WORST_CASE(Omega(n^1),?) * Step 1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: COND1(true(),z0,z1) -> c(COND2(gr(z0,0()),z0,z1),GR(z0,0())) COND2(false(),z0,z1) -> c4(COND3(gr(z1,0()),z0,z1),GR(z1,0())) COND2(true(),z0,z1) -> c1(COND1(or(gr(z0,0()),gr(z1,0())),p(z0),z1),OR(gr(z0,0()),gr(z1,0())),GR(z0,0())) COND2(true(),z0,z1) -> c2(COND1(or(gr(z0,0()),gr(z1,0())),p(z0),z1),OR(gr(z0,0()),gr(z1,0())),GR(z1,0())) COND2(true(),z0,z1) -> c3(COND1(or(gr(z0,0()),gr(z1,0())),p(z0),z1),P(z0)) COND3(false(),z0,z1) -> c8(COND1(or(gr(z0,0()),gr(z1,0())),z0,z1),OR(gr(z0,0()),gr(z1,0())),GR(z0,0())) COND3(false(),z0,z1) -> c9(COND1(or(gr(z0,0()),gr(z1,0())),z0,z1),OR(gr(z0,0()),gr(z1,0())),GR(z1,0())) COND3(true(),z0,z1) -> c5(COND1(or(gr(z0,0()),gr(z1,0())),z0,p(z1)),OR(gr(z0,0()),gr(z1,0())),GR(z0,0())) COND3(true(),z0,z1) -> c6(COND1(or(gr(z0,0()),gr(z1,0())),z0,p(z1)),OR(gr(z0,0()),gr(z1,0())),GR(z1,0())) COND3(true(),z0,z1) -> c7(COND1(or(gr(z0,0()),gr(z1,0())),z0,p(z1)),P(z1)) GR(0(),z0) -> c10() GR(s(z0),0()) -> c11() GR(s(z0),s(z1)) -> c12(GR(z0,z1)) OR(z0,true()) -> c15() OR(false(),false()) -> c13() OR(true(),z0) -> c14() P(0()) -> c16() P(s(z0)) -> c17() - Weak TRS: cond1(true(),z0,z1) -> cond2(gr(z0,0()),z0,z1) cond2(false(),z0,z1) -> cond3(gr(z1,0()),z0,z1) cond2(true(),z0,z1) -> cond1(or(gr(z0,0()),gr(z1,0())),p(z0),z1) cond3(false(),z0,z1) -> cond1(or(gr(z0,0()),gr(z1,0())),z0,z1) cond3(true(),z0,z1) -> cond1(or(gr(z0,0()),gr(z1,0())),z0,p(z1)) gr(0(),z0) -> false() gr(s(z0),0()) -> true() gr(s(z0),s(z1)) -> gr(z0,z1) or(z0,true()) -> true() or(false(),false()) -> false() or(true(),z0) -> true() p(0()) -> 0() p(s(z0)) -> z0 - Signature: {COND1/3,COND2/3,COND3/3,GR/2,OR/2,P/1,cond1/3,cond2/3,cond3/3,gr/2,or/2,p/1} / {0/0,c/2,c1/3,c10/0,c11/0 ,c12/1,c13/0,c14/0,c15/0,c16/0,c17/0,c2/3,c3/2,c4/2,c5/3,c6/3,c7/2,c8/3,c9/3,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {COND1,COND2,COND3,GR,OR,P,cond1,cond2,cond3,gr,or ,p} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: COND1(true(),z0,z1) -> c(COND2(gr(z0,0()),z0,z1),GR(z0,0())) COND2(false(),z0,z1) -> c4(COND3(gr(z1,0()),z0,z1),GR(z1,0())) COND2(true(),z0,z1) -> c1(COND1(or(gr(z0,0()),gr(z1,0())),p(z0),z1),OR(gr(z0,0()),gr(z1,0())),GR(z0,0())) COND2(true(),z0,z1) -> c2(COND1(or(gr(z0,0()),gr(z1,0())),p(z0),z1),OR(gr(z0,0()),gr(z1,0())),GR(z1,0())) COND2(true(),z0,z1) -> c3(COND1(or(gr(z0,0()),gr(z1,0())),p(z0),z1),P(z0)) COND3(false(),z0,z1) -> c8(COND1(or(gr(z0,0()),gr(z1,0())),z0,z1),OR(gr(z0,0()),gr(z1,0())),GR(z0,0())) COND3(false(),z0,z1) -> c9(COND1(or(gr(z0,0()),gr(z1,0())),z0,z1),OR(gr(z0,0()),gr(z1,0())),GR(z1,0())) COND3(true(),z0,z1) -> c5(COND1(or(gr(z0,0()),gr(z1,0())),z0,p(z1)),OR(gr(z0,0()),gr(z1,0())),GR(z0,0())) COND3(true(),z0,z1) -> c6(COND1(or(gr(z0,0()),gr(z1,0())),z0,p(z1)),OR(gr(z0,0()),gr(z1,0())),GR(z1,0())) COND3(true(),z0,z1) -> c7(COND1(or(gr(z0,0()),gr(z1,0())),z0,p(z1)),P(z1)) GR(0(),z0) -> c10() GR(s(z0),0()) -> c11() GR(s(z0),s(z1)) -> c12(GR(z0,z1)) OR(z0,true()) -> c15() OR(false(),false()) -> c13() OR(true(),z0) -> c14() P(0()) -> c16() P(s(z0)) -> c17() - Weak