MAYBE 915.68/297.14 MAYBE 915.68/297.14 915.68/297.14 We are left with following problem, upon which TcT provides the 915.68/297.14 certificate MAYBE. 915.68/297.14 915.68/297.14 Strict Trs: 915.68/297.14 { O(0()) -> 0() 915.68/297.14 , +(x, 0()) -> x 915.68/297.14 , +(x, +(y, z)) -> +(+(x, y), z) 915.68/297.14 , +(O(x), O(y)) -> O(+(x, y)) 915.68/297.14 , +(O(x), I(y)) -> I(+(x, y)) 915.68/297.14 , +(0(), x) -> x 915.68/297.14 , +(I(x), O(y)) -> I(+(x, y)) 915.68/297.14 , +(I(x), I(y)) -> O(+(+(x, y), I(0()))) 915.68/297.14 , -(x, 0()) -> x 915.68/297.14 , -(O(x), O(y)) -> O(-(x, y)) 915.68/297.14 , -(O(x), I(y)) -> I(-(-(x, y), I(1()))) 915.68/297.14 , -(0(), x) -> 0() 915.68/297.14 , -(I(x), O(y)) -> I(-(x, y)) 915.68/297.14 , -(I(x), I(y)) -> O(-(x, y)) 915.68/297.14 , not(true()) -> false() 915.68/297.14 , not(false()) -> true() 915.68/297.14 , and(x, true()) -> x 915.68/297.14 , and(x, false()) -> false() 915.68/297.14 , if(true(), x, y) -> x 915.68/297.14 , if(false(), x, y) -> y 915.68/297.14 , ge(x, 0()) -> true() 915.68/297.14 , ge(O(x), O(y)) -> ge(x, y) 915.68/297.14 , ge(O(x), I(y)) -> not(ge(y, x)) 915.68/297.14 , ge(0(), O(x)) -> ge(0(), x) 915.68/297.14 , ge(0(), I(x)) -> false() 915.68/297.14 , ge(I(x), O(y)) -> ge(x, y) 915.68/297.14 , ge(I(x), I(y)) -> ge(x, y) 915.68/297.14 , Log'(O(x)) -> if(ge(x, I(0())), +(Log'(x), I(0())), 0()) 915.68/297.14 , Log'(0()) -> 0() 915.68/297.14 , Log'(I(x)) -> +(Log'(x), I(0())) 915.68/297.14 , Log(x) -> -(Log'(x), I(0())) 915.68/297.14 , Val(L(x)) -> x 915.68/297.14 , Val(N(x, l(), r())) -> x 915.68/297.14 , Min(L(x)) -> x 915.68/297.14 , Min(N(x, l(), r())) -> Min(l()) 915.68/297.14 , Max(L(x)) -> x 915.68/297.14 , Max(N(x, l(), r())) -> Max(r()) 915.68/297.14 , BS(L(x)) -> true() 915.68/297.14 , BS(N(x, l(), r())) -> 915.68/297.14 and(and(ge(x, Max(l())), ge(Min(r()), x)), and(BS(l()), BS(r()))) 915.68/297.14 , Size(L(x)) -> I(0()) 915.68/297.14 , Size(N(x, l(), r())) -> +(+(Size(l()), Size(r())), I(1())) 915.68/297.14 , WB(L(x)) -> true() 915.68/297.14 , WB(N(x, l(), r())) -> 915.68/297.14 and(if(ge(Size(l()), Size(r())), 915.68/297.14 ge(I(0()), -(Size(l()), Size(r()))), 915.68/297.14 ge(I(0()), -(Size(r()), Size(l())))), 915.68/297.14 and(WB(l()), WB(r()))) } 915.68/297.14 Obligation: 915.68/297.14 runtime complexity 915.68/297.14 Answer: 915.68/297.14 MAYBE 915.68/297.14 915.68/297.14 None of the processors succeeded. 915.68/297.14 915.68/297.14 Details of failed attempt(s): 915.68/297.14 ----------------------------- 915.68/297.14 1) 'With Problem ... (timeout of 297 seconds)' failed due to the 915.68/297.14 following reason: 915.68/297.14 915.68/297.14 Computation stopped due to timeout after 297.0 seconds. 915.68/297.14 915.68/297.14 2) 'Best' failed due to the following reason: 915.68/297.14 915.68/297.14 None of the processors succeeded. 915.68/297.14 915.68/297.14 Details of failed attempt(s): 915.68/297.14 ----------------------------- 915.68/297.14 1) 'With Problem ... (timeout of 148 seconds) (timeout of 297 915.68/297.14 seconds)' failed due to the following reason: 915.68/297.14 915.68/297.14 Computation stopped due to timeout after 148.0 seconds. 915.68/297.14 915.68/297.14 2) 'Fastest (timeout of 24 seconds) (timeout of 297 seconds)' 915.68/297.14 failed due to the following reason: 915.68/297.14 915.68/297.14 None of the processors succeeded. 