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


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WORST_CASE(Omega(n^1), O(n^2))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(n^1, n^2).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 189 ms]
(4) CpxRelTRS
(5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) CompleteCoflocoProof [FINISHED, 446 ms]
    (14) BOUNDS(1, n^2)
(15) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxRelTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 0 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 501 ms]
    (22) BEST
        (23) proven lower bound
            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
            (25) BOUNDS(n^1, INF)
        (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 142 ms]
            (28) BOUNDS(1, INF)


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
   u21(ackout(X), Y) -> u22(ackin(Y, X))

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(s(x_1)) -> s(encArg(x_1))
   encArg(ackout(x_1)) -> ackout(encArg(x_1))
   encArg(u22(x_1)) -> u22(encArg(x_1))
   encArg(cons_ackin(x_1, x_2)) -> ackin(encArg(x_1), encArg(x_2))
   encArg(cons_u21(x_1, x_2)) -> u21(encArg(x_1), encArg(x_2))
   encode_ackin(x_1, x_2) -> ackin(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_u21(x_1, x_2) -> u21(encArg(x_1), encArg(x_2))
   encode_ackout(x_1) -> ackout(encArg(x_1))
   encode_u22(x_1) -> u22(encArg(x_1))


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

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


The TRS R consists of the following rules:

   ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
   u21(ackout(X), Y) -> u22(ackin(Y, X))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(ackout(x_1)) -> ackout(encArg(x_1))
   encArg(u22(x_1)) -> u22(encArg(x_1))
   encArg(cons_ackin(x_1, x_2)) -> ackin(encArg(x_1), encArg(x_2))
   encArg(cons_u21(x_1, x_2)) -> u21(encArg(x_1), encArg(x_2))
   encode_ackin(x_1, x_2) -> ackin(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_u21(x_1, x_2) -> u21(encArg(x_1), encArg(x_2))
   encode_ackout(x_1) -> ackout(encArg(x_1))
   encode_u22(x_1) -> u22(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(n^1, n^2).


The TRS R consists of the following rules:

   ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
   u21(ackout(X), Y) -> u22(ackin(Y, X))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(ackout(x_1)) -> ackout(encArg(x_1))
   encArg(u22(x_1)) -> u22(encArg(x_1))
   encArg(cons_ackin(x_1, x_2)) -> ackin(encArg(x_1), encArg(x_2))
   encArg(cons_u21(x_1, x_2)) -> u21(encArg(x_1), encArg(x_2))
   encode_ackin(x_1, x_2) -> ackin(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_u21(x_1, x_2) -> u21(encArg(x_1), encArg(x_2))
   encode_ackout(x_1) -> ackout(encArg(x_1))
   encode_u22(x_1) -> u22(encArg(x_1))

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

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

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X) [1]
   u21(ackout(X), Y) -> u22(ackin(Y, X)) [1]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(ackout(x_1)) -> ackout(encArg(x_1)) [0]
   encArg(u22(x_1)) -> u22(encArg(x_1)) [0]
   encArg(cons_ackin(x_1, x_2)) -> ackin(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_u21(x_1, x_2)) -> u21(encArg(x_1), encArg(x_2)) [0]
   encode_ackin(x_1, x_2) -> ackin(encArg(x_1), encArg(x_2)) [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_u21(x_1, x_2) -> u21(encArg(x_1), encArg(x_2)) [0]
   encode_ackout(x_1) -> ackout(encArg(x_1)) [0]
   encode_u22(x_1) -> u22(encArg(x_1)) [0]

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X) [1]
   u21(ackout(X), Y) -> u22(ackin(Y, X)) [1]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(ackout(x_1)) -> ackout(encArg(x_1)) [0]
   encArg(u22(x_1)) -> u22(encArg(x_1)) [0]
   encArg(cons_ackin(x_1, x_2)) -> ackin(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_u21(x_1, x_2)) -> u21(encArg(x_1), encArg(x_2)) [0]
   encode_ackin(x_1, x_2) -> ackin(encArg(x_1), encArg(x_2)) [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_u21(x_1, x_2) -> u21(encArg(x_1), encArg(x_2)) [0]
   encode_ackout(x_1) -> ackout(encArg(x_1)) [0]
   encode_u22(x_1) -> u22(encArg(x_1)) [0]

