/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(Omega(n^1), O(n^3))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 4 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 11 ms]
    (12) CpxRNTS
    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) IntTrsBoundProof [UPPER BOUND(ID), 189 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 58 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 373 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 146 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 463 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 186 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 180 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 56 ms]
    (38) CpxRNTS
    (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 200 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 32 ms]
    (44) CpxRNTS
    (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 970 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 376 ms]
    (50) CpxRNTS
    (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (52) CpxRNTS
    (53) IntTrsBoundProof [UPPER BOUND(ID), 1795 ms]
    (54) CpxRNTS
    (55) IntTrsBoundProof [UPPER BOUND(ID), 573 ms]
    (56) CpxRNTS
    (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (58) CpxRNTS
    (59) IntTrsBoundProof [UPPER BOUND(ID), 4624 ms]
    (60) CpxRNTS
    (61) IntTrsBoundProof [UPPER BOUND(ID), 1370 ms]
    (62) CpxRNTS
    (63) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (64) CpxRNTS
    (65) IntTrsBoundProof [UPPER BOUND(ID), 546 ms]
    (66) CpxRNTS
    (67) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (68) CpxRNTS
    (69) FinalProof [FINISHED, 0 ms]
    (70) BOUNDS(1, n^3)
(71) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(72) TRS for Loop Detection
    (73) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (74) BEST
        (75) proven lower bound
            (76) LowerBoundPropagationProof [FINISHED, 0 ms]
            (77) BOUNDS(n^1, INF)
        (78) TRS for Loop Detection


----------------------------------------

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   eq(0, 0) -> true
   eq(0, s(m)) -> false
   eq(s(n), 0) -> false
   eq(s(n), s(m)) -> eq(n, m)
   le(0, m) -> true
   le(s(n), 0) -> false
   le(s(n), s(m)) -> le(n, m)
   min(cons(x, nil)) -> x
   min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x)))
   if_min(true, cons(n, cons(m, x))) -> min(cons(n, x))
   if_min(false, cons(n, cons(m, x))) -> min(cons(m, x))
   replace(n, m, nil) -> nil
   replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x))
   if_replace(true, n, m, cons(k, x)) -> cons(m, x)
   if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x))
   empty(nil) -> true
   empty(cons(n, x)) -> false
   head(cons(n, x)) -> n
   tail(nil) -> nil
   tail(cons(n, x)) -> x
   sort(x) -> sortIter(x, nil)
   sortIter(x, y) -> if(empty(x), x, y, append(y, cons(min(x), nil)))
   if(true, x, y, z) -> y
   if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z)

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   eq(0, 0) -> true [1]
   eq(0, s(m)) -> false [1]
   eq(s(n), 0) -> false [1]
   eq(s(n), s(m)) -> eq(n, m) [1]
   le(0, m) -> true [1]
   le(s(n), 0) -> false [1]
   le(s(n), s(m)) -> le(n, m) [1]
   min(cons(x, nil)) -> x [1]
   min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x))) [1]
   if_min(true, cons(n, cons(m, x))) -> min(cons(n, x)) [1]
   if_min(false, cons(n, cons(m, x))) -> min(cons(m, x)) [1]
   replace(n, m, nil) -> nil [1]
   replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x)) [1]
   if_replace(true, n, m, cons(k, x)) -> cons(m, x) [1]
   if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x)) [1]
   empty(nil) -> true [1]
   empty(cons(n, x)) -> false [1]
   head(cons(n, x)) -> n [1]
   tail(nil) -> nil [1]
   tail(cons(n, x)) -> x [1]
   sort(x) -> sortIter(x, nil) [1]
   sortIter(x, y) -> if(empty(x), x, y, append(y, cons(min(x), nil))) [1]
   if(true, x, y, z) -> y [1]
   if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z) [1]

Rewrite Strategy: INNERMOST
----------------------------------------

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   eq(0, 0) -> true [1]
   eq(0, s(m)) -> false [1]
   eq(s(n), 0) -> false [1]
   eq(s(n), s(m)) -> eq(n, m) [1]
   le(0, m) -> true [1]
   le(s(n), 0) -> false [1]
   le(s(n), s(m)) -> le(n, m) [1]
   min(cons(x, nil)) -> x [1]
   min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x))) [1]
   if_min(true, cons(n, cons(m, x))) -> min(cons(n, x)) [1]
   if_min(false, cons(n, cons(m, x))) -> min(cons(m, x)) [1]
   replace(n, m, nil) -> nil [1]
   replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x)) [1]
   if_replace(true, n, m, cons(k, x)) -> cons(m, x) [1]
   if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x)) [1]
   empty(nil) -> true [1]
   empty(cons(n, x)) -> false [1]
   head(cons(n, x)) -> n [1]
   tail(nil) -> nil [1]
   tail(cons(n, x)) -> x [1]
   sort(x) -> sortIter(x, nil) [1]
   sortIter(x, y) -> if(empty(x), x, y, append(y, cons(min(x), nil))) [1]
   if(true, x, y, z) -> y [1]
   if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z) [1]

The TRS has the following type information:
eq :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
le :: 0:s -> 0:s -> true:false
min :: nil:cons:append -> 0:s
cons :: 0:s -> nil:cons:append -> nil:cons:append
nil :: nil:cons:append
if_min :: true:false -> nil:cons:append -> 0:s
replace :: 0:s -> 0:s -> nil:cons:append -> nil:cons:append
if_replace :: true:false -> 0:s -> 0:s -> nil:cons:append -> nil:cons:append
empty :: nil:cons:append -> true:false
head :: nil:cons:append -> 0:s
tail :: nil:cons:append -> nil:cons:append
sort :: nil:cons:append -> nil:cons:append
sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append
if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append
append :: nil:cons:append -> nil:cons:append -> nil:cons:append

Rewrite Strategy: INNERMOST
----------------------------------------

(5) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   sort_1
   sortIter_2
   if_4

(c) The following functions are completely defined:

   empty_1
   min_1
   replace_3
   head_1
   tail_1
   eq_2
   le_2
   if_replace_4
   if_min_2

Due to the following rules being added:

   empty(v0) -> null_empty [0]
   min(v0) -> 0 [0]
   replace(v0, v1, v2) -> nil [0]
   head(v0) -> 0 [0]
   tail(v0) -> nil [0]
   if_replace(v0, v1, v2, v3) -> nil [0]
   if_min(v0, v1) -> 0 [0]

And the following fresh constants:    null_empty

----------------------------------------

(6)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   eq(0, 0) -> true [1]
   eq(0, s(m)) -> false [1]
   eq(s(n), 0) -> false [1]
   eq(s(n), s(m)) -> eq(n, m) [1]
   le(0, m) -> true [1]
   le(s(n), 0) -> false [1]
   le(s(n), s(m)) -> le(n, m) [1]
   min(cons(x, nil)) -> x [1]
   min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x))) [1]
   if_min(true, cons(n, cons(m, x))) -> min(cons(n, x)) [1]
   if_min(false, cons(n, cons(m, x))) -> min(cons(m, x)) [1]
   replace(n, m, nil) -> nil [1]
   replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x)) [1]
   if_replace(true, n, m, cons(k, x)) -> cons(m, x) [1]
   if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x)) [1]
   empty(nil) -> true [1]
   empty(cons(n, x)) -> false [1]
   head(cons(n, x)) -> n [1]
   tail(nil) -> nil [1]
   tail(cons(n, x)) -> x [1]
   sort(x) -> sortIter(x, nil) [1]
   sortIter(x, y) -> if(empty(x), x, y, append(y, cons(min(x), nil))) [1]
   if(true, x, y, z) -> y [1]
   if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z) [1]
   empty(v0) -> null_empty [0]
   min(v0) -> 0 [0]
   replace(v0, v1, v2) -> nil [0]
   head(v0) -> 0 [0]
   tail(v0) -> nil [0]
   if_replace(v0, v1, v2, v3) -> nil [0]
   if_min(v0, v1) -> 0 [0]

The TRS has the following type information:
eq :: 0:s -> 0:s -> true:false:null_empty
0 :: 0:s
true :: true:false:null_empty
s :: 0:s -> 0:s
false :: true:false:null_empty
le :: 0:s -> 0:s -> true:false:null_empty
min :: nil:cons:append -> 0:s
cons :: 0:s -> nil:cons:append -> nil:cons:append
nil :: nil:cons:append
if_min :: true:false:null_empty -> nil:cons:append -> 0:s
replace :: 0:s -> 0:s -> nil:cons:append -> nil:cons:append
if_replace :: true:false:null_empty -> 0:s -> 0:s -> nil:cons:append -> nil:cons:append
empty :: nil:cons:append -> true:false:null_empty
head :: nil:cons:append -> 0:s
tail :: nil:cons:append -> nil:cons:append
sort :: nil:cons:append -> nil:cons:append
sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append
if :: true:false:null_empty -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append
append :: nil:cons:append -> nil:cons:append -> nil:cons:append
null_empty :: true:false:null_empty

