/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^1))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 183 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, 591 ms]
    (14) BOUNDS(1, n^1)
(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), 252 ms]
    (22) proven lower bound
    (23) LowerBoundPropagationProof [FINISHED, 0 ms]
    (24) BOUNDS(n^1, INF)


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

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


The TRS R consists of the following rules:

   f(x, y) -> x
   g(a) -> h(a, b, a)
   i(x) -> f(x, x)
   h(x, x, y) -> g(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(a) -> a
   encArg(b) -> b
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encArg(cons_h(x_1, x_2, x_3)) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a -> a
   encode_h(x_1, x_2, x_3) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_b -> b
   encode_i(x_1) -> i(encArg(x_1))


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

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


The TRS R consists of the following rules:

   f(x, y) -> x
   g(a) -> h(a, b, a)
   i(x) -> f(x, x)
   h(x, x, y) -> g(x)

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encArg(cons_h(x_1, x_2, x_3)) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a -> a
   encode_h(x_1, x_2, x_3) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_b -> b
   encode_i(x_1) -> i(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^1).


The TRS R consists of the following rules:

   f(x, y) -> x
   g(a) -> h(a, b, a)
   i(x) -> f(x, x)
   h(x, x, y) -> g(x)

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encArg(cons_h(x_1, x_2, x_3)) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a -> a
   encode_h(x_1, x_2, x_3) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_b -> b
   encode_i(x_1) -> i(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^1).


The TRS R consists of the following rules:

   f(x, y) -> x [1]
   g(a) -> h(a, b, a) [1]
   i(x) -> f(x, x) [1]
   h(x, x, y) -> g(x) [1]
   encArg(a) -> a [0]
   encArg(b) -> b [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_i(x_1)) -> i(encArg(x_1)) [0]
   encArg(cons_h(x_1, x_2, x_3)) -> h(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_a -> a [0]
   encode_h(x_1, x_2, x_3) -> h(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_b -> b [0]
   encode_i(x_1) -> i(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:

   f(x, y) -> x [1]
   g(a) -> h(a, b, a) [1]
   i(x) -> f(x, x) [1]
   h(x, x, y) -> g(x) [1]
   encArg(a) -> a [0]
   encArg(b) -> b [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_i(x_1)) -> i(encArg(x_1)) [0]
   encArg(cons_h(x_1, x_2, x_3)) -> h(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_a -> a [0]
   encode_h(x_1, x_2, x_3) -> h(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_b -> b [0]
   encode_i(x_1) -> i(encArg(x_1)) [0]

The TRS has the following type information:
f :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
g :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
a :: a:b:cons_f:cons_g:cons_i:cons_h
h :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
b :: a:b:cons_f:cons_g:cons_i:cons_h
i :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
encArg :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
cons_f :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
cons_g :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
cons_i :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
cons_h :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
encode_f :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
encode_g :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
encode_a :: a:b:cons_f:cons_g:cons_i:cons_h
encode_h :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
encode_b :: a:b:cons_f:cons_g:cons_i:cons_h
encode_i :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h

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_f(v0, v1) -> null_encode_f [0]
   encode_g(v0) -> null_encode_g [0]
   encode_a -> null_encode_a [0]
   encode_h(v0, v1, v2) -> null_encode_h [0]
   encode_b -> null_encode_b [0]
   encode_i(v0) -> null_encode_i [0]
   g(v0) -> null_g [0]
   h(v0, v1, v2) -> null_h [0]

And the following fresh constants:    null_encArg, null_encode_f, null_encode_g, null_encode_a, null_encode_h, null_encode_b, null_encode_i, null_g, null_h

