WORST_CASE(?,O(n^1))
proof of input_ixvOy1j2sN.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(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) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 107 ms]
(8) CdtProblem
(9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 25 ms]
(10) CdtProblem
(11) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) BOUNDS(1, 1)


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


The TRS R consists of the following rules:

   qsort(nil) -> nil
   qsort(.(x, y)) -> ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
   lowers(x, nil) -> nil
   lowers(x, .(y, z)) -> if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
   greaters(x, nil) -> nil
   greaters(x, .(y, z)) -> if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
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(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   qsort(nil) -> nil
   qsort(.(z0, z1)) -> ++(qsort(lowers(z0, z1)), .(z0, qsort(greaters(z0, z1))))
   lowers(z0, nil) -> nil
   lowers(z0, .(z1, z2)) -> if(<=(z1, z0), .(z1, lowers(z0, z2)), lowers(z0, z2))
   greaters(z0, nil) -> nil
   greaters(z0, .(z1, z2)) -> if(<=(z1, z0), greaters(z0, z2), .(z1, greaters(z0, z2)))
Tuples:
   QSORT(nil) -> c
   QSORT(.(z0, z1)) -> c1(QSORT(lowers(z0, z1)), LOWERS(z0, z1))
   QSORT(.(z0, z1)) -> c2(QSORT(greaters(z0, z1)), GREATERS(z0, z1))
   LOWERS(z0, nil) -> c3
   LOWERS(z0, .(z1, z2)) -> c4(LOWERS(z0, z2))
   LOWERS(z0, .(z1, z2)) -> c5(LOWERS(z0, z2))
   GREATERS(z0, nil) -> c6
   GREATERS(z0, .(z1, z2)) -> c7(GREATERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c8(GREATERS(z0, z2))
S tuples:
   QSORT(nil) -> c
   QSORT(.(z0, z1)) -> c1(QSORT(lowers(z0, z1)), LOWERS(z0, z1))
   QSORT(.(z0, z1)) -> c2(QSORT(greaters(z0, z1)), GREATERS(z0, z1))
   LOWERS(z0, nil) -> c3
   LOWERS(z0, .(z1, z2)) -> c4(LOWERS(z0, z2))
   LOWERS(z0, .(z1, z2)) -> c5(LOWERS(z0, z2))
   GREATERS(z0, nil) -> c6
   GREATERS(z0, .(z1, z2)) -> c7(GREATERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c8(GREATERS(z0, z2))
K tuples:none
Defined Rule Symbols:   qsort_1, lowers_2, greaters_2

Defined Pair Symbols:   QSORT_1, LOWERS_2, GREATERS_2

Compound Symbols:   c, c1_2, c2_2, c3, c4_1, c5_1, c6, c7_1, c8_1


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(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   GREATERS(z0, nil) -> c6
   LOWERS(z0, nil) -> c3
   QSORT(nil) -> c

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   qsort(nil) -> nil
   qsort(.(z0, z1)) -> ++(qsort(lowers(z0, z1)), .(z0, qsort(greaters(z0, z1))))
   lowers(z0, nil) -> nil
   lowers(z0, .(z1, z2)) -> if(<=(z1, z0), .(z1, lowers(z0, z2)), lowers(z0, z2))
   greaters(z0, nil) -> nil
   greaters(z0, .(z1, z2)) -> if(<=(z1, z0), greaters(z0, z2), .(z1, greaters(z0, z2)))
Tuples:
   QSORT(.(z0, z1)) -> c1(QSORT(lowers(z0, z1)), LOWERS(z0, z1))
   QSORT(.(z0, z1)) -> c2(QSORT(greaters(z0, z1)), GREATERS(z0, z1))
   LOWERS(z0, .(z1, z2)) -> c4(LOWERS(z0, z2))
   LOWERS(z0, .(z1, z2)) -> c5(LOWERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c7(GREATERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c8(GREATERS(z0, z2))
S tuples:
   QSORT(.(z0, z1)) -> c1(QSORT(lowers(z0, z1)), LOWERS(z0, z1))
   QSORT(.(z0, z1)) -> c2(QSORT(greaters(z0, z1)), GREATERS(z0, z1))
   LOWERS(z0, .(z1, z2)) -> c4(LOWERS(z0, z2))
   LOWERS(z0, .(z1, z2)) -> c5(LOWERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c7(GREATERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c8(GREATERS(z0, z2))
K tuples:none
Defined Rule Symbols:   qsort_1, lowers_2, greaters_2

