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


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(1, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 146 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 1 ms]
(6) CpxRelTRS
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) typed CpxTrs
    (9) OrderProof [LOWER BOUND(ID), 0 ms]
    (10) typed CpxTrs
    (11) RewriteLemmaProof [LOWER BOUND(ID), 478 ms]
    (12) BOUNDS(1, INF)
(13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(14) TRS for Loop Detection
(15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(16) CpxTRS
(17) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
(18) CpxRelTRS
    (19) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxWeightedTrs
    (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CpxTypedWeightedTrs
    (23) CompletionProof [UPPER BOUND(ID), 0 ms]
    (24) CpxTypedWeightedCompleteTrs
        (25) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
    (27) CompletionProof [UPPER BOUND(ID), 0 ms]
    (28) CpxTypedWeightedCompleteTrs
        (29) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (30) CpxTypedWeightedCompleteTrs
        (31) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 4 ms]
        (32) CpxRNTS
        (33) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (34) CpxRNTS
(35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(36) CdtProblem
    (37) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (40) CdtProblem
    (41) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) CdtProblem
    (43) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (50) CdtProblem
    (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem


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

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


The TRS R consists of the following rules:

   f(f(X, Y), Z) -> f(X, f(Y, Z))
   f(X, f(Y, Z)) -> f(Y, Y)

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

(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(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))


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

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


The TRS R consists of the following rules:

   f(f(X, Y), Z) -> f(X, f(Y, Z))
   f(X, f(Y, Z)) -> f(Y, Y)

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

   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

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

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

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


The TRS R consists of the following rules:

   f(f(X, Y), Z) -> f(X, f(Y, Z))
   f(X, f(Y, Z)) -> f(Y, Y)

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

   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

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

(5) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

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


The TRS R consists of the following rules:

   f(f(X, Y), Z) -> f(X, f(Y, Z))
   f(X, f(Y, Z)) -> f(Y, Y)

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

   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
f(f(X, Y), Z) -> f(X, f(Y, Z))
f(X, f(Y, Z)) -> f(Y, Y)
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

Types:
f :: f:encArg:encode_f -> f:encArg:encode_f -> f:encArg:encode_f
encArg :: cons_f -> f:encArg:encode_f
cons_f :: cons_f -> cons_f -> cons_f
encode_f :: cons_f -> cons_f -> f:encArg:encode_f
hole_f:encArg:encode_f1_0 :: f:encArg:encode_f
hole_cons_f2_0 :: cons_f
gen_cons_f3_0 :: Nat -> cons_f

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
f, encArg

They will be analysed ascendingly in the following order:
f < encArg

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

(10)
Obligation:
Innermost TRS:
Rules:
f(f(X, Y), Z) -> f(X, f(Y, Z))
f(X, f(Y, Z)) -> f(Y, Y)
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

Types:
f :: f:encArg:encode_f -> f:encArg:encode_f -> f:encArg:encode_f
encArg :: cons_f -> f:encArg:encode_f
cons_f :: cons_f -> cons_f -> cons_f
encode_f :: cons_f -> cons_f -> f:encArg:encode_f
hole_f:encArg:encode_f1_0 :: f:encArg:encode_f
hole_cons_f2_0 :: cons_f
gen_cons_f3_0 :: Nat -> cons_f


Generator Equations:
gen_cons_f3_0(0) <=> hole_cons_f2_0
gen_cons_f3_0(+(x, 1)) <=> cons_f(hole_cons_f2_0, gen_cons_f3_0(x))


The following defined symbols remain to be analysed:
f, encArg

They will be analysed ascendingly in the following order:
f < encArg

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_cons_f3_0(+(1, n16_0))) -> *4_0, rt in Omega(0)

Induction Base:
encArg(gen_cons_f3_0(+(1, 0)))

Induction Step:
encArg(gen_cons_f3_0(+(1, +(n16_0, 1)))) ->_R^Omega(0)
f(encArg(hole_cons_f2_0), encArg(gen_cons_f3_0(+(1, n16_0)))) ->_IH
f(encArg(hole_cons_f2_0), *4_0)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(12)
BOUNDS(1, INF)

