/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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), 178 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), 555 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 4 ms]
        (18) BOUNDS(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:

   h(f(x, y)) -> f(f(a, h(h(y))), x)

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(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(a) -> a
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a


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

(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:

   h(f(x, y)) -> f(f(a, h(h(y))), x)

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

   encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(a) -> a
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a

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:

   h(f(x, y)) -> f(f(a, h(h(y))), x)

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

   encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(a) -> a
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a

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:

   h(f(x, y)) -> f(f(a, h(h(y))), x)

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

   encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encArg(a) -> a
   encArg(cons_h(x_1)) -> h(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_a -> a

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

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

(8)
Obligation:
TRS:
Rules:
h(f(x, y)) -> f(f(a, h(h(y))), x)
encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(a) -> a
encArg(cons_h(x_1)) -> h(encArg(x_1))
encode_h(x_1) -> h(encArg(x_1))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_a -> a

Types:
h :: f:a:cons_h -> f:a:cons_h
f :: f:a:cons_h -> f:a:cons_h -> f:a:cons_h
a :: f:a:cons_h
encArg :: f:a:cons_h -> f:a:cons_h
cons_h :: f:a:cons_h -> f:a:cons_h
encode_h :: f:a:cons_h -> f:a:cons_h
encode_f :: f:a:cons_h -> f:a:cons_h -> f:a:cons_h
encode_a :: f:a:cons_h
hole_f:a:cons_h1_0 :: f:a:cons_h
gen_f:a:cons_h2_0 :: Nat -> f:a:cons_h

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

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

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

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

(10)
Obligation:
TRS:
Rules:
h(f(x, y)) -> f(f(a, h(h(y))), x)
encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(a) -> a
encArg(cons_h(x_1)) -> h(encArg(x_1))
encode_h(x_1) -> h(encArg(x_1))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_a -> a

Types:
h :: f:a:cons_h -> f:a:cons_h
f :: f:a:cons_h -> f:a:cons_h -> f:a:cons_h
a :: f:a:cons_h
encArg :: f:a:cons_h -> f:a:cons_h
cons_h :: f:a:cons_h -> f:a:cons_h
encode_h :: f:a:cons_h -> f:a:cons_h
encode_f :: f:a:cons_h -> f:a:cons_h -> f:a:cons_h
encode_a :: f:a:cons_h
hole_f:a:cons_h1_0 :: f:a:cons_h
gen_f:a:cons_h2_0 :: Nat -> f:a:cons_h


Generator Equations:
gen_f:a:cons_h2_0(0) <=> a
gen_f:a:cons_h2_0(+(x, 1)) <=> f(a, gen_f:a:cons_h2_0(x))


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

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

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
h(gen_f:a:cons_h2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
h(gen_f:a:cons_h2_0(+(1, 0)))

Induction Step:
h(gen_f:a:cons_h2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
f(f(a, h(h(gen_f:a:cons_h2_0(+(1, n4_0))))), a) ->_IH
f(f(a, h(*3_0)), a)

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

(12)
Complex Obligation (BEST)

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

(13)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
h(f(x, y)) -> f(f(a, h(h(y))), x)
encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(a) -> a
encArg(cons_h(x_1)) -> h(encArg(x_1))
encode_h(x_1) -> h(encArg(x_1))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_a -> a

Types:
h :: f:a:cons_h -> f:a:cons_h
f :: f:a:cons_h -> f:a:cons_h -> f:a:cons_h
a :: f:a:cons_h
encArg :: f:a:cons_h -> f:a:cons_h
cons_h :: f:a:cons_h -> f:a:cons_h
encode_h :: f:a:cons_h -> f:a:cons_h
encode_f :: f:a:cons_h -> f:a:cons_h -> f:a:cons_h
encode_a :: f:a:cons_h
hole_f:a:cons_h1_0 :: f:a:cons_h
gen_f:a:cons_h2_0 :: Nat -> f:a:cons_h


Generator Equations:
gen_f:a:cons_h2_0(0) <=> a
gen_f:a:cons_h2_0(+(x, 1)) <=> f(a, gen_f:a:cons_h2_0(x))


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

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

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
TRS:
Rules:
h(f(x, y)) -> f(f(a, h(h(y))), x)
encArg(f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encArg(a) -> a
encArg(cons_h(x_1)) -> h(encArg(x_1))
encode_h(x_1) -> h(encArg(x_1))
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_a -> a

Types:
h :: f:a:cons_h -> f:a:cons_h
f :: f:a:cons_h -> f:a:cons_h -> f:a:cons_h
a :: f:a:cons_h
encArg :: f:a:cons_h -> f:a:cons_h
cons_h :: f:a:cons_h -> f:a:cons_h
encode_h :: f:a:cons_h -> f:a:cons_h
encode_f :: f:a:cons_h -> f:a:cons_h -> f:a:cons_h
encode_a :: f:a:cons_h
hole_f:a:cons_h1_0 :: f:a:cons_h
gen_f:a:cons_h2_0 :: Nat -> f:a:cons_h


Lemmas:
h(gen_f:a:cons_h2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_f:a:cons_h2_0(0) <=> a
gen_f:a:cons_h2_0(+(x, 1)) <=> f(a, gen_f:a:cons_h2_0(x))


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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_f:a:cons_h2_0(n233_0)) -> gen_f:a:cons_h2_0(n233_0), rt in Omega(0)

Induction Base:
encArg(gen_f:a:cons_h2_0(0)) ->_R^Omega(0)
a

Induction Step:
encArg(gen_f:a:cons_h2_0(+(n233_0, 1))) ->_R^Omega(0)
f(encArg(a), encArg(gen_f:a:cons_h2_0(n233_0))) ->_R^Omega(0)
f(a, encArg(gen_f:a:cons_h2_0(n233_0))) ->_IH
f(a, gen_f:a:cons_h2_0(c234_0))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(18)
BOUNDS(1, INF)