WORST_CASE(Omega(n^1),O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 479 ms]
(2) CpxRelTRS
(3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(4) CdtProblem
    (5) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 82 ms]
    (10) CdtProblem
    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 32 ms]
    (12) CdtProblem
    (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 28 ms]
    (14) CdtProblem
    (15) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) BOUNDS(1, 1)
(17) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(18) CpxRelTRS
    (19) SlicingProof [LOWER BOUND(ID), 0 ms]
    (20) CpxRelTRS
    (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) typed CpxTrs
    (23) OrderProof [LOWER BOUND(ID), 0 ms]
    (24) typed CpxTrs
    (25) RewriteLemmaProof [LOWER BOUND(ID), 286 ms]
    (26) BEST
        (27) proven lower bound
            (28) LowerBoundPropagationProof [FINISHED, 0 ms]
            (29) BOUNDS(n^1, INF)
        (30) typed CpxTrs


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

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


The TRS R consists of the following rules:

   MSORT(nil) -> c
   MSORT(.(z0, z1)) -> c1(MIN(z0, z1))
   MSORT(.(z0, z1)) -> c2(MSORT(del(min(z0, z1), .(z0, z1))), DEL(min(z0, z1), .(z0, z1)), MIN(z0, z1))
   MIN(z0, nil) -> c3
   MIN(z0, .(z1, z2)) -> c4(MIN(z0, z2))
   MIN(z0, .(z1, z2)) -> c5(MIN(z1, z2))
   DEL(z0, nil) -> c6
   DEL(z0, .(z1, z2)) -> c7(DEL(z0, z2))

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

   msort(nil) -> nil
   msort(.(z0, z1)) -> .(min(z0, z1), msort(del(min(z0, z1), .(z0, z1))))
   min(z0, nil) -> z0
   min(z0, .(z1, z2)) -> if(<=(z0, z1), min(z0, z2), min(z1, z2))
   del(z0, nil) -> nil
   del(z0, .(z1, z2)) -> if(=(z0, z1), z2, .(z1, del(z0, z2)))

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

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

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


The TRS R consists of the following rules:

   MSORT(nil) -> c
   MSORT(.(z0, z1)) -> c1(MIN(z0, z1))
   MSORT(.(z0, z1)) -> c2(MSORT(del(min(z0, z1), .(z0, z1))), DEL(min(z0, z1), .(z0, z1)), MIN(z0, z1))
   MIN(z0, nil) -> c3
   MIN(z0, .(z1, z2)) -> c4(MIN(z0, z2))
   MIN(z0, .(z1, z2)) -> c5(MIN(z1, z2))
   DEL(z0, nil) -> c6
   DEL(z0, .(z1, z2)) -> c7(DEL(z0, z2))

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

   msort(nil) -> nil
   msort(.(z0, z1)) -> .(min(z0, z1), msort(del(min(z0, z1), .(z0, z1))))
   min(z0, nil) -> z0
   min(z0, .(z1, z2)) -> if(<=(z0, z1), min(z0, z2), min(z1, z2))
   del(z0, nil) -> nil
   del(z0, .(z1, z2)) -> if(=(z0, z1), z2, .(z1, del(z0, z2)))

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

(3) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   msort(nil) -> nil
   msort(.(z0, z1)) -> .(min(z0, z1), msort(del(min(z0, z1), .(z0, z1))))
   min(z0, nil) -> z0
   min(z0, .(z1, z2)) -> if(<=(z0, z1), min(z0, z2), min(z1, z2))
   del(z0, nil) -> nil
   del(z0, .(z1, z2)) -> if(=(z0, z1), z2, .(z1, del(z0, z2)))
   MSORT(nil) -> c
   MSORT(.(z0, z1)) -> c1(MIN(z0, z1))
   MSORT(.(z0, z1)) -> c2(MSORT(del(min(z0, z1), .(z0, z1))), DEL(min(z0, z1), .(z0, z1)), MIN(z0, z1))
   MIN(z0, nil) -> c3
   MIN(z0, .(z1, z2)) -> c4(MIN(z0, z2))
   MIN(z0, .(z1, z2)) -> c5(MIN(z1, z2))
   DEL(z0, nil) -> c6
   DEL(z0, .(z1, z2)) -> c7(DEL(z0, z2))
Tuples:
   MSORT'(nil) -> c8
   MSORT'(.(z0, z1)) -> c9(MIN'(z0, z1), MSORT'(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1))
   MIN'(z0, nil) -> c10
   MIN'(z0, .(z1, z2)) -> c11(MIN'(z0, z2), MIN'(z1, z2))
   DEL'(z0, nil) -> c12
   DEL'(z0, .(z1, z2)) -> c13(DEL'(z0, z2))
   MSORT''(nil) -> c14
   MSORT''(.(z0, z1)) -> c15(MIN''(z0, z1))
   MSORT''(.(z0, z1)) -> c16(MSORT''(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1), DEL''(min(z0, z1), .(z0, z1)), MIN'(z0, z1), MIN''(z0, z1))
   MIN''(z0, nil) -> c17
   MIN''(z0, .(z1, z2)) -> c18(MIN''(z0, z2))
   MIN''(z0, .(z1, z2)) -> c19(MIN''(z1, z2))
   DEL''(z0, nil) -> c20
   DEL''(z0, .(z1, z2)) -> c21(DEL''(z0, z2))
S tuples:
   MSORT''(nil) -> c14
   MSORT''(.(z0, z1)) -> c15(MIN''(z0, z1))
   MSORT''(.(z0, z1)) -> c16(MSORT''(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1), DEL''(min(z0, z1), .(z0, z1)), MIN'(z0, z1), MIN''(z0, z1))
   MIN''(z0, nil) -> c17
   MIN''(z0, .(z1, z2)) -> c18(MIN''(z0, z2))
   MIN''(z0, .(z1, z2)) -> c19(MIN''(z1, z2))
   DEL''(z0, nil) -> c20
   DEL''(z0, .(z1, z2)) -> c21(DEL''(z0, z2))
K tuples:none
Defined Rule Symbols:   MSORT_1, MIN_2, DEL_2, msort_1, min_2, del_2

