MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: double(0()) -> 0() double(s(x)) -> s(s(double(x))) greater(x,0()) -> first() greater(0(),s(y)) -> second() greater(s(x),s(y)) -> greater(x,y) if(false(),x,y,z) -> true() if(first(),x,y,z) -> if(le(s(x),y,s(z)),s(x),y,s(z)) if(second(),x,y,z) -> if(le(s(x),s(y),z),s(x),s(y),z) le(0(),y,z) -> greater(y,z) le(s(x),0(),z) -> false() le(s(x),s(y),0()) -> false() le(s(x),s(y),s(z)) -> le(x,y,z) triple(x) -> if(le(x,x,double(x)),x,0(),0()) - Signature: {double/1,greater/2,if/4,le/3,triple/1} / {0/0,false/0,first/0,s/1,second/0,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {double,greater,if,le,triple} and constructors {0,false ,first,s,second,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs double#(0()) -> c_1() double#(s(x)) -> c_2(double#(x)) greater#(x,0()) -> c_3() greater#(0(),s(y)) -> c_4() greater#(s(x),s(y)) -> c_5(greater#(x,y)) if#(false(),x,y,z) -> c_6() if#(first(),x,y,z) -> c_7(if#(le(s(x),y,s(z)),s(x),y,s(z)),le#(s(x),y,s(z))) if#(second(),x,y,z) -> c_8(if#(le(s(x),s(y),z),s(x),s(y),z),le#(s(x),s(y),z)) le#(0(),y,z) -> c_9(greater#(y,z)) le#(s(x),0(),z) -> c_10() le#(s(x),s(y),0()) -> c_11() le#(s(x),s(y),s(z)) -> c_12(le#(x,y,z)) triple#(x) -> c_13(if#(le(x,x,double(x)),x,0(),0()),le#(x,x,double(x)),double#(x)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: double#(0()) -> c_1() double#(s(x)) -> c_2(double#(x)) greater#(x,0()) -> c_3() greater#(0(),s(y)) -> c_4() greater#(s(x),s(y)) -> c_5(greater#(x,y)) if#(false(),x,y,z) -> c_6() if#(first(),x,y,z) -> c_7(if#(le(s(x),y,s(z)),s(x),y,s(z)),le#(s(x),y,s(z))) if#(second(),x,y,z) -> c_8(if#(le(s(x),s(y),z),s(x),s(y),z),le#(s(x),s(y),z)) le#(0(),y,z) -> c_9(greater#(y,z)) le#(s(x),0(),z) -> c_10() le#(s(x),s(y),0()) -> c_11() le#(s(x),s(y),s(z)) -> c_12(le#(x,y,z)) triple#(x) -> c_13(if#(le(x,x,double(x)),x,0(),0()),le#(x,x,double(x)),double#(x)) - Weak TRS: double(0()) -> 0() double(s(x)) -> s(s(double(x))) greater(x,0()) -> first() greater(0(),s(y)) -> second() greater(s(x),s(y)) -> greater(x,y) if(false(),x,y,z) -> true() if(first(),x,y,z) -> if(le(s(x),y,s(z)),s(x),y,s(z)) if(second(),x,y,z) -> if(le(s(x),s(y),z),s(x),s(y),z) le(0(),y,z) -> greater(y,z) le(s(x),0(),z) -> false() le(s(x),s(y),0()) -> false() le(s(x),s(y),s(z)) -> le(x,y,z) triple(x) -> if(le(x,x,double(x)),x,0(),0()) - Signature: {double/1,greater/2,if/4,le/3,triple/1,double#/1,greater#/2,if#/4,le#/3,triple#/1} / {0/0,false/0,first/0 ,s/1,second/0,true/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/0,c_11/0,c_12/1,c_13/3} - Obligation: innermost runtime complexity