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

(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:

   f(x, y) -> x [1]
   g(a) -> h(a, b, a) [1]
   i(x) -> f(x, x) [1]
   h(x, x, y) -> g(x) [1]
   encArg(a) -> a [0]
   encArg(b) -> b [0]
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) [0]
   encArg(cons_g(x_1)) -> g(encArg(x_1)) [0]
   encArg(cons_i(x_1)) -> i(encArg(x_1)) [0]
   encArg(cons_h(x_1, x_2, x_3)) -> h(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) [0]
   encode_g(x_1) -> g(encArg(x_1)) [0]
   encode_a -> a [0]
   encode_h(x_1, x_2, x_3) -> h(encArg(x_1), encArg(x_2), encArg(x_3)) [0]
   encode_b -> b [0]
   encode_i(x_1) -> i(encArg(x_1)) [0]
   encArg(v0) -> null_encArg [0]
   encode_f(v0, v1) -> null_encode_f [0]
   encode_g(v0) -> null_encode_g [0]
   encode_a -> null_encode_a [0]
   encode_h(v0, v1, v2) -> null_encode_h [0]
   encode_b -> null_encode_b [0]
   encode_i(v0) -> null_encode_i [0]
   g(v0) -> null_g [0]
   h(v0, v1, v2) -> null_h [0]

The TRS has the following type information:
f :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h -> a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h -> a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
g :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h -> a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
a :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
h :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h -> a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h -> a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h -> a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
b :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
i :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h -> a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
encArg :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h -> a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
cons_f :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h -> a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h -> a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
cons_g :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h -> a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
cons_i :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h -> a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
cons_h :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h -> a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h -> a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h -> a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
encode_f :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h -> a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h -> a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
encode_g :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h -> a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
encode_a :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
encode_h :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h -> a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h -> a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h -> a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
encode_b :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
encode_i :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h -> a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
null_encArg :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
null_encode_f :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
null_encode_g :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
null_encode_a :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
null_encode_h :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
null_encode_b :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
null_encode_i :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
null_g :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h
null_h :: a:b:cons_f:cons_g:cons_i:cons_h:null_encArg:null_encode_f:null_encode_g:null_encode_a:null_encode_h:null_encode_b:null_encode_i:null_g:null_h

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:

   a => 0
   b => 1
   null_encArg => 0
   null_encode_f => 0
   null_encode_g => 0
   null_encode_a => 0
   null_encode_h => 0
   null_encode_b => 0
   null_encode_i => 0
   null_g => 0
   null_h => 0

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

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

   encArg(z) -{ 0 }-> i(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> h(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, z = 1 + x_1 + x_2 + x_3, x_3 >= 0, x_2 >= 0
   encArg(z) -{ 0 }-> g(encArg(x_1)) :|: z = 1 + x_1, x_1 >= 0
   encArg(z) -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, z = 1 + x_1 + x_2, x_2 >= 0
   encArg(z) -{ 0 }-> 1 :|: z = 1
   encArg(z) -{ 0 }-> 0 :|: z = 0
   encArg(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_a -{ 0 }-> 0 :|: 
   encode_b -{ 0 }-> 1 :|: 
   encode_b -{ 0 }-> 0 :|: 
   encode_f(z, z') -{ 0 }-> f(encArg(x_1), encArg(x_2)) :|: x_1 >= 0, x_2 >= 0, z = x_1, z' = x_2
   encode_f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   encode_g(z) -{ 0 }-> g(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   encode_h(z, z', z'') -{ 0 }-> h(encArg(x_1), encArg(x_2), encArg(x_3)) :|: x_1 >= 0, x_3 >= 0, x_2 >= 0, z = x_1, z' = x_2, z'' = x_3
   encode_h(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   encode_i(z) -{ 0 }-> i(encArg(x_1)) :|: x_1 >= 0, z = x_1
   encode_i(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   f(z, z') -{ 1 }-> x :|: x >= 0, y >= 0, z = x, z' = y
   g(z) -{ 1 }-> h(0, 1, 0) :|: z = 0
   g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   h(z, z', z'') -{ 1 }-> g(x) :|: z' = x, z'' = y, x >= 0, y >= 0, z = x
   h(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   i(z) -{ 1 }-> f(x, x) :|: x >= 0, z = x

