MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { times(x, 0()) -> 0() , times(x, s(y)) -> plus(times(x, y), x) , plus(x, 0()) -> x , plus(x, s(y)) -> s(plus(x, y)) , plus(0(), x) -> x , plus(s(x), y) -> s(plus(x, y)) } 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: 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: { times(x, s(y)) -> plus(times(x, y), x) } 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(s) = {1}, Uargs(plus) = {1} TcT has computed the following constructor-restricted polynomial interpretation. [times](x1, x2) = x1*x2 + 2*x2^2 [0]() = 0 [s](x1) = 1 + x1 [plus](x1, x2) = x1 + x2 The following symbols are considered usable {times, plus} This order satisfies the following ordering constraints. [times(x, 0())] = >= = [0()] [times(x, s(y))] = x + x*y + 2 + 4*y + 2*y^2 > x*y + 2*y^2 + x = [plus(times(x, y), x)] [plus(x, 0())] = x >= x = [x] [plus(x, s(y))] = x + 1 + y >= 1 + x + y = [s(plus(x, y))] [plus(0(), x)] = x >= x = [x] [plus(s(x), y)] = 1 + x + y >= 1 + x + y = [s(plus(x, y))] We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { times(x, 0()) -> 0() , plus(x, 0()) -> x , plus(x, s(y)) -> s(plus(x, y)) , plus(0(), x) -> x , plus(s(x), y) -> s(plus(x, y)) } Weak Trs: { times(x, s(y)) -> plus(times(x, y), x) } Obligation: runtime complexity Answer: MAYBE We use the processor 'polynomial interpretation' to orient following rules strictly. Trs: { plus(x, 0()) -> x , plus(0(), x) -> x } 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(s) = {1}, Uargs(plus) = {1} TcT has computed the following constructor-restricted polynomial interpretation. [times](x1, x2) = x1*x2 + x2 [0]() = 0 [s](x1) = 1 + x1 [plus](x1, x2) = 1 + x1 + x2 The following symbols are considered usable {times, plus} This order satisfies the following ordering constraints. [times(x, 0())] = >= = [0()] [times(x, s(y))] = x + x*y + 1 + y >= 1 + x*y + y + x = [plus(times(x, y), x)] [plus(x, 0())] = 1 + x > x = [x] [plus(x, s(y))] = 2 + x + y >= 2 + x + y = [s(plus(x, y))] [plus(0(), x)] = 1 + x > x = [x] [plus(s(x), y)] = 2 + x + y >= 2 + x + y = [s(plus(x, y))] We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { times(x, 0()) -> 0() , plus(x, s(y)) -> s(plus(x, y)) , plus(s(x), y) -> s(plus(x, y)) } Weak Trs: { times(x, s(y)) -> plus(times(x, y), x) , plus(x, 0()) -> x , plus(0(), x) -> x } Obligation: runtime complexity Answer: MAYBE We use the processor 'polynomial interpretation' to orient following rules strictly. Trs: { times(x, 0()) -> 0() } 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(s) = {1}, Uargs(plus) = {1} TcT has computed the following constructor-restricted polynomial interpretation. [times](x1, x2) = 2*x1*x2 + x2^2 [0]() = 2 [s](x1) = 2 + x1 [plus](x1, x2) = x1 + x2 The following symbols are considered usable {times, plus} This order satisfies the following ordering constraints. [times(x, 0())] = 4*x + 4 > 2 = [0()] [times(x, s(y))] = 4*x + 2*x*y + 4 + 4*y + y^2 > 2*x*y + y^2 + x = [plus(times(x, y), x)] [plus(x, 0())] = x + 2 > x = [x] [plus(x, s(y))] = x + 2 + y >= 2 + x + y = [s(plus(x, y))] [plus(0(), x)] = 2 + x > x = [x] [plus(s(x), y)] = 2 + x + y >= 2 + x + y = [s(plus(x, y))] We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { plus(x, s(y)) -> s(plus(x, y)) , plus(s(x), y) -> s(plus(x, y)) } Weak Trs: { times(x, 0()) -> 0() , times(x, s(y)) -> plus(times(x, y), x) , plus(x, 0()) -> x , plus(0(), x) -> x } Obligation: runtime complexity Answer: MAYBE We use the processor 'polynomial interpretation' to orient following rules strictly. Trs: { plus(x, s(y)) -> s(plus(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(s) = {1}, Uargs(plus) = {1} TcT has computed the following constructor-restricted polynomial interpretation. [times](x1, x2) = 2*x1 + 2*x1*x2 + x2^2 [0]() = 0 [s](x1) = 1 + x1 [plus](x1, x2) = 1 + x1 + 2*x2 The following symbols are considered usable {times, plus} This order satisfies the following ordering constraints. [times(x, 0())] = 2*x >= = [0()] [times(x, s(y))] = 4*x + 2*x*y + 1 + 2*y + y^2 >= 1 + 4*x + 2*x*y + y^2 = [plus(times(x, y), x)] [plus(x, 0())] = 1 + x > x = [x] [plus(x, s(y))] = 3 + x + 2*y > 2 + x + 2*y = [s(plus(x, y))] [plus(0(), x)] = 1 + 2*x > x = [x] [plus(s(x), y)] = 2 + x + 2*y >= 2 + x + 2*y = [s(plus(x, y))] We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { plus(s(x), y) -> s(plus(x, y)) } Weak Trs: { times(x, 0()) -> 0() , times(x, s(y)) -> plus(times(x, y), x) , plus(x, 0()) -> x , plus(x, s(y)) -> s(plus(x, y)) , plus(0(), x) -> 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: 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: { times^#(x, 0()) -> c_1() , times^#(x, s(y)) -> c_2(plus^#(times(x, y), x)) , plus^#(x, 0()) -> c_3(x) , plus^#(x, s(y)) -> c_4(plus^#(x, y)) , plus^#(0(), x) -> c_5(x) , plus^#(s(x), y) -> c_6(plus^#(x, y)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { times^#(x, 0()) -> c_1() , times^#(x, s(y)) -> c_2(plus^#(times(x, y), x)) , plus^#(x, 0()) -> c_3(x) , plus^#(x, s(y)) -> c_4(plus^#(x, y)) , plus^#(0(), x) -> c_5(x) , plus^#(s(x), y) -> c_6(plus^#(x, y)) } Strict Trs: { times(x, 0()) -> 0() , times(x, s(y)) -> plus(times(x, y), x) , plus(x, 0()) -> x , plus(x, s(y)) -> s(plus(x, y)) , plus(0(), x) -> x , plus(s(x), y) -> s(plus(x, y)) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1} by applications of Pre({1}) = {3,5}. Here rules are labeled as follows: DPs: { 1: times^#(x, 0()) -> c_1() , 2: times^#(x, s(y)) -> c_2(plus^#(times(x, y), x)) , 3: plus^#(x, 0()) -> c_3(x) , 4: plus^#(x, s(y)) -> c_4(plus^#(x, y)) , 5: plus^#(0(), x) -> c_5(x) , 6: plus^#(s(x), y) -> c_6(plus^#(x, y)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { times^#(x, s(y)) -> c_2(plus^#(times(x, y), x)) , plus^#(x, 0()) -> c_3(x) , plus^#(x, s(y)) -> c_4(plus^#(x, y)) , plus^#(0(), x) -> c_5(x) , plus^#(s(x), y) -> c_6(plus^#(x, y)) } Strict Trs: { times(x, 0()) -> 0() , times(x, s(y)) -> plus(times(x, y), x) , plus(x, 0()) -> x , plus(x, s(y)) -> s(plus(x, y)) , plus(0(), x) -> x , plus(s(x), y) -> s(plus(x, y)) } Weak DPs: { times^#(x, 0()) -> c_1() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..