WORST_CASE(Omega(n^1),O(n^2)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^2)) + Considered Problem: - Strict TRS: IF(false(),z0,z1) -> c9() IF(true(),z0,z1) -> c8() LE(0(),z0) -> c2() LE(s(z0),0()) -> c3() LE(s(z0),s(z1)) -> c4(LE(z0,z1)) MINUS(z0,0()) -> c5() MINUS(z0,s(z1)) -> c6(IF(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))),LE(z0,s(z1))) MINUS(z0,s(z1)) -> c7(IF(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,P(minus(z0,p(s(z1)))) ,MINUS(z0,p(s(z1))) ,P(s(z1))) P(0()) -> c() P(s(z0)) -> c1() - Weak TRS: if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) minus(z0,0()) -> z0 minus(z0,s(z1)) -> if(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) p(0()) -> 0() p(s(z0)) -> z0 - Signature: {IF/3,LE/2,MINUS/2,P/1,if/3,le/2,minus/2,p/1} / {0/0,c/0,c1/0,c2/0,c3/0,c4/1,c5/0,c6/2,c7/4,c8/0,c9/0 ,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {IF,LE,MINUS,P,if,le,minus,p} and constructors {0,c,c1,c2 ,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: IF(false(),z0,z1) -> c9() IF(true(),z0,z1) -> c8() LE(0(),z0) -> c2() LE(s(z0),0()) -> c3() LE(s(z0),s(z1)) -> c4(LE(z0,z1)) MINUS(z0,0()) -> c5() MINUS(z0,s(z1)) -> c6(IF(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))),LE(z0,s(z1))) MINUS(z0,s(z1)) -> c7(IF(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,P(minus(z0,p(s(z1)))) ,MINUS(z0,p(s(z1))) ,P(s(z1))) P(0()) -> c() P(s(z0)) -> c1() - Weak TRS: if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) minus(z0,0()) -> z0 minus(z0,s(z1)) -> if(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) p(0()) -> 0() p(s(z0)) -> z0 - Signature: {IF/3,LE/2,MINUS/2,P/1,if/3,le/2,minus/2,p/1} / {0/0,c/0,c1/0,c2/0,c3/0,c4/1,c5/0,c6/2,c7/4,c8/0,c9/0 ,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {IF,LE,MINUS,P,if,le,minus,p} and constructors {0,c,c1,c2 ,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: IF(false(),z0,z1) -> c9() IF(true(),z0,z1) -> c8() LE(0(),z0) -> c2() LE(s(z0),0()) -> c3() LE(s(z0),s(z1)) -> c4(LE(z0,z1)) MINUS(z0,0()) -> c5() MINUS(z0,s(z1)) -> c6(IF(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))),LE(z0,s(z1))) MINUS(z0,s(z1)) -> c7(IF(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,P(minus(z0,p(s(z1)))) ,MINUS(z0,p(s(z1))) ,P(s(z1))) P(0()) -> c() P(s(z0)) -> c1() - Weak TRS: if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) minus(z0,0()) -> z0 minus(z0,s(z1)) -> if(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) p(0()) -> 0() p(s(z0)) -> z0 - Signature: {IF/3,LE/2,MINUS/2,P/1,if/3,le/2,minus/2,p/1} / {0/0,c/0,c1/0,c2/0,c3/0,c4/1,c5/0,c6/2,c7/4,c8/0,c9/0 ,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {IF,LE,MINUS,P,if,le,minus,p} and constructors {0,c,c1,c2 ,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: LE(x,y){x -> s(x),y -> s(y)} = LE(s(x),s(y)) ->^+ c4(LE(x,y)) = C[LE(x,y) = LE(x,y){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: IF(false(),z0,z1) -> c9() IF(true(),z0,z1) -> c8() LE(0(),z0) -> c2() LE(s(z0),0()) -> c3() LE(s(z0),s(z1)) -> c4(LE(z0,z1)) MINUS(z0,0()) -> c5() MINUS(z0,s(z1)) -> c6(IF(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))),LE(z0,s(z1))) MINUS(z0,s(z1)) -> c7(IF(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,P(minus(z0,p(s(z1)))) ,MINUS(z0,p(s(z1))) ,P(s(z1))) P(0()) -> c() P(s(z0)) -> c1() - Weak TRS: if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) minus(z0,0()) -> z0 minus(z0,s(z1)) -> if(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) p(0()) -> 0() p(s(z0)) -> z0 - Signature: {IF/3,LE/2,MINUS/2,P/1,if/3,le/2,minus/2,p/1} / {0/0,c/0,c1/0,c2/0,c3/0,c4/1,c5/0,c6/2,c7/4,c8/0,c9/0 ,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {IF,LE,MINUS,P,if,le,minus,p} and constructors {0,c,c1,c2 ,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs IF#(false(),z0,z1) -> c_1() IF#(true(),z0,z1) -> c_2() LE#(0(),z0) -> c_3() LE#(s(z0),0()) -> c_4() LE#(s(z0),s(z1)) -> c_5(LE#(z0,z1)) MINUS#(z0,0()) -> c_6() MINUS#(z0,s(z1)) -> c_7(IF#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,LE#(z0,s(z1))) MINUS#(z0,s(z1)) -> c_8(IF#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,P#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,MINUS#(z0,p(s(z1))) ,p#(s(z1)) ,P#(s(z1))) P#(0()) -> c_9() P#(s(z0)) -> c_10() Weak DPs if#(false(),z0,z1) -> c_11() if#(true(),z0,z1) -> c_12() le#(0(),z0) -> c_13() le#(s(z0),0()) -> c_14() le#(s(z0),s(z1)) -> c_15(le#(z0,z1)) minus#(z0,0()) -> c_16() minus#(z0,s(z1)) -> c_17(if#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1))) p#(0()) -> c_18() p#(s(z0)) -> c_19() and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: IF#(false(),z0,z1) -> c_1() IF#(true(),z0,z1) -> c_2() LE#(0(),z0) -> c_3() LE#(s(z0),0()) -> c_4() LE#(s(z0),s(z1)) -> c_5(LE#(z0,z1)) MINUS#(z0,0()) -> c_6() MINUS#(z0,s(z1)) -> c_7(IF#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,LE#(z0,s(z1))) MINUS#(z0,s(z1)) -> c_8(IF#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,P#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,MINUS#(z0,p(s(z1))) ,p#(s(z1)) ,P#(s(z1))) P#(0()) -> c_9() P#(s(z0)) -> c_10() - Weak DPs: if#(false(),z0,z1) -> c_11() if#(true(),z0,z1) -> c_12() le#(0(),z0) -> c_13() le#(s(z0),0()) -> c_14() le#(s(z0),s(z1)) -> c_15(le#(z0,z1)) minus#(z0,0()) -> c_16() minus#(z0,s(z1)) -> c_17(if#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1))) p#(0()) -> c_18() p#(s(z0)) -> c_19() - Weak TRS: IF(false(),z0,z1) -> c9() IF(true(),z0,z1) -> c8() LE(0(),z0) -> c2() LE(s(z0),0()) -> c3() LE(s(z0),s(z1)) -> c4(LE(z0,z1)) MINUS(z0,0()) -> c5() MINUS(z0,s(z1)) -> c6(IF(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))),LE(z0,s(z1))) MINUS(z0,s(z1)) -> c7(IF(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,P(minus(z0,p(s(z1)))) ,MINUS(z0,p(s(z1))) ,P(s(z1))) P(0()) -> c() P(s(z0)) -> c1() if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) minus(z0,0()) -> z0 minus(z0,s(z1)) -> if(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) p(0()) -> 0() p(s(z0)) -> z0 - Signature: {IF/3,LE/2,MINUS/2,P/1,if/3,le/2,minus/2,p/1,IF#/3,LE#/2,MINUS#/2,P#/1,if#/3,le#/2,minus#/2,p#/1} / {0/0,c/0 ,c1/0,c2/0,c3/0,c4/1,c5/0,c6/2,c7/4,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/6 ,c_8/11,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/0,c_17/5,c_18/0,c_19/0} - Obligation: innermost runtime complexity wrt. defined symbols {IF#,LE#,MINUS#,P#,if#,le#,minus#,p#} and constructors {0 ,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,4,6,9,10} by application of Pre({1,2,3,4,6,9,10}) = {5,7,8}. Here rules are labelled as follows: 1: IF#(false(),z0,z1) -> c_1() 2: IF#(true(),z0,z1) -> c_2() 3: LE#(0(),z0) -> c_3() 4: LE#(s(z0),0()) -> c_4() 5: LE#(s(z0),s(z1)) -> c_5(LE#(z0,z1)) 6: MINUS#(z0,0()) -> c_6() 7: MINUS#(z0,s(z1)) -> c_7(IF#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,LE#(z0,s(z1))) 8: MINUS#(z0,s(z1)) -> c_8(IF#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,P#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,MINUS#(z0,p(s(z1))) ,p#(s(z1)) ,P#(s(z1))) 9: P#(0()) -> c_9() 10: P#(s(z0)) -> c_10() 11: if#(false(),z0,z1) -> c_11() 12: if#(true(),z0,z1) -> c_12() 13: le#(0(),z0) -> c_13() 14: le#(s(z0),0()) -> c_14() 15: le#(s(z0),s(z1)) -> c_15(le#(z0,z1)) 16: minus#(z0,0()) -> c_16() 17: minus#(z0,s(z1)) -> c_17(if#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1))) 18: p#(0()) -> c_18() 19: p#(s(z0)) -> c_19() ** Step 1.b:3: RemoveWeakSuffixes. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: LE#(s(z0),s(z1)) -> c_5(LE#(z0,z1)) MINUS#(z0,s(z1)) -> c_7(IF#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,LE#(z0,s(z1))) MINUS#(z0,s(z1)) -> c_8(IF#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,P#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,MINUS#(z0,p(s(z1))) ,p#(s(z1)) ,P#(s(z1))) - Weak DPs: IF#(false(),z0,z1) -> c_1() IF#(true(),z0,z1) -> c_2() LE#(0(),z0) -> c_3() LE#(s(z0),0()) -> c_4() MINUS#(z0,0()) -> c_6() P#(0()) -> c_9() P#(s(z0)) -> c_10() if#(false(),z0,z1) -> c_11() if#(true(),z0,z1) -> c_12() le#(0(),z0) -> c_13() le#(s(z0),0()) -> c_14() le#(s(z0),s(z1)) -> c_15(le#(z0,z1)) minus#(z0,0()) -> c_16() minus#(z0,s(z1)) -> c_17(if#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1))) p#(0()) -> c_18() p#(s(z0)) -> c_19() - Weak TRS: IF(false(),z0,z1) -> c9() IF(true(),z0,z1) -> c8() LE(0(),z0) -> c2() LE(s(z0),0()) -> c3() LE(s(z0),s(z1)) -> c4(LE(z0,z1)) MINUS(z0,0()) -> c5() MINUS(z0,s(z1)) -> c6(IF(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))),LE(z0,s(z1))) MINUS(z0,s(z1)) -> c7(IF(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,P(minus(z0,p(s(z1)))) ,MINUS(z0,p(s(z1))) ,P(s(z1))) P(0()) -> c() P(s(z0)) -> c1() if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) minus(z0,0()) -> z0 minus(z0,s(z1)) -> if(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) p(0()) -> 0() p(s(z0)) -> z0 - Signature: {IF/3,LE/2,MINUS/2,P/1,if/3,le/2,minus/2,p/1,IF#/3,LE#/2,MINUS#/2,P#/1,if#/3,le#/2,minus#/2,p#/1} / {0/0,c/0 ,c1/0,c2/0,c3/0,c4/1,c5/0,c6/2,c7/4,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/6 ,c_8/11,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/0,c_17/5,c_18/0,c_19/0} - Obligation: innermost runtime complexity wrt. defined symbols {IF#,LE#,MINUS#,P#,if#,le#,minus#,p#} and constructors {0 ,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:LE#(s(z0),s(z1)) -> c_5(LE#(z0,z1)) -->_1 LE#(s(z0),0()) -> c_4():7 -->_1 LE#(0(),z0) -> c_3():6 -->_1 LE#(s(z0),s(z1)) -> c_5(LE#(z0,z1)):1 2:S:MINUS#(z0,s(z1)) -> c_7(IF#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,LE#(z0,s(z1))) -->_4 minus#(z0,s(z1)) -> c_17(if#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1))):17 -->_2 le#(s(z0),s(z1)) -> c_15(le#(z0,z1)):15 -->_5 p#(s(z0)) -> c_19():19 -->_3 p#(s(z0)) -> c_19():19 -->_3 p#(0()) -> c_18():18 -->_4 minus#(z0,0()) -> c_16():16 -->_2 le#(0(),z0) -> c_13():13 -->_6 LE#(0(),z0) -> c_3():6 -->_1 IF#(true(),z0,z1) -> c_2():5 -->_1 IF#(false(),z0,z1) -> c_1():4 -->_6 LE#(s(z0),s(z1)) -> c_5(LE#(z0,z1)):1 3:S:MINUS#(z0,s(z1)) -> c_8(IF#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,P#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,MINUS#(z0,p(s(z1))) ,p#(s(z1)) ,P#(s(z1))) -->_7 minus#(z0,s(z1)) -> c_17(if#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1))):17 -->_4 minus#(z0,s(z1)) -> c_17(if#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1))):17 -->_2 le#(s(z0),s(z1)) -> c_15(le#(z0,z1)):15 -->_10 p#(s(z0)) -> c_19():19 -->_8 p#(s(z0)) -> c_19():19 -->_5 p#(s(z0)) -> c_19():19 -->_3 p#(s(z0)) -> c_19():19 -->_3 p#(0()) -> c_18():18 -->_7 minus#(z0,0()) -> c_16():16 -->_4 minus#(z0,0()) -> c_16():16 -->_2 le#(0(),z0) -> c_13():13 -->_11 P#(s(z0)) -> c_10():10 -->_6 P#(s(z0)) -> c_10():10 -->_6 P#(0()) -> c_9():9 -->_9 MINUS#(z0,0()) -> c_6():8 -->_1 IF#(true(),z0,z1) -> c_2():5 -->_1 IF#(false(),z0,z1) -> c_1():4 -->_9 MINUS#(z0,s(z1)) -> c_8(IF#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,P#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,MINUS#(z0,p(s(z1))) ,p#(s(z1)) ,P#(s(z1))):3 -->_9 MINUS#(z0,s(z1)) -> c_7(IF#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,LE#(z0,s(z1))):2 