Defined Pair Symbols:   MSORT'_1, MIN'_2, DEL'_2, MSORT''_1, MIN''_2, DEL''_2

Compound Symbols:   c8, c9_4, c10, c11_2, c12, c13_1, c14, c15_1, c16_6, c17, c18_1, c19_1, c20, c21_1


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

(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   DEL'(z0, nil) -> c12
   MSORT'(nil) -> c8
   MIN'(z0, nil) -> c10
   MSORT''(nil) -> c14
   MIN''(z0, nil) -> c17
   DEL''(z0, nil) -> c20

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   msort(nil) -> nil
   msort(.(z0, z1)) -> .(min(z0, z1), msort(del(min(z0, z1), .(z0, z1))))
   min(z0, nil) -> z0
   min(z0, .(z1, z2)) -> if(<=(z0, z1), min(z0, z2), min(z1, z2))
   del(z0, nil) -> nil
   del(z0, .(z1, z2)) -> if(=(z0, z1), z2, .(z1, del(z0, z2)))
   MSORT(nil) -> c
   MSORT(.(z0, z1)) -> c1(MIN(z0, z1))
   MSORT(.(z0, z1)) -> c2(MSORT(del(min(z0, z1), .(z0, z1))), DEL(min(z0, z1), .(z0, z1)), MIN(z0, z1))
   MIN(z0, nil) -> c3
   MIN(z0, .(z1, z2)) -> c4(MIN(z0, z2))
   MIN(z0, .(z1, z2)) -> c5(MIN(z1, z2))
   DEL(z0, nil) -> c6
   DEL(z0, .(z1, z2)) -> c7(DEL(z0, z2))
Tuples:
   MSORT'(.(z0, z1)) -> c9(MIN'(z0, z1), MSORT'(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1))
   MIN'(z0, .(z1, z2)) -> c11(MIN'(z0, z2), MIN'(z1, z2))
   DEL'(z0, .(z1, z2)) -> c13(DEL'(z0, z2))
   MSORT''(.(z0, z1)) -> c15(MIN''(z0, z1))
   MSORT''(.(z0, z1)) -> c16(MSORT''(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1), DEL''(min(z0, z1), .(z0, z1)), MIN'(z0, z1), MIN''(z0, z1))
   MIN''(z0, .(z1, z2)) -> c18(MIN''(z0, z2))
   MIN''(z0, .(z1, z2)) -> c19(MIN''(z1, z2))
   DEL''(z0, .(z1, z2)) -> c21(DEL''(z0, z2))
S tuples:
   MSORT''(.(z0, z1)) -> c15(MIN''(z0, z1))
   MSORT''(.(z0, z1)) -> c16(MSORT''(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1), DEL''(min(z0, z1), .(z0, z1)), MIN'(z0, z1), MIN''(z0, z1))
   MIN''(z0, .(z1, z2)) -> c18(MIN''(z0, z2))
   MIN''(z0, .(z1, z2)) -> c19(MIN''(z1, z2))
   DEL''(z0, .(z1, z2)) -> c21(DEL''(z0, z2))
K tuples:none
Defined Rule Symbols:   MSORT_1, MIN_2, DEL_2, msort_1, min_2, del_2

