MAYBE

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict Trs:
  { eq(0(), 0()) -> true()
  , eq(0(), s(x)) -> false()
  , eq(s(x), 0()) -> false()
  , eq(s(x), s(y)) -> eq(x, y)
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , app(nil(), y) -> y
  , app(add(n, x), y) -> add(n, app(x, y))
  , min(add(n, nil())) -> n
  , min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
  , if_min(true(), add(n, add(m, x))) -> min(add(n, x))
  , if_min(false(), add(n, add(m, x))) -> min(add(m, x))
  , head(add(n, x)) -> n
  , tail(nil()) -> nil()
  , tail(add(n, x)) -> x
  , null(nil()) -> true()
  , null(add(n, x)) -> false()
  , rm(n, nil()) -> nil()
  , rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
  , if_rm(true(), n, add(m, x)) -> rm(n, x)
  , if_rm(false(), n, add(m, x)) -> add(m, rm(n, x))
  , minsort(x) -> mins(x, nil(), nil())
  , mins(x, y, z) -> if(null(x), x, y, z)
  , if(true(), x, y, z) -> z
  , if(false(), x, y, z) -> if2(eq(head(x), min(x)), x, y, z)
  , if2(true(), x, y, z) ->
    mins(app(rm(head(x), tail(x)), y),
         nil(),
         app(z, add(head(x), nil())))
  , if2(false(), x, y, z) -> mins(tail(x), add(head(x), y), z) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { eq^#(0(), 0()) -> c_1()
  , eq^#(0(), s(x)) -> c_2()
  , eq^#(s(x), 0()) -> c_3()
  , eq^#(s(x), s(y)) -> c_4(eq^#(x, y))
  , le^#(0(), y) -> c_5()
  , le^#(s(x), 0()) -> c_6()
  , le^#(s(x), s(y)) -> c_7(le^#(x, y))
  , app^#(nil(), y) -> c_8()
  , app^#(add(n, x), y) -> c_9(app^#(x, y))
  , min^#(add(n, nil())) -> c_10()
  , min^#(add(n, add(m, x))) ->
    c_11(if_min^#(le(n, m), add(n, add(m, x))), le^#(n, m))
  , if_min^#(true(), add(n, add(m, x))) -> c_12(min^#(add(n, x)))
  , if_min^#(false(), add(n, add(m, x))) -> c_13(min^#(add(m, x)))
  , head^#(add(n, x)) -> c_14()
  , tail^#(nil()) -> c_15()
  , tail^#(add(n, x)) -> c_16()
  , null^#(nil()) -> c_17()
  , null^#(add(n, x)) -> c_18()
  , rm^#(n, nil()) -> c_19()
  , rm^#(n, add(m, x)) ->
    c_20(if_rm^#(eq(n, m), n, add(m, x)), eq^#(n, m))
  , if_rm^#(true(), n, add(m, x)) -> c_21(rm^#(n, x))
  , if_rm^#(false(), n, add(m, x)) -> c_22(rm^#(n, x))
  , minsort^#(x) -> c_23(mins^#(x, nil(), nil()))
  , mins^#(x, y, z) -> c_24(if^#(null(x), x, y, z), null^#(x))
  , if^#(true(), x, y, z) -> c_25()
  , if^#(false(), x, y, z) ->
    c_26(if2^#(eq(head(x), min(x)), x, y, z),
         eq^#(head(x), min(x)),
         head^#(x),
         min^#(x))
  , if2^#(true(), x, y, z) ->
    c_27(mins^#(app(rm(head(x), tail(x)), y),
                nil(),
                app(z, add(head(x), nil()))),
         app^#(rm(head(x), tail(x)), y),
         rm^#(head(x), tail(x)),
         head^#(x),
         tail^#(x),
         app^#(z, add(head(x), nil())),
         head^#(x))
  , if2^#(false(), x, y, z) ->
    c_28(mins^#(tail(x), add(head(x), y), z), tail^#(x), head^#(x)) }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { eq^#(0(), 0()) -> c_1()