915.68/297.14 915.68/297.14 Details of failed attempt(s): 915.68/297.14 ----------------------------- 915.68/297.14 1) 'Bounds with minimal-enrichment and initial automaton 'match'' 915.68/297.14 failed due to the following reason: 915.68/297.14 915.68/297.14 match-boundness of the problem could not be verified. 915.68/297.14 915.68/297.14 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' 915.68/297.14 failed due to the following reason: 915.68/297.14 915.68/297.14 match-boundness of the problem could not be verified. 915.68/297.14 915.68/297.14 915.68/297.14 3) 'Best' failed due to the following reason: 915.68/297.14 915.68/297.14 None of the processors succeeded. 915.68/297.14 915.68/297.14 Details of failed attempt(s): 915.68/297.14 ----------------------------- 915.68/297.14 1) 'bsearch-popstar (timeout of 297 seconds)' failed due to the 915.68/297.14 following reason: 915.68/297.14 915.68/297.14 The processor is inapplicable, reason: 915.68/297.14 Processor only applicable for innermost runtime complexity analysis 915.68/297.14 915.68/297.14 2) 'Polynomial Path Order (PS) (timeout of 297 seconds)' failed due 915.68/297.14 to the following reason: 915.68/297.14 915.68/297.14 The processor is inapplicable, reason: 915.68/297.14 Processor only applicable for innermost runtime complexity analysis 915.68/297.14 915.68/297.14 915.68/297.14 915.68/297.14 3) 'Weak Dependency Pairs (timeout of 297 seconds)' failed due to 915.68/297.14 the following reason: 915.68/297.14 915.68/297.14 We add the following weak dependency pairs: 915.68/297.14 915.68/297.14 Strict DPs: 915.68/297.14 { O^#(0()) -> c_1() 915.68/297.14 , +^#(x, 0()) -> c_2(x) 915.68/297.14 , +^#(x, +(y, z)) -> c_3(+^#(+(x, y), z)) 915.68/297.14 , +^#(O(x), O(y)) -> c_4(O^#(+(x, y))) 915.68/297.14 , +^#(O(x), I(y)) -> c_5(+^#(x, y)) 915.68/297.14 , +^#(0(), x) -> c_6(x) 915.68/297.14 , +^#(I(x), O(y)) -> c_7(+^#(x, y)) 915.68/297.14 , +^#(I(x), I(y)) -> c_8(O^#(+(+(x, y), I(0())))) 915.68/297.14 , -^#(x, 0()) -> c_9(x) 915.68/297.14 , -^#(O(x), O(y)) -> c_10(O^#(-(x, y))) 915.68/297.14 , -^#(O(x), I(y)) -> c_11(-^#(-(x, y), I(1()))) 915.68/297.14 , -^#(0(), x) -> c_12() 915.68/297.14 , -^#(I(x), O(y)) -> c_13(-^#(x, y)) 915.68/297.14 , -^#(I(x), I(y)) -> c_14(O^#(-(x, y))) 915.68/297.14 , not^#(true()) -> c_15() 915.68/297.14 , not^#(false()) -> c_16() 915.68/297.14 , and^#(x, true()) -> c_17(x) 915.68/297.14 , and^#(x, false()) -> c_18() 915.68/297.14 , if^#(true(), x, y) -> c_19(x) 915.68/297.14 , if^#(false(), x, y) -> c_20(y) 915.68/297.14 , ge^#(x, 0()) -> c_21() 915.68/297.14 , ge^#(O(x), O(y)) -> c_22(ge^#(x, y)) 915.68/297.14 , ge^#(O(x), I(y)) -> c_23(not^#(ge(y, x))) 915.68/297.14 , ge^#(0(), O(x)) -> c_24(ge^#(0(), x)) 915.68/297.14 , ge^#(0(), I(x)) -> c_25() 915.68/297.14 , ge^#(I(x), O(y)) -> c_26(ge^#(x, y)) 915.68/297.14 , ge^#(I(x), I(y)) -> c_27(ge^#(x, y)) 915.68/297.14 , Log'^#(O(x)) -> 915.68/297.14 c_28(if^#(ge(x, I(0())), +(Log'(x), I(0())), 0())) 915.68/297.14 , Log'^#(0()) -> c_29() 915.68/297.14 , Log'^#(I(x)) -> c_30(+^#(Log'(x), I(0()))) 915.68/297.14 , Log^#(x) -> c_31(-^#(Log'(x), I(0()))) 915.68/297.14 , Val^#(L(x)) -> c_32(x) 915.68/297.14 , Val^#(N(x, l(), r())) -> c_33(x) 915.68/297.14 , Min^#(L(x)) -> c_34(x) 915.68/297.14 , Min^#(N(x, l(), r())) -> c_35(Min^#(l())) 915.68/297.14 , Max^#(L(x)) -> c_36(x) 915.68/297.14 , Max^#(N(x, l(), r())) -> c_37(Max^#(r())) 915.68/297.14 , BS^#(L(x)) -> c_38() 915.68/297.14 , BS^#(N(x, l(), r())) -> 915.68/297.14 c_39(and^#(and(ge(x, Max(l())), ge(Min(r()), x)), 915.68/297.14 and(BS(l()), BS(r())))) 915.68/297.14 , Size^#(L(x)) -> c_40() 915.68/297.14 , Size^#(N(x, l(), r())) -> 915.68/297.14 c_41(+^#(+(Size(l()), Size(r())), I(1()))) 915.68/297.14 , WB^#(L(x)) -> c_42() 915.68/297.14 , WB^#(N(x, l(), r())) -> 915.68/297.14 c_43(and^#(if(ge(Size(l()), Size(r())), 915.68/297.14 ge(I(0()), -(Size(l()), Size(r()))), 915.68/297.14 ge(I(0()), -(Size(r()), Size(l())))), 915.68/297.14 and(WB(l()), WB(r())))) } 915.68/297.14 915.68/297.14 and mark the set of starting terms. 