WORST_CASE(Omega(n^2),O(n^3))
proof of input_lJnOS9m47V.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxWeightedTrs
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedTrs
    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
    (15) NarrowingProof [BOTH BOUNDS(ID, ID), 2 ms]
    (16) CpxTypedWeightedCompleteTrs
    (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxRNTS
    (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 108 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 104 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 310 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 45 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 189 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 104 ms]
    (40) CpxRNTS
    (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 451 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 115 ms]
    (46) CpxRNTS
    (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 974 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 413 ms]
    (52) CpxRNTS
    (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (54) CpxRNTS
    (55) IntTrsBoundProof [UPPER BOUND(ID), 887 ms]
    (56) CpxRNTS
    (57) IntTrsBoundProof [UPPER BOUND(ID), 231 ms]
    (58) CpxRNTS
    (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (60) CpxRNTS
    (61) IntTrsBoundProof [UPPER BOUND(ID), 1039 ms]
    (62) CpxRNTS
    (63) IntTrsBoundProof [UPPER BOUND(ID), 605 ms]
    (64) CpxRNTS
    (65) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (66) CpxRNTS
    (67) IntTrsBoundProof [UPPER BOUND(ID), 1636 ms]
    (68) CpxRNTS
    (69) IntTrsBoundProof [UPPER BOUND(ID), 303 ms]
    (70) CpxRNTS
    (71) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (72) CpxRNTS
    (73) IntTrsBoundProof [UPPER BOUND(ID), 6214 ms]
    (74) CpxRNTS
    (75) IntTrsBoundProof [UPPER BOUND(ID), 1121 ms]
    (76) CpxRNTS
    (77) FinalProof [FINISHED, 0 ms]
    (78) BOUNDS(1, n^3)
(79) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 38 ms]
(80) CdtProblem
    (81) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CpxRelTRS
    (83) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CpxRelTRS
    (85) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) typed CpxTrs
    (87) OrderProof [LOWER BOUND(ID), 0 ms]
    (88) typed CpxTrs
    (89) RewriteLemmaProof [LOWER BOUND(ID), 433 ms]
    (90) BEST
        (91) proven lower bound
            (92) LowerBoundPropagationProof [FINISHED, 0 ms]
            (93) BOUNDS(n^1, INF)
        (94) typed CpxTrs
            (95) RewriteLemmaProof [LOWER BOUND(ID), 147 ms]
            (96) typed CpxTrs
            (97) RewriteLemmaProof [LOWER BOUND(ID), 45 ms]
            (98) typed CpxTrs
            (99) RewriteLemmaProof [LOWER BOUND(ID), 552 ms]
            (100) typed CpxTrs
            (101) RewriteLemmaProof [LOWER BOUND(ID), 2 ms]
            (102) typed CpxTrs
            (103) RewriteLemmaProof [LOWER BOUND(ID), 25 ms]
            (104) typed CpxTrs
            (105) RewriteLemmaProof [LOWER BOUND(ID), 621 ms]
            (106) BEST
                (107) proven lower bound
                    (108) LowerBoundPropagationProof [FINISHED, 0 ms]
                    (109) BOUNDS(n^2, INF)
                (110) typed CpxTrs
                    (111) RewriteLemmaProof [LOWER BOUND(ID), 58 ms]
                    (112) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

   eq(0, 0) -> true
   eq(0, s(Y)) -> false
   eq(s(X), 0) -> false
   eq(s(X), s(Y)) -> eq(X, Y)
   le(0, Y) -> true
   le(s(X), 0) -> false
   le(s(X), s(Y)) -> le(X, Y)
   min(cons(0, nil)) -> 0
   min(cons(s(N), nil)) -> s(N)
   min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L)))
   ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L))
   ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L))
   replace(N, M, nil) -> nil
   replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L))
   ifrepl(true, N, M, cons(K, L)) -> cons(M, L)
   ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L))
   selsort(nil) -> nil
   selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L))
   ifselsort(true, cons(N, L)) -> cons(N, selsort(L))
   ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L)))

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

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

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)))
Tuples:
   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)))
S tuples:
   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)))
K tuples:none
Defined Rule Symbols:   eq_2, le_2, min_1, ifmin_2, replace_3, ifrepl_4, selsort_1, ifselsort_2

Defined Pair Symbols:   EQ_2, LE_2, MIN_1, IFMIN_2, REPLACE_3, IFREPL_4, SELSORT_1, IFSELSORT_2

Compound Symbols:   c, c1, c2, c3_1, c4, c5, c6_1, c7, c8, c9_2, c10_1, c11_1, c12, c13_2, c14, c15_1, c16, c17_3, c18_1, c19_1, c20_3


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 10 trailing nodes:
   EQ(0, s(z0)) -> c1
   MIN(cons(s(z0), nil)) -> c8
   EQ(s(z0), 0) -> c2
   SELSORT(nil) -> c16
   LE(s(z0), 0) -> c5
   LE(0, z0) -> c4
   MIN(cons(0, nil)) -> c7
   REPLACE(z0, z1, nil) -> c12
   IFREPL(true, z0, z1, cons(z2, z3)) -> c14
   EQ(0, 0) -> c

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

(4)
Obligation:
Complexity Dependency Tuples Problem

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)))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   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, cons(z2, z3)) -> c13(IFREPL(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
   IFREPL(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
   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)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   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, cons(z2, z3)) -> c13(IFREPL(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
   IFREPL(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
   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)))
K tuples:none
Defined Rule Symbols:   eq_2, le_2, min_1, ifmin_2, replace_3, ifrepl_4, selsort_1, ifselsort_2

Defined Pair Symbols:   EQ_2, LE_2, MIN_1, IFMIN_2, REPLACE_3, IFREPL_4, SELSORT_1, IFSELSORT_2

Compound Symbols:   c3_1, c6_1, c9_2, c10_1, c11_1, c13_2, c15_1, c17_3, c18_1, c19_1, c20_3


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

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   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)))

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(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))
Tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   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, cons(z2, z3)) -> c13(IFREPL(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
   IFREPL(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
   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)))
S tuples:
   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   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, cons(z2, z3)) -> c13(IFREPL(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
   IFREPL(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
   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)))
K tuples:none
Defined Rule Symbols:   le_2, eq_2, min_1, ifmin_2, replace_3, ifrepl_4

Defined Pair Symbols:   EQ_2, LE_2, MIN_1, IFMIN_2, REPLACE_3, IFREPL_4, SELSORT_1, IFSELSORT_2

Compound Symbols:   c3_1, c6_1, c9_2, c10_1, c11_1, c13_2, c15_1, c17_3, c18_1, c19_1, c20_3


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

(7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1))
   LE(s(z0), s(z1)) -> c6(LE(z0, z1))
   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, cons(z2, z3)) -> c13(IFREPL(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
   IFREPL(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
   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:

   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   eq(0, 0) -> true
   eq(0, s(z0)) -> false
   eq(s(z0), 0) -> false
   eq(s(z0), s(z1)) -> eq(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))

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

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

(10)
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(s(z0), s(z1)) -> c3(EQ(z0, z1)) [1]
   LE(s(z0), s(z1)) -> c6(LE(z0, z1)) [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, cons(z2, z3)) -> c13(IFREPL(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) [1]
   IFREPL(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3)) [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]
   le(0, z0) -> true [0]
   le(s(z0), 0) -> false [0]
   le(s(z0), s(z1)) -> le(z0, z1) [0]
   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]
   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]

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

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

(12)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   EQ(s(z0), s(z1)) -> c3(EQ(z0, z1)) [1]
   LE(s(z0), s(z1)) -> c6(LE(z0, z1)) [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, cons(z2, z3)) -> c13(IFREPL(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) [1]
   IFREPL(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3)) [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]
   le(0, z0) -> true [0]
   le(s(z0), 0) -> false [0]
   le(s(z0), s(z1)) -> le(z0, z1) [0]
   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]
   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]

The TRS has the following type information:
EQ :: s:0 -> s:0 -> c3
s :: s:0 -> s:0
c3 :: c3 -> c3
LE :: s:0 -> s:0 -> c6
c6 :: c6 -> c6
MIN :: cons:nil -> c9
cons :: s:0 -> cons:nil -> cons:nil
c9 :: c10:c11 -> c6 -> c9
IFMIN :: true:false -> cons:nil -> c10:c11
le :: s:0 -> s:0 -> true:false
true :: true:false
c10 :: c9 -> c10:c11
false :: true:false
c11 :: c9 -> c10:c11
REPLACE :: s:0 -> s:0 -> cons:nil -> c13
c13 :: c15 -> c3 -> c13
IFREPL :: true:false -> s:0 -> s:0 -> cons:nil -> c15
eq :: s:0 -> s:0 -> true:false
c15 :: c13 -> c15
SELSORT :: cons:nil -> c17
c17 :: c18:c19:c20 -> c3 -> c9 -> c17
IFSELSORT :: true:false -> cons:nil -> c18:c19:c20
min :: cons:nil -> s:0
c18 :: c17 -> c18:c19:c20
c19 :: c9 -> c18:c19:c20
c20 :: c17 -> c13 -> c9 -> c18:c19:c20
replace :: s:0 -> s:0 -> cons:nil -> cons:nil
0 :: s:0
nil :: cons:nil
ifmin :: true:false -> cons:nil -> s:0
ifrepl :: true:false -> s:0 -> s:0 -> cons:nil -> cons:nil

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

(13) 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:

   le_2
   eq_2
   min_1
   ifmin_2
   replace_3
   ifrepl_4

Due to the following rules being added:

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

And the following fresh constants:    null_le, null_eq, const, const1, const2, const3, const4, const5, const6, const7

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

(14)
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(s(z0), s(z1)) -> c3(EQ(z0, z1)) [1]
   LE(s(z0), s(z1)) -> c6(LE(z0, z1)) [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, cons(z2, z3)) -> c13(IFREPL(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) [1]
   IFREPL(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3)) [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]
   le(0, z0) -> true [0]
   le(s(z0), 0) -> false [0]
   le(s(z0), s(z1)) -> le(z0, z1) [0]
   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]
   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]
   le(v0, v1) -> null_le [0]
   eq(v0, v1) -> null_eq [0]
   min(v0) -> 0 [0]
   ifmin(v0, v1) -> 0 [0]
   replace(v0, v1, v2) -> nil [0]
   ifrepl(v0, v1, v2, v3) -> nil [0]

The TRS has the following type information:
EQ :: s:0 -> s:0 -> c3
s :: s:0 -> s:0
c3 :: c3 -> c3
LE :: s:0 -> s:0 -> c6
c6 :: c6 -> c6
MIN :: cons:nil -> c9
cons :: s:0 -> cons:nil -> cons:nil
c9 :: c10:c11 -> c6 -> c9
IFMIN :: true:false:null_le:null_eq -> cons:nil -> c10:c11
le :: s:0 -> s:0 -> true:false:null_le:null_eq
true :: true:false:null_le:null_eq
c10 :: c9 -> c10:c11
false :: true:false:null_le:null_eq
c11 :: c9 -> c10:c11
REPLACE :: s:0 -> s:0 -> cons:nil -> c13
c13 :: c15 -> c3 -> c13
IFREPL :: true:false:null_le:null_eq -> s:0 -> s:0 -> cons:nil -> c15
eq :: s:0 -> s:0 -> true:false:null_le:null_eq
c15 :: c13 -> c15
SELSORT :: cons:nil -> c17
c17 :: c18:c19:c20 -> c3 -> c9 -> c17
IFSELSORT :: true:false:null_le:null_eq -> cons:nil -> c18:c19:c20
min :: cons:nil -> s:0
c18 :: c17 -> c18:c19:c20
c19 :: c9 -> c18:c19:c20
c20 :: c17 -> c13 -> c9 -> c18:c19:c20
replace :: s:0 -> s:0 -> cons:nil -> cons:nil
0 :: s:0
nil :: cons:nil
ifmin :: true:false:null_le:null_eq -> cons:nil -> s:0
ifrepl :: true:false:null_le:null_eq -> s:0 -> s:0 -> cons:nil -> cons:nil
null_le :: true:false:null_le:null_eq
null_eq :: true:false:null_le:null_eq
const :: c3
const1 :: c6
const2 :: c9
const3 :: c10:c11
const4 :: c13
const5 :: c15
const6 :: c17
const7 :: c18:c19:c20

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

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

(16)
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(s(z0), s(z1)) -> c3(EQ(z0, z1)) [1]
   LE(s(z0), s(z1)) -> c6(LE(z0, z1)) [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(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(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3)) [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]
   le(0, z0) -> true [0]
   le(s(z0), 0) -> false [0]
   le(s(z0), s(z1)) -> le(z0, z1) [0]
   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]
   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]
   le(v0, v1) -> null_le [0]
   eq(v0, v1) -> null_eq [0]
   min(v0) -> 0 [0]
   ifmin(v0, v1) -> 0 [0]
   replace(v0, v1, v2) -> nil [0]
   ifrepl(v0, v1, v2, v3) -> nil [0]

The TRS has the following type information:
EQ :: s:0 -> s:0 -> c3
s :: s:0 -> s:0
c3 :: c3 -> c3
LE :: s:0 -> s:0 -> c6
c6 :: c6 -> c6
MIN :: cons:nil -> c9
cons :: s:0 -> cons:nil -> cons:nil
c9 :: c10:c11 -> c6 -> c9
IFMIN :: true:false:null_le:null_eq -> cons:nil -> c10:c11
le :: s:0 -> s:0 -> true:false:null_le:null_eq
true :: true:false:null_le:null_eq
c10 :: c9 -> c10:c11
false :: true:false:null_le:null_eq
c11 :: c9 -> c10:c11
REPLACE :: s:0 -> s:0 -> cons:nil -> c13
c13 :: c15 -> c3 -> c13
IFREPL :: true:false:null_le:null_eq -> s:0 -> s:0 -> cons:nil -> c15
eq :: s:0 -> s:0 -> true:false:null_le:null_eq
c15 :: c13 -> c15
SELSORT :: cons:nil -> c17
c17 :: c18:c19:c20 -> c3 -> c9 -> c17
IFSELSORT :: true:false:null_le:null_eq -> cons:nil -> c18:c19:c20
min :: cons:nil -> s:0
c18 :: c17 -> c18:c19:c20
c19 :: c9 -> c18:c19:c20
c20 :: c17 -> c13 -> c9 -> c18:c19:c20
replace :: s:0 -> s:0 -> cons:nil -> cons:nil
0 :: s:0
nil :: cons:nil
ifmin :: true:false:null_le:null_eq -> cons:nil -> s:0
ifrepl :: true:false:null_le:null_eq -> s:0 -> s:0 -> cons:nil -> cons:nil
null_le :: true:false:null_le:null_eq
null_eq :: true:false:null_le:null_eq
const :: c3
const1 :: c6
const2 :: c9
const3 :: c10:c11
const4 :: c13
const5 :: c15
const6 :: c17
const7 :: c18:c19:c20

