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:
            DIV(z0,z1) -> c8(IF(ge(z1,s(0())),ge(z0,z1),z0,z1),GE(z1,s(0())))
            DIV(z0,z1) -> c9(IF(ge(z1,s(0())),ge(z0,z1),z0,z1),GE(z0,z1))
            GE(z0,0()) -> c()
            GE(0(),s(z0)) -> c1()
            GE(s(z0),s(z1)) -> c2(GE(z0,z1))
            ID_INC(z0) -> c6()
            ID_INC(z0) -> c7()
            IF(false(),z0,z1,z2) -> c10()
            IF(true(),false(),z0,z1) -> c11()
            IF(true(),true(),z0,z1) -> c12(ID_INC(div(minus(z0,z1),z1)),DIV(minus(z0,z1),z1),MINUS(z0,z1))
            MINUS(z0,0()) -> c3()
            MINUS(0(),z0) -> c4()
            MINUS(s(z0),s(z1)) -> c5(MINUS(z0,z1))
        - Weak TRS:
            div(z0,z1) -> if(ge(z1,s(0())),ge(z0,z1),z0,z1)
            ge(z0,0()) -> true()
            ge(0(),s(z0)) -> false()
            ge(s(z0),s(z1)) -> ge(z0,z1)
            id_inc(z0) -> z0
            id_inc(z0) -> s(z0)
            if(false(),z0,z1,z2) -> div_by_zero()
            if(true(),false(),z0,z1) -> 0()
            if(true(),true(),z0,z1) -> id_inc(div(minus(z0,z1),z1))
            minus(z0,0()) -> z0
            minus(0(),z0) -> 0()
            minus(s(z0),s(z1)) -> minus(z0,z1)
        - Signature:
            {DIV/2,GE/2,ID_INC/1,IF/4,MINUS/2,div/2,ge/2,id_inc/1,if/4,minus/2} / {0/0,c/0,c1/0,c10/0,c11/0,c12/3,c2/1
            ,c3/0,c4/0,c5/1,c6/0,c7/0,c8/2,c9/2,div_by_zero/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {DIV,GE,ID_INC,IF,MINUS,div,ge,id_inc,if
            ,minus} and constructors {0,c,c1,c10,c11,c12,c2,c3,c4,c5,c6,c7,c8,c9,div_by_zero,false,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            DIV(z0,z1) -> c8(IF(ge(z1,s(0())),ge(z0,z1),z0,z1),GE(z1,s(0())))
            DIV(z0,z1) -> c9(IF(ge(z1,s(0())),ge(z0,z1),z0,z1),GE(z0,z1))
            GE(z0,0()) -> c()
            GE(0(),s(z0)) -> c1()
            GE(s(z0),s(z1)) -> c2(GE(z0,z1))
            ID_INC(z0) -> c6()
            ID_INC(z0) -> c7()
            IF(false(),z0,z1,z2) -> c10()
            IF(true(),false(),z0,z1) -> c11()
            IF(true(),true(),z0,z1) -> c12(ID_INC(div(minus(z0,z1),z1)),DIV(minus(z0,z1),z1),MINUS(z0,z1))
            MINUS(z0,0()) -> c3()
            MINUS(0(),z0) -> c4()
            MINUS(s(z0),s(z1)) -> c5(MINUS(z0,z1))
        - Weak TRS:
            div(z0,z1) -> if(ge(z1,s(0())),ge(z0,z1),z0,z1)
            ge(z0,0()) -> true()
            ge(0(),s(z0)) -> false()
            ge(s(z0),s(z1)) -> ge(z0,z1)
            id_inc(z0) -> z0
            id_inc(z0) -> s(z0)
            if(false(),z0,z1,z2) -> div_by_zero()
            if(true(),false(),z0,z1) -> 0()
            if(true(),true(),z0,z1) -> id_inc(div(minus(z0,z1),z1))
            minus(z0,0()) -> z0
            minus(0(),z0) -> 0()
            minus(s(z0),s(z1)) -> minus(z0,z1)
        - Signature:
            {DIV/2,GE/2,ID_INC/1,IF/4,MINUS/2,div/2,ge/2,id_inc/1,if/4,minus/2} / {0/0,c/0,c1/0,c10/0,c11/0,c12/3,c2/1
            ,c3/0,c4/0,c5/1,c6/0,c7/0,c8/2,c9/2,div_by_zero/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {DIV,GE,ID_INC,IF,MINUS,div,ge,id_inc,if
            ,minus} and constructors {0,c,c1,c10,c11,c12,c2,c3,c4,c5,c6,c7,c8,c9,div_by_zero,false,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 3: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            DIV(z0,z1) -> c8(IF(ge(z1,s(0())),ge(z0,z1),z0,z1),GE(z1,s(0())))
            DIV(z0,z1) -> c9(IF(ge(z1,s(0())),ge(z0,z1),z0,z1),GE(z0,z1))
            GE(z0,0()) -> c()
            GE(0(),s(z0)) -> c1()
            GE(s(z0),s(z1)) -> c2(GE(z0,z1))
            ID_INC(z0) -> c6()
            ID_INC(z0) -> c7()
            IF(false(),z0,z1,z2) -> c10()
            IF(true(),false(),z0,z1) -> c11()
            IF(true(),true(),z0,z1) -> c12(ID_INC(div(minus(z0,z1),z1)),DIV(minus(z0,z1),z1),MINUS(z0,z1))
            MINUS(z0,0()) -> c3()
            MINUS(0(),z0) -> c4()
            MINUS(s(z0),s(z1)) -> c5(MINUS(z0,z1))
        - Weak TRS:
            div(z0,z1) -> if(ge(z1,s(0())),ge(z0,z1),z0,z1)
            ge(z0,0()) -> true()
            ge(0(),s(z0)) -> false()
            ge(s(z0),s(z1)) -> ge(z0,z1)
            id_inc(z0) -> z0
            id_inc(z0) -> s(z0)
            if(false(),z0,z1,z2) -> div_by_zero()
            if(true(),false(),z0,z1) -> 0()
            if(true(),true(),z0,z1) -> id_inc(div(minus(z0,z1),z1))
            minus(z0,0()) -> z0
            minus(0(),z0) -> 0()
            minus(s(z0),s(z1)) -> minus(z0,z1)
        - Signature:
            {DIV/2,GE/2,ID_INC/1,IF/4,MINUS/2,div/2,ge/2,id_inc/1,if/4,minus/2} / {0/0,c/0,c1/0,c10/0,c11/0,c12/3,c2/1
            ,c3/0,c4/0,c5/1,c6/0,c7/0,c8/2,c9/2,div_by_zero/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {DIV,GE,ID_INC,IF,MINUS,div,ge,id_inc,if
            ,minus} and constructors {0,c,c1,c10,c11,c12,c2,c3,c4,c5,c6,c7,c8,c9,div_by_zero,false,s,true}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          GE(x,y){x -> s(x),y -> s(y)} =
            GE(s(x),s(y)) ->^+ c2(GE(x,y))
              = C[GE(x,y) = GE(x,y){}]

WORST_CASE(Omega(n^1),?)