Defined Pair Symbols:   MSORT'_1, MIN'_2, DEL'_2, MSORT''_1, MIN''_2, DEL''_2

Compound Symbols:   c9_4, c11_2, c13_1, c15_1, c16_6, c18_1, c19_1, c21_1


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   msort(nil) -> nil
   msort(.(z0, z1)) -> .(min(z0, z1), msort(del(min(z0, z1), .(z0, z1))))
   MSORT(nil) -> c
   MSORT(.(z0, z1)) -> c1(MIN(z0, z1))
   MSORT(.(z0, z1)) -> c2(MSORT(del(min(z0, z1), .(z0, z1))), DEL(min(z0, z1), .(z0, z1)), MIN(z0, z1))
   MIN(z0, nil) -> c3
   MIN(z0, .(z1, z2)) -> c4(MIN(z0, z2))
   MIN(z0, .(z1, z2)) -> c5(MIN(z1, z2))
   DEL(z0, nil) -> c6
   DEL(z0, .(z1, z2)) -> c7(DEL(z0, z2))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, .(z1, z2)) -> if(=(z0, z1), z2, .(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, .(z1, z2)) -> if(<=(z0, z1), min(z0, z2), min(z1, z2))
Tuples:
   MSORT'(.(z0, z1)) -> c9(MIN'(z0, z1), MSORT'(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1))
   MIN'(z0, .(z1, z2)) -> c11(MIN'(z0, z2), MIN'(z1, z2))
   DEL'(z0, .(z1, z2)) -> c13(DEL'(z0, z2))
   MSORT''(.(z0, z1)) -> c15(MIN''(z0, z1))
   MSORT''(.(z0, z1)) -> c16(MSORT''(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1), DEL''(min(z0, z1), .(z0, z1)), MIN'(z0, z1), MIN''(z0, z1))
   MIN''(z0, .(z1, z2)) -> c18(MIN''(z0, z2))
   MIN''(z0, .(z1, z2)) -> c19(MIN''(z1, z2))
   DEL''(z0, .(z1, z2)) -> c21(DEL''(z0, z2))
S tuples:
   MSORT''(.(z0, z1)) -> c15(MIN''(z0, z1))
   MSORT''(.(z0, z1)) -> c16(MSORT''(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1), DEL''(min(z0, z1), .(z0, z1)), MIN'(z0, z1), MIN''(z0, z1))
   MIN''(z0, .(z1, z2)) -> c18(MIN''(z0, z2))
   MIN''(z0, .(z1, z2)) -> c19(MIN''(z1, z2))
   DEL''(z0, .(z1, z2)) -> c21(DEL''(z0, z2))
K tuples:none
Defined Rule Symbols:   del_2, min_2

Defined Pair Symbols:   MSORT'_1, MIN'_2, DEL'_2, MSORT''_1, MIN''_2, DEL''_2

Compound Symbols:   c9_4, c11_2, c13_1, c15_1, c16_6, c18_1, c19_1, c21_1


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

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MSORT''(.(z0, z1)) -> c15(MIN''(z0, z1))
   MIN''(z0, .(z1, z2)) -> c18(MIN''(z0, z2))
   MIN''(z0, .(z1, z2)) -> c19(MIN''(z1, z2))
We considered the (Usable) Rules:
   del(z0, .(z1, z2)) -> if(=(z0, z1), z2, .(z1, del(z0, z2)))
   min(z0, nil) -> z0
   min(z0, .(z1, z2)) -> if(<=(z0, z1), min(z0, z2), min(z1, z2))
And the Tuples:
   MSORT'(.(z0, z1)) -> c9(MIN'(z0, z1), MSORT'(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1))
   MIN'(z0, .(z1, z2)) -> c11(MIN'(z0, z2), MIN'(z1, z2))
   DEL'(z0, .(z1, z2)) -> c13(DEL'(z0, z2))
   MSORT''(.(z0, z1)) -> c15(MIN''(z0, z1))
   MSORT''(.(z0, z1)) -> c16(MSORT''(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1), DEL''(min(z0, z1), .(z0, z1)), MIN'(z0, z1), MIN''(z0, z1))
   MIN''(z0, .(z1, z2)) -> c18(MIN''(z0, z2))
   MIN''(z0, .(z1, z2)) -> c19(MIN''(z1, z2))
   DEL''(z0, .(z1, z2)) -> c21(DEL''(z0, z2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(.(x_1, x_2)) = [1] + x_1 + x_2
   POL(<=(x_1, x_2)) = x_1
   POL(=(x_1, x_2)) = 0
   POL(DEL'(x_1, x_2)) = [1] + x_1
   POL(DEL''(x_1, x_2)) = 0
   POL(MIN'(x_1, x_2)) = 0
   POL(MIN''(x_1, x_2)) = x_2
   POL(MSORT'(x_1)) = x_1
   POL(MSORT''(x_1)) = [1] + x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1, x_2, x_3, x_4, x_5, x_6)) = x_1 + x_2 + x_3 + x_4 + x_5 + x_6
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c9(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(del(x_1, x_2)) = 0
   POL(if(x_1, x_2, x_3)) = x_1
   POL(min(x_1, x_2)) = x_1
   POL(nil) = 0