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

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

   true => 2
   false => 1
   0 => 0
   nil => 0
   null_le => 0
   null_eq => 0
   const => 0
   const1 => 0
   const2 => 0
   const3 => 0
   const4 => 0
   const5 => 0
   const6 => 0
   const7 => 0

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

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

   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 }-> 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 + LE(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   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 }-> 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 }-> 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
   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


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

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

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

   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 }-> 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 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 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 }-> 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 }-> 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
   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


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

(21) 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 }

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

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

   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 }-> 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 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 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 }-> 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 }-> 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
   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

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

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

(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 }-> 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 }-> 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 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 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 }-> 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 }-> 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
   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

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

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

(25) 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: 0

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

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

   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 }-> 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 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 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 }-> 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 }-> 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
   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

Function symbols to be analyzed: {EQ}, {le}, {LE}, {eq}, {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: ?, size: O(1) [0]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   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 }-> 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 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 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 }-> 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 }-> 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
   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

Function symbols to be analyzed: {le}, {LE}, {eq}, {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]

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

(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') -{ z' }-> 1 + s :|: s >= 0, s <= 0, 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 }-> 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 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 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'') -{ 2 + z1'' }-> 1 + IFREPL(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s2 >= 0, s2 <= 0, 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)) + s' :|: s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 2 + z01 }-> 1 + IFREPL(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 0, 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)) + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 + z2 }-> 1 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 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, 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) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s9 >= 0, s9 <= 0, 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) + s4 + MIN(1 + 0 + 0) :|: s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(eq(1 + (z - 2), 0), 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 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 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 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, 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
   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

Function symbols to be analyzed: {le}, {LE}, {eq}, {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]

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

(31) 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

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

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

   EQ(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, 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 }-> 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 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 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'') -{ 2 + z1'' }-> 1 + IFREPL(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s2 >= 0, s2 <= 0, 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)) + s' :|: s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 2 + z01 }-> 1 + IFREPL(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 0, 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)) + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 + z2 }-> 1 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 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, 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) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s9 >= 0, s9 <= 0, 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) + s4 + MIN(1 + 0 + 0) :|: s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(eq(1 + (z - 2), 0), 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 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 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 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, 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
   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

Function symbols to be analyzed: {le}, {LE}, {eq}, {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  le: runtime: ?, size: O(1) [2]

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

(33) 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

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

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

   EQ(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, 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 }-> 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 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 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'') -{ 2 + z1'' }-> 1 + IFREPL(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s2 >= 0, s2 <= 0, 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)) + s' :|: s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 2 + z01 }-> 1 + IFREPL(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 0, 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)) + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 + z2 }-> 1 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 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, 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) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s9 >= 0, s9 <= 0, 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) + s4 + MIN(1 + 0 + 0) :|: s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(eq(1 + (z - 2), 0), 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 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 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 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, 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
   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

Function symbols to be analyzed: {LE}, {eq}, {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  le: runtime: O(1) [0], size: O(1) [2]

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

(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') -{ z' }-> 1 + s :|: s >= 0, s <= 0, 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 }-> 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 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 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'') -{ 2 + z1'' }-> 1 + IFREPL(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s2 >= 0, s2 <= 0, 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)) + s' :|: s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 2 + z01 }-> 1 + IFREPL(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 0, 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)) + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 + z2 }-> 1 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 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) -{ 1 }-> 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, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s9 >= 0, s9 <= 0, 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) -{ 1 }-> 1 + IFSELSORT(eq(0, 0), 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(eq(1 + (z - 2), 0), 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 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 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 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, 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
   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

Function symbols to be analyzed: {LE}, {eq}, {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  le: runtime: O(1) [0], size: O(1) [2]

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

(37) 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: 0

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

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

   EQ(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, 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 }-> 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 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 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'') -{ 2 + z1'' }-> 1 + IFREPL(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s2 >= 0, s2 <= 0, 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)) + s' :|: s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 2 + z01 }-> 1 + IFREPL(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 0, 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)) + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 + z2 }-> 1 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 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) -{ 1 }-> 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, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s9 >= 0, s9 <= 0, 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) -{ 1 }-> 1 + IFSELSORT(eq(0, 0), 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(eq(1 + (z - 2), 0), 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 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 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 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, 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
   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

Function symbols to be analyzed: {LE}, {eq}, {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: ?, size: O(1) [0]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   EQ(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, 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 }-> 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 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 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'') -{ 2 + z1'' }-> 1 + IFREPL(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s2 >= 0, s2 <= 0, 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)) + s' :|: s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 2 + z01 }-> 1 + IFREPL(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 0, 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)) + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 + z2 }-> 1 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 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) -{ 1 }-> 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, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s9 >= 0, s9 <= 0, 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) -{ 1 }-> 1 + IFSELSORT(eq(0, 0), 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(eq(1 + (z - 2), 0), 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 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 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 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, 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
   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

Function symbols to be analyzed: {eq}, {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]

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

(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') -{ z' }-> 1 + s :|: s >= 0, s <= 0, 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 }-> 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') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 2 + z1' }-> 1 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s24 :|: s24 >= 0, s24 <= 0, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s22 :|: s22 >= 0, s22 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s23 :|: s23 >= 0, s23 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s25 :|: s25 >= 0, s25 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 2 + z1'' }-> 1 + IFREPL(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s2 >= 0, s2 <= 0, 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)) + s' :|: s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 2 + z01 }-> 1 + IFREPL(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 0, 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)) + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 + z2 }-> 1 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 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) -{ 1 }-> 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, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s9 >= 0, s9 <= 0, 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) -{ 1 }-> 1 + IFSELSORT(eq(0, 0), 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(eq(1 + (z - 2), 0), 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 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 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 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, 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
   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

Function symbols to be analyzed: {eq}, {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]

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

(43) 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

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

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

   EQ(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, 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 }-> 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') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 2 + z1' }-> 1 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s24 :|: s24 >= 0, s24 <= 0, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s22 :|: s22 >= 0, s22 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s23 :|: s23 >= 0, s23 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s25 :|: s25 >= 0, s25 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 2 + z1'' }-> 1 + IFREPL(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s2 >= 0, s2 <= 0, 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)) + s' :|: s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 2 + z01 }-> 1 + IFREPL(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 0, 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)) + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 + z2 }-> 1 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 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) -{ 1 }-> 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, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s9 >= 0, s9 <= 0, 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) -{ 1 }-> 1 + IFSELSORT(eq(0, 0), 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(eq(1 + (z - 2), 0), 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 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 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 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, 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
   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

Function symbols to be analyzed: {eq}, {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  eq: runtime: ?, size: O(1) [2]

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

(45) 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

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

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

   EQ(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, 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 }-> 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') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 2 + z1' }-> 1 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s24 :|: s24 >= 0, s24 <= 0, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s22 :|: s22 >= 0, s22 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s23 :|: s23 >= 0, s23 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s25 :|: s25 >= 0, s25 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 2 + z1'' }-> 1 + IFREPL(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s2 >= 0, s2 <= 0, 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)) + s' :|: s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 2 + z01 }-> 1 + IFREPL(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 0, 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)) + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 + z2 }-> 1 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 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) -{ 1 }-> 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, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(eq(z0, 0), 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s9 >= 0, s9 <= 0, 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) -{ 1 }-> 1 + IFSELSORT(eq(0, 0), 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(eq(1 + (z - 2), 0), 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 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 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 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, 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
   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

Function symbols to be analyzed: {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  eq: runtime: O(1) [0], size: O(1) [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') -{ z' }-> 1 + s :|: s >= 0, s <= 0, 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 }-> 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') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 2 + z1' }-> 1 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s24 :|: s24 >= 0, s24 <= 0, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s22 :|: s22 >= 0, s22 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s23 :|: s23 >= 0, s23 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s25 :|: s25 >= 0, s25 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 2 + z1'' }-> 1 + IFREPL(s26, 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s26 >= 0, s26 <= 2, s2 >= 0, s2 <= 0, 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)) + s' :|: s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 2 + z01 }-> 1 + IFREPL(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 0, 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)) + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 + z2 }-> 1 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s27, 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s27 >= 0, s27 <= 2, s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(s28, 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s28 >= 0, s28 <= 2, s5 >= 0, s5 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s29, 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s29 >= 0, s29 <= 2, s6 >= 0, s6 <= 0, z - 2 >= 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s30 >= 0, s30 <= 2, s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s31, 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(s14, 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: s31 >= 0, s31 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s32, 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s32 >= 0, s32 <= 2, s9 >= 0, s9 <= 0, 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) -{ 1 }-> 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, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 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
   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(s34, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s34 >= 0, s34 <= 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

Function symbols to be analyzed: {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  eq: runtime: O(1) [0], size: O(1) [2]

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   EQ(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, 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 }-> 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') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 2 + z1' }-> 1 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s24 :|: s24 >= 0, s24 <= 0, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s22 :|: s22 >= 0, s22 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s23 :|: s23 >= 0, s23 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s25 :|: s25 >= 0, s25 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 2 + z1'' }-> 1 + IFREPL(s26, 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s26 >= 0, s26 <= 2, s2 >= 0, s2 <= 0, 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)) + s' :|: s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 2 + z01 }-> 1 + IFREPL(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 0, 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)) + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 + z2 }-> 1 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s27, 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s27 >= 0, s27 <= 2, s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(s28, 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s28 >= 0, s28 <= 2, s5 >= 0, s5 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s29, 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s29 >= 0, s29 <= 2, s6 >= 0, s6 <= 0, z - 2 >= 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s30 >= 0, s30 <= 2, s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s31, 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(s14, 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: s31 >= 0, s31 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s32, 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s32 >= 0, s32 <= 2, s9 >= 0, s9 <= 0, 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) -{ 1 }-> 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, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 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
   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(s34, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s34 >= 0, s34 <= 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

Function symbols to be analyzed: {REPLACE,IFREPL}, {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: ?, size: O(1) [0]
  IFREPL: runtime: ?, size: O(1) [1]

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

(51) 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: 2 + 3*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: 3 + 3*z4 + z4^2

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

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

   EQ(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, 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 }-> 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') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 2 + z1' }-> 1 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s24 :|: s24 >= 0, s24 <= 0, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s22 :|: s22 >= 0, s22 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s23 :|: s23 >= 0, s23 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s25 :|: s25 >= 0, s25 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 2 + z1'' }-> 1 + IFREPL(s26, 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s26 >= 0, s26 <= 2, s2 >= 0, s2 <= 0, 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)) + s' :|: s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 2 + z01 }-> 1 + IFREPL(1, 0, z', 1 + (1 + z01) + z3) + s'' :|: s'' >= 0, s'' <= 0, 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)) + s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 + z2 }-> 1 + IFREPL(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s27, 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s27 >= 0, s27 <= 2, s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(s28, 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s28 >= 0, s28 <= 2, s5 >= 0, s5 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s29, 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s29 >= 0, s29 <= 2, s6 >= 0, s6 <= 0, z - 2 >= 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s30 >= 0, s30 <= 2, s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s31, 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(s14, 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: s31 >= 0, s31 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s32, 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s32 >= 0, s32 <= 2, s9 >= 0, s9 <= 0, 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) -{ 1 }-> 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, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 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
   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(s34, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s34 >= 0, s34 <= 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

Function symbols to be analyzed: {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [2 + 3*z'' + z''^2], size: O(1) [0]
  IFREPL: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]

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

(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') -{ z' }-> 1 + s :|: s >= 0, s <= 0, 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) -{ 3 + 3*z3 + z3^2 }-> 1 + s40 :|: s40 >= 0, s40 <= 0, 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') -{ 7 + 5*z13 + 2*z13*z21 + z13^2 + 5*z21 + z21^2 }-> 1 + SELSORT(replace(ifmin(s17, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + s44 + MIN(1 + z0 + (1 + z13 + z21)) :|: s44 >= 0, s44 <= 0, s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 3 + 3*z1 + z1^2 }-> 1 + SELSORT(replace(0, z0, z1)) + s46 + MIN(1 + z0 + z1) :|: s46 >= 0, s46 <= 0, 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') -{ 3 }-> 1 + SELSORT(replace(0, 0, 0)) + s41 + MIN(1 + 0 + 0) :|: s41 >= 0, s41 <= 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 3 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + s45 + MIN(1 + (1 + (z' - 2)) + 0) :|: s45 >= 0, s45 <= 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 3 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s42 + MIN(1 + (1 + (z' - 2)) + 0) :|: s42 >= 0, s42 <= 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 3 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s43 + MIN(1 + (1 + (z' - 2)) + 0) :|: s43 >= 0, s43 <= 0, z = 1, z' - 2 >= 0
   LE(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 2 + z1' }-> 1 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s24 :|: s24 >= 0, s24 <= 0, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s22 :|: s22 >= 0, s22 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s23 :|: s23 >= 0, s23 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s25 :|: s25 >= 0, s25 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s35 + s' :|: s35 >= 0, s35 <= 1, s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z01 + 2*z01*z3 + z01^2 + 7*z3 + z3^2 }-> 1 + s36 + s'' :|: s36 >= 0, s36 <= 1, s'' >= 0, s'' <= 0, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s37 + s1 :|: s37 >= 0, s37 <= 1, s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z1'' + 2*z1''*z3 + z1''^2 + 7*z3 + z3^2 }-> 1 + s38 + s2 :|: s38 >= 0, s38 <= 1, s26 >= 0, s26 <= 2, s2 >= 0, s2 <= 0, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 8 + 6*z2 + 2*z2*z3 + z2^2 + 5*z3 + z3^2 }-> 1 + s39 + s3 :|: s39 >= 0, s39 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s27, 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s27 >= 0, s27 <= 2, s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(s28, 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s28 >= 0, s28 <= 2, s5 >= 0, s5 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s29, 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s29 >= 0, s29 <= 2, s6 >= 0, s6 <= 0, z - 2 >= 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s30 >= 0, s30 <= 2, s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s31, 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(s14, 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: s31 >= 0, s31 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s32, 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s32 >= 0, s32 <= 2, s9 >= 0, s9 <= 0, 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) -{ 1 }-> 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, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 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
   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(s34, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s34 >= 0, s34 <= 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

