** KEPT (pick-wt=4): 25 [] i(r($c2),$c2). ** KEPT (pick-wt=5): 26 [] i(k($c2,$c2),$c2). 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 10. >>>> Starting back demodulation with 12. >>>> Starting back demodulation with 14. >>>> Starting back demodulation with 16. >>>> Starting back demodulation with 19. >>>> Starting back demodulation with 21. >>>> Starting back demodulation with 23. ======= end of input processing ======= =========== start of search =========== ----> UNIT CONFLICT at 0.07 sec ----> 726 [binary,724.1,3.1] $F. Length of proof is 16. Level of proof is 8. ---------------- PROOF ---------------- 1 [] -i(x,y)|u(x,y)=y. 3 [] k($c2,c(k($c2,$c1)))!=c(k($c2,$c1)). 4 [] u(x,y)=u(y,x). 8 [] k(x,k(y,z))=k(k(x,y),z). 9 [copy,8,flip.1] k(k(x,y),z)=k(x,k(y,z)). 12,11 [] r(r(x))=x. 13 [] r(u(x,y))=u(r(x),r(y)). 15 [] r(k(x,y))=k(r(y),r(x)). 17 [] k(u(x,y),z)=u(k(x,z),k(y,z)). 18 [copy,17,flip.1] u(k(x,y),k(z,y))=k(u(x,z),y). 20 [] k(x,I)=x. 22 [] u(k(r(x),c(k(x,y))),c(y))=c(y). 24 [] i(I,$c2). 25 [] i(r($c2),$c2). 26 [] i(k($c2,$c2),$c2). 30 [hyper,24,1] u(I,$c2)=$c2. 33,32 [hyper,25,1] u(r($c2),$c2)=$c2. 42 [hyper,26,1] u(k($c2,$c2),$c2)=$c2. 45,44 [para_into,30.1.1,4.1.1] u($c2,I)=$c2. 54 [para_into,32.1.1,4.1.1] u($c2,r($c2))=$c2. 66,65 [para_into,13.1.1.1,54.1.1,demod,12,33] r($c2)=$c2. 82 [para_into,15.1.1.1,20.1.1,flip.1] k(r(I),r(x))=r(x). 153 [para_into,82.1.1.2,11.1.1,demod,12] k(r(I),x)=x. 159,158 [para_into,153.1.1,20.1.1] r(I)=I. 160 [back_demod,153,demod,159] k(I,x)=x. 189,188 [para_from,160.1.1,18.1.1.2] u(k(x,y),y)=k(u(x,I),y). 191 [back_demod,42,demod,189,45] k($c2,$c2)=$c2. 200 [para_from,191.1.1,9.1.1.1,flip.1] k($c2,k($c2,x))=k($c2,x). 724 [para_from,200.1.1,22.1.1.1.2.1,demod,66,189,45] k($c2,c(k($c2,x)))=c(k($c2,x)). 726 [binary,724.1,3.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 82 clauses generated 1698 clauses kept 626 clauses forward subsumed 1146 clauses back subsumed 20 Kbytes malloced 574 ----------- times (seconds) ----------- user CPU time 0.27 (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 Fri Nov 21 18:30:19 2003