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


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 168 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), 265 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:

   +(x, 0) -> x
   +(minus(x), x) -> 0
   minus(0) -> 0
   minus(minus(x)) -> x
   minus(+(x, y)) -> +(minus(y), minus(x))
   *(x, 1) -> x
   *(x, 0) -> 0
   *(x, +(y, z)) -> +(*(x, y), *(x, z))
   *(x, minus(y)) -> minus(*(x, y))

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(0) -> 0
   encArg(1) -> 1
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1)) -> minus(encArg(x_1))
   encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_minus(x_1) -> minus(encArg(x_1))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_1 -> 1


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

(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:

   +(x, 0) -> x
   +(minus(x), x) -> 0
   minus(0) -> 0
   minus(minus(x)) -> x
   minus(+(x, y)) -> +(minus(y), minus(x))
   *(x, 1) -> x
   *(x, 0) -> 0
   *(x, +(y, z)) -> +(*(x, y), *(x, z))
   *(x, minus(y)) -> minus(*(x, y))

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

   encArg(0) -> 0
   encArg(1) -> 1
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1)) -> minus(encArg(x_1))
   encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_minus(x_1) -> minus(encArg(x_1))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_1 -> 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, INF).


The TRS R consists of the following rules:

   +(x, 0) -> x
   +(minus(x), x) -> 0
   minus(0) -> 0
   minus(minus(x)) -> x
   minus(+(x, y)) -> +(minus(y), minus(x))
   *(x, 1) -> x
   *(x, 0) -> 0
   *(x, +(y, z)) -> +(*(x, y), *(x, z))
   *(x, minus(y)) -> minus(*(x, y))

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

   encArg(0) -> 0
   encArg(1) -> 1
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1)) -> minus(encArg(x_1))
   encArg(cons_*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_minus(x_1) -> minus(encArg(x_1))
   encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2))
   encode_1 -> 1

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

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

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


The TRS R consists of the following rules:

   +'(x, 0') -> x
   +'(minus(x), x) -> 0'
   minus(0') -> 0'
   minus(minus(x)) -> x
   minus(+'(x, y)) -> +'(minus(y), minus(x))
   *'(x, 1') -> x
   *'(x, 0') -> 0'
   *'(x, +'(y, z)) -> +'(*'(x, y), *'(x, z))
   *'(x, minus(y)) -> minus(*'(x, y))

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

   encArg(0') -> 0'
   encArg(1') -> 1'
   encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
   encArg(cons_minus(x_1)) -> minus(encArg(x_1))
   encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))
   encode_0 -> 0'
   encode_minus(x_1) -> minus(encArg(x_1))
   encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
   encode_1 -> 1'

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

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

(8)
Obligation:
Innermost TRS:
Rules:
+'(x, 0') -> x
+'(minus(x), x) -> 0'
minus(0') -> 0'
minus(minus(x)) -> x
minus(+'(x, y)) -> +'(minus(y), minus(x))
*'(x, 1') -> x
*'(x, 0') -> 0'
*'(x, +'(y, z)) -> +'(*'(x, y), *'(x, z))
*'(x, minus(y)) -> minus(*'(x, y))
encArg(0') -> 0'
encArg(1') -> 1'
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encArg(cons_minus(x_1)) -> minus(encArg(x_1))
encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_minus(x_1) -> minus(encArg(x_1))
encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
encode_1 -> 1'

Types:
+' :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
0' :: 0':1':cons_+:cons_minus:cons_*
minus :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
*' :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
1' :: 0':1':cons_+:cons_minus:cons_*
encArg :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
cons_+ :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
cons_minus :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
cons_* :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
encode_+ :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
encode_0 :: 0':1':cons_+:cons_minus:cons_*
encode_minus :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
encode_* :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
encode_1 :: 0':1':cons_+:cons_minus:cons_*
hole_0':1':cons_+:cons_minus:cons_*1_3 :: 0':1':cons_+:cons_minus:cons_*
gen_0':1':cons_+:cons_minus:cons_*2_3 :: Nat -> 0':1':cons_+:cons_minus:cons_*

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

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

They will be analysed ascendingly in the following order:
minus < *'
minus < encArg
*' < encArg