Function symbols to be analyzed: {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [2 + 3*z'' + z''^2], size: O(1) [0]
  IFREPL: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   EQ(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, 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) -{ 3 + 3*z3 + z3^2 }-> 1 + s40 :|: s40 >= 0, s40 <= 0, 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') -{ 7 + 5*z13 + 2*z13*z21 + z13^2 + 5*z21 + z21^2 }-> 1 + SELSORT(replace(ifmin(s17, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + s44 + MIN(1 + z0 + (1 + z13 + z21)) :|: s44 >= 0, s44 <= 0, s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 3 + 3*z1 + z1^2 }-> 1 + SELSORT(replace(0, z0, z1)) + s46 + MIN(1 + z0 + z1) :|: s46 >= 0, s46 <= 0, 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') -{ 3 }-> 1 + SELSORT(replace(0, 0, 0)) + s41 + MIN(1 + 0 + 0) :|: s41 >= 0, s41 <= 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 3 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + s45 + MIN(1 + (1 + (z' - 2)) + 0) :|: s45 >= 0, s45 <= 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 3 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s42 + MIN(1 + (1 + (z' - 2)) + 0) :|: s42 >= 0, s42 <= 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 3 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s43 + MIN(1 + (1 + (z' - 2)) + 0) :|: s43 >= 0, s43 <= 0, z = 1, z' - 2 >= 0
   LE(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 2 + z1' }-> 1 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s24 :|: s24 >= 0, s24 <= 0, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s22 :|: s22 >= 0, s22 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s23 :|: s23 >= 0, s23 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s25 :|: s25 >= 0, s25 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s35 + s' :|: s35 >= 0, s35 <= 1, s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z01 + 2*z01*z3 + z01^2 + 7*z3 + z3^2 }-> 1 + s36 + s'' :|: s36 >= 0, s36 <= 1, s'' >= 0, s'' <= 0, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s37 + s1 :|: s37 >= 0, s37 <= 1, s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z1'' + 2*z1''*z3 + z1''^2 + 7*z3 + z3^2 }-> 1 + s38 + s2 :|: s38 >= 0, s38 <= 1, s26 >= 0, s26 <= 2, s2 >= 0, s2 <= 0, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 8 + 6*z2 + 2*z2*z3 + z2^2 + 5*z3 + z3^2 }-> 1 + s39 + s3 :|: s39 >= 0, s39 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s27, 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s27 >= 0, s27 <= 2, s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(s28, 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s28 >= 0, s28 <= 2, s5 >= 0, s5 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s29, 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s29 >= 0, s29 <= 2, s6 >= 0, s6 <= 0, z - 2 >= 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s30 >= 0, s30 <= 2, s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s31, 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(s14, 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: s31 >= 0, s31 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s32, 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s32 >= 0, s32 <= 2, s9 >= 0, s9 <= 0, 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) -{ 1 }-> 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, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 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
   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(s34, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s34 >= 0, s34 <= 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

Function symbols to be analyzed: {min,ifmin}, {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [2 + 3*z'' + z''^2], size: O(1) [0]
  IFREPL: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]
  min: runtime: ?, size: O(n^1) [z]
  ifmin: runtime: ?, size: O(n^1) [z']

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   EQ(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, 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) -{ 3 + 3*z3 + z3^2 }-> 1 + s40 :|: s40 >= 0, s40 <= 0, 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') -{ 7 + 5*z13 + 2*z13*z21 + z13^2 + 5*z21 + z21^2 }-> 1 + SELSORT(replace(ifmin(s17, 1 + z0 + (1 + z13 + z21)), z0, 1 + z13 + z21)) + s44 + MIN(1 + z0 + (1 + z13 + z21)) :|: s44 >= 0, s44 <= 0, s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 3 + 3*z1 + z1^2 }-> 1 + SELSORT(replace(0, z0, z1)) + s46 + MIN(1 + z0 + z1) :|: s46 >= 0, s46 <= 0, 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') -{ 3 }-> 1 + SELSORT(replace(0, 0, 0)) + s41 + MIN(1 + 0 + 0) :|: s41 >= 0, s41 <= 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 3 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + s45 + MIN(1 + (1 + (z' - 2)) + 0) :|: s45 >= 0, s45 <= 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 3 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s42 + MIN(1 + (1 + (z' - 2)) + 0) :|: s42 >= 0, s42 <= 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 3 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s43 + MIN(1 + (1 + (z' - 2)) + 0) :|: s43 >= 0, s43 <= 0, z = 1, z' - 2 >= 0
   LE(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 2 + z1' }-> 1 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s24 :|: s24 >= 0, s24 <= 0, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s22 :|: s22 >= 0, s22 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s23 :|: s23 >= 0, s23 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s25 :|: s25 >= 0, s25 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s35 + s' :|: s35 >= 0, s35 <= 1, s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z01 + 2*z01*z3 + z01^2 + 7*z3 + z3^2 }-> 1 + s36 + s'' :|: s36 >= 0, s36 <= 1, s'' >= 0, s'' <= 0, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s37 + s1 :|: s37 >= 0, s37 <= 1, s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z1'' + 2*z1''*z3 + z1''^2 + 7*z3 + z3^2 }-> 1 + s38 + s2 :|: s38 >= 0, s38 <= 1, s26 >= 0, s26 <= 2, s2 >= 0, s2 <= 0, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 8 + 6*z2 + 2*z2*z3 + z2^2 + 5*z3 + z3^2 }-> 1 + s39 + s3 :|: s39 >= 0, s39 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s27, 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s27 >= 0, s27 <= 2, s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(s28, 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s28 >= 0, s28 <= 2, s5 >= 0, s5 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s29, 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s29 >= 0, s29 <= 2, s6 >= 0, s6 <= 0, z - 2 >= 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s30 >= 0, s30 <= 2, s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s31, 1 + z0 + (1 + z12 + z2'')) + EQ(z0, ifmin(s14, 1 + z0 + (1 + z12 + z2''))) + MIN(1 + z0 + (1 + z12 + z2'')) :|: s31 >= 0, s31 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s32, 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s32 >= 0, s32 <= 2, s9 >= 0, s9 <= 0, 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) -{ 1 }-> 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, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 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
   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(s34, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s34 >= 0, s34 <= 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

Function symbols to be analyzed: {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [2 + 3*z'' + z''^2], size: O(1) [0]
  IFREPL: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]
  min: runtime: O(1) [0], size: O(n^1) [z]
  ifmin: runtime: O(1) [0], size: O(n^1) [z']

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

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

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

   EQ(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, 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) -{ 3 + 3*z3 + z3^2 }-> 1 + s40 :|: s40 >= 0, s40 <= 0, 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') -{ 7 + 5*z13 + 2*z13*z21 + z13^2 + 5*z21 + z21^2 }-> 1 + SELSORT(replace(s55, z0, 1 + z13 + z21)) + s57 + MIN(1 + z0 + (1 + z13 + z21)) :|: s55 >= 0, s55 <= 1 + z0 + (1 + z13 + z21), s56 >= 0, s56 <= 1 + z0 + (1 + z13 + z21), s57 >= 0, s57 <= 0, 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') -{ 7 + 5*z13 + 2*z13*z21 + z13^2 + 5*z21 + z21^2 }-> 1 + SELSORT(replace(s58, z0, 1 + z13 + z21)) + s44 + MIN(1 + z0 + (1 + z13 + z21)) :|: s58 >= 0, s58 <= 1 + z0 + (1 + z13 + z21), s44 >= 0, s44 <= 0, s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 3 + 3*z1 + z1^2 }-> 1 + SELSORT(replace(0, z0, z1)) + s46 + MIN(1 + z0 + z1) :|: s46 >= 0, s46 <= 0, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 7 + 5*z14 + 2*z14*z22 + z14^2 + 5*z22 + z22^2 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + s60 + MIN(1 + z0 + (1 + z14 + z22)) :|: s59 >= 0, s59 <= 1 + z0 + (1 + z14 + z22), s60 >= 0, s60 <= 0, s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 3 }-> 1 + SELSORT(replace(0, 0, 0)) + s41 + MIN(1 + 0 + 0) :|: s41 >= 0, s41 <= 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 3 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + s45 + MIN(1 + (1 + (z' - 2)) + 0) :|: s45 >= 0, s45 <= 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 3 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s42 + MIN(1 + (1 + (z' - 2)) + 0) :|: s42 >= 0, s42 <= 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 3 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s43 + MIN(1 + (1 + (z' - 2)) + 0) :|: s43 >= 0, s43 <= 0, z = 1, z' - 2 >= 0
   LE(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 2 + z1' }-> 1 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s24 :|: s24 >= 0, s24 <= 0, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s22 :|: s22 >= 0, s22 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s23 :|: s23 >= 0, s23 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s25 :|: s25 >= 0, s25 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s35 + s' :|: s35 >= 0, s35 <= 1, s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z01 + 2*z01*z3 + z01^2 + 7*z3 + z3^2 }-> 1 + s36 + s'' :|: s36 >= 0, s36 <= 1, s'' >= 0, s'' <= 0, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s37 + s1 :|: s37 >= 0, s37 <= 1, s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z1'' + 2*z1''*z3 + z1''^2 + 7*z3 + z3^2 }-> 1 + s38 + s2 :|: s38 >= 0, s38 <= 1, s26 >= 0, s26 <= 2, s2 >= 0, s2 <= 0, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 8 + 6*z2 + 2*z2*z3 + z2^2 + 5*z3 + z3^2 }-> 1 + s39 + s3 :|: s39 >= 0, s39 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s27, 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s27 >= 0, s27 <= 2, s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(s28, 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s28 >= 0, s28 <= 2, s5 >= 0, s5 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s29, 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s29 >= 0, s29 <= 2, s6 >= 0, s6 <= 0, z - 2 >= 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s30 >= 0, s30 <= 2, s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 + s53 }-> 1 + IFSELSORT(s31, 1 + z0 + (1 + z12 + z2'')) + s54 + MIN(1 + z0 + (1 + z12 + z2'')) :|: s53 >= 0, s53 <= 1 + z0 + (1 + z12 + z2''), s54 >= 0, s54 <= 0, s31 >= 0, s31 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s32, 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s32 >= 0, s32 <= 2, s9 >= 0, s9 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 + s49 }-> 1 + IFSELSORT(s48, 1 + z0 + (1 + z11 + z2')) + s50 + MIN(1 + z0 + (1 + z11 + z2')) :|: s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 2, s49 >= 0, s49 <= 1 + z0 + (1 + z11 + z2'), s50 >= 0, s50 <= 0, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s52, 1 + z0 + (1 + z11 + z2')) + s7 + MIN(1 + z0 + (1 + z11 + z2')) :|: s51 >= 0, s51 <= 1 + z0 + (1 + z11 + z2'), s52 >= 0, s52 <= 2, s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 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 }-> s65 :|: s65 >= 0, s65 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> s66 :|: s66 >= 0, s66 <= 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
   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 }-> s61 :|: s61 >= 0, s61 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s62 :|: s62 >= 0, s62 <= 1 + (1 + z08) + (1 + 0 + z2), z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> s63 :|: s63 >= 0, s63 <= 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 }-> s64 :|: s64 >= 0, s64 <= 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(s34, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s34 >= 0, s34 <= 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

Function symbols to be analyzed: {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [2 + 3*z'' + z''^2], size: O(1) [0]
  IFREPL: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]
  min: runtime: O(1) [0], size: O(n^1) [z]
  ifmin: runtime: O(1) [0], size: O(n^1) [z']

