/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 166 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), 411 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), 66 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:

   g(tt, x, y) -> g(f(x, y), s(x), s(y))
   f(s(x), y) -> f(x, y)
   f(x, s(y)) -> f(x, y)
   f(0, 0) -> tt

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(tt) -> tt
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_g(x_1, x_2, x_3)) -> g(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2, x_3) -> g(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_tt -> tt
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_0 -> 0


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

(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:

   g(tt, x, y) -> g(f(x, y), s(x), s(y))
   f(s(x), y) -> f(x, y)
   f(x, s(y)) -> f(x, y)
   f(0, 0) -> tt

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

   encArg(tt) -> tt
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_g(x_1, x_2, x_3)) -> g(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2, x_3) -> g(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_tt -> tt
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_0 -> 0

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:

   g(tt, x, y) -> g(f(x, y), s(x), s(y))
   f(s(x), y) -> f(x, y)
   f(x, s(y)) -> f(x, y)
   f(0, 0) -> tt

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

   encArg(tt) -> tt
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0) -> 0
   encArg(cons_g(x_1, x_2, x_3)) -> g(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2, x_3) -> g(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_tt -> tt
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_0 -> 0

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:

   g(tt, x, y) -> g(f(x, y), s(x), s(y))
   f(s(x), y) -> f(x, y)
   f(x, s(y)) -> f(x, y)
   f(0', 0') -> tt

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

   encArg(tt) -> tt
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(0') -> 0'
   encArg(cons_g(x_1, x_2, x_3)) -> g(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
   encode_g(x_1, x_2, x_3) -> g(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_tt -> tt
   encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
   encode_s(x_1) -> s(encArg(x_1))
   encode_0 -> 0'

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

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

(8)
Obligation:
Innermost TRS:
Rules:
g(tt, x, y) -> g(f(x, y), s(x), s(y))
f(s(x), y) -> f(x, y)
f(x, s(y)) -> f(x, y)
f(0', 0') -> tt
encArg(tt) -> tt
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_g(x_1, x_2, x_3)) -> g(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encode_g(x_1, x_2, x_3) -> g(encArg(x_1), encArg(x_2), encArg(x_3))
encode_tt -> tt
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_s(x_1) -> s(encArg(x_1))
encode_0 -> 0'

Types:
g :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
tt :: tt:s:0':cons_g:cons_f
f :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
s :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
0' :: tt:s:0':cons_g:cons_f
encArg :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
cons_g :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
cons_f :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
encode_g :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
encode_tt :: tt:s:0':cons_g:cons_f
encode_f :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
encode_s :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
encode_0 :: tt:s:0':cons_g:cons_f
hole_tt:s:0':cons_g:cons_f1_4 :: tt:s:0':cons_g:cons_f
gen_tt:s:0':cons_g:cons_f2_4 :: Nat -> tt:s:0':cons_g:cons_f

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

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

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

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

(10)
Obligation:
Innermost TRS:
Rules:
g(tt, x, y) -> g(f(x, y), s(x), s(y))
f(s(x), y) -> f(x, y)
f(x, s(y)) -> f(x, y)
f(0', 0') -> tt
encArg(tt) -> tt
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_g(x_1, x_2, x_3)) -> g(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encode_g(x_1, x_2, x_3) -> g(encArg(x_1), encArg(x_2), encArg(x_3))
encode_tt -> tt
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_s(x_1) -> s(encArg(x_1))
encode_0 -> 0'

Types:
g :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
tt :: tt:s:0':cons_g:cons_f
f :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
s :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
0' :: tt:s:0':cons_g:cons_f
encArg :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
cons_g :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
cons_f :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
encode_g :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
encode_tt :: tt:s:0':cons_g:cons_f
encode_f :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
encode_s :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
encode_0 :: tt:s:0':cons_g:cons_f
hole_tt:s:0':cons_g:cons_f1_4 :: tt:s:0':cons_g:cons_f
gen_tt:s:0':cons_g:cons_f2_4 :: Nat -> tt:s:0':cons_g:cons_f


Generator Equations:
gen_tt:s:0':cons_g:cons_f2_4(0) <=> tt
gen_tt:s:0':cons_g:cons_f2_4(+(x, 1)) <=> s(gen_tt:s:0':cons_g:cons_f2_4(x))


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

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

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_tt:s:0':cons_g:cons_f2_4(+(1, n4_4)), gen_tt:s:0':cons_g:cons_f2_4(b)) -> *3_4, rt in Omega(n4_4)

