/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), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 174 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), 235 ms]
(12) proven lower bound
(13) LowerBoundPropagationProof [FINISHED, 0 ms]
(14) BOUNDS(n^1, INF)


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

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


The TRS R consists of the following rules:

   times(x, plus(y, 1)) -> plus(times(x, plus(y, times(1, 0))), x)
   times(x, 1) -> x
   plus(x, 0) -> x
   times(x, 0) -> 0

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(1) -> 1
   encArg(0) -> 0
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_0 -> 0


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

(2)
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:

   times(x, plus(y, 1)) -> plus(times(x, plus(y, times(1, 0))), x)
   times(x, 1) -> x
   plus(x, 0) -> x
   times(x, 0) -> 0

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

   encArg(1) -> 1
   encArg(0) -> 0
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_0 -> 0

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, INF).


The TRS R consists of the following rules:

   times(x, plus(y, 1)) -> plus(times(x, plus(y, times(1, 0))), x)
   times(x, 1) -> x
   plus(x, 0) -> x
   times(x, 0) -> 0

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

   encArg(1) -> 1
   encArg(0) -> 0
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_0 -> 0

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(n^1, INF).


The TRS R consists of the following rules:

   times(x, plus(y, 1')) -> plus(times(x, plus(y, times(1', 0'))), x)
   times(x, 1') -> x
   plus(x, 0') -> x
   times(x, 0') -> 0'

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

   encArg(1') -> 1'
   encArg(0') -> 0'
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_1 -> 1'
   encode_0 -> 0'

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

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

(8)
Obligation:
Innermost TRS:
Rules:
times(x, plus(y, 1')) -> plus(times(x, plus(y, times(1', 0'))), x)
times(x, 1') -> x
plus(x, 0') -> x
times(x, 0') -> 0'
encArg(1') -> 1'
encArg(0') -> 0'
encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
encode_1 -> 1'
encode_0 -> 0'

Types:
times :: 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus
plus :: 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus
1' :: 1':0':cons_times:cons_plus
0' :: 1':0':cons_times:cons_plus
encArg :: 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus
cons_times :: 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus
cons_plus :: 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus
encode_times :: 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus
encode_plus :: 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus
encode_1 :: 1':0':cons_times:cons_plus
encode_0 :: 1':0':cons_times:cons_plus
hole_1':0':cons_times:cons_plus1_3 :: 1':0':cons_times:cons_plus
gen_1':0':cons_times:cons_plus2_3 :: Nat -> 1':0':cons_times:cons_plus

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

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

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

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

(10)
Obligation:
Innermost TRS:
Rules:
times(x, plus(y, 1')) -> plus(times(x, plus(y, times(1', 0'))), x)
times(x, 1') -> x
plus(x, 0') -> x
times(x, 0') -> 0'
encArg(1') -> 1'
encArg(0') -> 0'
encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
encode_1 -> 1'
encode_0 -> 0'

Types:
times :: 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus
plus :: 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus
1' :: 1':0':cons_times:cons_plus
0' :: 1':0':cons_times:cons_plus
encArg :: 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus
cons_times :: 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus
cons_plus :: 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus
encode_times :: 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus
encode_plus :: 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus
encode_1 :: 1':0':cons_times:cons_plus
encode_0 :: 1':0':cons_times:cons_plus
hole_1':0':cons_times:cons_plus1_3 :: 1':0':cons_times:cons_plus
gen_1':0':cons_times:cons_plus2_3 :: Nat -> 1':0':cons_times:cons_plus


Generator Equations:
gen_1':0':cons_times:cons_plus2_3(0) <=> 1'
gen_1':0':cons_times:cons_plus2_3(+(x, 1)) <=> cons_times(1', gen_1':0':cons_times:cons_plus2_3(x))


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

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

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_1':0':cons_times:cons_plus2_3(n18_3)) -> gen_1':0':cons_times:cons_plus2_3(0), rt in Omega(n18_3)

Induction Base:
encArg(gen_1':0':cons_times:cons_plus2_3(0)) ->_R^Omega(0)
1'

Induction Step:
encArg(gen_1':0':cons_times:cons_plus2_3(+(n18_3, 1))) ->_R^Omega(0)
times(encArg(1'), encArg(gen_1':0':cons_times:cons_plus2_3(n18_3))) ->_R^Omega(0)
times(1', encArg(gen_1':0':cons_times:cons_plus2_3(n18_3))) ->_IH
times(1', gen_1':0':cons_times:cons_plus2_3(0)) ->_R^Omega(1)
1'

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

(12)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
times(x, plus(y, 1')) -> plus(times(x, plus(y, times(1', 0'))), x)
times(x, 1') -> x
plus(x, 0') -> x
times(x, 0') -> 0'
encArg(1') -> 1'
encArg(0') -> 0'
encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
encode_1 -> 1'
encode_0 -> 0'

Types:
times :: 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus
plus :: 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus
1' :: 1':0':cons_times:cons_plus
0' :: 1':0':cons_times:cons_plus
encArg :: 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus
cons_times :: 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus
cons_plus :: 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus
encode_times :: 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus
encode_plus :: 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus -> 1':0':cons_times:cons_plus
encode_1 :: 1':0':cons_times:cons_plus
encode_0 :: 1':0':cons_times:cons_plus
hole_1':0':cons_times:cons_plus1_3 :: 1':0':cons_times:cons_plus
gen_1':0':cons_times:cons_plus2_3 :: Nat -> 1':0':cons_times:cons_plus


Generator Equations:
gen_1':0':cons_times:cons_plus2_3(0) <=> 1'
gen_1':0':cons_times:cons_plus2_3(+(x, 1)) <=> cons_times(1', gen_1':0':cons_times:cons_plus2_3(x))


The following defined symbols remain to be analysed:
encArg
----------------------------------------

(13) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(14)
BOUNDS(n^1, INF)