WORST_CASE(Omega(n^2),O(n^3))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 848 ms]
(2) CpxRelTRS
(3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 326 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 169 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 345 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 116 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 203 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 26 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 464 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 116 ms]
    (40) CpxRNTS
    (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 2116 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 617 ms]
    (46) CpxRNTS
    (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 2953 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 798 ms]
    (52) CpxRNTS
    (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (54) CpxRNTS
    (55) IntTrsBoundProof [UPPER BOUND(ID), 965 ms]
    (56) CpxRNTS
    (57) IntTrsBoundProof [UPPER BOUND(ID), 195 ms]
    (58) CpxRNTS
    (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (60) CpxRNTS
    (61) IntTrsBoundProof [UPPER BOUND(ID), 1676 ms]
    (62) CpxRNTS
    (63) IntTrsBoundProof [UPPER BOUND(ID), 342 ms]
    (64) CpxRNTS
    (65) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (66) CpxRNTS
    (67) IntTrsBoundProof [UPPER BOUND(ID), 2495 ms]
    (68) CpxRNTS
    (69) IntTrsBoundProof [UPPER BOUND(ID), 487 ms]
    (70) CpxRNTS
    (71) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (72) CpxRNTS
    (73) IntTrsBoundProof [UPPER BOUND(ID), 605 ms]
    (74) CpxRNTS
    (75) IntTrsBoundProof [UPPER BOUND(ID), 115 ms]
    (76) CpxRNTS
    (77) FinalProof [FINISHED, 0 ms]
    (78) BOUNDS(1, n^3)
(79) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(80) CpxRelTRS
    (81) SlicingProof [LOWER BOUND(ID), 0 ms]
    (82) CpxRelTRS
    (83) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) typed CpxTrs
    (85) OrderProof [LOWER BOUND(ID), 22 ms]
    (86) typed CpxTrs
    (87) RewriteLemmaProof [LOWER BOUND(ID), 366 ms]
    (88) BEST
        (89) proven lower bound
            (90) LowerBoundPropagationProof [FINISHED, 0 ms]
            (91) BOUNDS(n^1, INF)
        (92) typed CpxTrs
            (93) RewriteLemmaProof [LOWER BOUND(ID), 99 ms]
            (94) typed CpxTrs
            (95) RewriteLemmaProof [LOWER BOUND(ID), 59 ms]
            (96) typed CpxTrs
            (97) RewriteLemmaProof [LOWER BOUND(ID), 611 ms]
            (98) typed CpxTrs
            (99) RewriteLemmaProof [LOWER BOUND(ID), 18 ms]
            (100) typed CpxTrs
            (101) RewriteLemmaProof [LOWER BOUND(ID), 97 ms]
            (102) typed CpxTrs
            (103) RewriteLemmaProof [LOWER BOUND(ID), 18.4 s]
            (104) BEST
                (105) proven lower bound
                    (106) LowerBoundPropagationProof [FINISHED, 0 ms]
                    (107) BOUNDS(n^2, INF)
                (108) typed CpxTrs
                    (109) RewriteLemmaProof [LOWER BOUND(ID), 1197 ms]
                    (110) BOUNDS(1, INF)


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^2, n^3).


The TRS R consists of the following rules:

   EQ(0, 0) -> c
   EQ(0, s(z0)) -> c1
   EQ(s(z0), 0) -> c2
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   LE(0, z0) -> c4
   LE(s(z0), 0) -> c5
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   MIN(cons(0, nil)) -> c7
   MIN(cons(s(z0), nil)) -> c8
   MIN(cons(z0, cons(z1, z2))) -> c9(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
   IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
   IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
   REPLACE(z0, z1, nil) -> c12
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
   IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c14
   IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
   SORT(nil) -> c16
   SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
   SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1)))

The (relative) TRS S consists of the following rules:

   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   min(cons(0, nil)) -> 0
   min(cons(s(z0), nil)) -> s(z0)
   min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
   if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
   if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
   replace(z0, z1, nil) -> nil
   replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
   if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
   if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
   sort(nil) -> nil
   sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))

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

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

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


The TRS R consists of the following rules:

   EQ(0, 0) -> c
   EQ(0, s(z0)) -> c1
   EQ(s(z0), 0) -> c2
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   LE(0, z0) -> c4
   LE(s(z0), 0) -> c5
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   MIN(cons(0, nil)) -> c7
   MIN(cons(s(z0), nil)) -> c8
   MIN(cons(z0, cons(z1, z2))) -> c9(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
   IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
   IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
   REPLACE(z0, z1, nil) -> c12
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
   IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c14
   IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
   SORT(nil) -> c16
   SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
   SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1)))

The (relative) TRS S consists of the following rules:

   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   min(cons(0, nil)) -> 0
   min(cons(s(z0), nil)) -> s(z0)
   min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
   if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
   if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
   replace(z0, z1, nil) -> nil
   replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
   if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
   if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
   sort(nil) -> nil
   sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))

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

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

(4)
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) -> c [1]
   EQ(0, s(z0)) -> c1 [1]
   EQ(s(z0), 0) -> c2 [1]
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) [1]
   LE(0, z0) -> c4 [1]
   LE(s(z0), 0) -> c5 [1]
   LE(s(z0), s(z1)) -> c6(LE(z0, z1)) [1]
   MIN(cons(0, nil)) -> c7 [1]
   MIN(cons(s(z0), nil)) -> c8 [1]
   MIN(cons(z0, cons(z1, z2))) -> c9(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1)) [1]
   IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2))) [1]
   IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2))) [1]
   REPLACE(z0, z1, nil) -> c12 [1]
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) [1]
   IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c14 [1]
   IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3)) [1]
   SORT(nil) -> c16 [1]
   SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1))) [1]
   SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1))) [1]
   eq(0, 0) -> true [0]
   eq(0, s(z0)) -> false [0]
   eq(s(z0), 0) -> false [0]
   eq(s(z0), s(z1)) -> eq(z0, z1) [0]
   le(0, z0) -> true [0]
   le(s(z0), 0) -> false [0]
   le(s(z0), s(z1)) -> le(z0, z1) [0]
   min(cons(0, nil)) -> 0 [0]
   min(cons(s(z0), nil)) -> s(z0) [0]
   min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2))) [0]
   if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) [0]
   if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) [0]
   replace(z0, z1, nil) -> nil [0]
   replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3)) [0]
   if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) [0]
   if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) [0]
   sort(nil) -> nil [0]
   sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1))) [0]

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

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

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   EQ(0, 0) -> c [1]
   EQ(0, s(z0)) -> c1 [1]
   EQ(s(z0), 0) -> c2 [1]
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) [1]
   LE(0, z0) -> c4 [1]
   LE(s(z0), 0) -> c5 [1]
   LE(s(z0), s(z1)) -> c6(LE(z0, z1)) [1]
   MIN(cons(0, nil)) -> c7 [1]
   MIN(cons(s(z0), nil)) -> c8 [1]
   MIN(cons(z0, cons(z1, z2))) -> c9(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1)) [1]
   IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2))) [1]
   IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2))) [1]
   REPLACE(z0, z1, nil) -> c12 [1]
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) [1]
   IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c14 [1]
   IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3)) [1]
   SORT(nil) -> c16 [1]
   SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1))) [1]
   SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1))) [1]
   eq(0, 0) -> true [0]
   eq(0, s(z0)) -> false [0]
   eq(s(z0), 0) -> false [0]
   eq(s(z0), s(z1)) -> eq(z0, z1) [0]
   le(0, z0) -> true [0]
   le(s(z0), 0) -> false [0]
   le(s(z0), s(z1)) -> le(z0, z1) [0]
   min(cons(0, nil)) -> 0 [0]
   min(cons(s(z0), nil)) -> s(z0) [0]
   min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2))) [0]
   if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) [0]
   if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) [0]
   replace(z0, z1, nil) -> nil [0]
   replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3)) [0]
   if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) [0]
   if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) [0]
   sort(nil) -> nil [0]
   sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1))) [0]

The TRS has the following type information:
EQ :: 0:s -> 0:s -> c:c1:c2:c3
0 :: 0:s
c :: c:c1:c2:c3
s :: 0:s -> 0:s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
LE :: 0:s -> 0:s -> c4:c5:c6
c4 :: c4:c5:c6
c5 :: c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
MIN :: nil:cons -> c7:c8:c9
cons :: 0:s -> nil:cons -> nil:cons
nil :: nil:cons
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c4:c5:c6 -> c7:c8:c9
IF_MIN :: true:false -> nil:cons -> c10:c11
le :: 0:s -> 0:s -> true:false
true :: true:false
c10 :: c7:c8:c9 -> c10:c11
false :: true:false
c11 :: c7:c8:c9 -> c10:c11
REPLACE :: 0:s -> 0:s -> nil:cons -> c12:c13
c12 :: c12:c13
c13 :: c14:c15 -> c:c1:c2:c3 -> c12:c13
IF_REPLACE :: true:false -> 0:s -> 0:s -> nil:cons -> c14:c15
eq :: 0:s -> 0:s -> true:false
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SORT :: nil:cons -> c16:c17:c18
c16 :: c16:c17:c18
c17 :: c7:c8:c9 -> c16:c17:c18
c18 :: c16:c17:c18 -> c12:c13 -> c7:c8:c9 -> c16:c17:c18
replace :: 0:s -> 0:s -> nil:cons -> nil:cons
min :: nil:cons -> 0:s
if_min :: true:false -> nil:cons -> 0:s
if_replace :: true:false -> 0:s -> 0:s -> nil:cons -> nil:cons
sort :: nil:cons -> nil:cons

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

(7) 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:

   EQ_2
   LE_2
   MIN_1
   IF_MIN_2
   REPLACE_3
   IF_REPLACE_4
   SORT_1

(c) The following functions are completely defined:

   eq_2
   le_2
   min_1
   if_min_2
   replace_3
   if_replace_4
   sort_1

Due to the following rules being added:

   eq(v0, v1) -> null_eq [0]
   le(v0, v1) -> null_le [0]
   min(v0) -> 0 [0]
   if_min(v0, v1) -> 0 [0]
   replace(v0, v1, v2) -> nil [0]
   if_replace(v0, v1, v2, v3) -> nil [0]
   sort(v0) -> nil [0]

And the following fresh constants:    null_eq, null_le, const

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

(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) -> c [1]
   EQ(0, s(z0)) -> c1 [1]
   EQ(s(z0), 0) -> c2 [1]
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) [1]
   LE(0, z0) -> c4 [1]
   LE(s(z0), 0) -> c5 [1]
   LE(s(z0), s(z1)) -> c6(LE(z0, z1)) [1]
   MIN(cons(0, nil)) -> c7 [1]
   MIN(cons(s(z0), nil)) -> c8 [1]
   MIN(cons(z0, cons(z1, z2))) -> c9(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1)) [1]
   IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2))) [1]
   IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2))) [1]
   REPLACE(z0, z1, nil) -> c12 [1]
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) [1]
   IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c14 [1]
   IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3)) [1]
   SORT(nil) -> c16 [1]
   SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1))) [1]
   SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1))) [1]
   eq(0, 0) -> true [0]
   eq(0, s(z0)) -> false [0]
   eq(s(z0), 0) -> false [0]
   eq(s(z0), s(z1)) -> eq(z0, z1) [0]
   le(0, z0) -> true [0]
   le(s(z0), 0) -> false [0]
   le(s(z0), s(z1)) -> le(z0, z1) [0]
   min(cons(0, nil)) -> 0 [0]
   min(cons(s(z0), nil)) -> s(z0) [0]
   min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2))) [0]
   if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) [0]
   if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) [0]
   replace(z0, z1, nil) -> nil [0]
   replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3)) [0]
   if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) [0]
   if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) [0]
   sort(nil) -> nil [0]
   sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1))) [0]
   eq(v0, v1) -> null_eq [0]
   le(v0, v1) -> null_le [0]
   min(v0) -> 0 [0]
   if_min(v0, v1) -> 0 [0]
   replace(v0, v1, v2) -> nil [0]
   if_replace(v0, v1, v2, v3) -> nil [0]
   sort(v0) -> nil [0]

The TRS has the following type information:
EQ :: 0:s -> 0:s -> c:c1:c2:c3
0 :: 0:s
c :: c:c1:c2:c3
s :: 0:s -> 0:s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
LE :: 0:s -> 0:s -> c4:c5:c6
c4 :: c4:c5:c6
c5 :: c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
MIN :: nil:cons -> c7:c8:c9
cons :: 0:s -> nil:cons -> nil:cons
nil :: nil:cons
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c4:c5:c6 -> c7:c8:c9
IF_MIN :: true:false:null_eq:null_le -> nil:cons -> c10:c11
le :: 0:s -> 0:s -> true:false:null_eq:null_le
true :: true:false:null_eq:null_le
c10 :: c7:c8:c9 -> c10:c11
false :: true:false:null_eq:null_le
c11 :: c7:c8:c9 -> c10:c11
REPLACE :: 0:s -> 0:s -> nil:cons -> c12:c13
c12 :: c12:c13
c13 :: c14:c15 -> c:c1:c2:c3 -> c12:c13
IF_REPLACE :: true:false:null_eq:null_le -> 0:s -> 0:s -> nil:cons -> c14:c15
eq :: 0:s -> 0:s -> true:false:null_eq:null_le
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SORT :: nil:cons -> c16:c17:c18
c16 :: c16:c17:c18
c17 :: c7:c8:c9 -> c16:c17:c18
c18 :: c16:c17:c18 -> c12:c13 -> c7:c8:c9 -> c16:c17:c18
replace :: 0:s -> 0:s -> nil:cons -> nil:cons
min :: nil:cons -> 0:s
if_min :: true:false:null_eq:null_le -> nil:cons -> 0:s
if_replace :: true:false:null_eq:null_le -> 0:s -> 0:s -> nil:cons -> nil:cons
sort :: nil:cons -> nil:cons
null_eq :: true:false:null_eq:null_le
null_le :: true:false:null_eq:null_le
const :: c10:c11

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

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

(10)
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) -> c [1]
   EQ(0, s(z0)) -> c1 [1]
   EQ(s(z0), 0) -> c2 [1]
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) [1]
   LE(0, z0) -> c4 [1]
   LE(s(z0), 0) -> c5 [1]
   LE(s(z0), s(z1)) -> c6(LE(z0, z1)) [1]
   MIN(cons(0, nil)) -> c7 [1]
   MIN(cons(s(z0), nil)) -> c8 [1]
   MIN(cons(0, cons(z1, z2))) -> c9(IF_MIN(true, cons(0, cons(z1, z2))), LE(0, z1)) [1]
   MIN(cons(s(z0'), cons(0, z2))) -> c9(IF_MIN(false, cons(s(z0'), cons(0, z2))), LE(s(z0'), 0)) [1]
   MIN(cons(s(z0''), cons(s(z1'), z2))) -> c9(IF_MIN(le(z0'', z1'), cons(s(z0''), cons(s(z1'), z2))), LE(s(z0''), s(z1'))) [1]
   MIN(cons(z0, cons(z1, z2))) -> c9(IF_MIN(null_le, cons(z0, cons(z1, z2))), LE(z0, z1)) [1]
   IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2))) [1]
   IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2))) [1]
   REPLACE(z0, z1, nil) -> c12 [1]
   REPLACE(0, z1, cons(0, z3)) -> c13(IF_REPLACE(true, 0, z1, cons(0, z3)), EQ(0, 0)) [1]
   REPLACE(0, z1, cons(s(z01), z3)) -> c13(IF_REPLACE(false, 0, z1, cons(s(z01), z3)), EQ(0, s(z01))) [1]
   REPLACE(s(z02), z1, cons(0, z3)) -> c13(IF_REPLACE(false, s(z02), z1, cons(0, z3)), EQ(s(z02), 0)) [1]
   REPLACE(s(z03), z1, cons(s(z1''), z3)) -> c13(IF_REPLACE(eq(z03, z1''), s(z03), z1, cons(s(z1''), z3)), EQ(s(z03), s(z1''))) [1]
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(null_eq, z0, z1, cons(z2, z3)), EQ(z0, z2)) [1]
   IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c14 [1]
   IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3)) [1]
   SORT(nil) -> c16 [1]
   SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1))) [1]
   SORT(cons(0, nil)) -> c18(SORT(replace(0, 0, nil)), REPLACE(0, 0, nil), MIN(cons(0, nil))) [1]
   SORT(cons(0, nil)) -> c18(SORT(replace(0, 0, nil)), REPLACE(0, 0, nil), MIN(cons(0, nil))) [1]
   SORT(cons(s(z04), nil)) -> c18(SORT(replace(s(z04), s(z04), nil)), REPLACE(s(z04), s(z04), nil), MIN(cons(s(z04), nil))) [1]
   SORT(cons(s(z04), nil)) -> c18(SORT(replace(s(z04), s(z04), nil)), REPLACE(0, s(z04), nil), MIN(cons(s(z04), nil))) [1]
   SORT(cons(z0, cons(z11, z2'))) -> c18(SORT(replace(if_min(le(z0, z11), cons(z0, cons(z11, z2'))), z0, cons(z11, z2'))), REPLACE(if_min(le(z0, z11), cons(z0, cons(z11, z2'))), z0, cons(z11, z2')), MIN(cons(z0, cons(z11, z2')))) [1]
   SORT(cons(z0, cons(z11, z2'))) -> c18(SORT(replace(if_min(le(z0, z11), cons(z0, cons(z11, z2'))), z0, cons(z11, z2'))), REPLACE(0, z0, cons(z11, z2')), MIN(cons(z0, cons(z11, z2')))) [1]
   SORT(cons(0, nil)) -> c18(SORT(replace(0, 0, nil)), REPLACE(0, 0, nil), MIN(cons(0, nil))) [1]
   SORT(cons(s(z05), nil)) -> c18(SORT(replace(0, s(z05), nil)), REPLACE(s(z05), s(z05), nil), MIN(cons(s(z05), nil))) [1]
   SORT(cons(z0, cons(z12, z2''))) -> c18(SORT(replace(0, z0, cons(z12, z2''))), REPLACE(if_min(le(z0, z12), cons(z0, cons(z12, z2''))), z0, cons(z12, z2'')), MIN(cons(z0, cons(z12, z2'')))) [1]
   SORT(cons(z0, z1)) -> c18(SORT(replace(0, z0, z1)), REPLACE(0, z0, z1), MIN(cons(z0, z1))) [1]
   eq(0, 0) -> true [0]
   eq(0, s(z0)) -> false [0]
   eq(s(z0), 0) -> false [0]
   eq(s(z0), s(z1)) -> eq(z0, z1) [0]
   le(0, z0) -> true [0]
   le(s(z0), 0) -> false [0]
   le(s(z0), s(z1)) -> le(z0, z1) [0]
   min(cons(0, nil)) -> 0 [0]
   min(cons(s(z0), nil)) -> s(z0) [0]
   min(cons(0, cons(z1, z2))) -> if_min(true, cons(0, cons(z1, z2))) [0]
   min(cons(s(z06), cons(0, z2))) -> if_min(false, cons(s(z06), cons(0, z2))) [0]
   min(cons(s(z07), cons(s(z13), z2))) -> if_min(le(z07, z13), cons(s(z07), cons(s(z13), z2))) [0]
   min(cons(z0, cons(z1, z2))) -> if_min(null_le, cons(z0, cons(z1, z2))) [0]
   if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) [0]
   if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) [0]
   replace(z0, z1, nil) -> nil [0]
   replace(0, z1, cons(0, z3)) -> if_replace(true, 0, z1, cons(0, z3)) [0]
   replace(0, z1, cons(s(z08), z3)) -> if_replace(false, 0, z1, cons(s(z08), z3)) [0]
   replace(s(z09), z1, cons(0, z3)) -> if_replace(false, s(z09), z1, cons(0, z3)) [0]
   replace(s(z010), z1, cons(s(z14), z3)) -> if_replace(eq(z010, z14), s(z010), z1, cons(s(z14), z3)) [0]
   replace(z0, z1, cons(z2, z3)) -> if_replace(null_eq, z0, z1, cons(z2, z3)) [0]
   if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) [0]
   if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) [0]
   sort(nil) -> nil [0]
   sort(cons(0, nil)) -> cons(min(cons(0, nil)), sort(replace(0, 0, nil))) [0]
   sort(cons(s(z011), nil)) -> cons(min(cons(s(z011), nil)), sort(replace(s(z011), s(z011), nil))) [0]
   sort(cons(z0, cons(z15, z21))) -> cons(min(cons(z0, cons(z15, z21))), sort(replace(if_min(le(z0, z15), cons(z0, cons(z15, z21))), z0, cons(z15, z21)))) [0]
   sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(0, z0, z1))) [0]
   eq(v0, v1) -> null_eq [0]
   le(v0, v1) -> null_le [0]
   min(v0) -> 0 [0]
   if_min(v0, v1) -> 0 [0]
   replace(v0, v1, v2) -> nil [0]
   if_replace(v0, v1, v2, v3) -> nil [0]
   sort(v0) -> nil [0]

The TRS has the following type information:
EQ :: 0:s -> 0:s -> c:c1:c2:c3
0 :: 0:s
c :: c:c1:c2:c3
s :: 0:s -> 0:s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
LE :: 0:s -> 0:s -> c4:c5:c6
c4 :: c4:c5:c6
c5 :: c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
MIN :: nil:cons -> c7:c8:c9
cons :: 0:s -> nil:cons -> nil:cons
nil :: nil:cons
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c4:c5:c6 -> c7:c8:c9
IF_MIN :: true:false:null_eq:null_le -> nil:cons -> c10:c11
le :: 0:s -> 0:s -> true:false:null_eq:null_le
true :: true:false:null_eq:null_le
c10 :: c7:c8:c9 -> c10:c11
false :: true:false:null_eq:null_le
c11 :: c7:c8:c9 -> c10:c11
REPLACE :: 0:s -> 0:s -> nil:cons -> c12:c13
c12 :: c12:c13
c13 :: c14:c15 -> c:c1:c2:c3 -> c12:c13
IF_REPLACE :: true:false:null_eq:null_le -> 0:s -> 0:s -> nil:cons -> c14:c15
eq :: 0:s -> 0:s -> true:false:null_eq:null_le
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SORT :: nil:cons -> c16:c17:c18
c16 :: c16:c17:c18
c17 :: c7:c8:c9 -> c16:c17:c18
c18 :: c16:c17:c18 -> c12:c13 -> c7:c8:c9 -> c16:c17:c18
replace :: 0:s -> 0:s -> nil:cons -> nil:cons
min :: nil:cons -> 0:s
if_min :: true:false:null_eq:null_le -> nil:cons -> 0:s
if_replace :: true:false:null_eq:null_le -> 0:s -> 0:s -> nil:cons -> nil:cons
sort :: nil:cons -> nil:cons
null_eq :: true:false:null_eq:null_le
null_le :: true:false:null_eq:null_le
const :: c10:c11

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

(11) 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
   c => 0
   c1 => 1
   c2 => 2
   c4 => 0
   c5 => 1
   nil => 0
   c7 => 0
   c8 => 1
   true => 2
   false => 1
   c12 => 0
   c14 => 0
   c16 => 0
   null_eq => 0
   null_le => 0
   const => 0

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z = 1 + z0, z0 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z0 >= 0, z' = 1 + z0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 + EQ(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0, z'' = z1
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z0, z1, z3) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0, z'' = z1
   LE(z, z') -{ 1 }-> 1 :|: z = 1 + z0, z0 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   LE(z, z') -{ 1 }-> 1 + LE(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   MIN(z) -{ 1 }-> 1 :|: z = 1 + (1 + z0) + 0, z0 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(le(z0'', z1'), 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + LE(1 + z0', 0) :|: z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + LE(z0, z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z = z0, z1 >= 0, z' = z1, z0 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(eq(z03, z1''), 1 + z03, z1, 1 + (1 + z1'') + z3) + EQ(1 + z03, 1 + z1'') :|: z1 >= 0, z'' = 1 + (1 + z1'') + z3, z' = z1, z = 1 + z03, z03 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(2, 0, z1, 1 + 0 + z3) + EQ(0, 0) :|: z1 >= 0, z'' = 1 + 0 + z3, z' = z1, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(1, 0, z1, 1 + (1 + z01) + z3) + EQ(0, 1 + z01) :|: z1 >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z' = z1, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(1, 1 + z02, z1, 1 + 0 + z3) + EQ(1 + z02, 0) :|: z1 >= 0, z = 1 + z02, z'' = 1 + 0 + z3, z02 >= 0, z' = z1, z3 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(0, z0, z1, 1 + z2 + z3) + EQ(z0, z2) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(le(z0, z12), 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + z05, 0)) + REPLACE(1 + z05, 1 + z05, 0) + MIN(1 + (1 + z05) + 0) :|: z = 1 + (1 + z05) + 0, z05 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + z04, 1 + z04, 0)) + REPLACE(0, 1 + z04, 0) + MIN(1 + (1 + z04) + 0) :|: z04 >= 0, z = 1 + (1 + z04) + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + z04, 1 + z04, 0)) + REPLACE(1 + z04, 1 + z04, 0) + MIN(1 + (1 + z04) + 0) :|: z04 >= 0, z = 1 + (1 + z04) + 0
   eq(z, z') -{ 0 }-> eq(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z0 >= 0, z' = 1 + z0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z = 1 + z0, z0 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z4 = v3, v2 >= 0, v3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z1 + z3 :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0, z'' = z1
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z0, z1, z3) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0, z'' = z1
   le(z, z') -{ 0 }-> le(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   le(z, z') -{ 0 }-> 2 :|: z0 >= 0, z = 0, z' = z0
   le(z, z') -{ 0 }-> 1 :|: z = 1 + z0, z0 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   min(z) -{ 0 }-> if_min(le(z07, z13), 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   min(z) -{ 0 }-> 1 + z0 :|: z = 1 + (1 + z0) + 0, z0 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z010, z14), 1 + z010, z1, 1 + (1 + z14) + z3) :|: z1 >= 0, z'' = 1 + (1 + z14) + z3, z' = z1, z010 >= 0, z14 >= 0, z3 >= 0, z = 1 + z010
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z1, 1 + 0 + z3) :|: z1 >= 0, z'' = 1 + 0 + z3, z' = z1, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z1, 1 + (1 + z08) + z3) :|: z08 >= 0, z1 >= 0, z' = z1, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + z09, z1, 1 + 0 + z3) :|: z1 >= 0, z'' = 1 + 0 + z3, z' = z1, z = 1 + z09, z09 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z0, z1, 1 + z2 + z3) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z = z0, z1 >= 0, z' = z1, z0 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(le(z0, z15), 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + z011) + 0) + sort(replace(1 + z011, 1 + z011, 0)) :|: z011 >= 0, z = 1 + (1 + z011) + 0