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, V6),0,[f(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V6),0,[g(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V6),0,[i(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V6),0,[h(V1, V, V6, Out)],[V1 >= 0,V >= 0,V6 >= 0]).
eq(start(V1, V, V6),0,[encArg(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V6),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V6),0,[fun1(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V6),0,[fun2(Out)],[]).
eq(start(V1, V, V6),0,[fun3(V1, V, V6, Out)],[V1 >= 0,V >= 0,V6 >= 0]).
eq(start(V1, V, V6),0,[fun4(Out)],[]).
eq(start(V1, V, V6),0,[fun5(V1, Out)],[V1 >= 0]).
eq(f(V1, V, Out),1,[],[Out = V3,V3 >= 0,V2 >= 0,V1 = V3,V = V2]).
eq(g(V1, Out),1,[h(0, 1, 0, Ret)],[Out = Ret,V1 = 0]).
eq(i(V1, Out),1,[f(V4, V4, Ret1)],[Out = Ret1,V4 >= 0,V1 = V4]).
eq(h(V1, V, V6, Out),1,[g(V5, Ret2)],[Out = Ret2,V = V5,V6 = V7,V5 >= 0,V7 >= 0,V1 = V5]).
eq(encArg(V1, Out),0,[],[Out = 0,V1 = 0]).
eq(encArg(V1, Out),0,[],[Out = 1,V1 = 1]).
eq(encArg(V1, Out),0,[encArg(V9, Ret0),encArg(V8, Ret11),f(Ret0, Ret11, Ret3)],[Out = Ret3,V9 >= 0,V1 = 1 + V8 + V9,V8 >= 0]).
eq(encArg(V1, Out),0,[encArg(V10, Ret01),g(Ret01, Ret4)],[Out = Ret4,V1 = 1 + V10,V10 >= 0]).
eq(encArg(V1, Out),0,[encArg(V11, Ret02),i(Ret02, Ret5)],[Out = Ret5,V1 = 1 + V11,V11 >= 0]).
eq(encArg(V1, Out),0,[encArg(V14, Ret03),encArg(V13, Ret12),encArg(V12, Ret21),h(Ret03, Ret12, Ret21, Ret6)],[Out = Ret6,V14 >= 0,V1 = 1 + V12 + V13 + V14,V12 >= 0,V13 >= 0]).
eq(fun(V1, V, Out),0,[encArg(V16, Ret04),encArg(V15, Ret13),f(Ret04, Ret13, Ret7)],[Out = Ret7,V16 >= 0,V15 >= 0,V1 = V16,V = V15]).
eq(fun1(V1, Out),0,[encArg(V17, Ret05),g(Ret05, Ret8)],[Out = Ret8,V17 >= 0,V1 = V17]).
eq(fun2(Out),0,[],[Out = 0]).
eq(fun3(V1, V, V6, Out),0,[encArg(V20, Ret06),encArg(V18, Ret14),encArg(V19, Ret22),h(Ret06, Ret14, Ret22, Ret9)],[Out = Ret9,V20 >= 0,V19 >= 0,V18 >= 0,V1 = V20,V = V18,V6 = V19]).
eq(fun4(Out),0,[],[Out = 1]).
eq(fun5(V1, Out),0,[encArg(V21, Ret07),i(Ret07, Ret10)],[Out = Ret10,V21 >= 0,V1 = V21]).
eq(encArg(V1, Out),0,[],[Out = 0,V22 >= 0,V1 = V22]).
eq(fun(V1, V, Out),0,[],[Out = 0,V24 >= 0,V23 >= 0,V1 = V24,V = V23]).
eq(fun1(V1, Out),0,[],[Out = 0,V25 >= 0,V1 = V25]).
eq(fun3(V1, V, V6, Out),0,[],[Out = 0,V26 >= 0,V6 = V28,V27 >= 0,V1 = V26,V = V27,V28 >= 0]).
eq(fun4(Out),0,[],[Out = 0]).
eq(fun5(V1, Out),0,[],[Out = 0,V29 >= 0,V1 = V29]).
eq(g(V1, Out),0,[],[Out = 0,V30 >= 0,V1 = V30]).
eq(h(V1, V, V6, Out),0,[],[Out = 0,V31 >= 0,V6 = V32,V33 >= 0,V1 = V31,V = V33,V32 >= 0]).
input_output_vars(f(V1,V,Out),[V1,V],[Out]).
input_output_vars(g(V1,Out),[V1],[Out]).
input_output_vars(i(V1,Out),[V1],[Out]).
input_output_vars(h(V1,V,V6,Out),[V1,V,V6],[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(Out),[],[Out]).
input_output_vars(fun3(V1,V,V6,Out),[V1,V,V6],[Out]).
input_output_vars(fun4(Out),[],[Out]).
input_output_vars(fun5(V1,Out),[V1],[Out]).