4:W:IF#(false(),z0,z1) -> c_1() 5:W:IF#(true(),z0,z1) -> c_2() 6:W:LE#(0(),z0) -> c_3() 7:W:LE#(s(z0),0()) -> c_4() 8:W:MINUS#(z0,0()) -> c_6() 9:W:P#(0()) -> c_9() 10:W:P#(s(z0)) -> c_10() 11:W:if#(false(),z0,z1) -> c_11() 12:W:if#(true(),z0,z1) -> c_12() 13:W:le#(0(),z0) -> c_13() 14:W:le#(s(z0),0()) -> c_14() 15:W:le#(s(z0),s(z1)) -> c_15(le#(z0,z1)) -->_1 le#(s(z0),s(z1)) -> c_15(le#(z0,z1)):15 -->_1 le#(s(z0),0()) -> c_14():14 -->_1 le#(0(),z0) -> c_13():13 16:W:minus#(z0,0()) -> c_16() 17:W:minus#(z0,s(z1)) -> c_17(if#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1))) -->_5 p#(s(z0)) -> c_19():19 -->_3 p#(s(z0)) -> c_19():19 -->_3 p#(0()) -> c_18():18 -->_4 minus#(z0,s(z1)) -> c_17(if#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1))):17 -->_4 minus#(z0,0()) -> c_16():16 -->_2 le#(s(z0),s(z1)) -> c_15(le#(z0,z1)):15 -->_2 le#(0(),z0) -> c_13():13 -->_1 if#(true(),z0,z1) -> c_12():12 -->_1 if#(false(),z0,z1) -> c_11():11 18:W:p#(0()) -> c_18() 19:W:p#(s(z0)) -> c_19() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: MINUS#(z0,0()) -> c_6() 9: P#(0()) -> c_9() 10: P#(s(z0)) -> c_10() 4: IF#(false(),z0,z1) -> c_1() 5: IF#(true(),z0,z1) -> c_2() 17: minus#(z0,s(z1)) -> c_17(if#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1))) 11: if#(false(),z0,z1) -> c_11() 12: if#(true(),z0,z1) -> c_12() 15: le#(s(z0),s(z1)) -> c_15(le#(z0,z1)) 13: le#(0(),z0) -> c_13() 14: le#(s(z0),0()) -> c_14() 16: minus#(z0,0()) -> c_16() 18: p#(0()) -> c_18() 19: p#(s(z0)) -> c_19() 6: LE#(0(),z0) -> c_3() 7: LE#(s(z0),0()) -> c_4() ** Step 1.b:4: SimplifyRHS. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: LE#(s(z0),s(z1)) -> c_5(LE#(z0,z1)) MINUS#(z0,s(z1)) -> c_7(IF#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,LE#(z0,s(z1))) MINUS#(z0,s(z1)) -> c_8(IF#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,P#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,MINUS#(z0,p(s(z1))) ,p#(s(z1)) ,P#(s(z1))) - Weak TRS: IF(false(),z0,z1) -> c9() IF(true(),z0,z1) -> c8() LE(0(),z0) -> c2() LE(s(z0),0()) -> c3() LE(s(z0),s(z1)) -> c4(LE(z0,z1)) MINUS(z0,0()) -> c5() MINUS(z0,s(z1)) -> c6(IF(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))),LE(z0,s(z1))) MINUS(z0,s(z1)) -> c7(IF(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,P(minus(z0,p(s(z1)))) ,MINUS(z0,p(s(z1))) ,P(s(z1))) P(0()) -> c() P(s(z0)) -> c1() if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) minus(z0,0()) -> z0 minus(z0,s(z1)) -> if(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) p(0()) -> 0() p(s(z0)) -> z0 - Signature: {IF/3,LE/2,MINUS/2,P/1,if/3,le/2,minus/2,p/1,IF#/3,LE#/2,MINUS#/2,P#/1,if#/3,le#/2,minus#/2,p#/1} / {0/0,c/0 ,c1/0,c2/0,c3/0,c4/1,c5/0,c6/2,c7/4,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/6 ,c_8/11,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/0,c_17/5,c_18/0,c_19/0} - Obligation: innermost runtime complexity wrt. defined symbols {IF#,LE#,MINUS#,P#,if#,le#,minus#,p#} and constructors {0 ,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:LE#(s(z0),s(z1)) -> c_5(LE#(z0,z1)) -->_1 LE#(s(z0),s(z1)) -> c_5(LE#(z0,z1)):1 2:S:MINUS#(z0,s(z1)) -> c_7(IF#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,LE#(z0,s(z1))) -->_6 LE#(s(z0),s(z1)) -> c_5(LE#(z0,z1)):1 3:S:MINUS#(z0,s(z1)) -> c_8(IF#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,P#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,MINUS#(z0,p(s(z1))) ,p#(s(z1)) ,P#(s(z1))) -->_9 MINUS#(z0,s(z1)) -> c_8(IF#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,P#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,MINUS#(z0,p(s(z1))) ,p#(s(z1)) ,P#(s(z1))):3 -->_9 MINUS#(z0,s(z1)) -> c_7(IF#(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,le#(z0,s(z1)) ,p#(minus(z0,p(s(z1)))) ,minus#(z0,p(s(z1))) ,p#(s(z1)) ,LE#(z0,s(z1))):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: MINUS#(z0,s(z1)) -> c_7(LE#(z0,s(z1))) MINUS#(z0,s(z1)) -> c_8(MINUS#(z0,p(s(z1)))) ** Step 1.b:5: UsableRules. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: LE#(s(z0),s(z1)) -> c_5(LE#(z0,z1)) MINUS#(z0,s(z1)) -> c_7(LE#(z0,s(z1))) MINUS#(z0,s(z1)) -> c_8(MINUS#(z0,p(s(z1)))) - Weak TRS: IF(false(),z0,z1) -> c9() IF(true(),z0,z1) -> c8() LE(0(),z0) -> c2() LE(s(z0),0()) -> c3() LE(s(z0),s(z1)) -> c4(LE(z0,z1)) MINUS(z0,0()) -> c5() MINUS(z0,s(z1)) -> c6(IF(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))),LE(z0,s(z1))) MINUS(z0,s(z1)) -> c7(IF(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) ,P(minus(z0,p(s(z1)))) ,MINUS(z0,p(s(z1))) ,P(s(z1))) P(0()) -> c() P(s(z0)) -> c1() if(false(),z0,z1) -> z1 if(true(),z0,z1) -> z0 le(0(),z0) -> true() le(s(z0),0()) -> false() le(s(z0),s(z1)) -> le(z0,z1) minus(z0,0()) -> z0 minus(z0,s(z1)) -> if(le(z0,s(z1)),0(),p(minus(z0,p(s(z1))))) p(0()) -> 0() p(s(z0)) -> z0 - Signature: {IF/3,LE/2,MINUS/2,P/1,if/3,le/2,minus/2,p/1,IF#/3,LE#/2,MINUS#/2,P#/1,if#/3,le#/2,minus#/2,p#/1} / {0/0,c/0 ,c1/0,c2/0,c3/0,c4/1,c5/0,c6/2,c7/4,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/1,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/0,c_17/5,c_18/0,c_19/0} - Obligation: innermost runtime complexity wrt. defined symbols {IF#,LE#,MINUS#,P#,if#,le#,minus#,p#} and constructors {0 ,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: p(s(z0)) -> z0 LE#(s(z0),s(z1)) -> c_5(LE#(z0,z1)) MINUS#(z0,s(z1)) -> c_7(LE#(z0,s(z1))) MINUS#(z0,s(z1)) -> c_8(MINUS#(z0,p(s(z1)))) ** Step 1.b:6: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: LE#(s(z0),s(z1)) -> c_5(LE#(z0,z1)) MINUS#(z0,s(z1)) -> c_7(LE#(z0,s(z1))) MINUS#(z0,s(z1)) -> c_8(MINUS#(z0,p(s(z1)))) - Weak