MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , pred(s(x)) -> x , minus(x, 0()) -> x , minus(x, s(y)) -> pred(minus(x, y)) , gcd(0(), y) -> y , gcd(s(x), 0()) -> s(x) , gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) , if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y)) , if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x)) } 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: The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(pred) = {1}, Uargs(gcd) = {1}, Uargs(if_gcd) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [le](x1, x2) = [1] [0] = [0] [true] = [1] [s](x1) = [1] x1 + [7] [false] = [1] [pred](x1) = [1] x1 + [7] [minus](x1, x2) = [1] x1 + [0] [gcd](x1, x2) = [1] x1 + [1] x2 + [1] [if_gcd](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] The following symbols are considered usable {le, pred, minus, gcd, if_gcd} The order satisfies the following ordering constraints: [le(0(), y)] = [1] >= [1] = [true()] [le(s(x), 0())] = [1] >= [1] = [false()] [le(s(x), s(y))] = [1] >= [1] = [le(x, y)] [pred(s(x))] = [1] x + [14] > [1] x + [0] = [x] [minus(x, 0())] = [1] x + [0] >= [1] x + [0] = [x] [minus(x, s(y))] = [1] x + [0] ? [1] x + [7] = [pred(minus(x, y))] [gcd(0(), y)] = [1] y + [1] > [1] y + [0] = [y] [gcd(s(x), 0())] = [1] x + [8] > [1] x + [7] = [s(x)] [gcd(s(x), s(y))] = [1] y + [1] x + [15] >= [1] y + [1] x + [15] = [if_gcd(le(y, x), s(x), s(y))] [if_gcd(true(), s(x), s(y))] = [1] y + [1] x + [15] > [1] y + [1] x + [8] = [gcd(minus(x, y), s(y))] [if_gcd(false(), s(x), s(y))] = [1] y + [1] x + [15] > [1] y + [1] x + [8] = [gcd(minus(y, x), s(x))] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , minus(x, 0()) -> x , minus(x, s(y)) -> pred(minus(x, y)) , gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) } Weak Trs: { pred(s(x)) -> x , gcd(0(), y) -> y , gcd(s(x), 0()) -> s(x) , if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y)) , if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x)) } Obligation: runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(pred) = {1}, Uargs(gcd) = {1}, Uargs(if_gcd) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [le](x1, x2) = [1] [0] = [7] [true] = [0] [s](x1) = [1] x1 + [4] [false] = [0] [pred](x1) = [1] x1 + [7] [minus](x1, x2) = [1] x1 + [0] [gcd](x1, x2) = [1] x1 + [1] x2 + [0] [if_gcd](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] The following symbols are considered usable {le, pred, minus, gcd, if_gcd} The order satisfies the following ordering constraints: [le(0(), y)] = [1] > [0] = [true()] [le(s(x), 0())] = [1] > [0] = [false()] [le(s(x), s(y))] = [1] >= [1] = [le(x, y)] [pred(s(x))] = [1] x + [11] > [1] x + [0] = [x] [minus(x, 0())] = [1] x + [0] >= [1] x + [0] = [x] [minus(x, s(y))] = [1] x + [0] ? [1] x + [7] = [pred(minus(x, y))] [gcd(0(), y)] = [1] y + [7] > [1] y + [0] = [y] [gcd(s(x), 0())] = [1] x + [11] > [1] x + [4] = [s(x)] [gcd(s(x), s(y))] = [1] y + [1] x + [8] ? [1] y + [1] x + [9] = [if_gcd(le(y, x), s(x), s(y))] [if_gcd(true(), s(x), s(y))] = [1] y + [1] x + [8] > [1] y + [1] x + [4] = [gcd(minus(x, y), s(y))] [if_gcd(false(), s(x), s(y))] = [1] y + [1] x + [8] > [1] y + [1] x + [4] = [gcd(minus(y, x), s(x))] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { le(s(x), s(y)) -> le(x, y) , minus(x, 0()) -> x , minus(x, s(y)) -> pred(minus(x, y)) , gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) } Weak Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , pred(s(x)) -> x , gcd(0(), y) -> y , gcd(s(x), 