The TRS has the following type information:
ackin :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
s :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
u21 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
ackout :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
u22 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encArg :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
cons_ackin :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
cons_u21 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_ackin :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_s :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_u21 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_ackout :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_u22 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21

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

(9) 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_ackin(v0, v1) -> null_encode_ackin [0]
   encode_s(v0) -> null_encode_s [0]
   encode_u21(v0, v1) -> null_encode_u21 [0]
   encode_ackout(v0) -> null_encode_ackout [0]
   encode_u22(v0) -> null_encode_u22 [0]
   ackin(v0, v1) -> null_ackin [0]
   u21(v0, v1) -> null_u21 [0]

And the following fresh constants:    null_encArg, null_encode_ackin, null_encode_s, null_encode_u21, null_encode_ackout, null_encode_u22, null_ackin, null_u21

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

(10)
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:

   ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X) [1]
   u21(ackout(X), Y) -> u22(ackin(Y, X)) [1]
   encArg(s(x_1)) -> s(encArg(x_1)) [0]
   encArg(ackout(x_1)) -> ackout(encArg(x_1)) [0]
   encArg(u22(x_1)) -> u22(encArg(x_1)) [0]
   encArg(cons_ackin(x_1, x_2)) -> ackin(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_u21(x_1, x_2)) -> u21(encArg(x_1), encArg(x_2)) [0]
   encode_ackin(x_1, x_2) -> ackin(encArg(x_1), encArg(x_2)) [0]
   encode_s(x_1) -> s(encArg(x_1)) [0]
   encode_u21(x_1, x_2) -> u21(encArg(x_1), encArg(x_2)) [0]
   encode_ackout(x_1) -> ackout(encArg(x_1)) [0]
   encode_u22(x_1) -> u22(encArg(x_1)) [0]
   encArg(v0) -> null_encArg [0]
   encode_ackin(v0, v1) -> null_encode_ackin [0]
   encode_s(v0) -> null_encode_s [0]
   encode_u21(v0, v1) -> null_encode_u21 [0]
   encode_ackout(v0) -> null_encode_ackout [0]
   encode_u22(v0) -> null_encode_u22 [0]
   ackin(v0, v1) -> null_ackin [0]
   u21(v0, v1) -> null_u21 [0]

The TRS has the following type information:
ackin :: s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21 -> s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21 -> s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21
s :: s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21 -> s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21
u21 :: s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21 -> s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21 -> s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21
ackout :: s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21 -> s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21
u22 :: s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21 -> s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21
encArg :: s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21 -> s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21
cons_ackin :: s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21 -> s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21 -> s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21
cons_u21 :: s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21 -> s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21 -> s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21
encode_ackin :: s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21 -> s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21 -> s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21
encode_s :: s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21 -> s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21
encode_u21 :: s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21 -> s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21 -> s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21
encode_ackout :: s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21 -> s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21
encode_u22 :: s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21 -> s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21
null_encArg :: s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21
null_encode_ackin :: s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21
null_encode_s :: s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21
null_encode_u21 :: s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21
null_encode_ackout :: s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21
null_encode_u22 :: s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21
null_ackin :: s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21
null_u21 :: s:ackout:u22:cons_ackin:cons_u21:null_encArg:null_encode_ackin:null_encode_s:null_encode_u21:null_encode_ackout:null_encode_u22:null_ackin:null_u21

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

(11) 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_ackin => 0
   null_encode_s => 0
   null_encode_u21 => 0
   null_encode_ackout => 0
   null_encode_u22 => 0
   null_ackin => 0
   null_u21 => 0

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

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

   ackin(z, z') -{ 1 }-> u21(ackin(1 + X, Y), X) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   ackin(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encArg(z) -{ 0 }-> u21(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> ackin(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encArg(z) -{ 0 }-> 1 + encArg(x_1) :|: z = 1 + x_1, x_1 >= 0
   encode_ackin(z, z') -{ 0 }-> ackin(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_ackin(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_ackout(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_ackout(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   encode_s(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_s(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   encode_u21(z, z') -{ 0 }-> u21(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_u21(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_u22(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_u22(z) -{ 0 }-> 1 + encArg(x_1) :|: x_1 >= 0, z = x_1
   u21(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   u21(z, z') -{ 1 }-> 1 + ackin(Y, X) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0

Only complete derivations are relevant for the runtime complexity.

