WORST_CASE(?,O(n^1))
proof of input_HwK14XjvGz.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxWeightedTrs
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxTypedWeightedTrs
(5) CompletionProof [UPPER BOUND(ID), 0 ms]
(6) CpxTypedWeightedCompleteTrs
(7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTypedWeightedCompleteTrs
(9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(10) CpxRNTS
(11) InliningProof [UPPER BOUND(ID), 77 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), 149 ms]
(20) CpxRNTS
(21) IntTrsBoundProof [UPPER BOUND(ID), 33 ms]
(22) CpxRNTS
(23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 415 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 126 ms]
(28) CpxRNTS
(29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(30) CpxRNTS
(31) IntTrsBoundProof [UPPER BOUND(ID), 631 ms]
(32) CpxRNTS
(33) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
(34) CpxRNTS
(35) FinalProof [FINISHED, 0 ms]
(36) BOUNDS(1, n^1)


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

(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:

   a__f(b, X, c) -> a__f(X, a__c, X)
   a__c -> b
   mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
   mark(c) -> a__c
   mark(b) -> b
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__c -> c

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

(1) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(2)
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(b, X, c) -> a__f(X, a__c, X) [1]
   a__c -> b [1]
   mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) [1]
   mark(c) -> a__c [1]
   mark(b) -> b [1]
   a__f(X1, X2, X3) -> f(X1, X2, X3) [1]
   a__c -> c [1]

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

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

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   a__f(b, X, c) -> a__f(X, a__c, X) [1]
   a__c -> b [1]
   mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) [1]
   mark(c) -> a__c [1]
   mark(b) -> b [1]
   a__f(X1, X2, X3) -> f(X1, X2, X3) [1]
   a__c -> c [1]

The TRS has the following type information:
a__f :: b:c:f -> b:c:f -> b:c:f -> b:c:f
b :: b:c:f
c :: b:c:f
a__c :: b:c:f
mark :: b:c:f -> b:c:f
f :: b:c:f -> b:c:f -> b:c:f -> b:c:f

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

(5) 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__c
   a__f_3

Due to the following rules being added:
none

And the following fresh constants: none

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

(6)
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(b, X, c) -> a__f(X, a__c, X) [1]
   a__c -> b [1]
   mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) [1]
   mark(c) -> a__c [1]
   mark(b) -> b [1]
   a__f(X1, X2, X3) -> f(X1, X2, X3) [1]
   a__c -> c [1]

The TRS has the following type information:
a__f :: b:c:f -> b:c:f -> b:c:f -> b:c:f
b :: b:c:f
c :: b:c:f
a__c :: b:c:f
mark :: b:c:f -> b:c:f
f :: b:c:f -> b:c:f -> b:c:f -> b:c:f

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

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

(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(b, X, c) -> a__f(X, b, X) [2]
   a__f(b, X, c) -> a__f(X, c, X) [2]
   a__c -> b [1]
   mark(f(X1, f(X1', X2', X3'), X3)) -> a__f(X1, a__f(X1', mark(X2'), X3'), X3) [2]
   mark(f(X1, c, X3)) -> a__f(X1, a__c, X3) [2]
   mark(f(X1, b, X3)) -> a__f(X1, b, X3) [2]
   mark(c) -> a__c [1]
   mark(b) -> b [1]
   a__f(X1, X2, X3) -> f(X1, X2, X3) [1]
   a__c -> c [1]

The TRS has the following type information:
a__f :: b:c:f -> b:c:f -> b:c:f -> b:c:f
b :: b:c:f
c :: b:c:f
a__c :: b:c:f
mark :: b:c:f -> b:c:f
f :: b:c:f -> b:c:f -> b:c:f -> b:c:f

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

(9) 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

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

(10)
Obligation:
Complexity RNTS consisting of the following rules:

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


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

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

   a__c -{ 1 }-> 0 :|: 
   a__c -{ 1 }-> 1 :|: 

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

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

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


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

(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:

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


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

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

   { a__c }
   { a__f }
   { mark }

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

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

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

Function symbols to be analyzed: {a__c}, {a__f}, {mark}

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

(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:

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

Function symbols to be analyzed: {a__c}, {a__f}, {mark}

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: a__c
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__c -{ 1 }-> 1 :|: 
   a__c -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 2 }-> a__f(z', 1, z') :|: z' >= 0, z = 0, z'' = 1
   a__f(z, z', z'') -{ 2 }-> a__f(z', 0, z') :|: z' >= 0, z = 0, z'' = 1
   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
   mark(z) -{ 2 }-> a__f(X1, a__f(X1', mark(X2'), X3'), X3) :|: X1 >= 0, X3' >= 0, X2' >= 0, X1' >= 0, X3 >= 0, z = 1 + X1 + (1 + X1' + X2' + X3') + X3
   mark(z) -{ 3 }-> a__f(X1, 1, X3) :|: X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0
   mark(z) -{ 2 }-> a__f(X1, 0, X3) :|: X1 >= 0, z = 1 + X1 + 0 + X3, X3 >= 0
   mark(z) -{ 3 }-> a__f(X1, 0, X3) :|: X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0
   mark(z) -{ 2 }-> 1 :|: z = 1
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 2 }-> 0 :|: z = 1

