WORST_CASE(?,O(n^1)) * Step 1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: ++(x,g(y,z)) -> g(++(x,y),z) ++(x,nil()) -> x f(x,g(y,z)) -> g(f(x,y),z) f(x,nil()) -> g(nil(),x) max(g(g(g(x,y),z),u())) -> max'(max(g(g(x,y),z)),u()) max(g(g(nil(),x),y)) -> max'(x,y) mem(x,max(x)) -> not(null(x)) mem(g(x,y),z) -> or(=(y,z),mem(x,z)) mem(nil(),y) -> false() null(g(x,y)) -> false() null(nil()) -> true() - Signature: {++/2,f/2,max/1,mem/2,null/1} / {=/2,false/0,g/2,max'/2,nil/0,not/1,or/2,true/0,u/0} - Obligation: innermost runtime complexity wrt. defined symbols {++,f,max,mem,null} and constructors {=,false,g,max',nil ,not,or,true,u} + 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(g) = {1}, uargs(max') = {1}, uargs(not) = {1}, uargs(or) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(++) = [1] x1 + [10] x2 + [0] p(=) = [1] x1 + [1] x2 + [0] p(f) = [1] x1 + [9] x2 + [0] p(false) = [0] p(g) = [1] x1 + [1] x2 + [1] p(max) = [8] x1 + [7] p(max') = [1] x1 + [1] x2 + [2] p(mem) = [11] x1 + [0] p(nil) = [0] p(not) = [1] x1 + [0] p(null) = [11] x1 + [0] p(or) = [1] x2 + [0] p(true) = [0] p(u) = [0] Following rules are strictly oriented: ++(x,g(y,z)) = [1] x + [10] y + [10] z + [10] > [1] x + [10] y + [1] z + [1] = g(++(x,y),z) f(x,g(y,z)) = [1] x + [9] y + [9] z + [9] > [1] x + [9] y + [1] z + [1] = g(f(x,y),z) max(g(g(g(x,y),z),u())) = [8] x + [8] y + [8] z + [31] > [8] x + [8] y + [8] z + [25] = max'(max(g(g(x,y),z)),u()) max(g(g(nil(),x),y)) = [8] x + [8] y + [23] > [1] x + [1] y + [2] = max'(x,y) mem(g(x,y),z) = [11] x + [11] y + [11] > [11] x + [0] = or(=(y,z),mem(x,z)) null(g(x,y)) = [11] x + [11] y + [11] > [0] = false() Following rules are (at-least) weakly oriented: ++(x,nil()) = [1] x + [0] >= [1] x + [0] = x f(x,nil()) = [1] x + [0] >= [1] x + [1] = g(nil(),x) mem(x,max(x)) = [11] x + [0] >= [11] x + [0] = not(null(x)) mem(nil(),y) = [0] >= [0] = false() null(nil()) = [0] >= [0] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: ++(x,nil()) -> x f(x,nil()) -> g(nil(),x) mem(x,max(x)) -> not(null(x)) mem(nil(),y) -> false() null(nil()) -> true() - Weak TRS: ++(x,g(y,z)) -> g(++(x,y),z) f(x,g(y,z)) -> g(f(x,y),z) max(g(g(g(x,y),z),u())) -> max'(max(g(g(x,y),z)),u()) max(g(g(nil(),x),y)) -> max'(x,y) mem(g(x,y),z) -> or(=(y,z),mem(x,z)) null(g(x,y)) -> false() - Signature: {++/2,f/2,max/1,mem/2,null/1} / {=/2,false/0,g/2,max'/2,nil/0,not/1,or/2,true/0,u/0} - Obligation: innermost runtime complexity wrt. defined symbols {++,f,max,mem,null} and constructors {=,false,g,max',nil ,not,or,true,u} + 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(g) = {1}, uargs(max') = {1}, uargs(not) = {1}, uargs(or) = {2} Following symbols are considered usable: {++,f,max,mem,null} TcT has computed the following interpretation: p(++) = [8] x1 + [2] x2 + [2] p(=) = [0] p(f) = [8] x1 + [2] x2 + [8] p(false) = [0] p(g) = [1] x1 + [1] x2 + [0] p(max) = [8] x1 + [0] p(max') = [1] x1 + [1] x2 + [0] p(mem) = [14] x1 + [2] x2 + [0] p(nil) = [2] p(not) = [1] x1 + [0] p(null) = [1] x1 + [0] p(or) = [1] x1 + [1] x2 + [0] p(true) = [2] p(u) = [0] Following rules are strictly oriented: ++(x,nil()) = [8] x + [6] > [1] x + [0] = x f(x,nil()) = [8] x + [12] > [1] x + [2] = g(nil(),x) mem(nil(),y) = [2] y + [28] > [0] = false() Following rules are (at-least) weakly oriented: ++(x,g(y,z)) = [8] x + [2] y + [2] z + [2] >= [8] x + [2] y + [1] z + [2] = g(++(x,y),z) f(x,g(y,z)) = [8] x + [2] y + [2] z + [8] >= [8] x + [2] y + [1] z + [8] = g(f(x,y),z) max(g(g(g(x,y),z),u())) = [8] x + [8] y + [8] z + [0] >= [8] x + [8] y + [8] z + [0] = max'(max(g(g(x,y),z)),u()) max(g(g(nil(),x),y)) = [8] x + [8] y + [16] >= [1] x + [1] y + [0] = max'(x,y) mem(x,max(x)) = [30] x + [0] >= [1] x + [0] = not(null(x)) mem(g(x,y),z) = [14] x + [14] y + [2] z + [0] >= [14] x + [2] z + [0] = or(=(y,z),mem(x,z)) null(g(x,y)) = [1] x + [1] y + [0] >= [0] = false() null(nil()) = [2] >= [2] = true() * Step 3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: mem(x,max(x)) -> not(null(x)) null(nil()) -> true() - Weak TRS: ++(x,g(y,z)) -> g(++(x,y),z) ++(x,nil()) -> x f(x,g(y,z)) -> g(f(x,y),z) f(x,nil()) -> g(nil(),x) max(g(g(g(x,y),z),u())) -> max'(max(g(g(x,y),z)),u()) max(g(g(nil(),x),y)) -> max'(x,y) mem(g(x,y),z) -> or(=(y,z),mem(x,z)) mem(nil(),y) -> false() null(g(x,y)) -> false() - Signature: {++/2,f/2,max/1,mem/2,null/1} / {=/2,false/0,g/2,max'/2,nil/0,not/1,or/2,true/0,u/0} - Obligation: innermost runtime complexity wrt. defined symbols {++,f,max,mem,null} and constructors {=,false,g,max',nil ,not,or,true,u} + 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(g) = {1}, uargs(max') = {1}, uargs(not) = {1}, uargs(or) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(++) = [4] x1 + [1] x2 + [0] p(=) = [1] x1 + [1] x2 + [0] p(f) = [1] x1 + [4] x2 + [3] p(false) = [3] p(g) = [1] x1 + [1] x2 + [3] p(max) = [1] x1 + [4] p(max') = [1] x1 + [2] p(mem) = [4] x1 + [3] p(nil) = [0] p(not) = [1] x1 + [0] p(null) = [4] x1 + [1] p(or) = [1] x2 + [0] p(true) = [0] p(u) = [1] Following rules are strictly oriented: mem(x,max(x)) = [4] x + [3] > [4] x + [1] = not(null(x)) null(nil()) = [1] > [0] = true() Following rules are (at-least) weakly oriented: ++(x,g(y,z)) = [4] x + [1] y + [1] z + [3] >= [4] x + [1] y + [1] z + [3] = g(++(x,y),z) ++(x,nil()) = [4] x + [0] >= [1] x + [0] = x f(x,g(y,z)) = [1] x + [4] y + [4] z + [15] >= [1] x + [4] y + [1] z + [6] = g(f(x,y),z) f(x,nil()) = [1] x + [3] >= [1] x + [3] = g(nil(),x) max(g(g(g(x,y),z),u())) = [1] x + [1] y + [1] z + [14] >= [1] x + [1] y + [1] z + [12] = max'(max(g(g(x,y),z)),u()) max(g(g(nil(),x),y)) = [1] x + [1] y + [10] >= [1] x + [2] = max'(x,y) mem(g(x,y),z) = [4] x + [4] y + [15] >= [4] x + [3] = or(=(y,z),mem(x,z)) mem(nil(),y) = [3] >= [3] = false() null(g(x,y)) = [4] x + [4] y + [13] >= [3] = false() 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: ++(x,g(y,z)) -> g(++(x,y),z) ++(x,nil()) -> x f(x,g(y,z)) -> g(f(x,y),z) f(x,nil()) -> g(nil(),x) max(g(g(g(x,y),z),u())) -> max'(max(g(g(x,y),z)),u()) max(g(g(nil(),x),y)) -> max'(x,y) mem(x,max(x)) -> not(null(x)) mem(g(x,y),z) -> or(=(y,z),mem(x,z)) mem(nil(),y) -> false() null(g(x,y)) -> false() null(nil()) -> true() - Signature: {++/2,f/2,max/1,mem/2,null/1} / {=/2,false/0,g/2,max'/2,nil/0,not/1,or/2,true/0,u/0} - Obligation: innermost runtime complexity wrt. defined symbols {++,f,max,mem,null} and constructors {=,false,g,max',nil ,not,or,true,u} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))