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

(10)
Obligation:
Innermost TRS:
Rules:
+'(x, 0') -> x
+'(minus(x), x) -> 0'
minus(0') -> 0'
minus(minus(x)) -> x
minus(+'(x, y)) -> +'(minus(y), minus(x))
*'(x, 1') -> x
*'(x, 0') -> 0'
*'(x, +'(y, z)) -> +'(*'(x, y), *'(x, z))
*'(x, minus(y)) -> minus(*'(x, y))
encArg(0') -> 0'
encArg(1') -> 1'
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encArg(cons_minus(x_1)) -> minus(encArg(x_1))
encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_minus(x_1) -> minus(encArg(x_1))
encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
encode_1 -> 1'

Types:
+' :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
0' :: 0':1':cons_+:cons_minus:cons_*
minus :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
*' :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
1' :: 0':1':cons_+:cons_minus:cons_*
encArg :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
cons_+ :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
cons_minus :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
cons_* :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
encode_+ :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
encode_0 :: 0':1':cons_+:cons_minus:cons_*
encode_minus :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
encode_* :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
encode_1 :: 0':1':cons_+:cons_minus:cons_*
hole_0':1':cons_+:cons_minus:cons_*1_3 :: 0':1':cons_+:cons_minus:cons_*
gen_0':1':cons_+:cons_minus:cons_*2_3 :: Nat -> 0':1':cons_+:cons_minus:cons_*


Generator Equations:
gen_0':1':cons_+:cons_minus:cons_*2_3(0) <=> 0'
gen_0':1':cons_+:cons_minus:cons_*2_3(+(x, 1)) <=> cons_+(0', gen_0':1':cons_+:cons_minus:cons_*2_3(x))


The following defined symbols remain to be analysed:
minus, *', encArg

They will be analysed ascendingly in the following order:
minus < *'
minus < encArg
*' < encArg

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':1':cons_+:cons_minus:cons_*2_3(n71_3)) -> gen_0':1':cons_+:cons_minus:cons_*2_3(0), rt in Omega(n71_3)

Induction Base:
encArg(gen_0':1':cons_+:cons_minus:cons_*2_3(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':1':cons_+:cons_minus:cons_*2_3(+(n71_3, 1))) ->_R^Omega(0)
+'(encArg(0'), encArg(gen_0':1':cons_+:cons_minus:cons_*2_3(n71_3))) ->_R^Omega(0)
+'(0', encArg(gen_0':1':cons_+:cons_minus:cons_*2_3(n71_3))) ->_IH
+'(0', gen_0':1':cons_+:cons_minus:cons_*2_3(0)) ->_R^Omega(1)
0'

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:
+'(x, 0') -> x
+'(minus(x), x) -> 0'
minus(0') -> 0'
minus(minus(x)) -> x
minus(+'(x, y)) -> +'(minus(y), minus(x))
*'(x, 1') -> x
*'(x, 0') -> 0'
*'(x, +'(y, z)) -> +'(*'(x, y), *'(x, z))
*'(x, minus(y)) -> minus(*'(x, y))
encArg(0') -> 0'
encArg(1') -> 1'
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encArg(cons_minus(x_1)) -> minus(encArg(x_1))
encArg(cons_*(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_minus(x_1) -> minus(encArg(x_1))
encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2))
encode_1 -> 1'

Types:
+' :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
0' :: 0':1':cons_+:cons_minus:cons_*
minus :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
*' :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
1' :: 0':1':cons_+:cons_minus:cons_*
encArg :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
cons_+ :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
cons_minus :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
cons_* :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
encode_+ :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
encode_0 :: 0':1':cons_+:cons_minus:cons_*
encode_minus :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
encode_* :: 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_* -> 0':1':cons_+:cons_minus:cons_*
encode_1 :: 0':1':cons_+:cons_minus:cons_*
hole_0':1':cons_+:cons_minus:cons_*1_3 :: 0':1':cons_+:cons_minus:cons_*
gen_0':1':cons_+:cons_minus:cons_*2_3 :: Nat -> 0':1':cons_+:cons_minus:cons_*


Generator Equations:
gen_0':1':cons_+:cons_minus:cons_*2_3(0) <=> 0'
gen_0':1':cons_+:cons_minus:cons_*2_3(+(x, 1)) <=> cons_+(0', gen_0':1':cons_+:cons_minus:cons_*2_3(x))


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

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

(14)
BOUNDS(n^1, INF)