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

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

(14)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(f(X, Y), Z) -> f(X, f(Y, Z))
   f(X, f(Y, Z)) -> f(Y, Y)

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

   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

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

(15) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(16)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(f(X, Y), Z) -> f(X, f(Y, Z))
   f(X, f(Y, Z)) -> f(Y, Y)
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))

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

(17) NonCtorToCtorProof (UPPER BOUND(ID))
transformed non-ctor to ctor-system
----------------------------------------

(18)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(X, c_f(Y, Z)) -> f(Y, Y)
   f(c_f(X, Y), Z) -> f(X, f(Y, Z))

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

   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   f(x0, x1) -> c_f(x0, x1)

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

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

(20)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(X, c_f(Y, Z)) -> f(Y, Y) [1]
   f(c_f(X, Y), Z) -> f(X, f(Y, Z)) [1]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   f(x0, x1) -> c_f(x0, x1) [0]

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

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

(22)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(X, c_f(Y, Z)) -> f(Y, Y) [1]
   f(c_f(X, Y), Z) -> f(X, f(Y, Z)) [1]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   f(x0, x1) -> c_f(x0, x1) [0]

The TRS has the following type information:
f :: c_f -> c_f -> c_f
c_f :: c_f -> c_f -> c_f
encArg :: cons_f -> c_f
cons_f :: cons_f -> cons_f -> cons_f
encode_f :: cons_f -> cons_f -> c_f

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

(23) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1) -> null_encode_f [0]
   f(v0, v1) -> null_f [0]

And the following fresh constants:    null_encArg, null_encode_f, null_f, const

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

(24)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(X, c_f(Y, Z)) -> f(Y, Y) [1]
   f(c_f(X, Y), Z) -> f(X, f(Y, Z)) [1]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   f(x0, x1) -> c_f(x0, x1) [0]
   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1) -> null_encode_f [0]
   f(v0, v1) -> null_f [0]

The TRS has the following type information:
f :: c_f:null_encArg:null_encode_f:null_f -> c_f:null_encArg:null_encode_f:null_f -> c_f:null_encArg:null_encode_f:null_f
c_f :: c_f:null_encArg:null_encode_f:null_f -> c_f:null_encArg:null_encode_f:null_f -> c_f:null_encArg:null_encode_f:null_f
encArg :: cons_f -> c_f:null_encArg:null_encode_f:null_f
cons_f :: cons_f -> cons_f -> cons_f
encode_f :: cons_f -> cons_f -> c_f:null_encArg:null_encode_f:null_f
null_encArg :: c_f:null_encArg:null_encode_f:null_f
null_encode_f :: c_f:null_encArg:null_encode_f:null_f
null_f :: c_f:null_encArg:null_encode_f:null_f
const :: cons_f

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

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

   null_encArg => 0
   null_encode_f => 0
   null_f => 0
   const => 0

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

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

   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 }-> 0 :|: v0 >= 0, z = v0
   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') -{ 1 }-> f(X, f(Y, Z)) :|: Z >= 0, Y >= 0, z = 1 + X + Y, X >= 0, z' = Z
   f(z, z') -{ 1 }-> f(Y, Y) :|: Z >= 0, Y >= 0, X >= 0, z = X, z' = 1 + Y + Z
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   f(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1

Only complete derivations are relevant for the runtime complexity.

