WORST_CASE(Omega(n^2),O(n^3))
proof of /export/starexec/sandbox/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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), 703 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), 7 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 6 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), 409 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 99 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 374 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 16 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 263 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 126 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 424 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 74 ms]
    (40) CpxRNTS
    (41) ResultPropagationProof [UPPER BOUND(ID), 2 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 2170 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 548 ms]
    (46) CpxRNTS
    (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 823 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 333 ms]
    (52) CpxRNTS
    (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (54) CpxRNTS
    (55) IntTrsBoundProof [UPPER BOUND(ID), 2974 ms]
    (56) CpxRNTS
    (57) IntTrsBoundProof [UPPER BOUND(ID), 728 ms]
    (58) CpxRNTS
    (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (60) CpxRNTS
    (61) IntTrsBoundProof [UPPER BOUND(ID), 1489 ms]
    (62) CpxRNTS
    (63) IntTrsBoundProof [UPPER BOUND(ID), 303 ms]
    (64) CpxRNTS
    (65) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (66) CpxRNTS
    (67) IntTrsBoundProof [UPPER BOUND(ID), 8540 ms]
    (68) CpxRNTS
    (69) IntTrsBoundProof [UPPER BOUND(ID), 1153 ms]
    (70) CpxRNTS
    (71) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (72) CpxRNTS
    (73) IntTrsBoundProof [UPPER BOUND(ID), 2081 ms]
    (74) CpxRNTS
    (75) IntTrsBoundProof [UPPER BOUND(ID), 213 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), 20 ms]
    (86) typed CpxTrs
    (87) RewriteLemmaProof [LOWER BOUND(ID), 387 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), 107 ms]
            (94) typed CpxTrs
            (95) RewriteLemmaProof [LOWER BOUND(ID), 27 ms]
            (96) typed CpxTrs
            (97) RewriteLemmaProof [LOWER BOUND(ID), 573 ms]
            (98) typed CpxTrs
            (99) RewriteLemmaProof [LOWER BOUND(ID), 48 ms]
            (100) typed CpxTrs
            (101) RewriteLemmaProof [LOWER BOUND(ID), 39 ms]
            (102) typed CpxTrs
            (103) RewriteLemmaProof [LOWER BOUND(ID), 536 ms]
            (104) BEST
                (105) proven lower bound
                    (106) LowerBoundPropagationProof [FINISHED, 0 ms]
                    (107) BOUNDS(n^2, INF)
                (108) typed CpxTrs
                    (109) RewriteLemmaProof [LOWER BOUND(ID), 62 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(IFMIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
   IFMIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
   IFMIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
   REPLACE(z0, z1, nil) -> c12
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IFREPL(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
   IFREPL(true, z0, z1, cons(z2, z3)) -> c14
   IFREPL(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
   SELSORT(nil) -> c16
   SELSORT(cons(z0, z1)) -> c17(IFSELSORT(eq(z0, min(cons(z0, z1))), cons(z0, z1)), EQ(z0, min(cons(z0, z1))), MIN(cons(z0, z1)))
   IFSELSORT(true, cons(z0, z1)) -> c18(SELSORT(z1))
   IFSELSORT(false, cons(z0, z1)) -> c19(MIN(cons(z0, z1)))
   IFSELSORT(false, cons(z0, z1)) -> c20(SELSORT(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))) -> ifmin(le(z0, z1), cons(z0, cons(z1, z2)))
   ifmin(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
   ifmin(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
   replace(z0, z1, nil) -> nil
   replace(z0, z1, cons(z2, z3)) -> ifrepl(eq(z0, z2), z0, z1, cons(z2, z3))
   ifrepl(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
   ifrepl(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
   selsort(nil) -> nil
   selsort(cons(z0, z1)) -> ifselsort(eq(z0, min(cons(z0, z1))), cons(z0, z1))
   ifselsort(true, cons(z0, z1)) -> cons(z0, selsort(z1))
   ifselsort(false, cons(z0, z1)) -> cons(min(cons(z0, z1)), selsort(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(IFMIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
   IFMIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
   IFMIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
   REPLACE(z0, z1, nil) -> c12
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IFREPL(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
   IFREPL(true, z0, z1, cons(z2, z3)) -> c14
   IFREPL(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
   SELSORT(nil) -> c16
   SELSORT(cons(z0, z1)) -> c17(IFSELSORT(eq(z0, min(cons(z0, z1))), cons(z0, z1)), EQ(z0, min(cons(z0, z1))), MIN(cons(z0, z1)))
   IFSELSORT(true, cons(z0, z1)) -> c18(SELSORT(z1))
   IFSELSORT(false, cons(z0, z1)) -> c19(MIN(cons(z0, z1)))
   IFSELSORT(false, cons(z0, z1)) -> c20(SELSORT(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))) -> ifmin(le(z0, z1), cons(z0, cons(z1, z2)))
   ifmin(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
   ifmin(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
   replace(z0, z1, nil) -> nil
   replace(z0, z1, cons(z2, z3)) -> ifrepl(eq(z0, z2), z0, z1, cons(z2, z3))
   ifrepl(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
   ifrepl(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
   selsort(nil) -> nil
   selsort(cons(z0, z1)) -> ifselsort(eq(z0, min(cons(z0, z1))), cons(z0, z1))
   ifselsort(true, cons(z0, z1)) -> cons(z0, selsort(z1))
   ifselsort(false, cons(z0, z1)) -> cons(min(cons(z0, z1)), selsort(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(IFMIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1)) [1]
   IFMIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2))) [1]
   IFMIN(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(IFREPL(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) [1]
   IFREPL(true, z0, z1, cons(z2, z3)) -> c14 [1]
   IFREPL(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3)) [1]
   SELSORT(nil) -> c16 [1]
   SELSORT(cons(z0, z1)) -> c17(IFSELSORT(eq(z0, min(cons(z0, z1))), cons(z0, z1)), EQ(z0, min(cons(z0, z1))), MIN(cons(z0, z1))) [1]
   IFSELSORT(true, cons(z0, z1)) -> c18(SELSORT(z1)) [1]
   IFSELSORT(false, cons(z0, z1)) -> c19(MIN(cons(z0, z1))) [1]
   IFSELSORT(false, cons(z0, z1)) -> c20(SELSORT(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))) -> ifmin(le(z0, z1), cons(z0, cons(z1, z2))) [0]
   ifmin(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) [0]
   ifmin(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) [0]
   replace(z0, z1, nil) -> nil [0]
   replace(z0, z1, cons(z2, z3)) -> ifrepl(eq(z0, z2), z0, z1, cons(z2, z3)) [0]
   ifrepl(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) [0]
   ifrepl(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) [0]
   selsort(nil) -> nil [0]
   selsort(cons(z0, z1)) -> ifselsort(eq(z0, min(cons(z0, z1))), cons(z0, z1)) [0]
   ifselsort(true, cons(z0, z1)) -> cons(z0, selsort(z1)) [0]
   ifselsort(false, cons(z0, z1)) -> cons(min(cons(z0, z1)), selsort(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(IFMIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1)) [1]
   IFMIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2))) [1]
   IFMIN(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(IFREPL(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) [1]
   IFREPL(true, z0, z1, cons(z2, z3)) -> c14 [1]
   IFREPL(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3)) [1]
   SELSORT(nil) -> c16 [1]
   SELSORT(cons(z0, z1)) -> c17(IFSELSORT(eq(z0, min(cons(z0, z1))), cons(z0, z1)), EQ(z0, min(cons(z0, z1))), MIN(cons(z0, z1))) [1]
   IFSELSORT(true, cons(z0, z1)) -> c18(SELSORT(z1)) [1]
   IFSELSORT(false, cons(z0, z1)) -> c19(MIN(cons(z0, z1))) [1]
   IFSELSORT(false, cons(z0, z1)) -> c20(SELSORT(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))) -> ifmin(le(z0, z1), cons(z0, cons(z1, z2))) [0]
   ifmin(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) [0]
   ifmin(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) [0]
   replace(z0, z1, nil) -> nil [0]
   replace(z0, z1, cons(z2, z3)) -> ifrepl(eq(z0, z2), z0, z1, cons(z2, z3)) [0]
   ifrepl(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) [0]
   ifrepl(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) [0]
   selsort(nil) -> nil [0]
   selsort(cons(z0, z1)) -> ifselsort(eq(z0, min(cons(z0, z1))), cons(z0, z1)) [0]
   ifselsort(true, cons(z0, z1)) -> cons(z0, selsort(z1)) [0]
   ifselsort(false, cons(z0, z1)) -> cons(min(cons(z0, z1)), selsort(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
IFMIN :: 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
IFREPL :: 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
SELSORT :: nil:cons -> c16:c17
c16 :: c16:c17
c17 :: c18:c19:c20 -> c:c1:c2:c3 -> c7:c8:c9 -> c16:c17
IFSELSORT :: true:false -> nil:cons -> c18:c19:c20
min :: nil:cons -> 0:s
c18 :: c16:c17 -> c18:c19:c20
c19 :: c7:c8:c9 -> c18:c19:c20
c20 :: c16:c17 -> c12:c13 -> c7:c8:c9 -> c18:c19:c20
replace :: 0:s -> 0:s -> nil:cons -> nil:cons
ifmin :: true:false -> nil:cons -> 0:s
ifrepl :: true:false -> 0:s -> 0:s -> nil:cons -> nil:cons
selsort :: nil:cons -> nil:cons
ifselsort :: true:false -> 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
   IFMIN_2
   REPLACE_3
   IFREPL_4
   SELSORT_1
   IFSELSORT_2

(c) The following functions are completely defined:

   eq_2
   le_2
   min_1
   ifmin_2
   replace_3
   ifrepl_4
   selsort_1
   ifselsort_2

Due to the following rules being added:

   eq(v0, v1) -> null_eq [0]
   le(v0, v1) -> null_le [0]
   min(v0) -> 0 [0]
   ifmin(v0, v1) -> 0 [0]
   replace(v0, v1, v2) -> nil [0]
   ifrepl(v0, v1, v2, v3) -> nil [0]
   selsort(v0) -> nil [0]
   ifselsort(v0, v1) -> nil [0]

And the following fresh constants:    null_eq, null_le, const, const1

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

(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(IFMIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1)) [1]
   IFMIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2))) [1]
   IFMIN(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(IFREPL(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) [1]
   IFREPL(true, z0, z1, cons(z2, z3)) -> c14 [1]
   IFREPL(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3)) [1]
   SELSORT(nil) -> c16 [1]
   SELSORT(cons(z0, z1)) -> c17(IFSELSORT(eq(z0, min(cons(z0, z1))), cons(z0, z1)), EQ(z0, min(cons(z0, z1))), MIN(cons(z0, z1))) [1]
   IFSELSORT(true, cons(z0, z1)) -> c18(SELSORT(z1)) [1]
   IFSELSORT(false, cons(z0, z1)) -> c19(MIN(cons(z0, z1))) [1]
   IFSELSORT(false, cons(z0, z1)) -> c20(SELSORT(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))) -> ifmin(le(z0, z1), cons(z0, cons(z1, z2))) [0]
   ifmin(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) [0]
   ifmin(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) [0]
   replace(z0, z1, nil) -> nil [0]
   replace(z0, z1, cons(z2, z3)) -> ifrepl(eq(z0, z2), z0, z1, cons(z2, z3)) [0]
   ifrepl(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) [0]
   ifrepl(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) [0]
   selsort(nil) -> nil [0]
   selsort(cons(z0, z1)) -> ifselsort(eq(z0, min(cons(z0, z1))), cons(z0, z1)) [0]
   ifselsort(true, cons(z0, z1)) -> cons(z0, selsort(z1)) [0]
   ifselsort(false, cons(z0, z1)) -> cons(min(cons(z0, z1)), selsort(replace(min(cons(z0, z1)), z0, z1))) [0]
   eq(v0, v1) -> null_eq [0]
   le(v0, v1) -> null_le [0]
   min(v0) -> 0 [0]
   ifmin(v0, v1) -> 0 [0]
   replace(v0, v1, v2) -> nil [0]
   ifrepl(v0, v1, v2, v3) -> nil [0]
   selsort(v0) -> nil [0]
   ifselsort(v0, v1) -> 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
IFMIN :: 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
IFREPL :: 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
SELSORT :: nil:cons -> c16:c17
c16 :: c16:c17
c17 :: c18:c19:c20 -> c:c1:c2:c3 -> c7:c8:c9 -> c16:c17
IFSELSORT :: true:false:null_eq:null_le -> nil:cons -> c18:c19:c20
min :: nil:cons -> 0:s
c18 :: c16:c17 -> c18:c19:c20
c19 :: c7:c8:c9 -> c18:c19:c20
c20 :: c16:c17 -> c12:c13 -> c7:c8:c9 -> c18:c19:c20
replace :: 0:s -> 0:s -> nil:cons -> nil:cons
ifmin :: true:false:null_eq:null_le -> nil:cons -> 0:s
ifrepl :: true:false:null_eq:null_le -> 0:s -> 0:s -> nil:cons -> nil:cons
selsort :: nil:cons -> nil:cons
ifselsort :: true:false:null_eq:null_le -> nil:cons -> nil:cons
null_eq :: true:false:null_eq:null_le
null_le :: true:false:null_eq:null_le
const :: c10:c11
const1 :: c18:c19:c20

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(IFMIN(true, cons(0, cons(z1, z2))), LE(0, z1)) [1]
   MIN(cons(s(z0'), cons(0, z2))) -> c9(IFMIN(false, cons(s(z0'), cons(0, z2))), LE(s(z0'), 0)) [1]
   MIN(cons(s(z0''), cons(s(z1'), z2))) -> c9(IFMIN(le(z0'', z1'), cons(s(z0''), cons(s(z1'), z2))), LE(s(z0''), s(z1'))) [1]
   MIN(cons(z0, cons(z1, z2))) -> c9(IFMIN(null_le, cons(z0, cons(z1, z2))), LE(z0, z1)) [1]
   IFMIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2))) [1]
   IFMIN(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(IFREPL(true, 0, z1, cons(0, z3)), EQ(0, 0)) [1]
   REPLACE(0, z1, cons(s(z01), z3)) -> c13(IFREPL(false, 0, z1, cons(s(z01), z3)), EQ(0, s(z01))) [1]
   REPLACE(s(z02), z1, cons(0, z3)) -> c13(IFREPL(false, s(z02), z1, cons(0, z3)), EQ(s(z02), 0)) [1]
   REPLACE(s(z03), z1, cons(s(z1''), z3)) -> c13(IFREPL(eq(z03, z1''), s(z03), z1, cons(s(z1''), z3)), EQ(s(z03), s(z1''))) [1]
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IFREPL(null_eq, z0, z1, cons(z2, z3)), EQ(z0, z2)) [1]
   IFREPL(true, z0, z1, cons(z2, z3)) -> c14 [1]
   IFREPL(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3)) [1]
   SELSORT(nil) -> c16 [1]
   SELSORT(cons(0, nil)) -> c17(IFSELSORT(eq(0, 0), cons(0, nil)), EQ(0, 0), MIN(cons(0, nil))) [1]
   SELSORT(cons(0, nil)) -> c17(IFSELSORT(eq(0, 0), cons(0, nil)), EQ(0, 0), MIN(cons(0, nil))) [1]
   SELSORT(cons(s(z04), nil)) -> c17(IFSELSORT(eq(s(z04), s(z04)), cons(s(z04), nil)), EQ(s(z04), s(z04)), MIN(cons(s(z04), nil))) [1]
   SELSORT(cons(s(z04), nil)) -> c17(IFSELSORT(eq(s(z04), s(z04)), cons(s(z04), nil)), EQ(s(z04), 0), MIN(cons(s(z04), nil))) [1]
   SELSORT(cons(z0, cons(z11, z2'))) -> c17(IFSELSORT(eq(z0, ifmin(le(z0, z11), cons(z0, cons(z11, z2')))), cons(z0, cons(z11, z2'))), EQ(z0, ifmin(le(z0, z11), cons(z0, cons(z11, z2')))), MIN(cons(z0, cons(z11, z2')))) [1]
   SELSORT(cons(z0, cons(z11, z2'))) -> c17(IFSELSORT(eq(z0, ifmin(le(z0, z11), cons(z0, cons(z11, z2')))), cons(z0, cons(z11, z2'))), EQ(z0, 0), MIN(cons(z0, cons(z11, z2')))) [1]
   SELSORT(cons(0, nil)) -> c17(IFSELSORT(eq(0, 0), cons(0, nil)), EQ(0, 0), MIN(cons(0, nil))) [1]
   SELSORT(cons(s(z05), nil)) -> c17(IFSELSORT(eq(s(z05), 0), cons(s(z05), nil)), EQ(s(z05), s(z05)), MIN(cons(s(z05), nil))) [1]
   SELSORT(cons(z0, cons(z12, z2''))) -> c17(IFSELSORT(eq(z0, 0), cons(z0, cons(z12, z2''))), EQ(z0, ifmin(le(z0, z12), cons(z0, cons(z12, z2'')))), MIN(cons(z0, cons(z12, z2'')))) [1]
   SELSORT(cons(z0, z1)) -> c17(IFSELSORT(eq(z0, 0), cons(z0, z1)), EQ(z0, 0), MIN(cons(z0, z1))) [1]
   IFSELSORT(true, cons(z0, z1)) -> c18(SELSORT(z1)) [1]
   IFSELSORT(false, cons(z0, z1)) -> c19(MIN(cons(z0, z1))) [1]
   IFSELSORT(false, cons(0, nil)) -> c20(SELSORT(replace(0, 0, nil)), REPLACE(0, 0, nil), MIN(cons(0, nil))) [1]
   IFSELSORT(false, cons(0, nil)) -> c20(SELSORT(replace(0, 0, nil)), REPLACE(0, 0, nil), MIN(cons(0, nil))) [1]
   IFSELSORT(false, cons(s(z06), nil)) -> c20(SELSORT(replace(s(z06), s(z06), nil)), REPLACE(s(z06), s(z06), nil), MIN(cons(s(z06), nil))) [1]
   IFSELSORT(false, cons(s(z06), nil)) -> c20(SELSORT(replace(s(z06), s(z06), nil)), REPLACE(0, s(z06), nil), MIN(cons(s(z06), nil))) [1]
   IFSELSORT(false, cons(z0, cons(z13, z21))) -> c20(SELSORT(replace(ifmin(le(z0, z13), cons(z0, cons(z13, z21))), z0, cons(z13, z21))), REPLACE(ifmin(le(z0, z13), cons(z0, cons(z13, z21))), z0, cons(z13, z21)), MIN(cons(z0, cons(z13, z21)))) [1]
   IFSELSORT(false, cons(z0, cons(z13, z21))) -> c20(SELSORT(replace(ifmin(le(z0, z13), cons(z0, cons(z13, z21))), z0, cons(z13, z21))), REPLACE(0, z0, cons(z13, z21)), MIN(cons(z0, cons(z13, z21)))) [1]
   IFSELSORT(false, cons(0, nil)) -> c20(SELSORT(replace(0, 0, nil)), REPLACE(0, 0, nil), MIN(cons(0, nil))) [1]
   IFSELSORT(false, cons(s(z07), nil)) -> c20(SELSORT(replace(0, s(z07), nil)), REPLACE(s(z07), s(z07), nil), MIN(cons(s(z07), nil))) [1]
   IFSELSORT(false, cons(z0, cons(z14, z22))) -> c20(SELSORT(replace(0, z0, cons(z14, z22))), REPLACE(ifmin(le(z0, z14), cons(z0, cons(z14, z22))), z0, cons(z14, z22)), MIN(cons(z0, cons(z14, z22)))) [1]
   IFSELSORT(false, cons(z0, z1)) -> c20(SELSORT(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))) -> ifmin(true, cons(0, cons(z1, z2))) [0]
   min(cons(s(z08), cons(0, z2))) -> ifmin(false, cons(s(z08), cons(0, z2))) [0]
   min(cons(s(z09), cons(s(z15), z2))) -> ifmin(le(z09, z15), cons(s(z09), cons(s(z15), z2))) [0]
   min(cons(z0, cons(z1, z2))) -> ifmin(null_le, cons(z0, cons(z1, z2))) [0]
   ifmin(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) [0]
   ifmin(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) [0]
   replace(z0, z1, nil) -> nil [0]
   replace(0, z1, cons(0, z3)) -> ifrepl(true, 0, z1, cons(0, z3)) [0]
   replace(0, z1, cons(s(z010), z3)) -> ifrepl(false, 0, z1, cons(s(z010), z3)) [0]
   replace(s(z011), z1, cons(0, z3)) -> ifrepl(false, s(z011), z1, cons(0, z3)) [0]
   replace(s(z012), z1, cons(s(z16), z3)) -> ifrepl(eq(z012, z16), s(z012), z1, cons(s(z16), z3)) [0]
   replace(z0, z1, cons(z2, z3)) -> ifrepl(null_eq, z0, z1, cons(z2, z3)) [0]
   ifrepl(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) [0]
   ifrepl(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) [0]
   selsort(nil) -> nil [0]
   selsort(cons(0, nil)) -> ifselsort(eq(0, 0), cons(0, nil)) [0]
   selsort(cons(s(z013), nil)) -> ifselsort(eq(s(z013), s(z013)), cons(s(z013), nil)) [0]
   selsort(cons(z0, cons(z17, z23))) -> ifselsort(eq(z0, ifmin(le(z0, z17), cons(z0, cons(z17, z23)))), cons(z0, cons(z17, z23))) [0]
   selsort(cons(z0, z1)) -> ifselsort(eq(z0, 0), cons(z0, z1)) [0]
   ifselsort(true, cons(z0, z1)) -> cons(z0, selsort(z1)) [0]
   ifselsort(false, cons(0, nil)) -> cons(min(cons(0, nil)), selsort(replace(0, 0, nil))) [0]
   ifselsort(false, cons(s(z014), nil)) -> cons(min(cons(s(z014), nil)), selsort(replace(s(z014), s(z014), nil))) [0]
   ifselsort(false, cons(z0, cons(z18, z24))) -> cons(min(cons(z0, cons(z18, z24))), selsort(replace(ifmin(le(z0, z18), cons(z0, cons(z18, z24))), z0, cons(z18, z24)))) [0]
   ifselsort(false, cons(z0, z1)) -> cons(min(cons(z0, z1)), selsort(replace(0, z0, z1))) [0]
   eq(v0, v1) -> null_eq [0]
   le(v0, v1) -> null_le [0]
   min(v0) -> 0 [0]
   ifmin(v0, v1) -> 0 [0]
   replace(v0, v1, v2) -> nil [0]
   ifrepl(v0, v1, v2, v3) -> nil [0]
   selsort(v0) -> nil [0]
   ifselsort(v0, v1) -> 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
IFMIN :: 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
IFREPL :: 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
SELSORT :: nil:cons -> c16:c17
c16 :: c16:c17
c17 :: c18:c19:c20 -> c:c1:c2:c3 -> c7:c8:c9 -> c16:c17
IFSELSORT :: true:false:null_eq:null_le -> nil:cons -> c18:c19:c20
min :: nil:cons -> 0:s
c18 :: c16:c17 -> c18:c19:c20
c19 :: c7:c8:c9 -> c18:c19:c20
c20 :: c16:c17 -> c12:c13 -> c7:c8:c9 -> c18:c19:c20
replace :: 0:s -> 0:s -> nil:cons -> nil:cons
ifmin :: true:false:null_eq:null_le -> nil:cons -> 0:s
ifrepl :: true:false:null_eq:null_le -> 0:s -> 0:s -> nil:cons -> nil:cons
selsort :: nil:cons -> nil:cons
ifselsort :: true:false:null_eq:null_le -> nil:cons -> nil:cons
null_eq :: true:false:null_eq:null_le
null_le :: true:false:null_eq:null_le
const :: c10:c11
const1 :: c18:c19:c20

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
   const1 => 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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0, z'' = z1
   IFREPL(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
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(0, z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + REPLACE(ifmin(le(z0, z14), 1 + z0 + (1 + z14 + z22)), z0, 1 + z14 + z22) + MIN(1 + z0 + (1 + z14 + z22)) :|: z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 1 + z07, 0)) + REPLACE(1 + z07, 1 + z07, 0) + MIN(1 + (1 + z07) + 0) :|: z07 >= 0, z = 1, z' = 1 + (1 + z07) + 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + z06, 1 + z06, 0)) + REPLACE(0, 1 + z06, 0) + MIN(1 + (1 + z06) + 0) :|: z' = 1 + (1 + z06) + 0, z = 1, z06 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + z06, 1 + z06, 0)) + REPLACE(1 + z06, 1 + z06, 0) + MIN(1 + (1 + z06) + 0) :|: z' = 1 + (1 + z06) + 0, z = 1, z06 >= 0
   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 + IFMIN(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 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(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 + IFMIN(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 + IFREPL(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 + IFREPL(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 + IFREPL(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 + IFREPL(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 + IFREPL(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
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))) + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, 0) + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + EQ(z0, 0) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(le(z0, z12), 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(0, 0), 1 + 0 + 0) + EQ(0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(1 + z04, 1 + z04), 1 + (1 + z04) + 0) + EQ(1 + z04, 0) + MIN(1 + (1 + z04) + 0) :|: z04 >= 0, z = 1 + (1 + z04) + 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(1 + z04, 1 + z04), 1 + (1 + z04) + 0) + EQ(1 + z04, 1 + z04) + MIN(1 + (1 + z04) + 0) :|: z04 >= 0, z = 1 + (1 + z04) + 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(1 + z05, 0), 1 + (1 + z05) + 0) + EQ(1 + z05, 1 + z05) + MIN(1 + (1 + z05) + 0) :|: z = 1 + (1 + z05) + 0, z05 >= 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
   ifmin(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z4 = v3, v2 >= 0, v3 >= 0
   ifrepl(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
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + z1) + selsort(replace(0, z0, z1)) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + (1 + z18 + z24)) + selsort(replace(ifmin(le(z0, z18), 1 + z0 + (1 + z18 + z24)), z0, 1 + z18 + z24)) :|: z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + 0 + 0) + selsort(replace(0, 0, 0)) :|: z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + (1 + z014) + 0) + selsort(replace(1 + z014, 1 + z014, 0)) :|: z = 1, z' = 1 + (1 + z014) + 0, z014 >= 0
   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 }-> ifmin(le(z09, z15), 1 + (1 + z09) + (1 + (1 + z15) + z2)) :|: z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> ifmin(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(1, 1 + (1 + z08) + (1 + 0 + z2)) :|: z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(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 }-> ifrepl(eq(z012, z16), 1 + z012, z1, 1 + (1 + z16) + z3) :|: z1 >= 0, z' = z1, z012 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z = 1 + z012, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z1, 1 + 0 + z3) :|: z1 >= 0, z'' = 1 + 0 + z3, z' = z1, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z1, 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z1 >= 0, z' = z1, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + z011, z1, 1 + 0 + z3) :|: z011 >= 0, z1 >= 0, z'' = 1 + 0 + z3, z' = z1, z = 1 + z011, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(eq(z0, ifmin(le(z0, z17), 1 + z0 + (1 + z17 + z23))), 1 + z0 + (1 + z17 + z23)) :|: z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> ifselsort(eq(z0, 0), 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(eq(0, 0), 1 + 0 + 0) :|: z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(eq(1 + z013, 1 + z013), 1 + (1 + z013) + 0) :|: z = 1 + (1 + z013) + 0, z013 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

(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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(0, z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + REPLACE(ifmin(le(z0, z14), 1 + z0 + (1 + z14 + z22)), z0, 1 + z14 + z22) + MIN(1 + z0 + (1 + z14 + z22)) :|: z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(0, 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 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 + IFMIN(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 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(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 + IFMIN(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 + IFREPL(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 + IFREPL(2, 0, z', 1 + 0 + (z'' - 1)) + EQ(0, 0) :|: z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IFREPL(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 + IFREPL(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 + IFREPL(0, z, z', 1 + z2 + z3) + EQ(z, z2) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))) + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, 0) + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + EQ(z0, 0) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(le(z0, z12), 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(0, 0), 1 + 0 + 0) + EQ(0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(1 + (z - 2), 0), 1 + (1 + (z - 2)) + 0) + EQ(1 + (z - 2), 1 + (z - 2)) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + EQ(1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + EQ(1 + (z - 2), 1 + (z - 2)) + 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
   ifmin(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + z1) + selsort(replace(0, z0, z1)) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + (1 + z18 + z24)) + selsort(replace(ifmin(le(z0, z18), 1 + z0 + (1 + z18 + z24)), z0, 1 + z18 + z24)) :|: z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + 0 + 0) + selsort(replace(0, 0, 0)) :|: z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + (1 + (z' - 2)) + 0) + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: z = 1, z' - 2 >= 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 }-> ifmin(le(z09, z15), 1 + (1 + z09) + (1 + (1 + z15) + z2)) :|: z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> ifmin(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(1, 1 + (1 + z08) + (1 + 0 + z2)) :|: z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(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 }-> ifrepl(eq(z - 1, z16), 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(eq(z0, ifmin(le(z0, z17), 1 + z0 + (1 + z17 + z23))), 1 + z0 + (1 + z17 + z23)) :|: z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> ifselsort(eq(z0, 0), 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(eq(0, 0), 1 + 0 + 0) :|: z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0


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

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

   { EQ }
   { le }
   { LE }
   { eq }
   { REPLACE, IFREPL }
   { min, ifmin }
   { IFMIN, MIN }
   { replace, ifrepl }
   { IFSELSORT, SELSORT }
   { ifselsort, selsort }

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

(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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(0, z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + REPLACE(ifmin(le(z0, z14), 1 + z0 + (1 + z14 + z22)), z0, 1 + z14 + z22) + MIN(1 + z0 + (1 + z14 + z22)) :|: z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(0, 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 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 + IFMIN(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 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(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 + IFMIN(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 + IFREPL(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 + IFREPL(2, 0, z', 1 + 0 + (z'' - 1)) + EQ(0, 0) :|: z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IFREPL(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 + IFREPL(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 + IFREPL(0, z, z', 1 + z2 + z3) + EQ(z, z2) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))) + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, 0) + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + EQ(z0, 0) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(le(z0, z12), 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(0, 0), 1 + 0 + 0) + EQ(0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(1 + (z - 2), 0), 1 + (1 + (z - 2)) + 0) + EQ(1 + (z - 2), 1 + (z - 2)) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + EQ(1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + EQ(1 + (z - 2), 1 + (z - 2)) + 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
   ifmin(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + z1) + selsort(replace(0, z0, z1)) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + (1 + z18 + z24)) + selsort(replace(ifmin(le(z0, z18), 1 + z0 + (1 + z18 + z24)), z0, 1 + z18 + z24)) :|: z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + 0 + 0) + selsort(replace(0, 0, 0)) :|: z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + (1 + (z' - 2)) + 0) + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: z = 1, z' - 2 >= 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 }-> ifmin(le(z09, z15), 1 + (1 + z09) + (1 + (1 + z15) + z2)) :|: z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> ifmin(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(1, 1 + (1 + z08) + (1 + 0 + z2)) :|: z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(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 }-> ifrepl(eq(z - 1, z16), 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(eq(z0, ifmin(le(z0, z17), 1 + z0 + (1 + z17 + z23))), 1 + z0 + (1 + z17 + z23)) :|: z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> ifselsort(eq(z0, 0), 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(eq(0, 0), 1 + 0 + 0) :|: z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {EQ}, {le}, {LE}, {eq}, {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}

