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)) , mod(0(), y) -> 0() , mod(s(x), 0()) -> 0() , mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) , if_mod(true(), s(x), s(y)) -> mod(minus(x, y), s(y)) , if_mod(false(), s(x), s(y)) -> 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(mod) = {1}, Uargs(if_mod) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [le](x1, x2) = [0] [0] = [0] [true] = [0] [s](x1) = [1] x1 + [0] [false] = [0] [pred](x1) = [1] x1 + [0] [minus](x1, x2) = [1] x1 + [1] [mod](x1, x2) = [1] x1 + [0] [if_mod](x1, x2, x3) = [1] x1 + [1] x2 + [0] The following symbols are considered usable {le, pred, minus, mod, if_mod} 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 + [0] >= [1] x + [0] = [x] [minus(x, 0())] = [1] x + [1] > [1] x + [0] = [x] [minus(x, s(y))] = [1] x + [1] >= [1] x + [1] = [pred(minus(x, y))] [mod(0(), y)] = [0] >= [0] = [0()] [mod(s(x), 0())] = [1] x + [0] >= [0] = [0()] [mod(s(x), s(y))] = [1] x + [0] >= [1] x + [0] = [if_mod(le(y, x), s(x), s(y))] [if_mod(true(), s(x), s(y))] = [1] x + [0] ? [1] x + [1] = [mod(minus(x, y), s(y))] [if_mod(false(), s(x), s(y))] = [1] x + [0] >= [1] x + [0] = [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) , pred(s(x)) -> x , minus(x, s(y)) -> pred(minus(x, y)) , mod(0(), y) -> 0() , mod(s(x), 0()) -> 0() , mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) , if_mod(true(), s(x), s(y)) -> mod(minus(x, y), s(y)) , if_mod(false(), s(x), s(y)) -> s(x) } Weak Trs: { minus(x, 0()) -> 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(mod) = {1}, Uargs(if_mod) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [le](x1, x2) = [1] [0] = [0] [true] = [0] [s](x1) = [1] x1 + [0] [false] = [0] [pred](x1) = [1] x1 + [0] [minus](x1, x2) = [1] x1 + [0] [mod](x1, x2) = [1] x1 + [0] [if_mod](x1, x2, x3) = [1] x1 + [1] x2 + [0] The following symbols are considered usable {le, pred, minus, mod, if_mod} 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 + [0] >= [1] x + [0] = [x] [minus(x, 0())] = [1] x + [0] >= [1] x + [0] = [x] [minus(x, s(y))] = [1] x + [0] >= [1] x + [0] = [pred(minus(x, y))] [mod(0(), y)] = [0] >= [0] = [0()] [mod(s(x), 0())] = [1] x + [0] >= [0] = [0()] [mod(s(x), s(y))] = [1] x + [0] ? [1] x + [1] = [if_mod(le(y, x), s(x), s(y))] [if_mod(true(), s(x), s(y))] = [1] x + [0] >= [1] x + [0] = [mod(minus(x, y), s(y))] [if_mod(false(), s(x), s(y))] = [1] x + [0] >= [1] x + [0] = [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) , pred(s(x)) -> x , minus(x, s(y)) -> pred(minus(x, y)) , mod(0(), y) -> 0() , mod(s(x), 0()) -> 0() , mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) , if_mod(true(), s(x), s(y)) -> mod(minus(x, y), s(y)) , if_mod(false(), s(x), s(y)) -> s(x) } Weak Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , minus(x, 0()) -> 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(mod) = {1}, Uargs(if_mod) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [le](x1, x2) = [4] [0] = [0] [true] = [1] [s](x1) = [1] x1 + [0] [false] = [0] [pred](x1) = [1] x1 + [0] [minus](x1, x2) = [1] x1 + [0] [mod](x1, x2) = [1] x1 + [0] [if_mod](x1, x2, x3) = [1] x1 + [1] x2 + [0] The following symbols are considered usable {le, pred, minus, mod, if_mod} The order satisfies the following ordering constraints: [le(0(), y)] = [4] > [1] = [true()] [le(s(x), 0())] = [4] > [0] = [false()] [le(s(x), s(y))] = [4] >= [4] = [le(x, y)] [pred(s(x))] = [1] x + [0] >= [1] x + [0] = [x] [minus(x, 0())] = [1] x + [0] >= [1] x + [0] = [x] [minus(x, s(y))] = [1] x + [0] >= [1] x + [0] = [pred(minus(x, y))] [mod(0(), y)] = [0] >= [0] = [0()] [mod(s(x), 