----- Otter 3.2, August 2001 ----- The process was started by ??? on ???, Fri Nov 21 22:50:17 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,7). assign(max_distinct_vars,2). 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("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("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("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("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)))). 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($c1))!=c(r($c1)). end_of_list. ------- end included file i.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=8): 3 [] -i(x,y)|i(r(x),r(y)). ** KEPT (pick-wt=7): 4 [] r(c($c1))!=c(r($c1)). ------------> process sos: ** KEPT (pick-wt=7): 5 [] u(x,y)=u(y,x). ** KEPT (pick-wt=11): 7 [copy,6,flip.1] u(u(x,y),z)=u(x,u(y,z)). ---> New Demodulator: 8 [new_demod,7] u(u(x,y),z)=u(x,u(y,z)). ** KEPT (pick-wt=14): 9 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ---> New Demodulator: 10 [new_demod,9] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. ** KEPT (pick-wt=11): 12 [copy,11,flip.1] k(k(x,y),z)=k(x,k(y,z)). ---> New Demodulator: 13 [new_demod,12] k(k(x,y),z)=k(x,k(y,z)). ** KEPT (pick-wt=5): 14 [] r(r(x))=x. ---> New Demodulator: 15 [new_demod,14] r(r(x))=x. ** KEPT (pick-wt=10): 16 [] r(u(x,y))=u(r(x),r(y)). ---> New Demodulator: 17 [new_demod,16] r(u(x,y))=u(r(x),r(y)). ** KEPT (pick-wt=10): 18 [] r(k(x,y))=k(r(y),r(x)). ---> New Demodulator: 19 [new_demod,18] r(k(x,y))=k(r(y),r(x)). ** 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=10): 25 [copy,24,flip.1] c(u(c(x),c(y)))=n(x,y). ---> New Demodulator: 26 [new_demod,25] c(u(c(x),c(y)))=n(x,y). ** KEPT (pick-wt=4): 28 [copy,27,flip.1] c(I)=D. ---> New Demodulator: 29 [new_demod,28] c(I)=D. ** KEPT (pick-wt=5): 31 [copy,30,flip.1] u(I,D)=U. ---> New Demodulator: 32 [new_demod,31] u(I,D)=U. ** KEPT (pick-wt=4): 34 [copy,33,flip.1] c(U)=Z. ---> New Demodulator: 35 [new_demod,34] c(U)=Z. ** KEPT (pick-wt=10): 37 [copy,36,flip.1] c(k(c(x),c(y)))=d(x,y). ---> New Demodulator: 38 [new_demod,37] c(k(c(x),c(y)))=d(x,y). ** KEPT (pick-wt=11): 40 [copy,39,demod,17] u(r(x),r(y))=u(r(x),r(y)). Following clause subsumed by 5 during input processing: 0 [copy,5,flip.1] u(x,y)=u(y,x). >>>> Starting back demodulation with 8. >>>> Starting back demodulation with 10. >>>> Starting back demodulation with 13. >>>> Starting back demodulation with 15. >>>> Starting back demodulation with 17. >>>> Starting back demodulation with 19. >>>> Starting back demodulation with 21. >>>> Starting back demodulation with 23. >>>> Starting back demodulation with 26. >> back demodulating 9 with 26. >>>> Starting back demodulation with 29. >>>> Starting back demodulation with 32. >>>> Starting back demodulation with 35. >>>> Starting back demodulation with 38. Following clause subsumed by 40 during input