/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (full) 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), 201 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), 291 ms]
(12) proven lower bound
(13) LowerBoundPropagationProof [FINISHED, 0 ms]
(14) BOUNDS(n^1, INF)


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

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


The TRS R consists of the following rules:

   a(lambda(x), y) -> lambda(a(x, 1))
   a(lambda(x), y) -> lambda(a(x, a(y, t)))
   a(a(x, y), z) -> a(x, a(y, z))
   lambda(x) -> x
   a(x, y) -> x
   a(x, y) -> y

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(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(t) -> t
   encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
   encArg(cons_lambda(x_1)) -> lambda(encArg(x_1))
   encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
   encode_lambda(x_1) -> lambda(encArg(x_1))
   encode_1 -> 1
   encode_t -> t


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

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


The TRS R consists of the following rules:

   a(lambda(x), y) -> lambda(a(x, 1))
   a(lambda(x), y) -> lambda(a(x, a(y, t)))
   a(a(x, y), z) -> a(x, a(y, z))
   lambda(x) -> x
   a(x, y) -> x
   a(x, y) -> y

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

   encArg(1) -> 1
   encArg(t) -> t
   encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
   encArg(cons_lambda(x_1)) -> lambda(encArg(x_1))
   encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
   encode_lambda(x_1) -> lambda(encArg(x_1))
   encode_1 -> 1
   encode_t -> t

Rewrite Strategy: FULL
----------------------------------------

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a(lambda(x), y) -> lambda(a(x, 1))
   a(lambda(x), y) -> lambda(a(x, a(y, t)))
   a(a(x, y), z) -> a(x, a(y, z))
   lambda(x) -> x
   a(x, y) -> x
   a(x, y) -> y

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

   encArg(1) -> 1
   encArg(t) -> t
   encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
   encArg(cons_lambda(x_1)) -> lambda(encArg(x_1))
   encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
   encode_lambda(x_1) -> lambda(encArg(x_1))
   encode_1 -> 1
   encode_t -> t

Rewrite Strategy: FULL
----------------------------------------

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

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


The TRS R consists of the following rules:

   a(lambda(x), y) -> lambda(a(x, 1'))
   a(lambda(x), y) -> lambda(a(x, a(y, t)))
   a(a(x, y), z) -> a(x, a(y, z))
   lambda(x) -> x
   a(x, y) -> x
   a(x, y) -> y

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

   encArg(1') -> 1'
   encArg(t) -> t
   encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
   encArg(cons_lambda(x_1)) -> lambda(encArg(x_1))
   encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
   encode_lambda(x_1) -> lambda(encArg(x_1))
   encode_1 -> 1'
   encode_t -> t

Rewrite Strategy: FULL
----------------------------------------

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

(8)
Obligation:
TRS:
Rules:
a(lambda(x), y) -> lambda(a(x, 1'))
a(lambda(x), y) -> lambda(a(x, a(y, t)))
a(a(x, y), z) -> a(x, a(y, z))
lambda(x) -> x
a(x, y) -> x
a(x, y) -> y
encArg(1') -> 1'
encArg(t) -> t
encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
encArg(cons_lambda(x_1)) -> lambda(encArg(x_1))
encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
encode_lambda(x_1) -> lambda(encArg(x_1))
encode_1 -> 1'
encode_t -> t

Types:
a :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
lambda :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
1' :: 1':t:cons_a:cons_lambda
t :: 1':t:cons_a:cons_lambda
encArg :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
cons_a :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
cons_lambda :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
encode_a :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
encode_lambda :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
encode_1 :: 1':t:cons_a:cons_lambda
encode_t :: 1':t:cons_a:cons_lambda
hole_1':t:cons_a:cons_lambda1_0 :: 1':t:cons_a:cons_lambda
gen_1':t:cons_a:cons_lambda2_0 :: Nat -> 1':t:cons_a:cons_lambda