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

(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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(0, z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + REPLACE(ifmin(le(z0, z14), 1 + z0 + (1 + z14 + z22)), z0, 1 + z14 + z22) + MIN(1 + z0 + (1 + z14 + z22)) :|: z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(0, 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 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 + IFMIN(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 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(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 + IFMIN(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 + IFREPL(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 + IFREPL(2, 0, z', 1 + 0 + (z'' - 1)) + EQ(0, 0) :|: z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IFREPL(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 + IFREPL(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 + IFREPL(0, z, z', 1 + z2 + z3) + EQ(z, z2) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))) + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, 0) + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + EQ(z0, 0) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(le(z0, z12), 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(0, 0), 1 + 0 + 0) + EQ(0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(1 + (z - 2), 0), 1 + (1 + (z - 2)) + 0) + EQ(1 + (z - 2), 1 + (z - 2)) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + EQ(1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + EQ(1 + (z - 2), 1 + (z - 2)) + 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
   ifmin(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + z1) + selsort(replace(0, z0, z1)) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + (1 + z18 + z24)) + selsort(replace(ifmin(le(z0, z18), 1 + z0 + (1 + z18 + z24)), z0, 1 + z18 + z24)) :|: z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + 0 + 0) + selsort(replace(0, 0, 0)) :|: z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + (1 + (z' - 2)) + 0) + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: z = 1, z' - 2 >= 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 }-> ifmin(le(z09, z15), 1 + (1 + z09) + (1 + (1 + z15) + z2)) :|: z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> ifmin(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(1, 1 + (1 + z08) + (1 + 0 + z2)) :|: z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(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 }-> ifrepl(eq(z - 1, z16), 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(eq(z0, ifmin(le(z0, z17), 1 + z0 + (1 + z17 + z23))), 1 + z0 + (1 + z17 + z23)) :|: z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> ifselsort(eq(z0, 0), 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(eq(0, 0), 1 + 0 + 0) :|: z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {EQ}, {le}, {LE}, {eq}, {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}

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

(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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(0, z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + REPLACE(ifmin(le(z0, z14), 1 + z0 + (1 + z14 + z22)), z0, 1 + z14 + z22) + MIN(1 + z0 + (1 + z14 + z22)) :|: z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(0, 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 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 + IFMIN(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 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(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 + IFMIN(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 + IFREPL(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 + IFREPL(2, 0, z', 1 + 0 + (z'' - 1)) + EQ(0, 0) :|: z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IFREPL(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 + IFREPL(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 + IFREPL(0, z, z', 1 + z2 + z3) + EQ(z, z2) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))) + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, 0) + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + EQ(z0, 0) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(le(z0, z12), 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(0, 0), 1 + 0 + 0) + EQ(0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(1 + (z - 2), 0), 1 + (1 + (z - 2)) + 0) + EQ(1 + (z - 2), 1 + (z - 2)) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + EQ(1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + EQ(1 + (z - 2), 1 + (z - 2)) + 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
   ifmin(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + z1) + selsort(replace(0, z0, z1)) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + (1 + z18 + z24)) + selsort(replace(ifmin(le(z0, z18), 1 + z0 + (1 + z18 + z24)), z0, 1 + z18 + z24)) :|: z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + 0 + 0) + selsort(replace(0, 0, 0)) :|: z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + (1 + (z' - 2)) + 0) + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: z = 1, z' - 2 >= 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 }-> ifmin(le(z09, z15), 1 + (1 + z09) + (1 + (1 + z15) + z2)) :|: z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> ifmin(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(1, 1 + (1 + z08) + (1 + 0 + z2)) :|: z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(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 }-> ifrepl(eq(z - 1, z16), 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(eq(z0, ifmin(le(z0, z17), 1 + z0 + (1 + z17 + z23))), 1 + z0 + (1 + z17 + z23)) :|: z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> ifselsort(eq(z0, 0), 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(eq(0, 0), 1 + 0 + 0) :|: z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {EQ}, {le}, {LE}, {eq}, {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}
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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(0, z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + REPLACE(ifmin(le(z0, z14), 1 + z0 + (1 + z14 + z22)), z0, 1 + z14 + z22) + MIN(1 + z0 + (1 + z14 + z22)) :|: z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(0, 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 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 + IFMIN(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 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(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 + IFMIN(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 + IFREPL(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 + IFREPL(2, 0, z', 1 + 0 + (z'' - 1)) + EQ(0, 0) :|: z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IFREPL(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 + IFREPL(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 + IFREPL(0, z, z', 1 + z2 + z3) + EQ(z, z2) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))) + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, 0) + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + EQ(z0, 0) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(le(z0, z12), 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(0, 0), 1 + 0 + 0) + EQ(0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(1 + (z - 2), 0), 1 + (1 + (z - 2)) + 0) + EQ(1 + (z - 2), 1 + (z - 2)) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + EQ(1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + EQ(1 + (z - 2), 1 + (z - 2)) + 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
   ifmin(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + z1) + selsort(replace(0, z0, z1)) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + (1 + z18 + z24)) + selsort(replace(ifmin(le(z0, z18), 1 + z0 + (1 + z18 + z24)), z0, 1 + z18 + z24)) :|: z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + 0 + 0) + selsort(replace(0, 0, 0)) :|: z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + (1 + (z' - 2)) + 0) + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: z = 1, z' - 2 >= 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 }-> ifmin(le(z09, z15), 1 + (1 + z09) + (1 + (1 + z15) + z2)) :|: z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> ifmin(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(1, 1 + (1 + z08) + (1 + 0 + z2)) :|: z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(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 }-> ifrepl(eq(z - 1, z16), 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(eq(z0, ifmin(le(z0, z17), 1 + z0 + (1 + z17 + z23))), 1 + z0 + (1 + z17 + z23)) :|: z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> ifselsort(eq(z0, 0), 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(eq(0, 0), 1 + 0 + 0) :|: z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {le}, {LE}, {eq}, {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}
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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(0, z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + REPLACE(ifmin(le(z0, z14), 1 + z0 + (1 + z14 + z22)), z0, 1 + z14 + z22) + MIN(1 + z0 + (1 + z14 + z22)) :|: z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(0, 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 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 + IFMIN(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 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(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 + IFMIN(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 + IFREPL(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 + IFREPL(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 + IFREPL(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 + IFREPL(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 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + s7 + MIN(1 + z0 + (1 + z11 + z2')) :|: s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))) + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(le(z0, z12), 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(0, 0), 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(eq(1 + (z - 2), 0), 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s6 >= 0, s6 <= 0 + 2, 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
   ifmin(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + z1) + selsort(replace(0, z0, z1)) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + (1 + z18 + z24)) + selsort(replace(ifmin(le(z0, z18), 1 + z0 + (1 + z18 + z24)), z0, 1 + z18 + z24)) :|: z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + 0 + 0) + selsort(replace(0, 0, 0)) :|: z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + (1 + (z' - 2)) + 0) + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: z = 1, z' - 2 >= 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 }-> ifmin(le(z09, z15), 1 + (1 + z09) + (1 + (1 + z15) + z2)) :|: z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> ifmin(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(1, 1 + (1 + z08) + (1 + 0 + z2)) :|: z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(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 }-> ifrepl(eq(z - 1, z16), 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(eq(z0, ifmin(le(z0, z17), 1 + z0 + (1 + z17 + z23))), 1 + z0 + (1 + z17 + z23)) :|: z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> ifselsort(eq(z0, 0), 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(eq(0, 0), 1 + 0 + 0) :|: z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {le}, {LE}, {eq}, {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}
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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(0, z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + REPLACE(ifmin(le(z0, z14), 1 + z0 + (1 + z14 + z22)), z0, 1 + z14 + z22) + MIN(1 + z0 + (1 + z14 + z22)) :|: z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(0, 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 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 + IFMIN(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 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(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 + IFMIN(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 + IFREPL(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 + IFREPL(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 + IFREPL(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 + IFREPL(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 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + s7 + MIN(1 + z0 + (1 + z11 + z2')) :|: s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))) + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(le(z0, z12), 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(0, 0), 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(eq(1 + (z - 2), 0), 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s6 >= 0, s6 <= 0 + 2, 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
   ifmin(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + z1) + selsort(replace(0, z0, z1)) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + (1 + z18 + z24)) + selsort(replace(ifmin(le(z0, z18), 1 + z0 + (1 + z18 + z24)), z0, 1 + z18 + z24)) :|: z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + 0 + 0) + selsort(replace(0, 0, 0)) :|: z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + (1 + (z' - 2)) + 0) + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: z = 1, z' - 2 >= 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 }-> ifmin(le(z09, z15), 1 + (1 + z09) + (1 + (1 + z15) + z2)) :|: z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> ifmin(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(1, 1 + (1 + z08) + (1 + 0 + z2)) :|: z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(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 }-> ifrepl(eq(z - 1, z16), 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(eq(z0, ifmin(le(z0, z17), 1 + z0 + (1 + z17 + z23))), 1 + z0 + (1 + z17 + z23)) :|: z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> ifselsort(eq(z0, 0), 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(eq(0, 0), 1 + 0 + 0) :|: z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {le}, {LE}, {eq}, {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}
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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(le(z0, z13), 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(0, z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + REPLACE(ifmin(le(z0, z14), 1 + z0 + (1 + z14 + z22)), z0, 1 + z14 + z22) + MIN(1 + z0 + (1 + z14 + z22)) :|: z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(0, 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 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 + IFMIN(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 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(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 + IFMIN(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 + IFREPL(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 + IFREPL(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 + IFREPL(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 + IFREPL(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 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + s7 + MIN(1 + z0 + (1 + z11 + z2')) :|: s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, ifmin(le(z0, z11), 1 + z0 + (1 + z11 + z2'))) + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(le(z0, z12), 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(0, 0), 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(eq(1 + (z - 2), 0), 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s6 >= 0, s6 <= 0 + 2, 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
   ifmin(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + z1) + selsort(replace(0, z0, z1)) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + (1 + z18 + z24)) + selsort(replace(ifmin(le(z0, z18), 1 + z0 + (1 + z18 + z24)), z0, 1 + z18 + z24)) :|: z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + 0 + 0) + selsort(replace(0, 0, 0)) :|: z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + (1 + (z' - 2)) + 0) + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: z = 1, z' - 2 >= 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 }-> ifmin(le(z09, z15), 1 + (1 + z09) + (1 + (1 + z15) + z2)) :|: z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> ifmin(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(1, 1 + (1 + z08) + (1 + 0 + z2)) :|: z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(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 }-> ifrepl(eq(z - 1, z16), 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(eq(z0, ifmin(le(z0, z17), 1 + z0 + (1 + z17 + z23))), 1 + z0 + (1 + z17 + z23)) :|: z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> ifselsort(eq(z0, 0), 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(eq(0, 0), 1 + 0 + 0) :|: z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {LE}, {eq}, {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}
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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(s15, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(ifmin(s16, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(s17, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(0, z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + REPLACE(ifmin(s18, 1 + z0 + (1 + z14 + z22)), z0, 1 + z14 + z22) + MIN(1 + z0 + (1 + z14 + z22)) :|: s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(0, 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 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 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(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 + IFMIN(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 + IFREPL(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 + IFREPL(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 + IFREPL(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 + IFREPL(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 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(s11, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, ifmin(s12, 1 + z0 + (1 + z11 + z2'))) + MIN(1 + z0 + (1 + z11 + z2')) :|: s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, ifmin(s13, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + s7 + MIN(1 + z0 + (1 + z11 + z2')) :|: s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(s14, 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(0, 0), 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(eq(1 + (z - 2), 0), 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s6 >= 0, s6 <= 0 + 2, 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
   ifmin(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + z1) + selsort(replace(0, z0, z1)) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + (1 + z18 + z24)) + selsort(replace(ifmin(s22, 1 + z0 + (1 + z18 + z24)), z0, 1 + z18 + z24)) :|: s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + 0 + 0) + selsort(replace(0, 0, 0)) :|: z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + (1 + (z' - 2)) + 0) + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: z = 1, z' - 2 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> ifmin(s20, 1 + (1 + z09) + (1 + (1 + z15) + z2)) :|: s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> ifmin(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(1, 1 + (1 + z08) + (1 + 0 + z2)) :|: z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(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 }-> ifrepl(eq(z - 1, z16), 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(eq(z0, ifmin(s21, 1 + z0 + (1 + z17 + z23))), 1 + z0 + (1 + z17 + z23)) :|: s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> ifselsort(eq(z0, 0), 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(eq(0, 0), 1 + 0 + 0) :|: z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {LE}, {eq}, {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}
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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(s15, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(ifmin(s16, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(s17, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(0, z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + REPLACE(ifmin(s18, 1 + z0 + (1 + z14 + z22)), z0, 1 + z14 + z22) + MIN(1 + z0 + (1 + z14 + z22)) :|: s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(0, 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 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 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(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 + IFMIN(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 + IFREPL(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 + IFREPL(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 + IFREPL(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 + IFREPL(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 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(s11, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, ifmin(s12, 1 + z0 + (1 + z11 + z2'))) + MIN(1 + z0 + (1 + z11 + z2')) :|: s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, ifmin(s13, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + s7 + MIN(1 + z0 + (1 + z11 + z2')) :|: s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(s14, 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(0, 0), 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(eq(1 + (z - 2), 0), 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s6 >= 0, s6 <= 0 + 2, 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
   ifmin(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + z1) + selsort(replace(0, z0, z1)) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + (1 + z18 + z24)) + selsort(replace(ifmin(s22, 1 + z0 + (1 + z18 + z24)), z0, 1 + z18 + z24)) :|: s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + 0 + 0) + selsort(replace(0, 0, 0)) :|: z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + (1 + (z' - 2)) + 0) + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: z = 1, z' - 2 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> ifmin(s20, 1 + (1 + z09) + (1 + (1 + z15) + z2)) :|: s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> ifmin(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(1, 1 + (1 + z08) + (1 + 0 + z2)) :|: z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(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 }-> ifrepl(eq(z - 1, z16), 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(eq(z0, ifmin(s21, 1 + z0 + (1 + z17 + z23))), 1 + z0 + (1 + z17 + z23)) :|: s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> ifselsort(eq(z0, 0), 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(eq(0, 0), 1 + 0 + 0) :|: z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {LE}, {eq}, {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}
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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(s15, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(ifmin(s16, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(s17, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(0, z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + REPLACE(ifmin(s18, 1 + z0 + (1 + z14 + z22)), z0, 1 + z14 + z22) + MIN(1 + z0 + (1 + z14 + z22)) :|: s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(0, 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 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 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(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 + IFMIN(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 + IFREPL(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 + IFREPL(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 + IFREPL(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 + IFREPL(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 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(s11, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, ifmin(s12, 1 + z0 + (1 + z11 + z2'))) + MIN(1 + z0 + (1 + z11 + z2')) :|: s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, ifmin(s13, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + s7 + MIN(1 + z0 + (1 + z11 + z2')) :|: s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(s14, 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(0, 0), 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(eq(1 + (z - 2), 0), 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s6 >= 0, s6 <= 0 + 2, 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
   ifmin(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + z1) + selsort(replace(0, z0, z1)) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + (1 + z18 + z24)) + selsort(replace(ifmin(s22, 1 + z0 + (1 + z18 + z24)), z0, 1 + z18 + z24)) :|: s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + 0 + 0) + selsort(replace(0, 0, 0)) :|: z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + (1 + (z' - 2)) + 0) + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: z = 1, z' - 2 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> ifmin(s20, 1 + (1 + z09) + (1 + (1 + z15) + z2)) :|: s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> ifmin(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(1, 1 + (1 + z08) + (1 + 0 + z2)) :|: z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(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 }-> ifrepl(eq(z - 1, z16), 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(eq(z0, ifmin(s21, 1 + z0 + (1 + z17 + z23))), 1 + z0 + (1 + z17 + z23)) :|: s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> ifselsort(eq(z0, 0), 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(eq(0, 0), 1 + 0 + 0) :|: z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {eq}, {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}
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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(s15, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(ifmin(s16, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(s17, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(0, z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + REPLACE(ifmin(s18, 1 + z0 + (1 + z14 + z22)), z0, 1 + z14 + z22) + MIN(1 + z0 + (1 + z14 + z22)) :|: s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(0, 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 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 + s23 :|: s23 >= 0, s23 <= 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 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s26 :|: s26 >= 0, s26 <= 1 + z1' + 1, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s24 :|: s24 >= 0, s24 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 3 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s25 :|: s25 >= 0, s25 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s27 :|: s27 >= 0, s27 <= 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 + IFREPL(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 + IFREPL(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 + IFREPL(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 + IFREPL(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 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(s11, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, ifmin(s12, 1 + z0 + (1 + z11 + z2'))) + MIN(1 + z0 + (1 + z11 + z2')) :|: s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, ifmin(s13, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + s7 + MIN(1 + z0 + (1 + z11 + z2')) :|: s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(s14, 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(0, 0), 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(eq(1 + (z - 2), 0), 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s6 >= 0, s6 <= 0 + 2, 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
   ifmin(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + z1) + selsort(replace(0, z0, z1)) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + (1 + z18 + z24)) + selsort(replace(ifmin(s22, 1 + z0 + (1 + z18 + z24)), z0, 1 + z18 + z24)) :|: s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + 0 + 0) + selsort(replace(0, 0, 0)) :|: z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + (1 + (z' - 2)) + 0) + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: z = 1, z' - 2 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> ifmin(s20, 1 + (1 + z09) + (1 + (1 + z15) + z2)) :|: s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> ifmin(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(1, 1 + (1 + z08) + (1 + 0 + z2)) :|: z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(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 }-> ifrepl(eq(z - 1, z16), 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(eq(z0, ifmin(s21, 1 + z0 + (1 + z17 + z23))), 1 + z0 + (1 + z17 + z23)) :|: s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> ifselsort(eq(z0, 0), 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(eq(0, 0), 1 + 0 + 0) :|: z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {eq}, {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}
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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(s15, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(ifmin(s16, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(s17, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(0, z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + REPLACE(ifmin(s18, 1 + z0 + (1 + z14 + z22)), z0, 1 + z14 + z22) + MIN(1 + z0 + (1 + z14 + z22)) :|: s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(0, 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 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 + s23 :|: s23 >= 0, s23 <= 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 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s26 :|: s26 >= 0, s26 <= 1 + z1' + 1, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s24 :|: s24 >= 0, s24 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 3 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s25 :|: s25 >= 0, s25 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s27 :|: s27 >= 0, s27 <= 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 + IFREPL(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 + IFREPL(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 + IFREPL(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 + IFREPL(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 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(s11, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, ifmin(s12, 1 + z0 + (1 + z11 + z2'))) + MIN(1 + z0 + (1 + z11 + z2')) :|: s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, ifmin(s13, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + s7 + MIN(1 + z0 + (1 + z11 + z2')) :|: s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(s14, 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(0, 0), 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(eq(1 + (z - 2), 0), 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s6 >= 0, s6 <= 0 + 2, 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
   ifmin(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + z1) + selsort(replace(0, z0, z1)) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + (1 + z18 + z24)) + selsort(replace(ifmin(s22, 1 + z0 + (1 + z18 + z24)), z0, 1 + z18 + z24)) :|: s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + 0 + 0) + selsort(replace(0, 0, 0)) :|: z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + (1 + (z' - 2)) + 0) + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: z = 1, z' - 2 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> ifmin(s20, 1 + (1 + z09) + (1 + (1 + z15) + z2)) :|: s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> ifmin(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(1, 1 + (1 + z08) + (1 + 0 + z2)) :|: z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(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 }-> ifrepl(eq(z - 1, z16), 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(eq(z0, ifmin(s21, 1 + z0 + (1 + z17 + z23))), 1 + z0 + (1 + z17 + z23)) :|: s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> ifselsort(eq(z0, 0), 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(eq(0, 0), 1 + 0 + 0) :|: z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {eq}, {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}
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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(s15, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(ifmin(s16, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(s17, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(0, z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + REPLACE(ifmin(s18, 1 + z0 + (1 + z14 + z22)), z0, 1 + z14 + z22) + MIN(1 + z0 + (1 + z14 + z22)) :|: s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(0, 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 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 + s23 :|: s23 >= 0, s23 <= 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 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s26 :|: s26 >= 0, s26 <= 1 + z1' + 1, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s24 :|: s24 >= 0, s24 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 3 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s25 :|: s25 >= 0, s25 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s27 :|: s27 >= 0, s27 <= 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 + IFREPL(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 + IFREPL(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 + IFREPL(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 + IFREPL(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 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(s11, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, ifmin(s12, 1 + z0 + (1 + z11 + z2'))) + MIN(1 + z0 + (1 + z11 + z2')) :|: s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, ifmin(s13, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + s7 + MIN(1 + z0 + (1 + z11 + z2')) :|: s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(s14, 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(0, 0), 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(eq(1 + (z - 2), 0), 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s6 >= 0, s6 <= 0 + 2, 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
   ifmin(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + z1) + selsort(replace(0, z0, z1)) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + (1 + z18 + z24)) + selsort(replace(ifmin(s22, 1 + z0 + (1 + z18 + z24)), z0, 1 + z18 + z24)) :|: s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + 0 + 0) + selsort(replace(0, 0, 0)) :|: z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + (1 + (z' - 2)) + 0) + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: z = 1, z' - 2 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> ifmin(s20, 1 + (1 + z09) + (1 + (1 + z15) + z2)) :|: s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> ifmin(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(1, 1 + (1 + z08) + (1 + 0 + z2)) :|: z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(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 }-> ifrepl(eq(z - 1, z16), 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(eq(z0, ifmin(s21, 1 + z0 + (1 + z17 + z23))), 1 + z0 + (1 + z17 + z23)) :|: s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> ifselsort(eq(z0, 0), 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(eq(0, 0), 1 + 0 + 0) :|: z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(eq(1 + (z - 2), 1 + (z - 2)), 1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}
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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(s15, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(ifmin(s16, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(s17, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(0, z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + REPLACE(ifmin(s18, 1 + z0 + (1 + z14 + z22)), z0, 1 + z14 + z22) + MIN(1 + z0 + (1 + z14 + z22)) :|: s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(0, 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 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 + s23 :|: s23 >= 0, s23 <= 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 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s26 :|: s26 >= 0, s26 <= 1 + z1' + 1, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s24 :|: s24 >= 0, s24 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 3 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s25 :|: s25 >= 0, s25 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s27 :|: s27 >= 0, s27 <= 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 + IFREPL(s28, 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s28 >= 0, s28 <= 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 + IFREPL(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 + IFREPL(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 + IFREPL(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 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s29, 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s29 >= 0, s29 <= 2, s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s30 >= 0, s30 <= 2, s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s31, 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s31 >= 0, s31 <= 2, s6 >= 0, s6 <= 0 + 2, z - 2 >= 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(s32, 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s32 >= 0, s32 <= 2, s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s33, 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(s14, 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: s33 >= 0, s33 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s34, 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s34 >= 0, s34 <= 2, s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(s11, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, ifmin(s12, 1 + z0 + (1 + z11 + z2'))) + MIN(1 + z0 + (1 + z11 + z2')) :|: s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, ifmin(s13, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + s7 + MIN(1 + z0 + (1 + z11 + z2')) :|: s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 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
   ifmin(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + z1) + selsort(replace(0, z0, z1)) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + (1 + z18 + z24)) + selsort(replace(ifmin(s22, 1 + z0 + (1 + z18 + z24)), z0, 1 + z18 + z24)) :|: s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + 0 + 0) + selsort(replace(0, 0, 0)) :|: z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + (1 + (z' - 2)) + 0) + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: z = 1, z' - 2 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> ifmin(s20, 1 + (1 + z09) + (1 + (1 + z15) + z2)) :|: s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> ifmin(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(1, 1 + (1 + z08) + (1 + 0 + z2)) :|: z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(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 }-> ifrepl(s36, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s36 >= 0, s36 <= 2, z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(s37, 1 + 0 + 0) :|: s37 >= 0, s37 <= 2, z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(s38, 1 + (1 + (z - 2)) + 0) :|: s38 >= 0, s38 <= 2, z - 2 >= 0
   selsort(z) -{ 0 }-> ifselsort(s39, 1 + z0 + z1) :|: s39 >= 0, s39 <= 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(eq(z0, ifmin(s21, 1 + z0 + (1 + z17 + z23))), 1 + z0 + (1 + z17 + z23)) :|: s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}
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: IFREPL
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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(s15, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(ifmin(s16, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(s17, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(0, z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + REPLACE(ifmin(s18, 1 + z0 + (1 + z14 + z22)), z0, 1 + z14 + z22) + MIN(1 + z0 + (1 + z14 + z22)) :|: s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(0, 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 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 + s23 :|: s23 >= 0, s23 <= 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 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s26 :|: s26 >= 0, s26 <= 1 + z1' + 1, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s24 :|: s24 >= 0, s24 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 3 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s25 :|: s25 >= 0, s25 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s27 :|: s27 >= 0, s27 <= 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 + IFREPL(s28, 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s28 >= 0, s28 <= 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 + IFREPL(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 + IFREPL(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 + IFREPL(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 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s29, 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s29 >= 0, s29 <= 2, s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s30 >= 0, s30 <= 2, s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s31, 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s31 >= 0, s31 <= 2, s6 >= 0, s6 <= 0 + 2, z - 2 >= 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(s32, 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s32 >= 0, s32 <= 2, s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s33, 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(s14, 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: s33 >= 0, s33 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s34, 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s34 >= 0, s34 <= 2, s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(s11, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, ifmin(s12, 1 + z0 + (1 + z11 + z2'))) + MIN(1 + z0 + (1 + z11 + z2')) :|: s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, ifmin(s13, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + s7 + MIN(1 + z0 + (1 + z11 + z2')) :|: s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 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
   ifmin(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + z1) + selsort(replace(0, z0, z1)) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + (1 + z18 + z24)) + selsort(replace(ifmin(s22, 1 + z0 + (1 + z18 + z24)), z0, 1 + z18 + z24)) :|: s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + 0 + 0) + selsort(replace(0, 0, 0)) :|: z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + (1 + (z' - 2)) + 0) + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: z = 1, z' - 2 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> ifmin(s20, 1 + (1 + z09) + (1 + (1 + z15) + z2)) :|: s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> ifmin(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(1, 1 + (1 + z08) + (1 + 0 + z2)) :|: z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(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 }-> ifrepl(s36, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s36 >= 0, s36 <= 2, z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(s37, 1 + 0 + 0) :|: s37 >= 0, s37 <= 2, z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(s38, 1 + (1 + (z - 2)) + 0) :|: s38 >= 0, s38 <= 2, z - 2 >= 0
   selsort(z) -{ 0 }-> ifselsort(s39, 1 + z0 + z1) :|: s39 >= 0, s39 <= 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(eq(z0, ifmin(s21, 1 + z0 + (1 + z17 + z23))), 1 + z0 + (1 + z17 + z23)) :|: s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}
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]
  IFREPL: 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: IFREPL
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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(s15, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(ifmin(s16, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(s17, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(0, z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + REPLACE(ifmin(s18, 1 + z0 + (1 + z14 + z22)), z0, 1 + z14 + z22) + MIN(1 + z0 + (1 + z14 + z22)) :|: s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(0, 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + REPLACE(1 + (z' - 2), 1 + (z' - 2), 0) + MIN(1 + (1 + (z' - 2)) + 0) :|: z = 1, z' - 2 >= 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 + s23 :|: s23 >= 0, s23 <= 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 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s26 :|: s26 >= 0, s26 <= 1 + z1' + 1, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s24 :|: s24 >= 0, s24 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 3 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s25 :|: s25 >= 0, s25 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s27 :|: s27 >= 0, s27 <= 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 + IFREPL(s28, 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s28 >= 0, s28 <= 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 + IFREPL(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 + IFREPL(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 + IFREPL(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 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= z2 + 2, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s29, 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s29 >= 0, s29 <= 2, s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s30 >= 0, s30 <= 2, s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s31, 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s31 >= 0, s31 <= 2, s6 >= 0, s6 <= 0 + 2, z - 2 >= 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(s32, 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s32 >= 0, s32 <= 2, s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s33, 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(s14, 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: s33 >= 0, s33 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s34, 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s34 >= 0, s34 <= 2, s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(s11, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, ifmin(s12, 1 + z0 + (1 + z11 + z2'))) + MIN(1 + z0 + (1 + z11 + z2')) :|: s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, ifmin(s13, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + s7 + MIN(1 + z0 + (1 + z11 + z2')) :|: s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 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
   ifmin(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + z1) + selsort(replace(0, z0, z1)) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + (1 + z18 + z24)) + selsort(replace(ifmin(s22, 1 + z0 + (1 + z18 + z24)), z0, 1 + z18 + z24)) :|: s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + 0 + 0) + selsort(replace(0, 0, 0)) :|: z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + (1 + (z' - 2)) + 0) + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: z = 1, z' - 2 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> ifmin(s20, 1 + (1 + z09) + (1 + (1 + z15) + z2)) :|: s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> ifmin(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(1, 1 + (1 + z08) + (1 + 0 + z2)) :|: z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(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 }-> ifrepl(s36, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s36 >= 0, s36 <= 2, z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(s37, 1 + 0 + 0) :|: s37 >= 0, s37 <= 2, z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(s38, 1 + (1 + (z - 2)) + 0) :|: s38 >= 0, s38 <= 2, z - 2 >= 0
   selsort(z) -{ 0 }-> ifselsort(s39, 1 + z0 + z1) :|: s39 >= 0, s39 <= 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(eq(z0, ifmin(s21, 1 + z0 + (1 + z17 + z23))), 1 + z0 + (1 + z17 + z23)) :|: s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}
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]
  IFREPL: 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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s45 :|: s45 >= 0, s45 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(s15, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(ifmin(s16, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 21 + 9*z13 + 2*z13*z21 + z13^2 + 9*z21 + z21^2 }-> 1 + SELSORT(replace(ifmin(s17, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + s49 + MIN(1 + z0 + (1 + z13 + z21)) :|: s49 >= 0, s49 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 13 + 7*z1 + z1^2 }-> 1 + SELSORT(replace(0, z0, z1)) + s51 + MIN(1 + z0 + z1) :|: s51 >= 0, s51 <= 7 * z1 + 6 + z1 * z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + REPLACE(ifmin(s18, 1 + z0 + (1 + z14 + z22)), z0, 1 + z14 + z22) + MIN(1 + z0 + (1 + z14 + z22)) :|: s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(0, 0, 0)) + s46 + MIN(1 + 0 + 0) :|: s46 >= 0, s46 <= 7 * 0 + 6 + 0 * 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + s50 + MIN(1 + (1 + (z' - 2)) + 0) :|: s50 >= 0, s50 <= 7 * 0 + 6 + 0 * 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s47 + MIN(1 + (1 + (z' - 2)) + 0) :|: s47 >= 0, s47 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s48 + MIN(1 + (1 + (z' - 2)) + 0) :|: s48 >= 0, s48 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 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 + s23 :|: s23 >= 0, s23 <= 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 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s26 :|: s26 >= 0, s26 <= 1 + z1' + 1, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s24 :|: s24 >= 0, s24 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 3 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s25 :|: s25 >= 0, s25 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s27 :|: s27 >= 0, s27 <= 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 + s40 + s' :|: s40 >= 0, s40 <= 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 + s41 + s'' :|: s41 >= 0, s41 <= 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 + s42 + s1 :|: s42 >= 0, s42 <= 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 + s43 + s2 :|: s43 >= 0, s43 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s28 >= 0, s28 <= 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 + s44 + s3 :|: s44 >= 0, s44 <= 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
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s29, 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s29 >= 0, s29 <= 2, s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s30 >= 0, s30 <= 2, s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s31, 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s31 >= 0, s31 <= 2, s6 >= 0, s6 <= 0 + 2, z - 2 >= 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(s32, 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s32 >= 0, s32 <= 2, s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s33, 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(s14, 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: s33 >= 0, s33 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s34, 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s34 >= 0, s34 <= 2, s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(s11, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, ifmin(s12, 1 + z0 + (1 + z11 + z2'))) + MIN(1 + z0 + (1 + z11 + z2')) :|: s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, ifmin(s13, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + s7 + MIN(1 + z0 + (1 + z11 + z2')) :|: s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 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
   ifmin(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + z1) + selsort(replace(0, z0, z1)) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + (1 + z18 + z24)) + selsort(replace(ifmin(s22, 1 + z0 + (1 + z18 + z24)), z0, 1 + z18 + z24)) :|: s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + 0 + 0) + selsort(replace(0, 0, 0)) :|: z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + (1 + (z' - 2)) + 0) + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: z = 1, z' - 2 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> ifmin(s20, 1 + (1 + z09) + (1 + (1 + z15) + z2)) :|: s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> ifmin(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(1, 1 + (1 + z08) + (1 + 0 + z2)) :|: z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(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 }-> ifrepl(s36, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s36 >= 0, s36 <= 2, z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(s37, 1 + 0 + 0) :|: s37 >= 0, s37 <= 2, z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(s38, 1 + (1 + (z - 2)) + 0) :|: s38 >= 0, s38 <= 2, z - 2 >= 0
   selsort(z) -{ 0 }-> ifselsort(s39, 1 + z0 + z1) :|: s39 >= 0, s39 <= 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(eq(z0, ifmin(s21, 1 + z0 + (1 + z17 + z23))), 1 + z0 + (1 + z17 + z23)) :|: s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}
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]
  IFREPL: 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 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: ifmin
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z'