  , eq^#(0(), s(x)) -> c_2()
  , eq^#(s(x), 0()) -> c_3()
  , eq^#(s(x), s(y)) -> c_4(eq^#(x, y))
  , le^#(0(), y) -> c_5()
  , le^#(s(x), 0()) -> c_6()
  , le^#(s(x), s(y)) -> c_7(le^#(x, y))
  , app^#(nil(), y) -> c_8()
  , app^#(add(n, x), y) -> c_9(app^#(x, y))
  , min^#(add(n, nil())) -> c_10()
  , min^#(add(n, add(m, x))) ->
    c_11(if_min^#(le(n, m), add(n, add(m, x))), le^#(n, m))
  , if_min^#(true(), add(n, add(m, x))) -> c_12(min^#(add(n, x)))
  , if_min^#(false(), add(n, add(m, x))) -> c_13(min^#(add(m, x)))
  , head^#(add(n, x)) -> c_14()
  , tail^#(nil()) -> c_15()
  , tail^#(add(n, x)) -> c_16()
  , null^#(nil()) -> c_17()
  , null^#(add(n, x)) -> c_18()
  , rm^#(n, nil()) -> c_19()
  , rm^#(n, add(m, x)) ->
    c_20(if_rm^#(eq(n, m), n, add(m, x)), eq^#(n, m))
  , if_rm^#(true(), n, add(m, x)) -> c_21(rm^#(n, x))
  , if_rm^#(false(), n, add(m, x)) -> c_22(rm^#(n, x))
  , minsort^#(x) -> c_23(mins^#(x, nil(), nil()))
  , mins^#(x, y, z) -> c_24(if^#(null(x), x, y, z), null^#(x))
  , if^#(true(), x, y, z) -> c_25()
  , if^#(false(), x, y, z) ->
    c_26(if2^#(eq(head(x), min(x)), x, y, z),
         eq^#(head(x), min(x)),
         head^#(x),
         min^#(x))
  , if2^#(true(), x, y, z) ->
    c_27(mins^#(app(rm(head(x), tail(x)), y),
                nil(),
                app(z, add(head(x), nil()))),
         app^#(rm(head(x), tail(x)), y),
         rm^#(head(x), tail(x)),
         head^#(x),
         tail^#(x),
         app^#(z, add(head(x), nil())),
         head^#(x))
  , if2^#(false(), x, y, z) ->
    c_28(mins^#(tail(x), add(head(x), y), z), tail^#(x), head^#(x)) }
Weak Trs:
  { eq(0(), 0()) -> true()
  , eq(0(), s(x)) -> false()
  , eq(s(x), 0()) -> false()
  , eq(s(x), s(y)) -> eq(x, y)
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , app(nil(), y) -> y
  , app(add(n, x), y) -> add(n, app(x, y))
  , min(add(n, nil())) -> n
  , min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
  , if_min(true(), add(n, add(m, x))) -> min(add(n, x))
  , if_min(false(), add(n, add(m, x))) -> min(add(m, x))
  , head(add(n, x)) -> n
  , tail(nil()) -> nil()
  , tail(add(n, x)) -> x
  , null(nil()) -> true()
  , null(add(n, x)) -> false()
  , rm(n, nil()) -> nil()
  , rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
  , if_rm(true(), n, add(m, x)) -> rm(n, x)
  , if_rm(false(), n, add(m, x)) -> add(m, rm(n, x))
  , minsort(x) -> mins(x, nil(), nil())
  , mins(x, y, z) -> if(null(x), x, y, z)
  , if(true(), x, y, z) -> z
  , if(false(), x, y, z) -> if2(eq(head(x), min(x)), x, y, z)
  , if2(true(), x, y, z) ->
    mins(app(rm(head(x), tail(x)), y),
         nil(),
         app(z, add(head(x), nil())))
  , if2(false(), x, y, z) -> mins(tail(x), add(head(x), y), z) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of
{1,2,3,5,6,8,10,14,15,16,17,18,19,25} by applications of
Pre({1,2,3,5,6,8,10,14,15,16,17,18,19,25}) =
{4,7,9,11,12,13,20,21,22,24,26,27,28}. Here rules are labeled as
follows:

  DPs:
    { 1: eq^#(0(), 0()) -> c_1()
    , 2: eq^#(0(), s(x)) -> c_2()
    , 3: eq^#(s(x), 0()) -> c_3()
    , 4: eq^#(s(x), s(y)) -> c_4(eq^#(x, y))
    , 5: le^#(0(), y) -> c_5()
    , 6: le^#(s(x), 0()) -> c_6()
    , 7: le^#(s(x), s(y)) -> c_7(le^#(x, y))
    , 8: app^#(nil(), y) -> c_8()
    , 9: app^#(add(n, x), y) -> c_9(app^#(x, y))
    , 10: min^#(add(n, nil())) -> c_10()
    , 11: min^#(add(n, add(m, x))) ->
          c_11(if_min^#(le(n, m), add(n, add(m, x))), le^#(n, m))
    , 12: if_min^#(true(), add(n, add(m, x))) -> c_12(min^#(add(n, x)))
    , 13: if_min^#(false(), add(n, add(m, x))) ->
          c_13(min^#(add(m, x)))
    , 14: head^#(add(n, x)) -> c_14()
    , 15: tail^#(nil()) -> c_15()
    , 16: tail^#(add(n, x)) -> c_16()
    , 17: null^#(nil()) -> c_17()
    , 18: null^#(add(n, x)) -> c_18()
    , 19: rm^#(n, nil()) -> c_19()
    , 20: rm^#(n, add(m, x)) ->
          c_20(if_rm^#(eq(n, m), n, add(m, x)), eq^#(n, m))
    , 21: if_rm^#(true(), n, add(m, x)) -> c_21(rm^#(n, x))
    , 22: if_rm^#(false(), n, add(m, x)) -> c_22(rm^#(n, x))
    , 23: minsort^#(x) -> c_23(mins^#(x, nil(), nil()))
    , 24: mins^#(x, y, z) -> c_24(if^#(null(x), x, y, z), null^#(x))
    , 25: if^#(true(), x, y, z) -> c_25()
    , 26: if^#(false(), x, y, z) ->
          c_26(if2^#(eq(head(x), min(x)), x, y, z),
               eq^#(head(x), min(x)),
               head^#(x),
               min^#(x))
    , 27: if2^#(true(), x, y, z) ->
          c_27(mins^#(app(rm(head(x), tail(x)), y),
                      nil(),
                      app(z, add(head(x), nil()))),
               app^#(rm(head(x), tail(x)), y),
               rm^#(head(x), tail(x)),
               head^#(x),
               tail^#(x),
               app^#(z, add(head(x), nil())),
               head^#(x))
    , 28: if2^#(false(), x, y, z) ->
          c_28(mins^#(tail(x), add(head(x), y), z), tail^#(x), head^#(x)) }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { eq^#(s(x), s(y)) -> c_4(eq^#(x, y))
  , le^#(s(x), s(y)) -> c_7(le^#(x, y))
  , app^#(add(n, x), y) -> c_9(app^#(x, y))
  , min^#(add(n, add(m, x))) ->
    c_11(if_min^#(le(n, m), add(n, add(m, x))), le^#(n, m))
  , if_min^#(true(), add(n, add(m, x))) -> c_12(min^#(add(n, x)))
  , if_min^#(false(), add(n, add(m, x))) -> c_13(min^#(add(m, x)))
  , rm^#(n, add(m, x)) ->
    c_20(if_rm^#(eq(n, m), n, add(m, x)), eq^#(n, m))
  , if_rm^#(true(), n, add(m, x)) -> c_21(rm^#(n, x))
  , if_rm^#(false(), n, add(m, x)) -> c_22(rm^#(n, x))
  , minsort^#(x) -> c_23(mins^#(x, nil(), nil()))
  , mins^#(x, y, z) -> c_24(if^#(null(x), x, y, z), null^#(x))
  , if^#(false(), x, y, z) ->
    c_26(if2^#(eq(head(x), min(x)), x, y, z),
         eq^#(head(x), min(x)),
         head^#(x),
         min^#(x))
  , if2^#(true(), x, y, z) ->
    c_27(mins^#(app(rm(head(x), tail(x)), y),
                nil(),
                app(z, add(head(x), nil()))),
         app^#(rm(head(x), tail(x)), y),
         rm^#(head(x), tail(x)),
         head^#(x),
         tail^#(x),
         app^#(z, add(head(x), nil())),
         head^#(x))
  , if2^#(false(), x, y, z) ->
    c_28(mins^#(tail(x), add(head(x), y), z), tail^#(x), head^#(x)) }
Weak DPs:
  { eq^#(0(), 0()) -> c_1()
  , eq^#(0(), s(x)) -> c_2()
  , eq^#(s(x), 0()) -> c_3()
  , le^#(0(), y) -> c_5()
  , le^#(s(x), 0()) -> c_6()
