/export/starexec/sandbox/solver/bin/starexec_run_tct_rci /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?,O(n^1)) * Step 1: Sum. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f0(x1,0(),x3,x4,x5) -> 0() f0(x1,S(x),x3,0(),x5) -> 0() f0(x1,S(x'),x3,S(x),x5) -> f1(x',S(x'),x,S(x),S(x)) f1(x1,x2,x3,x4,0()) -> 0() f1(x1,x2,x3,x4,S(x)) -> f2(x2,x1,x3,x4,x) f2(x1,x2,0(),x4,x5) -> 0() f2(x1,x2,S(x),0(),0()) -> 0() f2(x1,x2,S(x'),0(),S(x)) -> f3(x1,x2,x',0(),x) f2(x1,x2,S(x'),S(x),0()) -> 0() f2(x1,x2,S(x''),S(x'),S(x)) -> f5(x1,x2,S(x''),x',x) f3(x1,x2,x3,x4,0()) -> 0() f3(x1,x2,x3,x4,S(x)) -> f4(x1,x2,x4,x3,x) f4(0(),x2,x3,x4,x5) -> 0() f4(S(x),0(),x3,x4,0()) -> 0() f4(S(x'),0(),x3,x4,S(x)) -> f3(x',0(),x3,x4,x) f4(S(x'),S(x),x3,x4,0()) -> 0() f4(S(x''),S(x'),x3,x4,S(x)) -> f2(S(x''),x',x3,x4,x) f5(x1,x2,x3,x4,0()) -> 0() f5(x1,x2,x3,x4,S(x)) -> f6(x2,x1,x3,x4,x) f6(x1,x2,x3,x4,0()) -> 0() f6(x1,x2,x3,x4,S(x)) -> f0(x1,x2,x4,x3,x) - Signature: {f0/5,f1/5,f2/5,f3/5,f4/5,f5/5,f6/5} / {0/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {f0,f1,f2,f3,f4,f5,f6} and constructors {0,S} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DependencyPairs. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f0(x1,0(),x3,x4,x5) -> 0() f0(x1,S(x),x3,0(),x5) -> 0() f0(x1,S(x'),x3,S(x),x5) -> f1(x',S(x'),x,S(x),S(x)) f1(x1,x2,x3,x4,0()) -> 0() f1(x1,x2,x3,x4,S(x)) -> f2(x2,x1,x3,x4,x) f2(x1,x2,0(),x4,x5) -> 0() f2(x1,x2,S(x),0(),0()) -> 0() f2(x1,x2,S(x'),0(),S(x)) -> f3(x1,x2,x',0(),x) f2(x1,x2,S(x'),S(x),0()) -> 0() f2(x1,x2,S(x''),S(x'),S(x)) -> f5(x1,x2,S(x''),x',x) f3(x1,x2,x3,x4,0()) -> 0() f3(x1,x2,x3,x4,S(x)) -> f4(x1,x2,x4,x3,x) f4(0(),x2,x3,x4,x5) -> 0() f4(S(x),0(),x3,x4,0()) -> 0() f4(S(x'),0(),x3,x4,S(x)) -> f3(x',0(),x3,x4,x) f4(S(x'),S(x),x3,x4,0()) -> 0() f4(S(x''),S(x'),x3,x4,S(x)) -> f2(S(x''),x',x3,x4,x) f5(x1,x2,x3,x4,0()) -> 0() f5(x1,x2,x3,x4,S(x)) -> f6(x2,x1,x3,x4,x) f6(x1,x2,x3,x4,0()) -> 0() f6(x1,x2,x3,x4,S(x)) -> f0(x1,x2,x4,x3,x) - Signature: {f0/5,f1/5,f2/5,f3/5,f4/5,f5/5,f6/5} / {0/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {f0,f1,f2,f3,f4,f5,f6} and constructors {0,S} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs f0#(x1,0(),x3,x4,x5) -> c_1() f0#(x1,S(x),x3,0(),x5) -> c_2() f0#(x1,S(x'),x3,S(x),x5) -> c_3(f1#(x',S(x'),x,S(x),S(x))) f1#(x1,x2,x3,x4,0()) -> c_4() f1#(x1,x2,x3,x4,S(x)) -> c_5(f2#(x2,x1,x3,x4,x)) f2#(x1,x2,0(),x4,x5) -> c_6() f2#(x1,x2,S(x),0(),0()) -> c_7() f2#(x1,x2,S(x'),0(),S(x)) -> c_8(f3#(x1,x2,x',0(),x)) f2#(x1,x2,S(x'),S(x),0()) -> c_9() f2#(x1,x2,S(x''),S(x'),S(x)) -> c_10(f5#(x1,x2,S(x''),x',x)) f3#(x1,x2,x3,x4,0()) -> c_11() f3#(x1,x2,x3,x4,S(x)) -> c_12(f4#(x1,x2,x4,x3,x)) f4#(0(),x2,x3,x4,x5) -> c_13() f4#(S(x),0(),x3,x4,0()) -> c_14() f4#(S(x'),0(),x3,x4,S(x)) -> c_15(f3#(x',0(),x3,x4,x)) f4#(S(x'),S(x),x3,x4,0()) -> c_16() f4#(S(x''),S(x'),x3,x4,S(x)) -> c_17(f2#(S(x''),x',x3,x4,x)) f5#(x1,x2,x3,x4,0()) -> c_18() f5#(x1,x2,x3,x4,S(x)) -> c_19(f6#(x2,x1,x3,x4,x)) f6#(x1,x2,x3,x4,0()) -> c_20() f6#(x1,x2,x3,x4,S(x)) -> c_21(f0#(x1,x2,x4,x3,x)) Weak DPs and mark the set of starting terms. * Step 3: PredecessorEstimation. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f0#(x1,0(),x3,x4,x5) -> c_1() f0#(x1,S(x),x3,0(),x5) -> c_2() f0#(x1,S(x'),x3,S(x),x5) -> c_3(f1#(x',S(x'),x,S(x),S(x))) f1#(x1,x2,x3,x4,0()) -> c_4() f1#(x1,x2,x3,x4,S(x)) -> c_5(f2#(x2,x1,x3,x4,x)) f2#(x1,x2,0(),x4,x5) -> c_6() f2#(x1,x2,S(x),0(),0()) -> c_7() f2#(x1,x2,S(x'),0(),S(x)) -> c_8(f3#(x1,x2,x',0(),x)) f2#(x1,x2,S(x'),S(x),0()) -> c_9() f2#(x1,x2,S(x''),S(x'),S(x)) -> c_10(f5#(x1,x2,S(x''),x',x)) f3#(x1,x2,x3,x4,0()) -> c_11() f3#(x1,x2,x3,x4,S(x)) -> c_12(f4#(x1,x2,x4,x3,x)) f4#(0(),x2,x3,x4,x5) -> c_13() f4#(S(x),0(),x3,x4,0()) -> c_14() f4#(S(x'),0(),x3,x4,S(x)) -> c_15(f3#(x',0(),x3,x4,x)) f4#(S(x'),S(x),x3,x4,0()) -> c_16() f4#(S(x''),S(x'),x3,x4,S(x)) -> c_17(f2#(S(x''),x',x3,x4,x)) f5#(x1,x2,x3,x4,0()) -> c_18() f5#(x1,x2,x3,x4,S(x)) -> c_19(f6#(x2,x1,x3,x4,x)) f6#(x1,x2,x3,x4,0()) -> c_20() f6#(x1,x2,x3,x4,S(x)) -> c_21(f0#(x1,x2,x4,x3,x)) - Weak