wrt. defined symbols {double#,greater#,if#,le#,triple#} and constructors {0 ,false,first,s,second,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: double(0()) -> 0() double(s(x)) -> s(s(double(x))) greater(x,0()) -> first() greater(0(),s(y)) -> second() greater(s(x),s(y)) -> greater(x,y) le(0(),y,z) -> greater(y,z) le(s(x),0(),z) -> false() le(s(x),s(y),0()) -> false() le(s(x),s(y),s(z)) -> le(x,y,z) double#(0()) -> c_1() double#(s(x)) -> c_2(double#(x)) greater#(x,0()) -> c_3() greater#(0(),s(y)) -> c_4() greater#(s(x),s(y)) -> c_5(greater#(x,y)) if#(false(),x,y,z) -> c_6() if#(first(),x,y,z) -> c_7(if#(le(s(x),y,s(z)),s(x),y,s(z)),le#(s(x),y,s(z))) if#(second(),x,y,z) -> c_8(if#(le(s(x),s(y),z),s(x),s(y),z),le#(s(x),s(y),z)) le#(0(),y,z) -> c_9(greater#(y,z)) le#(s(x),0(),z) -> c_10() le#(s(x),s(y),0()) -> c_11() le#(s(x),s(y),s(z)) -> c_12(le#(x,y,z)) triple#(x) -> c_13(if#(le(x,x,double(x)),x,0(),0()),le#(x,x,double(x)),double#(x)) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: double#(0()) -> c_1() double#(s(x)) -> c_2(double#(x)) greater#(x,0()) -> c_3() greater#(0(),s(y)) -> c_4() greater#(s(x),s(y)) -> c_5(greater#(x,y)) if#(false(),x,y,z) -> c_6() if#(first(),x,y,z) -> c_7(if#(le(s(x),y,s(z)),s(x),y,s(z)),le#(s(x),y,s(z))) if#(second(),x,y,z) -> c_8(if#(le(s(x),s(y),z),s(x),s(y),z),le#(s(x),s(y),z)) le#(0(),y,z) -> c_9(greater#(y,z)) le#(s(x),0(),z) -> c_10() le#(s(x),s(y),0()) -> c_11() le#(s(x),s(y),s(z)) -> c_12(le#(x,y,z)) triple#(x) -> c_13(if#(le(x,x,double(x)),x,0(),0()),le#(x,x,double(x)),double#(x)) - Weak TRS: double(0()) -> 0() double(s(x)) -> s(s(double(x))) greater(x,0()) -> first() greater(0(),s(y)) -> second() greater(s(x),s(y)) -> greater(x,y) le(0(),y,z) -> greater(y,z) le(s(x),0(),z) -> false() le(s(x),s(y),0()) -> false() le(s(x),s(y),s(z)) -> le(x,y,z) - Signature: {double/1,greater/2,if/4,le/3,triple/1,double#/1,greater#/2,if#/4,le#/3,triple#/1} / {0/0,false/0,first/0 ,s/1,second/0,true/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/0,c_11/0,c_12/1,c_13/3} - Obligation: innermost runtime complexity wrt. defined symbols {double#,greater#,if#,le#,triple#} and constructors {0 ,false,first,s,second,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,4,6,10,11} by application of Pre({1,3,4,6,10,11}) = {2,5,7,8,9,12,13}. Here rules are labelled as follows: 1: double#(0()) -> c_1() 2: double#(s(x)) -> c_2(double#(x)) 3: greater#(x,0()) -> c_3() 4: greater#(0(),s(y)) -> c_4() 5: greater#(s(x),s(y)) -> c_5(greater#(x,y)) 6: if#(false(),x,y,z) -> c_6() 7: if#(first(),x,y,z) -> c_7(if#(le(s(x),y,s(z)),s(x),y,s(z)),le#(s(x),y,s(z))) 