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

(13) CompleteCoflocoProof (FINISHED)
Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:

eq(start(V1, V),0,[ackin(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[u21(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[encArg(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[fun1(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[fun2(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V),0,[fun3(V1, Out)],[V1 >= 0]).
eq(start(V1, V),0,[fun4(V1, Out)],[V1 >= 0]).
eq(ackin(V1, V, Out),1,[ackin(1 + X1, Y1, Ret0),u21(Ret0, X1, Ret)],[Out = Ret,V1 = 1 + X1,Y1 >= 0,V = 1 + Y1,X1 >= 0]).
eq(u21(V1, V, Out),1,[ackin(Y2, X2, Ret1)],[Out = 1 + Ret1,V1 = 1 + X2,V = Y2,Y2 >= 0,X2 >= 0]).
eq(encArg(V1, Out),0,[encArg(V2, Ret11)],[Out = 1 + Ret11,V1 = 1 + V2,V2 >= 0]).
eq(encArg(V1, Out),0,[encArg(V3, Ret01),encArg(V4, Ret12),ackin(Ret01, Ret12, Ret2)],[Out = Ret2,V3 >= 0,V1 = 1 + V3 + V4,V4 >= 0]).
eq(encArg(V1, Out),0,[encArg(V5, Ret02),encArg(V6, Ret13),u21(Ret02, Ret13, Ret3)],[Out = Ret3,V5 >= 0,V1 = 1 + V5 + V6,V6 >= 0]).
eq(fun(V1, V, Out),0,[encArg(V8, Ret03),encArg(V7, Ret14),ackin(Ret03, Ret14, Ret4)],[Out = Ret4,V8 >= 0,V7 >= 0,V1 = V8,V = V7]).
eq(fun1(V1, Out),0,[encArg(V9, Ret15)],[Out = 1 + Ret15,V9 >= 0,V1 = V9]).
eq(fun2(V1, V, Out),0,[encArg(V11, Ret04),encArg(V10, Ret16),u21(Ret04, Ret16, Ret5)],[Out = Ret5,V11 >= 0,V10 >= 0,V1 = V11,V = V10]).
eq(fun3(V1, Out),0,[encArg(V12, Ret17)],[Out = 1 + Ret17,V12 >= 0,V1 = V12]).
eq(fun4(V1, Out),0,[encArg(V13, Ret18)],[Out = 1 + Ret18,V13 >= 0,V1 = V13]).
eq(encArg(V1, Out),0,[],[Out = 0,V14 >= 0,V1 = V14]).
eq(fun(V1, V, Out),0,[],[Out = 0,V16 >= 0,V15 >= 0,V1 = V16,V = V15]).
eq(fun1(V1, Out),0,[],[Out = 0,V17 >= 0,V1 = V17]).
eq(fun2(V1, V, Out),0,[],[Out = 0,V18 >= 0,V19 >= 0,V1 = V18,V = V19]).
eq(fun3(V1, Out),0,[],[Out = 0,V20 >= 0,V1 = V20]).
eq(fun4(V1, Out),0,[],[Out = 0,V21 >= 0,V1 = V21]).
eq(ackin(V1, V, Out),0,[],[Out = 0,V22 >= 0,V23 >= 0,V1 = V22,V = V23]).
eq(u21(V1, V, Out),0,[],[Out = 0,V24 >= 0,V25 >= 0,V1 = V24,V = V25]).
input_output_vars(ackin(V1,V,Out),[V1,V],[Out]).
input_output_vars(u21(V1,V,Out),[V1,V],[Out]).
input_output_vars(encArg(V1,Out),[V1],[Out]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,Out),[V1],[Out]).
input_output_vars(fun2(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun3(V1,Out),[V1],[Out]).
input_output_vars(fun4(V1,Out),[V1],[Out]).


