/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 58 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), 461 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) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (26) CpxTypedWeightedCompleteTrs
        (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 1 ms]
        (28) CpxRNTS
        (29) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (30) CpxRNTS
    (31) CompletionProof [UPPER BOUND(ID), 0 ms]
    (32) CpxTypedWeightedCompleteTrs
        (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 7 ms]
        (34) CpxRNTS
(35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(36) CdtProblem
    (37) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (42) CdtProblem
    (43) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 9 ms]
    (44) CdtProblem
    (45) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 172 ms]
    (46) CdtProblem
    (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms]
    (54) CdtProblem
    (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms]
    (96) CdtProblem
    (97) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms]
    (100) CdtProblem
    (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 6 ms]
    (102) CdtProblem
    (103) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) 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:

   P(x1) -> Q(Q(p(x1)))
   p(p(x1)) -> q(q(x1))
   p(Q(Q(x1))) -> Q(Q(p(x1)))
   Q(p(q(x1))) -> q(p(Q(x1)))
   q(q(p(x1))) -> p(q(q(x1)))
   q(Q(x1)) -> x1
   Q(q(x1)) -> x1
   p(P(x1)) -> x1
   P(p(x1)) -> x1

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_P(x_1)) -> P(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_Q(x_1)) -> Q(encArg(x_1))
   encArg(cons_q(x_1)) -> q(encArg(x_1))
   encode_P(x_1) -> P(encArg(x_1))
   encode_Q(x_1) -> Q(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_q(x_1) -> q(encArg(x_1))


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

(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:

   P(x1) -> Q(Q(p(x1)))
   p(p(x1)) -> q(q(x1))
   p(Q(Q(x1))) -> Q(Q(p(x1)))
   Q(p(q(x1))) -> q(p(Q(x1)))
   q(q(p(x1))) -> p(q(q(x1)))
   q(Q(x1)) -> x1
   Q(q(x1)) -> x1
   p(P(x1)) -> x1
   P(p(x1)) -> x1

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

   encArg(cons_P(x_1)) -> P(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_Q(x_1)) -> Q(encArg(x_1))
   encArg(cons_q(x_1)) -> q(encArg(x_1))
   encode_P(x_1) -> P(encArg(x_1))
   encode_Q(x_1) -> Q(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_q(x_1) -> q(encArg(x_1))

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:

   P(x1) -> Q(Q(p(x1)))
   p(p(x1)) -> q(q(x1))
   p(Q(Q(x1))) -> Q(Q(p(x1)))
   Q(p(q(x1))) -> q(p(Q(x1)))
   q(q(p(x1))) -> p(q(q(x1)))
   q(Q(x1)) -> x1
   Q(q(x1)) -> x1
   p(P(x1)) -> x1
   P(p(x1)) -> x1

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

   encArg(cons_P(x_1)) -> P(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_Q(x_1)) -> Q(encArg(x_1))
   encArg(cons_q(x_1)) -> q(encArg(x_1))
   encode_P(x_1) -> P(encArg(x_1))
   encode_Q(x_1) -> Q(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_q(x_1) -> q(encArg(x_1))

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:

   P(x1) -> Q(Q(p(x1)))
   p(p(x1)) -> q(q(x1))
   p(Q(Q(x1))) -> Q(Q(p(x1)))
   Q(p(q(x1))) -> q(p(Q(x1)))
   q(q(p(x1))) -> p(q(q(x1)))
   q(Q(x1)) -> x1
   Q(q(x1)) -> x1
   p(P(x1)) -> x1
   P(p(x1)) -> x1

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

   encArg(cons_P(x_1)) -> P(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_Q(x_1)) -> Q(encArg(x_1))
   encArg(cons_q(x_1)) -> q(encArg(x_1))
   encode_P(x_1) -> P(encArg(x_1))
   encode_Q(x_1) -> Q(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_q(x_1) -> q(encArg(x_1))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
P(x1) -> Q(Q(p(x1)))
p(p(x1)) -> q(q(x1))
p(Q(Q(x1))) -> Q(Q(p(x1)))
Q(p(q(x1))) -> q(p(Q(x1)))
q(q(p(x1))) -> p(q(q(x1)))
q(Q(x1)) -> x1
Q(q(x1)) -> x1
p(P(x1)) -> x1
P(p(x1)) -> x1
encArg(cons_P(x_1)) -> P(encArg(x_1))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encArg(cons_Q(x_1)) -> Q(encArg(x_1))
encArg(cons_q(x_1)) -> q(encArg(x_1))
encode_P(x_1) -> P(encArg(x_1))
encode_Q(x_1) -> Q(encArg(x_1))
encode_p(x_1) -> p(encArg(x_1))
encode_q(x_1) -> q(encArg(x_1))

Types:
P :: p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q -> p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q
Q :: p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q -> p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q
p :: p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q -> p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q
q :: p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q -> p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q
encArg :: cons_P:cons_p:cons_Q:cons_q -> p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q
cons_P :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
cons_p :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
cons_Q :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
cons_q :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
encode_P :: cons_P:cons_p:cons_Q:cons_q -> p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q
encode_Q :: cons_P:cons_p:cons_Q:cons_q -> p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q
encode_p :: cons_P:cons_p:cons_Q:cons_q -> p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q
encode_q :: cons_P:cons_p:cons_Q:cons_q -> p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q
hole_p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q1_0 :: p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q
hole_cons_P:cons_p:cons_Q:cons_q2_0 :: cons_P:cons_p:cons_Q:cons_q
gen_cons_P:cons_p:cons_Q:cons_q3_0 :: Nat -> cons_P:cons_p:cons_Q:cons_q

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

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

They will be analysed ascendingly in the following order:
Q = p
Q = q
Q < encArg
p = q
p < encArg
q < encArg

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

(10)
Obligation:
Innermost TRS:
Rules:
P(x1) -> Q(Q(p(x1)))
p(p(x1)) -> q(q(x1))
p(Q(Q(x1))) -> Q(Q(p(x1)))
Q(p(q(x1))) -> q(p(Q(x1)))
q(q(p(x1))) -> p(q(q(x1)))
q(Q(x1)) -> x1
Q(q(x1)) -> x1
p(P(x1)) -> x1
P(p(x1)) -> x1
encArg(cons_P(x_1)) -> P(encArg(x_1))
encArg(cons_p(x_1)) -> p(encArg(x_1))
encArg(cons_Q(x_1)) -> Q(encArg(x_1))
encArg(cons_q(x_1)) -> q(encArg(x_1))
encode_P(x_1) -> P(encArg(x_1))
encode_Q(x_1) -> Q(encArg(x_1))
encode_p(x_1) -> p(encArg(x_1))
encode_q(x_1) -> q(encArg(x_1))

Types:
P :: p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q -> p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q
Q :: p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q -> p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q
p :: p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q -> p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q
q :: p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q -> p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q
encArg :: cons_P:cons_p:cons_Q:cons_q -> p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q
cons_P :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
cons_p :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
cons_Q :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
cons_q :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
encode_P :: cons_P:cons_p:cons_Q:cons_q -> p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q
encode_Q :: cons_P:cons_p:cons_Q:cons_q -> p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q
encode_p :: cons_P:cons_p:cons_Q:cons_q -> p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q
encode_q :: cons_P:cons_p:cons_Q:cons_q -> p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q
hole_p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q1_0 :: p:Q:P:q:encArg:encode_P:encode_Q:encode_p:encode_q
hole_cons_P:cons_p:cons_Q:cons_q2_0 :: cons_P:cons_p:cons_Q:cons_q
gen_cons_P:cons_p:cons_Q:cons_q3_0 :: Nat -> cons_P:cons_p:cons_Q:cons_q


Generator Equations:
gen_cons_P:cons_p:cons_Q:cons_q3_0(0) <=> hole_cons_P:cons_p:cons_Q:cons_q2_0
gen_cons_P:cons_p:cons_Q:cons_q3_0(+(x, 1)) <=> cons_P(gen_cons_P:cons_p:cons_Q:cons_q3_0(x))


The following defined symbols remain to be analysed:
q, Q, p, encArg

They will be analysed ascendingly in the following order:
Q = p
Q = q
Q < encArg
p = q
p < encArg
q < encArg

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_cons_P:cons_p:cons_Q:cons_q3_0(+(1, n25_0))) -> *4_0, rt in Omega(0)

Induction Base:
encArg(gen_cons_P:cons_p:cons_Q:cons_q3_0(+(1, 0)))

Induction Step:
encArg(gen_cons_P:cons_p:cons_Q:cons_q3_0(+(1, +(n25_0, 1)))) ->_R^Omega(0)
P(encArg(gen_cons_P:cons_p:cons_Q:cons_q3_0(+(1, n25_0)))) ->_IH
P(*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:

   P(x1) -> Q(Q(p(x1)))
   p(p(x1)) -> q(q(x1))
   p(Q(Q(x1))) -> Q(Q(p(x1)))
   Q(p(q(x1))) -> q(p(Q(x1)))
   q(q(p(x1))) -> p(q(q(x1)))
   q(Q(x1)) -> x1
   Q(q(x1)) -> x1
   p(P(x1)) -> x1
   P(p(x1)) -> x1

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

   encArg(cons_P(x_1)) -> P(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_Q(x_1)) -> Q(encArg(x_1))
   encArg(cons_q(x_1)) -> q(encArg(x_1))
   encode_P(x_1) -> P(encArg(x_1))
   encode_Q(x_1) -> Q(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_q(x_1) -> q(encArg(x_1))

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:

   P(x1) -> Q(Q(p(x1)))
   p(p(x1)) -> q(q(x1))
   p(Q(Q(x1))) -> Q(Q(p(x1)))
   Q(p(q(x1))) -> q(p(Q(x1)))
   q(q(p(x1))) -> p(q(q(x1)))
   q(Q(x1)) -> x1
   Q(q(x1)) -> x1
   p(P(x1)) -> x1
   P(p(x1)) -> x1
   encArg(cons_P(x_1)) -> P(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_Q(x_1)) -> Q(encArg(x_1))
   encArg(cons_q(x_1)) -> q(encArg(x_1))
   encode_P(x_1) -> P(encArg(x_1))
   encode_Q(x_1) -> Q(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_q(x_1) -> q(encArg(x_1))

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:

   P(x1) -> Q(Q(p(x1)))
   p(c_Q(c_Q(x1))) -> Q(Q(p(x1)))
   Q(c_q(x1)) -> x1
   q(c_Q(x1)) -> x1
   Q(c_p(c_q(x1))) -> q(p(Q(x1)))
   p(c_P(x1)) -> x1
   P(c_p(x1)) -> x1
   p(c_p(x1)) -> q(q(x1))
   q(c_q(c_p(x1))) -> p(q(q(x1)))

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

   encArg(cons_P(x_1)) -> P(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_Q(x_1)) -> Q(encArg(x_1))
   encArg(cons_q(x_1)) -> q(encArg(x_1))
   encode_P(x_1) -> P(encArg(x_1))
   encode_Q(x_1) -> Q(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_q(x_1) -> q(encArg(x_1))
   p(x0) -> c_p(x0)
   q(x0) -> c_q(x0)
   Q(x0) -> c_Q(x0)
   P(x0) -> c_P(x0)

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:

   P(x1) -> Q(Q(p(x1))) [1]
   p(c_Q(c_Q(x1))) -> Q(Q(p(x1))) [1]
   Q(c_q(x1)) -> x1 [1]
   q(c_Q(x1)) -> x1 [1]
   Q(c_p(c_q(x1))) -> q(p(Q(x1))) [1]
   p(c_P(x1)) -> x1 [1]
   P(c_p(x1)) -> x1 [1]
   p(c_p(x1)) -> q(q(x1)) [1]
   q(c_q(c_p(x1))) -> p(q(q(x1))) [1]
   encArg(cons_P(x_1)) -> P(encArg(x_1)) [0]
   encArg(cons_p(x_1)) -> p(encArg(x_1)) [0]
   encArg(cons_Q(x_1)) -> Q(encArg(x_1)) [0]
   encArg(cons_q(x_1)) -> q(encArg(x_1)) [0]
   encode_P(x_1) -> P(encArg(x_1)) [0]
   encode_Q(x_1) -> Q(encArg(x_1)) [0]
   encode_p(x_1) -> p(encArg(x_1)) [0]
   encode_q(x_1) -> q(encArg(x_1)) [0]
   p(x0) -> c_p(x0) [0]
   q(x0) -> c_q(x0) [0]
   Q(x0) -> c_Q(x0) [0]
   P(x0) -> c_P(x0) [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:

   P(x1) -> Q(Q(p(x1))) [1]
   p(c_Q(c_Q(x1))) -> Q(Q(p(x1))) [1]
   Q(c_q(x1)) -> x1 [1]
   q(c_Q(x1)) -> x1 [1]
   Q(c_p(c_q(x1))) -> q(p(Q(x1))) [1]
   p(c_P(x1)) -> x1 [1]
   P(c_p(x1)) -> x1 [1]
   p(c_p(x1)) -> q(q(x1)) [1]
   q(c_q(c_p(x1))) -> p(q(q(x1))) [1]
   encArg(cons_P(x_1)) -> P(encArg(x_1)) [0]
   encArg(cons_p(x_1)) -> p(encArg(x_1)) [0]
   encArg(cons_Q(x_1)) -> Q(encArg(x_1)) [0]
   encArg(cons_q(x_1)) -> q(encArg(x_1)) [0]
   encode_P(x_1) -> P(encArg(x_1)) [0]
   encode_Q(x_1) -> Q(encArg(x_1)) [0]
   encode_p(x_1) -> p(encArg(x_1)) [0]
   encode_q(x_1) -> q(encArg(x_1)) [0]
   p(x0) -> c_p(x0) [0]
   q(x0) -> c_q(x0) [0]
   Q(x0) -> c_Q(x0) [0]
   P(x0) -> c_P(x0) [0]

The TRS has the following type information:
P :: c_Q:c_q:c_p:c_P -> c_Q:c_q:c_p:c_P
Q :: c_Q:c_q:c_p:c_P -> c_Q:c_q:c_p:c_P
p :: c_Q:c_q:c_p:c_P -> c_Q:c_q:c_p:c_P
c_Q :: c_Q:c_q:c_p:c_P -> c_Q:c_q:c_p:c_P
c_q :: c_Q:c_q:c_p:c_P -> c_Q:c_q:c_p:c_P
q :: c_Q:c_q:c_p:c_P -> c_Q:c_q:c_p:c_P
c_p :: c_Q:c_q:c_p:c_P -> c_Q:c_q:c_p:c_P
c_P :: c_Q:c_q:c_p:c_P -> c_Q:c_q:c_p:c_P
encArg :: cons_P:cons_p:cons_Q:cons_q -> c_Q:c_q:c_p:c_P
cons_P :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
cons_p :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
cons_Q :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
cons_q :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
encode_P :: cons_P:cons_p:cons_Q:cons_q -> c_Q:c_q:c_p:c_P
encode_Q :: cons_P:cons_p:cons_Q:cons_q -> c_Q:c_q:c_p:c_P
encode_p :: cons_P:cons_p:cons_Q:cons_q -> c_Q:c_q:c_p:c_P
encode_q :: cons_P:cons_p:cons_Q:cons_q -> c_Q:c_q:c_p:c_P

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

(23) 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_P_1
   encode_Q_1
   encode_p_1
   encode_q_1
   p_1
   q_1
   Q_1
   P_1

Due to the following rules being added:

   encArg(v0) -> const [0]
   encode_P(v0) -> const [0]
   encode_Q(v0) -> const [0]
   encode_p(v0) -> const [0]
   encode_q(v0) -> const [0]
   p(v0) -> const [0]
   q(v0) -> const [0]
   Q(v0) -> const [0]
   P(v0) -> const [0]

And the following fresh constants:    const, const1

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

(24)
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:

   P(x1) -> Q(Q(p(x1))) [1]
   p(c_Q(c_Q(x1))) -> Q(Q(p(x1))) [1]
   Q(c_q(x1)) -> x1 [1]
   q(c_Q(x1)) -> x1 [1]
   Q(c_p(c_q(x1))) -> q(p(Q(x1))) [1]
   p(c_P(x1)) -> x1 [1]
   P(c_p(x1)) -> x1 [1]
   p(c_p(x1)) -> q(q(x1)) [1]
   q(c_q(c_p(x1))) -> p(q(q(x1))) [1]
   encArg(cons_P(x_1)) -> P(encArg(x_1)) [0]
   encArg(cons_p(x_1)) -> p(encArg(x_1)) [0]
   encArg(cons_Q(x_1)) -> Q(encArg(x_1)) [0]
   encArg(cons_q(x_1)) -> q(encArg(x_1)) [0]
   encode_P(x_1) -> P(encArg(x_1)) [0]
   encode_Q(x_1) -> Q(encArg(x_1)) [0]
   encode_p(x_1) -> p(encArg(x_1)) [0]
   encode_q(x_1) -> q(encArg(x_1)) [0]
   p(x0) -> c_p(x0) [0]
   q(x0) -> c_q(x0) [0]
   Q(x0) -> c_Q(x0) [0]
   P(x0) -> c_P(x0) [0]
   encArg(v0) -> const [0]
   encode_P(v0) -> const [0]
   encode_Q(v0) -> const [0]
   encode_p(v0) -> const [0]
   encode_q(v0) -> const [0]
   p(v0) -> const [0]
   q(v0) -> const [0]
   Q(v0) -> const [0]
   P(v0) -> const [0]

The TRS has the following type information:
P :: c_Q:c_q:c_p:c_P:const -> c_Q:c_q:c_p:c_P:const
Q :: c_Q:c_q:c_p:c_P:const -> c_Q:c_q:c_p:c_P:const
p :: c_Q:c_q:c_p:c_P:const -> c_Q:c_q:c_p:c_P:const
c_Q :: c_Q:c_q:c_p:c_P:const -> c_Q:c_q:c_p:c_P:const
c_q :: c_Q:c_q:c_p:c_P:const -> c_Q:c_q:c_p:c_P:const
q :: c_Q:c_q:c_p:c_P:const -> c_Q:c_q:c_p:c_P:const
c_p :: c_Q:c_q:c_p:c_P:const -> c_Q:c_q:c_p:c_P:const
c_P :: c_Q:c_q:c_p:c_P:const -> c_Q:c_q:c_p:c_P:const
encArg :: cons_P:cons_p:cons_Q:cons_q -> c_Q:c_q:c_p:c_P:const
cons_P :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
cons_p :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
cons_Q :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
cons_q :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
encode_P :: cons_P:cons_p:cons_Q:cons_q -> c_Q:c_q:c_p:c_P:const
encode_Q :: cons_P:cons_p:cons_Q:cons_q -> c_Q:c_q:c_p:c_P:const
encode_p :: cons_P:cons_p:cons_Q:cons_q -> c_Q:c_q:c_p:c_P:const
encode_q :: cons_P:cons_p:cons_Q:cons_q -> c_Q:c_q:c_p:c_P:const
const :: c_Q:c_q:c_p:c_P:const
const1 :: cons_P:cons_p:cons_Q:cons_q

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

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

(26)
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:

   P(c_Q(c_Q(x1'))) -> Q(Q(Q(Q(p(x1'))))) [2]
   P(c_P(x1'')) -> Q(Q(x1'')) [2]
   P(c_p(x11)) -> Q(Q(q(q(x11)))) [2]
   P(x1) -> Q(Q(c_p(x1))) [1]
   P(x1) -> Q(Q(const)) [1]
   p(c_Q(c_Q(c_Q(c_Q(x12))))) -> Q(Q(Q(Q(p(x12))))) [2]
   p(c_Q(c_Q(c_P(x13)))) -> Q(Q(x13)) [2]
   p(c_Q(c_Q(c_p(x14)))) -> Q(Q(q(q(x14)))) [2]
   p(c_Q(c_Q(x1))) -> Q(Q(c_p(x1))) [1]
   p(c_Q(c_Q(x1))) -> Q(Q(const)) [1]
   Q(c_q(x1)) -> x1 [1]
   q(c_Q(x1)) -> x1 [1]
   Q(c_p(c_q(c_q(x15)))) -> q(p(x15)) [2]
   Q(c_p(c_q(c_p(c_q(x16))))) -> q(p(q(p(Q(x16))))) [2]
   Q(c_p(c_q(x1))) -> q(p(c_Q(x1))) [1]
   Q(c_p(c_q(x1))) -> q(p(const)) [1]
   p(c_P(x1)) -> x1 [1]
   P(c_p(x1)) -> x1 [1]
   p(c_p(c_Q(x17))) -> q(x17) [2]
   p(c_p(c_q(c_p(x18)))) -> q(p(q(q(x18)))) [2]
   p(c_p(x1)) -> q(c_q(x1)) [1]
   p(c_p(x1)) -> q(const) [1]
   q(c_q(c_p(c_Q(x19)))) -> p(q(x19)) [2]
   q(c_q(c_p(c_q(c_p(x110))))) -> p(q(p(q(q(x110))))) [2]
   q(c_q(c_p(x1))) -> p(q(c_q(x1))) [1]
   q(c_q(c_p(x1))) -> p(q(const)) [1]
   encArg(cons_P(cons_P(x_1'))) -> P(P(encArg(x_1'))) [0]
   encArg(cons_P(cons_p(x_1''))) -> P(p(encArg(x_1''))) [0]
   encArg(cons_P(cons_Q(x_11))) -> P(Q(encArg(x_11))) [0]
   encArg(cons_P(cons_q(x_12))) -> P(q(encArg(x_12))) [0]
   encArg(cons_P(x_1)) -> P(const) [0]
   encArg(cons_p(cons_P(x_13))) -> p(P(encArg(x_13))) [0]
   encArg(cons_p(cons_p(x_14))) -> p(p(encArg(x_14))) [0]
   encArg(cons_p(cons_Q(x_15))) -> p(Q(encArg(x_15))) [0]
   encArg(cons_p(cons_q(x_16))) -> p(q(encArg(x_16))) [0]
   encArg(cons_p(x_1)) -> p(const) [0]
   encArg(cons_Q(cons_P(x_17))) -> Q(P(encArg(x_17))) [0]
   encArg(cons_Q(cons_p(x_18))) -> Q(p(encArg(x_18))) [0]
   encArg(cons_Q(cons_Q(x_19))) -> Q(Q(encArg(x_19))) [0]
   encArg(cons_Q(cons_q(x_110))) -> Q(q(encArg(x_110))) [0]
   encArg(cons_Q(x_1)) -> Q(const) [0]
   encArg(cons_q(cons_P(x_111))) -> q(P(encArg(x_111))) [0]
   encArg(cons_q(cons_p(x_112))) -> q(p(encArg(x_112))) [0]
   encArg(cons_q(cons_Q(x_113))) -> q(Q(encArg(x_113))) [0]
   encArg(cons_q(cons_q(x_114))) -> q(q(encArg(x_114))) [0]
   encArg(cons_q(x_1)) -> q(const) [0]
   encode_P(cons_P(x_115)) -> P(P(encArg(x_115))) [0]
   encode_P(cons_p(x_116)) -> P(p(encArg(x_116))) [0]
   encode_P(cons_Q(x_117)) -> P(Q(encArg(x_117))) [0]
   encode_P(cons_q(x_118)) -> P(q(encArg(x_118))) [0]
   encode_P(x_1) -> P(const) [0]
   encode_Q(cons_P(x_119)) -> Q(P(encArg(x_119))) [0]
   encode_Q(cons_p(x_120)) -> Q(p(encArg(x_120))) [0]
   encode_Q(cons_Q(x_121)) -> Q(Q(encArg(x_121))) [0]
   encode_Q(cons_q(x_122)) -> Q(q(encArg(x_122))) [0]
   encode_Q(x_1) -> Q(const) [0]
   encode_p(cons_P(x_123)) -> p(P(encArg(x_123))) [0]
   encode_p(cons_p(x_124)) -> p(p(encArg(x_124))) [0]
   encode_p(cons_Q(x_125)) -> p(Q(encArg(x_125))) [0]
   encode_p(cons_q(x_126)) -> p(q(encArg(x_126))) [0]
   encode_p(x_1) -> p(const) [0]
   encode_q(cons_P(x_127)) -> q(P(encArg(x_127))) [0]
   encode_q(cons_p(x_128)) -> q(p(encArg(x_128))) [0]
   encode_q(cons_Q(x_129)) -> q(Q(encArg(x_129))) [0]
   encode_q(cons_q(x_130)) -> q(q(encArg(x_130))) [0]
   encode_q(x_1) -> q(const) [0]
   p(x0) -> c_p(x0) [0]
   q(x0) -> c_q(x0) [0]
   Q(x0) -> c_Q(x0) [0]
   P(x0) -> c_P(x0) [0]
   encArg(v0) -> const [0]
   encode_P(v0) -> const [0]
   encode_Q(v0) -> const [0]
   encode_p(v0) -> const [0]
   encode_q(v0) -> const [0]
   p(v0) -> const [0]
   q(v0) -> const [0]
   Q(v0) -> const [0]
   P(v0) -> const [0]

The TRS has the following type information:
P :: c_Q:c_q:c_p:c_P:const -> c_Q:c_q:c_p:c_P:const
Q :: c_Q:c_q:c_p:c_P:const -> c_Q:c_q:c_p:c_P:const
p :: c_Q:c_q:c_p:c_P:const -> c_Q:c_q:c_p:c_P:const
c_Q :: c_Q:c_q:c_p:c_P:const -> c_Q:c_q:c_p:c_P:const
c_q :: c_Q:c_q:c_p:c_P:const -> c_Q:c_q:c_p:c_P:const
q :: c_Q:c_q:c_p:c_P:const -> c_Q:c_q:c_p:c_P:const
c_p :: c_Q:c_q:c_p:c_P:const -> c_Q:c_q:c_p:c_P:const
c_P :: c_Q:c_q:c_p:c_P:const -> c_Q:c_q:c_p:c_P:const
encArg :: cons_P:cons_p:cons_Q:cons_q -> c_Q:c_q:c_p:c_P:const
cons_P :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
cons_p :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
cons_Q :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
cons_q :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
encode_P :: cons_P:cons_p:cons_Q:cons_q -> c_Q:c_q:c_p:c_P:const
encode_Q :: cons_P:cons_p:cons_Q:cons_q -> c_Q:c_q:c_p:c_P:const
encode_p :: cons_P:cons_p:cons_Q:cons_q -> c_Q:c_q:c_p:c_P:const
encode_q :: cons_P:cons_p:cons_Q:cons_q -> c_Q:c_q:c_p:c_P:const
const :: c_Q:c_q:c_p:c_P:const
const1 :: cons_P:cons_p:cons_Q:cons_q

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

(27) 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

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

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

   P(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1
   P(z) -{ 2 }-> Q(Q(x1'')) :|: x1'' >= 0, z = 1 + x1''
   P(z) -{ 2 }-> Q(Q(q(q(x11)))) :|: z = 1 + x11, x11 >= 0
   P(z) -{ 2 }-> Q(Q(Q(Q(p(x1'))))) :|: x1' >= 0, z = 1 + (1 + x1')
   P(z) -{ 1 }-> Q(Q(0)) :|: x1 >= 0, z = x1
   P(z) -{ 1 }-> Q(Q(1 + x1)) :|: x1 >= 0, z = x1
   P(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   P(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0
   Q(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1
   Q(z) -{ 2 }-> q(p(x15)) :|: x15 >= 0, z = 1 + (1 + (1 + x15))
   Q(z) -{ 2 }-> q(p(q(p(Q(x16))))) :|: z = 1 + (1 + (1 + (1 + x16))), x16 >= 0
   Q(z) -{ 1 }-> q(p(0)) :|: z = 1 + (1 + x1), x1 >= 0
   Q(z) -{ 1 }-> q(p(1 + x1)) :|: z = 1 + (1 + x1), x1 >= 0
   Q(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   Q(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0
   encArg(z) -{ 0 }-> q(q(encArg(x_114))) :|: z = 1 + (1 + x_114), x_114 >= 0
   encArg(z) -{ 0 }-> q(p(encArg(x_112))) :|: x_112 >= 0, z = 1 + (1 + x_112)
   encArg(z) -{ 0 }-> q(Q(encArg(x_113))) :|: x_113 >= 0, z = 1 + (1 + x_113)
   encArg(z) -{ 0 }-> q(P(encArg(x_111))) :|: z = 1 + (1 + x_111), x_111 >= 0
   encArg(z) -{ 0 }-> q(0) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> p(q(encArg(x_16))) :|: x_16 >= 0, z = 1 + (1 + x_16)
   encArg(z) -{ 0 }-> p(p(encArg(x_14))) :|: x_14 >= 0, z = 1 + (1 + x_14)
   encArg(z) -{ 0 }-> p(Q(encArg(x_15))) :|: x_15 >= 0, z = 1 + (1 + x_15)
   encArg(z) -{ 0 }-> p(P(encArg(x_13))) :|: z = 1 + (1 + x_13), x_13 >= 0
   encArg(z) -{ 0 }-> p(0) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> Q(q(encArg(x_110))) :|: z = 1 + (1 + x_110), x_110 >= 0
   encArg(z) -{ 0 }-> Q(p(encArg(x_18))) :|: z = 1 + (1 + x_18), x_18 >= 0
   encArg(z) -{ 0 }-> Q(Q(encArg(x_19))) :|: z = 1 + (1 + x_19), x_19 >= 0
   encArg(z) -{ 0 }-> Q(P(encArg(x_17))) :|: x_17 >= 0, z = 1 + (1 + x_17)
   encArg(z) -{ 0 }-> Q(0) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> P(q(encArg(x_12))) :|: z = 1 + (1 + x_12), x_12 >= 0
   encArg(z) -{ 0 }-> P(p(encArg(x_1''))) :|: z = 1 + (1 + x_1''), x_1'' >= 0
   encArg(z) -{ 0 }-> P(Q(encArg(x_11))) :|: x_11 >= 0, z = 1 + (1 + x_11)
   encArg(z) -{ 0 }-> P(P(encArg(x_1'))) :|: z = 1 + (1 + x_1'), x_1' >= 0
   encArg(z) -{ 0 }-> P(0) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_P(z) -{ 0 }-> P(q(encArg(x_118))) :|: z = 1 + x_118, x_118 >= 0
   encode_P(z) -{ 0 }-> P(p(encArg(x_116))) :|: x_116 >= 0, z = 1 + x_116
   encode_P(z) -{ 0 }-> P(Q(encArg(x_117))) :|: x_117 >= 0, z = 1 + x_117
   encode_P(z) -{ 0 }-> P(P(encArg(x_115))) :|: x_115 >= 0, z = 1 + x_115
   encode_P(z) -{ 0 }-> P(0) :|: x_1 >= 0, z = x_1
   encode_P(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_Q(z) -{ 0 }-> Q(q(encArg(x_122))) :|: x_122 >= 0, z = 1 + x_122
   encode_Q(z) -{ 0 }-> Q(p(encArg(x_120))) :|: x_120 >= 0, z = 1 + x_120
   encode_Q(z) -{ 0 }-> Q(Q(encArg(x_121))) :|: z = 1 + x_121, x_121 >= 0
   encode_Q(z) -{ 0 }-> Q(P(encArg(x_119))) :|: z = 1 + x_119, x_119 >= 0
   encode_Q(z) -{ 0 }-> Q(0) :|: x_1 >= 0, z = x_1
   encode_Q(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_p(z) -{ 0 }-> p(q(encArg(x_126))) :|: z = 1 + x_126, x_126 >= 0
   encode_p(z) -{ 0 }-> p(p(encArg(x_124))) :|: x_124 >= 0, z = 1 + x_124
   encode_p(z) -{ 0 }-> p(Q(encArg(x_125))) :|: z = 1 + x_125, x_125 >= 0
   encode_p(z) -{ 0 }-> p(P(encArg(x_123))) :|: x_123 >= 0, z = 1 + x_123
   encode_p(z) -{ 0 }-> p(0) :|: x_1 >= 0, z = x_1
   encode_p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_q(z) -{ 0 }-> q(q(encArg(x_130))) :|: x_130 >= 0, z = 1 + x_130
   encode_q(z) -{ 0 }-> q(p(encArg(x_128))) :|: x_128 >= 0, z = 1 + x_128
   encode_q(z) -{ 0 }-> q(Q(encArg(x_129))) :|: z = 1 + x_129, x_129 >= 0
   encode_q(z) -{ 0 }-> q(P(encArg(x_127))) :|: x_127 >= 0, z = 1 + x_127
   encode_q(z) -{ 0 }-> q(0) :|: x_1 >= 0, z = x_1
   encode_q(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   p(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1
   p(z) -{ 2 }-> q(x17) :|: x17 >= 0, z = 1 + (1 + x17)
   p(z) -{ 2 }-> q(p(q(q(x18)))) :|: x18 >= 0, z = 1 + (1 + (1 + x18))
   p(z) -{ 1 }-> q(0) :|: x1 >= 0, z = 1 + x1
   p(z) -{ 1 }-> q(1 + x1) :|: x1 >= 0, z = 1 + x1
   p(z) -{ 2 }-> Q(Q(x13)) :|: z = 1 + (1 + (1 + x13)), x13 >= 0
   p(z) -{ 2 }-> Q(Q(q(q(x14)))) :|: z = 1 + (1 + (1 + x14)), x14 >= 0
   p(z) -{ 2 }-> Q(Q(Q(Q(p(x12))))) :|: x12 >= 0, z = 1 + (1 + (1 + (1 + x12)))
   p(z) -{ 1 }-> Q(Q(0)) :|: z = 1 + (1 + x1), x1 >= 0
   p(z) -{ 1 }-> Q(Q(1 + x1)) :|: z = 1 + (1 + x1), x1 >= 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   p(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0
   q(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1
   q(z) -{ 2 }-> p(q(x19)) :|: z = 1 + (1 + (1 + x19)), x19 >= 0
   q(z) -{ 2 }-> p(q(p(q(q(x110))))) :|: z = 1 + (1 + (1 + (1 + x110))), x110 >= 0
   q(z) -{ 1 }-> p(q(0)) :|: z = 1 + (1 + x1), x1 >= 0
   q(z) -{ 1 }-> p(q(1 + x1)) :|: z = 1 + (1 + x1), x1 >= 0
   q(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   q(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0


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

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

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

   P(z) -{ 2 }-> Q(Q(q(q(z - 1)))) :|: z - 1 >= 0
   P(z) -{ 2 }-> Q(Q(Q(Q(p(z - 2))))) :|: z - 2 >= 0
   P(z) -{ 1 }-> Q(Q(0)) :|: z >= 0
   P(z) -{ 2 }-> Q(Q(z - 1)) :|: z - 1 >= 0
   P(z) -{ 1 }-> Q(Q(1 + z)) :|: z >= 0
   P(z) -{ 0 }-> 0 :|: z >= 0
   P(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   P(z) -{ 0 }-> 1 + z :|: z >= 0
   Q(z) -{ 2 }-> q(p(q(p(Q(z - 4))))) :|: z - 4 >= 0
   Q(z) -{ 1 }-> q(p(0)) :|: z - 2 >= 0
   Q(z) -{ 2 }-> q(p(z - 3)) :|: z - 3 >= 0
   Q(z) -{ 1 }-> q(p(1 + (z - 2))) :|: z - 2 >= 0
   Q(z) -{ 0 }-> 0 :|: z >= 0
   Q(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   Q(z) -{ 0 }-> 1 + z :|: z >= 0
   encArg(z) -{ 0 }-> q(q(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> q(p(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> q(Q(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> q(P(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> q(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> p(q(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> p(p(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> p(Q(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> p(P(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> p(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> Q(q(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> Q(p(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> Q(Q(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> Q(P(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> Q(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> P(q(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> P(p(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> P(Q(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> P(P(encArg(z - 2))) :|: z - 2 >= 0
   encArg(z) -{ 0 }-> P(0) :|: z - 1 >= 0
   encArg(z) -{ 0 }-> 0 :|: z >= 0
   encode_P(z) -{ 0 }-> P(q(encArg(z - 1))) :|: z - 1 >= 0
   encode_P(z) -{ 0 }-> P(p(encArg(z - 1))) :|: z - 1 >= 0
   encode_P(z) -{ 0 }-> P(Q(encArg(z - 1))) :|: z - 1 >= 0
   encode_P(z) -{ 0 }-> P(P(encArg(z - 1))) :|: z - 1 >= 0
   encode_P(z) -{ 0 }-> P(0) :|: z >= 0
   encode_P(z) -{ 0 }-> 0 :|: z >= 0
   encode_Q(z) -{ 0 }-> Q(q(encArg(z - 1))) :|: z - 1 >= 0
   encode_Q(z) -{ 0 }-> Q(p(encArg(z - 1))) :|: z - 1 >= 0
   encode_Q(z) -{ 0 }-> Q(Q(encArg(z - 1))) :|: z - 1 >= 0
   encode_Q(z) -{ 0 }-> Q(P(encArg(z - 1))) :|: z - 1 >= 0
   encode_Q(z) -{ 0 }-> Q(0) :|: z >= 0
   encode_Q(z) -{ 0 }-> 0 :|: z >= 0
   encode_p(z) -{ 0 }-> p(q(encArg(z - 1))) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> p(p(encArg(z - 1))) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> p(Q(encArg(z - 1))) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> p(P(encArg(z - 1))) :|: z - 1 >= 0
   encode_p(z) -{ 0 }-> p(0) :|: z >= 0
   encode_p(z) -{ 0 }-> 0 :|: z >= 0
   encode_q(z) -{ 0 }-> q(q(encArg(z - 1))) :|: z - 1 >= 0
   encode_q(z) -{ 0 }-> q(p(encArg(z - 1))) :|: z - 1 >= 0
   encode_q(z) -{ 0 }-> q(Q(encArg(z - 1))) :|: z - 1 >= 0
   encode_q(z) -{ 0 }-> q(P(encArg(z - 1))) :|: z - 1 >= 0
   encode_q(z) -{ 0 }-> q(0) :|: z >= 0
   encode_q(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 2 }-> q(p(q(q(z - 3)))) :|: z - 3 >= 0
   p(z) -{ 1 }-> q(0) :|: z - 1 >= 0
   p(z) -{ 2 }-> q(z - 2) :|: z - 2 >= 0
   p(z) -{ 1 }-> q(1 + (z - 1)) :|: z - 1 >= 0
   p(z) -{ 2 }-> Q(Q(q(q(z - 3)))) :|: z - 3 >= 0
   p(z) -{ 2 }-> Q(Q(Q(Q(p(z - 4))))) :|: z - 4 >= 0
   p(z) -{ 1 }-> Q(Q(0)) :|: z - 2 >= 0
   p(z) -{ 2 }-> Q(Q(z - 3)) :|: z - 3 >= 0
   p(z) -{ 1 }-> Q(Q(1 + (z - 2))) :|: z - 2 >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   p(z) -{ 0 }-> 1 + z :|: z >= 0
   q(z) -{ 2 }-> p(q(p(q(q(z - 4))))) :|: z - 4 >= 0
   q(z) -{ 1 }-> p(q(0)) :|: z - 2 >= 0
   q(z) -{ 2 }-> p(q(z - 3)) :|: z - 3 >= 0
   q(z) -{ 1 }-> p(q(1 + (z - 2))) :|: z - 2 >= 0
   q(z) -{ 0 }-> 0 :|: z >= 0
   q(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   q(z) -{ 0 }-> 1 + z :|: z >= 0


