MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { O(0()) -> 0() , +(x, 0()) -> x , +(x, +(y, z)) -> +(+(x, y), z) , +(O(x), O(y)) -> O(+(x, y)) , +(O(x), I(y)) -> I(+(x, y)) , +(0(), x) -> x , +(I(x), O(y)) -> I(+(x, y)) , +(I(x), I(y)) -> O(+(+(x, y), I(0()))) , -(x, 0()) -> x , -(O(x), O(y)) -> O(-(x, y)) , -(O(x), I(y)) -> I(-(-(x, y), I(1()))) , -(0(), x) -> 0() , -(I(x), O(y)) -> I(-(x, y)) , -(I(x), I(y)) -> O(-(x, y)) , not(true()) -> false() , not(false()) -> true() , and(x, true()) -> x , and(x, false()) -> false() , if(true(), x, y) -> x , if(false(), x, y) -> y , ge(x, 0()) -> true() , ge(O(x), O(y)) -> ge(x, y) , ge(O(x), I(y)) -> not(ge(y, x)) , ge(0(), O(x)) -> ge(0(), x) , ge(0(), I(x)) -> false() , ge(I(x), O(y)) -> ge(x, y) , ge(I(x), I(y)) -> ge(x, y) , Log'(O(x)) -> if(ge(x, I(0())), +(Log'(x), I(0())), 0()) , Log'(0()) -> 0() , Log'(I(x)) -> +(Log'(x), I(0())) , Log(x) -> -(Log'(x), I(0())) , Val(L(x)) -> x , Val(N(x, l(), r())) -> x , Min(L(x)) -> x , Min(N(x, l(), r())) -> Min(l()) , Max(L(x)) -> x , Max(N(x, l(), r())) -> Max(r()) , BS(L(x)) -> true() , BS(N(x, l(), r())) -> and(and(ge(x, Max(l())), ge(Min(r()), x)), and(BS(l()), BS(r()))) , Size(L(x)) -> I(0()) , Size(N(x, l(), r())) -> +(+(Size(l()), Size(r())), I(1())) , WB(L(x)) -> true() , WB(N(x, l(), r())) -> and(if(ge(Size(l()), Size(r())), ge(I(0()), -(Size(l()), Size(r()))), ge(I(0()), -(Size(r()), Size(l())))), and(WB(l()), WB(r()))) } Obligation: runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 60.0 seconds. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 30 seconds) (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 30.0 seconds. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 2) 'bsearch-popstar (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 3) 'Fastest (timeout of 5 seconds) (timeout of 60 seconds)' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 3) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { O^#(0()) -> c_1() , +^#(x, 0()) -> c_2(x) , +^#(x, +(y, z)) -> c_3(+^#(+(x, y), z)) , +^#(O(x), O(y)) -> c_4(O^#(+(x, y))) , +^#(O(x), I(y)) -> c_5(+^#(x, y)) , +^#(0(), x) -> c_6(x) , +^#(I(x), O(y)) -> c_7(+^#(x, y)) , +^#(I(x), I(y)) -> c_8(O^#(+(+(x, y), I(0())))) , -^#(x, 0()) -> c_9(x) , -^#(O(x), O(y)) -> c_10(O^#(-(x, y))) , -^#(O(x), I(y)) -> c_11(-^#(-(x, y), I(1()))) , -^#(0(), x) -> c_12() , -^#(I(x), O(y)) -> c_13(-^#(x, y)) , -^#(I(x), I(y)) -> c_14(O^#(-(x, y))) , not^#(true()) -> c_15() , not^#(false()) -> c_16() , and^#(x, true()) -> c_17(x) , and^#(x, false()) -> c_18() , if^#(true(), x, y) -> c_19(x) , if^#(false(), x, y) -> c_20(y) , ge^#(x, 