/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^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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 221 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), 311 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), 76 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), 83 ms]
                (24) typed CpxTrs
                (25) RewriteLemmaProof [LOWER BOUND(ID), 48 ms]
                (26) 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:

   plus(x, 0) -> x
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0, y) -> 0
   times(s(0), y) -> y
   times(s(x), y) -> plus(y, times(x, y))
   div(0, y) -> 0
   div(x, y) -> quot(x, y, y)
   quot(0, s(y), z) -> 0
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0, s(z)) -> s(div(x, s(z)))
   div(div(x, y), z) -> div(x, times(y, z))

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(s(x_1)) -> s(encArg(x_1))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))


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

(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:

   plus(x, 0) -> x
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0, y) -> 0
   times(s(0), y) -> y
   times(s(x), y) -> plus(y, times(x, y))
   div(0, y) -> 0
   div(x, y) -> quot(x, y, y)
   quot(0, s(y), z) -> 0
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0, s(z)) -> s(div(x, s(z)))
   div(div(x, y), z) -> div(x, times(y, z))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))

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:

   plus(x, 0) -> x
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0, y) -> 0
   times(s(0), y) -> y
   times(s(x), y) -> plus(y, times(x, y))
   div(0, y) -> 0
   div(x, y) -> quot(x, y, y)
   quot(0, s(y), z) -> 0
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0, s(z)) -> s(div(x, s(z)))
   div(div(x, y), z) -> div(x, times(y, z))

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))

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:

   plus(x, 0') -> x
   plus(0', y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0', y) -> 0'
   times(s(0'), y) -> y
   times(s(x), y) -> plus(y, times(x, y))
   div(0', y) -> 0'
   div(x, y) -> quot(x, y, y)
   quot(0', s(y), z) -> 0'
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0', s(z)) -> s(div(x, s(z)))
   div(div(x, y), z) -> div(x, times(y, z))

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

   encArg(0') -> 0'
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_0 -> 0'
   encode_s(x_1) -> s(encArg(x_1))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
   encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
plus :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
0' :: 0':s:cons_plus:cons_times:cons_div:cons_quot
s :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
times :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
div :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
quot :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encArg :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_plus :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_times :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_div :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_quot :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_plus :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_0 :: 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_s :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_times :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_div :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_quot :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
hole_0':s:cons_plus:cons_times:cons_div:cons_quot1_4 :: 0':s:cons_plus:cons_times:cons_div:cons_quot
gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4 :: Nat -> 0':s:cons_plus:cons_times:cons_div:cons_quot

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
plus, times, div, quot, encArg

They will be analysed ascendingly in the following order:
plus < times
plus < encArg
times < div
times < encArg
div = quot
div < encArg
quot < encArg

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

(10)
Obligation:
Innermost TRS:
Rules:
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
plus :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
0' :: 0':s:cons_plus:cons_times:cons_div:cons_quot
s :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
times :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
div :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
quot :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encArg :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_plus :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_times :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_div :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_quot :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_plus :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_0 :: 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_s :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_times :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_div :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_quot :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
hole_0':s:cons_plus:cons_times:cons_div:cons_quot1_4 :: 0':s:cons_plus:cons_times:cons_div:cons_quot
gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4 :: Nat -> 0':s:cons_plus:cons_times:cons_div:cons_quot


Generator Equations:
gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(0) <=> 0'
gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(+(x, 1)) <=> s(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(x))


The following defined symbols remain to be analysed:
plus, times, div, quot, encArg

They will be analysed ascendingly in the following order:
plus < times
plus < encArg
times < div
times < encArg
div = quot
div < encArg
quot < encArg

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(n4_4), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(b)) -> gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(+(n4_4, b)), rt in Omega(1 + n4_4)

Induction Base:
plus(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(0), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(b)) ->_R^Omega(1)
gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(b)

Induction Step:
plus(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(+(n4_4, 1)), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(b)) ->_R^Omega(1)
s(plus(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(n4_4), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(b))) ->_IH
s(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(+(b, c5_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:
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
plus :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
0' :: 0':s:cons_plus:cons_times:cons_div:cons_quot
s :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
times :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
div :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
quot :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encArg :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_plus :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_times :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_div :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_quot :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_plus :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_0 :: 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_s :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_times :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_div :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_quot :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
hole_0':s:cons_plus:cons_times:cons_div:cons_quot1_4 :: 0':s:cons_plus:cons_times:cons_div:cons_quot
gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4 :: Nat -> 0':s:cons_plus:cons_times:cons_div:cons_quot


Generator Equations:
gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(0) <=> 0'
gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(+(x, 1)) <=> s(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(x))


The following defined symbols remain to be analysed:
plus, times, div, quot, encArg

They will be analysed ascendingly in the following order:
plus < times
plus < encArg
times < div
times < encArg
div = quot
div < encArg
quot < encArg

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
plus :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
0' :: 0':s:cons_plus:cons_times:cons_div:cons_quot
s :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
times :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
div :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
quot :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encArg :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_plus :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_times :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_div :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_quot :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_plus :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_0 :: 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_s :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_times :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_div :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_quot :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
hole_0':s:cons_plus:cons_times:cons_div:cons_quot1_4 :: 0':s:cons_plus:cons_times:cons_div:cons_quot
gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4 :: Nat -> 0':s:cons_plus:cons_times:cons_div:cons_quot