CoFloCo proof output:
Preprocessing Cost Relations
=====================================

#### Computed strongly connected components 
0. recursive [multiple] : [ackin/3,u21/3]
1. recursive [non_tail,multiple] : [encArg/2]
2. non_recursive  : [fun/3]
3. non_recursive  : [fun1/2]
4. non_recursive  : [fun2/3]
5. non_recursive  : [fun3/2]
6. non_recursive  : [fun4/2]
7. non_recursive  : [start/2]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into ackin/3
1. SCC is partially evaluated into encArg/2
2. SCC is partially evaluated into fun/3
3. SCC is partially evaluated into fun1/2
4. SCC is partially evaluated into fun2/3
5. SCC is partially evaluated into fun3/2
6. SCC is partially evaluated into fun4/2
7. SCC is partially evaluated into start/2

Control-Flow Refinement of Cost Relations
=====================================

### Specialization of cost equations ackin/3 
* CE 12 is refined into CE [29] 
* CE 11 is refined into CE [30] 
* CE 10 is refined into CE [31] 


### Cost equations --> "Loop" of ackin/3 
* CEs [31] --> Loop 14 
* CEs [30] --> Loop 15 
* CEs [29] --> Loop 16 

### Ranking functions of CR ackin(V1,V,Out) 

#### Partial ranking functions of CR ackin(V1,V,Out) 
* Partial RF of phase [14,15]:
  - RF of loop [14:1,15:1]:
    V depends on loops [15:2] 
  - RF of loop [15:2]:
    V1


### Specialization of cost equations encArg/2 
* CE 17 is refined into CE [32] 
* CE 14 is refined into CE [33] 
* CE 13 is refined into CE [34] 
* CE 16 is refined into CE [35] 
* CE 15 is refined into CE [36] 


### Cost equations --> "Loop" of encArg/2 
* CEs [36] --> Loop 17 
* CEs [33] --> Loop 18 
* CEs [34,35] --> Loop 19 
* CEs [32] --> Loop 20 

### Ranking functions of CR encArg(V1,Out) 
* RF of phase [17,18,19]: [V1]

#### Partial ranking functions of CR encArg(V1,Out) 
* Partial RF of phase [17,18,19]:
  - RF of loop [17:1,18:1,18:2,19:1,19:2]:
    V1


### Specialization of cost equations fun/3 
* CE 18 is refined into CE [37,38,39,40] 
* CE 19 is refined into CE [41] 


### Cost equations --> "Loop" of fun/3 
* CEs [37,38,39,40,41] --> Loop 21 

### Ranking functions of CR fun(V1,V,Out) 

#### Partial ranking functions of CR fun(V1,V,Out) 


### Specialization of cost equations fun1/2 
* CE 20 is refined into CE [42,43] 
* CE 21 is refined into CE [44] 


### Cost equations --> "Loop" of fun1/2 
* CEs [43] --> Loop 22 
* CEs [42] --> Loop 23 
* CEs [44] --> Loop 24 

### Ranking functions of CR fun1(V1,Out) 

#### Partial ranking functions of CR fun1(V1,Out) 


### Specialization of cost equations fun2/3 
* CE 22 is refined into CE [45,46,47,48] 
* CE 23 is refined into CE [49,50] 
* CE 24 is refined into CE [51] 


