/export/starexec/sandbox/solver/bin/starexec_run_tct_rc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) sel(0(),cons(X,Y)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) - Signature: {activate/1,from/1,sel/2} / {0/0,cons/2,n__from/1,s/1} - Obligation: runtime complexity wrt. defined symbols {activate,from,sel} and constructors {0,cons,n__from,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) sel(0(),cons(X,Y)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) - Signature: {activate/1,from/1,sel/2} / {0/0,cons/2,n__from/1,s/1} - Obligation: runtime complexity wrt. defined symbols {activate,from,sel} and constructors {0,cons,n__from,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) sel(0(),cons(X,Y)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) - Signature: {activate/1,from/1,sel/2} / {0/0,cons/2,n__from/1,s/1} - Obligation: runtime complexity wrt. defined symbols {activate,from,sel} and constructors {0,cons,n__from,s} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: sel(y,x){y -> s(y),x -> cons(u,x)} = sel(s(y),cons(u,x)) ->^+ sel(y,x) = C[sel(y,x) = sel(y,x){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) sel(0(),cons(X,Y)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) - Signature: {activate/1,from/1,sel/2} / {0/0,cons/2,n__from/1,s/1} - Obligation: runtime complexity wrt. defined symbols {activate,from,sel} and constructors {0,cons,n__from,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following weak dependency pairs: Strict DPs activate#(X) -> c_1(X) activate#(n__from(X)) -> c_2(from#(X)) from#(X) -> c_3(X,X) from#(X) -> c_4(X) sel#(0(),cons(X,Y)) -> c_5(X) sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) Weak DPs and mark the set of starting terms. ** Step 1.b:2: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1(X) activate#(n__from(X)) -> c_2(from#(X)) from#(X) -> c_3(X,X) from#(X) -> c_4(X) sel#(0(),cons(X,Y)) -> c_5(X) sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) - Strict TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) sel(0(),cons(X,Y)) -> X sel(s(X),cons(Y,Z)) -> sel(X,activate(Z)) - Signature: {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/2,c_4/1 ,c_5/1,c_6/1} - Obligation: runtime complexity wrt. defined symbols {activate#,from#,sel#} and constructors {0,cons,n__from,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) activate#(X) -> c_1(X) activate#(n__from(X)) -> c_2(from#(X)) from#(X) -> c_3(X,X) from#(X) -> c_4(X) sel#(0(),cons(X,Y)) -> c_5(X) sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) ** Step 1.b:3: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1(X) activate#(n__from(X)) -> c_2(from#(X)) from#(X) -> c_3(X,X) from#(X) -> c_4(X) sel#(0(),cons(X,Y)) -> c_5(X) sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) - Strict TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/2,c_4/1 ,c_5/1,c_6/1} - Obligation: runtime complexity wrt. defined symbols {activate#,from#,sel#} and constructors {0,cons,n__from,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(sel#) = {2}, uargs(c_2) = {1}, uargs(c_6) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [0] p(cons) = [1] x2 + [5] p(from) = [1] x1 + [0] p(n__from) = [1] x1 + [0] p(s) = [1] x1 + [0] p(sel) = [0] p(activate#) = [3] x1 + [0] p(from#) = [3] x1 + [0] p(sel#) = [1] x2 + [0] p(c_1) = [3] x1 + [0] p(c_2) = [1] x1 + [0] p(c_3) = [3] x1 + [0] p(c_4) = [3] x1 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] Following rules are strictly oriented: sel#(0(),cons(X,Y)) = [1] Y + [5] > [0] = c_5(X) sel#(s(X),cons(Y,Z)) = [1] Z + [5] > [1] Z + [0] = c_6(sel#(X,activate(Z))) Following rules are (at-least) weakly oriented: activate#(X) = [3] X + [0] >= [3] X + [0] = c_1(X) activate#(n__from(X)) = [3] X + [0] >= [3] X + [0] = c_2(from#(X)) from#(X) = [3] X + [0] >= [3] X + [0] = c_3(X,X) from#(X) = [3] X + [0] >= [3] X + [0] = c_4(X) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__from(X)) = [1] X + [0] >= [1] X + [0] = from(X) from(X) = [1] X + [0] >= [1] X + [5] = cons(X,n__from(s(X))) from(X) = [1] X + [0] >= [1] X + [0] = n__from(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:4: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1(X) activate#(n__from(X)) -> c_2(from#(X)) from#(X) -> c_3(X,X) from#(X) -> c_4(X) - Strict TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Weak DPs: sel#(0(),cons(X,Y)) -> c_5(X) sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) - Signature: {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/2,c_4/1 ,c_5/1,c_6/1} - Obligation: runtime complexity wrt. defined symbols {activate#,from#,sel#} and constructors {0,cons,n__from,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(sel#) = {2}, uargs(c_2) = {1}, uargs(c_6) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [2] p(cons) = [1] x2 + [7] p(from) = [0] p(n__from) = [1] p(s) = [1] x1 + [0] p(sel) = [4] p(activate#) = [1] x1 + [0] p(from#) = [0] p(sel#) = [2] x1 + [1] x2 + [8] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [15] p(c_6) = [1] x1 + [2] Following rules are strictly oriented: activate#(n__from(X)) = [1] > [0] = c_2(from#(X)) activate(X) = [1] X + [2] > [1] X + [0] = X activate(n__from(X)) = [3] > [0] = from(X) Following rules are (at-least) weakly oriented: activate#(X) = [1] X + [0] >= [0] = c_1(X) from#(X) = [0] >= [0] = c_3(X,X) from#(X) = [0] >= [0] = c_4(X) sel#(0(),cons(X,Y)) = [1] Y + [15] >= [15] = c_5(X) sel#(s(X),cons(Y,Z)) = [2] X + [1] Z + [15] >= [2] X + [1] Z + [12] = c_6(sel#(X,activate(Z))) from(X) = [0] >= [8] = cons(X,n__from(s(X))) from(X) = [0] >= [1] = n__from(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:5: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: activate#(X) -> c_1(X) from#(X) -> c_3(X,X) from#(X) -> c_4(X) - Strict TRS: from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Weak DPs: activate#(n__from(X)) -> c_2(from#(X)) sel#(0(),cons(X,Y)) -> c_5(X) sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) - Signature: {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/2,c_4/1 ,c_5/1,c_6/1} - Obligation: runtime complexity wrt. defined symbols {activate#,from#,sel#} and constructors {0,cons,n__from,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(sel#) = {2}, uargs(c_2) = {1}, uargs(c_6) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [0] p(cons) = [1] x2 + [0] p(from) = [1] x1 + [0] p(n__from) = [1] x1 + [0] p(s) = [1] x1 + [0] p(sel) = [0] p(activate#) = [1] p(from#) = [1] p(sel#) = [1] x2 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] Following rules are strictly oriented: activate#(X) = [1] > [0] = c_1(X) from#(X) = [1] > [0] = c_3(X,X) from#(X) = [1] > [0] = c_4(X) Following rules are (at-least) weakly oriented: activate#(n__from(X)) = [1] >= [1] = c_2(from#(X)) sel#(0(),cons(X,Y)) = [1] Y + [0] >= [0] = c_5(X) sel#(s(X),cons(Y,Z)) = [1] Z + [0] >= [1] Z + [0] = c_6(sel#(X,activate(Z))) activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__from(X)) = [1] X + [0] >= [1] X + [0] = from(X) from(X) = [1] X + [0] >= [1] X + [0] = cons(X,n__from(s(X))) from(X) = [1] X + [0] >= [1] X + [0] = n__from(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:6: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Weak DPs: activate#(X) -> c_1(X) activate#(n__from(X)) -> c_2(from#(X)) from#(X) -> c_3(X,X) from#(X) -> c_4(X) sel#(0(),cons(X,Y)) -> c_5(X) sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) - Signature: {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/2,c_4/1 ,c_5/1,c_6/1} - Obligation: runtime complexity wrt. defined symbols {activate#,from#,sel#} and constructors {0,cons,n__from,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(sel#) = {2}, uargs(c_2) = {1}, uargs(c_6) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [4] p(cons) = [1] x2 + [0] p(from) = [4] p(n__from) = [2] p(s) = [1] x1 + [1] p(sel) = [1] x1 + [1] x2 + [1] p(activate#) = [8] p(from#) = [1] p(sel#) = [4] x1 + [1] x2 + [0] p(c_1) = [8] p(c_2) = [1] x1 + [7] p(c_3) = [1] p(c_4) = [1] p(c_5) = [0] p(c_6) = [1] x1 + [0] Following rules are strictly oriented: from(X) = [4] > [2] = cons(X,n__from(s(X))) from(X) = [4] > [2] = n__from(X) Following rules are (at-least) weakly oriented: activate#(X) = [8] >= [8] = c_1(X) activate#(n__from(X)) = [8] >= [8] = c_2(from#(X)) from#(X) = [1] >= [1] = c_3(X,X) from#(X) = [1] >= [1] = c_4(X) sel#(0(),cons(X,Y)) = [1] Y + [0] >= [0] = c_5(X) sel#(s(X),cons(Y,Z)) = [4] X + [1] Z + [4] >= [4] X + [1] Z + [4] = c_6(sel#(X,activate(Z))) activate(X) = [1] X + [4] >= [1] X + [0] = X activate(n__from(X)) = [6] >= [4] = from(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:7: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: activate#(X) -> c_1(X) activate#(n__from(X)) -> c_2(from#(X)) from#(X) -> c_3(X,X) from#(X) -> c_4(X) sel#(0(),cons(X,Y)) -> c_5(X) sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z))) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(X) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) - Signature: {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/1,c_2/1,c_3/2,c_4/1 ,c_5/1,c_6/1} - Obligation: runtime complexity wrt. defined symbols {activate#,from#,sel#} and constructors {0,cons,n__from,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))