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

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 + EQ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(le(z0'', z1'), 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + LE(1 + z0', 0) :|: z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + LE(z0, z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + EQ(1 + (z - 1), 1 + z1'') :|: z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(2, 0, z', 1 + 0 + (z'' - 1)) + EQ(0, 0) :|: z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(1, 0, z', 1 + (1 + z01) + z3) + EQ(0, 1 + z01) :|: z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) + EQ(1 + (z - 1), 0) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(0, z, z', 1 + z2 + z3) + EQ(z, z2) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(le(z0, z12), 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(le(z07, z13), 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z14), 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(le(z0, z15), 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + (z - 2)) + 0) + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: z - 2 >= 0


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

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

   { EQ }
   { le }
   { LE }
   { eq }
   { REPLACE, IF_REPLACE }
   { IF_MIN, MIN }
   { min, if_min }
   { replace, if_replace }
   { SORT }
   { sort }

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 + EQ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(le(z0'', z1'), 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + LE(1 + z0', 0) :|: z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + LE(z0, z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + EQ(1 + (z - 1), 1 + z1'') :|: z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(2, 0, z', 1 + 0 + (z'' - 1)) + EQ(0, 0) :|: z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(1, 0, z', 1 + (1 + z01) + z3) + EQ(0, 1 + z01) :|: z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) + EQ(1 + (z - 1), 0) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(0, z, z', 1 + z2 + z3) + EQ(z, z2) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(le(z0, z12), 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(le(z07, z13), 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z14), 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(le(z0, z15), 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + (z - 2)) + 0) + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: z - 2 >= 0

Function symbols to be analyzed: {EQ}, {le}, {LE}, {eq}, {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}, {sort}

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

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 + EQ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(le(z0'', z1'), 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + LE(1 + z0', 0) :|: z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + LE(z0, z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + EQ(1 + (z - 1), 1 + z1'') :|: z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(2, 0, z', 1 + 0 + (z'' - 1)) + EQ(0, 0) :|: z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(1, 0, z', 1 + (1 + z01) + z3) + EQ(0, 1 + z01) :|: z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) + EQ(1 + (z - 1), 0) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(0, z, z', 1 + z2 + z3) + EQ(z, z2) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(le(z0, z12), 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(le(z07, z13), 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z14), 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(le(z0, z15), 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + (z - 2)) + 0) + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: z - 2 >= 0

Function symbols to be analyzed: {EQ}, {le}, {LE}, {eq}, {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}, {sort}

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 + EQ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(le(z0'', z1'), 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + LE(1 + z0', 0) :|: z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + LE(z0, z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + EQ(1 + (z - 1), 1 + z1'') :|: z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(2, 0, z', 1 + 0 + (z'' - 1)) + EQ(0, 0) :|: z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(1, 0, z', 1 + (1 + z01) + z3) + EQ(0, 1 + z01) :|: z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) + EQ(1 + (z - 1), 0) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(0, z, z', 1 + z2 + z3) + EQ(z, z2) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(le(z0, z12), 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(le(z07, z13), 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z14), 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(le(z0, z15), 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + (z - 2)) + 0) + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: z - 2 >= 0

Function symbols to be analyzed: {EQ}, {le}, {LE}, {eq}, {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: ?, size: O(n^1) [2 + z']

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

(21) 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'

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 + EQ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(le(z0'', z1'), 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + LE(1 + z0', 0) :|: z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + LE(z0, z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + EQ(1 + (z - 1), 1 + z1'') :|: z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(2, 0, z', 1 + 0 + (z'' - 1)) + EQ(0, 0) :|: z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(1, 0, z', 1 + (1 + z01) + z3) + EQ(0, 1 + z01) :|: z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) + EQ(1 + (z - 1), 0) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(0, z, z', 1 + z2 + z3) + EQ(z, z2) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(le(z0, z12), 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(le(z07, z13), 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z14), 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(le(z0, z15), 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + (z - 2)) + 0) + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: z - 2 >= 0

Function symbols to be analyzed: {le}, {LE}, {eq}, {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']

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

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(le(z0'', z1'), 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + LE(1 + z0', 0) :|: z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + LE(z0, z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 5 + z1'' }-> 1 + IF_REPLACE(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(2, 0, z', 1 + 0 + (z'' - 1)) + s' :|: s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 5 + z01 }-> 1 + IF_REPLACE(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) + s1 :|: s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 4 + z2 }-> 1 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(le(z0, z12), 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(le(z07, z13), 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z14), 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(le(z0, z15), 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + (z - 2)) + 0) + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: z - 2 >= 0

Function symbols to be analyzed: {le}, {LE}, {eq}, {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']

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

(25) 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

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(le(z0'', z1'), 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + LE(1 + z0', 0) :|: z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + LE(z0, z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 5 + z1'' }-> 1 + IF_REPLACE(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(2, 0, z', 1 + 0 + (z'' - 1)) + s' :|: s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 5 + z01 }-> 1 + IF_REPLACE(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) + s1 :|: s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 4 + z2 }-> 1 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(le(z0, z12), 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(le(z07, z13), 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z14), 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(le(z0, z15), 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + (z - 2)) + 0) + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: z - 2 >= 0

Function symbols to be analyzed: {le}, {LE}, {eq}, {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: ?, size: O(1) [2]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: le
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(le(z0'', z1'), 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + LE(1 + z0', 0) :|: z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + LE(z0, z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 5 + z1'' }-> 1 + IF_REPLACE(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(2, 0, z', 1 + 0 + (z'' - 1)) + s' :|: s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 5 + z01 }-> 1 + IF_REPLACE(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) + s1 :|: s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 4 + z2 }-> 1 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(le(z0, z12), 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(le(z07, z13), 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z14), 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(le(z0, z15), 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + (z - 2)) + 0) + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: z - 2 >= 0

Function symbols to be analyzed: {LE}, {eq}, {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]

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

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + LE(1 + z0', 0) :|: z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + LE(z0, z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 5 + z1'' }-> 1 + IF_REPLACE(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(2, 0, z', 1 + 0 + (z'' - 1)) + s' :|: s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 5 + z01 }-> 1 + IF_REPLACE(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) + s1 :|: s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 4 + z2 }-> 1 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z14), 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(s11, 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + (z - 2)) + 0) + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: z - 2 >= 0

Function symbols to be analyzed: {LE}, {eq}, {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + LE(1 + z0', 0) :|: z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + LE(z0, z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 5 + z1'' }-> 1 + IF_REPLACE(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(2, 0, z', 1 + 0 + (z'' - 1)) + s' :|: s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 5 + z01 }-> 1 + IF_REPLACE(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) + s1 :|: s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 4 + z2 }-> 1 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z14), 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(s11, 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + (z - 2)) + 0) + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: z - 2 >= 0

Function symbols to be analyzed: {LE}, {eq}, {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: ?, size: O(n^1) [1 + z']

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

(33) 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'

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + LE(1 + z0', 0) :|: z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + LE(z0, z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 5 + z1'' }-> 1 + IF_REPLACE(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(2, 0, z', 1 + 0 + (z'' - 1)) + s' :|: s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 5 + z01 }-> 1 + IF_REPLACE(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) + s1 :|: s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 4 + z2 }-> 1 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z14), 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(s11, 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + (z - 2)) + 0) + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: z - 2 >= 0

Function symbols to be analyzed: {eq}, {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']

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

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 4 + z1' }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s15 :|: s15 >= 0, s15 <= 1 + z1' + 1, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + s13 :|: s13 >= 0, s13 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 3 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s14 :|: s14 >= 0, s14 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + s16 :|: s16 >= 0, s16 <= z1 + 1, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 5 + z1'' }-> 1 + IF_REPLACE(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(2, 0, z', 1 + 0 + (z'' - 1)) + s' :|: s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 5 + z01 }-> 1 + IF_REPLACE(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) + s1 :|: s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 4 + z2 }-> 1 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z14), 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(s11, 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + (z - 2)) + 0) + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: z - 2 >= 0

Function symbols to be analyzed: {eq}, {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']

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

(37) 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

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 4 + z1' }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s15 :|: s15 >= 0, s15 <= 1 + z1' + 1, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + s13 :|: s13 >= 0, s13 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 3 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s14 :|: s14 >= 0, s14 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + s16 :|: s16 >= 0, s16 <= z1 + 1, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 5 + z1'' }-> 1 + IF_REPLACE(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(2, 0, z', 1 + 0 + (z'' - 1)) + s' :|: s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 5 + z01 }-> 1 + IF_REPLACE(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) + s1 :|: s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 4 + z2 }-> 1 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z14), 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(s11, 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + (z - 2)) + 0) + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: z - 2 >= 0

Function symbols to be analyzed: {eq}, {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  eq: runtime: ?, size: O(1) [2]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: eq
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 4 + z1' }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s15 :|: s15 >= 0, s15 <= 1 + z1' + 1, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + s13 :|: s13 >= 0, s13 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 3 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s14 :|: s14 >= 0, s14 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + s16 :|: s16 >= 0, s16 <= z1 + 1, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 5 + z1'' }-> 1 + IF_REPLACE(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(2, 0, z', 1 + 0 + (z'' - 1)) + s' :|: s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 5 + z01 }-> 1 + IF_REPLACE(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) + s1 :|: s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 4 + z2 }-> 1 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z14), 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(s11, 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + (z - 2)) + 0) + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: z - 2 >= 0

Function symbols to be analyzed: {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  eq: runtime: O(1) [0], size: O(1) [2]

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

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 4 + z1' }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s15 :|: s15 >= 0, s15 <= 1 + z1' + 1, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + s13 :|: s13 >= 0, s13 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 3 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s14 :|: s14 >= 0, s14 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + s16 :|: s16 >= 0, s16 <= z1 + 1, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 5 + z1'' }-> 1 + IF_REPLACE(s17, 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s17 >= 0, s17 <= 2, s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(2, 0, z', 1 + 0 + (z'' - 1)) + s' :|: s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 5 + z01 }-> 1 + IF_REPLACE(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) + s1 :|: s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 4 + z2 }-> 1 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> s18 :|: s18 >= 0, s18 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s19, 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: s19 >= 0, s19 <= 2, z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(s11, 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + (z - 2)) + 0) + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: z - 2 >= 0

Function symbols to be analyzed: {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  eq: runtime: O(1) [0], size: O(1) [2]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: REPLACE
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 6 + 7*z'' + z''^2

Computed SIZE bound using KoAT for: IF_REPLACE
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 7 + 7*z4 + z4^2

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 4 + z1' }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s15 :|: s15 >= 0, s15 <= 1 + z1' + 1, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + s13 :|: s13 >= 0, s13 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 3 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s14 :|: s14 >= 0, s14 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + s16 :|: s16 >= 0, s16 <= z1 + 1, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 5 + z1'' }-> 1 + IF_REPLACE(s17, 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s17 >= 0, s17 <= 2, s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(2, 0, z', 1 + 0 + (z'' - 1)) + s' :|: s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 5 + z01 }-> 1 + IF_REPLACE(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) + s1 :|: s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 4 + z2 }-> 1 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> s18 :|: s18 >= 0, s18 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s19, 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: s19 >= 0, s19 <= 2, z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(s11, 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + (z - 2)) + 0) + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: z - 2 >= 0

Function symbols to be analyzed: {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: ?, size: O(n^2) [6 + 7*z'' + z''^2]
  IF_REPLACE: runtime: ?, size: O(n^2) [7 + 7*z4 + z4^2]

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

(45) 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: 12 + 7*z'' + z''^2

Computed RUNTIME bound using KoAT for: IF_REPLACE
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 14 + 7*z4 + z4^2

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 4 + z1' }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s15 :|: s15 >= 0, s15 <= 1 + z1' + 1, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + s13 :|: s13 >= 0, s13 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 3 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s14 :|: s14 >= 0, s14 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + s16 :|: s16 >= 0, s16 <= z1 + 1, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 5 + z1'' }-> 1 + IF_REPLACE(s17, 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s17 >= 0, s17 <= 2, s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(2, 0, z', 1 + 0 + (z'' - 1)) + s' :|: s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 5 + z01 }-> 1 + IF_REPLACE(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 }-> 1 + IF_REPLACE(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) + s1 :|: s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 4 + z2 }-> 1 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> s18 :|: s18 >= 0, s18 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s19, 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: s19 >= 0, s19 <= 2, z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(s11, 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + (z - 2)) + 0) + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: z - 2 >= 0

Function symbols to be analyzed: {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [12 + 7*z'' + z''^2], size: O(n^2) [6 + 7*z'' + z''^2]
  IF_REPLACE: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]

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

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s25 :|: s25 >= 0, s25 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 4 + z1' }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s15 :|: s15 >= 0, s15 <= 1 + z1' + 1, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + s13 :|: s13 >= 0, s13 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 3 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s14 :|: s14 >= 0, s14 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + s16 :|: s16 >= 0, s16 <= z1 + 1, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s20 + s' :|: s20 >= 0, s20 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z01 + 2*z01*z3 + z01^2 + 11*z3 + z3^2 }-> 1 + s21 + s'' :|: s21 >= 0, s21 <= 7 * (1 + (1 + z01) + z3) + (1 + (1 + z01) + z3) * (1 + (1 + z01) + z3) + 7, s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s22 + s1 :|: s22 >= 0, s22 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z1'' + 2*z1''*z3 + z1''^2 + 11*z3 + z3^2 }-> 1 + s23 + s2 :|: s23 >= 0, s23 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s17 >= 0, s17 <= 2, s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 26 + 10*z2 + 2*z2*z3 + z2^2 + 9*z3 + z3^2 }-> 1 + s24 + s3 :|: s24 >= 0, s24 <= 7 * (1 + z2 + z3) + (1 + z2 + z3) * (1 + z2 + z3) + 7, s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 21 + 9*z11 + 2*z11*z2' + z11^2 + 9*z2' + z2'^2 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + s29 + MIN(1 + z0 + (1 + z11 + z2')) :|: s29 >= 0, s29 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 13 + 7*z1 + z1^2 }-> 1 + SORT(replace(0, z0, z1)) + s31 + MIN(1 + z0 + z1) :|: s31 >= 0, s31 <= 7 * z1 + 6 + z1 * z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 13 }-> 1 + SORT(replace(0, 0, 0)) + s26 + MIN(1 + 0 + 0) :|: s26 >= 0, s26 <= 7 * 0 + 6 + 0 * 0, z = 1 + 0 + 0
   SORT(z) -{ 13 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + s30 + MIN(1 + (1 + (z - 2)) + 0) :|: s30 >= 0, s30 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 13 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s27 + MIN(1 + (1 + (z - 2)) + 0) :|: s27 >= 0, s27 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 13 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s28 + MIN(1 + (1 + (z - 2)) + 0) :|: s28 >= 0, s28 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   eq(z, z') -{ 0 }-> s18 :|: s18 >= 0, s18 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s19, 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: s19 >= 0, s19 <= 2, z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(s11, 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + (z - 2)) + 0) + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: z - 2 >= 0

Function symbols to be analyzed: {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [12 + 7*z'' + z''^2], size: O(n^2) [6 + 7*z'' + z''^2]
  IF_REPLACE: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: IF_MIN
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 8 + 4*z' + z'^2

Computed SIZE bound using KoAT for: MIN
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 115 + 99*z + 26*z^2

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s25 :|: s25 >= 0, s25 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 4 + z1' }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s15 :|: s15 >= 0, s15 <= 1 + z1' + 1, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + s13 :|: s13 >= 0, s13 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 3 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s14 :|: s14 >= 0, s14 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + s16 :|: s16 >= 0, s16 <= z1 + 1, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s20 + s' :|: s20 >= 0, s20 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z01 + 2*z01*z3 + z01^2 + 11*z3 + z3^2 }-> 1 + s21 + s'' :|: s21 >= 0, s21 <= 7 * (1 + (1 + z01) + z3) + (1 + (1 + z01) + z3) * (1 + (1 + z01) + z3) + 7, s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s22 + s1 :|: s22 >= 0, s22 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z1'' + 2*z1''*z3 + z1''^2 + 11*z3 + z3^2 }-> 1 + s23 + s2 :|: s23 >= 0, s23 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s17 >= 0, s17 <= 2, s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 26 + 10*z2 + 2*z2*z3 + z2^2 + 9*z3 + z3^2 }-> 1 + s24 + s3 :|: s24 >= 0, s24 <= 7 * (1 + z2 + z3) + (1 + z2 + z3) * (1 + z2 + z3) + 7, s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 21 + 9*z11 + 2*z11*z2' + z11^2 + 9*z2' + z2'^2 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + s29 + MIN(1 + z0 + (1 + z11 + z2')) :|: s29 >= 0, s29 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 13 + 7*z1 + z1^2 }-> 1 + SORT(replace(0, z0, z1)) + s31 + MIN(1 + z0 + z1) :|: s31 >= 0, s31 <= 7 * z1 + 6 + z1 * z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 13 }-> 1 + SORT(replace(0, 0, 0)) + s26 + MIN(1 + 0 + 0) :|: s26 >= 0, s26 <= 7 * 0 + 6 + 0 * 0, z = 1 + 0 + 0
   SORT(z) -{ 13 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + s30 + MIN(1 + (1 + (z - 2)) + 0) :|: s30 >= 0, s30 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 13 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s27 + MIN(1 + (1 + (z - 2)) + 0) :|: s27 >= 0, s27 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 13 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s28 + MIN(1 + (1 + (z - 2)) + 0) :|: s28 >= 0, s28 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   eq(z, z') -{ 0 }-> s18 :|: s18 >= 0, s18 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s19, 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: s19 >= 0, s19 <= 2, z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(s11, 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + (z - 2)) + 0) + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: z - 2 >= 0

Function symbols to be analyzed: {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [12 + 7*z'' + z''^2], size: O(n^2) [6 + 7*z'' + z''^2]
  IF_REPLACE: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  IF_MIN: runtime: ?, size: O(n^2) [8 + 4*z' + z'^2]
  MIN: runtime: ?, size: O(n^2) [115 + 99*z + 26*z^2]

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: IF_MIN
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 10 + 6*z' + z'^2

Computed RUNTIME bound using KoAT for: MIN
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 154 + 119*z + 26*z^2

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s25 :|: s25 >= 0, s25 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 4 + z1' }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s15 :|: s15 >= 0, s15 <= 1 + z1' + 1, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + s13 :|: s13 >= 0, s13 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 3 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s14 :|: s14 >= 0, s14 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + s16 :|: s16 >= 0, s16 <= z1 + 1, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s20 + s' :|: s20 >= 0, s20 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z01 + 2*z01*z3 + z01^2 + 11*z3 + z3^2 }-> 1 + s21 + s'' :|: s21 >= 0, s21 <= 7 * (1 + (1 + z01) + z3) + (1 + (1 + z01) + z3) * (1 + (1 + z01) + z3) + 7, s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s22 + s1 :|: s22 >= 0, s22 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z1'' + 2*z1''*z3 + z1''^2 + 11*z3 + z3^2 }-> 1 + s23 + s2 :|: s23 >= 0, s23 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s17 >= 0, s17 <= 2, s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 26 + 10*z2 + 2*z2*z3 + z2^2 + 9*z3 + z3^2 }-> 1 + s24 + s3 :|: s24 >= 0, s24 <= 7 * (1 + z2 + z3) + (1 + z2 + z3) * (1 + z2 + z3) + 7, s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 21 + 9*z11 + 2*z11*z2' + z11^2 + 9*z2' + z2'^2 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + s29 + MIN(1 + z0 + (1 + z11 + z2')) :|: s29 >= 0, s29 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 13 + 7*z1 + z1^2 }-> 1 + SORT(replace(0, z0, z1)) + s31 + MIN(1 + z0 + z1) :|: s31 >= 0, s31 <= 7 * z1 + 6 + z1 * z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 13 }-> 1 + SORT(replace(0, 0, 0)) + s26 + MIN(1 + 0 + 0) :|: s26 >= 0, s26 <= 7 * 0 + 6 + 0 * 0, z = 1 + 0 + 0
   SORT(z) -{ 13 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + s30 + MIN(1 + (1 + (z - 2)) + 0) :|: s30 >= 0, s30 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 13 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s27 + MIN(1 + (1 + (z - 2)) + 0) :|: s27 >= 0, s27 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 13 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s28 + MIN(1 + (1 + (z - 2)) + 0) :|: s28 >= 0, s28 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   eq(z, z') -{ 0 }-> s18 :|: s18 >= 0, s18 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s19, 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: s19 >= 0, s19 <= 2, z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(s11, 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + (z - 2)) + 0) + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: z - 2 >= 0

Function symbols to be analyzed: {min,if_min}, {replace,if_replace}, {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [12 + 7*z'' + z''^2], size: O(n^2) [6 + 7*z'' + z''^2]
  IF_REPLACE: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  IF_MIN: runtime: O(n^2) [10 + 6*z' + z'^2], size: O(n^2) [8 + 4*z' + z'^2]
  MIN: runtime: O(n^2) [154 + 119*z + 26*z^2], size: O(n^2) [115 + 99*z + 26*z^2]

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

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 300 + 171*z0 + 52*z0*z2 + 26*z0^2 + 171*z2 + 26*z2^2 }-> 1 + s36 :|: s36 >= 0, s36 <= 99 * (1 + z0 + z2) + 26 * ((1 + z0 + z2) * (1 + z0 + z2)) + 115, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 300 + 171*z1 + 52*z1*z2 + 26*z1^2 + 171*z2 + 26*z2^2 }-> 1 + s37 :|: s37 >= 0, s37 <= 99 * (1 + z1 + z2) + 26 * ((1 + z1 + z2) * (1 + z1 + z2)) + 115, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s25 :|: s25 >= 0, s25 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 29 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s32 + s13 :|: s32 >= 0, s32 <= 8 + (1 + 0 + (1 + z1 + z2)) * (1 + 0 + (1 + z1 + z2)) + 4 * (1 + 0 + (1 + z1 + z2)), s13 >= 0, s13 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 40 + 12*z0' + 2*z0'*z2 + z0'^2 + 12*z2 + z2^2 }-> 1 + s33 + s14 :|: s33 >= 0, s33 <= 8 + (1 + (1 + z0') + (1 + 0 + z2)) * (1 + (1 + z0') + (1 + 0 + z2)) + 4 * (1 + (1 + z0') + (1 + 0 + z2)), s14 >= 0, s14 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 54 + 14*z0'' + 2*z0''*z1' + 2*z0''*z2 + z0''^2 + 15*z1' + 2*z1'*z2 + z1'^2 + 14*z2 + z2^2 }-> 1 + s34 + s15 :|: s34 >= 0, s34 <= 8 + (1 + (1 + z0'') + (1 + (1 + z1') + z2)) * (1 + (1 + z0'') + (1 + (1 + z1') + z2)) + 4 * (1 + (1 + z0'') + (1 + (1 + z1') + z2)), s15 >= 0, s15 <= 1 + z1' + 1, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 29 + 10*z0 + 2*z0*z1 + 2*z0*z2 + z0^2 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s35 + s16 :|: s35 >= 0, s35 <= 8 + (1 + z0 + (1 + z1 + z2)) * (1 + z0 + (1 + z1 + z2)) + 4 * (1 + z0 + (1 + z1 + z2)), s16 >= 0, s16 <= z1 + 1, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s20 + s' :|: s20 >= 0, s20 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z01 + 2*z01*z3 + z01^2 + 11*z3 + z3^2 }-> 1 + s21 + s'' :|: s21 >= 0, s21 <= 7 * (1 + (1 + z01) + z3) + (1 + (1 + z01) + z3) * (1 + (1 + z01) + z3) + 7, s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s22 + s1 :|: s22 >= 0, s22 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z1'' + 2*z1''*z3 + z1''^2 + 11*z3 + z3^2 }-> 1 + s23 + s2 :|: s23 >= 0, s23 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s17 >= 0, s17 <= 2, s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 26 + 10*z2 + 2*z2*z3 + z2^2 + 9*z3 + z3^2 }-> 1 + s24 + s3 :|: s24 >= 0, s24 <= 7 * (1 + z2 + z3) + (1 + z2 + z3) * (1 + z2 + z3) + 7, s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 300 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + s38 :|: s38 >= 0, s38 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 497 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 223*z11 + 52*z11*z2' + 26*z11^2 + 223*z2' + 26*z2'^2 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + s42 :|: s42 >= 0, s42 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 517 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 232*z11 + 54*z11*z2' + 27*z11^2 + 232*z2' + 27*z2'^2 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + s29 + s43 :|: s43 >= 0, s43 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s29 >= 0, s29 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 312 + 171*z0 + 52*z0*z1 + 26*z0^2 + 178*z1 + 27*z1^2 }-> 1 + SORT(replace(0, z0, z1)) + s31 + s46 :|: s46 >= 0, s46 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s31 >= 0, s31 <= 7 * z1 + 6 + z1 * z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 497 + 223*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 223*z12 + 52*z12*z2'' + 26*z12^2 + 223*z2'' + 26*z2''^2 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + s45 :|: s45 >= 0, s45 <= 99 * (1 + z0 + (1 + z12 + z2'')) + 26 * ((1 + z0 + (1 + z12 + z2'')) * (1 + z0 + (1 + z12 + z2''))) + 115, s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 312 }-> 1 + SORT(replace(0, 0, 0)) + s26 + s39 :|: s39 >= 0, s39 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s26 >= 0, s26 <= 7 * 0 + 6 + 0 * 0, z = 1 + 0 + 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + s30 + s44 :|: s44 >= 0, s44 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s30 >= 0, s30 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s27 + s40 :|: s40 >= 0, s40 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s27 >= 0, s27 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s28 + s41 :|: s41 >= 0, s41 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s28 >= 0, s28 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   eq(z, z') -{ 0 }-> s18 :|: s18 >= 0, s18 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s19, 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: s19 >= 0, s19 <= 2, z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(s11, 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + (z - 2)) + 0) + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: z - 2 >= 0