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

(27) 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:

   encArg_1
   encode_f_2
   f_2

Due to the following rules being added:

   encArg(v0) -> const [0]
   encode_f(v0, v1) -> const [0]
   f(v0, v1) -> const [0]

And the following fresh constants:    const, const1

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

(28)
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, c_f(Y, Z)) -> f(Y, Y) [1]
   f(c_f(X, Y), Z) -> f(X, f(Y, Z)) [1]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   f(x0, x1) -> c_f(x0, x1) [0]
   encArg(v0) -> const [0]
   encode_f(v0, v1) -> const [0]
   f(v0, v1) -> const [0]

The TRS has the following type information:
f :: c_f:const -> c_f:const -> c_f:const
c_f :: c_f:const -> c_f:const -> c_f:const
encArg :: cons_f -> c_f:const
cons_f :: cons_f -> cons_f -> cons_f
encode_f :: cons_f -> cons_f -> c_f:const
const :: c_f:const
const1 :: cons_f

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

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

(30)
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, c_f(Y, Z)) -> f(Y, Y) [1]
   f(c_f(X, Y), c_f(Y', Z')) -> f(X, f(Y', Y')) [2]
   f(c_f(X, c_f(X', Y'')), Z) -> f(X, f(X', f(Y'', Z))) [2]
   f(c_f(X, Y), Z) -> f(X, c_f(Y, Z)) [1]
   f(c_f(X, Y), Z) -> f(X, const) [1]
   encArg(cons_f(cons_f(x_1', x_2'), cons_f(x_1'', x_2''))) -> f(f(encArg(x_1'), encArg(x_2')), f(encArg(x_1''), encArg(x_2''))) [0]
   encArg(cons_f(cons_f(x_1', x_2'), x_2)) -> f(f(encArg(x_1'), encArg(x_2')), const) [0]
   encArg(cons_f(x_1, cons_f(x_11, x_21))) -> f(const, f(encArg(x_11), encArg(x_21))) [0]
   encArg(cons_f(x_1, x_2)) -> f(const, const) [0]
   encode_f(cons_f(x_12, x_22), cons_f(x_13, x_23)) -> f(f(encArg(x_12), encArg(x_22)), f(encArg(x_13), encArg(x_23))) [0]
   encode_f(cons_f(x_12, x_22), x_2) -> f(f(encArg(x_12), encArg(x_22)), const) [0]
   encode_f(x_1, cons_f(x_14, x_24)) -> f(const, f(encArg(x_14), encArg(x_24))) [0]
   encode_f(x_1, x_2) -> f(const, const) [0]
   f(x0, x1) -> c_f(x0, x1) [0]
   encArg(v0) -> const [0]
   encode_f(v0, v1) -> const [0]
   f(v0, v1) -> const [0]

The TRS has the following type information:
f :: c_f:const -> c_f:const -> c_f:const
c_f :: c_f:const -> c_f:const -> c_f:const
encArg :: cons_f -> c_f:const
cons_f :: cons_f -> cons_f -> cons_f
encode_f :: cons_f -> cons_f -> c_f:const
const :: c_f:const
const1 :: cons_f

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

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

   const => 0
   const1 => 0

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

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

   encArg(z) -{ 0 }-> f(f(encArg(x_1'), encArg(x_2')), f(encArg(x_1''), encArg(x_2''))) :|: x_1'' >= 0, x_2' >= 0, x_1' >= 0, x_2'' >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_1'' + x_2'')
   encArg(z) -{ 0 }-> f(f(encArg(x_1'), encArg(x_2')), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_11), encArg(x_21))) :|: x_11 >= 0, x_1 >= 0, z = 1 + x_1 + (1 + x_11 + x_21), x_21 >= 0
   encArg(z) -{ 0 }-> f(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_12), encArg(x_22)), f(encArg(x_13), encArg(x_23))) :|: z = 1 + x_12 + x_22, x_13 >= 0, x_23 >= 0, x_12 >= 0, z' = 1 + x_13 + x_23, x_22 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_12), encArg(x_22)), 0) :|: z = 1 + x_12 + x_22, x_2 >= 0, z' = x_2, x_12 >= 0, x_22 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_14), encArg(x_24))) :|: x_1 >= 0, x_14 >= 0, x_24 >= 0, z = x_1, z' = 1 + x_14 + x_24
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: 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(X, f(X', f(Y'', Z))) :|: Z >= 0, Y'' >= 0, X >= 0, z' = Z, X' >= 0, z = 1 + X + (1 + X' + Y'')
   f(z, z') -{ 2 }-> f(X, f(Y', Y')) :|: Y >= 0, Y' >= 0, Z' >= 0, z = 1 + X + Y, X >= 0, z' = 1 + Y' + Z'
   f(z, z') -{ 1 }-> f(X, 0) :|: Z >= 0, Y >= 0, z = 1 + X + Y, X >= 0, z' = Z
   f(z, z') -{ 1 }-> f(X, 1 + Y + Z) :|: Z >= 0, Y >= 0, z = 1 + X + Y, X >= 0, z' = Z
   f(z, z') -{ 1 }-> f(Y, Y) :|: Z >= 0, Y >= 0, X >= 0, z = X, z' = 1 + Y + Z
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   f(z, z') -{ 0 }-> 1 + x0 + x1 :|: z = x0, x0 >= 0, x1 >= 0, z' = x1


