WORST_CASE(Omega(n^1),O(n^2)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^2)) + Considered Problem: - Strict TRS: +'(z0,0()) -> c7() +'(z0,s(z1)) -> c8(+'(z0,z1)) DOUBLE(0()) -> c5() DOUBLE(s(z0)) -> c6(DOUBLE(z0)) SQR(0()) -> c() SQR(s(z0)) -> c1(+'(sqr(z0),s(double(z0))),SQR(z0)) SQR(s(z0)) -> c2(+'(sqr(z0),s(double(z0))),DOUBLE(z0)) SQR(s(z0)) -> c3(+'(sqr(z0),double(z0)),SQR(z0)) SQR(s(z0)) -> c4(+'(sqr(z0),double(z0)),DOUBLE(z0)) - Weak TRS: +(z0,0()) -> z0 +(z0,s(z1)) -> s(+(z0,z1)) double(0()) -> 0() double(s(z0)) -> s(s(double(z0))) sqr(0()) -> 0() sqr(s(z0)) -> +(sqr(z0),s(double(z0))) sqr(s(z0)) -> s(+(sqr(z0),double(z0))) - Signature: {+/2,+'/2,DOUBLE/1,SQR/1,double/1,sqr/1} / {0/0,c/0,c1/2,c2/2,c3/2,c4/2,c5/0,c6/1,c7/0,c8/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,+',DOUBLE,SQR,double,sqr} and constructors {0,c,c1,c2 ,c3,c4,c5,c6,c7,c8,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: +'(z0,0()) -> c7() +'(z0,s(z1)) -> c8(+'(z0,z1)) DOUBLE(0()) -> c5() DOUBLE(s(z0)) -> c6(DOUBLE(z0)) SQR(0()) -> c() SQR(s(z0)) -> c1(+'(sqr(z0),s(double(z0))),SQR(z0)) SQR(s(z0)) -> c2(+'(sqr(z0),s(double(z0))),DOUBLE(z0)) SQR(s(z0)) -> c3(+'(sqr(z0),double(z0)),SQR(z0)) SQR(s(z0)) -> c4(+'(sqr(z0),double(z0)),DOUBLE(z0)) - Weak TRS: +(z0,0()) -> z0 +(z0,s(z1)) -> s(+(z0,z1)) double(0()) -> 0() double(s(z0)) -> s(s(double(z0))) sqr(0()) -> 0() sqr(s(z0)) -> +(sqr(z0),s(double(z0))) sqr(s(z0)) -> s(+(sqr(z0),double(z0))) - Signature: {+/2,+'/2,DOUBLE/1,SQR/1,double/1,sqr/1} / {0/0,c/0,c1/2,c2/2,c3/2,c4/2,c5/0,c6/1,c7/0,c8/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,+',DOUBLE,SQR,double,sqr} and constructors {0,c,c1,c2 ,c3,c4,c5,c6,c7,c8,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: +'(z0,0()) -> c7() +'(z0,s(z1)) -> c8(+'(z0,z1)) DOUBLE(0()) -> c5() DOUBLE(s(z0)) -> c6(DOUBLE(z0)) SQR(0()) -> c() SQR(s(z0)) -> c1(+'(sqr(z0),s(double(z0))),SQR(z0)) SQR(s(z0)) -> c2(+'(sqr(z0),s(double(z0))),DOUBLE(z0)) SQR(s(z0)) -> c3(+'(sqr(z0),double(z0)),SQR(z0)) SQR(s(z0)) -> c4(+'(sqr(z0),double(z0)),DOUBLE(z0)) - Weak TRS: +(z0,0()) -> z0 +(z0,s(z1)) -> s(+(z0,z1)) double(0()) -> 0() double(s(z0)) -> s(s(double(z0))) sqr(0()) -> 0() sqr(s(z0)) -> +(sqr(z0),s(double(z0))) sqr(s(z0)) -> s(+(sqr(z0),double(z0))) - Signature: {+/2,+'/2,DOUBLE/1,SQR/1,double/1,sqr/1} / {0/0,c/0,c1/2,c2/2,c3/2,c4/2,c5/0,c6/1,c7/0,c8/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,+',DOUBLE,SQR,double,sqr} and constructors {0,c,c1,c2 ,c3,c4,c5,c6,c7,c8,s} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: +'(x,y){y -> s(y)} = +'(x,s(y)) ->^+ c8(+'(x,y)) = C[+'(x,y) = +'(x,y){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: +'(z0,0()) -> c7() +'(z0,s(z1)) -> c8(+'(z0,z1)) DOUBLE(0()) -> c5() DOUBLE(s(z0)) -> c6(DOUBLE(z0)) SQR(0()) -> c() SQR(s(z0)) -> c1(+'(sqr(z0),s(double(z0))),SQR(z0)) SQR(s(z0)) -> c2(+'(sqr(z0),s(double(z0))),DOUBLE(z0)) SQR(s(z0)) -> c3(+'(sqr(z0),double(z0)),SQR(z0)) SQR(s(z0)) -> c4(+'(sqr(z0),double(z0)),DOUBLE(z0)) - Weak TRS: +(z0,0()) -> z0 +(z0,s(z1)) -> s(+(z0,z1)) double(0()) -> 0() double(s(z0)) -> s(s(double(z0))) sqr(0()) -> 0() sqr(s(z0)) -> +(sqr(z0),s(double(z0))) sqr(s(z0)) -> s(+(sqr(z0),double(z0))) - Signature: {+/2,+'/2,DOUBLE/1,SQR/1,double/1,sqr/1} / {0/0,c/0,c1/2,c2/2,c3/2,c4/2,c5/0,c6/1,c7/0,c8/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,+',DOUBLE,SQR,double,sqr} and constructors {0,c,c1,c2 ,c3,c4,c5,c6,c7,c8,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs +'#(z0,0()) -> c_1() +'#(z0,s(z1)) -> c_2(+'#(z0,z1)) DOUBLE#(0()) -> c_3() DOUBLE#(s(z0)) -> c_4(DOUBLE#(z0)) SQR#(0()) -> c_5() SQR#(s(z0)) -> c_6(+'#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0),SQR#(z0)) SQR#(s(z0)) -> c_7(+'#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0),DOUBLE#(z0)) SQR#(s(z0)) -> c_8(+'#(sqr(z0),double(z0)),sqr#(z0),double#(z0),SQR#(z0)) SQR#(s(z0)) -> c_9(+'#(sqr(z0),double(z0)),sqr#(z0),double#(z0),DOUBLE#(z0)) Weak DPs +#(z0,0()) -> c_10() +#(z0,s(z1)) -> c_11(+#(z0,z1)) double#(0()) -> c_12() double#(s(z0)) -> c_13(double#(z0)) sqr#(0()) -> c_14() sqr#(s(z0)) -> c_15(+#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0)) sqr#(s(z0)) -> c_16(+#(sqr(z0),double(z0)),sqr#(z0),double#(z0)) and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: +'#(z0,0()) -> c_1() +'#(z0,s(z1)) -> c_2(+'#(z0,z1)) DOUBLE#(0()) -> c_3() DOUBLE#(s(z0)) -> c_4(DOUBLE#(z0)) SQR#(0()) -> c_5() SQR#(s(z0)) -> c_6(+'#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0),SQR#(z0)) SQR#(s(z0)) -> c_7(+'#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0),DOUBLE#(z0)) SQR#(s(z0)) -> c_8(+'#(sqr(z0),double(z0)),sqr#(z0),double#(z0),SQR#(z0)) SQR#(s(z0)) -> c_9(+'#(sqr(z0),double(z0)),sqr#(z0),double#(z0),DOUBLE#(z0)) - Weak DPs: +#(z0,0()) -> c_10() +#(z0,s(z1)) -> c_11(+#(z0,z1)) double#(0()) -> c_12() double#(s(z0)) -> c_13(double#(z0)) sqr#(0()) -> c_14() sqr#(s(z0)) -> c_15(+#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0)) sqr#(s(z0)) -> c_16(+#(sqr(z0),double(z0)),sqr#(z0),double#(z0)) - Weak TRS: +(z0,0()) -> z0 +(z0,s(z1)) -> s(+(z0,z1)) +'(z0,0()) -> c7() +'(z0,s(z1)) -> c8(+'(z0,z1)) DOUBLE(0()) -> c5() DOUBLE(s(z0)) -> c6(DOUBLE(z0)) SQR(0()) -> c() SQR(s(z0)) -> c1(+'(sqr(z0),s(double(z0))),SQR(z0)) SQR(s(z0)) -> c2(+'(sqr(z0),s(double(z0))),DOUBLE(z0)) SQR(s(z0)) -> c3(+'(sqr(z0),double(z0)),SQR(z0)) SQR(s(z0)) -> c4(+'(sqr(z0),double(z0)),DOUBLE(z0)) double(0()) -> 0() double(s(z0)) -> s(s(double(z0))) sqr(0()) -> 0() sqr(s(z0)) -> +(sqr(z0),s(double(z0))) sqr(s(z0)) -> s(+(sqr(z0),double(z0))) - Signature: {+/2,+'/2,DOUBLE/1,SQR/1,double/1,sqr/1,+#/2,+'#/2,DOUBLE#/1,SQR#/1,double#/1,sqr#/1} / {0/0,c/0,c1/2,c2/2 ,c3/2,c4/2,c5/0,c6/1,c7/0,c8/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/4,c_7/4,c_8/4,c_9/4,c_10/0,c_11/1 ,c_12/0,c_13/1,c_14/0,c_15/3,c_16/3} - Obligation: innermost runtime complexity wrt. defined symbols {+#,+'#,DOUBLE#,SQR#,double#,sqr#} and constructors {0,c ,c1,c2,c3,c4,c5,c6,c7,c8,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,5} by application of Pre({1,3,5}) = {2,4,6,7,8,9}. Here rules are labelled as follows: 1: +'#(z0,0()) -> c_1() 2: +'#(z0,s(z1)) -> c_2(+'#(z0,z1)) 3: DOUBLE#(0()) -> c_3() 4: DOUBLE#(s(z0)) -> c_4(DOUBLE#(z0)) 5: SQR#(0()) -> c_5() 6: SQR#(s(z0)) -> c_6(+'#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0),SQR#(z0)) 7: SQR#(s(z0)) -> c_7(+'#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0),DOUBLE#(z0)) 8: SQR#(s(z0)) -> c_8(+'#(sqr(z0),double(z0)),sqr#(z0),double#(z0),SQR#(z0)) 9: SQR#(s(z0)) -> c_9(+'#(sqr(z0),double(z0)),sqr#(z0),double#(z0),DOUBLE#(z0)) 10: +#(z0,0()) -> c_10() 11: +#(z0,s(z1)) -> c_11(+#(z0,z1)) 12: double#(0()) -> c_12() 13: double#(s(z0)) -> c_13(double#(z0)) 14: sqr#(0()) -> c_14() 15: sqr#(s(z0)) -> c_15(+#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0)) 16: sqr#(s(z0)) -> c_16(+#(sqr(z0),double(z0)),sqr#(z0),double#(z0)) ** Step 1.b:3: RemoveWeakSuffixes. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: +'#(z0,s(z1)) -> c_2(+'#(z0,z1)) DOUBLE#(s(z0)) -> c_4(DOUBLE#(z0)) SQR#(s(z0)) -> c_6(+'#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0),SQR#(z0)) SQR#(s(z0)) -> c_7(+'#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0),DOUBLE#(z0)) SQR#(s(z0)) -> c_8(+'#(sqr(z0),double(z0)),sqr#(z0),double#(z0),SQR#(z0)) SQR#(s(z0)) -> c_9(+'#(sqr(z0),double(z0)),sqr#(z0),double#(z0),DOUBLE#(z0)) - Weak DPs: +#(z0,0()) -> c_10() +#(z0,s(z1)) -> c_11(+#(z0,z1)) +'#(z0,0()) -> c_1() DOUBLE#(0()) -> c_3() SQR#(0()) -> c_5() double#(0()) -> c_12() double#(s(z0)) -> c_13(double#(z0)) sqr#(0()) -> c_14() sqr#(s(z0)) -> c_15(+#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0)) sqr#(s(z0)) -> c_16(+#(sqr(z0),double(z0)),sqr#(z0),double#(z0)) - Weak TRS: +(z0,0()) -> z0 +(z0,s(z1)) -> s(+(z0,z1)) +'(z0,0()) -> c7() +'(z0,s(z1)) -> c8(+'(z0,z1)) DOUBLE(0()) -> c5() DOUBLE(s(z0)) -> c6(DOUBLE(z0)) SQR(0()) -> c() SQR(s(z0)) -> c1(+'(sqr(z0),s(double(z0))),SQR(z0)) SQR(s(z0)) -> c2(+'(sqr(z0),s(double(z0))),DOUBLE(z0)) SQR(s(z0)) -> c3(+'(sqr(z0),double(z0)),SQR(z0)) SQR(s(z0)) -> c4(+'(sqr(z0),double(z0)),DOUBLE(z0)) double(0()) -> 0() double(s(z0)) -> s(s(double(z0))) sqr(0()) -> 0() sqr(s(z0)) -> +(sqr(z0),s(double(z0))) sqr(s(z0)) -> s(+(sqr(z0),double(z0))) - Signature: {+/2,+'/2,DOUBLE/1,SQR/1,double/1,sqr/1,+#/2,+'#/2,DOUBLE#/1,SQR#/1,double#/1,sqr#/1} / {0/0,c/0,c1/2,c2/2 ,c3/2,c4/2,c5/0,c6/1,c7/0,c8/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/4,c_7/4,c_8/4,c_9/4,c_10/0,c_11/1 ,c_12/0,c_13/1,c_14/0,c_15/3,c_16/3} - Obligation: innermost runtime complexity wrt. defined symbols {+#,+'#,DOUBLE#,SQR#,double#,sqr#} and constructors {0,c ,c1,c2,c3,c4,c5,c6,c7,c8,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:+'#(z0,s(z1)) -> c_2(+'#(z0,z1)) -->_1 +'#(z0,0()) -> c_1():9 -->_1 +'#(z0,s(z1)) -> c_2(+'#(z0,z1)):1 2:S:DOUBLE#(s(z0)) -> c_4(DOUBLE#(z0)) -->_1 DOUBLE#(0()) -> c_3():10 -->_1 DOUBLE#(s(z0)) -> c_4(DOUBLE#(z0)):2 3:S:SQR#(s(z0)) -> c_6(+'#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0),SQR#(z0)) -->_2 sqr#(s(z0)) -> c_16(+#(sqr(z0),double(z0)),sqr#(z0),double#(z0)):16 -->_2 sqr#(s(z0)) -> c_15(+#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0)):15 -->_3 double#(s(z0)) -> c_13(double#(z0)):13 -->_4 SQR#(s(z0)) -> c_9(+'#(sqr(z0),double(z0)),sqr#(z0),double#(z0),DOUBLE#(z0)):6 -->_4 SQR#(s(z0)) -> c_8(+'#(sqr(z0),double(z0)),sqr#(z0),double#(z0),SQR#(z0)):5 -->_4 SQR#(s(z0)) -> c_7(+'#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0),DOUBLE#(z0)):4 -->_2 sqr#(0()) -> c_14():14 -->_3 double#(0()) -> c_12():12 -->_4 SQR#(0()) -> c_5():11 -->_4 SQR#(s(z0)) -> c_6(+'#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0),SQR#(z0)):3 -->_1 +'#(z0,s(z1)) -> c_2(+'#(z0,z1)):1 4:S:SQR#(s(z0)) -> c_7(+'#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0),DOUBLE#(z0)) -->_2 sqr#(s(z0)) -> c_16(+#(sqr(z0),double(z0)),sqr#(z0),double#(z0)):16 -->_2 sqr#(s(z0)) -> c_15(+#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0)):15 -->_3 double#(s(z0)) -> c_13(double#(z0)):13 -->_2 sqr#(0()) -> c_14():14 -->_3 double#(0()) -> c_12():12 -->_4 DOUBLE#(0()) -> c_3():10 -->_4 DOUBLE#(s(z0)) -> c_4(DOUBLE#(z0)):2 -->_1 +'#(z0,s(z1)) -> c_2(+'#(z0,z1)):1 5:S:SQR#(s(z0)) -> c_8(+'#(sqr(z0),double(z0)),sqr#(z0),double#(z0),SQR#(z0)) -->_2 sqr#(s(z0)) -> c_16(+#(sqr(z0),double(z0)),sqr#(z0),double#(z0)):16 -->_2 sqr#(s(z0)) -> c_15(+#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0)):15 -->_3 double#(s(z0)) -> c_13(double#(z0)):13 -->_4 SQR#(s(z0)) -> c_9(+'#(sqr(z0),double(z0)),sqr#(z0),double#(z0),DOUBLE#(z0)):6 -->_2 sqr#(0()) -> c_14():14 -->_3 double#(0()) -> c_12():12 -->_4 SQR#(0()) -> c_5():11 -->_1 +'#(z0,0()) -> c_1():9 -->_4 SQR#(s(z0)) -> c_8(+'#(sqr(z0),double(z0)),sqr#(z0),double#(z0),SQR#(z0)):5 -->_4 SQR#(s(z0)) -> c_7(+'#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0),DOUBLE#(z0)):4 -->_4 SQR#(s(z0)) -> c_6(+'#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0),SQR#(z0)):3 -->_1 +'#(z0,s(z1)) -> c_2(+'#(z0,z1)):1 