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

(31) 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_P(v0) -> null_encode_P [0]
   encode_Q(v0) -> null_encode_Q [0]
   encode_p(v0) -> null_encode_p [0]
   encode_q(v0) -> null_encode_q [0]
   p(v0) -> null_p [0]
   q(v0) -> null_q [0]
   Q(v0) -> null_Q [0]
   P(v0) -> null_P [0]

And the following fresh constants:    null_encArg, null_encode_P, null_encode_Q, null_encode_p, null_encode_q, null_p, null_q, null_Q, null_P, const

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

(32)
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:

   P(x1) -> Q(Q(p(x1))) [1]
   p(c_Q(c_Q(x1))) -> Q(Q(p(x1))) [1]
   Q(c_q(x1)) -> x1 [1]
   q(c_Q(x1)) -> x1 [1]
   Q(c_p(c_q(x1))) -> q(p(Q(x1))) [1]
   p(c_P(x1)) -> x1 [1]
   P(c_p(x1)) -> x1 [1]
   p(c_p(x1)) -> q(q(x1)) [1]
   q(c_q(c_p(x1))) -> p(q(q(x1))) [1]
   encArg(cons_P(x_1)) -> P(encArg(x_1)) [0]
   encArg(cons_p(x_1)) -> p(encArg(x_1)) [0]
   encArg(cons_Q(x_1)) -> Q(encArg(x_1)) [0]
   encArg(cons_q(x_1)) -> q(encArg(x_1)) [0]
   encode_P(x_1) -> P(encArg(x_1)) [0]
   encode_Q(x_1) -> Q(encArg(x_1)) [0]
   encode_p(x_1) -> p(encArg(x_1)) [0]
   encode_q(x_1) -> q(encArg(x_1)) [0]
   p(x0) -> c_p(x0) [0]
   q(x0) -> c_q(x0) [0]
   Q(x0) -> c_Q(x0) [0]
   P(x0) -> c_P(x0) [0]
   encArg(v0) -> null_encArg [0]
   encode_P(v0) -> null_encode_P [0]
   encode_Q(v0) -> null_encode_Q [0]
   encode_p(v0) -> null_encode_p [0]
   encode_q(v0) -> null_encode_q [0]
   p(v0) -> null_p [0]
   q(v0) -> null_q [0]
   Q(v0) -> null_Q [0]
   P(v0) -> null_P [0]

The TRS has the following type information:
P :: c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P -> c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P
Q :: c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P -> c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P
p :: c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P -> c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P
c_Q :: c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P -> c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P
c_q :: c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P -> c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P
q :: c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P -> c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P
c_p :: c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P -> c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P
c_P :: c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P -> c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P
encArg :: cons_P:cons_p:cons_Q:cons_q -> c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P
cons_P :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
cons_p :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
cons_Q :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
cons_q :: cons_P:cons_p:cons_Q:cons_q -> cons_P:cons_p:cons_Q:cons_q
encode_P :: cons_P:cons_p:cons_Q:cons_q -> c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P
encode_Q :: cons_P:cons_p:cons_Q:cons_q -> c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P
encode_p :: cons_P:cons_p:cons_Q:cons_q -> c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P
encode_q :: cons_P:cons_p:cons_Q:cons_q -> c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P
null_encArg :: c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P
null_encode_P :: c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P
null_encode_Q :: c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P
null_encode_p :: c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P
null_encode_q :: c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P
null_p :: c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P
null_q :: c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P
null_Q :: c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P
null_P :: c_Q:c_q:c_p:c_P:null_encArg:null_encode_P:null_encode_Q:null_encode_p:null_encode_q:null_p:null_q:null_Q:null_P
const :: cons_P:cons_p:cons_Q:cons_q

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

(33) 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_P => 0
   null_encode_Q => 0
   null_encode_p => 0
   null_encode_q => 0
   null_p => 0
   null_q => 0
   null_Q => 0
   null_P => 0
   const => 0

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

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

   P(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1
   P(z) -{ 1 }-> Q(Q(p(x1))) :|: x1 >= 0, z = x1
   P(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   P(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0
   Q(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1
   Q(z) -{ 1 }-> q(p(Q(x1))) :|: z = 1 + (1 + x1), x1 >= 0
   Q(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   Q(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0
   encArg(z) -{ 0 }-> q(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> p(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> Q(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> P(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_P(z) -{ 0 }-> P(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_P(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_Q(z) -{ 0 }-> Q(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_Q(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_p(z) -{ 0 }-> p(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_q(z) -{ 0 }-> q(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_q(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   p(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1
   p(z) -{ 1 }-> q(q(x1)) :|: x1 >= 0, z = 1 + x1
   p(z) -{ 1 }-> Q(Q(p(x1))) :|: z = 1 + (1 + x1), x1 >= 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   p(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0
   q(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1
   q(z) -{ 1 }-> p(q(q(x1))) :|: z = 1 + (1 + x1), x1 >= 0
   q(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   q(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0

Only complete derivations are relevant for the runtime complexity.

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

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   encode_P(z0) -> P(encArg(z0))
   encode_Q(z0) -> Q(encArg(z0))
   encode_p(z0) -> p(encArg(z0))
   encode_q(z0) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   p(P(z0)) -> z0
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_p(z0)) -> c1(P''(encArg(z0)), ENCARG(z0))
   ENCARG(cons_Q(z0)) -> c2(Q'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_q(z0)) -> c3(Q''(encArg(z0)), ENCARG(z0))
   ENCODE_P(z0) -> c4(P'(encArg(z0)), ENCARG(z0))
   ENCODE_Q(z0) -> c5(Q'(encArg(z0)), ENCARG(z0))
   ENCODE_P'(z0) -> c6(P''(encArg(z0)), ENCARG(z0))
   ENCODE_Q'(z0) -> c7(Q''(encArg(z0)), ENCARG(z0))
   P'(z0) -> c8(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   P'(p(z0)) -> c9
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   P''(Q(Q(z0))) -> c11(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   P''(P(z0)) -> c12
   Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
S tuples:
   P'(z0) -> c8(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   P'(p(z0)) -> c9
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   P''(Q(Q(z0))) -> c11(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   P''(P(z0)) -> c12
   Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
K tuples:none
Defined Rule Symbols:   P_1, p_1, Q_1, q_1, encArg_1, encode_P_1, encode_Q_1, encode_p_1, encode_q_1

Defined Pair Symbols:   ENCARG_1, ENCODE_P_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1, P'_1, P''_1, Q'_1, Q''_1

Compound Symbols:   c_2, c1_2, c2_2, c3_2, c4_2, c5_2, c6_2, c7_2, c8_3, c9, c10_2, c11_3, c12, c13_3, c14, c15_3, c16


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

(37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   P''(P(z0)) -> c12

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   encode_P(z0) -> P(encArg(z0))
   encode_Q(z0) -> Q(encArg(z0))
   encode_p(z0) -> p(encArg(z0))
   encode_q(z0) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   p(P(z0)) -> z0
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_p(z0)) -> c1(P''(encArg(z0)), ENCARG(z0))
   ENCARG(cons_Q(z0)) -> c2(Q'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_q(z0)) -> c3(Q''(encArg(z0)), ENCARG(z0))
   ENCODE_P(z0) -> c4(P'(encArg(z0)), ENCARG(z0))
   ENCODE_Q(z0) -> c5(Q'(encArg(z0)), ENCARG(z0))
   ENCODE_P'(z0) -> c6(P''(encArg(z0)), ENCARG(z0))
   ENCODE_Q'(z0) -> c7(Q''(encArg(z0)), ENCARG(z0))
   P'(z0) -> c8(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   P'(p(z0)) -> c9
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   P''(Q(Q(z0))) -> c11(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
S tuples:
   P'(z0) -> c8(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   P'(p(z0)) -> c9
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   P''(Q(Q(z0))) -> c11(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
K tuples:none
Defined Rule Symbols:   P_1, p_1, Q_1, q_1, encArg_1, encode_P_1, encode_Q_1, encode_p_1, encode_q_1

Defined Pair Symbols:   ENCARG_1, ENCODE_P_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1, P'_1, P''_1, Q'_1, Q''_1

Compound Symbols:   c_2, c1_2, c2_2, c3_2, c4_2, c5_2, c6_2, c7_2, c8_3, c9, c10_2, c11_3, c13_3, c14, c15_3, c16


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

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   encode_P(z0) -> P(encArg(z0))
   encode_Q(z0) -> Q(encArg(z0))
   encode_p(z0) -> p(encArg(z0))
   encode_q(z0) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   p(P(z0)) -> z0
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_p(z0)) -> c1(P''(encArg(z0)), ENCARG(z0))
   ENCARG(cons_Q(z0)) -> c2(Q'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_q(z0)) -> c3(Q''(encArg(z0)), ENCARG(z0))
   P'(p(z0)) -> c9
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   P''(Q(Q(z0))) -> c11(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   ENCODE_P(z0) -> c12(ENCARG(z0))
   ENCODE_Q(z0) -> c12(Q'(encArg(z0)))
   ENCODE_Q(z0) -> c12(ENCARG(z0))
   ENCODE_P'(z0) -> c12(P''(encArg(z0)))
   ENCODE_P'(z0) -> c12(ENCARG(z0))
   ENCODE_Q'(z0) -> c12(Q''(encArg(z0)))
   ENCODE_Q'(z0) -> c12(ENCARG(z0))
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
S tuples:
   P'(p(z0)) -> c9
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   P''(Q(Q(z0))) -> c11(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
K tuples:none
Defined Rule Symbols:   P_1, p_1, Q_1, q_1, encArg_1, encode_P_1, encode_Q_1, encode_p_1, encode_q_1

Defined Pair Symbols:   ENCARG_1, P'_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c1_2, c2_2, c3_2, c9, c10_2, c11_3, c13_3, c14, c15_3, c16, c12_1


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

(41) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 4 leading nodes:
   ENCODE_P(z0) -> c12(ENCARG(z0))
   ENCODE_Q(z0) -> c12(ENCARG(z0))
   ENCODE_P'(z0) -> c12(ENCARG(z0))
   ENCODE_Q'(z0) -> c12(ENCARG(z0))

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   encode_P(z0) -> P(encArg(z0))
   encode_Q(z0) -> Q(encArg(z0))
   encode_p(z0) -> p(encArg(z0))
   encode_q(z0) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   p(P(z0)) -> z0
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_p(z0)) -> c1(P''(encArg(z0)), ENCARG(z0))
   ENCARG(cons_Q(z0)) -> c2(Q'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_q(z0)) -> c3(Q''(encArg(z0)), ENCARG(z0))
   P'(p(z0)) -> c9
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   P''(Q(Q(z0))) -> c11(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   ENCODE_Q(z0) -> c12(Q'(encArg(z0)))
   ENCODE_P'(z0) -> c12(P''(encArg(z0)))
   ENCODE_Q'(z0) -> c12(Q''(encArg(z0)))
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
S tuples:
   P'(p(z0)) -> c9
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   P''(Q(Q(z0))) -> c11(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
K tuples:none
Defined Rule Symbols:   P_1, p_1, Q_1, q_1, encArg_1, encode_P_1, encode_Q_1, encode_p_1, encode_q_1

Defined Pair Symbols:   ENCARG_1, P'_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c1_2, c2_2, c3_2, c9, c10_2, c11_3, c13_3, c14, c15_3, c16, c12_1


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

(43) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   encode_P(z0) -> P(encArg(z0))
   encode_Q(z0) -> Q(encArg(z0))
   encode_p(z0) -> p(encArg(z0))
   encode_q(z0) -> q(encArg(z0))
   p(P(z0)) -> z0

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_p(z0)) -> c1(P''(encArg(z0)), ENCARG(z0))
   ENCARG(cons_Q(z0)) -> c2(Q'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_q(z0)) -> c3(Q''(encArg(z0)), ENCARG(z0))
   P'(p(z0)) -> c9
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   P''(Q(Q(z0))) -> c11(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   ENCODE_Q(z0) -> c12(Q'(encArg(z0)))
   ENCODE_P'(z0) -> c12(P''(encArg(z0)))
   ENCODE_Q'(z0) -> c12(Q''(encArg(z0)))
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
S tuples:
   P'(p(z0)) -> c9
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   P''(Q(Q(z0))) -> c11(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
K tuples:none
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P'_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c1_2, c2_2, c3_2, c9, c10_2, c11_3, c13_3, c14, c15_3, c16, c12_1


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

(45) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   P'(p(z0)) -> c9
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
We considered the (Usable) Rules:none
And the Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_p(z0)) -> c1(P''(encArg(z0)), ENCARG(z0))
   ENCARG(cons_Q(z0)) -> c2(Q'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_q(z0)) -> c3(Q''(encArg(z0)), ENCARG(z0))
   P'(p(z0)) -> c9
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   P''(Q(Q(z0))) -> c11(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   ENCODE_Q(z0) -> c12(Q'(encArg(z0)))
   ENCODE_P'(z0) -> c12(P''(encArg(z0)))
   ENCODE_Q'(z0) -> c12(Q''(encArg(z0)))
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ENCARG(x_1)) = x_1
   POL(ENCODE_P(x_1)) = [1]
   POL(ENCODE_P'(x_1)) = x_1
   POL(ENCODE_Q(x_1)) = 0
   POL(ENCODE_Q'(x_1)) = x_1
   POL(P(x_1)) = [1] + x_1
   POL(P'(x_1)) = [1]
   POL(P''(x_1)) = 0
   POL(Q(x_1)) = [1]
   POL(Q'(x_1)) = 0
   POL(Q''(x_1)) = 0
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1)) = x_1
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c14) = 0
   POL(c15(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c16) = 0
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c9) = 0
   POL(cons_P(x_1)) = [1] + x_1
   POL(cons_Q(x_1)) = [1] + x_1
   POL(cons_p(x_1)) = [1] + x_1
   POL(cons_q(x_1)) = [1] + x_1
   POL(encArg(x_1)) = [1] + x_1
   POL(p(x_1)) = [1] + x_1
   POL(q(x_1)) = x_1

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_p(z0)) -> c1(P''(encArg(z0)), ENCARG(z0))
   ENCARG(cons_Q(z0)) -> c2(Q'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_q(z0)) -> c3(Q''(encArg(z0)), ENCARG(z0))
   P'(p(z0)) -> c9
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   P''(Q(Q(z0))) -> c11(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   ENCODE_Q(z0) -> c12(Q'(encArg(z0)))
   ENCODE_P'(z0) -> c12(P''(encArg(z0)))
   ENCODE_Q'(z0) -> c12(Q''(encArg(z0)))
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   P''(Q(Q(z0))) -> c11(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
K tuples:
   P'(p(z0)) -> c9
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P'_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c1_2, c2_2, c3_2, c9, c10_2, c11_3, c13_3, c14, c15_3, c16, c12_1


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

(47) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_p(z0)) -> c1(P''(encArg(z0)), ENCARG(z0)) by 
   ENCARG(cons_p(cons_P(z0))) -> c1(P''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_p(cons_Q(z0))) -> c1(P''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_p(cons_q(z0))) -> c1(P''(q(encArg(z0))), ENCARG(cons_q(z0)))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_Q(z0)) -> c2(Q'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_q(z0)) -> c3(Q''(encArg(z0)), ENCARG(z0))
   P'(p(z0)) -> c9
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   P''(Q(Q(z0))) -> c11(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   ENCODE_Q(z0) -> c12(Q'(encArg(z0)))
   ENCODE_P'(z0) -> c12(P''(encArg(z0)))
   ENCODE_Q'(z0) -> c12(Q''(encArg(z0)))
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_P(z0))) -> c1(P''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_p(cons_Q(z0))) -> c1(P''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_p(cons_q(z0))) -> c1(P''(q(encArg(z0))), ENCARG(cons_q(z0)))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   P''(Q(Q(z0))) -> c11(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
K tuples:
   P'(p(z0)) -> c9
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P'_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c2_2, c3_2, c9, c10_2, c11_3, c13_3, c14, c15_3, c16, c12_1, c1_2


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

(49) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   P'(p(z0)) -> c9

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_Q(z0)) -> c2(Q'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_q(z0)) -> c3(Q''(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   P''(Q(Q(z0))) -> c11(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   ENCODE_Q(z0) -> c12(Q'(encArg(z0)))
   ENCODE_P'(z0) -> c12(P''(encArg(z0)))
   ENCODE_Q'(z0) -> c12(Q''(encArg(z0)))
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_P(z0))) -> c1(P''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_p(cons_Q(z0))) -> c1(P''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_p(cons_q(z0))) -> c1(P''(q(encArg(z0))), ENCARG(cons_q(z0)))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   P''(Q(Q(z0))) -> c11(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1, P'_1

Compound Symbols:   c_2, c2_2, c3_2, c10_2, c11_3, c13_3, c14, c15_3, c16, c12_1, c1_2


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

(51) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_Q(z0)) -> c2(Q'(encArg(z0)), ENCARG(z0)) by 
   ENCARG(cons_Q(cons_P(z0))) -> c2(Q'(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_Q(cons_p(z0))) -> c2(Q'(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_Q(z0))) -> c2(Q'(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   ENCARG(cons_q(z0)) -> c3(Q''(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   P''(Q(Q(z0))) -> c11(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   ENCODE_Q(z0) -> c12(Q'(encArg(z0)))
   ENCODE_P'(z0) -> c12(P''(encArg(z0)))
   ENCODE_Q'(z0) -> c12(Q''(encArg(z0)))
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_P(z0))) -> c1(P''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_p(cons_Q(z0))) -> c1(P''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_p(cons_q(z0))) -> c1(P''(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_Q(cons_P(z0))) -> c2(Q'(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_Q(cons_p(z0))) -> c2(Q'(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_Q(z0))) -> c2(Q'(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   P''(Q(Q(z0))) -> c11(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1, P'_1

Compound Symbols:   c_2, c3_2, c10_2, c11_3, c13_3, c14, c15_3, c16, c12_1, c1_2, c2_2