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

(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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s45 :|: s45 >= 0, s45 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(s15, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(ifmin(s16, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 21 + 9*z13 + 2*z13*z21 + z13^2 + 9*z21 + z21^2 }-> 1 + SELSORT(replace(ifmin(s17, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + s49 + MIN(1 + z0 + (1 + z13 + z21)) :|: s49 >= 0, s49 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 13 + 7*z1 + z1^2 }-> 1 + SELSORT(replace(0, z0, z1)) + s51 + MIN(1 + z0 + z1) :|: s51 >= 0, s51 <= 7 * z1 + 6 + z1 * z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + REPLACE(ifmin(s18, 1 + z0 + (1 + z14 + z22)), z0, 1 + z14 + z22) + MIN(1 + z0 + (1 + z14 + z22)) :|: s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(0, 0, 0)) + s46 + MIN(1 + 0 + 0) :|: s46 >= 0, s46 <= 7 * 0 + 6 + 0 * 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + s50 + MIN(1 + (1 + (z' - 2)) + 0) :|: s50 >= 0, s50 <= 7 * 0 + 6 + 0 * 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s47 + MIN(1 + (1 + (z' - 2)) + 0) :|: s47 >= 0, s47 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s48 + MIN(1 + (1 + (z' - 2)) + 0) :|: s48 >= 0, s48 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 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 + s23 :|: s23 >= 0, s23 <= 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 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s26 :|: s26 >= 0, s26 <= 1 + z1' + 1, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s24 :|: s24 >= 0, s24 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 3 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s25 :|: s25 >= 0, s25 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s27 :|: s27 >= 0, s27 <= 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 + s40 + s' :|: s40 >= 0, s40 <= 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 + s41 + s'' :|: s41 >= 0, s41 <= 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 + s42 + s1 :|: s42 >= 0, s42 <= 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 + s43 + s2 :|: s43 >= 0, s43 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s28 >= 0, s28 <= 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 + s44 + s3 :|: s44 >= 0, s44 <= 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
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s29, 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s29 >= 0, s29 <= 2, s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s30 >= 0, s30 <= 2, s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s31, 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s31 >= 0, s31 <= 2, s6 >= 0, s6 <= 0 + 2, z - 2 >= 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(s32, 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s32 >= 0, s32 <= 2, s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s33, 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(s14, 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: s33 >= 0, s33 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s34, 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s34 >= 0, s34 <= 2, s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(s11, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, ifmin(s12, 1 + z0 + (1 + z11 + z2'))) + MIN(1 + z0 + (1 + z11 + z2')) :|: s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, ifmin(s13, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + s7 + MIN(1 + z0 + (1 + z11 + z2')) :|: s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 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
   ifmin(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + z1) + selsort(replace(0, z0, z1)) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + (1 + z18 + z24)) + selsort(replace(ifmin(s22, 1 + z0 + (1 + z18 + z24)), z0, 1 + z18 + z24)) :|: s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + 0 + 0) + selsort(replace(0, 0, 0)) :|: z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + (1 + (z' - 2)) + 0) + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: z = 1, z' - 2 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> ifmin(s20, 1 + (1 + z09) + (1 + (1 + z15) + z2)) :|: s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> ifmin(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(1, 1 + (1 + z08) + (1 + 0 + z2)) :|: z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(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 }-> ifrepl(s36, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s36 >= 0, s36 <= 2, z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(s37, 1 + 0 + 0) :|: s37 >= 0, s37 <= 2, z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(s38, 1 + (1 + (z - 2)) + 0) :|: s38 >= 0, s38 <= 2, z - 2 >= 0
   selsort(z) -{ 0 }-> ifselsort(s39, 1 + z0 + z1) :|: s39 >= 0, s39 <= 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(eq(z0, ifmin(s21, 1 + z0 + (1 + z17 + z23))), 1 + z0 + (1 + z17 + z23)) :|: s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}
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]
  IFREPL: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  min: runtime: ?, size: O(n^1) [z]
  ifmin: runtime: ?, size: O(n^1) [z']

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

(51) 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: ifmin
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

(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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s45 :|: s45 >= 0, s45 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(ifmin(s15, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + REPLACE(ifmin(s16, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21) + MIN(1 + z0 + (1 + z13 + z21)) :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 21 + 9*z13 + 2*z13*z21 + z13^2 + 9*z21 + z21^2 }-> 1 + SELSORT(replace(ifmin(s17, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + s49 + MIN(1 + z0 + (1 + z13 + z21)) :|: s49 >= 0, s49 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 13 + 7*z1 + z1^2 }-> 1 + SELSORT(replace(0, z0, z1)) + s51 + MIN(1 + z0 + z1) :|: s51 >= 0, s51 <= 7 * z1 + 6 + z1 * z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + REPLACE(ifmin(s18, 1 + z0 + (1 + z14 + z22)), z0, 1 + z14 + z22) + MIN(1 + z0 + (1 + z14 + z22)) :|: s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(0, 0, 0)) + s46 + MIN(1 + 0 + 0) :|: s46 >= 0, s46 <= 7 * 0 + 6 + 0 * 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + s50 + MIN(1 + (1 + (z' - 2)) + 0) :|: s50 >= 0, s50 <= 7 * 0 + 6 + 0 * 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s47 + MIN(1 + (1 + (z' - 2)) + 0) :|: s47 >= 0, s47 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s48 + MIN(1 + (1 + (z' - 2)) + 0) :|: s48 >= 0, s48 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 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 + s23 :|: s23 >= 0, s23 <= 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 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s26 :|: s26 >= 0, s26 <= 1 + z1' + 1, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s24 :|: s24 >= 0, s24 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 3 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s25 :|: s25 >= 0, s25 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s27 :|: s27 >= 0, s27 <= 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 + s40 + s' :|: s40 >= 0, s40 <= 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 + s41 + s'' :|: s41 >= 0, s41 <= 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 + s42 + s1 :|: s42 >= 0, s42 <= 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 + s43 + s2 :|: s43 >= 0, s43 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s28 >= 0, s28 <= 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 + s44 + s3 :|: s44 >= 0, s44 <= 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
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s29, 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s29 >= 0, s29 <= 2, s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s30 >= 0, s30 <= 2, s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s31, 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s31 >= 0, s31 <= 2, s6 >= 0, s6 <= 0 + 2, z - 2 >= 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(s32, 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s32 >= 0, s32 <= 2, s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s33, 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(s14, 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: s33 >= 0, s33 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s34, 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s34 >= 0, s34 <= 2, s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, ifmin(s11, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + EQ(z0, ifmin(s12, 1 + z0 + (1 + z11 + z2'))) + MIN(1 + z0 + (1 + z11 + z2')) :|: s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(eq(z0, ifmin(s13, 1 + z0 + (1 + z11 + z2'))), 1 + z0 + (1 + z11 + z2')) + s7 + MIN(1 + z0 + (1 + z11 + z2')) :|: s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 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
   ifmin(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + z1) + selsort(replace(0, z0, z1)) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + z0 + (1 + z18 + z24)) + selsort(replace(ifmin(s22, 1 + z0 + (1 + z18 + z24)), z0, 1 + z18 + z24)) :|: s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + 0 + 0) + selsort(replace(0, 0, 0)) :|: z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + min(1 + (1 + (z' - 2)) + 0) + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: z = 1, z' - 2 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> ifmin(s20, 1 + (1 + z09) + (1 + (1 + z15) + z2)) :|: s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> ifmin(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(1, 1 + (1 + z08) + (1 + 0 + z2)) :|: z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> ifmin(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 }-> ifrepl(s36, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s36 >= 0, s36 <= 2, z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(s37, 1 + 0 + 0) :|: s37 >= 0, s37 <= 2, z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(s38, 1 + (1 + (z - 2)) + 0) :|: s38 >= 0, s38 <= 2, z - 2 >= 0
   selsort(z) -{ 0 }-> ifselsort(s39, 1 + z0 + z1) :|: s39 >= 0, s39 <= 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(eq(z0, ifmin(s21, 1 + z0 + (1 + z17 + z23))), 1 + z0 + (1 + z17 + z23)) :|: s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}
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]
  IFREPL: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  min: runtime: O(1) [0], size: O(n^1) [z]
  ifmin: runtime: O(1) [0], size: O(n^1) [z']

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

(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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s45 :|: s45 >= 0, s45 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 21 + 9*z13 + 2*z13*z21 + z13^2 + 9*z21 + z21^2 }-> 1 + SELSORT(replace(s60, z0, 1 + z13 + z21)) + s62 + MIN(1 + z0 + (1 + z13 + z21)) :|: s60 >= 0, s60 <= 1 + z0 + (1 + z13 + z21), s61 >= 0, s61 <= 1 + z0 + (1 + z13 + z21), s62 >= 0, s62 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 21 + 9*z13 + 2*z13*z21 + z13^2 + 9*z21 + z21^2 }-> 1 + SELSORT(replace(s63, z0, 1 + z13 + z21)) + s49 + MIN(1 + z0 + (1 + z13 + z21)) :|: s63 >= 0, s63 <= 1 + z0 + (1 + z13 + z21), s49 >= 0, s49 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 13 + 7*z1 + z1^2 }-> 1 + SELSORT(replace(0, z0, z1)) + s51 + MIN(1 + z0 + z1) :|: s51 >= 0, s51 <= 7 * z1 + 6 + z1 * z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 21 + 9*z14 + 2*z14*z22 + z14^2 + 9*z22 + z22^2 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + s65 + MIN(1 + z0 + (1 + z14 + z22)) :|: s64 >= 0, s64 <= 1 + z0 + (1 + z14 + z22), s65 >= 0, s65 <= 7 * (1 + z14 + z22) + 6 + (1 + z14 + z22) * (1 + z14 + z22), s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(0, 0, 0)) + s46 + MIN(1 + 0 + 0) :|: s46 >= 0, s46 <= 7 * 0 + 6 + 0 * 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + s50 + MIN(1 + (1 + (z' - 2)) + 0) :|: s50 >= 0, s50 <= 7 * 0 + 6 + 0 * 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s47 + MIN(1 + (1 + (z' - 2)) + 0) :|: s47 >= 0, s47 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s48 + MIN(1 + (1 + (z' - 2)) + 0) :|: s48 >= 0, s48 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 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 + s23 :|: s23 >= 0, s23 <= 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 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s26 :|: s26 >= 0, s26 <= 1 + z1' + 1, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s24 :|: s24 >= 0, s24 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 3 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s25 :|: s25 >= 0, s25 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s27 :|: s27 >= 0, s27 <= 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 + s40 + s' :|: s40 >= 0, s40 <= 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 + s41 + s'' :|: s41 >= 0, s41 <= 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 + s42 + s1 :|: s42 >= 0, s42 <= 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 + s43 + s2 :|: s43 >= 0, s43 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s28 >= 0, s28 <= 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 + s44 + s3 :|: s44 >= 0, s44 <= 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
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s29, 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s29 >= 0, s29 <= 2, s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s30 >= 0, s30 <= 2, s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s31, 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s31 >= 0, s31 <= 2, s6 >= 0, s6 <= 0 + 2, z - 2 >= 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(s32, 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s32 >= 0, s32 <= 2, s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 4 + s58 }-> 1 + IFSELSORT(s33, 1 + z0 + (1 + z12 + z2'')) + s59 + MIN(1 + z0 + (1 + z12 + z2'')) :|: s58 >= 0, s58 <= 1 + z0 + (1 + z12 + z2''), s59 >= 0, s59 <= s58 + 2, s33 >= 0, s33 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s34, 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s34 >= 0, s34 <= 2, s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 4 + s54 }-> 1 + IFSELSORT(s53, 1 + z0 + (1 + z11 + z2')) + s55 + MIN(1 + z0 + (1 + z11 + z2')) :|: s52 >= 0, s52 <= 1 + z0 + (1 + z11 + z2'), s53 >= 0, s53 <= 2, s54 >= 0, s54 <= 1 + z0 + (1 + z11 + z2'), s55 >= 0, s55 <= s54 + 2, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s57, 1 + z0 + (1 + z11 + z2')) + s7 + MIN(1 + z0 + (1 + z11 + z2')) :|: s56 >= 0, s56 <= 1 + z0 + (1 + z11 + z2'), s57 >= 0, s57 <= 2, s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 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
   ifmin(z, z') -{ 0 }-> s70 :|: s70 >= 0, s70 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> s71 :|: s71 >= 0, s71 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s74 + selsort(replace(0, 0, 0)) :|: s74 >= 0, s74 <= 1 + 0 + 0, z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + s75 + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: s75 >= 0, s75 <= 1 + (1 + (z' - 2)) + 0, z = 1, z' - 2 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s76 + selsort(replace(s77, z0, 1 + z18 + z24)) :|: s76 >= 0, s76 <= 1 + z0 + (1 + z18 + z24), s77 >= 0, s77 <= 1 + z0 + (1 + z18 + z24), s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s78 + selsort(replace(0, z0, z1)) :|: s78 >= 0, s78 <= 1 + z0 + z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> s66 :|: s66 >= 0, s66 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s67 :|: s67 >= 0, s67 <= 1 + (1 + z08) + (1 + 0 + z2), z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> s68 :|: s68 >= 0, s68 <= 1 + (1 + z09) + (1 + (1 + z15) + z2), s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> s69 :|: s69 >= 0, s69 <= 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 }-> ifrepl(s36, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s36 >= 0, s36 <= 2, z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(s37, 1 + 0 + 0) :|: s37 >= 0, s37 <= 2, z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(s38, 1 + (1 + (z - 2)) + 0) :|: s38 >= 0, s38 <= 2, z - 2 >= 0
   selsort(z) -{ 0 }-> ifselsort(s39, 1 + z0 + z1) :|: s39 >= 0, s39 <= 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(s73, 1 + z0 + (1 + z17 + z23)) :|: s72 >= 0, s72 <= 1 + z0 + (1 + z17 + z23), s73 >= 0, s73 <= 2, s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}
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]
  IFREPL: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  min: runtime: O(1) [0], size: O(n^1) [z]
  ifmin: runtime: O(1) [0], size: O(n^1) [z']

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: IFMIN
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

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

(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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s45 :|: s45 >= 0, s45 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 21 + 9*z13 + 2*z13*z21 + z13^2 + 9*z21 + z21^2 }-> 1 + SELSORT(replace(s60, z0, 1 + z13 + z21)) + s62 + MIN(1 + z0 + (1 + z13 + z21)) :|: s60 >= 0, s60 <= 1 + z0 + (1 + z13 + z21), s61 >= 0, s61 <= 1 + z0 + (1 + z13 + z21), s62 >= 0, s62 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 21 + 9*z13 + 2*z13*z21 + z13^2 + 9*z21 + z21^2 }-> 1 + SELSORT(replace(s63, z0, 1 + z13 + z21)) + s49 + MIN(1 + z0 + (1 + z13 + z21)) :|: s63 >= 0, s63 <= 1 + z0 + (1 + z13 + z21), s49 >= 0, s49 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 13 + 7*z1 + z1^2 }-> 1 + SELSORT(replace(0, z0, z1)) + s51 + MIN(1 + z0 + z1) :|: s51 >= 0, s51 <= 7 * z1 + 6 + z1 * z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 21 + 9*z14 + 2*z14*z22 + z14^2 + 9*z22 + z22^2 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + s65 + MIN(1 + z0 + (1 + z14 + z22)) :|: s64 >= 0, s64 <= 1 + z0 + (1 + z14 + z22), s65 >= 0, s65 <= 7 * (1 + z14 + z22) + 6 + (1 + z14 + z22) * (1 + z14 + z22), s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(0, 0, 0)) + s46 + MIN(1 + 0 + 0) :|: s46 >= 0, s46 <= 7 * 0 + 6 + 0 * 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + s50 + MIN(1 + (1 + (z' - 2)) + 0) :|: s50 >= 0, s50 <= 7 * 0 + 6 + 0 * 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s47 + MIN(1 + (1 + (z' - 2)) + 0) :|: s47 >= 0, s47 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s48 + MIN(1 + (1 + (z' - 2)) + 0) :|: s48 >= 0, s48 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 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 + s23 :|: s23 >= 0, s23 <= 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 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s26 :|: s26 >= 0, s26 <= 1 + z1' + 1, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s24 :|: s24 >= 0, s24 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 3 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s25 :|: s25 >= 0, s25 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s27 :|: s27 >= 0, s27 <= 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 + s40 + s' :|: s40 >= 0, s40 <= 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 + s41 + s'' :|: s41 >= 0, s41 <= 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 + s42 + s1 :|: s42 >= 0, s42 <= 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 + s43 + s2 :|: s43 >= 0, s43 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s28 >= 0, s28 <= 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 + s44 + s3 :|: s44 >= 0, s44 <= 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
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s29, 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s29 >= 0, s29 <= 2, s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s30 >= 0, s30 <= 2, s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s31, 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s31 >= 0, s31 <= 2, s6 >= 0, s6 <= 0 + 2, z - 2 >= 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(s32, 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s32 >= 0, s32 <= 2, s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 4 + s58 }-> 1 + IFSELSORT(s33, 1 + z0 + (1 + z12 + z2'')) + s59 + MIN(1 + z0 + (1 + z12 + z2'')) :|: s58 >= 0, s58 <= 1 + z0 + (1 + z12 + z2''), s59 >= 0, s59 <= s58 + 2, s33 >= 0, s33 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s34, 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s34 >= 0, s34 <= 2, s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 4 + s54 }-> 1 + IFSELSORT(s53, 1 + z0 + (1 + z11 + z2')) + s55 + MIN(1 + z0 + (1 + z11 + z2')) :|: s52 >= 0, s52 <= 1 + z0 + (1 + z11 + z2'), s53 >= 0, s53 <= 2, s54 >= 0, s54 <= 1 + z0 + (1 + z11 + z2'), s55 >= 0, s55 <= s54 + 2, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s57, 1 + z0 + (1 + z11 + z2')) + s7 + MIN(1 + z0 + (1 + z11 + z2')) :|: s56 >= 0, s56 <= 1 + z0 + (1 + z11 + z2'), s57 >= 0, s57 <= 2, s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 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
   ifmin(z, z') -{ 0 }-> s70 :|: s70 >= 0, s70 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> s71 :|: s71 >= 0, s71 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s74 + selsort(replace(0, 0, 0)) :|: s74 >= 0, s74 <= 1 + 0 + 0, z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + s75 + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: s75 >= 0, s75 <= 1 + (1 + (z' - 2)) + 0, z = 1, z' - 2 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s76 + selsort(replace(s77, z0, 1 + z18 + z24)) :|: s76 >= 0, s76 <= 1 + z0 + (1 + z18 + z24), s77 >= 0, s77 <= 1 + z0 + (1 + z18 + z24), s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s78 + selsort(replace(0, z0, z1)) :|: s78 >= 0, s78 <= 1 + z0 + z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> s66 :|: s66 >= 0, s66 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s67 :|: s67 >= 0, s67 <= 1 + (1 + z08) + (1 + 0 + z2), z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> s68 :|: s68 >= 0, s68 <= 1 + (1 + z09) + (1 + (1 + z15) + z2), s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> s69 :|: s69 >= 0, s69 <= 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 }-> ifrepl(s36, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s36 >= 0, s36 <= 2, z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(s37, 1 + 0 + 0) :|: s37 >= 0, s37 <= 2, z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(s38, 1 + (1 + (z - 2)) + 0) :|: s38 >= 0, s38 <= 2, z - 2 >= 0
   selsort(z) -{ 0 }-> ifselsort(s39, 1 + z0 + z1) :|: s39 >= 0, s39 <= 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(s73, 1 + z0 + (1 + z17 + z23)) :|: s72 >= 0, s72 <= 1 + z0 + (1 + z17 + z23), s73 >= 0, s73 <= 2, s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}
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]
  IFREPL: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  min: runtime: O(1) [0], size: O(n^1) [z]
  ifmin: runtime: O(1) [0], size: O(n^1) [z']
  IFMIN: runtime: ?, size: O(n^2) [8 + 4*z' + z'^2]
  MIN: runtime: ?, size: O(n^2) [115 + 99*z + 26*z^2]

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: IFMIN
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

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

(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
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s45 :|: s45 >= 0, s45 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 21 + 9*z13 + 2*z13*z21 + z13^2 + 9*z21 + z21^2 }-> 1 + SELSORT(replace(s60, z0, 1 + z13 + z21)) + s62 + MIN(1 + z0 + (1 + z13 + z21)) :|: s60 >= 0, s60 <= 1 + z0 + (1 + z13 + z21), s61 >= 0, s61 <= 1 + z0 + (1 + z13 + z21), s62 >= 0, s62 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 21 + 9*z13 + 2*z13*z21 + z13^2 + 9*z21 + z21^2 }-> 1 + SELSORT(replace(s63, z0, 1 + z13 + z21)) + s49 + MIN(1 + z0 + (1 + z13 + z21)) :|: s63 >= 0, s63 <= 1 + z0 + (1 + z13 + z21), s49 >= 0, s49 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 13 + 7*z1 + z1^2 }-> 1 + SELSORT(replace(0, z0, z1)) + s51 + MIN(1 + z0 + z1) :|: s51 >= 0, s51 <= 7 * z1 + 6 + z1 * z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 21 + 9*z14 + 2*z14*z22 + z14^2 + 9*z22 + z22^2 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + s65 + MIN(1 + z0 + (1 + z14 + z22)) :|: s64 >= 0, s64 <= 1 + z0 + (1 + z14 + z22), s65 >= 0, s65 <= 7 * (1 + z14 + z22) + 6 + (1 + z14 + z22) * (1 + z14 + z22), s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(0, 0, 0)) + s46 + MIN(1 + 0 + 0) :|: s46 >= 0, s46 <= 7 * 0 + 6 + 0 * 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + s50 + MIN(1 + (1 + (z' - 2)) + 0) :|: s50 >= 0, s50 <= 7 * 0 + 6 + 0 * 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s47 + MIN(1 + (1 + (z' - 2)) + 0) :|: s47 >= 0, s47 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 13 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s48 + MIN(1 + (1 + (z' - 2)) + 0) :|: s48 >= 0, s48 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 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 + s23 :|: s23 >= 0, s23 <= 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 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s26 :|: s26 >= 0, s26 <= 1 + z1' + 1, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s24 :|: s24 >= 0, s24 <= z1 + 1, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 3 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s25 :|: s25 >= 0, s25 <= 0 + 1, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 3 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s27 :|: s27 >= 0, s27 <= 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 + s40 + s' :|: s40 >= 0, s40 <= 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 + s41 + s'' :|: s41 >= 0, s41 <= 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 + s42 + s1 :|: s42 >= 0, s42 <= 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 + s43 + s2 :|: s43 >= 0, s43 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s28 >= 0, s28 <= 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 + s44 + s3 :|: s44 >= 0, s44 <= 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
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s29, 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s29 >= 0, s29 <= 2, s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s30 >= 0, s30 <= 2, s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s31, 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s31 >= 0, s31 <= 2, s6 >= 0, s6 <= 0 + 2, z - 2 >= 0
   SELSORT(z) -{ 3 + z }-> 1 + IFSELSORT(s32, 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s32 >= 0, s32 <= 2, s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 4 + s58 }-> 1 + IFSELSORT(s33, 1 + z0 + (1 + z12 + z2'')) + s59 + MIN(1 + z0 + (1 + z12 + z2'')) :|: s58 >= 0, s58 <= 1 + z0 + (1 + z12 + z2''), s59 >= 0, s59 <= s58 + 2, s33 >= 0, s33 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s34, 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s34 >= 0, s34 <= 2, s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 4 + s54 }-> 1 + IFSELSORT(s53, 1 + z0 + (1 + z11 + z2')) + s55 + MIN(1 + z0 + (1 + z11 + z2')) :|: s52 >= 0, s52 <= 1 + z0 + (1 + z11 + z2'), s53 >= 0, s53 <= 2, s54 >= 0, s54 <= 1 + z0 + (1 + z11 + z2'), s55 >= 0, s55 <= s54 + 2, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 4 }-> 1 + IFSELSORT(s57, 1 + z0 + (1 + z11 + z2')) + s7 + MIN(1 + z0 + (1 + z11 + z2')) :|: s56 >= 0, s56 <= 1 + z0 + (1 + z11 + z2'), s57 >= 0, s57 <= 2, s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 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
   ifmin(z, z') -{ 0 }-> s70 :|: s70 >= 0, s70 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> s71 :|: s71 >= 0, s71 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s74 + selsort(replace(0, 0, 0)) :|: s74 >= 0, s74 <= 1 + 0 + 0, z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + s75 + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: s75 >= 0, s75 <= 1 + (1 + (z' - 2)) + 0, z = 1, z' - 2 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s76 + selsort(replace(s77, z0, 1 + z18 + z24)) :|: s76 >= 0, s76 <= 1 + z0 + (1 + z18 + z24), s77 >= 0, s77 <= 1 + z0 + (1 + z18 + z24), s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s78 + selsort(replace(0, z0, z1)) :|: s78 >= 0, s78 <= 1 + z0 + z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> s66 :|: s66 >= 0, s66 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s67 :|: s67 >= 0, s67 <= 1 + (1 + z08) + (1 + 0 + z2), z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> s68 :|: s68 >= 0, s68 <= 1 + (1 + z09) + (1 + (1 + z15) + z2), s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> s69 :|: s69 >= 0, s69 <= 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 }-> ifrepl(s36, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s36 >= 0, s36 <= 2, z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(s37, 1 + 0 + 0) :|: s37 >= 0, s37 <= 2, z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(s38, 1 + (1 + (z - 2)) + 0) :|: s38 >= 0, s38 <= 2, z - 2 >= 0
   selsort(z) -{ 0 }-> ifselsort(s39, 1 + z0 + z1) :|: s39 >= 0, s39 <= 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(s73, 1 + z0 + (1 + z17 + z23)) :|: s72 >= 0, s72 <= 1 + z0 + (1 + z17 + z23), s73 >= 0, s73 <= 2, s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}
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]
  IFREPL: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  min: runtime: O(1) [0], size: O(n^1) [z]
  ifmin: runtime: O(1) [0], size: O(n^1) [z']
  IFMIN: 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]