processing: 0 [copy,40,flip.1] u(r(x),r(y))=u(r(x),r(y)). >>>> Starting back demodulation with 42. ======= end of input processing ======= =========== start of search =========== Resetting weight limit to 10. sos_size=1949 Resetting weight limit to 9. sos_size=1960 ----> UNIT CONFLICT at 5.82 sec ----> 8311 [binary,8309.1,4.1] $F. Length of proof is 125. Level of proof is 32. ---------------- PROOF ---------------- 1 [] -i(x,y)|u(x,y)=y. 2 [] i(x,y)|u(x,y)!=y. 3 [] -i(x,y)|i(r(x),r(y)). 4 [] r(c($c1))!=c(r($c1)). 5 [] u(x,y)=u(y,x). 6 [] u(x,u(y,z))=u(u(x,y),z). 8,7 [copy,6,flip.1] u(u(x,y),z)=u(x,u(y,z)). 9 [] u(c(u(c(x),y)),c(u(c(x),c(y))))=x. 11 [] k(x,k(y,z))=k(k(x,y),z). 12 [copy,11,flip.1] k(k(x,y),z)=k(x,k(y,z)). 15,14 [] r(r(x))=x. 17,16 [] r(u(x,y))=u(r(x),r(y)). 19,18 [] r(k(x,y))=k(r(y),r(x)). 21,20 [] k(x,I)=x. 22 [] u(k(r(x),c(k(x,y))),c(y))=c(y). 24 [] n(x,y)=c(u(c(x),c(y))). 26,25 [copy,24,flip.1] c(u(c(x),c(y)))=n(x,y). 27 [] D=c(I). 29,28 [copy,27,flip.1] c(I)=D. 30 [] U=u(I,D). 32,31 [copy,30,flip.1] u(I,D)=U. 33 [] Z=c(U). 35,34 [copy,33,flip.1] c(U)=Z. 41 [back_demod,9,demod,26] u(c(u(c(x),y)),n(x,y))=x. 49,48 [para_into,31.1.1,5.1.1] u(D,I)=U. 51,50 [para_from,31.1.1,7.1.1.1] u(U,x)=u(I,u(D,x)). 52 [para_into,12.1.1.1,20.1.1,flip.1] k(x,k(I,y))=k(x,y). 55,54 [para_from,48.1.1,7.1.1.1,demod,51] u(I,u(D,x))=u(D,u(I,x)). 57,56 [back_demod,50,demod,55] u(U,x)=u(D,u(I,x)). 58 [para_into,56.1.1,5.1.1] u(x,U)=u(D,u(I,x)). 59 [copy,58,flip.1] u(D,u(I,x))=u(x,U). 60 [para_into,58.1.1,7.1.1] u(x,u(y,U))=u(D,u(I,u(x,y))). 61 [copy,60,flip.1] u(D,u(I,u(x,y)))=u(x,u(y,U)). 62 [para_from,58.1.1,7.1.1.1,demod,8,8,57] u(D,u(I,u(x,y)))=u(x,u(D,u(I,y))). 63 [copy,62,flip.1] u(x,u(D,u(I,y)))=u(D,u(I,u(x,y))). 68,67 [para_into,16.1.1.1,48.1.1] r(U)=u(r(D),r(I)). 73 [para_into,59.1.1.2,5.1.1] u(D,u(x,I))=u(x,U). 74 [para_into,59.1.1,5.1.1,demod,8] u(I,u(x,D))=u(x,U). 75 [copy,73,flip.1] u(x,U)=u(D,u(x,I)). 85 [para_from,74.1.1,59.1.1.2,demod,8] u(D,u(x,U))=u(x,u(D,U)). 86 [para_from,74.1.1,16.1.1.1,demod,17,68,17] u(r(x),u(r(D),r(I)))=u(r(I),u(r(x),r(D))). 88 [copy,85,flip.1] u(x,u(D,U))=u(D,u(x,U)). 89 [copy,86,flip.1] u(r(I),u(r(x),r(D)))=u(r(x),u(r(D),r(I))). 93 [para_into,18.1.1.1,20.1.1,flip.1] k(r(I),r(x))=r(x). 102,101 [para_into,93.1.1.2,14.1.1,demod,15] k(r(I),x)=x. 104,103 [para_into,101.1.1,52.1.1,demod,102,flip.1] k(I,x)=x. 106,105 [para_into,101.1.1,20.1.1] r(I)=I. 108,107 [back_demod,89,demod,106,106,flip.1] u(r(x),u(r(D),I))=u(I,u(r(x),r(D))). 114,113 [back_demod,67,demod,106] r(U)=u(r(D),I). 116 [para_into,22.1.1.1.1,105.1.1,demod,104,104] u(c(x),c(x))=c(x). 120 [para_into,22.1.1.1.2.1,20.1.1,demod,29,29] u(k(r(x),c(x)),D)=D. 133 [para_into,116.1.1.1,28.1.1,demod,29,29] u(D,D)=D. 137 [para_from,116.1.1,7.1.1.1,flip.1] u(c(x),u(c(x),y))=u(c(x),y). 143 [para_into,25.1.1.1.1,34.1.1] c(u(Z,c(x)))=n(U,x). 