WORST_CASE(Omega(n^2),O(n^3))
proof of input_fLs38VTIPM.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished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), 0 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), 220 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 74 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 299 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 44 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 99 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 42 ms]
    (40) CpxRNTS
    (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 471 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 75 ms]
    (46) CpxRNTS
    (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 943 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 432 ms]
    (52) CpxRNTS
    (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (54) CpxRNTS
    (55) IntTrsBoundProof [UPPER BOUND(ID), 1322 ms]
    (56) CpxRNTS
    (57) IntTrsBoundProof [UPPER BOUND(ID), 655 ms]
    (58) CpxRNTS
    (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (60) CpxRNTS
    (61) IntTrsBoundProof [UPPER BOUND(ID), 612 ms]
    (62) CpxRNTS
    (63) IntTrsBoundProof [UPPER BOUND(ID), 230 ms]
    (64) CpxRNTS
    (65) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (66) CpxRNTS
    (67) IntTrsBoundProof [UPPER BOUND(ID), 1676 ms]
    (68) CpxRNTS
    (69) IntTrsBoundProof [UPPER BOUND(ID), 283 ms]
    (70) CpxRNTS
    (71) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (72) CpxRNTS
    (73) IntTrsBoundProof [UPPER BOUND(ID), 1151 ms]
    (74) CpxRNTS
    (75) IntTrsBoundProof [UPPER BOUND(ID), 460 ms]
    (76) CpxRNTS
    (77) FinalProof [FINISHED, 0 ms]
    (78) BOUNDS(1, n^3)
(79) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 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), 6 ms]
    (88) typed CpxTrs
    (89) RewriteLemmaProof [LOWER BOUND(ID), 408 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), 146 ms]
            (96) typed CpxTrs
            (97) RewriteLemmaProof [LOWER BOUND(ID), 75 ms]
            (98) typed CpxTrs
            (99) RewriteLemmaProof [LOWER BOUND(ID), 634 ms]
            (100) typed CpxTrs
            (101) RewriteLemmaProof [LOWER BOUND(ID), 55 ms]
            (102) typed CpxTrs
            (103) RewriteLemmaProof [LOWER BOUND(ID), 81 ms]
            (104) typed CpxTrs
            (105) RewriteLemmaProof [LOWER BOUND(ID), 18.1 s]
            (106) BEST
                (107) proven lower bound
                    (108) LowerBoundPropagationProof [FINISHED, 0 ms]
                    (109) BOUNDS(n^2, INF)
                (110) typed CpxTrs
                    (111) RewriteLemmaProof [LOWER BOUND(ID), 1197 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(m)) -> false
   eq(s(n), 0) -> false
   eq(s(n), s(m)) -> eq(n, m)
   le(0, m) -> true
   le(s(n), 0) -> false
   le(s(n), s(m)) -> le(n, m)
   min(cons(0, nil)) -> 0
   min(cons(s(n), nil)) -> s(n)
   min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x)))
   if_min(true, cons(n, cons(m, x))) -> min(cons(n, x))
   if_min(false, cons(n, cons(m, x))) -> min(cons(m, x))
   replace(n, m, nil) -> nil
   replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x))
   if_replace(true, n, m, cons(k, x)) -> cons(m, x)
   if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x))
   sort(nil) -> nil
   sort(cons(n, x)) -> cons(min(cons(n, x)), sort(replace(min(cons(n, x)), n, x)))

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))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
   if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
   if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
   replace(z0, z1, nil) -> nil
   replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
   if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
   if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
   sort(nil) -> nil
   sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))
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(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
   IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
   IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
   REPLACE(z0, z1, nil) -> c12
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
   IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c14
   IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
   SORT(nil) -> c16
   SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
   SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1)))
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(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
   IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
   IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
   REPLACE(z0, z1, nil) -> c12
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
   IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c14
   IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
   SORT(nil) -> c16
   SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
   SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1)))
K tuples:none
Defined Rule Symbols:   eq_2, le_2, min_1, if_min_2, replace_3, if_replace_4, sort_1

Defined Pair Symbols:   EQ_2, LE_2, MIN_1, IF_MIN_2, REPLACE_3, IF_REPLACE_4, SORT_1

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_1, c18_3


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

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

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

(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))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
   if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
   if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
   replace(z0, z1, nil) -> nil
   replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
   if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
   if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
   sort(nil) -> nil
   sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))
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(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
   IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
   IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
   IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
   SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
   SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1)))
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(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
   IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
   IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
   IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
   SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
   SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1)))
K tuples:none
Defined Rule Symbols:   eq_2, le_2, min_1, if_min_2, replace_3, if_replace_4, sort_1

Defined Pair Symbols:   EQ_2, LE_2, MIN_1, IF_MIN_2, REPLACE_3, IF_REPLACE_4, SORT_1

Compound Symbols:   c3_1, c6_1, c9_2, c10_1, c11_1, c13_2, c15_1, c17_1, c18_3


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

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   sort(nil) -> nil
   sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(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)
   replace(z0, z1, nil) -> nil
   replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
   min(cons(0, nil)) -> 0
   min(cons(s(z0), nil)) -> s(z0)
   min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
   if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
   if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
   if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
   if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
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(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
   IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
   IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
   IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
   SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
   SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1)))
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(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
   IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
   IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
   IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
   SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
   SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1)))
K tuples:none
Defined Rule Symbols:   le_2, eq_2, replace_3, min_1, if_min_2, if_replace_4

Defined Pair Symbols:   EQ_2, LE_2, MIN_1, IF_MIN_2, REPLACE_3, IF_REPLACE_4, SORT_1

Compound Symbols:   c3_1, c6_1, c9_2, c10_1, c11_1, c13_2, c15_1, c17_1, c18_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(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
   IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
   IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
   IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
   SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
   SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1)))

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

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

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(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1)) [1]
   IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2))) [1]
   IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2))) [1]
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) [1]
   IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3)) [1]
   SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1))) [1]
   SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1))) [1]
   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]
   replace(z0, z1, nil) -> nil [0]
   replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3)) [0]
   min(cons(0, nil)) -> 0 [0]
   min(cons(s(z0), nil)) -> s(z0) [0]
   min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2))) [0]
   if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) [0]
   if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) [0]
   if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) [0]
   if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) [0]

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(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1)) [1]
   IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2))) [1]
   IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2))) [1]
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) [1]
   IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3)) [1]
   SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1))) [1]
   SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1))) [1]
   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]
   replace(z0, z1, nil) -> nil [0]
   replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3)) [0]
   min(cons(0, nil)) -> 0 [0]
   min(cons(s(z0), nil)) -> s(z0) [0]
   min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2))) [0]
   if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) [0]
   if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) [0]
   if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) [0]
   if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) [0]

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
IF_MIN :: 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
IF_REPLACE :: true:false -> s:0 -> s:0 -> cons:nil -> c15
eq :: s:0 -> s:0 -> true:false
c15 :: c13 -> c15
SORT :: cons:nil -> c17:c18
c17 :: c9 -> c17:c18
c18 :: c17:c18 -> c13 -> c9 -> c17:c18
replace :: s:0 -> s:0 -> cons:nil -> cons:nil
min :: cons:nil -> s:0
0 :: s:0
nil :: cons:nil
if_replace :: true:false -> s:0 -> s:0 -> cons:nil -> cons:nil
if_min :: true:false -> cons:nil -> s:0

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
   IF_MIN_2
   REPLACE_3
   IF_REPLACE_4
   SORT_1

(c) The following functions are completely defined:

   le_2
   eq_2
   replace_3
   min_1
   if_min_2
   if_replace_4

Due to the following rules being added:

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

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

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

(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(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1)) [1]
   IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2))) [1]
   IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2))) [1]
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2)) [1]
   IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3)) [1]
   SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1))) [1]
   SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1))) [1]
   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]
   replace(z0, z1, nil) -> nil [0]
   replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3)) [0]
   min(cons(0, nil)) -> 0 [0]
   min(cons(s(z0), nil)) -> s(z0) [0]
   min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2))) [0]
   if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) [0]
   if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) [0]
   if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) [0]
   if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) [0]
   le(v0, v1) -> null_le [0]
   eq(v0, v1) -> null_eq [0]
   replace(v0, v1, v2) -> nil [0]
   min(v0) -> 0 [0]
   if_min(v0, v1) -> 0 [0]
   if_replace(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
IF_MIN :: 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
IF_REPLACE :: 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
SORT :: cons:nil -> c17:c18
c17 :: c9 -> c17:c18
c18 :: c17:c18 -> c13 -> c9 -> c17:c18
replace :: s:0 -> s:0 -> cons:nil -> cons:nil
min :: cons:nil -> s:0
0 :: s:0
nil :: cons:nil
if_replace :: true:false:null_le:null_eq -> s:0 -> s:0 -> cons:nil -> cons:nil
if_min :: true:false:null_le:null_eq -> cons:nil -> s:0
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:c18

