----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Sat Nov 22 00:40:48 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_literals,1). 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("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("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("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("xiii.txt"). ------- start included file xiii.txt------- formula_list(usable). all x (k(x,Z)=Z). all x (k(Z,x)=Z). -(all x (d(x,U)=U)). end_of_list. -------> usable clausifies to: list(usable). 0 [] k(x,Z)=Z. 0 [] k(Z,x)=Z. 0 [] d($c1,U)!=U. end_of_list. ------- end included file xiii.txt------- SCAN INPUT: prop=0, horn=1, equality=1, symmetry=0, max_lits=2. This is a Horn set with equality. The strategy will be Knuth-Bendix and hyper_res, with positive clauses in sos and nonpositive clauses in usable. dependent: set(knuth_bendix). dependent: set(para_from). dependent: set(para_into). dependent: clear(para_from_right). dependent: clear(para_into_right). dependent: set(para_from_vars). dependent: set(eq_units_both_ways). dependent: set(dynamic_demod_all). dependent: set(dynamic_demod). dependent: set(order_eq). dependent: set(back_demod). dependent: set(lrpo). dependent: set(hyper_res). dependent: clear(order_hyper). ------------> process usable: ** KEPT (pick-wt=8): 1 [] -i(x,y)|u(x,y)=y. ** KEPT (pick-wt=8): 2 [] i(x,y)|u(x,y)!=y. ** KEPT (pick-wt=5): 3 [] d($c1,U)!=U. ------------> 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=14): 8 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ---> New Demodulator: 9 [new_demod,8] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ** KEPT (pick-wt=11): 11 [copy,10,flip.1] k(k(x,y),z)=k(x,k(y,z)). ---> New Demodulator: 12 [new_demod,11] k(k(x,y),z)=k(x,k(y,z)). ** KEPT (pick-wt=5): 13 [] r(r(x))=x. ---> New Demodulator: 14 [new_demod,13] r(r(x))=x. ** KEPT (pick-wt=10): 15 [] r(u(x,y))=u(r(x),r(y)). ---> New Demodulator: 16 [new_demod,15] r(u(x,y))=u(r(x),r(y)). ** KEPT (pick-wt=10): 17 [] r(k(x,y))=k(r(y),r(x)). ---> New Demodulator: 18 [new_demod,17] r(k(x,y))=k(r(y),r(x)). ** KEPT (pick-wt=13): 20 [copy,19,flip.1] u(k(x,y),k(z,y))=k(u(x,z),y). ---> New Demodulator: 21 [new_demod,20] u(k(x,y),k(z,y))=k(u(x,z),y). ** KEPT (pick-wt=5): 22 [] k(x,I)=x. ---> New Demodulator: 23 [new_demod,22] k(x,I)=x. ** KEPT (pick-wt=13): 24 [] u(k(r(x),c(k(x,y))),c(y))=c(y). ---> New Demodulator: 25 [new_demod,24] u(k(r(x),c(k(x,y))),c(y))=c(y). ** KEPT (pick-wt=4): 27 [copy,26,flip.1] c(I)=D. ---> New Demodulator: 28 [new_demod,27] c(I)=D. ** KEPT (pick-wt=5): 30 [copy,29,flip.1] u(I,D)=U. ---> New Demodulator: 31 [new_demod,30] u(I,D)=U. ** KEPT (pick-wt=4): 33 [copy,32,flip.1] c(U)=Z. ---> New Demodulator: 34 [new_demod,33] c(U)=Z. ** KEPT (pick-wt=10): 36 [copy,35,flip.1] c(k(c(x),c(y)))=d(x,y). ---> New Demodulator: 37 [new_demod,36] c(k(c(x),c(y)))=d(x,y). ** KEPT (pick-wt=5): 38 [] k(x,Z)=Z. ---> New Demodulator: 39 [new_demod,38] k(x,Z)=Z. ** KEPT (pick-wt=5): 40 [] k(Z,x)=Z. ---> New Demodulator: 41 [new_demod,40] k(Z,x)=Z. 