/export/starexec/sandbox2/solver/bin/starexec_run_tct_rc /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/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: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if,mark} and constructors {0,add ,and,cons,false,first,from,if,nil,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if,mark} and constructors {0,add ,and,cons,false,first,from,if,nil,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if,mark} and constructors {0,add ,and,cons,false,first,from,if,nil,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: mark(x){x -> add(x,y)} = mark(add(x,y)) ->^+ a__add(mark(x),y) = C[mark(x) = mark(x){}] ** Step 1.b:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if,mark} and constructors {0,add ,and,cons,false,first,from,if,nil,s,true} + 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(a__add) = {1}, uargs(a__and) = {1}, uargs(a__first) = {1,2}, uargs(a__if) = {1}, uargs(add) = {1}, uargs(and) = {1}, uargs(first) = {1,2}, uargs(if) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__add) = [1] x1 + [2] x2 + [0] p(a__and) = [1] x1 + [2] x2 + [0] p(a__first) = [1] x1 + [1] x2 + [2] p(a__from) = [0] p(a__if) = [1] x1 + [2] x2 + [2] x3 + [0] p(add) = [1] x1 + [1] x2 + [0] p(and) = [1] x1 + [1] x2 + [0] p(cons) = [0] p(false) = [0] p(first) = [1] x1 + [1] x2 + [1] p(from) = [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [2] x1 + [0] p(nil) = [0] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: a__first(X1,X2) = [1] X1 + [1] X2 + [2] > [1] X1 + [1] X2 + [1] = first(X1,X2) a__first(0(),X) = [1] X + [2] > [0] = nil() a__first(s(X),cons(Y,Z)) = [1] X + [2] > [0] = cons(Y,first(X,Z)) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [2] X2 + [0] >= [1] X1 + [1] X2 + [0] = add(X1,X2) a__add(0(),X) = [2] X + [0] >= [2] X + [0] = mark(X) a__add(s(X),Y) = [1] X + [2] Y + [0] >= [1] X + [1] Y + [0] = s(add(X,Y)) a__and(X1,X2) = [1] X1 + [2] X2 + [0] >= [1] X1 + [1] X2 + [0] = and(X1,X2) a__and(false(),Y) = [2] Y + [0] >= [0] = false() a__and(true(),X) = [2] X + [0] >= [2] X + [0] = mark(X) a__from(X) = [0] >= [0] = cons(X,from(s(X))) a__from(X) = [0] >= [0] = from(X) a__if(X1,X2,X3) = [1] X1 + [2] X2 + [2] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = if(X1,X2,X3) a__if(false(),X,Y) = [2] X + [2] Y + [0] >= [2] Y + [0] = mark(Y) a__if(true(),X,Y) = [2] X + [2] Y + [0] >= [2] X + [0] = mark(X) mark(0()) = [0] >= [0] = 0() mark(add(X1,X2)) = [2] X1 + [2] X2 + [0] >= [2] X1 + [2] X2 + [0] = a__add(mark(X1),X2) mark(and(X1,X2)) = [2] X1 + [2] X2 + [0] >= [2] X1 + [2] X2 + [0] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [0] = cons(X1,X2) mark(false()) = [0] >= [0] = false() mark(first(X1,X2)) = [2] X1 + [2] X2 + [2] >= [2] X1 + [2] X2 + [2] = a__first(mark(X1),mark(X2)) mark(from(X)) = [0] >= [0] = a__from(X) mark(if(X1,X2,X3)) = [2] X1 + [2] X2 + [2] X3 + [0] >= [2] X1 + [2] X2 + [2] X3 + [0] = a__if(mark(X1),X2,X3) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [2] X + [0] >= [1] X + [0] = s(X) mark(true()) = [0] >= [0] = true() ** Step 1.b:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Weak TRS: a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if,mark} and constructors {0,add ,and,cons,false,first,from,if,nil,s,true} + 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(a__add) = {1}, uargs(a__and) = {1}, uargs(a__first) = {1,2}, uargs(a__if) = {1}, uargs(add) = {1}, uargs(and) = {1}, uargs(first) = {1,2}, uargs(if) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [4] p(a__add) = [1] x1 + [4] x2 + [4] p(a__and) = [1] x1 + [4] x2 + [0] p(a__first) = [1] x1 + [1] x2 + [4] p(a__from) = [3] x1 + [0] p(a__if) = [1] x1 + [4] x2 + [4] x3 + [0] p(add) = [1] x1 + [1] x2 + [4] p(and) = [1] x1 + [1] x2 + [0] p(cons) = [0] p(false) = [0] p(first) = [1] x1 + [1] x2 + [1] p(from) = [1] x1 + [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [4] x1 + [0] p(nil) = [0] p(s) = [0] p(true) = [0] Following rules are strictly oriented: a__add(0(),X) = [4] X + [8] > [4] X + [0] = mark(X) a__add(s(X),Y) = [4] Y + [4] > [0] = s(add(X,Y)) mark(0()) = [16] > [4] = 0() mark(add(X1,X2)) = [4] X1 + [4] X2 + [16] > [4] X1 + [4] X2 + [4] = a__add(mark(X1),X2) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [4] X2 + [4] >= [1] X1 + [1] X2 + [4] = add(X1,X2) a__and(X1,X2) = [1] X1 + [4] X2 + [0] >= [1] X1 + [1] X2 + [0] = and(X1,X2) a__and(false(),Y) = [4] Y + [0] >= [0] = false() a__and(true(),X) = [4] X + [0] >= [4] X + [0] = mark(X) a__first(X1,X2) = [1] X1 + [1] X2 + [4] >= [1] X1 + [1] X2 + [1] = first(X1,X2) a__first(0(),X) = [1] X + [8] >= [0] = nil() a__first(s(X),cons(Y,Z)) = [4] >= [0] = cons(Y,first(X,Z)) a__from(X) = [3] X + [0] >= [0] = cons(X,from(s(X))) a__from(X) = [3] X + [0] >= [1] X + [0] = from(X) a__if(X1,X2,X3) = [1] X1 + [4] X2 + [4] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = if(X1,X2,X3) a__if(false(),X,Y) = [4] X + [4] Y + [0] >= [4] Y + [0] = mark(Y) a__if(true(),X,Y) = [4] X + [4] Y + [0] >= [4] X + [0] = mark(X) mark(and(X1,X2)) = [4] X1 + [4] X2 + [0] >= [4] X1 + [4] X2 + [0] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [0] = cons(X1,X2) mark(false()) = [0] >= [0] = false() mark(first(X1,X2)) = [4] X1 + [4] X2 + [4] >= [4] X1 + [4] X2 + [4] = a__first(mark(X1),mark(X2)) mark(from(X)) = [4] X + [0] >= [3] X + [0] = a__from(X) mark(if(X1,X2,X3)) = [4] X1 + [4] X2 + [4] X3 + [0] >= [4] X1 + [4] X2 + [4] X3 + [0] = a__if(mark(X1),X2,X3) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [0] = s(X) mark(true()) = [0] >= [0] = true() ** Step 1.b:3: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Weak TRS: a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if,mark} and constructors {0,add ,and,cons,false,first,from,if,nil,s,true} + 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(a__add) = {1}, uargs(a__and) = {1}, uargs(a__first) = {1,2}, uargs(a__if) = {1}, uargs(add) = {1}, uargs(and) = {1}, uargs(first) = {1,2}, uargs(if) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__add) = [1] x1 + [8] x2 + [0] p(a__and) = [1] x1 + [8] x2 + [0] p(a__first) = [1] x1 + [1] x2 + [0] p(a__from) = [1] x1 + [8] p(a__if) = [1] x1 + [8] x2 + [8] x3 + [0] p(add) = [1] x1 + [1] x2 + [0] p(and) = [1] x1 + [1] x2 + [0] p(cons) = [0] p(false) = [0] p(first) = [1] x1 + [1] x2 + [0] p(from) = [1] x1 + [1] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [8] x1 + [0] p(nil) = [0] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: a__from(X) = [1] X + [8] > [0] = cons(X,from(s(X))) a__from(X) = [1] X + [8] > [1] X + [1] = from(X) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [8] X2 + [0] >= [1] X1 + [1] X2 + [0] = add(X1,X2) a__add(0(),X) = [8] X + [0] >= [8] X + [0] = mark(X) a__add(s(X),Y) = [1] X + [8] Y + [0] >= [1] X + [1] Y + [0] = s(add(X,Y)) a__and(X1,X2) = [1] X1 + [8] X2 + [0] >= [1] X1 + [1] X2 + [0] = and(X1,X2) a__and(false(),Y) = [8] Y + [0] >= [0] = false() a__and(true(),X) = [8] X + [0] >= [8] X + [0] = mark(X) a__first(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = first(X1,X2) a__first(0(),X) = [1] X + [0] >= [0] = nil() a__first(s(X),cons(Y,Z)) = [1] X + [0] >= [0] = cons(Y,first(X,Z)) a__if(X1,X2,X3) = [1] X1 + [8] X2 + [8] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = if(X1,X2,X3) a__if(false(),X,Y) = [8] X + [8] Y + [0] >= [8] Y + [0] = mark(Y) a__if(true(),X,Y) = [8] X + [8] Y + [0] >= [8] X + [0] = mark(X) mark(0()) = [0] >= [0] = 0() mark(add(X1,X2)) = [8] X1 + [8] X2 + [0] >= [8] X1 + [8] X2 + [0] = a__add(mark(X1),X2) mark(and(X1,X2)) = [8] X1 + [8] X2 + [0] >= [8] X1 + [8] X2 + [0] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [0] = cons(X1,X2) mark(false()) = [0] >= [0] = false() mark(first(X1,X2)) = [8] X1 + [8] X2 + [0] >= [8] X1 + [8] X2 + [0] = a__first(mark(X1),mark(X2)) mark(from(X)) = [8] X + [8] >= [1] X + [8] = a__from(X) mark(if(X1,X2,X3)) = [8] X1 + [8] X2 + [8] X3 + [0] >= [8] X1 + [8] X2 + [8] X3 + [0] = a__if(mark(X1),X2,X3) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [8] X + [0] >= [1] X + [0] = s(X) mark(true()) = [0] >= [0] = true() ** Step 1.b:4: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Weak TRS: a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if,mark} and constructors {0,add ,and,cons,false,first,from,if,nil,s,true} + 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(a__add) = {1}, uargs(a__and) = {1}, uargs(a__first) = {1,2}, uargs(a__if) = {1}, uargs(add) = {1}, uargs(and) = {1}, uargs(first) = {1,2}, uargs(if) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(a__add) = [1] x1 + [4] x2 + [0] p(a__and) = [1] x1 + [4] x2 + [3] p(a__first) = [1] x1 + [1] x2 + [0] p(a__from) = [2] p(a__if) = [1] x1 + [4] x2 + [4] x3 + [6] p(add) = [1] x1 + [1] x2 + [0] p(and) = [1] x1 + [1] x2 + [2] p(cons) = [1] x2 + [1] p(false) = [0] p(first) = [1] x1 + [1] x2 + [0] p(from) = [1] p(if) = [1] x1 + [1] x2 + [1] x3 + [5] p(mark) = [4] x1 + [0] p(nil) = [2] p(s) = [1] x1 + [3] p(true) = [4] Following