8: if#(second(),x,y,z) -> c_8(if#(le(s(x),s(y),z),s(x),s(y),z),le#(s(x),s(y),z)) 9: le#(0(),y,z) -> c_9(greater#(y,z)) 10: le#(s(x),0(),z) -> c_10() 11: le#(s(x),s(y),0()) -> c_11() 12: le#(s(x),s(y),s(z)) -> c_12(le#(x,y,z)) 13: triple#(x) -> c_13(if#(le(x,x,double(x)),x,0(),0()),le#(x,x,double(x)),double#(x)) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: double#(s(x)) -> c_2(double#(x)) greater#(s(x),s(y)) -> c_5(greater#(x,y)) if#(first(),x,y,z) -> c_7(if#(le(s(x),y,s(z)),s(x),y,s(z)),le#(s(x),y,s(z))) if#(second(),x,y,z) -> c_8(if#(le(s(x),s(y),z),s(x),s(y),z),le#(s(x),s(y),z)) le#(0(),y,z) -> c_9(greater#(y,z)) le#(s(x),s(y),s(z)) -> c_12(le#(x,y,z)) triple#(x) -> c_13(if#(le(x,x,double(x)),x,0(),0()),le#(x,x,double(x)),double#(x)) - Weak DPs: double#(0()) -> c_1() greater#(x,0()) -> c_3() greater#(0(),s(y)) -> c_4() if#(false(),x,y,z) -> c_6() le#(s(x),0(),z) -> c_10() le#(s(x),s(y),0()) -> c_11() - Weak TRS: double(0()) -> 0() double(s(x)) -> s(s(double(x))) greater(x,0()) -> first() greater(0(),s(y)) -> second() greater(s(x),s(y)) -> greater(x,y) le(0(),y,z) -> greater(y,z) le(s(x),0(),z) -> false() le(s(x),s(y),0()) -> false() le(s(x),s(y),s(z)) -> le(x,y,z) - Signature: {double/1,greater/2,if/4,le/3,triple/1,double#/1,greater#/2,if#/4,le#/3,triple#/1} / {0/0,false/0,first/0 ,s/1,second/0,true/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/0,c_11/0,c_12/1,c_13/3} - Obligation: innermost runtime complexity wrt. defined symbols {double#,greater#,if#,le#,triple#} and constructors {0 ,false,first,s,second,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:double#(s(x)) -> c_2(double#(x)) -->_1 double#(0()) -> c_1():8 -->_1 double#(s(x)) -> c_2(double#(x)):1 2:S:greater#(s(x),s(y)) -> c_5(greater#(x,y)) -->_1 greater#(0(),s(y)) -> c_4():10 -->_1 greater#(x,0()) -> c_3():9 -->_1 greater#(s(x),s(y)) -> c_5(greater#(x,y)):2 3:S:if#(first(),x,y,z) -> c_7(if#(le(s(x),y,s(z)),s(x),y,s(z)),le#(s(x),y,s(z))) -->_2 le#(s(x),s(y),s(z)) -> c_12(le#(x,y,z)):6 -->_1 if#(second(),x,y,z) -> c_8(if#(le(s(x),s(y),z),s(x),s(y),z),le#(s(x),s(y),z)):4 -->_2 le#(s(x),0(),z) -> c_10():12 -->_1 if#(false(),x,y,z) -> c_6():11 -->_1 if#(first(),x,y,z) -> c_7(if#(le(s(x),y,s(z)),s(x),y,s(z)),le#(s(x),y,s(z))):3 4:S:if#(second(),x,y,z) -> c_8(if#(le(s(x),s(y),z),s(x),s(y),z),le#(s(x),s(y),z)) -->_2 le#(s(x),s(y),s(z)) -> c_12(le#(x,y,z)):6 -->_2 le#(s(x),s(y),0()) -> c_11():13 -->_1 if#(false(),x,y,z) -> c_6():11 -->_1 if#(second(),x,y,z) -> c_8(if#(le(s(x),s(y),z),s(x),s(y),z),le#(s(x),s(y),z)):4 -->_1 if#(first(),x,y,z) -> c_7(if#(le(s(x),y,s(z)),s(x),y,s(z)),le#(s(x),y,s(z))):3 