/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
    (6) CdtProblem
    (7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 38 ms]
    (12) CdtProblem
    (13) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) BOUNDS(1, 1)
(15) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(16) TRS for Loop Detection
    (17) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) TRS for Loop Detection


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


The TRS R consists of the following rules:

   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   ++(x, nil) -> x
   ++(x, g(y, z)) -> g(++(x, y), z)
   null(nil) -> true
   null(g(x, y)) -> false
   mem(nil, y) -> false
   mem(g(x, y), z) -> or(=(y, z), mem(x, z))
   mem(x, max(x)) -> not(null(x))
   max(g(g(nil, x), y)) -> max'(x, y)
   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
mem(x, max(x)) -> not(null(x))

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

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


The TRS R consists of the following rules:

   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   ++(x, nil) -> x
   ++(x, g(y, z)) -> g(++(x, y), z)
   null(nil) -> true
   null(g(x, y)) -> false
   mem(nil, y) -> false
   mem(g(x, y), z) -> or(=(y, z), mem(x, z))
   max(g(g(nil, x), y)) -> max'(x, y)
   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

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

(3) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, we have rc = irc.
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   ++(x, nil) -> x
   ++(x, g(y, z)) -> g(++(x, y), z)
   null(nil) -> true
   null(g(x, y)) -> false
   mem(nil, y) -> false
   mem(g(x, y), z) -> or(=(y, z), mem(x, z))
   max(g(g(nil, x), y)) -> max'(x, y)
   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

S is empty.
Rewrite Strategy: INNERMOST
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(5) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0, nil) -> g(nil, z0)
   f(z0, g(z1, z2)) -> g(f(z0, z1), z2)
   ++(z0, nil) -> z0
   ++(z0, g(z1, z2)) -> g(++(z0, z1), z2)
   null(nil) -> true
   null(g(z0, z1)) -> false
   mem(nil, z0) -> false
   mem(g(z0, z1), z2) -> or(=(z1, z2), mem(z0, z2))
   max(g(g(nil, z0), z1)) -> max'(z0, z1)
   max(g(g(g(z0, z1), z2), u)) -> max'(max(g(g(z0, z1), z2)), u)
Tuples:
   F(z0, nil) -> c
   F(z0, g(z1, z2)) -> c1(F(z0, z1))
   ++'(z0, nil) -> c2
   ++'(z0, g(z1, z2)) -> c3(++'(z0, z1))
   NULL(nil) -> c4
   NULL(g(z0, z1)) -> c5
   MEM(nil, z0) -> c6
   MEM(g(z0, z1), z2) -> c7(MEM(z0, z2))
   MAX(g(g(nil, z0), z1)) -> c8
   MAX(g(g(g(z0, z1), z2), u)) -> c9(MAX(g(g(z0, z1), z2)))
S tuples:
   F(z0, nil) -> c
   F(z0, g(z1, z2)) -> c1(F(z0, z1))
   ++'(z0, nil) -> c2
   ++'(z0, g(z1, z2)) -> c3(++'(z0, z1))
   NULL(nil) -> c4
   NULL(g(z0, z1)) -> c5
   MEM(nil, z0) -> c6
   MEM(g(z0, z1), z2) -> c7(MEM(z0, z2))
   MAX(g(g(nil, z0), z1)) -> c8
   MAX(g(g(g(z0, z1), z2), u)) -> c9(MAX(g(g(z0, z1), z2)))
K tuples:none
Defined Rule Symbols:   f_2, ++_2, null_1, mem_2, max_1

Defined Pair Symbols:   F_2, ++'_2, NULL_1, MEM_2, MAX_1

Compound Symbols:   c, c1_1, c2, c3_1, c4, c5, c6, c7_1, c8, c9_1


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(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   MAX(g(g(nil, z0), z1)) -> c8
   MEM(nil, z0) -> c6
   NULL(g(z0, z1)) -> c5
   ++'(z0, nil) -> c2
   F(z0, nil) -> c
   NULL(nil) -> c4

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0, nil) -> g(nil, z0)
   f(z0, g(z1, z2)) -> g(f(z0, z1), z2)
   ++(z0, nil) -> z0
   ++(z0, g(z1, z2)) -> g(++(z0, z1), z2)
   null(nil) -> true
   null(g(z0, z1)) -> false
   mem(nil, z0) -> false
   mem(g(z0, z1), z2) -> or(=(z1, z2), mem(z0, z2))
   max(g(g(nil, z0), z1)) -> max'(z0, z1)
   max(g(g(g(z0, z1), z2), u)) -> max'(max(g(g(z0, z1), z2)), u)
Tuples:
   F(z0, g(z1, z2)) -> c1(F(z0, z1))
   ++'(z0, g(z1, z2)) -> c3(++'(z0, z1))
   MEM(g(z0, z1), z2) -> c7(MEM(z0, z2))
   MAX(g(g(g(z0, z1), z2), u)) -> c9(MAX(g(g(z0, z1), z2)))
S tuples:
   F(z0, g(z1, z2)) -> c1(F(z0, z1))
   ++'(z0, g(z1, z2)) -> c3(++'(z0, z1))
   MEM(g(z0, z1), z2) -> c7(MEM(z0, z2))
   MAX(g(g(g(z0, z1), z2), u)) -> c9(MAX(g(g(z0, z1), z2)))
K tuples:none
Defined Rule Symbols:   f_2, ++_2, null_1, mem_2, max_1

