/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 44 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
    (7) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
    (8) BOUNDS(1, n^1)
(9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxRelTRS
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) typed CpxTrs
    (13) OrderProof [LOWER BOUND(ID), 4 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 183 ms]
    (16) proven lower bound
    (17) LowerBoundPropagationProof [FINISHED, 0 ms]
    (18) BOUNDS(n^1, INF)


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(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:

   c(b(a(X))) -> a(a(b(b(c(c(X))))))
   a(X) -> e
   b(X) -> e
   c(X) -> e

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(e) -> e
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_e -> e


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

(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:

   c(b(a(X))) -> a(a(b(b(c(c(X))))))
   a(X) -> e
   b(X) -> e
   c(X) -> e

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

   encArg(e) -> e
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_e -> e

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:

   c(b(a(X))) -> a(a(b(b(c(c(X))))))
   a(X) -> e
   b(X) -> e
   c(X) -> e

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

   encArg(e) -> e
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_e -> e

Rewrite Strategy: INNERMOST
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(5) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

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


The TRS R consists of the following rules:

   c(b(a(X))) -> a(a(b(b(c(c(X))))))
   a(X) -> e
   b(X) -> e
   c(X) -> e
   encArg(e) -> e
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_e -> e

S is empty.
Rewrite Strategy: INNERMOST
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(7) CpxTrsMatchBoundsProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 3. 
The certificate found is represented by the following graph.

"[45, 46, 47, 48, 49, 50, 51, 52]
{(45,46,[c_1|0, a_1|0, b_1|0, encArg_1|0, encode_c_1|0, encode_b_1|0, encode_a_1|0, encode_e|0, e|1]), (45,47,[c_1|1, a_1|1, b_1|1, e|2]), (45,48,[a_1|2, e|3]), (46,46,[e|0, cons_c_1|0, cons_a_1|0, cons_b_1|0]), (47,46,[encArg_1|1, e|1]), (47,47,[c_1|1, a_1|1, b_1|1, e|2]), (47,48,[a_1|2, e|3]), (48,49,[a_1|2, e|3]), (49,50,[b_1|2, e|3]), (50,51,[b_1|2, e|3]), (51,52,[c_1|2, e|3]), (52,47,[c_1|2, e|3]), (52,48,[c_1|2, a_1|2, e|3])}"
----------------------------------------

(8)
BOUNDS(1, n^1)

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

(10)
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:

   c(b(a(X))) -> a(a(b(b(c(c(X))))))
   a(X) -> e
   b(X) -> e
   c(X) -> e

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

   encArg(e) -> e
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_e -> e

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

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

(12)
Obligation:
Innermost TRS:
Rules:
c(b(a(X))) -> a(a(b(b(c(c(X))))))
a(X) -> e
b(X) -> e
c(X) -> e
encArg(e) -> e
encArg(cons_c(x_1)) -> c(encArg(x_1))
encArg(cons_a(x_1)) -> a(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_e -> e

Types:
c :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
b :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
a :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
e :: e:cons_c:cons_a:cons_b
encArg :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
cons_c :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
cons_a :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
cons_b :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
encode_c :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
encode_b :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
encode_a :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
encode_e :: e:cons_c:cons_a:cons_b
hole_e:cons_c:cons_a:cons_b1_0 :: e:cons_c:cons_a:cons_b
gen_e:cons_c:cons_a:cons_b2_0 :: Nat -> e:cons_c:cons_a:cons_b

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

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

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

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

(14)
Obligation:
Innermost TRS:
Rules:
c(b(a(X))) -> a(a(b(b(c(c(X))))))
a(X) -> e
b(X) -> e
c(X) -> e
encArg(e) -> e
encArg(cons_c(x_1)) -> c(encArg(x_1))
encArg(cons_a(x_1)) -> a(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_e -> e

Types:
c :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
b :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
a :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
e :: e:cons_c:cons_a:cons_b
encArg :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
cons_c :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
cons_a :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
cons_b :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
encode_c :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
encode_b :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
encode_a :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
encode_e :: e:cons_c:cons_a:cons_b
hole_e:cons_c:cons_a:cons_b1_0 :: e:cons_c:cons_a:cons_b
gen_e:cons_c:cons_a:cons_b2_0 :: Nat -> e:cons_c:cons_a:cons_b


Generator Equations:
gen_e:cons_c:cons_a:cons_b2_0(0) <=> e
gen_e:cons_c:cons_a:cons_b2_0(+(x, 1)) <=> cons_c(gen_e:cons_c:cons_a:cons_b2_0(x))


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

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

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(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_e:cons_c:cons_a:cons_b2_0(n10_0)) -> gen_e:cons_c:cons_a:cons_b2_0(0), rt in Omega(n10_0)

Induction Base:
encArg(gen_e:cons_c:cons_a:cons_b2_0(0)) ->_R^Omega(0)
e

Induction Step:
encArg(gen_e:cons_c:cons_a:cons_b2_0(+(n10_0, 1))) ->_R^Omega(0)
c(encArg(gen_e:cons_c:cons_a:cons_b2_0(n10_0))) ->_IH
c(gen_e:cons_c:cons_a:cons_b2_0(0)) ->_R^Omega(1)
e

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

(16)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
c(b(a(X))) -> a(a(b(b(c(c(X))))))
a(X) -> e
b(X) -> e
c(X) -> e
encArg(e) -> e
encArg(cons_c(x_1)) -> c(encArg(x_1))
encArg(cons_a(x_1)) -> a(encArg(x_1))
encArg(cons_b(x_1)) -> b(encArg(x_1))
encode_c(x_1) -> c(encArg(x_1))
encode_b(x_1) -> b(encArg(x_1))
encode_a(x_1) -> a(encArg(x_1))
encode_e -> e

Types:
c :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
b :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
a :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
e :: e:cons_c:cons_a:cons_b
encArg :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
cons_c :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
cons_a :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
cons_b :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
encode_c :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
encode_b :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
encode_a :: e:cons_c:cons_a:cons_b -> e:cons_c:cons_a:cons_b
encode_e :: e:cons_c:cons_a:cons_b
hole_e:cons_c:cons_a:cons_b1_0 :: e:cons_c:cons_a:cons_b
gen_e:cons_c:cons_a:cons_b2_0 :: Nat -> e:cons_c:cons_a:cons_b


Generator Equations:
gen_e:cons_c:cons_a:cons_b2_0(0) <=> e
gen_e:cons_c:cons_a:cons_b2_0(+(x, 1)) <=> cons_c(gen_e:cons_c:cons_a:cons_b2_0(x))


The following defined symbols remain to be analysed:
encArg
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(17) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(18)
BOUNDS(n^1, INF)