Rewrite Strategy: INNERMOST
----------------------------------------

(7) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(8)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   eq(0, 0) -> true [1]
   eq(0, s(m)) -> false [1]
   eq(s(n), 0) -> false [1]
   eq(s(n), s(m)) -> eq(n, m) [1]
   le(0, m) -> true [1]
   le(s(n), 0) -> false [1]
   le(s(n), s(m)) -> le(n, m) [1]
   min(cons(x, nil)) -> x [1]
   min(cons(0, cons(m, x))) -> if_min(true, cons(0, cons(m, x))) [2]
   min(cons(s(n'), cons(0, x))) -> if_min(false, cons(s(n'), cons(0, x))) [2]
   min(cons(s(n''), cons(s(m'), x))) -> if_min(le(n'', m'), cons(s(n''), cons(s(m'), x))) [2]
   if_min(true, cons(n, cons(m, x))) -> min(cons(n, x)) [1]
   if_min(false, cons(n, cons(m, x))) -> min(cons(m, x)) [1]
   replace(n, m, nil) -> nil [1]
   replace(0, m, cons(0, x)) -> if_replace(true, 0, m, cons(0, x)) [2]
   replace(0, m, cons(s(m''), x)) -> if_replace(false, 0, m, cons(s(m''), x)) [2]
   replace(s(n1), m, cons(0, x)) -> if_replace(false, s(n1), m, cons(0, x)) [2]
   replace(s(n2), m, cons(s(m1), x)) -> if_replace(eq(n2, m1), s(n2), m, cons(s(m1), x)) [2]
   if_replace(true, n, m, cons(k, x)) -> cons(m, x) [1]
   if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x)) [1]
   empty(nil) -> true [1]
   empty(cons(n, x)) -> false [1]
   head(cons(n, x)) -> n [1]
   tail(nil) -> nil [1]
   tail(cons(n, x)) -> x [1]
   sort(x) -> sortIter(x, nil) [1]
   sortIter(nil, y) -> if(true, nil, y, append(y, cons(0, nil))) [2]
   sortIter(cons(n3, nil), y) -> if(false, cons(n3, nil), y, append(y, cons(n3, nil))) [3]
   sortIter(cons(n3, cons(m2, x'')), y) -> if(false, cons(n3, cons(m2, x'')), y, append(y, cons(if_min(le(n3, m2), cons(n3, cons(m2, x''))), nil))) [3]
   sortIter(cons(n3, x'), y) -> if(false, cons(n3, x'), y, append(y, cons(0, nil))) [2]
   sortIter(cons(x1, nil), y) -> if(null_empty, cons(x1, nil), y, append(y, cons(x1, nil))) [2]
   sortIter(cons(n4, cons(m3, x2)), y) -> if(null_empty, cons(n4, cons(m3, x2)), y, append(y, cons(if_min(le(n4, m3), cons(n4, cons(m3, x2))), nil))) [2]
   sortIter(x, y) -> if(null_empty, x, y, append(y, cons(0, nil))) [1]
   if(true, x, y, z) -> y [1]
   if(false, cons(x3, nil), y, z) -> sortIter(replace(x3, x3, nil), z) [4]
   if(false, cons(x3, nil), y, z) -> sortIter(replace(x3, x3, nil), z) [3]
   if(false, cons(x3, nil), y, z) -> sortIter(replace(x3, 0, nil), z) [3]
   if(false, cons(x3, nil), y, z) -> sortIter(replace(x3, 0, nil), z) [2]
   if(false, cons(n5, cons(m4, x4)), y, z) -> sortIter(replace(if_min(le(n5, m4), cons(n5, cons(m4, x4))), n5, cons(m4, x4)), z) [4]
   if(false, cons(n5, cons(m4, x4)), y, z) -> sortIter(replace(if_min(le(n5, m4), cons(n5, cons(m4, x4))), n5, nil), z) [3]
   if(false, cons(n5, cons(m4, x4)), y, z) -> sortIter(replace(if_min(le(n5, m4), cons(n5, cons(m4, x4))), 0, cons(m4, x4)), z) [3]
   if(false, cons(n5, cons(m4, x4)), y, z) -> sortIter(replace(if_min(le(n5, m4), cons(n5, cons(m4, x4))), 0, nil), z) [2]
   if(false, cons(n6, x5), y, z) -> sortIter(replace(0, n6, x5), z) [3]
   if(false, cons(n6, x5), y, z) -> sortIter(replace(0, n6, nil), z) [2]
   if(false, nil, y, z) -> sortIter(replace(0, 0, nil), z) [2]
   if(false, cons(n7, x6), y, z) -> sortIter(replace(0, 0, x6), z) [2]
   if(false, x, y, z) -> sortIter(replace(0, 0, nil), z) [1]
   empty(v0) -> null_empty [0]
   min(v0) -> 0 [0]
   replace(v0, v1, v2) -> nil [0]
   head(v0) -> 0 [0]
   tail(v0) -> nil [0]
   if_replace(v0, v1, v2, v3) -> nil [0]
   if_min(v0, v1) -> 0 [0]

The TRS has the following type information:
eq :: 0:s -> 0:s -> true:false:null_empty
0 :: 0:s
true :: true:false:null_empty
s :: 0:s -> 0:s
false :: true:false:null_empty
le :: 0:s -> 0:s -> true:false:null_empty
min :: nil:cons:append -> 0:s
cons :: 0:s -> nil:cons:append -> nil:cons:append
nil :: nil:cons:append
if_min :: true:false:null_empty -> nil:cons:append -> 0:s
replace :: 0:s -> 0:s -> nil:cons:append -> nil:cons:append
if_replace :: true:false:null_empty -> 0:s -> 0:s -> nil:cons:append -> nil:cons:append
empty :: nil:cons:append -> true:false:null_empty
head :: nil:cons:append -> 0:s
tail :: nil:cons:append -> nil:cons:append
sort :: nil:cons:append -> nil:cons:append
sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append
if :: true:false:null_empty -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append
append :: nil:cons:append -> nil:cons:append -> nil:cons:append
null_empty :: true:false:null_empty

Rewrite Strategy: INNERMOST
----------------------------------------

(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   0 => 0
   true => 2
   false => 1
   nil => 0
   null_empty => 0

----------------------------------------

(10)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   eq(z', z'') -{ 1 }-> eq(n, m) :|: n >= 0, z'' = 1 + m, z' = 1 + n, m >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 1 + m, z' = 0, m >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, n >= 0, z' = 1 + n
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   if(z', z'', z1, z2) -{ 1 }-> y :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(x3, x3, 0), z) :|: z1 = y, z >= 0, z2 = z, y >= 0, z' = 1, z'' = 1 + x3 + 0, x3 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(x3, x3, 0), z) :|: z1 = y, z >= 0, z2 = z, y >= 0, z' = 1, z'' = 1 + x3 + 0, x3 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(x3, 0, 0), z) :|: z1 = y, z >= 0, z2 = z, y >= 0, z' = 1, z'' = 1 + x3 + 0, x3 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(x3, 0, 0), z) :|: z1 = y, z >= 0, z2 = z, y >= 0, z' = 1, z'' = 1 + x3 + 0, x3 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 0), z) :|: z1 = y, x4 >= 0, z >= 0, z2 = z, y >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z) :|: z1 = y, x4 >= 0, z >= 0, z2 = z, y >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 0), z) :|: z1 = y, x4 >= 0, z >= 0, z2 = z, y >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z) :|: z1 = y, x4 >= 0, z >= 0, z2 = z, y >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z) :|: z1 = y, x5 >= 0, z >= 0, z2 = z, n6 >= 0, y >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z) :|: z1 = y, x5 >= 0, z >= 0, z2 = z, n6 >= 0, y >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z) :|: z1 = y, z >= 0, z'' = 1 + n7 + x6, z2 = z, n7 >= 0, y >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z) :|: z'' = 0, z1 = y, z >= 0, z2 = z, y >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1
   if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(n, m, x) :|: n >= 0, z'' = n, z1 = m, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, m >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + m + x :|: n >= 0, z'' = n, z' = 2, z1 = m, z2 = 1 + k + x, x >= 0, k >= 0, m >= 0
   le(z', z'') -{ 1 }-> le(n, m) :|: n >= 0, z'' = 1 + m, z' = 1 + n, m >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' = m, z' = 0, m >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, n >= 0, z' = 1 + n
   min(z') -{ 1 }-> x :|: x >= 0, z' = 1 + x + 0
   min(z') -{ 2 }-> if_min(le(n'', m'), 1 + (1 + n'') + (1 + (1 + m') + x)) :|: z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   replace(z', z'', z1) -{ 2 }-> if_replace(eq(n2, m1), 1 + n2, m, 1 + (1 + m1) + x) :|: x >= 0, n2 >= 0, m1 >= 0, z'' = m, z' = 1 + n2, z1 = 1 + (1 + m1) + x, m >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, m, 1 + 0 + x) :|: z1 = 1 + 0 + x, x >= 0, z'' = m, z' = 0, m >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, m, 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z'' = m, z' = 0, m >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + n1, m, 1 + 0 + x) :|: z1 = 1 + 0 + x, x >= 0, n1 >= 0, z' = 1 + n1, z'' = m, m >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: n >= 0, z1 = 0, z' = n, z'' = m, m >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   sort(z') -{ 1 }-> sortIter(x, 0) :|: z' = x, x >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, y, 1 + y + (1 + 0 + 0)) :|: z'' = y, y >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', y, 1 + y + (1 + 0 + 0)) :|: z' = 1 + n3 + x', z'' = y, x' >= 0, y >= 0, n3 >= 0
   sortIter(z', z'') -{ 3 }-> if(1, 1 + n3 + 0, y, 1 + y + (1 + n3 + 0)) :|: z'' = y, y >= 0, z' = 1 + n3 + 0, n3 >= 0
   sortIter(z', z'') -{ 3 }-> if(1, 1 + n3 + (1 + m2 + x''), y, 1 + y + (1 + if_min(le(n3, m2), 1 + n3 + (1 + m2 + x'')) + 0)) :|: z'' = y, y >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 1 }-> if(0, x, y, 1 + y + (1 + 0 + 0)) :|: z' = x, z'' = y, x >= 0, y >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + n4 + (1 + m3 + x2), y, 1 + y + (1 + if_min(le(n4, m3), 1 + n4 + (1 + m3 + x2)) + 0)) :|: n4 >= 0, z'' = y, y >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + x1 + 0, y, 1 + y + (1 + x1 + 0)) :|: x1 >= 0, z'' = y, y >= 0, z' = 1 + x1 + 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0


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(11) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 2 }-> if_min(le(n'', m'), 1 + (1 + n'') + (1 + (1 + m') + x)) :|: z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(eq(z' - 1, m1), 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 3 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(le(n3, m2), 1 + n3 + (1 + m2 + x'')) + 0)) :|: z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(le(n4, m3), 1 + n4 + (1 + m3 + x2)) + 0)) :|: n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0


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(13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { empty }
   { le }
   { eq }
   { tail }
   { head }
   { min, if_min }
   { replace, if_replace }
   { sortIter, if }
   { sort }

----------------------------------------

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 2 }-> if_min(le(n'', m'), 1 + (1 + n'') + (1 + (1 + m') + x)) :|: z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(eq(z' - 1, m1), 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 3 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(le(n3, m2), 1 + n3 + (1 + m2 + x'')) + 0)) :|: z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(le(n4, m3), 1 + n4 + (1 + m3 + x2)) + 0)) :|: n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {empty}, {le}, {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort}

----------------------------------------

(15) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 2 }-> if_min(le(n'', m'), 1 + (1 + n'') + (1 + (1 + m') + x)) :|: z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(eq(z' - 1, m1), 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 3 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(le(n3, m2), 1 + n3 + (1 + m2 + x'')) + 0)) :|: z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(le(n4, m3), 1 + n4 + (1 + m3 + x2)) + 0)) :|: n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {empty}, {le}, {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort}

----------------------------------------

(17) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: empty
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