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

(61) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   EQ(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, 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) -{ 3 + 3*z3 + z3^2 }-> 1 + s40 :|: s40 >= 0, s40 <= 0, 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') -{ 7 + 5*z13 + 2*z13*z21 + z13^2 + 5*z21 + z21^2 }-> 1 + SELSORT(replace(s55, z0, 1 + z13 + z21)) + s57 + MIN(1 + z0 + (1 + z13 + z21)) :|: s55 >= 0, s55 <= 1 + z0 + (1 + z13 + z21), s56 >= 0, s56 <= 1 + z0 + (1 + z13 + z21), s57 >= 0, s57 <= 0, 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') -{ 7 + 5*z13 + 2*z13*z21 + z13^2 + 5*z21 + z21^2 }-> 1 + SELSORT(replace(s58, z0, 1 + z13 + z21)) + s44 + MIN(1 + z0 + (1 + z13 + z21)) :|: s58 >= 0, s58 <= 1 + z0 + (1 + z13 + z21), s44 >= 0, s44 <= 0, s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 3 + 3*z1 + z1^2 }-> 1 + SELSORT(replace(0, z0, z1)) + s46 + MIN(1 + z0 + z1) :|: s46 >= 0, s46 <= 0, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 7 + 5*z14 + 2*z14*z22 + z14^2 + 5*z22 + z22^2 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + s60 + MIN(1 + z0 + (1 + z14 + z22)) :|: s59 >= 0, s59 <= 1 + z0 + (1 + z14 + z22), s60 >= 0, s60 <= 0, s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 3 }-> 1 + SELSORT(replace(0, 0, 0)) + s41 + MIN(1 + 0 + 0) :|: s41 >= 0, s41 <= 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 3 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + s45 + MIN(1 + (1 + (z' - 2)) + 0) :|: s45 >= 0, s45 <= 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 3 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s42 + MIN(1 + (1 + (z' - 2)) + 0) :|: s42 >= 0, s42 <= 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 3 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s43 + MIN(1 + (1 + (z' - 2)) + 0) :|: s43 >= 0, s43 <= 0, z = 1, z' - 2 >= 0
   LE(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 2 + z1' }-> 1 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s24 :|: s24 >= 0, s24 <= 0, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s22 :|: s22 >= 0, s22 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s23 :|: s23 >= 0, s23 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s25 :|: s25 >= 0, s25 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s35 + s' :|: s35 >= 0, s35 <= 1, s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z01 + 2*z01*z3 + z01^2 + 7*z3 + z3^2 }-> 1 + s36 + s'' :|: s36 >= 0, s36 <= 1, s'' >= 0, s'' <= 0, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s37 + s1 :|: s37 >= 0, s37 <= 1, s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z1'' + 2*z1''*z3 + z1''^2 + 7*z3 + z3^2 }-> 1 + s38 + s2 :|: s38 >= 0, s38 <= 1, s26 >= 0, s26 <= 2, s2 >= 0, s2 <= 0, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 8 + 6*z2 + 2*z2*z3 + z2^2 + 5*z3 + z3^2 }-> 1 + s39 + s3 :|: s39 >= 0, s39 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s27, 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s27 >= 0, s27 <= 2, s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(s28, 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s28 >= 0, s28 <= 2, s5 >= 0, s5 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s29, 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s29 >= 0, s29 <= 2, s6 >= 0, s6 <= 0, z - 2 >= 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s30 >= 0, s30 <= 2, s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 + s53 }-> 1 + IFSELSORT(s31, 1 + z0 + (1 + z12 + z2'')) + s54 + MIN(1 + z0 + (1 + z12 + z2'')) :|: s53 >= 0, s53 <= 1 + z0 + (1 + z12 + z2''), s54 >= 0, s54 <= 0, s31 >= 0, s31 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s32, 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s32 >= 0, s32 <= 2, s9 >= 0, s9 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 + s49 }-> 1 + IFSELSORT(s48, 1 + z0 + (1 + z11 + z2')) + s50 + MIN(1 + z0 + (1 + z11 + z2')) :|: s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 2, s49 >= 0, s49 <= 1 + z0 + (1 + z11 + z2'), s50 >= 0, s50 <= 0, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s52, 1 + z0 + (1 + z11 + z2')) + s7 + MIN(1 + z0 + (1 + z11 + z2')) :|: s51 >= 0, s51 <= 1 + z0 + (1 + z11 + z2'), s52 >= 0, s52 <= 2, s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 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 }-> s65 :|: s65 >= 0, s65 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> s66 :|: s66 >= 0, s66 <= 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
   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 }-> s61 :|: s61 >= 0, s61 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s62 :|: s62 >= 0, s62 <= 1 + (1 + z08) + (1 + 0 + z2), z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> s63 :|: s63 >= 0, s63 <= 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 }-> s64 :|: s64 >= 0, s64 <= 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(s34, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s34 >= 0, s34 <= 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

Function symbols to be analyzed: {IFMIN,MIN}, {replace,ifrepl}, {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [2 + 3*z'' + z''^2], size: O(1) [0]
  IFREPL: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]
  min: runtime: O(1) [0], size: O(n^1) [z]
  ifmin: runtime: O(1) [0], size: O(n^1) [z']
  IFMIN: runtime: ?, size: O(1) [0]
  MIN: runtime: ?, size: O(1) [1]

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

(63) 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: 13 + 8*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: 178 + 139*z + 26*z^2

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

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

   EQ(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, 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) -{ 3 + 3*z3 + z3^2 }-> 1 + s40 :|: s40 >= 0, s40 <= 0, 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') -{ 7 + 5*z13 + 2*z13*z21 + z13^2 + 5*z21 + z21^2 }-> 1 + SELSORT(replace(s55, z0, 1 + z13 + z21)) + s57 + MIN(1 + z0 + (1 + z13 + z21)) :|: s55 >= 0, s55 <= 1 + z0 + (1 + z13 + z21), s56 >= 0, s56 <= 1 + z0 + (1 + z13 + z21), s57 >= 0, s57 <= 0, 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') -{ 7 + 5*z13 + 2*z13*z21 + z13^2 + 5*z21 + z21^2 }-> 1 + SELSORT(replace(s58, z0, 1 + z13 + z21)) + s44 + MIN(1 + z0 + (1 + z13 + z21)) :|: s58 >= 0, s58 <= 1 + z0 + (1 + z13 + z21), s44 >= 0, s44 <= 0, s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 3 + 3*z1 + z1^2 }-> 1 + SELSORT(replace(0, z0, z1)) + s46 + MIN(1 + z0 + z1) :|: s46 >= 0, s46 <= 0, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 7 + 5*z14 + 2*z14*z22 + z14^2 + 5*z22 + z22^2 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + s60 + MIN(1 + z0 + (1 + z14 + z22)) :|: s59 >= 0, s59 <= 1 + z0 + (1 + z14 + z22), s60 >= 0, s60 <= 0, s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 3 }-> 1 + SELSORT(replace(0, 0, 0)) + s41 + MIN(1 + 0 + 0) :|: s41 >= 0, s41 <= 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 3 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + s45 + MIN(1 + (1 + (z' - 2)) + 0) :|: s45 >= 0, s45 <= 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 3 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s42 + MIN(1 + (1 + (z' - 2)) + 0) :|: s42 >= 0, s42 <= 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 3 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s43 + MIN(1 + (1 + (z' - 2)) + 0) :|: s43 >= 0, s43 <= 0, z = 1, z' - 2 >= 0
   LE(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 2 + z1' }-> 1 + IFMIN(s10, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s24 :|: s24 >= 0, s24 <= 0, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(2, 1 + 0 + (1 + z1 + z2)) + s22 :|: s22 >= 0, s22 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IFMIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s23 :|: s23 >= 0, s23 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IFMIN(0, 1 + z0 + (1 + z1 + z2)) + s25 :|: s25 >= 0, s25 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s35 + s' :|: s35 >= 0, s35 <= 1, s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z01 + 2*z01*z3 + z01^2 + 7*z3 + z3^2 }-> 1 + s36 + s'' :|: s36 >= 0, s36 <= 1, s'' >= 0, s'' <= 0, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s37 + s1 :|: s37 >= 0, s37 <= 1, s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z1'' + 2*z1''*z3 + z1''^2 + 7*z3 + z3^2 }-> 1 + s38 + s2 :|: s38 >= 0, s38 <= 1, s26 >= 0, s26 <= 2, s2 >= 0, s2 <= 0, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 8 + 6*z2 + 2*z2*z3 + z2^2 + 5*z3 + z3^2 }-> 1 + s39 + s3 :|: s39 >= 0, s39 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s27, 1 + 0 + 0) + s4 + MIN(1 + 0 + 0) :|: s27 >= 0, s27 <= 2, s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(s28, 1 + (1 + (z - 2)) + 0) + s5 + MIN(1 + (1 + (z - 2)) + 0) :|: s28 >= 0, s28 <= 2, s5 >= 0, s5 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s29, 1 + (1 + (z - 2)) + 0) + s6 + MIN(1 + (1 + (z - 2)) + 0) :|: s29 >= 0, s29 <= 2, s6 >= 0, s6 <= 0, z - 2 >= 0
   SELSORT(z) -{ z }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s8 + MIN(1 + (1 + (z - 2)) + 0) :|: s30 >= 0, s30 <= 2, s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 1 + s53 }-> 1 + IFSELSORT(s31, 1 + z0 + (1 + z12 + z2'')) + s54 + MIN(1 + z0 + (1 + z12 + z2'')) :|: s53 >= 0, s53 <= 1 + z0 + (1 + z12 + z2''), s54 >= 0, s54 <= 0, s31 >= 0, s31 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s32, 1 + z0 + z1) + s9 + MIN(1 + z0 + z1) :|: s32 >= 0, s32 <= 2, s9 >= 0, s9 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 1 + s49 }-> 1 + IFSELSORT(s48, 1 + z0 + (1 + z11 + z2')) + s50 + MIN(1 + z0 + (1 + z11 + z2')) :|: s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 2, s49 >= 0, s49 <= 1 + z0 + (1 + z11 + z2'), s50 >= 0, s50 <= 0, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 1 }-> 1 + IFSELSORT(s52, 1 + z0 + (1 + z11 + z2')) + s7 + MIN(1 + z0 + (1 + z11 + z2')) :|: s51 >= 0, s51 <= 1 + z0 + (1 + z11 + z2'), s52 >= 0, s52 <= 2, s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 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 }-> s65 :|: s65 >= 0, s65 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> s66 :|: s66 >= 0, s66 <= 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
   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 }-> s61 :|: s61 >= 0, s61 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s62 :|: s62 >= 0, s62 <= 1 + (1 + z08) + (1 + 0 + z2), z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> s63 :|: s63 >= 0, s63 <= 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 }-> s64 :|: s64 >= 0, s64 <= 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(s34, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s34 >= 0, s34 <= 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

Function symbols to be analyzed: {replace,ifrepl}, {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [2 + 3*z'' + z''^2], size: O(1) [0]
  IFREPL: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]
  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) [13 + 8*z' + z'^2], size: O(1) [0]
  MIN: runtime: O(n^2) [178 + 139*z + 26*z^2], size: O(1) [1]

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

(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') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   IFMIN(z, z') -{ 344 + 191*z0 + 52*z0*z2 + 26*z0^2 + 191*z2 + 26*z2^2 }-> 1 + s71 :|: s71 >= 0, s71 <= 1, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 344 + 191*z1 + 52*z1*z2 + 26*z1^2 + 191*z2 + 26*z2^2 }-> 1 + s72 :|: s72 >= 0, s72 <= 1, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 3 + 3*z3 + z3^2 }-> 1 + s40 :|: s40 >= 0, s40 <= 0, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 344 + 191*z0 + 52*z0*z1 + 26*z0^2 + 191*z1 + 26*z1^2 }-> 1 + s81 :|: s81 >= 0, s81 <= 1, 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') -{ 567 + 243*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 248*z13 + 54*z13*z21 + 27*z13^2 + 248*z21 + 27*z21^2 }-> 1 + SELSORT(replace(s55, z0, 1 + z13 + z21)) + s57 + s85 :|: s85 >= 0, s85 <= 1, s55 >= 0, s55 <= 1 + z0 + (1 + z13 + z21), s56 >= 0, s56 <= 1 + z0 + (1 + z13 + z21), s57 >= 0, s57 <= 0, 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') -{ 567 + 243*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 248*z13 + 54*z13*z21 + 27*z13^2 + 248*z21 + 27*z21^2 }-> 1 + SELSORT(replace(s58, z0, 1 + z13 + z21)) + s44 + s86 :|: s86 >= 0, s86 <= 1, s58 >= 0, s58 <= 1 + z0 + (1 + z13 + z21), s44 >= 0, s44 <= 0, s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 346 + 191*z0 + 52*z0*z1 + 26*z0^2 + 194*z1 + 27*z1^2 }-> 1 + SELSORT(replace(0, z0, z1)) + s46 + s89 :|: s89 >= 0, s89 <= 1, s46 >= 0, s46 <= 0, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 567 + 243*z0 + 52*z0*z14 + 52*z0*z22 + 26*z0^2 + 248*z14 + 54*z14*z22 + 27*z14^2 + 248*z22 + 27*z22^2 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + s60 + s88 :|: s88 >= 0, s88 <= 1, s59 >= 0, s59 <= 1 + z0 + (1 + z14 + z22), s60 >= 0, s60 <= 0, s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 346 }-> 1 + SELSORT(replace(0, 0, 0)) + s41 + s82 :|: s82 >= 0, s82 <= 1, s41 >= 0, s41 <= 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 181 + 139*z' + 26*z'^2 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + s45 + s87 :|: s87 >= 0, s87 <= 1, s45 >= 0, s45 <= 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 181 + 139*z' + 26*z'^2 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s42 + s83 :|: s83 >= 0, s83 <= 1, s42 >= 0, s42 <= 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 181 + 139*z' + 26*z'^2 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s43 + s84 :|: s84 >= 0, s84 <= 1, s43 >= 0, s43 <= 0, z = 1, z' - 2 >= 0
   LE(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 34 + 13*z1 + 2*z1*z2 + z1^2 + 12*z2 + z2^2 }-> 1 + s67 + s22 :|: s67 >= 0, s67 <= 0, s22 >= 0, s22 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 47 + 14*z0' + 2*z0'*z2 + z0'^2 + 14*z2 + z2^2 }-> 1 + s68 + s23 :|: s68 >= 0, s68 <= 0, s23 >= 0, s23 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 63 + 16*z0'' + 2*z0''*z1' + 2*z0''*z2 + z0''^2 + 17*z1' + 2*z1'*z2 + z1'^2 + 16*z2 + z2^2 }-> 1 + s69 + s24 :|: s69 >= 0, s69 <= 0, s24 >= 0, s24 <= 0, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 34 + 12*z0 + 2*z0*z1 + 2*z0*z2 + z0^2 + 13*z1 + 2*z1*z2 + z1^2 + 12*z2 + z2^2 }-> 1 + s70 + s25 :|: s70 >= 0, s70 <= 0, s25 >= 0, s25 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s35 + s' :|: s35 >= 0, s35 <= 1, s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z01 + 2*z01*z3 + z01^2 + 7*z3 + z3^2 }-> 1 + s36 + s'' :|: s36 >= 0, s36 <= 1, s'' >= 0, s'' <= 0, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s37 + s1 :|: s37 >= 0, s37 <= 1, s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z1'' + 2*z1''*z3 + z1''^2 + 7*z3 + z3^2 }-> 1 + s38 + s2 :|: s38 >= 0, s38 <= 1, s26 >= 0, s26 <= 2, s2 >= 0, s2 <= 0, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 8 + 6*z2 + 2*z2*z3 + z2^2 + 5*z3 + z3^2 }-> 1 + s39 + s3 :|: s39 >= 0, s39 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 344 }-> 1 + IFSELSORT(s27, 1 + 0 + 0) + s4 + s73 :|: s73 >= 0, s73 <= 1, s27 >= 0, s27 <= 2, s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ 178 + 140*z + 26*z^2 }-> 1 + IFSELSORT(s28, 1 + (1 + (z - 2)) + 0) + s5 + s74 :|: s74 >= 0, s74 <= 1, s28 >= 0, s28 <= 2, s5 >= 0, s5 <= 0, z - 2 >= 0
   SELSORT(z) -{ 179 + 139*z + 26*z^2 }-> 1 + IFSELSORT(s29, 1 + (1 + (z - 2)) + 0) + s6 + s75 :|: s75 >= 0, s75 <= 1, s29 >= 0, s29 <= 2, s6 >= 0, s6 <= 0, z - 2 >= 0
   SELSORT(z) -{ 178 + 140*z + 26*z^2 }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s8 + s78 :|: s78 >= 0, s78 <= 1, s30 >= 0, s30 <= 2, s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 561 + s53 + 243*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 243*z12 + 52*z12*z2'' + 26*z12^2 + 243*z2'' + 26*z2''^2 }-> 1 + IFSELSORT(s31, 1 + z0 + (1 + z12 + z2'')) + s54 + s79 :|: s79 >= 0, s79 <= 1, s53 >= 0, s53 <= 1 + z0 + (1 + z12 + z2''), s54 >= 0, s54 <= 0, s31 >= 0, s31 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 344 + 191*z0 + 52*z0*z1 + 26*z0^2 + 191*z1 + 26*z1^2 }-> 1 + IFSELSORT(s32, 1 + z0 + z1) + s9 + s80 :|: s80 >= 0, s80 <= 1, s32 >= 0, s32 <= 2, s9 >= 0, s9 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 561 + s49 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 243*z11 + 52*z11*z2' + 26*z11^2 + 243*z2' + 26*z2'^2 }-> 1 + IFSELSORT(s48, 1 + z0 + (1 + z11 + z2')) + s50 + s76 :|: s76 >= 0, s76 <= 1, s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 2, s49 >= 0, s49 <= 1 + z0 + (1 + z11 + z2'), s50 >= 0, s50 <= 0, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 561 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 243*z11 + 52*z11*z2' + 26*z11^2 + 243*z2' + 26*z2'^2 }-> 1 + IFSELSORT(s52, 1 + z0 + (1 + z11 + z2')) + s7 + s77 :|: s77 >= 0, s77 <= 1, s51 >= 0, s51 <= 1 + z0 + (1 + z11 + z2'), s52 >= 0, s52 <= 2, s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 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 }-> s65 :|: s65 >= 0, s65 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> s66 :|: s66 >= 0, s66 <= 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
   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 }-> s61 :|: s61 >= 0, s61 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s62 :|: s62 >= 0, s62 <= 1 + (1 + z08) + (1 + 0 + z2), z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> s63 :|: s63 >= 0, s63 <= 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 }-> s64 :|: s64 >= 0, s64 <= 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(s34, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s34 >= 0, s34 <= 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

