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


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 159 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 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), 612 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), 0 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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 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:

   if(if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v))

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_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))


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

(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:

   if(if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v))

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

   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

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:

   if(if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v))

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

   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

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:

   if(if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v))

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

   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
if(if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v))
encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
if :: if:encArg:encode_if -> if:encArg:encode_if -> if:encArg:encode_if -> if:encArg:encode_if
encArg :: cons_if -> if:encArg:encode_if
cons_if :: cons_if -> cons_if -> cons_if -> cons_if
encode_if :: cons_if -> cons_if -> cons_if -> if:encArg:encode_if
hole_if:encArg:encode_if1_0 :: if:encArg:encode_if
hole_cons_if2_0 :: cons_if
gen_cons_if3_0 :: Nat -> cons_if

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

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

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

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

(10)
Obligation:
Innermost TRS:
Rules:
if(if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v))
encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
if :: if:encArg:encode_if -> if:encArg:encode_if -> if:encArg:encode_if -> if:encArg:encode_if
encArg :: cons_if -> if:encArg:encode_if
cons_if :: cons_if -> cons_if -> cons_if -> cons_if
encode_if :: cons_if -> cons_if -> cons_if -> if:encArg:encode_if
hole_if:encArg:encode_if1_0 :: if:encArg:encode_if
hole_cons_if2_0 :: cons_if
gen_cons_if3_0 :: Nat -> cons_if


Generator Equations:
gen_cons_if3_0(0) <=> hole_cons_if2_0
gen_cons_if3_0(+(x, 1)) <=> cons_if(hole_cons_if2_0, hole_cons_if2_0, gen_cons_if3_0(x))


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

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

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_cons_if3_0(+(1, n12_0))) -> *4_0, rt in Omega(0)

Induction Base:
encArg(gen_cons_if3_0(+(1, 0)))

Induction Step:
encArg(gen_cons_if3_0(+(1, +(n12_0, 1)))) ->_R^Omega(0)
if(encArg(hole_cons_if2_0), encArg(hole_cons_if2_0), encArg(gen_cons_if3_0(+(1, n12_0)))) ->_IH
if(encArg(hole_cons_if2_0), encArg(hole_cons_if2_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:

   if(if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v))

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

   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

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:

   if(if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

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:

   if(c_if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v))

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

   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   if(x0, x1, x2) -> c_if(x0, x1, x2)

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:

   if(c_if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v)) [1]
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   if(x0, x1, x2) -> c_if(x0, x1, x2) [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:

   if(c_if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v)) [1]
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   if(x0, x1, x2) -> c_if(x0, x1, x2) [0]

The TRS has the following type information:
if :: c_if -> c_if -> c_if -> c_if
c_if :: c_if -> c_if -> c_if -> c_if
encArg :: cons_if -> c_if
cons_if :: cons_if -> cons_if -> cons_if -> cons_if
encode_if :: cons_if -> cons_if -> cons_if -> c_if

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_if(v0, v1, v2) -> null_encode_if [0]
   if(v0, v1, v2) -> null_if [0]

And the following fresh constants:    null_encArg, null_encode_if, null_if, 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:

   if(c_if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v)) [1]
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   if(x0, x1, x2) -> c_if(x0, x1, x2) [0]
   encArg(v0) -> null_encArg [0]
   encode_if(v0, v1, v2) -> null_encode_if [0]
   if(v0, v1, v2) -> null_if [0]

The TRS has the following type information:
if :: c_if:null_encArg:null_encode_if:null_if -> c_if:null_encArg:null_encode_if:null_if -> c_if:null_encArg:null_encode_if:null_if -> c_if:null_encArg:null_encode_if:null_if
c_if :: c_if:null_encArg:null_encode_if:null_if -> c_if:null_encArg:null_encode_if:null_if -> c_if:null_encArg:null_encode_if:null_if -> c_if:null_encArg:null_encode_if:null_if
encArg :: cons_if -> c_if:null_encArg:null_encode_if:null_if
cons_if :: cons_if -> cons_if -> cons_if -> cons_if
encode_if :: cons_if -> cons_if -> cons_if -> c_if:null_encArg:null_encode_if:null_if
null_encArg :: c_if:null_encArg:null_encode_if:null_if
null_encode_if :: c_if:null_encArg:null_encode_if:null_if
null_if :: c_if:null_encArg:null_encode_if:null_if
const :: cons_if

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_if => 0
   null_if => 0
   const => 0

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

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

   encArg(z') -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z' = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   encode_if(z', z'', z1) -{ 0 }-> if(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z1 = x_3, z' = x_1, x_3 >= 0, x_2 >= 0, z'' = x_2
   encode_if(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   if(z', z'', z1) -{ 1 }-> if(x, if(y, u, v), if(z, u, v)) :|: z1 = v, z >= 0, v >= 0, x >= 0, y >= 0, z' = 1 + x + y + z, z'' = u, u >= 0
   if(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   if(z', z'', z1) -{ 0 }-> 1 + x0 + x1 + x2 :|: z'' = x1, x0 >= 0, x1 >= 0, z1 = x2, x2 >= 0, z' = x0

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_if_3
   if_3

Due to the following rules being added:

   encArg(v0) -> const [0]
   encode_if(v0, v1, v2) -> const [0]
   if(v0, v1, v2) -> 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:

   if(c_if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v)) [1]
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   if(x0, x1, x2) -> c_if(x0, x1, x2) [0]
   encArg(v0) -> const [0]
   encode_if(v0, v1, v2) -> const [0]
   if(v0, v1, v2) -> const [0]

The TRS has the following type information:
if :: c_if:const -> c_if:const -> c_if:const -> c_if:const
c_if :: c_if:const -> c_if:const -> c_if:const -> c_if:const
encArg :: cons_if -> c_if:const
cons_if :: cons_if -> cons_if -> cons_if -> cons_if
encode_if :: cons_if -> cons_if -> cons_if -> c_if:const
const :: c_if:const
const1 :: cons_if

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:

   if(c_if(x, c_if(x', y', z'), c_if(x'', y'', z'')), u, v) -> if(x, if(x', if(y', u, v), if(z', u, v)), if(x'', if(y'', u, v), if(z'', u, v))) [3]
   if(c_if(x, c_if(x', y', z'), z), u, v) -> if(x, if(x', if(y', u, v), if(z', u, v)), c_if(z, u, v)) [2]
   if(c_if(x, c_if(x', y', z'), z), u, v) -> if(x, if(x', if(y', u, v), if(z', u, v)), const) [2]
   if(c_if(x, y, c_if(x3, y1, z1)), u, v) -> if(x, c_if(y, u, v), if(x3, if(y1, u, v), if(z1, u, v))) [2]
   if(c_if(x, y, z), u, v) -> if(x, c_if(y, u, v), c_if(z, u, v)) [1]
   if(c_if(x, y, z), u, v) -> if(x, c_if(y, u, v), const) [1]
   if(c_if(x, y, c_if(x4, y2, z2)), u, v) -> if(x, const, if(x4, if(y2, u, v), if(z2, u, v))) [2]
   if(c_if(x, y, z), u, v) -> if(x, const, c_if(z, u, v)) [1]
   if(c_if(x, y, z), u, v) -> if(x, const, const) [1]
   encArg(cons_if(cons_if(x_1', x_2', x_3'), cons_if(x_1'', x_2'', x_3''), cons_if(x_12, x_22, x_32))) -> if(if(encArg(x_1'), encArg(x_2'), encArg(x_3')), if(encArg(x_1''), encArg(x_2''), encArg(x_3'')), if(encArg(x_12), encArg(x_22), encArg(x_32))) [0]
   encArg(cons_if(cons_if(x_1', x_2', x_3'), cons_if(x_1'', x_2'', x_3''), x_3)) -> if(if(encArg(x_1'), encArg(x_2'), encArg(x_3')), if(encArg(x_1''), encArg(x_2''), encArg(x_3'')), const) [0]
   encArg(cons_if(cons_if(x_1', x_2', x_3'), x_2, cons_if(x_13, x_23, x_33))) -> if(if(encArg(x_1'), encArg(x_2'), encArg(x_3')), const, if(encArg(x_13), encArg(x_23), encArg(x_33))) [0]
   encArg(cons_if(cons_if(x_1', x_2', x_3'), x_2, x_3)) -> if(if(encArg(x_1'), encArg(x_2'), encArg(x_3')), const, const) [0]
   encArg(cons_if(x_1, cons_if(x_11, x_21, x_31), cons_if(x_14, x_24, x_34))) -> if(const, if(encArg(x_11), encArg(x_21), encArg(x_31)), if(encArg(x_14), encArg(x_24), encArg(x_34))) [0]
   encArg(cons_if(x_1, cons_if(x_11, x_21, x_31), x_3)) -> if(const, if(encArg(x_11), encArg(x_21), encArg(x_31)), const) [0]
   encArg(cons_if(x_1, x_2, cons_if(x_15, x_25, x_35))) -> if(const, const, if(encArg(x_15), encArg(x_25), encArg(x_35))) [0]
   encArg(cons_if(x_1, x_2, x_3)) -> if(const, const, const) [0]
   encode_if(cons_if(x_16, x_26, x_36), cons_if(x_17, x_27, x_37), cons_if(x_19, x_29, x_39)) -> if(if(encArg(x_16), encArg(x_26), encArg(x_36)), if(encArg(x_17), encArg(x_27), encArg(x_37)), if(encArg(x_19), encArg(x_29), encArg(x_39))) [0]
   encode_if(cons_if(x_16, x_26, x_36), cons_if(x_17, x_27, x_37), x_3) -> if(if(encArg(x_16), encArg(x_26), encArg(x_36)), if(encArg(x_17), encArg(x_27), encArg(x_37)), const) [0]
   encode_if(cons_if(x_16, x_26, x_36), x_2, cons_if(x_110, x_210, x_310)) -> if(if(encArg(x_16), encArg(x_26), encArg(x_36)), const, if(encArg(x_110), encArg(x_210), encArg(x_310))) [0]
   encode_if(cons_if(x_16, x_26, x_36), x_2, x_3) -> if(if(encArg(x_16), encArg(x_26), encArg(x_36)), const, const) [0]
   encode_if(x_1, cons_if(x_18, x_28, x_38), cons_if(x_111, x_211, x_311)) -> if(const, if(encArg(x_18), encArg(x_28), encArg(x_38)), if(encArg(x_111), encArg(x_211), encArg(x_311))) [0]
   encode_if(x_1, cons_if(x_18, x_28, x_38), x_3) -> if(const, if(encArg(x_18), encArg(x_28), encArg(x_38)), const) [0]
   encode_if(x_1, x_2, cons_if(x_112, x_212, x_312)) -> if(const, const, if(encArg(x_112), encArg(x_212), encArg(x_312))) [0]
   encode_if(x_1, x_2, x_3) -> if(const, const, const) [0]
   if(x0, x1, x2) -> c_if(x0, x1, x2) [0]
   encArg(v0) -> const [0]
   encode_if(v0, v1, v2) -> const [0]
   if(v0, v1, v2) -> const [0]

