/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^5))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^5).

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 3157 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 3415 ms]
    (22) CpxRNTS
    (23) FinalProof [FINISHED, 0 ms]
    (24) BOUNDS(1, n^5)
(25) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(26) TRS for Loop Detection
    (27) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (28) BEST
        (29) proven lower bound
            (30) LowerBoundPropagationProof [FINISHED, 0 ms]
            (31) BOUNDS(n^1, INF)
        (32) TRS for Loop Detection


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^5).


The TRS R consists of the following rules:

   f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5)
   f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5)
   f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5)
   f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5)
   f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5)
   f(0, 0, 0, 0, 0) -> 0

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

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

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

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^5).


The TRS R consists of the following rules:

   f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5)
   f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5)
   f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5)
   f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5)
   f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5)
   f(0, 0, 0, 0, 0) -> 0

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

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^5).


The TRS R consists of the following rules:

   f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5) [1]
   f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5) [1]
   f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5) [1]
   f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5) [1]
   f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5) [1]
   f(0, 0, 0, 0, 0) -> 0 [1]

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

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

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5) [1]
   f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5) [1]
   f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5) [1]
   f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5) [1]
   f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5) [1]
   f(0, 0, 0, 0, 0) -> 0 [1]

The TRS has the following type information:
f :: s:0 -> s:0 -> s:0 -> s:0 -> s:0 -> s:0
s :: s:0 -> s:0
0 :: s:0

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

(7) 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:

   f_5

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants: none

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

(8)
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:

   f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5) [1]
   f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5) [1]
   f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5) [1]
   f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5) [1]
   f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5) [1]
   f(0, 0, 0, 0, 0) -> 0 [1]

The TRS has the following type information:
f :: s:0 -> s:0 -> s:0 -> s:0 -> s:0 -> s:0
s :: s:0 -> s:0
0 :: s:0

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

(10)
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:

   f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5) [1]
   f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5) [1]
   f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5) [1]
   f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5) [1]
   f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5) [1]
   f(0, 0, 0, 0, 0) -> 0 [1]

The TRS has the following type information:
f :: s:0 -> s:0 -> s:0 -> s:0 -> s:0 -> s:0
s :: s:0 -> s:0
0 :: s:0

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

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

   0 => 0

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z', z'', z1, z2) -{ 1 }-> f(x1, x2, x3, x4, x5) :|: z' = x2, z1 = x4, x1 >= 0, x4 >= 0, x5 >= 0, z = 1 + x1, z'' = x3, z2 = x5, x2 >= 0, x3 >= 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(x2, x2, x3, x4, x5) :|: z1 = x4, x4 >= 0, x5 >= 0, z' = 1 + x2, z'' = x3, z = 0, z2 = x5, x2 >= 0, x3 >= 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(x3, x3, x3, x4, x5) :|: z1 = x4, x4 >= 0, x5 >= 0, z'' = 1 + x3, z = 0, z2 = x5, x3 >= 0, z' = 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(x4, x4, x4, x4, x5) :|: z'' = 0, x4 >= 0, x5 >= 0, z1 = 1 + x4, z = 0, z2 = x5, z' = 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(x5, x5, x5, x5, x5) :|: z'' = 0, z1 = 0, x5 >= 0, z = 0, z2 = 1 + x5, z' = 0
   f(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z1 = 0, z2 = 0, z = 0, z' = 0


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

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

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z', z'', z1, z2) -{ 1 }-> f(z - 1, z', z'', z1, z2) :|: z - 1 >= 0, z1 >= 0, z2 >= 0, z' >= 0, z'' >= 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(z' - 1, z' - 1, z'', z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z' - 1 >= 0, z'' >= 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(z'' - 1, z'' - 1, z'' - 1, z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z'' - 1 >= 0, z' = 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(z1 - 1, z1 - 1, z1 - 1, z1 - 1, z2) :|: z'' = 0, z1 - 1 >= 0, z2 >= 0, z = 0, z' = 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(z2 - 1, z2 - 1, z2 - 1, z2 - 1, z2 - 1) :|: z'' = 0, z1 = 0, z2 - 1 >= 0, z = 0, z' = 0
   f(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z1 = 0, z2 = 0, z = 0, z' = 0