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, .(z1, z2)) -> if(=(z0, z1), z2, .(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, .(z1, z2)) -> if(<=(z0, z1), min(z0, z2), min(z1, z2))
Tuples:
   MSORT'(.(z0, z1)) -> c9(MIN'(z0, z1), MSORT'(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1))
   MIN'(z0, .(z1, z2)) -> c11(MIN'(z0, z2), MIN'(z1, z2))
   DEL'(z0, .(z1, z2)) -> c13(DEL'(z0, z2))
   MSORT''(.(z0, z1)) -> c15(MIN''(z0, z1))
   MSORT''(.(z0, z1)) -> c16(MSORT''(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1), DEL''(min(z0, z1), .(z0, z1)), MIN'(z0, z1), MIN''(z0, z1))
   MIN''(z0, .(z1, z2)) -> c18(MIN''(z0, z2))
   MIN''(z0, .(z1, z2)) -> c19(MIN''(z1, z2))
   DEL''(z0, .(z1, z2)) -> c21(DEL''(z0, z2))
S tuples:
   MSORT''(.(z0, z1)) -> c16(MSORT''(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1), DEL''(min(z0, z1), .(z0, z1)), MIN'(z0, z1), MIN''(z0, z1))
   DEL''(z0, .(z1, z2)) -> c21(DEL''(z0, z2))
K tuples:
   MSORT''(.(z0, z1)) -> c15(MIN''(z0, z1))
   MIN''(z0, .(z1, z2)) -> c18(MIN''(z0, z2))
   MIN''(z0, .(z1, z2)) -> c19(MIN''(z1, z2))
Defined Rule Symbols:   del_2, min_2

Defined Pair Symbols:   MSORT'_1, MIN'_2, DEL'_2, MSORT''_1, MIN''_2, DEL''_2

Compound Symbols:   c9_4, c11_2, c13_1, c15_1, c16_6, c18_1, c19_1, c21_1


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

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MSORT''(.(z0, z1)) -> c16(MSORT''(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1), DEL''(min(z0, z1), .(z0, z1)), MIN'(z0, z1), MIN''(z0, z1))
We considered the (Usable) Rules:
   del(z0, .(z1, z2)) -> if(=(z0, z1), z2, .(z1, del(z0, z2)))
   min(z0, nil) -> z0
   min(z0, .(z1, z2)) -> if(<=(z0, z1), min(z0, z2), min(z1, z2))
And the Tuples:
   MSORT'(.(z0, z1)) -> c9(MIN'(z0, z1), MSORT'(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1))
   MIN'(z0, .(z1, z2)) -> c11(MIN'(z0, z2), MIN'(z1, z2))
   DEL'(z0, .(z1, z2)) -> c13(DEL'(z0, z2))
   MSORT''(.(z0, z1)) -> c15(MIN''(z0, z1))
   MSORT''(.(z0, z1)) -> c16(MSORT''(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1), DEL''(min(z0, z1), .(z0, z1)), MIN'(z0, z1), MIN''(z0, z1))
   MIN''(z0, .(z1, z2)) -> c18(MIN''(z0, z2))
   MIN''(z0, .(z1, z2)) -> c19(MIN''(z1, z2))
   DEL''(z0, .(z1, z2)) -> c21(DEL''(z0, z2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(.(x_1, x_2)) = [1] + x_1 + x_2
   POL(<=(x_1, x_2)) = [1] + x_1 + x_2
   POL(=(x_1, x_2)) = 0
   POL(DEL'(x_1, x_2)) = x_1
   POL(DEL''(x_1, x_2)) = 0
   POL(MIN'(x_1, x_2)) = 0
   POL(MIN''(x_1, x_2)) = 0
   POL(MSORT'(x_1)) = x_1
   POL(MSORT''(x_1)) = [1] + x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1, x_2, x_3, x_4, x_5, x_6)) = x_1 + x_2 + x_3 + x_4 + x_5 + x_6
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c9(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(del(x_1, x_2)) = 0
   POL(if(x_1, x_2, x_3)) = x_1
   POL(min(x_1, x_2)) = x_1 + x_2
   POL(nil) = 0

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, .(z1, z2)) -> if(=(z0, z1), z2, .(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, .(z1, z2)) -> if(<=(z0, z1), min(z0, z2), min(z1, z2))
Tuples:
   MSORT'(.(z0, z1)) -> c9(MIN'(z0, z1), MSORT'(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1))
   MIN'(z0, .(z1, z2)) -> c11(MIN'(z0, z2), MIN'(z1, z2))
   DEL'(z0, .(z1, z2)) -> c13(DEL'(z0, z2))
   MSORT''(.(z0, z1)) -> c15(MIN''(z0, z1))
   MSORT''(.(z0, z1)) -> c16(MSORT''(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1), DEL''(min(z0, z1), .(z0, z1)), MIN'(z0, z1), MIN''(z0, z1))
   MIN''(z0, .(z1, z2)) -> c18(MIN''(z0, z2))
   MIN''(z0, .(z1, z2)) -> c19(MIN''(z1, z2))
   DEL''(z0, .(z1, z2)) -> c21(DEL''(z0, z2))
S tuples:
   DEL''(z0, .(z1, z2)) -> c21(DEL''(z0, z2))
K tuples:
   MSORT''(.(z0, z1)) -> c15(MIN''(z0, z1))
   MIN''(z0, .(z1, z2)) -> c18(MIN''(z0, z2))
   MIN''(z0, .(z1, z2)) -> c19(MIN''(z1, z2))
   MSORT''(.(z0, z1)) -> c16(MSORT''(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1), DEL''(min(z0, z1), .(z0, z1)), MIN'(z0, z1), MIN''(z0, z1))
Defined Rule Symbols:   del_2, min_2

