/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) 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) InliningProof [UPPER BOUND(ID), 99 ms]
    (14) CpxRNTS
    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 93 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 35 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 129 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 43 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 340 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 93 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 342 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 73 ms]
    (42) CpxRNTS
    (43) FinalProof [FINISHED, 0 ms]
    (44) BOUNDS(1, n^1)
(45) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(46) TRS for Loop Detection
    (47) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (48) BEST
        (49) proven lower bound
            (50) LowerBoundPropagationProof [FINISHED, 0 ms]
            (51) BOUNDS(n^1, INF)
        (52) TRS for Loop Detection


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

(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:

   a__f(X, g(X), Y) -> a__f(Y, Y, Y)
   a__g(b) -> c
   a__b -> c
   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3)
   mark(g(X)) -> a__g(mark(X))
   mark(b) -> a__b
   mark(c) -> c
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__g(X) -> g(X)
   a__b -> b

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

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

The duplicating contexts are:
a__f(X, g(X), [])


The defined contexts are:
a__g([])


As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
----------------------------------------

(2)
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:

   a__f(X, g(X), Y) -> a__f(Y, Y, Y)
   a__g(b) -> c
   a__b -> c
   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3)
   mark(g(X)) -> a__g(mark(X))
   mark(b) -> a__b
   mark(c) -> c
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__g(X) -> g(X)
   a__b -> b

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^1).


The TRS R consists of the following rules:

   a__f(X, g(X), Y) -> a__f(Y, Y, Y) [1]
   a__g(b) -> c [1]
   a__b -> c [1]
   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3) [1]
   mark(g(X)) -> a__g(mark(X)) [1]
   mark(b) -> a__b [1]
   mark(c) -> c [1]
   a__f(X1, X2, X3) -> f(X1, X2, X3) [1]
   a__g(X) -> g(X) [1]
   a__b -> b [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:

   a__f(X, g(X), Y) -> a__f(Y, Y, Y) [1]
   a__g(b) -> c [1]
   a__b -> c [1]
   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3) [1]
   mark(g(X)) -> a__g(mark(X)) [1]
   mark(b) -> a__b [1]
   mark(c) -> c [1]
   a__f(X1, X2, X3) -> f(X1, X2, X3) [1]
   a__g(X) -> g(X) [1]
   a__b -> b [1]

The TRS has the following type information:
a__f :: g:b:c:f -> g:b:c:f -> g:b:c:f -> g:b:c:f
g :: g:b:c:f -> g:b:c:f
a__g :: g:b:c:f -> g:b:c:f
b :: g:b:c:f
c :: g:b:c:f
a__b :: g:b:c:f
mark :: g:b:c:f -> g:b:c:f
f :: g:b:c:f -> g:b:c:f -> g:b:c:f -> g:b:c:f

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:
none

(c) The following functions are completely defined:

   mark_1
   a__b
   a__f_3
   a__g_1

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:

   a__f(X, g(X), Y) -> a__f(Y, Y, Y) [1]
   a__g(b) -> c [1]
   a__b -> c [1]
   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3) [1]
   mark(g(X)) -> a__g(mark(X)) [1]
   mark(b) -> a__b [1]
   mark(c) -> c [1]
   a__f(X1, X2, X3) -> f(X1, X2, X3) [1]
   a__g(X) -> g(X) [1]
   a__b -> b [1]

The TRS has the following type information:
a__f :: g:b:c:f -> g:b:c:f -> g:b:c:f -> g:b:c:f
g :: g:b:c:f -> g:b:c:f
a__g :: g:b:c:f -> g:b:c:f
b :: g:b:c:f
c :: g:b:c:f
a__b :: g:b:c:f
mark :: g:b:c:f -> g:b:c:f
f :: g:b:c:f -> g:b:c:f -> g:b:c:f -> g:b:c:f

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

(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:

   a__f(X, g(X), Y) -> a__f(Y, Y, Y) [1]
   a__g(b) -> c [1]
   a__b -> c [1]
   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3) [1]
   mark(g(f(X1', X2', X3'))) -> a__g(a__f(X1', X2', X3')) [2]
   mark(g(g(X'))) -> a__g(a__g(mark(X'))) [2]
   mark(g(b)) -> a__g(a__b) [2]
   mark(g(c)) -> a__g(c) [2]
   mark(b) -> a__b [1]
   mark(c) -> c [1]
   a__f(X1, X2, X3) -> f(X1, X2, X3) [1]
   a__g(X) -> g(X) [1]
   a__b -> b [1]