Defined Pair Symbols:   QSORT_1, LOWERS_2, GREATERS_2

Compound Symbols:   c1_2, c2_2, c4_1, c5_1, c7_1, c8_1


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(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   qsort(nil) -> nil
   qsort(.(z0, z1)) -> ++(qsort(lowers(z0, z1)), .(z0, qsort(greaters(z0, z1))))

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lowers(z0, nil) -> nil
   lowers(z0, .(z1, z2)) -> if(<=(z1, z0), .(z1, lowers(z0, z2)), lowers(z0, z2))
   greaters(z0, nil) -> nil
   greaters(z0, .(z1, z2)) -> if(<=(z1, z0), greaters(z0, z2), .(z1, greaters(z0, z2)))
Tuples:
   QSORT(.(z0, z1)) -> c1(QSORT(lowers(z0, z1)), LOWERS(z0, z1))
   QSORT(.(z0, z1)) -> c2(QSORT(greaters(z0, z1)), GREATERS(z0, z1))
   LOWERS(z0, .(z1, z2)) -> c4(LOWERS(z0, z2))
   LOWERS(z0, .(z1, z2)) -> c5(LOWERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c7(GREATERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c8(GREATERS(z0, z2))
S tuples:
   QSORT(.(z0, z1)) -> c1(QSORT(lowers(z0, z1)), LOWERS(z0, z1))
   QSORT(.(z0, z1)) -> c2(QSORT(greaters(z0, z1)), GREATERS(z0, z1))
   LOWERS(z0, .(z1, z2)) -> c4(LOWERS(z0, z2))
   LOWERS(z0, .(z1, z2)) -> c5(LOWERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c7(GREATERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c8(GREATERS(z0, z2))
K tuples:none
Defined Rule Symbols:   lowers_2, greaters_2

Defined Pair Symbols:   QSORT_1, LOWERS_2, GREATERS_2

Compound Symbols:   c1_2, c2_2, c4_1, c5_1, c7_1, c8_1


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(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   QSORT(.(z0, z1)) -> c1(QSORT(lowers(z0, z1)), LOWERS(z0, z1))
   QSORT(.(z0, z1)) -> c2(QSORT(greaters(z0, z1)), GREATERS(z0, z1))
We considered the (Usable) Rules:
   greaters(z0, nil) -> nil
   lowers(z0, .(z1, z2)) -> if(<=(z1, z0), .(z1, lowers(z0, z2)), lowers(z0, z2))
   greaters(z0, .(z1, z2)) -> if(<=(z1, z0), greaters(z0, z2), .(z1, greaters(z0, z2)))
   lowers(z0, nil) -> nil
And the Tuples:
   QSORT(.(z0, z1)) -> c1(QSORT(lowers(z0, z1)), LOWERS(z0, z1))
   QSORT(.(z0, z1)) -> c2(QSORT(greaters(z0, z1)), GREATERS(z0, z1))
   LOWERS(z0, .(z1, z2)) -> c4(LOWERS(z0, z2))
   LOWERS(z0, .(z1, z2)) -> c5(LOWERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c7(GREATERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c8(GREATERS(z0, z2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(.(x_1, x_2)) = [1] + x_1
   POL(<=(x_1, x_2)) = x_2
   POL(GREATERS(x_1, x_2)) = 0
   POL(LOWERS(x_1, x_2)) = 0
   POL(QSORT(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(greaters(x_1, x_2)) = x_1
   POL(if(x_1, x_2, x_3)) = x_1
   POL(lowers(x_1, x_2)) = x_1
   POL(nil) = 0

