tct-trs: Prelude.head: empty list tct-trs: Prelude.head: empty list tct-trs: Prelude.head: empty list tct-trs: Prelude.head: empty list WORST_CASE(Omega(n^1),?) * Step 1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: ADD(0(),z0) -> c11() ADD(s(z0),z1) -> c12(ADD(z0,z1)) COND1(true(),z0,z1) -> c(COND2(gr(z0,z1),z0,z1),GR(z0,z1)) COND2(false(),z0,z1) -> c3(COND3(eq(z0,z1),z0,z1),EQ(z0,z1)) COND2(true(),z0,z1) -> c1(COND1(gr(add(z0,z1),0()),p(z0),z1),GR(add(z0,z1),0()),ADD(z0,z1)) COND2(true(),z0,z1) -> c2(COND1(gr(add(z0,z1),0()),p(z0),z1),P(z0)) COND3(false(),z0,z1) -> c6(COND1(gr(add(z0,z1),0()),z0,p(z1)),GR(add(z0,z1),0()),ADD(z0,z1)) COND3(false(),z0,z1) -> c7(COND1(gr(add(z0,z1),0()),z0,p(z1)),P(z1)) COND3(true(),z0,z1) -> c4(COND1(gr(add(z0,z1),0()),p(z0),z1),GR(add(z0,z1),0()),ADD(z0,z1)) COND3(true(),z0,z1) -> c5(COND1(gr(add(z0,z1),0()),p(z0),z1),P(z0)) EQ(0(),0()) -> c13() EQ(0(),s(z0)) -> c14() EQ(s(z0),0()) -> c15() EQ(s(z0),s(z1)) -> c16(EQ(z0,z1)) GR(0(),z0) -> c8() GR(s(z0),0()) -> c9() GR(s(z0),s(z1)) -> c10(GR(z0,z1)) P(0()) -> c17() P(s(z0)) -> c18() - Weak TRS: add(0(),z0) -> z0 add(s(z0),z1) -> s(add(z0,z1)) cond1(true(),z0,z1) -> cond2(gr(z0,z1),z0,z1) cond2(false(),z0,z1) -> cond3(eq(z0,z1),z0,z1) cond2(true(),z0,z1) -> cond1(gr(add(z0,z1),0()),p(z0),z1) cond3(false(),z0,z1) -> cond1(gr(add(z0,z1),0()),z0,p(z1)) cond3(true(),z0,z1) -> cond1(gr(add(z0,z1),0()),p(z0),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: {ADD/2,COND1/3,COND2/3,COND3/3,EQ/2,GR/2,P/1,add/2,cond1/3,cond2/3,cond3/3,eq/2,gr/2,p/1} / {0/0,c/2,c1/3 ,c10/1,c11/0,c12/1,c13/0,c14/0,c15/0,c16/1,c17/0,c18/0,c2/2,c3/2,c4/3,c5/2,c6/3,c7/2,c8/0,c9/0,false/0,s/1 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {ADD,COND1,COND2,COND3,EQ,GR,P,add,cond1,cond2,cond3,eq,gr ,p} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,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: ADD(0(),z0) -> c11() ADD(s(z0),z1) -> c12(ADD(z0,z1)) COND1(true(),z0,z1) -> c(COND2(gr(z0,z1),z0,z1),GR(z0,z1)) COND2(false(),z0,z1) -> c3(COND3(eq(z0,z1),z0,z1),EQ(z0,z1)) COND2(true(),z0,z1) -> c1(COND1(gr(add(z0,z1),0()),p(z0),z1),GR(add(z0,z1),0()),ADD(z0,z1)) COND2(true(),z0,z1) -> c2(COND1(gr(add(z0,z1),0()),p(z0),z1),P(z0)) COND3(false(),z0,z1) -> c6(COND1(gr(add(z0,z1),0()),z0,p(z1)),GR(add(z0,z1),0()),ADD(z0,z1)) COND3(false(),z0,z1) -> c7(COND1(gr(add(z0,z1),0()),z0,p(z1)),P(z1)) COND3(true(),z0,z1) -> c4(COND1(gr(add(z0,z1),0()),p(z0),z1),GR(add(z0,z1),0()),ADD(z0,z1)) COND3(true(),z0,z1) -> c5(COND1(gr(add(z0,z1),0()),p(z0),z1),P(z0)) EQ(0(),0()) -> c13() EQ(0(),s(z0)) -> c14() EQ(s(z0),0()) -> c15() EQ(s(z0),s(z1)) -> c16(EQ(z0,z1)) GR(0(),z0) -> c8() GR(s(z0),0()) -> c9() GR(s(z0),s(z1)) -> c10(GR(z0,z1)) P(0()) -> c17() P(s(z0)) -> c18() - Weak