0()) -> c_21() , ge^#(O(x), O(y)) -> c_22(ge^#(x, y)) , ge^#(O(x), I(y)) -> c_23(not^#(ge(y, x))) , ge^#(0(), O(x)) -> c_24(ge^#(0(), x)) , ge^#(0(), I(x)) -> c_25() , ge^#(I(x), O(y)) -> c_26(ge^#(x, y)) , ge^#(I(x), I(y)) -> c_27(ge^#(x, y)) , Log'^#(O(x)) -> c_28(if^#(ge(x, I(0())), +(Log'(x), I(0())), 0())) , Log'^#(0()) -> c_29() , Log'^#(I(x)) -> c_30(+^#(Log'(x), I(0()))) , Log^#(x) -> c_31(-^#(Log'(x), I(0()))) , Val^#(L(x)) -> c_32(x) , Val^#(N(x, l(), r())) -> c_33(x) , Min^#(L(x)) -> c_34(x) , Min^#(N(x, l(), r())) -> c_35(Min^#(l())) , Max^#(L(x)) -> c_36(x) , Max^#(N(x, l(), r())) -> c_37(Max^#(r())) , BS^#(L(x)) -> c_38() , BS^#(N(x, l(), r())) -> c_39(and^#(and(ge(x, Max(l())), ge(Min(r()), x)), and(BS(l()), BS(r())))) , Size^#(L(x)) -> c_40() , Size^#(N(x, l(), r())) -> c_41(+^#(+(Size(l()), Size(r())), I(1()))) , WB^#(L(x)) -> c_42() , WB^#(N(x, l(), r())) -> c_43(and^#(if(ge(Size(l()), Size(r())), ge(I(0()), -(Size(l()), Size(r()))), ge(I(0()), -(Size(r()), Size(l())))), and(WB(l()), WB(r())))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { O^#(0()) -> c_1() , +^#(x, 0()) -> c_2(x) , +^#(x, +(y, z)) -> c_3(+^#(+(x, y), z)) , +^#(O(x), O(y)) -> c_4(O^#(+(x, y))) , +^#(O(x), I(y)) -> c_5(+^#(x, y)) , +^#(0(), x) -> c_6(x) , +^#(I(x), O(y)) -> c_7(+^#(x, y)) , +^#(I(x), I(y)) -> c_8(O^#(+(+(x, y), I(0())))) , -^#(x, 0()) -> c_9(x) , -^#(O(x), O(y)) -> c_10(O^#(-(x, y))) , -^#(O(x), I(y)) -> c_11(-^#(-(x, y), I(1()))) , -^#(0(), x) -> c_12() , -^#(I(x), O(y)) -> c_13(-^#(x, y)) , -^#(I(x), I(y)) -> c_14(O^#(-(x, y))) , not^#(true()) -> c_15() , not^#(false()) -> c_16() , and^#(x, true()) -> c_17(x) , and^#(x, false()) -> c_18() , if^#(true(), x, y) -> c_19(x) , if^#(false(), x, y) -> c_20(y) , ge^#(x, 0()) -> c_21() , ge^#(O(x), O(y)) -> c_22(ge^#(x, y)) , ge^#(O(x), I(y)) -> c_23(not^#(ge(y, x))) , ge^#(0(), O(x)) -> c_24(ge^#(0(), x)) , ge^#(0(), I(x)) -> c_25() , ge^#(I(x), O(y)) -> c_26(ge^#(x, y)) , ge^#(I(x), I(y)) -> c_27(ge^#(x, y)) , Log'^#(O(x)) -> c_28(if^#(ge(x, I(0())), +(Log'(x), I(0())), 