----------------------------------------

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 2 }-> if_min(le(n'', m'), 1 + (1 + n'') + (1 + (1 + m') + x)) :|: z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(eq(z' - 1, m1), 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 3 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(le(n3, m2), 1 + n3 + (1 + m2 + x'')) + 0)) :|: z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(le(n4, m3), 1 + n4 + (1 + m3 + x2)) + 0)) :|: n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {empty}, {le}, {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort}
Previous analysis results are:
  empty: runtime: ?, size: O(1) [2]

----------------------------------------

(19) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: empty
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 2 }-> if_min(le(n'', m'), 1 + (1 + n'') + (1 + (1 + m') + x)) :|: z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(eq(z' - 1, m1), 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 3 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(le(n3, m2), 1 + n3 + (1 + m2 + x'')) + 0)) :|: z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(le(n4, m3), 1 + n4 + (1 + m3 + x2)) + 0)) :|: n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {le}, {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]

----------------------------------------

(21) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 2 }-> if_min(le(n'', m'), 1 + (1 + n'') + (1 + (1 + m') + x)) :|: z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(eq(z' - 1, m1), 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 3 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(le(n3, m2), 1 + n3 + (1 + m2 + x'')) + 0)) :|: z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(le(n4, m3), 1 + n4 + (1 + m3 + x2)) + 0)) :|: n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {le}, {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]

----------------------------------------

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: le
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

----------------------------------------

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 2 }-> if_min(le(n'', m'), 1 + (1 + n'') + (1 + (1 + m') + x)) :|: z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(eq(z' - 1, m1), 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 3 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(le(n3, m2), 1 + n3 + (1 + m2 + x'')) + 0)) :|: z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(le(n4, m3), 1 + n4 + (1 + m3 + x2)) + 0)) :|: n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {le}, {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]
  le: runtime: ?, size: O(1) [2]

----------------------------------------

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: le
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z''

----------------------------------------

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(if_min(le(n5, m4), 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 2 }-> if_min(le(n'', m'), 1 + (1 + n'') + (1 + (1 + m') + x)) :|: z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(eq(z' - 1, m1), 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 3 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(le(n3, m2), 1 + n3 + (1 + m2 + x'')) + 0)) :|: z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(le(n4, m3), 1 + n4 + (1 + m3 + x2)) + 0)) :|: n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]

----------------------------------------

(27) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(eq(z' - 1, m1), 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]

----------------------------------------

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: eq
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

----------------------------------------

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(eq(z' - 1, m1), 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {eq}, {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  eq: runtime: ?, size: O(1) [2]

----------------------------------------

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: eq
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + z'

----------------------------------------

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 1 }-> eq(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(eq(z' - 1, m1), 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [2]

----------------------------------------

(33) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [2]

----------------------------------------

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: tail
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z'

----------------------------------------

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {tail}, {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [2]
  tail: runtime: ?, size: O(n^1) [z']

----------------------------------------

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: tail
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']

----------------------------------------

(39) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']

----------------------------------------

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: head
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z'

----------------------------------------

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {head}, {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  head: runtime: ?, size: O(n^1) [z']

----------------------------------------

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: head
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  head: runtime: O(1) [1], size: O(n^1) [z']

----------------------------------------

(45) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  head: runtime: O(1) [1], size: O(n^1) [z']

----------------------------------------

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: min
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z'

Computed SIZE bound using KoAT for: if_min
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z''

----------------------------------------

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {min,if_min}, {replace,if_replace}, {sortIter,if}, {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  head: runtime: O(1) [1], size: O(n^1) [z']
  min: runtime: ?, size: O(n^1) [z']
  if_min: runtime: ?, size: O(n^1) [z'']

----------------------------------------

(49) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: min
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 5 + 4*z' + z'^2

Computed RUNTIME bound using KoAT for: if_min
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 22 + 24*z'' + 8*z''^2

----------------------------------------

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 6 + m4 }-> sortIter(replace(if_min(s2, 1 + n5 + (1 + m4 + x4)), n5, 1 + m4 + x4), z2) :|: s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s3, 1 + n5 + (1 + m4 + x4)), n5, 0), z2) :|: s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 5 + m4 }-> sortIter(replace(if_min(s4, 1 + n5 + (1 + m4 + x4)), 0, 1 + m4 + x4), z2) :|: s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 4 + m4 }-> sortIter(replace(if_min(s5, 1 + n5 + (1 + m4 + x4)), 0, 0), z2) :|: s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if_min(z', z'') -{ 1 }-> min(1 + m + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 1 }-> min(1 + n + x) :|: z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 4 + m' }-> if_min(s', 1 + (1 + n'') + (1 + (1 + m') + x)) :|: s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 2 }-> if_min(2, 1 + 0 + (1 + m + x)) :|: x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 2 }-> if_min(1, 1 + (1 + n') + (1 + 0 + x)) :|: z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 5 + m2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + if_min(s'', 1 + n3 + (1 + m2 + x'')) + 0)) :|: s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 4 + m3 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + if_min(s1, 1 + n4 + (1 + m3 + x2)) + 0)) :|: s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {replace,if_replace}, {sortIter,if}, {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  head: runtime: O(1) [1], size: O(n^1) [z']
  min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z']
  if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z'']

----------------------------------------

(51) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(52)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 108 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s15, n5, 1 + m4 + x4), z2) :|: s15 >= 0, s15 <= 1 + n5 + (1 + m4 + x4), s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 107 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s16, n5, 0), z2) :|: s16 >= 0, s16 <= 1 + n5 + (1 + m4 + x4), s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 107 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s17, 0, 1 + m4 + x4), z2) :|: s17 >= 0, s17 <= 1 + n5 + (1 + m4 + x4), s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 106 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s18, 0, 0), z2) :|: s18 >= 0, s18 <= 1 + n5 + (1 + m4 + x4), s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s11 :|: s11 >= 0, s11 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s12 :|: s12 >= 0, s12 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 250 + 89*m' + 16*m'*n'' + 16*m'*x + 8*m'^2 + 88*n'' + 16*n''*x + 8*n''^2 + 88*x + 8*x^2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + n'') + (1 + (1 + m') + x), s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s8 :|: s8 >= 0, s8 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 168 + 72*n' + 16*n'*x + 8*n'^2 + 72*x + 8*x^2 }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + n') + (1 + 0 + x), z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 107 + 57*m2 + 16*m2*n3 + 16*m2*x'' + 8*m2^2 + 56*n3 + 16*n3*x'' + 8*n3^2 + 56*x'' + 8*x''^2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + s13 + 0)) :|: s13 >= 0, s13 <= 1 + n3 + (1 + m2 + x''), s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 106 + 57*m3 + 16*m3*n4 + 16*m3*x2 + 8*m3^2 + 56*n4 + 16*n4*x2 + 8*n4^2 + 56*x2 + 8*x2^2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + s14 + 0)) :|: s14 >= 0, s14 <= 1 + n4 + (1 + m3 + x2), s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {replace,if_replace}, {sortIter,if}, {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  head: runtime: O(1) [1], size: O(n^1) [z']
  min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z']
  if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z'']

----------------------------------------

(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: replace
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z'' + z1

Computed SIZE bound using CoFloCo for: if_replace
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z1 + z2

----------------------------------------

(54)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 108 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s15, n5, 1 + m4 + x4), z2) :|: s15 >= 0, s15 <= 1 + n5 + (1 + m4 + x4), s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 107 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s16, n5, 0), z2) :|: s16 >= 0, s16 <= 1 + n5 + (1 + m4 + x4), s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 107 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s17, 0, 1 + m4 + x4), z2) :|: s17 >= 0, s17 <= 1 + n5 + (1 + m4 + x4), s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 106 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s18, 0, 0), z2) :|: s18 >= 0, s18 <= 1 + n5 + (1 + m4 + x4), s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s11 :|: s11 >= 0, s11 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s12 :|: s12 >= 0, s12 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 250 + 89*m' + 16*m'*n'' + 16*m'*x + 8*m'^2 + 88*n'' + 16*n''*x + 8*n''^2 + 88*x + 8*x^2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + n'') + (1 + (1 + m') + x), s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s8 :|: s8 >= 0, s8 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 168 + 72*n' + 16*n'*x + 8*n'^2 + 72*x + 8*x^2 }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + n') + (1 + 0 + x), z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 107 + 57*m2 + 16*m2*n3 + 16*m2*x'' + 8*m2^2 + 56*n3 + 16*n3*x'' + 8*n3^2 + 56*x'' + 8*x''^2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + s13 + 0)) :|: s13 >= 0, s13 <= 1 + n3 + (1 + m2 + x''), s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 106 + 57*m3 + 16*m3*n4 + 16*m3*x2 + 8*m3^2 + 56*n4 + 16*n4*x2 + 8*n4^2 + 56*x2 + 8*x2^2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + s14 + 0)) :|: s14 >= 0, s14 <= 1 + n4 + (1 + m3 + x2), s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {replace,if_replace}, {sortIter,if}, {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  head: runtime: O(1) [1], size: O(n^1) [z']
  min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z']
  if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z'']
  replace: runtime: ?, size: O(n^1) [z'' + z1]
  if_replace: runtime: ?, size: O(n^1) [z1 + z2]

----------------------------------------

(55) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: replace
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 6 + z' + z'*z1 + 5*z1

Computed RUNTIME bound using KoAT for: if_replace
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 8 + z'' + z''*z2 + 5*z2

