* Step 1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: active(c()) -> mark(a()) active(c()) -> mark(b()) f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3)) f(mark(X1),X2,X3) -> mark(f(X1,X2,X3)) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) proper(c()) -> ok(c()) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,f/3,proper/1,top/1} / {a/0,b/0,c/0,mark/1,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,c,mark,ok} + 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(mark) = {1}, uargs(ok) = {1}, uargs(top) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [0] p(active) = [1] x1 + [0] p(b) = [11] p(c) = [9] p(f) = [1] x1 + [5] x2 + [0] p(mark) = [1] x1 + [4] p(ok) = [1] x1 + [5] p(proper) = [1] x1 + [0] p(top) = [1] x1 + [0] Following rules are strictly oriented: active(c()) = [9] > [4] = mark(a()) f(ok(X1),ok(X2),ok(X3)) = [1] X1 + [5] X2 + [30] > [1] X1 + [5] X2 + [5] = ok(f(X1,X2,X3)) top(mark(X)) = [1] X + [4] > [1] X + [0] = top(proper(X)) top(ok(X)) = [1] X + [5] > [1] X + [0] = top(active(X)) Following rules are (at-least) weakly oriented: active(c()) = [9] >= [15] = mark(b()) f(X1,X2,mark(X3)) = [1] X1 + [5] X2 + [0] >= [1] X1 + [5] X2 + [4] = mark(f(X1,X2,X3)) f(mark(X1),X2,X3) = [1] X1 + [5] X2 + [4] >= [1] X1 + [5] X2 + [4] = mark(f(X1,X2,X3)) proper(a()) = [0] >= [5] = ok(a()) proper(b()) = [11] >= [16] = ok(b()) proper(c()) = [9] >= [14] = ok(c()) 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: active(c()) -> mark(b()) f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3)) f(mark(X1),X2,X3) -> mark(f(X1,X2,X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) proper(c()) -> ok(c()) - Weak TRS: active(c()) -> mark(a()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,f/3,proper/1,top/1} / {a/0,b/0,c/0,mark/1,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,c,mark,ok} + 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(mark) = {1}, uargs(ok) = {1}, uargs(top) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [10] p(active) = [1] x1 + [9] p(b) = [0] p(c) = [11] p(f) = [1] x1 + [0] p(mark) = [1] x1 + [10] p(ok) = [1] x1 + [9] p(proper) = [1] x1 + [10] p(top) = [1] x1 + [0] Following rules are strictly oriented: active(c()) = [20] > [10] = mark(b()) proper(a()) = [20] > [19] = ok(a()) proper(b()) = [10] > [9] = ok(b()) proper(c()) = [21] > [20] = ok(c()) Following rules are (at-least) weakly oriented: active(c()) = [20] >= [20] = mark(a()) f(X1,X2,mark(X3)) = [1] X1 + [0] >= [1] X1 + [10] = mark(f(X1,X2,X3)) f(mark(X1),X2,X3) = [1] X1 + [10] >= [1] X1 + [10] = mark(f(X1,X2,X3)) f(ok(X1),ok(X2),ok(X3)) = [1] X1 + [9] >= [1] X1 + [9] = ok(f(X1,X2,X3)) top(mark(X)) = [1] X + [10] >= [1] X + [10] = top(proper(X)) top(ok(X)) = [1] X + [9] >= [1] X + [9] = top(active(X)) 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: f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3)) f(mark(X1),X2,X3) -> mark(f(X1,X2,X3)) - Weak TRS: active(c()) -> mark(a()) active(c()) -> mark(b()) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) proper(c()) -> ok(c()) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,f/3,proper/1,top/1} / {a/0,b/0,c/0,mark/1,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,c,mark,ok} + 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(mark) = {1}, uargs(ok) = {1}, uargs(top) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [1] p(active) = [1] x1 + [0] p(b) = [4] p(c) = [7] p(f) = [6] x1 + [9] x2 + [13] p(mark) = [1] x1 + [3] p(ok) = [1] x1 + [1] p(proper) = [1] x1 + [1] p(top) = [1] x1 + [0] Following rules are strictly oriented: f(mark(X1),X2,X3) = [6] X1 + [9] X2 + [31] > [6] X1 + [9] X2 + [16] = mark(f(X1,X2,X3)) Following rules are (at-least) weakly oriented: active(c()) = [7] >= [4] = mark(a()) active(c()) = [7] >= [7] = mark(b()) f(X1,X2,mark(X3)) = [6] X1 + [9] X2 + [13] >= [6] X1 + [9] X2 + [16] = mark(f(X1,X2,X3)) f(ok(X1),ok(X2),ok(X3)) = [6] X1 + [9] X2 + [28] >= [6] X1 + [9] X2 + [14] = ok(f(X1,X2,X3)) proper(a()) = [2] >= [2] = ok(a()) proper(b()) = [5] >= [5] = ok(b()) proper(c()) = [8] >= [8] = ok(c()) top(mark(X)) = [1] X + [3] >= [1] X + [1] = top(proper(X)) top(ok(X)) = [1] X + [1] >= [1] X + [0] = top(active(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3)) - Weak TRS: active(c()) -> mark(a()) active(c()) -> mark(b()) f(mark(X1),X2,X3) -> mark(f(X1,X2,X3)) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) proper(c()) -> ok(c()) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,f/3,proper/1,top/1} / {a/0,b/0,c/0,mark/1,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,c,mark,ok} + 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(mark) = {1}, uargs(ok) = {1}, uargs(top) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [0] p(active) = [1] x1 + [0] p(b) = [0] p(c) = [1] p(f) = [2] x1 + [4] x3 + [0] p(mark) = [1] x1 + [1] p(ok) = [1] x1 + [0] p(proper) = [1] p(top) = [1] x1 + [8] Following rules are strictly oriented: f(X1,X2,mark(X3)) = [2] X1 + [4] X3 + [4] > [2] X1 + [4] X3 + [1] = mark(f(X1,X2,X3)) Following rules are (at-least) weakly oriented: active(c()) = [1] >= [1] = mark(a()) active(c()) = [1] >= [1] = mark(b()) f(mark(X1),X2,X3) = [2] X1 + [4] X3 + [2] >= [2] X1 + [4] X3 + [1] = mark(f(X1,X2,X3)) f(ok(X1),ok(X2),ok(X3)) = [2] X1 + [4] X3 + [0] >= [2] X1 + [4] X3 + [0] = ok(f(X1,X2,X3)) proper(a()) = [1] >= [0] = ok(a()) proper(b()) = [1] >= [0] = ok(b()) proper(c()) = [1] >= [1] = ok(c()) top(mark(X)) = [1] X + [9] >= [9] = top(proper(X)) top(ok(X)) = [1] X + [8] >= [1] X + [8] = top(active(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: active(c()) -> mark(a()) active(c()) -> mark(b()) f(X1,X2,mark(X3)) -> mark(f(X1,X2,X3)) f(mark(X1),X2,X3) -> mark(f(X1,X2,X3)) f(ok(X1),ok(X2),ok(X3)) -> ok(f(X1,X2,X3)) proper(a()) -> ok(a()) proper(b()) -> ok(b()) proper(c()) -> ok(c()) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) - Signature: {active/1,f/3,proper/1,top/1} / {a/0,b/0,c/0,mark/1,ok/1} - Obligation: innermost runtime complexity wrt. defined symbols {active,f,proper,top} and constructors {a,b,c,mark,ok} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))