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

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

   { f }

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

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z', z'', z1, z2) -{ 1 }-> f(z - 1, z', z'', z1, z2) :|: z - 1 >= 0, z1 >= 0, z2 >= 0, z' >= 0, z'' >= 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(z' - 1, z' - 1, z'', z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z' - 1 >= 0, z'' >= 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(z'' - 1, z'' - 1, z'' - 1, z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z'' - 1 >= 0, z' = 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(z1 - 1, z1 - 1, z1 - 1, z1 - 1, z2) :|: z'' = 0, z1 - 1 >= 0, z2 >= 0, z = 0, z' = 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(z2 - 1, z2 - 1, z2 - 1, z2 - 1, z2 - 1) :|: z'' = 0, z1 = 0, z2 - 1 >= 0, z = 0, z' = 0
   f(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z1 = 0, z2 = 0, z = 0, z' = 0

Function symbols to be analyzed: {f}

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

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

   f(z, z', z'', z1, z2) -{ 1 }-> f(z - 1, z', z'', z1, z2) :|: z - 1 >= 0, z1 >= 0, z2 >= 0, z' >= 0, z'' >= 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(z' - 1, z' - 1, z'', z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z' - 1 >= 0, z'' >= 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(z'' - 1, z'' - 1, z'' - 1, z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z'' - 1 >= 0, z' = 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(z1 - 1, z1 - 1, z1 - 1, z1 - 1, z2) :|: z'' = 0, z1 - 1 >= 0, z2 >= 0, z = 0, z' = 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(z2 - 1, z2 - 1, z2 - 1, z2 - 1, z2 - 1) :|: z'' = 0, z1 = 0, z2 - 1 >= 0, z = 0, z' = 0
   f(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z1 = 0, z2 = 0, z = 0, z' = 0

Function symbols to be analyzed: {f}

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(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z, z', z'', z1, z2) -{ 1 }-> f(z - 1, z', z'', z1, z2) :|: z - 1 >= 0, z1 >= 0, z2 >= 0, z' >= 0, z'' >= 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(z' - 1, z' - 1, z'', z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z' - 1 >= 0, z'' >= 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(z'' - 1, z'' - 1, z'' - 1, z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z'' - 1 >= 0, z' = 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(z1 - 1, z1 - 1, z1 - 1, z1 - 1, z2) :|: z'' = 0, z1 - 1 >= 0, z2 >= 0, z = 0, z' = 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(z2 - 1, z2 - 1, z2 - 1, z2 - 1, z2 - 1) :|: z'' = 0, z1 = 0, z2 - 1 >= 0, z = 0, z' = 0
   f(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z1 = 0, z2 = 0, z = 0, z' = 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  f: runtime: ?, size: O(1) [0]

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^5) with polynomial bound: 1 + z + z' + z'*z'' + z'*z''*z1 + z'*z''*z1*z2 + z'*z''*z1*z2^2 + z'*z''*z1^2 + z'*z''*z2 + z'*z''*z2^2 + z'*z''*z2^3 + z'*z''^2 + z'*z1 + z'*z1*z2 + 3*z'*z1*z2^2 + 2*z'*z1*z2^3 + z'*z1^2 + 2*z'*z1^2*z2 + z'*z1^2*z2^2 + z'*z1^3 + z'*z2 + z'*z2^2 + 2*z'*z2^3 + z'*z2^4 + z'^2 + z'' + 2*z''*z1 + 5*z''*z1*z2 + 7*z''*z1*z2^2 + 4*z''*z1*z2^3 + 4*z''*z1^2 + 4*z''*z1^2*z2 + 2*z''*z1^2*z2^2 + 2*z''*z1^3 + 2*z''*z2 + 4*z''*z2^2 + 5*z''*z2^3 + 2*z''*z2^4 + 2*z''^2 + 2*z''^2*z1 + z''^2*z1*z2 + z''^2*z1*z2^2 + z''^2*z1^2 + 2*z''^2*z2 + z''^2*z2^2 + z''^2*z2^3 + z''^3 + z1 + 3*z1*z2 + 9*z1*z2^2 + 9*z1*z2^3 + 3*z1*z2^4 + 3*z1^2 + 6*z1^2*z2 + 7*z1^2*z2^2 + 3*z1^2*z2^3 + 3*z1^3 + 3*z1^3*z2 + z1^3*z2^2 + z1^4 + z2 + 4*z2^2 + 6*z2^3 + 4*z2^4 + z2^5