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

(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_q(z0)) -> c3(Q''(encArg(z0)), ENCARG(z0)) by 
   ENCARG(cons_q(cons_P(z0))) -> c3(Q''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_q(cons_p(z0))) -> c3(Q''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_q(cons_q(z0))) -> c3(Q''(q(encArg(z0))), ENCARG(cons_q(z0)))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   P''(Q(Q(z0))) -> c11(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   ENCODE_Q(z0) -> c12(Q'(encArg(z0)))
   ENCODE_P'(z0) -> c12(P''(encArg(z0)))
   ENCODE_Q'(z0) -> c12(Q''(encArg(z0)))
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_P(z0))) -> c1(P''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_p(cons_Q(z0))) -> c1(P''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_p(cons_q(z0))) -> c1(P''(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_Q(cons_P(z0))) -> c2(Q'(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_Q(cons_p(z0))) -> c2(Q'(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_Q(z0))) -> c2(Q'(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_P(z0))) -> c3(Q''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_q(cons_p(z0))) -> c3(Q''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_q(cons_q(z0))) -> c3(Q''(q(encArg(z0))), ENCARG(cons_q(z0)))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   P''(Q(Q(z0))) -> c11(Q'(Q(p(z0))), Q'(p(z0)), P''(z0))
   Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1, P'_1

Compound Symbols:   c_2, c10_2, c11_3, c13_3, c14, c15_3, c16, c12_1, c1_2, c2_2, c3_2


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

(55) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace P''(Q(Q(z0))) -> c11(Q'(Q(p(z0))), Q'(p(z0)), P''(z0)) by 
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   ENCODE_Q(z0) -> c12(Q'(encArg(z0)))
   ENCODE_P'(z0) -> c12(P''(encArg(z0)))
   ENCODE_Q'(z0) -> c12(Q''(encArg(z0)))
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_P(z0))) -> c1(P''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_p(cons_Q(z0))) -> c1(P''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_p(cons_q(z0))) -> c1(P''(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_Q(cons_P(z0))) -> c2(Q'(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_Q(cons_p(z0))) -> c2(Q'(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_Q(z0))) -> c2(Q'(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_P(z0))) -> c3(Q''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_q(cons_p(z0))) -> c3(Q''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_q(cons_q(z0))) -> c3(Q''(q(encArg(z0))), ENCARG(cons_q(z0)))
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1, P'_1

Compound Symbols:   c_2, c10_2, c13_3, c14, c15_3, c16, c12_1, c1_2, c2_2, c3_2, c11_3, c11_1


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

(57) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace Q'(p(q(z0))) -> c13(Q''(p(Q(z0))), P''(Q(z0)), Q'(z0)) by 
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(q(z0)))) -> c13(Q''(p(z0)), P''(Q(q(z0))), Q'(q(z0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   ENCODE_Q(z0) -> c12(Q'(encArg(z0)))
   ENCODE_P'(z0) -> c12(P''(encArg(z0)))
   ENCODE_Q'(z0) -> c12(Q''(encArg(z0)))
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_P(z0))) -> c1(P''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_p(cons_Q(z0))) -> c1(P''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_p(cons_q(z0))) -> c1(P''(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_Q(cons_P(z0))) -> c2(Q'(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_Q(cons_p(z0))) -> c2(Q'(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_Q(z0))) -> c2(Q'(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_P(z0))) -> c3(Q''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_q(cons_p(z0))) -> c3(Q''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_q(cons_q(z0))) -> c3(Q''(q(encArg(z0))), ENCARG(cons_q(z0)))
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(q(z0)))) -> c13(Q''(p(z0)), P''(Q(q(z0))), Q'(q(z0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(q(z0)))) -> c13(Q''(p(z0)), P''(Q(q(z0))), Q'(q(z0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1, P'_1

Compound Symbols:   c_2, c10_2, c14, c15_3, c16, c12_1, c1_2, c2_2, c3_2, c11_3, c11_1, c13_3, c13_1


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

(59) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   ENCODE_Q(z0) -> c12(Q'(encArg(z0)))
   ENCODE_P'(z0) -> c12(P''(encArg(z0)))
   ENCODE_Q'(z0) -> c12(Q''(encArg(z0)))
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_P(z0))) -> c1(P''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_p(cons_Q(z0))) -> c1(P''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_p(cons_q(z0))) -> c1(P''(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_Q(cons_P(z0))) -> c2(Q'(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_Q(cons_p(z0))) -> c2(Q'(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_Q(z0))) -> c2(Q'(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_P(z0))) -> c3(Q''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_q(cons_p(z0))) -> c3(Q''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_q(cons_q(z0))) -> c3(Q''(q(encArg(z0))), ENCARG(cons_q(z0)))
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0))
   Q''(Q(z0)) -> c16
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1, P'_1

Compound Symbols:   c_2, c10_2, c14, c15_3, c16, c12_1, c1_2, c2_2, c3_2, c11_3, c11_1, c13_3, c13_1, c13_2


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

(61) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace Q''(q(p(z0))) -> c15(P''(q(q(z0))), Q''(q(z0)), Q''(z0)) by 
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(Q(z0)))) -> c15(P''(q(z0)), Q''(q(Q(z0))), Q''(Q(z0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   ENCODE_Q(z0) -> c12(Q'(encArg(z0)))
   ENCODE_P'(z0) -> c12(P''(encArg(z0)))
   ENCODE_Q'(z0) -> c12(Q''(encArg(z0)))
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_P(z0))) -> c1(P''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_p(cons_Q(z0))) -> c1(P''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_p(cons_q(z0))) -> c1(P''(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_Q(cons_P(z0))) -> c2(Q'(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_Q(cons_p(z0))) -> c2(Q'(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_Q(z0))) -> c2(Q'(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_P(z0))) -> c3(Q''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_q(cons_p(z0))) -> c3(Q''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_q(cons_q(z0))) -> c3(Q''(q(encArg(z0))), ENCARG(cons_q(z0)))
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(Q(z0)))) -> c15(P''(q(z0)), Q''(q(Q(z0))), Q''(Q(z0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(Q(z0)))) -> c15(P''(q(z0)), Q''(q(Q(z0))), Q''(Q(z0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1, P'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_3, c11_1, c13_3, c13_1, c13_2, c15_3, c15_1


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

(63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   ENCODE_Q(z0) -> c12(Q'(encArg(z0)))
   ENCODE_P'(z0) -> c12(P''(encArg(z0)))
   ENCODE_Q'(z0) -> c12(Q''(encArg(z0)))
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_P(z0))) -> c1(P''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_p(cons_Q(z0))) -> c1(P''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_p(cons_q(z0))) -> c1(P''(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_Q(cons_P(z0))) -> c2(Q'(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_Q(cons_p(z0))) -> c2(Q'(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_Q(z0))) -> c2(Q'(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_P(z0))) -> c3(Q''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_q(cons_p(z0))) -> c3(Q''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_q(cons_q(z0))) -> c3(Q''(q(encArg(z0))), ENCARG(cons_q(z0)))
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1, P'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_3, c11_1, c13_3, c13_1, c13_2, c15_3, c15_1, c15_2


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

(65) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_Q(z0) -> c12(Q'(encArg(z0))) by 
   ENCODE_Q(cons_P(z0)) -> c12(Q'(P(encArg(z0))))
   ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0))))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   ENCODE_P'(z0) -> c12(P''(encArg(z0)))
   ENCODE_Q'(z0) -> c12(Q''(encArg(z0)))
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_P(z0))) -> c1(P''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_p(cons_Q(z0))) -> c1(P''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_p(cons_q(z0))) -> c1(P''(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_Q(cons_P(z0))) -> c2(Q'(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_Q(cons_p(z0))) -> c2(Q'(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_Q(z0))) -> c2(Q'(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_P(z0))) -> c3(Q''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_q(cons_p(z0))) -> c3(Q''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_q(cons_q(z0))) -> c3(Q''(q(encArg(z0))), ENCARG(cons_q(z0)))
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   ENCODE_Q(cons_P(z0)) -> c12(Q'(P(encArg(z0))))
   ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0))))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, ENCODE_P'_1, ENCODE_Q'_1, P'_1, ENCODE_Q_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_3, c11_1, c13_3, c13_1, c13_2, c15_3, c15_1, c15_2


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

(67) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_P'(z0) -> c12(P''(encArg(z0))) by 
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   ENCODE_Q'(z0) -> c12(Q''(encArg(z0)))
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_P(z0))) -> c1(P''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_p(cons_Q(z0))) -> c1(P''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_p(cons_q(z0))) -> c1(P''(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_Q(cons_P(z0))) -> c2(Q'(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_Q(cons_p(z0))) -> c2(Q'(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_Q(z0))) -> c2(Q'(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_P(z0))) -> c3(Q''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_q(cons_p(z0))) -> c3(Q''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_q(cons_q(z0))) -> c3(Q''(q(encArg(z0))), ENCARG(cons_q(z0)))
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   ENCODE_Q(cons_P(z0)) -> c12(Q'(P(encArg(z0))))
   ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0))))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, ENCODE_Q'_1, P'_1, ENCODE_Q_1, ENCODE_P'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_3, c11_1, c13_3, c13_1, c13_2, c15_3, c15_1, c15_2


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

(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_Q'(z0) -> c12(Q''(encArg(z0))) by 
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_P(z0))) -> c1(P''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_p(cons_Q(z0))) -> c1(P''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_p(cons_q(z0))) -> c1(P''(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_Q(cons_P(z0))) -> c2(Q'(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_Q(cons_p(z0))) -> c2(Q'(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_Q(z0))) -> c2(Q'(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_P(z0))) -> c3(Q''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_q(cons_p(z0))) -> c3(Q''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_q(cons_q(z0))) -> c3(Q''(q(encArg(z0))), ENCARG(cons_q(z0)))
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   ENCODE_Q(cons_P(z0)) -> c12(Q'(P(encArg(z0))))
   ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0))))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_3, c11_1, c13_3, c13_1, c13_2, c15_3, c15_1, c15_2


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

(71) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace P'(z0) -> c12(Q'(Q(p(z0)))) by 
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_P(z0))) -> c1(P''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_p(cons_Q(z0))) -> c1(P''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_p(cons_q(z0))) -> c1(P''(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_Q(cons_P(z0))) -> c2(Q'(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_Q(cons_p(z0))) -> c2(Q'(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_Q(z0))) -> c2(Q'(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_P(z0))) -> c3(Q''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_q(cons_p(z0))) -> c3(Q''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_q(cons_q(z0))) -> c3(Q''(q(encArg(z0))), ENCARG(cons_q(z0)))
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   ENCODE_Q(cons_P(z0)) -> c12(Q'(P(encArg(z0))))
   ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0))))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_3, c11_1, c13_3, c13_1, c13_2, c15_3, c15_1, c15_2


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

(73) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_p(cons_P(z0))) -> c1(P''(P(encArg(z0))), ENCARG(cons_P(z0))) by 
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_p(cons_Q(z0))) -> c1(P''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_p(cons_q(z0))) -> c1(P''(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_Q(cons_P(z0))) -> c2(Q'(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_Q(cons_p(z0))) -> c2(Q'(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_Q(z0))) -> c2(Q'(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_P(z0))) -> c3(Q''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_q(cons_p(z0))) -> c3(Q''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_q(cons_q(z0))) -> c3(Q''(q(encArg(z0))), ENCARG(cons_q(z0)))
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   ENCODE_Q(cons_P(z0)) -> c12(Q'(P(encArg(z0))))
   ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0))))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_3, c11_1, c13_3, c13_1, c13_2, c15_3, c15_1, c15_2, c1_1


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

(75) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_p(cons_Q(z0))) -> c1(P''(Q(encArg(z0))), ENCARG(cons_Q(z0))) by 
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_p(cons_q(z0))) -> c1(P''(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_Q(cons_P(z0))) -> c2(Q'(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_Q(cons_p(z0))) -> c2(Q'(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_Q(z0))) -> c2(Q'(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_P(z0))) -> c3(Q''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_q(cons_p(z0))) -> c3(Q''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_q(cons_q(z0))) -> c3(Q''(q(encArg(z0))), ENCARG(cons_q(z0)))
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   ENCODE_Q(cons_P(z0)) -> c12(Q'(P(encArg(z0))))
   ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0))))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_3, c11_1, c13_3, c13_1, c13_2, c15_3, c15_1, c15_2, c1_1


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

(77) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_p(cons_q(z0))) -> c1(P''(q(encArg(z0))), ENCARG(cons_q(z0))) by 
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_P(z0))) -> c2(Q'(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_Q(cons_p(z0))) -> c2(Q'(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_Q(z0))) -> c2(Q'(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_P(z0))) -> c3(Q''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_q(cons_p(z0))) -> c3(Q''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_q(cons_q(z0))) -> c3(Q''(q(encArg(z0))), ENCARG(cons_q(z0)))
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   ENCODE_Q(cons_P(z0)) -> c12(Q'(P(encArg(z0))))
   ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0))))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_3, c11_1, c13_3, c13_1, c13_2, c15_3, c15_1, c15_2, c1_1


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

(79) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_Q(cons_P(z0))) -> c2(Q'(P(encArg(z0))), ENCARG(cons_P(z0))) by 
   ENCARG(cons_Q(cons_P(x0))) -> c2(Q'(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_P(cons_P(z0)))) -> c2(Q'(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_Q(cons_P(cons_p(z0)))) -> c2(Q'(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_Q(cons_P(cons_Q(z0)))) -> c2(Q'(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_Q(cons_P(cons_q(z0)))) -> c2(Q'(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(ENCARG(cons_P(x0)))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_p(z0))) -> c2(Q'(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_Q(z0))) -> c2(Q'(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_P(z0))) -> c3(Q''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_q(cons_p(z0))) -> c3(Q''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_q(cons_q(z0))) -> c3(Q''(q(encArg(z0))), ENCARG(cons_q(z0)))
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   ENCODE_Q(cons_P(z0)) -> c12(Q'(P(encArg(z0))))
   ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0))))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(Q'(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_P(cons_P(z0)))) -> c2(Q'(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_Q(cons_P(cons_p(z0)))) -> c2(Q'(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_Q(cons_P(cons_Q(z0)))) -> c2(Q'(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_Q(cons_P(cons_q(z0)))) -> c2(Q'(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(ENCARG(cons_P(x0)))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_3, c11_1, c13_3, c13_1, c13_2, c15_3, c15_1, c15_2, c1_1, c2_1


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

(81) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_Q(cons_p(z0))) -> c2(Q'(p(encArg(z0))), ENCARG(cons_p(z0))) by 
   ENCARG(cons_Q(cons_p(cons_P(z0)))) -> c2(Q'(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_Q(cons_p(cons_p(z0)))) -> c2(Q'(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_Q(cons_p(cons_Q(z0)))) -> c2(Q'(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_Q(cons_p(cons_q(z0)))) -> c2(Q'(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_Q(z0))) -> c2(Q'(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_P(z0))) -> c3(Q''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_q(cons_p(z0))) -> c3(Q''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_q(cons_q(z0))) -> c3(Q''(q(encArg(z0))), ENCARG(cons_q(z0)))
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   ENCODE_Q(cons_P(z0)) -> c12(Q'(P(encArg(z0))))
   ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0))))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(Q'(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_P(cons_P(z0)))) -> c2(Q'(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_Q(cons_P(cons_p(z0)))) -> c2(Q'(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_Q(cons_P(cons_Q(z0)))) -> c2(Q'(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_Q(cons_P(cons_q(z0)))) -> c2(Q'(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_p(cons_P(z0)))) -> c2(Q'(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_Q(cons_p(cons_p(z0)))) -> c2(Q'(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_Q(cons_p(cons_Q(z0)))) -> c2(Q'(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_Q(cons_p(cons_q(z0)))) -> c2(Q'(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_3, c11_1, c13_3, c13_1, c13_2, c15_3, c15_1, c15_2, c1_1, c2_1


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

(83) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_Q(cons_Q(z0))) -> c2(Q'(Q(encArg(z0))), ENCARG(cons_Q(z0))) by 
   ENCARG(cons_Q(cons_Q(cons_P(z0)))) -> c2(Q'(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_Q(cons_Q(cons_p(z0)))) -> c2(Q'(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_Q(cons_Q(cons_Q(z0)))) -> c2(Q'(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_Q(cons_Q(cons_q(z0)))) -> c2(Q'(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_P(z0))) -> c3(Q''(P(encArg(z0))), ENCARG(cons_P(z0)))
   ENCARG(cons_q(cons_p(z0))) -> c3(Q''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_q(cons_q(z0))) -> c3(Q''(q(encArg(z0))), ENCARG(cons_q(z0)))
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   ENCODE_Q(cons_P(z0)) -> c12(Q'(P(encArg(z0))))
   ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0))))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(Q'(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_P(cons_P(z0)))) -> c2(Q'(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_Q(cons_P(cons_p(z0)))) -> c2(Q'(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_Q(cons_P(cons_Q(z0)))) -> c2(Q'(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_Q(cons_P(cons_q(z0)))) -> c2(Q'(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_p(cons_P(z0)))) -> c2(Q'(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_Q(cons_p(cons_p(z0)))) -> c2(Q'(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_Q(cons_p(cons_Q(z0)))) -> c2(Q'(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_Q(cons_p(cons_q(z0)))) -> c2(Q'(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_Q(cons_Q(cons_P(z0)))) -> c2(Q'(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_Q(cons_Q(cons_p(z0)))) -> c2(Q'(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_Q(cons_Q(cons_Q(z0)))) -> c2(Q'(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_Q(cons_Q(cons_q(z0)))) -> c2(Q'(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_3, c11_1, c13_3, c13_1, c13_2, c15_3, c15_1, c15_2, c1_1, c2_1


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

(85) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_q(cons_P(z0))) -> c3(Q''(P(encArg(z0))), ENCARG(cons_P(z0))) by 
   ENCARG(cons_q(cons_P(x0))) -> c3(Q''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_P(cons_P(z0)))) -> c3(Q''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_q(cons_P(cons_p(z0)))) -> c3(Q''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_q(cons_P(cons_Q(z0)))) -> c3(Q''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_q(cons_P(cons_q(z0)))) -> c3(Q''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(ENCARG(cons_P(x0)))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_p(z0))) -> c3(Q''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_q(cons_q(z0))) -> c3(Q''(q(encArg(z0))), ENCARG(cons_q(z0)))
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   ENCODE_Q(cons_P(z0)) -> c12(Q'(P(encArg(z0))))
   ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0))))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(Q'(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_P(cons_P(z0)))) -> c2(Q'(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_Q(cons_P(cons_p(z0)))) -> c2(Q'(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_Q(cons_P(cons_Q(z0)))) -> c2(Q'(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_Q(cons_P(cons_q(z0)))) -> c2(Q'(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_p(cons_P(z0)))) -> c2(Q'(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_Q(cons_p(cons_p(z0)))) -> c2(Q'(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_Q(cons_p(cons_Q(z0)))) -> c2(Q'(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_Q(cons_p(cons_q(z0)))) -> c2(Q'(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_Q(cons_Q(cons_P(z0)))) -> c2(Q'(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_Q(cons_Q(cons_p(z0)))) -> c2(Q'(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_Q(cons_Q(cons_Q(z0)))) -> c2(Q'(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_Q(cons_Q(cons_q(z0)))) -> c2(Q'(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(Q''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_P(cons_P(z0)))) -> c3(Q''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_q(cons_P(cons_p(z0)))) -> c3(Q''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_q(cons_P(cons_Q(z0)))) -> c3(Q''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_q(cons_P(cons_q(z0)))) -> c3(Q''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(ENCARG(cons_P(x0)))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_3, c11_1, c13_3, c13_1, c13_2, c15_3, c15_1, c15_2, c1_1, c2_1, c3_1