TRS: cond1(true(),z0,z1) -> cond2(gr(z0,0()),z0,z1) cond2(false(),z0,z1) -> cond3(gr(z1,0()),z0,z1) cond2(true(),z0,z1) -> cond1(or(gr(z0,0()),gr(z1,0())),p(z0),z1) cond3(false(),z0,z1) -> cond1(or(gr(z0,0()),gr(z1,0())),z0,z1) cond3(true(),z0,z1) -> cond1(or(gr(z0,0()),gr(z1,0())),z0,p(z1)) gr(0(),z0) -> false() gr(s(z0),0()) -> true() gr(s(z0),s(z1)) -> gr(z0,z1) or(z0,true()) -> true() or(false(),false()) -> false() or(true(),z0) -> true() p(0()) -> 0() p(s(z0)) -> z0 - Signature: {COND1/3,COND2/3,COND3/3,GR/2,OR/2,P/1,cond1/3,cond2/3,cond3/3,gr/2,or/2,p/1} / {0/0,c/2,c1/3,c10/0,c11/0 ,c12/1,c13/0,c14/0,c15/0,c16/0,c17/0,c2/3,c3/2,c4/2,c5/3,c6/3,c7/2,c8/3,c9/3,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {COND1,COND2,COND3,GR,OR,P,cond1,cond2,cond3,gr,or ,p} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 3: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: COND1(true(),z0,z1) -> c(COND2(gr(z0,0()),z0,z1),GR(z0,0())) COND2(false(),z0,z1) -> c4(COND3(gr(z1,0()),z0,z1),GR(z1,0())) COND2(true(),z0,z1) -> c1(COND1(or(gr(z0,0()),gr(z1,0())),p(z0),z1),OR(gr(z0,0()),gr(z1,0())),GR(z0,0())) COND2(true(),z0,z1) -> c2(COND1(or(gr(z0,0()),gr(z1,0())),p(z0),z1),OR(gr(z0,0()),gr(z1,0())),GR(z1,0())) COND2(true(),z0,z1) -> c3(COND1(or(gr(z0,0()),gr(z1,0())),p(z0),z1),P(z0)) COND3(false(),z0,z1) -> c8(COND1(or(gr(z0,0()),gr(z1,0())),z0,z1),OR(gr(z0,0()),gr(z1,0())),GR(z0,0())) COND3(false(),z0,z1) -> c9(COND1(or(gr(z0,0()),gr(z1,0())),z0,z1),OR(gr(z0,0()),gr(z1,0())),GR(z1,0())) COND3(true(),z0,z1) -> c5(COND1(or(gr(z0,0()),gr(z1,0())),z0,p(z1)),OR(gr(z0,0()),gr(z1,0())),GR(z0,0())) COND3(true(),z0,z1) -> c6(COND1(or(gr(z0,0()),gr(z1,0())),z0,p(z1)),OR(gr(z0,0()),gr(z1,0())),GR(z1,0())) COND3(true(),z0,z1) -> c7(COND1(or(gr(z0,0()),gr(z1,0())),z0,p(z1)),P(z1)) GR(0(),z0) -> c10() GR(s(z0),0()) -> c11() GR(s(z0),s(z1)) -> c12(GR(z0,z1)) OR(z0,true()) -> c15() OR(false(),false()) -> c13() OR(true(),z0) -> c14() P(0()) -> c16() P(s(z0)) -> c17() - Weak TRS: cond1(true(),z0,z1) -> cond2(gr(z0,0()),z0,z1) cond2(false(),z0,z1) -> cond3(gr(z1,0()),z0,z1) cond2(true(),z0,z1) -> cond1(or(gr(z0,0()),gr(z1,0())),p(z0),z1) cond3(false(),z0,z1) -> cond1(or(gr(z0,0()),gr(z1,0())),z0,z1) cond3(true(),z0,z1) -> cond1(or(gr(z0,0()),gr(z1,0())),z0,p(z1)) gr(0(),z0) -> false() gr(s(z0),0()) -> true() gr(s(z0),s(z1)) -> gr(z0,z1) or(z0,true()) -> true() or(false(),false()) -> false() or(true(),z0) -> true() p(0()) -> 0() p(s(z0)) -> z0 - Signature: {COND1/3,COND2/3,COND3/3,GR/2,OR/2,P/1,cond1/3,cond2/3,cond3/3,gr/2,or/2,p/1} / {0/0,c/2,c1/3,c10/0,c11/0 ,c12/1,c13/0,c14/0,c15/0,c16/0,c17/0,c2/3,c3/2,c4/2,c5/3,c6/3,c7/2,c8/3,c9/3,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {COND1,COND2,COND3,GR,OR,P,cond1,cond2,cond3,gr,or ,p} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,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: GR(x,y){x -> s(x),y -> s(y)} = GR(s(x),s(y)) ->^+ c12(GR(x,y)) = C[GR(x,y) = GR(x,y){}] WORST_CASE(Omega(n^1),?)