----------------------------------------

(56)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 108 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s15, n5, 1 + m4 + x4), z2) :|: s15 >= 0, s15 <= 1 + n5 + (1 + m4 + x4), s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 107 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s16, n5, 0), z2) :|: s16 >= 0, s16 <= 1 + n5 + (1 + m4 + x4), s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 107 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s17, 0, 1 + m4 + x4), z2) :|: s17 >= 0, s17 <= 1 + n5 + (1 + m4 + x4), s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 106 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 56*x4 + 8*x4^2 }-> sortIter(replace(s18, 0, 0), z2) :|: s18 >= 0, s18 <= 1 + n5 + (1 + m4 + x4), s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(0, n6, x5), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, n6, 0), z2) :|: x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, x6), z2) :|: z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(0, 0, 0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> sortIter(replace(0, 0, 0), z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 2 }-> sortIter(replace(z'' - 1, 0, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 4 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 3 }-> sortIter(replace(z'' - 1, z'' - 1, 0), z2) :|: z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s11 :|: s11 >= 0, s11 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s12 :|: s12 >= 0, s12 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + k + replace(z'', z1, x) :|: z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 250 + 89*m' + 16*m'*n'' + 16*m'*x + 8*m'^2 + 88*n'' + 16*n''*x + 8*n''^2 + 88*x + 8*x^2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + n'') + (1 + (1 + m') + x), s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s8 :|: s8 >= 0, s8 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 168 + 72*n' + 16*n'*x + 8*n'^2 + 72*x + 8*x^2 }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + n') + (1 + 0 + x), z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 4 + z' }-> if_replace(s7, 1 + (z' - 1), z'', 1 + (1 + m1) + x) :|: s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(2, 0, z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 0, z'', 1 + (1 + m'') + x) :|: m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 2 }-> if_replace(1, 1 + (z' - 1), z'', 1 + 0 + (z1 - 1)) :|: z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 107 + 57*m2 + 16*m2*n3 + 16*m2*x'' + 8*m2^2 + 56*n3 + 16*n3*x'' + 8*n3^2 + 56*x'' + 8*x''^2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + s13 + 0)) :|: s13 >= 0, s13 <= 1 + n3 + (1 + m2 + x''), s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 106 + 57*m3 + 16*m3*n4 + 16*m3*x2 + 8*m3^2 + 56*n4 + 16*n4*x2 + 8*n4^2 + 56*x2 + 8*x2^2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + s14 + 0)) :|: s14 >= 0, s14 <= 1 + n4 + (1 + m3 + x2), s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {sortIter,if}, {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  head: runtime: O(1) [1], size: O(n^1) [z']
  min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z']
  if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z'']
  replace: runtime: O(n^2) [6 + z' + z'*z1 + 5*z1], size: O(n^1) [z'' + z1]
  if_replace: runtime: O(n^2) [8 + z'' + z''*z2 + 5*z2], size: O(n^1) [z1 + z2]

----------------------------------------

(57) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(58)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 9 + z'' }-> sortIter(s24, z2) :|: s24 >= 0, s24 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 8 + z'' }-> sortIter(s25, z2) :|: s25 >= 0, s25 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 8 + z'' }-> sortIter(s26, z2) :|: s26 >= 0, s26 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 7 + z'' }-> sortIter(s27, z2) :|: s27 >= 0, s27 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 119 + 62*m4 + 16*m4*n5 + m4*s15 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s15 + s15*x4 + 61*x4 + 8*x4^2 }-> sortIter(s28, z2) :|: s28 >= 0, s28 <= n5 + (1 + m4 + x4), s15 >= 0, s15 <= 1 + n5 + (1 + m4 + x4), s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 113 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s16 + 56*x4 + 8*x4^2 }-> sortIter(s29, z2) :|: s29 >= 0, s29 <= n5 + 0, s16 >= 0, s16 <= 1 + n5 + (1 + m4 + x4), s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 118 + 62*m4 + 16*m4*n5 + m4*s17 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s17 + s17*x4 + 61*x4 + 8*x4^2 }-> sortIter(s30, z2) :|: s30 >= 0, s30 <= 0 + (1 + m4 + x4), s17 >= 0, s17 <= 1 + n5 + (1 + m4 + x4), s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 112 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s18 + 56*x4 + 8*x4^2 }-> sortIter(s31, z2) :|: s31 >= 0, s31 <= 0 + 0, s18 >= 0, s18 <= 1 + n5 + (1 + m4 + x4), s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 9 + 5*x5 }-> sortIter(s32, z2) :|: s32 >= 0, s32 <= n6 + x5, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 8 }-> sortIter(s33, z2) :|: s33 >= 0, s33 <= n6 + 0, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 8 }-> sortIter(s34, z2) :|: s34 >= 0, s34 <= 0 + 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 8 + 5*x6 }-> sortIter(s35, z2) :|: s35 >= 0, s35 <= 0 + x6, z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 7 }-> sortIter(s36, z2) :|: s36 >= 0, s36 <= 0 + 0, z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s11 :|: s11 >= 0, s11 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s12 :|: s12 >= 0, s12 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 7 + 5*x + x*z'' + z'' }-> 1 + k + s23 :|: s23 >= 0, s23 <= z1 + x, z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 250 + 89*m' + 16*m'*n'' + 16*m'*x + 8*m'^2 + 88*n'' + 16*n''*x + 8*n''^2 + 88*x + 8*x^2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + n'') + (1 + (1 + m') + x), s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s8 :|: s8 >= 0, s8 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 168 + 72*n' + 16*n'*x + 8*n'^2 + 72*x + 8*x^2 }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + n') + (1 + 0 + x), z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 10 + 5*z1 }-> s19 :|: s19 >= 0, s19 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 20 + 5*m'' + 5*x }-> s20 :|: s20 >= 0, s20 <= z'' + (1 + (1 + m'') + x), m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 10 + z' + z'*z1 + 5*z1 }-> s21 :|: s21 >= 0, s21 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 22 + 5*m1 + m1*z' + 5*x + x*z' + 4*z' }-> s22 :|: s22 >= 0, s22 <= z'' + (1 + (1 + m1) + x), s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 107 + 57*m2 + 16*m2*n3 + 16*m2*x'' + 8*m2^2 + 56*n3 + 16*n3*x'' + 8*n3^2 + 56*x'' + 8*x''^2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + s13 + 0)) :|: s13 >= 0, s13 <= 1 + n3 + (1 + m2 + x''), s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 106 + 57*m3 + 16*m3*n4 + 16*m3*x2 + 8*m3^2 + 56*n4 + 16*n4*x2 + 8*n4^2 + 56*x2 + 8*x2^2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + s14 + 0)) :|: s14 >= 0, s14 <= 1 + n4 + (1 + m3 + x2), s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {sortIter,if}, {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  head: runtime: O(1) [1], size: O(n^1) [z']
  min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z']
  if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z'']
  replace: runtime: O(n^2) [6 + z' + z'*z1 + 5*z1], size: O(n^1) [z'' + z1]
  if_replace: runtime: O(n^2) [8 + z'' + z''*z2 + 5*z2], size: O(n^1) [z1 + z2]

----------------------------------------

(59) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: sortIter
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 2*z' + z'^2 + z''

Computed SIZE bound using KoAT for: if
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 16*z'' + 8*z''^2 + z1 + 13*z2

----------------------------------------

(60)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 9 + z'' }-> sortIter(s24, z2) :|: s24 >= 0, s24 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 8 + z'' }-> sortIter(s25, z2) :|: s25 >= 0, s25 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 8 + z'' }-> sortIter(s26, z2) :|: s26 >= 0, s26 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 7 + z'' }-> sortIter(s27, z2) :|: s27 >= 0, s27 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 119 + 62*m4 + 16*m4*n5 + m4*s15 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s15 + s15*x4 + 61*x4 + 8*x4^2 }-> sortIter(s28, z2) :|: s28 >= 0, s28 <= n5 + (1 + m4 + x4), s15 >= 0, s15 <= 1 + n5 + (1 + m4 + x4), s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 113 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s16 + 56*x4 + 8*x4^2 }-> sortIter(s29, z2) :|: s29 >= 0, s29 <= n5 + 0, s16 >= 0, s16 <= 1 + n5 + (1 + m4 + x4), s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 118 + 62*m4 + 16*m4*n5 + m4*s17 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s17 + s17*x4 + 61*x4 + 8*x4^2 }-> sortIter(s30, z2) :|: s30 >= 0, s30 <= 0 + (1 + m4 + x4), s17 >= 0, s17 <= 1 + n5 + (1 + m4 + x4), s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 112 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s18 + 56*x4 + 8*x4^2 }-> sortIter(s31, z2) :|: s31 >= 0, s31 <= 0 + 0, s18 >= 0, s18 <= 1 + n5 + (1 + m4 + x4), s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 9 + 5*x5 }-> sortIter(s32, z2) :|: s32 >= 0, s32 <= n6 + x5, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 8 }-> sortIter(s33, z2) :|: s33 >= 0, s33 <= n6 + 0, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 8 }-> sortIter(s34, z2) :|: s34 >= 0, s34 <= 0 + 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 8 + 5*x6 }-> sortIter(s35, z2) :|: s35 >= 0, s35 <= 0 + x6, z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 7 }-> sortIter(s36, z2) :|: s36 >= 0, s36 <= 0 + 0, z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s11 :|: s11 >= 0, s11 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s12 :|: s12 >= 0, s12 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 7 + 5*x + x*z'' + z'' }-> 1 + k + s23 :|: s23 >= 0, s23 <= z1 + x, z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 250 + 89*m' + 16*m'*n'' + 16*m'*x + 8*m'^2 + 88*n'' + 16*n''*x + 8*n''^2 + 88*x + 8*x^2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + n'') + (1 + (1 + m') + x), s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s8 :|: s8 >= 0, s8 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 168 + 72*n' + 16*n'*x + 8*n'^2 + 72*x + 8*x^2 }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + n') + (1 + 0 + x), z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 10 + 5*z1 }-> s19 :|: s19 >= 0, s19 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 20 + 5*m'' + 5*x }-> s20 :|: s20 >= 0, s20 <= z'' + (1 + (1 + m'') + x), m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 10 + z' + z'*z1 + 5*z1 }-> s21 :|: s21 >= 0, s21 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 22 + 5*m1 + m1*z' + 5*x + x*z' + 4*z' }-> s22 :|: s22 >= 0, s22 <= z'' + (1 + (1 + m1) + x), s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 107 + 57*m2 + 16*m2*n3 + 16*m2*x'' + 8*m2^2 + 56*n3 + 16*n3*x'' + 8*n3^2 + 56*x'' + 8*x''^2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + s13 + 0)) :|: s13 >= 0, s13 <= 1 + n3 + (1 + m2 + x''), s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 106 + 57*m3 + 16*m3*n4 + 16*m3*x2 + 8*m3^2 + 56*n4 + 16*n4*x2 + 8*n4^2 + 56*x2 + 8*x2^2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + s14 + 0)) :|: s14 >= 0, s14 <= 1 + n4 + (1 + m3 + x2), s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {sortIter,if}, {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  head: runtime: O(1) [1], size: O(n^1) [z']
  min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z']
  if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z'']
  replace: runtime: O(n^2) [6 + z' + z'*z1 + 5*z1], size: O(n^1) [z'' + z1]
  if_replace: runtime: O(n^2) [8 + z'' + z''*z2 + 5*z2], size: O(n^1) [z1 + z2]
  sortIter: runtime: ?, size: O(n^2) [2*z' + z'^2 + z'']
  if: runtime: ?, size: O(n^2) [16*z'' + 8*z''^2 + z1 + 13*z2]