Function symbols to be analyzed: {replace,ifrepl}, {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [2 + 3*z'' + z''^2], size: O(1) [0]
  IFREPL: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]
  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) [13 + 8*z' + z'^2], size: O(1) [0]
  MIN: runtime: O(n^2) [178 + 139*z + 26*z^2], size: O(1) [1]

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

(67) 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

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

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

   EQ(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   IFMIN(z, z') -{ 344 + 191*z0 + 52*z0*z2 + 26*z0^2 + 191*z2 + 26*z2^2 }-> 1 + s71 :|: s71 >= 0, s71 <= 1, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 344 + 191*z1 + 52*z1*z2 + 26*z1^2 + 191*z2 + 26*z2^2 }-> 1 + s72 :|: s72 >= 0, s72 <= 1, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 3 + 3*z3 + z3^2 }-> 1 + s40 :|: s40 >= 0, s40 <= 0, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 344 + 191*z0 + 52*z0*z1 + 26*z0^2 + 191*z1 + 26*z1^2 }-> 1 + s81 :|: s81 >= 0, s81 <= 1, 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') -{ 567 + 243*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 248*z13 + 54*z13*z21 + 27*z13^2 + 248*z21 + 27*z21^2 }-> 1 + SELSORT(replace(s55, z0, 1 + z13 + z21)) + s57 + s85 :|: s85 >= 0, s85 <= 1, s55 >= 0, s55 <= 1 + z0 + (1 + z13 + z21), s56 >= 0, s56 <= 1 + z0 + (1 + z13 + z21), s57 >= 0, s57 <= 0, 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') -{ 567 + 243*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 248*z13 + 54*z13*z21 + 27*z13^2 + 248*z21 + 27*z21^2 }-> 1 + SELSORT(replace(s58, z0, 1 + z13 + z21)) + s44 + s86 :|: s86 >= 0, s86 <= 1, s58 >= 0, s58 <= 1 + z0 + (1 + z13 + z21), s44 >= 0, s44 <= 0, s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 346 + 191*z0 + 52*z0*z1 + 26*z0^2 + 194*z1 + 27*z1^2 }-> 1 + SELSORT(replace(0, z0, z1)) + s46 + s89 :|: s89 >= 0, s89 <= 1, s46 >= 0, s46 <= 0, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 567 + 243*z0 + 52*z0*z14 + 52*z0*z22 + 26*z0^2 + 248*z14 + 54*z14*z22 + 27*z14^2 + 248*z22 + 27*z22^2 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + s60 + s88 :|: s88 >= 0, s88 <= 1, s59 >= 0, s59 <= 1 + z0 + (1 + z14 + z22), s60 >= 0, s60 <= 0, s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 346 }-> 1 + SELSORT(replace(0, 0, 0)) + s41 + s82 :|: s82 >= 0, s82 <= 1, s41 >= 0, s41 <= 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 181 + 139*z' + 26*z'^2 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + s45 + s87 :|: s87 >= 0, s87 <= 1, s45 >= 0, s45 <= 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 181 + 139*z' + 26*z'^2 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s42 + s83 :|: s83 >= 0, s83 <= 1, s42 >= 0, s42 <= 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 181 + 139*z' + 26*z'^2 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s43 + s84 :|: s84 >= 0, s84 <= 1, s43 >= 0, s43 <= 0, z = 1, z' - 2 >= 0
   LE(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 34 + 13*z1 + 2*z1*z2 + z1^2 + 12*z2 + z2^2 }-> 1 + s67 + s22 :|: s67 >= 0, s67 <= 0, s22 >= 0, s22 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 47 + 14*z0' + 2*z0'*z2 + z0'^2 + 14*z2 + z2^2 }-> 1 + s68 + s23 :|: s68 >= 0, s68 <= 0, s23 >= 0, s23 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 63 + 16*z0'' + 2*z0''*z1' + 2*z0''*z2 + z0''^2 + 17*z1' + 2*z1'*z2 + z1'^2 + 16*z2 + z2^2 }-> 1 + s69 + s24 :|: s69 >= 0, s69 <= 0, s24 >= 0, s24 <= 0, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 34 + 12*z0 + 2*z0*z1 + 2*z0*z2 + z0^2 + 13*z1 + 2*z1*z2 + z1^2 + 12*z2 + z2^2 }-> 1 + s70 + s25 :|: s70 >= 0, s70 <= 0, s25 >= 0, s25 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s35 + s' :|: s35 >= 0, s35 <= 1, s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z01 + 2*z01*z3 + z01^2 + 7*z3 + z3^2 }-> 1 + s36 + s'' :|: s36 >= 0, s36 <= 1, s'' >= 0, s'' <= 0, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s37 + s1 :|: s37 >= 0, s37 <= 1, s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z1'' + 2*z1''*z3 + z1''^2 + 7*z3 + z3^2 }-> 1 + s38 + s2 :|: s38 >= 0, s38 <= 1, s26 >= 0, s26 <= 2, s2 >= 0, s2 <= 0, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 8 + 6*z2 + 2*z2*z3 + z2^2 + 5*z3 + z3^2 }-> 1 + s39 + s3 :|: s39 >= 0, s39 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 344 }-> 1 + IFSELSORT(s27, 1 + 0 + 0) + s4 + s73 :|: s73 >= 0, s73 <= 1, s27 >= 0, s27 <= 2, s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ 178 + 140*z + 26*z^2 }-> 1 + IFSELSORT(s28, 1 + (1 + (z - 2)) + 0) + s5 + s74 :|: s74 >= 0, s74 <= 1, s28 >= 0, s28 <= 2, s5 >= 0, s5 <= 0, z - 2 >= 0
   SELSORT(z) -{ 179 + 139*z + 26*z^2 }-> 1 + IFSELSORT(s29, 1 + (1 + (z - 2)) + 0) + s6 + s75 :|: s75 >= 0, s75 <= 1, s29 >= 0, s29 <= 2, s6 >= 0, s6 <= 0, z - 2 >= 0
   SELSORT(z) -{ 178 + 140*z + 26*z^2 }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s8 + s78 :|: s78 >= 0, s78 <= 1, s30 >= 0, s30 <= 2, s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 561 + s53 + 243*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 243*z12 + 52*z12*z2'' + 26*z12^2 + 243*z2'' + 26*z2''^2 }-> 1 + IFSELSORT(s31, 1 + z0 + (1 + z12 + z2'')) + s54 + s79 :|: s79 >= 0, s79 <= 1, s53 >= 0, s53 <= 1 + z0 + (1 + z12 + z2''), s54 >= 0, s54 <= 0, s31 >= 0, s31 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 344 + 191*z0 + 52*z0*z1 + 26*z0^2 + 191*z1 + 26*z1^2 }-> 1 + IFSELSORT(s32, 1 + z0 + z1) + s9 + s80 :|: s80 >= 0, s80 <= 1, s32 >= 0, s32 <= 2, s9 >= 0, s9 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 561 + s49 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 243*z11 + 52*z11*z2' + 26*z11^2 + 243*z2' + 26*z2'^2 }-> 1 + IFSELSORT(s48, 1 + z0 + (1 + z11 + z2')) + s50 + s76 :|: s76 >= 0, s76 <= 1, s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 2, s49 >= 0, s49 <= 1 + z0 + (1 + z11 + z2'), s50 >= 0, s50 <= 0, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 561 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 243*z11 + 52*z11*z2' + 26*z11^2 + 243*z2' + 26*z2'^2 }-> 1 + IFSELSORT(s52, 1 + z0 + (1 + z11 + z2')) + s7 + s77 :|: s77 >= 0, s77 <= 1, s51 >= 0, s51 <= 1 + z0 + (1 + z11 + z2'), s52 >= 0, s52 <= 2, s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 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 }-> s65 :|: s65 >= 0, s65 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> s66 :|: s66 >= 0, s66 <= 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
   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 }-> s61 :|: s61 >= 0, s61 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s62 :|: s62 >= 0, s62 <= 1 + (1 + z08) + (1 + 0 + z2), z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> s63 :|: s63 >= 0, s63 <= 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 }-> s64 :|: s64 >= 0, s64 <= 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(s34, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s34 >= 0, s34 <= 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

Function symbols to be analyzed: {replace,ifrepl}, {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [2 + 3*z'' + z''^2], size: O(1) [0]
  IFREPL: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]
  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) [13 + 8*z' + z'^2], size: O(1) [0]
  MIN: runtime: O(n^2) [178 + 139*z + 26*z^2], size: O(1) [1]
  replace: runtime: ?, size: O(n^1) [z' + z'']
  ifrepl: runtime: ?, size: O(n^1) [z'' + z4]

