/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 193 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), 5917 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), 72 ms]
        (18) BOUNDS(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:

   f(a) -> b
   f(c) -> d
   f(g(x, y)) -> g(f(x), f(y))
   f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
   g(x, x) -> h(e, x)

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(a) -> a
   encArg(b) -> b
   encArg(c) -> c
   encArg(d) -> d
   encArg(h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encArg(e) -> e
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_b -> b
   encode_c -> c
   encode_d -> d
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
   encode_e -> e


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

(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:

   f(a) -> b
   f(c) -> d
   f(g(x, y)) -> g(f(x), f(y))
   f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
   g(x, x) -> h(e, x)

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(c) -> c
   encArg(d) -> d
   encArg(h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encArg(e) -> e
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_b -> b
   encode_c -> c
   encode_d -> d
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
   encode_e -> e

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:

   f(a) -> b
   f(c) -> d
   f(g(x, y)) -> g(f(x), f(y))
   f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
   g(x, x) -> h(e, x)

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(c) -> c
   encArg(d) -> d
   encArg(h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encArg(e) -> e
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_b -> b
   encode_c -> c
   encode_d -> d
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
   encode_e -> e

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:

   f(a) -> b
   f(c) -> d
   f(g(x, y)) -> g(f(x), f(y))
   f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
   g(x, x) -> h(e, x)

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

   encArg(a) -> a
   encArg(b) -> b
   encArg(c) -> c
   encArg(d) -> d
   encArg(h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
   encArg(e) -> e
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a -> a
   encode_b -> b
   encode_c -> c
   encode_d -> d
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
   encode_e -> e

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

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

(8)
Obligation:
Innermost TRS:
Rules:
f(a) -> b
f(c) -> d
f(g(x, y)) -> g(f(x), f(y))
f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
g(x, x) -> h(e, x)
encArg(a) -> a
encArg(b) -> b
encArg(c) -> c
encArg(d) -> d
encArg(h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
encArg(e) -> e
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encode_f(x_1) -> f(encArg(x_1))
encode_a -> a
encode_b -> b
encode_c -> c
encode_d -> d
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
encode_e -> e

Types:
f :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
a :: a:b:c:d:h:e:cons_f:cons_g
b :: a:b:c:d:h:e:cons_f:cons_g
c :: a:b:c:d:h:e:cons_f:cons_g
d :: a:b:c:d:h:e:cons_f:cons_g
g :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
h :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
e :: a:b:c:d:h:e:cons_f:cons_g
encArg :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
cons_f :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
cons_g :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
encode_f :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
encode_a :: a:b:c:d:h:e:cons_f:cons_g
encode_b :: a:b:c:d:h:e:cons_f:cons_g
encode_c :: a:b:c:d:h:e:cons_f:cons_g
encode_d :: a:b:c:d:h:e:cons_f:cons_g
encode_g :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
encode_h :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
encode_e :: a:b:c:d:h:e:cons_f:cons_g
hole_a:b:c:d:h:e:cons_f:cons_g1_0 :: a:b:c:d:h:e:cons_f:cons_g
gen_a:b:c:d:h:e:cons_f:cons_g2_0 :: Nat -> a:b:c:d:h:e:cons_f:cons_g

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

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

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

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

(10)
Obligation:
Innermost TRS:
Rules:
f(a) -> b
f(c) -> d
f(g(x, y)) -> g(f(x), f(y))
f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
g(x, x) -> h(e, x)
encArg(a) -> a
encArg(b) -> b
encArg(c) -> c
encArg(d) -> d
encArg(h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
encArg(e) -> e
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encode_f(x_1) -> f(encArg(x_1))
encode_a -> a
encode_b -> b
encode_c -> c
encode_d -> d
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
encode_e -> e

Types:
f :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
a :: a:b:c:d:h:e:cons_f:cons_g
b :: a:b:c:d:h:e:cons_f:cons_g
c :: a:b:c:d:h:e:cons_f:cons_g
d :: a:b:c:d:h:e:cons_f:cons_g
g :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
h :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
e :: a:b:c:d:h:e:cons_f:cons_g
encArg :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
cons_f :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
cons_g :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
encode_f :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
encode_a :: a:b:c:d:h:e:cons_f:cons_g
encode_b :: a:b:c:d:h:e:cons_f:cons_g
encode_c :: a:b:c:d:h:e:cons_f:cons_g
encode_d :: a:b:c:d:h:e:cons_f:cons_g
encode_g :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
encode_h :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
encode_e :: a:b:c:d:h:e:cons_f:cons_g
hole_a:b:c:d:h:e:cons_f:cons_g1_0 :: a:b:c:d:h:e:cons_f:cons_g
gen_a:b:c:d:h:e:cons_f:cons_g2_0 :: Nat -> a:b:c:d:h:e:cons_f:cons_g


Generator Equations:
gen_a:b:c:d:h:e:cons_f:cons_g2_0(0) <=> a
gen_a:b:c:d:h:e:cons_f:cons_g2_0(+(x, 1)) <=> h(a, gen_a:b:c:d:h:e:cons_f:cons_g2_0(x))


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

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

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_a:b:c:d:h:e:cons_f:cons_g2_0(n4_0)) -> *3_0, rt in Omega(n4_0)

Induction Base:
f(gen_a:b:c:d:h:e:cons_f:cons_g2_0(0))