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

#### Computed strongly connected components 
0. non_recursive  : [f/3]
1. recursive  : [g/2,h/4]
2. non_recursive  : [i/2]
3. recursive [non_tail,multiple] : [encArg/2]
4. non_recursive  : [fun/3]
5. non_recursive  : [fun1/2]
6. non_recursive  : [fun2/1]
7. non_recursive  : [fun3/4]
8. non_recursive  : [fun4/1]
9. non_recursive  : [fun5/2]
10. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is completely evaluated into other SCCs
1. SCC is partially evaluated into g/2
2. SCC is completely evaluated into other SCCs
3. SCC is partially evaluated into encArg/2
4. SCC is partially evaluated into fun/3
5. SCC is partially evaluated into fun1/2
6. SCC is completely evaluated into other SCCs
7. SCC is partially evaluated into fun3/4
8. SCC is partially evaluated into fun4/1
9. SCC is partially evaluated into fun5/2
10. SCC is partially evaluated into start/3

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

### Specialization of cost equations g/2 
* CE 12 is refined into CE [32] 
* CE 13 is refined into CE [33] 


### Cost equations --> "Loop" of g/2 
* CEs [32,33] --> Loop 15 

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

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


### Specialization of cost equations encArg/2 
* CE 16 is refined into CE [34] 
* CE 17 is refined into CE [35] 
* CE 19 is refined into CE [36] 
* CE 20 is refined into CE [37] 
* CE 18 is refined into CE [38] 
* CE 15 is refined into CE [39] 
* CE 14 is refined into CE [40] 


### Cost equations --> "Loop" of encArg/2 
* CEs [39,40] --> Loop 16 
* CEs [38] --> Loop 17 
* CEs [37] --> Loop 18 
* CEs [36] --> Loop 19 
* CEs [34] --> Loop 20 
* CEs [35] --> Loop 21 

### Ranking functions of CR encArg(V1,Out) 
* RF of phase [16,17,18,19]: [V1]

#### Partial ranking functions of CR encArg(V1,Out) 
* Partial RF of phase [16,17,18,19]:
  - RF of loop [16:1,16:2,16:3,17:1,17:2,18:1,19:1]:
    V1


### Specialization of cost equations fun/3 
* CE 21 is refined into CE [41,42,43,44,45,46,47,48,49] 
* CE 22 is refined into CE [50] 


### Cost equations --> "Loop" of fun/3 
* CEs [47,48,49] --> Loop 22 
* CEs [44,45,46,50] --> Loop 23 
* CEs [41,42,43] --> Loop 24 

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

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


### Specialization of cost equations fun1/2 
* CE 23 is refined into CE [51,52,53] 
* CE 24 is refined into CE [54] 


### Cost equations --> "Loop" of fun1/2 
* CEs [51,52,53,54] --> Loop 25 

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

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


### Specialization of cost equations fun3/4 
* CE 25 is refined into CE [55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81] 
* CE 26 is refined into CE [82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102] 
* CE 27 is refined into CE [103] 


### Cost equations --> "Loop" of fun3/4 
* CEs [64,65,66,73,74,75,94,95,96] --> Loop 26 
* CEs [58,61,67,70,76,79,85,88,91,97,100] --> Loop 27 
* CEs [55,56,57,59,60,62,63,68,69,71,72,77,78,80,81,82,83,84,86,87,89,90,92,93,98,99,101,102,103] --> Loop 28 