The TRS has the following type information:
a__f :: g:b:c:f -> g:b:c:f -> g:b:c:f -> g:b:c:f
g :: g:b:c:f -> g:b:c:f
a__g :: g:b:c:f -> g:b:c:f
b :: g:b:c:f
c :: g:b:c:f
a__b :: g:b:c:f
mark :: g:b:c:f -> g:b:c:f
f :: g:b:c:f -> g:b:c:f -> g:b:c:f -> g:b:c:f

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:

   b => 0
   c => 1

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

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

   a__b -{ 1 }-> 1 :|: 
   a__b -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 1 }-> a__f(Y, Y, Y) :|: Y >= 0, z'' = Y, z' = 1 + X, X >= 0, z = X
   a__f(z, z', z'') -{ 1 }-> 1 + X1 + X2 + X3 :|: X1 >= 0, X3 >= 0, X2 >= 0, z = X1, z' = X2, z'' = X3
   a__g(z) -{ 1 }-> 1 :|: z = 0
   a__g(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   mark(z) -{ 2 }-> a__g(a__g(mark(X'))) :|: X' >= 0, z = 1 + (1 + X')
   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
   mark(z) -{ 2 }-> a__g(a__b) :|: z = 1 + 0
   mark(z) -{ 2 }-> a__g(1) :|: z = 1 + 1
   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
   mark(z) -{ 1 }-> a__b :|: z = 0
   mark(z) -{ 1 }-> 1 :|: z = 1


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

(13) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   a__g(z) -{ 1 }-> 1 :|: z = 0
   a__g(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   a__b -{ 1 }-> 0 :|: 
   a__b -{ 1 }-> 1 :|: 

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

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

   a__b -{ 1 }-> 1 :|: 
   a__b -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 1 }-> a__f(Y, Y, Y) :|: Y >= 0, z'' = Y, z' = 1 + X, X >= 0, z = X
   a__f(z, z', z'') -{ 1 }-> 1 + X1 + X2 + X3 :|: X1 >= 0, X3 >= 0, X2 >= 0, z = X1, z' = X2, z'' = X3
   a__g(z) -{ 1 }-> 1 :|: z = 0
   a__g(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   mark(z) -{ 2 }-> a__g(a__g(mark(X'))) :|: X' >= 0, z = 1 + (1 + X')
   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
   mark(z) -{ 3 }-> a__g(1) :|: z = 1 + 0
   mark(z) -{ 3 }-> a__g(0) :|: z = 1 + 0
   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
   mark(z) -{ 1 }-> 1 :|: z = 1
   mark(z) -{ 2 }-> 1 :|: z = 0
   mark(z) -{ 2 }-> 0 :|: z = 0
   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X


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

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

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

   a__b -{ 1 }-> 1 :|: 
   a__b -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 1 }-> a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
   a__g(z) -{ 1 }-> 1 :|: z = 0
   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
   mark(z) -{ 3 }-> a__g(1) :|: z = 1 + 0
   mark(z) -{ 3 }-> a__g(0) :|: z = 1 + 0
   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
   mark(z) -{ 1 }-> 1 :|: z = 1
   mark(z) -{ 2 }-> 1 :|: z = 0
   mark(z) -{ 2 }-> 0 :|: z = 0
   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X


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

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

   { a__b }
   { a__g }
   { a__f }
   { mark }

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

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

   a__b -{ 1 }-> 1 :|: 
   a__b -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 1 }-> a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
   a__g(z) -{ 1 }-> 1 :|: z = 0
   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
   mark(z) -{ 3 }-> a__g(1) :|: z = 1 + 0
   mark(z) -{ 3 }-> a__g(0) :|: z = 1 + 0
   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
   mark(z) -{ 1 }-> 1 :|: z = 1
   mark(z) -{ 2 }-> 1 :|: z = 0
   mark(z) -{ 2 }-> 0 :|: z = 0
   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X

Function symbols to be analyzed: {a__b}, {a__g}, {a__f}, {mark}

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

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

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

   a__b -{ 1 }-> 1 :|: 
   a__b -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 1 }-> a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
   a__g(z) -{ 1 }-> 1 :|: z = 0
   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
   mark(z) -{ 3 }-> a__g(1) :|: z = 1 + 0
   mark(z) -{ 3 }-> a__g(0) :|: z = 1 + 0
   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
   mark(z) -{ 1 }-> 1 :|: z = 1
   mark(z) -{ 2 }-> 1 :|: z = 0
   mark(z) -{ 2 }-> 0 :|: z = 0
   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X