0()) -> s(x) , if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y)) , if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x)) } Obligation: runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(pred) = {1}, Uargs(gcd) = {1}, Uargs(if_gcd) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [le](x1, x2) = [0] [0] = [7] [true] = [0] [s](x1) = [1] x1 + [4] [false] = [0] [pred](x1) = [1] x1 + [7] [minus](x1, x2) = [1] x1 + [1] [gcd](x1, x2) = [1] x1 + [1] x2 + [0] [if_gcd](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] The following symbols are considered usable {le, pred, minus, gcd, if_gcd} The order satisfies the following ordering constraints: [le(0(), y)] = [0] >= [0] = [true()] [le(s(x), 0())] = [0] >= [0] = [false()] [le(s(x), s(y))] = [0] >= [0] = [le(x, y)] [pred(s(x))] = [1] x + [11] > [1] x + [0] = [x] [minus(x, 0())] = [1] x + [1] > [1] x + [0] = [x] [minus(x, s(y))] = [1] x + [1] ? [1] x + [8] = [pred(minus(x, y))] [gcd(0(), y)] = [1] y + [7] > [1] y + [0] = [y] [gcd(s(x), 0())] = [1] x + [11] > [1] x + [4] = [s(x)] [gcd(s(x), s(y))] = [1] y + [1] x + [8] >= [1] y + [1] x + [8] = [if_gcd(le(y, x), s(x), s(y))] [if_gcd(true(), s(x), s(y))] = [1] y + [1] x + [8] > [1] y + [1] x + [5] = [gcd(minus(x, y), s(y))] [if_gcd(false(), s(x), s(y))] = [1] y + [1] x + [8] > [1] y + [1] x + [5] = [gcd(minus(y, x), s(x))] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { le(s(x), s(y)) -> le(x, y) , minus(x, s(y)) -> pred(minus(x, y)) , gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) } Weak Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , pred(s(x)) -> x , minus(x, 0()) -> x , gcd(0(), y) -> y , gcd(s(x), 0()) -> s(x) , if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y)) , if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x)) } Obligation: runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(pred) = {1}, Uargs(gcd) = {1}, Uargs(if_gcd) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [le](x1, x2) = [0] [0] = [7] [true] = [0] [s](x1) = [1] x1 + [4] [false] = [0] [pred](x1) = [1] x1 + [7] [minus](x1, x2) = [1] x1 + [0] [gcd](x1, x2) = [1] x1 + [1] x2 + [1] [if_gcd](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] The following symbols are considered usable {le, pred, minus, gcd, if_gcd} The order satisfies the following ordering constraints: [le(0(), y)] = [0] >= [0] = [true()] [le(s(x), 0())] = [0] >= [0] = [false()] [le(s(x), s(y))] = [0] >= [0] = [le(x, y)] [pred(s(x))] = [1] x + [11] > [1] x + [0] = [x] [minus(x, 0())] = [1] x + [0] >= [1] x + [0] = [x] [minus(x, s(y))] = [1] x + [0] ? [1] x + [7] = [pred(minus(x, y))] [gcd(0(), y)] = [1] y + [8] > [1] y + [0] = [y] [gcd(s(x), 0())] = [1] x + [12] > [1] x + [4] = [s(x)] [gcd(s(x), s(y))] = [1] y + [1] x + [9] > [1] y + [1] x + [8] = [if_gcd(le(y, x), s(x), s(y))] [if_gcd(true(), s(x), s(y))] = [1] y + [1] x + [8] > [1] y + [1] x + [5] = [gcd(minus(x, y), s(y))] [if_gcd(false(), s(x), s(y))] = [1] y + [1] x + [8] > [1] y + [1] x + [5] = [gcd(minus(y, x), s(x))] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { le(s(x), s(y)) -> le(x, y) , minus(x, s(y)) -> pred(minus(x, y)) } Weak Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , pred(s(x)) -> x , minus(x, 0()) -> x , gcd(0(), y) -> y , gcd(s(x), 0()) -> s(x) , gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) , if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y)) , if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x)) } Obligation: runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'Fastest' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem' failed due