6:S:SQR#(s(z0)) -> c_9(+'#(sqr(z0),double(z0)),sqr#(z0),double#(z0),DOUBLE#(z0)) -->_2 sqr#(s(z0)) -> c_16(+#(sqr(z0),double(z0)),sqr#(z0),double#(z0)):16 -->_2 sqr#(s(z0)) -> c_15(+#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0)):15 -->_3 double#(s(z0)) -> c_13(double#(z0)):13 -->_2 sqr#(0()) -> c_14():14 -->_3 double#(0()) -> c_12():12 -->_4 DOUBLE#(0()) -> c_3():10 -->_1 +'#(z0,0()) -> c_1():9 -->_4 DOUBLE#(s(z0)) -> c_4(DOUBLE#(z0)):2 -->_1 +'#(z0,s(z1)) -> c_2(+'#(z0,z1)):1 7:W:+#(z0,0()) -> c_10() 8:W:+#(z0,s(z1)) -> c_11(+#(z0,z1)) -->_1 +#(z0,s(z1)) -> c_11(+#(z0,z1)):8 -->_1 +#(z0,0()) -> c_10():7 9:W:+'#(z0,0()) -> c_1() 10:W:DOUBLE#(0()) -> c_3() 11:W:SQR#(0()) -> c_5() 12:W:double#(0()) -> c_12() 13:W:double#(s(z0)) -> c_13(double#(z0)) -->_1 double#(s(z0)) -> c_13(double#(z0)):13 -->_1 double#(0()) -> c_12():12 14:W:sqr#(0()) -> c_14() 15:W:sqr#(s(z0)) -> c_15(+#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0)) -->_2 sqr#(s(z0)) -> c_16(+#(sqr(z0),double(z0)),sqr#(z0),double#(z0)):16 -->_2 sqr#(s(z0)) -> c_15(+#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0)):15 -->_2 sqr#(0()) -> c_14():14 -->_3 double#(s(z0)) -> c_13(double#(z0)):13 -->_3 double#(0()) -> c_12():12 -->_1 +#(z0,s(z1)) -> c_11(+#(z0,z1)):8 16:W:sqr#(s(z0)) -> c_16(+#(sqr(z0),double(z0)),sqr#(z0),double#(z0)) -->_2 sqr#(s(z0)) -> c_16(+#(sqr(z0),double(z0)),sqr#(z0),double#(z0)):16 -->_2 sqr#(s(z0)) -> c_15(+#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0)):15 -->_2 sqr#(0()) -> c_14():14 -->_3 double#(s(z0)) -> c_13(double#(z0)):13 -->_3 double#(0()) -> c_12():12 -->_1 +#(z0,s(z1)) -> c_11(+#(z0,z1)):8 -->_1 +#(z0,0()) -> c_10():7 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 11: SQR#(0()) -> c_5() 16: sqr#(s(z0)) -> c_16(+#(sqr(z0),double(z0)),sqr#(z0),double#(z0)) 15: sqr#(s(z0)) -> c_15(+#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0)) 8: +#(z0,s(z1)) -> c_11(+#(z0,z1)) 7: +#(z0,0()) -> c_10() 13: double#(s(z0)) -> c_13(double#(z0)) 12: double#(0()) -> c_12() 14: sqr#(0()) -> c_14() 10: DOUBLE#(0()) -> c_3() 9: +'#(z0,0()) -> c_1() ** Step 1.b:4: SimplifyRHS. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: +'#(z0,s(z1)) -> c_2(+'#(z0,z1)) DOUBLE#(s(z0)) -> c_4(DOUBLE#(z0)) SQR#(s(z0)) -> c_6(+'#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0),SQR#(z0)) SQR#(s(z0)) -> c_7(+'#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0),DOUBLE#(z0)) SQR#(s(z0)) -> c_8(+'#(sqr(z0),double(z0)),sqr#(z0),double#(z0),SQR#(z0)) SQR#(s(z0)) -> c_9(+'#(sqr(z0),double(z0)),sqr#(z0),double#(z0),DOUBLE#(z0)) - Weak TRS: +(z0,0()) -> z0 +(z0,s(z1)) -> s(+(z0,z1)) +'(z0,0()) -> c7() +'(z0,s(z1)) -> c8(+'(z0,z1)) DOUBLE(0()) -> c5() DOUBLE(s(z0)) -> c6(DOUBLE(z0)) SQR(0()) -> c() SQR(s(z0)) -> c1(+'(sqr(z0),s(double(z0))),SQR(z0)) SQR(s(z0)) -> c2(+'(sqr(z0),s(double(z0))),DOUBLE(z0)) SQR(s(z0)) -> c3(+'(sqr(z0),double(z0)),SQR(z0)) SQR(s(z0)) -> c4(+'(sqr(z0),double(z0)),DOUBLE(z0)) double(0()) -> 0() double(s(z0)) -> s(s(double(z0))) sqr(0()) -> 0() sqr(s(z0)) -> +(sqr(z0),s(double(z0))) sqr(s(z0)) -> s(+(sqr(z0),double(z0))) - Signature: {+/2,+'/2,DOUBLE/1,SQR/1,double/1,sqr/1,+#/2,+'#/2,DOUBLE#/1,SQR#/1,double#/1,sqr#/1} / {0/0,c/0,c1/2,c2/2 ,c3/2,c4/2,c5/0,c6/1,c7/0,c8/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/4,c_7/4,c_8/4,c_9/4,c_10/0,c_11/1 ,c_12/0,c_13/1,c_14/0,c_15/3,c_16/3} - Obligation: innermost runtime complexity wrt. defined symbols {+#,+'#,DOUBLE#,SQR#,double#,sqr#} and constructors {0,c ,c1,c2,c3,c4,c5,c6,c7,c8,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:+'#(z0,s(z1)) -> c_2(+'#(z0,z1)) -->_1 +'#(z0,s(z1)) -> c_2(+'#(z0,z1)):1 2:S:DOUBLE#(s(z0)) -> c_4(DOUBLE#(z0)) -->_1 