/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 155 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), 269 ms]
(12) proven lower bound
(13) LowerBoundPropagationProof [FINISHED, 0 ms]
(14) 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, INF).


The TRS R consists of the following rules:

   .(1, x) -> x
   .(x, 1) -> x
   .(i(x), x) -> 1
   .(x, i(x)) -> 1
   i(1) -> 1
   i(i(x)) -> x
   .(i(y), .(y, z)) -> z
   .(y, .(i(y), z)) -> z
   .(.(x, y), z) -> .(x, .(y, z))
   i(.(x, y)) -> .(i(y), i(x))

S is empty.
Rewrite Strategy: INNERMOST
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(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(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   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, INF).


The TRS R consists of the following rules:

   .(1, x) -> x
   .(x, 1) -> x
   .(i(x), x) -> 1
   .(x, i(x)) -> 1
   i(1) -> 1
   i(i(x)) -> x
   .(i(y), .(y, z)) -> z
   .(y, .(i(y), z)) -> z
   .(.(x, y), z) -> .(x, .(y, z))
   i(.(x, y)) -> .(i(y), i(x))

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

   encArg(1) -> 1
   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_i(x_1) -> i(encArg(x_1))

Rewrite Strategy: INNERMOST
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(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
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(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:

   .(1, x) -> x
   .(x, 1) -> x
   .(i(x), x) -> 1
   .(x, i(x)) -> 1
   i(1) -> 1
   i(i(x)) -> x
   .(i(y), .(y, z)) -> z
   .(y, .(i(y), z)) -> z
   .(.(x, y), z) -> .(x, .(y, z))
   i(.(x, y)) -> .(i(y), i(x))

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

   encArg(1) -> 1
   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_1 -> 1
   encode_i(x_1) -> i(encArg(x_1))

Rewrite Strategy: INNERMOST
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(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:

   .(1', x) -> x
   .(x, 1') -> x
   .(i(x), x) -> 1'
   .(x, i(x)) -> 1'
   i(1') -> 1'
   i(i(x)) -> x
   .(i(y), .(y, z)) -> z
   .(y, .(i(y), z)) -> z
   .(.(x, y), z) -> .(x, .(y, z))
   i(.(x, y)) -> .(i(y), i(x))

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

   encArg(1') -> 1'
   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encArg(cons_i(x_1)) -> i(encArg(x_1))
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
   encode_1 -> 1'
   encode_i(x_1) -> i(encArg(x_1))

Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(8)
Obligation:
Innermost TRS:
Rules:
.(1', x) -> x
.(x, 1') -> x
.(i(x), x) -> 1'
.(x, i(x)) -> 1'
i(1') -> 1'
i(i(x)) -> x
.(i(y), .(y, z)) -> z
.(y, .(i(y), z)) -> z
.(.(x, y), z) -> .(x, .(y, z))
i(.(x, y)) -> .(i(y), i(x))
encArg(1') -> 1'
encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
encArg(cons_i(x_1)) -> i(encArg(x_1))
encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
encode_1 -> 1'
encode_i(x_1) -> i(encArg(x_1))

Types:
. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
1' :: 1':cons_.:cons_i
i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
encArg :: 1':cons_.:cons_i -> 1':cons_.:cons_i
cons_. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
cons_i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
encode_. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
encode_1 :: 1':cons_.:cons_i
encode_i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
hole_1':cons_.:cons_i1_0 :: 1':cons_.:cons_i
gen_1':cons_.:cons_i2_0 :: Nat -> 1':cons_.:cons_i

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

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

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(10)
Obligation:
Innermost TRS:
Rules:
.(1', x) -> x
.(x, 1') -> x
.(i(x), x) -> 1'
.(x, i(x)) -> 1'
i(1') -> 1'
i(i(x)) -> x
.(i(y), .(y, z)) -> z
.(y, .(i(y), z)) -> z
.(.(x, y), z) -> .(x, .(y, z))
i(.(x, y)) -> .(i(y), i(x))
encArg(1') -> 1'
encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
encArg(cons_i(x_1)) -> i(encArg(x_1))
encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
encode_1 -> 1'
encode_i(x_1) -> i(encArg(x_1))

Types:
. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
1' :: 1':cons_.:cons_i
i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
encArg :: 1':cons_.:cons_i -> 1':cons_.:cons_i
cons_. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
cons_i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
encode_. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
encode_1 :: 1':cons_.:cons_i
encode_i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
hole_1':cons_.:cons_i1_0 :: 1':cons_.:cons_i
gen_1':cons_.:cons_i2_0 :: Nat -> 1':cons_.:cons_i


Generator Equations:
gen_1':cons_.:cons_i2_0(0) <=> 1'
gen_1':cons_.:cons_i2_0(+(x, 1)) <=> cons_.(1', gen_1':cons_.:cons_i2_0(x))


The following defined symbols remain to be analysed:
., i, encArg

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

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(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_1':cons_.:cons_i2_0(n229_0)) -> gen_1':cons_.:cons_i2_0(0), rt in Omega(n229_0)

Induction Base:
encArg(gen_1':cons_.:cons_i2_0(0)) ->_R^Omega(0)
1'

Induction Step:
encArg(gen_1':cons_.:cons_i2_0(+(n229_0, 1))) ->_R^Omega(0)
.(encArg(1'), encArg(gen_1':cons_.:cons_i2_0(n229_0))) ->_R^Omega(0)
.(1', encArg(gen_1':cons_.:cons_i2_0(n229_0))) ->_IH
.(1', gen_1':cons_.:cons_i2_0(0)) ->_R^Omega(1)
gen_1':cons_.:cons_i2_0(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:
.(1', x) -> x
.(x, 1') -> x
.(i(x), x) -> 1'
.(x, i(x)) -> 1'
i(1') -> 1'
i(i(x)) -> x
.(i(y), .(y, z)) -> z
.(y, .(i(y), z)) -> z
.(.(x, y), z) -> .(x, .(y, z))
i(.(x, y)) -> .(i(y), i(x))
encArg(1') -> 1'
encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
encArg(cons_i(x_1)) -> i(encArg(x_1))
encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))
encode_1 -> 1'
encode_i(x_1) -> i(encArg(x_1))

Types:
. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
1' :: 1':cons_.:cons_i
i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
encArg :: 1':cons_.:cons_i -> 1':cons_.:cons_i
cons_. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
cons_i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
encode_. :: 1':cons_.:cons_i -> 1':cons_.:cons_i -> 1':cons_.:cons_i
encode_1 :: 1':cons_.:cons_i
encode_i :: 1':cons_.:cons_i -> 1':cons_.:cons_i
hole_1':cons_.:cons_i1_0 :: 1':cons_.:cons_i
gen_1':cons_.:cons_i2_0 :: Nat -> 1':cons_.:cons_i


Generator Equations:
gen_1':cons_.:cons_i2_0(0) <=> 1'
gen_1':cons_.:cons_i2_0(+(x, 1)) <=> cons_.(1', gen_1':cons_.:cons_i2_0(x))


The following defined symbols remain to be analysed:
encArg
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(13) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(14)
BOUNDS(n^1, INF)