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

(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
   IFMIN(z, z') -{ 300 + 171*z0 + 52*z0*z2 + 26*z0^2 + 171*z2 + 26*z2^2 }-> 1 + s83 :|: s83 >= 0, s83 <= 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
   IFMIN(z, z') -{ 300 + 171*z1 + 52*z1*z2 + 26*z1^2 + 171*z2 + 26*z2^2 }-> 1 + s84 :|: s84 >= 0, s84 <= 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
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s45 :|: s45 >= 0, s45 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 300 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + s93 :|: s93 >= 0, s93 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 517 + 223*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 232*z13 + 54*z13*z21 + 27*z13^2 + 232*z21 + 27*z21^2 }-> 1 + SELSORT(replace(s60, z0, 1 + z13 + z21)) + s62 + s97 :|: s97 >= 0, s97 <= 99 * (1 + z0 + (1 + z13 + z21)) + 26 * ((1 + z0 + (1 + z13 + z21)) * (1 + z0 + (1 + z13 + z21))) + 115, s60 >= 0, s60 <= 1 + z0 + (1 + z13 + z21), s61 >= 0, s61 <= 1 + z0 + (1 + z13 + z21), s62 >= 0, s62 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 517 + 223*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 232*z13 + 54*z13*z21 + 27*z13^2 + 232*z21 + 27*z21^2 }-> 1 + SELSORT(replace(s63, z0, 1 + z13 + z21)) + s49 + s98 :|: s98 >= 0, s98 <= 99 * (1 + z0 + (1 + z13 + z21)) + 26 * ((1 + z0 + (1 + z13 + z21)) * (1 + z0 + (1 + z13 + z21))) + 115, s63 >= 0, s63 <= 1 + z0 + (1 + z13 + z21), s49 >= 0, s49 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 312 + 171*z0 + 52*z0*z1 + 26*z0^2 + 178*z1 + 27*z1^2 }-> 1 + SELSORT(replace(0, z0, z1)) + s51 + s101 :|: s101 >= 0, s101 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s51 >= 0, s51 <= 7 * z1 + 6 + z1 * z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 517 + 223*z0 + 52*z0*z14 + 52*z0*z22 + 26*z0^2 + 232*z14 + 54*z14*z22 + 27*z14^2 + 232*z22 + 27*z22^2 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + s65 + s100 :|: s100 >= 0, s100 <= 99 * (1 + z0 + (1 + z14 + z22)) + 26 * ((1 + z0 + (1 + z14 + z22)) * (1 + z0 + (1 + z14 + z22))) + 115, s64 >= 0, s64 <= 1 + z0 + (1 + z14 + z22), s65 >= 0, s65 <= 7 * (1 + z14 + z22) + 6 + (1 + z14 + z22) * (1 + z14 + z22), s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 312 }-> 1 + SELSORT(replace(0, 0, 0)) + s46 + s94 :|: s94 >= 0, s94 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s46 >= 0, s46 <= 7 * 0 + 6 + 0 * 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 167 + 119*z' + 26*z'^2 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + s50 + s99 :|: s99 >= 0, s99 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s50 >= 0, s50 <= 7 * 0 + 6 + 0 * 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 167 + 119*z' + 26*z'^2 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s47 + s95 :|: s95 >= 0, s95 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s47 >= 0, s47 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 167 + 119*z' + 26*z'^2 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s48 + s96 :|: s96 >= 0, s96 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s48 >= 0, s48 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 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 + s23 :|: s23 >= 0, s23 <= 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 + s79 + s24 :|: s79 >= 0, s79 <= 8 + (1 + 0 + (1 + z1 + z2)) * (1 + 0 + (1 + z1 + z2)) + 4 * (1 + 0 + (1 + z1 + z2)), s24 >= 0, s24 <= 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 + s80 + s25 :|: s80 >= 0, s80 <= 8 + (1 + (1 + z0') + (1 + 0 + z2)) * (1 + (1 + z0') + (1 + 0 + z2)) + 4 * (1 + (1 + z0') + (1 + 0 + z2)), s25 >= 0, s25 <= 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 + s81 + s26 :|: s81 >= 0, s81 <= 8 + (1 + (1 + z0'') + (1 + (1 + z1') + z2)) * (1 + (1 + z0'') + (1 + (1 + z1') + z2)) + 4 * (1 + (1 + z0'') + (1 + (1 + z1') + z2)), s26 >= 0, s26 <= 1 + z1' + 1, s10 >= 0, s10 <= 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 + s82 + s27 :|: s82 >= 0, s82 <= 8 + (1 + z0 + (1 + z1 + z2)) * (1 + z0 + (1 + z1 + z2)) + 4 * (1 + z0 + (1 + z1 + z2)), s27 >= 0, s27 <= 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 + s40 + s' :|: s40 >= 0, s40 <= 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 + s41 + s'' :|: s41 >= 0, s41 <= 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 + s42 + s1 :|: s42 >= 0, s42 <= 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 + s43 + s2 :|: s43 >= 0, s43 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s28 >= 0, s28 <= 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 + s44 + s3 :|: s44 >= 0, s44 <= 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
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 303 }-> 1 + IFSELSORT(s29, 1 + 0 + 0) + s4 + s85 :|: s85 >= 0, s85 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s29 >= 0, s29 <= 2, s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 157 + 120*z + 26*z^2 }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s5 + s86 :|: s86 >= 0, s86 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s30 >= 0, s30 <= 2, s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 158 + 119*z + 26*z^2 }-> 1 + IFSELSORT(s31, 1 + (1 + (z - 2)) + 0) + s6 + s87 :|: s87 >= 0, s87 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s31 >= 0, s31 <= 2, s6 >= 0, s6 <= 0 + 2, z - 2 >= 0
   SELSORT(z) -{ 157 + 120*z + 26*z^2 }-> 1 + IFSELSORT(s32, 1 + (1 + (z - 2)) + 0) + s8 + s90 :|: s90 >= 0, s90 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s32 >= 0, s32 <= 2, s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 500 + s58 + 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 + IFSELSORT(s33, 1 + z0 + (1 + z12 + z2'')) + s59 + s91 :|: s91 >= 0, s91 <= 99 * (1 + z0 + (1 + z12 + z2'')) + 26 * ((1 + z0 + (1 + z12 + z2'')) * (1 + z0 + (1 + z12 + z2''))) + 115, s58 >= 0, s58 <= 1 + z0 + (1 + z12 + z2''), s59 >= 0, s59 <= s58 + 2, s33 >= 0, s33 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 303 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + IFSELSORT(s34, 1 + z0 + z1) + s9 + s92 :|: s92 >= 0, s92 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s34 >= 0, s34 <= 2, s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 500 + s54 + 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 + IFSELSORT(s53, 1 + z0 + (1 + z11 + z2')) + s55 + s88 :|: s88 >= 0, s88 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s52 >= 0, s52 <= 1 + z0 + (1 + z11 + z2'), s53 >= 0, s53 <= 2, s54 >= 0, s54 <= 1 + z0 + (1 + z11 + z2'), s55 >= 0, s55 <= s54 + 2, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 500 + 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 + IFSELSORT(s57, 1 + z0 + (1 + z11 + z2')) + s7 + s89 :|: s89 >= 0, s89 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s56 >= 0, s56 <= 1 + z0 + (1 + z11 + z2'), s57 >= 0, s57 <= 2, s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 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
   ifmin(z, z') -{ 0 }-> s70 :|: s70 >= 0, s70 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> s71 :|: s71 >= 0, s71 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s74 + selsort(replace(0, 0, 0)) :|: s74 >= 0, s74 <= 1 + 0 + 0, z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + s75 + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: s75 >= 0, s75 <= 1 + (1 + (z' - 2)) + 0, z = 1, z' - 2 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s76 + selsort(replace(s77, z0, 1 + z18 + z24)) :|: s76 >= 0, s76 <= 1 + z0 + (1 + z18 + z24), s77 >= 0, s77 <= 1 + z0 + (1 + z18 + z24), s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s78 + selsort(replace(0, z0, z1)) :|: s78 >= 0, s78 <= 1 + z0 + z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> s66 :|: s66 >= 0, s66 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s67 :|: s67 >= 0, s67 <= 1 + (1 + z08) + (1 + 0 + z2), z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> s68 :|: s68 >= 0, s68 <= 1 + (1 + z09) + (1 + (1 + z15) + z2), s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> s69 :|: s69 >= 0, s69 <= 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 }-> ifrepl(s36, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s36 >= 0, s36 <= 2, z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(s37, 1 + 0 + 0) :|: s37 >= 0, s37 <= 2, z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(s38, 1 + (1 + (z - 2)) + 0) :|: s38 >= 0, s38 <= 2, z - 2 >= 0
   selsort(z) -{ 0 }-> ifselsort(s39, 1 + z0 + z1) :|: s39 >= 0, s39 <= 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(s73, 1 + z0 + (1 + z17 + z23)) :|: s72 >= 0, s72 <= 1 + z0 + (1 + z17 + z23), s73 >= 0, s73 <= 2, s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}
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]
  IFREPL: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  min: runtime: O(1) [0], size: O(n^1) [z]
  ifmin: runtime: O(1) [0], size: O(n^1) [z']
  IFMIN: 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]

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

(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: ifrepl
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
   IFMIN(z, z') -{ 300 + 171*z0 + 52*z0*z2 + 26*z0^2 + 171*z2 + 26*z2^2 }-> 1 + s83 :|: s83 >= 0, s83 <= 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
   IFMIN(z, z') -{ 300 + 171*z1 + 52*z1*z2 + 26*z1^2 + 171*z2 + 26*z2^2 }-> 1 + s84 :|: s84 >= 0, s84 <= 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
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s45 :|: s45 >= 0, s45 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 300 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + s93 :|: s93 >= 0, s93 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 517 + 223*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 232*z13 + 54*z13*z21 + 27*z13^2 + 232*z21 + 27*z21^2 }-> 1 + SELSORT(replace(s60, z0, 1 + z13 + z21)) + s62 + s97 :|: s97 >= 0, s97 <= 99 * (1 + z0 + (1 + z13 + z21)) + 26 * ((1 + z0 + (1 + z13 + z21)) * (1 + z0 + (1 + z13 + z21))) + 115, s60 >= 0, s60 <= 1 + z0 + (1 + z13 + z21), s61 >= 0, s61 <= 1 + z0 + (1 + z13 + z21), s62 >= 0, s62 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 517 + 223*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 232*z13 + 54*z13*z21 + 27*z13^2 + 232*z21 + 27*z21^2 }-> 1 + SELSORT(replace(s63, z0, 1 + z13 + z21)) + s49 + s98 :|: s98 >= 0, s98 <= 99 * (1 + z0 + (1 + z13 + z21)) + 26 * ((1 + z0 + (1 + z13 + z21)) * (1 + z0 + (1 + z13 + z21))) + 115, s63 >= 0, s63 <= 1 + z0 + (1 + z13 + z21), s49 >= 0, s49 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 312 + 171*z0 + 52*z0*z1 + 26*z0^2 + 178*z1 + 27*z1^2 }-> 1 + SELSORT(replace(0, z0, z1)) + s51 + s101 :|: s101 >= 0, s101 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s51 >= 0, s51 <= 7 * z1 + 6 + z1 * z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 517 + 223*z0 + 52*z0*z14 + 52*z0*z22 + 26*z0^2 + 232*z14 + 54*z14*z22 + 27*z14^2 + 232*z22 + 27*z22^2 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + s65 + s100 :|: s100 >= 0, s100 <= 99 * (1 + z0 + (1 + z14 + z22)) + 26 * ((1 + z0 + (1 + z14 + z22)) * (1 + z0 + (1 + z14 + z22))) + 115, s64 >= 0, s64 <= 1 + z0 + (1 + z14 + z22), s65 >= 0, s65 <= 7 * (1 + z14 + z22) + 6 + (1 + z14 + z22) * (1 + z14 + z22), s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 312 }-> 1 + SELSORT(replace(0, 0, 0)) + s46 + s94 :|: s94 >= 0, s94 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s46 >= 0, s46 <= 7 * 0 + 6 + 0 * 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 167 + 119*z' + 26*z'^2 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + s50 + s99 :|: s99 >= 0, s99 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s50 >= 0, s50 <= 7 * 0 + 6 + 0 * 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 167 + 119*z' + 26*z'^2 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s47 + s95 :|: s95 >= 0, s95 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s47 >= 0, s47 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 167 + 119*z' + 26*z'^2 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s48 + s96 :|: s96 >= 0, s96 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s48 >= 0, s48 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 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 + s23 :|: s23 >= 0, s23 <= 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 + s79 + s24 :|: s79 >= 0, s79 <= 8 + (1 + 0 + (1 + z1 + z2)) * (1 + 0 + (1 + z1 + z2)) + 4 * (1 + 0 + (1 + z1 + z2)), s24 >= 0, s24 <= 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 + s80 + s25 :|: s80 >= 0, s80 <= 8 + (1 + (1 + z0') + (1 + 0 + z2)) * (1 + (1 + z0') + (1 + 0 + z2)) + 4 * (1 + (1 + z0') + (1 + 0 + z2)), s25 >= 0, s25 <= 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 + s81 + s26 :|: s81 >= 0, s81 <= 8 + (1 + (1 + z0'') + (1 + (1 + z1') + z2)) * (1 + (1 + z0'') + (1 + (1 + z1') + z2)) + 4 * (1 + (1 + z0'') + (1 + (1 + z1') + z2)), s26 >= 0, s26 <= 1 + z1' + 1, s10 >= 0, s10 <= 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 + s82 + s27 :|: s82 >= 0, s82 <= 8 + (1 + z0 + (1 + z1 + z2)) * (1 + z0 + (1 + z1 + z2)) + 4 * (1 + z0 + (1 + z1 + z2)), s27 >= 0, s27 <= 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 + s40 + s' :|: s40 >= 0, s40 <= 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 + s41 + s'' :|: s41 >= 0, s41 <= 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 + s42 + s1 :|: s42 >= 0, s42 <= 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 + s43 + s2 :|: s43 >= 0, s43 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s28 >= 0, s28 <= 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 + s44 + s3 :|: s44 >= 0, s44 <= 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
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 303 }-> 1 + IFSELSORT(s29, 1 + 0 + 0) + s4 + s85 :|: s85 >= 0, s85 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s29 >= 0, s29 <= 2, s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 157 + 120*z + 26*z^2 }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s5 + s86 :|: s86 >= 0, s86 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s30 >= 0, s30 <= 2, s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 158 + 119*z + 26*z^2 }-> 1 + IFSELSORT(s31, 1 + (1 + (z - 2)) + 0) + s6 + s87 :|: s87 >= 0, s87 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s31 >= 0, s31 <= 2, s6 >= 0, s6 <= 0 + 2, z - 2 >= 0
   SELSORT(z) -{ 157 + 120*z + 26*z^2 }-> 1 + IFSELSORT(s32, 1 + (1 + (z - 2)) + 0) + s8 + s90 :|: s90 >= 0, s90 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s32 >= 0, s32 <= 2, s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 500 + s58 + 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 + IFSELSORT(s33, 1 + z0 + (1 + z12 + z2'')) + s59 + s91 :|: s91 >= 0, s91 <= 99 * (1 + z0 + (1 + z12 + z2'')) + 26 * ((1 + z0 + (1 + z12 + z2'')) * (1 + z0 + (1 + z12 + z2''))) + 115, s58 >= 0, s58 <= 1 + z0 + (1 + z12 + z2''), s59 >= 0, s59 <= s58 + 2, s33 >= 0, s33 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 303 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + IFSELSORT(s34, 1 + z0 + z1) + s9 + s92 :|: s92 >= 0, s92 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s34 >= 0, s34 <= 2, s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 500 + s54 + 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 + IFSELSORT(s53, 1 + z0 + (1 + z11 + z2')) + s55 + s88 :|: s88 >= 0, s88 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s52 >= 0, s52 <= 1 + z0 + (1 + z11 + z2'), s53 >= 0, s53 <= 2, s54 >= 0, s54 <= 1 + z0 + (1 + z11 + z2'), s55 >= 0, s55 <= s54 + 2, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 500 + 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 + IFSELSORT(s57, 1 + z0 + (1 + z11 + z2')) + s7 + s89 :|: s89 >= 0, s89 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s56 >= 0, s56 <= 1 + z0 + (1 + z11 + z2'), s57 >= 0, s57 <= 2, s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 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
   ifmin(z, z') -{ 0 }-> s70 :|: s70 >= 0, s70 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> s71 :|: s71 >= 0, s71 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s74 + selsort(replace(0, 0, 0)) :|: s74 >= 0, s74 <= 1 + 0 + 0, z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + s75 + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: s75 >= 0, s75 <= 1 + (1 + (z' - 2)) + 0, z = 1, z' - 2 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s76 + selsort(replace(s77, z0, 1 + z18 + z24)) :|: s76 >= 0, s76 <= 1 + z0 + (1 + z18 + z24), s77 >= 0, s77 <= 1 + z0 + (1 + z18 + z24), s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s78 + selsort(replace(0, z0, z1)) :|: s78 >= 0, s78 <= 1 + z0 + z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> s66 :|: s66 >= 0, s66 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s67 :|: s67 >= 0, s67 <= 1 + (1 + z08) + (1 + 0 + z2), z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> s68 :|: s68 >= 0, s68 <= 1 + (1 + z09) + (1 + (1 + z15) + z2), s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> s69 :|: s69 >= 0, s69 <= 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 }-> ifrepl(s36, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s36 >= 0, s36 <= 2, z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(s37, 1 + 0 + 0) :|: s37 >= 0, s37 <= 2, z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(s38, 1 + (1 + (z - 2)) + 0) :|: s38 >= 0, s38 <= 2, z - 2 >= 0
   selsort(z) -{ 0 }-> ifselsort(s39, 1 + z0 + z1) :|: s39 >= 0, s39 <= 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(s73, 1 + z0 + (1 + z17 + z23)) :|: s72 >= 0, s72 <= 1 + z0 + (1 + z17 + z23), s73 >= 0, s73 <= 2, s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {replace,ifrepl}, {IFSELSORT,SELSORT}, {ifselsort,selsort}
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]
  IFREPL: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  min: runtime: O(1) [0], size: O(n^1) [z]
  ifmin: runtime: O(1) [0], size: O(n^1) [z']
  IFMIN: 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]
  replace: runtime: ?, size: O(n^1) [z' + z'']
  ifrepl: 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: ifrepl
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
   IFMIN(z, z') -{ 300 + 171*z0 + 52*z0*z2 + 26*z0^2 + 171*z2 + 26*z2^2 }-> 1 + s83 :|: s83 >= 0, s83 <= 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
   IFMIN(z, z') -{ 300 + 171*z1 + 52*z1*z2 + 26*z1^2 + 171*z2 + 26*z2^2 }-> 1 + s84 :|: s84 >= 0, s84 <= 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
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s45 :|: s45 >= 0, s45 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 300 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + s93 :|: s93 >= 0, s93 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 517 + 223*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 232*z13 + 54*z13*z21 + 27*z13^2 + 232*z21 + 27*z21^2 }-> 1 + SELSORT(replace(s60, z0, 1 + z13 + z21)) + s62 + s97 :|: s97 >= 0, s97 <= 99 * (1 + z0 + (1 + z13 + z21)) + 26 * ((1 + z0 + (1 + z13 + z21)) * (1 + z0 + (1 + z13 + z21))) + 115, s60 >= 0, s60 <= 1 + z0 + (1 + z13 + z21), s61 >= 0, s61 <= 1 + z0 + (1 + z13 + z21), s62 >= 0, s62 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 517 + 223*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 232*z13 + 54*z13*z21 + 27*z13^2 + 232*z21 + 27*z21^2 }-> 1 + SELSORT(replace(s63, z0, 1 + z13 + z21)) + s49 + s98 :|: s98 >= 0, s98 <= 99 * (1 + z0 + (1 + z13 + z21)) + 26 * ((1 + z0 + (1 + z13 + z21)) * (1 + z0 + (1 + z13 + z21))) + 115, s63 >= 0, s63 <= 1 + z0 + (1 + z13 + z21), s49 >= 0, s49 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 312 + 171*z0 + 52*z0*z1 + 26*z0^2 + 178*z1 + 27*z1^2 }-> 1 + SELSORT(replace(0, z0, z1)) + s51 + s101 :|: s101 >= 0, s101 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s51 >= 0, s51 <= 7 * z1 + 6 + z1 * z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 517 + 223*z0 + 52*z0*z14 + 52*z0*z22 + 26*z0^2 + 232*z14 + 54*z14*z22 + 27*z14^2 + 232*z22 + 27*z22^2 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + s65 + s100 :|: s100 >= 0, s100 <= 99 * (1 + z0 + (1 + z14 + z22)) + 26 * ((1 + z0 + (1 + z14 + z22)) * (1 + z0 + (1 + z14 + z22))) + 115, s64 >= 0, s64 <= 1 + z0 + (1 + z14 + z22), s65 >= 0, s65 <= 7 * (1 + z14 + z22) + 6 + (1 + z14 + z22) * (1 + z14 + z22), s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 312 }-> 1 + SELSORT(replace(0, 0, 0)) + s46 + s94 :|: s94 >= 0, s94 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s46 >= 0, s46 <= 7 * 0 + 6 + 0 * 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 167 + 119*z' + 26*z'^2 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + s50 + s99 :|: s99 >= 0, s99 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s50 >= 0, s50 <= 7 * 0 + 6 + 0 * 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 167 + 119*z' + 26*z'^2 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s47 + s95 :|: s95 >= 0, s95 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s47 >= 0, s47 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 167 + 119*z' + 26*z'^2 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s48 + s96 :|: s96 >= 0, s96 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s48 >= 0, s48 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 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 + s23 :|: s23 >= 0, s23 <= 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 + s79 + s24 :|: s79 >= 0, s79 <= 8 + (1 + 0 + (1 + z1 + z2)) * (1 + 0 + (1 + z1 + z2)) + 4 * (1 + 0 + (1 + z1 + z2)), s24 >= 0, s24 <= 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 + s80 + s25 :|: s80 >= 0, s80 <= 8 + (1 + (1 + z0') + (1 + 0 + z2)) * (1 + (1 + z0') + (1 + 0 + z2)) + 4 * (1 + (1 + z0') + (1 + 0 + z2)), s25 >= 0, s25 <= 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 + s81 + s26 :|: s81 >= 0, s81 <= 8 + (1 + (1 + z0'') + (1 + (1 + z1') + z2)) * (1 + (1 + z0'') + (1 + (1 + z1') + z2)) + 4 * (1 + (1 + z0'') + (1 + (1 + z1') + z2)), s26 >= 0, s26 <= 1 + z1' + 1, s10 >= 0, s10 <= 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 + s82 + s27 :|: s82 >= 0, s82 <= 8 + (1 + z0 + (1 + z1 + z2)) * (1 + z0 + (1 + z1 + z2)) + 4 * (1 + z0 + (1 + z1 + z2)), s27 >= 0, s27 <= 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 + s40 + s' :|: s40 >= 0, s40 <= 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 + s41 + s'' :|: s41 >= 0, s41 <= 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 + s42 + s1 :|: s42 >= 0, s42 <= 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 + s43 + s2 :|: s43 >= 0, s43 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s28 >= 0, s28 <= 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 + s44 + s3 :|: s44 >= 0, s44 <= 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
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 303 }-> 1 + IFSELSORT(s29, 1 + 0 + 0) + s4 + s85 :|: s85 >= 0, s85 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s29 >= 0, s29 <= 2, s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 157 + 120*z + 26*z^2 }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s5 + s86 :|: s86 >= 0, s86 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s30 >= 0, s30 <= 2, s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 158 + 119*z + 26*z^2 }-> 1 + IFSELSORT(s31, 1 + (1 + (z - 2)) + 0) + s6 + s87 :|: s87 >= 0, s87 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s31 >= 0, s31 <= 2, s6 >= 0, s6 <= 0 + 2, z - 2 >= 0
   SELSORT(z) -{ 157 + 120*z + 26*z^2 }-> 1 + IFSELSORT(s32, 1 + (1 + (z - 2)) + 0) + s8 + s90 :|: s90 >= 0, s90 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s32 >= 0, s32 <= 2, s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 500 + s58 + 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 + IFSELSORT(s33, 1 + z0 + (1 + z12 + z2'')) + s59 + s91 :|: s91 >= 0, s91 <= 99 * (1 + z0 + (1 + z12 + z2'')) + 26 * ((1 + z0 + (1 + z12 + z2'')) * (1 + z0 + (1 + z12 + z2''))) + 115, s58 >= 0, s58 <= 1 + z0 + (1 + z12 + z2''), s59 >= 0, s59 <= s58 + 2, s33 >= 0, s33 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 303 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + IFSELSORT(s34, 1 + z0 + z1) + s9 + s92 :|: s92 >= 0, s92 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s34 >= 0, s34 <= 2, s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 500 + s54 + 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 + IFSELSORT(s53, 1 + z0 + (1 + z11 + z2')) + s55 + s88 :|: s88 >= 0, s88 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s52 >= 0, s52 <= 1 + z0 + (1 + z11 + z2'), s53 >= 0, s53 <= 2, s54 >= 0, s54 <= 1 + z0 + (1 + z11 + z2'), s55 >= 0, s55 <= s54 + 2, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 500 + 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 + IFSELSORT(s57, 1 + z0 + (1 + z11 + z2')) + s7 + s89 :|: s89 >= 0, s89 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s56 >= 0, s56 <= 1 + z0 + (1 + z11 + z2'), s57 >= 0, s57 <= 2, s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 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
   ifmin(z, z') -{ 0 }-> s70 :|: s70 >= 0, s70 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> s71 :|: s71 >= 0, s71 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(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
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s74 + selsort(replace(0, 0, 0)) :|: s74 >= 0, s74 <= 1 + 0 + 0, z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + s75 + selsort(replace(1 + (z' - 2), 1 + (z' - 2), 0)) :|: s75 >= 0, s75 <= 1 + (1 + (z' - 2)) + 0, z = 1, z' - 2 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s76 + selsort(replace(s77, z0, 1 + z18 + z24)) :|: s76 >= 0, s76 <= 1 + z0 + (1 + z18 + z24), s77 >= 0, s77 <= 1 + z0 + (1 + z18 + z24), s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s78 + selsort(replace(0, z0, z1)) :|: s78 >= 0, s78 <= 1 + z0 + z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> s66 :|: s66 >= 0, s66 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s67 :|: s67 >= 0, s67 <= 1 + (1 + z08) + (1 + 0 + z2), z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> s68 :|: s68 >= 0, s68 <= 1 + (1 + z09) + (1 + (1 + z15) + z2), s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> s69 :|: s69 >= 0, s69 <= 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 }-> ifrepl(s36, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s36 >= 0, s36 <= 2, z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 0, z', 1 + (1 + z010) + z3) :|: z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> ifrepl(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
   selsort(z) -{ 0 }-> ifselsort(s37, 1 + 0 + 0) :|: s37 >= 0, s37 <= 2, z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(s38, 1 + (1 + (z - 2)) + 0) :|: s38 >= 0, s38 <= 2, z - 2 >= 0
   selsort(z) -{ 0 }-> ifselsort(s39, 1 + z0 + z1) :|: s39 >= 0, s39 <= 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(s73, 1 + z0 + (1 + z17 + z23)) :|: s72 >= 0, s72 <= 1 + z0 + (1 + z17 + z23), s73 >= 0, s73 <= 2, s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {IFSELSORT,SELSORT}, {ifselsort,selsort}
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]
  IFREPL: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  min: runtime: O(1) [0], size: O(n^1) [z]
  ifmin: runtime: O(1) [0], size: O(n^1) [z']
  IFMIN: 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]
  replace: runtime: O(1) [0], size: O(n^1) [z' + z'']
  ifrepl: 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
   IFMIN(z, z') -{ 300 + 171*z0 + 52*z0*z2 + 26*z0^2 + 171*z2 + 26*z2^2 }-> 1 + s83 :|: s83 >= 0, s83 <= 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
   IFMIN(z, z') -{ 300 + 171*z1 + 52*z1*z2 + 26*z1^2 + 171*z2 + 26*z2^2 }-> 1 + s84 :|: s84 >= 0, s84 <= 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
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s45 :|: s45 >= 0, s45 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 300 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + s93 :|: s93 >= 0, s93 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 312 }-> 1 + SELSORT(s102) + s46 + s94 :|: s102 >= 0, s102 <= 0 + 0, s94 >= 0, s94 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s46 >= 0, s46 <= 7 * 0 + 6 + 0 * 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 167 + 119*z' + 26*z'^2 }-> 1 + SELSORT(s103) + s47 + s95 :|: s103 >= 0, s103 <= 1 + (z' - 2) + 0, s95 >= 0, s95 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s47 >= 0, s47 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 167 + 119*z' + 26*z'^2 }-> 1 + SELSORT(s104) + s48 + s96 :|: s104 >= 0, s104 <= 1 + (z' - 2) + 0, s96 >= 0, s96 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s48 >= 0, s48 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 517 + 223*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 232*z13 + 54*z13*z21 + 27*z13^2 + 232*z21 + 27*z21^2 }-> 1 + SELSORT(s105) + s62 + s97 :|: s105 >= 0, s105 <= z0 + (1 + z13 + z21), s97 >= 0, s97 <= 99 * (1 + z0 + (1 + z13 + z21)) + 26 * ((1 + z0 + (1 + z13 + z21)) * (1 + z0 + (1 + z13 + z21))) + 115, s60 >= 0, s60 <= 1 + z0 + (1 + z13 + z21), s61 >= 0, s61 <= 1 + z0 + (1 + z13 + z21), s62 >= 0, s62 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 517 + 223*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 232*z13 + 54*z13*z21 + 27*z13^2 + 232*z21 + 27*z21^2 }-> 1 + SELSORT(s106) + s49 + s98 :|: s106 >= 0, s106 <= z0 + (1 + z13 + z21), s98 >= 0, s98 <= 99 * (1 + z0 + (1 + z13 + z21)) + 26 * ((1 + z0 + (1 + z13 + z21)) * (1 + z0 + (1 + z13 + z21))) + 115, s63 >= 0, s63 <= 1 + z0 + (1 + z13 + z21), s49 >= 0, s49 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 167 + 119*z' + 26*z'^2 }-> 1 + SELSORT(s107) + s50 + s99 :|: s107 >= 0, s107 <= 1 + (z' - 2) + 0, s99 >= 0, s99 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s50 >= 0, s50 <= 7 * 0 + 6 + 0 * 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 517 + 223*z0 + 52*z0*z14 + 52*z0*z22 + 26*z0^2 + 232*z14 + 54*z14*z22 + 27*z14^2 + 232*z22 + 27*z22^2 }-> 1 + SELSORT(s108) + s65 + s100 :|: s108 >= 0, s108 <= z0 + (1 + z14 + z22), s100 >= 0, s100 <= 99 * (1 + z0 + (1 + z14 + z22)) + 26 * ((1 + z0 + (1 + z14 + z22)) * (1 + z0 + (1 + z14 + z22))) + 115, s64 >= 0, s64 <= 1 + z0 + (1 + z14 + z22), s65 >= 0, s65 <= 7 * (1 + z14 + z22) + 6 + (1 + z14 + z22) * (1 + z14 + z22), s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 312 + 171*z0 + 52*z0*z1 + 26*z0^2 + 178*z1 + 27*z1^2 }-> 1 + SELSORT(s109) + s51 + s101 :|: s109 >= 0, s109 <= z0 + z1, s101 >= 0, s101 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s51 >= 0, s51 <= 7 * z1 + 6 + z1 * z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 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 + s23 :|: s23 >= 0, s23 <= 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 + s79 + s24 :|: s79 >= 0, s79 <= 8 + (1 + 0 + (1 + z1 + z2)) * (1 + 0 + (1 + z1 + z2)) + 4 * (1 + 0 + (1 + z1 + z2)), s24 >= 0, s24 <= 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 + s80 + s25 :|: s80 >= 0, s80 <= 8 + (1 + (1 + z0') + (1 + 0 + z2)) * (1 + (1 + z0') + (1 + 0 + z2)) + 4 * (1 + (1 + z0') + (1 + 0 + z2)), s25 >= 0, s25 <= 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 + s81 + s26 :|: s81 >= 0, s81 <= 8 + (1 + (1 + z0'') + (1 + (1 + z1') + z2)) * (1 + (1 + z0'') + (1 + (1 + z1') + z2)) + 4 * (1 + (1 + z0'') + (1 + (1 + z1') + z2)), s26 >= 0, s26 <= 1 + z1' + 1, s10 >= 0, s10 <= 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 + s82 + s27 :|: s82 >= 0, s82 <= 8 + (1 + z0 + (1 + z1 + z2)) * (1 + z0 + (1 + z1 + z2)) + 4 * (1 + z0 + (1 + z1 + z2)), s27 >= 0, s27 <= 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 + s40 + s' :|: s40 >= 0, s40 <= 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 + s41 + s'' :|: s41 >= 0, s41 <= 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 + s42 + s1 :|: s42 >= 0, s42 <= 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 + s43 + s2 :|: s43 >= 0, s43 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s28 >= 0, s28 <= 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 + s44 + s3 :|: s44 >= 0, s44 <= 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
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 303 }-> 1 + IFSELSORT(s29, 1 + 0 + 0) + s4 + s85 :|: s85 >= 0, s85 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s29 >= 0, s29 <= 2, s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 157 + 120*z + 26*z^2 }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s5 + s86 :|: s86 >= 0, s86 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s30 >= 0, s30 <= 2, s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 158 + 119*z + 26*z^2 }-> 1 + IFSELSORT(s31, 1 + (1 + (z - 2)) + 0) + s6 + s87 :|: s87 >= 0, s87 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s31 >= 0, s31 <= 2, s6 >= 0, s6 <= 0 + 2, z - 2 >= 0
   SELSORT(z) -{ 157 + 120*z + 26*z^2 }-> 1 + IFSELSORT(s32, 1 + (1 + (z - 2)) + 0) + s8 + s90 :|: s90 >= 0, s90 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s32 >= 0, s32 <= 2, s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 500 + s58 + 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 + IFSELSORT(s33, 1 + z0 + (1 + z12 + z2'')) + s59 + s91 :|: s91 >= 0, s91 <= 99 * (1 + z0 + (1 + z12 + z2'')) + 26 * ((1 + z0 + (1 + z12 + z2'')) * (1 + z0 + (1 + z12 + z2''))) + 115, s58 >= 0, s58 <= 1 + z0 + (1 + z12 + z2''), s59 >= 0, s59 <= s58 + 2, s33 >= 0, s33 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 303 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + IFSELSORT(s34, 1 + z0 + z1) + s9 + s92 :|: s92 >= 0, s92 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s34 >= 0, s34 <= 2, s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 500 + s54 + 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 + IFSELSORT(s53, 1 + z0 + (1 + z11 + z2')) + s55 + s88 :|: s88 >= 0, s88 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s52 >= 0, s52 <= 1 + z0 + (1 + z11 + z2'), s53 >= 0, s53 <= 2, s54 >= 0, s54 <= 1 + z0 + (1 + z11 + z2'), s55 >= 0, s55 <= s54 + 2, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 500 + 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 + IFSELSORT(s57, 1 + z0 + (1 + z11 + z2')) + s7 + s89 :|: s89 >= 0, s89 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s56 >= 0, s56 <= 1 + z0 + (1 + z11 + z2'), s57 >= 0, s57 <= 2, s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 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
   ifmin(z, z') -{ 0 }-> s70 :|: s70 >= 0, s70 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> s71 :|: s71 >= 0, s71 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z2 + s115 :|: s115 >= 0, s115 <= z'' + z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s74 + selsort(s116) :|: s116 >= 0, s116 <= 0 + 0, s74 >= 0, s74 <= 1 + 0 + 0, z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + s75 + selsort(s117) :|: s117 >= 0, s117 <= 1 + (z' - 2) + 0, s75 >= 0, s75 <= 1 + (1 + (z' - 2)) + 0, z = 1, z' - 2 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s76 + selsort(s118) :|: s118 >= 0, s118 <= z0 + (1 + z18 + z24), s76 >= 0, s76 <= 1 + z0 + (1 + z18 + z24), s77 >= 0, s77 <= 1 + z0 + (1 + z18 + z24), s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s78 + selsort(s119) :|: s119 >= 0, s119 <= z0 + z1, s78 >= 0, s78 <= 1 + z0 + z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> s66 :|: s66 >= 0, s66 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s67 :|: s67 >= 0, s67 <= 1 + (1 + z08) + (1 + 0 + z2), z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> s68 :|: s68 >= 0, s68 <= 1 + (1 + z09) + (1 + (1 + z15) + z2), s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> s69 :|: s69 >= 0, s69 <= 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 }-> s110 :|: s110 >= 0, s110 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s111 :|: s111 >= 0, s111 <= z' + (1 + (1 + z010) + z3), z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s112 :|: s112 >= 0, s112 <= z' + (1 + 0 + (z'' - 1)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s113 :|: s113 >= 0, s113 <= z' + (1 + (1 + z16) + z3), s36 >= 0, s36 <= 2, z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s114 :|: s114 >= 0, s114 <= 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
   selsort(z) -{ 0 }-> ifselsort(s37, 1 + 0 + 0) :|: s37 >= 0, s37 <= 2, z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(s38, 1 + (1 + (z - 2)) + 0) :|: s38 >= 0, s38 <= 2, z - 2 >= 0
   selsort(z) -{ 0 }-> ifselsort(s39, 1 + z0 + z1) :|: s39 >= 0, s39 <= 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(s73, 1 + z0 + (1 + z17 + z23)) :|: s72 >= 0, s72 <= 1 + z0 + (1 + z17 + z23), s73 >= 0, s73 <= 2, s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {IFSELSORT,SELSORT}, {ifselsort,selsort}
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]
  IFREPL: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  min: runtime: O(1) [0], size: O(n^1) [z]
  ifmin: runtime: O(1) [0], size: O(n^1) [z']
  IFMIN: 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]
  replace: runtime: O(1) [0], size: O(n^1) [z' + z'']
  ifrepl: runtime: O(1) [0], size: O(n^1) [z'' + z4]