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

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

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

   encArg(z) -{ 0 }-> f(f(encArg(x_1'), encArg(x_2')), f(encArg(x_1''), encArg(x_2''))) :|: x_1'' >= 0, x_2' >= 0, x_1' >= 0, x_2'' >= 0, z = 1 + (1 + x_1' + x_2') + (1 + x_1'' + x_2'')
   encArg(z) -{ 0 }-> f(f(encArg(x_1'), encArg(x_2')), 0) :|: z = 1 + (1 + x_1' + x_2') + x_2, x_2' >= 0, x_1' >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> f(0, f(encArg(x_11), encArg(x_21))) :|: x_11 >= 0, x_1 >= 0, z = 1 + x_1 + (1 + x_11 + x_21), x_21 >= 0
   encArg(z) -{ 0 }-> f(0, 0) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_12), encArg(x_22)), f(encArg(x_13), encArg(x_23))) :|: z = 1 + x_12 + x_22, x_13 >= 0, x_23 >= 0, x_12 >= 0, z' = 1 + x_13 + x_23, x_22 >= 0
   encode_f(z, z') -{ 0 }-> f(f(encArg(x_12), encArg(x_22)), 0) :|: z = 1 + x_12 + x_22, z' >= 0, x_12 >= 0, x_22 >= 0
   encode_f(z, z') -{ 0 }-> f(0, f(encArg(x_14), encArg(x_24))) :|: z >= 0, x_14 >= 0, x_24 >= 0, z' = 1 + x_14 + x_24
   encode_f(z, z') -{ 0 }-> f(0, 0) :|: z >= 0, z' >= 0
   encode_f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 2 }-> f(X, f(X', f(Y'', z'))) :|: z' >= 0, Y'' >= 0, X >= 0, X' >= 0, z = 1 + X + (1 + X' + Y'')
   f(z, z') -{ 2 }-> f(X, f(Y', Y')) :|: Y >= 0, Y' >= 0, Z' >= 0, z = 1 + X + Y, X >= 0, z' = 1 + Y' + Z'
   f(z, z') -{ 1 }-> f(X, 0) :|: z' >= 0, Y >= 0, z = 1 + X + Y, X >= 0
   f(z, z') -{ 1 }-> f(X, 1 + Y + z') :|: z' >= 0, Y >= 0, z = 1 + X + Y, X >= 0
   f(z, z') -{ 1 }-> f(Y, Y) :|: Z >= 0, Y >= 0, z >= 0, z' = 1 + Y + Z
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   f(z, z') -{ 0 }-> 1 + z + z' :|: z >= 0, z' >= 0


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

(35) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   f(f(z0, z1), z2) -> f(z0, f(z1, z2))
   f(z0, f(z1, z2)) -> f(z1, z1)
Tuples:
   ENCARG(cons_f(z0, z1)) -> c(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(f(z0, z1), z2) -> c2(F(z0, f(z1, z2)), F(z1, z2))
   F(z0, f(z1, z2)) -> c3(F(z1, z1))
S tuples:
   F(f(z0, z1), z2) -> c2(F(z0, f(z1, z2)), F(z1, z2))
   F(z0, f(z1, z2)) -> c3(F(z1, z1))
K tuples:none
Defined Rule Symbols:   f_2, encArg_1, encode_f_2