TRS: f0(x1,0(),x3,x4,x5) -> 0() f0(x1,S(x),x3,0(),x5) -> 0() f0(x1,S(x'),x3,S(x),x5) -> f1(x',S(x'),x,S(x),S(x)) f1(x1,x2,x3,x4,0()) -> 0() f1(x1,x2,x3,x4,S(x)) -> f2(x2,x1,x3,x4,x) f2(x1,x2,0(),x4,x5) -> 0() f2(x1,x2,S(x),0(),0()) -> 0() f2(x1,x2,S(x'),0(),S(x)) -> f3(x1,x2,x',0(),x) f2(x1,x2,S(x'),S(x),0()) -> 0() f2(x1,x2,S(x''),S(x'),S(x)) -> f5(x1,x2,S(x''),x',x) f3(x1,x2,x3,x4,0()) -> 0() f3(x1,x2,x3,x4,S(x)) -> f4(x1,x2,x4,x3,x) f4(0(),x2,x3,x4,x5) -> 0() f4(S(x),0(),x3,x4,0()) -> 0() f4(S(x'),0(),x3,x4,S(x)) -> f3(x',0(),x3,x4,x) f4(S(x'),S(x),x3,x4,0()) -> 0() f4(S(x''),S(x'),x3,x4,S(x)) -> f2(S(x''),x',x3,x4,x) f5(x1,x2,x3,x4,0()) -> 0() f5(x1,x2,x3,x4,S(x)) -> f6(x2,x1,x3,x4,x) f6(x1,x2,x3,x4,0()) -> 0() f6(x1,x2,x3,x4,S(x)) -> f0(x1,x2,x4,x3,x) - Signature: {f0/5,f1/5,f2/5,f3/5,f4/5,f5/5,f6/5,f0#/5,f1#/5,f2#/5,f3#/5,f4#/5,f5#/5,f6#/5} / {0/0,S/1,c_1/0,c_2/0,c_3/1 ,c_4/0,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/0,c_19/1 ,c_20/0,c_21/1} - Obligation: innermost runtime complexity wrt. defined symbols {f0#,f1#,f2#,f3#,f4#,f5#,f6#} and constructors {0,S} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,4,6,7,9,11,13,14,16,18,20} by application of Pre({1,2,4,6,7,9,11,13,14,16,18,20}) = {5,8,10,12,15,17,19,21}. Here rules are labelled as follows: 1: f0#(x1,0(),x3,x4,x5) -> c_1() 2: f0#(x1,S(x),x3,0(),x5) -> c_2() 3: f0#(x1,S(x'),x3,S(x),x5) -> c_3(f1#(x',S(x'),x,S(x),S(x))) 4: f1#(x1,x2,x3,x4,0()) -> c_4() 5: f1#(x1,x2,x3,x4,S(x)) -> c_5(f2#(x2,x1,x3,x4,x)) 6: f2#(x1,x2,0(),x4,x5) -> c_6() 7: f2#(x1,x2,S(x),0(),0()) -> c_7() 8: f2#(x1,x2,S(x'),0(),S(x)) -> c_8(f3#(x1,x2,x',0(),x)) 9: f2#(x1,x2,S(x'),S(x),0()) -> c_9() 10: f2#(x1,x2,S(x''),S(x'),S(x)) -> c_10(f5#(x1,x2,S(x''),x',x)) 11: f3#(x1,x2,x3,x4,0()) -> c_11() 12: f3#(x1,x2,x3,x4,S(x)) -> c_12(f4#(x1,x2,x4,x3,x)) 13: f4#(0(),x2,x3,x4,x5) -> c_13() 14: f4#(S(x),0(),x3,x4,0()) -> c_14() 15: f4#(S(x'),0(),x3,x4,S(x)) -> c_15(f3#(x',0(),x3,x4,x)) 16: f4#(S(x'),S(x),x3,x4,0()) -> c_16() 17: f4#(S(x''),S(x'),x3,x4,S(x)) -> c_17(f2#(S(x''),x',x3,x4,x)) 18: f5#(x1,x2,x3,x4,0()) -> c_18() 19: f5#(x1,x2,x3,x4,S(x)) -> c_19(f6#(x2,x1,x3,x4,x)) 20: f6#(x1,x2,x3,x4,0()) -> c_20() 21: f6#(x1,x2,x3,x4,S(x)) -> c_21(f0#(x1,x2,x4,x3,x)) * Step 4: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f0#(x1,S(x'),x3,S(x),x5) -> c_3(f1#(x',S(x'),x,S(x),S(x))) f1#(x1,x2,x3,x4,S(x)) -> c_5(f2#(x2,x1,x3,x4,x)) f2#(x1,x2,S(x'),0(),S(x)) -> c_8(f3#(x1,x2,x',0(),x)) f2#(x1,x2,S(x''),S(x'),S(x)) -> c_10(f5#(x1,x2,S(x''),x',x)) f3#(x1,x2,x3,x4,S(x)) -> c_12(f4#(x1,x2,x4,x3,x)) f4#(S(x'),0(),x3,x4,S(x)) -> c_15(f3#(x',0(),x3,x4,x)) f4#(S(x''),S(x'),x3,x4,S(x)) -> c_17(f2#(S(x''),x',x3,x4,x)) f5#(x1,x2,x3,x4,S(x)) -> c_19(f6#(x2,x1,x3,x4,x)) f6#(x1,x2,x3,x4,S(x)) -> c_21(f0#(x1,x2,x4,x3,x)) - Weak DPs: f0#(x1,0(),x3,x4,x5) -> c_1() f0#(x1,S(x),x3,0(),x5) -> c_2() f1#(x1,x2,x3,x4,0()) -> c_4() f2#(x1,x2,0(),x4,x5) -> c_6() f2#(x1,x2,S(x),0(),0()) -> c_7() f2#(x1,x2,S(x'),S(x),0()) -> c_9() f3#(x1,x2,x3,x4,0()) -> c_11() f4#(0(),x2,x3,x4,x5) -> c_13() f4#(S(x),0(),x3,x4,0()) -> c_14() f4#(S(x'),S(x),x3,x4,0()) -> c_16() f5#(x1,x2,x3,x4,0()) -> c_18() f6#(x1,x2,x3,x4,0()) -> c_20() - Weak TRS: f0(x1,0(),x3,x4,x5) -> 0() f0(x1,S(x),x3,0(),x5) -> 0() f0(x1,S(x'),x3,S(x),x5) -> f1(x',S(x'),x,S(x),S(x)) f1(x1,x2,x3,x4,0()) -> 0() f1(x1,x2,x3,x4,S(x)) -> f2(x2,x1,x3,x4,x) f2(x1,x2,0(),x4,x5) -> 0() f2(x1,x2,S(x),0(),0()) -> 0() f2(x1,x2,S(x'),0(),S(x)) -> f3(x1,x2,x',0(),x) f2(x1,x2,S(x'),S(x),0()) -> 0() f2(x1,x2,S(x''),S(x'),S(x)) -> f5(x1,x2,S(x''),x',x) f3(x1,x2,x3,x4,0()) -> 0() f3(x1,x2,x3,x4,S(x)) -> f4(x1,x2,x4,x3,x) f4(0(),x2,x3,x4,x5) -> 