Function symbols to be analyzed: {a__b}, {a__g}, {a__f}, {mark}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a__b -{ 1 }-> 1 :|: 
   a__b -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 1 }-> a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
   a__g(z) -{ 1 }-> 1 :|: z = 0
   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
   mark(z) -{ 3 }-> a__g(1) :|: z = 1 + 0
   mark(z) -{ 3 }-> a__g(0) :|: z = 1 + 0
   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
   mark(z) -{ 1 }-> 1 :|: z = 1
   mark(z) -{ 2 }-> 1 :|: z = 0
   mark(z) -{ 2 }-> 0 :|: z = 0
   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X

Function symbols to be analyzed: {a__b}, {a__g}, {a__f}, {mark}
Previous analysis results are:
  a__b: runtime: ?, size: O(1) [1]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   a__b -{ 1 }-> 1 :|: 
   a__b -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 1 }-> a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
   a__g(z) -{ 1 }-> 1 :|: z = 0
   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
   mark(z) -{ 3 }-> a__g(1) :|: z = 1 + 0
   mark(z) -{ 3 }-> a__g(0) :|: z = 1 + 0
   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
   mark(z) -{ 1 }-> 1 :|: z = 1
   mark(z) -{ 2 }-> 1 :|: z = 0
   mark(z) -{ 2 }-> 0 :|: z = 0
   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X

Function symbols to be analyzed: {a__g}, {a__f}, {mark}
Previous analysis results are:
  a__b: runtime: O(1) [1], size: O(1) [1]

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

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

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   a__b -{ 1 }-> 1 :|: 
   a__b -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 1 }-> a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
   a__g(z) -{ 1 }-> 1 :|: z = 0
   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
   mark(z) -{ 3 }-> a__g(1) :|: z = 1 + 0
   mark(z) -{ 3 }-> a__g(0) :|: z = 1 + 0
   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
   mark(z) -{ 1 }-> 1 :|: z = 1
   mark(z) -{ 2 }-> 1 :|: z = 0
   mark(z) -{ 2 }-> 0 :|: z = 0
   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X

Function symbols to be analyzed: {a__g}, {a__f}, {mark}
Previous analysis results are:
  a__b: runtime: O(1) [1], size: O(1) [1]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   a__b -{ 1 }-> 1 :|: 
   a__b -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 1 }-> a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
   a__g(z) -{ 1 }-> 1 :|: z = 0
   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
   mark(z) -{ 3 }-> a__g(1) :|: z = 1 + 0
   mark(z) -{ 3 }-> a__g(0) :|: z = 1 + 0
   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
   mark(z) -{ 1 }-> 1 :|: z = 1
   mark(z) -{ 2 }-> 1 :|: z = 0
   mark(z) -{ 2 }-> 0 :|: z = 0
   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X

Function symbols to be analyzed: {a__g}, {a__f}, {mark}
Previous analysis results are:
  a__b: runtime: O(1) [1], size: O(1) [1]
  a__g: runtime: ?, size: O(n^1) [1 + z]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   a__b -{ 1 }-> 1 :|: 
   a__b -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 1 }-> a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
   a__g(z) -{ 1 }-> 1 :|: z = 0
   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
   mark(z) -{ 3 }-> a__g(1) :|: z = 1 + 0
   mark(z) -{ 3 }-> a__g(0) :|: z = 1 + 0
   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
   mark(z) -{ 1 }-> 1 :|: z = 1
   mark(z) -{ 2 }-> 1 :|: z = 0
   mark(z) -{ 2 }-> 0 :|: z = 0
   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X

Function symbols to be analyzed: {a__f}, {mark}
Previous analysis results are:
  a__b: runtime: O(1) [1], size: O(1) [1]
  a__g: runtime: O(1) [1], size: O(n^1) [1 + z]

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

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

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   a__b -{ 1 }-> 1 :|: 
   a__b -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 1 }-> a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
   a__g(z) -{ 1 }-> 1 :|: z = 0
   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
   mark(z) -{ 4 }-> s :|: s >= 0, s <= 0 + 1, z = 1 + 0
   mark(z) -{ 4 }-> s' :|: s' >= 0, s' <= 1 + 1, z = 1 + 0
   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
   mark(z) -{ 1 }-> 1 :|: z = 1
   mark(z) -{ 2 }-> 1 :|: z = 0
   mark(z) -{ 2 }-> 0 :|: z = 0
   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X