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

(67) IntTrsBoundProof (UPPER BOUND(ID))

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

Computed SIZE bound using CoFloCo for: SELSORT
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
   IFMIN(z, z') -{ 300 + 171*z0 + 52*z0*z2 + 26*z0^2 + 171*z2 + 26*z2^2 }-> 1 + s83 :|: s83 >= 0, s83 <= 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
   IFMIN(z, z') -{ 300 + 171*z1 + 52*z1*z2 + 26*z1^2 + 171*z2 + 26*z2^2 }-> 1 + s84 :|: s84 >= 0, s84 <= 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
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s45 :|: s45 >= 0, s45 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 300 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + s93 :|: s93 >= 0, s93 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 312 }-> 1 + SELSORT(s102) + s46 + s94 :|: s102 >= 0, s102 <= 0 + 0, s94 >= 0, s94 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s46 >= 0, s46 <= 7 * 0 + 6 + 0 * 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 167 + 119*z' + 26*z'^2 }-> 1 + SELSORT(s103) + s47 + s95 :|: s103 >= 0, s103 <= 1 + (z' - 2) + 0, s95 >= 0, s95 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s47 >= 0, s47 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 167 + 119*z' + 26*z'^2 }-> 1 + SELSORT(s104) + s48 + s96 :|: s104 >= 0, s104 <= 1 + (z' - 2) + 0, s96 >= 0, s96 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s48 >= 0, s48 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 517 + 223*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 232*z13 + 54*z13*z21 + 27*z13^2 + 232*z21 + 27*z21^2 }-> 1 + SELSORT(s105) + s62 + s97 :|: s105 >= 0, s105 <= z0 + (1 + z13 + z21), s97 >= 0, s97 <= 99 * (1 + z0 + (1 + z13 + z21)) + 26 * ((1 + z0 + (1 + z13 + z21)) * (1 + z0 + (1 + z13 + z21))) + 115, s60 >= 0, s60 <= 1 + z0 + (1 + z13 + z21), s61 >= 0, s61 <= 1 + z0 + (1 + z13 + z21), s62 >= 0, s62 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 517 + 223*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 232*z13 + 54*z13*z21 + 27*z13^2 + 232*z21 + 27*z21^2 }-> 1 + SELSORT(s106) + s49 + s98 :|: s106 >= 0, s106 <= z0 + (1 + z13 + z21), s98 >= 0, s98 <= 99 * (1 + z0 + (1 + z13 + z21)) + 26 * ((1 + z0 + (1 + z13 + z21)) * (1 + z0 + (1 + z13 + z21))) + 115, s63 >= 0, s63 <= 1 + z0 + (1 + z13 + z21), s49 >= 0, s49 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 167 + 119*z' + 26*z'^2 }-> 1 + SELSORT(s107) + s50 + s99 :|: s107 >= 0, s107 <= 1 + (z' - 2) + 0, s99 >= 0, s99 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s50 >= 0, s50 <= 7 * 0 + 6 + 0 * 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 517 + 223*z0 + 52*z0*z14 + 52*z0*z22 + 26*z0^2 + 232*z14 + 54*z14*z22 + 27*z14^2 + 232*z22 + 27*z22^2 }-> 1 + SELSORT(s108) + s65 + s100 :|: s108 >= 0, s108 <= z0 + (1 + z14 + z22), s100 >= 0, s100 <= 99 * (1 + z0 + (1 + z14 + z22)) + 26 * ((1 + z0 + (1 + z14 + z22)) * (1 + z0 + (1 + z14 + z22))) + 115, s64 >= 0, s64 <= 1 + z0 + (1 + z14 + z22), s65 >= 0, s65 <= 7 * (1 + z14 + z22) + 6 + (1 + z14 + z22) * (1 + z14 + z22), s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 312 + 171*z0 + 52*z0*z1 + 26*z0^2 + 178*z1 + 27*z1^2 }-> 1 + SELSORT(s109) + s51 + s101 :|: s109 >= 0, s109 <= z0 + z1, s101 >= 0, s101 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s51 >= 0, s51 <= 7 * z1 + 6 + z1 * z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 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 + s23 :|: s23 >= 0, s23 <= 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 + s79 + s24 :|: s79 >= 0, s79 <= 8 + (1 + 0 + (1 + z1 + z2)) * (1 + 0 + (1 + z1 + z2)) + 4 * (1 + 0 + (1 + z1 + z2)), s24 >= 0, s24 <= 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 + s80 + s25 :|: s80 >= 0, s80 <= 8 + (1 + (1 + z0') + (1 + 0 + z2)) * (1 + (1 + z0') + (1 + 0 + z2)) + 4 * (1 + (1 + z0') + (1 + 0 + z2)), s25 >= 0, s25 <= 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 + s81 + s26 :|: s81 >= 0, s81 <= 8 + (1 + (1 + z0'') + (1 + (1 + z1') + z2)) * (1 + (1 + z0'') + (1 + (1 + z1') + z2)) + 4 * (1 + (1 + z0'') + (1 + (1 + z1') + z2)), s26 >= 0, s26 <= 1 + z1' + 1, s10 >= 0, s10 <= 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 + s82 + s27 :|: s82 >= 0, s82 <= 8 + (1 + z0 + (1 + z1 + z2)) * (1 + z0 + (1 + z1 + z2)) + 4 * (1 + z0 + (1 + z1 + z2)), s27 >= 0, s27 <= 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 + s40 + s' :|: s40 >= 0, s40 <= 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 + s41 + s'' :|: s41 >= 0, s41 <= 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 + s42 + s1 :|: s42 >= 0, s42 <= 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 + s43 + s2 :|: s43 >= 0, s43 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s28 >= 0, s28 <= 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 + s44 + s3 :|: s44 >= 0, s44 <= 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
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 303 }-> 1 + IFSELSORT(s29, 1 + 0 + 0) + s4 + s85 :|: s85 >= 0, s85 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s29 >= 0, s29 <= 2, s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 157 + 120*z + 26*z^2 }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s5 + s86 :|: s86 >= 0, s86 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s30 >= 0, s30 <= 2, s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 158 + 119*z + 26*z^2 }-> 1 + IFSELSORT(s31, 1 + (1 + (z - 2)) + 0) + s6 + s87 :|: s87 >= 0, s87 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s31 >= 0, s31 <= 2, s6 >= 0, s6 <= 0 + 2, z - 2 >= 0
   SELSORT(z) -{ 157 + 120*z + 26*z^2 }-> 1 + IFSELSORT(s32, 1 + (1 + (z - 2)) + 0) + s8 + s90 :|: s90 >= 0, s90 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s32 >= 0, s32 <= 2, s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 500 + s58 + 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 + IFSELSORT(s33, 1 + z0 + (1 + z12 + z2'')) + s59 + s91 :|: s91 >= 0, s91 <= 99 * (1 + z0 + (1 + z12 + z2'')) + 26 * ((1 + z0 + (1 + z12 + z2'')) * (1 + z0 + (1 + z12 + z2''))) + 115, s58 >= 0, s58 <= 1 + z0 + (1 + z12 + z2''), s59 >= 0, s59 <= s58 + 2, s33 >= 0, s33 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 303 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + IFSELSORT(s34, 1 + z0 + z1) + s9 + s92 :|: s92 >= 0, s92 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s34 >= 0, s34 <= 2, s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 500 + s54 + 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 + IFSELSORT(s53, 1 + z0 + (1 + z11 + z2')) + s55 + s88 :|: s88 >= 0, s88 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s52 >= 0, s52 <= 1 + z0 + (1 + z11 + z2'), s53 >= 0, s53 <= 2, s54 >= 0, s54 <= 1 + z0 + (1 + z11 + z2'), s55 >= 0, s55 <= s54 + 2, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 500 + 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 + IFSELSORT(s57, 1 + z0 + (1 + z11 + z2')) + s7 + s89 :|: s89 >= 0, s89 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s56 >= 0, s56 <= 1 + z0 + (1 + z11 + z2'), s57 >= 0, s57 <= 2, s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 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
   ifmin(z, z') -{ 0 }-> s70 :|: s70 >= 0, s70 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> s71 :|: s71 >= 0, s71 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z2 + s115 :|: s115 >= 0, s115 <= z'' + z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s74 + selsort(s116) :|: s116 >= 0, s116 <= 0 + 0, s74 >= 0, s74 <= 1 + 0 + 0, z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + s75 + selsort(s117) :|: s117 >= 0, s117 <= 1 + (z' - 2) + 0, s75 >= 0, s75 <= 1 + (1 + (z' - 2)) + 0, z = 1, z' - 2 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s76 + selsort(s118) :|: s118 >= 0, s118 <= z0 + (1 + z18 + z24), s76 >= 0, s76 <= 1 + z0 + (1 + z18 + z24), s77 >= 0, s77 <= 1 + z0 + (1 + z18 + z24), s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s78 + selsort(s119) :|: s119 >= 0, s119 <= z0 + z1, s78 >= 0, s78 <= 1 + z0 + z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> s66 :|: s66 >= 0, s66 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s67 :|: s67 >= 0, s67 <= 1 + (1 + z08) + (1 + 0 + z2), z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> s68 :|: s68 >= 0, s68 <= 1 + (1 + z09) + (1 + (1 + z15) + z2), s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> s69 :|: s69 >= 0, s69 <= 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 }-> s110 :|: s110 >= 0, s110 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s111 :|: s111 >= 0, s111 <= z' + (1 + (1 + z010) + z3), z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s112 :|: s112 >= 0, s112 <= z' + (1 + 0 + (z'' - 1)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s113 :|: s113 >= 0, s113 <= z' + (1 + (1 + z16) + z3), s36 >= 0, s36 <= 2, z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s114 :|: s114 >= 0, s114 <= 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
   selsort(z) -{ 0 }-> ifselsort(s37, 1 + 0 + 0) :|: s37 >= 0, s37 <= 2, z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(s38, 1 + (1 + (z - 2)) + 0) :|: s38 >= 0, s38 <= 2, z - 2 >= 0
   selsort(z) -{ 0 }-> ifselsort(s39, 1 + z0 + z1) :|: s39 >= 0, s39 <= 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(s73, 1 + z0 + (1 + z17 + z23)) :|: s72 >= 0, s72 <= 1 + z0 + (1 + z17 + z23), s73 >= 0, s73 <= 2, s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {IFSELSORT,SELSORT}, {ifselsort,selsort}
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]
  IFREPL: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  min: runtime: O(1) [0], size: O(n^1) [z]
  ifmin: runtime: O(1) [0], size: O(n^1) [z']
  IFMIN: 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]
  replace: runtime: O(1) [0], size: O(n^1) [z' + z'']
  ifrepl: runtime: O(1) [0], size: O(n^1) [z'' + z4]
  IFSELSORT: runtime: ?, size: INF
  SELSORT: runtime: ?, size: INF

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

(69) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: IFSELSORT
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 747 + 25592*z' + 18190*z'^2 + 3562*z'^3

Computed RUNTIME bound using KoAT for: SELSORT
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 560567 + 1494670*z + 1239132*z^2 + 327704*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
   IFMIN(z, z') -{ 300 + 171*z0 + 52*z0*z2 + 26*z0^2 + 171*z2 + 26*z2^2 }-> 1 + s83 :|: s83 >= 0, s83 <= 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
   IFMIN(z, z') -{ 300 + 171*z1 + 52*z1*z2 + 26*z1^2 + 171*z2 + 26*z2^2 }-> 1 + s84 :|: s84 >= 0, s84 <= 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
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s45 :|: s45 >= 0, s45 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 300 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + s93 :|: s93 >= 0, s93 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 1 }-> 1 + SELSORT(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 312 }-> 1 + SELSORT(s102) + s46 + s94 :|: s102 >= 0, s102 <= 0 + 0, s94 >= 0, s94 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s46 >= 0, s46 <= 7 * 0 + 6 + 0 * 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 167 + 119*z' + 26*z'^2 }-> 1 + SELSORT(s103) + s47 + s95 :|: s103 >= 0, s103 <= 1 + (z' - 2) + 0, s95 >= 0, s95 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s47 >= 0, s47 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 167 + 119*z' + 26*z'^2 }-> 1 + SELSORT(s104) + s48 + s96 :|: s104 >= 0, s104 <= 1 + (z' - 2) + 0, s96 >= 0, s96 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s48 >= 0, s48 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 517 + 223*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 232*z13 + 54*z13*z21 + 27*z13^2 + 232*z21 + 27*z21^2 }-> 1 + SELSORT(s105) + s62 + s97 :|: s105 >= 0, s105 <= z0 + (1 + z13 + z21), s97 >= 0, s97 <= 99 * (1 + z0 + (1 + z13 + z21)) + 26 * ((1 + z0 + (1 + z13 + z21)) * (1 + z0 + (1 + z13 + z21))) + 115, s60 >= 0, s60 <= 1 + z0 + (1 + z13 + z21), s61 >= 0, s61 <= 1 + z0 + (1 + z13 + z21), s62 >= 0, s62 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 517 + 223*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 232*z13 + 54*z13*z21 + 27*z13^2 + 232*z21 + 27*z21^2 }-> 1 + SELSORT(s106) + s49 + s98 :|: s106 >= 0, s106 <= z0 + (1 + z13 + z21), s98 >= 0, s98 <= 99 * (1 + z0 + (1 + z13 + z21)) + 26 * ((1 + z0 + (1 + z13 + z21)) * (1 + z0 + (1 + z13 + z21))) + 115, s63 >= 0, s63 <= 1 + z0 + (1 + z13 + z21), s49 >= 0, s49 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 167 + 119*z' + 26*z'^2 }-> 1 + SELSORT(s107) + s50 + s99 :|: s107 >= 0, s107 <= 1 + (z' - 2) + 0, s99 >= 0, s99 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s50 >= 0, s50 <= 7 * 0 + 6 + 0 * 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 517 + 223*z0 + 52*z0*z14 + 52*z0*z22 + 26*z0^2 + 232*z14 + 54*z14*z22 + 27*z14^2 + 232*z22 + 27*z22^2 }-> 1 + SELSORT(s108) + s65 + s100 :|: s108 >= 0, s108 <= z0 + (1 + z14 + z22), s100 >= 0, s100 <= 99 * (1 + z0 + (1 + z14 + z22)) + 26 * ((1 + z0 + (1 + z14 + z22)) * (1 + z0 + (1 + z14 + z22))) + 115, s64 >= 0, s64 <= 1 + z0 + (1 + z14 + z22), s65 >= 0, s65 <= 7 * (1 + z14 + z22) + 6 + (1 + z14 + z22) * (1 + z14 + z22), s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 312 + 171*z0 + 52*z0*z1 + 26*z0^2 + 178*z1 + 27*z1^2 }-> 1 + SELSORT(s109) + s51 + s101 :|: s109 >= 0, s109 <= z0 + z1, s101 >= 0, s101 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s51 >= 0, s51 <= 7 * z1 + 6 + z1 * z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 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 + s23 :|: s23 >= 0, s23 <= 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 + s79 + s24 :|: s79 >= 0, s79 <= 8 + (1 + 0 + (1 + z1 + z2)) * (1 + 0 + (1 + z1 + z2)) + 4 * (1 + 0 + (1 + z1 + z2)), s24 >= 0, s24 <= 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 + s80 + s25 :|: s80 >= 0, s80 <= 8 + (1 + (1 + z0') + (1 + 0 + z2)) * (1 + (1 + z0') + (1 + 0 + z2)) + 4 * (1 + (1 + z0') + (1 + 0 + z2)), s25 >= 0, s25 <= 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 + s81 + s26 :|: s81 >= 0, s81 <= 8 + (1 + (1 + z0'') + (1 + (1 + z1') + z2)) * (1 + (1 + z0'') + (1 + (1 + z1') + z2)) + 4 * (1 + (1 + z0'') + (1 + (1 + z1') + z2)), s26 >= 0, s26 <= 1 + z1' + 1, s10 >= 0, s10 <= 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 + s82 + s27 :|: s82 >= 0, s82 <= 8 + (1 + z0 + (1 + z1 + z2)) * (1 + z0 + (1 + z1 + z2)) + 4 * (1 + z0 + (1 + z1 + z2)), s27 >= 0, s27 <= 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 + s40 + s' :|: s40 >= 0, s40 <= 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 + s41 + s'' :|: s41 >= 0, s41 <= 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 + s42 + s1 :|: s42 >= 0, s42 <= 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 + s43 + s2 :|: s43 >= 0, s43 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s28 >= 0, s28 <= 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 + s44 + s3 :|: s44 >= 0, s44 <= 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
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 303 }-> 1 + IFSELSORT(s29, 1 + 0 + 0) + s4 + s85 :|: s85 >= 0, s85 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s29 >= 0, s29 <= 2, s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 157 + 120*z + 26*z^2 }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s5 + s86 :|: s86 >= 0, s86 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s30 >= 0, s30 <= 2, s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 158 + 119*z + 26*z^2 }-> 1 + IFSELSORT(s31, 1 + (1 + (z - 2)) + 0) + s6 + s87 :|: s87 >= 0, s87 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s31 >= 0, s31 <= 2, s6 >= 0, s6 <= 0 + 2, z - 2 >= 0
   SELSORT(z) -{ 157 + 120*z + 26*z^2 }-> 1 + IFSELSORT(s32, 1 + (1 + (z - 2)) + 0) + s8 + s90 :|: s90 >= 0, s90 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s32 >= 0, s32 <= 2, s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 500 + s58 + 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 + IFSELSORT(s33, 1 + z0 + (1 + z12 + z2'')) + s59 + s91 :|: s91 >= 0, s91 <= 99 * (1 + z0 + (1 + z12 + z2'')) + 26 * ((1 + z0 + (1 + z12 + z2'')) * (1 + z0 + (1 + z12 + z2''))) + 115, s58 >= 0, s58 <= 1 + z0 + (1 + z12 + z2''), s59 >= 0, s59 <= s58 + 2, s33 >= 0, s33 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 303 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + IFSELSORT(s34, 1 + z0 + z1) + s9 + s92 :|: s92 >= 0, s92 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s34 >= 0, s34 <= 2, s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 500 + s54 + 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 + IFSELSORT(s53, 1 + z0 + (1 + z11 + z2')) + s55 + s88 :|: s88 >= 0, s88 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s52 >= 0, s52 <= 1 + z0 + (1 + z11 + z2'), s53 >= 0, s53 <= 2, s54 >= 0, s54 <= 1 + z0 + (1 + z11 + z2'), s55 >= 0, s55 <= s54 + 2, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 500 + 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 + IFSELSORT(s57, 1 + z0 + (1 + z11 + z2')) + s7 + s89 :|: s89 >= 0, s89 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s56 >= 0, s56 <= 1 + z0 + (1 + z11 + z2'), s57 >= 0, s57 <= 2, s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 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
   ifmin(z, z') -{ 0 }-> s70 :|: s70 >= 0, s70 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> s71 :|: s71 >= 0, s71 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z2 + s115 :|: s115 >= 0, s115 <= z'' + z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s74 + selsort(s116) :|: s116 >= 0, s116 <= 0 + 0, s74 >= 0, s74 <= 1 + 0 + 0, z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + s75 + selsort(s117) :|: s117 >= 0, s117 <= 1 + (z' - 2) + 0, s75 >= 0, s75 <= 1 + (1 + (z' - 2)) + 0, z = 1, z' - 2 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s76 + selsort(s118) :|: s118 >= 0, s118 <= z0 + (1 + z18 + z24), s76 >= 0, s76 <= 1 + z0 + (1 + z18 + z24), s77 >= 0, s77 <= 1 + z0 + (1 + z18 + z24), s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s78 + selsort(s119) :|: s119 >= 0, s119 <= z0 + z1, s78 >= 0, s78 <= 1 + z0 + z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> s66 :|: s66 >= 0, s66 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s67 :|: s67 >= 0, s67 <= 1 + (1 + z08) + (1 + 0 + z2), z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> s68 :|: s68 >= 0, s68 <= 1 + (1 + z09) + (1 + (1 + z15) + z2), s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> s69 :|: s69 >= 0, s69 <= 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 }-> s110 :|: s110 >= 0, s110 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s111 :|: s111 >= 0, s111 <= z' + (1 + (1 + z010) + z3), z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s112 :|: s112 >= 0, s112 <= z' + (1 + 0 + (z'' - 1)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s113 :|: s113 >= 0, s113 <= z' + (1 + (1 + z16) + z3), s36 >= 0, s36 <= 2, z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s114 :|: s114 >= 0, s114 <= 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
   selsort(z) -{ 0 }-> ifselsort(s37, 1 + 0 + 0) :|: s37 >= 0, s37 <= 2, z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(s38, 1 + (1 + (z - 2)) + 0) :|: s38 >= 0, s38 <= 2, z - 2 >= 0
   selsort(z) -{ 0 }-> ifselsort(s39, 1 + z0 + z1) :|: s39 >= 0, s39 <= 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(s73, 1 + z0 + (1 + z17 + z23)) :|: s72 >= 0, s72 <= 1 + z0 + (1 + z17 + z23), s73 >= 0, s73 <= 2, s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {ifselsort,selsort}
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]
  IFREPL: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  min: runtime: O(1) [0], size: O(n^1) [z]
  ifmin: runtime: O(1) [0], size: O(n^1) [z']
  IFMIN: 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]
  replace: runtime: O(1) [0], size: O(n^1) [z' + z'']
  ifrepl: runtime: O(1) [0], size: O(n^1) [z'' + z4]
  IFSELSORT: runtime: O(n^3) [747 + 25592*z' + 18190*z'^2 + 3562*z'^3], size: INF
  SELSORT: runtime: O(n^3) [560567 + 1494670*z + 1239132*z^2 + 327704*z^3], size: INF

