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


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


WORST_CASE(Omega(n^2), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 182 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), 284 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), 783 ms]
        (18) BEST
            (19) proven lower bound
                (20) LowerBoundPropagationProof [FINISHED, 0 ms]
                (21) BOUNDS(n^2, INF)
            (22) typed CpxTrs
                (23) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
                (24) BOUNDS(1, INF)


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   f(0) -> 1
   f(s(x)) -> g(x, s(x))
   g(0, y) -> y
   g(s(x), y) -> g(x, +(y, s(x)))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   g(s(x), y) -> g(x, s(+(y, 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(0) -> 0
   encArg(1) -> 1
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_1 -> 1
   encode_s(x_1) -> s(encArg(x_1))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))


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

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


The TRS R consists of the following rules:

   f(0) -> 1
   f(s(x)) -> g(x, s(x))
   g(0, y) -> y
   g(s(x), y) -> g(x, +(y, s(x)))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   g(s(x), y) -> g(x, s(+(y, x)))

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

   encArg(0) -> 0
   encArg(1) -> 1
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_1 -> 1
   encode_s(x_1) -> s(encArg(x_1))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))

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^2, INF).


The TRS R consists of the following rules:

   f(0) -> 1
   f(s(x)) -> g(x, s(x))
   g(0, y) -> y
   g(s(x), y) -> g(x, +(y, s(x)))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   g(s(x), y) -> g(x, s(+(y, x)))

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

   encArg(0) -> 0
   encArg(1) -> 1
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0
   encode_1 -> 1
   encode_s(x_1) -> s(encArg(x_1))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2))

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^2, INF).


The TRS R consists of the following rules:

   f(0') -> 1'
   f(s(x)) -> g(x, s(x))
   g(0', y) -> y
   g(s(x), y) -> g(x, +'(y, s(x)))
   +'(x, 0') -> x
   +'(x, s(y)) -> s(+'(x, y))
   g(s(x), y) -> g(x, s(+'(y, x)))

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

   encArg(0') -> 0'
   encArg(1') -> 1'
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_f(x_1)) -> f(encArg(x_1))
   encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
   encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
   encode_f(x_1) -> f(encArg(x_1))
   encode_0 -> 0'
   encode_1 -> 1'
   encode_s(x_1) -> s(encArg(x_1))
   encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
   encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
f(0') -> 1'
f(s(x)) -> g(x, s(x))
g(0', y) -> y
g(s(x), y) -> g(x, +'(y, s(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
g(s(x), y) -> g(x, s(+'(y, x)))
encArg(0') -> 0'
encArg(1') -> 1'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encode_f(x_1) -> f(encArg(x_1))
encode_0 -> 0'
encode_1 -> 1'
encode_s(x_1) -> s(encArg(x_1))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))

Types:
f :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
0' :: 0':1':s:cons_f:cons_g:cons_+
1' :: 0':1':s:cons_f:cons_g:cons_+
s :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
g :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
+' :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encArg :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
cons_f :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
cons_g :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
cons_+ :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_f :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_0 :: 0':1':s:cons_f:cons_g:cons_+
encode_1 :: 0':1':s:cons_f:cons_g:cons_+
encode_s :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_g :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_+ :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
hole_0':1':s:cons_f:cons_g:cons_+1_3 :: 0':1':s:cons_f:cons_g:cons_+
gen_0':1':s:cons_f:cons_g:cons_+2_3 :: Nat -> 0':1':s:cons_f:cons_g:cons_+

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

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

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

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

(10)
Obligation:
Innermost TRS:
Rules:
f(0') -> 1'
f(s(x)) -> g(x, s(x))
g(0', y) -> y
g(s(x), y) -> g(x, +'(y, s(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
g(s(x), y) -> g(x, s(+'(y, x)))
encArg(0') -> 0'
encArg(1') -> 1'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encode_f(x_1) -> f(encArg(x_1))
encode_0 -> 0'
encode_1 -> 1'
encode_s(x_1) -> s(encArg(x_1))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))