----------------------------------------

(61) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: sortIter
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 254 + 2902*z' + 3584*z'^2 + 1452*z'^3

Computed RUNTIME bound using KoAT for: if
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 19200 + 87556*z'' + 128608*z''^2 + 69696*z''^3

----------------------------------------

(62)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 9 + z'' }-> sortIter(s24, z2) :|: s24 >= 0, s24 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 8 + z'' }-> sortIter(s25, z2) :|: s25 >= 0, s25 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 8 + z'' }-> sortIter(s26, z2) :|: s26 >= 0, s26 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 7 + z'' }-> sortIter(s27, z2) :|: s27 >= 0, s27 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 119 + 62*m4 + 16*m4*n5 + m4*s15 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s15 + s15*x4 + 61*x4 + 8*x4^2 }-> sortIter(s28, z2) :|: s28 >= 0, s28 <= n5 + (1 + m4 + x4), s15 >= 0, s15 <= 1 + n5 + (1 + m4 + x4), s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 113 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s16 + 56*x4 + 8*x4^2 }-> sortIter(s29, z2) :|: s29 >= 0, s29 <= n5 + 0, s16 >= 0, s16 <= 1 + n5 + (1 + m4 + x4), s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 118 + 62*m4 + 16*m4*n5 + m4*s17 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s17 + s17*x4 + 61*x4 + 8*x4^2 }-> sortIter(s30, z2) :|: s30 >= 0, s30 <= 0 + (1 + m4 + x4), s17 >= 0, s17 <= 1 + n5 + (1 + m4 + x4), s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 112 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s18 + 56*x4 + 8*x4^2 }-> sortIter(s31, z2) :|: s31 >= 0, s31 <= 0 + 0, s18 >= 0, s18 <= 1 + n5 + (1 + m4 + x4), s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 9 + 5*x5 }-> sortIter(s32, z2) :|: s32 >= 0, s32 <= n6 + x5, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 8 }-> sortIter(s33, z2) :|: s33 >= 0, s33 <= n6 + 0, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 8 }-> sortIter(s34, z2) :|: s34 >= 0, s34 <= 0 + 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 8 + 5*x6 }-> sortIter(s35, z2) :|: s35 >= 0, s35 <= 0 + x6, z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 7 }-> sortIter(s36, z2) :|: s36 >= 0, s36 <= 0 + 0, z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s11 :|: s11 >= 0, s11 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s12 :|: s12 >= 0, s12 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 7 + 5*x + x*z'' + z'' }-> 1 + k + s23 :|: s23 >= 0, s23 <= z1 + x, z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 250 + 89*m' + 16*m'*n'' + 16*m'*x + 8*m'^2 + 88*n'' + 16*n''*x + 8*n''^2 + 88*x + 8*x^2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + n'') + (1 + (1 + m') + x), s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s8 :|: s8 >= 0, s8 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 168 + 72*n' + 16*n'*x + 8*n'^2 + 72*x + 8*x^2 }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + n') + (1 + 0 + x), z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 10 + 5*z1 }-> s19 :|: s19 >= 0, s19 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 20 + 5*m'' + 5*x }-> s20 :|: s20 >= 0, s20 <= z'' + (1 + (1 + m'') + x), m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 10 + z' + z'*z1 + 5*z1 }-> s21 :|: s21 >= 0, s21 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 22 + 5*m1 + m1*z' + 5*x + x*z' + 4*z' }-> s22 :|: s22 >= 0, s22 <= z'' + (1 + (1 + m1) + x), s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 1 }-> sortIter(z', 0) :|: z' >= 0
   sortIter(z', z'') -{ 2 }-> if(2, 0, z'', 1 + z'' + (1 + 0 + 0)) :|: z'' >= 0, z' = 0
   sortIter(z', z'') -{ 2 }-> if(1, 1 + n3 + x', z'', 1 + z'' + (1 + 0 + 0)) :|: z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 107 + 57*m2 + 16*m2*n3 + 16*m2*x'' + 8*m2^2 + 56*n3 + 16*n3*x'' + 8*n3^2 + 56*x'' + 8*x''^2 }-> if(1, 1 + n3 + (1 + m2 + x''), z'', 1 + z'' + (1 + s13 + 0)) :|: s13 >= 0, s13 <= 1 + n3 + (1 + m2 + x''), s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 3 }-> if(1, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1 }-> if(0, z', z'', 1 + z'' + (1 + 0 + 0)) :|: z' >= 0, z'' >= 0
   sortIter(z', z'') -{ 106 + 57*m3 + 16*m3*n4 + 16*m3*x2 + 8*m3^2 + 56*n4 + 16*n4*x2 + 8*n4^2 + 56*x2 + 8*x2^2 }-> if(0, 1 + n4 + (1 + m3 + x2), z'', 1 + z'' + (1 + s14 + 0)) :|: s14 >= 0, s14 <= 1 + n4 + (1 + m3 + x2), s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 2 }-> if(0, 1 + (z' - 1) + 0, z'', 1 + z'' + (1 + (z' - 1) + 0)) :|: z' - 1 >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  head: runtime: O(1) [1], size: O(n^1) [z']
  min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z']
  if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z'']
  replace: runtime: O(n^2) [6 + z' + z'*z1 + 5*z1], size: O(n^1) [z'' + z1]
  if_replace: runtime: O(n^2) [8 + z'' + z''*z2 + 5*z2], size: O(n^1) [z1 + z2]
  sortIter: runtime: O(n^3) [254 + 2902*z' + 3584*z'^2 + 1452*z'^3], size: O(n^2) [2*z' + z'^2 + z'']
  if: runtime: O(n^3) [19200 + 87556*z'' + 128608*z''^2 + 69696*z''^3], size: O(n^2) [16*z'' + 8*z''^2 + z1 + 13*z2]

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(63) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(64)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 263 + 2902*s24 + 3584*s24^2 + 1452*s24^3 + z'' }-> s45 :|: s45 >= 0, s45 <= 2 * s24 + s24 * s24 + z2, s24 >= 0, s24 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 262 + 2902*s25 + 3584*s25^2 + 1452*s25^3 + z'' }-> s46 :|: s46 >= 0, s46 <= 2 * s25 + s25 * s25 + z2, s25 >= 0, s25 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 262 + 2902*s26 + 3584*s26^2 + 1452*s26^3 + z'' }-> s47 :|: s47 >= 0, s47 <= 2 * s26 + s26 * s26 + z2, s26 >= 0, s26 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 261 + 2902*s27 + 3584*s27^2 + 1452*s27^3 + z'' }-> s48 :|: s48 >= 0, s48 <= 2 * s27 + s27 * s27 + z2, s27 >= 0, s27 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 373 + 62*m4 + 16*m4*n5 + m4*s15 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s15 + s15*x4 + 2902*s28 + 3584*s28^2 + 1452*s28^3 + 61*x4 + 8*x4^2 }-> s49 :|: s49 >= 0, s49 <= 2 * s28 + s28 * s28 + z2, s28 >= 0, s28 <= n5 + (1 + m4 + x4), s15 >= 0, s15 <= 1 + n5 + (1 + m4 + x4), s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 367 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s16 + 2902*s29 + 3584*s29^2 + 1452*s29^3 + 56*x4 + 8*x4^2 }-> s50 :|: s50 >= 0, s50 <= 2 * s29 + s29 * s29 + z2, s29 >= 0, s29 <= n5 + 0, s16 >= 0, s16 <= 1 + n5 + (1 + m4 + x4), s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 372 + 62*m4 + 16*m4*n5 + m4*s17 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s17 + s17*x4 + 2902*s30 + 3584*s30^2 + 1452*s30^3 + 61*x4 + 8*x4^2 }-> s51 :|: s51 >= 0, s51 <= 2 * s30 + s30 * s30 + z2, s30 >= 0, s30 <= 0 + (1 + m4 + x4), s17 >= 0, s17 <= 1 + n5 + (1 + m4 + x4), s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 366 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s18 + 2902*s31 + 3584*s31^2 + 1452*s31^3 + 56*x4 + 8*x4^2 }-> s52 :|: s52 >= 0, s52 <= 2 * s31 + s31 * s31 + z2, s31 >= 0, s31 <= 0 + 0, s18 >= 0, s18 <= 1 + n5 + (1 + m4 + x4), s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 263 + 2902*s32 + 3584*s32^2 + 1452*s32^3 + 5*x5 }-> s53 :|: s53 >= 0, s53 <= 2 * s32 + s32 * s32 + z2, s32 >= 0, s32 <= n6 + x5, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 262 + 2902*s33 + 3584*s33^2 + 1452*s33^3 }-> s54 :|: s54 >= 0, s54 <= 2 * s33 + s33 * s33 + z2, s33 >= 0, s33 <= n6 + 0, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 262 + 2902*s34 + 3584*s34^2 + 1452*s34^3 }-> s55 :|: s55 >= 0, s55 <= 2 * s34 + s34 * s34 + z2, s34 >= 0, s34 <= 0 + 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 262 + 2902*s35 + 3584*s35^2 + 1452*s35^3 + 5*x6 }-> s56 :|: s56 >= 0, s56 <= 2 * s35 + s35 * s35 + z2, s35 >= 0, s35 <= 0 + x6, z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 261 + 2902*s36 + 3584*s36^2 + 1452*s36^3 }-> s57 :|: s57 >= 0, s57 <= 2 * s36 + s36 * s36 + z2, s36 >= 0, s36 <= 0 + 0, z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s11 :|: s11 >= 0, s11 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s12 :|: s12 >= 0, s12 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 7 + 5*x + x*z'' + z'' }-> 1 + k + s23 :|: s23 >= 0, s23 <= z1 + x, z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 250 + 89*m' + 16*m'*n'' + 16*m'*x + 8*m'^2 + 88*n'' + 16*n''*x + 8*n''^2 + 88*x + 8*x^2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + n'') + (1 + (1 + m') + x), s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s8 :|: s8 >= 0, s8 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 168 + 72*n' + 16*n'*x + 8*n'^2 + 72*x + 8*x^2 }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + n') + (1 + 0 + x), z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 10 + 5*z1 }-> s19 :|: s19 >= 0, s19 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 20 + 5*m'' + 5*x }-> s20 :|: s20 >= 0, s20 <= z'' + (1 + (1 + m'') + x), m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 10 + z' + z'*z1 + 5*z1 }-> s21 :|: s21 >= 0, s21 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 22 + 5*m1 + m1*z' + 5*x + x*z' + 4*z' }-> s22 :|: s22 >= 0, s22 <= z'' + (1 + (1 + m1) + x), s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 255 + 2902*z' + 3584*z'^2 + 1452*z'^3 }-> s37 :|: s37 >= 0, s37 <= 2 * z' + z' * z' + 0, z' >= 0
   sortIter(z', z'') -{ 19202 }-> s38 :|: s38 >= 0, s38 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * 0 + 8 * (0 * 0) + z'', z'' >= 0, z' = 0
   sortIter(z', z'') -{ 19203 + 87556*z' + 128608*z'^2 + 69696*z'^3 }-> s39 :|: s39 >= 0, s39 <= 13 * (1 + z'' + (1 + (z' - 1) + 0)) + 16 * (1 + (z' - 1) + 0) + 8 * ((1 + (z' - 1) + 0) * (1 + (z' - 1) + 0)) + z'', z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1266419 + 1438397*m2 + 1093584*m2*n3 + 418176*m2*n3*x'' + 209088*m2*n3^2 + 1093584*m2*x'' + 209088*m2*x''^2 + 546792*m2^2 + 209088*m2^2*n3 + 209088*m2^2*x'' + 69696*m2^3 + 1438396*n3 + 1093584*n3*x'' + 209088*n3*x''^2 + 546792*n3^2 + 209088*n3^2*x'' + 69696*n3^3 + 1438396*x'' + 546792*x''^2 + 69696*x''^3 }-> s40 :|: s40 >= 0, s40 <= 13 * (1 + z'' + (1 + s13 + 0)) + 16 * (1 + n3 + (1 + m2 + x'')) + 8 * ((1 + n3 + (1 + m2 + x'')) * (1 + n3 + (1 + m2 + x''))) + z'', s13 >= 0, s13 <= 1 + n3 + (1 + m2 + x''), s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 305062 + 553860*n3 + 675392*n3*x' + 209088*n3*x'^2 + 337696*n3^2 + 209088*n3^2*x' + 69696*n3^3 + 553860*x' + 337696*x'^2 + 69696*x'^3 }-> s41 :|: s41 >= 0, s41 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * (1 + n3 + x') + 8 * ((1 + n3 + x') * (1 + n3 + x')) + z'', z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 19202 + 87556*z' + 128608*z'^2 + 69696*z'^3 }-> s42 :|: s42 >= 0, s42 <= 13 * (1 + z'' + (1 + (z' - 1) + 0)) + 16 * (1 + (z' - 1) + 0) + 8 * ((1 + (z' - 1) + 0) * (1 + (z' - 1) + 0)) + z'', z' - 1 >= 0, z'' >= 0
   sortIter(z', z'') -{ 1266418 + 1438397*m3 + 1093584*m3*n4 + 418176*m3*n4*x2 + 209088*m3*n4^2 + 1093584*m3*x2 + 209088*m3*x2^2 + 546792*m3^2 + 209088*m3^2*n4 + 209088*m3^2*x2 + 69696*m3^3 + 1438396*n4 + 1093584*n4*x2 + 209088*n4*x2^2 + 546792*n4^2 + 209088*n4^2*x2 + 69696*n4^3 + 1438396*x2 + 546792*x2^2 + 69696*x2^3 }-> s43 :|: s43 >= 0, s43 <= 13 * (1 + z'' + (1 + s14 + 0)) + 16 * (1 + n4 + (1 + m3 + x2)) + 8 * ((1 + n4 + (1 + m3 + x2)) * (1 + n4 + (1 + m3 + x2))) + z'', s14 >= 0, s14 <= 1 + n4 + (1 + m3 + x2), s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 19201 + 87556*z' + 128608*z'^2 + 69696*z'^3 }-> s44 :|: s44 >= 0, s44 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * z' + 8 * (z' * z') + z'', z' >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  head: runtime: O(1) [1], size: O(n^1) [z']
  min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z']
  if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z'']
  replace: runtime: O(n^2) [6 + z' + z'*z1 + 5*z1], size: O(n^1) [z'' + z1]
  if_replace: runtime: O(n^2) [8 + z'' + z''*z2 + 5*z2], size: O(n^1) [z1 + z2]
  sortIter: runtime: O(n^3) [254 + 2902*z' + 3584*z'^2 + 1452*z'^3], size: O(n^2) [2*z' + z'^2 + z'']
  if: runtime: O(n^3) [19200 + 87556*z'' + 128608*z''^2 + 69696*z''^3], size: O(n^2) [16*z'' + 8*z''^2 + z1 + 13*z2]