### Ranking functions of CR fun3(V1,V,V6,Out) 

#### Partial ranking functions of CR fun3(V1,V,V6,Out) 


### Specialization of cost equations fun4/1 
* CE 28 is refined into CE [104] 
* CE 29 is refined into CE [105] 


### Cost equations --> "Loop" of fun4/1 
* CEs [104] --> Loop 29 
* CEs [105] --> Loop 30 

### Ranking functions of CR fun4(Out) 

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


### Specialization of cost equations fun5/2 
* CE 30 is refined into CE [106,107,108] 
* CE 31 is refined into CE [109] 


### Cost equations --> "Loop" of fun5/2 
* CEs [108] --> Loop 31 
* CEs [107,109] --> Loop 32 
* CEs [106] --> Loop 33 

### Ranking functions of CR fun5(V1,Out) 

#### Partial ranking functions of CR fun5(V1,Out) 


### Specialization of cost equations start/3 
* CE 1 is refined into CE [110] 
* CE 2 is refined into CE [111] 
* CE 3 is refined into CE [112] 
* CE 4 is refined into CE [113] 
* CE 5 is refined into CE [114] 
* CE 6 is refined into CE [115,116,117] 
* CE 7 is refined into CE [118,119,120] 
* CE 8 is refined into CE [121] 
* CE 9 is refined into CE [122,123] 
* CE 10 is refined into CE [124,125] 
* CE 11 is refined into CE [126,127,128] 


### Cost equations --> "Loop" of start/3 
* CEs [110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128] --> Loop 34 

### Ranking functions of CR start(V1,V,V6) 

#### Partial ranking functions of CR start(V1,V,V6) 


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

#### Cost of chains of g(V1,Out):
* Chain [15]: 1
  with precondition: [Out=0,V1>=0] 


#### Cost of chains of encArg(V1,Out):
* Chain [21]: 0
  with precondition: [V1=1,Out=1] 

* Chain [20]: 0
  with precondition: [Out=0,V1>=0] 

* Chain [multiple([16,17,18,19],[[21],[20]])]: 6*it(16)+0
  Such that:aux(1) =< V1
it(16) =< aux(1)

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


#### Cost of chains of fun(V1,V,Out):
* Chain [24]: 6*s(2)+1
  Such that:s(1) =< V
s(2) =< s(1)

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

* Chain [23]: 6*s(4)+1
  Such that:s(3) =< V
s(4) =< s(3)

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

* Chain [22]: 18*s(6)+6*s(12)+1
  Such that:s(11) =< V
aux(3) =< V1
s(6) =< aux(3)
s(12) =< s(11)

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


#### Cost of chains of fun1(V1,Out):
* Chain [25]: 6*s(14)+1
  Such that:s(13) =< V1
s(14) =< s(13)

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


#### Cost of chains of fun3(V1,V,V6,Out):
* Chain [28]: 78*s(16)+72*s(20)+48*s(34)+2
  Such that:aux(4) =< V1
aux(5) =< V
aux(6) =< V6
s(34) =< aux(4)
s(16) =< aux(6)
s(20) =< aux(5)

  with precondition: [Out=0,V1>=0,V>=0,V6>=0] 

* Chain [27]: 36*s(82)+24*s(86)+2
  Such that:aux(7) =< V1
aux(8) =< V
s(86) =< aux(7)
s(82) =< aux(8)

  with precondition: [V6=1,Out=0,V1>=0,V>=0] 

* Chain [26]: 18*s(102)+36*s(104)+2
  Such that:aux(9) =< V1
aux(10) =< V6
s(104) =< aux(9)
s(102) =< aux(10)

  with precondition: [V=1,Out=0,V1>=0,V6>=0] 


#### Cost of chains of fun4(Out):
* Chain [30]: 0
  with precondition: [Out=0] 

* Chain [29]: 0
  with precondition: [Out=1] 