Defined Pair Symbols:   ENCARG_1, ENCODE_F_2, F_2

Compound Symbols:   c_3, c1_3, c2_2, c3_1


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

(37) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   f(f(z0, z1), z2) -> f(z0, f(z1, z2))
   f(z0, f(z1, z2)) -> f(z1, z1)
Tuples:
   ENCARG(cons_f(z0, z1)) -> c(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(f(z0, z1), z2) -> c2(F(z0, f(z1, z2)), F(z1, z2))
   F(z0, f(z1, z2)) -> c3(F(z1, z1))
   ENCODE_F(z0, z1) -> c4(F(encArg(z0), encArg(z1)))
   ENCODE_F(z0, z1) -> c4(ENCARG(z0))
   ENCODE_F(z0, z1) -> c4(ENCARG(z1))
S tuples:
   F(f(z0, z1), z2) -> c2(F(z0, f(z1, z2)), F(z1, z2))
   F(z0, f(z1, z2)) -> c3(F(z1, z1))
K tuples:none
Defined Rule Symbols:   f_2, encArg_1, encode_f_2

Defined Pair Symbols:   ENCARG_1, F_2, ENCODE_F_2

Compound Symbols:   c_3, c2_2, c3_1, c4_1


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

(39) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   ENCODE_F(z0, z1) -> c4(ENCARG(z0))
   ENCODE_F(z0, z1) -> c4(ENCARG(z1))

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))
   f(f(z0, z1), z2) -> f(z0, f(z1, z2))
   f(z0, f(z1, z2)) -> f(z1, z1)
Tuples:
   ENCARG(cons_f(z0, z1)) -> c(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(f(z0, z1), z2) -> c2(F(z0, f(z1, z2)), F(z1, z2))
   F(z0, f(z1, z2)) -> c3(F(z1, z1))
   ENCODE_F(z0, z1) -> c4(F(encArg(z0), encArg(z1)))
S tuples:
   F(f(z0, z1), z2) -> c2(F(z0, f(z1, z2)), F(z1, z2))
   F(z0, f(z1, z2)) -> c3(F(z1, z1))
K tuples:none
Defined Rule Symbols:   f_2, encArg_1, encode_f_2

Defined Pair Symbols:   ENCARG_1, F_2, ENCODE_F_2

Compound Symbols:   c_3, c2_2, c3_1, c4_1


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

(41) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_f(z0, z1) -> f(encArg(z0), encArg(z1))

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   f(f(z0, z1), z2) -> f(z0, f(z1, z2))
   f(z0, f(z1, z2)) -> f(z1, z1)
Tuples:
   ENCARG(cons_f(z0, z1)) -> c(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1))
   F(f(z0, z1), z2) -> c2(F(z0, f(z1, z2)), F(z1, z2))
   F(z0, f(z1, z2)) -> c3(F(z1, z1))
   ENCODE_F(z0, z1) -> c4(F(encArg(z0), encArg(z1)))
S tuples:
   F(f(z0, z1), z2) -> c2(F(z0, f(z1, z2)), F(z1, z2))
   F(z0, f(z1, z2)) -> c3(F(z1, z1))
K tuples:none
Defined Rule Symbols:   encArg_1, f_2

Defined Pair Symbols:   ENCARG_1, F_2, ENCODE_F_2

Compound Symbols:   c_3, c2_2, c3_1, c4_1


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

(43) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_f(z0, z1)) -> c(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) by 
   ENCARG(cons_f(x0, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1)))
   ENCARG(cons_f(cons_f(z0, z1), x1)) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1))