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

(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
   IFMIN(z, z') -{ 300 + 171*z0 + 52*z0*z2 + 26*z0^2 + 171*z2 + 26*z2^2 }-> 1 + s83 :|: s83 >= 0, s83 <= 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
   IFMIN(z, z') -{ 300 + 171*z1 + 52*z1*z2 + 26*z1^2 + 171*z2 + 26*z2^2 }-> 1 + s84 :|: s84 >= 0, s84 <= 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
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s45 :|: s45 >= 0, s45 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 560568 + 1494670*z1 + 1239132*z1^2 + 327704*z1^3 }-> 1 + s128 :|: s128 >= 0, s128 <= inf6, z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 300 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + s93 :|: s93 >= 0, s93 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 560879 + 1494670*s102 + 1239132*s102^2 + 327704*s102^3 }-> 1 + s129 + s46 + s94 :|: s129 >= 0, s129 <= inf7, s102 >= 0, s102 <= 0 + 0, s94 >= 0, s94 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s46 >= 0, s46 <= 7 * 0 + 6 + 0 * 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 560734 + 1494670*s103 + 1239132*s103^2 + 327704*s103^3 + 119*z' + 26*z'^2 }-> 1 + s130 + s47 + s95 :|: s130 >= 0, s130 <= inf8, s103 >= 0, s103 <= 1 + (z' - 2) + 0, s95 >= 0, s95 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s47 >= 0, s47 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 560734 + 1494670*s104 + 1239132*s104^2 + 327704*s104^3 + 119*z' + 26*z'^2 }-> 1 + s131 + s48 + s96 :|: s131 >= 0, s131 <= inf9, s104 >= 0, s104 <= 1 + (z' - 2) + 0, s96 >= 0, s96 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s48 >= 0, s48 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 561084 + 1494670*s105 + 1239132*s105^2 + 327704*s105^3 + 223*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 232*z13 + 54*z13*z21 + 27*z13^2 + 232*z21 + 27*z21^2 }-> 1 + s132 + s62 + s97 :|: s132 >= 0, s132 <= inf10, s105 >= 0, s105 <= z0 + (1 + z13 + z21), s97 >= 0, s97 <= 99 * (1 + z0 + (1 + z13 + z21)) + 26 * ((1 + z0 + (1 + z13 + z21)) * (1 + z0 + (1 + z13 + z21))) + 115, s60 >= 0, s60 <= 1 + z0 + (1 + z13 + z21), s61 >= 0, s61 <= 1 + z0 + (1 + z13 + z21), s62 >= 0, s62 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 561084 + 1494670*s106 + 1239132*s106^2 + 327704*s106^3 + 223*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 232*z13 + 54*z13*z21 + 27*z13^2 + 232*z21 + 27*z21^2 }-> 1 + s133 + s49 + s98 :|: s133 >= 0, s133 <= inf11, s106 >= 0, s106 <= z0 + (1 + z13 + z21), s98 >= 0, s98 <= 99 * (1 + z0 + (1 + z13 + z21)) + 26 * ((1 + z0 + (1 + z13 + z21)) * (1 + z0 + (1 + z13 + z21))) + 115, s63 >= 0, s63 <= 1 + z0 + (1 + z13 + z21), s49 >= 0, s49 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 560734 + 1494670*s107 + 1239132*s107^2 + 327704*s107^3 + 119*z' + 26*z'^2 }-> 1 + s134 + s50 + s99 :|: s134 >= 0, s134 <= inf12, s107 >= 0, s107 <= 1 + (z' - 2) + 0, s99 >= 0, s99 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s50 >= 0, s50 <= 7 * 0 + 6 + 0 * 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 561084 + 1494670*s108 + 1239132*s108^2 + 327704*s108^3 + 223*z0 + 52*z0*z14 + 52*z0*z22 + 26*z0^2 + 232*z14 + 54*z14*z22 + 27*z14^2 + 232*z22 + 27*z22^2 }-> 1 + s135 + s65 + s100 :|: s135 >= 0, s135 <= inf13, s108 >= 0, s108 <= z0 + (1 + z14 + z22), s100 >= 0, s100 <= 99 * (1 + z0 + (1 + z14 + z22)) + 26 * ((1 + z0 + (1 + z14 + z22)) * (1 + z0 + (1 + z14 + z22))) + 115, s64 >= 0, s64 <= 1 + z0 + (1 + z14 + z22), s65 >= 0, s65 <= 7 * (1 + z14 + z22) + 6 + (1 + z14 + z22) * (1 + z14 + z22), s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 560879 + 1494670*s109 + 1239132*s109^2 + 327704*s109^3 + 171*z0 + 52*z0*z1 + 26*z0^2 + 178*z1 + 27*z1^2 }-> 1 + s136 + s51 + s101 :|: s136 >= 0, s136 <= inf14, s109 >= 0, s109 <= z0 + z1, s101 >= 0, s101 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s51 >= 0, s51 <= 7 * z1 + 6 + z1 * z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 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 + s23 :|: s23 >= 0, s23 <= 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 + s79 + s24 :|: s79 >= 0, s79 <= 8 + (1 + 0 + (1 + z1 + z2)) * (1 + 0 + (1 + z1 + z2)) + 4 * (1 + 0 + (1 + z1 + z2)), s24 >= 0, s24 <= 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 + s80 + s25 :|: s80 >= 0, s80 <= 8 + (1 + (1 + z0') + (1 + 0 + z2)) * (1 + (1 + z0') + (1 + 0 + z2)) + 4 * (1 + (1 + z0') + (1 + 0 + z2)), s25 >= 0, s25 <= 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 + s81 + s26 :|: s81 >= 0, s81 <= 8 + (1 + (1 + z0'') + (1 + (1 + z1') + z2)) * (1 + (1 + z0'') + (1 + (1 + z1') + z2)) + 4 * (1 + (1 + z0'') + (1 + (1 + z1') + z2)), s26 >= 0, s26 <= 1 + z1' + 1, s10 >= 0, s10 <= 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 + s82 + s27 :|: s82 >= 0, s82 <= 8 + (1 + z0 + (1 + z1 + z2)) * (1 + z0 + (1 + z1 + z2)) + 4 * (1 + z0 + (1 + z1 + z2)), s27 >= 0, s27 <= 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 + s40 + s' :|: s40 >= 0, s40 <= 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 + s41 + s'' :|: s41 >= 0, s41 <= 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 + s42 + s1 :|: s42 >= 0, s42 <= 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 + s43 + s2 :|: s43 >= 0, s43 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s28 >= 0, s28 <= 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 + s44 + s3 :|: s44 >= 0, s44 <= 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
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 48394 }-> 1 + s120 + s4 + s85 :|: s120 >= 0, s120 <= inf, s85 >= 0, s85 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s29 >= 0, s29 <= 2, s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 904 + 25712*z + 18216*z^2 + 3562*z^3 }-> 1 + s121 + s5 + s86 :|: s121 >= 0, s121 <= inf', s86 >= 0, s86 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s30 >= 0, s30 <= 2, s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 905 + 25711*z + 18216*z^2 + 3562*z^3 }-> 1 + s122 + s6 + s87 :|: s122 >= 0, s122 <= inf'', s87 >= 0, s87 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s31 >= 0, s31 <= 2, s6 >= 0, s6 <= 0 + 2, z - 2 >= 0
   SELSORT(z) -{ 153687 + s54 + 141319*z0 + 79176*z0*z11 + 21372*z0*z11*z2' + 10686*z0*z11^2 + 79176*z0*z2' + 10686*z0*z2'^2 + 39588*z0^2 + 10686*z0^2*z11 + 10686*z0^2*z2' + 3562*z0^3 + 141319*z11 + 79176*z11*z2' + 10686*z11*z2'^2 + 39588*z11^2 + 10686*z11^2*z2' + 3562*z11^3 + 141319*z2' + 39588*z2'^2 + 3562*z2'^3 }-> 1 + s123 + s55 + s88 :|: s123 >= 0, s123 <= inf1, s88 >= 0, s88 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s52 >= 0, s52 <= 1 + z0 + (1 + z11 + z2'), s53 >= 0, s53 <= 2, s54 >= 0, s54 <= 1 + z0 + (1 + z11 + z2'), s55 >= 0, s55 <= s54 + 2, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 153687 + 141319*z0 + 79176*z0*z11 + 21372*z0*z11*z2' + 10686*z0*z11^2 + 79176*z0*z2' + 10686*z0*z2'^2 + 39588*z0^2 + 10686*z0^2*z11 + 10686*z0^2*z2' + 3562*z0^3 + 141319*z11 + 79176*z11*z2' + 10686*z11*z2'^2 + 39588*z11^2 + 10686*z11^2*z2' + 3562*z11^3 + 141319*z2' + 39588*z2'^2 + 3562*z2'^3 }-> 1 + s124 + s7 + s89 :|: s124 >= 0, s124 <= inf2, s89 >= 0, s89 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s56 >= 0, s56 <= 1 + z0 + (1 + z11 + z2'), s57 >= 0, s57 <= 2, s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 904 + 25712*z + 18216*z^2 + 3562*z^3 }-> 1 + s125 + s8 + s90 :|: s125 >= 0, s125 <= inf3, s90 >= 0, s90 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s32 >= 0, s32 <= 2, s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 153687 + s58 + 141319*z0 + 79176*z0*z12 + 21372*z0*z12*z2'' + 10686*z0*z12^2 + 79176*z0*z2'' + 10686*z0*z2''^2 + 39588*z0^2 + 10686*z0^2*z12 + 10686*z0^2*z2'' + 3562*z0^3 + 141319*z12 + 79176*z12*z2'' + 10686*z12*z2''^2 + 39588*z12^2 + 10686*z12^2*z2'' + 3562*z12^3 + 141319*z2'' + 39588*z2''^2 + 3562*z2''^3 }-> 1 + s126 + s59 + s91 :|: s126 >= 0, s126 <= inf4, s91 >= 0, s91 <= 99 * (1 + z0 + (1 + z12 + z2'')) + 26 * ((1 + z0 + (1 + z12 + z2'')) * (1 + z0 + (1 + z12 + z2''))) + 115, s58 >= 0, s58 <= 1 + z0 + (1 + z12 + z2''), s59 >= 0, s59 <= s58 + 2, s33 >= 0, s33 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 48394 + 72829*z0 + 57804*z0*z1 + 10686*z0*z1^2 + 28902*z0^2 + 10686*z0^2*z1 + 3562*z0^3 + 72829*z1 + 28902*z1^2 + 3562*z1^3 }-> 1 + s127 + s9 + s92 :|: s127 >= 0, s127 <= inf5, s92 >= 0, s92 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s34 >= 0, s34 <= 2, s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eq(z, z') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 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
   ifmin(z, z') -{ 0 }-> s70 :|: s70 >= 0, s70 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> s71 :|: s71 >= 0, s71 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z2 + s115 :|: s115 >= 0, s115 <= z'' + z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s74 + selsort(s116) :|: s116 >= 0, s116 <= 0 + 0, s74 >= 0, s74 <= 1 + 0 + 0, z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + s75 + selsort(s117) :|: s117 >= 0, s117 <= 1 + (z' - 2) + 0, s75 >= 0, s75 <= 1 + (1 + (z' - 2)) + 0, z = 1, z' - 2 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s76 + selsort(s118) :|: s118 >= 0, s118 <= z0 + (1 + z18 + z24), s76 >= 0, s76 <= 1 + z0 + (1 + z18 + z24), s77 >= 0, s77 <= 1 + z0 + (1 + z18 + z24), s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s78 + selsort(s119) :|: s119 >= 0, s119 <= z0 + z1, s78 >= 0, s78 <= 1 + z0 + z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> s66 :|: s66 >= 0, s66 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s67 :|: s67 >= 0, s67 <= 1 + (1 + z08) + (1 + 0 + z2), z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> s68 :|: s68 >= 0, s68 <= 1 + (1 + z09) + (1 + (1 + z15) + z2), s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> s69 :|: s69 >= 0, s69 <= 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 }-> s110 :|: s110 >= 0, s110 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s111 :|: s111 >= 0, s111 <= z' + (1 + (1 + z010) + z3), z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s112 :|: s112 >= 0, s112 <= z' + (1 + 0 + (z'' - 1)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s113 :|: s113 >= 0, s113 <= z' + (1 + (1 + z16) + z3), s36 >= 0, s36 <= 2, z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s114 :|: s114 >= 0, s114 <= 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
   selsort(z) -{ 0 }-> ifselsort(s37, 1 + 0 + 0) :|: s37 >= 0, s37 <= 2, z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(s38, 1 + (1 + (z - 2)) + 0) :|: s38 >= 0, s38 <= 2, z - 2 >= 0
   selsort(z) -{ 0 }-> ifselsort(s39, 1 + z0 + z1) :|: s39 >= 0, s39 <= 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(s73, 1 + z0 + (1 + z17 + z23)) :|: s72 >= 0, s72 <= 1 + z0 + (1 + z17 + z23), s73 >= 0, s73 <= 2, s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {ifselsort,selsort}
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]
  IFREPL: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  min: runtime: O(1) [0], size: O(n^1) [z]
  ifmin: runtime: O(1) [0], size: O(n^1) [z']
  IFMIN: 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]
  replace: runtime: O(1) [0], size: O(n^1) [z' + z'']
  ifrepl: runtime: O(1) [0], size: O(n^1) [z'' + z4]
  IFSELSORT: runtime: O(n^3) [747 + 25592*z' + 18190*z'^2 + 3562*z'^3], size: INF
  SELSORT: runtime: O(n^3) [560567 + 1494670*z + 1239132*z^2 + 327704*z^3], size: INF

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

(73) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: ifselsort
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 3 + 3*z' + z'^2

Computed SIZE bound using KoAT for: selsort
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 30 + 34*z + 14*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
   IFMIN(z, z') -{ 300 + 171*z0 + 52*z0*z2 + 26*z0^2 + 171*z2 + 26*z2^2 }-> 1 + s83 :|: s83 >= 0, s83 <= 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
   IFMIN(z, z') -{ 300 + 171*z1 + 52*z1*z2 + 26*z1^2 + 171*z2 + 26*z2^2 }-> 1 + s84 :|: s84 >= 0, s84 <= 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
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s45 :|: s45 >= 0, s45 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 560568 + 1494670*z1 + 1239132*z1^2 + 327704*z1^3 }-> 1 + s128 :|: s128 >= 0, s128 <= inf6, z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 300 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + s93 :|: s93 >= 0, s93 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 560879 + 1494670*s102 + 1239132*s102^2 + 327704*s102^3 }-> 1 + s129 + s46 + s94 :|: s129 >= 0, s129 <= inf7, s102 >= 0, s102 <= 0 + 0, s94 >= 0, s94 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s46 >= 0, s46 <= 7 * 0 + 6 + 0 * 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 560734 + 1494670*s103 + 1239132*s103^2 + 327704*s103^3 + 119*z' + 26*z'^2 }-> 1 + s130 + s47 + s95 :|: s130 >= 0, s130 <= inf8, s103 >= 0, s103 <= 1 + (z' - 2) + 0, s95 >= 0, s95 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s47 >= 0, s47 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 560734 + 1494670*s104 + 1239132*s104^2 + 327704*s104^3 + 119*z' + 26*z'^2 }-> 1 + s131 + s48 + s96 :|: s131 >= 0, s131 <= inf9, s104 >= 0, s104 <= 1 + (z' - 2) + 0, s96 >= 0, s96 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s48 >= 0, s48 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 561084 + 1494670*s105 + 1239132*s105^2 + 327704*s105^3 + 223*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 232*z13 + 54*z13*z21 + 27*z13^2 + 232*z21 + 27*z21^2 }-> 1 + s132 + s62 + s97 :|: s132 >= 0, s132 <= inf10, s105 >= 0, s105 <= z0 + (1 + z13 + z21), s97 >= 0, s97 <= 99 * (1 + z0 + (1 + z13 + z21)) + 26 * ((1 + z0 + (1 + z13 + z21)) * (1 + z0 + (1 + z13 + z21))) + 115, s60 >= 0, s60 <= 1 + z0 + (1 + z13 + z21), s61 >= 0, s61 <= 1 + z0 + (1 + z13 + z21), s62 >= 0, s62 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 561084 + 1494670*s106 + 1239132*s106^2 + 327704*s106^3 + 223*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 232*z13 + 54*z13*z21 + 27*z13^2 + 232*z21 + 27*z21^2 }-> 1 + s133 + s49 + s98 :|: s133 >= 0, s133 <= inf11, s106 >= 0, s106 <= z0 + (1 + z13 + z21), s98 >= 0, s98 <= 99 * (1 + z0 + (1 + z13 + z21)) + 26 * ((1 + z0 + (1 + z13 + z21)) * (1 + z0 + (1 + z13 + z21))) + 115, s63 >= 0, s63 <= 1 + z0 + (1 + z13 + z21), s49 >= 0, s49 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 560734 + 1494670*s107 + 1239132*s107^2 + 327704*s107^3 + 119*z' + 26*z'^2 }-> 1 + s134 + s50 + s99 :|: s134 >= 0, s134 <= inf12, s107 >= 0, s107 <= 1 + (z' - 2) + 0, s99 >= 0, s99 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s50 >= 0, s50 <= 7 * 0 + 6 + 0 * 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 561084 + 1494670*s108 + 1239132*s108^2 + 327704*s108^3 + 223*z0 + 52*z0*z14 + 52*z0*z22 + 26*z0^2 + 232*z14 + 54*z14*z22 + 27*z14^2 + 232*z22 + 27*z22^2 }-> 1 + s135 + s65 + s100 :|: s135 >= 0, s135 <= inf13, s108 >= 0, s108 <= z0 + (1 + z14 + z22), s100 >= 0, s100 <= 99 * (1 + z0 + (1 + z14 + z22)) + 26 * ((1 + z0 + (1 + z14 + z22)) * (1 + z0 + (1 + z14 + z22))) + 115, s64 >= 0, s64 <= 1 + z0 + (1 + z14 + z22), s65 >= 0, s65 <= 7 * (1 + z14 + z22) + 6 + (1 + z14 + z22) * (1 + z14 + z22), s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 560879 + 1494670*s109 + 1239132*s109^2 + 327704*s109^3 + 171*z0 + 52*z0*z1 + 26*z0^2 + 178*z1 + 27*z1^2 }-> 1 + s136 + s51 + s101 :|: s136 >= 0, s136 <= inf14, s109 >= 0, s109 <= z0 + z1, s101 >= 0, s101 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s51 >= 0, s51 <= 7 * z1 + 6 + z1 * z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 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 + s23 :|: s23 >= 0, s23 <= 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 + s79 + s24 :|: s79 >= 0, s79 <= 8 + (1 + 0 + (1 + z1 + z2)) * (1 + 0 + (1 + z1 + z2)) + 4 * (1 + 0 + (1 + z1 + z2)), s24 >= 0, s24 <= 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 + s80 + s25 :|: s80 >= 0, s80 <= 8 + (1 + (1 + z0') + (1 + 0 + z2)) * (1 + (1 + z0') + (1 + 0 + z2)) + 4 * (1 + (1 + z0') + (1 + 0 + z2)), s25 >= 0, s25 <= 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 + s81 + s26 :|: s81 >= 0, s81 <= 8 + (1 + (1 + z0'') + (1 + (1 + z1') + z2)) * (1 + (1 + z0'') + (1 + (1 + z1') + z2)) + 4 * (1 + (1 + z0'') + (1 + (1 + z1') + z2)), s26 >= 0, s26 <= 1 + z1' + 1, s10 >= 0, s10 <= 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 + s82 + s27 :|: s82 >= 0, s82 <= 8 + (1 + z0 + (1 + z1 + z2)) * (1 + z0 + (1 + z1 + z2)) + 4 * (1 + z0 + (1 + z1 + z2)), s27 >= 0, s27 <= 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 + s40 + s' :|: s40 >= 0, s40 <= 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 + s41 + s'' :|: s41 >= 0, s41 <= 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 + s42 + s1 :|: s42 >= 0, s42 <= 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 + s43 + s2 :|: s43 >= 0, s43 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s28 >= 0, s28 <= 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 + s44 + s3 :|: s44 >= 0, s44 <= 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
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 48394 }-> 1 + s120 + s4 + s85 :|: s120 >= 0, s120 <= inf, s85 >= 0, s85 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s29 >= 0, s29 <= 2, s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 904 + 25712*z + 18216*z^2 + 3562*z^3 }-> 1 + s121 + s5 + s86 :|: s121 >= 0, s121 <= inf', s86 >= 0, s86 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s30 >= 0, s30 <= 2, s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 905 + 25711*z + 18216*z^2 + 3562*z^3 }-> 1 + s122 + s6 + s87 :|: s122 >= 0, s122 <= inf'', s87 >= 0, s87 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s31 >= 0, s31 <= 2, s6 >= 0, s6 <= 0 + 2, z - 2 >= 0
   SELSORT(z) -{ 153687 + s54 + 141319*z0 + 79176*z0*z11 + 21372*z0*z11*z2' + 10686*z0*z11^2 + 79176*z0*z2' + 10686*z0*z2'^2 + 39588*z0^2 + 10686*z0^2*z11 + 10686*z0^2*z2' + 3562*z0^3 + 141319*z11 + 79176*z11*z2' + 10686*z11*z2'^2 + 39588*z11^2 + 10686*z11^2*z2' + 3562*z11^3 + 141319*z2' + 39588*z2'^2 + 3562*z2'^3 }-> 1 + s123 + s55 + s88 :|: s123 >= 0, s123 <= inf1, s88 >= 0, s88 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s52 >= 0, s52 <= 1 + z0 + (1 + z11 + z2'), s53 >= 0, s53 <= 2, s54 >= 0, s54 <= 1 + z0 + (1 + z11 + z2'), s55 >= 0, s55 <= s54 + 2, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 153687 + 141319*z0 + 79176*z0*z11 + 21372*z0*z11*z2' + 10686*z0*z11^2 + 79176*z0*z2' + 10686*z0*z2'^2 + 39588*z0^2 + 10686*z0^2*z11 + 10686*z0^2*z2' + 3562*z0^3 + 141319*z11 + 79176*z11*z2' + 10686*z11*z2'^2 + 39588*z11^2 + 10686*z11^2*z2' + 3562*z11^3 + 141319*z2' + 39588*z2'^2 + 3562*z2'^3 }-> 1 + s124 + s7 + s89 :|: s124 >= 0, s124 <= inf2, s89 >= 0, s89 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s56 >= 0, s56 <= 1 + z0 + (1 + z11 + z2'), s57 >= 0, s57 <= 2, s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 904 + 25712*z + 18216*z^2 + 3562*z^3 }-> 1 + s125 + s8 + s90 :|: s125 >= 0, s125 <= inf3, s90 >= 0, s90 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s32 >= 0, s32 <= 2, s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 153687 + s58 + 141319*z0 + 79176*z0*z12 + 21372*z0*z12*z2'' + 10686*z0*z12^2 + 79176*z0*z2'' + 10686*z0*z2''^2 + 39588*z0^2 + 10686*z0^2*z12 + 10686*z0^2*z2'' + 3562*z0^3 + 141319*z12 + 79176*z12*z2'' + 10686*z12*z2''^2 + 39588*z12^2 + 10686*z12^2*z2'' + 3562*z12^3 + 141319*z2'' + 39588*z2''^2 + 3562*z2''^3 }-> 1 + s126 + s59 + s91 :|: s126 >= 0, s126 <= inf4, s91 >= 0, s91 <= 99 * (1 + z0 + (1 + z12 + z2'')) + 26 * ((1 + z0 + (1 + z12 + z2'')) * (1 + z0 + (1 + z12 + z2''))) + 115, s58 >= 0, s58 <= 1 + z0 + (1 + z12 + z2''), s59 >= 0, s59 <= s58 + 2, s33 >= 0, s33 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 48394 + 72829*z0 + 57804*z0*z1 + 10686*z0*z1^2 + 28902*z0^2 + 10686*z0^2*z1 + 3562*z0^3 + 72829*z1 + 28902*z1^2 + 3562*z1^3 }-> 1 + s127 + s9 + s92 :|: s127 >= 0, s127 <= inf5, s92 >= 0, s92 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s34 >= 0, s34 <= 2, s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eq(z, z') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 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
   ifmin(z, z') -{ 0 }-> s70 :|: s70 >= 0, s70 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> s71 :|: s71 >= 0, s71 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z2 + s115 :|: s115 >= 0, s115 <= z'' + z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s74 + selsort(s116) :|: s116 >= 0, s116 <= 0 + 0, s74 >= 0, s74 <= 1 + 0 + 0, z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + s75 + selsort(s117) :|: s117 >= 0, s117 <= 1 + (z' - 2) + 0, s75 >= 0, s75 <= 1 + (1 + (z' - 2)) + 0, z = 1, z' - 2 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s76 + selsort(s118) :|: s118 >= 0, s118 <= z0 + (1 + z18 + z24), s76 >= 0, s76 <= 1 + z0 + (1 + z18 + z24), s77 >= 0, s77 <= 1 + z0 + (1 + z18 + z24), s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s78 + selsort(s119) :|: s119 >= 0, s119 <= z0 + z1, s78 >= 0, s78 <= 1 + z0 + z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> s66 :|: s66 >= 0, s66 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s67 :|: s67 >= 0, s67 <= 1 + (1 + z08) + (1 + 0 + z2), z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> s68 :|: s68 >= 0, s68 <= 1 + (1 + z09) + (1 + (1 + z15) + z2), s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> s69 :|: s69 >= 0, s69 <= 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 }-> s110 :|: s110 >= 0, s110 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s111 :|: s111 >= 0, s111 <= z' + (1 + (1 + z010) + z3), z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s112 :|: s112 >= 0, s112 <= z' + (1 + 0 + (z'' - 1)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s113 :|: s113 >= 0, s113 <= z' + (1 + (1 + z16) + z3), s36 >= 0, s36 <= 2, z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s114 :|: s114 >= 0, s114 <= 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
   selsort(z) -{ 0 }-> ifselsort(s37, 1 + 0 + 0) :|: s37 >= 0, s37 <= 2, z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(s38, 1 + (1 + (z - 2)) + 0) :|: s38 >= 0, s38 <= 2, z - 2 >= 0
   selsort(z) -{ 0 }-> ifselsort(s39, 1 + z0 + z1) :|: s39 >= 0, s39 <= 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(s73, 1 + z0 + (1 + z17 + z23)) :|: s72 >= 0, s72 <= 1 + z0 + (1 + z17 + z23), s73 >= 0, s73 <= 2, s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {ifselsort,selsort}
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]
  IFREPL: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  min: runtime: O(1) [0], size: O(n^1) [z]
  ifmin: runtime: O(1) [0], size: O(n^1) [z']
  IFMIN: 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]
  replace: runtime: O(1) [0], size: O(n^1) [z' + z'']
  ifrepl: runtime: O(1) [0], size: O(n^1) [z'' + z4]
  IFSELSORT: runtime: O(n^3) [747 + 25592*z' + 18190*z'^2 + 3562*z'^3], size: INF
  SELSORT: runtime: O(n^3) [560567 + 1494670*z + 1239132*z^2 + 327704*z^3], size: INF
  ifselsort: runtime: ?, size: O(n^2) [3 + 3*z' + z'^2]
  selsort: runtime: ?, size: O(n^2) [30 + 34*z + 14*z^2]

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

(75) IntTrsBoundProof (UPPER BOUND(ID))

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

Computed RUNTIME bound using CoFloCo for: selsort
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
   IFMIN(z, z') -{ 300 + 171*z0 + 52*z0*z2 + 26*z0^2 + 171*z2 + 26*z2^2 }-> 1 + s83 :|: s83 >= 0, s83 <= 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
   IFMIN(z, z') -{ 300 + 171*z1 + 52*z1*z2 + 26*z1^2 + 171*z2 + 26*z2^2 }-> 1 + s84 :|: s84 >= 0, s84 <= 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
   IFREPL(z, z', z'', z4) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFREPL(z, z', z'', z4) -{ 13 + 7*z3 + z3^2 }-> 1 + s45 :|: s45 >= 0, s45 <= 7 * z3 + 6 + z3 * z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 560568 + 1494670*z1 + 1239132*z1^2 + 327704*z1^3 }-> 1 + s128 :|: s128 >= 0, s128 <= inf6, z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   IFSELSORT(z, z') -{ 300 + 171*z0 + 52*z0*z1 + 26*z0^2 + 171*z1 + 26*z1^2 }-> 1 + s93 :|: s93 >= 0, s93 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 560879 + 1494670*s102 + 1239132*s102^2 + 327704*s102^3 }-> 1 + s129 + s46 + s94 :|: s129 >= 0, s129 <= inf7, s102 >= 0, s102 <= 0 + 0, s94 >= 0, s94 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s46 >= 0, s46 <= 7 * 0 + 6 + 0 * 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 560734 + 1494670*s103 + 1239132*s103^2 + 327704*s103^3 + 119*z' + 26*z'^2 }-> 1 + s130 + s47 + s95 :|: s130 >= 0, s130 <= inf8, s103 >= 0, s103 <= 1 + (z' - 2) + 0, s95 >= 0, s95 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s47 >= 0, s47 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 560734 + 1494670*s104 + 1239132*s104^2 + 327704*s104^3 + 119*z' + 26*z'^2 }-> 1 + s131 + s48 + s96 :|: s131 >= 0, s131 <= inf9, s104 >= 0, s104 <= 1 + (z' - 2) + 0, s96 >= 0, s96 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s48 >= 0, s48 <= 7 * 0 + 6 + 0 * 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 561084 + 1494670*s105 + 1239132*s105^2 + 327704*s105^3 + 223*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 232*z13 + 54*z13*z21 + 27*z13^2 + 232*z21 + 27*z21^2 }-> 1 + s132 + s62 + s97 :|: s132 >= 0, s132 <= inf10, s105 >= 0, s105 <= z0 + (1 + z13 + z21), s97 >= 0, s97 <= 99 * (1 + z0 + (1 + z13 + z21)) + 26 * ((1 + z0 + (1 + z13 + z21)) * (1 + z0 + (1 + z13 + z21))) + 115, s60 >= 0, s60 <= 1 + z0 + (1 + z13 + z21), s61 >= 0, s61 <= 1 + z0 + (1 + z13 + z21), s62 >= 0, s62 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 561084 + 1494670*s106 + 1239132*s106^2 + 327704*s106^3 + 223*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 232*z13 + 54*z13*z21 + 27*z13^2 + 232*z21 + 27*z21^2 }-> 1 + s133 + s49 + s98 :|: s133 >= 0, s133 <= inf11, s106 >= 0, s106 <= z0 + (1 + z13 + z21), s98 >= 0, s98 <= 99 * (1 + z0 + (1 + z13 + z21)) + 26 * ((1 + z0 + (1 + z13 + z21)) * (1 + z0 + (1 + z13 + z21))) + 115, s63 >= 0, s63 <= 1 + z0 + (1 + z13 + z21), s49 >= 0, s49 <= 7 * (1 + z13 + z21) + 6 + (1 + z13 + z21) * (1 + z13 + z21), s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 560734 + 1494670*s107 + 1239132*s107^2 + 327704*s107^3 + 119*z' + 26*z'^2 }-> 1 + s134 + s50 + s99 :|: s134 >= 0, s134 <= inf12, s107 >= 0, s107 <= 1 + (z' - 2) + 0, s99 >= 0, s99 <= 99 * (1 + (1 + (z' - 2)) + 0) + 26 * ((1 + (1 + (z' - 2)) + 0) * (1 + (1 + (z' - 2)) + 0)) + 115, s50 >= 0, s50 <= 7 * 0 + 6 + 0 * 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 561084 + 1494670*s108 + 1239132*s108^2 + 327704*s108^3 + 223*z0 + 52*z0*z14 + 52*z0*z22 + 26*z0^2 + 232*z14 + 54*z14*z22 + 27*z14^2 + 232*z22 + 27*z22^2 }-> 1 + s135 + s65 + s100 :|: s135 >= 0, s135 <= inf13, s108 >= 0, s108 <= z0 + (1 + z14 + z22), s100 >= 0, s100 <= 99 * (1 + z0 + (1 + z14 + z22)) + 26 * ((1 + z0 + (1 + z14 + z22)) * (1 + z0 + (1 + z14 + z22))) + 115, s64 >= 0, s64 <= 1 + z0 + (1 + z14 + z22), s65 >= 0, s65 <= 7 * (1 + z14 + z22) + 6 + (1 + z14 + z22) * (1 + z14 + z22), s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 560879 + 1494670*s109 + 1239132*s109^2 + 327704*s109^3 + 171*z0 + 52*z0*z1 + 26*z0^2 + 178*z1 + 27*z1^2 }-> 1 + s136 + s51 + s101 :|: s136 >= 0, s136 <= inf14, s109 >= 0, s109 <= z0 + z1, s101 >= 0, s101 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s51 >= 0, s51 <= 7 * z1 + 6 + z1 * z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 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 + s23 :|: s23 >= 0, s23 <= 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 + s79 + s24 :|: s79 >= 0, s79 <= 8 + (1 + 0 + (1 + z1 + z2)) * (1 + 0 + (1 + z1 + z2)) + 4 * (1 + 0 + (1 + z1 + z2)), s24 >= 0, s24 <= 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 + s80 + s25 :|: s80 >= 0, s80 <= 8 + (1 + (1 + z0') + (1 + 0 + z2)) * (1 + (1 + z0') + (1 + 0 + z2)) + 4 * (1 + (1 + z0') + (1 + 0 + z2)), s25 >= 0, s25 <= 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 + s81 + s26 :|: s81 >= 0, s81 <= 8 + (1 + (1 + z0'') + (1 + (1 + z1') + z2)) * (1 + (1 + z0'') + (1 + (1 + z1') + z2)) + 4 * (1 + (1 + z0'') + (1 + (1 + z1') + z2)), s26 >= 0, s26 <= 1 + z1' + 1, s10 >= 0, s10 <= 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 + s82 + s27 :|: s82 >= 0, s82 <= 8 + (1 + z0 + (1 + z1 + z2)) * (1 + z0 + (1 + z1 + z2)) + 4 * (1 + z0 + (1 + z1 + z2)), s27 >= 0, s27 <= 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 + s40 + s' :|: s40 >= 0, s40 <= 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 + s41 + s'' :|: s41 >= 0, s41 <= 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 + s42 + s1 :|: s42 >= 0, s42 <= 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 + s43 + s2 :|: s43 >= 0, s43 <= 7 * (1 + (1 + z1'') + z3) + (1 + (1 + z1'') + z3) * (1 + (1 + z1'') + z3) + 7, s28 >= 0, s28 <= 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 + s44 + s3 :|: s44 >= 0, s44 <= 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
   SELSORT(z) -{ 1 }-> 0 :|: z = 0
   SELSORT(z) -{ 48394 }-> 1 + s120 + s4 + s85 :|: s120 >= 0, s120 <= inf, s85 >= 0, s85 <= 99 * (1 + 0 + 0) + 26 * ((1 + 0 + 0) * (1 + 0 + 0)) + 115, s29 >= 0, s29 <= 2, s4 >= 0, s4 <= 0 + 2, z = 1 + 0 + 0
   SELSORT(z) -{ 904 + 25712*z + 18216*z^2 + 3562*z^3 }-> 1 + s121 + s5 + s86 :|: s121 >= 0, s121 <= inf', s86 >= 0, s86 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s30 >= 0, s30 <= 2, s5 >= 0, s5 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 905 + 25711*z + 18216*z^2 + 3562*z^3 }-> 1 + s122 + s6 + s87 :|: s122 >= 0, s122 <= inf'', s87 >= 0, s87 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s31 >= 0, s31 <= 2, s6 >= 0, s6 <= 0 + 2, z - 2 >= 0
   SELSORT(z) -{ 153687 + s54 + 141319*z0 + 79176*z0*z11 + 21372*z0*z11*z2' + 10686*z0*z11^2 + 79176*z0*z2' + 10686*z0*z2'^2 + 39588*z0^2 + 10686*z0^2*z11 + 10686*z0^2*z2' + 3562*z0^3 + 141319*z11 + 79176*z11*z2' + 10686*z11*z2'^2 + 39588*z11^2 + 10686*z11^2*z2' + 3562*z11^3 + 141319*z2' + 39588*z2'^2 + 3562*z2'^3 }-> 1 + s123 + s55 + s88 :|: s123 >= 0, s123 <= inf1, s88 >= 0, s88 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s52 >= 0, s52 <= 1 + z0 + (1 + z11 + z2'), s53 >= 0, s53 <= 2, s54 >= 0, s54 <= 1 + z0 + (1 + z11 + z2'), s55 >= 0, s55 <= s54 + 2, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 153687 + 141319*z0 + 79176*z0*z11 + 21372*z0*z11*z2' + 10686*z0*z11^2 + 79176*z0*z2' + 10686*z0*z2'^2 + 39588*z0^2 + 10686*z0^2*z11 + 10686*z0^2*z2' + 3562*z0^3 + 141319*z11 + 79176*z11*z2' + 10686*z11*z2'^2 + 39588*z11^2 + 10686*z11^2*z2' + 3562*z11^3 + 141319*z2' + 39588*z2'^2 + 3562*z2'^3 }-> 1 + s124 + s7 + s89 :|: s124 >= 0, s124 <= inf2, s89 >= 0, s89 <= 99 * (1 + z0 + (1 + z11 + z2')) + 26 * ((1 + z0 + (1 + z11 + z2')) * (1 + z0 + (1 + z11 + z2'))) + 115, s56 >= 0, s56 <= 1 + z0 + (1 + z11 + z2'), s57 >= 0, s57 <= 2, s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0 + 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 904 + 25712*z + 18216*z^2 + 3562*z^3 }-> 1 + s125 + s8 + s90 :|: s125 >= 0, s125 <= inf3, s90 >= 0, s90 <= 99 * (1 + (1 + (z - 2)) + 0) + 26 * ((1 + (1 + (z - 2)) + 0) * (1 + (1 + (z - 2)) + 0)) + 115, s32 >= 0, s32 <= 2, s8 >= 0, s8 <= 1 + (z - 2) + 2, z - 2 >= 0
   SELSORT(z) -{ 153687 + s58 + 141319*z0 + 79176*z0*z12 + 21372*z0*z12*z2'' + 10686*z0*z12^2 + 79176*z0*z2'' + 10686*z0*z2''^2 + 39588*z0^2 + 10686*z0^2*z12 + 10686*z0^2*z2'' + 3562*z0^3 + 141319*z12 + 79176*z12*z2'' + 10686*z12*z2''^2 + 39588*z12^2 + 10686*z12^2*z2'' + 3562*z12^3 + 141319*z2'' + 39588*z2''^2 + 3562*z2''^3 }-> 1 + s126 + s59 + s91 :|: s126 >= 0, s126 <= inf4, s91 >= 0, s91 <= 99 * (1 + z0 + (1 + z12 + z2'')) + 26 * ((1 + z0 + (1 + z12 + z2'')) * (1 + z0 + (1 + z12 + z2''))) + 115, s58 >= 0, s58 <= 1 + z0 + (1 + z12 + z2''), s59 >= 0, s59 <= s58 + 2, s33 >= 0, s33 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 48394 + 72829*z0 + 57804*z0*z1 + 10686*z0*z1^2 + 28902*z0^2 + 10686*z0^2*z1 + 3562*z0^3 + 72829*z1 + 28902*z1^2 + 3562*z1^3 }-> 1 + s127 + s9 + s92 :|: s127 >= 0, s127 <= inf5, s92 >= 0, s92 <= 99 * (1 + z0 + z1) + 26 * ((1 + z0 + z1) * (1 + z0 + z1)) + 115, s34 >= 0, s34 <= 2, s9 >= 0, s9 <= 0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eq(z, z') -{ 0 }-> s35 :|: s35 >= 0, s35 <= 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
   ifmin(z, z') -{ 0 }-> s70 :|: s70 >= 0, s70 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> s71 :|: s71 >= 0, s71 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifrepl(z, z', z'', z4) -{ 0 }-> 1 + z2 + s115 :|: s115 >= 0, s115 <= z'' + z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   ifselsort(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s74 + selsort(s116) :|: s116 >= 0, s116 <= 0 + 0, s74 >= 0, s74 <= 1 + 0 + 0, z' = 1 + 0 + 0, z = 1
   ifselsort(z, z') -{ 0 }-> 1 + s75 + selsort(s117) :|: s117 >= 0, s117 <= 1 + (z' - 2) + 0, s75 >= 0, s75 <= 1 + (1 + (z' - 2)) + 0, z = 1, z' - 2 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s76 + selsort(s118) :|: s118 >= 0, s118 <= z0 + (1 + z18 + z24), s76 >= 0, s76 <= 1 + z0 + (1 + z18 + z24), s77 >= 0, s77 <= 1 + z0 + (1 + z18 + z24), s22 >= 0, s22 <= 2, z18 >= 0, z = 1, z' = 1 + z0 + (1 + z18 + z24), z0 >= 0, z24 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + s78 + selsort(s119) :|: s119 >= 0, s119 <= z0 + z1, s78 >= 0, s78 <= 1 + z0 + z1, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   ifselsort(z, z') -{ 0 }-> 1 + z0 + selsort(z1) :|: z = 2, z' = 1 + z0 + z1, z1 >= 0, z0 >= 0
   le(z, z') -{ 0 }-> s19 :|: s19 >= 0, s19 <= 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 }-> s66 :|: s66 >= 0, s66 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s67 :|: s67 >= 0, s67 <= 1 + (1 + z08) + (1 + 0 + z2), z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> s68 :|: s68 >= 0, s68 <= 1 + (1 + z09) + (1 + (1 + z15) + z2), s20 >= 0, s20 <= 2, z = 1 + (1 + z09) + (1 + (1 + z15) + z2), z15 >= 0, z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> s69 :|: s69 >= 0, s69 <= 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 }-> s110 :|: s110 >= 0, s110 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s111 :|: s111 >= 0, s111 <= z' + (1 + (1 + z010) + z3), z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s112 :|: s112 >= 0, s112 <= z' + (1 + 0 + (z'' - 1)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s113 :|: s113 >= 0, s113 <= z' + (1 + (1 + z16) + z3), s36 >= 0, s36 <= 2, z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s114 :|: s114 >= 0, s114 <= 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
   selsort(z) -{ 0 }-> ifselsort(s37, 1 + 0 + 0) :|: s37 >= 0, s37 <= 2, z = 1 + 0 + 0
   selsort(z) -{ 0 }-> ifselsort(s38, 1 + (1 + (z - 2)) + 0) :|: s38 >= 0, s38 <= 2, z - 2 >= 0
   selsort(z) -{ 0 }-> ifselsort(s39, 1 + z0 + z1) :|: s39 >= 0, s39 <= 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   selsort(z) -{ 0 }-> ifselsort(s73, 1 + z0 + (1 + z17 + z23)) :|: s72 >= 0, s72 <= 1 + z0 + (1 + z17 + z23), s73 >= 0, s73 <= 2, s21 >= 0, s21 <= 2, z = 1 + z0 + (1 + z17 + z23), z17 >= 0, z23 >= 0, z0 >= 0
   selsort(z) -{ 0 }-> 0 :|: z = 0
   selsort(z) -{ 0 }-> 0 :|: z >= 0