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(IF_MIN(true, cons(0, cons(z1, z2))), LE(0, z1)) [1]
   MIN(cons(s(z0'), cons(0, z2))) -> c9(IF_MIN(false, cons(s(z0'), cons(0, z2))), LE(s(z0'), 0)) [1]
   MIN(cons(s(z0''), cons(s(z1'), z2))) -> c9(IF_MIN(le(z0'', z1'), cons(s(z0''), cons(s(z1'), z2))), LE(s(z0''), s(z1'))) [1]
   MIN(cons(z0, cons(z1, z2))) -> c9(IF_MIN(null_le, cons(z0, cons(z1, z2))), LE(z0, z1)) [1]
   IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2))) [1]
   IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2))) [1]
   REPLACE(0, z1, cons(0, z3)) -> c13(IF_REPLACE(true, 0, z1, cons(0, z3)), EQ(0, 0)) [1]
   REPLACE(0, z1, cons(s(z01), z3)) -> c13(IF_REPLACE(false, 0, z1, cons(s(z01), z3)), EQ(0, s(z01))) [1]
   REPLACE(s(z02), z1, cons(0, z3)) -> c13(IF_REPLACE(false, s(z02), z1, cons(0, z3)), EQ(s(z02), 0)) [1]
   REPLACE(s(z03), z1, cons(s(z1''), z3)) -> c13(IF_REPLACE(eq(z03, z1''), s(z03), z1, cons(s(z1''), z3)), EQ(s(z03), s(z1''))) [1]
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(null_eq, z0, z1, cons(z2, z3)), EQ(z0, z2)) [1]
   IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3)) [1]
   SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1))) [1]
   SORT(cons(0, nil)) -> c18(SORT(replace(0, 0, nil)), REPLACE(0, 0, nil), MIN(cons(0, nil))) [1]
   SORT(cons(0, nil)) -> c18(SORT(replace(0, 0, nil)), REPLACE(0, 0, nil), MIN(cons(0, nil))) [1]
   SORT(cons(s(z04), nil)) -> c18(SORT(replace(s(z04), s(z04), nil)), REPLACE(s(z04), s(z04), nil), MIN(cons(s(z04), nil))) [1]
   SORT(cons(s(z04), nil)) -> c18(SORT(replace(s(z04), s(z04), nil)), REPLACE(0, s(z04), nil), MIN(cons(s(z04), nil))) [1]
   SORT(cons(z0, cons(z11, z2'))) -> c18(SORT(replace(if_min(le(z0, z11), cons(z0, cons(z11, z2'))), z0, cons(z11, z2'))), REPLACE(if_min(le(z0, z11), cons(z0, cons(z11, z2'))), z0, cons(z11, z2')), MIN(cons(z0, cons(z11, z2')))) [1]
   SORT(cons(z0, cons(z11, z2'))) -> c18(SORT(replace(if_min(le(z0, z11), cons(z0, cons(z11, z2'))), z0, cons(z11, z2'))), REPLACE(0, z0, cons(z11, z2')), MIN(cons(z0, cons(z11, z2')))) [1]
   SORT(cons(0, nil)) -> c18(SORT(replace(0, 0, nil)), REPLACE(0, 0, nil), MIN(cons(0, nil))) [1]
   SORT(cons(s(z05), nil)) -> c18(SORT(replace(0, s(z05), nil)), REPLACE(s(z05), s(z05), nil), MIN(cons(s(z05), nil))) [1]
   SORT(cons(z0, cons(z12, z2''))) -> c18(SORT(replace(0, z0, cons(z12, z2''))), REPLACE(if_min(le(z0, z12), cons(z0, cons(z12, z2''))), z0, cons(z12, z2'')), MIN(cons(z0, cons(z12, z2'')))) [1]
   SORT(cons(z0, z1)) -> c18(SORT(replace(0, z0, z1)), REPLACE(0, z0, z1), MIN(cons(z0, z1))) [1]
   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]
   replace(z0, z1, nil) -> nil [0]
   replace(0, z1, cons(0, z3)) -> if_replace(true, 0, z1, cons(0, z3)) [0]
   replace(0, z1, cons(s(z06), z3)) -> if_replace(false, 0, z1, cons(s(z06), z3)) [0]
   replace(s(z07), z1, cons(0, z3)) -> if_replace(false, s(z07), z1, cons(0, z3)) [0]
   replace(s(z08), z1, cons(s(z13), z3)) -> if_replace(eq(z08, z13), s(z08), z1, cons(s(z13), z3)) [0]
   replace(z0, z1, cons(z2, z3)) -> if_replace(null_eq, z0, z1, cons(z2, z3)) [0]
   min(cons(0, nil)) -> 0 [0]
   min(cons(s(z0), nil)) -> s(z0) [0]
   min(cons(0, cons(z1, z2))) -> if_min(true, cons(0, cons(z1, z2))) [0]
   min(cons(s(z09), cons(0, z2))) -> if_min(false, cons(s(z09), cons(0, z2))) [0]
   min(cons(s(z010), cons(s(z14), z2))) -> if_min(le(z010, z14), cons(s(z010), cons(s(z14), z2))) [0]
   min(cons(z0, cons(z1, z2))) -> if_min(null_le, cons(z0, cons(z1, z2))) [0]
   if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2)) [0]
   if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2)) [0]
   if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3) [0]
   if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3)) [0]
   le(v0, v1) -> null_le [0]
   eq(v0, v1) -> null_eq [0]
   replace(v0, v1, v2) -> nil [0]
   min(v0) -> 0 [0]
   if_min(v0, v1) -> 0 [0]
   if_replace(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
IF_MIN :: 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
IF_REPLACE :: 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
SORT :: cons:nil -> c17:c18
c17 :: c9 -> c17:c18
c18 :: c17:c18 -> c13 -> c9 -> c17:c18
replace :: s:0 -> s:0 -> cons:nil -> cons:nil
min :: cons:nil -> s:0
0 :: s:0
nil :: cons:nil
if_replace :: true:false:null_le:null_eq -> s:0 -> s:0 -> cons:nil -> cons:nil
if_min :: true:false:null_le:null_eq -> cons:nil -> s:0
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:c18

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

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

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


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

(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
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(le(z0'', z1'), 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + LE(1 + z0', 0) :|: z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + LE(z0, z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + EQ(1 + (z - 1), 1 + z1'') :|: z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(2, 0, z', 1 + 0 + (z'' - 1)) + EQ(0, 0) :|: z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(1, 0, z', 1 + (1 + z01) + z3) + EQ(0, 1 + z01) :|: z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) + EQ(1 + (z - 1), 0) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(0, z, z', 1 + z2 + z3) + EQ(z, z2) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(le(z0, z12), 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(le(z010, z14), 1 + (1 + z010) + (1 + (1 + z14) + z2)) :|: z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z09) + (1 + 0 + z2)) :|: z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z13), 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0


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

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

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

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

(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
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(le(z0'', z1'), 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + LE(1 + z0', 0) :|: z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + LE(z0, z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + EQ(1 + (z - 1), 1 + z1'') :|: z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(2, 0, z', 1 + 0 + (z'' - 1)) + EQ(0, 0) :|: z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(1, 0, z', 1 + (1 + z01) + z3) + EQ(0, 1 + z01) :|: z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) + EQ(1 + (z - 1), 0) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(0, z, z', 1 + z2 + z3) + EQ(z, z2) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(le(z0, z12), 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(le(z010, z14), 1 + (1 + z010) + (1 + (1 + z14) + z2)) :|: z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z09) + (1 + 0 + z2)) :|: z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z13), 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

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

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

(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
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(le(z0'', z1'), 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + LE(1 + z0', 0) :|: z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + LE(z0, z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + EQ(1 + (z - 1), 1 + z1'') :|: z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(2, 0, z', 1 + 0 + (z'' - 1)) + EQ(0, 0) :|: z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(1, 0, z', 1 + (1 + z01) + z3) + EQ(0, 1 + z01) :|: z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) + EQ(1 + (z - 1), 0) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(0, z, z', 1 + z2 + z3) + EQ(z, z2) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(le(z0, z12), 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(le(z010, z14), 1 + (1 + z010) + (1 + (1 + z14) + z2)) :|: z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z09) + (1 + 0 + z2)) :|: z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z13), 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

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

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

(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
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(le(z0'', z1'), 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + LE(1 + z0', 0) :|: z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + LE(z0, z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + EQ(1 + (z - 1), 1 + z1'') :|: z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(2, 0, z', 1 + 0 + (z'' - 1)) + EQ(0, 0) :|: z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(1, 0, z', 1 + (1 + z01) + z3) + EQ(0, 1 + z01) :|: z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) + EQ(1 + (z - 1), 0) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(0, z, z', 1 + z2 + z3) + EQ(z, z2) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(le(z0, z12), 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(le(z010, z14), 1 + (1 + z010) + (1 + (1 + z14) + z2)) :|: z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z09) + (1 + 0 + z2)) :|: z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z13), 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {EQ}, {le}, {LE}, {eq}, {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}
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
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(le(z0'', z1'), 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + LE(1 + z0', 0) :|: z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + LE(z0, z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(eq(z - 1, z1''), 1 + (z - 1), z', 1 + (1 + z1'') + z3) + EQ(1 + (z - 1), 1 + z1'') :|: z' >= 0, z'' = 1 + (1 + z1'') + z3, z - 1 >= 0, z3 >= 0, z1'' >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(2, 0, z', 1 + 0 + (z'' - 1)) + EQ(0, 0) :|: z' >= 0, z = 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(1, 0, z', 1 + (1 + z01) + z3) + EQ(0, 1 + z01) :|: z' >= 0, z'' = 1 + (1 + z01) + z3, z01 >= 0, z = 0, z3 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) + EQ(1 + (z - 1), 0) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   REPLACE(z, z', z'') -{ 1 }-> 1 + IF_REPLACE(0, z, z', 1 + z2 + z3) + EQ(z, z2) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(le(z0, z12), 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(le(z010, z14), 1 + (1 + z010) + (1 + (1 + z14) + z2)) :|: z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z09) + (1 + 0 + z2)) :|: z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z13), 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {le}, {LE}, {eq}, {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}
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
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(le(z0'', z1'), 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + LE(1 + z0', 0) :|: z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + LE(z0, z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 2 + z1'' }-> 1 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(le(z0, z12), 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(le(z010, z14), 1 + (1 + z010) + (1 + (1 + z14) + z2)) :|: z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z09) + (1 + 0 + z2)) :|: z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z13), 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {le}, {LE}, {eq}, {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}
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
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(le(z0'', z1'), 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + LE(1 + z0', 0) :|: z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + LE(z0, z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 2 + z1'' }-> 1 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(le(z0, z12), 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(le(z010, z14), 1 + (1 + z010) + (1 + (1 + z14) + z2)) :|: z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z09) + (1 + 0 + z2)) :|: z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z13), 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {le}, {LE}, {eq}, {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}
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
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(le(z0'', z1'), 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + LE(1 + z0', 0) :|: z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + LE(z0, z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 2 + z1'' }-> 1 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(le(z0, z11), 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(le(z0, z12), 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(le(z010, z14), 1 + (1 + z010) + (1 + (1 + z14) + z2)) :|: z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z09) + (1 + 0 + z2)) :|: z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z13), 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {LE}, {eq}, {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}
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
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + LE(1 + z0', 0) :|: z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + LE(z0, z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 2 + z1'' }-> 1 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z010) + (1 + (1 + z14) + z2)) :|: s10 >= 0, s10 <= 2, z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z09) + (1 + 0 + z2)) :|: z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z13), 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {LE}, {eq}, {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}
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
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + LE(1 + z0', 0) :|: z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + LE(z0, z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 2 + z1'' }-> 1 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z010) + (1 + (1 + z14) + z2)) :|: s10 >= 0, s10 <= 2, z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z09) + (1 + 0 + z2)) :|: z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z13), 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {LE}, {eq}, {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}
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
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ 1 }-> 1 + LE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + LE(1 + z0'', 1 + z1') :|: s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + LE(0, z1) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + LE(1 + z0', 0) :|: z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + LE(z0, z1) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 2 + z1'' }-> 1 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z010) + (1 + (1 + z14) + z2)) :|: s10 >= 0, s10 <= 2, z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z09) + (1 + 0 + z2)) :|: z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z13), 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {eq}, {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}
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
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 2 + z1' }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s14 :|: s14 >= 0, s14 <= 0, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + s12 :|: s12 >= 0, s12 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s13 :|: s13 >= 0, s13 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + s15 :|: s15 >= 0, s15 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 2 + z1'' }-> 1 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z010) + (1 + (1 + z14) + z2)) :|: s10 >= 0, s10 <= 2, z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z09) + (1 + 0 + z2)) :|: z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z13), 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {eq}, {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}
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
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 2 + z1' }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s14 :|: s14 >= 0, s14 <= 0, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + s12 :|: s12 >= 0, s12 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s13 :|: s13 >= 0, s13 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + s15 :|: s15 >= 0, s15 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 2 + z1'' }-> 1 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z010) + (1 + (1 + z14) + z2)) :|: s10 >= 0, s10 <= 2, z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z09) + (1 + 0 + z2)) :|: z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z13), 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {eq}, {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}
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
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 2 + z1' }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s14 :|: s14 >= 0, s14 <= 0, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + s12 :|: s12 >= 0, s12 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s13 :|: s13 >= 0, s13 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + s15 :|: s15 >= 0, s15 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 2 + z1'' }-> 1 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> eq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z010) + (1 + (1 + z14) + z2)) :|: s10 >= 0, s10 <= 2, z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z09) + (1 + 0 + z2)) :|: z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(eq(z - 1, z13), 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}
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
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 2 + z1' }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s14 :|: s14 >= 0, s14 <= 0, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + s12 :|: s12 >= 0, s12 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s13 :|: s13 >= 0, s13 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + s15 :|: s15 >= 0, s15 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 2 + z1'' }-> 1 + IF_REPLACE(s16, 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s16 >= 0, s16 <= 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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> s17 :|: s17 >= 0, s17 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z010) + (1 + (1 + z14) + z2)) :|: s10 >= 0, s10 <= 2, z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z09) + (1 + 0 + z2)) :|: z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s18, 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: s18 >= 0, s18 <= 2, z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}
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: IF_REPLACE
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
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 2 + z1' }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s14 :|: s14 >= 0, s14 <= 0, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + s12 :|: s12 >= 0, s12 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s13 :|: s13 >= 0, s13 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + s15 :|: s15 >= 0, s15 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 2 + z1'' }-> 1 + IF_REPLACE(s16, 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s16 >= 0, s16 <= 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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> s17 :|: s17 >= 0, s17 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z010) + (1 + (1 + z14) + z2)) :|: s10 >= 0, s10 <= 2, z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z09) + (1 + 0 + z2)) :|: z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s18, 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: s18 >= 0, s18 <= 2, z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {REPLACE,IF_REPLACE}, {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}
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]
  IF_REPLACE: 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: IF_REPLACE
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
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 1 }-> 1 + REPLACE(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 2 + z1' }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s14 :|: s14 >= 0, s14 <= 0, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + s12 :|: s12 >= 0, s12 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s13 :|: s13 >= 0, s13 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + s15 :|: s15 >= 0, s15 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 2 + z1'' }-> 1 + IF_REPLACE(s16, 1 + (z - 1), z', 1 + (1 + z1'') + z3) + s2 :|: s16 >= 0, s16 <= 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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(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 + IF_REPLACE(0, z, z', 1 + z2 + z3) + s3 :|: s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(0, z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, z1)) + REPLACE(0, z0, z1) + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 0, 0)) + REPLACE(0, 0, 0) + MIN(1 + 0 + 0) :|: z = 1 + 0 + 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(0, 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   SORT(z) -{ 1 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + REPLACE(1 + (z - 2), 1 + (z - 2), 0) + MIN(1 + (1 + (z - 2)) + 0) :|: z - 2 >= 0
   eq(z, z') -{ 0 }-> s17 :|: s17 >= 0, s17 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z010) + (1 + (1 + z14) + z2)) :|: s10 >= 0, s10 <= 2, z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z09) + (1 + 0 + z2)) :|: z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s18, 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: s18 >= 0, s18 <= 2, z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}
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]
  IF_REPLACE: 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
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 3 + 3*z3 + z3^2 }-> 1 + s24 :|: s24 >= 0, s24 <= 0, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 2 + z1' }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s14 :|: s14 >= 0, s14 <= 0, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + s12 :|: s12 >= 0, s12 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s13 :|: s13 >= 0, s13 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + s15 :|: s15 >= 0, s15 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s19 + s' :|: s19 >= 0, s19 <= 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 + s20 + s'' :|: s20 >= 0, s20 <= 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 + s21 + s1 :|: s21 >= 0, s21 <= 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 + s22 + s2 :|: s22 >= 0, s22 <= 1, s16 >= 0, s16 <= 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 + s23 + s3 :|: s23 >= 0, s23 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 7 + 5*z11 + 2*z11*z2' + z11^2 + 5*z2' + z2'^2 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + s28 + MIN(1 + z0 + (1 + z11 + z2')) :|: s28 >= 0, s28 <= 0, s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 3 + 3*z1 + z1^2 }-> 1 + SORT(replace(0, z0, z1)) + s30 + MIN(1 + z0 + z1) :|: s30 >= 0, s30 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 3 }-> 1 + SORT(replace(0, 0, 0)) + s25 + MIN(1 + 0 + 0) :|: s25 >= 0, s25 <= 0, z = 1 + 0 + 0
   SORT(z) -{ 3 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + s29 + MIN(1 + (1 + (z - 2)) + 0) :|: s29 >= 0, s29 <= 0, z - 2 >= 0
   SORT(z) -{ 3 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s26 + MIN(1 + (1 + (z - 2)) + 0) :|: s26 >= 0, s26 <= 0, z - 2 >= 0
   SORT(z) -{ 3 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s27 + MIN(1 + (1 + (z - 2)) + 0) :|: s27 >= 0, s27 <= 0, z - 2 >= 0
   eq(z, z') -{ 0 }-> s17 :|: s17 >= 0, s17 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z010) + (1 + (1 + z14) + z2)) :|: s10 >= 0, s10 <= 2, z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z09) + (1 + 0 + z2)) :|: z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s18, 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: s18 >= 0, s18 <= 2, z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}
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]
  IF_REPLACE: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: IF_MIN
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

