MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { pred(s(x)) -> x , minus(x, s(y)) -> pred(minus(x, y)) , minus(x, 0()) -> x , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) , quot(0(), s(y)) -> 0() } Obligation: runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) '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(s) = {1}, Uargs(minus) = {1}, Uargs(quot) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [pred](x1) = [1] x1 + [7] [s](x1) = [1] x1 + [0] [minus](x1, x2) = [1] x1 + [0] [0] = [7] [quot](x1, x2) = [1] x1 + [0] The following symbols are considered usable {pred, minus, quot} The order satisfies the following ordering constraints: [pred(s(x))] = [1] x + [7] > [1] x + [0] = [x] [minus(x, s(y))] = [1] x + [0] ? [1] x + [7] = [pred(minus(x, y))] [minus(x, 0())] = [1] x + [0] >= [1] x + [0] = [x] [quot(s(x), s(y))] = [1] x + [0] >= [1] x + [0] = [s(quot(minus(x, y), s(y)))] [quot(0(), s(y))] = [7] >= [7] = [0()] 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: { minus(x, s(y)) -> pred(minus(x, y)) , minus(x, 0()) -> x , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) , quot(0(), s(y)) -> 0() } Weak Trs: { pred(s(x)) -> 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(s) = {1}, Uargs(minus) = {1}, Uargs(quot) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [pred](x1) = [1] x1 + [5] [s](x1) = [1] x1 + [0] [minus](x1, x2) = [1] x1 + [1] [0] = [0] [quot](x1, x2) = [1] x1 + [0] The following symbols are considered usable {pred, minus, quot} The order satisfies the following ordering constraints: [pred(s(x))] = [1] x + [5] > [1] x + [0] = [x] [minus(x, s(y))] = [1] x + [1] ? [1] x + [6] = [pred(minus(x, y))] [minus(x, 0())] = [1] x + [1] > [1] x + [0] = [x] [quot(s(x), s(y))] = [1] x + [0] ? [1] x + [1] = [s(quot(minus(x, y), s(y)))] [quot(0(), s(y))] = [0] >= [0] = [0()] 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: { minus(x, s(y)) -> pred(minus(x, y)) , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) , quot(0(), s(y)) -> 0() } Weak Trs: { pred(s(x)) -> x , 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(s) = {1}, Uargs(minus) = {1}, Uargs(quot) = {1} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [pred](x1) = [1] x1 + [3] [s](x1) = [1] x1 + [0] [minus](x1, x2) = [1] x1 + [4] [0] = [0] [quot](x1, x2) = [1] x1 + [4] The following symbols are considered usable {pred, minus, quot} The order satisfies the following ordering constraints: [pred(s(x))] = [1] x + [3] > [1] x + [0] = [x] [minus(x, s(y))] = [1] x + [4] ? [1] x + [7] = [pred(minus(x, y))] [minus(x, 0())] = [1] x + [4] > [1] x + [0] = [x] [quot(s(x), s(y))] = [1] x + [4] ? [1] x + [8] = [s(quot(minus(x, y), s(y)))] [quot(0(), s(y))] = [4] > [0] = [0()] 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: { minus(x, s(y)) -> pred(minus(x, y)) , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) } Weak Trs: { pred(s(x)) -> x , minus(x, 0()) -> x , quot(0(), s(y)) -> 0() } Obligation: runtime complexity Answer: MAYBE We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. Trs: { quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^1)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are usable: Uargs(pred) = {1}, Uargs(s) = {1}, Uargs(minus) = {1}, Uargs(quot) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [pred](x1) = [1] x1 + [0] [s](x1) = [1] x1 + [4] [minus](x1, x2) = [1] x1 + [0] [0] = [4] [quot](x1, x2) = [2] x1 + [0] The following symbols are considered usable {pred, minus, quot} The order satisfies the following ordering constraints: [pred(s(x))] = [1] x + [4] > [1] x + [0] = [x] [minus(x, s(y))] = [1] x + [0] >= [1] x + [0] = [pred(minus(x, y))] [minus(x, 0())] = [1] x + [0] >= [1] x + [0] = [x] [quot(s(x), s(y))] = [2] x + [8] > [2] x + [4] = [s(quot(minus(x, y), s(y)))] [quot(0(), s(y))] = [8] > [4] = [0()] 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: { pred(s(x)) -> x , minus(x, 0()) -> x , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) , quot(0(), s(y)) -> 0() } 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: 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: 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: 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) '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) '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. 2) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { pred^#(s(x)) -> c_1(x) , minus^#(x, s(y)) -> c_2(pred^#(minus(x, y))) , minus^#(x, 0()) -> c_3(x) , quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y))) , quot^#(0(), s(y)) -> c_5() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { pred^#(s(x)) -> c_1(x) , minus^#(x, s(y)) -> c_2(pred^#(minus(x, y))) , minus^#(x, 0()) -> c_3(x) , quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y))) , quot^#(0(), s(y)) -> c_5() } Strict Trs: { pred(s(x)) -> x , minus(x, s(y)) -> pred(minus(x, y)) , minus(x, 0()) -> x , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) , quot(0(), s(y)) -> 0() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {5} by applications of Pre({5}) = {1,3,4}. Here rules are labeled as follows: DPs: { 1: pred^#(s(x)) -> c_1(x) , 2: minus^#(x, s(y)) -> c_2(pred^#(minus(x, y))) , 3: minus^#(x, 0()) -> c_3(x) , 4: quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y))) , 5: quot^#(0(), s(y)) -> c_5() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { pred^#(s(x)) -> c_1(x) , minus^#(x, s(y)) -> c_2(pred^#(minus(x, y))) , minus^#(x, 0()) -> c_3(x) , quot^#(s(x), s(y)) -> c_4(quot^#(minus(x, y), s(y))) } Strict Trs: { pred(s(x)) -> x , minus(x, s(y)) -> pred(minus(x, y)) , minus(x, 0()) -> x , quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) , quot(0(), s(y)) -> 0() } Weak DPs: { quot^#(0(), s(y)) -> c_5() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..