WORST_CASE(?,O(n^1)) * Step 1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__b() -> b() a__b() -> c() a__f(X,g(X),Y) -> a__f(Y,Y,Y) a__f(X1,X2,X3) -> f(X1,X2,X3) a__g(X) -> g(X) a__g(b()) -> c() mark(b()) -> a__b() mark(c()) -> c() mark(f(X1,X2,X3)) -> a__f(X1,X2,X3) mark(g(X)) -> a__g(mark(X)) - Signature: {a__b/0,a__f/3,a__g/1,mark/1} / {b/0,c/0,f/3,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__b,a__f,a__g,mark} and constructors {b,c,f,g} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs a__b#() -> c_1() a__b#() -> c_2() a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y)) a__f#(X1,X2,X3) -> c_4() a__g#(X) -> c_5() a__g#(b()) -> c_6() mark#(b()) -> c_7(a__b#()) mark#(c()) -> c_8() mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3)) mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__b#() -> c_1() a__b#() -> c_2() a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y)) a__f#(X1,X2,X3) -> c_4() a__g#(X) -> c_5() a__g#(b()) -> c_6() mark#(b()) -> c_7(a__b#()) mark#(c()) -> c_8() mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3)) mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X)) - Weak TRS: a__b() -> b() a__b() -> c() a__f(X,g(X),Y) -> a__f(Y,Y,Y) a__f(X1,X2,X3) -> f(X1,X2,X3) a__g(X) -> g(X) a__g(b()) -> c() mark(b()) -> a__b() mark(c()) -> c() mark(f(X1,X2,X3)) -> a__f(X1,X2,X3) mark(g(X)) -> a__g(mark(X)) - Signature: {a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,4,5,6,8} by application of Pre({1,2,4,5,6,8}) = {3,7,9,10}. Here rules are labelled as follows: 1: a__b#() -> c_1() 2: a__b#() -> c_2() 3: a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y)) 4: a__f#(X1,X2,X3) -> c_4() 5: a__g#(X) -> c_5() 6: a__g#(b()) -> c_6() 7: mark#(b()) -> c_7(a__b#()) 8: mark#(c()) -> c_8() 9: mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3)) 10: mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X)) * Step 3: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y)) mark#(b()) -> c_7(a__b#()) mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3)) mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X)) - Weak DPs: a__b#() -> c_1() a__b#() -> c_2() a__f#(X1,X2,X3) -> c_4() a__g#(X) -> c_5() a__g#(b()) -> c_6() mark#(c()) -> c_8() - Weak TRS: a__b() -> b() a__b() -> c() a__f(X,g(X),Y) -> a__f(Y,Y,Y) a__f(X1,X2,X3) -> f(X1,X2,X3) a__g(X) -> g(X) a__g(b()) -> c() mark(b()) -> a__b() mark(c()) -> c() mark(f(X1,X2,X3)) -> a__f(X1,X2,X3) mark(g(X)) -> a__g(mark(X)) - Signature: {a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2} by application of Pre({1,2}) = {3,4}. Here rules are labelled as follows: 1: a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y)) 2: mark#(b()) -> c_7(a__b#()) 3: mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3)) 4: mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X)) 5: a__b#() -> c_1() 6: a__b#() -> c_2() 7: a__f#(X1,X2,X3) -> c_4() 8: a__g#(X) -> c_5() 9: a__g#(b()) -> c_6() 10: mark#(c()) -> c_8() * Step 4: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3)) mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X)) - Weak DPs: a__b#() -> c_1() a__b#() -> c_2() a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y)) a__f#(X1,X2,X3) -> c_4() a__g#(X) -> c_5() a__g#(b()) -> c_6() mark#(b()) -> c_7(a__b#()) mark#(c()) -> c_8() - Weak TRS: a__b() -> b() a__b() -> c() a__f(X,g(X),Y) -> a__f(Y,Y,Y) a__f(X1,X2,X3) -> f(X1,X2,X3) a__g(X) -> g(X) a__g(b()) -> c() mark(b()) -> a__b() mark(c()) -> c() mark(f(X1,X2,X3)) -> a__f(X1,X2,X3) mark(g(X)) -> a__g(mark(X)) - Signature: {a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1} by application of Pre({1}) = {2}. Here rules are labelled as follows: 1: mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3)) 2: mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X)) 