TRS: p(s(z0)) -> z0 - Signature: {IF/3,LE/2,MINUS/2,P/1,if/3,le/2,minus/2,p/1,IF#/3,LE#/2,MINUS#/2,P#/1,if#/3,le#/2,minus#/2,p#/1} / {0/0,c/0 ,c1/0,c2/0,c3/0,c4/1,c5/0,c6/2,c7/4,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/1,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/0,c_17/5,c_18/0,c_19/0} - Obligation: innermost runtime complexity wrt. defined symbols {IF#,LE#,MINUS#,P#,if#,le#,minus#,p#} and constructors {0 ,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + 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_5) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1} Following symbols are considered usable: {IF#,LE#,MINUS#,P#,if#,le#,minus#,p#} TcT has computed the following interpretation: p(0) = [0] p(IF) = [0] p(LE) = [0] p(MINUS) = [0] p(P) = [0] p(c) = [0] p(c1) = [0] p(c2) = [0] p(c3) = [0] p(c4) = [1] x1 + [0] p(c5) = [0] p(c6) = [1] x1 + [1] x2 + [0] p(c7) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [0] p(c8) = [0] p(c9) = [0] p(false) = [0] p(if) = [0] p(le) = [0] p(minus) = [1] x1 + [0] p(p) = [0] p(s) = [0] p(true) = [0] p(IF#) = [2] x1 + [2] x2 + [2] x3 + [0] p(LE#) = [0] p(MINUS#) = [1] x1 + [2] p(P#) = [1] x1 + [1] p(if#) = [1] p(le#) = [1] x1 + [1] p(minus#) = [8] x1 + [0] p(p#) = [1] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [1] p(c_5) = [1] x1 + [0] p(c_6) = [2] p(c_7) = [4] x1 + [0] p(c_8) = [1] x1 + [0] p(c_9) = [1] p(c_10) = [0] p(c_11) = [0] p(c_12) = [2] p(c_13) = [4] p(c_14) = [1] p(c_15) = [4] p(c_16) = [1] p(c_17) = [1] x1 + [1] x3 + [1] x4 + [1] x5 + [1] p(c_18) = [8] p(c_19) = [1] Following rules are strictly oriented: MINUS#(z0,s(z1)) = [1] z0 + [2] > [0] = c_7(LE#(z0,s(z1))) Following rules are (at-least) weakly oriented: LE#(s(z0),s(z1)) = [0] >= [0] = c_5(LE#(z0,z1)) MINUS#(z0,s(z1)) = [1] z0 + [2] >= [1] z0 + [2] = c_8(MINUS#(z0,p(s(z1)))) ** Step 1.b:7: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: LE#(s(z0),s(z1)) -> c_5(LE#(z0,z1)) MINUS#(z0,s(z1)) -> c_8(MINUS#(z0,p(s(z1)))) - Weak DPs: MINUS#(z0,s(z1)) -> c_7(LE#(z0,s(z1))) - Weak TRS: p(s(z0)) -> z0 - Signature: {IF/3,LE/2,MINUS/2,P/1,if/3,le/2,minus/2,p/1,IF#/3,LE#/2,MINUS#/2,P#/1,if#/3,le#/2,minus#/2,p#/1} / {0/0,c/0 ,c1/0,c2/0,c3/0,c4/1,c5/0,c6/2,c7/4,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/1,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/0,c_17/5,c_18/0,c_19/0} - Obligation: innermost runtime complexity wrt. defined symbols {IF#,LE#,MINUS#,P#,if#,le#,minus#,p#} and constructors {0 ,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + 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_5) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1} Following symbols are considered usable: {IF#,LE#,MINUS#,P#,if#,le#,minus#,p#} TcT