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

(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
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 3 + 3*z3 + z3^2 }-> 1 + s24 :|: s24 >= 0, s24 <= 0, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 2 + z1' }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s14 :|: s14 >= 0, s14 <= 0, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + s12 :|: s12 >= 0, s12 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s13 :|: s13 >= 0, s13 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + s15 :|: s15 >= 0, s15 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s19 + s' :|: s19 >= 0, s19 <= 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 + s20 + s'' :|: s20 >= 0, s20 <= 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 + s21 + s1 :|: s21 >= 0, s21 <= 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 + s22 + s2 :|: s22 >= 0, s22 <= 1, s16 >= 0, s16 <= 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 + s23 + s3 :|: s23 >= 0, s23 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 7 + 5*z11 + 2*z11*z2' + z11^2 + 5*z2' + z2'^2 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + s28 + MIN(1 + z0 + (1 + z11 + z2')) :|: s28 >= 0, s28 <= 0, s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 3 + 3*z1 + z1^2 }-> 1 + SORT(replace(0, z0, z1)) + s30 + MIN(1 + z0 + z1) :|: s30 >= 0, s30 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 3 }-> 1 + SORT(replace(0, 0, 0)) + s25 + MIN(1 + 0 + 0) :|: s25 >= 0, s25 <= 0, z = 1 + 0 + 0
   SORT(z) -{ 3 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + s29 + MIN(1 + (1 + (z - 2)) + 0) :|: s29 >= 0, s29 <= 0, z - 2 >= 0
   SORT(z) -{ 3 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s26 + MIN(1 + (1 + (z - 2)) + 0) :|: s26 >= 0, s26 <= 0, z - 2 >= 0
   SORT(z) -{ 3 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s27 + MIN(1 + (1 + (z - 2)) + 0) :|: s27 >= 0, s27 <= 0, z - 2 >= 0
   eq(z, z') -{ 0 }-> s17 :|: s17 >= 0, s17 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z010) + (1 + (1 + z14) + z2)) :|: s10 >= 0, s10 <= 2, z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z09) + (1 + 0 + z2)) :|: z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s18, 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: s18 >= 0, s18 <= 2, z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {IF_MIN,MIN}, {min,if_min}, {replace,if_replace}, {SORT}
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]
  IF_REPLACE: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]
  IF_MIN: runtime: ?, size: O(1) [0]
  MIN: runtime: ?, size: O(1) [1]

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(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
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 1 }-> 1 + MIN(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 3 + 3*z3 + z3^2 }-> 1 + s24 :|: s24 >= 0, s24 <= 0, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 2 + z1' }-> 1 + IF_MIN(s4, 1 + (1 + z0'') + (1 + (1 + z1') + z2)) + s14 :|: s14 >= 0, s14 <= 0, s4 >= 0, s4 <= 2, z = 1 + (1 + z0'') + (1 + (1 + z1') + z2), z1' >= 0, z0'' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IF_MIN(2, 1 + 0 + (1 + z1 + z2)) + s12 :|: s12 >= 0, s12 <= 0, z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   MIN(z) -{ 1 }-> 1 + IF_MIN(1, 1 + (1 + z0') + (1 + 0 + z2)) + s13 :|: s13 >= 0, s13 <= 0, z = 1 + (1 + z0') + (1 + 0 + z2), z0' >= 0, z2 >= 0
   MIN(z) -{ 1 + z1 }-> 1 + IF_MIN(0, 1 + z0 + (1 + z1 + z2)) + s15 :|: s15 >= 0, s15 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s19 + s' :|: s19 >= 0, s19 <= 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 + s20 + s'' :|: s20 >= 0, s20 <= 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 + s21 + s1 :|: s21 >= 0, s21 <= 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 + s22 + s2 :|: s22 >= 0, s22 <= 1, s16 >= 0, s16 <= 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 + s23 + s3 :|: s23 >= 0, s23 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 1 }-> 1 + MIN(1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + MIN(1 + z0 + (1 + z11 + z2')) :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 7 + 5*z11 + 2*z11*z2' + z11^2 + 5*z2' + z2'^2 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + s28 + MIN(1 + z0 + (1 + z11 + z2')) :|: s28 >= 0, s28 <= 0, s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 3 + 3*z1 + z1^2 }-> 1 + SORT(replace(0, z0, z1)) + s30 + MIN(1 + z0 + z1) :|: s30 >= 0, s30 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 1 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + MIN(1 + z0 + (1 + z12 + z2'')) :|: s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 3 }-> 1 + SORT(replace(0, 0, 0)) + s25 + MIN(1 + 0 + 0) :|: s25 >= 0, s25 <= 0, z = 1 + 0 + 0
   SORT(z) -{ 3 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + s29 + MIN(1 + (1 + (z - 2)) + 0) :|: s29 >= 0, s29 <= 0, z - 2 >= 0
   SORT(z) -{ 3 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s26 + MIN(1 + (1 + (z - 2)) + 0) :|: s26 >= 0, s26 <= 0, z - 2 >= 0
   SORT(z) -{ 3 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s27 + MIN(1 + (1 + (z - 2)) + 0) :|: s27 >= 0, s27 <= 0, z - 2 >= 0
   eq(z, z') -{ 0 }-> s17 :|: s17 >= 0, s17 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z010) + (1 + (1 + z14) + z2)) :|: s10 >= 0, s10 <= 2, z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z09) + (1 + 0 + z2)) :|: z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s18, 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: s18 >= 0, s18 <= 2, z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {min,if_min}, {replace,if_replace}, {SORT}
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]
  IF_REPLACE: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]
  IF_MIN: 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]

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

(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
   IF_MIN(z, z') -{ 344 + 191*z0 + 52*z0*z2 + 26*z0^2 + 191*z2 + 26*z2^2 }-> 1 + s35 :|: s35 >= 0, s35 <= 1, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 344 + 191*z1 + 52*z1*z2 + 26*z1^2 + 191*z2 + 26*z2^2 }-> 1 + s36 :|: s36 >= 0, s36 <= 1, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 3 + 3*z3 + z3^2 }-> 1 + s24 :|: s24 >= 0, s24 <= 0, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 34 + 13*z1 + 2*z1*z2 + z1^2 + 12*z2 + z2^2 }-> 1 + s31 + s12 :|: s31 >= 0, s31 <= 0, s12 >= 0, s12 <= 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 + s32 + s13 :|: s32 >= 0, s32 <= 0, s13 >= 0, s13 <= 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 + s33 + s14 :|: s33 >= 0, s33 <= 0, s14 >= 0, s14 <= 0, s4 >= 0, s4 <= 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 + s34 + s15 :|: s34 >= 0, s34 <= 0, s15 >= 0, s15 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s19 + s' :|: s19 >= 0, s19 <= 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 + s20 + s'' :|: s20 >= 0, s20 <= 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 + s21 + s1 :|: s21 >= 0, s21 <= 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 + s22 + s2 :|: s22 >= 0, s22 <= 1, s16 >= 0, s16 <= 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 + s23 + s3 :|: s23 >= 0, s23 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 344 + 191*z0 + 52*z0*z1 + 26*z0^2 + 191*z1 + 26*z1^2 }-> 1 + s37 :|: s37 >= 0, s37 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(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 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + s41 :|: s41 >= 0, s41 <= 1, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 567 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 248*z11 + 54*z11*z2' + 27*z11^2 + 248*z2' + 27*z2'^2 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + s28 + s42 :|: s42 >= 0, s42 <= 1, s28 >= 0, s28 <= 0, s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 346 + 191*z0 + 52*z0*z1 + 26*z0^2 + 194*z1 + 27*z1^2 }-> 1 + SORT(replace(0, z0, z1)) + s30 + s45 :|: s45 >= 0, s45 <= 1, s30 >= 0, s30 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 561 + 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 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + s44 :|: s44 >= 0, s44 <= 1, s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 346 }-> 1 + SORT(replace(0, 0, 0)) + s25 + s38 :|: s38 >= 0, s38 <= 1, s25 >= 0, s25 <= 0, z = 1 + 0 + 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + s29 + s43 :|: s43 >= 0, s43 <= 1, s29 >= 0, s29 <= 0, z - 2 >= 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s26 + s39 :|: s39 >= 0, s39 <= 1, s26 >= 0, s26 <= 0, z - 2 >= 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s27 + s40 :|: s40 >= 0, s40 <= 1, s27 >= 0, s27 <= 0, z - 2 >= 0
   eq(z, z') -{ 0 }-> s17 :|: s17 >= 0, s17 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z010) + (1 + (1 + z14) + z2)) :|: s10 >= 0, s10 <= 2, z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z09) + (1 + 0 + z2)) :|: z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s18, 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: s18 >= 0, s18 <= 2, z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {min,if_min}, {replace,if_replace}, {SORT}
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]
  IF_REPLACE: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]
  IF_MIN: 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]