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

(87) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_q(cons_p(z0))) -> c3(Q''(p(encArg(z0))), ENCARG(cons_p(z0))) by 
   ENCARG(cons_q(cons_p(cons_P(z0)))) -> c3(Q''(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_q(cons_p(cons_p(z0)))) -> c3(Q''(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_q(cons_p(cons_Q(z0)))) -> c3(Q''(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_q(cons_p(cons_q(z0)))) -> c3(Q''(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   ENCARG(cons_q(cons_q(z0))) -> c3(Q''(q(encArg(z0))), ENCARG(cons_q(z0)))
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   ENCODE_Q(cons_P(z0)) -> c12(Q'(P(encArg(z0))))
   ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0))))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(Q'(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_P(cons_P(z0)))) -> c2(Q'(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_Q(cons_P(cons_p(z0)))) -> c2(Q'(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_Q(cons_P(cons_Q(z0)))) -> c2(Q'(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_Q(cons_P(cons_q(z0)))) -> c2(Q'(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_p(cons_P(z0)))) -> c2(Q'(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_Q(cons_p(cons_p(z0)))) -> c2(Q'(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_Q(cons_p(cons_Q(z0)))) -> c2(Q'(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_Q(cons_p(cons_q(z0)))) -> c2(Q'(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_Q(cons_Q(cons_P(z0)))) -> c2(Q'(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_Q(cons_Q(cons_p(z0)))) -> c2(Q'(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_Q(cons_Q(cons_Q(z0)))) -> c2(Q'(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_Q(cons_Q(cons_q(z0)))) -> c2(Q'(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(Q''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_P(cons_P(z0)))) -> c3(Q''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_q(cons_P(cons_p(z0)))) -> c3(Q''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_q(cons_P(cons_Q(z0)))) -> c3(Q''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_q(cons_P(cons_q(z0)))) -> c3(Q''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_p(cons_P(z0)))) -> c3(Q''(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_q(cons_p(cons_p(z0)))) -> c3(Q''(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_q(cons_p(cons_Q(z0)))) -> c3(Q''(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_q(cons_p(cons_q(z0)))) -> c3(Q''(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_3, c11_1, c13_3, c13_1, c13_2, c15_3, c15_1, c15_2, c1_1, c2_1, c3_1


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

(89) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCARG(cons_q(cons_q(z0))) -> c3(Q''(q(encArg(z0))), ENCARG(cons_q(z0))) by 
   ENCARG(cons_q(cons_q(cons_P(z0)))) -> c3(Q''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_q(cons_q(cons_p(z0)))) -> c3(Q''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_q(cons_q(cons_Q(z0)))) -> c3(Q''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_q(cons_q(cons_q(z0)))) -> c3(Q''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   ENCODE_Q(cons_P(z0)) -> c12(Q'(P(encArg(z0))))
   ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0))))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(Q'(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_P(cons_P(z0)))) -> c2(Q'(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_Q(cons_P(cons_p(z0)))) -> c2(Q'(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_Q(cons_P(cons_Q(z0)))) -> c2(Q'(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_Q(cons_P(cons_q(z0)))) -> c2(Q'(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_p(cons_P(z0)))) -> c2(Q'(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_Q(cons_p(cons_p(z0)))) -> c2(Q'(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_Q(cons_p(cons_Q(z0)))) -> c2(Q'(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_Q(cons_p(cons_q(z0)))) -> c2(Q'(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_Q(cons_Q(cons_P(z0)))) -> c2(Q'(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_Q(cons_Q(cons_p(z0)))) -> c2(Q'(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_Q(cons_Q(cons_Q(z0)))) -> c2(Q'(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_Q(cons_Q(cons_q(z0)))) -> c2(Q'(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(Q''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_P(cons_P(z0)))) -> c3(Q''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_q(cons_P(cons_p(z0)))) -> c3(Q''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_q(cons_P(cons_Q(z0)))) -> c3(Q''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_q(cons_P(cons_q(z0)))) -> c3(Q''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_p(cons_P(z0)))) -> c3(Q''(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_q(cons_p(cons_p(z0)))) -> c3(Q''(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_q(cons_p(cons_Q(z0)))) -> c3(Q''(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_q(cons_p(cons_q(z0)))) -> c3(Q''(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_q(cons_q(cons_P(z0)))) -> c3(Q''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_q(cons_q(cons_p(z0)))) -> c3(Q''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_q(cons_q(cons_Q(z0)))) -> c3(Q''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_q(cons_q(cons_q(z0)))) -> c3(Q''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_3, c11_1, c13_3, c13_1, c13_2, c15_3, c15_1, c15_2, c1_1, c2_1, c3_1


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

(91) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace P''(Q(Q(p(z0)))) -> c11(Q'(Q(q(q(z0)))), Q'(p(p(z0))), P''(p(z0))) by 
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   ENCODE_Q(cons_P(z0)) -> c12(Q'(P(encArg(z0))))
   ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0))))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(Q'(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_P(cons_P(z0)))) -> c2(Q'(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_Q(cons_P(cons_p(z0)))) -> c2(Q'(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_Q(cons_P(cons_Q(z0)))) -> c2(Q'(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_Q(cons_P(cons_q(z0)))) -> c2(Q'(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_p(cons_P(z0)))) -> c2(Q'(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_Q(cons_p(cons_p(z0)))) -> c2(Q'(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_Q(cons_p(cons_Q(z0)))) -> c2(Q'(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_Q(cons_p(cons_q(z0)))) -> c2(Q'(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_Q(cons_Q(cons_P(z0)))) -> c2(Q'(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_Q(cons_Q(cons_p(z0)))) -> c2(Q'(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_Q(cons_Q(cons_Q(z0)))) -> c2(Q'(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_Q(cons_Q(cons_q(z0)))) -> c2(Q'(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(Q''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_P(cons_P(z0)))) -> c3(Q''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_q(cons_P(cons_p(z0)))) -> c3(Q''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_q(cons_P(cons_Q(z0)))) -> c3(Q''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_q(cons_P(cons_q(z0)))) -> c3(Q''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_p(cons_P(z0)))) -> c3(Q''(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_q(cons_p(cons_p(z0)))) -> c3(Q''(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_q(cons_p(cons_Q(z0)))) -> c3(Q''(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_q(cons_p(cons_q(z0)))) -> c3(Q''(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_q(cons_q(cons_P(z0)))) -> c3(Q''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_q(cons_q(cons_p(z0)))) -> c3(Q''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_q(cons_q(cons_Q(z0)))) -> c3(Q''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_q(cons_q(cons_q(z0)))) -> c3(Q''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0))))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_3, c11_1, c13_3, c13_1, c13_2, c15_3, c15_1, c15_2, c1_1, c2_1, c3_1, c11_2


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

(93) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace P''(Q(Q(Q(Q(z0))))) -> c11(Q'(Q(Q(Q(p(z0))))), Q'(p(Q(Q(z0)))), P''(Q(Q(z0)))) by 
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   ENCODE_Q(cons_P(z0)) -> c12(Q'(P(encArg(z0))))
   ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0))))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(Q'(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_P(cons_P(z0)))) -> c2(Q'(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_Q(cons_P(cons_p(z0)))) -> c2(Q'(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_Q(cons_P(cons_Q(z0)))) -> c2(Q'(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_Q(cons_P(cons_q(z0)))) -> c2(Q'(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_p(cons_P(z0)))) -> c2(Q'(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_Q(cons_p(cons_p(z0)))) -> c2(Q'(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_Q(cons_p(cons_Q(z0)))) -> c2(Q'(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_Q(cons_p(cons_q(z0)))) -> c2(Q'(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_Q(cons_Q(cons_P(z0)))) -> c2(Q'(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_Q(cons_Q(cons_p(z0)))) -> c2(Q'(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_Q(cons_Q(cons_Q(z0)))) -> c2(Q'(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_Q(cons_Q(cons_q(z0)))) -> c2(Q'(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(Q''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_P(cons_P(z0)))) -> c3(Q''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_q(cons_P(cons_p(z0)))) -> c3(Q''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_q(cons_P(cons_Q(z0)))) -> c3(Q''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_q(cons_P(cons_q(z0)))) -> c3(Q''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_p(cons_P(z0)))) -> c3(Q''(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_q(cons_p(cons_p(z0)))) -> c3(Q''(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_q(cons_p(cons_Q(z0)))) -> c3(Q''(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_q(cons_p(cons_q(z0)))) -> c3(Q''(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_q(cons_q(cons_P(z0)))) -> c3(Q''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_q(cons_q(cons_p(z0)))) -> c3(Q''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_q(cons_q(cons_Q(z0)))) -> c3(Q''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_q(cons_q(cons_q(z0)))) -> c3(Q''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0))))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_1, c13_3, c13_1, c13_2, c15_3, c15_1, c15_2, c1_1, c2_1, c3_1, c11_3, c11_2


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

(95) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace Q'(p(q(p(q(z0))))) -> c13(Q''(p(q(p(Q(z0))))), P''(Q(p(q(z0)))), Q'(p(q(z0)))) by 
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(q(z0)))))) -> c13(Q''(p(q(p(z0)))), P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   ENCODE_Q(cons_P(z0)) -> c12(Q'(P(encArg(z0))))
   ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0))))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(Q'(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_P(cons_P(z0)))) -> c2(Q'(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_Q(cons_P(cons_p(z0)))) -> c2(Q'(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_Q(cons_P(cons_Q(z0)))) -> c2(Q'(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_Q(cons_P(cons_q(z0)))) -> c2(Q'(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_p(cons_P(z0)))) -> c2(Q'(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_Q(cons_p(cons_p(z0)))) -> c2(Q'(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_Q(cons_p(cons_Q(z0)))) -> c2(Q'(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_Q(cons_p(cons_q(z0)))) -> c2(Q'(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_Q(cons_Q(cons_P(z0)))) -> c2(Q'(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_Q(cons_Q(cons_p(z0)))) -> c2(Q'(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_Q(cons_Q(cons_Q(z0)))) -> c2(Q'(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_Q(cons_Q(cons_q(z0)))) -> c2(Q'(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(Q''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_P(cons_P(z0)))) -> c3(Q''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_q(cons_P(cons_p(z0)))) -> c3(Q''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_q(cons_P(cons_Q(z0)))) -> c3(Q''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_q(cons_P(cons_q(z0)))) -> c3(Q''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_p(cons_P(z0)))) -> c3(Q''(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_q(cons_p(cons_p(z0)))) -> c3(Q''(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_q(cons_p(cons_Q(z0)))) -> c3(Q''(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_q(cons_p(cons_q(z0)))) -> c3(Q''(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_q(cons_q(cons_P(z0)))) -> c3(Q''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_q(cons_q(cons_p(z0)))) -> c3(Q''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_q(cons_q(cons_Q(z0)))) -> c3(Q''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_q(cons_q(cons_q(z0)))) -> c3(Q''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(q(z0)))))) -> c13(Q''(p(q(p(z0)))), P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(q(z0)))))) -> c13(Q''(p(q(p(z0)))), P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_1, c13_1, c13_2, c15_3, c15_1, c15_2, c1_1, c2_1, c3_1, c11_3, c11_2, c13_3


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

(97) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   ENCODE_Q(cons_P(z0)) -> c12(Q'(P(encArg(z0))))
   ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0))))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(Q'(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_P(cons_P(z0)))) -> c2(Q'(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_Q(cons_P(cons_p(z0)))) -> c2(Q'(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_Q(cons_P(cons_Q(z0)))) -> c2(Q'(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_Q(cons_P(cons_q(z0)))) -> c2(Q'(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_p(cons_P(z0)))) -> c2(Q'(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_Q(cons_p(cons_p(z0)))) -> c2(Q'(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_Q(cons_p(cons_Q(z0)))) -> c2(Q'(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_Q(cons_p(cons_q(z0)))) -> c2(Q'(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_Q(cons_Q(cons_P(z0)))) -> c2(Q'(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_Q(cons_Q(cons_p(z0)))) -> c2(Q'(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_Q(cons_Q(cons_Q(z0)))) -> c2(Q'(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_Q(cons_Q(cons_q(z0)))) -> c2(Q'(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(Q''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_P(cons_P(z0)))) -> c3(Q''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_q(cons_P(cons_p(z0)))) -> c3(Q''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_q(cons_P(cons_Q(z0)))) -> c3(Q''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_q(cons_P(cons_q(z0)))) -> c3(Q''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_p(cons_P(z0)))) -> c3(Q''(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_q(cons_p(cons_p(z0)))) -> c3(Q''(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_q(cons_p(cons_Q(z0)))) -> c3(Q''(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_q(cons_p(cons_q(z0)))) -> c3(Q''(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_q(cons_q(cons_P(z0)))) -> c3(Q''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_q(cons_q(cons_p(z0)))) -> c3(Q''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_q(cons_q(cons_Q(z0)))) -> c3(Q''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_q(cons_q(cons_q(z0)))) -> c3(Q''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_1, c13_1, c13_2, c15_3, c15_1, c15_2, c1_1, c2_1, c3_1, c11_3, c11_2, c13_3


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

(99) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace Q'(p(q(q(z0)))) -> c13(P''(Q(q(z0))), Q'(q(z0))) by 
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   ENCODE_Q(cons_P(z0)) -> c12(Q'(P(encArg(z0))))
   ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0))))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(Q'(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_P(cons_P(z0)))) -> c2(Q'(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_Q(cons_P(cons_p(z0)))) -> c2(Q'(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_Q(cons_P(cons_Q(z0)))) -> c2(Q'(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_Q(cons_P(cons_q(z0)))) -> c2(Q'(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_p(cons_P(z0)))) -> c2(Q'(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_Q(cons_p(cons_p(z0)))) -> c2(Q'(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_Q(cons_p(cons_Q(z0)))) -> c2(Q'(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_Q(cons_p(cons_q(z0)))) -> c2(Q'(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_Q(cons_Q(cons_P(z0)))) -> c2(Q'(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_Q(cons_Q(cons_p(z0)))) -> c2(Q'(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_Q(cons_Q(cons_Q(z0)))) -> c2(Q'(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_Q(cons_Q(cons_q(z0)))) -> c2(Q'(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(Q''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_P(cons_P(z0)))) -> c3(Q''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_q(cons_P(cons_p(z0)))) -> c3(Q''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_q(cons_P(cons_Q(z0)))) -> c3(Q''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_q(cons_P(cons_q(z0)))) -> c3(Q''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_p(cons_P(z0)))) -> c3(Q''(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_q(cons_p(cons_p(z0)))) -> c3(Q''(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_q(cons_p(cons_Q(z0)))) -> c3(Q''(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_q(cons_p(cons_q(z0)))) -> c3(Q''(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_q(cons_q(cons_P(z0)))) -> c3(Q''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_q(cons_q(cons_p(z0)))) -> c3(Q''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_q(cons_q(cons_Q(z0)))) -> c3(Q''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_q(cons_q(cons_q(z0)))) -> c3(Q''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0))))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_1, c13_1, c15_3, c15_1, c15_2, c1_1, c2_1, c3_1, c11_3, c11_2, c13_3, c13_2


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

(101) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace Q''(q(p(q(p(z0))))) -> c15(P''(q(p(q(q(z0))))), Q''(q(q(p(z0)))), Q''(q(p(z0)))) by 
   Q''(q(p(q(p(q(p(z0))))))) -> c15(P''(q(p(q(p(q(q(z0))))))), Q''(q(q(p(q(p(z0)))))), Q''(q(p(q(p(z0))))))
   Q''(q(p(q(p(Q(z0)))))) -> c15(P''(q(p(q(z0)))), Q''(q(q(p(Q(z0))))), Q''(q(p(Q(z0)))))
   Q''(q(p(q(p(x0))))) -> c15(Q''(q(q(p(x0)))), Q''(q(p(x0))))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   ENCODE_Q(cons_P(z0)) -> c12(Q'(P(encArg(z0))))
   ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0))))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(Q'(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_P(cons_P(z0)))) -> c2(Q'(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_Q(cons_P(cons_p(z0)))) -> c2(Q'(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_Q(cons_P(cons_Q(z0)))) -> c2(Q'(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_Q(cons_P(cons_q(z0)))) -> c2(Q'(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_p(cons_P(z0)))) -> c2(Q'(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_Q(cons_p(cons_p(z0)))) -> c2(Q'(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_Q(cons_p(cons_Q(z0)))) -> c2(Q'(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_Q(cons_p(cons_q(z0)))) -> c2(Q'(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_Q(cons_Q(cons_P(z0)))) -> c2(Q'(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_Q(cons_Q(cons_p(z0)))) -> c2(Q'(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_Q(cons_Q(cons_Q(z0)))) -> c2(Q'(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_Q(cons_Q(cons_q(z0)))) -> c2(Q'(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(Q''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_P(cons_P(z0)))) -> c3(Q''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_q(cons_P(cons_p(z0)))) -> c3(Q''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_q(cons_P(cons_Q(z0)))) -> c3(Q''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_q(cons_P(cons_q(z0)))) -> c3(Q''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_p(cons_P(z0)))) -> c3(Q''(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_q(cons_p(cons_p(z0)))) -> c3(Q''(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_q(cons_p(cons_Q(z0)))) -> c3(Q''(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_q(cons_p(cons_q(z0)))) -> c3(Q''(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_q(cons_q(cons_P(z0)))) -> c3(Q''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_q(cons_q(cons_p(z0)))) -> c3(Q''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_q(cons_q(cons_Q(z0)))) -> c3(Q''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_q(cons_q(cons_q(z0)))) -> c3(Q''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
   Q''(q(p(q(p(q(p(z0))))))) -> c15(P''(q(p(q(p(q(q(z0))))))), Q''(q(q(p(q(p(z0)))))), Q''(q(p(q(p(z0))))))
   Q''(q(p(q(p(Q(z0)))))) -> c15(P''(q(p(q(z0)))), Q''(q(q(p(Q(z0))))), Q''(q(p(Q(z0)))))
   Q''(q(p(q(p(x0))))) -> c15(Q''(q(q(p(x0)))), Q''(q(p(x0))))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
   Q''(q(p(q(p(q(p(z0))))))) -> c15(P''(q(p(q(p(q(q(z0))))))), Q''(q(q(p(q(p(z0)))))), Q''(q(p(q(p(z0))))))
   Q''(q(p(q(p(Q(z0)))))) -> c15(P''(q(p(q(z0)))), Q''(q(q(p(Q(z0))))), Q''(q(p(Q(z0)))))
   Q''(q(p(q(p(x0))))) -> c15(Q''(q(q(p(x0)))), Q''(q(p(x0))))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_1, c13_1, c15_1, c15_2, c1_1, c2_1, c3_1, c11_3, c11_2, c13_3, c13_2, c15_3


