WORST_CASE(?, O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 163 ms]
(4) CpxRelTRS
(5) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxWeightedTrs
(9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxTypedWeightedTrs
(11) CompletionProof [UPPER BOUND(ID), 0 ms]
(12) CpxTypedWeightedCompleteTrs
(13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTypedWeightedCompleteTrs
(15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(16) CpxRNTS
(17) InliningProof [UPPER BOUND(ID), 36 ms]
(18) CpxRNTS
(19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CpxRNTS
(21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
(22) CpxRNTS
(23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 20 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
(28) CpxRNTS
(29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(30) CpxRNTS
(31) IntTrsBoundProof [UPPER BOUND(ID), 26 ms]
(32) CpxRNTS
(33) IntTrsBoundProof [UPPER BOUND(ID), 31 ms]
(34) CpxRNTS
(35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(36) CpxRNTS
(37) IntTrsBoundProof [UPPER BOUND(ID), 88 ms]
(38) CpxRNTS
(39) IntTrsBoundProof [UPPER BOUND(ID), 2 ms]
(40) CpxRNTS
(41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(42) CpxRNTS
(43) IntTrsBoundProof [UPPER BOUND(ID), 107 ms]
(44) CpxRNTS
(45) IntTrsBoundProof [UPPER BOUND(ID), 42 ms]
(46) CpxRNTS
(47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(48) CpxRNTS
(49) IntTrsBoundProof [UPPER BOUND(ID), 90 ms]
(50) CpxRNTS
(51) IntTrsBoundProof [UPPER BOUND(ID), 32 ms]
(52) CpxRNTS
(53) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(54) CpxRNTS
(55) IntTrsBoundProof [UPPER BOUND(ID), 159 ms]
(56) CpxRNTS
(57) IntTrsBoundProof [UPPER BOUND(ID), 125 ms]
(58) CpxRNTS
(59) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(60) CpxRNTS
(61) IntTrsBoundProof [UPPER BOUND(ID), 127 ms]
(62) CpxRNTS
(63) IntTrsBoundProof [UPPER BOUND(ID), 73 ms]
(64) CpxRNTS
(65) FinalProof [FINISHED, 0 ms]
(66) BOUNDS(1, n^1)


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

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


The TRS R consists of the following rules:

   f(x, x) -> f(a, b)
   b -> c

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

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(a) -> a
   encArg(c) -> c
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_b) -> b
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_b -> b
   encode_c -> c


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

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


The TRS R consists of the following rules:

   f(x, x) -> f(a, b)
   b -> c

The (relative) TRS S consists of the following rules:

   encArg(a) -> a
   encArg(c) -> c
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_b) -> b
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_b -> b
   encode_c -> c

Rewrite Strategy: FULL
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

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


The TRS R consists of the following rules:

   f(x, x) -> f(a, b)
   b -> c

The (relative) TRS S consists of the following rules:

   encArg(a) -> a
   encArg(c) -> c
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_b) -> b
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_b -> b
   encode_c -> c

Rewrite Strategy: FULL
----------------------------------------

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

As the TRS is a non-duplicating overlay system, we have rc = irc.
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   f(x, x) -> f(a, b)
   b -> c

The (relative) TRS S consists of the following rules:

   encArg(a) -> a
   encArg(c) -> c
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_b) -> b
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a
   encode_b -> b
   encode_c -> c

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

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

(8)
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:

   f(x, x) -> f(a, b) [1]
   b -> c [1]
   encArg(a) -> a [0]
   encArg(c) -> c [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_b) -> b [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_a -> a [0]
   encode_b -> b [0]
   encode_c -> c [0]

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

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

(10)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(x, x) -> f(a, b) [1]
   b -> c [1]
   encArg(a) -> a [0]
   encArg(c) -> c [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_b) -> b [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_a -> a [0]
   encode_b -> b [0]
   encode_c -> c [0]

The TRS has the following type information:
f :: a:c:cons_f:cons_b -> a:c:cons_f:cons_b -> a:c:cons_f:cons_b
a :: a:c:cons_f:cons_b
b :: a:c:cons_f:cons_b
c :: a:c:cons_f:cons_b
encArg :: a:c:cons_f:cons_b -> a:c:cons_f:cons_b
cons_f :: a:c:cons_f:cons_b -> a:c:cons_f:cons_b -> a:c:cons_f:cons_b
cons_b :: a:c:cons_f:cons_b
encode_f :: a:c:cons_f:cons_b -> a:c:cons_f:cons_b -> a:c:cons_f:cons_b
encode_a :: a:c:cons_f:cons_b
encode_b :: a:c:cons_f:cons_b
encode_c :: a:c:cons_f:cons_b