has computed the following interpretation: p(0) = [1] p(IF) = [1] x1 + [2] x3 + [1] p(LE) = [1] p(MINUS) = [1] x1 + [2] x2 + [0] p(P) = [1] x1 + [0] p(c) = [1] p(c1) = [4] p(c2) = [0] p(c3) = [1] p(c4) = [1] p(c5) = [0] p(c6) = [1] x2 + [1] p(c7) = [0] p(c8) = [0] p(c9) = [1] p(false) = [0] p(if) = [1] x3 + [4] p(le) = [2] p(minus) = [8] x1 + [1] p(p) = [0] p(s) = [1] x1 + [2] p(true) = [1] p(IF#) = [4] p(LE#) = [4] x1 + [2] p(MINUS#) = [8] x1 + [8] p(P#) = [8] x1 + [0] p(if#) = [1] x1 + [2] x2 + [0] p(le#) = [2] x1 + [0] p(minus#) = [4] x2 + [1] p(p#) = [1] x1 + [1] p(c_1) = [0] p(c_2) = [2] p(c_3) = [1] p(c_4) = [2] p(c_5) = [1] x1 + [6] p(c_6) = [1] p(c_7) = [2] x1 + [4] p(c_8) = [1] x1 + [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [2] p(c_12) = [1] p(c_13) = [1] p(c_14) = [0] p(c_15) = [1] p(c_16) = [1] p(c_17) = [1] x5 + [1] p(c_18) = [1] p(c_19) = [2] Following rules are strictly oriented: LE#(s(z0),s(z1)) = [4] z0 + [10] > [4] z0 + [8] = c_5(LE#(z0,z1)) Following rules are (at-least) weakly oriented: MINUS#(z0,s(z1)) = [8] z0 + [8] >= [8] z0 + [8] = c_7(LE#(z0,s(z1))) MINUS#(z0,s(z1)) = [8] z0 + [8] >= [8] z0 + [8] = c_8(MINUS#(z0,p(s(z1)))) ** Step 1.b:8: Ara. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: MINUS#(z0,s(z1)) -> c_8(MINUS#(z0,p(s(z1)))) - Weak DPs: LE#(s(z0),s(z1)) -> c_5(LE#(z0,z1)) MINUS#(z0,s(z1)) -> c_7(LE#(z0,s(z1))) - Weak TRS: p(s(z0)) -> z0 - Signature: {IF/3,LE/2,MINUS/2,P/1,if/3,le/2,minus/2,p/1,IF#/3,LE#/2,MINUS#/2,P#/1,if#/3,le#/2,minus#/2,p#/1} / {0/0,c/0 ,c1/0,c2/0,c3/0,c4/1,c5/0,c6/2,c7/4,c8/0,c9/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1 ,c_8/1,c_9/0,c_10/0,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/0,c_17/5,c_18/0,c_19/0} - Obligation: innermost runtime complexity wrt. defined symbols {IF#,LE#,MINUS#,P#,if#,le#,minus#,p#} and constructors {0 ,c,c1,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: Ara {minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1, isBestCase = False, mkCompletelyDefined = False, verboseOutput = False} + Details: Signatures used: ---------------- F (TrsFun "p") :: ["A"(0, 2)] -(0)-> "A"(1, 2) F (TrsFun "s") :: ["A"(3, 2)] -(1)-> "A"(1, 2) F (TrsFun "s") :: ["A"(2, 2)] -(0)-> "A"(0, 2) F (TrsFun "s") :: ["A"(1, 0)] -(1)-> "A"(1, 0) F (TrsFun "s") :: ["A"(1, 1)] -(0)-> "A"(0, 1) F (TrsFun "s") :: ["A"(2, 1)] -(1)-> "A"(1, 1) F (DpFun "LE") :: ["A"(1, 0) x "A"(0, 1)] -(0)-> "A"(0, 0) F (DpFun "MINUS") :: ["A"(1, 0) x "A"(1, 2)] -(0)-> "A"(0, 0) F (ComFun 5) :: ["A"(0, 0)] -(0)-> "A"(0, 0) F (ComFun 7) :: ["A"(0, 0)] -(0)-> "A"(0, 0) F (ComFun 8) :: ["A"(0, 0)] -(0)-> "A"(0, 0) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: MINUS#(z0,s(z1)) -> c_8(MINUS#(z0,p(s(z1)))) 2. Weak: WORST_CASE(Omega(n^1),O(n^2))