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

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

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

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

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

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

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

(25) 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: 2 + z + 2*z' + z''

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

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

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

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

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

(27) 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: 3

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

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

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

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

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(29) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(30)
Obligation:
Complexity RNTS consisting of the following rules:

   a__c -{ 1 }-> 1 :|: 
   a__c -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 5 }-> s :|: s >= 0, s <= z' + 2 * 0 + z' + 2, z' >= 0, z = 0, z'' = 1
   a__f(z, z', z'') -{ 5 }-> s' :|: s' >= 0, s' <= z' + 2 * 1 + z' + 2, z' >= 0, z = 0, z'' = 1
   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
   mark(z) -{ 5 }-> s'' :|: s'' >= 0, s'' <= X1 + 2 * 0 + X3 + 2, X1 >= 0, z = 1 + X1 + 0 + X3, X3 >= 0
   mark(z) -{ 6 }-> s1 :|: s1 >= 0, s1 <= X1 + 2 * 0 + X3 + 2, X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0
   mark(z) -{ 6 }-> s2 :|: s2 >= 0, s2 <= X1 + 2 * 1 + X3 + 2, X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0
   mark(z) -{ 2 }-> a__f(X1, a__f(X1', mark(X2'), X3'), X3) :|: X1 >= 0, X3' >= 0, X2' >= 0, X1' >= 0, X3 >= 0, z = 1 + X1 + (1 + X1' + X2' + X3') + X3
   mark(z) -{ 2 }-> 1 :|: z = 1
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 2 }-> 0 :|: z = 1

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

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(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: mark
after applying outer abstraction to obtain an ITS,
resulting in: EXP with polynomial bound: ?

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

   a__c -{ 1 }-> 1 :|: 
   a__c -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 5 }-> s :|: s >= 0, s <= z' + 2 * 0 + z' + 2, z' >= 0, z = 0, z'' = 1
   a__f(z, z', z'') -{ 5 }-> s' :|: s' >= 0, s' <= z' + 2 * 1 + z' + 2, z' >= 0, z = 0, z'' = 1
   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
   mark(z) -{ 5 }-> s'' :|: s'' >= 0, s'' <= X1 + 2 * 0 + X3 + 2, X1 >= 0, z = 1 + X1 + 0 + X3, X3 >= 0
   mark(z) -{ 6 }-> s1 :|: s1 >= 0, s1 <= X1 + 2 * 0 + X3 + 2, X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0
   mark(z) -{ 6 }-> s2 :|: s2 >= 0, s2 <= X1 + 2 * 1 + X3 + 2, X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0
   mark(z) -{ 2 }-> a__f(X1, a__f(X1', mark(X2'), X3'), X3) :|: X1 >= 0, X3' >= 0, X2' >= 0, X1' >= 0, X3 >= 0, z = 1 + X1 + (1 + X1' + X2' + X3') + X3
   mark(z) -{ 2 }-> 1 :|: z = 1
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 2 }-> 0 :|: z = 1

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

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(33) 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: 20 + 8*z

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

   a__c -{ 1 }-> 1 :|: 
   a__c -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 5 }-> s :|: s >= 0, s <= z' + 2 * 0 + z' + 2, z' >= 0, z = 0, z'' = 1
   a__f(z, z', z'') -{ 5 }-> s' :|: s' >= 0, s' <= z' + 2 * 1 + z' + 2, z' >= 0, z = 0, z'' = 1
   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
   mark(z) -{ 5 }-> s'' :|: s'' >= 0, s'' <= X1 + 2 * 0 + X3 + 2, X1 >= 0, z = 1 + X1 + 0 + X3, X3 >= 0
   mark(z) -{ 6 }-> s1 :|: s1 >= 0, s1 <= X1 + 2 * 0 + X3 + 2, X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0
   mark(z) -{ 6 }-> s2 :|: s2 >= 0, s2 <= X1 + 2 * 1 + X3 + 2, X1 >= 0, z = 1 + X1 + 1 + X3, X3 >= 0
   mark(z) -{ 2 }-> a__f(X1, a__f(X1', mark(X2'), X3'), X3) :|: X1 >= 0, X3' >= 0, X2' >= 0, X1' >= 0, X3 >= 0, z = 1 + X1 + (1 + X1' + X2' + X3') + X3
   mark(z) -{ 2 }-> 1 :|: z = 1
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 2 }-> 0 :|: z = 1

Function symbols to be analyzed: 
Previous analysis results are:
  a__c: runtime: O(1) [1], size: O(1) [1]
  a__f: runtime: O(1) [3], size: O(n^1) [2 + z + 2*z' + z'']
  mark: runtime: O(n^1) [20 + 8*z], size: EXP

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(35) FinalProof (FINISHED)
Computed overall runtime complexity
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(36)
BOUNDS(1, n^1)