Defined Pair Symbols:   MSORT'_1, MIN'_2, DEL'_2, MSORT''_1, MIN''_2, DEL''_2

Compound Symbols:   c9_4, c11_2, c13_1, c15_1, c16_6, c18_1, c19_1, c21_1


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

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   DEL''(z0, .(z1, z2)) -> c21(DEL''(z0, z2))
We considered the (Usable) Rules:
   del(z0, .(z1, z2)) -> if(=(z0, z1), z2, .(z1, del(z0, z2)))
And the Tuples:
   MSORT'(.(z0, z1)) -> c9(MIN'(z0, z1), MSORT'(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1))
   MIN'(z0, .(z1, z2)) -> c11(MIN'(z0, z2), MIN'(z1, z2))
   DEL'(z0, .(z1, z2)) -> c13(DEL'(z0, z2))
   MSORT''(.(z0, z1)) -> c15(MIN''(z0, z1))
   MSORT''(.(z0, z1)) -> c16(MSORT''(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1), DEL''(min(z0, z1), .(z0, z1)), MIN'(z0, z1), MIN''(z0, z1))
   MIN''(z0, .(z1, z2)) -> c18(MIN''(z0, z2))
   MIN''(z0, .(z1, z2)) -> c19(MIN''(z1, z2))
   DEL''(z0, .(z1, z2)) -> c21(DEL''(z0, z2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(.(x_1, x_2)) = [1] + x_1 + x_2
   POL(<=(x_1, x_2)) = [1] + x_1 + x_2
   POL(=(x_1, x_2)) = 0
   POL(DEL'(x_1, x_2)) = 0
   POL(DEL''(x_1, x_2)) = x_2
   POL(MIN'(x_1, x_2)) = 0
   POL(MIN''(x_1, x_2)) = 0
   POL(MSORT'(x_1)) = x_1
   POL(MSORT''(x_1)) = [1] + x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c15(x_1)) = x_1
   POL(c16(x_1, x_2, x_3, x_4, x_5, x_6)) = x_1 + x_2 + x_3 + x_4 + x_5 + x_6
   POL(c18(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c21(x_1)) = x_1
   POL(c9(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(del(x_1, x_2)) = 0
   POL(if(x_1, x_2, x_3)) = x_1
   POL(min(x_1, x_2)) = [1] + x_1 + x_2
   POL(nil) = 0

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   del(z0, .(z1, z2)) -> if(=(z0, z1), z2, .(z1, del(z0, z2)))
   del(z0, nil) -> nil
   min(z0, nil) -> z0
   min(z0, .(z1, z2)) -> if(<=(z0, z1), min(z0, z2), min(z1, z2))
Tuples:
   MSORT'(.(z0, z1)) -> c9(MIN'(z0, z1), MSORT'(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1))
   MIN'(z0, .(z1, z2)) -> c11(MIN'(z0, z2), MIN'(z1, z2))
   DEL'(z0, .(z1, z2)) -> c13(DEL'(z0, z2))
   MSORT''(.(z0, z1)) -> c15(MIN''(z0, z1))
   MSORT''(.(z0, z1)) -> c16(MSORT''(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1), DEL''(min(z0, z1), .(z0, z1)), MIN'(z0, z1), MIN''(z0, z1))
   MIN''(z0, .(z1, z2)) -> c18(MIN''(z0, z2))
   MIN''(z0, .(z1, z2)) -> c19(MIN''(z1, z2))
   DEL''(z0, .(z1, z2)) -> c21(DEL''(z0, z2))
S tuples:none
K tuples:
   MSORT''(.(z0, z1)) -> c15(MIN''(z0, z1))
   MIN''(z0, .(z1, z2)) -> c18(MIN''(z0, z2))
   MIN''(z0, .(z1, z2)) -> c19(MIN''(z1, z2))
   MSORT''(.(z0, z1)) -> c16(MSORT''(del(min(z0, z1), .(z0, z1))), DEL'(min(z0, z1), .(z0, z1)), MIN'(z0, z1), DEL''(min(z0, z1), .(z0, z1)), MIN'(z0, z1), MIN''(z0, z1))
   DEL''(z0, .(z1, z2)) -> c21(DEL''(z0, z2))
Defined Rule Symbols:   del_2, min_2

Defined Pair Symbols:   MSORT'_1, MIN'_2, DEL'_2, MSORT''_1, MIN''_2, DEL''_2

Compound Symbols:   c9_4, c11_2, c13_1, c15_1, c16_6, c18_1, c19_1, c21_1


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

(15) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
----------------------------------------

(16)
BOUNDS(1, 1)