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

(61) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

(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
   IF_MIN(z, z') -{ 344 + 191*z0 + 52*z0*z2 + 26*z0^2 + 191*z2 + 26*z2^2 }-> 1 + s35 :|: s35 >= 0, s35 <= 1, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 344 + 191*z1 + 52*z1*z2 + 26*z1^2 + 191*z2 + 26*z2^2 }-> 1 + s36 :|: s36 >= 0, s36 <= 1, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 3 + 3*z3 + z3^2 }-> 1 + s24 :|: s24 >= 0, s24 <= 0, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 34 + 13*z1 + 2*z1*z2 + z1^2 + 12*z2 + z2^2 }-> 1 + s31 + s12 :|: s31 >= 0, s31 <= 0, s12 >= 0, s12 <= 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 + s32 + s13 :|: s32 >= 0, s32 <= 0, s13 >= 0, s13 <= 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 + s33 + s14 :|: s33 >= 0, s33 <= 0, s14 >= 0, s14 <= 0, s4 >= 0, s4 <= 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 + s34 + s15 :|: s34 >= 0, s34 <= 0, s15 >= 0, s15 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s19 + s' :|: s19 >= 0, s19 <= 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 + s20 + s'' :|: s20 >= 0, s20 <= 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 + s21 + s1 :|: s21 >= 0, s21 <= 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 + s22 + s2 :|: s22 >= 0, s22 <= 1, s16 >= 0, s16 <= 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 + s23 + s3 :|: s23 >= 0, s23 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 344 + 191*z0 + 52*z0*z1 + 26*z0^2 + 191*z1 + 26*z1^2 }-> 1 + s37 :|: s37 >= 0, s37 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(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 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + s41 :|: s41 >= 0, s41 <= 1, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 567 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 248*z11 + 54*z11*z2' + 27*z11^2 + 248*z2' + 27*z2'^2 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + s28 + s42 :|: s42 >= 0, s42 <= 1, s28 >= 0, s28 <= 0, s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 346 + 191*z0 + 52*z0*z1 + 26*z0^2 + 194*z1 + 27*z1^2 }-> 1 + SORT(replace(0, z0, z1)) + s30 + s45 :|: s45 >= 0, s45 <= 1, s30 >= 0, s30 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 561 + 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 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + s44 :|: s44 >= 0, s44 <= 1, s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 346 }-> 1 + SORT(replace(0, 0, 0)) + s25 + s38 :|: s38 >= 0, s38 <= 1, s25 >= 0, s25 <= 0, z = 1 + 0 + 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + s29 + s43 :|: s43 >= 0, s43 <= 1, s29 >= 0, s29 <= 0, z - 2 >= 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s26 + s39 :|: s39 >= 0, s39 <= 1, s26 >= 0, s26 <= 0, z - 2 >= 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s27 + s40 :|: s40 >= 0, s40 <= 1, s27 >= 0, s27 <= 0, z - 2 >= 0
   eq(z, z') -{ 0 }-> s17 :|: s17 >= 0, s17 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z010) + (1 + (1 + z14) + z2)) :|: s10 >= 0, s10 <= 2, z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z09) + (1 + 0 + z2)) :|: z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s18, 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: s18 >= 0, s18 <= 2, z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {min,if_min}, {replace,if_replace}, {SORT}
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]
  IF_REPLACE: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]
  IF_MIN: 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]
  min: runtime: ?, size: O(n^1) [z]
  if_min: runtime: ?, size: O(n^1) [z']

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

(63) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

(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
   IF_MIN(z, z') -{ 344 + 191*z0 + 52*z0*z2 + 26*z0^2 + 191*z2 + 26*z2^2 }-> 1 + s35 :|: s35 >= 0, s35 <= 1, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 344 + 191*z1 + 52*z1*z2 + 26*z1^2 + 191*z2 + 26*z2^2 }-> 1 + s36 :|: s36 >= 0, s36 <= 1, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 3 + 3*z3 + z3^2 }-> 1 + s24 :|: s24 >= 0, s24 <= 0, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 34 + 13*z1 + 2*z1*z2 + z1^2 + 12*z2 + z2^2 }-> 1 + s31 + s12 :|: s31 >= 0, s31 <= 0, s12 >= 0, s12 <= 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 + s32 + s13 :|: s32 >= 0, s32 <= 0, s13 >= 0, s13 <= 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 + s33 + s14 :|: s33 >= 0, s33 <= 0, s14 >= 0, s14 <= 0, s4 >= 0, s4 <= 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 + s34 + s15 :|: s34 >= 0, s34 <= 0, s15 >= 0, s15 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s19 + s' :|: s19 >= 0, s19 <= 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 + s20 + s'' :|: s20 >= 0, s20 <= 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 + s21 + s1 :|: s21 >= 0, s21 <= 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 + s22 + s2 :|: s22 >= 0, s22 <= 1, s16 >= 0, s16 <= 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 + s23 + s3 :|: s23 >= 0, s23 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 344 + 191*z0 + 52*z0*z1 + 26*z0^2 + 191*z1 + 26*z1^2 }-> 1 + s37 :|: s37 >= 0, s37 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(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 + SORT(replace(if_min(s5, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + REPLACE(if_min(s6, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2') + s41 :|: s41 >= 0, s41 <= 1, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 567 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 248*z11 + 54*z11*z2' + 27*z11^2 + 248*z2' + 27*z2'^2 }-> 1 + SORT(replace(if_min(s7, 1 + z0 + (1 + z11 + z2')), z0, 1 + z11 + z2')) + s28 + s42 :|: s42 >= 0, s42 <= 1, s28 >= 0, s28 <= 0, s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 346 + 191*z0 + 52*z0*z1 + 26*z0^2 + 194*z1 + 27*z1^2 }-> 1 + SORT(replace(0, z0, z1)) + s30 + s45 :|: s45 >= 0, s45 <= 1, s30 >= 0, s30 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 561 + 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 + SORT(replace(0, z0, 1 + z12 + z2'')) + REPLACE(if_min(s8, 1 + z0 + (1 + z12 + z2'')), z0, 1 + z12 + z2'') + s44 :|: s44 >= 0, s44 <= 1, s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 346 }-> 1 + SORT(replace(0, 0, 0)) + s25 + s38 :|: s38 >= 0, s38 <= 1, s25 >= 0, s25 <= 0, z = 1 + 0 + 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + s29 + s43 :|: s43 >= 0, s43 <= 1, s29 >= 0, s29 <= 0, z - 2 >= 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s26 + s39 :|: s39 >= 0, s39 <= 1, s26 >= 0, s26 <= 0, z - 2 >= 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s27 + s40 :|: s40 >= 0, s40 <= 1, s27 >= 0, s27 <= 0, z - 2 >= 0
   eq(z, z') -{ 0 }-> s17 :|: s17 >= 0, s17 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> min(1 + z0 + z2) :|: z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> min(1 + z1 + z2) :|: z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> if_min(s10, 1 + (1 + z010) + (1 + (1 + z14) + z2)) :|: s10 >= 0, s10 <= 2, z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(2, 1 + 0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> if_min(1, 1 + (1 + z09) + (1 + 0 + z2)) :|: z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> if_min(0, 1 + z0 + (1 + z1 + z2)) :|: z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s18, 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: s18 >= 0, s18 <= 2, z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {replace,if_replace}, {SORT}
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]
  IF_REPLACE: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]
  IF_MIN: 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]
  min: runtime: O(1) [0], size: O(n^1) [z]
  if_min: runtime: O(1) [0], size: O(n^1) [z']

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

(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
   IF_MIN(z, z') -{ 344 + 191*z0 + 52*z0*z2 + 26*z0^2 + 191*z2 + 26*z2^2 }-> 1 + s35 :|: s35 >= 0, s35 <= 1, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 344 + 191*z1 + 52*z1*z2 + 26*z1^2 + 191*z2 + 26*z2^2 }-> 1 + s36 :|: s36 >= 0, s36 <= 1, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 3 + 3*z3 + z3^2 }-> 1 + s24 :|: s24 >= 0, s24 <= 0, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 34 + 13*z1 + 2*z1*z2 + z1^2 + 12*z2 + z2^2 }-> 1 + s31 + s12 :|: s31 >= 0, s31 <= 0, s12 >= 0, s12 <= 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 + s32 + s13 :|: s32 >= 0, s32 <= 0, s13 >= 0, s13 <= 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 + s33 + s14 :|: s33 >= 0, s33 <= 0, s14 >= 0, s14 <= 0, s4 >= 0, s4 <= 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 + s34 + s15 :|: s34 >= 0, s34 <= 0, s15 >= 0, s15 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s19 + s' :|: s19 >= 0, s19 <= 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 + s20 + s'' :|: s20 >= 0, s20 <= 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 + s21 + s1 :|: s21 >= 0, s21 <= 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 + s22 + s2 :|: s22 >= 0, s22 <= 1, s16 >= 0, s16 <= 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 + s23 + s3 :|: s23 >= 0, s23 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 344 + 191*z0 + 52*z0*z1 + 26*z0^2 + 191*z1 + 26*z1^2 }-> 1 + s37 :|: s37 >= 0, s37 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 567 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 248*z11 + 54*z11*z2' + 27*z11^2 + 248*z2' + 27*z2'^2 }-> 1 + SORT(replace(s46, z0, 1 + z11 + z2')) + s48 + s41 :|: s46 >= 0, s46 <= 1 + z0 + (1 + z11 + z2'), s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 0, s41 >= 0, s41 <= 1, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 567 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 248*z11 + 54*z11*z2' + 27*z11^2 + 248*z2' + 27*z2'^2 }-> 1 + SORT(replace(s49, z0, 1 + z11 + z2')) + s28 + s42 :|: s49 >= 0, s49 <= 1 + z0 + (1 + z11 + z2'), s42 >= 0, s42 <= 1, s28 >= 0, s28 <= 0, s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 346 + 191*z0 + 52*z0*z1 + 26*z0^2 + 194*z1 + 27*z1^2 }-> 1 + SORT(replace(0, z0, z1)) + s30 + s45 :|: s45 >= 0, s45 <= 1, s30 >= 0, s30 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 567 + 243*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 248*z12 + 54*z12*z2'' + 27*z12^2 + 248*z2'' + 27*z2''^2 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + s51 + s44 :|: s50 >= 0, s50 <= 1 + z0 + (1 + z12 + z2''), s51 >= 0, s51 <= 0, s44 >= 0, s44 <= 1, s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 346 }-> 1 + SORT(replace(0, 0, 0)) + s25 + s38 :|: s38 >= 0, s38 <= 1, s25 >= 0, s25 <= 0, z = 1 + 0 + 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + s29 + s43 :|: s43 >= 0, s43 <= 1, s29 >= 0, s29 <= 0, z - 2 >= 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s26 + s39 :|: s39 >= 0, s39 <= 1, s26 >= 0, s26 <= 0, z - 2 >= 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s27 + s40 :|: s40 >= 0, s40 <= 1, s27 >= 0, s27 <= 0, z - 2 >= 0
   eq(z, z') -{ 0 }-> s17 :|: s17 >= 0, s17 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> s56 :|: s56 >= 0, s56 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> s52 :|: s52 >= 0, s52 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 1 + (1 + z09) + (1 + 0 + z2), z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + (1 + z010) + (1 + (1 + z14) + z2), s10 >= 0, s10 <= 2, z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s18, 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: s18 >= 0, s18 <= 2, z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {replace,if_replace}, {SORT}
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]
  IF_REPLACE: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]
  IF_MIN: 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]
  min: runtime: O(1) [0], size: O(n^1) [z]
  if_min: runtime: O(1) [0], size: O(n^1) [z']