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

(103) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   ENCODE_Q(cons_P(z0)) -> c12(Q'(P(encArg(z0))))
   ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0))))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(Q'(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_P(cons_P(z0)))) -> c2(Q'(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_Q(cons_P(cons_p(z0)))) -> c2(Q'(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_Q(cons_P(cons_Q(z0)))) -> c2(Q'(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_Q(cons_P(cons_q(z0)))) -> c2(Q'(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_p(cons_P(z0)))) -> c2(Q'(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_Q(cons_p(cons_p(z0)))) -> c2(Q'(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_Q(cons_p(cons_Q(z0)))) -> c2(Q'(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_Q(cons_p(cons_q(z0)))) -> c2(Q'(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_Q(cons_Q(cons_P(z0)))) -> c2(Q'(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_Q(cons_Q(cons_p(z0)))) -> c2(Q'(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_Q(cons_Q(cons_Q(z0)))) -> c2(Q'(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_Q(cons_Q(cons_q(z0)))) -> c2(Q'(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(Q''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_P(cons_P(z0)))) -> c3(Q''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_q(cons_P(cons_p(z0)))) -> c3(Q''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_q(cons_P(cons_Q(z0)))) -> c3(Q''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_q(cons_P(cons_q(z0)))) -> c3(Q''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_p(cons_P(z0)))) -> c3(Q''(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_q(cons_p(cons_p(z0)))) -> c3(Q''(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_q(cons_p(cons_Q(z0)))) -> c3(Q''(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_q(cons_p(cons_q(z0)))) -> c3(Q''(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_q(cons_q(cons_P(z0)))) -> c3(Q''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_q(cons_q(cons_p(z0)))) -> c3(Q''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_q(cons_q(cons_Q(z0)))) -> c3(Q''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_q(cons_q(cons_q(z0)))) -> c3(Q''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
   Q''(q(p(q(p(q(p(z0))))))) -> c15(P''(q(p(q(p(q(q(z0))))))), Q''(q(q(p(q(p(z0)))))), Q''(q(p(q(p(z0))))))
   Q''(q(p(q(p(x0))))) -> c15(Q''(q(q(p(x0)))), Q''(q(p(x0))))
   Q''(q(p(q(p(Q(z0)))))) -> c15(Q''(q(q(p(Q(z0))))), Q''(q(p(Q(z0)))))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0)))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
   Q''(q(p(q(p(q(p(z0))))))) -> c15(P''(q(p(q(p(q(q(z0))))))), Q''(q(q(p(q(p(z0)))))), Q''(q(p(q(p(z0))))))
   Q''(q(p(q(p(x0))))) -> c15(Q''(q(q(p(x0)))), Q''(q(p(x0))))
   Q''(q(p(q(p(Q(z0)))))) -> c15(Q''(q(q(p(Q(z0))))), Q''(q(p(Q(z0)))))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_1, c13_1, c15_1, c15_2, c1_1, c2_1, c3_1, c11_3, c11_2, c13_3, c13_2, c15_3


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

(105) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace Q''(q(p(Q(z0)))) -> c15(Q''(q(Q(z0))), Q''(Q(z0))) by 
   Q''(q(p(Q(z0)))) -> c15(Q''(z0), Q''(Q(z0)))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   ENCODE_Q(cons_P(z0)) -> c12(Q'(P(encArg(z0))))
   ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0))))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(Q'(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_P(cons_P(z0)))) -> c2(Q'(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_Q(cons_P(cons_p(z0)))) -> c2(Q'(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_Q(cons_P(cons_Q(z0)))) -> c2(Q'(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_Q(cons_P(cons_q(z0)))) -> c2(Q'(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_p(cons_P(z0)))) -> c2(Q'(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_Q(cons_p(cons_p(z0)))) -> c2(Q'(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_Q(cons_p(cons_Q(z0)))) -> c2(Q'(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_Q(cons_p(cons_q(z0)))) -> c2(Q'(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_Q(cons_Q(cons_P(z0)))) -> c2(Q'(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_Q(cons_Q(cons_p(z0)))) -> c2(Q'(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_Q(cons_Q(cons_Q(z0)))) -> c2(Q'(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_Q(cons_Q(cons_q(z0)))) -> c2(Q'(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(Q''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_P(cons_P(z0)))) -> c3(Q''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_q(cons_P(cons_p(z0)))) -> c3(Q''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_q(cons_P(cons_Q(z0)))) -> c3(Q''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_q(cons_P(cons_q(z0)))) -> c3(Q''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_p(cons_P(z0)))) -> c3(Q''(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_q(cons_p(cons_p(z0)))) -> c3(Q''(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_q(cons_p(cons_Q(z0)))) -> c3(Q''(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_q(cons_p(cons_q(z0)))) -> c3(Q''(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_q(cons_q(cons_P(z0)))) -> c3(Q''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_q(cons_q(cons_p(z0)))) -> c3(Q''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_q(cons_q(cons_Q(z0)))) -> c3(Q''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_q(cons_q(cons_q(z0)))) -> c3(Q''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
   Q''(q(p(q(p(q(p(z0))))))) -> c15(P''(q(p(q(p(q(q(z0))))))), Q''(q(q(p(q(p(z0)))))), Q''(q(p(q(p(z0))))))
   Q''(q(p(q(p(x0))))) -> c15(Q''(q(q(p(x0)))), Q''(q(p(x0))))
   Q''(q(p(q(p(Q(z0)))))) -> c15(Q''(q(q(p(Q(z0))))), Q''(q(p(Q(z0)))))
   Q''(q(p(Q(z0)))) -> c15(Q''(z0), Q''(Q(z0)))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
   Q''(q(p(q(p(q(p(z0))))))) -> c15(P''(q(p(q(p(q(q(z0))))))), Q''(q(q(p(q(p(z0)))))), Q''(q(p(q(p(z0))))))
   Q''(q(p(q(p(x0))))) -> c15(Q''(q(q(p(x0)))), Q''(q(p(x0))))
   Q''(q(p(q(p(Q(z0)))))) -> c15(Q''(q(q(p(Q(z0))))), Q''(q(p(Q(z0)))))
   Q''(q(p(Q(z0)))) -> c15(Q''(z0), Q''(Q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_1, c13_1, c15_1, c1_1, c2_1, c3_1, c11_3, c11_2, c13_3, c13_2, c15_3, c15_2


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

(107) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_Q(cons_P(z0)) -> c12(Q'(P(encArg(z0)))) by 
   ENCODE_Q(cons_P(x0)) -> c12(Q'(Q(Q(p(encArg(x0))))))
   ENCODE_Q(cons_P(cons_P(z0))) -> c12(Q'(P(P(encArg(z0)))))
   ENCODE_Q(cons_P(cons_p(z0))) -> c12(Q'(P(p(encArg(z0)))))
   ENCODE_Q(cons_P(cons_Q(z0))) -> c12(Q'(P(Q(encArg(z0)))))
   ENCODE_Q(cons_P(cons_q(z0))) -> c12(Q'(P(q(encArg(z0)))))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0))))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(Q'(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_P(cons_P(z0)))) -> c2(Q'(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_Q(cons_P(cons_p(z0)))) -> c2(Q'(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_Q(cons_P(cons_Q(z0)))) -> c2(Q'(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_Q(cons_P(cons_q(z0)))) -> c2(Q'(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_p(cons_P(z0)))) -> c2(Q'(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_Q(cons_p(cons_p(z0)))) -> c2(Q'(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_Q(cons_p(cons_Q(z0)))) -> c2(Q'(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_Q(cons_p(cons_q(z0)))) -> c2(Q'(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_Q(cons_Q(cons_P(z0)))) -> c2(Q'(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_Q(cons_Q(cons_p(z0)))) -> c2(Q'(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_Q(cons_Q(cons_Q(z0)))) -> c2(Q'(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_Q(cons_Q(cons_q(z0)))) -> c2(Q'(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(Q''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_P(cons_P(z0)))) -> c3(Q''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_q(cons_P(cons_p(z0)))) -> c3(Q''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_q(cons_P(cons_Q(z0)))) -> c3(Q''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_q(cons_P(cons_q(z0)))) -> c3(Q''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_p(cons_P(z0)))) -> c3(Q''(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_q(cons_p(cons_p(z0)))) -> c3(Q''(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_q(cons_p(cons_Q(z0)))) -> c3(Q''(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_q(cons_p(cons_q(z0)))) -> c3(Q''(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_q(cons_q(cons_P(z0)))) -> c3(Q''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_q(cons_q(cons_p(z0)))) -> c3(Q''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_q(cons_q(cons_Q(z0)))) -> c3(Q''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_q(cons_q(cons_q(z0)))) -> c3(Q''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
   Q''(q(p(q(p(q(p(z0))))))) -> c15(P''(q(p(q(p(q(q(z0))))))), Q''(q(q(p(q(p(z0)))))), Q''(q(p(q(p(z0))))))
   Q''(q(p(q(p(x0))))) -> c15(Q''(q(q(p(x0)))), Q''(q(p(x0))))
   Q''(q(p(q(p(Q(z0)))))) -> c15(Q''(q(q(p(Q(z0))))), Q''(q(p(Q(z0)))))
   Q''(q(p(Q(z0)))) -> c15(Q''(z0), Q''(Q(z0)))
   ENCODE_Q(cons_P(x0)) -> c12(Q'(Q(Q(p(encArg(x0))))))
   ENCODE_Q(cons_P(cons_P(z0))) -> c12(Q'(P(P(encArg(z0)))))
   ENCODE_Q(cons_P(cons_p(z0))) -> c12(Q'(P(p(encArg(z0)))))
   ENCODE_Q(cons_P(cons_Q(z0))) -> c12(Q'(P(Q(encArg(z0)))))
   ENCODE_Q(cons_P(cons_q(z0))) -> c12(Q'(P(q(encArg(z0)))))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
   Q''(q(p(q(p(q(p(z0))))))) -> c15(P''(q(p(q(p(q(q(z0))))))), Q''(q(q(p(q(p(z0)))))), Q''(q(p(q(p(z0))))))
   Q''(q(p(q(p(x0))))) -> c15(Q''(q(q(p(x0)))), Q''(q(p(x0))))
   Q''(q(p(q(p(Q(z0)))))) -> c15(Q''(q(q(p(Q(z0))))), Q''(q(p(Q(z0)))))
   Q''(q(p(Q(z0)))) -> c15(Q''(z0), Q''(Q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_1, c13_1, c15_1, c1_1, c2_1, c3_1, c11_3, c11_2, c13_3, c13_2, c15_3, c15_2


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

(109) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_Q(cons_p(z0)) -> c12(Q'(p(encArg(z0)))) by 
   ENCODE_Q(cons_p(cons_P(z0))) -> c12(Q'(p(P(encArg(z0)))))
   ENCODE_Q(cons_p(cons_p(z0))) -> c12(Q'(p(p(encArg(z0)))))
   ENCODE_Q(cons_p(cons_Q(z0))) -> c12(Q'(p(Q(encArg(z0)))))
   ENCODE_Q(cons_p(cons_q(z0))) -> c12(Q'(p(q(encArg(z0)))))

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0))))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(Q'(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_P(cons_P(z0)))) -> c2(Q'(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_Q(cons_P(cons_p(z0)))) -> c2(Q'(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_Q(cons_P(cons_Q(z0)))) -> c2(Q'(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_Q(cons_P(cons_q(z0)))) -> c2(Q'(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_p(cons_P(z0)))) -> c2(Q'(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_Q(cons_p(cons_p(z0)))) -> c2(Q'(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_Q(cons_p(cons_Q(z0)))) -> c2(Q'(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_Q(cons_p(cons_q(z0)))) -> c2(Q'(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_Q(cons_Q(cons_P(z0)))) -> c2(Q'(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_Q(cons_Q(cons_p(z0)))) -> c2(Q'(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_Q(cons_Q(cons_Q(z0)))) -> c2(Q'(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_Q(cons_Q(cons_q(z0)))) -> c2(Q'(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(Q''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_P(cons_P(z0)))) -> c3(Q''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_q(cons_P(cons_p(z0)))) -> c3(Q''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_q(cons_P(cons_Q(z0)))) -> c3(Q''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_q(cons_P(cons_q(z0)))) -> c3(Q''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_p(cons_P(z0)))) -> c3(Q''(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_q(cons_p(cons_p(z0)))) -> c3(Q''(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_q(cons_p(cons_Q(z0)))) -> c3(Q''(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_q(cons_p(cons_q(z0)))) -> c3(Q''(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_q(cons_q(cons_P(z0)))) -> c3(Q''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_q(cons_q(cons_p(z0)))) -> c3(Q''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_q(cons_q(cons_Q(z0)))) -> c3(Q''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_q(cons_q(cons_q(z0)))) -> c3(Q''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
   Q''(q(p(q(p(q(p(z0))))))) -> c15(P''(q(p(q(p(q(q(z0))))))), Q''(q(q(p(q(p(z0)))))), Q''(q(p(q(p(z0))))))
   Q''(q(p(q(p(x0))))) -> c15(Q''(q(q(p(x0)))), Q''(q(p(x0))))
   Q''(q(p(q(p(Q(z0)))))) -> c15(Q''(q(q(p(Q(z0))))), Q''(q(p(Q(z0)))))
   Q''(q(p(Q(z0)))) -> c15(Q''(z0), Q''(Q(z0)))
   ENCODE_Q(cons_P(x0)) -> c12(Q'(Q(Q(p(encArg(x0))))))
   ENCODE_Q(cons_P(cons_P(z0))) -> c12(Q'(P(P(encArg(z0)))))
   ENCODE_Q(cons_P(cons_p(z0))) -> c12(Q'(P(p(encArg(z0)))))
   ENCODE_Q(cons_P(cons_Q(z0))) -> c12(Q'(P(Q(encArg(z0)))))
   ENCODE_Q(cons_P(cons_q(z0))) -> c12(Q'(P(q(encArg(z0)))))
   ENCODE_Q(cons_p(cons_P(z0))) -> c12(Q'(p(P(encArg(z0)))))
   ENCODE_Q(cons_p(cons_p(z0))) -> c12(Q'(p(p(encArg(z0)))))
   ENCODE_Q(cons_p(cons_Q(z0))) -> c12(Q'(p(Q(encArg(z0)))))
   ENCODE_Q(cons_p(cons_q(z0))) -> c12(Q'(p(q(encArg(z0)))))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
   Q''(q(p(q(p(q(p(z0))))))) -> c15(P''(q(p(q(p(q(q(z0))))))), Q''(q(q(p(q(p(z0)))))), Q''(q(p(q(p(z0))))))
   Q''(q(p(q(p(x0))))) -> c15(Q''(q(q(p(x0)))), Q''(q(p(x0))))
   Q''(q(p(q(p(Q(z0)))))) -> c15(Q''(q(q(p(Q(z0))))), Q''(q(p(Q(z0)))))
   Q''(q(p(Q(z0)))) -> c15(Q''(z0), Q''(Q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_1, c13_1, c15_1, c1_1, c2_1, c3_1, c11_3, c11_2, c13_3, c13_2, c15_3, c15_2


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

(111) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_Q(cons_Q(z0)) -> c12(Q'(Q(encArg(z0)))) by 
   ENCODE_Q(cons_Q(cons_P(z0))) -> c12(Q'(Q(P(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_p(z0))) -> c12(Q'(Q(p(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_Q(z0))) -> c12(Q'(Q(Q(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_q(z0))) -> c12(Q'(Q(q(encArg(z0)))))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(Q'(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_P(cons_P(z0)))) -> c2(Q'(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_Q(cons_P(cons_p(z0)))) -> c2(Q'(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_Q(cons_P(cons_Q(z0)))) -> c2(Q'(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_Q(cons_P(cons_q(z0)))) -> c2(Q'(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_p(cons_P(z0)))) -> c2(Q'(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_Q(cons_p(cons_p(z0)))) -> c2(Q'(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_Q(cons_p(cons_Q(z0)))) -> c2(Q'(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_Q(cons_p(cons_q(z0)))) -> c2(Q'(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_Q(cons_Q(cons_P(z0)))) -> c2(Q'(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_Q(cons_Q(cons_p(z0)))) -> c2(Q'(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_Q(cons_Q(cons_Q(z0)))) -> c2(Q'(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_Q(cons_Q(cons_q(z0)))) -> c2(Q'(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(Q''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_P(cons_P(z0)))) -> c3(Q''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_q(cons_P(cons_p(z0)))) -> c3(Q''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_q(cons_P(cons_Q(z0)))) -> c3(Q''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_q(cons_P(cons_q(z0)))) -> c3(Q''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_p(cons_P(z0)))) -> c3(Q''(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_q(cons_p(cons_p(z0)))) -> c3(Q''(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_q(cons_p(cons_Q(z0)))) -> c3(Q''(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_q(cons_p(cons_q(z0)))) -> c3(Q''(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_q(cons_q(cons_P(z0)))) -> c3(Q''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_q(cons_q(cons_p(z0)))) -> c3(Q''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_q(cons_q(cons_Q(z0)))) -> c3(Q''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_q(cons_q(cons_q(z0)))) -> c3(Q''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
   Q''(q(p(q(p(q(p(z0))))))) -> c15(P''(q(p(q(p(q(q(z0))))))), Q''(q(q(p(q(p(z0)))))), Q''(q(p(q(p(z0))))))
   Q''(q(p(q(p(x0))))) -> c15(Q''(q(q(p(x0)))), Q''(q(p(x0))))
   Q''(q(p(q(p(Q(z0)))))) -> c15(Q''(q(q(p(Q(z0))))), Q''(q(p(Q(z0)))))
   Q''(q(p(Q(z0)))) -> c15(Q''(z0), Q''(Q(z0)))
   ENCODE_Q(cons_P(x0)) -> c12(Q'(Q(Q(p(encArg(x0))))))
   ENCODE_Q(cons_P(cons_P(z0))) -> c12(Q'(P(P(encArg(z0)))))
   ENCODE_Q(cons_P(cons_p(z0))) -> c12(Q'(P(p(encArg(z0)))))
   ENCODE_Q(cons_P(cons_Q(z0))) -> c12(Q'(P(Q(encArg(z0)))))
   ENCODE_Q(cons_P(cons_q(z0))) -> c12(Q'(P(q(encArg(z0)))))
   ENCODE_Q(cons_p(cons_P(z0))) -> c12(Q'(p(P(encArg(z0)))))
   ENCODE_Q(cons_p(cons_p(z0))) -> c12(Q'(p(p(encArg(z0)))))
   ENCODE_Q(cons_p(cons_Q(z0))) -> c12(Q'(p(Q(encArg(z0)))))
   ENCODE_Q(cons_p(cons_q(z0))) -> c12(Q'(p(q(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_P(z0))) -> c12(Q'(Q(P(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_p(z0))) -> c12(Q'(Q(p(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_Q(z0))) -> c12(Q'(Q(Q(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_q(z0))) -> c12(Q'(Q(q(encArg(z0)))))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
   Q''(q(p(q(p(q(p(z0))))))) -> c15(P''(q(p(q(p(q(q(z0))))))), Q''(q(q(p(q(p(z0)))))), Q''(q(p(q(p(z0))))))
   Q''(q(p(q(p(x0))))) -> c15(Q''(q(q(p(x0)))), Q''(q(p(x0))))
   Q''(q(p(q(p(Q(z0)))))) -> c15(Q''(q(q(p(Q(z0))))), Q''(q(p(Q(z0)))))
   Q''(q(p(Q(z0)))) -> c15(Q''(z0), Q''(Q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_1, c13_1, c15_1, c1_1, c2_1, c3_1, c11_3, c11_2, c13_3, c13_2, c15_3, c15_2