0())) , Log'^#(0()) -> c_29() , Log'^#(I(x)) -> c_30(+^#(Log'(x), I(0()))) , Log^#(x) -> c_31(-^#(Log'(x), I(0()))) , Val^#(L(x)) -> c_32(x) , Val^#(N(x, l(), r())) -> c_33(x) , Min^#(L(x)) -> c_34(x) , Min^#(N(x, l(), r())) -> c_35(Min^#(l())) , Max^#(L(x)) -> c_36(x) , Max^#(N(x, l(), r())) -> c_37(Max^#(r())) , BS^#(L(x)) -> c_38() , BS^#(N(x, l(), r())) -> c_39(and^#(and(ge(x, Max(l())), ge(Min(r()), x)), and(BS(l()), BS(r())))) , Size^#(L(x)) -> c_40() , Size^#(N(x, l(), r())) -> c_41(+^#(+(Size(l()), Size(r())), I(1()))) , WB^#(L(x)) -> c_42() , WB^#(N(x, l(), r())) -> c_43(and^#(if(ge(Size(l()), Size(r())), ge(I(0()), -(Size(l()), Size(r()))), ge(I(0()), -(Size(r()), Size(l())))), and(WB(l()), WB(r())))) } Strict Trs: { O(0()) -> 0() , +(x, 0()) -> x , +(x, +(y, z)) -> +(+(x, y), z) , +(O(x), O(y)) -> O(+(x, y)) , +(O(x), I(y)) -> I(+(x, y)) , +(0(), x) -> x , +(I(x), O(y)) -> I(+(x, y)) , +(I(x), I(y)) -> O(+(+(x, y), I(0()))) , -(x, 0()) -> x , -(O(x), O(y)) -> O(-(x, y)) , -(O(x), I(y)) -> I(-(-(x, y), I(1()))) , -(0(), x) -> 0() , -(I(x), O(y)) -> I(-(x, y)) , -(I(x), I(y)) -> O(-(x, y)) , not(true()) -> false() , not(false()) -> true() , and(x, true()) -> x , and(x, false()) -> false() , if(true(), x, y) -> x , if(false(), x, y) -> y , ge(x, 0()) -> true() , ge(O(x), O(y)) -> ge(x, y) , ge(O(x), I(y)) -> not(ge(y, x)) , ge(0(), O(x)) -> ge(0(), x) , ge(0(), I(x)) -> false() , ge(I(x), O(y)) -> ge(x, y) , ge(I(x), I(y)) -> ge(x, y) , Log'(O(x)) -> if(ge(x, I(0())), +(Log'(x), I(0())), 0()) , Log'(0()) -> 0() , Log'(I(x)) -> +(Log'(x), I(0())) , Log(x) -> -(Log'(x), I(0())) , Val(L(x)) -> x , Val(N(x, l(), r())) -> x , Min(L(x)) -> x , Min(N(x, l(), r())) -> Min(l()) , Max(L(x)) -> x , Max(N(x, l(), r())) -> Max(r()) , BS(L(x)) -> true() , BS(N(x, l(), r())) -> and(and(ge(x, Max(l())), ge(Min(r()), x)), and(BS(l()), BS(r()))) , Size(L(x)) -> I(0()) , Size(N(x, l(), r())) -> +(+(Size(l()), Size(r())), I(1())) , WB(L(x)) -> true() , WB(N(x, l(), r())) -> and(if(ge(Size(l()), Size(r())), ge(I(0()), -(Size(l()), Size(r()))), ge(I(0()), -(Size(r()), Size(l())))), and(WB(l()), WB(r()))) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,12,15,16,18,21,25,29,35,37,38,39,40,41,42,43} by applications of Pre({1,12,15,16,18,21,25,29,35,37,38,39,40,41,42,43}) = {2,4,6,8,9,10,11,13,14,17,19,20,22,23,24,26,27,31,32,33,34,36}. Here rules are labeled as follows: DPs: { 1: O^#(0()) -> c_1() , 2: +^#(x, 0()) -> c_2(x) , 3: +^#(x, +(y, z)) -> c_3(+^#(+(x, y), z)) , 4: +^#(O(x), O(y)) -> c_4(O^#(+(x, y))) , 5: +^#(O(x), I(y)) -> c_5(+^#(x, y)) , 6: +^#(0(), x) -> c_6(x) , 7: +^#(I(x), O(y)) -> c_7(+^#(x, y)) , 8: +^#(I(x), I(y)) -> c_8(O^#(+(+(x, y), I(0())))) , 9: -^#(x, 0()) -> c_9(x) , 10: -^#(O(x), O(y)) -> c_10(O^#(-(x, y))) , 