### Cost equations --> "Loop" of fun2/3 
* CEs [49,50] --> Loop 25 
* CEs [45,46,47,48,51] --> Loop 26 

### Ranking functions of CR fun2(V1,V,Out) 

#### Partial ranking functions of CR fun2(V1,V,Out) 


### Specialization of cost equations fun3/2 
* CE 25 is refined into CE [52,53] 
* CE 26 is refined into CE [54] 


### Cost equations --> "Loop" of fun3/2 
* CEs [53] --> Loop 27 
* CEs [52] --> Loop 28 
* CEs [54] --> Loop 29 

### Ranking functions of CR fun3(V1,Out) 

#### Partial ranking functions of CR fun3(V1,Out) 


### Specialization of cost equations fun4/2 
* CE 27 is refined into CE [55,56] 
* CE 28 is refined into CE [57] 


### Cost equations --> "Loop" of fun4/2 
* CEs [56] --> Loop 30 
* CEs [55] --> Loop 31 
* CEs [57] --> Loop 32 

### Ranking functions of CR fun4(V1,Out) 

#### Partial ranking functions of CR fun4(V1,Out) 


### Specialization of cost equations start/2 
* CE 1 is refined into CE [58] 
* CE 2 is refined into CE [59] 
* CE 3 is refined into CE [60] 
* CE 4 is refined into CE [61,62] 
* CE 5 is refined into CE [63] 
* CE 6 is refined into CE [64,65,66] 
* CE 7 is refined into CE [67,68] 
* CE 8 is refined into CE [69,70,71] 
* CE 9 is refined into CE [72,73,74] 


### Cost equations --> "Loop" of start/2 
* CEs [58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74] --> Loop 33 

### Ranking functions of CR start(V1,V) 

#### Partial ranking functions of CR start(V1,V) 


Computing Bounds
=====================================

#### Cost of chains of ackin(V1,V,Out):
* Chain [16]: 0
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [multiple([14,15],[[16]])]: 1*it(14)+0
  Such that:it(14) =< V

  with precondition: [Out=0,V1>=1,V>=1] 


#### Cost of chains of encArg(V1,Out):
* Chain [20]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [multiple([17,18,19],[[20]])]: 1*it(18)+1*s(6)+1*s(7)+0
  Such that:aux(7) =< V1
aux(8) =< 2/3*V1
it(17) =< aux(7)
it(18) =< aux(7)
it(18) =< aux(8)
aux(3) =< aux(7)
s(7) =< it(17)*aux(3)
s(6) =< it(18)*aux(7)

  with precondition: [V1>=1,Out>=0,V1>=Out] 


#### Cost of chains of fun(V1,V,Out):
* Chain [21]: 2*s(12)+2*s(14)+2*s(15)+2*s(16)+2*s(20)+2*s(22)+2*s(23)+0
  Such that:aux(11) =< V1
aux(12) =< 2/3*V1
aux(13) =< V
aux(14) =< 2/3*V
s(20) =< aux(11)
s(20) =< aux(12)
s(21) =< aux(11)
s(22) =< aux(11)*s(21)
s(23) =< s(20)*aux(11)
s(16) =< aux(13)
s(12) =< aux(13)
s(12) =< aux(14)
s(13) =< aux(13)
s(14) =< aux(13)*s(13)
s(15) =< s(12)*aux(13)

  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of fun1(V1,Out):
* Chain [24]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [23]: 0
  with precondition: [Out=1,V1>=0] 

* Chain [22]: 1*s(43)+1*s(45)+1*s(46)+0
  Such that:s(40) =< V1
s(41) =< 2/3*V1
s(43) =< s(40)
s(43) =< s(41)
s(44) =< s(40)
s(45) =< s(40)*s(44)
s(46) =< s(43)*s(40)

  with precondition: [V1>=1,Out>=1,V1+1>=Out] 


#### Cost of chains of fun2(V1,V,Out):
* Chain [26]: 2*s(50)+2*s(52)+2*s(53)+2*s(57)+2*s(59)+2*s(60)+0
  Such that:aux(15) =< V1
aux(16) =< 2/3*V1
aux(17) =< V
aux(18) =< 2/3*V
s(57) =< aux(15)
s(57) =< aux(16)
s(58) =< aux(15)
s(59) =< aux(15)*s(58)
s(60) =< s(57)*aux(15)
s(50) =< aux(17)
s(50) =< aux(18)
s(51) =< aux(17)
s(52) =< aux(17)*s(51)
s(53) =< s(50)*aux(17)

