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:
            ID(0()) -> c5()
            ID(s(z0)) -> c6(ID(z0))
            IF2(false(),z0,z1,z2) -> c17(IF3(z0,z1,z2))
            IF2(true(),z0,z1,z2) -> c16()
            IF3(false(),z0,z1) -> c19()
            IF3(true(),z0,z1) -> c18(MOD(minus(z0,z1),s(z1)),MINUS(z0,z1))
            IF_MOD(false(),z0,z1,z2,z3) -> c15(IF2(z0,z1,z2,z3))
            IF_MOD(true(),z0,z1,z2,z3) -> c14()
            LE(0(),z0) -> c()
            LE(s(z0),0()) -> c1()
            LE(s(z0),s(z1)) -> c2(LE(z0,z1))
            MINUS(z0,0()) -> c7()
            MINUS(s(z0),s(z1)) -> c8(MINUS(z0,z1))
            MOD(z0,z1) -> c10(IF_MOD(zero(z0),zero(z1),le(z1,z0),id(z0),id(z1)),ZERO(z1))
            MOD(z0,z1) -> c11(IF_MOD(zero(z0),zero(z1),le(z1,z0),id(z0),id(z1)),LE(z1,z0))
            MOD(z0,z1) -> c12(IF_MOD(zero(z0),zero(z1),le(z1,z0),id(z0),id(z1)),ID(z0))
            MOD(z0,z1) -> c13(IF_MOD(zero(z0),zero(z1),le(z1,z0),id(z0),id(z1)),ID(z1))
            MOD(z0,z1) -> c9(IF_MOD(zero(z0),zero(z1),le(z1,z0),id(z0),id(z1)),ZERO(z0))
            ZERO(0()) -> c3()
            ZERO(s(z0)) -> c4()
        - Weak TRS:
            id(0()) -> 0()
            id(s(z0)) -> s(id(z0))
            if2(false(),z0,z1,z2) -> if3(z0,z1,z2)
            if2(true(),z0,z1,z2) -> 0()
            if3(false(),z0,z1) -> z0
            if3(true(),z0,z1) -> mod(minus(z0,z1),s(z1))
            if_mod(false(),z0,z1,z2,z3) -> if2(z0,z1,z2,z3)
            if_mod(true(),z0,z1,z2,z3) -> 0()
            le(0(),z0) -> true()
            le(s(z0),0()) -> false()
            le(s(z0),s(z1)) -> le(z0,z1)
            minus(z0,0()) -> z0
            minus(s(z0),s(z1)) -> minus(z0,z1)
            mod(z0,z1) -> if_mod(zero(z0),zero(z1),le(z1,z0),id(z0),id(z1))
            zero(0()) -> true()
            zero(s(z0)) -> false()
        - Signature:
            {ID/1,IF2/4,IF3/3,IF_MOD/5,LE/2,MINUS/2,MOD/2,ZERO/1,id/1,if2/4,if3/3,if_mod/5,le/2,minus/2,mod/2
            ,zero/1} / {0/0,c/0,c1/0,c10/2,c11/2,c12/2,c13/2,c14/0,c15/1,c16/0,c17/1,c18/2,c19/0,c2/1,c3/0,c4/0,c5/0
            ,c6/1,c7/0,c8/1,c9/2,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ID,IF2,IF3,IF_MOD,LE,MINUS,MOD,ZERO,id,if2,if3,if_mod,le
            ,minus,mod,zero} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,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:
            ID(0()) -> c5()
            ID(s(z0)) -> c6(ID(z0))
            IF2(false(),z0,z1,z2) -> c17(IF3(z0,z1,z2))
            IF2(true(),z0,z1,z2) -> c16()
            IF3(false(),z0,z1) -> c19()
            IF3(true(),z0,z1) -> c18(MOD(minus(z0,z1),s(z1)),MINUS(z0,z1))
            IF_MOD(false(),z0,z1,z2,z3) -> c15(IF2(z0,z1,z2,z3))
            IF_MOD(true(),z0,z1,z2,z3) -> c14()
            LE(0(),z0) -> c()
            LE(s(z0),0()) -> c1()
            LE(s(z0),s(z1)) -> c2(LE(z0,z1))
            MINUS(z0,0()) -> c7()
            MINUS(s(z0),s(z1)) -> c8(MINUS(z0,z1))
            MOD(z0,z1) -> c10(IF_MOD(zero(z0),zero(z1),le(z1,z0),id(z0),id(z1)),ZERO(z1))
            MOD(z0,z1) -> c11(IF_MOD(zero(z0),zero(z1),le(z1,z0),id(z0),id(z1)),LE(z1,z0))
            MOD(z0,z1) -> c12(IF_MOD(zero(z0),zero(z1),le(z1,z0),id(z0),id(z1)),ID(z0))
            MOD(z0,z1) -> c13(IF_MOD(zero(z0),zero(z1),le(z1,z0),id(z0),id(z1)),ID(z1))
            MOD(z0,z1) -> c9(IF_MOD(zero(z0),zero(z1),le(z1,z0),id(z0),id(z1)),ZERO(z0))
            ZERO(0()) -> c3()
            ZERO(s(z0)) -> c4()
        - Weak TRS:
            id(0()) -> 0()
            id(s(z0)) -> s(id(z0))