Function symbols to be analyzed: {min,if_min}, {replace,if_replace}, {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [12 + 7*z'' + z''^2], size: O(n^2) [6 + 7*z'' + z''^2]
  IF_REPLACE: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  IF_MIN: runtime: O(n^2) [10 + 6*z' + z'^2], size: O(n^2) [8 + 4*z' + z'^2]
  MIN: runtime: O(n^2) [154 + 119*z + 26*z^2], size: O(n^2) [115 + 99*z + 26*z^2]

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

(55) 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'

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 300 + 171*z0 + 52*z0*z2 + 26*z0^2 + 171*z2 + 26*z2^2 }-> 1 + s36 :|: s36 >= 0, s36 <= 99 * (1 + z0 + z2) + 26 * ((1 + z0 + z2) * (1 + z0 + z2)) + 115, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 300 + 171*z1 + 52*z1*z2 + 26*z1^2 + 171*z2 + 26*z2^2 }-> 1 + s37 :|: s37 >= 0, s37 <= 99 * (1 + z1 + z2) + 26 * ((1 + z1 + z2) * (1 + z1 + z2)) + 115, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s25 :|: s25 >= 0, s25 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 29 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s32 + s13 :|: s32 >= 0, s32 <= 8 + (1 + 0 + (1 + z1 + z2)) * (1 + 0 + (1 + z1 + z2)) + 4 * (1 + 0 + (1 + z1 + z2)), s13 >= 0, s13 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 40 + 12*z0' + 2*z0'*z2 + z0'^2 + 12*z2 + z2^2 }-> 1 + s33 + s14 :|: s33 >= 0, s33 <= 8 + (1 + (1 + z0') + (1 + 0 + z2)) * (1 + (1 + z0') + (1 + 0 + z2)) + 4 * (1 + (1 + z0') + (1 + 0 + z2)), s14 >= 0, s14 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 54 + 14*z0'' + 2*z0''*z1' + 2*z0''*z2 + z0''^2 + 15*z1' + 2*z1'*z2 + z1'^2 + 14*z2 + z2^2 }-> 1 + s34 + s15 :|: s34 >= 0, s34 <= 8 + (1 + (1 + z0'') + (1 + (1 + z1') + z2)) * (1 + (1 + z0'') + (1 + (1 + z1') + z2)) + 4 * (1 + (1 + z0'') + (1 + (1 + z1') + z2)), s15 >= 0, s15 <= 1 + z1' + 1, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 29 + 10*z0 + 2*z0*z1 + 2*z0*z2 + z0^2 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s35 + s16 :|: s35 >= 0, s35 <= 8 + (1 + z0 + (1 + z1 + z2)) * (1 + z0 + (1 + z1 + z2)) + 4 * (1 + z0 + (1 + z1 + z2)), s16 >= 0, s16 <= z1 + 1, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s20 + s' :|: s20 >= 0, s20 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z01 + 2*z01*z3 + z01^2 + 11*z3 + z3^2 }-> 1 + s21 + s'' :|: s21 >= 0, s21 <= 7 * (1 + (1 + z01) + z3) + (1 + (1 + z01) + z3) * (1 + (1 + z01) + z3) + 7, s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s22 + s1 :|: s22 >= 0, s22 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z1'' + 2*z1''*z3 + z1''^2 + 11*z3 + z3^2 }-> 1 + s23 + s2 :|: s23 >= 0, s23 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s17 >= 0, s17 <= 2, s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 26 + 10*z2 + 2*z2*z3 + z2^2 + 9*z3 + z3^2 }-> 1 + s24 + s3 :|: s24 >= 0, s24 <= 7 * (1 + z2 + z3) + (1 + z2 + z3) * (1 + z2 + z3) + 7, s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 300 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + s38 :|: s38 >= 0, s38 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 497 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 223*z11 + 52*z11*z2' + 26*z11^2 + 223*z2' + 26*z2'^2 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + s42 :|: s42 >= 0, s42 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 517 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 232*z11 + 54*z11*z2' + 27*z11^2 + 232*z2' + 27*z2'^2 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + s29 + s43 :|: s43 >= 0, s43 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s29 >= 0, s29 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 312 + 171*z0 + 52*z0*z1 + 26*z0^2 + 178*z1 + 27*z1^2 }-> 1 + SORT(replace(0, z0, z1)) + s31 + s46 :|: s46 >= 0, s46 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s31 >= 0, s31 <= 7 * z1 + 6 + z1 * z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 497 + 223*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 223*z12 + 52*z12*z2'' + 26*z12^2 + 223*z2'' + 26*z2''^2 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + s45 :|: s45 >= 0, s45 <= 99 * (1 + z0 + (1 + z12 + z2'')) + 26 * ((1 + z0 + (1 + z12 + z2'')) * (1 + z0 + (1 + z12 + z2''))) + 115, s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 312 }-> 1 + SORT(replace(0, 0, 0)) + s26 + s39 :|: s39 >= 0, s39 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s26 >= 0, s26 <= 7 * 0 + 6 + 0 * 0, z = 1 + 0 + 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + s30 + s44 :|: s44 >= 0, s44 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s30 >= 0, s30 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s27 + s40 :|: s40 >= 0, s40 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s27 >= 0, s27 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s28 + s41 :|: s41 >= 0, s41 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s28 >= 0, s28 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   eq(z, z') -{ 0 }-> s18 :|: s18 >= 0, s18 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s19, 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: s19 >= 0, s19 <= 2, z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(s11, 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + (z - 2)) + 0) + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: z - 2 >= 0

Function symbols to be analyzed: {min,if_min}, {replace,if_replace}, {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [12 + 7*z'' + z''^2], size: O(n^2) [6 + 7*z'' + z''^2]
  IF_REPLACE: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  IF_MIN: runtime: O(n^2) [10 + 6*z' + z'^2], size: O(n^2) [8 + 4*z' + z'^2]
  MIN: runtime: O(n^2) [154 + 119*z + 26*z^2], size: O(n^2) [115 + 99*z + 26*z^2]
  min: runtime: ?, size: O(n^1) [z]
  if_min: runtime: ?, size: O(n^1) [z']

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: min
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed RUNTIME bound using CoFloCo for: if_min
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 300 + 171*z0 + 52*z0*z2 + 26*z0^2 + 171*z2 + 26*z2^2 }-> 1 + s36 :|: s36 >= 0, s36 <= 99 * (1 + z0 + z2) + 26 * ((1 + z0 + z2) * (1 + z0 + z2)) + 115, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 300 + 171*z1 + 52*z1*z2 + 26*z1^2 + 171*z2 + 26*z2^2 }-> 1 + s37 :|: s37 >= 0, s37 <= 99 * (1 + z1 + z2) + 26 * ((1 + z1 + z2) * (1 + z1 + z2)) + 115, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s25 :|: s25 >= 0, s25 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 29 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s32 + s13 :|: s32 >= 0, s32 <= 8 + (1 + 0 + (1 + z1 + z2)) * (1 + 0 + (1 + z1 + z2)) + 4 * (1 + 0 + (1 + z1 + z2)), s13 >= 0, s13 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 40 + 12*z0' + 2*z0'*z2 + z0'^2 + 12*z2 + z2^2 }-> 1 + s33 + s14 :|: s33 >= 0, s33 <= 8 + (1 + (1 + z0') + (1 + 0 + z2)) * (1 + (1 + z0') + (1 + 0 + z2)) + 4 * (1 + (1 + z0') + (1 + 0 + z2)), s14 >= 0, s14 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 54 + 14*z0'' + 2*z0''*z1' + 2*z0''*z2 + z0''^2 + 15*z1' + 2*z1'*z2 + z1'^2 + 14*z2 + z2^2 }-> 1 + s34 + s15 :|: s34 >= 0, s34 <= 8 + (1 + (1 + z0'') + (1 + (1 + z1') + z2)) * (1 + (1 + z0'') + (1 + (1 + z1') + z2)) + 4 * (1 + (1 + z0'') + (1 + (1 + z1') + z2)), s15 >= 0, s15 <= 1 + z1' + 1, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 29 + 10*z0 + 2*z0*z1 + 2*z0*z2 + z0^2 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s35 + s16 :|: s35 >= 0, s35 <= 8 + (1 + z0 + (1 + z1 + z2)) * (1 + z0 + (1 + z1 + z2)) + 4 * (1 + z0 + (1 + z1 + z2)), s16 >= 0, s16 <= z1 + 1, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s20 + s' :|: s20 >= 0, s20 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z01 + 2*z01*z3 + z01^2 + 11*z3 + z3^2 }-> 1 + s21 + s'' :|: s21 >= 0, s21 <= 7 * (1 + (1 + z01) + z3) + (1 + (1 + z01) + z3) * (1 + (1 + z01) + z3) + 7, s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s22 + s1 :|: s22 >= 0, s22 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z1'' + 2*z1''*z3 + z1''^2 + 11*z3 + z3^2 }-> 1 + s23 + s2 :|: s23 >= 0, s23 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s17 >= 0, s17 <= 2, s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 26 + 10*z2 + 2*z2*z3 + z2^2 + 9*z3 + z3^2 }-> 1 + s24 + s3 :|: s24 >= 0, s24 <= 7 * (1 + z2 + z3) + (1 + z2 + z3) * (1 + z2 + z3) + 7, s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 300 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + s38 :|: s38 >= 0, s38 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 497 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 223*z11 + 52*z11*z2' + 26*z11^2 + 223*z2' + 26*z2'^2 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + s42 :|: s42 >= 0, s42 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 517 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 232*z11 + 54*z11*z2' + 27*z11^2 + 232*z2' + 27*z2'^2 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + s29 + s43 :|: s43 >= 0, s43 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s29 >= 0, s29 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 312 + 171*z0 + 52*z0*z1 + 26*z0^2 + 178*z1 + 27*z1^2 }-> 1 + SORT(replace(0, z0, z1)) + s31 + s46 :|: s46 >= 0, s46 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s31 >= 0, s31 <= 7 * z1 + 6 + z1 * z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 497 + 223*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 223*z12 + 52*z12*z2'' + 26*z12^2 + 223*z2'' + 26*z2''^2 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + s45 :|: s45 >= 0, s45 <= 99 * (1 + z0 + (1 + z12 + z2'')) + 26 * ((1 + z0 + (1 + z12 + z2'')) * (1 + z0 + (1 + z12 + z2''))) + 115, s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 312 }-> 1 + SORT(replace(0, 0, 0)) + s26 + s39 :|: s39 >= 0, s39 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s26 >= 0, s26 <= 7 * 0 + 6 + 0 * 0, z = 1 + 0 + 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + s30 + s44 :|: s44 >= 0, s44 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s30 >= 0, s30 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s27 + s40 :|: s40 >= 0, s40 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s27 >= 0, s27 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s28 + s41 :|: s41 >= 0, s41 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s28 >= 0, s28 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   eq(z, z') -{ 0 }-> s18 :|: s18 >= 0, s18 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z07) + (1 + (1 + z13) + z2)) :|: s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z06) + (1 + 0 + z2)) :|: z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s19, 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: s19 >= 0, s19 <= 2, z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + min(1 + z0 + z1) + sort(replace(0, z0, z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   sort(z) -{ 0 }-> 1 + min(1 + z0 + (1 + z15 + z21)) + sort(replace(if_min(s11, 1 + z0 + (1 + z15 + z21)), z0, 1 + z15 + z21)) :|: s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + min(1 + 0 + 0) + sort(replace(0, 0, 0)) :|: z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + min(1 + (1 + (z - 2)) + 0) + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: z - 2 >= 0

Function symbols to be analyzed: {replace,if_replace}, {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [12 + 7*z'' + z''^2], size: O(n^2) [6 + 7*z'' + z''^2]
  IF_REPLACE: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  IF_MIN: runtime: O(n^2) [10 + 6*z' + z'^2], size: O(n^2) [8 + 4*z' + z'^2]
  MIN: runtime: O(n^2) [154 + 119*z + 26*z^2], size: O(n^2) [115 + 99*z + 26*z^2]
  min: runtime: O(1) [0], size: O(n^1) [z]
  if_min: runtime: O(1) [0], size: O(n^1) [z']

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

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 300 + 171*z0 + 52*z0*z2 + 26*z0^2 + 171*z2 + 26*z2^2 }-> 1 + s36 :|: s36 >= 0, s36 <= 99 * (1 + z0 + z2) + 26 * ((1 + z0 + z2) * (1 + z0 + z2)) + 115, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 300 + 171*z1 + 52*z1*z2 + 26*z1^2 + 171*z2 + 26*z2^2 }-> 1 + s37 :|: s37 >= 0, s37 <= 99 * (1 + z1 + z2) + 26 * ((1 + z1 + z2) * (1 + z1 + z2)) + 115, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s25 :|: s25 >= 0, s25 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 29 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s32 + s13 :|: s32 >= 0, s32 <= 8 + (1 + 0 + (1 + z1 + z2)) * (1 + 0 + (1 + z1 + z2)) + 4 * (1 + 0 + (1 + z1 + z2)), s13 >= 0, s13 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 40 + 12*z0' + 2*z0'*z2 + z0'^2 + 12*z2 + z2^2 }-> 1 + s33 + s14 :|: s33 >= 0, s33 <= 8 + (1 + (1 + z0') + (1 + 0 + z2)) * (1 + (1 + z0') + (1 + 0 + z2)) + 4 * (1 + (1 + z0') + (1 + 0 + z2)), s14 >= 0, s14 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 54 + 14*z0'' + 2*z0''*z1' + 2*z0''*z2 + z0''^2 + 15*z1' + 2*z1'*z2 + z1'^2 + 14*z2 + z2^2 }-> 1 + s34 + s15 :|: s34 >= 0, s34 <= 8 + (1 + (1 + z0'') + (1 + (1 + z1') + z2)) * (1 + (1 + z0'') + (1 + (1 + z1') + z2)) + 4 * (1 + (1 + z0'') + (1 + (1 + z1') + z2)), s15 >= 0, s15 <= 1 + z1' + 1, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 29 + 10*z0 + 2*z0*z1 + 2*z0*z2 + z0^2 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s35 + s16 :|: s35 >= 0, s35 <= 8 + (1 + z0 + (1 + z1 + z2)) * (1 + z0 + (1 + z1 + z2)) + 4 * (1 + z0 + (1 + z1 + z2)), s16 >= 0, s16 <= z1 + 1, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s20 + s' :|: s20 >= 0, s20 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z01 + 2*z01*z3 + z01^2 + 11*z3 + z3^2 }-> 1 + s21 + s'' :|: s21 >= 0, s21 <= 7 * (1 + (1 + z01) + z3) + (1 + (1 + z01) + z3) * (1 + (1 + z01) + z3) + 7, s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s22 + s1 :|: s22 >= 0, s22 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z1'' + 2*z1''*z3 + z1''^2 + 11*z3 + z3^2 }-> 1 + s23 + s2 :|: s23 >= 0, s23 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s17 >= 0, s17 <= 2, s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 26 + 10*z2 + 2*z2*z3 + z2^2 + 9*z3 + z3^2 }-> 1 + s24 + s3 :|: s24 >= 0, s24 <= 7 * (1 + z2 + z3) + (1 + z2 + z3) * (1 + z2 + z3) + 7, s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 300 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + s38 :|: s38 >= 0, s38 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 517 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 232*z11 + 54*z11*z2' + 27*z11^2 + 232*z2' + 27*z2'^2 }-> 1 + SORT(replace(s47, z0, 1 + z11 + z2')) + s49 + s42 :|: s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 1 + z0 + (1 + z11 + z2'), s49 >= 0, s49 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s42 >= 0, s42 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 517 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 232*z11 + 54*z11*z2' + 27*z11^2 + 232*z2' + 27*z2'^2 }-> 1 + SORT(replace(s50, z0, 1 + z11 + z2')) + s29 + s43 :|: s50 >= 0, s50 <= 1 + z0 + (1 + z11 + z2'), s43 >= 0, s43 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s29 >= 0, s29 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 312 + 171*z0 + 52*z0*z1 + 26*z0^2 + 178*z1 + 27*z1^2 }-> 1 + SORT(replace(0, z0, z1)) + s31 + s46 :|: s46 >= 0, s46 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s31 >= 0, s31 <= 7 * z1 + 6 + z1 * z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 517 + 223*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 232*z12 + 54*z12*z2'' + 27*z12^2 + 232*z2'' + 27*z2''^2 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + s52 + s45 :|: s51 >= 0, s51 <= 1 + z0 + (1 + z12 + z2''), s52 >= 0, s52 <= 7 * (1 + z12 + z2'') + 6 + (1 + z12 + z2'') * (1 + z12 + z2''), s45 >= 0, s45 <= 99 * (1 + z0 + (1 + z12 + z2'')) + 26 * ((1 + z0 + (1 + z12 + z2'')) * (1 + z0 + (1 + z12 + z2''))) + 115, s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 312 }-> 1 + SORT(replace(0, 0, 0)) + s26 + s39 :|: s39 >= 0, s39 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s26 >= 0, s26 <= 7 * 0 + 6 + 0 * 0, z = 1 + 0 + 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + s30 + s44 :|: s44 >= 0, s44 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s30 >= 0, s30 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s27 + s40 :|: s40 >= 0, s40 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s27 >= 0, s27 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s28 + s41 :|: s41 >= 0, s41 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s28 >= 0, s28 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   eq(z, z') -{ 0 }-> s18 :|: s18 >= 0, s18 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + (1 + z06) + (1 + 0 + z2), z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + (1 + z07) + (1 + (1 + z13) + z2), s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> s56 :|: s56 >= 0, s56 <= 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s19, 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: s19 >= 0, s19 <= 2, z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + s59 + sort(replace(0, 0, 0)) :|: s59 >= 0, s59 <= 1 + 0 + 0, z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + s60 + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: s60 >= 0, s60 <= 1 + (1 + (z - 2)) + 0, z - 2 >= 0
   sort(z) -{ 0 }-> 1 + s61 + sort(replace(s62, z0, 1 + z15 + z21)) :|: s61 >= 0, s61 <= 1 + z0 + (1 + z15 + z21), s62 >= 0, s62 <= 1 + z0 + (1 + z15 + z21), s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + s63 + sort(replace(0, z0, z1)) :|: s63 >= 0, s63 <= 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Function symbols to be analyzed: {replace,if_replace}, {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [12 + 7*z'' + z''^2], size: O(n^2) [6 + 7*z'' + z''^2]
  IF_REPLACE: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  IF_MIN: runtime: O(n^2) [10 + 6*z' + z'^2], size: O(n^2) [8 + 4*z' + z'^2]
  MIN: runtime: O(n^2) [154 + 119*z + 26*z^2], size: O(n^2) [115 + 99*z + 26*z^2]
  min: runtime: O(1) [0], size: O(n^1) [z]
  if_min: runtime: O(1) [0], size: O(n^1) [z']

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

(61) IntTrsBoundProof (UPPER BOUND(ID))

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

Computed SIZE bound using CoFloCo for: if_replace
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z'' + z4

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 300 + 171*z0 + 52*z0*z2 + 26*z0^2 + 171*z2 + 26*z2^2 }-> 1 + s36 :|: s36 >= 0, s36 <= 99 * (1 + z0 + z2) + 26 * ((1 + z0 + z2) * (1 + z0 + z2)) + 115, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 300 + 171*z1 + 52*z1*z2 + 26*z1^2 + 171*z2 + 26*z2^2 }-> 1 + s37 :|: s37 >= 0, s37 <= 99 * (1 + z1 + z2) + 26 * ((1 + z1 + z2) * (1 + z1 + z2)) + 115, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s25 :|: s25 >= 0, s25 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 29 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s32 + s13 :|: s32 >= 0, s32 <= 8 + (1 + 0 + (1 + z1 + z2)) * (1 + 0 + (1 + z1 + z2)) + 4 * (1 + 0 + (1 + z1 + z2)), s13 >= 0, s13 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 40 + 12*z0' + 2*z0'*z2 + z0'^2 + 12*z2 + z2^2 }-> 1 + s33 + s14 :|: s33 >= 0, s33 <= 8 + (1 + (1 + z0') + (1 + 0 + z2)) * (1 + (1 + z0') + (1 + 0 + z2)) + 4 * (1 + (1 + z0') + (1 + 0 + z2)), s14 >= 0, s14 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 54 + 14*z0'' + 2*z0''*z1' + 2*z0''*z2 + z0''^2 + 15*z1' + 2*z1'*z2 + z1'^2 + 14*z2 + z2^2 }-> 1 + s34 + s15 :|: s34 >= 0, s34 <= 8 + (1 + (1 + z0'') + (1 + (1 + z1') + z2)) * (1 + (1 + z0'') + (1 + (1 + z1') + z2)) + 4 * (1 + (1 + z0'') + (1 + (1 + z1') + z2)), s15 >= 0, s15 <= 1 + z1' + 1, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 29 + 10*z0 + 2*z0*z1 + 2*z0*z2 + z0^2 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s35 + s16 :|: s35 >= 0, s35 <= 8 + (1 + z0 + (1 + z1 + z2)) * (1 + z0 + (1 + z1 + z2)) + 4 * (1 + z0 + (1 + z1 + z2)), s16 >= 0, s16 <= z1 + 1, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s20 + s' :|: s20 >= 0, s20 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z01 + 2*z01*z3 + z01^2 + 11*z3 + z3^2 }-> 1 + s21 + s'' :|: s21 >= 0, s21 <= 7 * (1 + (1 + z01) + z3) + (1 + (1 + z01) + z3) * (1 + (1 + z01) + z3) + 7, s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s22 + s1 :|: s22 >= 0, s22 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z1'' + 2*z1''*z3 + z1''^2 + 11*z3 + z3^2 }-> 1 + s23 + s2 :|: s23 >= 0, s23 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s17 >= 0, s17 <= 2, s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 26 + 10*z2 + 2*z2*z3 + z2^2 + 9*z3 + z3^2 }-> 1 + s24 + s3 :|: s24 >= 0, s24 <= 7 * (1 + z2 + z3) + (1 + z2 + z3) * (1 + z2 + z3) + 7, s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 300 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + s38 :|: s38 >= 0, s38 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 517 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 232*z11 + 54*z11*z2' + 27*z11^2 + 232*z2' + 27*z2'^2 }-> 1 + SORT(replace(s47, z0, 1 + z11 + z2')) + s49 + s42 :|: s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 1 + z0 + (1 + z11 + z2'), s49 >= 0, s49 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s42 >= 0, s42 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 517 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 232*z11 + 54*z11*z2' + 27*z11^2 + 232*z2' + 27*z2'^2 }-> 1 + SORT(replace(s50, z0, 1 + z11 + z2')) + s29 + s43 :|: s50 >= 0, s50 <= 1 + z0 + (1 + z11 + z2'), s43 >= 0, s43 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s29 >= 0, s29 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 312 + 171*z0 + 52*z0*z1 + 26*z0^2 + 178*z1 + 27*z1^2 }-> 1 + SORT(replace(0, z0, z1)) + s31 + s46 :|: s46 >= 0, s46 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s31 >= 0, s31 <= 7 * z1 + 6 + z1 * z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 517 + 223*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 232*z12 + 54*z12*z2'' + 27*z12^2 + 232*z2'' + 27*z2''^2 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + s52 + s45 :|: s51 >= 0, s51 <= 1 + z0 + (1 + z12 + z2''), s52 >= 0, s52 <= 7 * (1 + z12 + z2'') + 6 + (1 + z12 + z2'') * (1 + z12 + z2''), s45 >= 0, s45 <= 99 * (1 + z0 + (1 + z12 + z2'')) + 26 * ((1 + z0 + (1 + z12 + z2'')) * (1 + z0 + (1 + z12 + z2''))) + 115, s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 312 }-> 1 + SORT(replace(0, 0, 0)) + s26 + s39 :|: s39 >= 0, s39 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s26 >= 0, s26 <= 7 * 0 + 6 + 0 * 0, z = 1 + 0 + 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + s30 + s44 :|: s44 >= 0, s44 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s30 >= 0, s30 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s27 + s40 :|: s40 >= 0, s40 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s27 >= 0, s27 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s28 + s41 :|: s41 >= 0, s41 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s28 >= 0, s28 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   eq(z, z') -{ 0 }-> s18 :|: s18 >= 0, s18 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + (1 + z06) + (1 + 0 + z2), z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + (1 + z07) + (1 + (1 + z13) + z2), s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> s56 :|: s56 >= 0, s56 <= 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s19, 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: s19 >= 0, s19 <= 2, z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + s59 + sort(replace(0, 0, 0)) :|: s59 >= 0, s59 <= 1 + 0 + 0, z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + s60 + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: s60 >= 0, s60 <= 1 + (1 + (z - 2)) + 0, z - 2 >= 0
   sort(z) -{ 0 }-> 1 + s61 + sort(replace(s62, z0, 1 + z15 + z21)) :|: s61 >= 0, s61 <= 1 + z0 + (1 + z15 + z21), s62 >= 0, s62 <= 1 + z0 + (1 + z15 + z21), s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + s63 + sort(replace(0, z0, z1)) :|: s63 >= 0, s63 <= 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Function symbols to be analyzed: {replace,if_replace}, {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [12 + 7*z'' + z''^2], size: O(n^2) [6 + 7*z'' + z''^2]
  IF_REPLACE: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  IF_MIN: runtime: O(n^2) [10 + 6*z' + z'^2], size: O(n^2) [8 + 4*z' + z'^2]
  MIN: runtime: O(n^2) [154 + 119*z + 26*z^2], size: O(n^2) [115 + 99*z + 26*z^2]
  min: runtime: O(1) [0], size: O(n^1) [z]
  if_min: runtime: O(1) [0], size: O(n^1) [z']
  replace: runtime: ?, size: O(n^1) [z' + z'']
  if_replace: runtime: ?, size: O(n^1) [z'' + z4]