  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [25]: 2*s(78)+2*s(80)+2*s(81)+2*s(82)+1*s(93)+1*s(95)+1*s(96)+1
  Such that:s(90) =< V
s(91) =< 2/3*V
aux(21) =< V1
aux(22) =< 2/3*V1
s(82) =< aux(21)
s(78) =< aux(21)
s(78) =< aux(22)
s(79) =< aux(21)
s(80) =< aux(21)*s(79)
s(81) =< s(78)*aux(21)
s(93) =< s(90)
s(93) =< s(91)
s(94) =< s(90)
s(95) =< s(90)*s(94)
s(96) =< s(93)*s(90)

  with precondition: [Out=1,V1>=1,V>=0] 


#### Cost of chains of fun3(V1,Out):
* Chain [29]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [28]: 0
  with precondition: [Out=1,V1>=0] 

* Chain [27]: 1*s(101)+1*s(103)+1*s(104)+0
  Such that:s(98) =< V1
s(99) =< 2/3*V1
s(101) =< s(98)
s(101) =< s(99)
s(102) =< s(98)
s(103) =< s(98)*s(102)
s(104) =< s(101)*s(98)

  with precondition: [V1>=1,Out>=1,V1+1>=Out] 


#### Cost of chains of fun4(V1,Out):
* Chain [32]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [31]: 0
  with precondition: [Out=1,V1>=0] 

* Chain [30]: 1*s(108)+1*s(110)+1*s(111)+0
  Such that:s(105) =< V1
s(106) =< 2/3*V1
s(108) =< s(105)
s(108) =< s(106)
s(109) =< s(105)
s(110) =< s(105)*s(109)
s(111) =< s(108)*s(105)

  with precondition: [V1>=1,Out>=1,V1+1>=Out] 


#### Cost of chains of start(V1,V):
* Chain [33]: 3*s(112)+3*s(113)+10*s(117)+10*s(119)+10*s(120)+5*s(130)+5*s(132)+5*s(133)+1
  Such that:aux(23) =< V1
aux(24) =< 2/3*V1
aux(25) =< V
aux(26) =< 2/3*V
s(112) =< aux(23)
s(113) =< aux(25)
s(117) =< aux(23)
s(117) =< aux(24)
s(118) =< aux(23)
s(119) =< aux(23)*s(118)
s(120) =< s(117)*aux(23)
s(130) =< aux(25)
s(130) =< aux(26)
s(131) =< aux(25)
s(132) =< aux(25)*s(131)
s(133) =< s(130)*aux(25)

  with precondition: [V1>=0] 


Closed-form bounds of start(V1,V): 
-------------------------------------
* Chain [33] with precondition: [V1>=0] 
    - Upper bound: 13*V1+1+20*V1*V1+nat(V)*8+nat(V)*10*nat(V) 
    - Complexity: n^2 

### Maximum cost of start(V1,V): 13*V1+1+20*V1*V1+nat(V)*8+nat(V)*10*nat(V) 
Asymptotic class: n^2 
* Total analysis performed in 364 ms.


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

(14)
BOUNDS(1, n^2)

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

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

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


The TRS R consists of the following rules:

   ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
   u21(ackout(X), Y) -> u22(ackin(Y, X))

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

   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(ackout(x_1)) -> ackout(encArg(x_1))
   encArg(u22(x_1)) -> u22(encArg(x_1))
   encArg(cons_ackin(x_1, x_2)) -> ackin(encArg(x_1), encArg(x_2))
   encArg(cons_u21(x_1, x_2)) -> u21(encArg(x_1), encArg(x_2))
   encode_ackin(x_1, x_2) -> ackin(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_u21(x_1, x_2) -> u21(encArg(x_1), encArg(x_2))
   encode_ackout(x_1) -> ackout(encArg(x_1))
   encode_u22(x_1) -> u22(encArg(x_1))

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

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

(18)
Obligation:
Innermost TRS:
Rules:
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(ackout(x_1)) -> ackout(encArg(x_1))
encArg(u22(x_1)) -> u22(encArg(x_1))
encArg(cons_ackin(x_1, x_2)) -> ackin(encArg(x_1), encArg(x_2))
encArg(cons_u21(x_1, x_2)) -> u21(encArg(x_1), encArg(x_2))
encode_ackin(x_1, x_2) -> ackin(encArg(x_1), encArg(x_2))
encode_s(x_1) -> s(encArg(x_1))
encode_u21(x_1, x_2) -> u21(encArg(x_1), encArg(x_2))
encode_ackout(x_1) -> ackout(encArg(x_1))
encode_u22(x_1) -> u22(encArg(x_1))

