WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: f(a()) -> b() f(c()) -> d() f(g(x,y)) -> g(f(x),f(y)) f(h(x,y)) -> g(h(y,f(x)),h(x,f(y))) g(x,x) -> h(e(),x) - Signature: {f/1,g/2} / {a/0,b/0,c/0,d/0,e/0,h/2} - Obligation: innermost runtime complexity wrt. defined symbols {f,g} and constructors {a,b,c,d,e,h} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: f(a()) -> b() f(c()) -> d() f(g(x,y)) -> g(f(x),f(y)) f(h(x,y)) -> g(h(y,f(x)),h(x,f(y))) g(x,x) -> h(e(),x) - Signature: {f/1,g/2} / {a/0,b/0,c/0,d/0,e/0,h/2} - Obligation: innermost runtime complexity wrt. defined symbols {f,g} and constructors {a,b,c,d,e,h} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: f(x){x -> h(x,y)} = f(h(x,y)) ->^+ g(h(y,f(x)),h(x,f(y))) = C[f(x) = f(x){}] ** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(a()) -> b() f(c()) -> d() f(g(x,y)) -> g(f(x),f(y)) f(h(x,y)) -> g(h(y,f(x)),h(x,f(y))) g(x,x) -> h(e(),x) - Signature: {f/1,g/2} / {a/0,b/0,c/0,d/0,e/0,h/2} - Obligation: innermost runtime complexity wrt. defined symbols {f,g} and constructors {a,b,c,d,e,h} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs f#(a()) -> c_1() f#(c()) -> c_2() f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)) f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)) g#(x,x) -> c_5() Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(a()) -> c_1() f#(c()) -> c_2() f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)) f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)) g#(x,x) -> c_5() - Weak TRS: f(a()) -> b() f(c()) -> d() f(g(x,y)) -> g(f(x),f(y)) f(h(x,y)) -> g(h(y,f(x)),h(x,f(y))) g(x,x) -> h(e(),x) - Signature: {f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/3,c_4/3,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,b,c,d,e,h} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,5} by application of Pre({1,2,5}) = {3,4}. Here rules are labelled as follows: 1: f#(a()) -> c_1() 2: f#(c()) -> c_2() 3: f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)) 4: f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)) 5: g#(x,x) -> c_5() ** Step 1.b:3: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)) f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)) - Weak DPs: f#(a()) -> c_1() f#(c()) -> c_2() g#(x,x) -> c_5() - Weak TRS: f(a()) -> b() f(c()) -> d() f(g(x,y)) -> g(f(x),f(y)) f(h(x,y)) -> g(h(y,f(x)),h(x,f(y))) g(x,x) -> h(e(),x) - Signature: {f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/3,c_4/3,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,b,c,d,e,h} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)) -->_3 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2 -->_2 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2 -->_1 g#(x,x) -> c_5():5 -->_3 f#(c()) -> c_2():4 -->_2 f#(c()) -> c_2():4 -->_3 f#(a()) -> c_1():3 -->_2 f#(a()) -> c_1():3 -->_3 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1 -->_2 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1 2:S:f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)) -->_1 g#(x,x) -> c_5():5 -->_3 f#(c()) -> c_2():4 -->_2 f#(c()) -> c_2():4 -->_3 f#(a()) -> c_1():3 -->_2 f#(a()) -> c_1():3 -->_3 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2 -->_2 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2 -->_3 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1 -->_2 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1 3:W:f#(a()) -> c_1() 4:W:f#(c()) -> c_2() 5:W:g#(x,x) -> c_5() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: f#(a()) -> c_1() 4: f#(c()) -> c_2() 5: g#(x,x) -> c_5() ** Step 1.b:4: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)) f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)) - Weak TRS: f(a()) -> b() f(c()) -> d() f(g(x,y)) -> g(f(x),f(y)) f(h(x,y)) -> g(h(y,f(x)),h(x,f(y))) g(x,x) -> h(e(),x) - Signature: {f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/3,c_4/3,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,b,c,d,e,h} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)) -->_3 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2 -->_2 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2 -->_3 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1 -->_2 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1 2:S:f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)) -->_3 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2 -->_2 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2 -->_3 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1 -->_2 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: f#(g(x,y)) -> c_3(f#(x),f#(y)) f#(h(x,y)) -> c_4(f#(x),f#(y)) ** Step 1.b:5: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(g(x,y)) -> c_3(f#(x),f#(y)) f#(h(x,y)) -> c_4(f#(x),f#(y)) - Weak TRS: f(a()) -> b() f(c()) -> d() f(g(x,y)) -> g(f(x),f(y)) f(h(x,y)) -> g(h(y,f(x)),h(x,f(y))) g(x,x) -> h(e(),x) - Signature: {f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/2,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,b,c,d,e,h} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: f#(g(x,y)) -> c_3(f#(x),f#(y)) f#(h(x,y)) -> c_4(f#(x),f#(y)) ** Step 1.b:6: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(g(x,y)) -> c_3(f#(x),f#(y)) f#(h(x,y)) -> c_4(f#(x),f#(y)) - Signature: {f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/2,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,b,c,d,e,h} + 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_3) = {1,2}, uargs(c_4) = {1,2} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [0] p(b) = [8] p(c) = [1] p(d) = [0] p(e) = [1] p(f) = [1] x1 + [1] p(g) = [2] x1 + [1] x2 + [0] p(h) = [1] x1 + [1] x2 + [1] p(f#) = [4] x1 + [2] p(g#) = [1] x1 + [2] x2 + [0] p(c_1) = [1] p(c_2) = [0] p(c_3) = [1] x1 + [1] x2 + [14] p(c_4) = [1] x1 + [1] x2 + [0] p(c_5) = [1] Following rules are strictly oriented: f#(h(x,y)) = [4] x + [4] y + [6] > [4] x + [4] y + [4] = c_4(f#(x),f#(y)) Following rules are (at-least) weakly oriented: f#(g(x,y)) = [8] x + [4] y + [2] >= [4] x + [4] y + [18] = c_3(f#(x),f#(y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:7: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(g(x,y)) -> c_3(f#(x),f#(y)) - Weak DPs: f#(h(x,y)) -> c_4(f#(x),f#(y)) - Signature: {f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/2,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,b,c,d,e,h} + 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(c_3) = {1,2}, uargs(c_4) = {1,2} Following symbols are considered usable: {f#,g#} TcT has computed the following interpretation: p(a) = [1] p(b) = [1] p(c) = [0] p(d) = [1] p(e) = [1] p(f) = [2] p(g) = [8] x1 + [8] x2 + [8] p(h) = [1] x1 + [1] x2 + [1] p(f#) = [2] x1 + [0] p(g#) = [1] x1 + [1] p(c_1) = [2] p(c_2) = [1] p(c_3) = [8] x1 + [4] x2 + [8] p(c_4) = [1] x1 + [1] x2 + [2] p(c_5) = [0] Following rules are strictly oriented: f#(g(x,y)) = [16] x + [16] y + [16] > [16] x + [8] y + [8] = c_3(f#(x),f#(y)) Following rules are (at-least) weakly oriented: f#(h(x,y)) = [2] x + [2] y + [2] >= [2] x + [2] y + [2] = c_4(f#(x),f#(y)) ** Step 1.b:8: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: f#(g(x,y)) -> c_3(f#(x),f#(y)) f#(h(x,y)) -> c_4(f#(x),f#(y)) - Signature: {f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/2,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,b,c,d,e,h} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))