The TRS has the following type information:
if :: c_if:const -> c_if:const -> c_if:const -> c_if:const
c_if :: c_if:const -> c_if:const -> c_if:const -> c_if:const
encArg :: cons_if -> c_if:const
cons_if :: cons_if -> cons_if -> cons_if -> cons_if
encode_if :: cons_if -> cons_if -> cons_if -> c_if:const
const :: c_if:const
const1 :: cons_if

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(z3) -{ 0 }-> if(if(encArg(x_1'), encArg(x_2'), encArg(x_3')), if(encArg(x_1''), encArg(x_2''), encArg(x_3'')), if(encArg(x_12), encArg(x_22), encArg(x_32))) :|: x_32 >= 0, x_1'' >= 0, z3 = 1 + (1 + x_1' + x_2' + x_3') + (1 + x_1'' + x_2'' + x_3'') + (1 + x_12 + x_22 + x_32), x_3' >= 0, x_2' >= 0, x_1' >= 0, x_2'' >= 0, x_12 >= 0, x_3'' >= 0, x_22 >= 0
   encArg(z3) -{ 0 }-> if(if(encArg(x_1'), encArg(x_2'), encArg(x_3')), if(encArg(x_1''), encArg(x_2''), encArg(x_3'')), 0) :|: z3 = 1 + (1 + x_1' + x_2' + x_3') + (1 + x_1'' + x_2'' + x_3'') + x_3, x_1'' >= 0, x_3' >= 0, x_2' >= 0, x_1' >= 0, x_3 >= 0, x_2'' >= 0, x_3'' >= 0
   encArg(z3) -{ 0 }-> if(if(encArg(x_1'), encArg(x_2'), encArg(x_3')), 0, if(encArg(x_13), encArg(x_23), encArg(x_33))) :|: x_3' >= 0, x_2' >= 0, x_13 >= 0, x_1' >= 0, z3 = 1 + (1 + x_1' + x_2' + x_3') + x_2 + (1 + x_13 + x_23 + x_33), x_23 >= 0, x_2 >= 0, x_33 >= 0
   encArg(z3) -{ 0 }-> if(if(encArg(x_1'), encArg(x_2'), encArg(x_3')), 0, 0) :|: x_3' >= 0, x_2' >= 0, x_1' >= 0, x_3 >= 0, x_2 >= 0, z3 = 1 + (1 + x_1' + x_2' + x_3') + x_2 + x_3
   encArg(z3) -{ 0 }-> if(0, if(encArg(x_11), encArg(x_21), encArg(x_31)), if(encArg(x_14), encArg(x_24), encArg(x_34))) :|: x_11 >= 0, x_1 >= 0, x_14 >= 0, x_34 >= 0, x_24 >= 0, z3 = 1 + x_1 + (1 + x_11 + x_21 + x_31) + (1 + x_14 + x_24 + x_34), x_31 >= 0, x_21 >= 0
   encArg(z3) -{ 0 }-> if(0, if(encArg(x_11), encArg(x_21), encArg(x_31)), 0) :|: x_11 >= 0, x_1 >= 0, z3 = 1 + x_1 + (1 + x_11 + x_21 + x_31) + x_3, x_3 >= 0, x_31 >= 0, x_21 >= 0
   encArg(z3) -{ 0 }-> if(0, 0, if(encArg(x_15), encArg(x_25), encArg(x_35))) :|: x_15 >= 0, z3 = 1 + x_1 + x_2 + (1 + x_15 + x_25 + x_35), x_1 >= 0, x_35 >= 0, x_25 >= 0, x_2 >= 0
   encArg(z3) -{ 0 }-> if(0, 0, 0) :|: x_1 >= 0, x_3 >= 0, x_2 >= 0, z3 = 1 + x_1 + x_2 + x_3
   encArg(z3) -{ 0 }-> 0 :|: v0 >= 0, z3 = v0
   encode_if(z3, z4, z5) -{ 0 }-> if(if(encArg(x_16), encArg(x_26), encArg(x_36)), if(encArg(x_17), encArg(x_27), encArg(x_37)), if(encArg(x_19), encArg(x_29), encArg(x_39))) :|: x_36 >= 0, x_16 >= 0, x_26 >= 0, x_17 >= 0, z3 = 1 + x_16 + x_26 + x_36, z5 = 1 + x_19 + x_29 + x_39, x_27 >= 0, x_37 >= 0, z4 = 1 + x_17 + x_27 + x_37, x_19 >= 0, x_29 >= 0, x_39 >= 0
   encode_if(z3, z4, z5) -{ 0 }-> if(if(encArg(x_16), encArg(x_26), encArg(x_36)), if(encArg(x_17), encArg(x_27), encArg(x_37)), 0) :|: x_36 >= 0, x_16 >= 0, x_26 >= 0, x_17 >= 0, z3 = 1 + x_16 + x_26 + x_36, x_3 >= 0, x_27 >= 0, x_37 >= 0, z4 = 1 + x_17 + x_27 + x_37, z5 = x_3
   encode_if(z3, z4, z5) -{ 0 }-> if(if(encArg(x_16), encArg(x_26), encArg(x_36)), 0, if(encArg(x_110), encArg(x_210), encArg(x_310))) :|: x_36 >= 0, z4 = x_2, x_16 >= 0, x_26 >= 0, z3 = 1 + x_16 + x_26 + x_36, z5 = 1 + x_110 + x_210 + x_310, x_2 >= 0, x_310 >= 0, x_110 >= 0, x_210 >= 0
   encode_if(z3, z4, z5) -{ 0 }-> if(if(encArg(x_16), encArg(x_26), encArg(x_36)), 0, 0) :|: x_36 >= 0, z4 = x_2, x_16 >= 0, x_26 >= 0, z3 = 1 + x_16 + x_26 + x_36, x_3 >= 0, x_2 >= 0, z5 = x_3
   encode_if(z3, z4, z5) -{ 0 }-> if(0, if(encArg(x_18), encArg(x_28), encArg(x_38)), if(encArg(x_111), encArg(x_211), encArg(x_311))) :|: x_311 >= 0, x_1 >= 0, z3 = x_1, x_38 >= 0, z5 = 1 + x_111 + x_211 + x_311, x_18 >= 0, x_28 >= 0, x_211 >= 0, z4 = 1 + x_18 + x_28 + x_38, x_111 >= 0
   encode_if(z3, z4, z5) -{ 0 }-> if(0, if(encArg(x_18), encArg(x_28), encArg(x_38)), 0) :|: x_1 >= 0, z3 = x_1, x_38 >= 0, x_3 >= 0, x_18 >= 0, x_28 >= 0, z4 = 1 + x_18 + x_28 + x_38, z5 = x_3
   encode_if(z3, z4, z5) -{ 0 }-> if(0, 0, if(encArg(x_112), encArg(x_212), encArg(x_312))) :|: x_1 >= 0, z4 = x_2, x_212 >= 0, x_312 >= 0, z3 = x_1, x_2 >= 0, z5 = 1 + x_112 + x_212 + x_312, x_112 >= 0
   encode_if(z3, z4, z5) -{ 0 }-> if(0, 0, 0) :|: x_1 >= 0, z4 = x_2, z3 = x_1, x_3 >= 0, x_2 >= 0, z5 = x_3
   encode_if(z3, z4, z5) -{ 0 }-> 0 :|: z4 = v1, v0 >= 0, z3 = v0, v1 >= 0, z5 = v2, v2 >= 0
   if(z3, z4, z5) -{ 3 }-> if(x, if(x', if(y', u, v), if(z', u, v)), if(x'', if(y'', u, v), if(z'', u, v))) :|: z5 = v, v >= 0, z'' >= 0, x >= 0, x' >= 0, z' >= 0, z3 = 1 + x + (1 + x' + y' + z') + (1 + x'' + y'' + z''), y' >= 0, y'' >= 0, x'' >= 0, z4 = u, u >= 0
   if(z3, z4, z5) -{ 2 }-> if(x, if(x', if(y', u, v), if(z', u, v)), 0) :|: z5 = v, z >= 0, v >= 0, x >= 0, x' >= 0, z' >= 0, y' >= 0, z3 = 1 + x + (1 + x' + y' + z') + z, z4 = u, u >= 0
   if(z3, z4, z5) -{ 2 }-> if(x, if(x', if(y', u, v), if(z', u, v)), 1 + z + u + v) :|: z5 = v, z >= 0, v >= 0, x >= 0, x' >= 0, z' >= 0, y' >= 0, z3 = 1 + x + (1 + x' + y' + z') + z, z4 = u, u >= 0
   if(z3, z4, z5) -{ 2 }-> if(x, 0, if(x4, if(y2, u, v), if(z2, u, v))) :|: z5 = v, x4 >= 0, v >= 0, x >= 0, y >= 0, y2 >= 0, z3 = 1 + x + y + (1 + x4 + y2 + z2), z4 = u, z2 >= 0, u >= 0
   if(z3, z4, z5) -{ 1 }-> if(x, 0, 0) :|: z5 = v, z >= 0, v >= 0, x >= 0, y >= 0, z3 = 1 + x + y + z, z4 = u, u >= 0
   if(z3, z4, z5) -{ 1 }-> if(x, 0, 1 + z + u + v) :|: z5 = v, z >= 0, v >= 0, x >= 0, y >= 0, z3 = 1 + x + y + z, z4 = u, u >= 0
   if(z3, z4, z5) -{ 2 }-> if(x, 1 + y + u + v, if(x3, if(y1, u, v), if(z1, u, v))) :|: y1 >= 0, z5 = v, z1 >= 0, v >= 0, z3 = 1 + x + y + (1 + x3 + y1 + z1), x >= 0, y >= 0, z4 = u, x3 >= 0, u >= 0
   if(z3, z4, z5) -{ 1 }-> if(x, 1 + y + u + v, 0) :|: z5 = v, z >= 0, v >= 0, x >= 0, y >= 0, z3 = 1 + x + y + z, z4 = u, u >= 0
   if(z3, z4, z5) -{ 1 }-> if(x, 1 + y + u + v, 1 + z + u + v) :|: z5 = v, z >= 0, v >= 0, x >= 0, y >= 0, z3 = 1 + x + y + z, z4 = u, u >= 0
   if(z3, z4, z5) -{ 0 }-> 0 :|: z4 = v1, v0 >= 0, z3 = v0, v1 >= 0, z5 = v2, v2 >= 0
   if(z3, z4, z5) -{ 0 }-> 1 + x0 + x1 + x2 :|: z4 = x1, x0 >= 0, x1 >= 0, z3 = x0, z5 = x2, x2 >= 0