----------------------------------------

(65) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: sort
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 2*z' + z'^2

----------------------------------------

(66)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 263 + 2902*s24 + 3584*s24^2 + 1452*s24^3 + z'' }-> s45 :|: s45 >= 0, s45 <= 2 * s24 + s24 * s24 + z2, s24 >= 0, s24 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 262 + 2902*s25 + 3584*s25^2 + 1452*s25^3 + z'' }-> s46 :|: s46 >= 0, s46 <= 2 * s25 + s25 * s25 + z2, s25 >= 0, s25 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 262 + 2902*s26 + 3584*s26^2 + 1452*s26^3 + z'' }-> s47 :|: s47 >= 0, s47 <= 2 * s26 + s26 * s26 + z2, s26 >= 0, s26 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 261 + 2902*s27 + 3584*s27^2 + 1452*s27^3 + z'' }-> s48 :|: s48 >= 0, s48 <= 2 * s27 + s27 * s27 + z2, s27 >= 0, s27 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 373 + 62*m4 + 16*m4*n5 + m4*s15 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s15 + s15*x4 + 2902*s28 + 3584*s28^2 + 1452*s28^3 + 61*x4 + 8*x4^2 }-> s49 :|: s49 >= 0, s49 <= 2 * s28 + s28 * s28 + z2, s28 >= 0, s28 <= n5 + (1 + m4 + x4), s15 >= 0, s15 <= 1 + n5 + (1 + m4 + x4), s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 367 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s16 + 2902*s29 + 3584*s29^2 + 1452*s29^3 + 56*x4 + 8*x4^2 }-> s50 :|: s50 >= 0, s50 <= 2 * s29 + s29 * s29 + z2, s29 >= 0, s29 <= n5 + 0, s16 >= 0, s16 <= 1 + n5 + (1 + m4 + x4), s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 372 + 62*m4 + 16*m4*n5 + m4*s17 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s17 + s17*x4 + 2902*s30 + 3584*s30^2 + 1452*s30^3 + 61*x4 + 8*x4^2 }-> s51 :|: s51 >= 0, s51 <= 2 * s30 + s30 * s30 + z2, s30 >= 0, s30 <= 0 + (1 + m4 + x4), s17 >= 0, s17 <= 1 + n5 + (1 + m4 + x4), s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 366 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s18 + 2902*s31 + 3584*s31^2 + 1452*s31^3 + 56*x4 + 8*x4^2 }-> s52 :|: s52 >= 0, s52 <= 2 * s31 + s31 * s31 + z2, s31 >= 0, s31 <= 0 + 0, s18 >= 0, s18 <= 1 + n5 + (1 + m4 + x4), s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 263 + 2902*s32 + 3584*s32^2 + 1452*s32^3 + 5*x5 }-> s53 :|: s53 >= 0, s53 <= 2 * s32 + s32 * s32 + z2, s32 >= 0, s32 <= n6 + x5, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 262 + 2902*s33 + 3584*s33^2 + 1452*s33^3 }-> s54 :|: s54 >= 0, s54 <= 2 * s33 + s33 * s33 + z2, s33 >= 0, s33 <= n6 + 0, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 262 + 2902*s34 + 3584*s34^2 + 1452*s34^3 }-> s55 :|: s55 >= 0, s55 <= 2 * s34 + s34 * s34 + z2, s34 >= 0, s34 <= 0 + 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 262 + 2902*s35 + 3584*s35^2 + 1452*s35^3 + 5*x6 }-> s56 :|: s56 >= 0, s56 <= 2 * s35 + s35 * s35 + z2, s35 >= 0, s35 <= 0 + x6, z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 261 + 2902*s36 + 3584*s36^2 + 1452*s36^3 }-> s57 :|: s57 >= 0, s57 <= 2 * s36 + s36 * s36 + z2, s36 >= 0, s36 <= 0 + 0, z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s11 :|: s11 >= 0, s11 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s12 :|: s12 >= 0, s12 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 7 + 5*x + x*z'' + z'' }-> 1 + k + s23 :|: s23 >= 0, s23 <= z1 + x, z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 250 + 89*m' + 16*m'*n'' + 16*m'*x + 8*m'^2 + 88*n'' + 16*n''*x + 8*n''^2 + 88*x + 8*x^2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + n'') + (1 + (1 + m') + x), s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s8 :|: s8 >= 0, s8 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 168 + 72*n' + 16*n'*x + 8*n'^2 + 72*x + 8*x^2 }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + n') + (1 + 0 + x), z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 10 + 5*z1 }-> s19 :|: s19 >= 0, s19 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 20 + 5*m'' + 5*x }-> s20 :|: s20 >= 0, s20 <= z'' + (1 + (1 + m'') + x), m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 10 + z' + z'*z1 + 5*z1 }-> s21 :|: s21 >= 0, s21 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 22 + 5*m1 + m1*z' + 5*x + x*z' + 4*z' }-> s22 :|: s22 >= 0, s22 <= z'' + (1 + (1 + m1) + x), s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 255 + 2902*z' + 3584*z'^2 + 1452*z'^3 }-> s37 :|: s37 >= 0, s37 <= 2 * z' + z' * z' + 0, z' >= 0
   sortIter(z', z'') -{ 19202 }-> s38 :|: s38 >= 0, s38 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * 0 + 8 * (0 * 0) + z'', z'' >= 0, z' = 0
   sortIter(z', z'') -{ 19203 + 87556*z' + 128608*z'^2 + 69696*z'^3 }-> s39 :|: s39 >= 0, s39 <= 13 * (1 + z'' + (1 + (z' - 1) + 0)) + 16 * (1 + (z' - 1) + 0) + 8 * ((1 + (z' - 1) + 0) * (1 + (z' - 1) + 0)) + z'', z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1266419 + 1438397*m2 + 1093584*m2*n3 + 418176*m2*n3*x'' + 209088*m2*n3^2 + 1093584*m2*x'' + 209088*m2*x''^2 + 546792*m2^2 + 209088*m2^2*n3 + 209088*m2^2*x'' + 69696*m2^3 + 1438396*n3 + 1093584*n3*x'' + 209088*n3*x''^2 + 546792*n3^2 + 209088*n3^2*x'' + 69696*n3^3 + 1438396*x'' + 546792*x''^2 + 69696*x''^3 }-> s40 :|: s40 >= 0, s40 <= 13 * (1 + z'' + (1 + s13 + 0)) + 16 * (1 + n3 + (1 + m2 + x'')) + 8 * ((1 + n3 + (1 + m2 + x'')) * (1 + n3 + (1 + m2 + x''))) + z'', s13 >= 0, s13 <= 1 + n3 + (1 + m2 + x''), s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 305062 + 553860*n3 + 675392*n3*x' + 209088*n3*x'^2 + 337696*n3^2 + 209088*n3^2*x' + 69696*n3^3 + 553860*x' + 337696*x'^2 + 69696*x'^3 }-> s41 :|: s41 >= 0, s41 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * (1 + n3 + x') + 8 * ((1 + n3 + x') * (1 + n3 + x')) + z'', z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 19202 + 87556*z' + 128608*z'^2 + 69696*z'^3 }-> s42 :|: s42 >= 0, s42 <= 13 * (1 + z'' + (1 + (z' - 1) + 0)) + 16 * (1 + (z' - 1) + 0) + 8 * ((1 + (z' - 1) + 0) * (1 + (z' - 1) + 0)) + z'', z' - 1 >= 0, z'' >= 0
   sortIter(z', z'') -{ 1266418 + 1438397*m3 + 1093584*m3*n4 + 418176*m3*n4*x2 + 209088*m3*n4^2 + 1093584*m3*x2 + 209088*m3*x2^2 + 546792*m3^2 + 209088*m3^2*n4 + 209088*m3^2*x2 + 69696*m3^3 + 1438396*n4 + 1093584*n4*x2 + 209088*n4*x2^2 + 546792*n4^2 + 209088*n4^2*x2 + 69696*n4^3 + 1438396*x2 + 546792*x2^2 + 69696*x2^3 }-> s43 :|: s43 >= 0, s43 <= 13 * (1 + z'' + (1 + s14 + 0)) + 16 * (1 + n4 + (1 + m3 + x2)) + 8 * ((1 + n4 + (1 + m3 + x2)) * (1 + n4 + (1 + m3 + x2))) + z'', s14 >= 0, s14 <= 1 + n4 + (1 + m3 + x2), s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 19201 + 87556*z' + 128608*z'^2 + 69696*z'^3 }-> s44 :|: s44 >= 0, s44 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * z' + 8 * (z' * z') + z'', z' >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: {sort}
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  head: runtime: O(1) [1], size: O(n^1) [z']
  min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z']
  if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z'']
  replace: runtime: O(n^2) [6 + z' + z'*z1 + 5*z1], size: O(n^1) [z'' + z1]
  if_replace: runtime: O(n^2) [8 + z'' + z''*z2 + 5*z2], size: O(n^1) [z1 + z2]
  sortIter: runtime: O(n^3) [254 + 2902*z' + 3584*z'^2 + 1452*z'^3], size: O(n^2) [2*z' + z'^2 + z'']
  if: runtime: O(n^3) [19200 + 87556*z'' + 128608*z''^2 + 69696*z''^3], size: O(n^2) [16*z'' + 8*z''^2 + z1 + 13*z2]
  sort: runtime: ?, size: O(n^2) [2*z' + z'^2]