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

(69) 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

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

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

   EQ(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   IFMIN(z, z') -{ 344 + 191*z0 + 52*z0*z2 + 26*z0^2 + 191*z2 + 26*z2^2 }-> 1 + s71 :|: s71 >= 0, s71 <= 1, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 344 + 191*z1 + 52*z1*z2 + 26*z1^2 + 191*z2 + 26*z2^2 }-> 1 + s72 :|: s72 >= 0, s72 <= 1, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 3 + 3*z3 + z3^2 }-> 1 + s40 :|: s40 >= 0, s40 <= 0, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 344 + 191*z0 + 52*z0*z1 + 26*z0^2 + 191*z1 + 26*z1^2 }-> 1 + s81 :|: s81 >= 0, s81 <= 1, 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') -{ 567 + 243*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 248*z13 + 54*z13*z21 + 27*z13^2 + 248*z21 + 27*z21^2 }-> 1 + SELSORT(replace(s55, z0, 1 + z13 + z21)) + s57 + s85 :|: s85 >= 0, s85 <= 1, s55 >= 0, s55 <= 1 + z0 + (1 + z13 + z21), s56 >= 0, s56 <= 1 + z0 + (1 + z13 + z21), s57 >= 0, s57 <= 0, 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') -{ 567 + 243*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 248*z13 + 54*z13*z21 + 27*z13^2 + 248*z21 + 27*z21^2 }-> 1 + SELSORT(replace(s58, z0, 1 + z13 + z21)) + s44 + s86 :|: s86 >= 0, s86 <= 1, s58 >= 0, s58 <= 1 + z0 + (1 + z13 + z21), s44 >= 0, s44 <= 0, s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 346 + 191*z0 + 52*z0*z1 + 26*z0^2 + 194*z1 + 27*z1^2 }-> 1 + SELSORT(replace(0, z0, z1)) + s46 + s89 :|: s89 >= 0, s89 <= 1, s46 >= 0, s46 <= 0, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   IFSELSORT(z, z') -{ 567 + 243*z0 + 52*z0*z14 + 52*z0*z22 + 26*z0^2 + 248*z14 + 54*z14*z22 + 27*z14^2 + 248*z22 + 27*z22^2 }-> 1 + SELSORT(replace(0, z0, 1 + z14 + z22)) + s60 + s88 :|: s88 >= 0, s88 <= 1, s59 >= 0, s59 <= 1 + z0 + (1 + z14 + z22), s60 >= 0, s60 <= 0, s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 346 }-> 1 + SELSORT(replace(0, 0, 0)) + s41 + s82 :|: s82 >= 0, s82 <= 1, s41 >= 0, s41 <= 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 181 + 139*z' + 26*z'^2 }-> 1 + SELSORT(replace(0, 1 + (z' - 2), 0)) + s45 + s87 :|: s87 >= 0, s87 <= 1, s45 >= 0, s45 <= 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 181 + 139*z' + 26*z'^2 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s42 + s83 :|: s83 >= 0, s83 <= 1, s42 >= 0, s42 <= 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 181 + 139*z' + 26*z'^2 }-> 1 + SELSORT(replace(1 + (z' - 2), 1 + (z' - 2), 0)) + s43 + s84 :|: s84 >= 0, s84 <= 1, s43 >= 0, s43 <= 0, z = 1, z' - 2 >= 0
   LE(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 34 + 13*z1 + 2*z1*z2 + z1^2 + 12*z2 + z2^2 }-> 1 + s67 + s22 :|: s67 >= 0, s67 <= 0, s22 >= 0, s22 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 47 + 14*z0' + 2*z0'*z2 + z0'^2 + 14*z2 + z2^2 }-> 1 + s68 + s23 :|: s68 >= 0, s68 <= 0, s23 >= 0, s23 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 63 + 16*z0'' + 2*z0''*z1' + 2*z0''*z2 + z0''^2 + 17*z1' + 2*z1'*z2 + z1'^2 + 16*z2 + z2^2 }-> 1 + s69 + s24 :|: s69 >= 0, s69 <= 0, s24 >= 0, s24 <= 0, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 34 + 12*z0 + 2*z0*z1 + 2*z0*z2 + z0^2 + 13*z1 + 2*z1*z2 + z1^2 + 12*z2 + z2^2 }-> 1 + s70 + s25 :|: s70 >= 0, s70 <= 0, s25 >= 0, s25 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s35 + s' :|: s35 >= 0, s35 <= 1, s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z01 + 2*z01*z3 + z01^2 + 7*z3 + z3^2 }-> 1 + s36 + s'' :|: s36 >= 0, s36 <= 1, s'' >= 0, s'' <= 0, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s37 + s1 :|: s37 >= 0, s37 <= 1, s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z1'' + 2*z1''*z3 + z1''^2 + 7*z3 + z3^2 }-> 1 + s38 + s2 :|: s38 >= 0, s38 <= 1, s26 >= 0, s26 <= 2, s2 >= 0, s2 <= 0, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 8 + 6*z2 + 2*z2*z3 + z2^2 + 5*z3 + z3^2 }-> 1 + s39 + s3 :|: s39 >= 0, s39 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 344 }-> 1 + IFSELSORT(s27, 1 + 0 + 0) + s4 + s73 :|: s73 >= 0, s73 <= 1, s27 >= 0, s27 <= 2, s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ 178 + 140*z + 26*z^2 }-> 1 + IFSELSORT(s28, 1 + (1 + (z - 2)) + 0) + s5 + s74 :|: s74 >= 0, s74 <= 1, s28 >= 0, s28 <= 2, s5 >= 0, s5 <= 0, z - 2 >= 0
   SELSORT(z) -{ 179 + 139*z + 26*z^2 }-> 1 + IFSELSORT(s29, 1 + (1 + (z - 2)) + 0) + s6 + s75 :|: s75 >= 0, s75 <= 1, s29 >= 0, s29 <= 2, s6 >= 0, s6 <= 0, z - 2 >= 0
   SELSORT(z) -{ 178 + 140*z + 26*z^2 }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s8 + s78 :|: s78 >= 0, s78 <= 1, s30 >= 0, s30 <= 2, s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 561 + s53 + 243*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 243*z12 + 52*z12*z2'' + 26*z12^2 + 243*z2'' + 26*z2''^2 }-> 1 + IFSELSORT(s31, 1 + z0 + (1 + z12 + z2'')) + s54 + s79 :|: s79 >= 0, s79 <= 1, s53 >= 0, s53 <= 1 + z0 + (1 + z12 + z2''), s54 >= 0, s54 <= 0, s31 >= 0, s31 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 344 + 191*z0 + 52*z0*z1 + 26*z0^2 + 191*z1 + 26*z1^2 }-> 1 + IFSELSORT(s32, 1 + z0 + z1) + s9 + s80 :|: s80 >= 0, s80 <= 1, s32 >= 0, s32 <= 2, s9 >= 0, s9 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 561 + s49 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 243*z11 + 52*z11*z2' + 26*z11^2 + 243*z2' + 26*z2'^2 }-> 1 + IFSELSORT(s48, 1 + z0 + (1 + z11 + z2')) + s50 + s76 :|: s76 >= 0, s76 <= 1, s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 2, s49 >= 0, s49 <= 1 + z0 + (1 + z11 + z2'), s50 >= 0, s50 <= 0, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 561 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 243*z11 + 52*z11*z2' + 26*z11^2 + 243*z2' + 26*z2'^2 }-> 1 + IFSELSORT(s52, 1 + z0 + (1 + z11 + z2')) + s7 + s77 :|: s77 >= 0, s77 <= 1, s51 >= 0, s51 <= 1 + z0 + (1 + z11 + z2'), s52 >= 0, s52 <= 2, s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 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 }-> s65 :|: s65 >= 0, s65 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> s66 :|: s66 >= 0, s66 <= 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
   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 }-> s61 :|: s61 >= 0, s61 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s62 :|: s62 >= 0, s62 <= 1 + (1 + z08) + (1 + 0 + z2), z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> s63 :|: s63 >= 0, s63 <= 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 }-> s64 :|: s64 >= 0, s64 <= 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(s34, 1 + (z - 1), z', 1 + (1 + z16) + z3) :|: s34 >= 0, s34 <= 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

Function symbols to be analyzed: {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [2 + 3*z'' + z''^2], size: O(1) [0]
  IFREPL: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]
  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) [13 + 8*z' + z'^2], size: O(1) [0]
  MIN: runtime: O(n^2) [178 + 139*z + 26*z^2], size: O(1) [1]
  replace: runtime: O(1) [0], size: O(n^1) [z' + z'']
  ifrepl: runtime: O(1) [0], size: O(n^1) [z'' + z4]

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

(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') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   IFMIN(z, z') -{ 344 + 191*z0 + 52*z0*z2 + 26*z0^2 + 191*z2 + 26*z2^2 }-> 1 + s71 :|: s71 >= 0, s71 <= 1, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 344 + 191*z1 + 52*z1*z2 + 26*z1^2 + 191*z2 + 26*z2^2 }-> 1 + s72 :|: s72 >= 0, s72 <= 1, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 3 + 3*z3 + z3^2 }-> 1 + s40 :|: s40 >= 0, s40 <= 0, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 344 + 191*z0 + 52*z0*z1 + 26*z0^2 + 191*z1 + 26*z1^2 }-> 1 + s81 :|: s81 >= 0, s81 <= 1, 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') -{ 346 }-> 1 + SELSORT(s90) + s41 + s82 :|: s90 >= 0, s90 <= 0 + 0, s82 >= 0, s82 <= 1, s41 >= 0, s41 <= 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 181 + 139*z' + 26*z'^2 }-> 1 + SELSORT(s91) + s42 + s83 :|: s91 >= 0, s91 <= 1 + (z' - 2) + 0, s83 >= 0, s83 <= 1, s42 >= 0, s42 <= 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 181 + 139*z' + 26*z'^2 }-> 1 + SELSORT(s92) + s43 + s84 :|: s92 >= 0, s92 <= 1 + (z' - 2) + 0, s84 >= 0, s84 <= 1, s43 >= 0, s43 <= 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 567 + 243*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 248*z13 + 54*z13*z21 + 27*z13^2 + 248*z21 + 27*z21^2 }-> 1 + SELSORT(s93) + s57 + s85 :|: s93 >= 0, s93 <= z0 + (1 + z13 + z21), s85 >= 0, s85 <= 1, s55 >= 0, s55 <= 1 + z0 + (1 + z13 + z21), s56 >= 0, s56 <= 1 + z0 + (1 + z13 + z21), s57 >= 0, s57 <= 0, 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') -{ 567 + 243*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 248*z13 + 54*z13*z21 + 27*z13^2 + 248*z21 + 27*z21^2 }-> 1 + SELSORT(s94) + s44 + s86 :|: s94 >= 0, s94 <= z0 + (1 + z13 + z21), s86 >= 0, s86 <= 1, s58 >= 0, s58 <= 1 + z0 + (1 + z13 + z21), s44 >= 0, s44 <= 0, s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 181 + 139*z' + 26*z'^2 }-> 1 + SELSORT(s95) + s45 + s87 :|: s95 >= 0, s95 <= 1 + (z' - 2) + 0, s87 >= 0, s87 <= 1, s45 >= 0, s45 <= 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 567 + 243*z0 + 52*z0*z14 + 52*z0*z22 + 26*z0^2 + 248*z14 + 54*z14*z22 + 27*z14^2 + 248*z22 + 27*z22^2 }-> 1 + SELSORT(s96) + s60 + s88 :|: s96 >= 0, s96 <= z0 + (1 + z14 + z22), s88 >= 0, s88 <= 1, s59 >= 0, s59 <= 1 + z0 + (1 + z14 + z22), s60 >= 0, s60 <= 0, s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 346 + 191*z0 + 52*z0*z1 + 26*z0^2 + 194*z1 + 27*z1^2 }-> 1 + SELSORT(s97) + s46 + s89 :|: s97 >= 0, s97 <= z0 + z1, s89 >= 0, s89 <= 1, s46 >= 0, s46 <= 0, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   LE(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 34 + 13*z1 + 2*z1*z2 + z1^2 + 12*z2 + z2^2 }-> 1 + s67 + s22 :|: s67 >= 0, s67 <= 0, s22 >= 0, s22 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 47 + 14*z0' + 2*z0'*z2 + z0'^2 + 14*z2 + z2^2 }-> 1 + s68 + s23 :|: s68 >= 0, s68 <= 0, s23 >= 0, s23 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 63 + 16*z0'' + 2*z0''*z1' + 2*z0''*z2 + z0''^2 + 17*z1' + 2*z1'*z2 + z1'^2 + 16*z2 + z2^2 }-> 1 + s69 + s24 :|: s69 >= 0, s69 <= 0, s24 >= 0, s24 <= 0, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 34 + 12*z0 + 2*z0*z1 + 2*z0*z2 + z0^2 + 13*z1 + 2*z1*z2 + z1^2 + 12*z2 + z2^2 }-> 1 + s70 + s25 :|: s70 >= 0, s70 <= 0, s25 >= 0, s25 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s35 + s' :|: s35 >= 0, s35 <= 1, s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z01 + 2*z01*z3 + z01^2 + 7*z3 + z3^2 }-> 1 + s36 + s'' :|: s36 >= 0, s36 <= 1, s'' >= 0, s'' <= 0, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s37 + s1 :|: s37 >= 0, s37 <= 1, s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z1'' + 2*z1''*z3 + z1''^2 + 7*z3 + z3^2 }-> 1 + s38 + s2 :|: s38 >= 0, s38 <= 1, s26 >= 0, s26 <= 2, s2 >= 0, s2 <= 0, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 8 + 6*z2 + 2*z2*z3 + z2^2 + 5*z3 + z3^2 }-> 1 + s39 + s3 :|: s39 >= 0, s39 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 344 }-> 1 + IFSELSORT(s27, 1 + 0 + 0) + s4 + s73 :|: s73 >= 0, s73 <= 1, s27 >= 0, s27 <= 2, s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ 178 + 140*z + 26*z^2 }-> 1 + IFSELSORT(s28, 1 + (1 + (z - 2)) + 0) + s5 + s74 :|: s74 >= 0, s74 <= 1, s28 >= 0, s28 <= 2, s5 >= 0, s5 <= 0, z - 2 >= 0
   SELSORT(z) -{ 179 + 139*z + 26*z^2 }-> 1 + IFSELSORT(s29, 1 + (1 + (z - 2)) + 0) + s6 + s75 :|: s75 >= 0, s75 <= 1, s29 >= 0, s29 <= 2, s6 >= 0, s6 <= 0, z - 2 >= 0
   SELSORT(z) -{ 178 + 140*z + 26*z^2 }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s8 + s78 :|: s78 >= 0, s78 <= 1, s30 >= 0, s30 <= 2, s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 561 + s53 + 243*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 243*z12 + 52*z12*z2'' + 26*z12^2 + 243*z2'' + 26*z2''^2 }-> 1 + IFSELSORT(s31, 1 + z0 + (1 + z12 + z2'')) + s54 + s79 :|: s79 >= 0, s79 <= 1, s53 >= 0, s53 <= 1 + z0 + (1 + z12 + z2''), s54 >= 0, s54 <= 0, s31 >= 0, s31 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 344 + 191*z0 + 52*z0*z1 + 26*z0^2 + 191*z1 + 26*z1^2 }-> 1 + IFSELSORT(s32, 1 + z0 + z1) + s9 + s80 :|: s80 >= 0, s80 <= 1, s32 >= 0, s32 <= 2, s9 >= 0, s9 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 561 + s49 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 243*z11 + 52*z11*z2' + 26*z11^2 + 243*z2' + 26*z2'^2 }-> 1 + IFSELSORT(s48, 1 + z0 + (1 + z11 + z2')) + s50 + s76 :|: s76 >= 0, s76 <= 1, s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 2, s49 >= 0, s49 <= 1 + z0 + (1 + z11 + z2'), s50 >= 0, s50 <= 0, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 561 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 243*z11 + 52*z11*z2' + 26*z11^2 + 243*z2' + 26*z2'^2 }-> 1 + IFSELSORT(s52, 1 + z0 + (1 + z11 + z2')) + s7 + s77 :|: s77 >= 0, s77 <= 1, s51 >= 0, s51 <= 1 + z0 + (1 + z11 + z2'), s52 >= 0, s52 <= 2, s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 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 }-> s65 :|: s65 >= 0, s65 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> s66 :|: s66 >= 0, s66 <= 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 + s103 :|: s103 >= 0, s103 <= z'' + z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 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 }-> s61 :|: s61 >= 0, s61 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s62 :|: s62 >= 0, s62 <= 1 + (1 + z08) + (1 + 0 + z2), z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> s63 :|: s63 >= 0, s63 <= 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 }-> s64 :|: s64 >= 0, s64 <= 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 }-> s100 :|: s100 >= 0, s100 <= z' + (1 + 0 + (z'' - 1)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s101 :|: s101 >= 0, s101 <= z' + (1 + (1 + z16) + z3), s34 >= 0, s34 <= 2, z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s102 :|: s102 >= 0, s102 <= z' + (1 + z2 + z3), z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s98 :|: s98 >= 0, s98 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s99 :|: s99 >= 0, s99 <= z' + (1 + (1 + z010) + z3), z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 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