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

(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(z3) -{ 0 }-> if(if(encArg(x_1'), encArg(x_2'), encArg(x_3')), if(encArg(x_1''), encArg(x_2''), encArg(x_3'')), if(encArg(x_12), encArg(x_22), encArg(x_32))) :|: x_32 >= 0, x_1'' >= 0, z3 = 1 + (1 + x_1' + x_2' + x_3') + (1 + x_1'' + x_2'' + x_3'') + (1 + x_12 + x_22 + x_32), x_3' >= 0, x_2' >= 0, x_1' >= 0, x_2'' >= 0, x_12 >= 0, x_3'' >= 0, x_22 >= 0
   encArg(z3) -{ 0 }-> if(if(encArg(x_1'), encArg(x_2'), encArg(x_3')), if(encArg(x_1''), encArg(x_2''), encArg(x_3'')), 0) :|: z3 = 1 + (1 + x_1' + x_2' + x_3') + (1 + x_1'' + x_2'' + x_3'') + x_3, x_1'' >= 0, x_3' >= 0, x_2' >= 0, x_1' >= 0, x_3 >= 0, x_2'' >= 0, x_3'' >= 0
   encArg(z3) -{ 0 }-> if(if(encArg(x_1'), encArg(x_2'), encArg(x_3')), 0, if(encArg(x_13), encArg(x_23), encArg(x_33))) :|: x_3' >= 0, x_2' >= 0, x_13 >= 0, x_1' >= 0, z3 = 1 + (1 + x_1' + x_2' + x_3') + x_2 + (1 + x_13 + x_23 + x_33), x_23 >= 0, x_2 >= 0, x_33 >= 0
   encArg(z3) -{ 0 }-> if(if(encArg(x_1'), encArg(x_2'), encArg(x_3')), 0, 0) :|: x_3' >= 0, x_2' >= 0, x_1' >= 0, x_3 >= 0, x_2 >= 0, z3 = 1 + (1 + x_1' + x_2' + x_3') + x_2 + x_3
   encArg(z3) -{ 0 }-> if(0, if(encArg(x_11), encArg(x_21), encArg(x_31)), if(encArg(x_14), encArg(x_24), encArg(x_34))) :|: x_11 >= 0, x_1 >= 0, x_14 >= 0, x_34 >= 0, x_24 >= 0, z3 = 1 + x_1 + (1 + x_11 + x_21 + x_31) + (1 + x_14 + x_24 + x_34), x_31 >= 0, x_21 >= 0
   encArg(z3) -{ 0 }-> if(0, if(encArg(x_11), encArg(x_21), encArg(x_31)), 0) :|: x_11 >= 0, x_1 >= 0, z3 = 1 + x_1 + (1 + x_11 + x_21 + x_31) + x_3, x_3 >= 0, x_31 >= 0, x_21 >= 0
   encArg(z3) -{ 0 }-> if(0, 0, if(encArg(x_15), encArg(x_25), encArg(x_35))) :|: x_15 >= 0, z3 = 1 + x_1 + x_2 + (1 + x_15 + x_25 + x_35), x_1 >= 0, x_35 >= 0, x_25 >= 0, x_2 >= 0
   encArg(z3) -{ 0 }-> if(0, 0, 0) :|: x_1 >= 0, x_3 >= 0, x_2 >= 0, z3 = 1 + x_1 + x_2 + x_3
   encArg(z3) -{ 0 }-> 0 :|: z3 >= 0
   encode_if(z3, z4, z5) -{ 0 }-> if(if(encArg(x_16), encArg(x_26), encArg(x_36)), if(encArg(x_17), encArg(x_27), encArg(x_37)), if(encArg(x_19), encArg(x_29), encArg(x_39))) :|: x_36 >= 0, x_16 >= 0, x_26 >= 0, x_17 >= 0, z3 = 1 + x_16 + x_26 + x_36, z5 = 1 + x_19 + x_29 + x_39, x_27 >= 0, x_37 >= 0, z4 = 1 + x_17 + x_27 + x_37, x_19 >= 0, x_29 >= 0, x_39 >= 0
   encode_if(z3, z4, z5) -{ 0 }-> if(if(encArg(x_16), encArg(x_26), encArg(x_36)), if(encArg(x_17), encArg(x_27), encArg(x_37)), 0) :|: x_36 >= 0, x_16 >= 0, x_26 >= 0, x_17 >= 0, z3 = 1 + x_16 + x_26 + x_36, z5 >= 0, x_27 >= 0, x_37 >= 0, z4 = 1 + x_17 + x_27 + x_37
   encode_if(z3, z4, z5) -{ 0 }-> if(if(encArg(x_16), encArg(x_26), encArg(x_36)), 0, if(encArg(x_110), encArg(x_210), encArg(x_310))) :|: x_36 >= 0, x_16 >= 0, x_26 >= 0, z3 = 1 + x_16 + x_26 + x_36, z5 = 1 + x_110 + x_210 + x_310, z4 >= 0, x_310 >= 0, x_110 >= 0, x_210 >= 0
   encode_if(z3, z4, z5) -{ 0 }-> if(if(encArg(x_16), encArg(x_26), encArg(x_36)), 0, 0) :|: x_36 >= 0, x_16 >= 0, x_26 >= 0, z3 = 1 + x_16 + x_26 + x_36, z5 >= 0, z4 >= 0
   encode_if(z3, z4, z5) -{ 0 }-> if(0, if(encArg(x_18), encArg(x_28), encArg(x_38)), if(encArg(x_111), encArg(x_211), encArg(x_311))) :|: x_311 >= 0, z3 >= 0, x_38 >= 0, z5 = 1 + x_111 + x_211 + x_311, x_18 >= 0, x_28 >= 0, x_211 >= 0, z4 = 1 + x_18 + x_28 + x_38, x_111 >= 0
   encode_if(z3, z4, z5) -{ 0 }-> if(0, if(encArg(x_18), encArg(x_28), encArg(x_38)), 0) :|: z3 >= 0, x_38 >= 0, z5 >= 0, x_18 >= 0, x_28 >= 0, z4 = 1 + x_18 + x_28 + x_38
   encode_if(z3, z4, z5) -{ 0 }-> if(0, 0, if(encArg(x_112), encArg(x_212), encArg(x_312))) :|: z3 >= 0, x_212 >= 0, x_312 >= 0, z4 >= 0, z5 = 1 + x_112 + x_212 + x_312, x_112 >= 0
   encode_if(z3, z4, z5) -{ 0 }-> if(0, 0, 0) :|: z3 >= 0, z5 >= 0, z4 >= 0