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

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

(18)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   MSORT(nil) -> c
   MSORT(.(z0, z1)) -> c1(MIN(z0, z1))
   MSORT(.(z0, z1)) -> c2(MSORT(del(min(z0, z1), .(z0, z1))), DEL(min(z0, z1), .(z0, z1)), MIN(z0, z1))
   MIN(z0, nil) -> c3
   MIN(z0, .(z1, z2)) -> c4(MIN(z0, z2))
   MIN(z0, .(z1, z2)) -> c5(MIN(z1, z2))
   DEL(z0, nil) -> c6
   DEL(z0, .(z1, z2)) -> c7(DEL(z0, z2))

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

   msort(nil) -> nil
   msort(.(z0, z1)) -> .(min(z0, z1), msort(del(min(z0, z1), .(z0, z1))))
   min(z0, nil) -> z0
   min(z0, .(z1, z2)) -> if(<=(z0, z1), min(z0, z2), min(z1, z2))
   del(z0, nil) -> nil
   del(z0, .(z1, z2)) -> if(='(z0, z1), z2, .(z1, del(z0, z2)))

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

(19) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
MIN/0
if/0
<=/0
<=/1
='/0
='/1

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

(20)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   MSORT(nil) -> c
   MSORT(.(z0, z1)) -> c1(MIN(z1))
   MSORT(.(z0, z1)) -> c2(MSORT(del(min(z0, z1), .(z0, z1))), DEL(min(z0, z1), .(z0, z1)), MIN(z1))
   MIN(nil) -> c3
   MIN(.(z1, z2)) -> c4(MIN(z2))
   MIN(.(z1, z2)) -> c5(MIN(z2))
   DEL(z0, nil) -> c6
   DEL(z0, .(z1, z2)) -> c7(DEL(z0, z2))

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

   msort(nil) -> nil
   msort(.(z0, z1)) -> .(min(z0, z1), msort(del(min(z0, z1), .(z0, z1))))
   min(z0, nil) -> z0
   min(z0, .(z1, z2)) -> if(min(z0, z2), min(z1, z2))
   del(z0, nil) -> nil
   del(z0, .(z1, z2)) -> if(z2, .(z1, del(z0, z2)))

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

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

(22)
Obligation:
Innermost TRS:
Rules:
MSORT(nil) -> c
MSORT(.(z0, z1)) -> c1(MIN(z1))
MSORT(.(z0, z1)) -> c2(MSORT(del(min(z0, z1), .(z0, z1))), DEL(min(z0, z1), .(z0, z1)), MIN(z1))
MIN(nil) -> c3
MIN(.(z1, z2)) -> c4(MIN(z2))
MIN(.(z1, z2)) -> c5(MIN(z2))
DEL(z0, nil) -> c6
DEL(z0, .(z1, z2)) -> c7(DEL(z0, z2))
msort(nil) -> nil
msort(.(z0, z1)) -> .(min(z0, z1), msort(del(min(z0, z1), .(z0, z1))))
min(z0, nil) -> z0
min(z0, .(z1, z2)) -> if(min(z0, z2), min(z1, z2))
del(z0, nil) -> nil
del(z0, .(z1, z2)) -> if(z2, .(z1, del(z0, z2)))

Types:
MSORT :: nil:.:if -> c:c1:c2
nil :: nil:.:if
c :: c:c1:c2
. :: nil:.:if -> nil:.:if -> nil:.:if
c1 :: c3:c4:c5 -> c:c1:c2
MIN :: nil:.:if -> c3:c4:c5
c2 :: c:c1:c2 -> c6:c7 -> c3:c4:c5 -> c:c1:c2
del :: nil:.:if -> nil:.:if -> nil:.:if
min :: nil:.:if -> nil:.:if -> nil:.:if
DEL :: nil:.:if -> nil:.:if -> c6:c7
c3 :: c3:c4:c5
c4 :: c3:c4:c5 -> c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
msort :: nil:.:if -> nil:.:if
if :: nil:.:if -> nil:.:if -> nil:.:if
hole_c:c1:c21_8 :: c:c1:c2
hole_nil:.:if2_8 :: nil:.:if
hole_c3:c4:c53_8 :: c3:c4:c5
hole_c6:c74_8 :: c6:c7
gen_c:c1:c25_8 :: Nat -> c:c1:c2
gen_nil:.:if6_8 :: Nat -> nil:.:if
gen_c3:c4:c57_8 :: Nat -> c3:c4:c5
gen_c6:c78_8 :: Nat -> c6:c7

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

(23) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
MSORT, MIN, del, min, DEL, msort

They will be analysed ascendingly in the following order:
MIN < MSORT
del < MSORT
min < MSORT
DEL < MSORT
del < msort
min < msort