5:S:le#(0(),y,z) -> c_9(greater#(y,z)) -->_1 greater#(0(),s(y)) -> c_4():10 -->_1 greater#(x,0()) -> c_3():9 -->_1 greater#(s(x),s(y)) -> c_5(greater#(x,y)):2 6:S:le#(s(x),s(y),s(z)) -> c_12(le#(x,y,z)) -->_1 le#(s(x),s(y),0()) -> c_11():13 -->_1 le#(s(x),0(),z) -> c_10():12 -->_1 le#(s(x),s(y),s(z)) -> c_12(le#(x,y,z)):6 -->_1 le#(0(),y,z) -> c_9(greater#(y,z)):5 7:S:triple#(x) -> c_13(if#(le(x,x,double(x)),x,0(),0()),le#(x,x,double(x)),double#(x)) -->_2 le#(s(x),s(y),0()) -> c_11():13 -->_1 if#(false(),x,y,z) -> c_6():11 -->_3 double#(0()) -> c_1():8 -->_2 le#(s(x),s(y),s(z)) -> c_12(le#(x,y,z)):6 -->_2 le#(0(),y,z) -> c_9(greater#(y,z)):5 -->_1 if#(second(),x,y,z) -> c_8(if#(le(s(x),s(y),z),s(x),s(y),z),le#(s(x),s(y),z)):4 -->_1 if#(first(),x,y,z) -> c_7(if#(le(s(x),y,s(z)),s(x),y,s(z)),le#(s(x),y,s(z))):3 -->_3 double#(s(x)) -> c_2(double#(x)):1 8:W:double#(0()) -> c_1() 9:W:greater#(x,0()) -> c_3() 10:W:greater#(0(),s(y)) -> c_4() 11:W:if#(false(),x,y,z) -> c_6() 12:W:le#(s(x),0(),z) -> c_10() 13:W:le#(s(x),s(y),0()) -> c_11() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 11: if#(false(),x,y,z) -> c_6() 12: le#(s(x),0(),z) -> c_10() 13: le#(s(x),s(y),0()) -> c_11() 9: greater#(x,0()) -> c_3() 10: greater#(0(),s(y)) -> c_4() 8: double#(0()) -> c_1() * Step 5: NaturalMI MAYBE + Considered Problem: - Strict DPs: double#(s(x)) -> c_2(double#(x)) greater#(s(x),s(y)) -> c_5(greater#(x,y)) if#(first(),x,y,z) -> c_7(if#(le(s(x),y,s(z)),s(x),y,s(z)),le#(s(x),y,s(z))) if#(second(),x,y,z) -> c_8(if#(le(s(x),s(y),z),s(x),s(y),z),le#(s(x),s(y),z)) le#(0(),y,z) -> c_9(greater#(y,z)) le#(s(x),s(y),s(z)) -> c_12(le#(x,y,z)) triple#(x) -> c_13(if#(le(x,x,double(x)),x,0(),0()),le#(x,x,double(x)),double#(x)) - Weak TRS: double(0()) -> 0() double(s(x)) -> s(s(double(x))) greater(x,0()) -> first() greater(0(),s(y)) -> second() greater(s(x),s(y)) -> greater(x,y) le(0(),y,z) -> greater(y,z) le(s(x),0(),z) -> false() le(s(x),s(y),0()) -> false() le(s(x),s(y),s(z)) -> le(x,y,z) - Signature: {double/1,greater/2,if/4,le/3,triple/1,double#/1,greater#/2,if#/4,le#/3,triple#/1} / {0/0,false/0,first/0 ,s/1,second/0,true/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/0,c_11/0,c_12/1,c_13/3} - Obligation: innermost runtime complexity wrt. defined symbols {double#,greater#,if#,le#,triple#} and constructors {0 ,false,first,s,second,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_5) = {1}, uargs(c_7) = {1,2}, uargs(c_8) = {1,2}, uargs(c_9) = {1}, uargs(c_12) = {1}, uargs(c_13) = {1,2,3} Following symbols are considered usable: {greater,le,double#,greater#,if#,le#,triple#} TcT