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

(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: if_replace
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
   IF_MIN(z, z') -{ 344 + 191*z0 + 52*z0*z2 + 26*z0^2 + 191*z2 + 26*z2^2 }-> 1 + s35 :|: s35 >= 0, s35 <= 1, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 344 + 191*z1 + 52*z1*z2 + 26*z1^2 + 191*z2 + 26*z2^2 }-> 1 + s36 :|: s36 >= 0, s36 <= 1, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 3 + 3*z3 + z3^2 }-> 1 + s24 :|: s24 >= 0, s24 <= 0, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 34 + 13*z1 + 2*z1*z2 + z1^2 + 12*z2 + z2^2 }-> 1 + s31 + s12 :|: s31 >= 0, s31 <= 0, s12 >= 0, s12 <= 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 + s32 + s13 :|: s32 >= 0, s32 <= 0, s13 >= 0, s13 <= 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 + s33 + s14 :|: s33 >= 0, s33 <= 0, s14 >= 0, s14 <= 0, s4 >= 0, s4 <= 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 + s34 + s15 :|: s34 >= 0, s34 <= 0, s15 >= 0, s15 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s19 + s' :|: s19 >= 0, s19 <= 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 + s20 + s'' :|: s20 >= 0, s20 <= 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 + s21 + s1 :|: s21 >= 0, s21 <= 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 + s22 + s2 :|: s22 >= 0, s22 <= 1, s16 >= 0, s16 <= 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 + s23 + s3 :|: s23 >= 0, s23 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 344 + 191*z0 + 52*z0*z1 + 26*z0^2 + 191*z1 + 26*z1^2 }-> 1 + s37 :|: s37 >= 0, s37 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 567 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 248*z11 + 54*z11*z2' + 27*z11^2 + 248*z2' + 27*z2'^2 }-> 1 + SORT(replace(s46, z0, 1 + z11 + z2')) + s48 + s41 :|: s46 >= 0, s46 <= 1 + z0 + (1 + z11 + z2'), s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 0, s41 >= 0, s41 <= 1, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 567 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 248*z11 + 54*z11*z2' + 27*z11^2 + 248*z2' + 27*z2'^2 }-> 1 + SORT(replace(s49, z0, 1 + z11 + z2')) + s28 + s42 :|: s49 >= 0, s49 <= 1 + z0 + (1 + z11 + z2'), s42 >= 0, s42 <= 1, s28 >= 0, s28 <= 0, s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 346 + 191*z0 + 52*z0*z1 + 26*z0^2 + 194*z1 + 27*z1^2 }-> 1 + SORT(replace(0, z0, z1)) + s30 + s45 :|: s45 >= 0, s45 <= 1, s30 >= 0, s30 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 567 + 243*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 248*z12 + 54*z12*z2'' + 27*z12^2 + 248*z2'' + 27*z2''^2 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + s51 + s44 :|: s50 >= 0, s50 <= 1 + z0 + (1 + z12 + z2''), s51 >= 0, s51 <= 0, s44 >= 0, s44 <= 1, s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 346 }-> 1 + SORT(replace(0, 0, 0)) + s25 + s38 :|: s38 >= 0, s38 <= 1, s25 >= 0, s25 <= 0, z = 1 + 0 + 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + s29 + s43 :|: s43 >= 0, s43 <= 1, s29 >= 0, s29 <= 0, z - 2 >= 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s26 + s39 :|: s39 >= 0, s39 <= 1, s26 >= 0, s26 <= 0, z - 2 >= 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s27 + s40 :|: s40 >= 0, s40 <= 1, s27 >= 0, s27 <= 0, z - 2 >= 0
   eq(z, z') -{ 0 }-> s17 :|: s17 >= 0, s17 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> s56 :|: s56 >= 0, s56 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> s52 :|: s52 >= 0, s52 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 1 + (1 + z09) + (1 + 0 + z2), z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + (1 + z010) + (1 + (1 + z14) + z2), s10 >= 0, s10 <= 2, z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s18, 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: s18 >= 0, s18 <= 2, z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {replace,if_replace}, {SORT}
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]
  IF_REPLACE: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]
  IF_MIN: 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]
  min: runtime: O(1) [0], size: O(n^1) [z]
  if_min: runtime: O(1) [0], size: O(n^1) [z']
  replace: runtime: ?, size: O(n^1) [z' + z'']
  if_replace: runtime: ?, size: O(n^1) [z'' + z4]

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

(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: if_replace
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
   IF_MIN(z, z') -{ 344 + 191*z0 + 52*z0*z2 + 26*z0^2 + 191*z2 + 26*z2^2 }-> 1 + s35 :|: s35 >= 0, s35 <= 1, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 344 + 191*z1 + 52*z1*z2 + 26*z1^2 + 191*z2 + 26*z2^2 }-> 1 + s36 :|: s36 >= 0, s36 <= 1, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 3 + 3*z3 + z3^2 }-> 1 + s24 :|: s24 >= 0, s24 <= 0, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 34 + 13*z1 + 2*z1*z2 + z1^2 + 12*z2 + z2^2 }-> 1 + s31 + s12 :|: s31 >= 0, s31 <= 0, s12 >= 0, s12 <= 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 + s32 + s13 :|: s32 >= 0, s32 <= 0, s13 >= 0, s13 <= 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 + s33 + s14 :|: s33 >= 0, s33 <= 0, s14 >= 0, s14 <= 0, s4 >= 0, s4 <= 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 + s34 + s15 :|: s34 >= 0, s34 <= 0, s15 >= 0, s15 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s19 + s' :|: s19 >= 0, s19 <= 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 + s20 + s'' :|: s20 >= 0, s20 <= 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 + s21 + s1 :|: s21 >= 0, s21 <= 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 + s22 + s2 :|: s22 >= 0, s22 <= 1, s16 >= 0, s16 <= 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 + s23 + s3 :|: s23 >= 0, s23 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 344 + 191*z0 + 52*z0*z1 + 26*z0^2 + 191*z1 + 26*z1^2 }-> 1 + s37 :|: s37 >= 0, s37 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 567 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 248*z11 + 54*z11*z2' + 27*z11^2 + 248*z2' + 27*z2'^2 }-> 1 + SORT(replace(s46, z0, 1 + z11 + z2')) + s48 + s41 :|: s46 >= 0, s46 <= 1 + z0 + (1 + z11 + z2'), s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 0, s41 >= 0, s41 <= 1, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 567 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 248*z11 + 54*z11*z2' + 27*z11^2 + 248*z2' + 27*z2'^2 }-> 1 + SORT(replace(s49, z0, 1 + z11 + z2')) + s28 + s42 :|: s49 >= 0, s49 <= 1 + z0 + (1 + z11 + z2'), s42 >= 0, s42 <= 1, s28 >= 0, s28 <= 0, s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 346 + 191*z0 + 52*z0*z1 + 26*z0^2 + 194*z1 + 27*z1^2 }-> 1 + SORT(replace(0, z0, z1)) + s30 + s45 :|: s45 >= 0, s45 <= 1, s30 >= 0, s30 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 567 + 243*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 248*z12 + 54*z12*z2'' + 27*z12^2 + 248*z2'' + 27*z2''^2 }-> 1 + SORT(replace(0, z0, 1 + z12 + z2'')) + s51 + s44 :|: s50 >= 0, s50 <= 1 + z0 + (1 + z12 + z2''), s51 >= 0, s51 <= 0, s44 >= 0, s44 <= 1, s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 346 }-> 1 + SORT(replace(0, 0, 0)) + s25 + s38 :|: s38 >= 0, s38 <= 1, s25 >= 0, s25 <= 0, z = 1 + 0 + 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(replace(0, 1 + (z - 2), 0)) + s29 + s43 :|: s43 >= 0, s43 <= 1, s29 >= 0, s29 <= 0, z - 2 >= 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s26 + s39 :|: s39 >= 0, s39 <= 1, s26 >= 0, s26 <= 0, z - 2 >= 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(replace(1 + (z - 2), 1 + (z - 2), 0)) + s27 + s40 :|: s40 >= 0, s40 <= 1, s27 >= 0, s27 <= 0, z - 2 >= 0
   eq(z, z') -{ 0 }-> s17 :|: s17 >= 0, s17 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> s56 :|: s56 >= 0, s56 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + replace(z', z'', z3) :|: z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> s52 :|: s52 >= 0, s52 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 1 + (1 + z09) + (1 + 0 + z2), z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + (1 + z010) + (1 + (1 + z14) + z2), s10 >= 0, s10 <= 2, z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> s55 :|: s55 >= 0, s55 <= 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   min(z) -{ 0 }-> 0 :|: z = 1 + 0 + 0
   min(z) -{ 0 }-> 0 :|: z >= 0
   min(z) -{ 0 }-> 1 + (z - 2) :|: z - 2 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(s18, 1 + (z - 1), z', 1 + (1 + z13) + z3) :|: s18 >= 0, s18 <= 2, z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(2, 0, z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 0, z', 1 + (1 + z06) + z3) :|: z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(1, 1 + (z - 1), z', 1 + 0 + (z'' - 1)) :|: z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> if_replace(0, z, z', 1 + z2 + z3) :|: z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z'' = 0, z' >= 0, z >= 0
   replace(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0

Function symbols to be analyzed: {SORT}
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]
  IF_REPLACE: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]
  IF_MIN: 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]
  min: runtime: O(1) [0], size: O(n^1) [z]
  if_min: runtime: O(1) [0], size: O(n^1) [z']
  replace: runtime: O(1) [0], size: O(n^1) [z' + z'']
  if_replace: runtime: O(1) [0], size: O(n^1) [z'' + z4]