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]
  IFREPL: runtime: O(n^2) [14 + 7*z4 + z4^2], size: O(n^2) [7 + 7*z4 + z4^2]
  min: runtime: O(1) [0], size: O(n^1) [z]
  ifmin: runtime: O(1) [0], size: O(n^1) [z']
  IFMIN: 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]
  replace: runtime: O(1) [0], size: O(n^1) [z' + z'']
  ifrepl: runtime: O(1) [0], size: O(n^1) [z'' + z4]
  IFSELSORT: runtime: O(n^3) [747 + 25592*z' + 18190*z'^2 + 3562*z'^3], size: INF
  SELSORT: runtime: O(n^3) [560567 + 1494670*z + 1239132*z^2 + 327704*z^3], size: INF
  ifselsort: runtime: O(1) [0], size: O(n^2) [3 + 3*z' + z'^2]
  selsort: runtime: O(1) [0], size: O(n^2) [30 + 34*z + 14*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(IFMIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
   IFMIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
   IFMIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
   REPLACE(z0, z1, nil) -> c12
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IFREPL(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
   IFREPL(true, z0, z1, cons(z2, z3)) -> c14
   IFREPL(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
   SELSORT(nil) -> c16
   SELSORT(cons(z0, z1)) -> c17(IFSELSORT(eq(z0, min(cons(z0, z1))), cons(z0, z1)), EQ(z0, min(cons(z0, z1))), MIN(cons(z0, z1)))
   IFSELSORT(true, cons(z0, z1)) -> c18(SELSORT(z1))
   IFSELSORT(false, cons(z0, z1)) -> c19(MIN(cons(z0, z1)))
   IFSELSORT(false, cons(z0, z1)) -> c20(SELSORT(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))) -> ifmin(le(z0, z1), cons(z0, cons(z1, z2)))
   ifmin(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
   ifmin(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
   replace(z0, z1, nil) -> nil
   replace(z0, z1, cons(z2, z3)) -> ifrepl(eq(z0, z2), z0, z1, cons(z2, z3))
   ifrepl(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
   ifrepl(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
   selsort(nil) -> nil
   selsort(cons(z0, z1)) -> ifselsort(eq(z0, min(cons(z0, z1))), cons(z0, z1))
   ifselsort(true, cons(z0, z1)) -> cons(z0, selsort(z1))
   ifselsort(false, cons(z0, z1)) -> cons(min(cons(z0, z1)), selsort(replace(min(cons(z0, z1)), z0, z1)))

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

(81) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
REPLACE/1
IFREPL/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(IFMIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
   IFMIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
   IFMIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
   REPLACE(z0, nil) -> c12
   REPLACE(z0, cons(z2, z3)) -> c13(IFREPL(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
   IFREPL(true, z0, cons(z2, z3)) -> c14
   IFREPL(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
   SELSORT(nil) -> c16
   SELSORT(cons(z0, z1)) -> c17(IFSELSORT(eq(z0, min(cons(z0, z1))), cons(z0, z1)), EQ(z0, min(cons(z0, z1))), MIN(cons(z0, z1)))
   IFSELSORT(true, cons(z0, z1)) -> c18(SELSORT(z1))
   IFSELSORT(false, cons(z0, z1)) -> c19(MIN(cons(z0, z1)))
   IFSELSORT(false, cons(z0, z1)) -> c20(SELSORT(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))) -> ifmin(le(z0, z1), cons(z0, cons(z1, z2)))
   ifmin(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
   ifmin(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
   replace(z0, z1, nil) -> nil
   replace(z0, z1, cons(z2, z3)) -> ifrepl(eq(z0, z2), z0, z1, cons(z2, z3))
   ifrepl(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
   ifrepl(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
   selsort(nil) -> nil
   selsort(cons(z0, z1)) -> ifselsort(eq(z0, min(cons(z0, z1))), cons(z0, z1))
   ifselsort(true, cons(z0, z1)) -> cons(z0, selsort(z1))
   ifselsort(false, cons(z0, z1)) -> cons(min(cons(z0, z1)), selsort(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(IFMIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IFMIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IFMIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, nil) -> c12
REPLACE(z0, cons(z2, z3)) -> c13(IFREPL(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
IFREPL(true, z0, cons(z2, z3)) -> c14
IFREPL(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
SELSORT(nil) -> c16
SELSORT(cons(z0, z1)) -> c17(IFSELSORT(eq(z0, min(cons(z0, z1))), cons(z0, z1)), EQ(z0, min(cons(z0, z1))), MIN(cons(z0, z1)))
IFSELSORT(true, cons(z0, z1)) -> c18(SELSORT(z1))
IFSELSORT(false, cons(z0, z1)) -> c19(MIN(cons(z0, z1)))
IFSELSORT(false, cons(z0, z1)) -> c20(SELSORT(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))) -> ifmin(le(z0, z1), cons(z0, cons(z1, z2)))
ifmin(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
ifmin(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> ifrepl(eq(z0, z2), z0, z1, cons(z2, z3))
ifrepl(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
ifrepl(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
selsort(nil) -> nil
selsort(cons(z0, z1)) -> ifselsort(eq(z0, min(cons(z0, z1))), cons(z0, z1))
ifselsort(true, cons(z0, z1)) -> cons(z0, selsort(z1))
ifselsort(false, cons(z0, z1)) -> cons(min(cons(z0, z1)), selsort(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
IFMIN :: 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
IFREPL :: true:false -> 0':s -> nil:cons -> c14:c15
eq :: 0':s -> 0':s -> true:false
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SELSORT :: nil:cons -> c16:c17
c16 :: c16:c17
c17 :: c18:c19:c20 -> c:c1:c2:c3 -> c7:c8:c9 -> c16:c17
IFSELSORT :: true:false -> nil:cons -> c18:c19:c20
min :: nil:cons -> 0':s
c18 :: c16:c17 -> c18:c19:c20
c19 :: c7:c8:c9 -> c18:c19:c20
c20 :: c16:c17 -> c12:c13 -> c7:c8:c9 -> c18:c19:c20
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
ifmin :: true:false -> nil:cons -> 0':s
ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
selsort :: nil:cons -> nil:cons
ifselsort :: true:false -> nil:cons -> nil:cons
hole_c:c1:c2:c31_21 :: c:c1:c2:c3
hole_0':s2_21 :: 0':s
hole_c4:c5:c63_21 :: c4:c5:c6
hole_c7:c8:c94_21 :: c7:c8:c9
hole_nil:cons5_21 :: nil:cons
hole_c10:c116_21 :: c10:c11
hole_true:false7_21 :: true:false
hole_c12:c138_21 :: c12:c13
hole_c14:c159_21 :: c14:c15
hole_c16:c1710_21 :: c16:c17
hole_c18:c19:c2011_21 :: c18:c19:c20
gen_c:c1:c2:c312_21 :: Nat -> c:c1:c2:c3
gen_0':s13_21 :: Nat -> 0':s
gen_c4:c5:c614_21 :: Nat -> c4:c5:c6
gen_nil:cons15_21 :: Nat -> nil:cons

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

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

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

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

(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(IFMIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IFMIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IFMIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, nil) -> c12
REPLACE(z0, cons(z2, z3)) -> c13(IFREPL(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
IFREPL(true, z0, cons(z2, z3)) -> c14
IFREPL(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
SELSORT(nil) -> c16
SELSORT(cons(z0, z1)) -> c17(IFSELSORT(eq(z0, min(cons(z0, z1))), cons(z0, z1)), EQ(z0, min(cons(z0, z1))), MIN(cons(z0, z1)))
IFSELSORT(true, cons(z0, z1)) -> c18(SELSORT(z1))
IFSELSORT(false, cons(z0, z1)) -> c19(MIN(cons(z0, z1)))
IFSELSORT(false, cons(z0, z1)) -> c20(SELSORT(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))) -> ifmin(le(z0, z1), cons(z0, cons(z1, z2)))
ifmin(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
ifmin(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> ifrepl(eq(z0, z2), z0, z1, cons(z2, z3))
ifrepl(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
ifrepl(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
selsort(nil) -> nil
selsort(cons(z0, z1)) -> ifselsort(eq(z0, min(cons(z0, z1))), cons(z0, z1))
ifselsort(true, cons(z0, z1)) -> cons(z0, selsort(z1))
ifselsort(false, cons(z0, z1)) -> cons(min(cons(z0, z1)), selsort(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
IFMIN :: 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
IFREPL :: true:false -> 0':s -> nil:cons -> c14:c15
eq :: 0':s -> 0':s -> true:false
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SELSORT :: nil:cons -> c16:c17
c16 :: c16:c17
c17 :: c18:c19:c20 -> c:c1:c2:c3 -> c7:c8:c9 -> c16:c17
IFSELSORT :: true:false -> nil:cons -> c18:c19:c20
min :: nil:cons -> 0':s
c18 :: c16:c17 -> c18:c19:c20
c19 :: c7:c8:c9 -> c18:c19:c20
c20 :: c16:c17 -> c12:c13 -> c7:c8:c9 -> c18:c19:c20
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
ifmin :: true:false -> nil:cons -> 0':s
ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
selsort :: nil:cons -> nil:cons
ifselsort :: true:false -> nil:cons -> nil:cons
hole_c:c1:c2:c31_21 :: c:c1:c2:c3
hole_0':s2_21 :: 0':s
hole_c4:c5:c63_21 :: c4:c5:c6
hole_c7:c8:c94_21 :: c7:c8:c9
hole_nil:cons5_21 :: nil:cons
hole_c10:c116_21 :: c10:c11
hole_true:false7_21 :: true:false
hole_c12:c138_21 :: c12:c13
hole_c14:c159_21 :: c14:c15
hole_c16:c1710_21 :: c16:c17
hole_c18:c19:c2011_21 :: c18:c19:c20
gen_c:c1:c2:c312_21 :: Nat -> c:c1:c2:c3
gen_0':s13_21 :: Nat -> 0':s
gen_c4:c5:c614_21 :: Nat -> c4:c5:c6
gen_nil:cons15_21 :: Nat -> nil:cons


Generator Equations:
gen_c:c1:c2:c312_21(0) <=> c
gen_c:c1:c2:c312_21(+(x, 1)) <=> c3(gen_c:c1:c2:c312_21(x))
gen_0':s13_21(0) <=> 0'
gen_0':s13_21(+(x, 1)) <=> s(gen_0':s13_21(x))
gen_c4:c5:c614_21(0) <=> c4
gen_c4:c5:c614_21(+(x, 1)) <=> c6(gen_c4:c5:c614_21(x))
gen_nil:cons15_21(0) <=> nil
gen_nil:cons15_21(+(x, 1)) <=> cons(0', gen_nil:cons15_21(x))


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

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

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

(87) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
EQ(gen_0':s13_21(n17_21), gen_0':s13_21(n17_21)) -> gen_c:c1:c2:c312_21(n17_21), rt in Omega(1 + n17_21)

Induction Base:
EQ(gen_0':s13_21(0), gen_0':s13_21(0)) ->_R^Omega(1)
c

Induction Step:
EQ(gen_0':s13_21(+(n17_21, 1)), gen_0':s13_21(+(n17_21, 1))) ->_R^Omega(1)
c3(EQ(gen_0':s13_21(n17_21), gen_0':s13_21(n17_21))) ->_IH
c3(gen_c:c1:c2:c312_21(c18_21))

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(IFMIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IFMIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IFMIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, nil) -> c12
REPLACE(z0, cons(z2, z3)) -> c13(IFREPL(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
IFREPL(true, z0, cons(z2, z3)) -> c14
IFREPL(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
SELSORT(nil) -> c16
SELSORT(cons(z0, z1)) -> c17(IFSELSORT(eq(z0, min(cons(z0, z1))), cons(z0, z1)), EQ(z0, min(cons(z0, z1))), MIN(cons(z0, z1)))
IFSELSORT(true, cons(z0, z1)) -> c18(SELSORT(z1))
IFSELSORT(false, cons(z0, z1)) -> c19(MIN(cons(z0, z1)))
IFSELSORT(false, cons(z0, z1)) -> c20(SELSORT(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))) -> ifmin(le(z0, z1), cons(z0, cons(z1, z2)))
ifmin(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
ifmin(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> ifrepl(eq(z0, z2), z0, z1, cons(z2, z3))
ifrepl(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
ifrepl(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
selsort(nil) -> nil
selsort(cons(z0, z1)) -> ifselsort(eq(z0, min(cons(z0, z1))), cons(z0, z1))
ifselsort(true, cons(z0, z1)) -> cons(z0, selsort(z1))
ifselsort(false, cons(z0, z1)) -> cons(min(cons(z0, z1)), selsort(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
IFMIN :: 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
IFREPL :: true:false -> 0':s -> nil:cons -> c14:c15
eq :: 0':s -> 0':s -> true:false
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SELSORT :: nil:cons -> c16:c17
c16 :: c16:c17
c17 :: c18:c19:c20 -> c:c1:c2:c3 -> c7:c8:c9 -> c16:c17
IFSELSORT :: true:false -> nil:cons -> c18:c19:c20
min :: nil:cons -> 0':s
c18 :: c16:c17 -> c18:c19:c20
c19 :: c7:c8:c9 -> c18:c19:c20
c20 :: c16:c17 -> c12:c13 -> c7:c8:c9 -> c18:c19:c20
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
ifmin :: true:false -> nil:cons -> 0':s
ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
selsort :: nil:cons -> nil:cons
ifselsort :: true:false -> nil:cons -> nil:cons
hole_c:c1:c2:c31_21 :: c:c1:c2:c3
hole_0':s2_21 :: 0':s
hole_c4:c5:c63_21 :: c4:c5:c6
hole_c7:c8:c94_21 :: c7:c8:c9
hole_nil:cons5_21 :: nil:cons
hole_c10:c116_21 :: c10:c11
hole_true:false7_21 :: true:false
hole_c12:c138_21 :: c12:c13
hole_c14:c159_21 :: c14:c15
hole_c16:c1710_21 :: c16:c17
hole_c18:c19:c2011_21 :: c18:c19:c20
gen_c:c1:c2:c312_21 :: Nat -> c:c1:c2:c3
gen_0':s13_21 :: Nat -> 0':s
gen_c4:c5:c614_21 :: Nat -> c4:c5:c6
gen_nil:cons15_21 :: Nat -> nil:cons


Generator Equations:
gen_c:c1:c2:c312_21(0) <=> c
gen_c:c1:c2:c312_21(+(x, 1)) <=> c3(gen_c:c1:c2:c312_21(x))
gen_0':s13_21(0) <=> 0'
gen_0':s13_21(+(x, 1)) <=> s(gen_0':s13_21(x))
gen_c4:c5:c614_21(0) <=> c4
gen_c4:c5:c614_21(+(x, 1)) <=> c6(gen_c4:c5:c614_21(x))
gen_nil:cons15_21(0) <=> nil
gen_nil:cons15_21(+(x, 1)) <=> cons(0', gen_nil:cons15_21(x))


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

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

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

(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(IFMIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IFMIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IFMIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, nil) -> c12
REPLACE(z0, cons(z2, z3)) -> c13(IFREPL(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
IFREPL(true, z0, cons(z2, z3)) -> c14
IFREPL(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
SELSORT(nil) -> c16
SELSORT(cons(z0, z1)) -> c17(IFSELSORT(eq(z0, min(cons(z0, z1))), cons(z0, z1)), EQ(z0, min(cons(z0, z1))), MIN(cons(z0, z1)))
IFSELSORT(true, cons(z0, z1)) -> c18(SELSORT(z1))
IFSELSORT(false, cons(z0, z1)) -> c19(MIN(cons(z0, z1)))
IFSELSORT(false, cons(z0, z1)) -> c20(SELSORT(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))) -> ifmin(le(z0, z1), cons(z0, cons(z1, z2)))
ifmin(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
ifmin(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> ifrepl(eq(z0, z2), z0, z1, cons(z2, z3))
ifrepl(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
ifrepl(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
selsort(nil) -> nil
selsort(cons(z0, z1)) -> ifselsort(eq(z0, min(cons(z0, z1))), cons(z0, z1))
ifselsort(true, cons(z0, z1)) -> cons(z0, selsort(z1))
ifselsort(false, cons(z0, z1)) -> cons(min(cons(z0, z1)), selsort(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
IFMIN :: 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
IFREPL :: true:false -> 0':s -> nil:cons -> c14:c15
eq :: 0':s -> 0':s -> true:false
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SELSORT :: nil:cons -> c16:c17
c16 :: c16:c17
c17 :: c18:c19:c20 -> c:c1:c2:c3 -> c7:c8:c9 -> c16:c17
IFSELSORT :: true:false -> nil:cons -> c18:c19:c20
min :: nil:cons -> 0':s
c18 :: c16:c17 -> c18:c19:c20
c19 :: c7:c8:c9 -> c18:c19:c20
c20 :: c16:c17 -> c12:c13 -> c7:c8:c9 -> c18:c19:c20
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
ifmin :: true:false -> nil:cons -> 0':s
ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
selsort :: nil:cons -> nil:cons
ifselsort :: true:false -> nil:cons -> nil:cons
hole_c:c1:c2:c31_21 :: c:c1:c2:c3
hole_0':s2_21 :: 0':s
hole_c4:c5:c63_21 :: c4:c5:c6
hole_c7:c8:c94_21 :: c7:c8:c9
hole_nil:cons5_21 :: nil:cons
hole_c10:c116_21 :: c10:c11
hole_true:false7_21 :: true:false
hole_c12:c138_21 :: c12:c13
hole_c14:c159_21 :: c14:c15
hole_c16:c1710_21 :: c16:c17
hole_c18:c19:c2011_21 :: c18:c19:c20
gen_c:c1:c2:c312_21 :: Nat -> c:c1:c2:c3
gen_0':s13_21 :: Nat -> 0':s
gen_c4:c5:c614_21 :: Nat -> c4:c5:c6
gen_nil:cons15_21 :: Nat -> nil:cons


Lemmas:
EQ(gen_0':s13_21(n17_21), gen_0':s13_21(n17_21)) -> gen_c:c1:c2:c312_21(n17_21), rt in Omega(1 + n17_21)


Generator Equations:
gen_c:c1:c2:c312_21(0) <=> c
gen_c:c1:c2:c312_21(+(x, 1)) <=> c3(gen_c:c1:c2:c312_21(x))
gen_0':s13_21(0) <=> 0'
gen_0':s13_21(+(x, 1)) <=> s(gen_0':s13_21(x))
gen_c4:c5:c614_21(0) <=> c4
gen_c4:c5:c614_21(+(x, 1)) <=> c6(gen_c4:c5:c614_21(x))
gen_nil:cons15_21(0) <=> nil
gen_nil:cons15_21(+(x, 1)) <=> cons(0', gen_nil:cons15_21(x))


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

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

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

(93) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LE(gen_0':s13_21(n1041_21), gen_0':s13_21(n1041_21)) -> gen_c4:c5:c614_21(n1041_21), rt in Omega(1 + n1041_21)

Induction Base:
LE(gen_0':s13_21(0), gen_0':s13_21(0)) ->_R^Omega(1)
c4

Induction Step:
LE(gen_0':s13_21(+(n1041_21, 1)), gen_0':s13_21(+(n1041_21, 1))) ->_R^Omega(1)
c6(LE(gen_0':s13_21(n1041_21), gen_0':s13_21(n1041_21))) ->_IH
c6(gen_c4:c5:c614_21(c1042_21))

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(IFMIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IFMIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IFMIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, nil) -> c12
REPLACE(z0, cons(z2, z3)) -> c13(IFREPL(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
IFREPL(true, z0, cons(z2, z3)) -> c14
IFREPL(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
SELSORT(nil) -> c16
SELSORT(cons(z0, z1)) -> c17(IFSELSORT(eq(z0, min(cons(z0, z1))), cons(z0, z1)), EQ(z0, min(cons(z0, z1))), MIN(cons(z0, z1)))
IFSELSORT(true, cons(z0, z1)) -> c18(SELSORT(z1))
IFSELSORT(false, cons(z0, z1)) -> c19(MIN(cons(z0, z1)))
IFSELSORT(false, cons(z0, z1)) -> c20(SELSORT(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))) -> ifmin(le(z0, z1), cons(z0, cons(z1, z2)))
ifmin(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
ifmin(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> ifrepl(eq(z0, z2), z0, z1, cons(z2, z3))
ifrepl(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
ifrepl(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
selsort(nil) -> nil
selsort(cons(z0, z1)) -> ifselsort(eq(z0, min(cons(z0, z1))), cons(z0, z1))
ifselsort(true, cons(z0, z1)) -> cons(z0, selsort(z1))
ifselsort(false, cons(z0, z1)) -> cons(min(cons(z0, z1)), selsort(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
IFMIN :: 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
IFREPL :: true:false -> 0':s -> nil:cons -> c14:c15
eq :: 0':s -> 0':s -> true:false
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SELSORT :: nil:cons -> c16:c17
c16 :: c16:c17
c17 :: c18:c19:c20 -> c:c1:c2:c3 -> c7:c8:c9 -> c16:c17
IFSELSORT :: true:false -> nil:cons -> c18:c19:c20
min :: nil:cons -> 0':s
c18 :: c16:c17 -> c18:c19:c20
c19 :: c7:c8:c9 -> c18:c19:c20
c20 :: c16:c17 -> c12:c13 -> c7:c8:c9 -> c18:c19:c20
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
ifmin :: true:false -> nil:cons -> 0':s
ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
selsort :: nil:cons -> nil:cons
ifselsort :: true:false -> nil:cons -> nil:cons
hole_c:c1:c2:c31_21 :: c:c1:c2:c3
hole_0':s2_21 :: 0':s
hole_c4:c5:c63_21 :: c4:c5:c6
hole_c7:c8:c94_21 :: c7:c8:c9
hole_nil:cons5_21 :: nil:cons
hole_c10:c116_21 :: c10:c11
hole_true:false7_21 :: true:false
hole_c12:c138_21 :: c12:c13
hole_c14:c159_21 :: c14:c15
hole_c16:c1710_21 :: c16:c17
hole_c18:c19:c2011_21 :: c18:c19:c20
gen_c:c1:c2:c312_21 :: Nat -> c:c1:c2:c3
gen_0':s13_21 :: Nat -> 0':s
gen_c4:c5:c614_21 :: Nat -> c4:c5:c6
gen_nil:cons15_21 :: Nat -> nil:cons


Lemmas:
EQ(gen_0':s13_21(n17_21), gen_0':s13_21(n17_21)) -> gen_c:c1:c2:c312_21(n17_21), rt in Omega(1 + n17_21)
LE(gen_0':s13_21(n1041_21), gen_0':s13_21(n1041_21)) -> gen_c4:c5:c614_21(n1041_21), rt in Omega(1 + n1041_21)


Generator Equations:
gen_c:c1:c2:c312_21(0) <=> c
gen_c:c1:c2:c312_21(+(x, 1)) <=> c3(gen_c:c1:c2:c312_21(x))
gen_0':s13_21(0) <=> 0'
gen_0':s13_21(+(x, 1)) <=> s(gen_0':s13_21(x))
gen_c4:c5:c614_21(0) <=> c4
gen_c4:c5:c614_21(+(x, 1)) <=> c6(gen_c4:c5:c614_21(x))
gen_nil:cons15_21(0) <=> nil
gen_nil:cons15_21(+(x, 1)) <=> cons(0', gen_nil:cons15_21(x))


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

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

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

(95) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s13_21(n1839_21), gen_0':s13_21(n1839_21)) -> true, rt in Omega(0)

Induction Base:
le(gen_0':s13_21(0), gen_0':s13_21(0)) ->_R^Omega(0)
true

Induction Step:
le(gen_0':s13_21(+(n1839_21, 1)), gen_0':s13_21(+(n1839_21, 1))) ->_R^Omega(0)
le(gen_0':s13_21(n1839_21), gen_0':s13_21(n1839_21)) ->_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(IFMIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IFMIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IFMIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, nil) -> c12
REPLACE(z0, cons(z2, z3)) -> c13(IFREPL(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
IFREPL(true, z0, cons(z2, z3)) -> c14
IFREPL(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
SELSORT(nil) -> c16
SELSORT(cons(z0, z1)) -> c17(IFSELSORT(eq(z0, min(cons(z0, z1))), cons(z0, z1)), EQ(z0, min(cons(z0, z1))), MIN(cons(z0, z1)))
IFSELSORT(true, cons(z0, z1)) -> c18(SELSORT(z1))
IFSELSORT(false, cons(z0, z1)) -> c19(MIN(cons(z0, z1)))
IFSELSORT(false, cons(z0, z1)) -> c20(SELSORT(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))) -> ifmin(le(z0, z1), cons(z0, cons(z1, z2)))
ifmin(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
ifmin(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> ifrepl(eq(z0, z2), z0, z1, cons(z2, z3))
ifrepl(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
ifrepl(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
selsort(nil) -> nil
selsort(cons(z0, z1)) -> ifselsort(eq(z0, min(cons(z0, z1))), cons(z0, z1))
ifselsort(true, cons(z0, z1)) -> cons(z0, selsort(z1))
ifselsort(false, cons(z0, z1)) -> cons(min(cons(z0, z1)), selsort(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
IFMIN :: 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
IFREPL :: true:false -> 0':s -> nil:cons -> c14:c15
eq :: 0':s -> 0':s -> true:false
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SELSORT :: nil:cons -> c16:c17
c16 :: c16:c17
c17 :: c18:c19:c20 -> c:c1:c2:c3 -> c7:c8:c9 -> c16:c17
IFSELSORT :: true:false -> nil:cons -> c18:c19:c20
min :: nil:cons -> 0':s
c18 :: c16:c17 -> c18:c19:c20
c19 :: c7:c8:c9 -> c18:c19:c20
c20 :: c16:c17 -> c12:c13 -> c7:c8:c9 -> c18:c19:c20
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
ifmin :: true:false -> nil:cons -> 0':s
ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
selsort :: nil:cons -> nil:cons
ifselsort :: true:false -> nil:cons -> nil:cons
hole_c:c1:c2:c31_21 :: c:c1:c2:c3
hole_0':s2_21 :: 0':s
hole_c4:c5:c63_21 :: c4:c5:c6
hole_c7:c8:c94_21 :: c7:c8:c9
hole_nil:cons5_21 :: nil:cons
hole_c10:c116_21 :: c10:c11
hole_true:false7_21 :: true:false
hole_c12:c138_21 :: c12:c13
hole_c14:c159_21 :: c14:c15
hole_c16:c1710_21 :: c16:c17
hole_c18:c19:c2011_21 :: c18:c19:c20
gen_c:c1:c2:c312_21 :: Nat -> c:c1:c2:c3
gen_0':s13_21 :: Nat -> 0':s
gen_c4:c5:c614_21 :: Nat -> c4:c5:c6
gen_nil:cons15_21 :: Nat -> nil:cons