rules are strictly oriented: a__and(X1,X2) = [1] X1 + [4] X2 + [3] > [1] X1 + [1] X2 + [2] = and(X1,X2) a__and(false(),Y) = [4] Y + [3] > [0] = false() a__and(true(),X) = [4] X + [7] > [4] X + [0] = mark(X) a__if(X1,X2,X3) = [1] X1 + [4] X2 + [4] X3 + [6] > [1] X1 + [1] X2 + [1] X3 + [5] = if(X1,X2,X3) a__if(false(),X,Y) = [4] X + [4] Y + [6] > [4] Y + [0] = mark(Y) a__if(true(),X,Y) = [4] X + [4] Y + [10] > [4] X + [0] = mark(X) mark(and(X1,X2)) = [4] X1 + [4] X2 + [8] > [4] X1 + [4] X2 + [3] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [4] X2 + [4] > [1] X2 + [1] = cons(X1,X2) mark(from(X)) = [4] > [2] = a__from(X) mark(if(X1,X2,X3)) = [4] X1 + [4] X2 + [4] X3 + [20] > [4] X1 + [4] X2 + [4] X3 + [6] = a__if(mark(X1),X2,X3) mark(nil()) = [8] > [2] = nil() mark(s(X)) = [4] X + [12] > [1] X + [3] = s(X) mark(true()) = [16] > [4] = true() Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [4] X2 + [0] >= [1] X1 + [1] X2 + [0] = add(X1,X2) a__add(0(),X) = [4] X + [2] >= [4] X + [0] = mark(X) a__add(s(X),Y) = [1] X + [4] Y + [3] >= [1] X + [1] Y + [3] = s(add(X,Y)) a__first(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = first(X1,X2) a__first(0(),X) = [1] X + [2] >= [2] = nil() a__first(s(X),cons(Y,Z)) = [1] X + [1] Z + [4] >= [1] X + [1] Z + [1] = cons(Y,first(X,Z)) a__from(X) = [2] >= [2] = cons(X,from(s(X))) a__from(X) = [2] >= [1] = from(X) mark(0()) = [8] >= [2] = 0() mark(add(X1,X2)) = [4] X1 + [4] X2 + [0] >= [4] X1 + [4] X2 + [0] = a__add(mark(X1),X2) mark(false()) = [0] >= [0] = false() mark(first(X1,X2)) = [4] X1 + [4] X2 + [0] >= [4] X1 + [4] X2 + [0] = a__first(mark(X1),mark(X2)) ** Step 1.b:5: NaturalPI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) - Weak TRS: a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if,mark} and constructors {0,add ,and,cons,false,first,from,if,nil,s,true} + 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(a__add) = {1}, uargs(a__and) = {1}, uargs(a__first) = {1,2}, uargs(a__if) = {1}, uargs(add) = {1}, uargs(and) = {1}, uargs(first) = {1,2}, uargs(if) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = 2 p(a__add) = x1 + 4*x2 p(a__and) = 1 + x1 + 4*x2 p(a__first) = x1 + x2 p(a__from) = 5 + 4*x1 p(a__if) = 2 + x1 + 4*x2 + 4*x3 p(add) = x1 + x2 p(and) = 1 + x1 + x2 p(cons) = 1 + x2 p(false) = 1 p(first) = x1 + x2 p(from) = 2 + x1 p(if) = 2 + x1 + x2 + x3 p(mark) = 4*x1 p(nil) = 1 p(s) = 2 + x1 p(true) = 2 Following rules are strictly oriented: mark(false()) = 4 > 1 = false() Following rules are (at-least) weakly oriented: a__add(X1,X2) = X1 + 4*X2 >= X1 + X2 = add(X1,X2) a__add(0(),X) = 2 + 4*X >= 4*X = mark(X) a__add(s(X),Y) = 2 + X + 4*Y >= 2 + X + Y = s(add(X,Y)) a__and(X1,X2) = 1 + X1 + 4*X2 >= 1 + X1 + X2 = and(X1,X2) a__and(false(),Y) = 2 + 4*Y >= 1 = false() a__and(true(),X) = 3 + 4*X >= 4*X = mark(X) a__first(X1,X2) = X1 + X2 >= X1 + X2 = first(X1,X2) a__first(0(),X) = 2 + X >= 1 = nil() a__first(s(X),cons(Y,Z)) = 3 + X + Z >= 1 + X + Z = cons(Y,first(X,Z)) a__from(X) = 