11: -^#(O(x), I(y)) -> c_11(-^#(-(x, y), I(1()))) , 12: -^#(0(), x) -> c_12() , 13: -^#(I(x), O(y)) -> c_13(-^#(x, y)) , 14: -^#(I(x), I(y)) -> c_14(O^#(-(x, y))) , 15: not^#(true()) -> c_15() , 16: not^#(false()) -> c_16() , 17: and^#(x, true()) -> c_17(x) , 18: and^#(x, false()) -> c_18() , 19: if^#(true(), x, y) -> c_19(x) , 20: if^#(false(), x, y) -> c_20(y) , 21: ge^#(x, 0()) -> c_21() , 22: ge^#(O(x), O(y)) -> c_22(ge^#(x, y)) , 23: ge^#(O(x), I(y)) -> c_23(not^#(ge(y, x))) , 24: ge^#(0(), O(x)) -> c_24(ge^#(0(), x)) , 25: ge^#(0(), I(x)) -> c_25() , 26: ge^#(I(x), O(y)) -> c_26(ge^#(x, y)) , 27: ge^#(I(x), I(y)) -> c_27(ge^#(x, y)) , 28: Log'^#(O(x)) -> c_28(if^#(ge(x, I(0())), +(Log'(x), I(0())), 0())) , 29: Log'^#(0()) -> c_29() , 30: Log'^#(I(x)) -> c_30(+^#(Log'(x), I(0()))) , 31: Log^#(x) -> c_31(-^#(Log'(x), I(0()))) , 32: Val^#(L(x)) -> c_32(x) , 33: Val^#(N(x, l(), r())) -> c_33(x) , 34: Min^#(L(x)) -> c_34(x) , 35: Min^#(N(x, l(), r())) -> c_35(Min^#(l())) , 36: Max^#(L(x)) -> c_36(x) , 37: Max^#(N(x, l(), r())) -> c_37(Max^#(r())) , 38: BS^#(L(x)) -> c_38() , 39: BS^#(N(x, l(), r())) -> c_39(and^#(and(ge(x, Max(l())), ge(Min(r()), x)), and(BS(l()), BS(r())))) , 40: Size^#(L(x)) -> c_40() , 41: Size^#(N(x, l(), r())) -> c_41(+^#(+(Size(l()), Size(r())), I(1()))) , 42: WB^#(L(x)) -> c_42() , 43: WB^#(N(x, l(), r())) -> c_43(and^#(if(ge(Size(l()), Size(r())), ge(I(0()), -(Size(l()), Size(r()))), ge(I(0()), -(Size(r()), Size(l())))), and(WB(l()), WB(r())))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { +^#(x, 0()) -> c_2(x) , +^#(x, +(y, z)) -> c_3(+^#(+(x, y), z)) , +^#(O(x), O(y)) -> c_4(O^#(+(x, y))) , +^#(O(x), I(y)) -> c_5(+^#(x, y)) , +^#(0(), x) -> c_6(x) , +^#(I(x), O(y)) -> c_7(+^#(x, y)) , +^#(I(x), I(y)) -> c_8(O^#(+(+(x, y), I(0())))) , -^#(x, 0()) -> c_9(x) , -^#(O(x), O(y)) -> c_10(O^#(-(x, y))) , -^#(O(x), I(y)) -> c_11(-^#(-(x, y), I(1()))) , -^#(I(x), O(y)) -> c_13(-^#(x, y)) , -^#(I(x), I(y)) -> c_14(O^#(-(x, y))) , and^#(x, true()) -> c_17(x) , if^#(true(), x, y) -> c_19(x) , if^#(false(), x, y) -> c_20(y) , ge^#(O(x), O(y)) -> c_22(ge^#(x, y)) , ge^#(O(x), I(y)) -> c_23(not^#(ge(y, x))) , ge^#(0(), O(x)) -> c_24(ge^#(0(), x)) , ge^#(I(x), O(y)) -> c_26(ge^#(x, y)) , ge^#(I(x), I(y)) -> c_27(ge^#(x, y)) , Log'^#(O(x)) -> c_28(if^#(ge(x, I(0())), +(Log'(x), I(0())), 0())) , Log'^#(I(x)) -> c_30(+^#(Log'(x), I(0()))) , Log^#(x) -> c_31(-^#(Log'(x), I(0()))) , Val^#(L(x)) -> c_32(x) , Val^#(N(x, l(), r())) -> c_33(x) , Min^#(L(x)) -> c_34(x) , Max^#(L(x)) -> c_36(x) } Strict