Induction Base:
f(gen_tt:s:0':cons_g:cons_f2_4(+(1, 0)), gen_tt:s:0':cons_g:cons_f2_4(b))

Induction Step:
f(gen_tt:s:0':cons_g:cons_f2_4(+(1, +(n4_4, 1))), gen_tt:s:0':cons_g:cons_f2_4(b)) ->_R^Omega(1)
f(gen_tt:s:0':cons_g:cons_f2_4(+(1, n4_4)), gen_tt:s:0':cons_g:cons_f2_4(b)) ->_IH
*3_4

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:
g(tt, x, y) -> g(f(x, y), s(x), s(y))
f(s(x), y) -> f(x, y)
f(x, s(y)) -> f(x, y)
f(0', 0') -> tt
encArg(tt) -> tt
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_g(x_1, x_2, x_3)) -> g(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encode_g(x_1, x_2, x_3) -> g(encArg(x_1), encArg(x_2), encArg(x_3))
encode_tt -> tt
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_s(x_1) -> s(encArg(x_1))
encode_0 -> 0'

Types:
g :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
tt :: tt:s:0':cons_g:cons_f
f :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
s :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
0' :: tt:s:0':cons_g:cons_f
encArg :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
cons_g :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
cons_f :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
encode_g :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
encode_tt :: tt:s:0':cons_g:cons_f
encode_f :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
encode_s :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
encode_0 :: tt:s:0':cons_g:cons_f
hole_tt:s:0':cons_g:cons_f1_4 :: tt:s:0':cons_g:cons_f
gen_tt:s:0':cons_g:cons_f2_4 :: Nat -> tt:s:0':cons_g:cons_f


Generator Equations:
gen_tt:s:0':cons_g:cons_f2_4(0) <=> tt
gen_tt:s:0':cons_g:cons_f2_4(+(x, 1)) <=> s(gen_tt:s:0':cons_g:cons_f2_4(x))


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

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

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
g(tt, x, y) -> g(f(x, y), s(x), s(y))
f(s(x), y) -> f(x, y)
f(x, s(y)) -> f(x, y)
f(0', 0') -> tt
encArg(tt) -> tt
encArg(s(x_1)) -> s(encArg(x_1))
encArg(0') -> 0'
encArg(cons_g(x_1, x_2, x_3)) -> g(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2))
encode_g(x_1, x_2, x_3) -> g(encArg(x_1), encArg(x_2), encArg(x_3))
encode_tt -> tt
encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2))
encode_s(x_1) -> s(encArg(x_1))
encode_0 -> 0'

Types:
g :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
tt :: tt:s:0':cons_g:cons_f
f :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
s :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
0' :: tt:s:0':cons_g:cons_f
encArg :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
cons_g :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
cons_f :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
encode_g :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
encode_tt :: tt:s:0':cons_g:cons_f
encode_f :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
encode_s :: tt:s:0':cons_g:cons_f -> tt:s:0':cons_g:cons_f
encode_0 :: tt:s:0':cons_g:cons_f
hole_tt:s:0':cons_g:cons_f1_4 :: tt:s:0':cons_g:cons_f
gen_tt:s:0':cons_g:cons_f2_4 :: Nat -> tt:s:0':cons_g:cons_f


Lemmas:
f(gen_tt:s:0':cons_g:cons_f2_4(+(1, n4_4)), gen_tt:s:0':cons_g:cons_f2_4(b)) -> *3_4, rt in Omega(n4_4)


Generator Equations:
gen_tt:s:0':cons_g:cons_f2_4(0) <=> tt
gen_tt:s:0':cons_g:cons_f2_4(+(x, 1)) <=> s(gen_tt:s:0':cons_g:cons_f2_4(x))


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

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

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_tt:s:0':cons_g:cons_f2_4(n1237_4)) -> gen_tt:s:0':cons_g:cons_f2_4(n1237_4), rt in Omega(0)

Induction Base:
encArg(gen_tt:s:0':cons_g:cons_f2_4(0)) ->_R^Omega(0)
tt

Induction Step:
encArg(gen_tt:s:0':cons_g:cons_f2_4(+(n1237_4, 1))) ->_R^Omega(0)
s(encArg(gen_tt:s:0':cons_g:cons_f2_4(n1237_4))) ->_IH
s(gen_tt:s:0':cons_g:cons_f2_4(c1238_4))

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

(18)
BOUNDS(1, INF)