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

(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
   IF_MIN(z, z') -{ 344 + 191*z0 + 52*z0*z2 + 26*z0^2 + 191*z2 + 26*z2^2 }-> 1 + s35 :|: s35 >= 0, s35 <= 1, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 344 + 191*z1 + 52*z1*z2 + 26*z1^2 + 191*z2 + 26*z2^2 }-> 1 + s36 :|: s36 >= 0, s36 <= 1, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 3 + 3*z3 + z3^2 }-> 1 + s24 :|: s24 >= 0, s24 <= 0, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 34 + 13*z1 + 2*z1*z2 + z1^2 + 12*z2 + z2^2 }-> 1 + s31 + s12 :|: s31 >= 0, s31 <= 0, s12 >= 0, s12 <= 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 + s32 + s13 :|: s32 >= 0, s32 <= 0, s13 >= 0, s13 <= 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 + s33 + s14 :|: s33 >= 0, s33 <= 0, s14 >= 0, s14 <= 0, s4 >= 0, s4 <= 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 + s34 + s15 :|: s34 >= 0, s34 <= 0, s15 >= 0, s15 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s19 + s' :|: s19 >= 0, s19 <= 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 + s20 + s'' :|: s20 >= 0, s20 <= 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 + s21 + s1 :|: s21 >= 0, s21 <= 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 + s22 + s2 :|: s22 >= 0, s22 <= 1, s16 >= 0, s16 <= 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 + s23 + s3 :|: s23 >= 0, s23 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 344 + 191*z0 + 52*z0*z1 + 26*z0^2 + 191*z1 + 26*z1^2 }-> 1 + s37 :|: s37 >= 0, s37 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 346 }-> 1 + SORT(s58) + s25 + s38 :|: s58 >= 0, s58 <= 0 + 0, s38 >= 0, s38 <= 1, s25 >= 0, s25 <= 0, z = 1 + 0 + 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(s59) + s26 + s39 :|: s59 >= 0, s59 <= 1 + (z - 2) + 0, s39 >= 0, s39 <= 1, s26 >= 0, s26 <= 0, z - 2 >= 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(s60) + s27 + s40 :|: s60 >= 0, s60 <= 1 + (z - 2) + 0, s40 >= 0, s40 <= 1, s27 >= 0, s27 <= 0, z - 2 >= 0
   SORT(z) -{ 567 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 248*z11 + 54*z11*z2' + 27*z11^2 + 248*z2' + 27*z2'^2 }-> 1 + SORT(s61) + s48 + s41 :|: s61 >= 0, s61 <= z0 + (1 + z11 + z2'), s46 >= 0, s46 <= 1 + z0 + (1 + z11 + z2'), s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 0, s41 >= 0, s41 <= 1, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 567 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 248*z11 + 54*z11*z2' + 27*z11^2 + 248*z2' + 27*z2'^2 }-> 1 + SORT(s62) + s28 + s42 :|: s62 >= 0, s62 <= z0 + (1 + z11 + z2'), s49 >= 0, s49 <= 1 + z0 + (1 + z11 + z2'), s42 >= 0, s42 <= 1, s28 >= 0, s28 <= 0, s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(s63) + s29 + s43 :|: s63 >= 0, s63 <= 1 + (z - 2) + 0, s43 >= 0, s43 <= 1, s29 >= 0, s29 <= 0, z - 2 >= 0
   SORT(z) -{ 567 + 243*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 248*z12 + 54*z12*z2'' + 27*z12^2 + 248*z2'' + 27*z2''^2 }-> 1 + SORT(s64) + s51 + s44 :|: s64 >= 0, s64 <= z0 + (1 + z12 + z2''), s50 >= 0, s50 <= 1 + z0 + (1 + z12 + z2''), s51 >= 0, s51 <= 0, s44 >= 0, s44 <= 1, s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 346 + 191*z0 + 52*z0*z1 + 26*z0^2 + 194*z1 + 27*z1^2 }-> 1 + SORT(s65) + s30 + s45 :|: s65 >= 0, s65 <= z0 + z1, s45 >= 0, s45 <= 1, s30 >= 0, s30 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eq(z, z') -{ 0 }-> s17 :|: s17 >= 0, s17 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> s56 :|: s56 >= 0, s56 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + s71 :|: s71 >= 0, s71 <= z'' + z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> s52 :|: s52 >= 0, s52 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 1 + (1 + z09) + (1 + 0 + z2), z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + (1 + z010) + (1 + (1 + z14) + z2), s10 >= 0, s10 <= 2, z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> s55 :|: s55 >= 0, s55 <= 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 }-> s66 :|: s66 >= 0, s66 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s67 :|: s67 >= 0, s67 <= z' + (1 + (1 + z06) + z3), z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s68 :|: s68 >= 0, s68 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + (1 + (1 + z13) + z3), s18 >= 0, s18 <= 2, z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s70 :|: s70 >= 0, s70 <= 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: {SORT}
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]
  IF_REPLACE: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]
  IF_MIN: 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]
  min: runtime: O(1) [0], size: O(n^1) [z]
  if_min: runtime: O(1) [0], size: O(n^1) [z']
  replace: runtime: O(1) [0], size: O(n^1) [z' + z'']
  if_replace: runtime: O(1) [0], size: O(n^1) [z'' + z4]

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

(73) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: SORT
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2*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
   IF_MIN(z, z') -{ 344 + 191*z0 + 52*z0*z2 + 26*z0^2 + 191*z2 + 26*z2^2 }-> 1 + s35 :|: s35 >= 0, s35 <= 1, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 344 + 191*z1 + 52*z1*z2 + 26*z1^2 + 191*z2 + 26*z2^2 }-> 1 + s36 :|: s36 >= 0, s36 <= 1, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 3 + 3*z3 + z3^2 }-> 1 + s24 :|: s24 >= 0, s24 <= 0, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 34 + 13*z1 + 2*z1*z2 + z1^2 + 12*z2 + z2^2 }-> 1 + s31 + s12 :|: s31 >= 0, s31 <= 0, s12 >= 0, s12 <= 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 + s32 + s13 :|: s32 >= 0, s32 <= 0, s13 >= 0, s13 <= 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 + s33 + s14 :|: s33 >= 0, s33 <= 0, s14 >= 0, s14 <= 0, s4 >= 0, s4 <= 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 + s34 + s15 :|: s34 >= 0, s34 <= 0, s15 >= 0, s15 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s19 + s' :|: s19 >= 0, s19 <= 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 + s20 + s'' :|: s20 >= 0, s20 <= 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 + s21 + s1 :|: s21 >= 0, s21 <= 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 + s22 + s2 :|: s22 >= 0, s22 <= 1, s16 >= 0, s16 <= 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 + s23 + s3 :|: s23 >= 0, s23 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 344 + 191*z0 + 52*z0*z1 + 26*z0^2 + 191*z1 + 26*z1^2 }-> 1 + s37 :|: s37 >= 0, s37 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 346 }-> 1 + SORT(s58) + s25 + s38 :|: s58 >= 0, s58 <= 0 + 0, s38 >= 0, s38 <= 1, s25 >= 0, s25 <= 0, z = 1 + 0 + 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(s59) + s26 + s39 :|: s59 >= 0, s59 <= 1 + (z - 2) + 0, s39 >= 0, s39 <= 1, s26 >= 0, s26 <= 0, z - 2 >= 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(s60) + s27 + s40 :|: s60 >= 0, s60 <= 1 + (z - 2) + 0, s40 >= 0, s40 <= 1, s27 >= 0, s27 <= 0, z - 2 >= 0
   SORT(z) -{ 567 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 248*z11 + 54*z11*z2' + 27*z11^2 + 248*z2' + 27*z2'^2 }-> 1 + SORT(s61) + s48 + s41 :|: s61 >= 0, s61 <= z0 + (1 + z11 + z2'), s46 >= 0, s46 <= 1 + z0 + (1 + z11 + z2'), s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 0, s41 >= 0, s41 <= 1, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 567 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 248*z11 + 54*z11*z2' + 27*z11^2 + 248*z2' + 27*z2'^2 }-> 1 + SORT(s62) + s28 + s42 :|: s62 >= 0, s62 <= z0 + (1 + z11 + z2'), s49 >= 0, s49 <= 1 + z0 + (1 + z11 + z2'), s42 >= 0, s42 <= 1, s28 >= 0, s28 <= 0, s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(s63) + s29 + s43 :|: s63 >= 0, s63 <= 1 + (z - 2) + 0, s43 >= 0, s43 <= 1, s29 >= 0, s29 <= 0, z - 2 >= 0
   SORT(z) -{ 567 + 243*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 248*z12 + 54*z12*z2'' + 27*z12^2 + 248*z2'' + 27*z2''^2 }-> 1 + SORT(s64) + s51 + s44 :|: s64 >= 0, s64 <= z0 + (1 + z12 + z2''), s50 >= 0, s50 <= 1 + z0 + (1 + z12 + z2''), s51 >= 0, s51 <= 0, s44 >= 0, s44 <= 1, s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 346 + 191*z0 + 52*z0*z1 + 26*z0^2 + 194*z1 + 27*z1^2 }-> 1 + SORT(s65) + s30 + s45 :|: s65 >= 0, s65 <= z0 + z1, s45 >= 0, s45 <= 1, s30 >= 0, s30 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eq(z, z') -{ 0 }-> s17 :|: s17 >= 0, s17 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> s56 :|: s56 >= 0, s56 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + s71 :|: s71 >= 0, s71 <= z'' + z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> s52 :|: s52 >= 0, s52 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 1 + (1 + z09) + (1 + 0 + z2), z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + (1 + z010) + (1 + (1 + z14) + z2), s10 >= 0, s10 <= 2, z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> s55 :|: s55 >= 0, s55 <= 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 }-> s66 :|: s66 >= 0, s66 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s67 :|: s67 >= 0, s67 <= z' + (1 + (1 + z06) + z3), z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s68 :|: s68 >= 0, s68 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + (1 + (1 + z13) + z3), s18 >= 0, s18 <= 2, z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s70 :|: s70 >= 0, s70 <= 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: {SORT}
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]
  IF_REPLACE: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]
  IF_MIN: 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]
  min: runtime: O(1) [0], size: O(n^1) [z]
  if_min: runtime: O(1) [0], size: O(n^1) [z']
  replace: runtime: O(1) [0], size: O(n^1) [z' + z'']
  if_replace: runtime: O(1) [0], size: O(n^1) [z'' + z4]
  SORT: runtime: ?, size: O(n^1) [2*z]