Types:
f :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
0' :: 0':1':s:cons_f:cons_g:cons_+
1' :: 0':1':s:cons_f:cons_g:cons_+
s :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
g :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
+' :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encArg :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
cons_f :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
cons_g :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
cons_+ :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_f :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_0 :: 0':1':s:cons_f:cons_g:cons_+
encode_1 :: 0':1':s:cons_f:cons_g:cons_+
encode_s :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_g :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_+ :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
hole_0':1':s:cons_f:cons_g:cons_+1_3 :: 0':1':s:cons_f:cons_g:cons_+
gen_0':1':s:cons_f:cons_g:cons_+2_3 :: Nat -> 0':1':s:cons_f:cons_g:cons_+


Generator Equations:
gen_0':1':s:cons_f:cons_g:cons_+2_3(0) <=> 0'
gen_0':1':s:cons_f:cons_g:cons_+2_3(+(x, 1)) <=> s(gen_0':1':s:cons_f:cons_g:cons_+2_3(x))


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

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

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':1':s:cons_f:cons_g:cons_+2_3(a), gen_0':1':s:cons_f:cons_g:cons_+2_3(n4_3)) -> gen_0':1':s:cons_f:cons_g:cons_+2_3(+(n4_3, a)), rt in Omega(1 + n4_3)

Induction Base:
+'(gen_0':1':s:cons_f:cons_g:cons_+2_3(a), gen_0':1':s:cons_f:cons_g:cons_+2_3(0)) ->_R^Omega(1)
gen_0':1':s:cons_f:cons_g:cons_+2_3(a)

Induction Step:
+'(gen_0':1':s:cons_f:cons_g:cons_+2_3(a), gen_0':1':s:cons_f:cons_g:cons_+2_3(+(n4_3, 1))) ->_R^Omega(1)
s(+'(gen_0':1':s:cons_f:cons_g:cons_+2_3(a), gen_0':1':s:cons_f:cons_g:cons_+2_3(n4_3))) ->_IH
s(gen_0':1':s:cons_f:cons_g:cons_+2_3(+(a, c5_3)))

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(0') -> 1'
f(s(x)) -> g(x, s(x))
g(0', y) -> y
g(s(x), y) -> g(x, +'(y, s(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
g(s(x), y) -> g(x, s(+'(y, x)))
encArg(0') -> 0'
encArg(1') -> 1'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encode_f(x_1) -> f(encArg(x_1))
encode_0 -> 0'
encode_1 -> 1'
encode_s(x_1) -> s(encArg(x_1))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))

Types:
f :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
0' :: 0':1':s:cons_f:cons_g:cons_+
1' :: 0':1':s:cons_f:cons_g:cons_+
s :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
g :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
+' :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encArg :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
cons_f :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
cons_g :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
cons_+ :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_f :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_0 :: 0':1':s:cons_f:cons_g:cons_+
encode_1 :: 0':1':s:cons_f:cons_g:cons_+
encode_s :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_g :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_+ :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
hole_0':1':s:cons_f:cons_g:cons_+1_3 :: 0':1':s:cons_f:cons_g:cons_+
gen_0':1':s:cons_f:cons_g:cons_+2_3 :: Nat -> 0':1':s:cons_f:cons_g:cons_+


Generator Equations:
gen_0':1':s:cons_f:cons_g:cons_+2_3(0) <=> 0'
gen_0':1':s:cons_f:cons_g:cons_+2_3(+(x, 1)) <=> s(gen_0':1':s:cons_f:cons_g:cons_+2_3(x))