  , app^#(nil(), y) -> c_8()
  , min^#(add(n, nil())) -> c_10()
  , head^#(add(n, x)) -> c_14()
  , tail^#(nil()) -> c_15()
  , tail^#(add(n, x)) -> c_16()
  , null^#(nil()) -> c_17()
  , null^#(add(n, x)) -> c_18()
  , rm^#(n, nil()) -> c_19()
  , if^#(true(), x, y, z) -> c_25() }
Weak Trs:
  { eq(0(), 0()) -> true()
  , eq(0(), s(x)) -> false()
  , eq(s(x), 0()) -> false()
  , eq(s(x), s(y)) -> eq(x, y)
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , app(nil(), y) -> y
  , app(add(n, x), y) -> add(n, app(x, y))
  , min(add(n, nil())) -> n
  , min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
  , if_min(true(), add(n, add(m, x))) -> min(add(n, x))
  , if_min(false(), add(n, add(m, x))) -> min(add(m, x))
  , head(add(n, x)) -> n
  , tail(nil()) -> nil()
  , tail(add(n, x)) -> x
  , null(nil()) -> true()
  , null(add(n, x)) -> false()
  , rm(n, nil()) -> nil()
  , rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
  , if_rm(true(), n, add(m, x)) -> rm(n, x)
  , if_rm(false(), n, add(m, x)) -> add(m, rm(n, x))
  , minsort(x) -> mins(x, nil(), nil())
  , mins(x, y, z) -> if(null(x), x, y, z)
  , if(true(), x, y, z) -> z
  , if(false(), x, y, z) -> if2(eq(head(x), min(x)), x, y, z)
  , if2(true(), x, y, z) ->
    mins(app(rm(head(x), tail(x)), y),
         nil(),
         app(z, add(head(x), nil())))
  , if2(false(), x, y, z) -> mins(tail(x), add(head(x), y), z) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ eq^#(0(), 0()) -> c_1()
, eq^#(0(), s(x)) -> c_2()
, eq^#(s(x), 0()) -> c_3()
, le^#(0(), y) -> c_5()
, le^#(s(x), 0()) -> c_6()
, app^#(nil(), y) -> c_8()
, min^#(add(n, nil())) -> c_10()
, head^#(add(n, x)) -> c_14()
, tail^#(nil()) -> c_15()
, tail^#(add(n, x)) -> c_16()
, null^#(nil()) -> c_17()
, null^#(add(n, x)) -> c_18()
, rm^#(n, nil()) -> c_19()
, if^#(true(), x, y, z) -> c_25() }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { eq^#(s(x), s(y)) -> c_4(eq^#(x, y))
  , le^#(s(x), s(y)) -> c_7(le^#(x, y))
  , app^#(add(n, x), y) -> c_9(app^#(x, y))
  , min^#(add(n, add(m, x))) ->
    c_11(if_min^#(le(n, m), add(n, add(m, x))), le^#(n, m))
  , if_min^#(true(), add(n, add(m, x))) -> c_12(min^#(add(n, x)))
  , if_min^#(false(), add(n, add(m, x))) -> c_13(min^#(add(m, x)))
  , rm^#(n, add(m, x)) ->
    c_20(if_rm^#(eq(n, m), n, add(m, x)), eq^#(n, m))
  , if_rm^#(true(), n, add(m, x)) -> c_21(rm^#(n, x))
  , if_rm^#(false(), n, add(m, x)) -> c_22(rm^#(n, x))
  , minsort^#(x) -> c_23(mins^#(x, nil(), nil()))
  , mins^#(x, y, z) -> c_24(if^#(null(x), x, y, z), null^#(x))
  , if^#(false(), x, y, z) ->
    c_26(if2^#(eq(head(x), min(x)), x, y, z),
         eq^#(head(x), min(x)),
         head^#(x),
         min^#(x))
  , if2^#(true(), x, y, z) ->
    c_27(mins^#(app(rm(head(x), tail(x)), y),
                nil(),
                app(z, add(head(x), nil()))),
         app^#(rm(head(x), tail(x)), y),
         rm^#(head(x), tail(x)),
         head^#(x),
         tail^#(x),
         app^#(z, add(head(x), nil())),
         head^#(x))
  , if2^#(false(), x, y, z) ->
    c_28(mins^#(tail(x), add(head(x), y), z), tail^#(x), head^#(x)) }
Weak Trs:
  { eq(0(), 0()) -> true()
  , eq(0(), s(x)) -> false()
  , eq(s(x), 0()) -> false()
  , eq(s(x), s(y)) -> eq(x, y)
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , app(nil(), y) -> y
  , app(add(n, x), y) -> add(n, app(x, y))