Lemmas:
EQ(gen_0':s13_21(n17_21), gen_0':s13_21(n17_21)) -> gen_c:c1:c2:c312_21(n17_21), rt in Omega(1 + n17_21)
LE(gen_0':s13_21(n1041_21), gen_0':s13_21(n1041_21)) -> gen_c4:c5:c614_21(n1041_21), rt in Omega(1 + n1041_21)
le(gen_0':s13_21(n1839_21), gen_0':s13_21(n1839_21)) -> true, rt in Omega(0)


Generator Equations:
gen_c:c1:c2:c312_21(0) <=> c
gen_c:c1:c2:c312_21(+(x, 1)) <=> c3(gen_c:c1:c2:c312_21(x))
gen_0':s13_21(0) <=> 0'
gen_0':s13_21(+(x, 1)) <=> s(gen_0':s13_21(x))
gen_c4:c5:c614_21(0) <=> c4
gen_c4:c5:c614_21(+(x, 1)) <=> c6(gen_c4:c5:c614_21(x))
gen_nil:cons15_21(0) <=> nil
gen_nil:cons15_21(+(x, 1)) <=> cons(0', gen_nil:cons15_21(x))


The following defined symbols remain to be analysed:
MIN, REPLACE, eq, SELSORT, min, replace, selsort

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

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

(97) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MIN(gen_nil:cons15_21(+(1, n2326_21))) -> *16_21, rt in Omega(n2326_21)

Induction Base:
MIN(gen_nil:cons15_21(+(1, 0)))

Induction Step:
MIN(gen_nil:cons15_21(+(1, +(n2326_21, 1)))) ->_R^Omega(1)
c9(IFMIN(le(0', 0'), cons(0', cons(0', gen_nil:cons15_21(n2326_21)))), LE(0', 0')) ->_L^Omega(0)
c9(IFMIN(true, cons(0', cons(0', gen_nil:cons15_21(n2326_21)))), LE(0', 0')) ->_R^Omega(1)
c9(c10(MIN(cons(0', gen_nil:cons15_21(n2326_21)))), LE(0', 0')) ->_IH
c9(c10(*16_21), LE(0', 0')) ->_L^Omega(1)
c9(c10(*16_21), gen_c4:c5:c614_21(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(IFMIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IFMIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IFMIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, nil) -> c12
REPLACE(z0, cons(z2, z3)) -> c13(IFREPL(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
IFREPL(true, z0, cons(z2, z3)) -> c14
IFREPL(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
SELSORT(nil) -> c16
SELSORT(cons(z0, z1)) -> c17(IFSELSORT(eq(z0, min(cons(z0, z1))), cons(z0, z1)), EQ(z0, min(cons(z0, z1))), MIN(cons(z0, z1)))
IFSELSORT(true, cons(z0, z1)) -> c18(SELSORT(z1))
IFSELSORT(false, cons(z0, z1)) -> c19(MIN(cons(z0, z1)))
IFSELSORT(false, cons(z0, z1)) -> c20(SELSORT(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))) -> ifmin(le(z0, z1), cons(z0, cons(z1, z2)))
ifmin(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
ifmin(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> ifrepl(eq(z0, z2), z0, z1, cons(z2, z3))
ifrepl(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
ifrepl(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
selsort(nil) -> nil
selsort(cons(z0, z1)) -> ifselsort(eq(z0, min(cons(z0, z1))), cons(z0, z1))
ifselsort(true, cons(z0, z1)) -> cons(z0, selsort(z1))
ifselsort(false, cons(z0, z1)) -> cons(min(cons(z0, z1)), selsort(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
IFMIN :: 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
IFREPL :: true:false -> 0':s -> nil:cons -> c14:c15
eq :: 0':s -> 0':s -> true:false
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SELSORT :: nil:cons -> c16:c17
c16 :: c16:c17
c17 :: c18:c19:c20 -> c:c1:c2:c3 -> c7:c8:c9 -> c16:c17
IFSELSORT :: true:false -> nil:cons -> c18:c19:c20
min :: nil:cons -> 0':s
c18 :: c16:c17 -> c18:c19:c20
c19 :: c7:c8:c9 -> c18:c19:c20
c20 :: c16:c17 -> c12:c13 -> c7:c8:c9 -> c18:c19:c20
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
ifmin :: true:false -> nil:cons -> 0':s
ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
selsort :: nil:cons -> nil:cons
ifselsort :: true:false -> nil:cons -> nil:cons
hole_c:c1:c2:c31_21 :: c:c1:c2:c3
hole_0':s2_21 :: 0':s
hole_c4:c5:c63_21 :: c4:c5:c6
hole_c7:c8:c94_21 :: c7:c8:c9
hole_nil:cons5_21 :: nil:cons
hole_c10:c116_21 :: c10:c11
hole_true:false7_21 :: true:false
hole_c12:c138_21 :: c12:c13
hole_c14:c159_21 :: c14:c15
hole_c16:c1710_21 :: c16:c17
hole_c18:c19:c2011_21 :: c18:c19:c20
gen_c:c1:c2:c312_21 :: Nat -> c:c1:c2:c3
gen_0':s13_21 :: Nat -> 0':s
gen_c4:c5:c614_21 :: Nat -> c4:c5:c6
gen_nil:cons15_21 :: Nat -> nil:cons


Lemmas:
EQ(gen_0':s13_21(n17_21), gen_0':s13_21(n17_21)) -> gen_c:c1:c2:c312_21(n17_21), rt in Omega(1 + n17_21)
LE(gen_0':s13_21(n1041_21), gen_0':s13_21(n1041_21)) -> gen_c4:c5:c614_21(n1041_21), rt in Omega(1 + n1041_21)
le(gen_0':s13_21(n1839_21), gen_0':s13_21(n1839_21)) -> true, rt in Omega(0)
MIN(gen_nil:cons15_21(+(1, n2326_21))) -> *16_21, rt in Omega(n2326_21)


Generator Equations:
gen_c:c1:c2:c312_21(0) <=> c
gen_c:c1:c2:c312_21(+(x, 1)) <=> c3(gen_c:c1:c2:c312_21(x))
gen_0':s13_21(0) <=> 0'
gen_0':s13_21(+(x, 1)) <=> s(gen_0':s13_21(x))
gen_c4:c5:c614_21(0) <=> c4
gen_c4:c5:c614_21(+(x, 1)) <=> c6(gen_c4:c5:c614_21(x))
gen_nil:cons15_21(0) <=> nil
gen_nil:cons15_21(+(x, 1)) <=> cons(0', gen_nil:cons15_21(x))


The following defined symbols remain to be analysed:
eq, REPLACE, SELSORT, min, replace, selsort

They will be analysed ascendingly in the following order:
eq < REPLACE
REPLACE < SELSORT
eq < SELSORT
eq < replace
eq < selsort
min < SELSORT
replace < SELSORT
min < selsort
replace < selsort

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

(99) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
eq(gen_0':s13_21(n6767_21), gen_0':s13_21(n6767_21)) -> true, rt in Omega(0)

Induction Base:
eq(gen_0':s13_21(0), gen_0':s13_21(0)) ->_R^Omega(0)
true

Induction Step:
eq(gen_0':s13_21(+(n6767_21, 1)), gen_0':s13_21(+(n6767_21, 1))) ->_R^Omega(0)
eq(gen_0':s13_21(n6767_21), gen_0':s13_21(n6767_21)) ->_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(IFMIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IFMIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IFMIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, nil) -> c12
REPLACE(z0, cons(z2, z3)) -> c13(IFREPL(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
IFREPL(true, z0, cons(z2, z3)) -> c14
IFREPL(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
SELSORT(nil) -> c16
SELSORT(cons(z0, z1)) -> c17(IFSELSORT(eq(z0, min(cons(z0, z1))), cons(z0, z1)), EQ(z0, min(cons(z0, z1))), MIN(cons(z0, z1)))
IFSELSORT(true, cons(z0, z1)) -> c18(SELSORT(z1))
IFSELSORT(false, cons(z0, z1)) -> c19(MIN(cons(z0, z1)))
IFSELSORT(false, cons(z0, z1)) -> c20(SELSORT(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))) -> ifmin(le(z0, z1), cons(z0, cons(z1, z2)))
ifmin(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
ifmin(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> ifrepl(eq(z0, z2), z0, z1, cons(z2, z3))
ifrepl(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
ifrepl(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
selsort(nil) -> nil
selsort(cons(z0, z1)) -> ifselsort(eq(z0, min(cons(z0, z1))), cons(z0, z1))
ifselsort(true, cons(z0, z1)) -> cons(z0, selsort(z1))
ifselsort(false, cons(z0, z1)) -> cons(min(cons(z0, z1)), selsort(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
IFMIN :: 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
IFREPL :: true:false -> 0':s -> nil:cons -> c14:c15
eq :: 0':s -> 0':s -> true:false
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SELSORT :: nil:cons -> c16:c17
c16 :: c16:c17
c17 :: c18:c19:c20 -> c:c1:c2:c3 -> c7:c8:c9 -> c16:c17
IFSELSORT :: true:false -> nil:cons -> c18:c19:c20
min :: nil:cons -> 0':s
c18 :: c16:c17 -> c18:c19:c20
c19 :: c7:c8:c9 -> c18:c19:c20
c20 :: c16:c17 -> c12:c13 -> c7:c8:c9 -> c18:c19:c20
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
ifmin :: true:false -> nil:cons -> 0':s
ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
selsort :: nil:cons -> nil:cons
ifselsort :: true:false -> nil:cons -> nil:cons
hole_c:c1:c2:c31_21 :: c:c1:c2:c3
hole_0':s2_21 :: 0':s
hole_c4:c5:c63_21 :: c4:c5:c6
hole_c7:c8:c94_21 :: c7:c8:c9
hole_nil:cons5_21 :: nil:cons
hole_c10:c116_21 :: c10:c11
hole_true:false7_21 :: true:false
hole_c12:c138_21 :: c12:c13
hole_c14:c159_21 :: c14:c15
hole_c16:c1710_21 :: c16:c17
hole_c18:c19:c2011_21 :: c18:c19:c20
gen_c:c1:c2:c312_21 :: Nat -> c:c1:c2:c3
gen_0':s13_21 :: Nat -> 0':s
gen_c4:c5:c614_21 :: Nat -> c4:c5:c6
gen_nil:cons15_21 :: Nat -> nil:cons


Lemmas:
EQ(gen_0':s13_21(n17_21), gen_0':s13_21(n17_21)) -> gen_c:c1:c2:c312_21(n17_21), rt in Omega(1 + n17_21)
LE(gen_0':s13_21(n1041_21), gen_0':s13_21(n1041_21)) -> gen_c4:c5:c614_21(n1041_21), rt in Omega(1 + n1041_21)
le(gen_0':s13_21(n1839_21), gen_0':s13_21(n1839_21)) -> true, rt in Omega(0)
MIN(gen_nil:cons15_21(+(1, n2326_21))) -> *16_21, rt in Omega(n2326_21)
eq(gen_0':s13_21(n6767_21), gen_0':s13_21(n6767_21)) -> true, rt in Omega(0)


Generator Equations:
gen_c:c1:c2:c312_21(0) <=> c
gen_c:c1:c2:c312_21(+(x, 1)) <=> c3(gen_c:c1:c2:c312_21(x))
gen_0':s13_21(0) <=> 0'
gen_0':s13_21(+(x, 1)) <=> s(gen_0':s13_21(x))
gen_c4:c5:c614_21(0) <=> c4
gen_c4:c5:c614_21(+(x, 1)) <=> c6(gen_c4:c5:c614_21(x))
gen_nil:cons15_21(0) <=> nil
gen_nil:cons15_21(+(x, 1)) <=> cons(0', gen_nil:cons15_21(x))


The following defined symbols remain to be analysed:
REPLACE, SELSORT, min, replace, selsort

They will be analysed ascendingly in the following order:
REPLACE < SELSORT
min < SELSORT
replace < SELSORT
min < selsort
replace < selsort

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

(101) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
min(gen_nil:cons15_21(+(1, n8012_21))) -> gen_0':s13_21(0), rt in Omega(0)

Induction Base:
min(gen_nil:cons15_21(+(1, 0))) ->_R^Omega(0)
0'

Induction Step:
min(gen_nil:cons15_21(+(1, +(n8012_21, 1)))) ->_R^Omega(0)
ifmin(le(0', 0'), cons(0', cons(0', gen_nil:cons15_21(n8012_21)))) ->_L^Omega(0)
ifmin(true, cons(0', cons(0', gen_nil:cons15_21(n8012_21)))) ->_R^Omega(0)
min(cons(0', gen_nil:cons15_21(n8012_21))) ->_IH
gen_0':s13_21(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(IFMIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IFMIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IFMIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, nil) -> c12
REPLACE(z0, cons(z2, z3)) -> c13(IFREPL(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
IFREPL(true, z0, cons(z2, z3)) -> c14
IFREPL(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
SELSORT(nil) -> c16
SELSORT(cons(z0, z1)) -> c17(IFSELSORT(eq(z0, min(cons(z0, z1))), cons(z0, z1)), EQ(z0, min(cons(z0, z1))), MIN(cons(z0, z1)))
IFSELSORT(true, cons(z0, z1)) -> c18(SELSORT(z1))
IFSELSORT(false, cons(z0, z1)) -> c19(MIN(cons(z0, z1)))
IFSELSORT(false, cons(z0, z1)) -> c20(SELSORT(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))) -> ifmin(le(z0, z1), cons(z0, cons(z1, z2)))
ifmin(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
ifmin(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> ifrepl(eq(z0, z2), z0, z1, cons(z2, z3))
ifrepl(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
ifrepl(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
selsort(nil) -> nil
selsort(cons(z0, z1)) -> ifselsort(eq(z0, min(cons(z0, z1))), cons(z0, z1))
ifselsort(true, cons(z0, z1)) -> cons(z0, selsort(z1))
ifselsort(false, cons(z0, z1)) -> cons(min(cons(z0, z1)), selsort(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
IFMIN :: 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
IFREPL :: true:false -> 0':s -> nil:cons -> c14:c15
eq :: 0':s -> 0':s -> true:false
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SELSORT :: nil:cons -> c16:c17
c16 :: c16:c17
c17 :: c18:c19:c20 -> c:c1:c2:c3 -> c7:c8:c9 -> c16:c17
IFSELSORT :: true:false -> nil:cons -> c18:c19:c20
min :: nil:cons -> 0':s
c18 :: c16:c17 -> c18:c19:c20
c19 :: c7:c8:c9 -> c18:c19:c20
c20 :: c16:c17 -> c12:c13 -> c7:c8:c9 -> c18:c19:c20
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
ifmin :: true:false -> nil:cons -> 0':s
ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
selsort :: nil:cons -> nil:cons
ifselsort :: true:false -> nil:cons -> nil:cons
hole_c:c1:c2:c31_21 :: c:c1:c2:c3
hole_0':s2_21 :: 0':s
hole_c4:c5:c63_21 :: c4:c5:c6
hole_c7:c8:c94_21 :: c7:c8:c9
hole_nil:cons5_21 :: nil:cons
hole_c10:c116_21 :: c10:c11
hole_true:false7_21 :: true:false
hole_c12:c138_21 :: c12:c13
hole_c14:c159_21 :: c14:c15
hole_c16:c1710_21 :: c16:c17
hole_c18:c19:c2011_21 :: c18:c19:c20
gen_c:c1:c2:c312_21 :: Nat -> c:c1:c2:c3
gen_0':s13_21 :: Nat -> 0':s
gen_c4:c5:c614_21 :: Nat -> c4:c5:c6
gen_nil:cons15_21 :: Nat -> nil:cons


Lemmas:
EQ(gen_0':s13_21(n17_21), gen_0':s13_21(n17_21)) -> gen_c:c1:c2:c312_21(n17_21), rt in Omega(1 + n17_21)
LE(gen_0':s13_21(n1041_21), gen_0':s13_21(n1041_21)) -> gen_c4:c5:c614_21(n1041_21), rt in Omega(1 + n1041_21)
le(gen_0':s13_21(n1839_21), gen_0':s13_21(n1839_21)) -> true, rt in Omega(0)
MIN(gen_nil:cons15_21(+(1, n2326_21))) -> *16_21, rt in Omega(n2326_21)
eq(gen_0':s13_21(n6767_21), gen_0':s13_21(n6767_21)) -> true, rt in Omega(0)
min(gen_nil:cons15_21(+(1, n8012_21))) -> gen_0':s13_21(0), rt in Omega(0)


Generator Equations:
gen_c:c1:c2:c312_21(0) <=> c
gen_c:c1:c2:c312_21(+(x, 1)) <=> c3(gen_c:c1:c2:c312_21(x))
gen_0':s13_21(0) <=> 0'
gen_0':s13_21(+(x, 1)) <=> s(gen_0':s13_21(x))
gen_c4:c5:c614_21(0) <=> c4
gen_c4:c5:c614_21(+(x, 1)) <=> c6(gen_c4:c5:c614_21(x))
gen_nil:cons15_21(0) <=> nil
gen_nil:cons15_21(+(x, 1)) <=> cons(0', gen_nil:cons15_21(x))


The following defined symbols remain to be analysed:
replace, SELSORT, selsort

They will be analysed ascendingly in the following order:
replace < SELSORT
replace < selsort

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

(103) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
SELSORT(gen_nil:cons15_21(n9351_21)) -> *16_21, rt in Omega(n9351_21 + n9351_21^2)

Induction Base:
SELSORT(gen_nil:cons15_21(0))

Induction Step:
SELSORT(gen_nil:cons15_21(+(n9351_21, 1))) ->_R^Omega(1)
c17(IFSELSORT(eq(0', min(cons(0', gen_nil:cons15_21(n9351_21)))), cons(0', gen_nil:cons15_21(n9351_21))), EQ(0', min(cons(0', gen_nil:cons15_21(n9351_21)))), MIN(cons(0', gen_nil:cons15_21(n9351_21)))) ->_L^Omega(0)
c17(IFSELSORT(eq(0', gen_0':s13_21(0)), cons(0', gen_nil:cons15_21(n9351_21))), EQ(0', min(cons(0', gen_nil:cons15_21(n9351_21)))), MIN(cons(0', gen_nil:cons15_21(n9351_21)))) ->_L^Omega(0)
c17(IFSELSORT(true, cons(0', gen_nil:cons15_21(n9351_21))), EQ(0', min(cons(0', gen_nil:cons15_21(n9351_21)))), MIN(cons(0', gen_nil:cons15_21(n9351_21)))) ->_R^Omega(1)
c17(c18(SELSORT(gen_nil:cons15_21(n9351_21))), EQ(0', min(cons(0', gen_nil:cons15_21(n9351_21)))), MIN(cons(0', gen_nil:cons15_21(n9351_21)))) ->_IH
c17(c18(*16_21), EQ(0', min(cons(0', gen_nil:cons15_21(n9351_21)))), MIN(cons(0', gen_nil:cons15_21(n9351_21)))) ->_L^Omega(0)
c17(c18(*16_21), EQ(0', gen_0':s13_21(0)), MIN(cons(0', gen_nil:cons15_21(n9351_21)))) ->_L^Omega(1)
c17(c18(*16_21), gen_c:c1:c2:c312_21(0), MIN(cons(0', gen_nil:cons15_21(n9351_21)))) ->_L^Omega(n9351_21)
c17(c18(*16_21), gen_c:c1:c2:c312_21(0), *16_21)

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(IFMIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IFMIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IFMIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, nil) -> c12
REPLACE(z0, cons(z2, z3)) -> c13(IFREPL(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
IFREPL(true, z0, cons(z2, z3)) -> c14
IFREPL(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
SELSORT(nil) -> c16
SELSORT(cons(z0, z1)) -> c17(IFSELSORT(eq(z0, min(cons(z0, z1))), cons(z0, z1)), EQ(z0, min(cons(z0, z1))), MIN(cons(z0, z1)))
IFSELSORT(true, cons(z0, z1)) -> c18(SELSORT(z1))
IFSELSORT(false, cons(z0, z1)) -> c19(MIN(cons(z0, z1)))
IFSELSORT(false, cons(z0, z1)) -> c20(SELSORT(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))) -> ifmin(le(z0, z1), cons(z0, cons(z1, z2)))
ifmin(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
ifmin(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> ifrepl(eq(z0, z2), z0, z1, cons(z2, z3))
ifrepl(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
ifrepl(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
selsort(nil) -> nil
selsort(cons(z0, z1)) -> ifselsort(eq(z0, min(cons(z0, z1))), cons(z0, z1))
ifselsort(true, cons(z0, z1)) -> cons(z0, selsort(z1))
ifselsort(false, cons(z0, z1)) -> cons(min(cons(z0, z1)), selsort(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
IFMIN :: 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
IFREPL :: true:false -> 0':s -> nil:cons -> c14:c15
eq :: 0':s -> 0':s -> true:false
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SELSORT :: nil:cons -> c16:c17
c16 :: c16:c17
c17 :: c18:c19:c20 -> c:c1:c2:c3 -> c7:c8:c9 -> c16:c17
IFSELSORT :: true:false -> nil:cons -> c18:c19:c20
min :: nil:cons -> 0':s
c18 :: c16:c17 -> c18:c19:c20
c19 :: c7:c8:c9 -> c18:c19:c20
c20 :: c16:c17 -> c12:c13 -> c7:c8:c9 -> c18:c19:c20
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
ifmin :: true:false -> nil:cons -> 0':s
ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
selsort :: nil:cons -> nil:cons
ifselsort :: true:false -> nil:cons -> nil:cons
hole_c:c1:c2:c31_21 :: c:c1:c2:c3
hole_0':s2_21 :: 0':s
hole_c4:c5:c63_21 :: c4:c5:c6
hole_c7:c8:c94_21 :: c7:c8:c9
hole_nil:cons5_21 :: nil:cons
hole_c10:c116_21 :: c10:c11
hole_true:false7_21 :: true:false
hole_c12:c138_21 :: c12:c13
hole_c14:c159_21 :: c14:c15
hole_c16:c1710_21 :: c16:c17
hole_c18:c19:c2011_21 :: c18:c19:c20
gen_c:c1:c2:c312_21 :: Nat -> c:c1:c2:c3
gen_0':s13_21 :: Nat -> 0':s
gen_c4:c5:c614_21 :: Nat -> c4:c5:c6
gen_nil:cons15_21 :: Nat -> nil:cons


Lemmas:
EQ(gen_0':s13_21(n17_21), gen_0':s13_21(n17_21)) -> gen_c:c1:c2:c312_21(n17_21), rt in Omega(1 + n17_21)
LE(gen_0':s13_21(n1041_21), gen_0':s13_21(n1041_21)) -> gen_c4:c5:c614_21(n1041_21), rt in Omega(1 + n1041_21)
le(gen_0':s13_21(n1839_21), gen_0':s13_21(n1839_21)) -> true, rt in Omega(0)
MIN(gen_nil:cons15_21(+(1, n2326_21))) -> *16_21, rt in Omega(n2326_21)
eq(gen_0':s13_21(n6767_21), gen_0':s13_21(n6767_21)) -> true, rt in Omega(0)
min(gen_nil:cons15_21(+(1, n8012_21))) -> gen_0':s13_21(0), rt in Omega(0)


Generator Equations:
gen_c:c1:c2:c312_21(0) <=> c
gen_c:c1:c2:c312_21(+(x, 1)) <=> c3(gen_c:c1:c2:c312_21(x))
gen_0':s13_21(0) <=> 0'
gen_0':s13_21(+(x, 1)) <=> s(gen_0':s13_21(x))
gen_c4:c5:c614_21(0) <=> c4
gen_c4:c5:c614_21(+(x, 1)) <=> c6(gen_c4:c5:c614_21(x))
gen_nil:cons15_21(0) <=> nil
gen_nil:cons15_21(+(x, 1)) <=> cons(0', gen_nil:cons15_21(x))


The following defined symbols remain to be analysed:
SELSORT, selsort
----------------------------------------

(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(IFMIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IFMIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IFMIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, nil) -> c12
REPLACE(z0, cons(z2, z3)) -> c13(IFREPL(eq(z0, z2), z0, cons(z2, z3)), EQ(z0, z2))
IFREPL(true, z0, cons(z2, z3)) -> c14
IFREPL(false, z0, cons(z2, z3)) -> c15(REPLACE(z0, z3))
SELSORT(nil) -> c16
SELSORT(cons(z0, z1)) -> c17(IFSELSORT(eq(z0, min(cons(z0, z1))), cons(z0, z1)), EQ(z0, min(cons(z0, z1))), MIN(cons(z0, z1)))
IFSELSORT(true, cons(z0, z1)) -> c18(SELSORT(z1))
IFSELSORT(false, cons(z0, z1)) -> c19(MIN(cons(z0, z1)))
IFSELSORT(false, cons(z0, z1)) -> c20(SELSORT(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))) -> ifmin(le(z0, z1), cons(z0, cons(z1, z2)))
ifmin(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
ifmin(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> ifrepl(eq(z0, z2), z0, z1, cons(z2, z3))
ifrepl(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
ifrepl(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
selsort(nil) -> nil
selsort(cons(z0, z1)) -> ifselsort(eq(z0, min(cons(z0, z1))), cons(z0, z1))
ifselsort(true, cons(z0, z1)) -> cons(z0, selsort(z1))
ifselsort(false, cons(z0, z1)) -> cons(min(cons(z0, z1)), selsort(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
IFMIN :: 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
IFREPL :: true:false -> 0':s -> nil:cons -> c14:c15
eq :: 0':s -> 0':s -> true:false
c14 :: c14:c15
c15 :: c12:c13 -> c14:c15
SELSORT :: nil:cons -> c16:c17
c16 :: c16:c17
c17 :: c18:c19:c20 -> c:c1:c2:c3 -> c7:c8:c9 -> c16:c17
IFSELSORT :: true:false -> nil:cons -> c18:c19:c20
min :: nil:cons -> 0':s
c18 :: c16:c17 -> c18:c19:c20
c19 :: c7:c8:c9 -> c18:c19:c20
c20 :: c16:c17 -> c12:c13 -> c7:c8:c9 -> c18:c19:c20
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
ifmin :: true:false -> nil:cons -> 0':s
ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
selsort :: nil:cons -> nil:cons
ifselsort :: true:false -> nil:cons -> nil:cons
hole_c:c1:c2:c31_21 :: c:c1:c2:c3
hole_0':s2_21 :: 0':s
hole_c4:c5:c63_21 :: c4:c5:c6
hole_c7:c8:c94_21 :: c7:c8:c9
hole_nil:cons5_21 :: nil:cons
hole_c10:c116_21 :: c10:c11
hole_true:false7_21 :: true:false
hole_c12:c138_21 :: c12:c13
hole_c14:c159_21 :: c14:c15
hole_c16:c1710_21 :: c16:c17
hole_c18:c19:c2011_21 :: c18:c19:c20
gen_c:c1:c2:c312_21 :: Nat -> c:c1:c2:c3
gen_0':s13_21 :: Nat -> 0':s
gen_c4:c5:c614_21 :: Nat -> c4:c5:c6
gen_nil:cons15_21 :: Nat -> nil:cons


Lemmas:
EQ(gen_0':s13_21(n17_21), gen_0':s13_21(n17_21)) -> gen_c:c1:c2:c312_21(n17_21), rt in Omega(1 + n17_21)
LE(gen_0':s13_21(n1041_21), gen_0':s13_21(n1041_21)) -> gen_c4:c5:c614_21(n1041_21), rt in Omega(1 + n1041_21)
le(gen_0':s13_21(n1839_21), gen_0':s13_21(n1839_21)) -> true, rt in Omega(0)
MIN(gen_nil:cons15_21(+(1, n2326_21))) -> *16_21, rt in Omega(n2326_21)
eq(gen_0':s13_21(n6767_21), gen_0':s13_21(n6767_21)) -> true, rt in Omega(0)
min(gen_nil:cons15_21(+(1, n8012_21))) -> gen_0':s13_21(0), rt in Omega(0)
SELSORT(gen_nil:cons15_21(n9351_21)) -> *16_21, rt in Omega(n9351_21 + n9351_21^2)


Generator Equations:
gen_c:c1:c2:c312_21(0) <=> c
gen_c:c1:c2:c312_21(+(x, 1)) <=> c3(gen_c:c1:c2:c312_21(x))
gen_0':s13_21(0) <=> 0'
gen_0':s13_21(+(x, 1)) <=> s(gen_0':s13_21(x))
gen_c4:c5:c614_21(0) <=> c4
gen_c4:c5:c614_21(+(x, 1)) <=> c6(gen_c4:c5:c614_21(x))
gen_nil:cons15_21(0) <=> nil
gen_nil:cons15_21(+(x, 1)) <=> cons(0', gen_nil:cons15_21(x))


The following defined symbols remain to be analysed:
selsort
----------------------------------------

(109) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
selsort(gen_nil:cons15_21(n19292_21)) -> gen_nil:cons15_21(n19292_21), rt in Omega(0)

Induction Base:
selsort(gen_nil:cons15_21(0)) ->_R^Omega(0)
nil

Induction Step:
selsort(gen_nil:cons15_21(+(n19292_21, 1))) ->_R^Omega(0)
ifselsort(eq(0', min(cons(0', gen_nil:cons15_21(n19292_21)))), cons(0', gen_nil:cons15_21(n19292_21))) ->_L^Omega(0)
ifselsort(eq(0', gen_0':s13_21(0)), cons(0', gen_nil:cons15_21(n19292_21))) ->_L^Omega(0)
ifselsort(true, cons(0', gen_nil:cons15_21(n19292_21))) ->_R^Omega(0)
cons(0', selsort(gen_nil:cons15_21(n19292_21))) ->_IH
cons(0', gen_nil:cons15_21(c19293_21))

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

(110)
BOUNDS(1, INF)