Types:
ackin :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
s :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
u21 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
ackout :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
u22 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encArg :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
cons_ackin :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
cons_u21 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_ackin :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_s :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_u21 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_ackout :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_u22 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
hole_s:ackout:u22:cons_ackin:cons_u211_0 :: s:ackout:u22:cons_ackin:cons_u21
gen_s:ackout:u22:cons_ackin:cons_u212_0 :: Nat -> s:ackout:u22:cons_ackin:cons_u21

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

(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
ackin, u21, encArg

They will be analysed ascendingly in the following order:
ackin = u21
ackin < encArg
u21 < encArg

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

(20)
Obligation:
Innermost TRS:
Rules:
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(ackout(x_1)) -> ackout(encArg(x_1))
encArg(u22(x_1)) -> u22(encArg(x_1))
encArg(cons_ackin(x_1, x_2)) -> ackin(encArg(x_1), encArg(x_2))
encArg(cons_u21(x_1, x_2)) -> u21(encArg(x_1), encArg(x_2))
encode_ackin(x_1, x_2) -> ackin(encArg(x_1), encArg(x_2))
encode_s(x_1) -> s(encArg(x_1))
encode_u21(x_1, x_2) -> u21(encArg(x_1), encArg(x_2))
encode_ackout(x_1) -> ackout(encArg(x_1))
encode_u22(x_1) -> u22(encArg(x_1))

Types:
ackin :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
s :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
u21 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
ackout :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
u22 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encArg :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
cons_ackin :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
cons_u21 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_ackin :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_s :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_u21 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_ackout :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_u22 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
hole_s:ackout:u22:cons_ackin:cons_u211_0 :: s:ackout:u22:cons_ackin:cons_u21
gen_s:ackout:u22:cons_ackin:cons_u212_0 :: Nat -> s:ackout:u22:cons_ackin:cons_u21


Generator Equations:
gen_s:ackout:u22:cons_ackin:cons_u212_0(0) <=> hole_s:ackout:u22:cons_ackin:cons_u211_0
gen_s:ackout:u22:cons_ackin:cons_u212_0(+(x, 1)) <=> s(gen_s:ackout:u22:cons_ackin:cons_u212_0(x))


The following defined symbols remain to be analysed:
u21, ackin, encArg

They will be analysed ascendingly in the following order:
ackin = u21
ackin < encArg
u21 < encArg

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ackin(gen_s:ackout:u22:cons_ackin:cons_u212_0(1), gen_s:ackout:u22:cons_ackin:cons_u212_0(+(1, n9_0))) -> *3_0, rt in Omega(n9_0)

Induction Base:
ackin(gen_s:ackout:u22:cons_ackin:cons_u212_0(1), gen_s:ackout:u22:cons_ackin:cons_u212_0(+(1, 0)))

Induction Step:
ackin(gen_s:ackout:u22:cons_ackin:cons_u212_0(1), gen_s:ackout:u22:cons_ackin:cons_u212_0(+(1, +(n9_0, 1)))) ->_R^Omega(1)
u21(ackin(s(gen_s:ackout:u22:cons_ackin:cons_u212_0(0)), gen_s:ackout:u22:cons_ackin:cons_u212_0(+(1, n9_0))), gen_s:ackout:u22:cons_ackin:cons_u212_0(0)) ->_IH
u21(*3_0, gen_s:ackout:u22:cons_ackin:cons_u212_0(0))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(22)
Complex Obligation (BEST)

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(23)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(ackout(x_1)) -> ackout(encArg(x_1))
encArg(u22(x_1)) -> u22(encArg(x_1))
encArg(cons_ackin(x_1, x_2)) -> ackin(encArg(x_1), encArg(x_2))
encArg(cons_u21(x_1, x_2)) -> u21(encArg(x_1), encArg(x_2))
encode_ackin(x_1, x_2) -> ackin(encArg(x_1), encArg(x_2))
encode_s(x_1) -> s(encArg(x_1))
encode_u21(x_1, x_2) -> u21(encArg(x_1), encArg(x_2))
encode_ackout(x_1) -> ackout(encArg(x_1))
encode_u22(x_1) -> u22(encArg(x_1))