#### Cost of chains of fun5(V1,Out):
* Chain [33]: 2
  with precondition: [V1=1,Out=1] 

* Chain [32]: 2
  with precondition: [Out=0,V1>=0] 

* Chain [31]: 6*s(130)+2
  Such that:s(129) =< V1
s(130) =< s(129)

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


#### Cost of chains of start(V1,V,V6):
* Chain [34]: 144*s(132)+126*s(134)+96*s(147)+2
  Such that:s(145) =< V6
aux(13) =< V1
aux(14) =< V
s(132) =< aux(13)
s(134) =< aux(14)
s(147) =< s(145)

  with precondition: [] 


Closed-form bounds of start(V1,V,V6): 
-------------------------------------
* Chain [34] with precondition: [] 
    - Upper bound: nat(V1)*144+2+nat(V)*126+nat(V6)*96 
    - Complexity: n 

### Maximum cost of start(V1,V,V6): nat(V1)*144+2+nat(V)*126+nat(V6)*96 
Asymptotic class: n 
* Total analysis performed in 494 ms.


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

(14)
BOUNDS(1, n^1)

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

(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:

   f(x, y) -> x
   g(a) -> h(a, b, a)
   i(x) -> f(x, x)
   h(x, x, y) -> g(x)

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(cons_g(x_1)) -> g(encArg(x_1))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encArg(cons_h(x_1, x_2, x_3)) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1) -> g(encArg(x_1))
   encode_a -> a
   encode_h(x_1, x_2, x_3) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_b -> b
   encode_i(x_1) -> i(encArg(x_1))

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

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

(18)
Obligation:
Innermost TRS:
Rules:
f(x, y) -> x
g(a) -> h(a, b, a)
i(x) -> f(x, x)
h(x, x, y) -> g(x)
encArg(a) -> a
encArg(b) -> b
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(cons_g(x_1)) -> g(encArg(x_1))
encArg(cons_i(x_1)) -> i(encArg(x_1))
encArg(cons_h(x_1, x_2, x_3)) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_g(x_1) -> g(encArg(x_1))
encode_a -> a
encode_h(x_1, x_2, x_3) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
encode_b -> b
encode_i(x_1) -> i(encArg(x_1))

Types:
f :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
g :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
a :: a:b:cons_f:cons_g:cons_i:cons_h
h :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
b :: a:b:cons_f:cons_g:cons_i:cons_h
i :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
encArg :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
cons_f :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
cons_g :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
cons_i :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
cons_h :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
encode_f :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
encode_g :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
encode_a :: a:b:cons_f:cons_g:cons_i:cons_h
encode_h :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
encode_b :: a:b:cons_f:cons_g:cons_i:cons_h
encode_i :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
hole_a:b:cons_f:cons_g:cons_i:cons_h1_0 :: a:b:cons_f:cons_g:cons_i:cons_h
gen_a:b:cons_f:cons_g:cons_i:cons_h2_0 :: Nat -> a:b:cons_f:cons_g:cons_i:cons_h

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

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

They will be analysed ascendingly in the following order:
g = h
g < encArg
h < encArg

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

(20)
Obligation:
Innermost TRS:
Rules:
f(x, y) -> x
g(a) -> h(a, b, a)
i(x) -> f(x, x)
h(x, x, y) -> g(x)
encArg(a) -> a
encArg(b) -> b
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(cons_g(x_1)) -> g(encArg(x_1))
encArg(cons_i(x_1)) -> i(encArg(x_1))
encArg(cons_h(x_1, x_2, x_3)) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_g(x_1) -> g(encArg(x_1))
encode_a -> a
encode_h(x_1, x_2, x_3) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
encode_b -> b
encode_i(x_1) -> i(encArg(x_1))