Function symbols to be analyzed: {a__f}, {mark}
Previous analysis results are:
  a__b: runtime: O(1) [1], size: O(1) [1]
  a__g: runtime: O(1) [1], size: O(n^1) [1 + z]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: a__f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z + z' + 3*z''

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

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   a__b -{ 1 }-> 1 :|: 
   a__b -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 1 }-> a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
   a__g(z) -{ 1 }-> 1 :|: z = 0
   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
   mark(z) -{ 4 }-> s :|: s >= 0, s <= 0 + 1, z = 1 + 0
   mark(z) -{ 4 }-> s' :|: s' >= 0, s' <= 1 + 1, z = 1 + 0
   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
   mark(z) -{ 1 }-> 1 :|: z = 1
   mark(z) -{ 2 }-> 1 :|: z = 0
   mark(z) -{ 2 }-> 0 :|: z = 0
   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X

Function symbols to be analyzed: {a__f}, {mark}
Previous analysis results are:
  a__b: runtime: O(1) [1], size: O(1) [1]
  a__g: runtime: O(1) [1], size: O(n^1) [1 + z]
  a__f: runtime: ?, size: O(n^1) [1 + z + z' + 3*z'']

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: a__f
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   a__b -{ 1 }-> 1 :|: 
   a__b -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 1 }-> a__f(z'', z'', z'') :|: z'' >= 0, z' - 1 >= 0, z = z' - 1
   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
   a__g(z) -{ 1 }-> 1 :|: z = 0
   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
   mark(z) -{ 4 }-> s :|: s >= 0, s <= 0 + 1, z = 1 + 0
   mark(z) -{ 4 }-> s' :|: s' >= 0, s' <= 1 + 1, z = 1 + 0
   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
   mark(z) -{ 2 }-> a__g(a__f(X1', X2', X3')) :|: X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
   mark(z) -{ 1 }-> a__f(X1, X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
   mark(z) -{ 1 }-> 1 :|: z = 1
   mark(z) -{ 2 }-> 1 :|: z = 0
   mark(z) -{ 2 }-> 0 :|: z = 0
   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X

Function symbols to be analyzed: {mark}
Previous analysis results are:
  a__b: runtime: O(1) [1], size: O(1) [1]
  a__g: runtime: O(1) [1], size: O(n^1) [1 + z]
  a__f: runtime: O(1) [2], size: O(n^1) [1 + z + z' + 3*z'']

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

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

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   a__b -{ 1 }-> 1 :|: 
   a__b -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 3 }-> s'' :|: s'' >= 0, s'' <= z'' + z'' + 3 * z'' + 1, z'' >= 0, z' - 1 >= 0, z = z' - 1
   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
   a__g(z) -{ 1 }-> 1 :|: z = 0
   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
   mark(z) -{ 4 }-> s :|: s >= 0, s <= 0 + 1, z = 1 + 0
   mark(z) -{ 4 }-> s' :|: s' >= 0, s' <= 1 + 1, z = 1 + 0
   mark(z) -{ 3 }-> s1 :|: s1 >= 0, s1 <= X1 + X2 + 3 * X3 + 1, X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
   mark(z) -{ 5 }-> s3 :|: s2 >= 0, s2 <= X1' + X2' + 3 * X3' + 1, s3 >= 0, s3 <= s2 + 1, X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
   mark(z) -{ 1 }-> 1 :|: z = 1
   mark(z) -{ 2 }-> 1 :|: z = 0
   mark(z) -{ 2 }-> 0 :|: z = 0
   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X

Function symbols to be analyzed: {mark}
Previous analysis results are:
  a__b: runtime: O(1) [1], size: O(1) [1]
  a__g: runtime: O(1) [1], size: O(n^1) [1 + z]
  a__f: runtime: O(1) [2], size: O(n^1) [1 + z + z' + 3*z'']

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: mark
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + 3*z