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

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

   f(z, z', z'', z1, z2) -{ 1 }-> f(z - 1, z', z'', z1, z2) :|: z - 1 >= 0, z1 >= 0, z2 >= 0, z' >= 0, z'' >= 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(z' - 1, z' - 1, z'', z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z' - 1 >= 0, z'' >= 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(z'' - 1, z'' - 1, z'' - 1, z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z'' - 1 >= 0, z' = 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(z1 - 1, z1 - 1, z1 - 1, z1 - 1, z2) :|: z'' = 0, z1 - 1 >= 0, z2 >= 0, z = 0, z' = 0
   f(z, z', z'', z1, z2) -{ 1 }-> f(z2 - 1, z2 - 1, z2 - 1, z2 - 1, z2 - 1) :|: z'' = 0, z1 = 0, z2 - 1 >= 0, z = 0, z' = 0
   f(z, z', z'', z1, z2) -{ 1 }-> 0 :|: z'' = 0, z1 = 0, z2 = 0, z = 0, z' = 0

Function symbols to be analyzed: 
Previous analysis results are:
  f: runtime: O(n^5) [1 + z + z' + z'*z'' + z'*z''*z1 + z'*z''*z1*z2 + z'*z''*z1*z2^2 + z'*z''*z1^2 + z'*z''*z2 + z'*z''*z2^2 + z'*z''*z2^3 + z'*z''^2 + z'*z1 + z'*z1*z2 + 3*z'*z1*z2^2 + 2*z'*z1*z2^3 + z'*z1^2 + 2*z'*z1^2*z2 + z'*z1^2*z2^2 + z'*z1^3 + z'*z2 + z'*z2^2 + 2*z'*z2^3 + z'*z2^4 + z'^2 + z'' + 2*z''*z1 + 5*z''*z1*z2 + 7*z''*z1*z2^2 + 4*z''*z1*z2^3 + 4*z''*z1^2 + 4*z''*z1^2*z2 + 2*z''*z1^2*z2^2 + 2*z''*z1^3 + 2*z''*z2 + 4*z''*z2^2 + 5*z''*z2^3 + 2*z''*z2^4 + 2*z''^2 + 2*z''^2*z1 + z''^2*z1*z2 + z''^2*z1*z2^2 + z''^2*z1^2 + 2*z''^2*z2 + z''^2*z2^2 + z''^2*z2^3 + z''^3 + z1 + 3*z1*z2 + 9*z1*z2^2 + 9*z1*z2^3 + 3*z1*z2^4 + 3*z1^2 + 6*z1^2*z2 + 7*z1^2*z2^2 + 3*z1^2*z2^3 + 3*z1^3 + 3*z1^3*z2 + z1^3*z2^2 + z1^4 + z2 + 4*z2^2 + 6*z2^3 + 4*z2^4 + z2^5], size: O(1) [0]

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

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

(24)
BOUNDS(1, n^5)

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(25) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(26)
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^5).


The TRS R consists of the following rules:

   f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5)
   f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5)
   f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5)
   f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5)
   f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5)
   f(0, 0, 0, 0, 0) -> 0

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

(27) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

f(s(x1), x2, x3, x4, x5) ->^+ f(x1, x2, x3, x4, x5)

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

The pumping substitution is [x1 / s(x1)].

The result substitution is [ ].




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(28)
Complex Obligation (BEST)

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(29)
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^5).


The TRS R consists of the following rules:

   f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5)
   f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5)
   f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5)
   f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5)
   f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5)
   f(0, 0, 0, 0, 0) -> 0

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

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

(31)
BOUNDS(n^1, INF)

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(32)
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^5).


The TRS R consists of the following rules:

   f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5)
   f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5)
   f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5)
   f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5)
   f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5)
   f(0, 0, 0, 0, 0) -> 0

S is empty.
Rewrite Strategy: FULL