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

(63) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: replace
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed RUNTIME bound using CoFloCo for: if_replace
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 300 + 171*z0 + 52*z0*z2 + 26*z0^2 + 171*z2 + 26*z2^2 }-> 1 + s36 :|: s36 >= 0, s36 <= 99 * (1 + z0 + z2) + 26 * ((1 + z0 + z2) * (1 + z0 + z2)) + 115, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 300 + 171*z1 + 52*z1*z2 + 26*z1^2 + 171*z2 + 26*z2^2 }-> 1 + s37 :|: s37 >= 0, s37 <= 99 * (1 + z1 + z2) + 26 * ((1 + z1 + z2) * (1 + z1 + z2)) + 115, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s25 :|: s25 >= 0, s25 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 29 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s32 + s13 :|: s32 >= 0, s32 <= 8 + (1 + 0 + (1 + z1 + z2)) * (1 + 0 + (1 + z1 + z2)) + 4 * (1 + 0 + (1 + z1 + z2)), s13 >= 0, s13 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 40 + 12*z0' + 2*z0'*z2 + z0'^2 + 12*z2 + z2^2 }-> 1 + s33 + s14 :|: s33 >= 0, s33 <= 8 + (1 + (1 + z0') + (1 + 0 + z2)) * (1 + (1 + z0') + (1 + 0 + z2)) + 4 * (1 + (1 + z0') + (1 + 0 + z2)), s14 >= 0, s14 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 54 + 14*z0'' + 2*z0''*z1' + 2*z0''*z2 + z0''^2 + 15*z1' + 2*z1'*z2 + z1'^2 + 14*z2 + z2^2 }-> 1 + s34 + s15 :|: s34 >= 0, s34 <= 8 + (1 + (1 + z0'') + (1 + (1 + z1') + z2)) * (1 + (1 + z0'') + (1 + (1 + z1') + z2)) + 4 * (1 + (1 + z0'') + (1 + (1 + z1') + z2)), s15 >= 0, s15 <= 1 + z1' + 1, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 29 + 10*z0 + 2*z0*z1 + 2*z0*z2 + z0^2 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s35 + s16 :|: s35 >= 0, s35 <= 8 + (1 + z0 + (1 + z1 + z2)) * (1 + z0 + (1 + z1 + z2)) + 4 * (1 + z0 + (1 + z1 + z2)), s16 >= 0, s16 <= z1 + 1, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s20 + s' :|: s20 >= 0, s20 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z01 + 2*z01*z3 + z01^2 + 11*z3 + z3^2 }-> 1 + s21 + s'' :|: s21 >= 0, s21 <= 7 * (1 + (1 + z01) + z3) + (1 + (1 + z01) + z3) * (1 + (1 + z01) + z3) + 7, s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s22 + s1 :|: s22 >= 0, s22 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z1'' + 2*z1''*z3 + z1''^2 + 11*z3 + z3^2 }-> 1 + s23 + s2 :|: s23 >= 0, s23 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s17 >= 0, s17 <= 2, s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 26 + 10*z2 + 2*z2*z3 + z2^2 + 9*z3 + z3^2 }-> 1 + s24 + s3 :|: s24 >= 0, s24 <= 7 * (1 + z2 + z3) + (1 + z2 + z3) * (1 + z2 + z3) + 7, s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 300 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + s38 :|: s38 >= 0, s38 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 517 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 232*z11 + 54*z11*z2' + 27*z11^2 + 232*z2' + 27*z2'^2 }-> 1 + SORT(replace(s47, z0, 1 + z11 + z2')) + s49 + s42 :|: s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 1 + z0 + (1 + z11 + z2'), s49 >= 0, s49 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s42 >= 0, s42 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 517 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 232*z11 + 54*z11*z2' + 27*z11^2 + 232*z2' + 27*z2'^2 }-> 1 + SORT(replace(s50, z0, 1 + z11 + z2')) + s29 + s43 :|: s50 >= 0, s50 <= 1 + z0 + (1 + z11 + z2'), s43 >= 0, s43 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s29 >= 0, s29 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 312 + 171*z0 + 52*z0*z1 + 26*z0^2 + 178*z1 + 27*z1^2 }-> 1 + SORT(replace(0, z0, z1)) + s31 + s46 :|: s46 >= 0, s46 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s31 >= 0, s31 <= 7 * z1 + 6 + z1 * z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 517 + 223*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 232*z12 + 54*z12*z2'' + 27*z12^2 + 232*z2'' + 27*z2''^2 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + s52 + s45 :|: s51 >= 0, s51 <= 1 + z0 + (1 + z12 + z2''), s52 >= 0, s52 <= 7 * (1 + z12 + z2'') + 6 + (1 + z12 + z2'') * (1 + z12 + z2''), s45 >= 0, s45 <= 99 * (1 + z0 + (1 + z12 + z2'')) + 26 * ((1 + z0 + (1 + z12 + z2'')) * (1 + z0 + (1 + z12 + z2''))) + 115, s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 312 }-> 1 + SORT(replace(0, 0, 0)) + s26 + s39 :|: s39 >= 0, s39 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s26 >= 0, s26 <= 7 * 0 + 6 + 0 * 0, z = 1 + 0 + 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + s30 + s44 :|: s44 >= 0, s44 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s30 >= 0, s30 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s27 + s40 :|: s40 >= 0, s40 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s27 >= 0, s27 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s28 + s41 :|: s41 >= 0, s41 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s28 >= 0, s28 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   eq(z, z') -{ 0 }-> s18 :|: s18 >= 0, s18 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + (1 + z06) + (1 + 0 + z2), z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + (1 + z07) + (1 + (1 + z13) + z2), s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> s56 :|: s56 >= 0, s56 <= 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s19, 1 + (z - 1), z', 1 + (1 + z14) + z3) :|: s19 >= 0, s19 <= 2, z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z08) + z3) :|: z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + s59 + sort(replace(0, 0, 0)) :|: s59 >= 0, s59 <= 1 + 0 + 0, z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + s60 + sort(replace(1 + (z - 2), 1 + (z - 2), 0)) :|: s60 >= 0, s60 <= 1 + (1 + (z - 2)) + 0, z - 2 >= 0
   sort(z) -{ 0 }-> 1 + s61 + sort(replace(s62, z0, 1 + z15 + z21)) :|: s61 >= 0, s61 <= 1 + z0 + (1 + z15 + z21), s62 >= 0, s62 <= 1 + z0 + (1 + z15 + z21), s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + s63 + sort(replace(0, z0, z1)) :|: s63 >= 0, s63 <= 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Function symbols to be analyzed: {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [12 + 7*z'' + z''^2], size: O(n^2) [6 + 7*z'' + z''^2]
  IF_REPLACE: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  IF_MIN: runtime: O(n^2) [10 + 6*z' + z'^2], size: O(n^2) [8 + 4*z' + z'^2]
  MIN: runtime: O(n^2) [154 + 119*z + 26*z^2], size: O(n^2) [115 + 99*z + 26*z^2]
  min: runtime: O(1) [0], size: O(n^1) [z]
  if_min: runtime: O(1) [0], size: O(n^1) [z']
  replace: runtime: O(1) [0], size: O(n^1) [z' + z'']
  if_replace: runtime: O(1) [0], size: O(n^1) [z'' + z4]

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

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 300 + 171*z0 + 52*z0*z2 + 26*z0^2 + 171*z2 + 26*z2^2 }-> 1 + s36 :|: s36 >= 0, s36 <= 99 * (1 + z0 + z2) + 26 * ((1 + z0 + z2) * (1 + z0 + z2)) + 115, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 300 + 171*z1 + 52*z1*z2 + 26*z1^2 + 171*z2 + 26*z2^2 }-> 1 + s37 :|: s37 >= 0, s37 <= 99 * (1 + z1 + z2) + 26 * ((1 + z1 + z2) * (1 + z1 + z2)) + 115, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s25 :|: s25 >= 0, s25 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 29 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s32 + s13 :|: s32 >= 0, s32 <= 8 + (1 + 0 + (1 + z1 + z2)) * (1 + 0 + (1 + z1 + z2)) + 4 * (1 + 0 + (1 + z1 + z2)), s13 >= 0, s13 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 40 + 12*z0' + 2*z0'*z2 + z0'^2 + 12*z2 + z2^2 }-> 1 + s33 + s14 :|: s33 >= 0, s33 <= 8 + (1 + (1 + z0') + (1 + 0 + z2)) * (1 + (1 + z0') + (1 + 0 + z2)) + 4 * (1 + (1 + z0') + (1 + 0 + z2)), s14 >= 0, s14 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 54 + 14*z0'' + 2*z0''*z1' + 2*z0''*z2 + z0''^2 + 15*z1' + 2*z1'*z2 + z1'^2 + 14*z2 + z2^2 }-> 1 + s34 + s15 :|: s34 >= 0, s34 <= 8 + (1 + (1 + z0'') + (1 + (1 + z1') + z2)) * (1 + (1 + z0'') + (1 + (1 + z1') + z2)) + 4 * (1 + (1 + z0'') + (1 + (1 + z1') + z2)), s15 >= 0, s15 <= 1 + z1' + 1, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 29 + 10*z0 + 2*z0*z1 + 2*z0*z2 + z0^2 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s35 + s16 :|: s35 >= 0, s35 <= 8 + (1 + z0 + (1 + z1 + z2)) * (1 + z0 + (1 + z1 + z2)) + 4 * (1 + z0 + (1 + z1 + z2)), s16 >= 0, s16 <= z1 + 1, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s20 + s' :|: s20 >= 0, s20 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z01 + 2*z01*z3 + z01^2 + 11*z3 + z3^2 }-> 1 + s21 + s'' :|: s21 >= 0, s21 <= 7 * (1 + (1 + z01) + z3) + (1 + (1 + z01) + z3) * (1 + (1 + z01) + z3) + 7, s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s22 + s1 :|: s22 >= 0, s22 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z1'' + 2*z1''*z3 + z1''^2 + 11*z3 + z3^2 }-> 1 + s23 + s2 :|: s23 >= 0, s23 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s17 >= 0, s17 <= 2, s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 26 + 10*z2 + 2*z2*z3 + z2^2 + 9*z3 + z3^2 }-> 1 + s24 + s3 :|: s24 >= 0, s24 <= 7 * (1 + z2 + z3) + (1 + z2 + z3) * (1 + z2 + z3) + 7, s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 300 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + s38 :|: s38 >= 0, s38 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 312 }-> 1 + SORT(s64) + s26 + s39 :|: s64 >= 0, s64 <= 0 + 0, s39 >= 0, s39 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s26 >= 0, s26 <= 7 * 0 + 6 + 0 * 0, z = 1 + 0 + 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(s65) + s27 + s40 :|: s65 >= 0, s65 <= 1 + (z - 2) + 0, s40 >= 0, s40 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s27 >= 0, s27 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(s66) + s28 + s41 :|: s66 >= 0, s66 <= 1 + (z - 2) + 0, s41 >= 0, s41 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s28 >= 0, s28 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 517 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 232*z11 + 54*z11*z2' + 27*z11^2 + 232*z2' + 27*z2'^2 }-> 1 + SORT(s67) + s49 + s42 :|: s67 >= 0, s67 <= z0 + (1 + z11 + z2'), s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 1 + z0 + (1 + z11 + z2'), s49 >= 0, s49 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s42 >= 0, s42 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 517 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 232*z11 + 54*z11*z2' + 27*z11^2 + 232*z2' + 27*z2'^2 }-> 1 + SORT(s68) + s29 + s43 :|: s68 >= 0, s68 <= z0 + (1 + z11 + z2'), s50 >= 0, s50 <= 1 + z0 + (1 + z11 + z2'), s43 >= 0, s43 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s29 >= 0, s29 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(s69) + s30 + s44 :|: s69 >= 0, s69 <= 1 + (z - 2) + 0, s44 >= 0, s44 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s30 >= 0, s30 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 517 + 223*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 232*z12 + 54*z12*z2'' + 27*z12^2 + 232*z2'' + 27*z2''^2 }-> 1 + SORT(s70) + s52 + s45 :|: s70 >= 0, s70 <= z0 + (1 + z12 + z2''), s51 >= 0, s51 <= 1 + z0 + (1 + z12 + z2''), s52 >= 0, s52 <= 7 * (1 + z12 + z2'') + 6 + (1 + z12 + z2'') * (1 + z12 + z2''), s45 >= 0, s45 <= 99 * (1 + z0 + (1 + z12 + z2'')) + 26 * ((1 + z0 + (1 + z12 + z2'')) * (1 + z0 + (1 + z12 + z2''))) + 115, s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 312 + 171*z0 + 52*z0*z1 + 26*z0^2 + 178*z1 + 27*z1^2 }-> 1 + SORT(s71) + s31 + s46 :|: s71 >= 0, s71 <= z0 + z1, s46 >= 0, s46 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s31 >= 0, s31 <= 7 * z1 + 6 + z1 * z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eq(z, z') -{ 0 }-> s18 :|: s18 >= 0, s18 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + s77 :|: s77 >= 0, s77 <= z'' + z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + (1 + z06) + (1 + 0 + z2), z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + (1 + z07) + (1 + (1 + z13) + z2), s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> s56 :|: s56 >= 0, s56 <= 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> s72 :|: s72 >= 0, s72 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s73 :|: s73 >= 0, s73 <= z' + (1 + (1 + z08) + z3), z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s74 :|: s74 >= 0, s74 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s75 :|: s75 >= 0, s75 <= z' + (1 + (1 + z14) + z3), s19 >= 0, s19 <= 2, z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s76 :|: s76 >= 0, s76 <= z' + (1 + z2 + z3), z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + s59 + sort(s78) :|: s78 >= 0, s78 <= 0 + 0, s59 >= 0, s59 <= 1 + 0 + 0, z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + s60 + sort(s79) :|: s79 >= 0, s79 <= 1 + (z - 2) + 0, s60 >= 0, s60 <= 1 + (1 + (z - 2)) + 0, z - 2 >= 0
   sort(z) -{ 0 }-> 1 + s61 + sort(s80) :|: s80 >= 0, s80 <= z0 + (1 + z15 + z21), s61 >= 0, s61 <= 1 + z0 + (1 + z15 + z21), s62 >= 0, s62 <= 1 + z0 + (1 + z15 + z21), s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + s63 + sort(s81) :|: s81 >= 0, s81 <= z0 + z1, s63 >= 0, s63 <= 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Function symbols to be analyzed: {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [12 + 7*z'' + z''^2], size: O(n^2) [6 + 7*z'' + z''^2]
  IF_REPLACE: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  IF_MIN: runtime: O(n^2) [10 + 6*z' + z'^2], size: O(n^2) [8 + 4*z' + z'^2]
  MIN: runtime: O(n^2) [154 + 119*z + 26*z^2], size: O(n^2) [115 + 99*z + 26*z^2]
  min: runtime: O(1) [0], size: O(n^1) [z]
  if_min: runtime: O(1) [0], size: O(n^1) [z']
  replace: runtime: O(1) [0], size: O(n^1) [z' + z'']
  if_replace: runtime: O(1) [0], size: O(n^1) [z'' + z4]

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

(67) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: SORT
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 300 + 171*z0 + 52*z0*z2 + 26*z0^2 + 171*z2 + 26*z2^2 }-> 1 + s36 :|: s36 >= 0, s36 <= 99 * (1 + z0 + z2) + 26 * ((1 + z0 + z2) * (1 + z0 + z2)) + 115, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 300 + 171*z1 + 52*z1*z2 + 26*z1^2 + 171*z2 + 26*z2^2 }-> 1 + s37 :|: s37 >= 0, s37 <= 99 * (1 + z1 + z2) + 26 * ((1 + z1 + z2) * (1 + z1 + z2)) + 115, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s25 :|: s25 >= 0, s25 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 29 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s32 + s13 :|: s32 >= 0, s32 <= 8 + (1 + 0 + (1 + z1 + z2)) * (1 + 0 + (1 + z1 + z2)) + 4 * (1 + 0 + (1 + z1 + z2)), s13 >= 0, s13 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 40 + 12*z0' + 2*z0'*z2 + z0'^2 + 12*z2 + z2^2 }-> 1 + s33 + s14 :|: s33 >= 0, s33 <= 8 + (1 + (1 + z0') + (1 + 0 + z2)) * (1 + (1 + z0') + (1 + 0 + z2)) + 4 * (1 + (1 + z0') + (1 + 0 + z2)), s14 >= 0, s14 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 54 + 14*z0'' + 2*z0''*z1' + 2*z0''*z2 + z0''^2 + 15*z1' + 2*z1'*z2 + z1'^2 + 14*z2 + z2^2 }-> 1 + s34 + s15 :|: s34 >= 0, s34 <= 8 + (1 + (1 + z0'') + (1 + (1 + z1') + z2)) * (1 + (1 + z0'') + (1 + (1 + z1') + z2)) + 4 * (1 + (1 + z0'') + (1 + (1 + z1') + z2)), s15 >= 0, s15 <= 1 + z1' + 1, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 29 + 10*z0 + 2*z0*z1 + 2*z0*z2 + z0^2 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s35 + s16 :|: s35 >= 0, s35 <= 8 + (1 + z0 + (1 + z1 + z2)) * (1 + z0 + (1 + z1 + z2)) + 4 * (1 + z0 + (1 + z1 + z2)), s16 >= 0, s16 <= z1 + 1, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s20 + s' :|: s20 >= 0, s20 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z01 + 2*z01*z3 + z01^2 + 11*z3 + z3^2 }-> 1 + s21 + s'' :|: s21 >= 0, s21 <= 7 * (1 + (1 + z01) + z3) + (1 + (1 + z01) + z3) * (1 + (1 + z01) + z3) + 7, s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s22 + s1 :|: s22 >= 0, s22 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z1'' + 2*z1''*z3 + z1''^2 + 11*z3 + z3^2 }-> 1 + s23 + s2 :|: s23 >= 0, s23 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s17 >= 0, s17 <= 2, s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 26 + 10*z2 + 2*z2*z3 + z2^2 + 9*z3 + z3^2 }-> 1 + s24 + s3 :|: s24 >= 0, s24 <= 7 * (1 + z2 + z3) + (1 + z2 + z3) * (1 + z2 + z3) + 7, s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 300 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + s38 :|: s38 >= 0, s38 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 312 }-> 1 + SORT(s64) + s26 + s39 :|: s64 >= 0, s64 <= 0 + 0, s39 >= 0, s39 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s26 >= 0, s26 <= 7 * 0 + 6 + 0 * 0, z = 1 + 0 + 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(s65) + s27 + s40 :|: s65 >= 0, s65 <= 1 + (z - 2) + 0, s40 >= 0, s40 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s27 >= 0, s27 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(s66) + s28 + s41 :|: s66 >= 0, s66 <= 1 + (z - 2) + 0, s41 >= 0, s41 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s28 >= 0, s28 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 517 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 232*z11 + 54*z11*z2' + 27*z11^2 + 232*z2' + 27*z2'^2 }-> 1 + SORT(s67) + s49 + s42 :|: s67 >= 0, s67 <= z0 + (1 + z11 + z2'), s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 1 + z0 + (1 + z11 + z2'), s49 >= 0, s49 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s42 >= 0, s42 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 517 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 232*z11 + 54*z11*z2' + 27*z11^2 + 232*z2' + 27*z2'^2 }-> 1 + SORT(s68) + s29 + s43 :|: s68 >= 0, s68 <= z0 + (1 + z11 + z2'), s50 >= 0, s50 <= 1 + z0 + (1 + z11 + z2'), s43 >= 0, s43 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s29 >= 0, s29 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(s69) + s30 + s44 :|: s69 >= 0, s69 <= 1 + (z - 2) + 0, s44 >= 0, s44 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s30 >= 0, s30 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 517 + 223*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 232*z12 + 54*z12*z2'' + 27*z12^2 + 232*z2'' + 27*z2''^2 }-> 1 + SORT(s70) + s52 + s45 :|: s70 >= 0, s70 <= z0 + (1 + z12 + z2''), s51 >= 0, s51 <= 1 + z0 + (1 + z12 + z2''), s52 >= 0, s52 <= 7 * (1 + z12 + z2'') + 6 + (1 + z12 + z2'') * (1 + z12 + z2''), s45 >= 0, s45 <= 99 * (1 + z0 + (1 + z12 + z2'')) + 26 * ((1 + z0 + (1 + z12 + z2'')) * (1 + z0 + (1 + z12 + z2''))) + 115, s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 312 + 171*z0 + 52*z0*z1 + 26*z0^2 + 178*z1 + 27*z1^2 }-> 1 + SORT(s71) + s31 + s46 :|: s71 >= 0, s71 <= z0 + z1, s46 >= 0, s46 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s31 >= 0, s31 <= 7 * z1 + 6 + z1 * z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eq(z, z') -{ 0 }-> s18 :|: s18 >= 0, s18 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + s77 :|: s77 >= 0, s77 <= z'' + z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + (1 + z06) + (1 + 0 + z2), z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + (1 + z07) + (1 + (1 + z13) + z2), s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> s56 :|: s56 >= 0, s56 <= 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> s72 :|: s72 >= 0, s72 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s73 :|: s73 >= 0, s73 <= z' + (1 + (1 + z08) + z3), z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s74 :|: s74 >= 0, s74 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s75 :|: s75 >= 0, s75 <= z' + (1 + (1 + z14) + z3), s19 >= 0, s19 <= 2, z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s76 :|: s76 >= 0, s76 <= z' + (1 + z2 + z3), z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + s59 + sort(s78) :|: s78 >= 0, s78 <= 0 + 0, s59 >= 0, s59 <= 1 + 0 + 0, z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + s60 + sort(s79) :|: s79 >= 0, s79 <= 1 + (z - 2) + 0, s60 >= 0, s60 <= 1 + (1 + (z - 2)) + 0, z - 2 >= 0
   sort(z) -{ 0 }-> 1 + s61 + sort(s80) :|: s80 >= 0, s80 <= z0 + (1 + z15 + z21), s61 >= 0, s61 <= 1 + z0 + (1 + z15 + z21), s62 >= 0, s62 <= 1 + z0 + (1 + z15 + z21), s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + s63 + sort(s81) :|: s81 >= 0, s81 <= z0 + z1, s63 >= 0, s63 <= 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Function symbols to be analyzed: {SORT}, {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [12 + 7*z'' + z''^2], size: O(n^2) [6 + 7*z'' + z''^2]
  IF_REPLACE: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  IF_MIN: runtime: O(n^2) [10 + 6*z' + z'^2], size: O(n^2) [8 + 4*z' + z'^2]
  MIN: runtime: O(n^2) [154 + 119*z + 26*z^2], size: O(n^2) [115 + 99*z + 26*z^2]
  min: runtime: O(1) [0], size: O(n^1) [z]
  if_min: runtime: O(1) [0], size: O(n^1) [z']
  replace: runtime: O(1) [0], size: O(n^1) [z' + z'']
  if_replace: runtime: O(1) [0], size: O(n^1) [z'' + z4]
  SORT: runtime: ?, size: INF