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(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lowers(z0, nil) -> nil
   lowers(z0, .(z1, z2)) -> if(<=(z1, z0), .(z1, lowers(z0, z2)), lowers(z0, z2))
   greaters(z0, nil) -> nil
   greaters(z0, .(z1, z2)) -> if(<=(z1, z0), greaters(z0, z2), .(z1, greaters(z0, z2)))
Tuples:
   QSORT(.(z0, z1)) -> c1(QSORT(lowers(z0, z1)), LOWERS(z0, z1))
   QSORT(.(z0, z1)) -> c2(QSORT(greaters(z0, z1)), GREATERS(z0, z1))
   LOWERS(z0, .(z1, z2)) -> c4(LOWERS(z0, z2))
   LOWERS(z0, .(z1, z2)) -> c5(LOWERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c7(GREATERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c8(GREATERS(z0, z2))
S tuples:
   LOWERS(z0, .(z1, z2)) -> c4(LOWERS(z0, z2))
   LOWERS(z0, .(z1, z2)) -> c5(LOWERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c7(GREATERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c8(GREATERS(z0, z2))
K tuples:
   QSORT(.(z0, z1)) -> c1(QSORT(lowers(z0, z1)), LOWERS(z0, z1))
   QSORT(.(z0, z1)) -> c2(QSORT(greaters(z0, z1)), GREATERS(z0, z1))
Defined Rule Symbols:   lowers_2, greaters_2

Defined Pair Symbols:   QSORT_1, LOWERS_2, GREATERS_2

Compound Symbols:   c1_2, c2_2, c4_1, c5_1, c7_1, c8_1


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(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LOWERS(z0, .(z1, z2)) -> c4(LOWERS(z0, z2))
   LOWERS(z0, .(z1, z2)) -> c5(LOWERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c7(GREATERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c8(GREATERS(z0, z2))
We considered the (Usable) Rules:
   greaters(z0, nil) -> nil
   lowers(z0, .(z1, z2)) -> if(<=(z1, z0), .(z1, lowers(z0, z2)), lowers(z0, z2))
   greaters(z0, .(z1, z2)) -> if(<=(z1, z0), greaters(z0, z2), .(z1, greaters(z0, z2)))
   lowers(z0, nil) -> nil
And the Tuples:
   QSORT(.(z0, z1)) -> c1(QSORT(lowers(z0, z1)), LOWERS(z0, z1))
   QSORT(.(z0, z1)) -> c2(QSORT(greaters(z0, z1)), GREATERS(z0, z1))
   LOWERS(z0, .(z1, z2)) -> c4(LOWERS(z0, z2))
   LOWERS(z0, .(z1, z2)) -> c5(LOWERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c7(GREATERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c8(GREATERS(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_2
   POL(GREATERS(x_1, x_2)) = [1] + x_2
   POL(LOWERS(x_1, x_2)) = [1] + x_2
   POL(QSORT(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(greaters(x_1, x_2)) = x_1
   POL(if(x_1, x_2, x_3)) = x_1
   POL(lowers(x_1, x_2)) = x_1
   POL(nil) = 0

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(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lowers(z0, nil) -> nil
   lowers(z0, .(z1, z2)) -> if(<=(z1, z0), .(z1, lowers(z0, z2)), lowers(z0, z2))
   greaters(z0, nil) -> nil
   greaters(z0, .(z1, z2)) -> if(<=(z1, z0), greaters(z0, z2), .(z1, greaters(z0, z2)))
Tuples:
   QSORT(.(z0, z1)) -> c1(QSORT(lowers(z0, z1)), LOWERS(z0, z1))
   QSORT(.(z0, z1)) -> c2(QSORT(greaters(z0, z1)), GREATERS(z0, z1))
   LOWERS(z0, .(z1, z2)) -> c4(LOWERS(z0, z2))
   LOWERS(z0, .(z1, z2)) -> c5(LOWERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c7(GREATERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c8(GREATERS(z0, z2))
S tuples:none
K tuples:
   QSORT(.(z0, z1)) -> c1(QSORT(lowers(z0, z1)), LOWERS(z0, z1))
   QSORT(.(z0, z1)) -> c2(QSORT(greaters(z0, z1)), GREATERS(z0, z1))
   LOWERS(z0, .(z1, z2)) -> c4(LOWERS(z0, z2))
   LOWERS(z0, .(z1, z2)) -> c5(LOWERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c7(GREATERS(z0, z2))
   GREATERS(z0, .(z1, z2)) -> c8(GREATERS(z0, z2))
Defined Rule Symbols:   lowers_2, greaters_2

Defined Pair Symbols:   QSORT_1, LOWERS_2, GREATERS_2

Compound Symbols:   c1_2, c2_2, c4_1, c5_1, c7_1, c8_1


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(11) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(12)
BOUNDS(1, 1)