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

(11) 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:

   b
   f_2
   encArg_1
   encode_f_2
   encode_a
   encode_b
   encode_c

Due to the following rules being added:

   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1) -> null_encode_f [0]
   encode_a -> null_encode_a [0]
   encode_b -> null_encode_b [0]
   encode_c -> null_encode_c [0]
   f(v0, v1) -> null_f [0]

And the following fresh constants:    null_encArg, null_encode_f, null_encode_a, null_encode_b, null_encode_c, null_f

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

(12)
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(x, x) -> f(a, b) [1]
   b -> c [1]
   encArg(a) -> a [0]
   encArg(c) -> c [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_b) -> b [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_a -> a [0]
   encode_b -> b [0]
   encode_c -> c [0]
   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1) -> null_encode_f [0]
   encode_a -> null_encode_a [0]
   encode_b -> null_encode_b [0]
   encode_c -> null_encode_c [0]
   f(v0, v1) -> null_f [0]

The TRS has the following type information:
f :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
a :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
b :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
c :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
encArg :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
cons_f :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
cons_b :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
encode_f :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
encode_a :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
encode_b :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
encode_c :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
null_encArg :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
null_encode_f :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
null_encode_a :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
null_encode_b :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
null_encode_c :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
null_f :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f

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

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

(14)
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(x, x) -> f(a, c) [2]
   b -> c [1]
   encArg(a) -> a [0]
   encArg(c) -> c [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_b) -> b [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_a -> a [0]
   encode_b -> b [0]
   encode_c -> c [0]
   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1) -> null_encode_f [0]
   encode_a -> null_encode_a [0]
   encode_b -> null_encode_b [0]
   encode_c -> null_encode_c [0]
   f(v0, v1) -> null_f [0]

The TRS has the following type information:
f :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
a :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
b :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
c :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
encArg :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
cons_f :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
cons_b :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
encode_f :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f -> a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
encode_a :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
encode_b :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
encode_c :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
null_encArg :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
null_encode_f :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
null_encode_a :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
null_encode_b :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
null_encode_c :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f
null_f :: a:c:cons_f:cons_b:null_encArg:null_encode_f:null_encode_a:null_encode_b:null_encode_c:null_f

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

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

   a => 0
   c => 1
   cons_b => 2
   null_encArg => 0
   null_encode_f => 0
   null_encode_a => 0
   null_encode_b => 0
   null_encode_c => 0
   null_f => 0

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

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

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> b :|: z = 2
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 0 }-> b :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   f(z, z') -{ 2 }-> f(0, 1) :|: z' = x, x >= 0, z = x
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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

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

   b -{ 1 }-> 1 :|: 

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

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

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   f(z, z') -{ 2 }-> f(0, 1) :|: z' = x, x >= 0, z = x
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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

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

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

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 2 }-> f(0, 1) :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0


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

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

   { encode_a }
   { encode_c }
   { f }
   { encode_b }
   { b }
   { encArg }
   { encode_f }

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

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

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 2 }-> f(0, 1) :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_a}, {encode_c}, {f}, {encode_b}, {b}, {encArg}, {encode_f}

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

(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:

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 2 }-> f(0, 1) :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_a}, {encode_c}, {f}, {encode_b}, {b}, {encArg}, {encode_f}

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 2 }-> f(0, 1) :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 2 }-> f(0, 1) :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_c}, {f}, {encode_b}, {b}, {encArg}, {encode_f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]

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

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

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

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 2 }-> f(0, 1) :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_c}, {f}, {encode_b}, {b}, {encArg}, {encode_f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 2 }-> f(0, 1) :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

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

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 2 }-> f(0, 1) :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {f}, {encode_b}, {b}, {encArg}, {encode_f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  encode_c: runtime: O(1) [0], size: O(1) [1]

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

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

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

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 2 }-> f(0, 1) :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {f}, {encode_b}, {b}, {encArg}, {encode_f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  encode_c: runtime: O(1) [0], size: O(1) [1]

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

(37) 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

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

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

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 2 }-> f(0, 1) :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

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

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 2 }-> f(0, 1) :|: z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_b}, {b}, {encArg}, {encode_f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  encode_c: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(1) [2], size: O(1) [0]

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

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

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

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 4 }-> s :|: s >= 0, s <= 0, z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_b}, {b}, {encArg}, {encode_f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  encode_c: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(1) [2], size: O(1) [0]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 4 }-> s :|: s >= 0, s <= 0, z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_b}, {b}, {encArg}, {encode_f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  encode_c: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(1) [2], size: O(1) [0]
  encode_b: runtime: ?, size: O(1) [1]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 4 }-> s :|: s >= 0, s <= 0, z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {b}, {encArg}, {encode_f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  encode_c: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(1) [2], size: O(1) [0]
  encode_b: runtime: O(1) [1], size: O(1) [1]