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(69) 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: 301 + 3018*z + 2871*z^2 + 897*z^3

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

(70)
Obligation:
Complexity RNTS consisting of the following rules:

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 300 + 171*z0 + 52*z0*z2 + 26*z0^2 + 171*z2 + 26*z2^2 }-> 1 + s36 :|: s36 >= 0, s36 <= 99 * (1 + z0 + z2) + 26 * ((1 + z0 + z2) * (1 + z0 + z2)) + 115, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 300 + 171*z1 + 52*z1*z2 + 26*z1^2 + 171*z2 + 26*z2^2 }-> 1 + s37 :|: s37 >= 0, s37 <= 99 * (1 + z1 + z2) + 26 * ((1 + z1 + z2) * (1 + z1 + z2)) + 115, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s25 :|: s25 >= 0, s25 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 29 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s32 + s13 :|: s32 >= 0, s32 <= 8 + (1 + 0 + (1 + z1 + z2)) * (1 + 0 + (1 + z1 + z2)) + 4 * (1 + 0 + (1 + z1 + z2)), s13 >= 0, s13 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 40 + 12*z0' + 2*z0'*z2 + z0'^2 + 12*z2 + z2^2 }-> 1 + s33 + s14 :|: s33 >= 0, s33 <= 8 + (1 + (1 + z0') + (1 + 0 + z2)) * (1 + (1 + z0') + (1 + 0 + z2)) + 4 * (1 + (1 + z0') + (1 + 0 + z2)), s14 >= 0, s14 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 54 + 14*z0'' + 2*z0''*z1' + 2*z0''*z2 + z0''^2 + 15*z1' + 2*z1'*z2 + z1'^2 + 14*z2 + z2^2 }-> 1 + s34 + s15 :|: s34 >= 0, s34 <= 8 + (1 + (1 + z0'') + (1 + (1 + z1') + z2)) * (1 + (1 + z0'') + (1 + (1 + z1') + z2)) + 4 * (1 + (1 + z0'') + (1 + (1 + z1') + z2)), s15 >= 0, s15 <= 1 + z1' + 1, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 29 + 10*z0 + 2*z0*z1 + 2*z0*z2 + z0^2 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s35 + s16 :|: s35 >= 0, s35 <= 8 + (1 + z0 + (1 + z1 + z2)) * (1 + z0 + (1 + z1 + z2)) + 4 * (1 + z0 + (1 + z1 + z2)), s16 >= 0, s16 <= z1 + 1, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s20 + s' :|: s20 >= 0, s20 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z01 + 2*z01*z3 + z01^2 + 11*z3 + z3^2 }-> 1 + s21 + s'' :|: s21 >= 0, s21 <= 7 * (1 + (1 + z01) + z3) + (1 + (1 + z01) + z3) * (1 + (1 + z01) + z3) + 7, s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s22 + s1 :|: s22 >= 0, s22 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z1'' + 2*z1''*z3 + z1''^2 + 11*z3 + z3^2 }-> 1 + s23 + s2 :|: s23 >= 0, s23 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s17 >= 0, s17 <= 2, s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 26 + 10*z2 + 2*z2*z3 + z2^2 + 9*z3 + z3^2 }-> 1 + s24 + s3 :|: s24 >= 0, s24 <= 7 * (1 + z2 + z3) + (1 + z2 + z3) * (1 + z2 + z3) + 7, s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 300 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + s38 :|: s38 >= 0, s38 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 312 }-> 1 + SORT(s64) + s26 + s39 :|: s64 >= 0, s64 <= 0 + 0, s39 >= 0, s39 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s26 >= 0, s26 <= 7 * 0 + 6 + 0 * 0, z = 1 + 0 + 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(s65) + s27 + s40 :|: s65 >= 0, s65 <= 1 + (z - 2) + 0, s40 >= 0, s40 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s27 >= 0, s27 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(s66) + s28 + s41 :|: s66 >= 0, s66 <= 1 + (z - 2) + 0, s41 >= 0, s41 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s28 >= 0, s28 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 517 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 232*z11 + 54*z11*z2' + 27*z11^2 + 232*z2' + 27*z2'^2 }-> 1 + SORT(s67) + s49 + s42 :|: s67 >= 0, s67 <= z0 + (1 + z11 + z2'), s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 1 + z0 + (1 + z11 + z2'), s49 >= 0, s49 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s42 >= 0, s42 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 517 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 232*z11 + 54*z11*z2' + 27*z11^2 + 232*z2' + 27*z2'^2 }-> 1 + SORT(s68) + s29 + s43 :|: s68 >= 0, s68 <= z0 + (1 + z11 + z2'), s50 >= 0, s50 <= 1 + z0 + (1 + z11 + z2'), s43 >= 0, s43 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s29 >= 0, s29 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 167 + 119*z + 26*z^2 }-> 1 + SORT(s69) + s30 + s44 :|: s69 >= 0, s69 <= 1 + (z - 2) + 0, s44 >= 0, s44 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s30 >= 0, s30 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 517 + 223*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 232*z12 + 54*z12*z2'' + 27*z12^2 + 232*z2'' + 27*z2''^2 }-> 1 + SORT(s70) + s52 + s45 :|: s70 >= 0, s70 <= z0 + (1 + z12 + z2''), s51 >= 0, s51 <= 1 + z0 + (1 + z12 + z2''), s52 >= 0, s52 <= 7 * (1 + z12 + z2'') + 6 + (1 + z12 + z2'') * (1 + z12 + z2''), s45 >= 0, s45 <= 99 * (1 + z0 + (1 + z12 + z2'')) + 26 * ((1 + z0 + (1 + z12 + z2'')) * (1 + z0 + (1 + z12 + z2''))) + 115, s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 312 + 171*z0 + 52*z0*z1 + 26*z0^2 + 178*z1 + 27*z1^2 }-> 1 + SORT(s71) + s31 + s46 :|: s71 >= 0, s71 <= z0 + z1, s46 >= 0, s46 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s31 >= 0, s31 <= 7 * z1 + 6 + z1 * z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eq(z, z') -{ 0 }-> s18 :|: s18 >= 0, s18 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + s77 :|: s77 >= 0, s77 <= z'' + z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + (1 + z06) + (1 + 0 + z2), z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + (1 + z07) + (1 + (1 + z13) + z2), s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> s56 :|: s56 >= 0, s56 <= 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> s72 :|: s72 >= 0, s72 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s73 :|: s73 >= 0, s73 <= z' + (1 + (1 + z08) + z3), z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s74 :|: s74 >= 0, s74 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s75 :|: s75 >= 0, s75 <= z' + (1 + (1 + z14) + z3), s19 >= 0, s19 <= 2, z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s76 :|: s76 >= 0, s76 <= z' + (1 + z2 + z3), z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + s59 + sort(s78) :|: s78 >= 0, s78 <= 0 + 0, s59 >= 0, s59 <= 1 + 0 + 0, z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + s60 + sort(s79) :|: s79 >= 0, s79 <= 1 + (z - 2) + 0, s60 >= 0, s60 <= 1 + (1 + (z - 2)) + 0, z - 2 >= 0
   sort(z) -{ 0 }-> 1 + s61 + sort(s80) :|: s80 >= 0, s80 <= z0 + (1 + z15 + z21), s61 >= 0, s61 <= 1 + z0 + (1 + z15 + z21), s62 >= 0, s62 <= 1 + z0 + (1 + z15 + z21), s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + s63 + sort(s81) :|: s81 >= 0, s81 <= z0 + z1, s63 >= 0, s63 <= 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Function symbols to be analyzed: {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [12 + 7*z'' + z''^2], size: O(n^2) [6 + 7*z'' + z''^2]
  IF_REPLACE: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  IF_MIN: runtime: O(n^2) [10 + 6*z' + z'^2], size: O(n^2) [8 + 4*z' + z'^2]
  MIN: runtime: O(n^2) [154 + 119*z + 26*z^2], size: O(n^2) [115 + 99*z + 26*z^2]
  min: runtime: O(1) [0], size: O(n^1) [z]
  if_min: runtime: O(1) [0], size: O(n^1) [z']
  replace: runtime: O(1) [0], size: O(n^1) [z' + z'']
  if_replace: runtime: O(1) [0], size: O(n^1) [z'' + z4]
  SORT: runtime: O(n^3) [301 + 3018*z + 2871*z^2 + 897*z^3], size: INF

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

(72)
Obligation:
Complexity RNTS consisting of the following rules:

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 300 + 171*z0 + 52*z0*z2 + 26*z0^2 + 171*z2 + 26*z2^2 }-> 1 + s36 :|: s36 >= 0, s36 <= 99 * (1 + z0 + z2) + 26 * ((1 + z0 + z2) * (1 + z0 + z2)) + 115, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 300 + 171*z1 + 52*z1*z2 + 26*z1^2 + 171*z2 + 26*z2^2 }-> 1 + s37 :|: s37 >= 0, s37 <= 99 * (1 + z1 + z2) + 26 * ((1 + z1 + z2) * (1 + z1 + z2)) + 115, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s25 :|: s25 >= 0, s25 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 29 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s32 + s13 :|: s32 >= 0, s32 <= 8 + (1 + 0 + (1 + z1 + z2)) * (1 + 0 + (1 + z1 + z2)) + 4 * (1 + 0 + (1 + z1 + z2)), s13 >= 0, s13 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 40 + 12*z0' + 2*z0'*z2 + z0'^2 + 12*z2 + z2^2 }-> 1 + s33 + s14 :|: s33 >= 0, s33 <= 8 + (1 + (1 + z0') + (1 + 0 + z2)) * (1 + (1 + z0') + (1 + 0 + z2)) + 4 * (1 + (1 + z0') + (1 + 0 + z2)), s14 >= 0, s14 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 54 + 14*z0'' + 2*z0''*z1' + 2*z0''*z2 + z0''^2 + 15*z1' + 2*z1'*z2 + z1'^2 + 14*z2 + z2^2 }-> 1 + s34 + s15 :|: s34 >= 0, s34 <= 8 + (1 + (1 + z0'') + (1 + (1 + z1') + z2)) * (1 + (1 + z0'') + (1 + (1 + z1') + z2)) + 4 * (1 + (1 + z0'') + (1 + (1 + z1') + z2)), s15 >= 0, s15 <= 1 + z1' + 1, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 29 + 10*z0 + 2*z0*z1 + 2*z0*z2 + z0^2 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s35 + s16 :|: s35 >= 0, s35 <= 8 + (1 + z0 + (1 + z1 + z2)) * (1 + z0 + (1 + z1 + z2)) + 4 * (1 + z0 + (1 + z1 + z2)), s16 >= 0, s16 <= z1 + 1, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s20 + s' :|: s20 >= 0, s20 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z01 + 2*z01*z3 + z01^2 + 11*z3 + z3^2 }-> 1 + s21 + s'' :|: s21 >= 0, s21 <= 7 * (1 + (1 + z01) + z3) + (1 + (1 + z01) + z3) * (1 + (1 + z01) + z3) + 7, s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s22 + s1 :|: s22 >= 0, s22 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z1'' + 2*z1''*z3 + z1''^2 + 11*z3 + z3^2 }-> 1 + s23 + s2 :|: s23 >= 0, s23 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s17 >= 0, s17 <= 2, s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 26 + 10*z2 + 2*z2*z3 + z2^2 + 9*z3 + z3^2 }-> 1 + s24 + s3 :|: s24 >= 0, s24 <= 7 * (1 + z2 + z3) + (1 + z2 + z3) * (1 + z2 + z3) + 7, s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 300 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + s38 :|: s38 >= 0, s38 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 613 + 3018*s64 + 2871*s64^2 + 897*s64^3 }-> 1 + s82 + s26 + s39 :|: s82 >= 0, s82 <= inf, s64 >= 0, s64 <= 0 + 0, s39 >= 0, s39 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s26 >= 0, s26 <= 7 * 0 + 6 + 0 * 0, z = 1 + 0 + 0
   SORT(z) -{ 468 + 3018*s65 + 2871*s65^2 + 897*s65^3 + 119*z + 26*z^2 }-> 1 + s83 + s27 + s40 :|: s83 >= 0, s83 <= inf', s65 >= 0, s65 <= 1 + (z - 2) + 0, s40 >= 0, s40 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s27 >= 0, s27 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 468 + 3018*s66 + 2871*s66^2 + 897*s66^3 + 119*z + 26*z^2 }-> 1 + s84 + s28 + s41 :|: s84 >= 0, s84 <= inf'', s66 >= 0, s66 <= 1 + (z - 2) + 0, s41 >= 0, s41 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s28 >= 0, s28 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 818 + 3018*s67 + 2871*s67^2 + 897*s67^3 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 232*z11 + 54*z11*z2' + 27*z11^2 + 232*z2' + 27*z2'^2 }-> 1 + s85 + s49 + s42 :|: s85 >= 0, s85 <= inf1, s67 >= 0, s67 <= z0 + (1 + z11 + z2'), s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 1 + z0 + (1 + z11 + z2'), s49 >= 0, s49 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s42 >= 0, s42 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 818 + 3018*s68 + 2871*s68^2 + 897*s68^3 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 232*z11 + 54*z11*z2' + 27*z11^2 + 232*z2' + 27*z2'^2 }-> 1 + s86 + s29 + s43 :|: s86 >= 0, s86 <= inf2, s68 >= 0, s68 <= z0 + (1 + z11 + z2'), s50 >= 0, s50 <= 1 + z0 + (1 + z11 + z2'), s43 >= 0, s43 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s29 >= 0, s29 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 468 + 3018*s69 + 2871*s69^2 + 897*s69^3 + 119*z + 26*z^2 }-> 1 + s87 + s30 + s44 :|: s87 >= 0, s87 <= inf3, s69 >= 0, s69 <= 1 + (z - 2) + 0, s44 >= 0, s44 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s30 >= 0, s30 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 818 + 3018*s70 + 2871*s70^2 + 897*s70^3 + 223*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 232*z12 + 54*z12*z2'' + 27*z12^2 + 232*z2'' + 27*z2''^2 }-> 1 + s88 + s52 + s45 :|: s88 >= 0, s88 <= inf4, s70 >= 0, s70 <= z0 + (1 + z12 + z2''), s51 >= 0, s51 <= 1 + z0 + (1 + z12 + z2''), s52 >= 0, s52 <= 7 * (1 + z12 + z2'') + 6 + (1 + z12 + z2'') * (1 + z12 + z2''), s45 >= 0, s45 <= 99 * (1 + z0 + (1 + z12 + z2'')) + 26 * ((1 + z0 + (1 + z12 + z2'')) * (1 + z0 + (1 + z12 + z2''))) + 115, s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 613 + 3018*s71 + 2871*s71^2 + 897*s71^3 + 171*z0 + 52*z0*z1 + 26*z0^2 + 178*z1 + 27*z1^2 }-> 1 + s89 + s31 + s46 :|: s89 >= 0, s89 <= inf5, s71 >= 0, s71 <= z0 + z1, s46 >= 0, s46 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s31 >= 0, s31 <= 7 * z1 + 6 + z1 * z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eq(z, z') -{ 0 }-> s18 :|: s18 >= 0, s18 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + s77 :|: s77 >= 0, s77 <= z'' + z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + (1 + z06) + (1 + 0 + z2), z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + (1 + z07) + (1 + (1 + z13) + z2), s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> s56 :|: s56 >= 0, s56 <= 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> s72 :|: s72 >= 0, s72 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s73 :|: s73 >= 0, s73 <= z' + (1 + (1 + z08) + z3), z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s74 :|: s74 >= 0, s74 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s75 :|: s75 >= 0, s75 <= z' + (1 + (1 + z14) + z3), s19 >= 0, s19 <= 2, z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s76 :|: s76 >= 0, s76 <= z' + (1 + z2 + z3), z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + s59 + sort(s78) :|: s78 >= 0, s78 <= 0 + 0, s59 >= 0, s59 <= 1 + 0 + 0, z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + s60 + sort(s79) :|: s79 >= 0, s79 <= 1 + (z - 2) + 0, s60 >= 0, s60 <= 1 + (1 + (z - 2)) + 0, z - 2 >= 0
   sort(z) -{ 0 }-> 1 + s61 + sort(s80) :|: s80 >= 0, s80 <= z0 + (1 + z15 + z21), s61 >= 0, s61 <= 1 + z0 + (1 + z15 + z21), s62 >= 0, s62 <= 1 + z0 + (1 + z15 + z21), s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + s63 + sort(s81) :|: s81 >= 0, s81 <= z0 + z1, s63 >= 0, s63 <= 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Function symbols to be analyzed: {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [12 + 7*z'' + z''^2], size: O(n^2) [6 + 7*z'' + z''^2]
  IF_REPLACE: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  IF_MIN: runtime: O(n^2) [10 + 6*z' + z'^2], size: O(n^2) [8 + 4*z' + z'^2]
  MIN: runtime: O(n^2) [154 + 119*z + 26*z^2], size: O(n^2) [115 + 99*z + 26*z^2]
  min: runtime: O(1) [0], size: O(n^1) [z]
  if_min: runtime: O(1) [0], size: O(n^1) [z']
  replace: runtime: O(1) [0], size: O(n^1) [z' + z'']
  if_replace: runtime: O(1) [0], size: O(n^1) [z'' + z4]
  SORT: runtime: O(n^3) [301 + 3018*z + 2871*z^2 + 897*z^3], size: INF

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

(73) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(74)
Obligation:
Complexity RNTS consisting of the following rules:

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 300 + 171*z0 + 52*z0*z2 + 26*z0^2 + 171*z2 + 26*z2^2 }-> 1 + s36 :|: s36 >= 0, s36 <= 99 * (1 + z0 + z2) + 26 * ((1 + z0 + z2) * (1 + z0 + z2)) + 115, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 300 + 171*z1 + 52*z1*z2 + 26*z1^2 + 171*z2 + 26*z2^2 }-> 1 + s37 :|: s37 >= 0, s37 <= 99 * (1 + z1 + z2) + 26 * ((1 + z1 + z2) * (1 + z1 + z2)) + 115, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s25 :|: s25 >= 0, s25 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 29 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s32 + s13 :|: s32 >= 0, s32 <= 8 + (1 + 0 + (1 + z1 + z2)) * (1 + 0 + (1 + z1 + z2)) + 4 * (1 + 0 + (1 + z1 + z2)), s13 >= 0, s13 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 40 + 12*z0' + 2*z0'*z2 + z0'^2 + 12*z2 + z2^2 }-> 1 + s33 + s14 :|: s33 >= 0, s33 <= 8 + (1 + (1 + z0') + (1 + 0 + z2)) * (1 + (1 + z0') + (1 + 0 + z2)) + 4 * (1 + (1 + z0') + (1 + 0 + z2)), s14 >= 0, s14 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 54 + 14*z0'' + 2*z0''*z1' + 2*z0''*z2 + z0''^2 + 15*z1' + 2*z1'*z2 + z1'^2 + 14*z2 + z2^2 }-> 1 + s34 + s15 :|: s34 >= 0, s34 <= 8 + (1 + (1 + z0'') + (1 + (1 + z1') + z2)) * (1 + (1 + z0'') + (1 + (1 + z1') + z2)) + 4 * (1 + (1 + z0'') + (1 + (1 + z1') + z2)), s15 >= 0, s15 <= 1 + z1' + 1, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 29 + 10*z0 + 2*z0*z1 + 2*z0*z2 + z0^2 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s35 + s16 :|: s35 >= 0, s35 <= 8 + (1 + z0 + (1 + z1 + z2)) * (1 + z0 + (1 + z1 + z2)) + 4 * (1 + z0 + (1 + z1 + z2)), s16 >= 0, s16 <= z1 + 1, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s20 + s' :|: s20 >= 0, s20 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z01 + 2*z01*z3 + z01^2 + 11*z3 + z3^2 }-> 1 + s21 + s'' :|: s21 >= 0, s21 <= 7 * (1 + (1 + z01) + z3) + (1 + (1 + z01) + z3) * (1 + (1 + z01) + z3) + 7, s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s22 + s1 :|: s22 >= 0, s22 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z1'' + 2*z1''*z3 + z1''^2 + 11*z3 + z3^2 }-> 1 + s23 + s2 :|: s23 >= 0, s23 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s17 >= 0, s17 <= 2, s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 26 + 10*z2 + 2*z2*z3 + z2^2 + 9*z3 + z3^2 }-> 1 + s24 + s3 :|: s24 >= 0, s24 <= 7 * (1 + z2 + z3) + (1 + z2 + z3) * (1 + z2 + z3) + 7, s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 300 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + s38 :|: s38 >= 0, s38 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 613 + 3018*s64 + 2871*s64^2 + 897*s64^3 }-> 1 + s82 + s26 + s39 :|: s82 >= 0, s82 <= inf, s64 >= 0, s64 <= 0 + 0, s39 >= 0, s39 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s26 >= 0, s26 <= 7 * 0 + 6 + 0 * 0, z = 1 + 0 + 0
   SORT(z) -{ 468 + 3018*s65 + 2871*s65^2 + 897*s65^3 + 119*z + 26*z^2 }-> 1 + s83 + s27 + s40 :|: s83 >= 0, s83 <= inf', s65 >= 0, s65 <= 1 + (z - 2) + 0, s40 >= 0, s40 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s27 >= 0, s27 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 468 + 3018*s66 + 2871*s66^2 + 897*s66^3 + 119*z + 26*z^2 }-> 1 + s84 + s28 + s41 :|: s84 >= 0, s84 <= inf'', s66 >= 0, s66 <= 1 + (z - 2) + 0, s41 >= 0, s41 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s28 >= 0, s28 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 818 + 3018*s67 + 2871*s67^2 + 897*s67^3 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 232*z11 + 54*z11*z2' + 27*z11^2 + 232*z2' + 27*z2'^2 }-> 1 + s85 + s49 + s42 :|: s85 >= 0, s85 <= inf1, s67 >= 0, s67 <= z0 + (1 + z11 + z2'), s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 1 + z0 + (1 + z11 + z2'), s49 >= 0, s49 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s42 >= 0, s42 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 818 + 3018*s68 + 2871*s68^2 + 897*s68^3 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 232*z11 + 54*z11*z2' + 27*z11^2 + 232*z2' + 27*z2'^2 }-> 1 + s86 + s29 + s43 :|: s86 >= 0, s86 <= inf2, s68 >= 0, s68 <= z0 + (1 + z11 + z2'), s50 >= 0, s50 <= 1 + z0 + (1 + z11 + z2'), s43 >= 0, s43 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s29 >= 0, s29 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 468 + 3018*s69 + 2871*s69^2 + 897*s69^3 + 119*z + 26*z^2 }-> 1 + s87 + s30 + s44 :|: s87 >= 0, s87 <= inf3, s69 >= 0, s69 <= 1 + (z - 2) + 0, s44 >= 0, s44 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s30 >= 0, s30 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 818 + 3018*s70 + 2871*s70^2 + 897*s70^3 + 223*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 232*z12 + 54*z12*z2'' + 27*z12^2 + 232*z2'' + 27*z2''^2 }-> 1 + s88 + s52 + s45 :|: s88 >= 0, s88 <= inf4, s70 >= 0, s70 <= z0 + (1 + z12 + z2''), s51 >= 0, s51 <= 1 + z0 + (1 + z12 + z2''), s52 >= 0, s52 <= 7 * (1 + z12 + z2'') + 6 + (1 + z12 + z2'') * (1 + z12 + z2''), s45 >= 0, s45 <= 99 * (1 + z0 + (1 + z12 + z2'')) + 26 * ((1 + z0 + (1 + z12 + z2'')) * (1 + z0 + (1 + z12 + z2''))) + 115, s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 613 + 3018*s71 + 2871*s71^2 + 897*s71^3 + 171*z0 + 52*z0*z1 + 26*z0^2 + 178*z1 + 27*z1^2 }-> 1 + s89 + s31 + s46 :|: s89 >= 0, s89 <= inf5, s71 >= 0, s71 <= z0 + z1, s46 >= 0, s46 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s31 >= 0, s31 <= 7 * z1 + 6 + z1 * z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eq(z, z') -{ 0 }-> s18 :|: s18 >= 0, s18 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + s77 :|: s77 >= 0, s77 <= z'' + z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + (1 + z06) + (1 + 0 + z2), z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + (1 + z07) + (1 + (1 + z13) + z2), s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> s56 :|: s56 >= 0, s56 <= 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> s72 :|: s72 >= 0, s72 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s73 :|: s73 >= 0, s73 <= z' + (1 + (1 + z08) + z3), z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s74 :|: s74 >= 0, s74 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s75 :|: s75 >= 0, s75 <= z' + (1 + (1 + z14) + z3), s19 >= 0, s19 <= 2, z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s76 :|: s76 >= 0, s76 <= z' + (1 + z2 + z3), z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + s59 + sort(s78) :|: s78 >= 0, s78 <= 0 + 0, s59 >= 0, s59 <= 1 + 0 + 0, z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + s60 + sort(s79) :|: s79 >= 0, s79 <= 1 + (z - 2) + 0, s60 >= 0, s60 <= 1 + (1 + (z - 2)) + 0, z - 2 >= 0
   sort(z) -{ 0 }-> 1 + s61 + sort(s80) :|: s80 >= 0, s80 <= z0 + (1 + z15 + z21), s61 >= 0, s61 <= 1 + z0 + (1 + z15 + z21), s62 >= 0, s62 <= 1 + z0 + (1 + z15 + z21), s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + s63 + sort(s81) :|: s81 >= 0, s81 <= z0 + z1, s63 >= 0, s63 <= 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Function symbols to be analyzed: {sort}
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [12 + 7*z'' + z''^2], size: O(n^2) [6 + 7*z'' + z''^2]
  IF_REPLACE: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  IF_MIN: runtime: O(n^2) [10 + 6*z' + z'^2], size: O(n^2) [8 + 4*z' + z'^2]
  MIN: runtime: O(n^2) [154 + 119*z + 26*z^2], size: O(n^2) [115 + 99*z + 26*z^2]
  min: runtime: O(1) [0], size: O(n^1) [z]
  if_min: runtime: O(1) [0], size: O(n^1) [z']
  replace: runtime: O(1) [0], size: O(n^1) [z' + z'']
  if_replace: runtime: O(1) [0], size: O(n^1) [z'' + z4]
  SORT: runtime: O(n^3) [301 + 3018*z + 2871*z^2 + 897*z^3], size: INF
  sort: runtime: ?, size: O(n^2) [z + z^2]

