WORST_CASE(?,O(n^1)) * Step 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: innermost 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: {append,hd,ifappend,is_empty,tl} TcT has computed the following interpretation: p(append) = [2] x1 + [1] x2 + [0] p(cons) = [1] x1 + [1] x2 + [0] p(false) = [5] p(hd) = [1] x1 + [4] p(ifappend) = [1] x2 + [2] x3 + [0] p(is_empty) = [2] x1 + [9] p(nil) = [5] p(tl) = [4] x1 + [5] p(true) = [1] Following rules are strictly oriented: hd(cons(x,l)) = [1] l + [1] x + [4] > [1] x + [0] = x ifappend(l1,l2,nil()) = [1] l2 + [10] > [1] l2 + [0] = l2 is_empty(cons(x,l)) = [2] l + [2] x + [9] > [5] = false() is_empty(nil()) = [19] > [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) = [2] l1 + [1] l2 + [0] >= [2] l1 + [1] l2 + [0] = ifappend(l1,l2,l1) ifappend(l1,l2,cons(x,l)) = [2] l + [1] l2 + [2] x + [0] >= [2] l + [1] l2 + [1] x + [0] = cons(x,append(l,l2)) * Step 2: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: append(l1,l2) -> ifappend(l1,l2,l1) ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2)) - Weak TRS: hd(cons(x,l)) -> x 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: innermost 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: {append,hd,ifappend,is_empty,tl} TcT has computed the following interpretation: p(append) = [2] x1 + [2] x2 + [1] p(cons) = [1] x1 + [1] x2 + [8] p(false) = [0] p(hd) = [1] x1 + [0] p(ifappend) = [2] x2 + [2] x3 + [1] p(is_empty) = [0] p(nil) = [5] p(tl) = [1] x1 + [8] p(true) = [0] Following rules are strictly oriented: ifappend(l1,l2,cons(x,l)) = [2] l + [2] l2 + [2] x + [17] > [2] l + [2] l2 + [1] x + [9] = cons(x,append(l,l2)) Following rules are (at-least) weakly oriented: append(l1,l2) = [2] l1 + [2] l2 + [1] >= [2] l1 + [2] l2 + [1] = ifappend(l1,l2,l1) hd(cons(x,l)) = [1] l + [1] x + [8] >= [1] x + [0] = x ifappend(l1,l2,nil()) = [2] l2 + [11] >= [1] l2 + [0] = l2 is_empty(cons(x,l)) = [0] >= [0] = false() is_empty(nil()) = [0] >= [0] = true() tl(cons(x,l)) = [1] l + [1] x + [16] >= [1] l + [0] = l * Step 3: 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: innermost 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: {append,hd,ifappend,is_empty,tl} TcT has computed the following interpretation: p(append) = [8] x1 + [1] x2 + [7] p(cons) = [1] x1 + [1] x2 + [1] p(false) = [0] p(hd) = [14] x1 + [0] p(ifappend) = [1] x2 + [8] x3 + [5] p(is_empty) = [1] x1 + [15] p(nil) = [1] p(tl) = [5] x1 + [7] p(true) = [0] Following rules are strictly oriented: append(l1,l2) = [8] l1 + [1] l2 + [7] > [8] l1 + [1] l2 + [5] = ifappend(l1,l2,l1) Following rules are (at-least) weakly oriented: hd(cons(x,l)) = [14] l + [14] x + [14] >= [1] x + [0] = x ifappend(l1,l2,cons(x,l)) = [8] l + [1] l2 + [8] x + [13] >= [8] l + [1] l2 + [1] x + [8] = cons(x,append(l,l2)) ifappend(l1,l2,nil()) = [1] l2 + [13] >= [1] l2 + [0] = l2 is_empty(cons(x,l)) = [1] l + [1] x + [16] >= [0] = false() is_empty(nil()) = [16] >= [0] = true() tl(cons(x,l)) = [5] l + [5] x + [12] >= [1] l + [0] = l * Step 4: 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: innermost 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(?,O(n^1))