   encode_if(z3, z4, z5) -{ 0 }-> 0 :|: z3 >= 0, z4 >= 0, z5 >= 0
   if(z3, z4, z5) -{ 3 }-> if(x, if(x', if(y', z4, z5), if(z', z4, z5)), if(x'', if(y'', z4, z5), if(z'', z4, z5))) :|: z5 >= 0, z'' >= 0, x >= 0, x' >= 0, z' >= 0, z3 = 1 + x + (1 + x' + y' + z') + (1 + x'' + y'' + z''), y' >= 0, y'' >= 0, x'' >= 0, z4 >= 0
   if(z3, z4, z5) -{ 2 }-> if(x, if(x', if(y', z4, z5), if(z', z4, z5)), 0) :|: z >= 0, z5 >= 0, x >= 0, x' >= 0, z' >= 0, y' >= 0, z3 = 1 + x + (1 + x' + y' + z') + z, z4 >= 0
   if(z3, z4, z5) -{ 2 }-> if(x, if(x', if(y', z4, z5), if(z', z4, z5)), 1 + z + z4 + z5) :|: z >= 0, z5 >= 0, x >= 0, x' >= 0, z' >= 0, y' >= 0, z3 = 1 + x + (1 + x' + y' + z') + z, z4 >= 0
   if(z3, z4, z5) -{ 2 }-> if(x, 0, if(x4, if(y2, z4, z5), if(z2, z4, z5))) :|: x4 >= 0, z5 >= 0, x >= 0, y >= 0, y2 >= 0, z3 = 1 + x + y + (1 + x4 + y2 + z2), z2 >= 0, z4 >= 0
   if(z3, z4, z5) -{ 1 }-> if(x, 0, 0) :|: z >= 0, z5 >= 0, x >= 0, y >= 0, z3 = 1 + x + y + z, z4 >= 0
   if(z3, z4, z5) -{ 1 }-> if(x, 0, 1 + z + z4 + z5) :|: z >= 0, z5 >= 0, x >= 0, y >= 0, z3 = 1 + x + y + z, z4 >= 0
   if(z3, z4, z5) -{ 2 }-> if(x, 1 + y + z4 + z5, if(x3, if(y1, z4, z5), if(z1, z4, z5))) :|: y1 >= 0, z1 >= 0, z5 >= 0, z3 = 1 + x + y + (1 + x3 + y1 + z1), x >= 0, y >= 0, x3 >= 0, z4 >= 0
   if(z3, z4, z5) -{ 1 }-> if(x, 1 + y + z4 + z5, 0) :|: z >= 0, z5 >= 0, x >= 0, y >= 0, z3 = 1 + x + y + z, z4 >= 0
   if(z3, z4, z5) -{ 1 }-> if(x, 1 + y + z4 + z5, 1 + z + z4 + z5) :|: z >= 0, z5 >= 0, x >= 0, y >= 0, z3 = 1 + x + y + z, z4 >= 0
   if(z3, z4, z5) -{ 0 }-> 0 :|: z3 >= 0, z4 >= 0, z5 >= 0
   if(z3, z4, z5) -{ 0 }-> 1 + z3 + z4 + z5 :|: z3 >= 0, z4 >= 0, z5 >= 0


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

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2))
   encode_if(z0, z1, z2) -> if(encArg(z0), encArg(z1), encArg(z2))
   if(if(z0, z1, z2), z3, z4) -> if(z0, if(z1, z3, z4), if(z2, z3, z4))
Tuples:
   ENCARG(cons_if(z0, z1, z2)) -> c(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   ENCODE_IF(z0, z1, z2) -> c1(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   IF(if(z0, z1, z2), z3, z4) -> c2(IF(z0, if(z1, z3, z4), if(z2, z3, z4)), IF(z1, z3, z4), IF(z2, z3, z4))
S tuples:
   IF(if(z0, z1, z2), z3, z4) -> c2(IF(z0, if(z1, z3, z4), if(z2, z3, z4)), IF(z1, z3, z4), IF(z2, z3, z4))
K tuples:none
Defined Rule Symbols:   if_3, encArg_1, encode_if_3

Defined Pair Symbols:   ENCARG_1, ENCODE_IF_3, IF_3

Compound Symbols:   c_4, c1_4, c2_3


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

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2))
   encode_if(z0, z1, z2) -> if(encArg(z0), encArg(z1), encArg(z2))
   if(if(z0, z1, z2), z3, z4) -> if(z0, if(z1, z3, z4), if(z2, z3, z4))
Tuples:
   ENCARG(cons_if(z0, z1, z2)) -> c(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   IF(if(z0, z1, z2), z3, z4) -> c2(IF(z0, if(z1, z3, z4), if(z2, z3, z4)), IF(z1, z3, z4), IF(z2, z3, z4))
   ENCODE_IF(z0, z1, z2) -> c3(IF(encArg(z0), encArg(z1), encArg(z2)))
   ENCODE_IF(z0, z1, z2) -> c3(ENCARG(z0))
   ENCODE_IF(z0, z1, z2) -> c3(ENCARG(z1))
   ENCODE_IF(z0, z1, z2) -> c3(ENCARG(z2))
S tuples:
   IF(if(z0, z1, z2), z3, z4) -> c2(IF(z0, if(z1, z3, z4), if(z2, z3, z4)), IF(z1, z3, z4), IF(z2, z3, z4))
K tuples:none
Defined Rule Symbols:   if_3, encArg_1, encode_if_3

Defined Pair Symbols:   ENCARG_1, IF_3, ENCODE_IF_3

Compound Symbols:   c_4, c2_3, c3_1


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

(39) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 3 leading nodes:
   ENCODE_IF(z0, z1, z2) -> c3(ENCARG(z0))
   ENCODE_IF(z0, z1, z2) -> c3(ENCARG(z1))
   ENCODE_IF(z0, z1, z2) -> c3(ENCARG(z2))