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

(75) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: sort
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

(76)
Obligation:
Complexity RNTS consisting of the following rules:

   EQ(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0
   EQ(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   EQ(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   EQ(z, z') -{ 3 + z' }-> 1 + s :|: s >= 0, s <= z' - 1 + 2, z' - 1 >= 0, z - 1 >= 0
   IF_MIN(z, z') -{ 300 + 171*z0 + 52*z0*z2 + 26*z0^2 + 171*z2 + 26*z2^2 }-> 1 + s36 :|: s36 >= 0, s36 <= 99 * (1 + z0 + z2) + 26 * ((1 + z0 + z2) * (1 + z0 + z2)) + 115, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 300 + 171*z1 + 52*z1*z2 + 26*z1^2 + 171*z2 + 26*z2^2 }-> 1 + s37 :|: s37 >= 0, s37 <= 99 * (1 + z1 + z2) + 26 * ((1 + z1 + z2) * (1 + z1 + z2)) + 115, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s25 :|: s25 >= 0, s25 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   LE(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   LE(z, z') -{ 2 + z' }-> 1 + s12 :|: s12 >= 0, s12 <= z' - 1 + 1, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 :|: z - 2 >= 0
   MIN(z) -{ 1 }-> 0 :|: z = 1 + 0 + 0
   MIN(z) -{ 29 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s32 + s13 :|: s32 >= 0, s32 <= 8 + (1 + 0 + (1 + z1 + z2)) * (1 + 0 + (1 + z1 + z2)) + 4 * (1 + 0 + (1 + z1 + z2)), s13 >= 0, s13 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 40 + 12*z0' + 2*z0'*z2 + z0'^2 + 12*z2 + z2^2 }-> 1 + s33 + s14 :|: s33 >= 0, s33 <= 8 + (1 + (1 + z0') + (1 + 0 + z2)) * (1 + (1 + z0') + (1 + 0 + z2)) + 4 * (1 + (1 + z0') + (1 + 0 + z2)), s14 >= 0, s14 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 54 + 14*z0'' + 2*z0''*z1' + 2*z0''*z2 + z0''^2 + 15*z1' + 2*z1'*z2 + z1'^2 + 14*z2 + z2^2 }-> 1 + s34 + s15 :|: s34 >= 0, s34 <= 8 + (1 + (1 + z0'') + (1 + (1 + z1') + z2)) * (1 + (1 + z0'') + (1 + (1 + z1') + z2)) + 4 * (1 + (1 + z0'') + (1 + (1 + z1') + z2)), s15 >= 0, s15 <= 1 + z1' + 1, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 29 + 10*z0 + 2*z0*z1 + 2*z0*z2 + z0^2 + 11*z1 + 2*z1*z2 + z1^2 + 10*z2 + z2^2 }-> 1 + s35 + s16 :|: s35 >= 0, s35 <= 8 + (1 + z0 + (1 + z1 + z2)) * (1 + z0 + (1 + z1 + z2)) + 4 * (1 + z0 + (1 + z1 + z2)), s16 >= 0, s16 <= z1 + 1, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s20 + s' :|: s20 >= 0, s20 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s' >= 0, s' <= 0 + 2, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z01 + 2*z01*z3 + z01^2 + 11*z3 + z3^2 }-> 1 + s21 + s'' :|: s21 >= 0, s21 <= 7 * (1 + (1 + z01) + z3) + (1 + (1 + z01) + z3) * (1 + (1 + z01) + z3) + 7, s'' >= 0, s'' <= 1 + z01 + 2, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 18 + 7*z'' + z''^2 }-> 1 + s22 + s1 :|: s22 >= 0, s22 <= 7 * (1 + 0 + (z'' - 1)) + (1 + 0 + (z'' - 1)) * (1 + 0 + (z'' - 1)) + 7, s1 >= 0, s1 <= 0 + 2, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 37 + 12*z1'' + 2*z1''*z3 + z1''^2 + 11*z3 + z3^2 }-> 1 + s23 + s2 :|: s23 >= 0, s23 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s17 >= 0, s17 <= 2, s2 >= 0, s2 <= 1 + z1'' + 2, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 26 + 10*z2 + 2*z2*z3 + z2^2 + 9*z3 + z3^2 }-> 1 + s24 + s3 :|: s24 >= 0, s24 <= 7 * (1 + z2 + z3) + (1 + z2 + z3) * (1 + z2 + z3) + 7, s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 0 :|: z = 0
   SORT(z) -{ 300 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + s38 :|: s38 >= 0, s38 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 613 + 3018*s64 + 2871*s64^2 + 897*s64^3 }-> 1 + s82 + s26 + s39 :|: s82 >= 0, s82 <= inf, s64 >= 0, s64 <= 0 + 0, s39 >= 0, s39 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s26 >= 0, s26 <= 7 * 0 + 6 + 0 * 0, z = 1 + 0 + 0
   SORT(z) -{ 468 + 3018*s65 + 2871*s65^2 + 897*s65^3 + 119*z + 26*z^2 }-> 1 + s83 + s27 + s40 :|: s83 >= 0, s83 <= inf', s65 >= 0, s65 <= 1 + (z - 2) + 0, s40 >= 0, s40 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s27 >= 0, s27 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 468 + 3018*s66 + 2871*s66^2 + 897*s66^3 + 119*z + 26*z^2 }-> 1 + s84 + s28 + s41 :|: s84 >= 0, s84 <= inf'', s66 >= 0, s66 <= 1 + (z - 2) + 0, s41 >= 0, s41 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s28 >= 0, s28 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 818 + 3018*s67 + 2871*s67^2 + 897*s67^3 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 232*z11 + 54*z11*z2' + 27*z11^2 + 232*z2' + 27*z2'^2 }-> 1 + s85 + s49 + s42 :|: s85 >= 0, s85 <= inf1, s67 >= 0, s67 <= z0 + (1 + z11 + z2'), s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 1 + z0 + (1 + z11 + z2'), s49 >= 0, s49 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s42 >= 0, s42 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 818 + 3018*s68 + 2871*s68^2 + 897*s68^3 + 223*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 232*z11 + 54*z11*z2' + 27*z11^2 + 232*z2' + 27*z2'^2 }-> 1 + s86 + s29 + s43 :|: s86 >= 0, s86 <= inf2, s68 >= 0, s68 <= z0 + (1 + z11 + z2'), s50 >= 0, s50 <= 1 + z0 + (1 + z11 + z2'), s43 >= 0, s43 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s29 >= 0, s29 <= 7 * (1 + z11 + z2') + 6 + (1 + z11 + z2') * (1 + z11 + z2'), s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 468 + 3018*s69 + 2871*s69^2 + 897*s69^3 + 119*z + 26*z^2 }-> 1 + s87 + s30 + s44 :|: s87 >= 0, s87 <= inf3, s69 >= 0, s69 <= 1 + (z - 2) + 0, s44 >= 0, s44 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s30 >= 0, s30 <= 7 * 0 + 6 + 0 * 0, z - 2 >= 0
   SORT(z) -{ 818 + 3018*s70 + 2871*s70^2 + 897*s70^3 + 223*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 232*z12 + 54*z12*z2'' + 27*z12^2 + 232*z2'' + 27*z2''^2 }-> 1 + s88 + s52 + s45 :|: s88 >= 0, s88 <= inf4, s70 >= 0, s70 <= z0 + (1 + z12 + z2''), s51 >= 0, s51 <= 1 + z0 + (1 + z12 + z2''), s52 >= 0, s52 <= 7 * (1 + z12 + z2'') + 6 + (1 + z12 + z2'') * (1 + z12 + z2''), s45 >= 0, s45 <= 99 * (1 + z0 + (1 + z12 + z2'')) + 26 * ((1 + z0 + (1 + z12 + z2'')) * (1 + z0 + (1 + z12 + z2''))) + 115, s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 613 + 3018*s71 + 2871*s71^2 + 897*s71^3 + 171*z0 + 52*z0*z1 + 26*z0^2 + 178*z1 + 27*z1^2 }-> 1 + s89 + s31 + s46 :|: s89 >= 0, s89 <= inf5, s71 >= 0, s71 <= z0 + z1, s46 >= 0, s46 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s31 >= 0, s31 <= 7 * z1 + 6 + z1 * z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eq(z, z') -{ 0 }-> s18 :|: s18 >= 0, s18 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + s77 :|: s77 >= 0, s77 <= z'' + z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + (1 + z06) + (1 + 0 + z2), z = 1 + (1 + z06) + (1 + 0 + z2), z06 >= 0, z2 >= 0
   min(z) -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + (1 + z07) + (1 + (1 + z13) + z2), s10 >= 0, s10 <= 2, z07 >= 0, z = 1 + (1 + z07) + (1 + (1 + z13) + z2), z13 >= 0, z2 >= 0
   min(z) -{ 0 }-> s56 :|: s56 >= 0, s56 <= 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> s72 :|: s72 >= 0, s72 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s73 :|: s73 >= 0, s73 <= z' + (1 + (1 + z08) + z3), z08 >= 0, z' >= 0, z = 0, z'' = 1 + (1 + z08) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s74 :|: s74 >= 0, s74 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s75 :|: s75 >= 0, s75 <= z' + (1 + (1 + z14) + z3), s19 >= 0, s19 <= 2, z' >= 0, z'' = 1 + (1 + z14) + z3, z - 1 >= 0, z14 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s76 :|: s76 >= 0, s76 <= z' + (1 + z2 + z3), z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   sort(z) -{ 0 }-> 0 :|: z = 0
   sort(z) -{ 0 }-> 0 :|: z >= 0
   sort(z) -{ 0 }-> 1 + s59 + sort(s78) :|: s78 >= 0, s78 <= 0 + 0, s59 >= 0, s59 <= 1 + 0 + 0, z = 1 + 0 + 0
   sort(z) -{ 0 }-> 1 + s60 + sort(s79) :|: s79 >= 0, s79 <= 1 + (z - 2) + 0, s60 >= 0, s60 <= 1 + (1 + (z - 2)) + 0, z - 2 >= 0
   sort(z) -{ 0 }-> 1 + s61 + sort(s80) :|: s80 >= 0, s80 <= z0 + (1 + z15 + z21), s61 >= 0, s61 <= 1 + z0 + (1 + z15 + z21), s62 >= 0, s62 <= 1 + z0 + (1 + z15 + z21), s11 >= 0, s11 <= 2, z = 1 + z0 + (1 + z15 + z21), z21 >= 0, z15 >= 0, z0 >= 0
   sort(z) -{ 0 }-> 1 + s63 + sort(s81) :|: s81 >= 0, s81 <= z0 + z1, s63 >= 0, s63 <= 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Function symbols to be analyzed: 
Previous analysis results are:
  EQ: runtime: O(n^1) [3 + z'], size: O(n^1) [2 + z']
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [2 + z'], size: O(n^1) [1 + z']
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [12 + 7*z'' + z''^2], size: O(n^2) [6 + 7*z'' + z''^2]
  IF_REPLACE: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  IF_MIN: runtime: O(n^2) [10 + 6*z' + z'^2], size: O(n^2) [8 + 4*z' + z'^2]
  MIN: runtime: O(n^2) [154 + 119*z + 26*z^2], size: O(n^2) [115 + 99*z + 26*z^2]
  min: runtime: O(1) [0], size: O(n^1) [z]
  if_min: runtime: O(1) [0], size: O(n^1) [z']
  replace: runtime: O(1) [0], size: O(n^1) [z' + z'']
  if_replace: runtime: O(1) [0], size: O(n^1) [z'' + z4]
  SORT: runtime: O(n^3) [301 + 3018*z + 2871*z^2 + 897*z^3], size: INF
  sort: runtime: O(1) [0], size: O(n^2) [z + z^2]

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

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

(78)
BOUNDS(1, n^3)

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

(79) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(80)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   EQ(0', 0') -> c
   EQ(0', s(z0)) -> c1
   EQ(s(z0), 0') -> c2
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   LE(0', z0) -> c4
   LE(s(z0), 0') -> c5
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   MIN(cons(0', nil)) -> c7
   MIN(cons(s(z0), nil)) -> c8
   MIN(cons(z0, cons(z1, z2))) -> c9(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
   IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
   IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
   REPLACE(z0, z1, nil) -> c12
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
   IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c14
   IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
   SORT(nil) -> c16
   SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
   SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1)))

The (relative) TRS S consists of the following rules:

   eq(0', 0') -> true
   eq(0', s(z0)) -> false
   eq(s(z0), 0') -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0', z0) -> true
   le(s(z0), 0') -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   min(cons(0', nil)) -> 0'
   min(cons(s(z0), nil)) -> s(z0)
   min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
   if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
   if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
   replace(z0, z1, nil) -> nil
   replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
   if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
   if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
   sort(nil) -> nil
   sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))

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

(81) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
REPLACE/1
IF_REPLACE/2

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

(82)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   EQ(0', 0') -> c
   EQ(0', s(z0)) -> c1
   EQ(s(z0), 0') -> c2
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   LE(0', z0) -> c4
   LE(s(z0), 0') -> c5
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   MIN(cons(0', nil)) -> c7
   MIN(cons(s(z0), nil)) -> c8
   MIN(cons(z0, cons(z1, z2))) -> c9(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
   IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
   IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
   REPLACE(z0, nil) -> c12
   REPLACE(z0, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
   IF_REPLACE(true, z0, cons(z2, z3)) -> c14
   IF_REPLACE(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
   SORT(nil) -> c16
   SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
   SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z1), MIN(cons(z0, z1)))

The (relative) TRS S consists of the following rules:

   eq(0', 0') -> true
   eq(0', s(z0)) -> false
   eq(s(z0), 0') -> false
   eq(s(z0), s(z1)) -> eq(z0, z1)
   le(0', z0) -> true
   le(s(z0), 0') -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   min(cons(0', nil)) -> 0'
   min(cons(s(z0), nil)) -> s(z0)
   min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
   if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
   if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
   replace(z0, z1, nil) -> nil
   replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
   if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
   if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
   sort(nil) -> nil
   sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))

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

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

(84)
Obligation:
Innermost TRS:
Rules:
EQ(0', 0') -> c
EQ(0', s(z0)) -> c1
EQ(s(z0), 0') -> c2
EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
LE(0', z0) -> c4
LE(s(z0), 0') -> c5
LE(s(z0), s(z1)) -> c6(LE(z0, z1))
MIN(cons(0', nil)) -> c7
MIN(cons(s(z0), nil)) -> c8
MIN(cons(z0, cons(z1, z2))) -> c9(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, nil) -> c12
REPLACE(z0, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
IF_REPLACE(true, z0, cons(z2, z3)) -> c14
IF_REPLACE(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
SORT(nil) -> c16
SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z1), MIN(cons(z0, z1)))
eq(0', 0') -> true
eq(0', s(z0)) -> false
eq(s(z0), 0') -> false
eq(s(z0), s(z1)) -> eq(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
min(cons(0', nil)) -> 0'
min(cons(s(z0), nil)) -> s(z0)
min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
sort(nil) -> nil
sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))

Types:
EQ :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
LE :: 0':s -> 0':s -> c4:c5:c6
c4 :: c4:c5:c6
c5 :: c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
MIN :: nil:cons -> c7:c8:c9
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c4:c5:c6 -> c7:c8:c9
IF_MIN :: true:false -> nil:cons -> c10:c11
le :: 0':s -> 0':s -> true:false
true :: true:false
c10 :: c7:c8:c9 -> c10:c11
false :: true:false
c11 :: c7:c8:c9 -> c10:c11
REPLACE :: 0':s -> nil:cons -> c12:c13
c12 :: c12:c13
c13 :: c14:c15 -> c:c1:c2:c3 -> c12:c13
IF_REPLACE :: true:false -> 0':s -> nil:cons -> c14:c15
eq :: 0':s -> 0':s -> true:false
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SORT :: nil:cons -> c16:c17:c18
c16 :: c16:c17:c18
c17 :: c7:c8:c9 -> c16:c17:c18
c18 :: c16:c17:c18 -> c12:c13 -> c7:c8:c9 -> c16:c17:c18
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
min :: nil:cons -> 0':s
if_min :: true:false -> nil:cons -> 0':s
if_replace :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
sort :: nil:cons -> nil:cons
hole_c:c1:c2:c31_19 :: c:c1:c2:c3
hole_0':s2_19 :: 0':s
hole_c4:c5:c63_19 :: c4:c5:c6
hole_c7:c8:c94_19 :: c7:c8:c9
hole_nil:cons5_19 :: nil:cons
hole_c10:c116_19 :: c10:c11
hole_true:false7_19 :: true:false
hole_c12:c138_19 :: c12:c13
hole_c14:c159_19 :: c14:c15
hole_c16:c17:c1810_19 :: c16:c17:c18
gen_c:c1:c2:c311_19 :: Nat -> c:c1:c2:c3
gen_0':s12_19 :: Nat -> 0':s
gen_c4:c5:c613_19 :: Nat -> c4:c5:c6
gen_nil:cons14_19 :: Nat -> nil:cons
gen_c16:c17:c1815_19 :: Nat -> c16:c17:c18

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

(85) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
EQ, LE, MIN, le, REPLACE, eq, SORT, replace, min, sort

They will be analysed ascendingly in the following order:
EQ < REPLACE
LE < MIN
le < MIN
MIN < SORT
le < min
eq < REPLACE
REPLACE < SORT
eq < replace
replace < SORT
min < SORT
replace < sort
min < sort

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

(86)
Obligation:
Innermost TRS:
Rules:
EQ(0', 0') -> c
EQ(0', s(z0)) -> c1
EQ(s(z0), 0') -> c2
EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
LE(0', z0) -> c4
LE(s(z0), 0') -> c5
LE(s(z0), s(z1)) -> c6(LE(z0, z1))
MIN(cons(0', nil)) -> c7
MIN(cons(s(z0), nil)) -> c8
MIN(cons(z0, cons(z1, z2))) -> c9(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, nil) -> c12
REPLACE(z0, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
IF_REPLACE(true, z0, cons(z2, z3)) -> c14
IF_REPLACE(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
SORT(nil) -> c16
SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z1), MIN(cons(z0, z1)))
eq(0', 0') -> true
eq(0', s(z0)) -> false
eq(s(z0), 0') -> false
eq(s(z0), s(z1)) -> eq(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
min(cons(0', nil)) -> 0'
min(cons(s(z0), nil)) -> s(z0)
min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
sort(nil) -> nil
sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))

Types:
EQ :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
LE :: 0':s -> 0':s -> c4:c5:c6
c4 :: c4:c5:c6
c5 :: c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
MIN :: nil:cons -> c7:c8:c9
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c4:c5:c6 -> c7:c8:c9
IF_MIN :: true:false -> nil:cons -> c10:c11
le :: 0':s -> 0':s -> true:false
true :: true:false
c10 :: c7:c8:c9 -> c10:c11
false :: true:false
c11 :: c7:c8:c9 -> c10:c11
REPLACE :: 0':s -> nil:cons -> c12:c13
c12 :: c12:c13
c13 :: c14:c15 -> c:c1:c2:c3 -> c12:c13
IF_REPLACE :: true:false -> 0':s -> nil:cons -> c14:c15
eq :: 0':s -> 0':s -> true:false
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SORT :: nil:cons -> c16:c17:c18
c16 :: c16:c17:c18
c17 :: c7:c8:c9 -> c16:c17:c18
c18 :: c16:c17:c18 -> c12:c13 -> c7:c8:c9 -> c16:c17:c18
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
min :: nil:cons -> 0':s
if_min :: true:false -> nil:cons -> 0':s
if_replace :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
sort :: nil:cons -> nil:cons
hole_c:c1:c2:c31_19 :: c:c1:c2:c3
hole_0':s2_19 :: 0':s
hole_c4:c5:c63_19 :: c4:c5:c6
hole_c7:c8:c94_19 :: c7:c8:c9
hole_nil:cons5_19 :: nil:cons
hole_c10:c116_19 :: c10:c11
hole_true:false7_19 :: true:false
hole_c12:c138_19 :: c12:c13
hole_c14:c159_19 :: c14:c15
hole_c16:c17:c1810_19 :: c16:c17:c18
gen_c:c1:c2:c311_19 :: Nat -> c:c1:c2:c3
gen_0':s12_19 :: Nat -> 0':s
gen_c4:c5:c613_19 :: Nat -> c4:c5:c6
gen_nil:cons14_19 :: Nat -> nil:cons
gen_c16:c17:c1815_19 :: Nat -> c16:c17:c18


Generator Equations:
gen_c:c1:c2:c311_19(0) <=> c
gen_c:c1:c2:c311_19(+(x, 1)) <=> c3(gen_c:c1:c2:c311_19(x))
gen_0':s12_19(0) <=> 0'
gen_0':s12_19(+(x, 1)) <=> s(gen_0':s12_19(x))
gen_c4:c5:c613_19(0) <=> c4
gen_c4:c5:c613_19(+(x, 1)) <=> c6(gen_c4:c5:c613_19(x))
gen_nil:cons14_19(0) <=> nil
gen_nil:cons14_19(+(x, 1)) <=> cons(0', gen_nil:cons14_19(x))
gen_c16:c17:c1815_19(0) <=> c16
gen_c16:c17:c1815_19(+(x, 1)) <=> c18(gen_c16:c17:c1815_19(x), c12, c7)


The following defined symbols remain to be analysed:
EQ, LE, MIN, le, REPLACE, eq, SORT, replace, min, sort

They will be analysed ascendingly in the following order:
EQ < REPLACE
LE < MIN
le < MIN
MIN < SORT
le < min
eq < REPLACE
REPLACE < SORT
eq < replace
replace < SORT
min < SORT
replace < sort
min < sort

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

(87) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
EQ(gen_0':s12_19(n17_19), gen_0':s12_19(n17_19)) -> gen_c:c1:c2:c311_19(n17_19), rt in Omega(1 + n17_19)

Induction Base:
EQ(gen_0':s12_19(0), gen_0':s12_19(0)) ->_R^Omega(1)
c

Induction Step:
EQ(gen_0':s12_19(+(n17_19, 1)), gen_0':s12_19(+(n17_19, 1))) ->_R^Omega(1)
c3(EQ(gen_0':s12_19(n17_19), gen_0':s12_19(n17_19))) ->_IH
c3(gen_c:c1:c2:c311_19(c18_19))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(88)
Complex Obligation (BEST)

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

(89)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
EQ(0', 0') -> c
EQ(0', s(z0)) -> c1
EQ(s(z0), 0') -> c2
EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
LE(0', z0) -> c4
LE(s(z0), 0') -> c5
LE(s(z0), s(z1)) -> c6(LE(z0, z1))
MIN(cons(0', nil)) -> c7
MIN(cons(s(z0), nil)) -> c8
MIN(cons(z0, cons(z1, z2))) -> c9(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, nil) -> c12
REPLACE(z0, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
IF_REPLACE(true, z0, cons(z2, z3)) -> c14
IF_REPLACE(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
SORT(nil) -> c16
SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z1), MIN(cons(z0, z1)))
eq(0', 0') -> true
eq(0', s(z0)) -> false
eq(s(z0), 0') -> false
eq(s(z0), s(z1)) -> eq(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
min(cons(0', nil)) -> 0'
min(cons(s(z0), nil)) -> s(z0)
min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
sort(nil) -> nil
sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))