Function symbols to be analyzed: {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [2 + 3*z'' + z''^2], size: O(1) [0]
  IFREPL: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]
  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) [13 + 8*z' + z'^2], size: O(1) [0]
  MIN: runtime: O(n^2) [178 + 139*z + 26*z^2], size: O(1) [1]
  replace: runtime: O(1) [0], size: O(n^1) [z' + z'']
  ifrepl: runtime: O(1) [0], size: O(n^1) [z'' + z4]

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

(73) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: IFSELSORT
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 4*z'

Computed SIZE bound using KoAT for: SELSORT
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 6*z

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

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

   EQ(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   IFMIN(z, z') -{ 344 + 191*z0 + 52*z0*z2 + 26*z0^2 + 191*z2 + 26*z2^2 }-> 1 + s71 :|: s71 >= 0, s71 <= 1, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 344 + 191*z1 + 52*z1*z2 + 26*z1^2 + 191*z2 + 26*z2^2 }-> 1 + s72 :|: s72 >= 0, s72 <= 1, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 3 + 3*z3 + z3^2 }-> 1 + s40 :|: s40 >= 0, s40 <= 0, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 344 + 191*z0 + 52*z0*z1 + 26*z0^2 + 191*z1 + 26*z1^2 }-> 1 + s81 :|: s81 >= 0, s81 <= 1, 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') -{ 346 }-> 1 + SELSORT(s90) + s41 + s82 :|: s90 >= 0, s90 <= 0 + 0, s82 >= 0, s82 <= 1, s41 >= 0, s41 <= 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 181 + 139*z' + 26*z'^2 }-> 1 + SELSORT(s91) + s42 + s83 :|: s91 >= 0, s91 <= 1 + (z' - 2) + 0, s83 >= 0, s83 <= 1, s42 >= 0, s42 <= 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 181 + 139*z' + 26*z'^2 }-> 1 + SELSORT(s92) + s43 + s84 :|: s92 >= 0, s92 <= 1 + (z' - 2) + 0, s84 >= 0, s84 <= 1, s43 >= 0, s43 <= 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 567 + 243*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 248*z13 + 54*z13*z21 + 27*z13^2 + 248*z21 + 27*z21^2 }-> 1 + SELSORT(s93) + s57 + s85 :|: s93 >= 0, s93 <= z0 + (1 + z13 + z21), s85 >= 0, s85 <= 1, s55 >= 0, s55 <= 1 + z0 + (1 + z13 + z21), s56 >= 0, s56 <= 1 + z0 + (1 + z13 + z21), s57 >= 0, s57 <= 0, 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') -{ 567 + 243*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 248*z13 + 54*z13*z21 + 27*z13^2 + 248*z21 + 27*z21^2 }-> 1 + SELSORT(s94) + s44 + s86 :|: s94 >= 0, s94 <= z0 + (1 + z13 + z21), s86 >= 0, s86 <= 1, s58 >= 0, s58 <= 1 + z0 + (1 + z13 + z21), s44 >= 0, s44 <= 0, s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 181 + 139*z' + 26*z'^2 }-> 1 + SELSORT(s95) + s45 + s87 :|: s95 >= 0, s95 <= 1 + (z' - 2) + 0, s87 >= 0, s87 <= 1, s45 >= 0, s45 <= 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 567 + 243*z0 + 52*z0*z14 + 52*z0*z22 + 26*z0^2 + 248*z14 + 54*z14*z22 + 27*z14^2 + 248*z22 + 27*z22^2 }-> 1 + SELSORT(s96) + s60 + s88 :|: s96 >= 0, s96 <= z0 + (1 + z14 + z22), s88 >= 0, s88 <= 1, s59 >= 0, s59 <= 1 + z0 + (1 + z14 + z22), s60 >= 0, s60 <= 0, s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 346 + 191*z0 + 52*z0*z1 + 26*z0^2 + 194*z1 + 27*z1^2 }-> 1 + SELSORT(s97) + s46 + s89 :|: s97 >= 0, s97 <= z0 + z1, s89 >= 0, s89 <= 1, s46 >= 0, s46 <= 0, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   LE(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 34 + 13*z1 + 2*z1*z2 + z1^2 + 12*z2 + z2^2 }-> 1 + s67 + s22 :|: s67 >= 0, s67 <= 0, s22 >= 0, s22 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 47 + 14*z0' + 2*z0'*z2 + z0'^2 + 14*z2 + z2^2 }-> 1 + s68 + s23 :|: s68 >= 0, s68 <= 0, s23 >= 0, s23 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 63 + 16*z0'' + 2*z0''*z1' + 2*z0''*z2 + z0''^2 + 17*z1' + 2*z1'*z2 + z1'^2 + 16*z2 + z2^2 }-> 1 + s69 + s24 :|: s69 >= 0, s69 <= 0, s24 >= 0, s24 <= 0, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 34 + 12*z0 + 2*z0*z1 + 2*z0*z2 + z0^2 + 13*z1 + 2*z1*z2 + z1^2 + 12*z2 + z2^2 }-> 1 + s70 + s25 :|: s70 >= 0, s70 <= 0, s25 >= 0, s25 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s35 + s' :|: s35 >= 0, s35 <= 1, s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z01 + 2*z01*z3 + z01^2 + 7*z3 + z3^2 }-> 1 + s36 + s'' :|: s36 >= 0, s36 <= 1, s'' >= 0, s'' <= 0, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s37 + s1 :|: s37 >= 0, s37 <= 1, s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z1'' + 2*z1''*z3 + z1''^2 + 7*z3 + z3^2 }-> 1 + s38 + s2 :|: s38 >= 0, s38 <= 1, s26 >= 0, s26 <= 2, s2 >= 0, s2 <= 0, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 8 + 6*z2 + 2*z2*z3 + z2^2 + 5*z3 + z3^2 }-> 1 + s39 + s3 :|: s39 >= 0, s39 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 344 }-> 1 + IFSELSORT(s27, 1 + 0 + 0) + s4 + s73 :|: s73 >= 0, s73 <= 1, s27 >= 0, s27 <= 2, s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ 178 + 140*z + 26*z^2 }-> 1 + IFSELSORT(s28, 1 + (1 + (z - 2)) + 0) + s5 + s74 :|: s74 >= 0, s74 <= 1, s28 >= 0, s28 <= 2, s5 >= 0, s5 <= 0, z - 2 >= 0
   SELSORT(z) -{ 179 + 139*z + 26*z^2 }-> 1 + IFSELSORT(s29, 1 + (1 + (z - 2)) + 0) + s6 + s75 :|: s75 >= 0, s75 <= 1, s29 >= 0, s29 <= 2, s6 >= 0, s6 <= 0, z - 2 >= 0
   SELSORT(z) -{ 178 + 140*z + 26*z^2 }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s8 + s78 :|: s78 >= 0, s78 <= 1, s30 >= 0, s30 <= 2, s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 561 + s53 + 243*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 243*z12 + 52*z12*z2'' + 26*z12^2 + 243*z2'' + 26*z2''^2 }-> 1 + IFSELSORT(s31, 1 + z0 + (1 + z12 + z2'')) + s54 + s79 :|: s79 >= 0, s79 <= 1, s53 >= 0, s53 <= 1 + z0 + (1 + z12 + z2''), s54 >= 0, s54 <= 0, s31 >= 0, s31 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 344 + 191*z0 + 52*z0*z1 + 26*z0^2 + 191*z1 + 26*z1^2 }-> 1 + IFSELSORT(s32, 1 + z0 + z1) + s9 + s80 :|: s80 >= 0, s80 <= 1, s32 >= 0, s32 <= 2, s9 >= 0, s9 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 561 + s49 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 243*z11 + 52*z11*z2' + 26*z11^2 + 243*z2' + 26*z2'^2 }-> 1 + IFSELSORT(s48, 1 + z0 + (1 + z11 + z2')) + s50 + s76 :|: s76 >= 0, s76 <= 1, s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 2, s49 >= 0, s49 <= 1 + z0 + (1 + z11 + z2'), s50 >= 0, s50 <= 0, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 561 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 243*z11 + 52*z11*z2' + 26*z11^2 + 243*z2' + 26*z2'^2 }-> 1 + IFSELSORT(s52, 1 + z0 + (1 + z11 + z2')) + s7 + s77 :|: s77 >= 0, s77 <= 1, s51 >= 0, s51 <= 1 + z0 + (1 + z11 + z2'), s52 >= 0, s52 <= 2, s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 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 }-> s65 :|: s65 >= 0, s65 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> s66 :|: s66 >= 0, s66 <= 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 + s103 :|: s103 >= 0, s103 <= z'' + z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 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 }-> s61 :|: s61 >= 0, s61 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s62 :|: s62 >= 0, s62 <= 1 + (1 + z08) + (1 + 0 + z2), z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> s63 :|: s63 >= 0, s63 <= 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 }-> s64 :|: s64 >= 0, s64 <= 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 }-> s100 :|: s100 >= 0, s100 <= z' + (1 + 0 + (z'' - 1)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s101 :|: s101 >= 0, s101 <= z' + (1 + (1 + z16) + z3), s34 >= 0, s34 <= 2, z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s102 :|: s102 >= 0, s102 <= z' + (1 + z2 + z3), z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s98 :|: s98 >= 0, s98 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s99 :|: s99 >= 0, s99 <= z' + (1 + (1 + z010) + z3), z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 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

Function symbols to be analyzed: {IFSELSORT,SELSORT}
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [2 + 3*z'' + z''^2], size: O(1) [0]
  IFREPL: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]
  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) [13 + 8*z' + z'^2], size: O(1) [0]
  MIN: runtime: O(n^2) [178 + 139*z + 26*z^2], size: O(1) [1]
  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: O(n^1) [4*z']
  SELSORT: runtime: ?, size: O(n^1) [6*z]

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

(75) 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: 830 + 27872*z' + 19254*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: 594694 + 1569430*z + 1275308*z^2 + 327704*z^3