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2))
   encode_if(z0, z1, z2) -> if(encArg(z0), encArg(z1), encArg(z2))
   if(if(z0, z1, z2), z3, z4) -> if(z0, if(z1, z3, z4), if(z2, z3, z4))
Tuples:
   ENCARG(cons_if(z0, z1, z2)) -> c(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   IF(if(z0, z1, z2), z3, z4) -> c2(IF(z0, if(z1, z3, z4), if(z2, z3, z4)), IF(z1, z3, z4), IF(z2, z3, z4))
   ENCODE_IF(z0, z1, z2) -> c3(IF(encArg(z0), encArg(z1), encArg(z2)))
S tuples:
   IF(if(z0, z1, z2), z3, z4) -> c2(IF(z0, if(z1, z3, z4), if(z2, z3, z4)), IF(z1, z3, z4), IF(z2, z3, z4))
K tuples:none
Defined Rule Symbols:   if_3, encArg_1, encode_if_3

Defined Pair Symbols:   ENCARG_1, IF_3, ENCODE_IF_3

Compound Symbols:   c_4, c2_3, c3_1


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

(41) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_if(z0, z1, z2) -> if(encArg(z0), encArg(z1), encArg(z2))

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2))
   if(if(z0, z1, z2), z3, z4) -> if(z0, if(z1, z3, z4), if(z2, z3, z4))
Tuples:
   ENCARG(cons_if(z0, z1, z2)) -> c(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2))
   IF(if(z0, z1, z2), z3, z4) -> c2(IF(z0, if(z1, z3, z4), if(z2, z3, z4)), IF(z1, z3, z4), IF(z2, z3, z4))
   ENCODE_IF(z0, z1, z2) -> c3(IF(encArg(z0), encArg(z1), encArg(z2)))
S tuples:
   IF(if(z0, z1, z2), z3, z4) -> c2(IF(z0, if(z1, z3, z4), if(z2, z3, z4)), IF(z1, z3, z4), IF(z2, z3, z4))
K tuples:none
Defined Rule Symbols:   encArg_1, if_3

Defined Pair Symbols:   ENCARG_1, IF_3, ENCODE_IF_3

Compound Symbols:   c_4, c2_3, c3_1


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

(43) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_if(z0, z1, z2)) -> c(IF(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) by 
   ENCARG(cons_if(x0, x1, cons_if(z0, z1, z2))) -> c(IF(encArg(x0), encArg(x1), if(encArg(z0), encArg(z1), encArg(z2))), ENCARG(x0), ENCARG(x1), ENCARG(cons_if(z0, z1, z2)))
   ENCARG(cons_if(x0, cons_if(z0, z1, z2), x2)) -> c(IF(encArg(x0), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x2)), ENCARG(x0), ENCARG(cons_if(z0, z1, z2)), ENCARG(x2))
   ENCARG(cons_if(cons_if(z0, z1, z2), x1, x2)) -> c(IF(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x1), encArg(x2)), ENCARG(cons_if(z0, z1, z2)), ENCARG(x1), ENCARG(x2))

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2))
   if(if(z0, z1, z2), z3, z4) -> if(z0, if(z1, z3, z4), if(z2, z3, z4))
Tuples:
   IF(if(z0, z1, z2), z3, z4) -> c2(IF(z0, if(z1, z3, z4), if(z2, z3, z4)), IF(z1, z3, z4), IF(z2, z3, z4))
   ENCODE_IF(z0, z1, z2) -> c3(IF(encArg(z0), encArg(z1), encArg(z2)))
   ENCARG(cons_if(x0, x1, cons_if(z0, z1, z2))) -> c(IF(encArg(x0), encArg(x1), if(encArg(z0), encArg(z1), encArg(z2))), ENCARG(x0), ENCARG(x1), ENCARG(cons_if(z0, z1, z2)))
   ENCARG(cons_if(x0, cons_if(z0, z1, z2), x2)) -> c(IF(encArg(x0), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x2)), ENCARG(x0), ENCARG(cons_if(z0, z1, z2)), ENCARG(x2))
   ENCARG(cons_if(cons_if(z0, z1, z2), x1, x2)) -> c(IF(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x1), encArg(x2)), ENCARG(cons_if(z0, z1, z2)), ENCARG(x1), ENCARG(x2))
S tuples:
   IF(if(z0, z1, z2), z3, z4) -> c2(IF(z0, if(z1, z3, z4), if(z2, z3, z4)), IF(z1, z3, z4), IF(z2, z3, z4))
K tuples:none
Defined Rule Symbols:   encArg_1, if_3

Defined Pair Symbols:   IF_3, ENCODE_IF_3, ENCARG_1

Compound Symbols:   c2_3, c3_1, c_4


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

(45) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_IF(z0, z1, z2) -> c3(IF(encArg(z0), encArg(z1), encArg(z2))) by 
   ENCODE_IF(x0, x1, cons_if(z0, z1, z2)) -> c3(IF(encArg(x0), encArg(x1), if(encArg(z0), encArg(z1), encArg(z2))))
   ENCODE_IF(x0, cons_if(z0, z1, z2), x2) -> c3(IF(encArg(x0), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x2)))
   ENCODE_IF(cons_if(z0, z1, z2), x1, x2) -> c3(IF(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x1), encArg(x2)))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2))
   if(if(z0, z1, z2), z3, z4) -> if(z0, if(z1, z3, z4), if(z2, z3, z4))
Tuples:
   IF(if(z0, z1, z2), z3, z4) -> c2(IF(z0, if(z1, z3, z4), if(z2, z3, z4)), IF(z1, z3, z4), IF(z2, z3, z4))
   ENCARG(cons_if(x0, x1, cons_if(z0, z1, z2))) -> c(IF(encArg(x0), encArg(x1), if(encArg(z0), encArg(z1), encArg(z2))), ENCARG(x0), ENCARG(x1), ENCARG(cons_if(z0, z1, z2)))
   ENCARG(cons_if(x0, cons_if(z0, z1, z2), x2)) -> c(IF(encArg(x0), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x2)), ENCARG(x0), ENCARG(cons_if(z0, z1, z2)), ENCARG(x2))
   ENCARG(cons_if(cons_if(z0, z1, z2), x1, x2)) -> c(IF(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x1), encArg(x2)), ENCARG(cons_if(z0, z1, z2)), ENCARG(x1), ENCARG(x2))
   ENCODE_IF(x0, x1, cons_if(z0, z1, z2)) -> c3(IF(encArg(x0), encArg(x1), if(encArg(z0), encArg(z1), encArg(z2))))
   ENCODE_IF(x0, cons_if(z0, z1, z2), x2) -> c3(IF(encArg(x0), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x2)))
   ENCODE_IF(cons_if(z0, z1, z2), x1, x2) -> c3(IF(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x1), encArg(x2)))
S tuples:
   IF(if(z0, z1, z2), z3, z4) -> c2(IF(z0, if(z1, z3, z4), if(z2, z3, z4)), IF(z1, z3, z4), IF(z2, z3, z4))
K tuples:none
Defined Rule Symbols:   encArg_1, if_3