0() f4(S(x),0(),x3,x4,0()) -> 0() f4(S(x'),0(),x3,x4,S(x)) -> f3(x',0(),x3,x4,x) f4(S(x'),S(x),x3,x4,0()) -> 0() f4(S(x''),S(x'),x3,x4,S(x)) -> f2(S(x''),x',x3,x4,x) f5(x1,x2,x3,x4,0()) -> 0() f5(x1,x2,x3,x4,S(x)) -> f6(x2,x1,x3,x4,x) f6(x1,x2,x3,x4,0()) -> 0() f6(x1,x2,x3,x4,S(x)) -> f0(x1,x2,x4,x3,x) - Signature: {f0/5,f1/5,f2/5,f3/5,f4/5,f5/5,f6/5,f0#/5,f1#/5,f2#/5,f3#/5,f4#/5,f5#/5,f6#/5} / {0/0,S/1,c_1/0,c_2/0,c_3/1 ,c_4/0,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/0,c_19/1 ,c_20/0,c_21/1} - Obligation: innermost runtime complexity wrt. defined symbols {f0#,f1#,f2#,f3#,f4#,f5#,f6#} and constructors {0,S} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:f0#(x1,S(x'),x3,S(x),x5) -> c_3(f1#(x',S(x'),x,S(x),S(x))) -->_1 f1#(x1,x2,x3,x4,S(x)) -> c_5(f2#(x2,x1,x3,x4,x)):2 2:S:f1#(x1,x2,x3,x4,S(x)) -> c_5(f2#(x2,x1,x3,x4,x)) -->_1 f2#(x1,x2,S(x''),S(x'),S(x)) -> c_10(f5#(x1,x2,S(x''),x',x)):4 -->_1 f2#(x1,x2,S(x'),0(),S(x)) -> c_8(f3#(x1,x2,x',0(),x)):3 -->_1 f2#(x1,x2,S(x'),S(x),0()) -> c_9():15 -->_1 f2#(x1,x2,S(x),0(),0()) -> c_7():14 -->_1 f2#(x1,x2,0(),x4,x5) -> c_6():13 3:S:f2#(x1,x2,S(x'),0(),S(x)) -> c_8(f3#(x1,x2,x',0(),x)) -->_1 f3#(x1,x2,x3,x4,S(x)) -> c_12(f4#(x1,x2,x4,x3,x)):5 -->_1 f3#(x1,x2,x3,x4,0()) -> c_11():16 4:S:f2#(x1,x2,S(x''),S(x'),S(x)) -> c_10(f5#(x1,x2,S(x''),x',x)) -->_1 f5#(x1,x2,x3,x4,S(x)) -> c_19(f6#(x2,x1,x3,x4,x)):8 -->_1 f5#(x1,x2,x3,x4,0()) -> c_18():20 5:S:f3#(x1,x2,x3,x4,S(x)) -> c_12(f4#(x1,x2,x4,x3,x)) -->_1 f4#(S(x''),S(x'),x3,x4,S(x)) -> c_17(f2#(S(x''),x',x3,x4,x)):7 -->_1 f4#(S(x'),0(),x3,x4,S(x)) -> c_15(f3#(x',0(),x3,x4,x)):6 -->_1 f4#(S(x'),S(x),x3,x4,0()) -> c_16():19 -->_1 f4#(S(x),0(),x3,x4,0()) -> c_14():18 -->_1 f4#(0(),x2,x3,x4,x5) -> c_13():17 6:S:f4#(S(x'),0(),x3,x4,S(x)) -> c_15(f3#(x',0(),x3,x4,x)) -->_1 f3#(x1,x2,x3,x4,0()) -> c_11():16 -->_1 f3#(x1,x2,x3,x4,S(x)) -> c_12(f4#(x1,x2,x4,x3,x)):5 7:S:f4#(S(x''),S(x'),x3,x4,S(x)) -> c_17(f2#(S(x''),x',x3,x4,x)) -->_1 f2#(x1,x2,S(x'),S(x),0()) -> c_9():15 -->_1 f2#(x1,x2,S(x),0(),0()) -> c_7():14 -->_1 f2#(x1,x2,0(),x4,x5) -> c_6():13 -->_1 f2#(x1,x2,S(x''),S(x'),S(x)) -> c_10(f5#(x1,x2,S(x''),x',x)):4 -->_1 f2#(x1,x2,S(x'),0(),S(x)) -> c_8(f3#(x1,x2,x',0(),x)):3 8:S:f5#(x1,x2,x3,x4,S(x)) -> c_19(f6#(x2,x1,x3,x4,x)) -->_1 f6#(x1,x2,x3,x4,S(x)) -> c_21(f0#(x1,x2,x4,x3,x)):9 -->_1 f6#(x1,x2,x3,x4,0()) -> c_20():21 9:S:f6#(x1,x2,x3,x4,S(x)) -> c_21(f0#(x1,x2,x4,x3,x)) -->_1 f0#(x1,S(x),x3,0(),x5) -> c_2():11 -->_1 f0#(x1,0(),x3,x4,x5) -> c_1():10 -->_1 f0#(x1,S(x'),x3,S(x),x5) -> c_3(f1#(x',S(x'),x,S(x),S(x))):1 10:W:f0#(x1,0(),x3,x4,x5) -> c_1() 11:W:f0#(x1,S(x),x3,0(),x5) -> c_2() 12:W:f1#(x1,x2,x3,x4,0()) -> c_4() 13:W:f2#(x1,x2,0(),x4,x5) -> c_6() 14:W:f2#(x1,x2,S(x),0(),0()) -> c_7() 15:W:f2#(x1,x2,S(x'),S(x),0()) -> c_9() 16:W:f3#(x1,x2,x3,x4,0()) -> c_11() 17:W:f4#(0(),x2,x3,x4,x5) -> c_13() 18:W:f4#(S(x),0(),x3,x4,0()) -> c_14() 19:W:f4#(S(x'),S(x),x3,x4,0()) -> c_16() 20:W:f5#(x1,x2,x3,x4,0()) -> c_18() 21:W:f6#(x1,x2,x3,x4,0()) -> c_20() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 12: f1#(x1,x2,x3,x4,0()) -> c_4() 17: f4#(0(),x2,x3,x4,x5) -> c_13() 18: f4#(S(x),0(),x3,x4,0()) -> c_14() 19: f4#(S(x'),S(x),x3,x4,0()) -> c_16() 16: f3#(x1,x2,x3,x4,0()) -> c_11() 13: f2#(x1,x2,0(),x4,x5) -> c_6() 14: f2#(x1,x2,S(x),0(),0()) -> c_7() 15: f2#(x1,x2,S(x'),S(x),0()) -> c_9() 20: f5#(x1,x2,x3,x4,0()) -> c_18() 21: f6#(x1,x2,x3,x4,0()) -> c_20() 10: f0#(x1,0(),x3,x4,x5) -> c_1() 11: f0#(x1,S(x),x3,0(),x5) -> c_2() * Step 5: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f0#(x1,S(x'),x3,S(x),x5) -> c_3(f1#(x',S(x'),x,S(x),S(x))) f1#(x1,x2,x3,x4,S(x)) -> c_5(f2#(x2,x1,x3,x4,x)) f2#(x1,x2,S(x'),0(),S(x)) -> c_8(f3#(x1,x2,x',0(),x)) f2#(x1,x2,S(x''),S(x'),S(x)) -> c_10(f5#(x1,x2,S(x''),x',x)) f3#(x1,x2,x3,x4,S(x)) -> c_12(f4#(x1,x2,x4,x3,x)) f4#(S(x'),0(),x3,x4,S(x)) -> c_15(f3#(x',0(),x3,x4,x)) f4#(S(x''),S(x'),x3,x4,S(x)) -> c_17(f2#(S(x''),x',x3,x4,x)) f5#(x1,x2,x3,x4,S(x)) -> c_19(f6#(x2,x1,x3,x4,x)) f6#(x1,x2,x3,x4,S(x)) -> c_21(f0#(x1,x2,x4,x3,x)) - Weak