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

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

   EQ(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   IFMIN(z, z') -{ 344 + 191*z0 + 52*z0*z2 + 26*z0^2 + 191*z2 + 26*z2^2 }-> 1 + s71 :|: s71 >= 0, s71 <= 1, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IFMIN(z, z') -{ 344 + 191*z1 + 52*z1*z2 + 26*z1^2 + 191*z2 + 26*z2^2 }-> 1 + s72 :|: s72 >= 0, s72 <= 1, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IFREPL(z, z', z'', z4) -{ 3 + 3*z3 + z3^2 }-> 1 + s40 :|: s40 >= 0, s40 <= 0, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IFSELSORT(z, z') -{ 344 + 191*z0 + 52*z0*z1 + 26*z0^2 + 191*z1 + 26*z1^2 }-> 1 + s81 :|: s81 >= 0, s81 <= 1, 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') -{ 346 }-> 1 + SELSORT(s90) + s41 + s82 :|: s90 >= 0, s90 <= 0 + 0, s82 >= 0, s82 <= 1, s41 >= 0, s41 <= 0, z' = 1 + 0 + 0, z = 1
   IFSELSORT(z, z') -{ 181 + 139*z' + 26*z'^2 }-> 1 + SELSORT(s91) + s42 + s83 :|: s91 >= 0, s91 <= 1 + (z' - 2) + 0, s83 >= 0, s83 <= 1, s42 >= 0, s42 <= 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 181 + 139*z' + 26*z'^2 }-> 1 + SELSORT(s92) + s43 + s84 :|: s92 >= 0, s92 <= 1 + (z' - 2) + 0, s84 >= 0, s84 <= 1, s43 >= 0, s43 <= 0, z = 1, z' - 2 >= 0
   IFSELSORT(z, z') -{ 567 + 243*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 248*z13 + 54*z13*z21 + 27*z13^2 + 248*z21 + 27*z21^2 }-> 1 + SELSORT(s93) + s57 + s85 :|: s93 >= 0, s93 <= z0 + (1 + z13 + z21), s85 >= 0, s85 <= 1, s55 >= 0, s55 <= 1 + z0 + (1 + z13 + z21), s56 >= 0, s56 <= 1 + z0 + (1 + z13 + z21), s57 >= 0, s57 <= 0, 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') -{ 567 + 243*z0 + 52*z0*z13 + 52*z0*z21 + 26*z0^2 + 248*z13 + 54*z13*z21 + 27*z13^2 + 248*z21 + 27*z21^2 }-> 1 + SELSORT(s94) + s44 + s86 :|: s94 >= 0, s94 <= z0 + (1 + z13 + z21), s86 >= 0, s86 <= 1, s58 >= 0, s58 <= 1 + z0 + (1 + z13 + z21), s44 >= 0, s44 <= 0, s17 >= 0, s17 <= 2, z21 >= 0, z = 1, z' = 1 + z0 + (1 + z13 + z21), z0 >= 0, z13 >= 0
   IFSELSORT(z, z') -{ 181 + 139*z' + 26*z'^2 }-> 1 + SELSORT(s95) + s45 + s87 :|: s95 >= 0, s95 <= 1 + (z' - 2) + 0, s87 >= 0, s87 <= 1, s45 >= 0, s45 <= 0, z' - 2 >= 0, z = 1
   IFSELSORT(z, z') -{ 567 + 243*z0 + 52*z0*z14 + 52*z0*z22 + 26*z0^2 + 248*z14 + 54*z14*z22 + 27*z14^2 + 248*z22 + 27*z22^2 }-> 1 + SELSORT(s96) + s60 + s88 :|: s96 >= 0, s96 <= z0 + (1 + z14 + z22), s88 >= 0, s88 <= 1, s59 >= 0, s59 <= 1 + z0 + (1 + z14 + z22), s60 >= 0, s60 <= 0, s18 >= 0, s18 <= 2, z' = 1 + z0 + (1 + z14 + z22), z = 1, z0 >= 0, z22 >= 0, z14 >= 0
   IFSELSORT(z, z') -{ 346 + 191*z0 + 52*z0*z1 + 26*z0^2 + 194*z1 + 27*z1^2 }-> 1 + SELSORT(s97) + s46 + s89 :|: s97 >= 0, s97 <= z0 + z1, s89 >= 0, s89 <= 1, s46 >= 0, s46 <= 0, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0
   LE(z, z') -{ z' }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 34 + 13*z1 + 2*z1*z2 + z1^2 + 12*z2 + z2^2 }-> 1 + s67 + s22 :|: s67 >= 0, s67 <= 0, s22 >= 0, s22 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 47 + 14*z0' + 2*z0'*z2 + z0'^2 + 14*z2 + z2^2 }-> 1 + s68 + s23 :|: s68 >= 0, s68 <= 0, s23 >= 0, s23 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 63 + 16*z0'' + 2*z0''*z1' + 2*z0''*z2 + z0''^2 + 17*z1' + 2*z1'*z2 + z1'^2 + 16*z2 + z2^2 }-> 1 + s69 + s24 :|: s69 >= 0, s69 <= 0, s24 >= 0, s24 <= 0, s10 >= 0, s10 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 34 + 12*z0 + 2*z0*z1 + 2*z0*z2 + z0^2 + 13*z1 + 2*z1*z2 + z1^2 + 12*z2 + z2^2 }-> 1 + s70 + s25 :|: s70 >= 0, s70 <= 0, s25 >= 0, s25 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s35 + s' :|: s35 >= 0, s35 <= 1, s' >= 0, s' <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z01 + 2*z01*z3 + z01^2 + 7*z3 + z3^2 }-> 1 + s36 + s'' :|: s36 >= 0, s36 <= 1, s'' >= 0, s'' <= 0, z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s37 + s1 :|: s37 >= 0, s37 <= 1, s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 15 + 8*z1'' + 2*z1''*z3 + z1''^2 + 7*z3 + z3^2 }-> 1 + s38 + s2 :|: s38 >= 0, s38 <= 1, s26 >= 0, s26 <= 2, s2 >= 0, s2 <= 0, z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 8 + 6*z2 + 2*z2*z3 + z2^2 + 5*z3 + z3^2 }-> 1 + s39 + s3 :|: s39 >= 0, s39 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SELSORT(z) -{ 344 }-> 1 + IFSELSORT(s27, 1 + 0 + 0) + s4 + s73 :|: s73 >= 0, s73 <= 1, s27 >= 0, s27 <= 2, s4 >= 0, s4 <= 0, z = 1 + 0 + 0
   SELSORT(z) -{ 178 + 140*z + 26*z^2 }-> 1 + IFSELSORT(s28, 1 + (1 + (z - 2)) + 0) + s5 + s74 :|: s74 >= 0, s74 <= 1, s28 >= 0, s28 <= 2, s5 >= 0, s5 <= 0, z - 2 >= 0
   SELSORT(z) -{ 179 + 139*z + 26*z^2 }-> 1 + IFSELSORT(s29, 1 + (1 + (z - 2)) + 0) + s6 + s75 :|: s75 >= 0, s75 <= 1, s29 >= 0, s29 <= 2, s6 >= 0, s6 <= 0, z - 2 >= 0
   SELSORT(z) -{ 178 + 140*z + 26*z^2 }-> 1 + IFSELSORT(s30, 1 + (1 + (z - 2)) + 0) + s8 + s78 :|: s78 >= 0, s78 <= 1, s30 >= 0, s30 <= 2, s8 >= 0, s8 <= 0, z - 2 >= 0
   SELSORT(z) -{ 561 + s53 + 243*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 243*z12 + 52*z12*z2'' + 26*z12^2 + 243*z2'' + 26*z2''^2 }-> 1 + IFSELSORT(s31, 1 + z0 + (1 + z12 + z2'')) + s54 + s79 :|: s79 >= 0, s79 <= 1, s53 >= 0, s53 <= 1 + z0 + (1 + z12 + z2''), s54 >= 0, s54 <= 0, s31 >= 0, s31 <= 2, s14 >= 0, s14 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SELSORT(z) -{ 344 + 191*z0 + 52*z0*z1 + 26*z0^2 + 191*z1 + 26*z1^2 }-> 1 + IFSELSORT(s32, 1 + z0 + z1) + s9 + s80 :|: s80 >= 0, s80 <= 1, s32 >= 0, s32 <= 2, s9 >= 0, s9 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SELSORT(z) -{ 561 + s49 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 243*z11 + 52*z11*z2' + 26*z11^2 + 243*z2' + 26*z2'^2 }-> 1 + IFSELSORT(s48, 1 + z0 + (1 + z11 + z2')) + s50 + s76 :|: s76 >= 0, s76 <= 1, s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 2, s49 >= 0, s49 <= 1 + z0 + (1 + z11 + z2'), s50 >= 0, s50 <= 0, s11 >= 0, s11 <= 2, s12 >= 0, s12 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SELSORT(z) -{ 561 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 243*z11 + 52*z11*z2' + 26*z11^2 + 243*z2' + 26*z2'^2 }-> 1 + IFSELSORT(s52, 1 + z0 + (1 + z11 + z2')) + s7 + s77 :|: s77 >= 0, s77 <= 1, s51 >= 0, s51 <= 1 + z0 + (1 + z11 + z2'), s52 >= 0, s52 <= 2, s13 >= 0, s13 <= 2, s7 >= 0, s7 <= 0, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   eq(z, z') -{ 0 }-> s33 :|: s33 >= 0, s33 <= 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 }-> s65 :|: s65 >= 0, s65 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   ifmin(z, z') -{ 0 }-> s66 :|: s66 >= 0, s66 <= 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 + s103 :|: s103 >= 0, s103 <= z'' + z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 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 }-> s61 :|: s61 >= 0, s61 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s62 :|: s62 >= 0, s62 <= 1 + (1 + z08) + (1 + 0 + z2), z08 >= 0, z = 1 + (1 + z08) + (1 + 0 + z2), z2 >= 0
   min(z) -{ 0 }-> s63 :|: s63 >= 0, s63 <= 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 }-> s64 :|: s64 >= 0, s64 <= 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 }-> s100 :|: s100 >= 0, s100 <= z' + (1 + 0 + (z'' - 1)), z - 1 >= 0, z' >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s101 :|: s101 >= 0, s101 <= z' + (1 + (1 + z16) + z3), s34 >= 0, s34 <= 2, z' >= 0, z - 1 >= 0, z16 >= 0, z'' = 1 + (1 + z16) + z3, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s102 :|: s102 >= 0, s102 <= z' + (1 + z2 + z3), z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s98 :|: s98 >= 0, s98 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s99 :|: s99 >= 0, s99 <= z' + (1 + (1 + z010) + z3), z'' = 1 + (1 + z010) + z3, z' >= 0, z = 0, z010 >= 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

Function symbols to be analyzed: 
Previous analysis results are:
  EQ: runtime: O(n^1) [z'], size: O(1) [0]
  le: runtime: O(1) [0], size: O(1) [2]
  LE: runtime: O(n^1) [z'], size: O(1) [0]
  eq: runtime: O(1) [0], size: O(1) [2]
  REPLACE: runtime: O(n^2) [2 + 3*z'' + z''^2], size: O(1) [0]
  IFREPL: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]
  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) [13 + 8*z' + z'^2], size: O(1) [0]
  MIN: runtime: O(n^2) [178 + 139*z + 26*z^2], size: O(1) [1]
  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) [830 + 27872*z' + 19254*z'^2 + 3562*z'^3], size: O(n^1) [4*z']
  SELSORT: runtime: O(n^3) [594694 + 1569430*z + 1275308*z^2 + 327704*z^3], size: O(n^1) [6*z]

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

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

(78)
BOUNDS(1, n^3)

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

(79) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(80)
Obligation:
Complexity Dependency Tuples Problem

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)))
Tuples:
   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)))
S tuples:
   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)))
K tuples:none
Defined Rule Symbols:   eq_2, le_2, min_1, ifmin_2, replace_3, ifrepl_4, selsort_1, ifselsort_2

Defined Pair Symbols:   EQ_2, LE_2, MIN_1, IFMIN_2, REPLACE_3, IFREPL_4, SELSORT_1, IFSELSORT_2

Compound Symbols:   c, c1, c2, c3_1, c4, c5, c6_1, c7, c8, c9_2, c10_1, c11_1, c12, c13_2, c14, c15_1, c16, c17_3, c18_1, c19_1, c20_3


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

(81) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(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, 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
----------------------------------------

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

(84)
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
----------------------------------------

(85) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(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, 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)))
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 -> 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
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

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

(87) 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

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

(88)
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, 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)))
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 -> 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
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

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

(89) 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).
----------------------------------------

(90)
Complex Obligation (BEST)

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

(91)
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, 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)))
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 -> 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
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

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

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

(93)
BOUNDS(n^1, INF)

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

(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, 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)))
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 -> 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
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

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

(95) 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).
----------------------------------------

(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, 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)))
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 -> 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
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

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

(97) 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).
----------------------------------------

(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, 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)))
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 -> 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
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

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

(99) 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).
----------------------------------------

(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, 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)))
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 -> 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
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

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

(101) 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).
----------------------------------------

(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, 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)))
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 -> 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
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

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

(103) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
min(gen_nil:cons15_21(+(1, n8126_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, +(n8126_21, 1)))) ->_R^Omega(0)
ifmin(le(0', 0'), cons(0', cons(0', gen_nil:cons15_21(n8126_21)))) ->_L^Omega(0)
ifmin(true, cons(0', cons(0', gen_nil:cons15_21(n8126_21)))) ->_R^Omega(0)
min(cons(0', gen_nil:cons15_21(n8126_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).
----------------------------------------

(104)
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, 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)))
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 -> 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
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, n8126_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

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

(105) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
SELSORT(gen_nil:cons15_21(n9465_21)) -> *16_21, rt in Omega(n9465_21 + n9465_21^2)

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

Induction Step:
SELSORT(gen_nil:cons15_21(+(n9465_21, 1))) ->_R^Omega(1)
c17(IFSELSORT(eq(0', min(cons(0', gen_nil:cons15_21(n9465_21)))), cons(0', gen_nil:cons15_21(n9465_21))), EQ(0', min(cons(0', gen_nil:cons15_21(n9465_21)))), MIN(cons(0', gen_nil:cons15_21(n9465_21)))) ->_L^Omega(0)
c17(IFSELSORT(eq(0', gen_0':s13_21(0)), cons(0', gen_nil:cons15_21(n9465_21))), EQ(0', min(cons(0', gen_nil:cons15_21(n9465_21)))), MIN(cons(0', gen_nil:cons15_21(n9465_21)))) ->_L^Omega(0)
c17(IFSELSORT(true, cons(0', gen_nil:cons15_21(n9465_21))), EQ(0', min(cons(0', gen_nil:cons15_21(n9465_21)))), MIN(cons(0', gen_nil:cons15_21(n9465_21)))) ->_R^Omega(1)
c17(c18(SELSORT(gen_nil:cons15_21(n9465_21))), EQ(0', min(cons(0', gen_nil:cons15_21(n9465_21)))), MIN(cons(0', gen_nil:cons15_21(n9465_21)))) ->_IH
c17(c18(*16_21), EQ(0', min(cons(0', gen_nil:cons15_21(n9465_21)))), MIN(cons(0', gen_nil:cons15_21(n9465_21)))) ->_L^Omega(0)
c17(c18(*16_21), EQ(0', gen_0':s13_21(0)), MIN(cons(0', gen_nil:cons15_21(n9465_21)))) ->_L^Omega(1)
c17(c18(*16_21), gen_c:c1:c2:c312_21(0), MIN(cons(0', gen_nil:cons15_21(n9465_21)))) ->_L^Omega(n9465_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).
----------------------------------------

(106)
Complex Obligation (BEST)

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

(107)
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, 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)))
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 -> 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
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, n8126_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
----------------------------------------

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

(109)
BOUNDS(n^2, INF)

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

(110)
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, 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)))
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 -> 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
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, n8126_21))) -> gen_0':s13_21(0), rt in Omega(0)
SELSORT(gen_nil:cons15_21(n9465_21)) -> *16_21, rt in Omega(n9465_21 + n9465_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
----------------------------------------

(111) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
selsort(gen_nil:cons15_21(n19406_21)) -> gen_nil:cons15_21(n19406_21), rt in Omega(0)

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

Induction Step:
selsort(gen_nil:cons15_21(+(n19406_21, 1))) ->_R^Omega(0)
ifselsort(eq(0', min(cons(0', gen_nil:cons15_21(n19406_21)))), cons(0', gen_nil:cons15_21(n19406_21))) ->_L^Omega(0)
ifselsort(eq(0', gen_0':s13_21(0)), cons(0', gen_nil:cons15_21(n19406_21))) ->_L^Omega(0)
ifselsort(true, cons(0', gen_nil:cons15_21(n19406_21))) ->_R^Omega(0)
cons(0', selsort(gen_nil:cons15_21(n19406_21))) ->_IH
cons(0', gen_nil:cons15_21(c19407_21))

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

(112)
BOUNDS(1, INF)