Defined Pair Symbols:   IF_3, ENCARG_1, ENCODE_IF_3

Compound Symbols:   c2_3, c_4, c3_1


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

(47) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_if(x0, x1, cons_if(z0, z1, z2))) -> c(IF(encArg(x0), encArg(x1), if(encArg(z0), encArg(z1), encArg(z2))), ENCARG(x0), ENCARG(x1), ENCARG(cons_if(z0, z1, z2))) by 
   ENCARG(cons_if(x0, x1, cons_if(x2, x3, cons_if(z0, z1, z2)))) -> c(IF(encArg(x0), encArg(x1), if(encArg(x2), encArg(x3), if(encArg(z0), encArg(z1), encArg(z2)))), ENCARG(x0), ENCARG(x1), ENCARG(cons_if(x2, x3, cons_if(z0, z1, z2))))
   ENCARG(cons_if(x0, x1, cons_if(x2, cons_if(z0, z1, z2), x4))) -> c(IF(encArg(x0), encArg(x1), if(encArg(x2), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x4))), ENCARG(x0), ENCARG(x1), ENCARG(cons_if(x2, cons_if(z0, z1, z2), x4)))
   ENCARG(cons_if(x0, x1, cons_if(cons_if(z0, z1, z2), x3, x4))) -> c(IF(encArg(x0), encArg(x1), if(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x3), encArg(x4))), ENCARG(x0), ENCARG(x1), ENCARG(cons_if(cons_if(z0, z1, z2), x3, x4)))
   ENCARG(cons_if(x0, cons_if(z0, z1, z2), cons_if(x2, x3, x4))) -> c(IF(encArg(x0), if(encArg(z0), encArg(z1), encArg(z2)), if(encArg(x2), encArg(x3), encArg(x4))), ENCARG(x0), ENCARG(cons_if(z0, z1, z2)), ENCARG(cons_if(x2, x3, x4)))
   ENCARG(cons_if(cons_if(z0, z1, z2), x1, cons_if(x2, x3, x4))) -> c(IF(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x1), if(encArg(x2), encArg(x3), encArg(x4))), ENCARG(cons_if(z0, z1, z2)), ENCARG(x1), ENCARG(cons_if(x2, x3, x4)))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2))
   if(if(z0, z1, z2), z3, z4) -> if(z0, if(z1, z3, z4), if(z2, z3, z4))
Tuples:
   IF(if(z0, z1, z2), z3, z4) -> c2(IF(z0, if(z1, z3, z4), if(z2, z3, z4)), IF(z1, z3, z4), IF(z2, z3, z4))
   ENCARG(cons_if(x0, cons_if(z0, z1, z2), x2)) -> c(IF(encArg(x0), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x2)), ENCARG(x0), ENCARG(cons_if(z0, z1, z2)), ENCARG(x2))
   ENCARG(cons_if(cons_if(z0, z1, z2), x1, x2)) -> c(IF(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x1), encArg(x2)), ENCARG(cons_if(z0, z1, z2)), ENCARG(x1), ENCARG(x2))
   ENCODE_IF(x0, x1, cons_if(z0, z1, z2)) -> c3(IF(encArg(x0), encArg(x1), if(encArg(z0), encArg(z1), encArg(z2))))
   ENCODE_IF(x0, cons_if(z0, z1, z2), x2) -> c3(IF(encArg(x0), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x2)))
   ENCODE_IF(cons_if(z0, z1, z2), x1, x2) -> c3(IF(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x1), encArg(x2)))
   ENCARG(cons_if(x0, x1, cons_if(x2, x3, cons_if(z0, z1, z2)))) -> c(IF(encArg(x0), encArg(x1), if(encArg(x2), encArg(x3), if(encArg(z0), encArg(z1), encArg(z2)))), ENCARG(x0), ENCARG(x1), ENCARG(cons_if(x2, x3, cons_if(z0, z1, z2))))
   ENCARG(cons_if(x0, x1, cons_if(x2, cons_if(z0, z1, z2), x4))) -> c(IF(encArg(x0), encArg(x1), if(encArg(x2), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x4))), ENCARG(x0), ENCARG(x1), ENCARG(cons_if(x2, cons_if(z0, z1, z2), x4)))
   ENCARG(cons_if(x0, x1, cons_if(cons_if(z0, z1, z2), x3, x4))) -> c(IF(encArg(x0), encArg(x1), if(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x3), encArg(x4))), ENCARG(x0), ENCARG(x1), ENCARG(cons_if(cons_if(z0, z1, z2), x3, x4)))
   ENCARG(cons_if(x0, cons_if(z0, z1, z2), cons_if(x2, x3, x4))) -> c(IF(encArg(x0), if(encArg(z0), encArg(z1), encArg(z2)), if(encArg(x2), encArg(x3), encArg(x4))), ENCARG(x0), ENCARG(cons_if(z0, z1, z2)), ENCARG(cons_if(x2, x3, x4)))
   ENCARG(cons_if(cons_if(z0, z1, z2), x1, cons_if(x2, x3, x4))) -> c(IF(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x1), if(encArg(x2), encArg(x3), encArg(x4))), ENCARG(cons_if(z0, z1, z2)), ENCARG(x1), ENCARG(cons_if(x2, x3, x4)))
S tuples:
   IF(if(z0, z1, z2), z3, z4) -> c2(IF(z0, if(z1, z3, z4), if(z2, z3, z4)), IF(z1, z3, z4), IF(z2, z3, z4))
K tuples:none
Defined Rule Symbols:   encArg_1, if_3

Defined Pair Symbols:   IF_3, ENCARG_1, ENCODE_IF_3

Compound Symbols:   c2_3, c_4, c3_1


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

(49) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_if(x0, cons_if(z0, z1, z2), x2)) -> c(IF(encArg(x0), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x2)), ENCARG(x0), ENCARG(cons_if(z0, z1, z2)), ENCARG(x2)) by 
   ENCARG(cons_if(x0, cons_if(x1, x2, x3), cons_if(z0, z1, z2))) -> c(IF(encArg(x0), if(encArg(x1), encArg(x2), encArg(x3)), if(encArg(z0), encArg(z1), encArg(z2))), ENCARG(x0), ENCARG(cons_if(x1, x2, x3)), ENCARG(cons_if(z0, z1, z2)))
   ENCARG(cons_if(x0, cons_if(x1, x2, cons_if(z0, z1, z2)), x4)) -> c(IF(encArg(x0), if(encArg(x1), encArg(x2), if(encArg(z0), encArg(z1), encArg(z2))), encArg(x4)), ENCARG(x0), ENCARG(cons_if(x1, x2, cons_if(z0, z1, z2))), ENCARG(x4))
   ENCARG(cons_if(x0, cons_if(x1, cons_if(z0, z1, z2), x3), x4)) -> c(IF(encArg(x0), if(encArg(x1), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x3)), encArg(x4)), ENCARG(x0), ENCARG(cons_if(x1, cons_if(z0, z1, z2), x3)), ENCARG(x4))
   ENCARG(cons_if(x0, cons_if(cons_if(z0, z1, z2), x2, x3), x4)) -> c(IF(encArg(x0), if(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x2), encArg(x3)), encArg(x4)), ENCARG(x0), ENCARG(cons_if(cons_if(z0, z1, z2), x2, x3)), ENCARG(x4))
   ENCARG(cons_if(cons_if(z0, z1, z2), cons_if(x1, x2, x3), x4)) -> c(IF(if(encArg(z0), encArg(z1), encArg(z2)), if(encArg(x1), encArg(x2), encArg(x3)), encArg(x4)), ENCARG(cons_if(z0, z1, z2)), ENCARG(cons_if(x1, x2, x3)), ENCARG(x4))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2))
   if(if(z0, z1, z2), z3, z4) -> if(z0, if(z1, z3, z4), if(z2, z3, z4))
Tuples:
   IF(if(z0, z1, z2), z3, z4) -> c2(IF(z0, if(z1, z3, z4), if(z2, z3, z4)), IF(z1, z3, z4), IF(z2, z3, z4))
   ENCARG(cons_if(cons_if(z0, z1, z2), x1, x2)) -> c(IF(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x1), encArg(x2)), ENCARG(cons_if(z0, z1, z2)), ENCARG(x1), ENCARG(x2))
   ENCODE_IF(x0, x1, cons_if(z0, z1, z2)) -> c3(IF(encArg(x0), encArg(x1), if(encArg(z0), encArg(z1), encArg(z2))))
   ENCODE_IF(x0, cons_if(z0, z1, z2), x2) -> c3(IF(encArg(x0), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x2)))
   ENCODE_IF(cons_if(z0, z1, z2), x1, x2) -> c3(IF(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x1), encArg(x2)))
   ENCARG(cons_if(x0, x1, cons_if(x2, x3, cons_if(z0, z1, z2)))) -> c(IF(encArg(x0), encArg(x1), if(encArg(x2), encArg(x3), if(encArg(z0), encArg(z1), encArg(z2)))), ENCARG(x0), ENCARG(x1), ENCARG(cons_if(x2, x3, cons_if(z0, z1, z2))))
   ENCARG(cons_if(x0, x1, cons_if(x2, cons_if(z0, z1, z2), x4))) -> c(IF(encArg(x0), encArg(x1), if(encArg(x2), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x4))), ENCARG(x0), ENCARG(x1), ENCARG(cons_if(x2, cons_if(z0, z1, z2), x4)))
   ENCARG(cons_if(x0, x1, cons_if(cons_if(z0, z1, z2), x3, x4))) -> c(IF(encArg(x0), encArg(x1), if(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x3), encArg(x4))), ENCARG(x0), ENCARG(x1), ENCARG(cons_if(cons_if(z0, z1, z2), x3, x4)))
   ENCARG(cons_if(x0, cons_if(z0, z1, z2), cons_if(x2, x3, x4))) -> c(IF(encArg(x0), if(encArg(z0), encArg(z1), encArg(z2)), if(encArg(x2), encArg(x3), encArg(x4))), ENCARG(x0), ENCARG(cons_if(z0, z1, z2)), ENCARG(cons_if(x2, x3, x4)))
   ENCARG(cons_if(cons_if(z0, z1, z2), x1, cons_if(x2, x3, x4))) -> c(IF(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x1), if(encArg(x2), encArg(x3), encArg(x4))), ENCARG(cons_if(z0, z1, z2)), ENCARG(x1), ENCARG(cons_if(x2, x3, x4)))
   ENCARG(cons_if(x0, cons_if(x1, x2, cons_if(z0, z1, z2)), x4)) -> c(IF(encArg(x0), if(encArg(x1), encArg(x2), if(encArg(z0), encArg(z1), encArg(z2))), encArg(x4)), ENCARG(x0), ENCARG(cons_if(x1, x2, cons_if(z0, z1, z2))), ENCARG(x4))
   ENCARG(cons_if(x0, cons_if(x1, cons_if(z0, z1, z2), x3), x4)) -> c(IF(encArg(x0), if(encArg(x1), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x3)), encArg(x4)), ENCARG(x0), ENCARG(cons_if(x1, cons_if(z0, z1, z2), x3)), ENCARG(x4))
   ENCARG(cons_if(x0, cons_if(cons_if(z0, z1, z2), x2, x3), x4)) -> c(IF(encArg(x0), if(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x2), encArg(x3)), encArg(x4)), ENCARG(x0), ENCARG(cons_if(cons_if(z0, z1, z2), x2, x3)), ENCARG(x4))
   ENCARG(cons_if(cons_if(z0, z1, z2), cons_if(x1, x2, x3), x4)) -> c(IF(if(encArg(z0), encArg(z1), encArg(z2)), if(encArg(x1), encArg(x2), encArg(x3)), encArg(x4)), ENCARG(cons_if(z0, z1, z2)), ENCARG(cons_if(x1, x2, x3)), ENCARG(x4))
S tuples:
   IF(if(z0, z1, z2), z3, z4) -> c2(IF(z0, if(z1, z3, z4), if(z2, z3, z4)), IF(z1, z3, z4), IF(z2, z3, z4))
K tuples:none
Defined Rule Symbols:   encArg_1, if_3