915.68/297.14 915.68/297.14 We are left with following problem, upon which TcT provides the 915.68/297.14 certificate MAYBE. 915.68/297.14 915.68/297.14 Strict DPs: 915.68/297.14 { O^#(0()) -> c_1() 915.68/297.14 , +^#(x, 0()) -> c_2(x) 915.68/297.14 , +^#(x, +(y, z)) -> c_3(+^#(+(x, y), z)) 915.68/297.14 , +^#(O(x), O(y)) -> c_4(O^#(+(x, y))) 915.68/297.14 , +^#(O(x), I(y)) -> c_5(+^#(x, y)) 915.68/297.14 , +^#(0(), x) -> c_6(x) 915.68/297.14 , +^#(I(x), O(y)) -> c_7(+^#(x, y)) 915.68/297.14 , +^#(I(x), I(y)) -> c_8(O^#(+(+(x, y), I(0())))) 915.68/297.14 , -^#(x, 0()) -> c_9(x) 915.68/297.14 , -^#(O(x), O(y)) -> c_10(O^#(-(x, y))) 915.68/297.14 , -^#(O(x), I(y)) -> c_11(-^#(-(x, y), I(1()))) 915.68/297.14 , -^#(0(), x) -> c_12() 915.68/297.14 , -^#(I(x), O(y)) -> c_13(-^#(x, y)) 915.68/297.14 , -^#(I(x), I(y)) -> c_14(O^#(-(x, y))) 915.68/297.14 , not^#(true()) -> c_15() 915.68/297.14 , not^#(false()) -> c_16() 915.68/297.14 , and^#(x, true()) -> c_17(x) 915.68/297.14 , and^#(x, false()) -> c_18() 915.68/297.14 , if^#(true(), x, y) -> c_19(x) 915.68/297.14 , if^#(false(), x, y) -> c_20(y) 915.68/297.14 , ge^#(x, 0()) -> c_21() 915.68/297.14 , ge^#(O(x), O(y)) -> c_22(ge^#(x, y)) 915.68/297.14 , ge^#(O(x), I(y)) -> c_23(not^#(ge(y, x))) 915.68/297.14 , ge^#(0(), O(x)) -> c_24(ge^#(0(), x)) 915.68/297.14 , ge^#(0(), I(x)) -> c_25() 915.68/297.14 , ge^#(I(x), O(y)) -> c_26(ge^#(x, y)) 915.68/297.14 , ge^#(I(x), I(y)) -> c_27(ge^#(x, y)) 915.68/297.14 , Log'^#(O(x)) -> 915.68/297.14 c_28(if^#(ge(x, I(0())), +(Log'(x), I(0())), 0())) 915.68/297.14 , Log'^#(0()) -> c_29() 915.68/297.14 , Log'^#(I(x)) -> c_30(+^#(Log'(x), I(0()))) 915.68/297.14 , Log^#(x) -> c_31(-^#(Log'(x), I(0()))) 915.68/297.14 , Val^#(L(x)) -> c_32(x) 915.68/297.14 , Val^#(N(x, l(), r())) -> c_33(x) 915.68/297.14 , Min^#(L(x)) -> c_34(x) 915.68/297.14 , Min^#(N(x, l(), r())) -> c_35(Min^#(l())) 915.68/297.14 , Max^#(L(x)) -> c_36(x) 915.68/297.14 , Max^#(N(x, l(), r())) -> c_37(Max^#(r())) 915.68/297.14 , BS^#(L(x)) -> c_38() 915.68/297.14 , BS^#(N(x, l(), r())) -> 915.68/297.14 c_39(and^#(and(ge(x, Max(l())), ge(Min(r()), x)), 915.68/297.14 and(BS(l()), BS(r())))) 915.68/297.14 , Size^#(L(x)) -> c_40() 915.68/297.14 , Size^#(N(x, l(), r())) -> 915.68/297.14 c_41(+^#(+(Size(l()), Size(r())), I(1()))) 915.68/297.14 , WB^#(L(x)) -> c_42() 915.68/297.14 , WB^#(N(x, l(), r())) -> 915.68/297.14 c_43(and^#(if(ge(Size(l()), Size(r())), 915.68/297.14 ge(I(0()), -(Size(l()), Size(r()))), 915.68/297.14 ge(I(0()), -(Size(r()), Size(l())))), 915.68/297.14 and(WB(l()), WB(r())))) } 915.68/297.14 Strict Trs: 915.68/297.14 { O(0()) -> 0() 915.68/297.14 , +(x, 0()) -> x 915.68/297.14 , +(x, +(y, z)) -> +(+(x, y), z) 915.68/297.14 , +(O(x), O(y)) -> O(+(x, y)) 915.68/297.14 , +(O(x), I(y)) -> I(+(x, y)) 915.68/297.14 , +(0(), x) -> x 915.68/297.14 , +(I(x), O(y)) -> I(+(x, y)) 915.68/297.14 , +(I(x), I(y)) -> O(+(+(x, y), I(0()))) 915.68/297.14 , -(x, 0()) -> x 915.68/297.14 , -(O(x), O(y)) -> O(-(x, y)) 915.68/297.14 , -(O(x), I(y)) -> I(-(-(x, y), I(1()))) 915.68/297.14 , -(0(), x) -> 0() 915.68/297.14 , -(I(x), O(y)) -> I(-(x, y)) 915.68/297.14 , -(I(x), I(y)) -> O(-(x, y)) 915.68/297.14 , not(true()) -> false() 915.68/297.14 , not(false()) -> true() 915.68/297.14 , and(x, true()) -> x 915.68/297.14 , and(x, false()) -> false() 915.68/297.14 , if(true(), x, y) -> x 915.68/297.14 , if(false(), x, y) -> y 915.68/297.14 , ge(x, 0()) -> true() 915.68/297.14 , ge(O(x), O(y)) -> ge(x, y) 915.68/297.14 , ge(O(x), I(y)) -> not(ge(y, x)) 915.68/297.14 , ge(0(), O(x)) -> ge(0(), x) 915.68/297.14 , ge(0(), I(x)) -> false() 915.68/297.14 , ge(I(x), O(y)) -> ge(x, y) 915.68/297.14 , ge(I(x), I(y)) -> ge(x, y) 915.68/297.14 , Log'(O(x)) -> if(ge(x, I(0())), +(Log'(x), I(0())), 0()) 915.68/297.14 , Log'(0()) -> 0() 915.68/297.14 , Log'(I(x)) -> +(Log'(x), I(0())) 915.68/297.14 , Log(x) -> -(Log'(x), I(0())) 915.68/297.14 , Val(L(x)) -> x 915.68/297.14 , Val(N(x, l(), r())) -> x 915.68/297.14 , Min(L(x)) -> x 915.68/297.14 , Min(N(x, l(), r())) -> Min(l()) 915.68/297.14 , Max(L(x)) -> x 915.68/297.14 , Max(N(x, l(), r())) -> Max(r()) 915.68/297.14 , BS(L(x)) -> true() 915.68/297.14 , BS(N(x, l(), r())) -> 915.68/297.14 and(and(ge(x, Max(l())), ge(Min(r()), x)), and(BS(l()), BS(r()))) 915.68/297.14 , Size(L(x)) -> I(0()) 915.68/297.14 , Size(N(x, l(), r())) -> +(+(Size(l()), Size(r())), I(1())) 915.68/297.14 , WB(L(x)) -> true() 915.68/297.14 , WB(N(x, l(), r())) -> 915.68/297.14 and(if(ge(Size(l()), Size(r())), 915.68/297.14 ge(I(0()), -(Size(l()), Size(r()))), 915.68/297.14 ge(I(0()), -(Size(r()), Size(l())))), 915.68/297.14 and(WB(l()), WB(r()))) } 915.68/297.14 Obligation: 915.68/297.14 runtime complexity 915.68/297.14 Answer: 915.68/297.14 MAYBE 915.68/297.14 915.68/297.14 We estimate the number of application of 915.68/297.14 {1,12,15,16,18,21,25,29,35,37,38,39,40,41,42,43} by applications of 915.68/297.14 Pre({1,12,15,16,18,21,25,29,35,37,38,39,40,41,42,43}) = 915.68/297.14 {2,4,6,8,9,10,11,13,14,17,19,20,22,23,24,26,27,31,32,33,34,36}. 915.68/297.14 Here rules are labeled as follows: 915.68/297.14 915.68/297.14 DPs: 915.68/297.14 { 1: O^#(0()) -> c_1() 915.68/297.14 , 2: +^#(x, 0()) -> c_2(x) 915.68/297.14 , 3: +^#(x, +(y, z)) -> c_3(+^#(+(x, y), z)) 915.68/297.14 , 4: +^#(O(x), O(y)) -> c_4(O^#(+(x, y))) 915.68/297.14 , 5: +^#(O(x), I(y)) -> c_5(+^#(x, y)) 915.68/297.14 , 6: +^#(0(), x) -> c_6(x) 915.68/297.14 , 7: +^#(I(x), O(y)) -> c_7(+^#(x, y)) 915.68/297.14 , 8: +^#(I(x), I(y)) -> c_8(O^#(+(+(x, y), I(0())))) 915.68/297.14 , 9: -^#(x, 0()) -> c_9(x) 915.68/297.14 , 10: -^#(O(x), O(y)) -> c_10(O^#(-(x, y))) 915.68/297.14 , 11: -^#(O(x), I(y)) -> c_11(-^#(-(x, y), I(1()))) 915.68/297.14 , 12: -^#(0(), x) -> c_12() 915.68/297.14 , 13: -^#(I(x), O(y)) -> c_13(-^#(x, y)) 915.68/297.14 , 14: -^#(I(x), I(y)) -> c_14(O^#(-(x, y))) 915.68/297.14 , 15: not^#(true()) -> c_15() 915.68/297.14 , 16: not^#(false()) -> c_16() 915.68/297.14 , 17: and^#(x, true()) -> c_17(x) 915.68/297.14 , 18: and^#(x, false()) -> c_18() 915.68/297.14 , 19: if^#(true(), x, y) -> c_19(x) 915.68/297.14 , 20: if^#(false(), x, y) -> c_20(y) 915.68/297.14 , 21: ge^#(x, 0()) -> c_21() 915.68/297.14 , 22: ge^#(O(x), O(y)) -> c_22(ge^#(x, y)) 915.68/297.14 , 23: ge^#(O(x), I(y)) -> c_23(not^#(ge(y, x))) 915.68/297.14 , 24: ge^#(0(), O(x)) -> c_24(ge^#(0(), x)) 915.68/297.14 , 25: ge^#(0(), I(x)) -> c_25() 915.68/297.14 , 