5 + 4*X >= 5 + X = cons(X,from(s(X))) a__from(X) = 5 + 4*X >= 2 + X = from(X) a__if(X1,X2,X3) = 2 + X1 + 4*X2 + 4*X3 >= 2 + X1 + X2 + X3 = if(X1,X2,X3) a__if(false(),X,Y) = 3 + 4*X + 4*Y >= 4*Y = mark(Y) a__if(true(),X,Y) = 4 + 4*X + 4*Y >= 4*X = mark(X) mark(0()) = 8 >= 2 = 0() mark(add(X1,X2)) = 4*X1 + 4*X2 >= 4*X1 + 4*X2 = a__add(mark(X1),X2) mark(and(X1,X2)) = 4 + 4*X1 + 4*X2 >= 1 + 4*X1 + 4*X2 = a__and(mark(X1),X2) mark(cons(X1,X2)) = 4 + 4*X2 >= 1 + X2 = cons(X1,X2) mark(first(X1,X2)) = 4*X1 + 4*X2 >= 4*X1 + 4*X2 = a__first(mark(X1),mark(X2)) mark(from(X)) = 8 + 4*X >= 5 + 4*X = a__from(X) mark(if(X1,X2,X3)) = 8 + 4*X1 + 4*X2 + 4*X3 >= 2 + 4*X1 + 4*X2 + 4*X3 = a__if(mark(X1),X2,X3) mark(nil()) = 4 >= 1 = nil() mark(s(X)) = 8 + 4*X >= 2 + X = s(X) mark(true()) = 8 >= 2 = true() ** Step 1.b:6: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) - Weak TRS: a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if,mark} and constructors {0,add ,and,cons,false,first,from,if,nil,s,true} + 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(a__add) = {1}, uargs(a__and) = {1}, uargs(a__first) = {1,2}, uargs(a__if) = {1}, uargs(add) = {1}, uargs(and) = {1}, uargs(first) = {1,2}, uargs(if) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__add) = [1] x1 + [8] x2 + [0] p(a__and) = [1] x1 + [8] x2 + [4] p(a__first) = [1] x1 + [1] x2 + [7] p(a__from) = [1] x1 + [0] p(a__if) = [1] x1 + [8] x2 + [8] x3 + [12] p(add) = [1] x1 + [1] x2 + [0] p(and) = [1] x1 + [1] x2 + [1] p(cons) = [0] p(false) = [1] p(first) = [1] x1 + [1] x2 + [1] p(from) = [1] x1 + [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [2] p(mark) = [8] x1 + [0] p(nil) = [0] p(s) = [3] p(true) = [0] Following rules are strictly oriented: mark(first(X1,X2)) = [8] X1 + [8] X2 + [8] > [8] X1 + [8] X2 + [7] = a__first(mark(X1),mark(X2)) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [8] X2 + [0] >= [1] X1 + [1] X2 + [0] = add(X1,X2) a__add(0(),X) = [8] X + [0] >= [8] X + [0] = mark(X) a__add(s(X),Y) = [8] Y + [3] >= [3] = s(add(X,Y)) a__and(X1,X2) = [1] X1 + [8] X2 + [4] >= [1] X1 + [1] X2 + [1] = and(X1,X2) a__and(false(),Y) = [8] Y + [5] >= [1] = false() a__and(true(),X) = [8] X + [4] >= [8] X + [0] = mark(X) a__first(X1,X2) = [1] X1 + [1] X2 + [7] >= [1] X1 + [1] X2 + [1] = first(X1,X2) a__first(0(),X) = [1] X + [7] >= [0] = nil() a__first(s(X),cons(Y,Z)) = [10] >= [0] = cons(Y,first(X,Z)) a__from(X) = [1] X + [0] >= [0] = cons(X,from(s(X))) a__from(X) = [1] X + [0] >= [1] X + [0] = from(X) a__if(X1,X2,X3) = [1] X1 + [8] X2 + [8] X3 + [12] >= [1] X1 + [1] X2 + [1] X3 + [2] = if(X1,X2,X3) a__if(false(),X,Y) = [8] X + [8] Y + [13] >= [8] Y + [0] = mark(Y) a__if(true(),X,Y) = [8] X + [8] Y + [12] >= [8] X + [0] = mark(X) mark(0()) = [0] >= [0] = 0() mark(add(X1,X2)) = [8] X1 + [8] X2 + [0] >= [8] X1 + [8] X2 + [0] = a__add(mark(X1),X2) mark(and(X1,X2)) = [8] X1 + [8] X2 + [8] >= [8] X1 + [8] X2 + [4] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [0] = cons(X1,X2) mark(false()) = [8] >= [1] = false() mark(from(X)) = [8] X + [0] >= [1] X + [0] = a__from(X) mark(if(X1,X2,X3)) = [8] X1 + [8] X2 + [8] X3 + [16] >= [8] X1 + [8] X2 + [8] X3 + [12] = a__if(mark(X1),X2,X3) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [24] >= [3] = s(X) mark(true()) = [0] >= [0] = true() ** Step 1.b:7: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) - Weak TRS: a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if,mark} and constructors {0,add ,and,cons,false,first,from,if,nil,s,true} + 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(a__add) = {1}, uargs(a__and) = {1}, uargs(a__first) = {1,2}, uargs(a__if) = {1}, uargs(add) = {1}, uargs(and) = {1}, uargs(first) = {1,2}, uargs(if) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [5] p(a__add) = [1] x1 + [2] x2 + [8] p(a__and) = [1] x1 + [2] x2 + [8] p(a__first) = [1] x1 + [1] x2 + [0] p(a__from) = [9] p(a__if) = [1] x1 + [2] x2 + [2] x3 + [15] p(add) = [1] x1 + [1] x2 + [4] p(and) = [1] x1 + [1] x2 + [4] p(cons) = [0] p(false) = [4] p(first) = [1] x1 + [1] x2 + [0] p(from) = [9] p(if) = [1] x1 + [1] x2 + [1] x3 + [12] p(mark) = [2] x1 + [0] p(nil) = [3] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: a__add(X1,X2) = [1] X1 + [2] X2 + [8] > [1] X1 + [1] X2 + [4] = add(X1,X2) Following rules are (at-least) weakly oriented: a__add(0(),X) = [2] X + [13] >= [2] X + [0] = mark(X) a__add(s(X),Y) = [1] X + [2] Y + [8] >= [1] X + [1] Y + [4] = s(add(X,Y)) a__and(X1,X2) = [1] X1 + [2] X2 + [8] >= [1] X1 + [1] X2 + [4] = and(X1,X2) a__and(false(),Y) = [2] Y + [12] >= [4] = false() a__and(true(),X) = [2] X + [8] >= [2] X + [0] = mark(X) a__first(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = first(X1,X2) a__first(0(),X) = [1] X + [5] >= [3] = nil() a__first(s(X),cons(Y,Z)) = [1] X + [0] >= [0] = cons(Y,first(X,Z)) a__from(X) = [9] >= [0] = cons(X,from(s(X))) a__from(X) = [9] >= [9] = from(X) a__if(X1,X2,X3) = [1] X1 + [2] X2 + [2] X3 + [15] >= [1] X1 + [1] X2 + [1] X3 + [12] = if(X1,X2,X3) a__if(false(),X,Y) = [2] X + [2] Y + [19] >= [2] Y + [0] = mark(Y) a__if(true(),X,Y) = [2] X + [2] Y + [15] >= [2] X + [0] = mark(X) mark(0()) = [10] >= [5] = 0() mark(add(X1,X2)) = [2] X1 + [2] X2 + [8] >= [2] X1 + [2] X2 + [8] = a__add(mark(X1),X2) mark(and(X1,X2)) = [2] X1 + [2] X2 + [8] >= [2] X1 + [2] X2 + [8] = a__and(mark(X1),X2) mark(cons(X1,X2)) = [0] >= [0] = cons(X1,X2) mark(false()) = [8] >= [4] = false() mark(first(X1,X2)) = [2] X1 + [2] X2 + [0] >= [2] X1 + [2] X2 + [0] = a__first(mark(X1),mark(X2)) mark(from(X)) = [18] >= [9] = a__from(X) mark(if(X1,X2,X3)) = [2] X1 + [2] X2 + [2] X3 + [24] >= [2] X1 + [2] X2 + [2] X3 + [15] = a__if(mark(X1),X2,X3) mark(nil()) = [6] >= [3] = nil() mark(s(X)) = [2] X + [0] >= [1] X + [0] = s(X) mark(true()) = [0] >= [0] = true() ** Step 1.b:8: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if,mark} and constructors {0,add ,and,cons,false,first,from,if,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))