to the following reason: We use the processor 'polynomial interpretation' to orient following rules strictly. Trs: { le(s(x), s(y)) -> le(x, y) } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^2)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are considered usable: Uargs(pred) = {1}, Uargs(gcd) = {1}, Uargs(if_gcd) = {1} TcT has computed the following constructor-restricted polynomial interpretation. [le](x1, x2) = x1 [0]() = 0 [true]() = 0 [s](x1) = 1 + x1 [false]() = 0 [pred](x1) = x1 [minus](x1, x2) = x1 [gcd](x1, x2) = x1 + x1*x2 + 2*x2 [if_gcd](x1, x2, x3) = 1 + x1 + x2 + x2*x3 + x3 The following symbols are considered usable {le, pred, minus, gcd, if_gcd} This order satisfies the following ordering constraints. [le(0(), y)] = >= = [true()] [le(s(x), 0())] = 1 + x > = [false()] [le(s(x), s(y))] = 1 + x > x = [le(x, y)] [pred(s(x))] = 1 + x > x = [x] [minus(x, 0())] = x >= x = [x] [minus(x, s(y))] = x >= x = [pred(minus(x, y))] [gcd(0(), y)] = 2*y >= y = [y] [gcd(s(x), 0())] = 1 + x >= 1 + x = [s(x)] [gcd(s(x), s(y))] = 4 + 2*x + 3*y + x*y >= 4 + 3*y + 2*x + x*y = [if_gcd(le(y, x), s(x), s(y))] [if_gcd(true(), s(x), s(y))] = 4 + 2*x + 2*y + x*y > 2*x + x*y + 2 + 2*y = [gcd(minus(x, y), s(y))] [if_gcd(false(), s(x), s(y))] = 4 + 2*x + 2*y + x*y > 2*y + y*x + 2 + 2*x = [gcd(minus(y, x), s(x))] We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { minus(x, s(y)) -> pred(minus(x, y)) } Weak Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , pred(s(x)) -> x , minus(x, 0()) -> x , gcd(0(), y) -> y , gcd(s(x), 0()) -> s(x) , gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) , if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y)) , if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x)) } Obligation: runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: We use the processor 'matrix interpretation of dimension 2' to orient following rules strictly. Trs: { le(s(x), s(y)) -> le(x, y) } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^2)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are usable: Uargs(pred) = {1}, Uargs(gcd) = {1}, Uargs(if_gcd) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [le](x1, x2) = [0 0] x1 + [0 1] x2 + [2] [4 0] [4 0] [0] [0] = [0] [0] [true] = [0] [0] [s](x1) = [1 1] x1 + [0] [0 1] [2] [false] = [0] [0] [pred](x1) = [1 0] x1 + [0] [0 1] [0] [minus](x1, x2) = [1 0] x1 + [0] [0 2] [0] [gcd](x1, x2) = [2 1] x1 + [2 0] x2 + [0] [5 4] [5 3] [0] [if_gcd](x1, x2, x3) = [1 0] x1 + [2 0] x2 + [2 0] x3 + [0] [0 0] [5 4] [5 3] [0] The following symbols are considered usable {le, pred, minus, gcd, if_gcd} The order satisfies the following ordering constraints: [le(0(), y)] = [0 1] y + [2] [4 0] [0] > [0] [0] = [true()] [le(s(x), 0())] = [0 0] x + [2] [4 4] [0] > [0] [0] = [false()] [le(s(x), s(y))] = [0 1] y + [0 0] x + [4] [4 4] [4 4] [0] > [0 1] y + [0 0] x + [2] [4 0] [4 0] [0] = [le(x, y)] [pred(s(x))] = [1 1] x + [0] [0 1] [2] >= [1 0] x + [0] [0 1] [0] = [x] [minus(x, 0())] = [1 0] x + [0] [0 2] [0] >= [1 0] x + [0] [0 1] [0] = [x] [minus(x, s(y))] = [1 0] x + [0] [0 2] [0] >= [1 0] x + [0] [0 2] [0] = [pred(minus(x, y))] [gcd(0(), y)] = [2 0] y + [0] [5 3] [0] >= [1 0] y + [0] [0 1] [0] = [y] [gcd(s(x), 0())] = [2 3] x + [2] [5 9] [8] > [1 1] x + [0] [0 1] [2] = [s(x)] [gcd(s(x), s(y))] = [2 2] y + [2 3] x + [2] [5 8] [5 9] [14] >= [2 2] y + [2 3] x + [2] [5 8] [5 9] [14] = [if_gcd(le(y, x), s(x), s(y))] [if_gcd(true(), s(x), s(y))] = [2 2] y + [2 2] x + [0] [5 8] [5 9] [14] >= [2 2] y + [2 2] x + [0] [5 8] [5 