  , min(add(n, nil())) -> n
  , min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
  , if_min(true(), add(n, add(m, x))) -> min(add(n, x))
  , if_min(false(), add(n, add(m, x))) -> min(add(m, x))
  , head(add(n, x)) -> n
  , tail(nil()) -> nil()
  , tail(add(n, x)) -> x
  , null(nil()) -> true()
  , null(add(n, x)) -> false()
  , rm(n, nil()) -> nil()
  , rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
  , if_rm(true(), n, add(m, x)) -> rm(n, x)
  , if_rm(false(), n, add(m, x)) -> add(m, rm(n, x))
  , minsort(x) -> mins(x, nil(), nil())
  , mins(x, y, z) -> if(null(x), x, y, z)
  , if(true(), x, y, z) -> z
  , if(false(), x, y, z) -> if2(eq(head(x), min(x)), x, y, z)
  , if2(true(), x, y, z) ->
    mins(app(rm(head(x), tail(x)), y),
         nil(),
         app(z, add(head(x), nil())))
  , if2(false(), x, y, z) -> mins(tail(x), add(head(x), y), z) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { mins^#(x, y, z) -> c_24(if^#(null(x), x, y, z), null^#(x))
  , if^#(false(), x, y, z) ->
    c_26(if2^#(eq(head(x), min(x)), x, y, z),
         eq^#(head(x), min(x)),
         head^#(x),
         min^#(x))
  , if2^#(true(), x, y, z) ->
    c_27(mins^#(app(rm(head(x), tail(x)), y),
                nil(),
                app(z, add(head(x), nil()))),
         app^#(rm(head(x), tail(x)), y),
         rm^#(head(x), tail(x)),
         head^#(x),
         tail^#(x),
         app^#(z, add(head(x), nil())),
         head^#(x))
  , if2^#(false(), x, y, z) ->
    c_28(mins^#(tail(x), add(head(x), y), z), tail^#(x), head^#(x)) }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { eq^#(s(x), s(y)) -> c_1(eq^#(x, y))
  , le^#(s(x), s(y)) -> c_2(le^#(x, y))
  , app^#(add(n, x), y) -> c_3(app^#(x, y))
  , min^#(add(n, add(m, x))) ->
    c_4(if_min^#(le(n, m), add(n, add(m, x))), le^#(n, m))
  , if_min^#(true(), add(n, add(m, x))) -> c_5(min^#(add(n, x)))
  , if_min^#(false(), add(n, add(m, x))) -> c_6(min^#(add(m, x)))
  , rm^#(n, add(m, x)) ->
    c_7(if_rm^#(eq(n, m), n, add(m, x)), eq^#(n, m))
  , if_rm^#(true(), n, add(m, x)) -> c_8(rm^#(n, x))
  , if_rm^#(false(), n, add(m, x)) -> c_9(rm^#(n, x))
  , minsort^#(x) -> c_10(mins^#(x, nil(), nil()))
  , mins^#(x, y, z) -> c_11(if^#(null(x), x, y, z))
  , if^#(false(), x, y, z) ->
    c_12(if2^#(eq(head(x), min(x)), x, y, z),
         eq^#(head(x), min(x)),
         min^#(x))
  , if2^#(true(), x, y, z) ->
    c_13(mins^#(app(rm(head(x), tail(x)), y),
                nil(),
                app(z, add(head(x), nil()))),
         app^#(rm(head(x), tail(x)), y),
         rm^#(head(x), tail(x)),
         app^#(z, add(head(x), nil())))
  , if2^#(false(), x, y, z) ->
    c_14(mins^#(tail(x), add(head(x), y), z)) }
Weak Trs:
  { eq(0(), 0()) -> true()
  , eq(0(), s(x)) -> false()
  , eq(s(x), 0()) -> false()
  , eq(s(x), s(y)) -> eq(x, y)
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , app(nil(), y) -> y
  , app(add(n, x), y) -> add(n, app(x, y))
  , min(add(n, nil())) -> n
  , min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
  , if_min(true(), add(n, add(m, x))) -> min(add(n, x))
  , if_min(false(), add(n, add(m, x))) -> min(add(m, x))
  , head(add(n, x)) -> n
  , tail(nil()) -> nil()
  , tail(add(n, x)) -> x
  , null(nil()) -> true()
  , null(add(n, x)) -> false()
  , rm(n, nil()) -> nil()
  , rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