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

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

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 4 }-> s :|: s >= 0, s <= 0, z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {b}, {encArg}, {encode_f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  encode_c: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(1) [2], size: O(1) [0]
  encode_b: runtime: O(1) [1], size: O(1) [1]

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 4 }-> s :|: s >= 0, s <= 0, z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {b}, {encArg}, {encode_f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  encode_c: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(1) [2], size: O(1) [0]
  encode_b: runtime: O(1) [1], size: O(1) [1]
  b: runtime: ?, size: O(1) [1]

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(52)
Obligation:
Complexity RNTS consisting of the following rules:

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 4 }-> s :|: s >= 0, s <= 0, z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encArg}, {encode_f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  encode_c: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(1) [2], size: O(1) [0]
  encode_b: runtime: O(1) [1], size: O(1) [1]
  b: runtime: O(1) [1], size: O(1) [1]

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

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

(54)
Obligation:
Complexity RNTS consisting of the following rules:

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 4 }-> s :|: s >= 0, s <= 0, z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encArg}, {encode_f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  encode_c: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(1) [2], size: O(1) [0]
  encode_b: runtime: O(1) [1], size: O(1) [1]
  b: runtime: O(1) [1], size: O(1) [1]

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(56)
Obligation:
Complexity RNTS consisting of the following rules:

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 4 }-> s :|: s >= 0, s <= 0, z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encArg}, {encode_f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  encode_c: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(1) [2], size: O(1) [0]
  encode_b: runtime: O(1) [1], size: O(1) [1]
  b: runtime: O(1) [1], size: O(1) [1]
  encArg: runtime: ?, size: O(1) [1]

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(58)
Obligation:
Complexity RNTS consisting of the following rules:

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(z), encArg(z')) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 4 }-> s :|: s >= 0, s <= 0, z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  encode_c: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(1) [2], size: O(1) [0]
  encode_b: runtime: O(1) [1], size: O(1) [1]
  b: runtime: O(1) [1], size: O(1) [1]
  encArg: runtime: O(n^1) [1 + 3*z], size: O(1) [1]

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

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

(60)
Obligation:
Complexity RNTS consisting of the following rules:

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 4 + 3*x_1 + 3*x_2 }-> s1 :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 4 + 3*z + 3*z' }-> s4 :|: s2 >= 0, s2 <= 1, s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 4 }-> s :|: s >= 0, s <= 0, z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  encode_c: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(1) [2], size: O(1) [0]
  encode_b: runtime: O(1) [1], size: O(1) [1]
  b: runtime: O(1) [1], size: O(1) [1]
  encArg: runtime: O(n^1) [1 + 3*z], size: O(1) [1]

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

(61) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(62)
Obligation:
Complexity RNTS consisting of the following rules:

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 4 + 3*x_1 + 3*x_2 }-> s1 :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 4 + 3*z + 3*z' }-> s4 :|: s2 >= 0, s2 <= 1, s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 4 }-> s :|: s >= 0, s <= 0, z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {encode_f}
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  encode_c: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(1) [2], size: O(1) [0]
  encode_b: runtime: O(1) [1], size: O(1) [1]
  b: runtime: O(1) [1], size: O(1) [1]
  encArg: runtime: O(n^1) [1 + 3*z], size: O(1) [1]
  encode_f: runtime: ?, size: O(1) [0]

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

(63) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: encode_f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 4 + 3*z + 3*z'

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

(64)
Obligation:
Complexity RNTS consisting of the following rules:

   b -{ 1 }-> 1 :|: 
   encArg(z) -{ 4 + 3*x_1 + 3*x_2 }-> s1 :|: s' >= 0, s' <= 1, s'' >= 0, s'' <= 1, s1 >= 0, s1 <= 0, x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 1 }-> 1 :|: z = 2
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 1 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_c -{ 0 }-> 1 :|: 
   encode_c -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 4 + 3*z + 3*z' }-> s4 :|: s2 >= 0, s2 <= 1, s3 >= 0, s3 <= 1, s4 >= 0, s4 <= 0, z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 4 }-> s :|: s >= 0, s <= 0, z' >= 0, z = z'
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  encode_a: runtime: O(1) [0], size: O(1) [0]
  encode_c: runtime: O(1) [0], size: O(1) [1]
  f: runtime: O(1) [2], size: O(1) [0]
  encode_b: runtime: O(1) [1], size: O(1) [1]
  b: runtime: O(1) [1], size: O(1) [1]
  encArg: runtime: O(n^1) [1 + 3*z], size: O(1) [1]
  encode_f: runtime: O(n^1) [4 + 3*z + 3*z'], size: O(1) [0]

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

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

(66)
BOUNDS(1, n^1)