Trs: { O(0()) -> 0() , +(x, 0()) -> x , +(x, +(y, z)) -> +(+(x, y), z) , +(O(x), O(y)) -> O(+(x, y)) , +(O(x), I(y)) -> I(+(x, y)) , +(0(), x) -> x , +(I(x), O(y)) -> I(+(x, y)) , +(I(x), I(y)) -> O(+(+(x, y), I(0()))) , -(x, 0()) -> x , -(O(x), O(y)) -> O(-(x, y)) , -(O(x), I(y)) -> I(-(-(x, y), I(1()))) , -(0(), x) -> 0() , -(I(x), O(y)) -> I(-(x, y)) , -(I(x), I(y)) -> O(-(x, y)) , not(true()) -> false() , not(false()) -> true() , and(x, true()) -> x , and(x, false()) -> false() , if(true(), x, y) -> x , if(false(), x, y) -> y , ge(x, 0()) -> true() , ge(O(x), O(y)) -> ge(x, y) , ge(O(x), I(y)) -> not(ge(y, x)) , ge(0(), O(x)) -> ge(0(), x) , ge(0(), I(x)) -> false() , ge(I(x), O(y)) -> ge(x, y) , ge(I(x), I(y)) -> ge(x, y) , Log'(O(x)) -> if(ge(x, I(0())), +(Log'(x), I(0())), 0()) , Log'(0()) -> 0() , Log'(I(x)) -> +(Log'(x), I(0())) , Log(x) -> -(Log'(x), I(0())) , Val(L(x)) -> x , Val(N(x, l(), r())) -> x , Min(L(x)) -> x , Min(N(x, l(), r())) -> Min(l()) , Max(L(x)) -> x , Max(N(x, l(), r())) -> Max(r()) , BS(L(x)) -> true() , BS(N(x, l(), r())) -> and(and(ge(x, Max(l())), ge(Min(r()), x)), and(BS(l()), BS(r()))) , Size(L(x)) -> I(0()) , Size(N(x, l(), r())) -> +(+(Size(l()), Size(r())), I(1())) , WB(L(x)) -> true() , WB(N(x, l(), r())) -> and(if(ge(Size(l()), Size(r())), ge(I(0()), -(Size(l()), Size(r()))), ge(I(0()), -(Size(r()), Size(l())))), and(WB(l()), WB(r()))) } Weak DPs: { O^#(0()) -> c_1() , -^#(0(), x) -> c_12() , not^#(true()) -> c_15() , not^#(false()) -> c_16() , and^#(x, false()) -> c_18() , ge^#(x, 0()) -> c_21() , ge^#(0(), I(x)) -> c_25() , Log'^#(0()) -> c_29() , Min^#(N(x, l(), r())) -> c_35(Min^#(l())) , Max^#(N(x, l(), r())) -> c_37(Max^#(r())) , BS^#(L(x)) -> c_38() , BS^#(N(x, l(), r())) -> c_39(and^#(and(ge(x, Max(l())), ge(Min(r()), x)), and(BS(l()), BS(r())))) , Size^#(L(x)) -> c_40() , Size^#(N(x, l(), r())) -> c_41(+^#(+(Size(l()), Size(r())), I(1()))) , WB^#(L(x)) -> c_42() , WB^#(N(x, l(), r())) -> c_43(and^#(if(ge(Size(l()), Size(r())), ge(I(0()), -(Size(l()), Size(r()))), ge(I(0()), -(Size(r()), Size(l())))), and(WB(l()), WB(r())))) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {3,7,9,12,17} by applications of Pre({3,7,9,12,17}) = {1,2,4,5,6,8,10,11,13,14,15,16,19,20,22,23,24,25,26,27}. Here rules are labeled as follows: DPs: { 1: +^#(x, 0()) -> c_2(x) , 2: +^#(x, +(y, z)) -> c_3(+^#(+(x, y), z)) , 3: +^#(O(x), O(y)) -> c_4(O^#(+(x, y))) , 