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