151 [para_into,25.1.1.1,116.1.1] c(c(x))=n(x,x). 153 [para_into,25.1.1.1,5.1.1,demod,26] n(x,y)=n(y,x). 163,162 [para_from,133.1.1,74.1.1.2,demod,32,flip.1] u(D,U)=U. 169,168 [back_demod,88,demod,163,flip.1] u(D,u(x,U))=u(x,U). 187 [para_into,41.1.1.1.1.1,34.1.1] u(c(u(Z,x)),n(U,x))=U. 189 [para_into,41.1.1.1.1.1,28.1.1] u(c(u(D,x)),n(I,x))=I. 194,193 [para_into,41.1.1.1.1,58.1.1] u(c(u(D,u(I,c(x)))),n(x,U))=x. 195 [para_into,41.1.1.1.1,41.1.1] u(c(x),n(u(c(x),y),n(x,y)))=u(c(x),y). 211 [para_into,151.1.1.1,28.1.1] c(D)=n(I,I). 282 [hyper,120,2] i(k(r(x),c(x)),D). 301 [para_into,120.1.1,5.1.1] u(D,k(r(x),c(x)))=D. 309 [hyper,282,3,demod,19,15] i(k(r(c(x)),x),r(D)). 322 [para_into,309.1.1.1.1,211.1.1] i(k(r(n(I,I)),D),r(D)). 374 [para_from,63.1.1,41.1.1.1.1] u(c(u(D,u(I,u(c(x),y)))),n(x,u(D,u(I,y))))=x. 408 [para_from,143.1.1,41.1.1.1.1.1] u(c(u(n(U,x),y)),n(u(Z,c(x)),y))=u(Z,c(x)). 834,833 [para_into,107.1.1.1,105.1.1,demod,106] u(I,u(r(D),I))=u(I,u(I,r(D))). 995 [para_into,301.1.1.2.2,211.1.1] u(D,k(r(D),n(I,I)))=D. 1222 [hyper,322,1] u(k(r(n(I,I)),D),r(D))=r(D). 1511 [hyper,137,2] i(c(x),u(c(x),y)). 1577 [para_into,1511.1.2,41.1.1] i(c(u(c(x),y)),x). 1596 [para_into,1577.1.1.1.1,143.1.1] i(c(u(n(U,x),y)),u(Z,c(x))). 1597 [para_into,1577.1.1.1.1,34.1.1] i(c(u(Z,x)),U). 1598 [para_into,1577.1.1.1.1,28.1.1] i(c(u(D,x)),I). 1607 [para_into,1577.1.1,25.1.1] i(n(x,y),x). 1610,1609 [hyper,1607,1] u(n(x,y),x)=x. 1611 [para_into,1607.1.1,153.1.1] i(n(x,y),y). 1612 [hyper,1611,3] i(r(n(x,y)),r(y)). 1614,1613 [hyper,1611,1] u(n(x,y),y)=y. 1621,1620 [hyper,1597,1] u(c(u(Z,x)),U)=U. 1623 [para_into,1597.1.1.1,75.1.1] i(c(u(D,u(Z,I))),U). 1631 [para_into,1598.1.1.1,162.1.1,demod,35] i(Z,I). 1637,1636 [hyper,1631,1] u(Z,I)=I. 1638 [back_demod,1623,demod,1637,49,35] i(Z,U). 1648 [hyper,1638,1] u(Z,U)=U. 1692 [para_into,1636.1.1,5.1.1] u(I,Z)=I. 1703 [para_from,1648.1.1,16.1.1.1,demod,114,114,108,flip.1] u(I,u(r(Z),r(D)))=u(r(D),I). 1715 [para_from,1692.1.1,16.1.1.1,demod,106,106,flip.1] u(I,r(Z))=I. 1750 [para_into,187.1.1,5.1.1] u(n(U,x),c(u(Z,x)))=U. 1756 [para_from,187.1.1,61.1.1.2.2,demod,169,1610,1621] u(I,U)=U. 1760 [para_from,187.1.1,7.1.1.1,demod,57,flip.1] u(c(u(Z,x)),u(n(U,x),y))=u(D,u(I,y)). 1788,1787 [para_into,1756.1.1,75.1.1] u(D,u(I,I))=U. 1793,1792 [para_from,1756.1.1,16.1.1.1,demod,114,106,114,834] u(r(D),I)=u(I,u(I,r(D))). 1800 [back_demod,1703,demod,1793] u(I,u(r(Z),r(D)))=u(I,u(I,r(D))). 1880 [back_demod,113,demod,1793] r(U)=u(I,u(I,r(D))). 1930 [para_from,189.1.1,62.1.1.2.2,demod,1788,flip.1] u(c(u(D,x)),u(D,u(I,n(I,x))))=U. 1947,1946 [para_from,1715.1.1,7.1.1.1,flip.1] u(I,u(r(Z),x))=u(I,x). 1949,1948 [back_demod,1800,demod,1947,flip.1] u(I,u(I,r(D)))=u(I,r(D)). 1953 [back_demod,1880,demod,1949] r(U)=u(I,r(D)). 2063 [para_into,193.1.1.2,153.1.1] u(c(u(D,u(I,c(x)))),n(U,x))=x. 2067 [para_from,193.1.1,137.1.1.2,demod,194] u(c(u(D,u(I,c(x)))),x)=x. 2073 [para_from,193.1.1,61.1.1.2.2,demod,1614,flip.1] u(c(u(D,u(I,c(x)))),U)=u(D,u(I,x)). 