WORST_CASE(?,O(n^1)) * Step 1: Sum. 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: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: NaturalPI. 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: 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} Following symbols are considered usable: {append,hd,ifappend,is_empty,tl} TcT has computed the following interpretation: p(append) = 1 + x1 + 2*x2 p(cons) = 1 + x1 + x2 p(false) = 0 p(hd) = 6 + 2*x1 p(ifappend) = 1 + 2*x2 + x3 p(is_empty) = 15 p(nil) = 0 p(tl) = 12 + 2*x1 p(true) = 11 Following rules are strictly oriented: hd(cons(x,l)) = 8 + 2*l + 2*x > x = x ifappend(l1,l2,nil()) = 1 + 2*l2 > l2 = l2 is_empty(cons(x,l)) = 15 > 0 = false() is_empty(nil()) = 15 > 11 = true() tl(cons(x,l)) = 14 + 2*l + 2*x > l = l Following rules are (at-least) weakly oriented: append(l1,l2) = 1 + l1 + 2*l2 >= 1 + l1 + 2*l2 = ifappend(l1,l2,l1) ifappend(l1,l2,cons(x,l)) = 2 + l + 2*l2 + x >= 2 + l + 2*l2 + x = cons(x,append(l,l2)) * Step 3: 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) = [4] x1 + [1] x2 + [4] p(cons) = [1] x1 + [1] x2 + [4] p(false) = [5] p(hd) = [4] x1 + [0] p(ifappend) = [1] x2 + [4] x3 + [0] p(is_empty) = [1] x1 + [1] p(nil) = [0] p(tl) = [1] x1 + [1] p(true) = [0] Following rules are strictly oriented: append(l1,l2) = [4] l1 + [1] l2 + [4] > [4] l1 + [1] l2 + [0] = ifappend(l1,l2,l1) ifappend(l1,l2,cons(x,l)) = [4] l + [1] l2 + [4] x + [16] > [4] l + [1] l2 + [1] x + [8] = cons(x,append(l,l2)) Following rules are (at-least) weakly oriented: hd(cons(x,l)) = [4] l + [4] x + [16] >= [1] x + [0] = x ifappend(l1,l2,nil()) = [1] l2 + [0] >= [1] l2 + [0] = l2 is_empty(cons(x,l)) = [1] l + [1] x + [5] >= [5] = false() is_empty(nil()) = [1] >= [0] = true() tl(cons(x,l)) = [1] l + [1] x + [5] >= [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))