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

(75) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: SORT
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 344 + 3318*z + 3123*z^2 + 897*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
   IF_MIN(z, z') -{ 344 + 191*z0 + 52*z0*z2 + 26*z0^2 + 191*z2 + 26*z2^2 }-> 1 + s35 :|: s35 >= 0, s35 <= 1, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   IF_MIN(z, z') -{ 344 + 191*z1 + 52*z1*z2 + 26*z1^2 + 191*z2 + 26*z2^2 }-> 1 + s36 :|: s36 >= 0, s36 <= 1, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   IF_REPLACE(z, z', z'', z4) -{ 3 + 3*z3 + z3^2 }-> 1 + s24 :|: s24 >= 0, s24 <= 0, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   LE(z, z') -{ z' }-> 1 + s11 :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z) -{ 34 + 13*z1 + 2*z1*z2 + z1^2 + 12*z2 + z2^2 }-> 1 + s31 + s12 :|: s31 >= 0, s31 <= 0, s12 >= 0, s12 <= 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 + s32 + s13 :|: s32 >= 0, s32 <= 0, s13 >= 0, s13 <= 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 + s33 + s14 :|: s33 >= 0, s33 <= 0, s14 >= 0, s14 <= 0, s4 >= 0, s4 <= 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 + s34 + s15 :|: s34 >= 0, s34 <= 0, s15 >= 0, s15 <= 0, z1 >= 0, z = 1 + z0 + (1 + z1 + z2), z0 >= 0, z2 >= 0
   REPLACE(z, z', z'') -{ 4 + 3*z'' + z''^2 }-> 1 + s19 + s' :|: s19 >= 0, s19 <= 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 + s20 + s'' :|: s20 >= 0, s20 <= 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 + s21 + s1 :|: s21 >= 0, s21 <= 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 + s22 + s2 :|: s22 >= 0, s22 <= 1, s16 >= 0, s16 <= 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 + s23 + s3 :|: s23 >= 0, s23 <= 1, s3 >= 0, s3 <= 0, z' >= 0, z >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   SORT(z) -{ 344 + 191*z0 + 52*z0*z1 + 26*z0^2 + 191*z1 + 26*z1^2 }-> 1 + s37 :|: s37 >= 0, s37 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SORT(z) -{ 346 }-> 1 + SORT(s58) + s25 + s38 :|: s58 >= 0, s58 <= 0 + 0, s38 >= 0, s38 <= 1, s25 >= 0, s25 <= 0, z = 1 + 0 + 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(s59) + s26 + s39 :|: s59 >= 0, s59 <= 1 + (z - 2) + 0, s39 >= 0, s39 <= 1, s26 >= 0, s26 <= 0, z - 2 >= 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(s60) + s27 + s40 :|: s60 >= 0, s60 <= 1 + (z - 2) + 0, s40 >= 0, s40 <= 1, s27 >= 0, s27 <= 0, z - 2 >= 0
   SORT(z) -{ 567 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 248*z11 + 54*z11*z2' + 27*z11^2 + 248*z2' + 27*z2'^2 }-> 1 + SORT(s61) + s48 + s41 :|: s61 >= 0, s61 <= z0 + (1 + z11 + z2'), s46 >= 0, s46 <= 1 + z0 + (1 + z11 + z2'), s47 >= 0, s47 <= 1 + z0 + (1 + z11 + z2'), s48 >= 0, s48 <= 0, s41 >= 0, s41 <= 1, s5 >= 0, s5 <= 2, s6 >= 0, s6 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 567 + 243*z0 + 52*z0*z11 + 52*z0*z2' + 26*z0^2 + 248*z11 + 54*z11*z2' + 27*z11^2 + 248*z2' + 27*z2'^2 }-> 1 + SORT(s62) + s28 + s42 :|: s62 >= 0, s62 <= z0 + (1 + z11 + z2'), s49 >= 0, s49 <= 1 + z0 + (1 + z11 + z2'), s42 >= 0, s42 <= 1, s28 >= 0, s28 <= 0, s7 >= 0, s7 <= 2, z = 1 + z0 + (1 + z11 + z2'), z11 >= 0, z2' >= 0, z0 >= 0
   SORT(z) -{ 181 + 139*z + 26*z^2 }-> 1 + SORT(s63) + s29 + s43 :|: s63 >= 0, s63 <= 1 + (z - 2) + 0, s43 >= 0, s43 <= 1, s29 >= 0, s29 <= 0, z - 2 >= 0
   SORT(z) -{ 567 + 243*z0 + 52*z0*z12 + 52*z0*z2'' + 26*z0^2 + 248*z12 + 54*z12*z2'' + 27*z12^2 + 248*z2'' + 27*z2''^2 }-> 1 + SORT(s64) + s51 + s44 :|: s64 >= 0, s64 <= z0 + (1 + z12 + z2''), s50 >= 0, s50 <= 1 + z0 + (1 + z12 + z2''), s51 >= 0, s51 <= 0, s44 >= 0, s44 <= 1, s8 >= 0, s8 <= 2, z0 >= 0, z12 >= 0, z = 1 + z0 + (1 + z12 + z2''), z2'' >= 0
   SORT(z) -{ 346 + 191*z0 + 52*z0*z1 + 26*z0^2 + 194*z1 + 27*z1^2 }-> 1 + SORT(s65) + s30 + s45 :|: s65 >= 0, s65 <= z0 + z1, s45 >= 0, s45 <= 1, s30 >= 0, s30 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   eq(z, z') -{ 0 }-> s17 :|: s17 >= 0, s17 <= 2, z' - 1 >= 0, z - 1 >= 0
   eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
   eq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0
   eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_min(z, z') -{ 0 }-> s56 :|: s56 >= 0, s56 <= 1 + z0 + z2, z = 2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 1 + z1 + z2, z' = 1 + z0 + (1 + z1 + z2), z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   if_min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z'' + z3 :|: z = 2, z'' >= 0, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   if_replace(z, z', z'', z4) -{ 0 }-> 1 + z2 + s71 :|: s71 >= 0, s71 <= z'' + z3, z'' >= 0, z = 1, z' >= 0, z4 = 1 + z2 + z3, z2 >= 0, z3 >= 0
   le(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 0 }-> 2 :|: z' >= 0, z = 0
   le(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
   le(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z) -{ 0 }-> s52 :|: s52 >= 0, s52 <= 1 + 0 + (1 + z1 + z2), z1 >= 0, z = 1 + 0 + (1 + z1 + z2), z2 >= 0
   min(z) -{ 0 }-> s53 :|: s53 >= 0, s53 <= 1 + (1 + z09) + (1 + 0 + z2), z = 1 + (1 + z09) + (1 + 0 + z2), z09 >= 0, z2 >= 0
   min(z) -{ 0 }-> s54 :|: s54 >= 0, s54 <= 1 + (1 + z010) + (1 + (1 + z14) + z2), s10 >= 0, s10 <= 2, z = 1 + (1 + z010) + (1 + (1 + z14) + z2), z010 >= 0, z14 >= 0, z2 >= 0
   min(z) -{ 0 }-> s55 :|: s55 >= 0, s55 <= 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 }-> s66 :|: s66 >= 0, s66 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z = 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s67 :|: s67 >= 0, s67 <= z' + (1 + (1 + z06) + z3), z'' = 1 + (1 + z06) + z3, z' >= 0, z06 >= 0, z = 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s68 :|: s68 >= 0, s68 <= z' + (1 + 0 + (z'' - 1)), z' >= 0, z - 1 >= 0, z'' - 1 >= 0
   replace(z, z', z'') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + (1 + (1 + z13) + z3), s18 >= 0, s18 <= 2, z - 1 >= 0, z' >= 0, z'' = 1 + (1 + z13) + z3, z13 >= 0, z3 >= 0
   replace(z, z', z'') -{ 0 }-> s70 :|: s70 >= 0, s70 <= 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: 
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]
  IF_REPLACE: runtime: O(n^2) [3 + 3*z4 + z4^2], size: O(1) [1]
  IF_MIN: 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]
  min: runtime: O(1) [0], size: O(n^1) [z]
  if_min: runtime: O(1) [0], size: O(n^1) [z']
  replace: runtime: O(1) [0], size: O(n^1) [z' + z'']
  if_replace: runtime: O(1) [0], size: O(n^1) [z'' + z4]
  SORT: runtime: O(n^3) [344 + 3318*z + 3123*z^2 + 897*z^3], size: O(n^1) [2*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))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
   if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
   if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
   replace(z0, z1, nil) -> nil
   replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
   if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
   if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
   sort(nil) -> nil
   sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))
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(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
   IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
   IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
   REPLACE(z0, z1, nil) -> c12
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
   IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c14
   IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
   SORT(nil) -> c16
   SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
   SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1)))
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(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
   IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
   IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
   REPLACE(z0, z1, nil) -> c12
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
   IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c14
   IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
   SORT(nil) -> c16
   SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
   SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1)))
K tuples:none
Defined Rule Symbols:   eq_2, le_2, min_1, if_min_2, replace_3, if_replace_4, sort_1

Defined Pair Symbols:   EQ_2, LE_2, MIN_1, IF_MIN_2, REPLACE_3, IF_REPLACE_4, SORT_1

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_1, c18_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(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
   IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
   IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
   REPLACE(z0, z1, nil) -> c12
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
   IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c14
   IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
   SORT(nil) -> c16
   SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
   SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1)))

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

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

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

(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(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
   IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
   IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
   REPLACE(z0, z1, nil) -> c12
   REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
   IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c14
   IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
   SORT(nil) -> c16
   SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
   SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1)))

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

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

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

(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(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, z1, nil) -> c12
REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c14
IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
SORT(nil) -> c16
SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1)))
eq(0', 0') -> true
eq(0', s(z0)) -> false
eq(s(z0), 0') -> false
eq(s(z0), s(z1)) -> eq(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
min(cons(0', nil)) -> 0'
min(cons(s(z0), nil)) -> s(z0)
min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
sort(nil) -> nil
sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))

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

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

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

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

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

(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(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, z1, nil) -> c12
REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c14
IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
SORT(nil) -> c16
SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1)))
eq(0', 0') -> true
eq(0', s(z0)) -> false
eq(s(z0), 0') -> false
eq(s(z0), s(z1)) -> eq(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
min(cons(0', nil)) -> 0'
min(cons(s(z0), nil)) -> s(z0)
min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
sort(nil) -> nil
sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))

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


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


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

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

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

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

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

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

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

(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(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, z1, nil) -> c12
REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c14
IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
SORT(nil) -> c16
SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1)))
eq(0', 0') -> true
eq(0', s(z0)) -> false
eq(s(z0), 0') -> false
eq(s(z0), s(z1)) -> eq(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
min(cons(0', nil)) -> 0'
min(cons(s(z0), nil)) -> s(z0)
min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
sort(nil) -> nil
sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))

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


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


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

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

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

(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(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, z1, nil) -> c12
REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c14
IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
SORT(nil) -> c16
SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1)))
eq(0', 0') -> true
eq(0', s(z0)) -> false
eq(s(z0), 0') -> false
eq(s(z0), s(z1)) -> eq(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
min(cons(0', nil)) -> 0'
min(cons(s(z0), nil)) -> s(z0)
min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
sort(nil) -> nil
sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))

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


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


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


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

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

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

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

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

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

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

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

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


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


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


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

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

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

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

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

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

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

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

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


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


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


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

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

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

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

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

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

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

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

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


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


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


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

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

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

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

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

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

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

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

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


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


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


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

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

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

(103) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
min(gen_nil:cons14_19(+(1, n9513_19))) -> gen_0':s12_19(0), rt in Omega(0)

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

Induction Step:
min(gen_nil:cons14_19(+(1, +(n9513_19, 1)))) ->_R^Omega(0)
if_min(le(0', 0'), cons(0', cons(0', gen_nil:cons14_19(n9513_19)))) ->_L^Omega(0)
if_min(true, cons(0', cons(0', gen_nil:cons14_19(n9513_19)))) ->_R^Omega(0)
min(cons(0', gen_nil:cons14_19(n9513_19))) ->_IH
gen_0':s12_19(0)

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

(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(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, z1, nil) -> c12
REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c14
IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
SORT(nil) -> c16
SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1)))
eq(0', 0') -> true
eq(0', s(z0)) -> false
eq(s(z0), 0') -> false
eq(s(z0), s(z1)) -> eq(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
min(cons(0', nil)) -> 0'
min(cons(s(z0), nil)) -> s(z0)
min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
sort(nil) -> nil
sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))