26: ge^#(I(x), O(y)) -> c_26(ge^#(x, y)) 915.68/297.14 , 27: ge^#(I(x), I(y)) -> c_27(ge^#(x, y)) 915.68/297.14 , 28: Log'^#(O(x)) -> 915.68/297.14 c_28(if^#(ge(x, I(0())), +(Log'(x), I(0())), 0())) 915.68/297.14 , 29: Log'^#(0()) -> c_29() 915.68/297.14 , 30: Log'^#(I(x)) -> c_30(+^#(Log'(x), I(0()))) 915.68/297.14 , 31: Log^#(x) -> c_31(-^#(Log'(x), I(0()))) 915.68/297.14 , 32: Val^#(L(x)) -> c_32(x) 915.68/297.14 , 33: Val^#(N(x, l(), r())) -> c_33(x) 915.68/297.14 , 34: Min^#(L(x)) -> c_34(x) 915.68/297.14 , 35: Min^#(N(x, l(), r())) -> c_35(Min^#(l())) 915.68/297.14 , 36: Max^#(L(x)) -> c_36(x) 915.68/297.14 , 37: Max^#(N(x, l(), r())) -> c_37(Max^#(r())) 915.68/297.14 , 38: BS^#(L(x)) -> c_38() 915.68/297.14 , 39: BS^#(N(x, l(), r())) -> 915.68/297.14 c_39(and^#(and(ge(x, Max(l())), ge(Min(r()), x)), 915.68/297.14 and(BS(l()), BS(r())))) 915.68/297.14 , 40: Size^#(L(x)) -> c_40() 915.68/297.14 , 41: Size^#(N(x, l(), r())) -> 915.68/297.14 c_41(+^#(+(Size(l()), Size(r())), I(1()))) 915.68/297.14 , 42: WB^#(L(x)) -> c_42() 915.68/297.14 , 43: WB^#(N(x, l(), r())) -> 915.68/297.14 c_43(and^#(if(ge(Size(l()), Size(r())), 915.68/297.14 ge(I(0()), -(Size(l()), Size(r()))), 915.68/297.14 ge(I(0()), -(Size(r()), Size(l())))), 915.68/297.14 and(WB(l()), WB(r())))) } 915.68/297.14 915.68/297.14 We are left with following problem, upon which TcT provides the 915.68/297.14 certificate MAYBE. 915.68/297.14 915.68/297.14 Strict DPs: 915.68/297.14 { +^#(x, 0()) -> c_2(x) 915.68/297.14 , +^#(x, +(y, z)) -> c_3(+^#(+(x, y), z)) 915.68/297.14 , +^#(O(x), O(y)) -> c_4(O^#(+(x, y))) 915.68/297.14 , +^#(O(x), I(y)) -> c_5(+^#(x, y)) 915.68/297.14 , +^#(0(), x) -> c_6(x) 915.68/297.14 , +^#(I(x), O(y)) -> c_7(+^#(x, y)) 915.68/297.14 , +^#(I(x), I(y)) -> c_8(O^#(+(+(x, y), I(0())))) 915.68/297.14 , -^#(x, 0()) -> c_9(x) 915.68/297.14 , -^#(O(x), O(y)) -> c_10(O^#(-(x, y))) 915.68/297.14 , -^#(O(x), I(y)) -> c_11(-^#(-(x, y), I(1()))) 915.68/297.14 , -^#(I(x), O(y)) -> c_13(-^#(x, y)) 915.68/297.14 , -^#(I(x), I(y)) -> c_14(O^#(-(x, y))) 915.68/297.14 , and^#(x, true()) -> c_17(x) 915.68/297.14 , if^#(true(), x, y) -> c_19(x) 915.68/297.14 , if^#(false(), x, y) -> c_20(y) 915.68/297.14 , ge^#(O(x), O(y)) -> c_22(ge^#(x, y)) 915.68/297.14 , ge^#(O(x), I(y)) -> c_23(not^#(ge(y, x))) 915.68/297.14 , ge^#(0(), O(x)) -> c_24(ge^#(0(), x)) 915.68/297.14 , ge^#(I(x), O(y)) -> c_26(ge^#(x, y)) 915.68/297.14 , ge^#(I(x), I(y)) -> c_27(ge^#(x, y)) 915.68/297.14 , Log'^#(O(x)) -> 915.68/297.14 c_28(if^#(ge(x, I(0())), +(Log'(x), I(0())), 0())) 915.68/297.14 , Log'^#(I(x)) -> c_30(+^#(Log'(x), I(0()))) 915.68/297.14 , Log^#(x) -> c_31(-^#(Log'(x), I(0()))) 915.68/297.14 , Val^#(L(x)) -> c_32(x) 915.68/297.14 , Val^#(N(x, l(), r())) -> c_33(x) 915.68/297.14 , Min^#(L(x)) -> c_34(x) 915.68/297.14 , Max^#(L(x)) -> c_36(x) } 915.68/297.14 Strict Trs: 915.68/297.14 { O(0()) -> 0() 915.68/297.14 , +(x, 0()) -> x 915.68/297.14 , +(x, +(y, z)) -> +(+(x, y), z) 915.68/297.14 , +(O(x), O(y)) -> O(+(x, y)) 915.68/297.14 , +(O(x), I(y)) -> I(+(x, y)) 915.68/297.14 , +(0(), x) -> x 915.68/297.14 , +(I(x), O(y)) -> I(+(x, y)) 915.68/297.14 , +(I(x), I(y)) -> O(+(+(x, y), I(0()))) 915.68/297.14 , -(x, 0()) -> x 915.68/297.14 , -(O(x), O(y)) -> O(-(x, y)) 915.68/297.14 , -(O(x), I(y)) -> I(-(-(x, y), I(1()))) 915.68/297.14 , -(0(), x) -> 0() 915.68/297.14 , -(I(x), O(y)) -> I(-(x, y)) 915.68/297.14 , -(I(x), I(y)) -> O(-(x, y)) 915.68/297.14 , not(true()) -> false() 