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

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

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
f(0') -> 1'
f(s(x)) -> g(x, s(x))
g(0', y) -> y
g(s(x), y) -> g(x, +'(y, s(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
g(s(x), y) -> g(x, s(+'(y, x)))
encArg(0') -> 0'
encArg(1') -> 1'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encode_f(x_1) -> f(encArg(x_1))
encode_0 -> 0'
encode_1 -> 1'
encode_s(x_1) -> s(encArg(x_1))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))

Types:
f :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
0' :: 0':1':s:cons_f:cons_g:cons_+
1' :: 0':1':s:cons_f:cons_g:cons_+
s :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
g :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
+' :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encArg :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
cons_f :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
cons_g :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
cons_+ :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_f :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_0 :: 0':1':s:cons_f:cons_g:cons_+
encode_1 :: 0':1':s:cons_f:cons_g:cons_+
encode_s :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_g :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_+ :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
hole_0':1':s:cons_f:cons_g:cons_+1_3 :: 0':1':s:cons_f:cons_g:cons_+
gen_0':1':s:cons_f:cons_g:cons_+2_3 :: Nat -> 0':1':s:cons_f:cons_g:cons_+


Lemmas:
+'(gen_0':1':s:cons_f:cons_g:cons_+2_3(a), gen_0':1':s:cons_f:cons_g:cons_+2_3(n4_3)) -> gen_0':1':s:cons_f:cons_g:cons_+2_3(+(n4_3, a)), rt in Omega(1 + n4_3)


Generator Equations:
gen_0':1':s:cons_f:cons_g:cons_+2_3(0) <=> 0'
gen_0':1':s:cons_f:cons_g:cons_+2_3(+(x, 1)) <=> s(gen_0':1':s:cons_f:cons_g:cons_+2_3(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:
g(gen_0':1':s:cons_f:cons_g:cons_+2_3(n825_3), gen_0':1':s:cons_f:cons_g:cons_+2_3(b)) -> *3_3, rt in Omega(n825_3 + n825_3^2)

Induction Base:
g(gen_0':1':s:cons_f:cons_g:cons_+2_3(0), gen_0':1':s:cons_f:cons_g:cons_+2_3(b))

Induction Step:
g(gen_0':1':s:cons_f:cons_g:cons_+2_3(+(n825_3, 1)), gen_0':1':s:cons_f:cons_g:cons_+2_3(b)) ->_R^Omega(1)
g(gen_0':1':s:cons_f:cons_g:cons_+2_3(n825_3), +'(gen_0':1':s:cons_f:cons_g:cons_+2_3(b), s(gen_0':1':s:cons_f:cons_g:cons_+2_3(n825_3)))) ->_L^Omega(2 + n825_3)
g(gen_0':1':s:cons_f:cons_g:cons_+2_3(n825_3), gen_0':1':s:cons_f:cons_g:cons_+2_3(+(+(n825_3, 1), b))) ->_IH
*3_3

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
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(18)
Complex Obligation (BEST)

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(19)
Obligation:
Proved the lower bound n^2 for the following obligation:

Innermost TRS:
Rules:
f(0') -> 1'
f(s(x)) -> g(x, s(x))
g(0', y) -> y
g(s(x), y) -> g(x, +'(y, s(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
g(s(x), y) -> g(x, s(+'(y, x)))
encArg(0') -> 0'
encArg(1') -> 1'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encode_f(x_1) -> f(encArg(x_1))
encode_0 -> 0'
encode_1 -> 1'
encode_s(x_1) -> s(encArg(x_1))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))

Types:
f :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
0' :: 0':1':s:cons_f:cons_g:cons_+
1' :: 0':1':s:cons_f:cons_g:cons_+
s :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
g :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
+' :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encArg :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
cons_f :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
cons_g :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
cons_+ :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_f :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_0 :: 0':1':s:cons_f:cons_g:cons_+
encode_1 :: 0':1':s:cons_f:cons_g:cons_+
encode_s :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_g :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_+ :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
hole_0':1':s:cons_f:cons_g:cons_+1_3 :: 0':1':s:cons_f:cons_g:cons_+
gen_0':1':s:cons_f:cons_g:cons_+2_3 :: Nat -> 0':1':s:cons_f:cons_g:cons_+