2122 [para_into,195.1.1.2.1,193.1.1,demod,194] u(c(u(D,u(I,c(x)))),n(x,n(u(D,u(I,c(x))),n(x,U))))=x. 2265,2264 [para_into,1609.1.1,5.1.1] u(x,n(x,y))=x. 2275,2274 [back_demod,1930,demod,2265,49] u(c(u(D,x)),U)=U. 2277,2276 [back_demod,2073,demod,2275,flip.1] u(D,u(I,x))=U. 2282 [back_demod,2122,demod,2277,35,2277] u(Z,n(x,n(U,n(x,U))))=x. 2289,2288 [back_demod,2067,demod,2277,35] u(Z,x)=x. 2293,2292 [back_demod,2063,demod,2277,35,2289] n(U,x)=x. 2326 [back_demod,1760,demod,2289,2293,2277] u(c(x),u(x,y))=U. 2340,2339 [back_demod,374,demod,2277,35,2277,2289] n(x,U)=x. 2356,2355 [back_demod,2282,demod,2340,2293,2289] n(x,x)=x. 2370 [back_demod,1750,demod,2293,2289] u(x,c(x))=U. 2374 [back_demod,1596,demod,2293,2289] i(c(u(x,y)),c(x)). 2407 [back_demod,408,demod,2293,2289,2289] u(c(u(x,y)),n(c(x),y))=c(x). 2559,2558 [back_demod,1222,demod,2356,106,104] u(D,r(D))=r(D). 2579,2578 [back_demod,995,demod,2356,21,2559] r(D)=D. 2596,2595 [back_demod,151,demod,2356] c(c(x))=x. 2685,2684 [back_demod,1953,demod,2579,32] r(U)=U. 2909 [para_from,2370.1.1,16.1.1.1,demod,2685,flip.1] u(r(x),r(c(x)))=U. 3003 [para_into,1612.1.2,14.1.1] i(r(n(x,r(y))),y). 3202 [hyper,3003,1] u(r(n(x,r(y))),y)=y. 3460 [para_into,2326.1.1.2,5.1.1] u(c(x),u(y,x))=U. 3546,3545 [para_into,2909.1.1.1,14.1.1] u(x,r(c(r(x))))=U. 3844 [para_from,3545.1.1,195.1.1.2.1,demod,2293,3546] u(c(x),n(x,r(c(r(c(x))))))=U. 5026 [para_from,3202.1.1,2374.1.1.1] i(c(x),c(r(n(y,r(x))))). 7049 [para_into,5026.1.1,2595.1.1] i(x,c(r(n(y,r(c(x)))))). 7298 [para_into,2407.1.1.1.1,3460.1.1,demod,35,2596,2289,2596] n(x,u(y,x))=x. 7320 [para_into,7298.1.1,153.1.1] n(u(x,y),y)=y. 7335,7334 [para_into,7320.1.1.1,2264.1.1] n(x,n(x,y))=n(x,y). 8267 [para_from,3844.1.1,2407.1.1.1.1,demod,35,2596,7335,2289,2596] n(x,r(c(r(c(x)))))=x. 8269 [para_into,8267.1.1,153.1.1] n(r(c(r(c(x)))),x)=x. 8271 [para_from,8267.1.1,7049.1.2.1.1] i(r(c(x)),c(r(x))). 8295 [para_into,8271.1.2.1,14.1.1] i(r(c(r(x))),c(x)). 8298 [para_into,8295.1.2,2595.1.1] i(r(c(r(c(x)))),x). 8301,8300 [hyper,8298,1] u(r(c(r(c(x)))),x)=x. 8305 [para_from,8269.1.1,2264.1.1.2,demod,8301,flip.1] r(c(r(c(x))))=x. 8307 [para_into,8305.1.1.1.1.1,2595.1.1] r(c(r(x)))=c(x). 8309 [para_into,8307.1.1.1.1,14.1.1] r(c(x))=c(r(x)). 8311 [binary,8309.1,4.1] $F. ------------ end of proof ------------- Search stopped by max_proofs option. ============ end of search ============ -------------- statistics ------------- clauses given 2364 clauses generated 306691 clauses kept 5447 clauses forward subsumed 209665 clauses back subsumed 284 Kbytes malloced 4885 ----------- times (seconds) ----------- user CPU time 6.02 (0 hr, 0 min, 6 sec) system CPU time 0.00 (0 hr, 0 min, 0 sec) wall-clock time 6 (0 hr, 0 min, 6 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 22:50:23 2003