DOUBLE#(s(z0)) -> c_4(DOUBLE#(z0)):2 3:S:SQR#(s(z0)) -> c_6(+'#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0),SQR#(z0)) -->_4 SQR#(s(z0)) -> c_9(+'#(sqr(z0),double(z0)),sqr#(z0),double#(z0),DOUBLE#(z0)):6 -->_4 SQR#(s(z0)) -> c_8(+'#(sqr(z0),double(z0)),sqr#(z0),double#(z0),SQR#(z0)):5 -->_4 SQR#(s(z0)) -> c_7(+'#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0),DOUBLE#(z0)):4 -->_4 SQR#(s(z0)) -> c_6(+'#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0),SQR#(z0)):3 -->_1 +'#(z0,s(z1)) -> c_2(+'#(z0,z1)):1 4:S:SQR#(s(z0)) -> c_7(+'#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0),DOUBLE#(z0)) -->_4 DOUBLE#(s(z0)) -> c_4(DOUBLE#(z0)):2 -->_1 +'#(z0,s(z1)) -> c_2(+'#(z0,z1)):1 5:S:SQR#(s(z0)) -> c_8(+'#(sqr(z0),double(z0)),sqr#(z0),double#(z0),SQR#(z0)) -->_4 SQR#(s(z0)) -> c_9(+'#(sqr(z0),double(z0)),sqr#(z0),double#(z0),DOUBLE#(z0)):6 -->_4 SQR#(s(z0)) -> c_8(+'#(sqr(z0),double(z0)),sqr#(z0),double#(z0),SQR#(z0)):5 -->_4 SQR#(s(z0)) -> c_7(+'#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0),DOUBLE#(z0)):4 -->_4 SQR#(s(z0)) -> c_6(+'#(sqr(z0),s(double(z0))),sqr#(z0),double#(z0),SQR#(z0)):3 -->_1 +'#(z0,s(z1)) -> c_2(+'#(z0,z1)):1 6:S:SQR#(s(z0)) -> c_9(+'#(sqr(z0),double(z0)),sqr#(z0),double#(z0),DOUBLE#(z0)) -->_4 DOUBLE#(s(z0)) -> c_4(DOUBLE#(z0)):2 -->_1 +'#(z0,s(z1)) -> c_2(+'#(z0,z1)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: SQR#(s(z0)) -> c_6(+'#(sqr(z0),s(double(z0))),SQR#(z0)) SQR#(s(z0)) -> c_7(+'#(sqr(z0),s(double(z0))),DOUBLE#(z0)) SQR#(s(z0)) -> c_8(+'#(sqr(z0),double(z0)),SQR#(z0)) SQR#(s(z0)) -> c_9(+'#(sqr(z0),double(z0)),DOUBLE#(z0)) ** Step 1.b:5: UsableRules. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: +'#(z0,s(z1)) -> c_2(+'#(z0,z1)) DOUBLE#(s(z0)) -> c_4(DOUBLE#(z0)) SQR#(s(z0)) -> c_6(+'#(sqr(z0),s(double(z0))),SQR#(z0)) SQR#(s(z0)) -> c_7(+'#(sqr(z0),s(double(z0))),DOUBLE#(z0)) SQR#(s(z0)) -> c_8(+'#(sqr(z0),double(z0)),SQR#(z0)) SQR#(s(z0)) -> c_9(+'#(sqr(z0),double(z0)),DOUBLE#(z0)) - Weak TRS: +(z0,0()) -> z0 +(z0,s(z1)) -> s(+(z0,z1)) +'(z0,0()) -> c7() +'(z0,s(z1)) -> c8(+'(z0,z1)) DOUBLE(0()) -> c5() DOUBLE(s(z0)) -> c6(DOUBLE(z0)) SQR(0()) -> c() SQR(s(z0)) -> c1(+'(sqr(z0),s(double(z0))),SQR(z0)) SQR(s(z0)) -> c2(+'(sqr(z0),s(double(z0))),DOUBLE(z0)) SQR(s(z0)) -> c3(+'(sqr(z0),double(z0)),SQR(z0)) SQR(s(z0)) -> c4(+'(sqr(z0),double(z0)),DOUBLE(z0)) double(0()) -> 0() double(s(z0)) -> s(s(double(z0))) sqr(0()) -> 0() sqr(s(z0)) -> +(sqr(z0),s(double(z0))) sqr(s(z0)) -> s(+(sqr(z0),double(z0))) - Signature: {+/2,+'/2,DOUBLE/1,SQR/1,double/1,sqr/1,+#/2,+'#/2,DOUBLE#/1,SQR#/1,double#/1,sqr#/1} / {0/0,c/0,c1/2,c2/2 ,c3/2,c4/2,c5/0,c6/1,c7/0,c8/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2,c_7/2,c_8/2,c_9/2,c_10/0,c_11/1 ,c_12/0,c_13/1,c_14/0,c_15/3,c_16/3} - Obligation: innermost runtime complexity wrt. defined symbols {+#,+'#,DOUBLE#,SQR#,double#,sqr#} and constructors {0,c ,c1,c2,c3,c4,c5,c6,c7,c8,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: +(z0,0()) -> z0 +(z0,s(z1)) -> s(+(z0,z1)) double(0()) -> 0() double(s(z0)) -> s(s(double(z0))) sqr(0()) -> 0() sqr(s(z0)) -> +(sqr(z0),s(double(z0))) sqr(s(z0)) -> s(+(sqr(z0),double(z0))) +'#(z0,s(z1)) -> c_2(+'#(z0,z1)) DOUBLE#(s(z0)) -> c_4(DOUBLE#(z0)) SQR#(s(z0)) -> c_6(+'#(sqr(z0),s(double(z0))),SQR#(z0)) SQR#(s(z0)) -> c_7(+'#(sqr(z0),s(double(z0))),DOUBLE#(z0)) SQR#(s(z0)) -> c_8(+'#(sqr(z0),double(z0)),SQR#(z0)) SQR#(s(z0)) -> c_9(+'#(sqr(z0),double(z0)),DOUBLE#(z0)) ** Step 1.b:6: DecomposeDG. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: +'#(z0,s(z1)) -> c_2(+'#(z0,z1)) DOUBLE#(s(z0)) -> c_4(DOUBLE#(z0)) SQR#(s(z0)) -> c_6(+'#(sqr(z0),s(double(z0))),SQR#(z0)) SQR#(s(z0)) -> c_7(+'#(sqr(z0),s(double(z0))),DOUBLE#(z0)) SQR#(s(z0)) -> c_8(+'#(sqr(z0),double(z0)),SQR#(z0)) SQR#(s(z0)) -> c_9(+'#(sqr(z0),double(z0)),DOUBLE#(z0)) - Weak TRS: +(z0,0()) -> z0 +(z0,s(z1)) -> s(+(z0,z1)) double(0()) -> 0() double(s(z0)) -> s(s(double(z0))) sqr(0()) -> 0() sqr(s(z0)) -> +(sqr(z0),s(double(z0))) sqr(s(z0)) -> s(+(sqr(z0),double(z0))) - Signature: {+/2,+'/2,DOUBLE/1,SQR/1,double/1,sqr/1,+#/2,+'#/2,DOUBLE#/1,SQR#/1,double#/1,sqr#/1} / {0/0,c/0,c1/2,c2/2 ,c3/2,c4/2,c5/0,c6/1,c7/0,c8/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2,c_7/2,c_8/2,c_9/2,c_10/0,c_11/1 ,c_12/0,c_13/1,c_14/0,c_15/3,c_16/3} - Obligation: innermost runtime complexity wrt. defined symbols {+#,+'#,DOUBLE#,SQR#,double#,sqr#} and constructors {0,c ,c1,c2,c3,c4,c5,c6,c7,c8,s} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component SQR#(s(z0)) -> c_6(+'#(sqr(z0),s(double(z0))),SQR#(z0)) SQR#(s(z0)) -> c_8(+'#(sqr(z0),double(z0)),SQR#(z0)) and a lower component +'#(z0,s(z1)) -> c_2(+'#(z0,z1)) DOUBLE#(s(z0)) -> c_4(DOUBLE#(z0)) SQR#(s(z0)) -> c_7(+'#(sqr(z0),s(double(z0))),DOUBLE#(z0)) SQR#(s(z0)) -> c_9(+'#(sqr(z0),double(z0)),DOUBLE#(z0)) Further, following extension rules are added to the lower component. SQR#(s(z0)) -> +'#(sqr(z0),double(z0)) SQR#(s(z0)) -> +'#(sqr(z0),s(double(z0))) SQR#(s(z0)) -> SQR#(z0) *** Step 1.b:6.a:1: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: SQR#(s(z0)) -> c_6(+'#(sqr(z0),s(double(z0))),SQR#(z0)) SQR#(s(z0)) -> c_8(+'#(sqr(z0),double(z0)),SQR#(z0)) - Weak TRS: +(z0,0()) -> z0 +(z0,s(z1)) -> s(+(z0,z1)) double(0()) -> 0() double(s(z0)) -> s(s(double(z0))) sqr(0()) -> 0() sqr(s(z0)) -> +(sqr(z0),s(double(z0))) sqr(s(z0)) -> s(+(sqr(z0),double(z0))) - Signature: {+/2,+'/2,DOUBLE/1,SQR/1,double/1,sqr/1,+#/2,+'#/2,DOUBLE#/1,SQR#/1,double#/1,sqr#/1} / {0/0,c/0,c1/2,c2/2 ,c3/2,c4/2,c5/0,c6/1,c7/0,c8/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2,c_7/2,c_8/2,c_9/2,c_10/0,c_11/1 ,c_12/0,c_13/1,c_14/0,c_15/3,c_16/3} - Obligation: innermost runtime complexity wrt. defined symbols {+#,+'#,DOUBLE#,SQR#,double#,sqr#} and constructors {0,c ,c1,c2,c3,c4,c5,c6,c7,c8,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:SQR#(s(z0)) -> c_6(+'#(sqr(z0),s(double(z0))),SQR#(z0)) -->_2 SQR#(s(z0)) -> c_8(+'#(sqr(z0),double(z0)),SQR#(z0)):2 -->_2 SQR#(s(z0)) -> c_6(+'#(sqr(z0),s(double(z0))),SQR#(z0)):1 2:S:SQR#(s(z0)) -> c_8(+'#(sqr(z0),double(z0)),SQR#(z0)) -->_2 SQR#(s(z0)) -> c_8(+'#(sqr(z0),double(z0)),SQR#(z0)):2 -->_2 SQR#(s(z0)) -> c_6(+'#(sqr(z0),s(double(z0))),SQR#(z0)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: SQR#(s(z0)) -> c_6(SQR#(z0)) SQR#(s(z0)) -> c_8(SQR#(z0)) *** Step 1.b:6.a:2: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: SQR#(s(z0)) -> c_6(SQR#(z0)) SQR#(s(z0)) -> c_8(SQR#(z0)) - Weak TRS: +(z0,0()) -> z0 +(z0,s(z1)) -> s(+(z0,z1)) double(0()) -> 0() double(s(z0)) -> s(s(double(z0))) sqr(0()) -> 0() sqr(s(z0)) -> +(sqr(z0),s(double(z0))) sqr(s(z0)) -> s(+(sqr(z0),double(z0))) - Signature: {+/2,+'/2,DOUBLE/1,SQR/1,double/1,sqr/1,+#/2,+'#/2,DOUBLE#/1,SQR#/1,double#/1,sqr#/1} / {0/0,c/0,c1/2,c2/2 ,c3/2,c4/2,c5/0,c6/1,c7/0,c8/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/2,c_8/1,c_9/2,c_10/0,c_11/1 ,c_12/0,c_13/1,c_14/0,c_15/3,c_16/3} - Obligation: innermost runtime complexity wrt. defined symbols {+#,+'#,DOUBLE#,SQR#,double#,sqr#} and constructors {0,c ,c1,c2,c3,c4,c5,c6,c7,c8,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: SQR#(s(z0)) -> c_6(SQR#(z0)) SQR#(s(z0)) -> c_8(SQR#(z0)) *** Step 1.b:6.a:3: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: SQR#(s(z0)) -> c_6(SQR#(z0)) SQR#(s(z0)) -> c_8(SQR#(z0)) - Signature: {+/2,+'/2,DOUBLE/1,SQR/1,double/1,sqr/1,+#/2,+'#/2,DOUBLE#/1,SQR#/1,double#/1,sqr#/1} / {0/0,c/0,c1/2,c2/2 ,c3/2,c4/2,c5/0,c6/1,c7/0,c8/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/2,c_8/1,c_9/2,c_10/0,c_11/1 ,c_12/0,c_13/1,c_14/0,c_15/3,c_16/3} - Obligation: innermost runtime complexity wrt. defined symbols {+#,+'#,DOUBLE#,SQR#,double#,sqr#} and constructors {0,c ,c1,c2,c3,c4,c5,c6,c7,c8,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_6) = {1}, uargs(c_8) = {1} Following symbols are considered usable: {+#,+'#,DOUBLE#,SQR#,double#,sqr#} TcT has computed the following interpretation: p(+) = [1] p(+') = [0] p(0) = [0] p(DOUBLE) = [0] p(SQR) = [0] p(c) = [0] p(c1) = [1] x1 + [1] x2 + [0] p(c2) = [0] p(c3) = [1] p(c4) = [1] x1 + [1] p(c5) = [1] p(c6) = [8] p(c7) = [1] p(c8) = [1] p(double) = [1] p(s) = [1] x1 + [2] p(sqr) = [2] x1 + [8] p(+#) = [1] x1 + [2] x2 + [0] p(+'#) = [0] p(DOUBLE#) = [1] p(SQR#) = [1] x1 + [0] p(double#) = [1] p(sqr#) = [4] x1 + [0] p(c_1) = [2] p(c_2) = [0] p(c_3) = [2] p(c_4) = [1] x1 + [1] p(c_5) = [2] p(c_6) = [1] x1 + [0] p(c_7) = [1] p(c_8) = [1] x1 + [2] p(c_9) = [1] x1 + [1] p(c_10) = [1] p(c_11) = [0] p(c_12) = [0] p(c_13) = [1] x1 + [1] p(c_14) = [1] p(c_15) = [1] x3 + [2] p(c_16) = [2] x2 + [1] x3 + [1] Following rules are strictly oriented: SQR#(s(z0)) = [1] z0 + [2] > [1] z0 + [0] = c_6(SQR#(z0)) Following rules are (at-least) weakly oriented: SQR#(s(z0)) = [1] z0 + [2] >= [1] z0 + [2] = c_8(SQR#(z0)) *** Step 1.b:6.a:4: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: SQR#(s(z0)) -> c_8(SQR#(z0)) - Weak DPs: SQR#(s(z0)) -> c_6(SQR#(z0)) - Signature: {+/2,+'/2,DOUBLE/1,SQR/1,double/1,sqr/1,+#/2,+'#/2,DOUBLE#/1,SQR#/1,double#/1,sqr#/1} / {0/0,c/0,c1/2,c2/2 ,c3/2,c4/2,c5/0,c6/1,c7/0,c8/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/2,c_8/1,c_9/2,c_10/0,c_11/1 ,c_12/0,c_13/1,c_14/0,c_15/3,c_16/3} - Obligation: innermost runtime complexity wrt. defined symbols {+#,+'#,DOUBLE#,SQR#,double#,sqr#} and constructors {0,c ,c1,c2,c3,c4,c5,c6,c7,c8,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_6) = {1}, uargs(c_8) = {1} Following symbols are considered usable: {+#,+'#,DOUBLE#,SQR#,double#,sqr#} TcT has computed the following interpretation: p(+) = [2] x2 + [0] p(+') = [0] p(0) = [0] p(DOUBLE) = [0] p(SQR) = [1] x1 + [0] p(c) = [0] p(c1) = [1] p(c2) = [0] p(c3) = [2] p(c4) = [1] p(c5) = [1] p(c6) = [0] p(c7) = [1] p(c8) = [1] p(double) = [1] p(s) = [1] x1 + [5] p(sqr) = [2] p(+#) = [1] x2 + [4] p(+'#) = [1] x1 + [8] p(DOUBLE#) = [1] p(SQR#) = [5] x1 + [0] p(double#) = [1] p(sqr#) = [1] x1 + [1] p(c_1) = [2] p(c_2) = [1] x1 + [0] p(c_3) = [2] p(c_4) = [0] p(c_5) = [1] p(c_6) = [1] x1 + [9] p(c_7) = [2] x2 + [0] p(c_8) = [1] x1 + [12] p(c_9) = [2] x1 + [1] x2 + [0] p(c_10) = [0] p(c_11) = [1] p(c_12) = [0] p(c_13) = [1] x1 + [1] p(c_14) = [0] p(c_15) = [8] x1 + [1] x2 + [1] x3 + [1] p(c_16) = [1] x1 + [2] x2 + [8] x3 + [8] Following rules are strictly oriented: SQR#(s(z0)) = [5] z0 + [25] > [5] z0 + [12] = c_8(SQR#(z0)) Following rules are (at-least) weakly oriented: SQR#(s(z0)) = [5] z0 + [25] >= [5] z0 + [9] = c_6(SQR#(z0)) *** Step 1.b:6.a:5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: SQR#(s(z0)) -> c_6(SQR#(z0)) SQR#(s(z0)) -> c_8(SQR#(z0)) - Signature: {+/2,+'/2,DOUBLE/1,SQR/1,double/1,sqr/1,+#/2,+'#/2,DOUBLE#/1,SQR#/1,double#/1,sqr#/1} / {0/0,c/0,c1/2,c2/2 ,c3/2,c4/2,c5/0,c6/1,c7/0,c8/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/2,c_8/1,c_9/2,c_10/0,c_11/1 ,c_12/0,c_13/1,c_14/0,c_15/3,c_16/3} - Obligation: innermost runtime complexity wrt. defined symbols {+#,+'#,DOUBLE#,SQR#,double#,sqr#} and constructors {0,c ,c1,c2,c3,c4,c5,c6,c7,c8,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 