8] [6] = [gcd(minus(x, y), s(y))] [if_gcd(false(), s(x), s(y))] = [2 2] y + [2 2] x + [0] [5 8] [5 9] [14] >= [2 2] y + [2 2] x + [0] [5 8] [5 8] [6] = [gcd(minus(y, x), s(x))] We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { minus(x, s(y)) -> pred(minus(x, y)) } Weak Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , pred(s(x)) -> x , minus(x, 0()) -> x , gcd(0(), y) -> y , gcd(s(x), 0()) -> s(x) , gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) , if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y)) , if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x)) } Obligation: runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'WithProblem' failed due to the following reason: Empty strict component of the problem is NOT empty. 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: { le^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() , le^#(s(x), s(y)) -> c_3(le^#(x, y)) , pred^#(s(x)) -> c_4(x) , minus^#(x, 0()) -> c_5(x) , minus^#(x, s(y)) -> c_6(pred^#(minus(x, y))) , gcd^#(0(), y) -> c_7(y) , gcd^#(s(x), 0()) -> c_8(x) , gcd^#(s(x), s(y)) -> c_9(if_gcd^#(le(y, x), s(x), s(y))) , if_gcd^#(true(), s(x), s(y)) -> c_10(gcd^#(minus(x, y), s(y))) , if_gcd^#(false(), s(x), s(y)) -> c_11(gcd^#(minus(y, x), s(x))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { le^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() , le^#(s(x), s(y)) -> c_3(le^#(x, y)) , pred^#(s(x)) -> c_4(x) , minus^#(x, 0()) -> c_5(x) , minus^#(x, s(y)) -> c_6(pred^#(minus(x, y))) , gcd^#(0(), y) -> c_7(y) , gcd^#(s(x), 0()) -> c_8(x) , gcd^#(s(x), s(y)) -> c_9(if_gcd^#(le(y, x), s(x), s(y))) , if_gcd^#(true(), s(x), s(y)) -> c_10(gcd^#(minus(x, y), s(y))) , if_gcd^#(false(), s(x), s(y)) -> c_11(gcd^#(minus(y, x), s(x))) } Strict Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , pred(s(x)) -> x , minus(x, 0()) -> x , minus(x, s(y)) -> pred(minus(x, y)) , gcd(0(), y) -> y , gcd(s(x), 0()) -> s(x) , gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) , if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y)) , if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x)) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,2} by applications of Pre({1,2}) = {3,4,5,7,8}. Here rules are labeled as follows: DPs: { 1: le^#(0(), y) -> c_1() , 2: le^#(s(x), 0()) -> c_2() , 3: le^#(s(x), s(y)) -> c_3(le^#(x, y)) , 4: pred^#(s(x)) -> c_4(x) , 5: minus^#(x, 0()) -> c_5(x) , 6: minus^#(x, s(y)) -> c_6(pred^#(minus(x, y))) , 7: gcd^#(0(), y) -> c_7(y) , 8: gcd^#(s(x), 0()) -> c_8(x) , 9: gcd^#(s(x), s(y)) -> c_9(if_gcd^#(le(y, x), s(x), s(y))) , 10: if_gcd^#(true(), s(x), s(y)) -> c_10(gcd^#(minus(x, y), s(y))) , 11: if_gcd^#(false(), s(x), s(y)) -> c_11(gcd^#(minus(y, x), s(x))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { le^#(s(x), s(y)) -> c_3(le^#(x, y)) , pred^#(s(x)) -> c_4(x) , minus^#(x, 0()) -> c_5(x) , minus^#(x, s(y)) -> c_6(pred^#(minus(x, y))) , gcd^#(0(), y) -> c_7(y) , gcd^#(s(x), 0()) -> c_8(x) , gcd^#(s(x), s(y)) -> c_9(if_gcd^#(le(y, x), s(x), s(y))) , if_gcd^#(true(), s(x), s(y)) -> c_10(gcd^#(minus(x, y), s(y))) , if_gcd^#(false(), s(x), s(y)) -> c_11(gcd^#(minus(y, x), s(x))) } Strict Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , pred(s(x)) -> x , minus(x, 0()) -> x , minus(x, s(y)) -> pred(minus(x, y)) , gcd(0(), y) -> y , gcd(s(x), 0()) -> s(x) , gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) , if_gcd(true(), s(x), s(y)) -> gcd(minus(x, y), s(y)) , if_gcd(false(), s(x), s(y)) -> gcd(minus(y, x), s(x)) } Weak DPs: { le^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..