4: +^#(O(x), I(y)) -> c_5(+^#(x, y)) , 5: +^#(0(), x) -> c_6(x) , 6: +^#(I(x), O(y)) -> c_7(+^#(x, y)) , 7: +^#(I(x), I(y)) -> c_8(O^#(+(+(x, y), I(0())))) , 8: -^#(x, 0()) -> c_9(x) , 9: -^#(O(x), O(y)) -> c_10(O^#(-(x, y))) , 10: -^#(O(x), I(y)) -> c_11(-^#(-(x, y), I(1()))) , 11: -^#(I(x), O(y)) -> c_13(-^#(x, y)) , 12: -^#(I(x), I(y)) -> c_14(O^#(-(x, y))) , 13: and^#(x, true()) -> c_17(x) , 14: if^#(true(), x, y) -> c_19(x) , 15: if^#(false(), x, y) -> c_20(y) , 16: ge^#(O(x), O(y)) -> c_22(ge^#(x, y)) , 17: ge^#(O(x), I(y)) -> c_23(not^#(ge(y, x))) , 18: ge^#(0(), O(x)) -> c_24(ge^#(0(), x)) , 19: ge^#(I(x), O(y)) -> c_26(ge^#(x, y)) , 20: ge^#(I(x), I(y)) -> c_27(ge^#(x, y)) , 21: Log'^#(O(x)) -> c_28(if^#(ge(x, I(0())), +(Log'(x), I(0())), 0())) , 22: Log'^#(I(x)) -> c_30(+^#(Log'(x), I(0()))) , 23: Log^#(x) -> c_31(-^#(Log'(x), I(0()))) , 24: Val^#(L(x)) -> c_32(x) , 25: Val^#(N(x, l(), r())) -> c_33(x) , 26: Min^#(L(x)) -> c_34(x) , 27: Max^#(L(x)) -> c_36(x) , 28: O^#(0()) -> c_1() , 29: -^#(0(), x) -> c_12() , 30: not^#(true()) -> c_15() , 31: not^#(false()) -> c_16() , 32: and^#(x, false()) -> c_18() , 33: ge^#(x, 0()) -> c_21() , 34: ge^#(0(), I(x)) -> c_25() , 35: Log'^#(0()) -> c_29() , 36: Min^#(N(x, l(), r())) -> c_35(Min^#(l())) , 37: Max^#(N(x, l(), r())) -> c_37(Max^#(r())) , 38: BS^#(L(x)) -> c_38() , 39: BS^#(N(x, l(), r())) -> c_39(and^#(and(ge(x, Max(l())), ge(Min(r()), x)), and(BS(l()), BS(r())))) , 40: Size^#(L(x)) -> c_40() , 41: Size^#(N(x, l(), r())) -> c_41(+^#(+(Size(l()), Size(r())), I(1()))) , 42: WB^#(L(x)) -> c_42() , 43: WB^#(N(x, l(), r())) -> c_43(and^#(if(ge(Size(l()), Size(r())), ge(I(0()), -(Size(l()), Size(r()))), ge(I(0()), -(Size(r()), Size(l())))), and(WB(l()), WB(r())))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { +^#(x, 0()) -> c_2(x) , +^#(x, +(y, z)) -> c_3(+^#(+(x, y), z)) , +^#(O(x), I(y)) -> c_5(+^#(x, y)) , +^#(0(), x) -> c_6(x) , +^#(I(x), O(y)) -> c_7(+^#(x, y)) , -^#(x, 0()) -> c_9(x) , -^#(O(x), I(y)) -> c_11(-^#(-(x, y), I(1()))) , -^#(I(x), O(y)) -> c_13(-^#(x, y)) , and^#(x, true()) -> c_17(x) , if^#(true(), x, y) -> c_19(x) , if^#(false(), x, y) -> c_20(y) , ge^#(O(x), O(y)) -> c_22(ge^#(x, y)) , ge^#(0(), O(x)) -> c_24(ge^#(0(), x)) , ge^#(I(x), O(y)) -> c_26(ge^#(x, y)) , ge^#(I(x), I(y)) -> c_27(ge^#(x, y)) , Log'^#(O(x)) -> c_28(if^#(ge(x, I(0())), +(Log'(x), I(0())), 