915.68/297.14 , not(false()) -> true() 915.68/297.14 , and(x, true()) -> x 915.68/297.14 , and(x, false()) -> false() 915.68/297.14 , if(true(), x, y) -> x 915.68/297.14 , if(false(), x, y) -> y 915.68/297.14 , ge(x, 0()) -> true() 915.68/297.14 , ge(O(x), O(y)) -> ge(x, y) 915.68/297.14 , ge(O(x), I(y)) -> not(ge(y, x)) 915.68/297.14 , ge(0(), O(x)) -> ge(0(), x) 915.68/297.14 , ge(0(), I(x)) -> false() 915.68/297.14 , ge(I(x), O(y)) -> ge(x, y) 915.68/297.14 , ge(I(x), I(y)) -> ge(x, y) 915.68/297.14 , Log'(O(x)) -> if(ge(x, I(0())), +(Log'(x), I(0())), 0()) 915.68/297.14 , Log'(0()) -> 0() 915.68/297.14 , Log'(I(x)) -> +(Log'(x), I(0())) 915.68/297.14 , Log(x) -> -(Log'(x), I(0())) 915.68/297.14 , Val(L(x)) -> x 915.68/297.14 , Val(N(x, l(), r())) -> x 915.68/297.14 , Min(L(x)) -> x 915.68/297.14 , Min(N(x, l(), r())) -> Min(l()) 915.68/297.14 , Max(L(x)) -> x 915.68/297.14 , Max(N(x, l(), r())) -> Max(r()) 915.68/297.14 , BS(L(x)) -> true() 915.68/297.14 , BS(N(x, l(), r())) -> 915.68/297.14 and(and(ge(x, Max(l())), ge(Min(r()), x)), and(BS(l()), BS(r()))) 915.68/297.14 , Size(L(x)) -> I(0()) 915.68/297.14 , Size(N(x, l(), r())) -> +(+(Size(l()), Size(r())), I(1())) 915.68/297.14 , WB(L(x)) -> true() 915.68/297.14 , WB(N(x, l(), r())) -> 915.68/297.14 and(if(ge(Size(l()), Size(r())), 915.68/297.14 ge(I(0()), -(Size(l()), Size(r()))), 915.68/297.14 ge(I(0()), -(Size(r()), Size(l())))), 915.68/297.14 and(WB(l()), WB(r()))) } 915.68/297.14 Weak DPs: 915.68/297.14 { O^#(0()) -> c_1() 915.68/297.14 , -^#(0(), x) -> c_12() 915.68/297.15 , not^#(true()) -> c_15() 915.68/297.15 , not^#(false()) -> c_16() 915.68/297.15 , and^#(x, false()) -> c_18() 915.68/297.15 , ge^#(x, 0()) -> c_21() 915.68/297.15 , ge^#(0(), I(x)) -> c_25() 915.68/297.15 , Log'^#(0()) -> c_29() 915.68/297.15 , Min^#(N(x, l(), r())) -> c_35(Min^#(l())) 915.68/297.15 , Max^#(N(x, l(), r())) -> c_37(Max^#(r())) 915.68/297.15 , BS^#(L(x)) -> c_38() 915.68/297.15 , BS^#(N(x, l(), r())) -> 915.68/297.15 c_39(and^#(and(ge(x, Max(l())), ge(Min(r()), x)), 915.68/297.15 and(BS(l()), BS(r())))) 915.68/297.15 , Size^#(L(x)) -> c_40() 915.68/297.15 , Size^#(N(x, l(), r())) -> 915.68/297.15 c_41(+^#(+(Size(l()), Size(r())), I(1()))) 915.68/297.15 , WB^#(L(x)) -> c_42() 915.68/297.15 , WB^#(N(x, l(), r())) -> 915.68/297.15 c_43(and^#(if(ge(Size(l()), Size(r())), 915.68/297.15 ge(I(0()), -(Size(l()), Size(r()))), 915.68/297.15 ge(I(0()), -(Size(r()), Size(l())))), 915.68/297.15 and(WB(l()), WB(r())))) } 915.68/297.15 Obligation: 915.68/297.15 runtime complexity 915.68/297.15 Answer: 915.68/297.15 MAYBE 915.68/297.15 915.68/297.15 We estimate the number of application of {3,7,9,12,17} by 915.68/297.15 applications of Pre({3,7,9,12,17}) = 915.68/297.15 {1,2,4,5,6,8,10,11,13,14,15,16,19,20,22,23,24,25,26,27}. Here rules 915.68/297.15 are labeled as follows: 915.68/297.15 915.68/297.15 DPs: 915.68/297.15 { 1: +^#(x, 0()) -> c_2(x) 915.68/297.15 , 2: +^#(x, +(y, z)) -> c_3(+^#(+(x, y), z)) 915.68/297.15 , 3: +^#(O(x), O(y)) -> c_4(O^#(+(x, y))) 915.68/297.15 , 4: +^#(O(x), I(y)) -> c_5(+^#(x, y)) 915.68/297.15 , 5: +^#(0(), x) -> c_6(x) 915.68/297.15 , 6: +^#(I(x), O(y)) -> c_7(+^#(x, y)) 915.68/297.15 , 7: +^#(I(x), I(y)) -> c_8(O^#(+(+(x, y), I(0())))) 915.68/297.15 , 8: -^#(x, 0()) -> c_9(x) 915.68/297.15 , 9: -^#(O(x), O(y)) -> c_10(O^#(-(x, y))) 915.68/297.15 , 10: -^#(O(x), I(y)) -> c_11(-^#(-(x, y), I(1()))) 915.68/297.15 , 11: -^#(I(x), O(y)) -> c_13(-^#(x, y)) 