/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: append(l1,l2) -> ifappend(l1,l2,l1) hd(cons(x,l)) -> x ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2)) ifappend(l1,l2,nil()) -> l2 is_empty(cons(x,l)) -> false() is_empty(nil()) -> true() tl(cons(x,l)) -> l - Signature: {append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0} - Obligation: runtime complexity wrt. defined symbols {append,hd,ifappend,is_empty,tl} and constructors {cons,false,nil ,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: append(l1,l2) -> ifappend(l1,l2,l1) hd(cons(x,l)) -> x ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2)) ifappend(l1,l2,nil()) -> l2 is_empty(cons(x,l)) -> false() is_empty(nil()) -> true() tl(cons(x,l)) -> l - Signature: {append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0} - Obligation: runtime complexity wrt. defined symbols {append,hd,ifappend,is_empty,tl} and constructors {cons,false,nil ,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: append(l1,l2) -> ifappend(l1,l2,l1) hd(cons(x,l)) -> x ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2)) ifappend(l1,l2,nil()) -> l2 is_empty(cons(x,l)) -> false() is_empty(nil()) -> true() tl(cons(x,l)) -> l - Signature: {append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0} - Obligation: runtime complexity wrt. defined symbols {append,hd,ifappend,is_empty,tl} and constructors {cons,false,nil ,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: append(x_6,z){x_6 -> cons(v,x_6)} = append(cons(v,x_6),z) ->^+ cons(v,append(x_6,z)) = C[append(x_6,z) = append(x_6,z){}] ** Step 1.b:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: append(l1,l2) -> ifappend(l1,l2,l1) hd(cons(x,l)) -> x ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2)) ifappend(l1,l2,nil()) -> l2 is_empty(cons(x,l)) -> false() is_empty(nil()) -> true() tl(cons(x,l)) -> l - Signature: {append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0} - Obligation: runtime complexity wrt. defined symbols {append,hd,ifappend,is_empty,tl} and constructors {cons,false,nil ,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(cons) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(append) = [4] x1 + [8] x2 + [2] p(cons) = [1] x1 + [1] x2 + [1] p(false) = [4] p(hd) = [4] x1 + [4] p(ifappend) = [8] x2 + [4] x3 + [2] p(is_empty) = [1] x1 + [14] p(nil) = [2] p(tl) = [4] x1 + [1] p(true) = [1] Following rules are strictly oriented: hd(cons(x,l)) = [4] l + [4] x + [8] > [1] x + [0] = x ifappend(l1,l2,cons(x,l)) = [4] l + [8] l2 + [4] x + [6] > [4] l + [8] l2 + [1] x + [3] = cons(x,append(l,l2)) ifappend(l1,l2,nil()) = [8] l2 + [10] > [1] l2 + [0] = l2 is_empty(cons(x,l)) = [1] l + [1] x + [15] > [4] = false() is_empty(nil()) = [16] > [1] = true() tl(cons(x,l)) = [4] l + [4] x + [5] > [1] l + [0] = l Following rules are (at-least) weakly oriented: append(l1,l2) = [4] l1 + [8] l2 + [2] >= [4] l1 + [8] l2 + [2] = ifappend(l1,l2,l1) ** Step 1.b:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: append(l1,l2) -> ifappend(l1,l2,l1) - Weak TRS: hd(cons(x,l)) -> x ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2)) ifappend(l1,l2,nil()) -> l2 is_empty(cons(x,l)) -> false() is_empty(nil()) -> true() tl(cons(x,l)) -> l - Signature: {append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0} - Obligation: runtime complexity wrt. defined symbols {append,hd,ifappend,is_empty,tl} and constructors {cons,false,nil ,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(cons) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(append) = [2] x1 + [8] x2 + [8] p(cons) = [1] x1 + [1] x2 + [8] p(false) = [1] p(hd) = [2] x1 + [11] p(ifappend) = [8] x2 + [2] x3 + [0] p(is_empty) = [2] x1 + [0] p(nil) = [9] p(tl) = [2] x1 + [6] p(true) = [1] Following rules are strictly oriented: append(l1,l2) = [2] l1 + [8] l2 + [8] > [2] l1 + [8] l2 + [0] = ifappend(l1,l2,l1) Following rules are (at-least) weakly oriented: hd(cons(x,l)) = [2] l + [2] x + [27] >= [1] x + [0] = x ifappend(l1,l2,cons(x,l)) = [2] l + [8] l2 + [2] x + [16] >= [2] l + [8] l2 + [1] x + [16] = cons(x,append(l,l2)) ifappend(l1,l2,nil()) = [8] l2 + [18] >= [1] l2 + [0] = l2 is_empty(cons(x,l)) = [2] l + [2] x + [16] >= [1] = false() is_empty(nil()) = [18] >= [1] = true() tl(cons(x,l)) = [2] l + [2] x + [22] >= [1] l + [0] = l ** Step 1.b:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: append(l1,l2) -> ifappend(l1,l2,l1) hd(cons(x,l)) -> x ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2)) ifappend(l1,l2,nil()) -> l2 is_empty(cons(x,l)) -> false() is_empty(nil()) -> true() tl(cons(x,l)) -> l - Signature: {append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0} - Obligation: runtime complexity wrt. defined symbols {append,hd,ifappend,is_empty,tl} and constructors {cons,false,nil ,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))