Types:
EQ :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
LE :: 0':s -> 0':s -> c4:c5:c6
c4 :: c4:c5:c6
c5 :: c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
MIN :: nil:cons -> c7:c8:c9
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c4:c5:c6 -> c7:c8:c9
IF_MIN :: true:false -> nil:cons -> c10:c11
le :: 0':s -> 0':s -> true:false
true :: true:false
c10 :: c7:c8:c9 -> c10:c11
false :: true:false
c11 :: c7:c8:c9 -> c10:c11
REPLACE :: 0':s -> nil:cons -> c12:c13
c12 :: c12:c13
c13 :: c14:c15 -> c:c1:c2:c3 -> c12:c13
IF_REPLACE :: true:false -> 0':s -> nil:cons -> c14:c15
eq :: 0':s -> 0':s -> true:false
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SORT :: nil:cons -> c16:c17:c18
c16 :: c16:c17:c18
c17 :: c7:c8:c9 -> c16:c17:c18
c18 :: c16:c17:c18 -> c12:c13 -> c7:c8:c9 -> c16:c17:c18
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
min :: nil:cons -> 0':s
if_min :: true:false -> nil:cons -> 0':s
if_replace :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
sort :: nil:cons -> nil:cons
hole_c:c1:c2:c31_19 :: c:c1:c2:c3
hole_0':s2_19 :: 0':s
hole_c4:c5:c63_19 :: c4:c5:c6
hole_c7:c8:c94_19 :: c7:c8:c9
hole_nil:cons5_19 :: nil:cons
hole_c10:c116_19 :: c10:c11
hole_true:false7_19 :: true:false
hole_c12:c138_19 :: c12:c13
hole_c14:c159_19 :: c14:c15
hole_c16:c17:c1810_19 :: c16:c17:c18
gen_c:c1:c2:c311_19 :: Nat -> c:c1:c2:c3
gen_0':s12_19 :: Nat -> 0':s
gen_c4:c5:c613_19 :: Nat -> c4:c5:c6
gen_nil:cons14_19 :: Nat -> nil:cons
gen_c16:c17:c1815_19 :: Nat -> c16:c17:c18


Generator Equations:
gen_c:c1:c2:c311_19(0) <=> c
gen_c:c1:c2:c311_19(+(x, 1)) <=> c3(gen_c:c1:c2:c311_19(x))
gen_0':s12_19(0) <=> 0'
gen_0':s12_19(+(x, 1)) <=> s(gen_0':s12_19(x))
gen_c4:c5:c613_19(0) <=> c4
gen_c4:c5:c613_19(+(x, 1)) <=> c6(gen_c4:c5:c613_19(x))
gen_nil:cons14_19(0) <=> nil
gen_nil:cons14_19(+(x, 1)) <=> cons(0', gen_nil:cons14_19(x))
gen_c16:c17:c1815_19(0) <=> c16
gen_c16:c17:c1815_19(+(x, 1)) <=> c18(gen_c16:c17:c1815_19(x), c12, c7)


The following defined symbols remain to be analysed:
EQ, LE, MIN, le, REPLACE, eq, SORT, replace, min, sort

They will be analysed ascendingly in the following order:
EQ < REPLACE
LE < MIN
le < MIN
MIN < SORT
le < min
eq < REPLACE
REPLACE < SORT
eq < replace
replace < SORT
min < SORT
replace < sort
min < sort

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

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

(91)
BOUNDS(n^1, INF)

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

(92)
Obligation:
Innermost TRS:
Rules:
EQ(0', 0') -> c
EQ(0', s(z0)) -> c1
EQ(s(z0), 0') -> c2
EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
LE(0', z0) -> c4
LE(s(z0), 0') -> c5
LE(s(z0), s(z1)) -> c6(LE(z0, z1))
MIN(cons(0', nil)) -> c7
MIN(cons(s(z0), nil)) -> c8
MIN(cons(z0, cons(z1, z2))) -> c9(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, nil) -> c12
REPLACE(z0, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
IF_REPLACE(true, z0, cons(z2, z3)) -> c14
IF_REPLACE(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
SORT(nil) -> c16
SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z1), MIN(cons(z0, z1)))
eq(0', 0') -> true
eq(0', s(z0)) -> false
eq(s(z0), 0') -> false
eq(s(z0), s(z1)) -> eq(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
min(cons(0', nil)) -> 0'
min(cons(s(z0), nil)) -> s(z0)
min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
sort(nil) -> nil
sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))

Types:
EQ :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
LE :: 0':s -> 0':s -> c4:c5:c6
c4 :: c4:c5:c6
c5 :: c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
MIN :: nil:cons -> c7:c8:c9
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c4:c5:c6 -> c7:c8:c9
IF_MIN :: true:false -> nil:cons -> c10:c11
le :: 0':s -> 0':s -> true:false
true :: true:false
c10 :: c7:c8:c9 -> c10:c11
false :: true:false
c11 :: c7:c8:c9 -> c10:c11
REPLACE :: 0':s -> nil:cons -> c12:c13
c12 :: c12:c13
c13 :: c14:c15 -> c:c1:c2:c3 -> c12:c13
IF_REPLACE :: true:false -> 0':s -> nil:cons -> c14:c15
eq :: 0':s -> 0':s -> true:false
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SORT :: nil:cons -> c16:c17:c18
c16 :: c16:c17:c18
c17 :: c7:c8:c9 -> c16:c17:c18
c18 :: c16:c17:c18 -> c12:c13 -> c7:c8:c9 -> c16:c17:c18
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
min :: nil:cons -> 0':s
if_min :: true:false -> nil:cons -> 0':s
if_replace :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
sort :: nil:cons -> nil:cons
hole_c:c1:c2:c31_19 :: c:c1:c2:c3
hole_0':s2_19 :: 0':s
hole_c4:c5:c63_19 :: c4:c5:c6
hole_c7:c8:c94_19 :: c7:c8:c9
hole_nil:cons5_19 :: nil:cons
hole_c10:c116_19 :: c10:c11
hole_true:false7_19 :: true:false
hole_c12:c138_19 :: c12:c13
hole_c14:c159_19 :: c14:c15
hole_c16:c17:c1810_19 :: c16:c17:c18
gen_c:c1:c2:c311_19 :: Nat -> c:c1:c2:c3
gen_0':s12_19 :: Nat -> 0':s
gen_c4:c5:c613_19 :: Nat -> c4:c5:c6
gen_nil:cons14_19 :: Nat -> nil:cons
gen_c16:c17:c1815_19 :: Nat -> c16:c17:c18


Lemmas:
EQ(gen_0':s12_19(n17_19), gen_0':s12_19(n17_19)) -> gen_c:c1:c2:c311_19(n17_19), rt in Omega(1 + n17_19)


Generator Equations:
gen_c:c1:c2:c311_19(0) <=> c
gen_c:c1:c2:c311_19(+(x, 1)) <=> c3(gen_c:c1:c2:c311_19(x))
gen_0':s12_19(0) <=> 0'
gen_0':s12_19(+(x, 1)) <=> s(gen_0':s12_19(x))
gen_c4:c5:c613_19(0) <=> c4
gen_c4:c5:c613_19(+(x, 1)) <=> c6(gen_c4:c5:c613_19(x))
gen_nil:cons14_19(0) <=> nil
gen_nil:cons14_19(+(x, 1)) <=> cons(0', gen_nil:cons14_19(x))
gen_c16:c17:c1815_19(0) <=> c16
gen_c16:c17:c1815_19(+(x, 1)) <=> c18(gen_c16:c17:c1815_19(x), c12, c7)


The following defined symbols remain to be analysed:
LE, MIN, le, REPLACE, eq, SORT, replace, min, sort

They will be analysed ascendingly in the following order:
LE < MIN
le < MIN
MIN < SORT
le < min
eq < REPLACE
REPLACE < SORT
eq < replace
replace < SORT
min < SORT
replace < sort
min < sort

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

(93) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LE(gen_0':s12_19(n1023_19), gen_0':s12_19(n1023_19)) -> gen_c4:c5:c613_19(n1023_19), rt in Omega(1 + n1023_19)

Induction Base:
LE(gen_0':s12_19(0), gen_0':s12_19(0)) ->_R^Omega(1)
c4

Induction Step:
LE(gen_0':s12_19(+(n1023_19, 1)), gen_0':s12_19(+(n1023_19, 1))) ->_R^Omega(1)
c6(LE(gen_0':s12_19(n1023_19), gen_0':s12_19(n1023_19))) ->_IH
c6(gen_c4:c5:c613_19(c1024_19))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(94)
Obligation:
Innermost TRS:
Rules:
EQ(0', 0') -> c
EQ(0', s(z0)) -> c1
EQ(s(z0), 0') -> c2
EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
LE(0', z0) -> c4
LE(s(z0), 0') -> c5
LE(s(z0), s(z1)) -> c6(LE(z0, z1))
MIN(cons(0', nil)) -> c7
MIN(cons(s(z0), nil)) -> c8
MIN(cons(z0, cons(z1, z2))) -> c9(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, nil) -> c12
REPLACE(z0, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
IF_REPLACE(true, z0, cons(z2, z3)) -> c14
IF_REPLACE(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
SORT(nil) -> c16
SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z1), MIN(cons(z0, z1)))
eq(0', 0') -> true
eq(0', s(z0)) -> false
eq(s(z0), 0') -> false
eq(s(z0), s(z1)) -> eq(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
min(cons(0', nil)) -> 0'
min(cons(s(z0), nil)) -> s(z0)
min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
sort(nil) -> nil
sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))

Types:
EQ :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
LE :: 0':s -> 0':s -> c4:c5:c6
c4 :: c4:c5:c6
c5 :: c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
MIN :: nil:cons -> c7:c8:c9
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c4:c5:c6 -> c7:c8:c9
IF_MIN :: true:false -> nil:cons -> c10:c11
le :: 0':s -> 0':s -> true:false
true :: true:false
c10 :: c7:c8:c9 -> c10:c11
false :: true:false
c11 :: c7:c8:c9 -> c10:c11
REPLACE :: 0':s -> nil:cons -> c12:c13
c12 :: c12:c13
c13 :: c14:c15 -> c:c1:c2:c3 -> c12:c13
IF_REPLACE :: true:false -> 0':s -> nil:cons -> c14:c15
eq :: 0':s -> 0':s -> true:false
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SORT :: nil:cons -> c16:c17:c18
c16 :: c16:c17:c18
c17 :: c7:c8:c9 -> c16:c17:c18
c18 :: c16:c17:c18 -> c12:c13 -> c7:c8:c9 -> c16:c17:c18
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
min :: nil:cons -> 0':s
if_min :: true:false -> nil:cons -> 0':s
if_replace :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
sort :: nil:cons -> nil:cons
hole_c:c1:c2:c31_19 :: c:c1:c2:c3
hole_0':s2_19 :: 0':s
hole_c4:c5:c63_19 :: c4:c5:c6
hole_c7:c8:c94_19 :: c7:c8:c9
hole_nil:cons5_19 :: nil:cons
hole_c10:c116_19 :: c10:c11
hole_true:false7_19 :: true:false
hole_c12:c138_19 :: c12:c13
hole_c14:c159_19 :: c14:c15
hole_c16:c17:c1810_19 :: c16:c17:c18
gen_c:c1:c2:c311_19 :: Nat -> c:c1:c2:c3
gen_0':s12_19 :: Nat -> 0':s
gen_c4:c5:c613_19 :: Nat -> c4:c5:c6
gen_nil:cons14_19 :: Nat -> nil:cons
gen_c16:c17:c1815_19 :: Nat -> c16:c17:c18


Lemmas:
EQ(gen_0':s12_19(n17_19), gen_0':s12_19(n17_19)) -> gen_c:c1:c2:c311_19(n17_19), rt in Omega(1 + n17_19)
LE(gen_0':s12_19(n1023_19), gen_0':s12_19(n1023_19)) -> gen_c4:c5:c613_19(n1023_19), rt in Omega(1 + n1023_19)


Generator Equations:
gen_c:c1:c2:c311_19(0) <=> c
gen_c:c1:c2:c311_19(+(x, 1)) <=> c3(gen_c:c1:c2:c311_19(x))
gen_0':s12_19(0) <=> 0'
gen_0':s12_19(+(x, 1)) <=> s(gen_0':s12_19(x))
gen_c4:c5:c613_19(0) <=> c4
gen_c4:c5:c613_19(+(x, 1)) <=> c6(gen_c4:c5:c613_19(x))
gen_nil:cons14_19(0) <=> nil
gen_nil:cons14_19(+(x, 1)) <=> cons(0', gen_nil:cons14_19(x))
gen_c16:c17:c1815_19(0) <=> c16
gen_c16:c17:c1815_19(+(x, 1)) <=> c18(gen_c16:c17:c1815_19(x), c12, c7)


The following defined symbols remain to be analysed:
le, MIN, REPLACE, eq, SORT, replace, min, sort

They will be analysed ascendingly in the following order:
le < MIN
MIN < SORT
le < min
eq < REPLACE
REPLACE < SORT
eq < replace
replace < SORT
min < SORT
replace < sort
min < sort

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

(95) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s12_19(n1803_19), gen_0':s12_19(n1803_19)) -> true, rt in Omega(0)

Induction Base:
le(gen_0':s12_19(0), gen_0':s12_19(0)) ->_R^Omega(0)
true

Induction Step:
le(gen_0':s12_19(+(n1803_19, 1)), gen_0':s12_19(+(n1803_19, 1))) ->_R^Omega(0)
le(gen_0':s12_19(n1803_19), gen_0':s12_19(n1803_19)) ->_IH
true

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(96)
Obligation:
Innermost TRS:
Rules:
EQ(0', 0') -> c
EQ(0', s(z0)) -> c1
EQ(s(z0), 0') -> c2
EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
LE(0', z0) -> c4
LE(s(z0), 0') -> c5
LE(s(z0), s(z1)) -> c6(LE(z0, z1))
MIN(cons(0', nil)) -> c7
MIN(cons(s(z0), nil)) -> c8
MIN(cons(z0, cons(z1, z2))) -> c9(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, nil) -> c12
REPLACE(z0, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
IF_REPLACE(true, z0, cons(z2, z3)) -> c14
IF_REPLACE(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
SORT(nil) -> c16
SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z1), MIN(cons(z0, z1)))
eq(0', 0') -> true
eq(0', s(z0)) -> false
eq(s(z0), 0') -> false
eq(s(z0), s(z1)) -> eq(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
min(cons(0', nil)) -> 0'
min(cons(s(z0), nil)) -> s(z0)
min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
sort(nil) -> nil
sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))

Types:
EQ :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
LE :: 0':s -> 0':s -> c4:c5:c6
c4 :: c4:c5:c6
c5 :: c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
MIN :: nil:cons -> c7:c8:c9
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c4:c5:c6 -> c7:c8:c9
IF_MIN :: true:false -> nil:cons -> c10:c11
le :: 0':s -> 0':s -> true:false
true :: true:false
c10 :: c7:c8:c9 -> c10:c11
false :: true:false
c11 :: c7:c8:c9 -> c10:c11
REPLACE :: 0':s -> nil:cons -> c12:c13
c12 :: c12:c13
c13 :: c14:c15 -> c:c1:c2:c3 -> c12:c13
IF_REPLACE :: true:false -> 0':s -> nil:cons -> c14:c15
eq :: 0':s -> 0':s -> true:false
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SORT :: nil:cons -> c16:c17:c18
c16 :: c16:c17:c18
c17 :: c7:c8:c9 -> c16:c17:c18
c18 :: c16:c17:c18 -> c12:c13 -> c7:c8:c9 -> c16:c17:c18
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
min :: nil:cons -> 0':s
if_min :: true:false -> nil:cons -> 0':s
if_replace :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
sort :: nil:cons -> nil:cons
hole_c:c1:c2:c31_19 :: c:c1:c2:c3
hole_0':s2_19 :: 0':s
hole_c4:c5:c63_19 :: c4:c5:c6
hole_c7:c8:c94_19 :: c7:c8:c9
hole_nil:cons5_19 :: nil:cons
hole_c10:c116_19 :: c10:c11
hole_true:false7_19 :: true:false
hole_c12:c138_19 :: c12:c13
hole_c14:c159_19 :: c14:c15
hole_c16:c17:c1810_19 :: c16:c17:c18
gen_c:c1:c2:c311_19 :: Nat -> c:c1:c2:c3
gen_0':s12_19 :: Nat -> 0':s
gen_c4:c5:c613_19 :: Nat -> c4:c5:c6
gen_nil:cons14_19 :: Nat -> nil:cons
gen_c16:c17:c1815_19 :: Nat -> c16:c17:c18


Lemmas:
EQ(gen_0':s12_19(n17_19), gen_0':s12_19(n17_19)) -> gen_c:c1:c2:c311_19(n17_19), rt in Omega(1 + n17_19)
LE(gen_0':s12_19(n1023_19), gen_0':s12_19(n1023_19)) -> gen_c4:c5:c613_19(n1023_19), rt in Omega(1 + n1023_19)
le(gen_0':s12_19(n1803_19), gen_0':s12_19(n1803_19)) -> true, rt in Omega(0)


Generator Equations:
gen_c:c1:c2:c311_19(0) <=> c
gen_c:c1:c2:c311_19(+(x, 1)) <=> c3(gen_c:c1:c2:c311_19(x))
gen_0':s12_19(0) <=> 0'
gen_0':s12_19(+(x, 1)) <=> s(gen_0':s12_19(x))
gen_c4:c5:c613_19(0) <=> c4
gen_c4:c5:c613_19(+(x, 1)) <=> c6(gen_c4:c5:c613_19(x))
gen_nil:cons14_19(0) <=> nil
gen_nil:cons14_19(+(x, 1)) <=> cons(0', gen_nil:cons14_19(x))
gen_c16:c17:c1815_19(0) <=> c16
gen_c16:c17:c1815_19(+(x, 1)) <=> c18(gen_c16:c17:c1815_19(x), c12, c7)


The following defined symbols remain to be analysed:
MIN, REPLACE, eq, SORT, replace, min, sort

They will be analysed ascendingly in the following order:
MIN < SORT
eq < REPLACE
REPLACE < SORT
eq < replace
replace < SORT
min < SORT
replace < sort
min < sort

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

(97) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MIN(gen_nil:cons14_19(+(1, n2270_19))) -> *16_19, rt in Omega(n2270_19)

Induction Base:
MIN(gen_nil:cons14_19(+(1, 0)))

Induction Step:
MIN(gen_nil:cons14_19(+(1, +(n2270_19, 1)))) ->_R^Omega(1)
c9(IF_MIN(le(0', 0'), cons(0', cons(0', gen_nil:cons14_19(n2270_19)))), LE(0', 0')) ->_L^Omega(0)
c9(IF_MIN(true, cons(0', cons(0', gen_nil:cons14_19(n2270_19)))), LE(0', 0')) ->_R^Omega(1)
c9(c10(MIN(cons(0', gen_nil:cons14_19(n2270_19)))), LE(0', 0')) ->_IH
c9(c10(*16_19), LE(0', 0')) ->_L^Omega(1)
c9(c10(*16_19), gen_c4:c5:c613_19(0))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(98)
Obligation:
Innermost TRS:
Rules:
EQ(0', 0') -> c
EQ(0', s(z0)) -> c1
EQ(s(z0), 0') -> c2
EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
LE(0', z0) -> c4
LE(s(z0), 0') -> c5
LE(s(z0), s(z1)) -> c6(LE(z0, z1))
MIN(cons(0', nil)) -> c7
MIN(cons(s(z0), nil)) -> c8
MIN(cons(z0, cons(z1, z2))) -> c9(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, nil) -> c12
REPLACE(z0, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
IF_REPLACE(true, z0, cons(z2, z3)) -> c14
IF_REPLACE(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
SORT(nil) -> c16
SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z1), MIN(cons(z0, z1)))
eq(0', 0') -> true
eq(0', s(z0)) -> false
eq(s(z0), 0') -> false
eq(s(z0), s(z1)) -> eq(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
min(cons(0', nil)) -> 0'
min(cons(s(z0), nil)) -> s(z0)
min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
sort(nil) -> nil
sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))

Types:
EQ :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
LE :: 0':s -> 0':s -> c4:c5:c6
c4 :: c4:c5:c6
c5 :: c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
MIN :: nil:cons -> c7:c8:c9
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c4:c5:c6 -> c7:c8:c9
IF_MIN :: true:false -> nil:cons -> c10:c11
le :: 0':s -> 0':s -> true:false
true :: true:false
c10 :: c7:c8:c9 -> c10:c11
false :: true:false
c11 :: c7:c8:c9 -> c10:c11
REPLACE :: 0':s -> nil:cons -> c12:c13
c12 :: c12:c13
c13 :: c14:c15 -> c:c1:c2:c3 -> c12:c13
IF_REPLACE :: true:false -> 0':s -> nil:cons -> c14:c15
eq :: 0':s -> 0':s -> true:false
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SORT :: nil:cons -> c16:c17:c18
c16 :: c16:c17:c18
c17 :: c7:c8:c9 -> c16:c17:c18
c18 :: c16:c17:c18 -> c12:c13 -> c7:c8:c9 -> c16:c17:c18
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
min :: nil:cons -> 0':s
if_min :: true:false -> nil:cons -> 0':s
if_replace :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
sort :: nil:cons -> nil:cons
hole_c:c1:c2:c31_19 :: c:c1:c2:c3
hole_0':s2_19 :: 0':s
hole_c4:c5:c63_19 :: c4:c5:c6
hole_c7:c8:c94_19 :: c7:c8:c9
hole_nil:cons5_19 :: nil:cons
hole_c10:c116_19 :: c10:c11
hole_true:false7_19 :: true:false
hole_c12:c138_19 :: c12:c13
hole_c14:c159_19 :: c14:c15
hole_c16:c17:c1810_19 :: c16:c17:c18
gen_c:c1:c2:c311_19 :: Nat -> c:c1:c2:c3
gen_0':s12_19 :: Nat -> 0':s
gen_c4:c5:c613_19 :: Nat -> c4:c5:c6
gen_nil:cons14_19 :: Nat -> nil:cons
gen_c16:c17:c1815_19 :: Nat -> c16:c17:c18


Lemmas:
EQ(gen_0':s12_19(n17_19), gen_0':s12_19(n17_19)) -> gen_c:c1:c2:c311_19(n17_19), rt in Omega(1 + n17_19)
LE(gen_0':s12_19(n1023_19), gen_0':s12_19(n1023_19)) -> gen_c4:c5:c613_19(n1023_19), rt in Omega(1 + n1023_19)
le(gen_0':s12_19(n1803_19), gen_0':s12_19(n1803_19)) -> true, rt in Omega(0)
MIN(gen_nil:cons14_19(+(1, n2270_19))) -> *16_19, rt in Omega(n2270_19)


Generator Equations:
gen_c:c1:c2:c311_19(0) <=> c
gen_c:c1:c2:c311_19(+(x, 1)) <=> c3(gen_c:c1:c2:c311_19(x))
gen_0':s12_19(0) <=> 0'
gen_0':s12_19(+(x, 1)) <=> s(gen_0':s12_19(x))
gen_c4:c5:c613_19(0) <=> c4
gen_c4:c5:c613_19(+(x, 1)) <=> c6(gen_c4:c5:c613_19(x))
gen_nil:cons14_19(0) <=> nil
gen_nil:cons14_19(+(x, 1)) <=> cons(0', gen_nil:cons14_19(x))
gen_c16:c17:c1815_19(0) <=> c16
gen_c16:c17:c1815_19(+(x, 1)) <=> c18(gen_c16:c17:c1815_19(x), c12, c7)


The following defined symbols remain to be analysed:
eq, REPLACE, SORT, replace, min, sort

They will be analysed ascendingly in the following order:
eq < REPLACE
REPLACE < SORT
eq < replace
replace < SORT
min < SORT
replace < sort
min < sort

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

(99) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
eq(gen_0':s12_19(n7487_19), gen_0':s12_19(n7487_19)) -> true, rt in Omega(0)

Induction Base:
eq(gen_0':s12_19(0), gen_0':s12_19(0)) ->_R^Omega(0)
true

Induction Step:
eq(gen_0':s12_19(+(n7487_19, 1)), gen_0':s12_19(+(n7487_19, 1))) ->_R^Omega(0)
eq(gen_0':s12_19(n7487_19), gen_0':s12_19(n7487_19)) ->_IH
true

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(100)
Obligation:
Innermost TRS:
Rules:
EQ(0', 0') -> c
EQ(0', s(z0)) -> c1
EQ(s(z0), 0') -> c2
EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
LE(0', z0) -> c4
LE(s(z0), 0') -> c5
LE(s(z0), s(z1)) -> c6(LE(z0, z1))
MIN(cons(0', nil)) -> c7
MIN(cons(s(z0), nil)) -> c8
MIN(cons(z0, cons(z1, z2))) -> c9(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, nil) -> c12
REPLACE(z0, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
IF_REPLACE(true, z0, cons(z2, z3)) -> c14
IF_REPLACE(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
SORT(nil) -> c16
SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z1), MIN(cons(z0, z1)))
eq(0', 0') -> true
eq(0', s(z0)) -> false
eq(s(z0), 0') -> false
eq(s(z0), s(z1)) -> eq(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
min(cons(0', nil)) -> 0'
min(cons(s(z0), nil)) -> s(z0)
min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
sort(nil) -> nil
sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))