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

(113) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_P'(cons_P(z0)) -> c12(P''(P(encArg(z0)))) by 
   ENCODE_P'(cons_P(x0)) -> c12(P''(Q(Q(p(encArg(x0))))))
   ENCODE_P'(cons_P(cons_P(z0))) -> c12(P''(P(P(encArg(z0)))))
   ENCODE_P'(cons_P(cons_p(z0))) -> c12(P''(P(p(encArg(z0)))))
   ENCODE_P'(cons_P(cons_Q(z0))) -> c12(P''(P(Q(encArg(z0)))))
   ENCODE_P'(cons_P(cons_q(z0))) -> c12(P''(P(q(encArg(z0)))))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(Q'(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_P(cons_P(z0)))) -> c2(Q'(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_Q(cons_P(cons_p(z0)))) -> c2(Q'(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_Q(cons_P(cons_Q(z0)))) -> c2(Q'(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_Q(cons_P(cons_q(z0)))) -> c2(Q'(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_p(cons_P(z0)))) -> c2(Q'(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_Q(cons_p(cons_p(z0)))) -> c2(Q'(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_Q(cons_p(cons_Q(z0)))) -> c2(Q'(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_Q(cons_p(cons_q(z0)))) -> c2(Q'(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_Q(cons_Q(cons_P(z0)))) -> c2(Q'(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_Q(cons_Q(cons_p(z0)))) -> c2(Q'(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_Q(cons_Q(cons_Q(z0)))) -> c2(Q'(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_Q(cons_Q(cons_q(z0)))) -> c2(Q'(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(Q''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_P(cons_P(z0)))) -> c3(Q''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_q(cons_P(cons_p(z0)))) -> c3(Q''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_q(cons_P(cons_Q(z0)))) -> c3(Q''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_q(cons_P(cons_q(z0)))) -> c3(Q''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_p(cons_P(z0)))) -> c3(Q''(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_q(cons_p(cons_p(z0)))) -> c3(Q''(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_q(cons_p(cons_Q(z0)))) -> c3(Q''(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_q(cons_p(cons_q(z0)))) -> c3(Q''(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_q(cons_q(cons_P(z0)))) -> c3(Q''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_q(cons_q(cons_p(z0)))) -> c3(Q''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_q(cons_q(cons_Q(z0)))) -> c3(Q''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_q(cons_q(cons_q(z0)))) -> c3(Q''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
   Q''(q(p(q(p(q(p(z0))))))) -> c15(P''(q(p(q(p(q(q(z0))))))), Q''(q(q(p(q(p(z0)))))), Q''(q(p(q(p(z0))))))
   Q''(q(p(q(p(x0))))) -> c15(Q''(q(q(p(x0)))), Q''(q(p(x0))))
   Q''(q(p(q(p(Q(z0)))))) -> c15(Q''(q(q(p(Q(z0))))), Q''(q(p(Q(z0)))))
   Q''(q(p(Q(z0)))) -> c15(Q''(z0), Q''(Q(z0)))
   ENCODE_Q(cons_P(x0)) -> c12(Q'(Q(Q(p(encArg(x0))))))
   ENCODE_Q(cons_P(cons_P(z0))) -> c12(Q'(P(P(encArg(z0)))))
   ENCODE_Q(cons_P(cons_p(z0))) -> c12(Q'(P(p(encArg(z0)))))
   ENCODE_Q(cons_P(cons_Q(z0))) -> c12(Q'(P(Q(encArg(z0)))))
   ENCODE_Q(cons_P(cons_q(z0))) -> c12(Q'(P(q(encArg(z0)))))
   ENCODE_Q(cons_p(cons_P(z0))) -> c12(Q'(p(P(encArg(z0)))))
   ENCODE_Q(cons_p(cons_p(z0))) -> c12(Q'(p(p(encArg(z0)))))
   ENCODE_Q(cons_p(cons_Q(z0))) -> c12(Q'(p(Q(encArg(z0)))))
   ENCODE_Q(cons_p(cons_q(z0))) -> c12(Q'(p(q(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_P(z0))) -> c12(Q'(Q(P(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_p(z0))) -> c12(Q'(Q(p(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_Q(z0))) -> c12(Q'(Q(Q(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_q(z0))) -> c12(Q'(Q(q(encArg(z0)))))
   ENCODE_P'(cons_P(x0)) -> c12(P''(Q(Q(p(encArg(x0))))))
   ENCODE_P'(cons_P(cons_P(z0))) -> c12(P''(P(P(encArg(z0)))))
   ENCODE_P'(cons_P(cons_p(z0))) -> c12(P''(P(p(encArg(z0)))))
   ENCODE_P'(cons_P(cons_Q(z0))) -> c12(P''(P(Q(encArg(z0)))))
   ENCODE_P'(cons_P(cons_q(z0))) -> c12(P''(P(q(encArg(z0)))))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
   Q''(q(p(q(p(q(p(z0))))))) -> c15(P''(q(p(q(p(q(q(z0))))))), Q''(q(q(p(q(p(z0)))))), Q''(q(p(q(p(z0))))))
   Q''(q(p(q(p(x0))))) -> c15(Q''(q(q(p(x0)))), Q''(q(p(x0))))
   Q''(q(p(q(p(Q(z0)))))) -> c15(Q''(q(q(p(Q(z0))))), Q''(q(p(Q(z0)))))
   Q''(q(p(Q(z0)))) -> c15(Q''(z0), Q''(Q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_1, c13_1, c15_1, c1_1, c2_1, c3_1, c11_3, c11_2, c13_3, c13_2, c15_3, c15_2


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

(115) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_P'(cons_Q(z0)) -> c12(P''(Q(encArg(z0)))) by 
   ENCODE_P'(cons_Q(cons_P(z0))) -> c12(P''(Q(P(encArg(z0)))))
   ENCODE_P'(cons_Q(cons_p(z0))) -> c12(P''(Q(p(encArg(z0)))))
   ENCODE_P'(cons_Q(cons_Q(z0))) -> c12(P''(Q(Q(encArg(z0)))))
   ENCODE_P'(cons_Q(cons_q(z0))) -> c12(P''(Q(q(encArg(z0)))))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(Q'(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_P(cons_P(z0)))) -> c2(Q'(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_Q(cons_P(cons_p(z0)))) -> c2(Q'(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_Q(cons_P(cons_Q(z0)))) -> c2(Q'(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_Q(cons_P(cons_q(z0)))) -> c2(Q'(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_p(cons_P(z0)))) -> c2(Q'(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_Q(cons_p(cons_p(z0)))) -> c2(Q'(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_Q(cons_p(cons_Q(z0)))) -> c2(Q'(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_Q(cons_p(cons_q(z0)))) -> c2(Q'(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_Q(cons_Q(cons_P(z0)))) -> c2(Q'(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_Q(cons_Q(cons_p(z0)))) -> c2(Q'(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_Q(cons_Q(cons_Q(z0)))) -> c2(Q'(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_Q(cons_Q(cons_q(z0)))) -> c2(Q'(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(Q''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_P(cons_P(z0)))) -> c3(Q''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_q(cons_P(cons_p(z0)))) -> c3(Q''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_q(cons_P(cons_Q(z0)))) -> c3(Q''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_q(cons_P(cons_q(z0)))) -> c3(Q''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_p(cons_P(z0)))) -> c3(Q''(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_q(cons_p(cons_p(z0)))) -> c3(Q''(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_q(cons_p(cons_Q(z0)))) -> c3(Q''(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_q(cons_p(cons_q(z0)))) -> c3(Q''(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_q(cons_q(cons_P(z0)))) -> c3(Q''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_q(cons_q(cons_p(z0)))) -> c3(Q''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_q(cons_q(cons_Q(z0)))) -> c3(Q''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_q(cons_q(cons_q(z0)))) -> c3(Q''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
   Q''(q(p(q(p(q(p(z0))))))) -> c15(P''(q(p(q(p(q(q(z0))))))), Q''(q(q(p(q(p(z0)))))), Q''(q(p(q(p(z0))))))
   Q''(q(p(q(p(x0))))) -> c15(Q''(q(q(p(x0)))), Q''(q(p(x0))))
   Q''(q(p(q(p(Q(z0)))))) -> c15(Q''(q(q(p(Q(z0))))), Q''(q(p(Q(z0)))))
   Q''(q(p(Q(z0)))) -> c15(Q''(z0), Q''(Q(z0)))
   ENCODE_Q(cons_P(x0)) -> c12(Q'(Q(Q(p(encArg(x0))))))
   ENCODE_Q(cons_P(cons_P(z0))) -> c12(Q'(P(P(encArg(z0)))))
   ENCODE_Q(cons_P(cons_p(z0))) -> c12(Q'(P(p(encArg(z0)))))
   ENCODE_Q(cons_P(cons_Q(z0))) -> c12(Q'(P(Q(encArg(z0)))))
   ENCODE_Q(cons_P(cons_q(z0))) -> c12(Q'(P(q(encArg(z0)))))
   ENCODE_Q(cons_p(cons_P(z0))) -> c12(Q'(p(P(encArg(z0)))))
   ENCODE_Q(cons_p(cons_p(z0))) -> c12(Q'(p(p(encArg(z0)))))
   ENCODE_Q(cons_p(cons_Q(z0))) -> c12(Q'(p(Q(encArg(z0)))))
   ENCODE_Q(cons_p(cons_q(z0))) -> c12(Q'(p(q(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_P(z0))) -> c12(Q'(Q(P(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_p(z0))) -> c12(Q'(Q(p(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_Q(z0))) -> c12(Q'(Q(Q(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_q(z0))) -> c12(Q'(Q(q(encArg(z0)))))
   ENCODE_P'(cons_P(x0)) -> c12(P''(Q(Q(p(encArg(x0))))))
   ENCODE_P'(cons_P(cons_P(z0))) -> c12(P''(P(P(encArg(z0)))))
   ENCODE_P'(cons_P(cons_p(z0))) -> c12(P''(P(p(encArg(z0)))))
   ENCODE_P'(cons_P(cons_Q(z0))) -> c12(P''(P(Q(encArg(z0)))))
   ENCODE_P'(cons_P(cons_q(z0))) -> c12(P''(P(q(encArg(z0)))))
   ENCODE_P'(cons_Q(cons_P(z0))) -> c12(P''(Q(P(encArg(z0)))))
   ENCODE_P'(cons_Q(cons_p(z0))) -> c12(P''(Q(p(encArg(z0)))))
   ENCODE_P'(cons_Q(cons_Q(z0))) -> c12(P''(Q(Q(encArg(z0)))))
   ENCODE_P'(cons_Q(cons_q(z0))) -> c12(P''(Q(q(encArg(z0)))))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
   Q''(q(p(q(p(q(p(z0))))))) -> c15(P''(q(p(q(p(q(q(z0))))))), Q''(q(q(p(q(p(z0)))))), Q''(q(p(q(p(z0))))))
   Q''(q(p(q(p(x0))))) -> c15(Q''(q(q(p(x0)))), Q''(q(p(x0))))
   Q''(q(p(q(p(Q(z0)))))) -> c15(Q''(q(q(p(Q(z0))))), Q''(q(p(Q(z0)))))
   Q''(q(p(Q(z0)))) -> c15(Q''(z0), Q''(Q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_1, c13_1, c15_1, c1_1, c2_1, c3_1, c11_3, c11_2, c13_3, c13_2, c15_3, c15_2


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

(117) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_P'(cons_q(z0)) -> c12(P''(q(encArg(z0)))) by 
   ENCODE_P'(cons_q(cons_P(z0))) -> c12(P''(q(P(encArg(z0)))))
   ENCODE_P'(cons_q(cons_p(z0))) -> c12(P''(q(p(encArg(z0)))))
   ENCODE_P'(cons_q(cons_Q(z0))) -> c12(P''(q(Q(encArg(z0)))))
   ENCODE_P'(cons_q(cons_q(z0))) -> c12(P''(q(q(encArg(z0)))))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(Q'(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_P(cons_P(z0)))) -> c2(Q'(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_Q(cons_P(cons_p(z0)))) -> c2(Q'(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_Q(cons_P(cons_Q(z0)))) -> c2(Q'(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_Q(cons_P(cons_q(z0)))) -> c2(Q'(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_p(cons_P(z0)))) -> c2(Q'(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_Q(cons_p(cons_p(z0)))) -> c2(Q'(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_Q(cons_p(cons_Q(z0)))) -> c2(Q'(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_Q(cons_p(cons_q(z0)))) -> c2(Q'(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_Q(cons_Q(cons_P(z0)))) -> c2(Q'(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_Q(cons_Q(cons_p(z0)))) -> c2(Q'(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_Q(cons_Q(cons_Q(z0)))) -> c2(Q'(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_Q(cons_Q(cons_q(z0)))) -> c2(Q'(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(Q''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_P(cons_P(z0)))) -> c3(Q''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_q(cons_P(cons_p(z0)))) -> c3(Q''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_q(cons_P(cons_Q(z0)))) -> c3(Q''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_q(cons_P(cons_q(z0)))) -> c3(Q''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_p(cons_P(z0)))) -> c3(Q''(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_q(cons_p(cons_p(z0)))) -> c3(Q''(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_q(cons_p(cons_Q(z0)))) -> c3(Q''(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_q(cons_p(cons_q(z0)))) -> c3(Q''(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_q(cons_q(cons_P(z0)))) -> c3(Q''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_q(cons_q(cons_p(z0)))) -> c3(Q''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_q(cons_q(cons_Q(z0)))) -> c3(Q''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_q(cons_q(cons_q(z0)))) -> c3(Q''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
   Q''(q(p(q(p(q(p(z0))))))) -> c15(P''(q(p(q(p(q(q(z0))))))), Q''(q(q(p(q(p(z0)))))), Q''(q(p(q(p(z0))))))
   Q''(q(p(q(p(x0))))) -> c15(Q''(q(q(p(x0)))), Q''(q(p(x0))))
   Q''(q(p(q(p(Q(z0)))))) -> c15(Q''(q(q(p(Q(z0))))), Q''(q(p(Q(z0)))))
   Q''(q(p(Q(z0)))) -> c15(Q''(z0), Q''(Q(z0)))
   ENCODE_Q(cons_P(x0)) -> c12(Q'(Q(Q(p(encArg(x0))))))
   ENCODE_Q(cons_P(cons_P(z0))) -> c12(Q'(P(P(encArg(z0)))))
   ENCODE_Q(cons_P(cons_p(z0))) -> c12(Q'(P(p(encArg(z0)))))
   ENCODE_Q(cons_P(cons_Q(z0))) -> c12(Q'(P(Q(encArg(z0)))))
   ENCODE_Q(cons_P(cons_q(z0))) -> c12(Q'(P(q(encArg(z0)))))
   ENCODE_Q(cons_p(cons_P(z0))) -> c12(Q'(p(P(encArg(z0)))))
   ENCODE_Q(cons_p(cons_p(z0))) -> c12(Q'(p(p(encArg(z0)))))
   ENCODE_Q(cons_p(cons_Q(z0))) -> c12(Q'(p(Q(encArg(z0)))))
   ENCODE_Q(cons_p(cons_q(z0))) -> c12(Q'(p(q(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_P(z0))) -> c12(Q'(Q(P(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_p(z0))) -> c12(Q'(Q(p(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_Q(z0))) -> c12(Q'(Q(Q(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_q(z0))) -> c12(Q'(Q(q(encArg(z0)))))
   ENCODE_P'(cons_P(x0)) -> c12(P''(Q(Q(p(encArg(x0))))))
   ENCODE_P'(cons_P(cons_P(z0))) -> c12(P''(P(P(encArg(z0)))))
   ENCODE_P'(cons_P(cons_p(z0))) -> c12(P''(P(p(encArg(z0)))))
   ENCODE_P'(cons_P(cons_Q(z0))) -> c12(P''(P(Q(encArg(z0)))))
   ENCODE_P'(cons_P(cons_q(z0))) -> c12(P''(P(q(encArg(z0)))))
   ENCODE_P'(cons_Q(cons_P(z0))) -> c12(P''(Q(P(encArg(z0)))))
   ENCODE_P'(cons_Q(cons_p(z0))) -> c12(P''(Q(p(encArg(z0)))))
   ENCODE_P'(cons_Q(cons_Q(z0))) -> c12(P''(Q(Q(encArg(z0)))))
   ENCODE_P'(cons_Q(cons_q(z0))) -> c12(P''(Q(q(encArg(z0)))))
   ENCODE_P'(cons_q(cons_P(z0))) -> c12(P''(q(P(encArg(z0)))))
   ENCODE_P'(cons_q(cons_p(z0))) -> c12(P''(q(p(encArg(z0)))))
   ENCODE_P'(cons_q(cons_Q(z0))) -> c12(P''(q(Q(encArg(z0)))))
   ENCODE_P'(cons_q(cons_q(z0))) -> c12(P''(q(q(encArg(z0)))))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
   Q''(q(p(q(p(q(p(z0))))))) -> c15(P''(q(p(q(p(q(q(z0))))))), Q''(q(q(p(q(p(z0)))))), Q''(q(p(q(p(z0))))))
   Q''(q(p(q(p(x0))))) -> c15(Q''(q(q(p(x0)))), Q''(q(p(x0))))
   Q''(q(p(q(p(Q(z0)))))) -> c15(Q''(q(q(p(Q(z0))))), Q''(q(p(Q(z0)))))
   Q''(q(p(Q(z0)))) -> c15(Q''(z0), Q''(Q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_1, c13_1, c15_1, c1_1, c2_1, c3_1, c11_3, c11_2, c13_3, c13_2, c15_3, c15_2