Lemmas:
+'(gen_0':1':s:cons_f:cons_g:cons_+2_3(a), gen_0':1':s:cons_f:cons_g:cons_+2_3(n4_3)) -> gen_0':1':s:cons_f:cons_g:cons_+2_3(+(n4_3, a)), rt in Omega(1 + n4_3)


Generator Equations:
gen_0':1':s:cons_f:cons_g:cons_+2_3(0) <=> 0'
gen_0':1':s:cons_f:cons_g:cons_+2_3(+(x, 1)) <=> s(gen_0':1':s:cons_f:cons_g:cons_+2_3(x))


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

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

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(20) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(21)
BOUNDS(n^2, INF)

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(22)
Obligation:
Innermost TRS:
Rules:
f(0') -> 1'
f(s(x)) -> g(x, s(x))
g(0', y) -> y
g(s(x), y) -> g(x, +'(y, s(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
g(s(x), y) -> g(x, s(+'(y, x)))
encArg(0') -> 0'
encArg(1') -> 1'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_f(x_1)) -> f(encArg(x_1))
encArg(cons_g(x_1, x_2)) -> g(encArg(x_1), encArg(x_2))
encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2))
encode_f(x_1) -> f(encArg(x_1))
encode_0 -> 0'
encode_1 -> 1'
encode_s(x_1) -> s(encArg(x_1))
encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2))
encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2))

Types:
f :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
0' :: 0':1':s:cons_f:cons_g:cons_+
1' :: 0':1':s:cons_f:cons_g:cons_+
s :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
g :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
+' :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encArg :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
cons_f :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
cons_g :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
cons_+ :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_f :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_0 :: 0':1':s:cons_f:cons_g:cons_+
encode_1 :: 0':1':s:cons_f:cons_g:cons_+
encode_s :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_g :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
encode_+ :: 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+ -> 0':1':s:cons_f:cons_g:cons_+
hole_0':1':s:cons_f:cons_g:cons_+1_3 :: 0':1':s:cons_f:cons_g:cons_+
gen_0':1':s:cons_f:cons_g:cons_+2_3 :: Nat -> 0':1':s:cons_f:cons_g:cons_+


Lemmas:
+'(gen_0':1':s:cons_f:cons_g:cons_+2_3(a), gen_0':1':s:cons_f:cons_g:cons_+2_3(n4_3)) -> gen_0':1':s:cons_f:cons_g:cons_+2_3(+(n4_3, a)), rt in Omega(1 + n4_3)
g(gen_0':1':s:cons_f:cons_g:cons_+2_3(n825_3), gen_0':1':s:cons_f:cons_g:cons_+2_3(b)) -> *3_3, rt in Omega(n825_3 + n825_3^2)


Generator Equations:
gen_0':1':s:cons_f:cons_g:cons_+2_3(0) <=> 0'
gen_0':1':s:cons_f:cons_g:cons_+2_3(+(x, 1)) <=> s(gen_0':1':s:cons_f:cons_g:cons_+2_3(x))


The following defined symbols remain to be analysed:
encArg
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(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':1':s:cons_f:cons_g:cons_+2_3(n3879_3)) -> gen_0':1':s:cons_f:cons_g:cons_+2_3(n3879_3), rt in Omega(0)

Induction Base:
encArg(gen_0':1':s:cons_f:cons_g:cons_+2_3(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':1':s:cons_f:cons_g:cons_+2_3(+(n3879_3, 1))) ->_R^Omega(0)
s(encArg(gen_0':1':s:cons_f:cons_g:cons_+2_3(n3879_3))) ->_IH
s(gen_0':1':s:cons_f:cons_g:cons_+2_3(c3880_3))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(24)
BOUNDS(1, INF)