/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__add(X1,X2)) -> add(X1,X2) activate(n__from(X)) -> from(X) activate(n__fst(X1,X2)) -> fst(X1,X2) activate(n__len(X)) -> len(X) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) fst(X1,X2) -> n__fst(X1,X2) fst(0(),Z) -> nil() fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z))) len(X) -> n__len(X) len(cons(X,Z)) -> s(n__len(activate(Z))) len(nil()) -> 0() - Signature: {activate/1,add/2,from/1,fst/2,len/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,nil/0,s/1} - Obligation: runtime complexity wrt. defined symbols {activate,add,from,fst,len} and constructors {0,cons,n__add,n__from ,n__fst,n__len,nil,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__add(X1,X2)) -> add(X1,X2) activate(n__from(X)) -> from(X) activate(n__fst(X1,X2)) -> fst(X1,X2) activate(n__len(X)) -> len(X) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) fst(X1,X2) -> n__fst(X1,X2) fst(0(),Z) -> nil() fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z))) len(X) -> n__len(X) len(cons(X,Z)) -> s(n__len(activate(Z))) len(nil()) -> 0() - Signature: {activate/1,add/2,from/1,fst/2,len/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,nil/0,s/1} - Obligation: runtime complexity wrt. defined symbols {activate,add,from,fst,len} and constructors {0,cons,n__add,n__from ,n__fst,n__len,nil,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__add(X1,X2)) -> add(X1,X2) activate(n__from(X)) -> from(X) activate(n__fst(X1,X2)) -> fst(X1,X2) activate(n__len(X)) -> len(X) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) fst(X1,X2) -> n__fst(X1,X2) fst(0(),Z) -> nil() fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z))) len(X) -> n__len(X) len(cons(X,Z)) -> s(n__len(activate(Z))) len(nil()) -> 0() - Signature: {activate/1,add/2,from/1,fst/2,len/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,nil/0,s/1} - Obligation: runtime complexity wrt. defined symbols {activate,add,from,fst,len} and constructors {0,cons,n__add,n__from ,n__fst,n__len,nil,s} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: activate(x){x -> n__add(s(x),z)} = activate(n__add(s(x),z)) ->^+ s(n__add(activate(x),z)) = C[activate(x) = activate(x){}] ** Step 1.b:1: NaturalPI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(X1,X2) activate(n__from(X)) -> from(X) activate(n__fst(X1,X2)) -> fst(X1,X2) activate(n__len(X)) -> len(X) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) fst(X1,X2) -> n__fst(X1,X2) fst(0(),Z) -> nil() fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z))) len(X) -> n__len(X) len(cons(X,Z)) -> s(n__len(activate(Z))) len(nil()) -> 0() - Signature: {activate/1,add/2,from/1,fst/2,len/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,nil/0,s/1} - Obligation: runtime complexity wrt. defined symbols {activate,add,from,fst,len} and constructors {0,cons,n__add,n__from ,n__fst,n__len,nil,s} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(cons) = {2}, uargs(n__add) = {1}, uargs(n__fst) = {1,2}, uargs(n__len) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = 0 p(activate) = 5 + 4*x1 p(add) = 6 + 4*x1 + 4*x2 p(cons) = 3 + x2 p(from) = 7 + 2*x1 p(fst) = 2 + 4*x1 + 4*x2 p(len) = 1 + 4*x1 p(n__add) = 1 + x1 + x2 p(n__from) = 2 + x1 p(n__fst) = x1 + x2 p(n__len) = x1 p(nil) = 0 p(s) = x1 Following rules are strictly oriented: activate(X) = 5 + 4*X > X = X activate(n__add(X1,X2)) = 9 + 4*X1 + 4*X2 > 6 + 4*X1 + 4*X2 = add(X1,X2) activate(n__from(X)) = 13 + 4*X > 7 + 2*X = from(X) activate(n__fst(X1,X2)) = 5 + 4*X1 + 4*X2 > 2 + 4*X1 + 4*X2 = fst(X1,X2) activate(n__len(X)) = 5 + 4*X > 1 + 4*X = len(X) add(X1,X2) = 6 + 4*X1 + 4*X2 > 1 + X1 + X2 = n__add(X1,X2) add(0(),X) = 6 + 4*X > X = X from(X) = 7 + 2*X > 5 + X = cons(X,n__from(s(X))) from(X) = 7 + 2*X > 2 + X = n__from(X) fst(X1,X2) = 2 + 4*X1 + 4*X2 > X1 + X2 = n__fst(X1,X2) fst(0(),Z) = 2 + 4*Z > 0 = nil() fst(s(X),cons(Y,Z)) = 14 + 4*X + 4*Z > 13 + 4*X + 4*Z = cons(Y,n__fst(activate(X),activate(Z))) len(X) = 1 + 4*X > X = n__len(X) len(cons(X,Z)) = 13 + 4*Z > 5 + 4*Z = s(n__len(activate(Z))) len(nil()) = 1 > 0 = 0() Following rules are (at-least) weakly oriented: add(s(X),Y) = 6 + 4*X + 4*Y >= 6 + 4*X + Y = s(n__add(activate(X),Y)) ** Step 1.b:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: add(s(X),Y) -> s(n__add(activate(X),Y)) - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(X1,X2) activate(n__from(X)) -> from(X) activate(n__fst(X1,X2)) -> fst(X1,X2) activate(n__len(X)) -> len(X) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) fst(X1,X2) -> n__fst(X1,X2) fst(0(),Z) -> nil() fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z))) len(X) -> n__len(X) len(cons(X,Z)) -> s(n__len(activate(Z))) len(nil()) -> 0() - Signature: {activate/1,add/2,from/1,fst/2,len/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,nil/0,s/1} - Obligation: runtime complexity wrt. defined symbols {activate,add,from,fst,len} and constructors {0,cons,n__add,n__from ,n__fst,n__len,nil,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(cons) = {2}, uargs(n__add) = {1}, uargs(n__fst) = {1,2}, uargs(n__len) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [10] p(activate) = [2] x1 + [0] p(add) = [2] x1 + [2] x2 + [0] p(cons) = [1] x2 + [7] p(from) = [15] p(fst) = [2] x1 + [2] x2 + [0] p(len) = [2] x1 + [0] p(n__add) = [1] x1 + [1] x2 + [0] p(n__from) = [8] p(n__fst) = [1] x1 + [1] x2 + [0] p(n__len) = [1] x1 + [0] p(nil) = [8] p(s) = [1] x1 + [8] Following rules are strictly oriented: add(s(X),Y) = [2] X + [2] Y + [16] > [2] X + [1] Y + [8] = s(n__add(activate(X),Y)) Following rules are (at-least) weakly oriented: activate(X) = [2] X + [0] >= [1] X + [0] = X activate(n__add(X1,X2)) = [2] X1 + [2] X2 + [0] >= [2] X1 + [2] X2 + [0] = add(X1,X2) activate(n__from(X)) = [16] >= [15] = from(X) activate(n__fst(X1,X2)) = [2] X1 + [2] X2 + [0] >= [2] X1 + [2] X2 + [0] = fst(X1,X2) activate(n__len(X)) = [2] X + [0] >= [2] X + [0] = len(X) add(X1,X2) = [2] X1 + [2] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__add(X1,X2) add(0(),X) = [2] X + [20] >= [1] X + [0] = X from(X) = [15] >= [15] = cons(X,n__from(s(X))) from(X) = [15] >= [8] = n__from(X) fst(X1,X2) = [2] X1 + [2] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__fst(X1,X2) fst(0(),Z) = [2] Z + [20] >= [8] = nil() fst(s(X),cons(Y,Z)) = [2] X + [2] Z + [30] >= [2] X + [2] Z + [7] = cons(Y,n__fst(activate(X),activate(Z))) len(X) = [2] X + [0] >= [1] X + [0] = n__len(X) len(cons(X,Z)) = [2] Z + [14] >= [2] Z + [8] = s(n__len(activate(Z))) len(nil()) = [16] >= [10] = 0() ** Step 1.b:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(X1,X2) activate(n__from(X)) -> from(X) activate(n__fst(X1,X2)) -> fst(X1,X2) activate(n__len(X)) -> len(X) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) from(X) -> cons(X,n__from(s(X))) from(X) -> n__from(X) fst(X1,X2) -> n__fst(X1,X2) fst(0(),Z) -> nil() fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z))) len(X) -> n__len(X) len(cons(X,Z)) -> s(n__len(activate(Z))) len(nil()) -> 0() - Signature: {activate/1,add/2,from/1,fst/2,len/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,nil/0,s/1} - Obligation: runtime complexity wrt. defined symbols {activate,add,from,fst,len} and constructors {0,cons,n__add,n__from ,n__fst,n__len,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))