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

(119) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_Q'(cons_P(z0)) -> c12(Q''(P(encArg(z0)))) by 
   ENCODE_Q'(cons_P(x0)) -> c12(Q''(Q(Q(p(encArg(x0))))))
   ENCODE_Q'(cons_P(cons_P(z0))) -> c12(Q''(P(P(encArg(z0)))))
   ENCODE_Q'(cons_P(cons_p(z0))) -> c12(Q''(P(p(encArg(z0)))))
   ENCODE_Q'(cons_P(cons_Q(z0))) -> c12(Q''(P(Q(encArg(z0)))))
   ENCODE_Q'(cons_P(cons_q(z0))) -> c12(Q''(P(q(encArg(z0)))))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(Q'(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_P(cons_P(z0)))) -> c2(Q'(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_Q(cons_P(cons_p(z0)))) -> c2(Q'(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_Q(cons_P(cons_Q(z0)))) -> c2(Q'(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_Q(cons_P(cons_q(z0)))) -> c2(Q'(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_p(cons_P(z0)))) -> c2(Q'(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_Q(cons_p(cons_p(z0)))) -> c2(Q'(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_Q(cons_p(cons_Q(z0)))) -> c2(Q'(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_Q(cons_p(cons_q(z0)))) -> c2(Q'(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_Q(cons_Q(cons_P(z0)))) -> c2(Q'(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_Q(cons_Q(cons_p(z0)))) -> c2(Q'(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_Q(cons_Q(cons_Q(z0)))) -> c2(Q'(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_Q(cons_Q(cons_q(z0)))) -> c2(Q'(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(Q''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_P(cons_P(z0)))) -> c3(Q''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_q(cons_P(cons_p(z0)))) -> c3(Q''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_q(cons_P(cons_Q(z0)))) -> c3(Q''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_q(cons_P(cons_q(z0)))) -> c3(Q''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_p(cons_P(z0)))) -> c3(Q''(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_q(cons_p(cons_p(z0)))) -> c3(Q''(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_q(cons_p(cons_Q(z0)))) -> c3(Q''(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_q(cons_p(cons_q(z0)))) -> c3(Q''(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_q(cons_q(cons_P(z0)))) -> c3(Q''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_q(cons_q(cons_p(z0)))) -> c3(Q''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_q(cons_q(cons_Q(z0)))) -> c3(Q''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_q(cons_q(cons_q(z0)))) -> c3(Q''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
   Q''(q(p(q(p(q(p(z0))))))) -> c15(P''(q(p(q(p(q(q(z0))))))), Q''(q(q(p(q(p(z0)))))), Q''(q(p(q(p(z0))))))
   Q''(q(p(q(p(x0))))) -> c15(Q''(q(q(p(x0)))), Q''(q(p(x0))))
   Q''(q(p(q(p(Q(z0)))))) -> c15(Q''(q(q(p(Q(z0))))), Q''(q(p(Q(z0)))))
   Q''(q(p(Q(z0)))) -> c15(Q''(z0), Q''(Q(z0)))
   ENCODE_Q(cons_P(x0)) -> c12(Q'(Q(Q(p(encArg(x0))))))
   ENCODE_Q(cons_P(cons_P(z0))) -> c12(Q'(P(P(encArg(z0)))))
   ENCODE_Q(cons_P(cons_p(z0))) -> c12(Q'(P(p(encArg(z0)))))
   ENCODE_Q(cons_P(cons_Q(z0))) -> c12(Q'(P(Q(encArg(z0)))))
   ENCODE_Q(cons_P(cons_q(z0))) -> c12(Q'(P(q(encArg(z0)))))
   ENCODE_Q(cons_p(cons_P(z0))) -> c12(Q'(p(P(encArg(z0)))))
   ENCODE_Q(cons_p(cons_p(z0))) -> c12(Q'(p(p(encArg(z0)))))
   ENCODE_Q(cons_p(cons_Q(z0))) -> c12(Q'(p(Q(encArg(z0)))))
   ENCODE_Q(cons_p(cons_q(z0))) -> c12(Q'(p(q(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_P(z0))) -> c12(Q'(Q(P(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_p(z0))) -> c12(Q'(Q(p(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_Q(z0))) -> c12(Q'(Q(Q(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_q(z0))) -> c12(Q'(Q(q(encArg(z0)))))
   ENCODE_P'(cons_P(x0)) -> c12(P''(Q(Q(p(encArg(x0))))))
   ENCODE_P'(cons_P(cons_P(z0))) -> c12(P''(P(P(encArg(z0)))))
   ENCODE_P'(cons_P(cons_p(z0))) -> c12(P''(P(p(encArg(z0)))))
   ENCODE_P'(cons_P(cons_Q(z0))) -> c12(P''(P(Q(encArg(z0)))))
   ENCODE_P'(cons_P(cons_q(z0))) -> c12(P''(P(q(encArg(z0)))))
   ENCODE_P'(cons_Q(cons_P(z0))) -> c12(P''(Q(P(encArg(z0)))))
   ENCODE_P'(cons_Q(cons_p(z0))) -> c12(P''(Q(p(encArg(z0)))))
   ENCODE_P'(cons_Q(cons_Q(z0))) -> c12(P''(Q(Q(encArg(z0)))))
   ENCODE_P'(cons_Q(cons_q(z0))) -> c12(P''(Q(q(encArg(z0)))))
   ENCODE_P'(cons_q(cons_P(z0))) -> c12(P''(q(P(encArg(z0)))))
   ENCODE_P'(cons_q(cons_p(z0))) -> c12(P''(q(p(encArg(z0)))))
   ENCODE_P'(cons_q(cons_Q(z0))) -> c12(P''(q(Q(encArg(z0)))))
   ENCODE_P'(cons_q(cons_q(z0))) -> c12(P''(q(q(encArg(z0)))))
   ENCODE_Q'(cons_P(x0)) -> c12(Q''(Q(Q(p(encArg(x0))))))
   ENCODE_Q'(cons_P(cons_P(z0))) -> c12(Q''(P(P(encArg(z0)))))
   ENCODE_Q'(cons_P(cons_p(z0))) -> c12(Q''(P(p(encArg(z0)))))
   ENCODE_Q'(cons_P(cons_Q(z0))) -> c12(Q''(P(Q(encArg(z0)))))
   ENCODE_Q'(cons_P(cons_q(z0))) -> c12(Q''(P(q(encArg(z0)))))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
   Q''(q(p(q(p(q(p(z0))))))) -> c15(P''(q(p(q(p(q(q(z0))))))), Q''(q(q(p(q(p(z0)))))), Q''(q(p(q(p(z0))))))
   Q''(q(p(q(p(x0))))) -> c15(Q''(q(q(p(x0)))), Q''(q(p(x0))))
   Q''(q(p(q(p(Q(z0)))))) -> c15(Q''(q(q(p(Q(z0))))), Q''(q(p(Q(z0)))))
   Q''(q(p(Q(z0)))) -> c15(Q''(z0), Q''(Q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_1, c13_1, c15_1, c1_1, c2_1, c3_1, c11_3, c11_2, c13_3, c13_2, c15_3, c15_2


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

(121) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace ENCODE_Q'(cons_p(z0)) -> c12(Q''(p(encArg(z0)))) by 
   ENCODE_Q'(cons_p(cons_P(z0))) -> c12(Q''(p(P(encArg(z0)))))
   ENCODE_Q'(cons_p(cons_p(z0))) -> c12(Q''(p(p(encArg(z0)))))
   ENCODE_Q'(cons_p(cons_Q(z0))) -> c12(Q''(p(Q(encArg(z0)))))
   ENCODE_Q'(cons_p(cons_q(z0))) -> c12(Q''(p(q(encArg(z0)))))

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

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   encArg(cons_P(z0)) -> P(encArg(z0))
   encArg(cons_p(z0)) -> p(encArg(z0))
   encArg(cons_Q(z0)) -> Q(encArg(z0))
   encArg(cons_q(z0)) -> q(encArg(z0))
   P(z0) -> Q(Q(p(z0)))
   P(p(z0)) -> z0
   p(p(z0)) -> q(q(z0))
   p(Q(Q(z0))) -> Q(Q(p(z0)))
   Q(p(q(z0))) -> q(p(Q(z0)))
   Q(q(z0)) -> z0
   q(q(p(z0))) -> p(q(q(z0)))
   q(Q(z0)) -> z0
Tuples:
   ENCARG(cons_P(z0)) -> c(P'(encArg(z0)), ENCARG(z0))
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   ENCODE_P(z0) -> c12(P'(encArg(z0)))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
   ENCARG(cons_p(cons_p(z0))) -> c1(P''(p(encArg(z0))), ENCARG(cons_p(z0)))
   ENCARG(cons_Q(cons_q(z0))) -> c2(Q'(q(encArg(z0))), ENCARG(cons_q(z0)))
   ENCARG(cons_q(cons_Q(z0))) -> c3(Q''(Q(encArg(z0))), ENCARG(cons_Q(z0)))
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   ENCODE_Q(cons_q(z0)) -> c12(Q'(q(encArg(z0))))
   ENCODE_P'(cons_p(z0)) -> c12(P''(p(encArg(z0))))
   ENCODE_Q'(cons_Q(z0)) -> c12(Q''(Q(encArg(z0))))
   ENCODE_Q'(cons_q(z0)) -> c12(Q''(q(encArg(z0))))
   P'(q(z0)) -> c12(Q'(q(p(Q(z0)))))
   P'(p(z0)) -> c12(Q'(Q(q(q(z0)))))
   P'(Q(Q(z0))) -> c12(Q'(Q(Q(Q(p(z0))))))
   ENCARG(cons_p(cons_P(x0))) -> c1(P''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_P(cons_P(z0)))) -> c1(P''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_p(cons_P(cons_p(z0)))) -> c1(P''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_p(cons_P(cons_Q(z0)))) -> c1(P''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_p(cons_P(cons_q(z0)))) -> c1(P''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_p(cons_P(x0))) -> c1(ENCARG(cons_P(x0)))
   ENCARG(cons_p(cons_Q(cons_P(z0)))) -> c1(P''(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_p(cons_Q(cons_p(z0)))) -> c1(P''(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_p(cons_Q(cons_Q(z0)))) -> c1(P''(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_p(cons_Q(cons_q(z0)))) -> c1(P''(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_p(cons_q(cons_P(z0)))) -> c1(P''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_p(cons_q(cons_p(z0)))) -> c1(P''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_p(cons_q(cons_Q(z0)))) -> c1(P''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_p(cons_q(cons_q(z0)))) -> c1(P''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(Q'(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_P(cons_P(z0)))) -> c2(Q'(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_Q(cons_P(cons_p(z0)))) -> c2(Q'(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_Q(cons_P(cons_Q(z0)))) -> c2(Q'(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_Q(cons_P(cons_q(z0)))) -> c2(Q'(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_Q(cons_P(x0))) -> c2(ENCARG(cons_P(x0)))
   ENCARG(cons_Q(cons_p(cons_P(z0)))) -> c2(Q'(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_Q(cons_p(cons_p(z0)))) -> c2(Q'(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_Q(cons_p(cons_Q(z0)))) -> c2(Q'(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_Q(cons_p(cons_q(z0)))) -> c2(Q'(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_Q(cons_Q(cons_P(z0)))) -> c2(Q'(Q(P(encArg(z0)))), ENCARG(cons_Q(cons_P(z0))))
   ENCARG(cons_Q(cons_Q(cons_p(z0)))) -> c2(Q'(Q(p(encArg(z0)))), ENCARG(cons_Q(cons_p(z0))))
   ENCARG(cons_Q(cons_Q(cons_Q(z0)))) -> c2(Q'(Q(Q(encArg(z0)))), ENCARG(cons_Q(cons_Q(z0))))
   ENCARG(cons_Q(cons_Q(cons_q(z0)))) -> c2(Q'(Q(q(encArg(z0)))), ENCARG(cons_Q(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(Q''(Q(Q(p(encArg(x0))))), ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_P(cons_P(z0)))) -> c3(Q''(P(P(encArg(z0)))), ENCARG(cons_P(cons_P(z0))))
   ENCARG(cons_q(cons_P(cons_p(z0)))) -> c3(Q''(P(p(encArg(z0)))), ENCARG(cons_P(cons_p(z0))))
   ENCARG(cons_q(cons_P(cons_Q(z0)))) -> c3(Q''(P(Q(encArg(z0)))), ENCARG(cons_P(cons_Q(z0))))
   ENCARG(cons_q(cons_P(cons_q(z0)))) -> c3(Q''(P(q(encArg(z0)))), ENCARG(cons_P(cons_q(z0))))
   ENCARG(cons_q(cons_P(x0))) -> c3(ENCARG(cons_P(x0)))
   ENCARG(cons_q(cons_p(cons_P(z0)))) -> c3(Q''(p(P(encArg(z0)))), ENCARG(cons_p(cons_P(z0))))
   ENCARG(cons_q(cons_p(cons_p(z0)))) -> c3(Q''(p(p(encArg(z0)))), ENCARG(cons_p(cons_p(z0))))
   ENCARG(cons_q(cons_p(cons_Q(z0)))) -> c3(Q''(p(Q(encArg(z0)))), ENCARG(cons_p(cons_Q(z0))))
   ENCARG(cons_q(cons_p(cons_q(z0)))) -> c3(Q''(p(q(encArg(z0)))), ENCARG(cons_p(cons_q(z0))))
   ENCARG(cons_q(cons_q(cons_P(z0)))) -> c3(Q''(q(P(encArg(z0)))), ENCARG(cons_q(cons_P(z0))))
   ENCARG(cons_q(cons_q(cons_p(z0)))) -> c3(Q''(q(p(encArg(z0)))), ENCARG(cons_q(cons_p(z0))))
   ENCARG(cons_q(cons_q(cons_Q(z0)))) -> c3(Q''(q(Q(encArg(z0)))), ENCARG(cons_q(cons_Q(z0))))
   ENCARG(cons_q(cons_q(cons_q(z0)))) -> c3(Q''(q(q(encArg(z0)))), ENCARG(cons_q(cons_q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
   Q''(q(p(q(p(q(p(z0))))))) -> c15(P''(q(p(q(p(q(q(z0))))))), Q''(q(q(p(q(p(z0)))))), Q''(q(p(q(p(z0))))))
   Q''(q(p(q(p(x0))))) -> c15(Q''(q(q(p(x0)))), Q''(q(p(x0))))
   Q''(q(p(q(p(Q(z0)))))) -> c15(Q''(q(q(p(Q(z0))))), Q''(q(p(Q(z0)))))
   Q''(q(p(Q(z0)))) -> c15(Q''(z0), Q''(Q(z0)))
   ENCODE_Q(cons_P(x0)) -> c12(Q'(Q(Q(p(encArg(x0))))))
   ENCODE_Q(cons_P(cons_P(z0))) -> c12(Q'(P(P(encArg(z0)))))
   ENCODE_Q(cons_P(cons_p(z0))) -> c12(Q'(P(p(encArg(z0)))))
   ENCODE_Q(cons_P(cons_Q(z0))) -> c12(Q'(P(Q(encArg(z0)))))
   ENCODE_Q(cons_P(cons_q(z0))) -> c12(Q'(P(q(encArg(z0)))))
   ENCODE_Q(cons_p(cons_P(z0))) -> c12(Q'(p(P(encArg(z0)))))
   ENCODE_Q(cons_p(cons_p(z0))) -> c12(Q'(p(p(encArg(z0)))))
   ENCODE_Q(cons_p(cons_Q(z0))) -> c12(Q'(p(Q(encArg(z0)))))
   ENCODE_Q(cons_p(cons_q(z0))) -> c12(Q'(p(q(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_P(z0))) -> c12(Q'(Q(P(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_p(z0))) -> c12(Q'(Q(p(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_Q(z0))) -> c12(Q'(Q(Q(encArg(z0)))))
   ENCODE_Q(cons_Q(cons_q(z0))) -> c12(Q'(Q(q(encArg(z0)))))
   ENCODE_P'(cons_P(x0)) -> c12(P''(Q(Q(p(encArg(x0))))))
   ENCODE_P'(cons_P(cons_P(z0))) -> c12(P''(P(P(encArg(z0)))))
   ENCODE_P'(cons_P(cons_p(z0))) -> c12(P''(P(p(encArg(z0)))))
   ENCODE_P'(cons_P(cons_Q(z0))) -> c12(P''(P(Q(encArg(z0)))))
   ENCODE_P'(cons_P(cons_q(z0))) -> c12(P''(P(q(encArg(z0)))))
   ENCODE_P'(cons_Q(cons_P(z0))) -> c12(P''(Q(P(encArg(z0)))))
   ENCODE_P'(cons_Q(cons_p(z0))) -> c12(P''(Q(p(encArg(z0)))))
   ENCODE_P'(cons_Q(cons_Q(z0))) -> c12(P''(Q(Q(encArg(z0)))))
   ENCODE_P'(cons_Q(cons_q(z0))) -> c12(P''(Q(q(encArg(z0)))))
   ENCODE_P'(cons_q(cons_P(z0))) -> c12(P''(q(P(encArg(z0)))))
   ENCODE_P'(cons_q(cons_p(z0))) -> c12(P''(q(p(encArg(z0)))))
   ENCODE_P'(cons_q(cons_Q(z0))) -> c12(P''(q(Q(encArg(z0)))))
   ENCODE_P'(cons_q(cons_q(z0))) -> c12(P''(q(q(encArg(z0)))))
   ENCODE_Q'(cons_P(x0)) -> c12(Q''(Q(Q(p(encArg(x0))))))
   ENCODE_Q'(cons_P(cons_P(z0))) -> c12(Q''(P(P(encArg(z0)))))
   ENCODE_Q'(cons_P(cons_p(z0))) -> c12(Q''(P(p(encArg(z0)))))
   ENCODE_Q'(cons_P(cons_Q(z0))) -> c12(Q''(P(Q(encArg(z0)))))
   ENCODE_Q'(cons_P(cons_q(z0))) -> c12(Q''(P(q(encArg(z0)))))
   ENCODE_Q'(cons_p(cons_P(z0))) -> c12(Q''(p(P(encArg(z0)))))
   ENCODE_Q'(cons_p(cons_p(z0))) -> c12(Q''(p(p(encArg(z0)))))
   ENCODE_Q'(cons_p(cons_Q(z0))) -> c12(Q''(p(Q(encArg(z0)))))
   ENCODE_Q'(cons_p(cons_q(z0))) -> c12(Q''(p(q(encArg(z0)))))
S tuples:
   P''(p(z0)) -> c10(Q''(q(z0)), Q''(z0))
   Q'(q(z0)) -> c14
   Q''(Q(z0)) -> c16
   P''(Q(Q(x0))) -> c11(Q'(p(x0)))
   Q'(p(q(x0))) -> c13(P''(Q(x0)))
   Q''(q(p(x0))) -> c15(Q''(q(x0)))
   P''(Q(Q(p(x0)))) -> c11(Q'(q(x0)), Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(p(Q(z0))))) -> c11(Q'(Q(q(z0))), Q'(p(p(Q(z0)))), P''(p(Q(z0))))
   P''(Q(Q(p(x0)))) -> c11(Q'(p(p(x0))), P''(p(x0)))
   P''(Q(Q(Q(Q(p(z0)))))) -> c11(Q'(Q(Q(Q(q(q(z0)))))), Q'(p(Q(Q(p(z0))))), P''(Q(Q(p(z0)))))
   P''(Q(Q(Q(Q(Q(Q(z0))))))) -> c11(Q'(Q(Q(Q(Q(Q(p(z0))))))), Q'(p(Q(Q(Q(Q(z0)))))), P''(Q(Q(Q(Q(z0))))))
   P''(Q(Q(Q(Q(x0))))) -> c11(Q'(p(Q(Q(x0)))), P''(Q(Q(x0))))
   Q'(p(q(p(q(p(q(z0))))))) -> c13(Q''(p(q(p(q(p(Q(z0))))))), P''(Q(p(q(p(q(z0)))))), Q'(p(q(p(q(z0))))))
   Q'(p(q(p(q(x0))))) -> c13(P''(Q(p(q(x0)))), Q'(p(q(x0))))
   Q'(p(q(p(q(q(z0)))))) -> c13(P''(Q(p(q(q(z0))))), Q'(p(q(q(z0)))))
   Q'(p(q(q(z0)))) -> c13(P''(z0), Q'(q(z0)))
   Q''(q(p(q(p(q(p(z0))))))) -> c15(P''(q(p(q(p(q(q(z0))))))), Q''(q(q(p(q(p(z0)))))), Q''(q(p(q(p(z0))))))
   Q''(q(p(q(p(x0))))) -> c15(Q''(q(q(p(x0)))), Q''(q(p(x0))))
   Q''(q(p(q(p(Q(z0)))))) -> c15(Q''(q(q(p(Q(z0))))), Q''(q(p(Q(z0)))))
   Q''(q(p(Q(z0)))) -> c15(Q''(z0), Q''(Q(z0)))
K tuples:
   P'(z0) -> c12(Q'(Q(p(z0))))
   P'(z0) -> c12(Q'(p(z0)))
   P'(z0) -> c12(P''(z0))
Defined Rule Symbols:   encArg_1, P_1, p_1, Q_1, q_1

Defined Pair Symbols:   ENCARG_1, P''_1, Q'_1, Q''_1, ENCODE_P_1, P'_1, ENCODE_Q_1, ENCODE_P'_1, ENCODE_Q'_1

Compound Symbols:   c_2, c10_2, c14, c16, c12_1, c1_2, c2_2, c3_2, c11_1, c13_1, c15_1, c1_1, c2_1, c3_1, c11_3, c11_2, c13_3, c13_2, c15_3, c15_2