  , if_rm(true(), n, add(m, x)) -> rm(n, x)
  , if_rm(false(), n, add(m, x)) -> add(m, rm(n, x))
  , minsort(x) -> mins(x, nil(), nil())
  , mins(x, y, z) -> if(null(x), x, y, z)
  , if(true(), x, y, z) -> z
  , if(false(), x, y, z) -> if2(eq(head(x), min(x)), x, y, z)
  , if2(true(), x, y, z) ->
    mins(app(rm(head(x), tail(x)), y),
         nil(),
         app(z, add(head(x), nil())))
  , if2(false(), x, y, z) -> mins(tail(x), add(head(x), y), z) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { eq(0(), 0()) -> true()
    , eq(0(), s(x)) -> false()
    , eq(s(x), 0()) -> false()
    , eq(s(x), s(y)) -> eq(x, y)
    , le(0(), y) -> true()
    , le(s(x), 0()) -> false()
    , le(s(x), s(y)) -> le(x, y)
    , app(nil(), y) -> y
    , app(add(n, x), y) -> add(n, app(x, y))
    , min(add(n, nil())) -> n
    , min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
    , if_min(true(), add(n, add(m, x))) -> min(add(n, x))
    , if_min(false(), add(n, add(m, x))) -> min(add(m, x))
    , head(add(n, x)) -> n
    , tail(nil()) -> nil()
    , tail(add(n, x)) -> x
    , null(nil()) -> true()
    , null(add(n, x)) -> false()
    , rm(n, nil()) -> nil()
    , rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
    , if_rm(true(), n, add(m, x)) -> rm(n, x)
    , if_rm(false(), n, add(m, x)) -> add(m, rm(n, x)) }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { eq^#(s(x), s(y)) -> c_1(eq^#(x, y))
  , le^#(s(x), s(y)) -> c_2(le^#(x, y))
  , app^#(add(n, x), y) -> c_3(app^#(x, y))
  , min^#(add(n, add(m, x))) ->
    c_4(if_min^#(le(n, m), add(n, add(m, x))), le^#(n, m))
  , if_min^#(true(), add(n, add(m, x))) -> c_5(min^#(add(n, x)))
  , if_min^#(false(), add(n, add(m, x))) -> c_6(min^#(add(m, x)))
  , rm^#(n, add(m, x)) ->
    c_7(if_rm^#(eq(n, m), n, add(m, x)), eq^#(n, m))
  , if_rm^#(true(), n, add(m, x)) -> c_8(rm^#(n, x))
  , if_rm^#(false(), n, add(m, x)) -> c_9(rm^#(n, x))
  , minsort^#(x) -> c_10(mins^#(x, nil(), nil()))
  , mins^#(x, y, z) -> c_11(if^#(null(x), x, y, z))
  , if^#(false(), x, y, z) ->
    c_12(if2^#(eq(head(x), min(x)), x, y, z),
         eq^#(head(x), min(x)),
         min^#(x))
  , if2^#(true(), x, y, z) ->
    c_13(mins^#(app(rm(head(x), tail(x)), y),
                nil(),
                app(z, add(head(x), nil()))),
         app^#(rm(head(x), tail(x)), y),
         rm^#(head(x), tail(x)),
         app^#(z, add(head(x), nil())))
  , if2^#(false(), x, y, z) ->
    c_14(mins^#(tail(x), add(head(x), y), z)) }
Weak Trs:
  { eq(0(), 0()) -> true()
  , eq(0(), s(x)) -> false()
  , eq(s(x), 0()) -> false()
  , eq(s(x), s(y)) -> eq(x, y)
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , app(nil(), y) -> y
  , app(add(n, x), y) -> add(n, app(x, y))
  , min(add(n, nil())) -> n
  , min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
  , if_min(true(), add(n, add(m, x))) -> min(add(n, x))
  , if_min(false(), add(n, add(m, x))) -> min(add(m, x))
  , head(add(n, x)) -> n
  , tail(nil()) -> nil()
  , tail(add(n, x)) -> x
  , null(nil()) -> true()
  , null(add(n, x)) -> false()
  , rm(n, nil()) -> nil()
  , rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
  , if_rm(true(), n, add(m, x)) -> rm(n, x)
  , if_rm(false(), n, add(m, x)) -> add(m, rm(n, x)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Consider the dependency graph