0())] = [1] x + [0] >= [0] = [0()] [mod(s(x), s(y))] = [1] x + [0] ? [1] x + [4] = [if_mod(le(y, x), s(x), s(y))] [if_mod(true(), s(x), s(y))] = [1] x + [1] > [1] x + [0] = [mod(minus(x, y), s(y))] [if_mod(false(), s(x), s(y))] = [1] x + [0] >= [1] x + [0] = [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) , pred(s(x)) -> x , minus(x, s(y)) -> pred(minus(x, y)) , mod(0(), y) -> 0() , mod(s(x), 0()) -> 0() , mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) , if_mod(false(), s(x), s(y)) -> s(x) } Weak Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , minus(x, 0()) -> x , if_mod(true(), s(x), s(y)) -> mod(minus(x, y), s(y)) } 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(mod) = {1}, Uargs(if_mod) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [le](x1, x2) = [4] [0] = [0] [true] = [0] [s](x1) = [1] x1 + [0] [false] = [4] [pred](x1) = [1] x1 + [0] [minus](x1, x2) = [1] x1 + [0] [mod](x1, x2) = [1] x1 + [0] [if_mod](x1, x2, x3) = [1] x1 + [1] x2 + [0] The following symbols are considered usable {le, pred, minus, mod, if_mod} The order satisfies the following ordering constraints: [le(0(), y)] = [4] > [0] = [true()] [le(s(x), 0())] = [4] >= [4] = [false()] [le(s(x), s(y))] = [4] >= [4] = [le(x, y)] [pred(s(x))] = [1] x + [0] >= [1] x + [0] = [x] [minus(x, 0())] = [1] x + [0] >= [1] x + [0] = [x] [minus(x, s(y))] = [1] x + [0] >= [1] x + [0] = [pred(minus(x, y))] [mod(0(), y)] = [0] >= [0] = [0()] [mod(s(x), 0())] = [1] x + [0] >= [0] = [0()] [mod(s(x), s(y))] = [1] x + [0] ? [1] x + [4] = [if_mod(le(y, x), s(x), s(y))] [if_mod(true(), s(x), s(y))] = [1] x + [0] >= [1] x + [0] = [mod(minus(x, y), s(y))] [if_mod(false(), s(x), s(y))] = [1] x + [4] > [1] x + [0] = [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) , pred(s(x)) -> x , minus(x, s(y)) -> pred(minus(x, y)) , mod(0(), y) -> 0() , mod(s(x), 0()) -> 0() , mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) } Weak Trs: { le(0(), y) -> true() , le(s(x), 0()) -> false() , minus(x, 0()) -> x , if_mod(true(), s(x), s(y)) -> mod(minus(x, y), s(y)) , if_mod(false(), s(x), s(y)) -> 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(mod) = {1}, Uargs(if_mod) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [le](x1, x2) = [0] [0] = [0] [true] = [0] [s](x1) = [1] x1 + [1] [false] = [0] [pred](x1) = [1] x1 + [0] [minus](x1, x2) = [1] x1 + [0] [mod](x1, x2) = [1] x1 + [0] [if_mod](x1, x2, x3) = [1] x1 + [1] x2 + [0] The following symbols are considered usable {le, pred, minus, mod, if_mod} 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 + [1] > [1] x + [0] = [x] [minus(x, 0())] = [1] x + [0] >= [1] x + [0] = [x] [minus(x, s(y))] = [1] x + [0] >= [1] x + [0] = [pred(minus(x, y))] [mod(0(), y)] = [0] >= [0] = [0()] [mod(s(x), 0())] = [1] x + [1] > [0] = [0()] [mod(s(x), s(y))] = [1] x + [1] >= [1] x + [1] = [if_mod(le(y, x), s(x), s(y))] [if_mod(true(), s(x), s(y))] = [1] x + [1] > [1] x + [0] = [mod(minus(x, y), s(y))] [if_mod(false(), s(x), s(y))] = [1] x + [1] >= [1] x + [1] = [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)) , mod(0(), y) -> 0() , mod(s(x), s(y)) -> if_mod(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 , mod(s(x), 0()) -> 0() , if_mod(true(), s(x), s(y)) -> mod(minus(x, y), s(y)) , if_mod(false(), s(x), s(y)) -> 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(mod) = {1}, Uargs(if_mod) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [le](x1, x2) = [4] [0] = [0] [true] = [4] [s](x1) = [1] x1 + [4] [false] = [0] [pred](x1) = [1] x1 + [7] [minus](x1, x2) = [1] x1 + [4] [mod](x1, x2) = [1] x1 + [4] [if_mod](x1, x2, x3) = [1] x1 + [1] x2 + [0] The following symbols are considered usable {le, pred, minus, mod, if_mod} The order satisfies the following ordering constraints: [le(0(), y)] = [4] >= [4] = [true()] [le(s(x), 0())] = [4] > [0] = [false()] [le(s(x), s(y))] = [4] >= [4] = [le(x, y)] [pred(s(x))] = [1] x + [11] > [1] x + [0] = [x] [minus(x, 0())] = [1] x + [4] > [1] x + [0] = [x] [minus(x, s(y))] = [1] x + [4] ? [1] x + [11] = [pred(minus(x, y))] [mod(0(), y)] = [4] > [0] = [0()] [mod(s(x), 0())] = [1] x + [8] > [0] = [0()] [mod(s(x), s(y))] = [1] x + [8] >= [1] x + [8] = [if_mod(le(y, x), s(x), s(y))] [if_mod(true(), s(x), s(y))] = [1] x + [8] >= [1] x + [8] = [mod(minus(x, y), s(y))] [if_mod(false(), s(x), s(y))] = [1] x + [4] >= [1] x + [4] = [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)) , mod(s(x), s(y)) -> if_mod(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 , mod(0(), y) -> 0() , mod(s(x), 0()) -> 0() , if_mod(true(), s(x), s(y)) -> mod(minus(x, y), s(y)) , if_mod(false(), s(x), s(y)) -> 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(mod) = {1}, Uargs(if_mod) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [le](x1, x2) = [0] [0] = [0] [true] = [0] [s](x1) = [1] x1 + [1] [false] = [0] [pred](x1) = [1] x1 + [7] [minus](x1, x2) = [1] x1 + [0] [mod](x1, x2) = [1] x1 + [3] [if_mod](x1, x2, x3) = [1] x1 + [1] x2 + [2] The following symbols are considered usable {le, pred, minus, mod, if_mod} 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 + [8] > [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))] [mod(0(), y)] = [3] > [0] = [0()] [mod(s(x), 0())] = [1] x + [4] > [0] = [0()] [mod(s(x), s(y))] = [1] x + [4] > [1] x + [3] = [if_mod(le(y, x), s(x), s(y))] [if_mod(true(), s(x), s(y))] = [1] x + [3] >= [1] x + [3] = [mod(minus(x, y), s(y))] [if_mod(false(), s(x), s(y))] = [1] x + [3] > [1] x + [1] = [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 , mod(0(), y) -> 0() , mod(s(x), 0()) -> 0() , mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) , if_mod(true(), s(x), s(y)) -> mod(minus(x, y), s(y)) , if_mod(false(), s(x), s(y)) -> 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(mod) = {1}, Uargs(if_mod) = {1} TcT has computed the following constructor-restricted polynomial interpretation. [le](x1, x2) = 2 + 2*x2 [0]() = 0 [true]() = 0 [s](x1) = 1 + x1 [false]() = 0 [pred](x1) = x1 [minus](x1, x2) = x1 [mod](x1, x2) = 1 + 2*x1 + x1^2 [if_mod](x1, x2, x3) = 1 + x1 + x2^2 The following symbols are considered usable {le, pred, minus, mod, if_mod} This order satisfies the following ordering constraints. [le(0(), y)] = 2 + 2*y > = [true()] [le(s(x), 0())] = 2 > = [false()] [le(s(x), s(y))] = 4 + 2*y > 2 + 2*y = [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))] [mod(0(), y)] = 1 > = [0()] [mod(s(x), 0())] = 4 + 4*x + x^2 > = [0()] [mod(s(x), s(y))] = 4 + 4*x + x^2 >= 4 + 4*x + x^2 = [if_mod(le(y, x), s(x), s(y))] [if_mod(true(), s(x), s(y))] = 2 + 2*x + x^2 > 1 + 2*x + x^2 = [mod(minus(x, y), s(y))] [if_mod(false(), s(x), s(y))] = 2 + 2*x + x^2 > 1 + 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 , mod(0(), y) -> 0() , mod(s(x), 0()) -> 0() , mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) , if_mod(true(), s(x), s(y)) -> mod(minus(x, y), s(y)) , if_mod(false(), s(x), s(y)) -> 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(mod) = {1}, Uargs(if_mod) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [le](x1, x2) = [0 1] x2 + [2] [0 0] [1] [0] = [0] [0] [true] = [2] [1] [s](x1) = [1 2] x1 + [1] [0 1] [1] [false] = [0] [0] [pred](x1) = [1 0] x1 + [0] [0 1] [0] [minus](x1, x2) = [1 0] x1 + [0] [0 1] [0] [mod](x1, x2) = [2 3] x1 + [7] [2 4] [0] [if_mod](x1, x2, x3) = [2 1] x1 + [2 1] x2 + [0] [0 0] [2 0] [4] The following symbols are considered usable {le, pred, minus, mod, if_mod} The order satisfies the following ordering constraints: [le(0(), y)] = [0 1] y + [2] [0 0] [1] >= [2] [1] = [true()] [le(s(x), 0())] = [2] [1] > [0] [0] = [false()] [le(s(x), s(y))] = [0 1] y + [3] [0 0] [1] > [0 1] y + [2] [0 0] [1] = [le(x, y)] [pred(s(x))] = [1 2] x + [1] [0 1] [1] > [1 0] x + [0] [0 1] [0] = [x] [minus(x, 0())] = [1 0] x + [0] [0 1] [0] >= [1 0] x + [0] [0 1] [0] = [x] [minus(x, s(y))] = [1 0] x + [0] [0 1] [0] >= [1 0] x + [0] [0 1] [0] = [pred(minus(x, y))] [mod(0(), y)] = [7] [0] > [0] [0] = [0()] [mod(s(x), 0())] = [2 7] x + [12] [2 8] [6] > [0] [0] = [0()] [mod(s(x), s(y))] = [2 7] x + [12] [2 8] [6] > [2 7] x + [8] [2 4] [6] = [if_mod(le(y, x), s(x), s(y))] [if_mod(true(), s(x), s(y))] = [2 5] x + [8] [2 4] [6] > [2 3] x + [7] [2 4] [0] = [mod(minus(x, y), s(y))] [if_mod(false(), s(x), s(y))] = [2 5] x + [3] [2 4] [6] > [1 2] x + [1] [0 1] [1] = [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 , mod(0(), y) -> 0() , mod(s(x), 0()) -> 0() , mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) , if_mod(true(), s(x), s(y)) -> mod(minus(x, y), s(y)) , if_mod(false(), s(x), s(y)) -> 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) '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 minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 3) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) '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 2) '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 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))) , mod^#(0(), y) -> c_7() , mod^#(s(x), 0()) -> c_8() , mod^#(s(x), s(y)) -> c_9(if_mod^#(le(y, x), s(x), s(y))) , if_mod^#(true(), s(x), s(y)) -> c_10(mod^#(minus(x, y), s(y))) , if_mod^#(false(), s(x), s(y)) -> c_11(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))) , mod^#(0(), y) -> c_7() , mod^#(s(x), 0()) -> c_8() , mod^#(s(x), s(y)) -> c_9(if_mod^#(le(y, x), s(x), s(y))) , if_mod^#(true(), s(x), s(y)) -> c_10(mod^#(minus(x, y), s(y))) , if_mod^#(false(), s(x), s(y)) -> c_11(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)) , mod(0(), y) -> 0() , mod(s(x), 0()) -> 0() , mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) , if_mod(true(), s(x), s(y)) -> mod(minus(x, y), s(y)) , if_mod(false(), s(x), s(y)) -> s(x) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,2,7,8} by applications of Pre({1,2,7,8}) = {3,4,5,10,11}. 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: mod^#(0(), y) -> c_7() , 8: mod^#(s(x), 0()) -> c_8() , 9: mod^#(s(x), s(y)) -> c_9(if_mod^#(le(y, x), s(x), s(y))) , 10: if_mod^#(true(), s(x), s(y)) -> c_10(mod^#(minus(x, y), s(y))) , 11: if_mod^#(false(), s(x), s(y)) -> c_11(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))) , mod^#(s(x), s(y)) -> c_9(if_mod^#(le(y, x), s(x), s(y))) , if_mod^#(true(), s(x), s(y)) -> c_10(mod^#(minus(x, y), s(y))) , if_mod^#(false(), s(x), s(y)) -> c_11(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)) , mod(0(), y) -> 0() , mod(s(x), 0()) -> 0() , mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y)) , if_mod(true(), s(x), s(y)) -> mod(minus(x, y), s(y)) , if_mod(false(), s(x), s(y)) -> s(x) } Weak DPs: { le^#(0(), y) -> c_1() , le^#(s(x), 0()) -> c_2() , mod^#(0(), y) -> c_7() , mod^#(s(x), 0()) -> c_8() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..