Lemmas:
plus(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(n4_4), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(b)) -> gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(+(n4_4, b)), rt in Omega(1 + n4_4)


Generator Equations:
gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(0) <=> 0'
gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(+(x, 1)) <=> s(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(x))


The following defined symbols remain to be analysed:
times, div, quot, encArg

They will be analysed ascendingly in the following order:
times < div
times < encArg
div = quot
div < encArg
quot < encArg

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
times(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(n985_4), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(b)) -> gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(*(n985_4, b)), rt in Omega(1 + b*n985_4 + n985_4)

Induction Base:
times(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(0), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(b)) ->_R^Omega(1)
0'

Induction Step:
times(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(+(n985_4, 1)), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(b)) ->_R^Omega(1)
plus(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(b), times(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(n985_4), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(b))) ->_IH
plus(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(b), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(*(c986_4, b))) ->_L^Omega(1 + b)
gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(+(b, *(n985_4, b)))

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(18)
Complex Obligation (BEST)

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

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

Innermost TRS:
Rules:
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
plus :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
0' :: 0':s:cons_plus:cons_times:cons_div:cons_quot
s :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
times :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
div :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
quot :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encArg :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_plus :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_times :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_div :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_quot :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_plus :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_0 :: 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_s :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_times :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_div :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_quot :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
hole_0':s:cons_plus:cons_times:cons_div:cons_quot1_4 :: 0':s:cons_plus:cons_times:cons_div:cons_quot
gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4 :: Nat -> 0':s:cons_plus:cons_times:cons_div:cons_quot


Lemmas:
plus(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(n4_4), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(b)) -> gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(+(n4_4, b)), rt in Omega(1 + n4_4)


Generator Equations:
gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(0) <=> 0'
gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(+(x, 1)) <=> s(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(x))


The following defined symbols remain to be analysed:
times, div, quot, encArg

They will be analysed ascendingly in the following order:
times < div
times < encArg
div = quot
div < encArg
quot < encArg

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

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

(21)
BOUNDS(n^2, INF)

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

(22)
Obligation:
Innermost TRS:
Rules:
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
plus :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
0' :: 0':s:cons_plus:cons_times:cons_div:cons_quot
s :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
times :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
div :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
quot :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encArg :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_plus :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_times :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_div :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_quot :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_plus :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_0 :: 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_s :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_times :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_div :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_quot :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
hole_0':s:cons_plus:cons_times:cons_div:cons_quot1_4 :: 0':s:cons_plus:cons_times:cons_div:cons_quot
gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4 :: Nat -> 0':s:cons_plus:cons_times:cons_div:cons_quot


Lemmas:
plus(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(n4_4), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(b)) -> gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(+(n4_4, b)), rt in Omega(1 + n4_4)
times(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(n985_4), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(b)) -> gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(*(n985_4, b)), rt in Omega(1 + b*n985_4 + n985_4)


Generator Equations:
gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(0) <=> 0'
gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(+(x, 1)) <=> s(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(x))


The following defined symbols remain to be analysed:
quot, div, encArg

They will be analysed ascendingly in the following order:
div = quot
div < encArg
quot < encArg

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
quot(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(n2306_4), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(+(1, n2306_4)), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(c)) -> gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(0), rt in Omega(1 + n2306_4)

Induction Base:
quot(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(0), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(+(1, 0)), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(c)) ->_R^Omega(1)
0'

Induction Step:
quot(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(+(n2306_4, 1)), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(+(1, +(n2306_4, 1))), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(c)) ->_R^Omega(1)
quot(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(n2306_4), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(+(1, n2306_4)), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(c)) ->_IH
gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(24)
Obligation:
Innermost TRS:
Rules:
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2))
encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))
encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2))
encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
plus :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
0' :: 0':s:cons_plus:cons_times:cons_div:cons_quot
s :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
times :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
div :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
quot :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encArg :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_plus :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_times :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_div :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
cons_quot :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_plus :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_0 :: 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_s :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_times :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_div :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
encode_quot :: 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot -> 0':s:cons_plus:cons_times:cons_div:cons_quot
hole_0':s:cons_plus:cons_times:cons_div:cons_quot1_4 :: 0':s:cons_plus:cons_times:cons_div:cons_quot
gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4 :: Nat -> 0':s:cons_plus:cons_times:cons_div:cons_quot


Lemmas:
plus(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(n4_4), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(b)) -> gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(+(n4_4, b)), rt in Omega(1 + n4_4)
times(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(n985_4), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(b)) -> gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(*(n985_4, b)), rt in Omega(1 + b*n985_4 + n985_4)
quot(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(n2306_4), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(+(1, n2306_4)), gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(c)) -> gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(0), rt in Omega(1 + n2306_4)


Generator Equations:
gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(0) <=> 0'
gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(+(x, 1)) <=> s(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(x))


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

They will be analysed ascendingly in the following order:
div = quot
div < encArg
quot < encArg

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(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(n3415_4)) -> gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(n3415_4), rt in Omega(0)

Induction Base:
encArg(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(+(n3415_4, 1))) ->_R^Omega(0)
s(encArg(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(n3415_4))) ->_IH
s(gen_0':s:cons_plus:cons_times:cons_div:cons_quot2_4(c3416_4))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(26)
BOUNDS(1, INF)