Types:
ackin :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
s :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
u21 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
ackout :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
u22 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encArg :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
cons_ackin :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
cons_u21 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_ackin :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_s :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_u21 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_ackout :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_u22 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
hole_s:ackout:u22:cons_ackin:cons_u211_0 :: s:ackout:u22:cons_ackin:cons_u21
gen_s:ackout:u22:cons_ackin:cons_u212_0 :: Nat -> s:ackout:u22:cons_ackin:cons_u21


Generator Equations:
gen_s:ackout:u22:cons_ackin:cons_u212_0(0) <=> hole_s:ackout:u22:cons_ackin:cons_u211_0
gen_s:ackout:u22:cons_ackin:cons_u212_0(+(x, 1)) <=> s(gen_s:ackout:u22:cons_ackin:cons_u212_0(x))


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

They will be analysed ascendingly in the following order:
ackin = u21
ackin < encArg
u21 < encArg

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(24) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(25)
BOUNDS(n^1, INF)

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(26)
Obligation:
Innermost TRS:
Rules:
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
encArg(s(x_1)) -> s(encArg(x_1))
encArg(ackout(x_1)) -> ackout(encArg(x_1))
encArg(u22(x_1)) -> u22(encArg(x_1))
encArg(cons_ackin(x_1, x_2)) -> ackin(encArg(x_1), encArg(x_2))
encArg(cons_u21(x_1, x_2)) -> u21(encArg(x_1), encArg(x_2))
encode_ackin(x_1, x_2) -> ackin(encArg(x_1), encArg(x_2))
encode_s(x_1) -> s(encArg(x_1))
encode_u21(x_1, x_2) -> u21(encArg(x_1), encArg(x_2))
encode_ackout(x_1) -> ackout(encArg(x_1))
encode_u22(x_1) -> u22(encArg(x_1))

Types:
ackin :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
s :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
u21 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
ackout :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
u22 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encArg :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
cons_ackin :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
cons_u21 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_ackin :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_s :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_u21 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_ackout :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
encode_u22 :: s:ackout:u22:cons_ackin:cons_u21 -> s:ackout:u22:cons_ackin:cons_u21
hole_s:ackout:u22:cons_ackin:cons_u211_0 :: s:ackout:u22:cons_ackin:cons_u21
gen_s:ackout:u22:cons_ackin:cons_u212_0 :: Nat -> s:ackout:u22:cons_ackin:cons_u21


Lemmas:
ackin(gen_s:ackout:u22:cons_ackin:cons_u212_0(1), gen_s:ackout:u22:cons_ackin:cons_u212_0(+(1, n9_0))) -> *3_0, rt in Omega(n9_0)


Generator Equations:
gen_s:ackout:u22:cons_ackin:cons_u212_0(0) <=> hole_s:ackout:u22:cons_ackin:cons_u211_0
gen_s:ackout:u22:cons_ackin:cons_u212_0(+(x, 1)) <=> s(gen_s:ackout:u22:cons_ackin:cons_u212_0(x))


The following defined symbols remain to be analysed:
u21, encArg

They will be analysed ascendingly in the following order:
ackin = u21
ackin < encArg
u21 < encArg

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(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_s:ackout:u22:cons_ackin:cons_u212_0(+(1, n451_0))) -> *3_0, rt in Omega(0)

Induction Base:
encArg(gen_s:ackout:u22:cons_ackin:cons_u212_0(+(1, 0)))

Induction Step:
encArg(gen_s:ackout:u22:cons_ackin:cons_u212_0(+(1, +(n451_0, 1)))) ->_R^Omega(0)
s(encArg(gen_s:ackout:u22:cons_ackin:cons_u212_0(+(1, n451_0)))) ->_IH
s(*3_0)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(28)
BOUNDS(1, INF)