  1: eq^#(s(x), s(y)) -> c_1(eq^#(x, y))
     -->_1 eq^#(s(x), s(y)) -> c_1(eq^#(x, y)) :1
  
  2: le^#(s(x), s(y)) -> c_2(le^#(x, y))
     -->_1 le^#(s(x), s(y)) -> c_2(le^#(x, y)) :2
  
  3: app^#(add(n, x), y) -> c_3(app^#(x, y))
     -->_1 app^#(add(n, x), y) -> c_3(app^#(x, y)) :3
  
  4: min^#(add(n, add(m, x))) ->
     c_4(if_min^#(le(n, m), add(n, add(m, x))), le^#(n, m))
     -->_1 if_min^#(false(), add(n, add(m, x))) ->
           c_6(min^#(add(m, x))) :6
     -->_1 if_min^#(true(), add(n, add(m, x))) ->
           c_5(min^#(add(n, x))) :5
     -->_2 le^#(s(x), s(y)) -> c_2(le^#(x, y)) :2
  
  5: if_min^#(true(), add(n, add(m, x))) -> c_5(min^#(add(n, x)))
     -->_1 min^#(add(n, add(m, x))) ->
           c_4(if_min^#(le(n, m), add(n, add(m, x))), le^#(n, m)) :4
  
  6: if_min^#(false(), add(n, add(m, x))) -> c_6(min^#(add(m, x)))
     -->_1 min^#(add(n, add(m, x))) ->
           c_4(if_min^#(le(n, m), add(n, add(m, x))), le^#(n, m)) :4
  
  7: rm^#(n, add(m, x)) ->
     c_7(if_rm^#(eq(n, m), n, add(m, x)), eq^#(n, m))
     -->_1 if_rm^#(false(), n, add(m, x)) -> c_9(rm^#(n, x)) :9
     -->_1 if_rm^#(true(), n, add(m, x)) -> c_8(rm^#(n, x)) :8
     -->_2 eq^#(s(x), s(y)) -> c_1(eq^#(x, y)) :1
  
  8: if_rm^#(true(), n, add(m, x)) -> c_8(rm^#(n, x))
     -->_1 rm^#(n, add(m, x)) ->
           c_7(if_rm^#(eq(n, m), n, add(m, x)), eq^#(n, m)) :7
  
  9: if_rm^#(false(), n, add(m, x)) -> c_9(rm^#(n, x))
     -->_1 rm^#(n, add(m, x)) ->
           c_7(if_rm^#(eq(n, m), n, add(m, x)), eq^#(n, m)) :7
  
  10: minsort^#(x) -> c_10(mins^#(x, nil(), nil()))
     -->_1 mins^#(x, y, z) -> c_11(if^#(null(x), x, y, z)) :11
  
  11: mins^#(x, y, z) -> c_11(if^#(null(x), x, y, z))
     -->_1 if^#(false(), x, y, z) ->
           c_12(if2^#(eq(head(x), min(x)), x, y, z),
                eq^#(head(x), min(x)),
                min^#(x)) :12
  
  12: if^#(false(), x, y, z) ->
      c_12(if2^#(eq(head(x), min(x)), x, y, z),
           eq^#(head(x), min(x)),
           min^#(x))
     -->_1 if2^#(false(), x, y, z) ->
           c_14(mins^#(tail(x), add(head(x), y), z)) :14
     -->_1 if2^#(true(), x, y, z) ->
           c_13(mins^#(app(rm(head(x), tail(x)), y),
                       nil(),
                       app(z, add(head(x), nil()))),
                app^#(rm(head(x), tail(x)), y),
                rm^#(head(x), tail(x)),
                app^#(z, add(head(x), nil()))) :13
     -->_3 min^#(add(n, add(m, x))) ->
           c_4(if_min^#(le(n, m), add(n, add(m, x))), le^#(n, m)) :4
     -->_2 eq^#(s(x), s(y)) -> c_1(eq^#(x, y)) :1
  