915.68/297.15 , 12: -^#(I(x), I(y)) -> c_14(O^#(-(x, y))) 915.68/297.15 , 13: and^#(x, true()) -> c_17(x) 915.68/297.15 , 14: if^#(true(), x, y) -> c_19(x) 915.68/297.15 , 15: if^#(false(), x, y) -> c_20(y) 915.68/297.15 , 16: ge^#(O(x), O(y)) -> c_22(ge^#(x, y)) 915.68/297.15 , 17: ge^#(O(x), I(y)) -> c_23(not^#(ge(y, x))) 915.68/297.15 , 18: ge^#(0(), O(x)) -> c_24(ge^#(0(), x)) 915.68/297.15 , 19: ge^#(I(x), O(y)) -> c_26(ge^#(x, y)) 915.68/297.15 , 20: ge^#(I(x), I(y)) -> c_27(ge^#(x, y)) 915.68/297.15 , 21: Log'^#(O(x)) -> 915.68/297.15 c_28(if^#(ge(x, I(0())), +(Log'(x), I(0())), 0())) 915.68/297.15 , 22: Log'^#(I(x)) -> c_30(+^#(Log'(x), I(0()))) 915.68/297.15 , 23: Log^#(x) -> c_31(-^#(Log'(x), I(0()))) 915.68/297.15 , 24: Val^#(L(x)) -> c_32(x) 915.68/297.15 , 25: Val^#(N(x, l(), r())) -> c_33(x) 915.68/297.15 , 26: Min^#(L(x)) -> c_34(x) 915.68/297.15 , 27: Max^#(L(x)) -> c_36(x) 915.68/297.15 , 28: O^#(0()) -> c_1() 915.68/297.15 , 29: -^#(0(), x) -> c_12() 915.68/297.15 , 30: not^#(true()) -> c_15() 915.68/297.15 , 31: not^#(false()) -> c_16() 915.68/297.15 , 32: and^#(x, false()) -> c_18() 915.68/297.15 , 33: ge^#(x, 0()) -> c_21() 915.68/297.15 , 34: ge^#(0(), I(x)) -> c_25() 915.68/297.15 , 35: Log'^#(0()) -> c_29() 915.68/297.15 , 36: Min^#(N(x, l(), r())) -> c_35(Min^#(l())) 915.68/297.15 , 37: Max^#(N(x, l(), r())) -> c_37(Max^#(r())) 915.68/297.15 , 38: BS^#(L(x)) -> c_38() 915.68/297.15 , 39: BS^#(N(x, l(), r())) -> 915.68/297.15 c_39(and^#(and(ge(x, Max(l())), ge(Min(r()), x)), 915.68/297.15 and(BS(l()), BS(r())))) 915.68/297.15 , 40: Size^#(L(x)) -> c_40() 915.68/297.15 , 41: Size^#(N(x, l(), r())) -> 915.68/297.15 c_41(+^#(+(Size(l()), Size(r())), I(1()))) 915.68/297.15 , 42: WB^#(L(x)) -> c_42() 915.68/297.15 , 43: WB^#(N(x, l(), r())) -> 915.68/297.15 c_43(and^#(if(ge(Size(l()), Size(r())), 915.68/297.15 ge(I(0()), -(Size(l()), Size(r()))), 915.68/297.15 ge(I(0()), -(Size(r()), Size(l())))), 915.68/297.15 and(WB(l()), WB(r())))) } 915.68/297.15 915.68/297.15 We are left with following problem, upon which TcT provides the 915.68/297.15 certificate MAYBE. 915.68/297.15 915.68/297.15 Strict DPs: 915.68/297.15 { +^#(x, 0()) -> c_2(x) 915.68/297.15 , +^#(x, +(y, z)) -> c_3(+^#(+(x, y), z)) 915.68/297.15 , +^#(O(x), I(y)) -> c_5(+^#(x, y)) 915.68/297.15 , +^#(0(), x) -> c_6(x) 915.68/297.15 , +^#(I(x), O(y)) -> c_7(+^#(x, y)) 915.68/297.15 , -^#(x, 0()) -> c_9(x) 915.68/297.15 , -^#(O(x), I(y)) -> c_11(-^#(-(x, y), I(1()))) 915.68/297.15 , -^#(I(x), O(y)) -> c_13(-^#(x, y)) 915.68/297.15 , and^#(x, true()) -> c_17(x) 915.68/297.15 , if^#(true(), x, y) -> c_19(x) 915.68/297.15 , if^#(false(), x, y) -> c_20(y) 915.68/297.15 , ge^#(O(x), O(y)) -> c_22(ge^#(x, y)) 915.68/297.15 , ge^#(0(), O(x)) -> c_24(ge^#(0(), x)) 915.68/297.15 , ge^#(I(x), O(y)) -> c_26(ge^#(x, y)) 915.68/297.15 , ge^#(I(x), I(y)) -> c_27(ge^#(x, y)) 915.68/297.15 , Log'^#(O(x)) -> 915.68/297.15 c_28(if^#(ge(x, I(0())), +(Log'(x), I(0())), 0())) 915.68/297.15 , Log'^#(I(x)) -> c_30(+^#(Log'(x), I(0()))) 915.68/297.15 , Log^#(x) -> c_31(-^#(Log'(x), I(0()))) 915.68/297.15 , Val^#(L(x)) -> c_32(x) 915.68/297.15 , Val^#(N(x, l(), r())) -> c_33(x) 915.68/297.15 , Min^#(L(x)) -> c_34(x) 915.68/297.15 , Max^#(L(x)) -> c_36(x) } 915.68/297.15 Strict Trs: 915.68/297.15 { O(0()) -> 0() 915.68/297.15 , +(x, 0()) -> x 915.68/297.15 , +(x, +(y, z)) -> +(+(x, y), z) 915.68/297.15 , +(O(x), O(y)) -> O(+(x, y)) 915.68/297.15 , +(O(x), I(y)) -> I(+(x, y)) 915.68/297.15 , +(0(), x) -> x 915.68/297.15 , +(I(x), O(y)) -> I(+(x, y)) 915.68/297.15 , +(I(x), I(y)) -> O(+(+(x, y), I(0()))) 915.68/297.15 , -(x, 0()) -> x 915.68/297.15 , -(O(x), O(y)) -> O(-(x, y)) 915.68/297.15 , -(O(x), I(y)) -> I(-(-(x, y), I(1()))) 915.68/297.15 , -(0(), x) -> 0() 915.68/297.15 , -(I(x), O(y)) -> I(-(x, y)) 915.68/297.15 , -(I(x), I(y)) -> O(-(x, y)) 915.68/297.15 , not(true()) -> false() 915.68/297.15 , not(false()) -> true() 915.68/297.15 , and(x, true()) -> x 915.68/297.15 , and(x, false()) -> false() 915.68/297.15 , if(true(), x, y) -> x 915.68/297.15 , if(false(), x, y) -> y 915.68/297.15 , ge(x, 0()) -> true() 915.68/297.15 , ge(O(x), O(y)) -> ge(x, y) 915.68/297.15 , ge(O(x), I(y)) -> not(ge(y, x)) 915.68/297.15 , ge(0(), O(x)) -> ge(0(), x) 915.68/297.15 , ge(0(), I(x)) -> false() 915.68/297.15 , ge(I(x), O(y)) -> ge(x, y) 915.68/297.15 , ge(I(x), I(y)) -> ge(x, y) 915.68/297.15 , Log'(O(x)) -> if(ge(x, I(0())), +(Log'(x), I(0())), 0()) 915.68/297.15 , Log'(0()) -> 0() 915.68/297.15 , Log'(I(x)) -> +(Log'(x), I(0())) 915.68/297.15 , Log(x) -> -(Log'(x), I(0())) 915.68/297.15 , Val(L(x)) -> x 915.68/297.15 , Val(N(x, l(), r())) -> x 915.68/297.15 , Min(L(x)) -> x 915.68/297.15 , Min(N(x, l(), r())) -> Min(l()) 915.68/297.15 , Max(L(x)) -> x 915.68/297.15 , Max(N(x, l(), r())) -> Max(r()) 915.68/297.15 , BS(L(x)) -> true() 915.68/297.15 , BS(N(x, l(), r())) -> 915.68/297.15 and(and(ge(x, Max(l())), ge(Min(r()), x)), and(BS(l()), BS(r()))) 915.68/297.15 , Size(L(x)) -> I(0()) 915.68/297.15 , Size(N(x, l(), r())) -> +(+(Size(l()), Size(r())), I(1())) 915.68/297.15 , WB(L(x)) -> true() 915.68/297.15 , WB(N(x, l(), r())) -> 915.68/297.15 and(if(ge(Size(l()), Size(r())), 915.68/297.15 ge(I(0()), -(Size(l()), Size(r()))), 915.68/297.15 ge(I(0()), -(Size(r()), Size(l())))), 915.68/297.15 and(WB(l()), WB(r()))) } 915.68/297.15 Weak DPs: 915.68/297.15 { O^#(0()) -> c_1() 915.68/297.15 , +^#(O(x), O(y)) -> c_4(O^#(+(x, y))) 915.68/297.15 , +^#(I(x), I(y)) -> c_8(O^#(+(+(x, y), I(0())))) 915.68/297.15 , -^#(O(x), O(y)) -> c_10(O^#(-(x, y))) 915.68/297.15 , -^#(0(), x) -> c_12() 915.68/297.15 , -^#(I(x), I(y)) -> c_14(O^#(-(x, y))) 915.68/297.15 , not^#(true()) -> c_15() 915.68/297.15 , not^#(false()) -> c_16() 915.68/297.15 , and^#(x, false()) -> c_18() 915.68/297.15 , ge^#(x, 0()) -> c_21() 915.68/297.15 , ge^#(O(x), I(y)) -> c_23(not^#(ge(y, x))) 915.68/297.15 , ge^#(0(), I(x)) -> c_25() 915.68/297.15 , Log'^#(0()) -> c_29() 915.68/297.15 , Min^#(N(x, l(), r())) -> c_35(Min^#(l())) 915.68/297.15 , Max^#(N(x, l(), r())) -> c_37(Max^#(r())) 915.68/297.15 , BS^#(L(x)) -> c_38() 915.68/297.15 , BS^#(N(x, l(), r())) -> 915.68/297.15 c_39(and^#(and(ge(x, Max(l())), ge(Min(r()), x)), 915.68/297.15 and(BS(l()), BS(r())))) 915.68/297.15 , Size^#(L(x)) -> c_40() 915.68/297.15 , Size^#(N(x, l(), r())) -> 915.68/297.15 c_41(+^#(+(Size(l()), Size(r())), I(1()))) 915.68/297.15 , WB^#(L(x)) -> c_42() 915.68/297.15 , WB^#(N(x, l(), r())) -> 915.68/297.15 c_43(and^#(if(ge(Size(l()), Size(r())), 915.68/297.15 ge(I(0()), -(Size(l()), Size(r()))), 915.68/297.15 ge(I(0()), -(Size(r()), Size(l())))), 915.68/297.15 and(WB(l()), WB(r())))) } 915.68/297.15 Obligation: 915.68/297.15 runtime complexity 915.68/297.15 Answer: 915.68/297.15 MAYBE 915.68/297.15 915.68/297.15 Empty strict component of the problem is NOT empty. 915.68/297.15 915.68/297.15 915.68/297.15 Arrrr.. 915.84/297.27 EOF