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

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   a__b -{ 1 }-> 1 :|: 
   a__b -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 3 }-> s'' :|: s'' >= 0, s'' <= z'' + z'' + 3 * z'' + 1, z'' >= 0, z' - 1 >= 0, z = z' - 1
   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
   a__g(z) -{ 1 }-> 1 :|: z = 0
   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
   mark(z) -{ 4 }-> s :|: s >= 0, s <= 0 + 1, z = 1 + 0
   mark(z) -{ 4 }-> s' :|: s' >= 0, s' <= 1 + 1, z = 1 + 0
   mark(z) -{ 3 }-> s1 :|: s1 >= 0, s1 <= X1 + X2 + 3 * X3 + 1, X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
   mark(z) -{ 5 }-> s3 :|: s2 >= 0, s2 <= X1' + X2' + 3 * X3' + 1, s3 >= 0, s3 <= s2 + 1, X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
   mark(z) -{ 1 }-> 1 :|: z = 1
   mark(z) -{ 2 }-> 1 :|: z = 0
   mark(z) -{ 2 }-> 0 :|: z = 0
   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X

Function symbols to be analyzed: {mark}
Previous analysis results are:
  a__b: runtime: O(1) [1], size: O(1) [1]
  a__g: runtime: O(1) [1], size: O(n^1) [1 + z]
  a__f: runtime: O(1) [2], size: O(n^1) [1 + z + z' + 3*z'']
  mark: runtime: ?, size: O(n^1) [1 + 3*z]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: mark
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 22 + 4*z

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

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   a__b -{ 1 }-> 1 :|: 
   a__b -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 3 }-> s'' :|: s'' >= 0, s'' <= z'' + z'' + 3 * z'' + 1, z'' >= 0, z' - 1 >= 0, z = z' - 1
   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
   a__g(z) -{ 1 }-> 1 :|: z = 0
   a__g(z) -{ 1 }-> 1 + z :|: z >= 0
   mark(z) -{ 4 }-> s :|: s >= 0, s <= 0 + 1, z = 1 + 0
   mark(z) -{ 4 }-> s' :|: s' >= 0, s' <= 1 + 1, z = 1 + 0
   mark(z) -{ 3 }-> s1 :|: s1 >= 0, s1 <= X1 + X2 + 3 * X3 + 1, X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
   mark(z) -{ 5 }-> s3 :|: s2 >= 0, s2 <= X1' + X2' + 3 * X3' + 1, s3 >= 0, s3 <= s2 + 1, X3' >= 0, X2' >= 0, X1' >= 0, z = 1 + (1 + X1' + X2' + X3')
   mark(z) -{ 2 }-> a__g(a__g(mark(z - 2))) :|: z - 2 >= 0
   mark(z) -{ 1 }-> 1 :|: z = 1
   mark(z) -{ 2 }-> 1 :|: z = 0
   mark(z) -{ 2 }-> 0 :|: z = 0
   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X

Function symbols to be analyzed: 
Previous analysis results are:
  a__b: runtime: O(1) [1], size: O(1) [1]
  a__g: runtime: O(1) [1], size: O(n^1) [1 + z]
  a__f: runtime: O(1) [2], size: O(n^1) [1 + z + z' + 3*z'']
  mark: runtime: O(n^1) [22 + 4*z], size: O(n^1) [1 + 3*z]

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

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

(44)
BOUNDS(1, n^1)

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

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

(46)
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:

   a__f(X, g(X), Y) -> a__f(Y, Y, Y)
   a__g(b) -> c
   a__b -> c
   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3)
   mark(g(X)) -> a__g(mark(X))
   mark(b) -> a__b
   mark(c) -> c
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__g(X) -> g(X)
   a__b -> b

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

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

The rewrite sequence

mark(g(X)) ->^+ a__g(mark(X))

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

The pumping substitution is [X / g(X)].

The result substitution is [ ].




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

(48)
Complex Obligation (BEST)

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

(49)
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:

   a__f(X, g(X), Y) -> a__f(Y, Y, Y)
   a__g(b) -> c
   a__b -> c
   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3)
   mark(g(X)) -> a__g(mark(X))
   mark(b) -> a__b
   mark(c) -> c
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__g(X) -> g(X)
   a__b -> b

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

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

(51)
BOUNDS(n^1, INF)

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

(52)
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:

   a__f(X, g(X), Y) -> a__f(Y, Y, Y)
   a__g(b) -> c
   a__b -> c
   mark(f(X1, X2, X3)) -> a__f(X1, X2, X3)
   mark(g(X)) -> a__g(mark(X))
   mark(b) -> a__b
   mark(c) -> c
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__g(X) -> g(X)
   a__b -> b

S is empty.
Rewrite Strategy: FULL