  13: if2^#(true(), x, y, z) ->
      c_13(mins^#(app(rm(head(x), tail(x)), y),
                  nil(),
                  app(z, add(head(x), nil()))),
           app^#(rm(head(x), tail(x)), y),
           rm^#(head(x), tail(x)),
           app^#(z, add(head(x), nil())))
     -->_1 mins^#(x, y, z) -> c_11(if^#(null(x), x, y, z)) :11
     -->_3 rm^#(n, add(m, x)) ->
           c_7(if_rm^#(eq(n, m), n, add(m, x)), eq^#(n, m)) :7
     -->_4 app^#(add(n, x), y) -> c_3(app^#(x, y)) :3
     -->_2 app^#(add(n, x), y) -> c_3(app^#(x, y)) :3
  
  14: if2^#(false(), x, y, z) ->
      c_14(mins^#(tail(x), add(head(x), y), z))
     -->_1 mins^#(x, y, z) -> c_11(if^#(null(x), x, y, z)) :11
  

Following roots of the dependency graph are removed, as the
considered set of starting terms is closed under reduction with
respect to these rules (modulo compound contexts).

  { minsort^#(x) -> c_10(mins^#(x, nil(), nil())) }


We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { eq^#(s(x), s(y)) -> c_1(eq^#(x, y))
  , le^#(s(x), s(y)) -> c_2(le^#(x, y))
  , app^#(add(n, x), y) -> c_3(app^#(x, y))
  , min^#(add(n, add(m, x))) ->
    c_4(if_min^#(le(n, m), add(n, add(m, x))), le^#(n, m))
  , if_min^#(true(), add(n, add(m, x))) -> c_5(min^#(add(n, x)))
  , if_min^#(false(), add(n, add(m, x))) -> c_6(min^#(add(m, x)))
  , rm^#(n, add(m, x)) ->
    c_7(if_rm^#(eq(n, m), n, add(m, x)), eq^#(n, m))
  , if_rm^#(true(), n, add(m, x)) -> c_8(rm^#(n, x))
  , if_rm^#(false(), n, add(m, x)) -> c_9(rm^#(n, x))
  , mins^#(x, y, z) -> c_11(if^#(null(x), x, y, z))
  , if^#(false(), x, y, z) ->
    c_12(if2^#(eq(head(x), min(x)), x, y, z),
         eq^#(head(x), min(x)),
         min^#(x))
  , if2^#(true(), x, y, z) ->
    c_13(mins^#(app(rm(head(x), tail(x)), y),
                nil(),
                app(z, add(head(x), nil()))),
         app^#(rm(head(x), tail(x)), y),
         rm^#(head(x), tail(x)),
         app^#(z, add(head(x), nil())))
  , if2^#(false(), x, y, z) ->
    c_14(mins^#(tail(x), add(head(x), y), z)) }
Weak Trs:
  { eq(0(), 0()) -> true()
  , eq(0(), s(x)) -> false()
  , eq(s(x), 0()) -> false()
  , eq(s(x), s(y)) -> eq(x, y)
  , le(0(), y) -> true()
  , le(s(x), 0()) -> false()
  , le(s(x), s(y)) -> le(x, y)
  , app(nil(), y) -> y
  , app(add(n, x), y) -> add(n, app(x, y))
  , min(add(n, nil())) -> n
  , min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x)))
  , if_min(true(), add(n, add(m, x))) -> min(add(n, x))
  , if_min(false(), add(n, add(m, x))) -> min(add(m, x))
  , head(add(n, x)) -> n
  , tail(nil()) -> nil()
  , tail(add(n, x)) -> x
  , null(nil()) -> true()
  , null(add(n, x)) -> false()
  , rm(n, nil()) -> nil()
  , rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x))
  , if_rm(true(), n, add(m, x)) -> rm(n, x)
  , if_rm(false(), n, add(m, x)) -> add(m, rm(n, x)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..