TRS: add(0(),z0) -> z0 add(s(z0),z1) -> s(add(z0,z1)) cond1(true(),z0,z1) -> cond2(gr(z0,z1),z0,z1) cond2(false(),z0,z1) -> cond3(eq(z0,z1),z0,z1) cond2(true(),z0,z1) -> cond1(gr(add(z0,z1),0()),p(z0),z1) cond3(false(),z0,z1) -> cond1(gr(add(z0,z1),0()),z0,p(z1)) cond3(true(),z0,z1) -> cond1(gr(add(z0,z1),0()),p(z0),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: {ADD/2,COND1/3,COND2/3,COND3/3,EQ/2,GR/2,P/1,add/2,cond1/3,cond2/3,cond3/3,eq/2,gr/2,p/1} / {0/0,c/2,c1/3 ,c10/1,c11/0,c12/1,c13/0,c14/0,c15/0,c16/1,c17/0,c18/0,c2/2,c3/2,c4/3,c5/2,c6/3,c7/2,c8/0,c9/0,false/0,s/1 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {ADD,COND1,COND2,COND3,EQ,GR,P,add,cond1,cond2,cond3,eq,gr ,p} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,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: ADD(0(),z0) -> c11() ADD(s(z0),z1) -> c12(ADD(z0,z1)) COND1(true(),z0,z1) -> c(COND2(gr(z0,z1),z0,z1),GR(z0,z1)) COND2(false(),z0,z1) -> c3(COND3(eq(z0,z1),z0,z1),EQ(z0,z1)) COND2(true(),z0,z1) -> c1(COND1(gr(add(z0,z1),0()),p(z0),z1),GR(add(z0,z1),0()),ADD(z0,z1)) COND2(true(),z0,z1) -> c2(COND1(gr(add(z0,z1),0()),p(z0),z1),P(z0)) COND3(false(),z0,z1) -> c6(COND1(gr(add(z0,z1),0()),z0,p(z1)),GR(add(z0,z1),0()),ADD(z0,z1)) COND3(false(),z0,z1) -> c7(COND1(gr(add(z0,z1),0()),z0,p(z1)),P(z1)) COND3(true(),z0,z1) -> c4(COND1(gr(add(z0,z1),0()),p(z0),z1),GR(add(z0,z1),0()),ADD(z0,z1)) COND3(true(),z0,z1) -> c5(COND1(gr(add(z0,z1),0()),p(z0),z1),P(z0)) EQ(0(),0()) -> c13() EQ(0(),s(z0)) -> c14() EQ(s(z0),0()) -> c15() EQ(s(z0),s(z1)) -> c16(EQ(z0,z1)) GR(0(),z0) -> c8() GR(s(z0),0()) -> c9() GR(s(z0),s(z1)) -> c10(GR(z0,z1)) P(0()) -> c17() P(s(z0)) -> c18() - Weak TRS: add(0(),z0) -> z0 add(s(z0),z1) -> s(add(z0,z1)) cond1(true(),z0,z1) -> cond2(gr(z0,z1),z0,z1) cond2(false(),z0,z1) -> cond3(eq(z0,z1),z0,z1) cond2(true(),z0,z1) -> cond1(gr(add(z0,z1),0()),p(z0),z1) cond3(false(),z0,z1) -> cond1(gr(add(z0,z1),0()),z0,p(z1)) cond3(true(),z0,z1) -> cond1(gr(add(z0,z1),0()),p(z0),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: {ADD/2,COND1/3,COND2/3,COND3/3,EQ/2,GR/2,P/1,add/2,cond1/3,cond2/3,cond3/3,eq/2,gr/2,p/1} / {0/0,c/2,c1/3 ,c10/1,c11/0,c12/1,c13/0,c14/0,c15/0,c16/1,c17/0,c18/0,c2/2,c3/2,c4/3,c5/2,c6/3,c7/2,c8/0,c9/0,false/0,s/1 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {ADD,COND1,COND2,COND3,EQ,GR,P,add,cond1,cond2,cond3,eq,gr ,p} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,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: ADD(x,y){x -> s(x)} = ADD(s(x),y) ->^+ c12(ADD(x,y)) = C[ADD(x,y) = ADD(x,y){}] WORST_CASE(Omega(n^1),?)