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   f(f(z0, z1), z2) -> f(z0, f(z1, z2))
   f(z0, f(z1, z2)) -> f(z1, z1)
Tuples:
   F(f(z0, z1), z2) -> c2(F(z0, f(z1, z2)), F(z1, z2))
   F(z0, f(z1, z2)) -> c3(F(z1, z1))
   ENCODE_F(z0, z1) -> c4(F(encArg(z0), encArg(z1)))
   ENCARG(cons_f(x0, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1)))
   ENCARG(cons_f(cons_f(z0, z1), x1)) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1))
S tuples:
   F(f(z0, z1), z2) -> c2(F(z0, f(z1, z2)), F(z1, z2))
   F(z0, f(z1, z2)) -> c3(F(z1, z1))
K tuples:none
Defined Rule Symbols:   encArg_1, f_2

Defined Pair Symbols:   F_2, ENCODE_F_2, ENCARG_1

Compound Symbols:   c2_2, c3_1, c4_1, c_3


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

(45) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_F(z0, z1) -> c4(F(encArg(z0), encArg(z1))) by 
   ENCODE_F(x0, cons_f(z0, z1)) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))))
   ENCODE_F(cons_f(z0, z1), x1) -> c4(F(f(encArg(z0), encArg(z1)), encArg(x1)))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   f(f(z0, z1), z2) -> f(z0, f(z1, z2))
   f(z0, f(z1, z2)) -> f(z1, z1)
Tuples:
   F(f(z0, z1), z2) -> c2(F(z0, f(z1, z2)), F(z1, z2))
   F(z0, f(z1, z2)) -> c3(F(z1, z1))
   ENCARG(cons_f(x0, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1)))
   ENCARG(cons_f(cons_f(z0, z1), x1)) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1))
   ENCODE_F(x0, cons_f(z0, z1)) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))))
   ENCODE_F(cons_f(z0, z1), x1) -> c4(F(f(encArg(z0), encArg(z1)), encArg(x1)))
S tuples:
   F(f(z0, z1), z2) -> c2(F(z0, f(z1, z2)), F(z1, z2))
   F(z0, f(z1, z2)) -> c3(F(z1, z1))
K tuples:none
Defined Rule Symbols:   encArg_1, f_2

Defined Pair Symbols:   F_2, ENCARG_1, ENCODE_F_2

Compound Symbols:   c2_2, c3_1, c_3, c4_1


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

(47) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(z0, f(z1, z2)) -> c3(F(z1, z1)) by 
   F(z0, f(f(y0, y1), z2)) -> c3(F(f(y0, y1), f(y0, y1)))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   f(f(z0, z1), z2) -> f(z0, f(z1, z2))
   f(z0, f(z1, z2)) -> f(z1, z1)
Tuples:
   F(f(z0, z1), z2) -> c2(F(z0, f(z1, z2)), F(z1, z2))
   ENCARG(cons_f(x0, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1)))
   ENCARG(cons_f(cons_f(z0, z1), x1)) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1))
   ENCODE_F(x0, cons_f(z0, z1)) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))))
   ENCODE_F(cons_f(z0, z1), x1) -> c4(F(f(encArg(z0), encArg(z1)), encArg(x1)))
   F(z0, f(f(y0, y1), z2)) -> c3(F(f(y0, y1), f(y0, y1)))
S tuples:
   F(f(z0, z1), z2) -> c2(F(z0, f(z1, z2)), F(z1, z2))
   F(z0, f(f(y0, y1), z2)) -> c3(F(f(y0, y1), f(y0, y1)))
K tuples:none
Defined Rule Symbols:   encArg_1, f_2

Defined Pair Symbols:   F_2, ENCARG_1, ENCODE_F_2

Compound Symbols:   c2_2, c_3, c4_1, c3_1


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

(49) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   F(z0, f(f(y0, y1), z2)) -> c3(F(f(y0, y1), f(y0, y1)))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   f(f(z0, z1), z2) -> f(z0, f(z1, z2))
   f(z0, f(z1, z2)) -> f(z1, z1)
Tuples:
   F(f(z0, z1), z2) -> c2(F(z0, f(z1, z2)), F(z1, z2))
   ENCARG(cons_f(x0, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1)))
   ENCARG(cons_f(cons_f(z0, z1), x1)) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1))
   ENCODE_F(x0, cons_f(z0, z1)) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))))
   ENCODE_F(cons_f(z0, z1), x1) -> c4(F(f(encArg(z0), encArg(z1)), encArg(x1)))
S tuples:
   F(f(z0, z1), z2) -> c2(F(z0, f(z1, z2)), F(z1, z2))
K tuples:none
Defined Rule Symbols:   encArg_1, f_2

