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: WeightGap 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: 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 + [1] x2 + [6] p(=) = [1] x1 + [1] x2 + [0] p(f) = [1] x1 + [3] x2 + [1] p(false) = [0] p(g) = [1] x1 + [1] x2 + [0] p(max) = [2] x1 + [0] p(max') = [1] x1 + [0] p(mem) = [4] x1 + [2] x2 + [2] p(nil) = [2] p(not) = [1] x1 + [4] p(null) = [1] x1 + [4] p(or) = [1] x2 + [0] p(true) = [1] p(u) = [4] Following rules are strictly oriented: ++(x,nil()) = [1] x + [8] > [1] x + [0] = x f(x,nil()) = [1] x + [7] > [1] x + [2] = g(nil(),x) mem(nil(),y) = [2] y + [10] > [0] = false() null(nil()) = [6] > [1] = true() Following rules are (at-least) weakly oriented: ++(x,g(y,z)) = [1] x + [1] y + [1] z + [6] >= [1] x + [1] y + [1] z + [6] = g(++(x,y),z) f(x,g(y,z)) = [1] x + [3] y + [3] z + [1] >= [1] x + [3] y + [1] z + [1] = g(f(x,y),z) max(g(g(g(x,y),z),u())) = [2] x + [2] y + [2] z + [8] >= [2] x + [2] y + [2] z + [0] = max'(max(g(g(x,y),z)),u()) max(g(g(nil(),x),y)) = [2] x + [2] y + [4] >= [1] x + [0] = max'(x,y) mem(x,max(x)) = [8] x + [2] >= [1] x + [8] = not(null(x)) mem(g(x,y),z) = [4] x + [4] y + [2] z + [2] >= [4] x + [2] z + [2] = or(=(y,z),mem(x,z)) null(g(x,y)) = [1] x + [1] y + [4] >= [0] = false() 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: mem(x,max(x)) -> not(null(x)) - 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() 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 + [2] x2 + [7] p(=) = [2] p(f) = [1] x1 + [4] x2 + [3] p(false) = [0] p(g) = [1] x1 + [1] x2 + [2] p(max) = [1] x1 + [1] p(max') = [1] x1 + [2] p(mem) = [7] x2 + [0] p(nil) = [2] p(not) = [1] x1 + [1] p(null) = [4] x1 + [1] p(or) = [1] x2 + [0] p(true) = [0] p(u) = [0] Following rules are strictly oriented: mem(x,max(x)) = [7] x + [7] > [4] x + [2] = not(null(x)) Following rules are (at-least) weakly oriented: ++(x,g(y,z)) = [1] x + [2] y + [2] z + [11] >= [1] x + [2] y + [1] z + [9] = g(++(x,y),z) ++(x,nil()) = [1] x + [11] >= [1] x + [0] = x f(x,g(y,z)) = [1] x + [4] y + [4] z + [11] >= [1] x + [4] y + [1] z + [5] = g(f(x,y),z) f(x,nil()) = [1] x + [11] >= [1] x + [4] = g(nil(),x) max(g(g(g(x,y),z),u())) = [1] x + [1] y + [1] z + [7] >= [1] x + [1] y + [1] z + [7] = max'(max(g(g(x,y),z)),u()) max(g(g(nil(),x),y)) = [1] x + [1] y + [7] >= [1] x + [2] = max'(x,y) mem(g(x,y),z) = [7] z + [0] >= [7] z + [0] = or(=(y,z),mem(x,z)) mem(nil(),y) = [7] y + [0] >= [0] = false() null(g(x,y)) = [4] x + [4] y + [9] >= [0] = false() null(nil()) = [9] >= [0] = true() 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))