            if2(false(),z0,z1,z2) -> if3(z0,z1,z2)
            if2(true(),z0,z1,z2) -> 0()
            if3(false(),z0,z1) -> z0
            if3(true(),z0,z1) -> mod(minus(z0,z1),s(z1))
            if_mod(false(),z0,z1,z2,z3) -> if2(z0,z1,z2,z3)
            if_mod(true(),z0,z1,z2,z3) -> 0()
            le(0(),z0) -> true()
            le(s(z0),0()) -> false()
            le(s(z0),s(z1)) -> le(z0,z1)
            minus(z0,0()) -> z0
            minus(s(z0),s(z1)) -> minus(z0,z1)
            mod(z0,z1) -> if_mod(zero(z0),zero(z1),le(z1,z0),id(z0),id(z1))
            zero(0()) -> true()
            zero(s(z0)) -> false()
        - Signature:
            {ID/1,IF2/4,IF3/3,IF_MOD/5,LE/2,MINUS/2,MOD/2,ZERO/1,id/1,if2/4,if3/3,if_mod/5,le/2,minus/2,mod/2
            ,zero/1} / {0/0,c/0,c1/0,c10/2,c11/2,c12/2,c13/2,c14/0,c15/1,c16/0,c17/1,c18/2,c19/0,c2/1,c3/0,c4/0,c5/0
            ,c6/1,c7/0,c8/1,c9/2,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ID,IF2,IF3,IF_MOD,LE,MINUS,MOD,ZERO,id,if2,if3,if_mod,le
            ,minus,mod,zero} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,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:
            ID(0()) -> c5()
            ID(s(z0)) -> c6(ID(z0))
            IF2(false(),z0,z1,z2) -> c17(IF3(z0,z1,z2))
            IF2(true(),z0,z1,z2) -> c16()
            IF3(false(),z0,z1) -> c19()
            IF3(true(),z0,z1) -> c18(MOD(minus(z0,z1),s(z1)),MINUS(z0,z1))
            IF_MOD(false(),z0,z1,z2,z3) -> c15(IF2(z0,z1,z2,z3))
            IF_MOD(true(),z0,z1,z2,z3) -> c14()
            LE(0(),z0) -> c()
            LE(s(z0),0()) -> c1()
            LE(s(z0),s(z1)) -> c2(LE(z0,z1))
            MINUS(z0,0()) -> c7()
            MINUS(s(z0),s(z1)) -> c8(MINUS(z0,z1))
            MOD(z0,z1) -> c10(IF_MOD(zero(z0),zero(z1),le(z1,z0),id(z0),id(z1)),ZERO(z1))
            MOD(z0,z1) -> c11(IF_MOD(zero(z0),zero(z1),le(z1,z0),id(z0),id(z1)),LE(z1,z0))
            MOD(z0,z1) -> c12(IF_MOD(zero(z0),zero(z1),le(z1,z0),id(z0),id(z1)),ID(z0))
            MOD(z0,z1) -> c13(IF_MOD(zero(z0),zero(z1),le(z1,z0),id(z0),id(z1)),ID(z1))
            MOD(z0,z1) -> c9(IF_MOD(zero(z0),zero(z1),le(z1,z0),id(z0),id(z1)),ZERO(z0))
            ZERO(0()) -> c3()
            ZERO(s(z0)) -> c4()
        - Weak TRS:
            id(0()) -> 0()
            id(s(z0)) -> s(id(z0))
            if2(false(),z0,z1,z2) -> if3(z0,z1,z2)
            if2(true(),z0,z1,z2) -> 0()
            if3(false(),z0,z1) -> z0
            if3(true(),z0,z1) -> mod(minus(z0,z1),s(z1))
            if_mod(false(),z0,z1,z2,z3) -> if2(z0,z1,z2,z3)
            if_mod(true(),z0,z1,z2,z3) -> 0()
            le(0(),z0) -> true()
            le(s(z0),0()) -> false()
            le(s(z0),s(z1)) -> le(z0,z1)
            minus(z0,0()) -> z0
            minus(s(z0),s(z1)) -> minus(z0,z1)
            mod(z0,z1) -> if_mod(zero(z0),zero(z1),le(z1,z0),id(z0),id(z1))
            zero(0()) -> true()
            zero(s(z0)) -> false()
        - Signature:
            {ID/1,IF2/4,IF3/3,IF_MOD/5,LE/2,MINUS/2,MOD/2,ZERO/1,id/1,if2/4,if3/3,if_mod/5,le/2,minus/2,mod/2
            ,zero/1} / {0/0,c/0,c1/0,c10/2,c11/2,c12/2,c13/2,c14/0,c15/1,c16/0,c17/1,c18/2,c19/0,c2/1,c3/0,c4/0,c5/0
            ,c6/1,c7/0,c8/1,c9/2,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {ID,IF2,IF3,IF_MOD,LE,MINUS,MOD,ZERO,id,if2,if3,if_mod,le
            ,minus,mod,zero} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,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:
          ID(x){x -> s(x)} =
            ID(s(x)) ->^+ c6(ID(x))
              = C[ID(x) = ID(x){}]

WORST_CASE(Omega(n^1),?)