Types:
EQ :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
LE :: 0':s -> 0':s -> c4:c5:c6
c4 :: c4:c5:c6
c5 :: c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
MIN :: nil:cons -> c7:c8:c9
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c4:c5:c6 -> c7:c8:c9
IF_MIN :: true:false -> nil:cons -> c10:c11
le :: 0':s -> 0':s -> true:false
true :: true:false
c10 :: c7:c8:c9 -> c10:c11
false :: true:false
c11 :: c7:c8:c9 -> c10:c11
REPLACE :: 0':s -> nil:cons -> c12:c13
c12 :: c12:c13
c13 :: c14:c15 -> c:c1:c2:c3 -> c12:c13
IF_REPLACE :: true:false -> 0':s -> nil:cons -> c14:c15
eq :: 0':s -> 0':s -> true:false
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SORT :: nil:cons -> c16:c17:c18
c16 :: c16:c17:c18
c17 :: c7:c8:c9 -> c16:c17:c18
c18 :: c16:c17:c18 -> c12:c13 -> c7:c8:c9 -> c16:c17:c18
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
min :: nil:cons -> 0':s
if_min :: true:false -> nil:cons -> 0':s
if_replace :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
sort :: nil:cons -> nil:cons
hole_c:c1:c2:c31_19 :: c:c1:c2:c3
hole_0':s2_19 :: 0':s
hole_c4:c5:c63_19 :: c4:c5:c6
hole_c7:c8:c94_19 :: c7:c8:c9
hole_nil:cons5_19 :: nil:cons
hole_c10:c116_19 :: c10:c11
hole_true:false7_19 :: true:false
hole_c12:c138_19 :: c12:c13
hole_c14:c159_19 :: c14:c15
hole_c16:c17:c1810_19 :: c16:c17:c18
gen_c:c1:c2:c311_19 :: Nat -> c:c1:c2:c3
gen_0':s12_19 :: Nat -> 0':s
gen_c4:c5:c613_19 :: Nat -> c4:c5:c6
gen_nil:cons14_19 :: Nat -> nil:cons
gen_c16:c17:c1815_19 :: Nat -> c16:c17:c18


Lemmas:
EQ(gen_0':s12_19(n17_19), gen_0':s12_19(n17_19)) -> gen_c:c1:c2:c311_19(n17_19), rt in Omega(1 + n17_19)
LE(gen_0':s12_19(n1023_19), gen_0':s12_19(n1023_19)) -> gen_c4:c5:c613_19(n1023_19), rt in Omega(1 + n1023_19)
le(gen_0':s12_19(n1803_19), gen_0':s12_19(n1803_19)) -> true, rt in Omega(0)
MIN(gen_nil:cons14_19(+(1, n2270_19))) -> *16_19, rt in Omega(n2270_19)
eq(gen_0':s12_19(n7487_19), gen_0':s12_19(n7487_19)) -> true, rt in Omega(0)


Generator Equations:
gen_c:c1:c2:c311_19(0) <=> c
gen_c:c1:c2:c311_19(+(x, 1)) <=> c3(gen_c:c1:c2:c311_19(x))
gen_0':s12_19(0) <=> 0'
gen_0':s12_19(+(x, 1)) <=> s(gen_0':s12_19(x))
gen_c4:c5:c613_19(0) <=> c4
gen_c4:c5:c613_19(+(x, 1)) <=> c6(gen_c4:c5:c613_19(x))
gen_nil:cons14_19(0) <=> nil
gen_nil:cons14_19(+(x, 1)) <=> cons(0', gen_nil:cons14_19(x))
gen_c16:c17:c1815_19(0) <=> c16
gen_c16:c17:c1815_19(+(x, 1)) <=> c18(gen_c16:c17:c1815_19(x), c12, c7)


The following defined symbols remain to be analysed:
REPLACE, SORT, replace, min, sort

They will be analysed ascendingly in the following order:
REPLACE < SORT
replace < SORT
min < SORT
replace < sort
min < sort

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

(101) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
min(gen_nil:cons14_19(+(1, n9373_19))) -> gen_0':s12_19(0), rt in Omega(0)

Induction Base:
min(gen_nil:cons14_19(+(1, 0))) ->_R^Omega(0)
0'

Induction Step:
min(gen_nil:cons14_19(+(1, +(n9373_19, 1)))) ->_R^Omega(0)
if_min(le(0', 0'), cons(0', cons(0', gen_nil:cons14_19(n9373_19)))) ->_L^Omega(0)
if_min(true, cons(0', cons(0', gen_nil:cons14_19(n9373_19)))) ->_R^Omega(0)
min(cons(0', gen_nil:cons14_19(n9373_19))) ->_IH
gen_0':s12_19(0)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(102)
Obligation:
Innermost TRS:
Rules:
EQ(0', 0') -> c
EQ(0', s(z0)) -> c1
EQ(s(z0), 0') -> c2
EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
LE(0', z0) -> c4
LE(s(z0), 0') -> c5
LE(s(z0), s(z1)) -> c6(LE(z0, z1))
MIN(cons(0', nil)) -> c7
MIN(cons(s(z0), nil)) -> c8
MIN(cons(z0, cons(z1, z2))) -> c9(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, nil) -> c12
REPLACE(z0, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
IF_REPLACE(true, z0, cons(z2, z3)) -> c14
IF_REPLACE(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
SORT(nil) -> c16
SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z1), MIN(cons(z0, z1)))
eq(0', 0') -> true
eq(0', s(z0)) -> false
eq(s(z0), 0') -> false
eq(s(z0), s(z1)) -> eq(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
min(cons(0', nil)) -> 0'
min(cons(s(z0), nil)) -> s(z0)
min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
sort(nil) -> nil
sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))

Types:
EQ :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
LE :: 0':s -> 0':s -> c4:c5:c6
c4 :: c4:c5:c6
c5 :: c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
MIN :: nil:cons -> c7:c8:c9
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c4:c5:c6 -> c7:c8:c9
IF_MIN :: true:false -> nil:cons -> c10:c11
le :: 0':s -> 0':s -> true:false
true :: true:false
c10 :: c7:c8:c9 -> c10:c11
false :: true:false
c11 :: c7:c8:c9 -> c10:c11
REPLACE :: 0':s -> nil:cons -> c12:c13
c12 :: c12:c13
c13 :: c14:c15 -> c:c1:c2:c3 -> c12:c13
IF_REPLACE :: true:false -> 0':s -> nil:cons -> c14:c15
eq :: 0':s -> 0':s -> true:false
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SORT :: nil:cons -> c16:c17:c18
c16 :: c16:c17:c18
c17 :: c7:c8:c9 -> c16:c17:c18
c18 :: c16:c17:c18 -> c12:c13 -> c7:c8:c9 -> c16:c17:c18
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
min :: nil:cons -> 0':s
if_min :: true:false -> nil:cons -> 0':s
if_replace :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
sort :: nil:cons -> nil:cons
hole_c:c1:c2:c31_19 :: c:c1:c2:c3
hole_0':s2_19 :: 0':s
hole_c4:c5:c63_19 :: c4:c5:c6
hole_c7:c8:c94_19 :: c7:c8:c9
hole_nil:cons5_19 :: nil:cons
hole_c10:c116_19 :: c10:c11
hole_true:false7_19 :: true:false
hole_c12:c138_19 :: c12:c13
hole_c14:c159_19 :: c14:c15
hole_c16:c17:c1810_19 :: c16:c17:c18
gen_c:c1:c2:c311_19 :: Nat -> c:c1:c2:c3
gen_0':s12_19 :: Nat -> 0':s
gen_c4:c5:c613_19 :: Nat -> c4:c5:c6
gen_nil:cons14_19 :: Nat -> nil:cons
gen_c16:c17:c1815_19 :: Nat -> c16:c17:c18


Lemmas:
EQ(gen_0':s12_19(n17_19), gen_0':s12_19(n17_19)) -> gen_c:c1:c2:c311_19(n17_19), rt in Omega(1 + n17_19)
LE(gen_0':s12_19(n1023_19), gen_0':s12_19(n1023_19)) -> gen_c4:c5:c613_19(n1023_19), rt in Omega(1 + n1023_19)
le(gen_0':s12_19(n1803_19), gen_0':s12_19(n1803_19)) -> true, rt in Omega(0)
MIN(gen_nil:cons14_19(+(1, n2270_19))) -> *16_19, rt in Omega(n2270_19)
eq(gen_0':s12_19(n7487_19), gen_0':s12_19(n7487_19)) -> true, rt in Omega(0)
min(gen_nil:cons14_19(+(1, n9373_19))) -> gen_0':s12_19(0), rt in Omega(0)


Generator Equations:
gen_c:c1:c2:c311_19(0) <=> c
gen_c:c1:c2:c311_19(+(x, 1)) <=> c3(gen_c:c1:c2:c311_19(x))
gen_0':s12_19(0) <=> 0'
gen_0':s12_19(+(x, 1)) <=> s(gen_0':s12_19(x))
gen_c4:c5:c613_19(0) <=> c4
gen_c4:c5:c613_19(+(x, 1)) <=> c6(gen_c4:c5:c613_19(x))
gen_nil:cons14_19(0) <=> nil
gen_nil:cons14_19(+(x, 1)) <=> cons(0', gen_nil:cons14_19(x))
gen_c16:c17:c1815_19(0) <=> c16
gen_c16:c17:c1815_19(+(x, 1)) <=> c18(gen_c16:c17:c1815_19(x), c12, c7)


The following defined symbols remain to be analysed:
SORT, sort
----------------------------------------

(103) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
SORT(gen_nil:cons14_19(+(1, n10227_19))) -> *16_19, rt in Omega(n10227_19 + n10227_19^2)

Induction Base:
SORT(gen_nil:cons14_19(+(1, 0)))

Induction Step:
SORT(gen_nil:cons14_19(+(1, +(n10227_19, 1)))) ->_R^Omega(1)
c18(SORT(replace(min(cons(0', gen_nil:cons14_19(+(1, n10227_19)))), 0', gen_nil:cons14_19(+(1, n10227_19)))), REPLACE(min(cons(0', gen_nil:cons14_19(+(1, n10227_19)))), gen_nil:cons14_19(+(1, n10227_19))), MIN(cons(0', gen_nil:cons14_19(+(1, n10227_19))))) ->_L^Omega(0)
c18(SORT(replace(gen_0':s12_19(0), 0', gen_nil:cons14_19(+(1, n10227_19)))), REPLACE(min(cons(0', gen_nil:cons14_19(+(1, n10227_19)))), gen_nil:cons14_19(+(1, n10227_19))), MIN(cons(0', gen_nil:cons14_19(+(1, n10227_19))))) ->_R^Omega(0)
c18(SORT(if_replace(eq(gen_0':s12_19(0), 0'), gen_0':s12_19(0), 0', cons(0', gen_nil:cons14_19(n10227_19)))), REPLACE(min(cons(0', gen_nil:cons14_19(+(1, n10227_19)))), gen_nil:cons14_19(+(1, n10227_19))), MIN(cons(0', gen_nil:cons14_19(+(1, n10227_19))))) ->_L^Omega(0)
c18(SORT(if_replace(true, gen_0':s12_19(0), 0', cons(0', gen_nil:cons14_19(n10227_19)))), REPLACE(min(cons(0', gen_nil:cons14_19(+(1, n10227_19)))), gen_nil:cons14_19(+(1, n10227_19))), MIN(cons(0', gen_nil:cons14_19(+(1, n10227_19))))) ->_R^Omega(0)
c18(SORT(cons(0', gen_nil:cons14_19(n10227_19))), REPLACE(min(cons(0', gen_nil:cons14_19(+(1, n10227_19)))), gen_nil:cons14_19(+(1, n10227_19))), MIN(cons(0', gen_nil:cons14_19(+(1, n10227_19))))) ->_IH
c18(*16_19, REPLACE(min(cons(0', gen_nil:cons14_19(+(1, n10227_19)))), gen_nil:cons14_19(+(1, n10227_19))), MIN(cons(0', gen_nil:cons14_19(+(1, n10227_19))))) ->_L^Omega(0)
c18(*16_19, REPLACE(gen_0':s12_19(0), gen_nil:cons14_19(+(1, n10227_19))), MIN(cons(0', gen_nil:cons14_19(+(1, n10227_19))))) ->_R^Omega(1)
c18(*16_19, c13(IF_REPLACE(eq(gen_0':s12_19(0), 0'), gen_0':s12_19(0), cons(0', gen_nil:cons14_19(n10227_19))), EQ(gen_0':s12_19(0), 0')), MIN(cons(0', gen_nil:cons14_19(+(1, n10227_19))))) ->_L^Omega(0)
c18(*16_19, c13(IF_REPLACE(true, gen_0':s12_19(0), cons(0', gen_nil:cons14_19(n10227_19))), EQ(gen_0':s12_19(0), 0')), MIN(cons(0', gen_nil:cons14_19(+(1, n10227_19))))) ->_R^Omega(1)
c18(*16_19, c13(c14, EQ(gen_0':s12_19(0), 0')), MIN(cons(0', gen_nil:cons14_19(+(1, n10227_19))))) ->_L^Omega(1)
c18(*16_19, c13(c14, gen_c:c1:c2:c311_19(0)), MIN(cons(0', gen_nil:cons14_19(+(1, n10227_19))))) ->_R^Omega(1)
c18(*16_19, c13(c14, gen_c:c1:c2:c311_19(0)), c9(IF_MIN(le(0', 0'), cons(0', cons(0', gen_nil:cons14_19(n10227_19)))), LE(0', 0'))) ->_L^Omega(0)
c18(*16_19, c13(c14, gen_c:c1:c2:c311_19(0)), c9(IF_MIN(true, cons(0', cons(0', gen_nil:cons14_19(n10227_19)))), LE(0', 0'))) ->_R^Omega(1)
c18(*16_19, c13(c14, gen_c:c1:c2:c311_19(0)), c9(c10(MIN(cons(0', gen_nil:cons14_19(n10227_19)))), LE(0', 0'))) ->_L^Omega(n10227_19)
c18(*16_19, c13(c14, gen_c:c1:c2:c311_19(0)), c9(c10(*16_19), LE(0', 0'))) ->_L^Omega(1)
c18(*16_19, c13(c14, gen_c:c1:c2:c311_19(0)), c9(c10(*16_19), gen_c4:c5:c613_19(0)))

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(104)
Complex Obligation (BEST)

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

(105)
Obligation:
Proved the lower bound n^2 for the following obligation:

Innermost TRS:
Rules:
EQ(0', 0') -> c
EQ(0', s(z0)) -> c1
EQ(s(z0), 0') -> c2
EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
LE(0', z0) -> c4
LE(s(z0), 0') -> c5
LE(s(z0), s(z1)) -> c6(LE(z0, z1))
MIN(cons(0', nil)) -> c7
MIN(cons(s(z0), nil)) -> c8
MIN(cons(z0, cons(z1, z2))) -> c9(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, nil) -> c12
REPLACE(z0, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
IF_REPLACE(true, z0, cons(z2, z3)) -> c14
IF_REPLACE(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
SORT(nil) -> c16
SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z1), MIN(cons(z0, z1)))
eq(0', 0') -> true
eq(0', s(z0)) -> false
eq(s(z0), 0') -> false
eq(s(z0), s(z1)) -> eq(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
min(cons(0', nil)) -> 0'
min(cons(s(z0), nil)) -> s(z0)
min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
sort(nil) -> nil
sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))

Types:
EQ :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
LE :: 0':s -> 0':s -> c4:c5:c6
c4 :: c4:c5:c6
c5 :: c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
MIN :: nil:cons -> c7:c8:c9
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c4:c5:c6 -> c7:c8:c9
IF_MIN :: true:false -> nil:cons -> c10:c11
le :: 0':s -> 0':s -> true:false
true :: true:false
c10 :: c7:c8:c9 -> c10:c11
false :: true:false
c11 :: c7:c8:c9 -> c10:c11
REPLACE :: 0':s -> nil:cons -> c12:c13
c12 :: c12:c13
c13 :: c14:c15 -> c:c1:c2:c3 -> c12:c13
IF_REPLACE :: true:false -> 0':s -> nil:cons -> c14:c15
eq :: 0':s -> 0':s -> true:false
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SORT :: nil:cons -> c16:c17:c18
c16 :: c16:c17:c18
c17 :: c7:c8:c9 -> c16:c17:c18
c18 :: c16:c17:c18 -> c12:c13 -> c7:c8:c9 -> c16:c17:c18
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
min :: nil:cons -> 0':s
if_min :: true:false -> nil:cons -> 0':s
if_replace :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
sort :: nil:cons -> nil:cons
hole_c:c1:c2:c31_19 :: c:c1:c2:c3
hole_0':s2_19 :: 0':s
hole_c4:c5:c63_19 :: c4:c5:c6
hole_c7:c8:c94_19 :: c7:c8:c9
hole_nil:cons5_19 :: nil:cons
hole_c10:c116_19 :: c10:c11
hole_true:false7_19 :: true:false
hole_c12:c138_19 :: c12:c13
hole_c14:c159_19 :: c14:c15
hole_c16:c17:c1810_19 :: c16:c17:c18
gen_c:c1:c2:c311_19 :: Nat -> c:c1:c2:c3
gen_0':s12_19 :: Nat -> 0':s
gen_c4:c5:c613_19 :: Nat -> c4:c5:c6
gen_nil:cons14_19 :: Nat -> nil:cons
gen_c16:c17:c1815_19 :: Nat -> c16:c17:c18


Lemmas:
EQ(gen_0':s12_19(n17_19), gen_0':s12_19(n17_19)) -> gen_c:c1:c2:c311_19(n17_19), rt in Omega(1 + n17_19)
LE(gen_0':s12_19(n1023_19), gen_0':s12_19(n1023_19)) -> gen_c4:c5:c613_19(n1023_19), rt in Omega(1 + n1023_19)
le(gen_0':s12_19(n1803_19), gen_0':s12_19(n1803_19)) -> true, rt in Omega(0)
MIN(gen_nil:cons14_19(+(1, n2270_19))) -> *16_19, rt in Omega(n2270_19)
eq(gen_0':s12_19(n7487_19), gen_0':s12_19(n7487_19)) -> true, rt in Omega(0)
min(gen_nil:cons14_19(+(1, n9373_19))) -> gen_0':s12_19(0), rt in Omega(0)


Generator Equations:
gen_c:c1:c2:c311_19(0) <=> c
gen_c:c1:c2:c311_19(+(x, 1)) <=> c3(gen_c:c1:c2:c311_19(x))
gen_0':s12_19(0) <=> 0'
gen_0':s12_19(+(x, 1)) <=> s(gen_0':s12_19(x))
gen_c4:c5:c613_19(0) <=> c4
gen_c4:c5:c613_19(+(x, 1)) <=> c6(gen_c4:c5:c613_19(x))
gen_nil:cons14_19(0) <=> nil
gen_nil:cons14_19(+(x, 1)) <=> cons(0', gen_nil:cons14_19(x))
gen_c16:c17:c1815_19(0) <=> c16
gen_c16:c17:c1815_19(+(x, 1)) <=> c18(gen_c16:c17:c1815_19(x), c12, c7)


The following defined symbols remain to be analysed:
SORT, sort
----------------------------------------

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

(107)
BOUNDS(n^2, INF)

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

(108)
Obligation:
Innermost TRS:
Rules:
EQ(0', 0') -> c
EQ(0', s(z0)) -> c1
EQ(s(z0), 0') -> c2
EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
LE(0', z0) -> c4
LE(s(z0), 0') -> c5
LE(s(z0), s(z1)) -> c6(LE(z0, z1))
MIN(cons(0', nil)) -> c7
MIN(cons(s(z0), nil)) -> c8
MIN(cons(z0, cons(z1, z2))) -> c9(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, nil) -> c12
REPLACE(z0, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
IF_REPLACE(true, z0, cons(z2, z3)) -> c14
IF_REPLACE(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
SORT(nil) -> c16
SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z1), MIN(cons(z0, z1)))
eq(0', 0') -> true
eq(0', s(z0)) -> false
eq(s(z0), 0') -> false
eq(s(z0), s(z1)) -> eq(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
min(cons(0', nil)) -> 0'
min(cons(s(z0), nil)) -> s(z0)
min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
sort(nil) -> nil
sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))

Types:
EQ :: 0':s -> 0':s -> c:c1:c2:c3
0' :: 0':s
c :: c:c1:c2:c3
s :: 0':s -> 0':s
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
LE :: 0':s -> 0':s -> c4:c5:c6
c4 :: c4:c5:c6
c5 :: c4:c5:c6
c6 :: c4:c5:c6 -> c4:c5:c6
MIN :: nil:cons -> c7:c8:c9
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c10:c11 -> c4:c5:c6 -> c7:c8:c9
IF_MIN :: true:false -> nil:cons -> c10:c11
le :: 0':s -> 0':s -> true:false
true :: true:false
c10 :: c7:c8:c9 -> c10:c11
false :: true:false
c11 :: c7:c8:c9 -> c10:c11
REPLACE :: 0':s -> nil:cons -> c12:c13
c12 :: c12:c13
c13 :: c14:c15 -> c:c1:c2:c3 -> c12:c13
IF_REPLACE :: true:false -> 0':s -> nil:cons -> c14:c15
eq :: 0':s -> 0':s -> true:false
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SORT :: nil:cons -> c16:c17:c18
c16 :: c16:c17:c18
c17 :: c7:c8:c9 -> c16:c17:c18
c18 :: c16:c17:c18 -> c12:c13 -> c7:c8:c9 -> c16:c17:c18
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
min :: nil:cons -> 0':s
if_min :: true:false -> nil:cons -> 0':s
if_replace :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
sort :: nil:cons -> nil:cons
hole_c:c1:c2:c31_19 :: c:c1:c2:c3
hole_0':s2_19 :: 0':s
hole_c4:c5:c63_19 :: c4:c5:c6
hole_c7:c8:c94_19 :: c7:c8:c9
hole_nil:cons5_19 :: nil:cons
hole_c10:c116_19 :: c10:c11
hole_true:false7_19 :: true:false
hole_c12:c138_19 :: c12:c13
hole_c14:c159_19 :: c14:c15
hole_c16:c17:c1810_19 :: c16:c17:c18
gen_c:c1:c2:c311_19 :: Nat -> c:c1:c2:c3
gen_0':s12_19 :: Nat -> 0':s
gen_c4:c5:c613_19 :: Nat -> c4:c5:c6
gen_nil:cons14_19 :: Nat -> nil:cons
gen_c16:c17:c1815_19 :: Nat -> c16:c17:c18


Lemmas:
EQ(gen_0':s12_19(n17_19), gen_0':s12_19(n17_19)) -> gen_c:c1:c2:c311_19(n17_19), rt in Omega(1 + n17_19)
LE(gen_0':s12_19(n1023_19), gen_0':s12_19(n1023_19)) -> gen_c4:c5:c613_19(n1023_19), rt in Omega(1 + n1023_19)
le(gen_0':s12_19(n1803_19), gen_0':s12_19(n1803_19)) -> true, rt in Omega(0)
MIN(gen_nil:cons14_19(+(1, n2270_19))) -> *16_19, rt in Omega(n2270_19)
eq(gen_0':s12_19(n7487_19), gen_0':s12_19(n7487_19)) -> true, rt in Omega(0)
min(gen_nil:cons14_19(+(1, n9373_19))) -> gen_0':s12_19(0), rt in Omega(0)
SORT(gen_nil:cons14_19(+(1, n10227_19))) -> *16_19, rt in Omega(n10227_19 + n10227_19^2)


Generator Equations:
gen_c:c1:c2:c311_19(0) <=> c
gen_c:c1:c2:c311_19(+(x, 1)) <=> c3(gen_c:c1:c2:c311_19(x))
gen_0':s12_19(0) <=> 0'
gen_0':s12_19(+(x, 1)) <=> s(gen_0':s12_19(x))
gen_c4:c5:c613_19(0) <=> c4
gen_c4:c5:c613_19(+(x, 1)) <=> c6(gen_c4:c5:c613_19(x))
gen_nil:cons14_19(0) <=> nil
gen_nil:cons14_19(+(x, 1)) <=> cons(0', gen_nil:cons14_19(x))
gen_c16:c17:c1815_19(0) <=> c16
gen_c16:c17:c1815_19(+(x, 1)) <=> c18(gen_c16:c17:c1815_19(x), c12, c7)


The following defined symbols remain to be analysed:
sort
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(109) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sort(gen_nil:cons14_19(+(1, n400026_19))) -> *16_19, rt in Omega(0)

Induction Base:
sort(gen_nil:cons14_19(+(1, 0)))

Induction Step:
sort(gen_nil:cons14_19(+(1, +(n400026_19, 1)))) ->_R^Omega(0)
cons(min(cons(0', gen_nil:cons14_19(+(1, n400026_19)))), sort(replace(min(cons(0', gen_nil:cons14_19(+(1, n400026_19)))), 0', gen_nil:cons14_19(+(1, n400026_19))))) ->_L^Omega(0)
cons(gen_0':s12_19(0), sort(replace(min(cons(0', gen_nil:cons14_19(+(1, n400026_19)))), 0', gen_nil:cons14_19(+(1, n400026_19))))) ->_L^Omega(0)
cons(gen_0':s12_19(0), sort(replace(gen_0':s12_19(0), 0', gen_nil:cons14_19(+(1, n400026_19))))) ->_R^Omega(0)
cons(gen_0':s12_19(0), sort(if_replace(eq(gen_0':s12_19(0), 0'), gen_0':s12_19(0), 0', cons(0', gen_nil:cons14_19(n400026_19))))) ->_L^Omega(0)
cons(gen_0':s12_19(0), sort(if_replace(true, gen_0':s12_19(0), 0', cons(0', gen_nil:cons14_19(n400026_19))))) ->_R^Omega(0)
cons(gen_0':s12_19(0), sort(cons(0', gen_nil:cons14_19(n400026_19)))) ->_IH
cons(gen_0':s12_19(0), *16_19)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(110)
BOUNDS(1, INF)