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

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

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

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

(10)
Obligation:
TRS:
Rules:
a(lambda(x), y) -> lambda(a(x, 1'))
a(lambda(x), y) -> lambda(a(x, a(y, t)))
a(a(x, y), z) -> a(x, a(y, z))
lambda(x) -> x
a(x, y) -> x
a(x, y) -> y
encArg(1') -> 1'
encArg(t) -> t
encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
encArg(cons_lambda(x_1)) -> lambda(encArg(x_1))
encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
encode_lambda(x_1) -> lambda(encArg(x_1))
encode_1 -> 1'
encode_t -> t

Types:
a :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
lambda :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
1' :: 1':t:cons_a:cons_lambda
t :: 1':t:cons_a:cons_lambda
encArg :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
cons_a :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
cons_lambda :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
encode_a :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
encode_lambda :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
encode_1 :: 1':t:cons_a:cons_lambda
encode_t :: 1':t:cons_a:cons_lambda
hole_1':t:cons_a:cons_lambda1_0 :: 1':t:cons_a:cons_lambda
gen_1':t:cons_a:cons_lambda2_0 :: Nat -> 1':t:cons_a:cons_lambda


Generator Equations:
gen_1':t:cons_a:cons_lambda2_0(0) <=> 1'
gen_1':t:cons_a:cons_lambda2_0(+(x, 1)) <=> cons_a(1', gen_1':t:cons_a:cons_lambda2_0(x))


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

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

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_1':t:cons_a:cons_lambda2_0(n50_0)) -> gen_1':t:cons_a:cons_lambda2_0(0), rt in Omega(n50_0)

Induction Base:
encArg(gen_1':t:cons_a:cons_lambda2_0(0)) ->_R^Omega(0)
1'

Induction Step:
encArg(gen_1':t:cons_a:cons_lambda2_0(+(n50_0, 1))) ->_R^Omega(0)
a(encArg(1'), encArg(gen_1':t:cons_a:cons_lambda2_0(n50_0))) ->_R^Omega(0)
a(1', encArg(gen_1':t:cons_a:cons_lambda2_0(n50_0))) ->_IH
a(1', gen_1':t:cons_a:cons_lambda2_0(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:

TRS:
Rules:
a(lambda(x), y) -> lambda(a(x, 1'))
a(lambda(x), y) -> lambda(a(x, a(y, t)))
a(a(x, y), z) -> a(x, a(y, z))
lambda(x) -> x
a(x, y) -> x
a(x, y) -> y
encArg(1') -> 1'
encArg(t) -> t
encArg(cons_a(x_1, x_2)) -> a(encArg(x_1), encArg(x_2))
encArg(cons_lambda(x_1)) -> lambda(encArg(x_1))
encode_a(x_1, x_2) -> a(encArg(x_1), encArg(x_2))
encode_lambda(x_1) -> lambda(encArg(x_1))
encode_1 -> 1'
encode_t -> t

Types:
a :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
lambda :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
1' :: 1':t:cons_a:cons_lambda
t :: 1':t:cons_a:cons_lambda
encArg :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
cons_a :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
cons_lambda :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
encode_a :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
encode_lambda :: 1':t:cons_a:cons_lambda -> 1':t:cons_a:cons_lambda
encode_1 :: 1':t:cons_a:cons_lambda
encode_t :: 1':t:cons_a:cons_lambda
hole_1':t:cons_a:cons_lambda1_0 :: 1':t:cons_a:cons_lambda
gen_1':t:cons_a:cons_lambda2_0 :: Nat -> 1':t:cons_a:cons_lambda


Generator Equations:
gen_1':t:cons_a:cons_lambda2_0(0) <=> 1'
gen_1':t:cons_a:cons_lambda2_0(+(x, 1)) <=> cons_a(1', gen_1':t:cons_a:cons_lambda2_0(x))


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

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

(14)
BOUNDS(n^1, INF)