WORST_CASE(?,O(n^1)) * Step 1: WeightGap 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: 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) = [1] x1 + [2] x2 + [0] p(cons) = [1] x1 + [1] x2 + [3] p(false) = [0] p(hd) = [2] x1 + [0] p(ifappend) = [2] x2 + [1] x3 + [0] p(is_empty) = [0] p(nil) = [0] p(tl) = [7] x1 + [0] p(true) = [0] Following rules are strictly oriented: hd(cons(x,l)) = [2] l + [2] x + [6] > [1] x + [0] = x tl(cons(x,l)) = [7] l + [7] x + [21] > [1] l + [0] = l Following rules are (at-least) weakly oriented: append(l1,l2) = [1] l1 + [2] l2 + [0] >= [1] l1 + [2] l2 + [0] = ifappend(l1,l2,l1) ifappend(l1,l2,cons(x,l)) = [1] l + [2] l2 + [1] x + [3] >= [1] l + [2] l2 + [1] x + [3] = cons(x,append(l,l2)) ifappend(l1,l2,nil()) = [2] l2 + [0] >= [1] l2 + [0] = l2 is_empty(cons(x,l)) = [0] >= [0] = false() is_empty(nil()) = [0] >= [0] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap 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)) ifappend(l1,l2,nil()) -> l2 is_empty(cons(x,l)) -> false() is_empty(nil()) -> true() - Weak TRS: hd(cons(x,l)) -> x 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: 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) = [3] x1 + [1] x2 + [0] p(cons) = [1] x1 + [1] x2 + [1] p(false) = [0] p(hd) = [1] x1 + [12] p(ifappend) = [1] x2 + [3] x3 + [0] p(is_empty) = [8] p(nil) = [7] p(tl) = [8] x1 + [8] p(true) = [0] Following rules are strictly oriented: ifappend(l1,l2,cons(x,l)) = [3] l + [1] l2 + [3] x + [3] > [3] l + [1] l2 + [1] x + [1] = cons(x,append(l,l2)) ifappend(l1,l2,nil()) = [1] l2 + [21] > [1] l2 + [0] = l2 is_empty(cons(x,l)) = [8] > [0] = false() is_empty(nil()) = [8] > [0] = true() Following rules are (at-least) weakly oriented: append(l1,l2) = [3] l1 + [1] l2 + [0] >= [3] l1 + [1] l2 + [0] = ifappend(l1,l2,l1) hd(cons(x,l)) = [1] l + [1] x + [13] >= [1] x + [0] = x tl(cons(x,l)) = [8] l + [8] x + [16] >= [1] l + [0] = l Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: 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 + [2] x2 + [11] p(cons) = [1] x1 + [1] x2 + [2] p(false) = [10] p(hd) = [4] x1 + [10] p(ifappend) = [2] x2 + [4] x3 + [5] p(is_empty) = [4] x1 + [2] p(nil) = [0] p(tl) = [1] x1 + [0] p(true) = [2] Following rules are strictly oriented: append(l1,l2) = [4] l1 + [2] l2 + [11] > [4] l1 + [2] l2 + [5] = ifappend(l1,l2,l1) Following rules are (at-least) weakly oriented: hd(cons(x,l)) = [4] l + [4] x + [18] >= [1] x + [0] = x ifappend(l1,l2,cons(x,l)) = [4] l + [2] l2 + [4] x + [13] >= [4] l + [2] l2 + [1] x + [13] = cons(x,append(l,l2)) ifappend(l1,l2,nil()) = [2] l2 + [5] >= [1] l2 + [0] = l2 is_empty(cons(x,l)) = [4] l + [4] x + [10] >= [10] = false() is_empty(nil()) = [2] >= [2] = true() tl(cons(x,l)) = [1] l + [1] x + [2] >= [1] l + [0] = l Further, it can be verified that all rules not oriented are covered by the weightgap condition. * 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))