Defined Pair Symbols:   IF_3, ENCARG_1, ENCODE_IF_3

Compound Symbols:   c2_3, c_4, c3_1


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

(51) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_IF(x0, x1, cons_if(z0, z1, z2)) -> c3(IF(encArg(x0), encArg(x1), if(encArg(z0), encArg(z1), encArg(z2)))) by 
   ENCODE_IF(x0, x1, cons_if(x2, x3, cons_if(z0, z1, z2))) -> c3(IF(encArg(x0), encArg(x1), if(encArg(x2), encArg(x3), if(encArg(z0), encArg(z1), encArg(z2)))))
   ENCODE_IF(x0, x1, cons_if(x2, cons_if(z0, z1, z2), x4)) -> c3(IF(encArg(x0), encArg(x1), if(encArg(x2), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x4))))
   ENCODE_IF(x0, x1, cons_if(cons_if(z0, z1, z2), x3, x4)) -> c3(IF(encArg(x0), encArg(x1), if(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x3), encArg(x4))))
   ENCODE_IF(x0, cons_if(z0, z1, z2), cons_if(x2, x3, x4)) -> c3(IF(encArg(x0), if(encArg(z0), encArg(z1), encArg(z2)), if(encArg(x2), encArg(x3), encArg(x4))))
   ENCODE_IF(cons_if(z0, z1, z2), x1, cons_if(x2, x3, x4)) -> c3(IF(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x1), if(encArg(x2), encArg(x3), encArg(x4))))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2))
   if(if(z0, z1, z2), z3, z4) -> if(z0, if(z1, z3, z4), if(z2, z3, z4))
Tuples:
   IF(if(z0, z1, z2), z3, z4) -> c2(IF(z0, if(z1, z3, z4), if(z2, z3, z4)), IF(z1, z3, z4), IF(z2, z3, z4))
   ENCARG(cons_if(cons_if(z0, z1, z2), x1, x2)) -> c(IF(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x1), encArg(x2)), ENCARG(cons_if(z0, z1, z2)), ENCARG(x1), ENCARG(x2))
   ENCODE_IF(x0, cons_if(z0, z1, z2), x2) -> c3(IF(encArg(x0), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x2)))
   ENCODE_IF(cons_if(z0, z1, z2), x1, x2) -> c3(IF(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x1), encArg(x2)))
   ENCARG(cons_if(x0, x1, cons_if(x2, x3, cons_if(z0, z1, z2)))) -> c(IF(encArg(x0), encArg(x1), if(encArg(x2), encArg(x3), if(encArg(z0), encArg(z1), encArg(z2)))), ENCARG(x0), ENCARG(x1), ENCARG(cons_if(x2, x3, cons_if(z0, z1, z2))))
   ENCARG(cons_if(x0, x1, cons_if(x2, cons_if(z0, z1, z2), x4))) -> c(IF(encArg(x0), encArg(x1), if(encArg(x2), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x4))), ENCARG(x0), ENCARG(x1), ENCARG(cons_if(x2, cons_if(z0, z1, z2), x4)))
   ENCARG(cons_if(x0, x1, cons_if(cons_if(z0, z1, z2), x3, x4))) -> c(IF(encArg(x0), encArg(x1), if(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x3), encArg(x4))), ENCARG(x0), ENCARG(x1), ENCARG(cons_if(cons_if(z0, z1, z2), x3, x4)))
   ENCARG(cons_if(x0, cons_if(z0, z1, z2), cons_if(x2, x3, x4))) -> c(IF(encArg(x0), if(encArg(z0), encArg(z1), encArg(z2)), if(encArg(x2), encArg(x3), encArg(x4))), ENCARG(x0), ENCARG(cons_if(z0, z1, z2)), ENCARG(cons_if(x2, x3, x4)))
   ENCARG(cons_if(cons_if(z0, z1, z2), x1, cons_if(x2, x3, x4))) -> c(IF(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x1), if(encArg(x2), encArg(x3), encArg(x4))), ENCARG(cons_if(z0, z1, z2)), ENCARG(x1), ENCARG(cons_if(x2, x3, x4)))
   ENCARG(cons_if(x0, cons_if(x1, x2, cons_if(z0, z1, z2)), x4)) -> c(IF(encArg(x0), if(encArg(x1), encArg(x2), if(encArg(z0), encArg(z1), encArg(z2))), encArg(x4)), ENCARG(x0), ENCARG(cons_if(x1, x2, cons_if(z0, z1, z2))), ENCARG(x4))
   ENCARG(cons_if(x0, cons_if(x1, cons_if(z0, z1, z2), x3), x4)) -> c(IF(encArg(x0), if(encArg(x1), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x3)), encArg(x4)), ENCARG(x0), ENCARG(cons_if(x1, cons_if(z0, z1, z2), x3)), ENCARG(x4))
   ENCARG(cons_if(x0, cons_if(cons_if(z0, z1, z2), x2, x3), x4)) -> c(IF(encArg(x0), if(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x2), encArg(x3)), encArg(x4)), ENCARG(x0), ENCARG(cons_if(cons_if(z0, z1, z2), x2, x3)), ENCARG(x4))
   ENCARG(cons_if(cons_if(z0, z1, z2), cons_if(x1, x2, x3), x4)) -> c(IF(if(encArg(z0), encArg(z1), encArg(z2)), if(encArg(x1), encArg(x2), encArg(x3)), encArg(x4)), ENCARG(cons_if(z0, z1, z2)), ENCARG(cons_if(x1, x2, x3)), ENCARG(x4))
   ENCODE_IF(x0, x1, cons_if(x2, x3, cons_if(z0, z1, z2))) -> c3(IF(encArg(x0), encArg(x1), if(encArg(x2), encArg(x3), if(encArg(z0), encArg(z1), encArg(z2)))))
   ENCODE_IF(x0, x1, cons_if(x2, cons_if(z0, z1, z2), x4)) -> c3(IF(encArg(x0), encArg(x1), if(encArg(x2), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x4))))
   ENCODE_IF(x0, x1, cons_if(cons_if(z0, z1, z2), x3, x4)) -> c3(IF(encArg(x0), encArg(x1), if(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x3), encArg(x4))))
   ENCODE_IF(x0, cons_if(z0, z1, z2), cons_if(x2, x3, x4)) -> c3(IF(encArg(x0), if(encArg(z0), encArg(z1), encArg(z2)), if(encArg(x2), encArg(x3), encArg(x4))))
   ENCODE_IF(cons_if(z0, z1, z2), x1, cons_if(x2, x3, x4)) -> c3(IF(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x1), if(encArg(x2), encArg(x3), encArg(x4))))
S tuples:
   IF(if(z0, z1, z2), z3, z4) -> c2(IF(z0, if(z1, z3, z4), if(z2, z3, z4)), IF(z1, z3, z4), IF(z2, z3, z4))
K tuples:none
Defined Rule Symbols:   encArg_1, if_3