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 9. >>>> Starting back demodulation with 12. >>>> Starting back demodulation with 14. >>>> Starting back demodulation with 16. >>>> Starting back demodulation with 18. >>>> Starting back demodulation with 21. >>>> Starting back demodulation with 23. >>>> Starting back demodulation with 25. >>>> Starting back demodulation with 28. >>>> Starting back demodulation with 31. >>>> Starting back demodulation with 34. >>>> Starting back demodulation with 37. >>>> Starting back demodulation with 39. >>>> Starting back demodulation with 41. ======= end of input processing ======= =========== start of search =========== ----> UNIT CONFLICT at 0.07 sec ----> 980 [binary,978.1,333.1] $F. Length of proof is 36. Level of proof is 8. ---------------- PROOF ---------------- 3 [] d($c1,U)!=U. 4 [] u(x,y)=u(y,x). 5 [] u(x,u(y,z))=u(u(x,y),z). 7,6 [copy,5,flip.1] u(u(x,y),z)=u(x,u(y,z)). 9,8 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. 10 [] k(x,k(y,z))=k(k(x,y),z). 11 [copy,10,flip.1] k(k(x,y),z)=k(x,k(y,z)). 14,13 [] r(r(x))=x. 17 [] r(k(x,y))=k(r(y),r(x)). 19 [] k(u(x,y),z)=u(k(x,z),k(y,z)). 20 [copy,19,flip.1] u(k(x,y),k(z,y))=k(u(x,z),y). 23,22 [] k(x,I)=x. 24 [] u(k(r(x),c(k(x,y))),c(y))=c(y). 26 [] D=c(I). 28,27 [copy,26,flip.1] c(I)=D. 29 [] U=u(I,D). 30 [copy,29,flip.1] u(I,D)=U. 32 [] Z=c(U). 34,33 [copy,32,flip.1] c(U)=Z. 35 [] d(x,y)=c(k(c(x),c(y))). 36 [copy,35,flip.1] c(k(c(x),c(y)))=d(x,y). 39,38 [] k(x,Z)=Z. 41,40 [] k(Z,x)=Z. 45 [para_into,6.1.1.1,4.1.1,demod,7] u(x,u(y,z))=u(y,u(x,z)). 50 [para_into,30.1.1,4.1.1] u(D,I)=U. 56 [para_into,8.1.1.1.1.1,27.1.1,demod,28] u(c(u(D,x)),c(u(D,c(x))))=I. 66 [para_into,8.1.1.2.1,8.1.1] u(c(u(c(u(c(x),y)),u(c(x),c(y)))),c(x))=u(c(x),y). 72 [para_from,8.1.1,6.1.1.1,flip.1] u(c(u(c(x),y)),u(c(u(c(x),c(y))),z))=u(x,z). 76 [para_into,11.1.1.1,22.1.1,flip.1] k(x,k(I,y))=k(x,y). 119 [para_into,17.1.1.1,40.1.1,flip.1] k(r(x),r(Z))=r(Z). 123 [para_into,17.1.1.1,22.1.1,flip.1] k(r(I),r(x))=r(x). 129 [para_into,119.1.1.1,13.1.1] k(x,r(Z))=r(Z). 131 [para_into,129.1.1,40.1.1,flip.1] r(Z)=Z. 140,139 [para_into,123.1.1.2,13.1.1,demod,14] k(r(I),x)=x. 143 [para_into,20.1.1.1,38.1.1,demod,39,39] u(Z,Z)=Z. 160,159 [para_into,139.1.1,76.1.1,demod,140,flip.1] k(I,x)=x. 161 [para_into,139.1.1,22.1.1] r(I)=I. 169,168 [para_into,24.1.1.1.1,161.1.1,demod,160,160] u(c(x),c(x))=c(x). 171,170 [para_into,24.1.1.1.1,131.1.1,demod,41,41] u(Z,c(x))=c(x). 204,203 [para_into,170.1.1,4.1.1] u(c(x),Z)=c(x). 220 [para_into,36.1.1.1.1,33.1.1,demod,41] c(Z)=d(U,x). 225 [para_into,36.1.1.1.2,33.1.1,demod,39] c(Z)=d(x,U). 229 [copy,225,flip.1] d(x,U)=c(Z). 259 [para_into,45.1.1.2,143.1.1,flip.1] u(Z,u(x,Z))=u(x,Z). 331 [para_from,229.1.1,3.1.1] c(Z)!=U. 333 [para_into,331.1.1,220.1.1] d(U,x)!=U. 379 [para_into,56.1.1.1.1,50.1.1,demod,34,28,171] c(u(D,D))=I. 430,429 [para_from,379.1.1,36.1.1.1.2,demod,23] c(c(x))=d(x,u(D,D)). 582 [para_into,66.1.1.1.1.1.1.1,33.1.1,demod,34,171,34,204,34] c(u(c(u(Z,x)),c(x)))=u(Z,x). 940,939 [para_into,72.1.1.2,203.1.1,demod,9,flip.1] u(x,Z)=x. 961,960 [back_demod,259,demod,940,940] u(Z,x)=x. 978 [back_demod,582,demod,961,169,430,961] d(x,u(D,D))=x. 980 [binary,978.1,333.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 91 clauses generated 1261 clauses kept 562 clauses forward subsumed 785 clauses back subsumed 25 Kbytes malloced 798 ----------- 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 Sat Nov 22 00:40:49 2003