TRS: f0(x1,0(),x3,x4,x5) -> 0() f0(x1,S(x),x3,0(),x5) -> 0() f0(x1,S(x'),x3,S(x),x5) -> f1(x',S(x'),x,S(x),S(x)) f1(x1,x2,x3,x4,0()) -> 0() f1(x1,x2,x3,x4,S(x)) -> f2(x2,x1,x3,x4,x) f2(x1,x2,0(),x4,x5) -> 0() f2(x1,x2,S(x),0(),0()) -> 0() f2(x1,x2,S(x'),0(),S(x)) -> f3(x1,x2,x',0(),x) f2(x1,x2,S(x'),S(x),0()) -> 0() f2(x1,x2,S(x''),S(x'),S(x)) -> f5(x1,x2,S(x''),x',x) f3(x1,x2,x3,x4,0()) -> 0() f3(x1,x2,x3,x4,S(x)) -> f4(x1,x2,x4,x3,x) f4(0(),x2,x3,x4,x5) -> 0() f4(S(x),0(),x3,x4,0()) -> 0() f4(S(x'),0(),x3,x4,S(x)) -> f3(x',0(),x3,x4,x) f4(S(x'),S(x),x3,x4,0()) -> 0() f4(S(x''),S(x'),x3,x4,S(x)) -> f2(S(x''),x',x3,x4,x) f5(x1,x2,x3,x4,0()) -> 0() f5(x1,x2,x3,x4,S(x)) -> f6(x2,x1,x3,x4,x) f6(x1,x2,x3,x4,0()) -> 0() f6(x1,x2,x3,x4,S(x)) -> f0(x1,x2,x4,x3,x) - Signature: {f0/5,f1/5,f2/5,f3/5,f4/5,f5/5,f6/5,f0#/5,f1#/5,f2#/5,f3#/5,f4#/5,f5#/5,f6#/5} / {0/0,S/1,c_1/0,c_2/0,c_3/1 ,c_4/0,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/0,c_19/1 ,c_20/0,c_21/1} - Obligation: innermost runtime complexity wrt. defined symbols {f0#,f1#,f2#,f3#,f4#,f5#,f6#} and constructors {0,S} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: f0#(x1,S(x'),x3,S(x),x5) -> c_3(f1#(x',S(x'),x,S(x),S(x))) f1#(x1,x2,x3,x4,S(x)) -> c_5(f2#(x2,x1,x3,x4,x)) f2#(x1,x2,S(x'),0(),S(x)) -> c_8(f3#(x1,x2,x',0(),x)) f2#(x1,x2,S(x''),S(x'),S(x)) -> c_10(f5#(x1,x2,S(x''),x',x)) f3#(x1,x2,x3,x4,S(x)) -> c_12(f4#(x1,x2,x4,x3,x)) f4#(S(x'),0(),x3,x4,S(x)) -> c_15(f3#(x',0(),x3,x4,x)) f4#(S(x''),S(x'),x3,x4,S(x)) -> c_17(f2#(S(x''),x',x3,x4,x)) f5#(x1,x2,x3,x4,S(x)) -> c_19(f6#(x2,x1,x3,x4,x)) f6#(x1,x2,x3,x4,S(x)) -> c_21(f0#(x1,x2,x4,x3,x)) * Step 6: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f0#(x1,S(x'),x3,S(x),x5) -> c_3(f1#(x',S(x'),x,S(x),S(x))) f1#(x1,x2,x3,x4,S(x)) -> c_5(f2#(x2,x1,x3,x4,x)) f2#(x1,x2,S(x'),0(),S(x)) -> c_8(f3#(x1,x2,x',0(),x)) f2#(x1,x2,S(x''),S(x'),S(x)) -> c_10(f5#(x1,x2,S(x''),x',x)) f3#(x1,x2,x3,x4,S(x)) -> c_12(f4#(x1,x2,x4,x3,x)) f4#(S(x'),0(),x3,x4,S(x)) -> c_15(f3#(x',0(),x3,x4,x)) f4#(S(x''),S(x'),x3,x4,S(x)) -> c_17(f2#(S(x''),x',x3,x4,x)) f5#(x1,x2,x3,x4,S(x)) -> c_19(f6#(x2,x1,x3,x4,x)) f6#(x1,x2,x3,x4,S(x)) -> c_21(f0#(x1,x2,x4,x3,x)) - Signature: {f0/5,f1/5,f2/5,f3/5,f4/5,f5/5,f6/5,f0#/5,f1#/5,f2#/5,f3#/5,f4#/5,f5#/5,f6#/5} / {0/0,S/1,c_1/0,c_2/0,c_3/1 ,c_4/0,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/0,c_19/1 ,c_20/0,c_21/1} - Obligation: innermost runtime complexity wrt. defined symbols {f0#,f1#,f2#,f3#,f4#,f5#,f6#} and constructors {0,S} + 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}, uargs(c_5) = {1}, uargs(c_8) = {1}, uargs(c_10) = {1}, uargs(c_12) = {1}, uargs(c_15) = {1}, uargs(c_17) = {1}, uargs(c_19) = {1}, uargs(c_21) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(S) = [1] x1 + [1] p(f0) = [0] p(f1) = [0] p(f2) = [0] p(f3) = [0] p(f4) = [0] p(f5) = [0] p(f6) = [0] p(f0#) = [1] x2 + [4] x4 + [6] p(f1#) = [1] x2 + [4] x3 + [0] p(f2#) = [1] x1 + [4] x3 + [1] p(f3#) = [1] x1 + [4] x3 + [4] x4 + [0] p(f4#) = [1] x1 + [4] x3 + [4] x4 + [0] p(f5#) = [1] x1 + [4] x3 + [0] p(f6#) = [1] x2 + [4] x3 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [1] x1 + [0] p(c_9) = [0] p(c_10) = [1] x1 + [0] p(c_11) = [0] p(c_12) = [1] x1 + [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [1] x1 + [0] p(c_16) = [0] p(c_17) = [1] x1 + [0] p(c_18) = [0] p(c_19) = [1] x1 + [0] p(c_20) = [0] p(c_21) = [1] x1 + [3] Following rules are strictly oriented: f0#(x1,S(x'),x3,S(x),x5) = [4] x + [1] x' + [11] > [4] x + [1] x' + [1] = c_3(f1#(x',S(x'),x,S(x),S(x))) f2#(x1,x2,S(x'),0(),S(x)) = [4] x' + [1] x1 + [5] > [4] x' + [1] x1 + [0] = c_8(f3#(x1,x2,x',0(),x)) f2#(x1,x2,S(x''),S(x'),S(x)) = [4] x'' + [1] x1 + [5] > [4] x'' + [1] x1 + [4] = c_10(f5#(x1,x2,S(x''),x',x)) f4#(S(x'),0(),x3,x4,S(x)) = [1] x' + [4] x3 + [4] x4 + [1] > [1] x' + [4] x3 + [4] x4 + [0] = c_15(f3#(x',0(),x3,x4,x)) Following rules are (at-least) weakly oriented: f1#(x1,x2,x3,x4,S(x)) = [1] x2 + [4] x3 + [0] >= [1] x2 + [4] x3 + [1] = c_5(f2#(x2,x1,x3,x4,x)) f3#(x1,x2,x3,x4,S(x)) = [1] x1 + [4] x3 + [4] x4 + [0] >= [1] x1 + [4] x3 + [4] x4 + [0] = c_12(f4#(x1,x2,x4,x3,x)) f4#(S(x''),S(x'),x3,x4,S(x)) = [1] x'' + [4] x3 + [4] x4 + [1] >= [1] x'' + [4] x3 + [2] = c_17(f2#(S(x''),x',x3,x4,x)) f5#(x1,x2,x3,x4,S(x)) = [1] x1 + [4] x3 + [0] >= [1] x1 + [4] x3 + [0] = c_19(f6#(x2,x1,x3,x4,x)) f6#(x1,x2,x3,x4,S(x)) = [1] x2 + [4] x3 + [0] >= [1] x2 + [4] x3 + [9] = c_21(f0#(x1,x2,x4,x3,x)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f1#(x1,x2,x3,x4,S(x)) -> c_5(f2#(x2,x1,x3,x4,x)) f3#(x1,x2,x3,x4,S(x)) -> c_12(f4#(x1,x2,x4,x3,x)) f4#(S(x''),S(x'),x3,x4,S(x)) -> c_17(f2#(S(x''),x',x3,x4,x)) f5#(x1,x2,x3,x4,S(x)) -> c_19(f6#(x2,x1,x3,x4,x)) f6#(x1,x2,x3,x4,S(x)) -> c_21(f0#(x1,x2,x4,x3,x)) - Weak DPs: f0#(x1,S(x'),x3,S(x),x5) -> c_3(f1#(x',S(x'),x,S(x),S(x))) f2#(x1,x2,S(x'),0(),S(x)) -> c_8(f3#(x1,x2,x',0(),x)) f2#(x1,x2,S(x''),S(x'),S(x)) -> c_10(f5#(x1,x2,S(x''),x',x)) f4#(S(x'),0(),x3,x4,S(x)) -> c_15(f3#(x',0(),x3,x4,x)) - Signature: {f0/5,f1/5,f2/5,f3/5,f4/5,f5/5,f6/5,f0#/5,f1#/5,f2#/5,f3#/5,f4#/5,f5#/5,f6#/5} / {0/0,S/1,c_1/0,c_2/0,c_3/1 ,c_4/0,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/0,c_19/1 ,c_20/0,c_21/1} - Obligation: innermost runtime complexity wrt. defined symbols {f0#,f1#,f2#,f3#,f4#,f5#,f6#} and constructors {0,S} + 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}, uargs(c_5) = {1}, uargs(c_8) = {1}, uargs(c_10) = {1}, uargs(c_12) = {1}, uargs(c_15) = {1}, uargs(c_17) = {1}, uargs(c_19) = {1}, uargs(c_21) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(S) = [0] p(f0) = [0] p(f1) = [0] p(f2) = [0] p(f3) = [1] x5 + [0] p(f4) = [0] p(f5) = [0] p(f6) = [1] x2 + [2] x3 + [0] p(f0#) = [1] p(f1#) = [2] x4 + [0] p(f2#) = [1] p(f3#) = [0] p(f4#) = [4] p(f5#) = [0] p(f6#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] p(c_5) = [1] x1 + [7] p(c_6) = [1] p(c_7) = [4] p(c_8) = [1] x1 + [0] p(c_9) = [1] p(c_10) = [1] x1 + [0] p(c_11) = [1] p(c_12) = [1] x1 + [5] p(c_13) = [4] p(c_14) = [4] p(c_15) = [1] x1 + [4] p(c_16) = [1] p(c_17) = [1] x1 + [2] p(c_18) = [0] p(c_19) = [1] x1 + [1] p(c_20) = [0] p(c_21) = [1] x1 + [0] Following rules are strictly oriented: f4#(S(x''),S(x'),x3,x4,S(x)) = [4] > [3] = c_17(f2#(S(x''),x',x3,x4,x)) Following rules are (at-least) weakly oriented: f0#(x1,S(x'),x3,S(x),x5) = [1] >= [0] = c_3(f1#(x',S(x'),x,S(x),S(x))) f1#(x1,x2,x3,x4,S(x)) = [2] x4 + [0] >= [8] = c_5(f2#(x2,x1,x3,x4,x)) f2#(x1,x2,S(x'),0(),S(x)) = [1] >= [0] = c_8(f3#(x1,x2,x',0(),x)) f2#(x1,x2,S(x''),S(x'),S(x)) = [1] >= [0] = c_10(f5#(x1,x2,S(x''),x',x)) f3#(x1,x2,x3,x4,S(x)) = [0] >= [9] = c_12(f4#(x1,x2,x4,x3,x)) f4#(S(x'),0(),x3,x4,S(x)) = [4] >= [4] = c_15(f3#(x',0(),x3,x4,x)) f5#(x1,x2,x3,x4,S(x)) = [0] >= [1] = c_19(f6#(x2,x1,x3,x4,x)) f6#(x1,x2,x3,x4,S(x)) = [0] >= [1] = c_21(f0#(x1,x2,x4,x3,x)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 8: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f1#(x1,x2,x3,x4,S(x)) -> c_5(f2#(x2,x1,x3,x4,x)) f3#(x1,x2,x3,x4,S(x)) -> c_12(f4#(x1,x2,x4,x3,x)) f5#(x1,x2,x3,x4,S(x)) -> c_19(f6#(x2,x1,x3,x4,x)) f6#(x1,x2,x3,x4,S(x)) -> c_21(f0#(x1,x2,x4,x3,x)) - Weak DPs: f0#(x1,S(x'),x3,S(x),x5) -> c_3(f1#(x',S(x'),x,S(x),S(x))) f2#(x1,x2,S(x'),0(),S(x)) -> c_8(f3#(x1,x2,x',0(),x)) f2#(x1,x2,S(x''),S(x'),S(x)) -> c_10(f5#(x1,x2,S(x''),x',x)) f4#(S(x'),0(),x3,x4,S(x)) -> c_15(f3#(x',0(),x3,x4,x)) f4#(S(x''),S(x'),x3,x4,S(x)) -> c_17(f2#(S(x''),x',x3,x4,x)) - Signature: {f0/5,f1/5,f2/5,f3/5,f4/5,f5/5,f6/5,f0#/5,f1#/5,f2#/5,f3#/5,f4#/5,f5#/5,f6#/5} / {0/0,S/1,c_1/0,c_2/0,c_3/1 ,c_4/0,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/0,c_19/1 ,c_20/0,c_21/1} - Obligation: innermost runtime complexity wrt. defined symbols {f0#,f1#,f2#,f3#,f4#,f5#,f6#} and constructors {0,S} + 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}, uargs(c_5) = {1}, uargs(c_8) = {1}, uargs(c_10) = {1}, uargs(c_12) = {1}, uargs(c_15) = {1}, uargs(c_17) = {1}, uargs(c_19) = {1}, uargs(c_21) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(S) = [0] p(f0) = [0] p(f1) = [0] p(f2) = [0] p(f3) = [0] p(f4) = [0] p(f5) = [0] p(f6) = [0] p(f0#) = [1] x2 + [0] p(f1#) = [5] x2 + [1] x4 + [0] p(f2#) = [4] x1 + [1] p(f3#) = [1] p(f4#) = [3] p(f5#) = [3] x1 + [0] p(f6#) = [3] x2 + [1] p(c_1) = [0] p(c_2) = [4] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [1] x1 + [1] p(c_6) = [0] p(c_7) = [0] p(c_8) = [1] x1 + [0] p(c_9) = [0] p(c_10) = [1] x1 + [0] p(c_11) = [1] p(c_12) = [1] x1 + [0] p(c_13) = [1] p(c_14) = [2] p(c_15) = [1] x1 + [0] p(c_16) = [0] p(c_17) = [1] x1 + [2] p(c_18) = [0] p(c_19) = [1] x1 + [3] p(c_20) = [2] p(c_21) = [1] x1 + [0] Following rules are strictly oriented: f6#(x1,x2,x3,x4,S(x)) = [3] x2 + [1] > [1] x2 + [0] = c_21(f0#(x1,x2,x4,x3,x)) Following rules are (at-least) weakly oriented: f0#(x1,S(x'),x3,S(x),x5) = [0] >= [0] = c_3(f1#(x',S(x'),x,S(x),S(x))) f1#(x1,x2,x3,x4,S(x)) = [5] x2 + [1] x4 + [0] >= [4] x2 + [2] = c_5(f2#(x2,x1,x3,x4,x)) f2#(x1,x2,S(x'),0(),S(x)) = [4] x1 + [1] >= [1] = c_8(f3#(x1,x2,x',0(),x)) f2#(x1,x2,S(x''),S(x'),S(x)) = [4] x1 + [1] >= [3] x1 + [0] = c_10(f5#(x1,x2,S(x''),x',x)) f3#(x1,x2,x3,x4,S(x)) = [1] >= [3] = c_12(f4#(x1,x2,x4,x3,x)) f4#(S(x'),0(),x3,x4,S(x)) = [3] >= [1] = c_15(f3#(x',0(),x3,x4,x)) f4#(S(x''),S(x'),x3,x4,S(x)) = [3] >= [3] = c_17(f2#(S(x''),x',x3,x4,x)) f5#(x1,x2,x3,x4,S(x)) = [3] x1 + [0] >= [3] x1 + [4] = c_19(f6#(x2,x1,x3,x4,x)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 9: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f1#(x1,x2,x3,x4,S(x)) -> c_5(f2#(x2,x1,x3,x4,x)) f3#(x1,x2,x3,x4,S(x)) -> c_12(f4#(x1,x2,x4,x3,x)) f5#(x1,x2,x3,x4,S(x)) -> c_19(f6#(x2,x1,x3,x4,x)) - Weak DPs: f0#(x1,S(x'),x3,S(x),x5) -> c_3(f1#(x',S(x'),x,S(x),S(x))) f2#(x1,x2,S(x'),0(),S(x)) -> c_8(f3#(x1,x2,x',0(),x)) f2#(x1,x2,S(x''),S(x'),S(x)) -> c_10(f5#(x1,x2,S(x''),x',x)) f4#(S(x'),0(),x3,x4,S(x)) -> c_15(f3#(x',0(),x3,x4,x)) f4#(S(x''),S(x'),x3,x4,S(x)) -> c_17(f2#(S(x''),x',x3,x4,x)) f6#(x1,x2,x3,x4,S(x)) -> c_21(f0#(x1,x2,x4,x3,x)) - Signature: {f0/5,f1/5,f2/5,f3/5,f4/5,f5/5,f6/5,f0#/5,f1#/5,f2#/5,f3#/5,f4#/5,f5#/5,f6#/5} / {0/0,S/1,c_1/0,c_2/0,c_3/1 ,c_4/0,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/0,c_19/1 ,c_20/0,c_21/1} - Obligation: innermost runtime complexity wrt. defined symbols {f0#,f1#,f2#,f3#,f4#,f5#,f6#} and constructors {0,S} + 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}, uargs(c_5) = {1}, uargs(c_8) = {1}, uargs(c_10) = {1}, uargs(c_12) = {1}, uargs(c_15) = {1}, uargs(c_17) = {1}, uargs(c_19) = {1}, uargs(c_21) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(S) = [0] p(f0) = [0] p(f1) = [0] p(f2) = [0] p(f3) = [0] p(f4) = [0] p(f5) = [1] x1 + [4] x3 + [4] x5 + [0] p(f6) = [1] x1 + [4] x4 + [4] x5 + [1] p(f0#) = [0] p(f1#) = [4] x2 + [2] x5 + [0] p(f2#) = [1] p(f3#) = [1] p(f4#) = [3] p(f5#) = [1] x3 + [1] p(f6#) = [1] x3 + [0] p(c_1) = [0] p(c_2) = [1] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [1] x1 + [7] p(c_6) = [0] p(c_7) = [0] p(c_8) = [1] x1 + [0] p(c_9) = [1] p(c_10) = [1] x1 + [0] p(c_11) = [0] p(c_12) = [1] x1 + [3] p(c_13) = [4] p(c_14) = [1] p(c_15) = [1] x1 + [0] p(c_16) = [1] p(c_17) = [1] x1 + [2] p(c_18) = [0] p(c_19) = [1] x1 + [0] p(c_20) = [0] p(c_21) = [1] x1 + [0] Following rules are strictly oriented: f5#(x1,x2,x3,x4,S(x)) = [1] x3 + [1] > [1] x3 + [0] = c_19(f6#(x2,x1,x3,x4,x)) Following rules are (at-least) weakly oriented: f0#(x1,S(x'),x3,S(x),x5) = [0] >= [0] = c_3(f1#(x',S(x'),x,S(x),S(x))) f1#(x1,x2,x3,x4,S(x)) = [4] x2 + [0] >= [8] = c_5(f2#(x2,x1,x3,x4,x)) f2#(x1,x2,S(x'),0(),S(x)) = [1] >= [1] = c_8(f3#(x1,x2,x',0(),x)) f2#(x1,x2,S(x''),S(x'),S(x)) = [1] >= [1] = c_10(f5#(x1,x2,S(x''),x',x)) f3#(x1,x2,x3,x4,S(x)) = [1] >= [6] = c_12(f4#(x1,x2,x4,x3,x)) f4#(S(x'),0(),x3,x4,S(x)) = [3] >= [1] = c_15(f3#(x',0(),x3,x4,x)) f4#(S(x''),S(x'),x3,x4,S(x)) = [3] >= [3] = c_17(f2#(S(x''),x',x3,x4,x)) f6#(x1,x2,x3,x4,S(x)) = [1] x3 + [0] >= [0] = c_21(f0#(x1,x2,x4,x3,x)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 10: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f1#(x1,x2,x3,x4,S(x)) -> c_5(f2#(x2,x1,x3,x4,x)) f3#(x1,x2,x3,x4,S(x)) -> c_12(f4#(x1,x2,x4,x3,x)) - Weak DPs: f0#(x1,S(x'),x3,S(x),x5) -> c_3(f1#(x',S(x'),x,S(x),S(x))) f2#(x1,x2,S(x'),0(),S(x)) -> c_8(f3#(x1,x2,x',0(),x)) f2#(x1,x2,S(x''),S(x'),S(x)) -> c_10(f5#(x1,x2,S(x''),x',x)) f4#(S(x'),0(),x3,x4,S(x)) -> c_15(f3#(x',0(),x3,x4,x)) f4#(S(x''),S(x'),x3,x4,S(x)) -> c_17(f2#(S(x''),x',x3,x4,x)) f5#(x1,x2,x3,x4,S(x)) -> c_19(f6#(x2,x1,x3,x4,x)) f6#(x1,x2,x3,x4,S(x)) -> c_21(f0#(x1,x2,x4,x3,x)) - Signature: {f0/5,f1/5,f2/5,f3/5,f4/5,f5/5,f6/5,f0#/5,f1#/5,f2#/5,f3#/5,f4#/5,f5#/5,f6#/5} / {0/0,S/1,c_1/0,c_2/0,c_3/1 ,c_4/0,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/0,c_19/1 ,c_20/0,c_21/1} - Obligation: innermost runtime complexity wrt. defined symbols {f0#,f1#,f2#,f3#,f4#,f5#,f6#} and constructors {0,S} + 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}, uargs(c_5) = {1}, uargs(c_8) = {1}, uargs(c_10) = {1}, uargs(c_12) = {1}, uargs(c_15) = {1}, uargs(c_17) = {1}, uargs(c_19) = {1}, uargs(c_21) = {1} Following symbols are considered usable: {f0#,f1#,f2#,f3#,f4#,f5#,f6#} TcT has computed the following interpretation: p(0) = [2] p(S) = [1] x1 + [3] p(f0) = [1] x2 + [1] x4 + [2] x5 + [0] p(f1) = [1] x2 + [1] x5 + [1] p(f2) = [1] x2 + [2] x3 + [1] p(f3) = [1] x4 + [1] p(f4) = [4] x5 + [1] p(f5) = [4] x2 + [4] x3 + [1] x4 + [1] x5 + [4] p(f6) = [2] x1 + [4] x2 + [2] x3 + [1] x4 + [0] p(f0#) = [4] x4 + [0] p(f1#) = [4] x3 + [2] p(f2#) = [4] x3 + [1] p(f3#) = [4] x3 + [4] x4 + [1] p(f4#) = [4] x3 + [4] x4 + [1] p(f5#) = [4] x3 + [0] p(f6#) = [4] x3 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [6] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] p(c_7) = [0] p(c_8) = [1] x1 + [4] p(c_9) = [1] p(c_10) = [1] x1 + [1] p(c_11) = [1] p(c_12) = [1] x1 + [0] p(c_13) = [1] p(c_14) = [1] p(c_15) = [1] x1 + [0] p(c_16) = [0] p(c_17) = [1] x1 + [0] p(c_18) = [0] p(c_19) = [1] x1 + [0] p(c_20) = [2] p(c_21) = [1] x1 + [0] Following rules are strictly oriented: f1#(x1,x2,x3,x4,S(x)) = [4] x3 + [2] > [4] x3 + [1] = c_5(f2#(x2,x1,x3,x4,x)) Following rules are (at-least) weakly oriented: f0#(x1,S(x'),x3,S(x),x5) = [4] x + [12] >= [4] x + [8] = c_3(f1#(x',S(x'),x,S(x),S(x))) f2#(x1,x2,S(x'),0(),S(x)) = [4] x' + [13] >= [4] x' + [13] = c_8(f3#(x1,x2,x',0(),x)) f2#(x1,x2,S(x''),S(x'),S(x)) = [4] x'' + [13] >= [4] x'' + [13] = c_10(f5#(x1,x2,S(x''),x',x)) f3#(x1,x2,x3,x4,S(x)) = [4] x3 + [4] x4 + [1] >= [4] x3 + [4] x4 + [1] = c_12(f4#(x1,x2,x4,x3,x)) f4#(S(x'),0(),x3,x4,S(x)) = [4] x3 + [4] x4 + [1] >= [4] x3 + [4] x4 + [1] = c_15(f3#(x',0(),x3,x4,x)) f4#(S(x''),S(x'),x3,x4,S(x)) = [4] x3 + [4] x4 + [1] >= [4] x3 + [1] = c_17(f2#(S(x''),x',x3,x4,x)) f5#(x1,x2,x3,x4,S(x)) = [4] x3 + [0] >= [4] x3 + [0] = c_19(f6#(x2,x1,x3,x4,x)) f6#(x1,x2,x3,x4,S(x)) = [4] x3 + [0] >= [4] x3 + [0] = c_21(f0#(x1,x2,x4,x3,x)) * Step 11: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f3#(x1,x2,x3,x4,S(x)) -> c_12(f4#(x1,x2,x4,x3,x)) - Weak DPs: f0#(x1,S(x'),x3,S(x),x5) -> c_3(f1#(x',S(x'),x,S(x),S(x))) f1#(x1,x2,x3,x4,S(x)) -> c_5(f2#(x2,x1,x3,x4,x)) f2#(x1,x2,S(x'),0(),S(x)) -> c_8(f3#(x1,x2,x',0(),x)) f2#(x1,x2,S(x''),S(x'),S(x)) -> c_10(f5#(x1,x2,S(x''),x',x)) f4#(S(x'),0(),x3,x4,S(x)) -> c_15(f3#(x',0(),x3,x4,x)) f4#(S(x''),S(x'),x3,x4,S(x)) -> c_17(f2#(S(x''),x',x3,x4,x)) f5#(x1,x2,x3,x4,S(x)) -> c_19(f6#(x2,x1,x3,x4,x)) f6#(x1,x2,x3,x4,S(x)) -> c_21(f0#(x1,x2,x4,x3,x)) - Signature: {f0/5,f1/5,f2/5,f3/5,f4/5,f5/5,f6/5,f0#/5,f1#/5,f2#/5,f3#/5,f4#/5,f5#/5,f6#/5} / {0/0,S/1,c_1/0,c_2/0,c_3/1 ,c_4/0,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/0,c_19/1 ,c_20/0,c_21/1} - Obligation: innermost runtime complexity wrt. defined symbols {f0#,f1#,f2#,f3#,f4#,f5#,f6#} and constructors {0,S} + 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}, uargs(c_5) = {1}, uargs(c_8) = {1}, uargs(c_10) = {1}, uargs(c_12) = {1}, uargs(c_15) = {1}, uargs(c_17) = {1}, uargs(c_19) = {1}, uargs(c_21) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(S) = [1] x1 + [2] p(f0) = [1] x1 + [2] x2 + [0] p(f1) = [4] x1 + [4] x2 + [0] p(f2) = [1] x1 + [2] x2 + [1] x3 + [1] x4 + [4] p(f3) = [4] x1 + [1] x2 + [1] x3 + [1] x5 + [2] p(f4) = [2] x1 + [1] x2 + [4] x5 + [0] p(f5) = [4] x1 + [4] x2 + [1] x3 + [1] x4 + [1] p(f6) = [1] x4 + [4] x5 + [0] p(f0#) = [4] x2 + [2] x4 + [0] p(f1#) = [4] x2 + [2] x3 + [1] p(f2#) = [4] x1 + [2] x3 + [0] p(f3#) = [4] x1 + [2] x3 + [2] x4 + [1] p(f4#) = [4] x1 + [2] x3 + [2] x4 + [0] p(f5#) = [4] x1 + [2] x3 + [0] p(f6#) = [4] x2 + [2] x3 + [0] p(c_1) = [0] p(c_2) = [4] p(c_3) = [1] x1 + [1] p(c_4) = [1] p(c_5) = [1] x1 + [0] p(c_6) = [0] p(c_7) = [2] p(c_8) = [1] x1 + [0] p(c_9) = [1] p(c_10) = [1] x1 + [0] p(c_11) = [1] p(c_12) = [1] x1 + [0] p(c_13) = [2] p(c_14) = [1] p(c_15) = [1] x1 + [7] p(c_16) = [1] p(c_17) = [1] x1 + [0] p(c_18) = [2] p(c_19) = [1] x1 + [0] p(c_20) = [4] p(c_21) = [1] x1 + [0] Following rules are strictly oriented: f3#(x1,x2,x3,x4,S(x)) = [4] x1 + [2] x3 + [2] x4 + [1] > [4] x1 + [2] x3 + [2] x4 + [0] = c_12(f4#(x1,x2,x4,x3,x)) Following rules are (at-least) weakly oriented: f0#(x1,S(x'),x3,S(x),x5) = [2] x + [4] x' + [12] >= [2] x + [4] x' + [10] = c_3(f1#(x',S(x'),x,S(x),S(x))) f1#(x1,x2,x3,x4,S(x)) = [4] x2 + [2] x3 + [1] >= [4] x2 + [2] x3 + [0] = c_5(f2#(x2,x1,x3,x4,x)) f2#(x1,x2,S(x'),0(),S(x)) = [2] x' + [4] x1 + [4] >= [2] x' + [4] x1 + [1] = c_8(f3#(x1,x2,x',0(),x)) f2#(x1,x2,S(x''),S(x'),S(x)) = [2] x'' + [4] x1 + [4] >= [2] x'' + [4] x1 + [4] = c_10(f5#(x1,x2,S(x''),x',x)) f4#(S(x'),0(),x3,x4,S(x)) = [4] x' + [2] x3 + [2] x4 + [8] >= [4] x' + [2] x3 + [2] x4 + [8] = c_15(f3#(x',0(),x3,x4,x)) f4#(S(x''),S(x'),x3,x4,S(x)) = [4] x'' + [2] x3 + [2] x4 + [8] >= [4] x'' + [2] x3 + [8] = c_17(f2#(S(x''),x',x3,x4,x)) f5#(x1,x2,x3,x4,S(x)) = [4] x1 + [2] x3 + [0] >= [4] x1 + [2] x3 + [0] = c_19(f6#(x2,x1,x3,x4,x)) f6#(x1,x2,x3,x4,S(x)) = [4] x2 + [2] x3 + [0] >= [4] x2 + [2] x3 + [0] = c_21(f0#(x1,x2,x4,x3,x)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 12: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: f0#(x1,S(x'),x3,S(x),x5) -> c_3(f1#(x',S(x'),x,S(x),S(x))) f1#(x1,x2,x3,x4,S(x)) -> c_5(f2#(x2,x1,x3,x4,x)) f2#(x1,x2,S(x'),0(),S(x)) -> c_8(f3#(x1,x2,x',0(),x)) f2#(x1,x2,S(x''),S(x'),S(x)) -> c_10(f5#(x1,x2,S(x''),x',x)) f3#(x1,x2,x3,x4,S(x)) -> c_12(f4#(x1,x2,x4,x3,x)) f4#(S(x'),0(),x3,x4,S(x)) -> c_15(f3#(x',0(),x3,x4,x)) f4#(S(x''),S(x'),x3,x4,S(x)) -> c_17(f2#(S(x''),x',x3,x4,x)) f5#(x1,x2,x3,x4,S(x)) -> c_19(f6#(x2,x1,x3,x4,x)) f6#(x1,x2,x3,x4,S(x)) -> c_21(f0#(x1,x2,x4,x3,x)) - Signature: {f0/5,f1/5,f2/5,f3/5,f4/5,f5/5,f6/5,f0#/5,f1#/5,f2#/5,f3#/5,f4#/5,f5#/5,f6#/5} / {0/0,S/1,c_1/0,c_2/0,c_3/1 ,c_4/0,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0,c_12/1,c_13/0,c_14/0,c_15/1,c_16/0,c_17/1,c_18/0,c_19/1 ,c_20/0,c_21/1} - Obligation: innermost runtime complexity wrt. defined symbols {f0#,f1#,f2#,f3#,f4#,f5#,f6#} and constructors {0,S} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))