0())) , Log'^#(I(x)) -> c_30(+^#(Log'(x), I(0()))) , Log^#(x) -> c_31(-^#(Log'(x), I(0()))) , Val^#(L(x)) -> c_32(x) , Val^#(N(x, l(), r())) -> c_33(x) , Min^#(L(x)) -> c_34(x) , Max^#(L(x)) -> c_36(x) } Strict Trs: { O(0()) -> 0() , +(x, 0()) -> x , +(x, +(y, z)) -> +(+(x, y), z) , +(O(x), O(y)) -> O(+(x, y)) , +(O(x), I(y)) -> I(+(x, y)) , +(0(), x) -> x , +(I(x), O(y)) -> I(+(x, y)) , +(I(x), I(y)) -> O(+(+(x, y), I(0()))) , -(x, 0()) -> x , -(O(x), O(y)) -> O(-(x, y)) , -(O(x), I(y)) -> I(-(-(x, y), I(1()))) , -(0(), x) -> 0() , -(I(x), O(y)) -> I(-(x, y)) , -(I(x), I(y)) -> O(-(x, y)) , not(true()) -> false() , not(false()) -> true() , and(x, true()) -> x , and(x, false()) -> false() , if(true(), x, y) -> x , if(false(), x, y) -> y , ge(x, 0()) -> true() , ge(O(x), O(y)) -> ge(x, y) , ge(O(x), I(y)) -> not(ge(y, x)) , ge(0(), O(x)) -> ge(0(), x) , ge(0(), I(x)) -> false() , ge(I(x), O(y)) -> ge(x, y) , ge(I(x), I(y)) -> ge(x, y) , Log'(O(x)) -> if(ge(x, I(0())), +(Log'(x), I(0())), 0()) , Log'(0()) -> 0() , Log'(I(x)) -> +(Log'(x), I(0())) , Log(x) -> -(Log'(x), I(0())) , Val(L(x)) -> x , Val(N(x, l(), r())) -> x , Min(L(x)) -> x , Min(N(x, l(), r())) -> Min(l()) , Max(L(x)) -> x , Max(N(x, l(), r())) -> Max(r()) , BS(L(x)) -> true() , BS(N(x, l(), r())) -> and(and(ge(x, Max(l())), ge(Min(r()), x)), and(BS(l()), BS(r()))) , Size(L(x)) -> I(0()) , Size(N(x, l(), r())) -> +(+(Size(l()), Size(r())), I(1())) , WB(L(x)) -> true() , WB(N(x, l(), r())) -> and(if(ge(Size(l()), Size(r())), ge(I(0()), -(Size(l()), Size(r()))), ge(I(0()), -(Size(r()), Size(l())))), and(WB(l()), WB(r()))) } Weak DPs: { O^#(0()) -> c_1() , +^#(O(x), O(y)) -> c_4(O^#(+(x, y))) , +^#(I(x), I(y)) -> c_8(O^#(+(+(x, y), I(0())))) , -^#(O(x), O(y)) -> c_10(O^#(-(x, y))) , -^#(0(), x) -> c_12() , -^#(I(x), I(y)) -> c_14(O^#(-(x, y))) , not^#(true()) -> c_15() , not^#(false()) -> c_16() , and^#(x, false()) -> c_18() , ge^#(x, 0()) -> c_21() , ge^#(O(x), I(y)) -> c_23(not^#(ge(y, x))) , ge^#(0(), I(x)) -> c_25() , Log'^#(0()) -> c_29() , Min^#(N(x, l(), r())) -> c_35(Min^#(l())) , Max^#(N(x, l(), r())) -> c_37(Max^#(r())) , BS^#(L(x)) -> c_38() , BS^#(N(x, l(), r())) -> c_39(and^#(and(ge(x, Max(l())), ge(Min(r()), x)), and(BS(l()), BS(r())))) , Size^#(L(x)) -> c_40() , Size^#(N(x, l(), r())) -> c_41(+^#(+(Size(l()), Size(r())), I(1()))) , WB^#(L(x)) -> c_42() , WB^#(N(x, l(), r())) -> c_43(and^#(if(ge(Size(l()), Size(r())), ge(I(0()), -(Size(l()), Size(r()))), ge(I(0()), -(Size(r()), Size(l())))), and(WB(l()), WB(r())))) } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..