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


Lemmas:
EQ(gen_0':s12_19(n17_19), gen_0':s12_19(n17_19)) -> gen_c:c1:c2:c311_19(n17_19), rt in Omega(1 + n17_19)
LE(gen_0':s12_19(n1023_19), gen_0':s12_19(n1023_19)) -> gen_c4:c5:c613_19(n1023_19), rt in Omega(1 + n1023_19)
le(gen_0':s12_19(n1803_19), gen_0':s12_19(n1803_19)) -> true, rt in Omega(0)
MIN(gen_nil:cons14_19(+(1, n2270_19))) -> *16_19, rt in Omega(n2270_19)
eq(gen_0':s12_19(n7487_19), gen_0':s12_19(n7487_19)) -> true, rt in Omega(0)
min(gen_nil:cons14_19(+(1, n9513_19))) -> gen_0':s12_19(0), rt in Omega(0)


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


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

(105) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
SORT(gen_nil:cons14_19(+(1, n10367_19))) -> *16_19, rt in Omega(n10367_19 + n10367_19^2)

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

Induction Step:
SORT(gen_nil:cons14_19(+(1, +(n10367_19, 1)))) ->_R^Omega(1)
c18(SORT(replace(min(cons(0', gen_nil:cons14_19(+(1, n10367_19)))), 0', gen_nil:cons14_19(+(1, n10367_19)))), REPLACE(min(cons(0', gen_nil:cons14_19(+(1, n10367_19)))), 0', gen_nil:cons14_19(+(1, n10367_19))), MIN(cons(0', gen_nil:cons14_19(+(1, n10367_19))))) ->_L^Omega(0)
c18(SORT(replace(gen_0':s12_19(0), 0', gen_nil:cons14_19(+(1, n10367_19)))), REPLACE(min(cons(0', gen_nil:cons14_19(+(1, n10367_19)))), 0', gen_nil:cons14_19(+(1, n10367_19))), MIN(cons(0', gen_nil:cons14_19(+(1, n10367_19))))) ->_R^Omega(0)
c18(SORT(if_replace(eq(gen_0':s12_19(0), 0'), gen_0':s12_19(0), 0', cons(0', gen_nil:cons14_19(n10367_19)))), REPLACE(min(cons(0', gen_nil:cons14_19(+(1, n10367_19)))), 0', gen_nil:cons14_19(+(1, n10367_19))), MIN(cons(0', gen_nil:cons14_19(+(1, n10367_19))))) ->_L^Omega(0)
c18(SORT(if_replace(true, gen_0':s12_19(0), 0', cons(0', gen_nil:cons14_19(n10367_19)))), REPLACE(min(cons(0', gen_nil:cons14_19(+(1, n10367_19)))), 0', gen_nil:cons14_19(+(1, n10367_19))), MIN(cons(0', gen_nil:cons14_19(+(1, n10367_19))))) ->_R^Omega(0)
c18(SORT(cons(0', gen_nil:cons14_19(n10367_19))), REPLACE(min(cons(0', gen_nil:cons14_19(+(1, n10367_19)))), 0', gen_nil:cons14_19(+(1, n10367_19))), MIN(cons(0', gen_nil:cons14_19(+(1, n10367_19))))) ->_IH
c18(*16_19, REPLACE(min(cons(0', gen_nil:cons14_19(+(1, n10367_19)))), 0', gen_nil:cons14_19(+(1, n10367_19))), MIN(cons(0', gen_nil:cons14_19(+(1, n10367_19))))) ->_L^Omega(0)
c18(*16_19, REPLACE(gen_0':s12_19(0), 0', gen_nil:cons14_19(+(1, n10367_19))), MIN(cons(0', gen_nil:cons14_19(+(1, n10367_19))))) ->_R^Omega(1)
c18(*16_19, c13(IF_REPLACE(eq(gen_0':s12_19(0), 0'), gen_0':s12_19(0), 0', cons(0', gen_nil:cons14_19(n10367_19))), EQ(gen_0':s12_19(0), 0')), MIN(cons(0', gen_nil:cons14_19(+(1, n10367_19))))) ->_L^Omega(0)
c18(*16_19, c13(IF_REPLACE(true, gen_0':s12_19(0), 0', cons(0', gen_nil:cons14_19(n10367_19))), EQ(gen_0':s12_19(0), 0')), MIN(cons(0', gen_nil:cons14_19(+(1, n10367_19))))) ->_R^Omega(1)
c18(*16_19, c13(c14, EQ(gen_0':s12_19(0), 0')), MIN(cons(0', gen_nil:cons14_19(+(1, n10367_19))))) ->_L^Omega(1)
c18(*16_19, c13(c14, gen_c:c1:c2:c311_19(0)), MIN(cons(0', gen_nil:cons14_19(+(1, n10367_19))))) ->_R^Omega(1)
c18(*16_19, c13(c14, gen_c:c1:c2:c311_19(0)), c9(IF_MIN(le(0', 0'), cons(0', cons(0', gen_nil:cons14_19(n10367_19)))), LE(0', 0'))) ->_L^Omega(0)
c18(*16_19, c13(c14, gen_c:c1:c2:c311_19(0)), c9(IF_MIN(true, cons(0', cons(0', gen_nil:cons14_19(n10367_19)))), LE(0', 0'))) ->_R^Omega(1)
c18(*16_19, c13(c14, gen_c:c1:c2:c311_19(0)), c9(c10(MIN(cons(0', gen_nil:cons14_19(n10367_19)))), LE(0', 0'))) ->_L^Omega(n10367_19)
c18(*16_19, c13(c14, gen_c:c1:c2:c311_19(0)), c9(c10(*16_19), LE(0', 0'))) ->_L^Omega(1)
c18(*16_19, c13(c14, gen_c:c1:c2:c311_19(0)), c9(c10(*16_19), gen_c4:c5:c613_19(0)))

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

(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(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, z1, nil) -> c12
REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c14
IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
SORT(nil) -> c16
SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1)))
eq(0', 0') -> true
eq(0', s(z0)) -> false
eq(s(z0), 0') -> false
eq(s(z0), s(z1)) -> eq(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
min(cons(0', nil)) -> 0'
min(cons(s(z0), nil)) -> s(z0)
min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
sort(nil) -> nil
sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))

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


Lemmas:
EQ(gen_0':s12_19(n17_19), gen_0':s12_19(n17_19)) -> gen_c:c1:c2:c311_19(n17_19), rt in Omega(1 + n17_19)
LE(gen_0':s12_19(n1023_19), gen_0':s12_19(n1023_19)) -> gen_c4:c5:c613_19(n1023_19), rt in Omega(1 + n1023_19)
le(gen_0':s12_19(n1803_19), gen_0':s12_19(n1803_19)) -> true, rt in Omega(0)
MIN(gen_nil:cons14_19(+(1, n2270_19))) -> *16_19, rt in Omega(n2270_19)
eq(gen_0':s12_19(n7487_19), gen_0':s12_19(n7487_19)) -> true, rt in Omega(0)
min(gen_nil:cons14_19(+(1, n9513_19))) -> gen_0':s12_19(0), rt in Omega(0)


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


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

(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(IF_MIN(le(z0, z1), cons(z0, cons(z1, z2))), LE(z0, z1))
IF_MIN(true, cons(z0, cons(z1, z2))) -> c10(MIN(cons(z0, z2)))
IF_MIN(false, cons(z0, cons(z1, z2))) -> c11(MIN(cons(z1, z2)))
REPLACE(z0, z1, nil) -> c12
REPLACE(z0, z1, cons(z2, z3)) -> c13(IF_REPLACE(eq(z0, z2), z0, z1, cons(z2, z3)), EQ(z0, z2))
IF_REPLACE(true, z0, z1, cons(z2, z3)) -> c14
IF_REPLACE(false, z0, z1, cons(z2, z3)) -> c15(REPLACE(z0, z1, z3))
SORT(nil) -> c16
SORT(cons(z0, z1)) -> c17(MIN(cons(z0, z1)))
SORT(cons(z0, z1)) -> c18(SORT(replace(min(cons(z0, z1)), z0, z1)), REPLACE(min(cons(z0, z1)), z0, z1), MIN(cons(z0, z1)))
eq(0', 0') -> true
eq(0', s(z0)) -> false
eq(s(z0), 0') -> false
eq(s(z0), s(z1)) -> eq(z0, z1)
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
min(cons(0', nil)) -> 0'
min(cons(s(z0), nil)) -> s(z0)
min(cons(z0, cons(z1, z2))) -> if_min(le(z0, z1), cons(z0, cons(z1, z2)))
if_min(true, cons(z0, cons(z1, z2))) -> min(cons(z0, z2))
if_min(false, cons(z0, cons(z1, z2))) -> min(cons(z1, z2))
replace(z0, z1, nil) -> nil
replace(z0, z1, cons(z2, z3)) -> if_replace(eq(z0, z2), z0, z1, cons(z2, z3))
if_replace(true, z0, z1, cons(z2, z3)) -> cons(z1, z3)
if_replace(false, z0, z1, cons(z2, z3)) -> cons(z2, replace(z0, z1, z3))
sort(nil) -> nil
sort(cons(z0, z1)) -> cons(min(cons(z0, z1)), sort(replace(min(cons(z0, z1)), z0, z1)))

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


Lemmas:
EQ(gen_0':s12_19(n17_19), gen_0':s12_19(n17_19)) -> gen_c:c1:c2:c311_19(n17_19), rt in Omega(1 + n17_19)
LE(gen_0':s12_19(n1023_19), gen_0':s12_19(n1023_19)) -> gen_c4:c5:c613_19(n1023_19), rt in Omega(1 + n1023_19)
le(gen_0':s12_19(n1803_19), gen_0':s12_19(n1803_19)) -> true, rt in Omega(0)
MIN(gen_nil:cons14_19(+(1, n2270_19))) -> *16_19, rt in Omega(n2270_19)
eq(gen_0':s12_19(n7487_19), gen_0':s12_19(n7487_19)) -> true, rt in Omega(0)
min(gen_nil:cons14_19(+(1, n9513_19))) -> gen_0':s12_19(0), rt in Omega(0)
SORT(gen_nil:cons14_19(+(1, n10367_19))) -> *16_19, rt in Omega(n10367_19 + n10367_19^2)


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


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

(111) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sort(gen_nil:cons14_19(+(1, n401090_19))) -> *16_19, rt in Omega(0)

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

Induction Step:
sort(gen_nil:cons14_19(+(1, +(n401090_19, 1)))) ->_R^Omega(0)
cons(min(cons(0', gen_nil:cons14_19(+(1, n401090_19)))), sort(replace(min(cons(0', gen_nil:cons14_19(+(1, n401090_19)))), 0', gen_nil:cons14_19(+(1, n401090_19))))) ->_L^Omega(0)
cons(gen_0':s12_19(0), sort(replace(min(cons(0', gen_nil:cons14_19(+(1, n401090_19)))), 0', gen_nil:cons14_19(+(1, n401090_19))))) ->_L^Omega(0)
cons(gen_0':s12_19(0), sort(replace(gen_0':s12_19(0), 0', gen_nil:cons14_19(+(1, n401090_19))))) ->_R^Omega(0)
cons(gen_0':s12_19(0), sort(if_replace(eq(gen_0':s12_19(0), 0'), gen_0':s12_19(0), 0', cons(0', gen_nil:cons14_19(n401090_19))))) ->_L^Omega(0)
cons(gen_0':s12_19(0), sort(if_replace(true, gen_0':s12_19(0), 0', cons(0', gen_nil:cons14_19(n401090_19))))) ->_R^Omega(0)
cons(gen_0':s12_19(0), sort(cons(0', gen_nil:cons14_19(n401090_19)))) ->_IH
cons(gen_0':s12_19(0), *16_19)

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

(112)
BOUNDS(1, INF)