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

(24)
Obligation:
Innermost TRS:
Rules:
MSORT(nil) -> c
MSORT(.(z0, z1)) -> c1(MIN(z1))
MSORT(.(z0, z1)) -> c2(MSORT(del(min(z0, z1), .(z0, z1))), DEL(min(z0, z1), .(z0, z1)), MIN(z1))
MIN(nil) -> c3
MIN(.(z1, z2)) -> c4(MIN(z2))
MIN(.(z1, z2)) -> c5(MIN(z2))
DEL(z0, nil) -> c6
DEL(z0, .(z1, z2)) -> c7(DEL(z0, z2))
msort(nil) -> nil
msort(.(z0, z1)) -> .(min(z0, z1), msort(del(min(z0, z1), .(z0, z1))))
min(z0, nil) -> z0
min(z0, .(z1, z2)) -> if(min(z0, z2), min(z1, z2))
del(z0, nil) -> nil
del(z0, .(z1, z2)) -> if(z2, .(z1, del(z0, z2)))

Types:
MSORT :: nil:.:if -> c:c1:c2
nil :: nil:.:if
c :: c:c1:c2
. :: nil:.:if -> nil:.:if -> nil:.:if
c1 :: c3:c4:c5 -> c:c1:c2
MIN :: nil:.:if -> c3:c4:c5
c2 :: c:c1:c2 -> c6:c7 -> c3:c4:c5 -> c:c1:c2
del :: nil:.:if -> nil:.:if -> nil:.:if
min :: nil:.:if -> nil:.:if -> nil:.:if
DEL :: nil:.:if -> nil:.:if -> c6:c7
c3 :: c3:c4:c5
c4 :: c3:c4:c5 -> c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
msort :: nil:.:if -> nil:.:if
if :: nil:.:if -> nil:.:if -> nil:.:if
hole_c:c1:c21_8 :: c:c1:c2
hole_nil:.:if2_8 :: nil:.:if
hole_c3:c4:c53_8 :: c3:c4:c5
hole_c6:c74_8 :: c6:c7
gen_c:c1:c25_8 :: Nat -> c:c1:c2
gen_nil:.:if6_8 :: Nat -> nil:.:if
gen_c3:c4:c57_8 :: Nat -> c3:c4:c5
gen_c6:c78_8 :: Nat -> c6:c7


Generator Equations:
gen_c:c1:c25_8(0) <=> c
gen_c:c1:c25_8(+(x, 1)) <=> c2(gen_c:c1:c25_8(x), c6, c3)
gen_nil:.:if6_8(0) <=> nil
gen_nil:.:if6_8(+(x, 1)) <=> .(nil, gen_nil:.:if6_8(x))
gen_c3:c4:c57_8(0) <=> c3
gen_c3:c4:c57_8(+(x, 1)) <=> c4(gen_c3:c4:c57_8(x))
gen_c6:c78_8(0) <=> c6
gen_c6:c78_8(+(x, 1)) <=> c7(gen_c6:c78_8(x))


The following defined symbols remain to be analysed:
MIN, MSORT, del, min, DEL, msort

They will be analysed ascendingly in the following order:
MIN < MSORT
del < MSORT
min < MSORT
DEL < MSORT
del < msort
min < msort

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MIN(gen_nil:.:if6_8(n10_8)) -> gen_c3:c4:c57_8(n10_8), rt in Omega(1 + n10_8)

Induction Base:
MIN(gen_nil:.:if6_8(0)) ->_R^Omega(1)
c3

Induction Step:
MIN(gen_nil:.:if6_8(+(n10_8, 1))) ->_R^Omega(1)
c4(MIN(gen_nil:.:if6_8(n10_8))) ->_IH
c4(gen_c3:c4:c57_8(c11_8))

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

(26)
Complex Obligation (BEST)

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

(27)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
MSORT(nil) -> c
MSORT(.(z0, z1)) -> c1(MIN(z1))
MSORT(.(z0, z1)) -> c2(MSORT(del(min(z0, z1), .(z0, z1))), DEL(min(z0, z1), .(z0, z1)), MIN(z1))
MIN(nil) -> c3
MIN(.(z1, z2)) -> c4(MIN(z2))
MIN(.(z1, z2)) -> c5(MIN(z2))
DEL(z0, nil) -> c6
DEL(z0, .(z1, z2)) -> c7(DEL(z0, z2))
msort(nil) -> nil
msort(.(z0, z1)) -> .(min(z0, z1), msort(del(min(z0, z1), .(z0, z1))))
min(z0, nil) -> z0
min(z0, .(z1, z2)) -> if(min(z0, z2), min(z1, z2))
del(z0, nil) -> nil
del(z0, .(z1, z2)) -> if(z2, .(z1, del(z0, z2)))

Types:
MSORT :: nil:.:if -> c:c1:c2
nil :: nil:.:if
c :: c:c1:c2
. :: nil:.:if -> nil:.:if -> nil:.:if
c1 :: c3:c4:c5 -> c:c1:c2
MIN :: nil:.:if -> c3:c4:c5
c2 :: c:c1:c2 -> c6:c7 -> c3:c4:c5 -> c:c1:c2
del :: nil:.:if -> nil:.:if -> nil:.:if
min :: nil:.:if -> nil:.:if -> nil:.:if
DEL :: nil:.:if -> nil:.:if -> c6:c7
c3 :: c3:c4:c5
c4 :: c3:c4:c5 -> c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
msort :: nil:.:if -> nil:.:if
if :: nil:.:if -> nil:.:if -> nil:.:if
hole_c:c1:c21_8 :: c:c1:c2
hole_nil:.:if2_8 :: nil:.:if
hole_c3:c4:c53_8 :: c3:c4:c5
hole_c6:c74_8 :: c6:c7
gen_c:c1:c25_8 :: Nat -> c:c1:c2
gen_nil:.:if6_8 :: Nat -> nil:.:if
gen_c3:c4:c57_8 :: Nat -> c3:c4:c5
gen_c6:c78_8 :: Nat -> c6:c7