Defined Pair Symbols:   F_2, ++'_2, MEM_2, MAX_1

Compound Symbols:   c1_1, c3_1, c7_1, c9_1


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(9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(z0, nil) -> g(nil, z0)
   f(z0, g(z1, z2)) -> g(f(z0, z1), z2)
   ++(z0, nil) -> z0
   ++(z0, g(z1, z2)) -> g(++(z0, z1), z2)
   null(nil) -> true
   null(g(z0, z1)) -> false
   mem(nil, z0) -> false
   mem(g(z0, z1), z2) -> or(=(z1, z2), mem(z0, z2))
   max(g(g(nil, z0), z1)) -> max'(z0, z1)
   max(g(g(g(z0, z1), z2), u)) -> max'(max(g(g(z0, z1), z2)), u)

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(z0, g(z1, z2)) -> c1(F(z0, z1))
   ++'(z0, g(z1, z2)) -> c3(++'(z0, z1))
   MEM(g(z0, z1), z2) -> c7(MEM(z0, z2))
   MAX(g(g(g(z0, z1), z2), u)) -> c9(MAX(g(g(z0, z1), z2)))
S tuples:
   F(z0, g(z1, z2)) -> c1(F(z0, z1))
   ++'(z0, g(z1, z2)) -> c3(++'(z0, z1))
   MEM(g(z0, z1), z2) -> c7(MEM(z0, z2))
   MAX(g(g(g(z0, z1), z2), u)) -> c9(MAX(g(g(z0, z1), z2)))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_2, ++'_2, MEM_2, MAX_1

Compound Symbols:   c1_1, c3_1, c7_1, c9_1


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(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(z0, g(z1, z2)) -> c1(F(z0, z1))
   ++'(z0, g(z1, z2)) -> c3(++'(z0, z1))
   MEM(g(z0, z1), z2) -> c7(MEM(z0, z2))
   MAX(g(g(g(z0, z1), z2), u)) -> c9(MAX(g(g(z0, z1), z2)))
We considered the (Usable) Rules:none
And the Tuples:
   F(z0, g(z1, z2)) -> c1(F(z0, z1))
   ++'(z0, g(z1, z2)) -> c3(++'(z0, z1))
   MEM(g(z0, z1), z2) -> c7(MEM(z0, z2))
   MAX(g(g(g(z0, z1), z2), u)) -> c9(MAX(g(g(z0, z1), z2)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(++'(x_1, x_2)) = x_2
   POL(F(x_1, x_2)) = x_2
   POL(MAX(x_1)) = x_1
   POL(MEM(x_1, x_2)) = x_1
   POL(c1(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(g(x_1, x_2)) = [1] + x_1 + x_2
   POL(u) = [1]

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(z0, g(z1, z2)) -> c1(F(z0, z1))
   ++'(z0, g(z1, z2)) -> c3(++'(z0, z1))
   MEM(g(z0, z1), z2) -> c7(MEM(z0, z2))
   MAX(g(g(g(z0, z1), z2), u)) -> c9(MAX(g(g(z0, z1), z2)))
S tuples:none
K tuples:
   F(z0, g(z1, z2)) -> c1(F(z0, z1))
   ++'(z0, g(z1, z2)) -> c3(++'(z0, z1))
   MEM(g(z0, z1), z2) -> c7(MEM(z0, z2))
   MAX(g(g(g(z0, z1), z2), u)) -> c9(MAX(g(g(z0, z1), z2)))
Defined Rule Symbols:none

Defined Pair Symbols:   F_2, ++'_2, MEM_2, MAX_1

Compound Symbols:   c1_1, c3_1, c7_1, c9_1


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(13) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(14)
BOUNDS(1, 1)

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(15) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(16)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   ++(x, nil) -> x
   ++(x, g(y, z)) -> g(++(x, y), z)
   null(nil) -> true
   null(g(x, y)) -> false
   mem(nil, y) -> false
   mem(g(x, y), z) -> or(=(y, z), mem(x, z))
   mem(x, max(x)) -> not(null(x))
   max(g(g(nil, x), y)) -> max'(x, y)
   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

S is empty.
Rewrite Strategy: FULL
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(17) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

++(x, g(y, z)) ->^+ g(++(x, y), z)

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [y / g(y, z)].

The result substitution is [ ].




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(18)
Complex Obligation (BEST)

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(19)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   ++(x, nil) -> x
   ++(x, g(y, z)) -> g(++(x, y), z)
   null(nil) -> true
   null(g(x, y)) -> false
   mem(nil, y) -> false
   mem(g(x, y), z) -> or(=(y, z), mem(x, z))
   mem(x, max(x)) -> not(null(x))
   max(g(g(nil, x), y)) -> max'(x, y)
   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

S is empty.
Rewrite Strategy: FULL
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(20) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(21)
BOUNDS(n^1, INF)

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(22)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   ++(x, nil) -> x
   ++(x, g(y, z)) -> g(++(x, y), z)
   null(nil) -> true
   null(g(x, y)) -> false
   mem(nil, y) -> false
   mem(g(x, y), z) -> or(=(y, z), mem(x, z))
   mem(x, max(x)) -> not(null(x))
   max(g(g(nil, x), y)) -> max'(x, y)
   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

S is empty.
Rewrite Strategy: FULL