1.b:6.b:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: +'#(z0,s(z1)) -> c_2(+'#(z0,z1)) DOUBLE#(s(z0)) -> c_4(DOUBLE#(z0)) SQR#(s(z0)) -> c_7(+'#(sqr(z0),s(double(z0))),DOUBLE#(z0)) SQR#(s(z0)) -> c_9(+'#(sqr(z0),double(z0)),DOUBLE#(z0)) - Weak DPs: SQR#(s(z0)) -> +'#(sqr(z0),double(z0)) SQR#(s(z0)) -> +'#(sqr(z0),s(double(z0))) SQR#(s(z0)) -> SQR#(z0) - Weak TRS: +(z0,0()) -> z0 +(z0,s(z1)) -> s(+(z0,z1)) double(0()) -> 0() double(s(z0)) -> s(s(double(z0))) sqr(0()) -> 0() sqr(s(z0)) -> +(sqr(z0),s(double(z0))) sqr(s(z0)) -> s(+(sqr(z0),double(z0))) - Signature: {+/2,+'/2,DOUBLE/1,SQR/1,double/1,sqr/1,+#/2,+'#/2,DOUBLE#/1,SQR#/1,double#/1,sqr#/1} / {0/0,c/0,c1/2,c2/2 ,c3/2,c4/2,c5/0,c6/1,c7/0,c8/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2,c_7/2,c_8/2,c_9/2,c_10/0,c_11/1 ,c_12/0,c_13/1,c_14/0,c_15/3,c_16/3} - Obligation: innermost runtime complexity wrt. defined symbols {+#,+'#,DOUBLE#,SQR#,double#,sqr#} and constructors {0,c ,c1,c2,c3,c4,c5,c6,c7,c8,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_2) = {1}, uargs(c_4) = {1}, uargs(c_7) = {1,2}, uargs(c_9) = {1,2} Following symbols are considered usable: {double,+#,+'#,DOUBLE#,SQR#,double#,sqr#} TcT has computed the following interpretation: p(+) = [0] p(+') = [0] p(0) = [0] p(DOUBLE) = [0] p(SQR) = [0] p(c) = [0] p(c1) = [1] x1 + [1] x2 + [0] p(c2) = [1] x1 + [1] x2 + [0] p(c3) = [1] x1 + [1] x2 + [0] p(c4) = [1] x2 + [0] p(c5) = [0] p(c6) = [1] x1 + [0] p(c7) = [0] p(c8) = [1] x1 + [0] p(double) = [2] x1 + [0] p(s) = [1] x1 + [3] p(sqr) = [0] p(+#) = [0] p(+'#) = [1] x2 + [0] p(DOUBLE#) = [4] x1 + [5] p(SQR#) = [8] x1 + [5] p(double#) = [0] p(sqr#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [1] p(c_3) = [0] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [1] x1 + [1] x2 + [0] p(c_8) = [1] x2 + [0] p(c_9) = [1] x1 + [1] x2 + [3] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [1] x1 + [8] x2 + [2] x3 + [0] p(c_16) = [2] x1 + [0] Following rules are strictly oriented: +'#(z0,s(z1)) = [1] z1 + [3] > [1] z1 + [1] = c_2(+'#(z0,z1)) DOUBLE#(s(z0)) = [4] z0 + [17] > [4] z0 + [5] = c_4(DOUBLE#(z0)) SQR#(s(z0)) = [8] z0 + [29] > [6] z0 + [8] = c_7(+'#(sqr(z0),s(double(z0))),DOUBLE#(z0)) SQR#(s(z0)) = [8] z0 + [29] > [6] z0 + [8] = c_9(+'#(sqr(z0),double(z0)),DOUBLE#(z0)) Following rules are (at-least) weakly oriented: SQR#(s(z0)) = [8] z0 + [29] >= [2] z0 + [0] = +'#(sqr(z0),double(z0)) SQR#(s(z0)) = [8] z0 + [29] >= [2] z0 + [3] = +'#(sqr(z0),s(double(z0))) SQR#(s(z0)) = [8] z0 + [29] >= [8] z0 + [5] = SQR#(z0) double(0()) = [0] >= [0] = 0() double(s(z0)) = [2] z0 + [6] >= [2] z0 + [6] = s(s(double(z0))) *** Step 1.b:6.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: +'#(z0,s(z1)) -> c_2(+'#(z0,z1)) DOUBLE#(s(z0)) -> c_4(DOUBLE#(z0)) SQR#(s(z0)) -> +'#(sqr(z0),double(z0)) SQR#(s(z0)) -> +'#(sqr(z0),s(double(z0))) SQR#(s(z0)) -> SQR#(z0) SQR#(s(z0)) -> c_7(+'#(sqr(z0),s(double(z0))),DOUBLE#(z0)) SQR#(s(z0)) -> c_9(+'#(sqr(z0),double(z0)),DOUBLE#(z0)) - Weak TRS: +(z0,0()) -> z0 +(z0,s(z1)) -> s(+(z0,z1)) double(0()) -> 0() double(s(z0)) -> s(s(double(z0))) sqr(0()) -> 0() sqr(s(z0)) -> +(sqr(z0),s(double(z0))) sqr(s(z0)) -> s(+(sqr(z0),double(z0))) - Signature: {+/2,+'/2,DOUBLE/1,SQR/1,double/1,sqr/1,+#/2,+'#/2,DOUBLE#/1,SQR#/1,double#/1,sqr#/1} / {0/0,c/0,c1/2,c2/2 ,c3/2,c4/2,c5/0,c6/1,c7/0,c8/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2,c_7/2,c_8/2,c_9/2,c_10/0,c_11/1 ,c_12/0,c_13/1,c_14/0,c_15/3,c_16/3} - Obligation: innermost runtime complexity wrt. defined symbols {+#,+'#,DOUBLE#,SQR#,double#,sqr#} and constructors {0,c ,c1,c2,c3,c4,c5,c6,c7,c8,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^2))