Generator Equations:
gen_c:c1:c25_8(0) <=> c
gen_c:c1:c25_8(+(x, 1)) <=> c2(gen_c:c1:c25_8(x), c6, c3)
gen_nil:.:if6_8(0) <=> nil
gen_nil:.:if6_8(+(x, 1)) <=> .(nil, gen_nil:.:if6_8(x))
gen_c3:c4:c57_8(0) <=> c3
gen_c3:c4:c57_8(+(x, 1)) <=> c4(gen_c3:c4:c57_8(x))
gen_c6:c78_8(0) <=> c6
gen_c6:c78_8(+(x, 1)) <=> c7(gen_c6:c78_8(x))


The following defined symbols remain to be analysed:
MIN, MSORT, del, min, DEL, msort

They will be analysed ascendingly in the following order:
MIN < MSORT
del < MSORT
min < MSORT
DEL < MSORT
del < msort
min < msort

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

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

(29)
BOUNDS(n^1, INF)

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

(30)
Obligation:
Innermost TRS:
Rules:
MSORT(nil) -> c
MSORT(.(z0, z1)) -> c1(MIN(z1))
MSORT(.(z0, z1)) -> c2(MSORT(del(min(z0, z1), .(z0, z1))), DEL(min(z0, z1), .(z0, z1)), MIN(z1))
MIN(nil) -> c3
MIN(.(z1, z2)) -> c4(MIN(z2))
MIN(.(z1, z2)) -> c5(MIN(z2))
DEL(z0, nil) -> c6
DEL(z0, .(z1, z2)) -> c7(DEL(z0, z2))
msort(nil) -> nil
msort(.(z0, z1)) -> .(min(z0, z1), msort(del(min(z0, z1), .(z0, z1))))
min(z0, nil) -> z0
min(z0, .(z1, z2)) -> if(min(z0, z2), min(z1, z2))
del(z0, nil) -> nil
del(z0, .(z1, z2)) -> if(z2, .(z1, del(z0, z2)))

Types:
MSORT :: nil:.:if -> c:c1:c2
nil :: nil:.:if
c :: c:c1:c2
. :: nil:.:if -> nil:.:if -> nil:.:if
c1 :: c3:c4:c5 -> c:c1:c2
MIN :: nil:.:if -> c3:c4:c5
c2 :: c:c1:c2 -> c6:c7 -> c3:c4:c5 -> c:c1:c2
del :: nil:.:if -> nil:.:if -> nil:.:if
min :: nil:.:if -> nil:.:if -> nil:.:if
DEL :: nil:.:if -> nil:.:if -> c6:c7
c3 :: c3:c4:c5
c4 :: c3:c4:c5 -> c3:c4:c5
c5 :: c3:c4:c5 -> c3:c4:c5
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
msort :: nil:.:if -> nil:.:if
if :: nil:.:if -> nil:.:if -> nil:.:if
hole_c:c1:c21_8 :: c:c1:c2
hole_nil:.:if2_8 :: nil:.:if
hole_c3:c4:c53_8 :: c3:c4:c5
hole_c6:c74_8 :: c6:c7
gen_c:c1:c25_8 :: Nat -> c:c1:c2
gen_nil:.:if6_8 :: Nat -> nil:.:if
gen_c3:c4:c57_8 :: Nat -> c3:c4:c5
gen_c6:c78_8 :: Nat -> c6:c7


Lemmas:
MIN(gen_nil:.:if6_8(n10_8)) -> gen_c3:c4:c57_8(n10_8), rt in Omega(1 + n10_8)


Generator Equations:
gen_c:c1:c25_8(0) <=> c
gen_c:c1:c25_8(+(x, 1)) <=> c2(gen_c:c1:c25_8(x), c6, c3)
gen_nil:.:if6_8(0) <=> nil
gen_nil:.:if6_8(+(x, 1)) <=> .(nil, gen_nil:.:if6_8(x))
gen_c3:c4:c57_8(0) <=> c3
gen_c3:c4:c57_8(+(x, 1)) <=> c4(gen_c3:c4:c57_8(x))
gen_c6:c78_8(0) <=> c6
gen_c6:c78_8(+(x, 1)) <=> c7(gen_c6:c78_8(x))


The following defined symbols remain to be analysed:
del, MSORT, min, DEL, msort

They will be analysed ascendingly in the following order:
del < MSORT
min < MSORT
DEL < MSORT
del < msort
min < msort