Induction Step:
f(gen_a:b:c:d:h:e:cons_f:cons_g2_0(+(n4_0, 1))) ->_R^Omega(1)
g(h(gen_a:b:c:d:h:e:cons_f:cons_g2_0(n4_0), f(a)), h(a, f(gen_a:b:c:d:h:e:cons_f:cons_g2_0(n4_0)))) ->_R^Omega(1)
g(h(gen_a:b:c:d:h:e:cons_f:cons_g2_0(n4_0), b), h(a, f(gen_a:b:c:d:h:e:cons_f:cons_g2_0(n4_0)))) ->_IH
g(h(gen_a:b:c:d:h:e:cons_f:cons_g2_0(n4_0), b), h(a, *3_0))

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:

Innermost TRS:
Rules:
f(a) -> b
f(c) -> d
f(g(x, y)) -> g(f(x), f(y))
f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
g(x, x) -> h(e, x)
encArg(a) -> a
encArg(b) -> b
encArg(c) -> c
encArg(d) -> d
encArg(h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
encArg(e) -> e
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encode_f(x_1) -> f(encArg(x_1))
encode_a -> a
encode_b -> b
encode_c -> c
encode_d -> d
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
encode_e -> e

Types:
f :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
a :: a:b:c:d:h:e:cons_f:cons_g
b :: a:b:c:d:h:e:cons_f:cons_g
c :: a:b:c:d:h:e:cons_f:cons_g
d :: a:b:c:d:h:e:cons_f:cons_g
g :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
h :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
e :: a:b:c:d:h:e:cons_f:cons_g
encArg :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
cons_f :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
cons_g :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
encode_f :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
encode_a :: a:b:c:d:h:e:cons_f:cons_g
encode_b :: a:b:c:d:h:e:cons_f:cons_g
encode_c :: a:b:c:d:h:e:cons_f:cons_g
encode_d :: a:b:c:d:h:e:cons_f:cons_g
encode_g :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
encode_h :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
encode_e :: a:b:c:d:h:e:cons_f:cons_g
hole_a:b:c:d:h:e:cons_f:cons_g1_0 :: a:b:c:d:h:e:cons_f:cons_g
gen_a:b:c:d:h:e:cons_f:cons_g2_0 :: Nat -> a:b:c:d:h:e:cons_f:cons_g


Generator Equations:
gen_a:b:c:d:h:e:cons_f:cons_g2_0(0) <=> a
gen_a:b:c:d:h:e:cons_f:cons_g2_0(+(x, 1)) <=> h(a, gen_a:b:c:d:h:e:cons_f:cons_g2_0(x))


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

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

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
f(a) -> b
f(c) -> d
f(g(x, y)) -> g(f(x), f(y))
f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
g(x, x) -> h(e, x)
encArg(a) -> a
encArg(b) -> b
encArg(c) -> c
encArg(d) -> d
encArg(h(x_1, x_2)) -> h(encArg(x_1), encArg(x_2))
encArg(e) -> e
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encode_f(x_1) -> f(encArg(x_1))
encode_a -> a
encode_b -> b
encode_c -> c
encode_d -> d
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_h(x_1, x_2) -> h(encArg(x_1), encArg(x_2))
encode_e -> e

Types:
f :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
a :: a:b:c:d:h:e:cons_f:cons_g
b :: a:b:c:d:h:e:cons_f:cons_g
c :: a:b:c:d:h:e:cons_f:cons_g
d :: a:b:c:d:h:e:cons_f:cons_g
g :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
h :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
e :: a:b:c:d:h:e:cons_f:cons_g
encArg :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
cons_f :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
cons_g :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
encode_f :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
encode_a :: a:b:c:d:h:e:cons_f:cons_g
encode_b :: a:b:c:d:h:e:cons_f:cons_g
encode_c :: a:b:c:d:h:e:cons_f:cons_g
encode_d :: a:b:c:d:h:e:cons_f:cons_g
encode_g :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
encode_h :: a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g -> a:b:c:d:h:e:cons_f:cons_g
encode_e :: a:b:c:d:h:e:cons_f:cons_g
hole_a:b:c:d:h:e:cons_f:cons_g1_0 :: a:b:c:d:h:e:cons_f:cons_g
gen_a:b:c:d:h:e:cons_f:cons_g2_0 :: Nat -> a:b:c:d:h:e:cons_f:cons_g


Lemmas:
f(gen_a:b:c:d:h:e:cons_f:cons_g2_0(n4_0)) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_a:b:c:d:h:e:cons_f:cons_g2_0(0) <=> a
gen_a:b:c:d:h:e:cons_f:cons_g2_0(+(x, 1)) <=> h(a, gen_a:b:c:d:h:e:cons_f:cons_g2_0(x))


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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_a:b:c:d:h:e:cons_f:cons_g2_0(n67866_0)) -> gen_a:b:c:d:h:e:cons_f:cons_g2_0(n67866_0), rt in Omega(0)

Induction Base:
encArg(gen_a:b:c:d:h:e:cons_f:cons_g2_0(0)) ->_R^Omega(0)
a

Induction Step:
encArg(gen_a:b:c:d:h:e:cons_f:cons_g2_0(+(n67866_0, 1))) ->_R^Omega(0)
h(encArg(a), encArg(gen_a:b:c:d:h:e:cons_f:cons_g2_0(n67866_0))) ->_R^Omega(0)
h(a, encArg(gen_a:b:c:d:h:e:cons_f:cons_g2_0(n67866_0))) ->_IH
h(a, gen_a:b:c:d:h:e:cons_f:cons_g2_0(c67867_0))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(18)
BOUNDS(1, INF)