has computed the following interpretation: p(0) = [0] p(double) = [1] x1 + [4] p(false) = [1] p(first) = [1] p(greater) = [1] p(if) = [1] x1 + [1] x2 + [4] x3 + [0] p(le) = [1] p(s) = [0] p(second) = [1] p(triple) = [2] x1 + [0] p(true) = [4] p(double#) = [0] p(greater#) = [0] p(if#) = [4] x1 + [0] p(le#) = [0] p(triple#) = [5] p(c_1) = [0] p(c_2) = [2] x1 + [0] p(c_3) = [1] p(c_4) = [0] p(c_5) = [4] x1 + [0] p(c_6) = [2] p(c_7) = [1] x1 + [4] x2 + [0] p(c_8) = [1] x1 + [1] x2 + [0] p(c_9) = [2] x1 + [0] p(c_10) = [1] p(c_11) = [0] p(c_12) = [4] x1 + [0] p(c_13) = [1] x1 + [2] x2 + [2] x3 + [0] Following rules are strictly oriented: triple#(x) = [5] > [4] = c_13(if#(le(x,x,double(x)),x,0(),0()),le#(x,x,double(x)),double#(x)) Following rules are (at-least) weakly oriented: double#(s(x)) = [0] >= [0] = c_2(double#(x)) greater#(s(x),s(y)) = [0] >= [0] = c_5(greater#(x,y)) if#(first(),x,y,z) = [4] >= [4] = c_7(if#(le(s(x),y,s(z)),s(x),y,s(z)),le#(s(x),y,s(z))) if#(second(),x,y,z) = [4] >= [4] = c_8(if#(le(s(x),s(y),z),s(x),s(y),z),le#(s(x),s(y),z)) le#(0(),y,z) = [0] >= [0] = c_9(greater#(y,z)) le#(s(x),s(y),s(z)) = [0] >= [0] = c_12(le#(x,y,z)) greater(x,0()) = [1] >= [1] = first() greater(0(),s(y)) = [1] >= [1] = second() greater(s(x),s(y)) = [1] >= [1] = greater(x,y) le(0(),y,z) = [1] >= [1] = greater(y,z) le(s(x),0(),z) = [1] >= [1] = false() le(s(x),s(y),0()) = [1] >= [1] = false() le(s(x),s(y),s(z)) = [1] >= [1] = le(x,y,z) * Step 6: Failure MAYBE + Considered Problem: - Strict DPs: double#(s(x)) -> c_2(double#(x)) greater#(s(x),s(y)) -> c_5(greater#(x,y)) if#(first(),x,y,z) -> c_7(if#(le(s(x),y,s(z)),s(x),y,s(z)),le#(s(x),y,s(z))) if#(second(),x,y,z) -> c_8(if#(le(s(x),s(y),z),s(x),s(y),z),le#(s(x),s(y),z)) le#(0(),y,z) -> c_9(greater#(y,z)) le#(s(x),s(y),s(z)) -> c_12(le#(x,y,z)) - Weak DPs: triple#(x) -> c_13(if#(le(x,x,double(x)),x,0(),0()),le#(x,x,double(x)),double#(x)) - Weak TRS: double(0()) -> 0() double(s(x)) -> s(s(double(x))) greater(x,0()) -> first() greater(0(),s(y)) -> second() greater(s(x),s(y)) -> greater(x,y) le(0(),y,z) -> greater(y,z) le(s(x),0(),z) -> false() le(s(x),s(y),0()) -> false() le(s(x),s(y),s(z)) -> le(x,y,z) - Signature: {double/1,greater/2,if/4,le/3,triple/1,double#/1,greater#/2,if#/4,le#/3,triple#/1} / {0/0,false/0,first/0 ,s/1,second/0,true/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/2,c_9/1,c_10/0,c_11/0,c_12/1,c_13/3} - Obligation: innermost runtime complexity wrt. defined symbols {double#,greater#,if#,le#,triple#} and constructors {0 ,false,first,s,second,true} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE