MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { and(true(), X) -> X , and(false(), Y) -> false() , if(true(), X, Y) -> X , if(false(), X, Y) -> Y , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , first(0(), X) -> nil() , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) , from(X) -> cons(X, from(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: We add the following weak dependency pairs: Strict DPs: { and^#(true(), X) -> c_1(X) , and^#(false(), Y) -> c_2() , if^#(true(), X, Y) -> c_3(X) , if^#(false(), X, Y) -> c_4(Y) , add^#(0(), X) -> c_5(X) , add^#(s(X), Y) -> c_6(add^#(X, Y)) , first^#(0(), X) -> c_7() , first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z)) , from^#(X) -> c_9(X, from^#(s(X))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { and^#(true(), X) -> c_1(X) , and^#(false(), Y) -> c_2() , if^#(true(), X, Y) -> c_3(X) , if^#(false(), X, Y) -> c_4(Y) , add^#(0(), X) -> c_5(X) , add^#(s(X), Y) -> c_6(add^#(X, Y)) , first^#(0(), X) -> c_7() , first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z)) , from^#(X) -> c_9(X, from^#(s(X))) } Strict Trs: { and(true(), X) -> X , and(false(), Y) -> false() , if(true(), X, Y) -> X , if(false(), X, Y) -> Y , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , first(0(), X) -> nil() , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) , from(X) -> cons(X, from(s(X))) } Obligation: runtime complexity Answer: MAYBE No rule is usable, rules are removed from the input problem. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { and^#(true(), X) -> c_1(X) , and^#(false(), Y) -> c_2() , if^#(true(), X, Y) -> c_3(X) , if^#(false(), X, Y) -> c_4(Y) , add^#(0(), X) -> c_5(X) , add^#(s(X), Y) -> c_6(add^#(X, Y)) , first^#(0(), X) -> c_7() , first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z)) , from^#(X) -> c_9(X, from^#(s(X))) } Obligation: runtime complexity Answer: MAYBE The weightgap principle applies (using the following constant growth matrix-interpretation) The following argument positions are usable: Uargs(c_6) = {1}, Uargs(c_8) = {2}, Uargs(c_9) = {2} TcT has computed the following constructor-restricted matrix interpretation. [true] = [0] [0] [false] = [0] [0] [0] = [0] [0] [s](x1) = [1 0] x1 + [0] [0 0] [0] [cons](x1, x2) = [1 2] x1 + [1 0] x2 + [0] [0 1] [0 0] [0] [and^#](x1, x2) = [0 0] x1 + [1 1] x2 + [0] [1 2] [1 1] [2] [c_1](x1) = [1 1] x1 + [1] [1 1] [1] [c_2] = [1] [2] [if^#](x1, x2, x3) = [0 0] x1 + [1 1] x2 + [1 1] x3 + [0] [1 1] [2 1] [1 1] [2] [c_3](x1) = [1 1] x1 + [1] [2 1] [1] [c_4](x1) = [1 1] x1 + [1] [1 1] [1] [add^#](x1, x2) = [0 0] x1 + [2 1] x2 + [0] [2 2] [1 1] [1] [c_5](x1) = [1 1] x1 + [1] [1 1] [1] [c_6](x1) = [1 0] x1 + [2] [0 1] [0] [first^#](x1, x2) = [1] [2] [c_7] = [0] [1] [c_8](x1, x2) = [1 0] x2 + [1] [0 1] [0] [from^#](x1) = [1] [2] [c_9](x1, x2) = [0 0] x1 + [1 0] x2 + [1] [2 2] [0 1] [1] The following symbols are considered usable {and^#, if^#, add^#, first^#, from^#} The order satisfies the following ordering constraints: [and^#(true(), X)] = [1 1] X + [0] [1 1] [2] ? [1 1] X + [1] [1 1] [1] = [c_1(X)] [and^#(false(), Y)] = [1 1] Y + [0] [1 1] [2] ? [1] [2] = [c_2()] [if^#(true(), X, Y)] = [1 1] X + [1 1] Y + [0] [2 1] [1 1] [2] ? [1 1] X + [1] [2 1] [1] = [c_3(X)] [if^#(false(), X, Y)] = [1 1] X + [1 1] Y + [0] [2 1] [1 1] [2] ? [1 1] Y + [1] [1 1] [1] = [c_4(Y)] [add^#(0(), X)] = [2 1] X + [0] [1 1] [1] ? [1 1] X + [1] [1 1] [1] = [c_5(X)] [add^#(s(X), Y)] = [0 0] X + [2 1] Y + [0] [2 0] [1 1] [1] ? [0 0] X + [2 1] Y + [2] [2 2] [1 1] [1] = [c_6(add^#(X, Y))] [first^#(0(), X)] = [1] [2] > [0] [1] = [c_7()] [first^#(s(X), cons(Y, Z))] = [1] [2] ? [2] [2] = [c_8(Y, first^#(X, Z))] [from^#(X)] = [1] [2] ? [0 0] X + [2] [2 2] [3] = [c_9(X, from^#(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 DPs: { and^#(true(), X) -> c_1(X) , and^#(false(), Y) -> c_2() , if^#(true(), X, Y) -> c_3(X) , if^#(false(), X, Y) -> c_4(Y) , add^#(0(), X) -> c_5(X) , add^#(s(X), Y) -> c_6(add^#(X, Y)) , first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z)) , from^#(X) -> c_9(X, from^#(s(X))) } Weak DPs: { first^#(0(), X) -> c_7() } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {2} by applications of Pre({2}) = {1,3,4,5,7,8}. Here rules are labeled as follows: DPs: { 1: and^#(true(), X) -> c_1(X) , 2: and^#(false(), Y) -> c_2() , 3: if^#(true(), X, Y) -> c_3(X) , 4: if^#(false(), X, Y) -> c_4(Y) , 5: add^#(0(), X) -> c_5(X) , 6: add^#(s(X), Y) -> c_6(add^#(X, Y)) , 7: first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z)) , 8: from^#(X) -> c_9(X, from^#(s(X))) , 9: first^#(0(), X) -> c_7() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { and^#(true(), X) -> c_1(X) , if^#(true(), X, Y) -> c_3(X) , if^#(false(), X, Y) -> c_4(Y) , add^#(0(), X) -> c_5(X) , add^#(s(X), Y) -> c_6(add^#(X, Y)) , first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z)) , from^#(X) -> c_9(X, from^#(s(X))) } Weak DPs: { and^#(false(), Y) -> c_2() , first^#(0(), X) -> c_7() } Obligation: runtime complexity Answer: MAYBE The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { and^#(false(), Y) -> c_2() , first^#(0(), X) -> c_7() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { and^#(true(), X) -> c_1(X) , if^#(true(), X, Y) -> c_3(X) , if^#(false(), X, Y) -> c_4(Y) , add^#(0(), X) -> c_5(X) , add^#(s(X), Y) -> c_6(add^#(X, Y)) , first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z)) , from^#(X) -> c_9(X, from^#(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) 'Inspecting Problem...' failed due to the following reason: We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 1: and^#(true(), X) -> c_1(X) , 2: if^#(true(), X, Y) -> c_3(X) , 3: if^#(false(), X, Y) -> c_4(Y) , 6: first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_6) = {1}, Uargs(c_8) = {2}, Uargs(c_9) = {2} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [and](x1, x2) = [7] x1 + [7] x2 + [0] [true] = [5] [false] = [7] [if](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [add](x1, x2) = [7] x1 + [7] x2 + [0] [0] = [7] [s](x1) = [0] [first](x1, x2) = [7] x1 + [7] x2 + [0] [nil] = [0] [cons](x1, x2) = [1] x1 + [1] x2 + [4] [from](x1) = [7] x1 + [0] [and^#](x1, x2) = [3] x1 + [7] x2 + [0] [c_1](x1) = [7] x1 + [6] [c_2] = [0] [if^#](x1, x2, x3) = [7] x2 + [7] x3 + [5] [c_3](x1) = [7] x1 + [4] [c_4](x1) = [7] x1 + [3] [add^#](x1, x2) = [0] [c_5](x1) = [0] [c_6](x1) = [4] x1 + [0] [first^#](x1, x2) = [2] x2 + [0] [c_7] = [0] [c_8](x1, x2) = [1] x1 + [1] x2 + [1] [from^#](x1) = [0] [c_9](x1, x2) = [2] x2 + [0] The following symbols are considered usable {and^#, if^#, add^#, first^#, from^#} The order satisfies the following ordering constraints: [and^#(true(), X)] = [7] X + [15] > [7] X + [6] = [c_1(X)] [if^#(true(), X, Y)] = [7] X + [7] Y + [5] > [7] X + [4] = [c_3(X)] [if^#(false(), X, Y)] = [7] X + [7] Y + [5] > [7] Y + [3] = [c_4(Y)] [add^#(0(), X)] = [0] >= [0] = [c_5(X)] [add^#(s(X), Y)] = [0] >= [0] = [c_6(add^#(X, Y))] [first^#(s(X), cons(Y, Z))] = [2] Y + [2] Z + [8] > [1] Y + [2] Z + [1] = [c_8(Y, first^#(X, Z))] [from^#(X)] = [0] >= [0] = [c_9(X, from^#(s(X)))] The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { add^#(0(), X) -> c_5(X) , add^#(s(X), Y) -> c_6(add^#(X, Y)) , from^#(X) -> c_9(X, from^#(s(X))) } Weak DPs: { and^#(true(), X) -> c_1(X) , if^#(true(), X, Y) -> c_3(X) , if^#(false(), X, Y) -> c_4(Y) , first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z)) } Obligation: runtime complexity Answer: MAYBE We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 1: add^#(0(), X) -> c_5(X) , 4: and^#(true(), X) -> c_1(X) , 5: if^#(true(), X, Y) -> c_3(X) , 6: if^#(false(), X, Y) -> c_4(Y) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_6) = {1}, Uargs(c_8) = {2}, Uargs(c_9) = {2} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [and](x1, x2) = [7] x1 + [7] x2 + [0] [true] = [1] [false] = [1] [if](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [add](x1, x2) = [7] x1 + [7] x2 + [0] [0] = [4] [s](x1) = [1] x1 + [0] [first](x1, x2) = [7] x1 + [7] x2 + [0] [nil] = [0] [cons](x1, x2) = [1] x1 + [1] x2 + [7] [from](x1) = [7] x1 + [0] [and^#](x1, x2) = [7] x1 + [7] x2 + [7] [c_1](x1) = [7] x1 + [4] [c_2] = [0] [if^#](x1, x2, x3) = [1] x1 + [7] x2 + [7] x3 + [6] [c_3](x1) = [7] x1 + [5] [c_4](x1) = [7] x1 + [6] [add^#](x1, x2) = [2] x1 + [7] x2 + [0] [c_5](x1) = [7] x1 + [6] [c_6](x1) = [1] x1 + [0] [first^#](x1, x2) = [0] [c_7] = [0] [c_8](x1, x2) = [2] x2 + [0] [from^#](x1) = [0] [c_9](x1, x2) = [2] x2 + [0] The following symbols are considered usable {and^#, if^#, add^#, first^#, from^#} The order satisfies the following ordering constraints: [and^#(true(), X)] = [7] X + [14] > [7] X + [4] = [c_1(X)] [if^#(true(), X, Y)] = [7] X + [7] Y + [7] > [7] X + [5] = [c_3(X)] [if^#(false(), X, Y)] = [7] X + [7] Y + [7] > [7] Y + [6] = [c_4(Y)] [add^#(0(), X)] = [7] X + [8] > [7] X + [6] = [c_5(X)] [add^#(s(X), Y)] = [2] X + [7] Y + [0] >= [2] X + [7] Y + [0] = [c_6(add^#(X, Y))] [first^#(s(X), cons(Y, Z))] = [0] >= [0] = [c_8(Y, first^#(X, Z))] [from^#(X)] = [0] >= [0] = [c_9(X, from^#(s(X)))] The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { add^#(s(X), Y) -> c_6(add^#(X, Y)) , from^#(X) -> c_9(X, from^#(s(X))) } Weak DPs: { and^#(true(), X) -> c_1(X) , if^#(true(), X, Y) -> c_3(X) , if^#(false(), X, Y) -> c_4(Y) , add^#(0(), X) -> c_5(X) , first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z)) } Obligation: runtime complexity Answer: MAYBE We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 1: add^#(s(X), Y) -> c_6(add^#(X, Y)) , 3: and^#(true(), X) -> c_1(X) , 4: if^#(true(), X, Y) -> c_3(X) , 5: if^#(false(), X, Y) -> c_4(Y) , 6: add^#(0(), X) -> c_5(X) , 7: first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_6) = {1}, Uargs(c_8) = {2}, Uargs(c_9) = {2} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [and](x1, x2) = [7] x1 + [7] x2 + [0] [true] = [1] [false] = [1] [if](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [add](x1, x2) = [7] x1 + [7] x2 + [0] [0] = [2] [s](x1) = [1] x1 + [2] [first](x1, x2) = [7] x1 + [7] x2 + [0] [nil] = [0] [cons](x1, x2) = [1] x1 + [1] x2 + [2] [from](x1) = [7] x1 + [0] [and^#](x1, x2) = [7] x1 + [7] x2 + [7] [c_1](x1) = [7] x1 + [5] [c_2] = [0] [if^#](x1, x2, x3) = [1] x1 + [7] x2 + [7] x3 + [4] [c_3](x1) = [7] x1 + [4] [c_4](x1) = [7] x1 + [4] [add^#](x1, x2) = [4] x1 + [7] x2 + [0] [c_5](x1) = [7] x1 + [6] [c_6](x1) = [1] x1 + [5] [first^#](x1, x2) = [4] x2 + [0] [c_7] = [0] [c_8](x1, x2) = [4] x1 + [1] x2 + [5] [from^#](x1) = [0] [c_9](x1, x2) = [1] x2 + [0] The following symbols are considered usable {and^#, if^#, add^#, first^#, from^#} The order satisfies the following ordering constraints: [and^#(true(), X)] = [7] X + [14] > [7] X + [5] = [c_1(X)] [if^#(true(), X, Y)] = [7] X + [7] Y + [5] > [7] X + [4] = [c_3(X)] [if^#(false(), X, Y)] = [7] X + [7] Y + [5] > [7] Y + [4] = [c_4(Y)] [add^#(0(), X)] = [7] X + [8] > [7] X + [6] = [c_5(X)] [add^#(s(X), Y)] = [4] X + [7] Y + [8] > [4] X + [7] Y + [5] = [c_6(add^#(X, Y))] [first^#(s(X), cons(Y, Z))] = [4] Y + [4] Z + [8] > [4] Y + [4] Z + [5] = [c_8(Y, first^#(X, Z))] [from^#(X)] = [0] >= [0] = [c_9(X, from^#(s(X)))] The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { from^#(X) -> c_9(X, from^#(s(X))) } Weak DPs: { and^#(true(), X) -> c_1(X) , if^#(true(), X, Y) -> c_3(X) , if^#(false(), X, Y) -> c_4(Y) , add^#(0(), X) -> c_5(X) , add^#(s(X), Y) -> c_6(add^#(X, Y)) , first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z)) } 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) '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) 'Polynomial Path Order (PS)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 2) 'Fastest (timeout of 5 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) 'Polynomial Path Order (PS)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 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(s) = {1}, Uargs(cons) = {2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [and](x1, x2) = [1] x2 + [0] [true] = [0] [false] = [1] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0] [add](x1, x2) = [1] x2 + [0] [0] = [7] [s](x1) = [1] x1 + [0] [first](x1, x2) = [1] x2 + [0] [nil] = [7] [cons](x1, x2) = [1] x2 + [0] [from](x1) = [1] x1 + [0] The following symbols are considered usable {and, if, add, first, from} The order satisfies the following ordering constraints: [and(true(), X)] = [1] X + [0] >= [1] X + [0] = [X] [and(false(), Y)] = [1] Y + [0] ? [1] = [false()] [if(true(), X, Y)] = [1] X + [1] Y + [0] >= [1] X + [0] = [X] [if(false(), X, Y)] = [1] X + [1] Y + [1] > [1] Y + [0] = [Y] [add(0(), X)] = [1] X + [0] >= [1] X + [0] = [X] [add(s(X), Y)] = [1] Y + [0] >= [1] Y + [0] = [s(add(X, Y))] [first(0(), X)] = [1] X + [0] ? [7] = [nil()] [first(s(X), cons(Y, Z))] = [1] Z + [0] >= [1] Z + [0] = [cons(Y, first(X, Z))] [from(X)] = [1] X + [0] >= [1] X + [0] = [cons(X, from(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: { and(true(), X) -> X , and(false(), Y) -> false() , if(true(), X, Y) -> X , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , first(0(), X) -> nil() , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) , from(X) -> cons(X, from(s(X))) } Weak Trs: { if(false(), X, Y) -> Y } Obligation: runtime complexity Answer: MAYBE The weightgap principle applies (using the following nonconstant growth matrix-interpretation) The following argument positions are usable: Uargs(s) = {1}, Uargs(cons) = {2} TcT has computed the following matrix interpretation satisfying not(EDA) and not(IDA(1)). [and](x1, x2) = [1] x1 + [1] x2 + [7] [true] = [7] [false] = [7] [if](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [7] [add](x1, x2) = [1] x1 + [1] x2 + [7] [0] = [7] [s](x1) = [1] x1 + [7] [first](x1, x2) = [1] x1 + [1] x2 + [7] [nil] = [5] [cons](x1, x2) = [1] x2 + [1] [from](x1) = [1] x1 + [7] The following symbols are considered usable {and, if, add, first, from} The order satisfies the following ordering constraints: [and(true(), X)] = [1] X + [14] > [1] X + [0] = [X] [and(false(), Y)] = [1] Y + [14] > [7] = [false()] [if(true(), X, Y)] = [1] X + [1] Y + [14] > [1] X + [0] = [X] [if(false(), X, Y)] = [1] X + [1] Y + [14] > [1] Y + [0] = [Y] [add(0(), X)] = [1] X + [14] > [1] X + [0] = [X] [add(s(X), Y)] = [1] X + [1] Y + [14] >= [1] X + [1] Y + [14] = [s(add(X, Y))] [first(0(), X)] = [1] X + [14] > [5] = [nil()] [first(s(X), cons(Y, Z))] = [1] X + [1] Z + [15] > [1] X + [1] Z + [8] = [cons(Y, first(X, Z))] [from(X)] = [1] X + [7] ? [1] X + [15] = [cons(X, from(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: { add(s(X), Y) -> s(add(X, Y)) , from(X) -> cons(X, from(s(X))) } Weak Trs: { and(true(), X) -> X , and(false(), Y) -> false() , if(true(), X, Y) -> X , if(false(), X, Y) -> Y , add(0(), X) -> X , first(0(), X) -> nil() , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) } Obligation: runtime complexity Answer: MAYBE We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. Trs: { add(s(X), Y) -> s(add(X, 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(s) = {1}, Uargs(cons) = {2} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [and](x1, x2) = [7] x2 + [1] [true] = [0] [false] = [0] [if](x1, x2, x3) = [7] x2 + [7] x3 + [5] [add](x1, x2) = [4] x1 + [7] x2 + [0] [0] = [0] [s](x1) = [1] x1 + [2] [first](x1, x2) = [7] x2 + [0] [nil] = [0] [cons](x1, x2) = [1] x2 + [0] [from](x1) = [0] The following symbols are considered usable {and, if, add, first, from} The order satisfies the following ordering constraints: [and(true(), X)] = [7] X + [1] > [1] X + [0] = [X] [and(false(), Y)] = [7] Y + [1] > [0] = [false()] [if(true(), X, Y)] = [7] X + [7] Y + [5] > [1] X + [0] = [X] [if(false(), X, Y)] = [7] X + [7] Y + [5] > [1] Y + [0] = [Y] [add(0(), X)] = [7] X + [0] >= [1] X + [0] = [X] [add(s(X), Y)] = [4] X + [7] Y + [8] > [4] X + [7] Y + [2] = [s(add(X, Y))] [first(0(), X)] = [7] X + [0] >= [0] = [nil()] [first(s(X), cons(Y, Z))] = [7] Z + [0] >= [7] Z + [0] = [cons(Y, first(X, Z))] [from(X)] = [0] >= [0] = [cons(X, from(s(X)))] We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { from(X) -> cons(X, from(s(X))) } Weak Trs: { and(true(), X) -> X , and(false(), Y) -> false() , if(true(), X, Y) -> X , if(false(), X, Y) -> Y , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , first(0(), X) -> nil() , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) } 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) '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) '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) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { and^#(true(), X) -> c_1(X) , and^#(false(), Y) -> c_2() , if^#(true(), X, Y) -> c_3(X) , if^#(false(), X, Y) -> c_4(Y) , add^#(0(), X) -> c_5(X) , add^#(s(X), Y) -> c_6(add^#(X, Y)) , first^#(0(), X) -> c_7() , first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z)) , from^#(X) -> c_9(X, from^#(s(X))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { and^#(true(), X) -> c_1(X) , and^#(false(), Y) -> c_2() , if^#(true(), X, Y) -> c_3(X) , if^#(false(), X, Y) -> c_4(Y) , add^#(0(), X) -> c_5(X) , add^#(s(X), Y) -> c_6(add^#(X, Y)) , first^#(0(), X) -> c_7() , first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z)) , from^#(X) -> c_9(X, from^#(s(X))) } Strict Trs: { and(true(), X) -> X , and(false(), Y) -> false() , if(true(), X, Y) -> X , if(false(), X, Y) -> Y , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , first(0(), X) -> nil() , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) , from(X) -> cons(X, from(s(X))) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {2,7} by applications of Pre({2,7}) = {1,3,4,5,8,9}. Here rules are labeled as follows: DPs: { 1: and^#(true(), X) -> c_1(X) , 2: and^#(false(), Y) -> c_2() , 3: if^#(true(), X, Y) -> c_3(X) , 4: if^#(false(), X, Y) -> c_4(Y) , 5: add^#(0(), X) -> c_5(X) , 6: add^#(s(X), Y) -> c_6(add^#(X, Y)) , 7: first^#(0(), X) -> c_7() , 8: first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z)) , 9: from^#(X) -> c_9(X, from^#(s(X))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { and^#(true(), X) -> c_1(X) , if^#(true(), X, Y) -> c_3(X) , if^#(false(), X, Y) -> c_4(Y) , add^#(0(), X) -> c_5(X) , add^#(s(X), Y) -> c_6(add^#(X, Y)) , first^#(s(X), cons(Y, Z)) -> c_8(Y, first^#(X, Z)) , from^#(X) -> c_9(X, from^#(s(X))) } Strict Trs: { and(true(), X) -> X , and(false(), Y) -> false() , if(true(), X, Y) -> X , if(false(), X, Y) -> Y , add(0(), X) -> X , add(s(X), Y) -> s(add(X, Y)) , first(0(), X) -> nil() , first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) , from(X) -> cons(X, from(s(X))) } Weak DPs: { and^#(false(), Y) -> c_2() , first^#(0(), X) -> c_7() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..