Defined Pair Symbols:   IF_3, ENCARG_1, ENCODE_IF_3

Compound Symbols:   c2_3, c_4, c3_1


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

(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_IF(x0, cons_if(z0, z1, z2), x2) -> c3(IF(encArg(x0), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x2))) by 
   ENCODE_IF(x0, cons_if(x1, x2, x3), cons_if(z0, z1, z2)) -> c3(IF(encArg(x0), if(encArg(x1), encArg(x2), encArg(x3)), if(encArg(z0), encArg(z1), encArg(z2))))
   ENCODE_IF(x0, cons_if(x1, x2, cons_if(z0, z1, z2)), x4) -> c3(IF(encArg(x0), if(encArg(x1), encArg(x2), if(encArg(z0), encArg(z1), encArg(z2))), encArg(x4)))
   ENCODE_IF(x0, cons_if(x1, cons_if(z0, z1, z2), x3), x4) -> c3(IF(encArg(x0), if(encArg(x1), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x3)), encArg(x4)))
   ENCODE_IF(x0, cons_if(cons_if(z0, z1, z2), x2, x3), x4) -> c3(IF(encArg(x0), if(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x2), encArg(x3)), encArg(x4)))
   ENCODE_IF(cons_if(z0, z1, z2), cons_if(x1, x2, x3), x4) -> c3(IF(if(encArg(z0), encArg(z1), encArg(z2)), if(encArg(x1), encArg(x2), encArg(x3)), encArg(x4)))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_if(z0, z1, z2)) -> if(encArg(z0), encArg(z1), encArg(z2))
   if(if(z0, z1, z2), z3, z4) -> if(z0, if(z1, z3, z4), if(z2, z3, z4))
Tuples:
   IF(if(z0, z1, z2), z3, z4) -> c2(IF(z0, if(z1, z3, z4), if(z2, z3, z4)), IF(z1, z3, z4), IF(z2, z3, z4))
   ENCARG(cons_if(cons_if(z0, z1, z2), x1, x2)) -> c(IF(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x1), encArg(x2)), ENCARG(cons_if(z0, z1, z2)), ENCARG(x1), ENCARG(x2))
   ENCODE_IF(cons_if(z0, z1, z2), x1, x2) -> c3(IF(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x1), encArg(x2)))
   ENCARG(cons_if(x0, x1, cons_if(x2, x3, cons_if(z0, z1, z2)))) -> c(IF(encArg(x0), encArg(x1), if(encArg(x2), encArg(x3), if(encArg(z0), encArg(z1), encArg(z2)))), ENCARG(x0), ENCARG(x1), ENCARG(cons_if(x2, x3, cons_if(z0, z1, z2))))
   ENCARG(cons_if(x0, x1, cons_if(x2, cons_if(z0, z1, z2), x4))) -> c(IF(encArg(x0), encArg(x1), if(encArg(x2), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x4))), ENCARG(x0), ENCARG(x1), ENCARG(cons_if(x2, cons_if(z0, z1, z2), x4)))
   ENCARG(cons_if(x0, x1, cons_if(cons_if(z0, z1, z2), x3, x4))) -> c(IF(encArg(x0), encArg(x1), if(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x3), encArg(x4))), ENCARG(x0), ENCARG(x1), ENCARG(cons_if(cons_if(z0, z1, z2), x3, x4)))
   ENCARG(cons_if(x0, cons_if(z0, z1, z2), cons_if(x2, x3, x4))) -> c(IF(encArg(x0), if(encArg(z0), encArg(z1), encArg(z2)), if(encArg(x2), encArg(x3), encArg(x4))), ENCARG(x0), ENCARG(cons_if(z0, z1, z2)), ENCARG(cons_if(x2, x3, x4)))
   ENCARG(cons_if(cons_if(z0, z1, z2), x1, cons_if(x2, x3, x4))) -> c(IF(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x1), if(encArg(x2), encArg(x3), encArg(x4))), ENCARG(cons_if(z0, z1, z2)), ENCARG(x1), ENCARG(cons_if(x2, x3, x4)))
   ENCARG(cons_if(x0, cons_if(x1, x2, cons_if(z0, z1, z2)), x4)) -> c(IF(encArg(x0), if(encArg(x1), encArg(x2), if(encArg(z0), encArg(z1), encArg(z2))), encArg(x4)), ENCARG(x0), ENCARG(cons_if(x1, x2, cons_if(z0, z1, z2))), ENCARG(x4))
   ENCARG(cons_if(x0, cons_if(x1, cons_if(z0, z1, z2), x3), x4)) -> c(IF(encArg(x0), if(encArg(x1), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x3)), encArg(x4)), ENCARG(x0), ENCARG(cons_if(x1, cons_if(z0, z1, z2), x3)), ENCARG(x4))
   ENCARG(cons_if(x0, cons_if(cons_if(z0, z1, z2), x2, x3), x4)) -> c(IF(encArg(x0), if(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x2), encArg(x3)), encArg(x4)), ENCARG(x0), ENCARG(cons_if(cons_if(z0, z1, z2), x2, x3)), ENCARG(x4))
   ENCARG(cons_if(cons_if(z0, z1, z2), cons_if(x1, x2, x3), x4)) -> c(IF(if(encArg(z0), encArg(z1), encArg(z2)), if(encArg(x1), encArg(x2), encArg(x3)), encArg(x4)), ENCARG(cons_if(z0, z1, z2)), ENCARG(cons_if(x1, x2, x3)), ENCARG(x4))
   ENCODE_IF(x0, x1, cons_if(x2, x3, cons_if(z0, z1, z2))) -> c3(IF(encArg(x0), encArg(x1), if(encArg(x2), encArg(x3), if(encArg(z0), encArg(z1), encArg(z2)))))
   ENCODE_IF(x0, x1, cons_if(x2, cons_if(z0, z1, z2), x4)) -> c3(IF(encArg(x0), encArg(x1), if(encArg(x2), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x4))))
   ENCODE_IF(x0, x1, cons_if(cons_if(z0, z1, z2), x3, x4)) -> c3(IF(encArg(x0), encArg(x1), if(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x3), encArg(x4))))
   ENCODE_IF(x0, cons_if(z0, z1, z2), cons_if(x2, x3, x4)) -> c3(IF(encArg(x0), if(encArg(z0), encArg(z1), encArg(z2)), if(encArg(x2), encArg(x3), encArg(x4))))
   ENCODE_IF(cons_if(z0, z1, z2), x1, cons_if(x2, x3, x4)) -> c3(IF(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x1), if(encArg(x2), encArg(x3), encArg(x4))))
   ENCODE_IF(x0, cons_if(x1, x2, cons_if(z0, z1, z2)), x4) -> c3(IF(encArg(x0), if(encArg(x1), encArg(x2), if(encArg(z0), encArg(z1), encArg(z2))), encArg(x4)))
   ENCODE_IF(x0, cons_if(x1, cons_if(z0, z1, z2), x3), x4) -> c3(IF(encArg(x0), if(encArg(x1), if(encArg(z0), encArg(z1), encArg(z2)), encArg(x3)), encArg(x4)))
   ENCODE_IF(x0, cons_if(cons_if(z0, z1, z2), x2, x3), x4) -> c3(IF(encArg(x0), if(if(encArg(z0), encArg(z1), encArg(z2)), encArg(x2), encArg(x3)), encArg(x4)))
   ENCODE_IF(cons_if(z0, z1, z2), cons_if(x1, x2, x3), x4) -> c3(IF(if(encArg(z0), encArg(z1), encArg(z2)), if(encArg(x1), encArg(x2), encArg(x3)), encArg(x4)))
S tuples:
   IF(if(z0, z1, z2), z3, z4) -> c2(IF(z0, if(z1, z3, z4), if(z2, z3, z4)), IF(z1, z3, z4), IF(z2, z3, z4))
K tuples:none
Defined Rule Symbols:   encArg_1, if_3

Defined Pair Symbols:   IF_3, ENCARG_1, ENCODE_IF_3

Compound Symbols:   c2_3, c_4, c3_1