Defined Pair Symbols:   F_2, ENCARG_1, ENCODE_F_2

Compound Symbols:   c2_2, c_3, c4_1


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

(51) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_f(x0, cons_f(z0, z1))) -> c(F(encArg(x0), f(encArg(z0), encArg(z1))), ENCARG(x0), ENCARG(cons_f(z0, z1))) by 
   ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1))))
   ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2)))
   ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2)))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   f(f(z0, z1), z2) -> f(z0, f(z1, z2))
   f(z0, f(z1, z2)) -> f(z1, z1)
Tuples:
   F(f(z0, z1), z2) -> c2(F(z0, f(z1, z2)), F(z1, z2))
   ENCARG(cons_f(cons_f(z0, z1), x1)) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1))
   ENCODE_F(x0, cons_f(z0, z1)) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1))))
   ENCODE_F(cons_f(z0, z1), x1) -> c4(F(f(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1))))
   ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2)))
   ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2)))
S tuples:
   F(f(z0, z1), z2) -> c2(F(z0, f(z1, z2)), F(z1, z2))
K tuples:none
Defined Rule Symbols:   encArg_1, f_2

Defined Pair Symbols:   F_2, ENCARG_1, ENCODE_F_2

Compound Symbols:   c2_2, c_3, c4_1


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

(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_F(x0, cons_f(z0, z1)) -> c4(F(encArg(x0), f(encArg(z0), encArg(z1)))) by 
   ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))))
   ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c4(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))))
   ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c4(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1))
   f(f(z0, z1), z2) -> f(z0, f(z1, z2))
   f(z0, f(z1, z2)) -> f(z1, z1)
Tuples:
   F(f(z0, z1), z2) -> c2(F(z0, f(z1, z2)), F(z1, z2))
   ENCARG(cons_f(cons_f(z0, z1), x1)) -> c(F(f(encArg(z0), encArg(z1)), encArg(x1)), ENCARG(cons_f(z0, z1)), ENCARG(x1))
   ENCODE_F(cons_f(z0, z1), x1) -> c4(F(f(encArg(z0), encArg(z1)), encArg(x1)))
   ENCARG(cons_f(x0, cons_f(x1, cons_f(z0, z1)))) -> c(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))), ENCARG(x0), ENCARG(cons_f(x1, cons_f(z0, z1))))
   ENCARG(cons_f(x0, cons_f(cons_f(z0, z1), x2))) -> c(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))), ENCARG(x0), ENCARG(cons_f(cons_f(z0, z1), x2)))
   ENCARG(cons_f(cons_f(z0, z1), cons_f(x1, x2))) -> c(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))), ENCARG(cons_f(z0, z1)), ENCARG(cons_f(x1, x2)))
   ENCODE_F(x0, cons_f(x1, cons_f(z0, z1))) -> c4(F(encArg(x0), f(encArg(x1), f(encArg(z0), encArg(z1)))))
   ENCODE_F(x0, cons_f(cons_f(z0, z1), x2)) -> c4(F(encArg(x0), f(f(encArg(z0), encArg(z1)), encArg(x2))))
   ENCODE_F(cons_f(z0, z1), cons_f(x1, x2)) -> c4(F(f(encArg(z0), encArg(z1)), f(encArg(x1), encArg(x2))))
S tuples:
   F(f(z0, z1), z2) -> c2(F(z0, f(z1, z2)), F(z1, z2))
K tuples:none
Defined Rule Symbols:   encArg_1, f_2

Defined Pair Symbols:   F_2, ENCARG_1, ENCODE_F_2

Compound Symbols:   c2_2, c_3, c4_1