----------------------------------------

(67) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: sort
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 255 + 2902*z' + 3584*z'^2 + 1452*z'^3

----------------------------------------

(68)
Obligation:
Complexity RNTS consisting of the following rules:

   empty(z') -{ 1 }-> 2 :|: z' = 0
   empty(z') -{ 1 }-> 1 :|: n >= 0, z' = 1 + n + x, x >= 0
   empty(z') -{ 0 }-> 0 :|: z' >= 0
   eq(z', z'') -{ 3 + z' }-> s6 :|: s6 >= 0, s6 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = 0
   eq(z', z'') -{ 1 }-> 1 :|: z' = 0, z'' - 1 >= 0
   eq(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 263 + 2902*s24 + 3584*s24^2 + 1452*s24^3 + z'' }-> s45 :|: s45 >= 0, s45 <= 2 * s24 + s24 * s24 + z2, s24 >= 0, s24 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 262 + 2902*s25 + 3584*s25^2 + 1452*s25^3 + z'' }-> s46 :|: s46 >= 0, s46 <= 2 * s25 + s25 * s25 + z2, s25 >= 0, s25 <= z'' - 1 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 262 + 2902*s26 + 3584*s26^2 + 1452*s26^3 + z'' }-> s47 :|: s47 >= 0, s47 <= 2 * s26 + s26 * s26 + z2, s26 >= 0, s26 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 261 + 2902*s27 + 3584*s27^2 + 1452*s27^3 + z'' }-> s48 :|: s48 >= 0, s48 <= 2 * s27 + s27 * s27 + z2, s27 >= 0, s27 <= 0 + 0, z2 >= 0, z1 >= 0, z' = 1, z'' - 1 >= 0
   if(z', z'', z1, z2) -{ 373 + 62*m4 + 16*m4*n5 + m4*s15 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s15 + s15*x4 + 2902*s28 + 3584*s28^2 + 1452*s28^3 + 61*x4 + 8*x4^2 }-> s49 :|: s49 >= 0, s49 <= 2 * s28 + s28 * s28 + z2, s28 >= 0, s28 <= n5 + (1 + m4 + x4), s15 >= 0, s15 <= 1 + n5 + (1 + m4 + x4), s2 >= 0, s2 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 367 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s16 + 2902*s29 + 3584*s29^2 + 1452*s29^3 + 56*x4 + 8*x4^2 }-> s50 :|: s50 >= 0, s50 <= 2 * s29 + s29 * s29 + z2, s29 >= 0, s29 <= n5 + 0, s16 >= 0, s16 <= 1 + n5 + (1 + m4 + x4), s3 >= 0, s3 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 372 + 62*m4 + 16*m4*n5 + m4*s17 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + 2*s17 + s17*x4 + 2902*s30 + 3584*s30^2 + 1452*s30^3 + 61*x4 + 8*x4^2 }-> s51 :|: s51 >= 0, s51 <= 2 * s30 + s30 * s30 + z2, s30 >= 0, s30 <= 0 + (1 + m4 + x4), s17 >= 0, s17 <= 1 + n5 + (1 + m4 + x4), s4 >= 0, s4 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 366 + 57*m4 + 16*m4*n5 + 16*m4*x4 + 8*m4^2 + 56*n5 + 16*n5*x4 + 8*n5^2 + s18 + 2902*s31 + 3584*s31^2 + 1452*s31^3 + 56*x4 + 8*x4^2 }-> s52 :|: s52 >= 0, s52 <= 2 * s31 + s31 * s31 + z2, s31 >= 0, s31 <= 0 + 0, s18 >= 0, s18 <= 1 + n5 + (1 + m4 + x4), s5 >= 0, s5 <= 2, x4 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n5 + (1 + m4 + x4), n5 >= 0, z' = 1, m4 >= 0
   if(z', z'', z1, z2) -{ 263 + 2902*s32 + 3584*s32^2 + 1452*s32^3 + 5*x5 }-> s53 :|: s53 >= 0, s53 <= 2 * s32 + s32 * s32 + z2, s32 >= 0, s32 <= n6 + x5, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 262 + 2902*s33 + 3584*s33^2 + 1452*s33^3 }-> s54 :|: s54 >= 0, s54 <= 2 * s33 + s33 * s33 + z2, s33 >= 0, s33 <= n6 + 0, x5 >= 0, z2 >= 0, n6 >= 0, z1 >= 0, z'' = 1 + n6 + x5, z' = 1
   if(z', z'', z1, z2) -{ 262 + 2902*s34 + 3584*s34^2 + 1452*s34^3 }-> s55 :|: s55 >= 0, s55 <= 2 * s34 + s34 * s34 + z2, s34 >= 0, s34 <= 0 + 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 262 + 2902*s35 + 3584*s35^2 + 1452*s35^3 + 5*x6 }-> s56 :|: s56 >= 0, s56 <= 2 * s35 + s35 * s35 + z2, s35 >= 0, s35 <= 0 + x6, z2 >= 0, z'' = 1 + n7 + x6, n7 >= 0, z1 >= 0, x6 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 261 + 2902*s36 + 3584*s36^2 + 1452*s36^3 }-> s57 :|: s57 >= 0, s57 <= 2 * s36 + s36 * s36 + z2, s36 >= 0, s36 <= 0 + 0, z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if_min(z', z'') -{ 11 + 6*n + 2*n*x + n^2 + 6*x + x^2 }-> s11 :|: s11 >= 0, s11 <= 1 + n + x, z'' = 1 + n + (1 + m + x), n >= 0, z' = 2, x >= 0, m >= 0
   if_min(z', z'') -{ 11 + 6*m + 2*m*x + m^2 + 6*x + x^2 }-> s12 :|: s12 >= 0, s12 <= 1 + m + x, z'' = 1 + n + (1 + m + x), n >= 0, x >= 0, z' = 1, m >= 0
   if_min(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   if_replace(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if_replace(z', z'', z1, z2) -{ 7 + 5*x + x*z'' + z'' }-> 1 + k + s23 :|: s23 >= 0, s23 <= z1 + x, z'' >= 0, z2 = 1 + k + x, x >= 0, z' = 1, k >= 0, z1 >= 0
   if_replace(z', z'', z1, z2) -{ 1 }-> 1 + z1 + x :|: z'' >= 0, z' = 2, z2 = 1 + k + x, x >= 0, k >= 0, z1 >= 0
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z' = 0, z'' >= 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   min(z') -{ 250 + 89*m' + 16*m'*n'' + 16*m'*x + 8*m'^2 + 88*n'' + 16*n''*x + 8*n''^2 + 88*x + 8*x^2 }-> s10 :|: s10 >= 0, s10 <= 1 + (1 + n'') + (1 + (1 + m') + x), s' >= 0, s' <= 2, z' = 1 + (1 + n'') + (1 + (1 + m') + x), x >= 0, m' >= 0, n'' >= 0
   min(z') -{ 104 + 56*m + 16*m*x + 8*m^2 + 56*x + 8*x^2 }-> s8 :|: s8 >= 0, s8 <= 1 + 0 + (1 + m + x), x >= 0, z' = 1 + 0 + (1 + m + x), m >= 0
   min(z') -{ 168 + 72*n' + 16*n'*x + 8*n'^2 + 72*x + 8*x^2 }-> s9 :|: s9 >= 0, s9 <= 1 + (1 + n') + (1 + 0 + x), z' = 1 + (1 + n') + (1 + 0 + x), x >= 0, n' >= 0
   min(z') -{ 0 }-> 0 :|: z' >= 0
   min(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   replace(z', z'', z1) -{ 10 + 5*z1 }-> s19 :|: s19 >= 0, s19 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 20 + 5*m'' + 5*x }-> s20 :|: s20 >= 0, s20 <= z'' + (1 + (1 + m'') + x), m'' >= 0, x >= 0, z1 = 1 + (1 + m'') + x, z' = 0, z'' >= 0
   replace(z', z'', z1) -{ 10 + z' + z'*z1 + 5*z1 }-> s21 :|: s21 >= 0, s21 <= z'' + (1 + 0 + (z1 - 1)), z1 - 1 >= 0, z' - 1 >= 0, z'' >= 0
   replace(z', z'', z1) -{ 22 + 5*m1 + m1*z' + 5*x + x*z' + 4*z' }-> s22 :|: s22 >= 0, s22 <= z'' + (1 + (1 + m1) + x), s7 >= 0, s7 <= 2, x >= 0, z' - 1 >= 0, m1 >= 0, z1 = 1 + (1 + m1) + x, z'' >= 0
   replace(z', z'', z1) -{ 1 }-> 0 :|: z' >= 0, z1 = 0, z'' >= 0
   replace(z', z'', z1) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0
   sort(z') -{ 255 + 2902*z' + 3584*z'^2 + 1452*z'^3 }-> s37 :|: s37 >= 0, s37 <= 2 * z' + z' * z' + 0, z' >= 0
   sortIter(z', z'') -{ 19202 }-> s38 :|: s38 >= 0, s38 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * 0 + 8 * (0 * 0) + z'', z'' >= 0, z' = 0
   sortIter(z', z'') -{ 19203 + 87556*z' + 128608*z'^2 + 69696*z'^3 }-> s39 :|: s39 >= 0, s39 <= 13 * (1 + z'' + (1 + (z' - 1) + 0)) + 16 * (1 + (z' - 1) + 0) + 8 * ((1 + (z' - 1) + 0) * (1 + (z' - 1) + 0)) + z'', z'' >= 0, z' - 1 >= 0
   sortIter(z', z'') -{ 1266419 + 1438397*m2 + 1093584*m2*n3 + 418176*m2*n3*x'' + 209088*m2*n3^2 + 1093584*m2*x'' + 209088*m2*x''^2 + 546792*m2^2 + 209088*m2^2*n3 + 209088*m2^2*x'' + 69696*m2^3 + 1438396*n3 + 1093584*n3*x'' + 209088*n3*x''^2 + 546792*n3^2 + 209088*n3^2*x'' + 69696*n3^3 + 1438396*x'' + 546792*x''^2 + 69696*x''^3 }-> s40 :|: s40 >= 0, s40 <= 13 * (1 + z'' + (1 + s13 + 0)) + 16 * (1 + n3 + (1 + m2 + x'')) + 8 * ((1 + n3 + (1 + m2 + x'')) * (1 + n3 + (1 + m2 + x''))) + z'', s13 >= 0, s13 <= 1 + n3 + (1 + m2 + x''), s'' >= 0, s'' <= 2, z'' >= 0, x'' >= 0, m2 >= 0, n3 >= 0, z' = 1 + n3 + (1 + m2 + x'')
   sortIter(z', z'') -{ 305062 + 553860*n3 + 675392*n3*x' + 209088*n3*x'^2 + 337696*n3^2 + 209088*n3^2*x' + 69696*n3^3 + 553860*x' + 337696*x'^2 + 69696*x'^3 }-> s41 :|: s41 >= 0, s41 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * (1 + n3 + x') + 8 * ((1 + n3 + x') * (1 + n3 + x')) + z'', z' = 1 + n3 + x', x' >= 0, z'' >= 0, n3 >= 0
   sortIter(z', z'') -{ 19202 + 87556*z' + 128608*z'^2 + 69696*z'^3 }-> s42 :|: s42 >= 0, s42 <= 13 * (1 + z'' + (1 + (z' - 1) + 0)) + 16 * (1 + (z' - 1) + 0) + 8 * ((1 + (z' - 1) + 0) * (1 + (z' - 1) + 0)) + z'', z' - 1 >= 0, z'' >= 0
   sortIter(z', z'') -{ 1266418 + 1438397*m3 + 1093584*m3*n4 + 418176*m3*n4*x2 + 209088*m3*n4^2 + 1093584*m3*x2 + 209088*m3*x2^2 + 546792*m3^2 + 209088*m3^2*n4 + 209088*m3^2*x2 + 69696*m3^3 + 1438396*n4 + 1093584*n4*x2 + 209088*n4*x2^2 + 546792*n4^2 + 209088*n4^2*x2 + 69696*n4^3 + 1438396*x2 + 546792*x2^2 + 69696*x2^3 }-> s43 :|: s43 >= 0, s43 <= 13 * (1 + z'' + (1 + s14 + 0)) + 16 * (1 + n4 + (1 + m3 + x2)) + 8 * ((1 + n4 + (1 + m3 + x2)) * (1 + n4 + (1 + m3 + x2))) + z'', s14 >= 0, s14 <= 1 + n4 + (1 + m3 + x2), s1 >= 0, s1 <= 2, n4 >= 0, z'' >= 0, z' = 1 + n4 + (1 + m3 + x2), x2 >= 0, m3 >= 0
   sortIter(z', z'') -{ 19201 + 87556*z' + 128608*z'^2 + 69696*z'^3 }-> s44 :|: s44 >= 0, s44 <= 13 * (1 + z'' + (1 + 0 + 0)) + 16 * z' + 8 * (z' * z') + z'', z' >= 0, z'' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  empty: runtime: O(1) [1], size: O(1) [2]
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  eq: runtime: O(n^1) [3 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  head: runtime: O(1) [1], size: O(n^1) [z']
  min: runtime: O(n^2) [5 + 4*z' + z'^2], size: O(n^1) [z']
  if_min: runtime: O(n^2) [22 + 24*z'' + 8*z''^2], size: O(n^1) [z'']
  replace: runtime: O(n^2) [6 + z' + z'*z1 + 5*z1], size: O(n^1) [z'' + z1]
  if_replace: runtime: O(n^2) [8 + z'' + z''*z2 + 5*z2], size: O(n^1) [z1 + z2]
  sortIter: runtime: O(n^3) [254 + 2902*z' + 3584*z'^2 + 1452*z'^3], size: O(n^2) [2*z' + z'^2 + z'']
  if: runtime: O(n^3) [19200 + 87556*z'' + 128608*z''^2 + 69696*z''^3], size: O(n^2) [16*z'' + 8*z''^2 + z1 + 13*z2]
  sort: runtime: O(n^3) [255 + 2902*z' + 3584*z'^2 + 1452*z'^3], size: O(n^2) [2*z' + z'^2]