3: a__b#() -> c_1() 4: a__b#() -> c_2() 5: a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y)) 6: a__f#(X1,X2,X3) -> c_4() 7: a__g#(X) -> c_5() 8: a__g#(b()) -> c_6() 9: mark#(b()) -> c_7(a__b#()) 10: mark#(c()) -> c_8() * Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X)) - Weak DPs: a__b#() -> c_1() a__b#() -> c_2() a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y)) a__f#(X1,X2,X3) -> c_4() a__g#(X) -> c_5() a__g#(b()) -> c_6() mark#(b()) -> c_7(a__b#()) mark#(c()) -> c_8() mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3)) - Weak TRS: a__b() -> b() a__b() -> c() a__f(X,g(X),Y) -> a__f(Y,Y,Y) a__f(X1,X2,X3) -> f(X1,X2,X3) a__g(X) -> g(X) a__g(b()) -> c() mark(b()) -> a__b() mark(c()) -> c() mark(f(X1,X2,X3)) -> a__f(X1,X2,X3) mark(g(X)) -> a__g(mark(X)) - Signature: {a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X)) -->_2 mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3)):10 -->_2 mark#(b()) -> c_7(a__b#()):8 -->_2 mark#(c()) -> c_8():9 -->_1 a__g#(b()) -> c_6():7 -->_1 a__g#(X) -> c_5():6 -->_2 mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X)):1 2:W:a__b#() -> c_1() 3:W:a__b#() -> c_2() 4:W:a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y)) -->_1 a__f#(X1,X2,X3) -> c_4():5 5:W:a__f#(X1,X2,X3) -> c_4() 6:W:a__g#(X) -> c_5() 7:W:a__g#(b()) -> c_6() 8:W:mark#(b()) -> c_7(a__b#()) -->_1 a__b#() -> c_2():3 -->_1 a__b#() -> c_1():2 9:W:mark#(c()) -> c_8() 10:W:mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3)) -->_1 a__f#(X1,X2,X3) -> c_4():5 -->_1 a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: a__g#(X) -> c_5() 7: a__g#(b()) -> c_6() 9: mark#(c()) -> c_8() 8: mark#(b()) -> c_7(a__b#()) 2: a__b#() -> c_1() 3: a__b#() -> c_2() 10: mark#(f(X1,X2,X3)) -> c_9(a__f#(X1,X2,X3)) 4: a__f#(X,g(X),Y) -> c_3(a__f#(Y,Y,Y)) 5: a__f#(X1,X2,X3) -> c_4() * Step 6: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X)) - Weak TRS: a__b() -> b() a__b() -> c() a__f(X,g(X),Y) -> a__f(Y,Y,Y) a__f(X1,X2,X3) -> f(X1,X2,X3) a__g(X) -> g(X) a__g(b()) -> c() mark(b()) -> a__b() mark(c()) -> c() mark(f(X1,X2,X3)) -> a__f(X1,X2,X3) mark(g(X)) -> a__g(mark(X)) - Signature: {a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X)) -->_2 mark#(g(X)) -> c_10(a__g#(mark(X)),mark#(X)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: mark#(g(X)) -> c_10(mark#(X)) * Step 7: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mark#(g(X)) -> c_10(mark#(X)) - Weak TRS: a__b() -> b() a__b() -> c() a__f(X,g(X),Y) -> a__f(Y,Y,Y) a__f(X1,X2,X3) -> f(X1,X2,X3) a__g(X) -> g(X) a__g(b()) -> c() mark(b()) -> a__b() mark(c()) -> c() mark(f(X1,X2,X3)) -> a__f(X1,X2,X3) mark(g(X)) -> a__g(mark(X)) - Signature: {a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: mark#(g(X)) -> c_10(mark#(X)) * Step 8: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mark#(g(X)) -> c_10(mark#(X)) - Signature: {a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_10) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(a__b) = [0] p(a__f) = [0] p(a__g) = [0] p(b) = [0] p(c) = [0] p(f) = [1] x1 + [1] x2 + [1] x3 + [0] p(g) = [1] x1 + [1] p(mark) = [0] p(a__b#) = [0] p(a__f#) = [0] p(a__g#) = [0] p(mark#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] x1 + [0] Following rules are strictly oriented: mark#(g(X)) = [1] X + [1] > [1] X + [0] = c_10(mark#(X)) Following rules are (at-least) weakly oriented: Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 9: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: mark#(g(X)) -> c_10(mark#(X)) - Signature: {a__b/0,a__f/3,a__g/1,mark/1,a__b#/0,a__f#/3,a__g#/1,mark#/1} / {b/0,c/0,f/3,g/1,c_1/0,c_2/0,c_3/1,c_4/0 ,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__b#,a__f#,a__g#,mark#} and constructors {b,c,f,g} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))