Types:
f :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
g :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
a :: a:b:cons_f:cons_g:cons_i:cons_h
h :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
b :: a:b:cons_f:cons_g:cons_i:cons_h
i :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
encArg :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
cons_f :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
cons_g :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
cons_i :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
cons_h :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
encode_f :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
encode_g :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
encode_a :: a:b:cons_f:cons_g:cons_i:cons_h
encode_h :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
encode_b :: a:b:cons_f:cons_g:cons_i:cons_h
encode_i :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
hole_a:b:cons_f:cons_g:cons_i:cons_h1_0 :: a:b:cons_f:cons_g:cons_i:cons_h
gen_a:b:cons_f:cons_g:cons_i:cons_h2_0 :: Nat -> a:b:cons_f:cons_g:cons_i:cons_h


Generator Equations:
gen_a:b:cons_f:cons_g:cons_i:cons_h2_0(0) <=> a
gen_a:b:cons_f:cons_g:cons_i:cons_h2_0(+(x, 1)) <=> cons_f(a, gen_a:b:cons_f:cons_g:cons_i:cons_h2_0(x))


The following defined symbols remain to be analysed:
h, g, encArg

They will be analysed ascendingly in the following order:
g = h
g < encArg
h < encArg

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(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_a:b:cons_f:cons_g:cons_i:cons_h2_0(n54_0)) -> gen_a:b:cons_f:cons_g:cons_i:cons_h2_0(0), rt in Omega(n54_0)

Induction Base:
encArg(gen_a:b:cons_f:cons_g:cons_i:cons_h2_0(0)) ->_R^Omega(0)
a

Induction Step:
encArg(gen_a:b:cons_f:cons_g:cons_i:cons_h2_0(+(n54_0, 1))) ->_R^Omega(0)
f(encArg(a), encArg(gen_a:b:cons_f:cons_g:cons_i:cons_h2_0(n54_0))) ->_R^Omega(0)
f(a, encArg(gen_a:b:cons_f:cons_g:cons_i:cons_h2_0(n54_0))) ->_IH
f(a, gen_a:b:cons_f:cons_g:cons_i:cons_h2_0(0)) ->_R^Omega(1)
a

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

Innermost TRS:
Rules:
f(x, y) -> x
g(a) -> h(a, b, a)
i(x) -> f(x, x)
h(x, x, y) -> g(x)
encArg(a) -> a
encArg(b) -> b
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(cons_g(x_1)) -> g(encArg(x_1))
encArg(cons_i(x_1)) -> i(encArg(x_1))
encArg(cons_h(x_1, x_2, x_3)) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_g(x_1) -> g(encArg(x_1))
encode_a -> a
encode_h(x_1, x_2, x_3) -> h(encArg(x_1), encArg(x_2), encArg(x_3))
encode_b -> b
encode_i(x_1) -> i(encArg(x_1))

Types:
f :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
g :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
a :: a:b:cons_f:cons_g:cons_i:cons_h
h :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
b :: a:b:cons_f:cons_g:cons_i:cons_h
i :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
encArg :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
cons_f :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
cons_g :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
cons_i :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
cons_h :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
encode_f :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
encode_g :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
encode_a :: a:b:cons_f:cons_g:cons_i:cons_h
encode_h :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
encode_b :: a:b:cons_f:cons_g:cons_i:cons_h
encode_i :: a:b:cons_f:cons_g:cons_i:cons_h -> a:b:cons_f:cons_g:cons_i:cons_h
hole_a:b:cons_f:cons_g:cons_i:cons_h1_0 :: a:b:cons_f:cons_g:cons_i:cons_h
gen_a:b:cons_f:cons_g:cons_i:cons_h2_0 :: Nat -> a:b:cons_f:cons_g:cons_i:cons_h


Generator Equations:
gen_a:b:cons_f:cons_g:cons_i:cons_h2_0(0) <=> a
gen_a:b:cons_f:cons_g:cons_i:cons_h2_0(+(x, 1)) <=> cons_f(a, gen_a:b:cons_f:cons_g:cons_i:cons_h2_0(x))


The following defined symbols remain to be analysed:
encArg
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(23) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(24)
BOUNDS(n^1, INF)