----------------------------------------

(69) FinalProof (FINISHED)
Computed overall runtime complexity
----------------------------------------

(70)
BOUNDS(1, n^3)

----------------------------------------

(71) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(72)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   eq(0, 0) -> true
   eq(0, s(m)) -> false
   eq(s(n), 0) -> false
   eq(s(n), s(m)) -> eq(n, m)
   le(0, m) -> true
   le(s(n), 0) -> false
   le(s(n), s(m)) -> le(n, m)
   min(cons(x, nil)) -> x
   min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x)))
   if_min(true, cons(n, cons(m, x))) -> min(cons(n, x))
   if_min(false, cons(n, cons(m, x))) -> min(cons(m, x))
   replace(n, m, nil) -> nil
   replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x))
   if_replace(true, n, m, cons(k, x)) -> cons(m, x)
   if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x))
   empty(nil) -> true
   empty(cons(n, x)) -> false
   head(cons(n, x)) -> n
   tail(nil) -> nil
   tail(cons(n, x)) -> x
   sort(x) -> sortIter(x, nil)
   sortIter(x, y) -> if(empty(x), x, y, append(y, cons(min(x), nil)))
   if(true, x, y, z) -> y
   if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z)

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(73) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

eq(s(n), s(m)) ->^+ eq(n, m)

gives rise to a decreasing loop by considering the right hand sides subterm at position [].

The pumping substitution is [n / s(n), m / s(m)].

The result substitution is [ ].




----------------------------------------

(74)
Complex Obligation (BEST)

----------------------------------------

(75)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   eq(0, 0) -> true
   eq(0, s(m)) -> false
   eq(s(n), 0) -> false
   eq(s(n), s(m)) -> eq(n, m)
   le(0, m) -> true
   le(s(n), 0) -> false
   le(s(n), s(m)) -> le(n, m)
   min(cons(x, nil)) -> x
   min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x)))
   if_min(true, cons(n, cons(m, x))) -> min(cons(n, x))
   if_min(false, cons(n, cons(m, x))) -> min(cons(m, x))
   replace(n, m, nil) -> nil
   replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x))
   if_replace(true, n, m, cons(k, x)) -> cons(m, x)
   if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x))
   empty(nil) -> true
   empty(cons(n, x)) -> false
   head(cons(n, x)) -> n
   tail(nil) -> nil
   tail(cons(n, x)) -> x
   sort(x) -> sortIter(x, nil)
   sortIter(x, y) -> if(empty(x), x, y, append(y, cons(min(x), nil)))
   if(true, x, y, z) -> y
   if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z)

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(76) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(77)
BOUNDS(n^1, INF)

----------------------------------------

(78)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   eq(0, 0) -> true
   eq(0, s(m)) -> false
   eq(s(n), 0) -> false
   eq(s(n), s(m)) -> eq(n, m)
   le(0, m) -> true
   le(s(n), 0) -> false
   le(s(n), s(m)) -> le(n, m)
   min(cons(x, nil)) -> x
   min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x)))
   if_min(true, cons(n, cons(m, x))) -> min(cons(n, x))
   if_min(false, cons(n, cons(m, x))) -> min(cons(m, x))
   replace(n, m, nil) -> nil
   replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x))
   if_replace(true, n, m, cons(k, x)) -> cons(m, x)
   if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x))
   empty(nil) -> true
   empty(cons(n, x)) -> false
   head(cons(n, x)) -> n
   tail(nil) -> nil
   tail(cons(n, x)) -> x
   sort(x) -> sortIter(x, nil)
   sortIter(x, y) -> if(empty(x), x, y, append(y, cons(min(x), nil)))
   if(true, x, y, z) -> y
   if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z)

S is empty.
Rewrite Strategy: INNERMOST