/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 634 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) TRS for Loop Detection


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

(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:

   car(cons(x, l)) -> x
   cddr(nil) -> nil
   cddr(cons(x, nil)) -> nil
   cddr(cons(x, cons(y, l))) -> l
   cadr(cons(x, cons(y, l))) -> y
   isZero(0) -> true
   isZero(s(x)) -> false
   plus(x, y) -> ifplus(isZero(x), x, y)
   ifplus(true, x, y) -> y
   ifplus(false, x, y) -> s(plus(p(x), y))
   times(x, y) -> iftimes(isZero(x), x, y)
   iftimes(true, x, y) -> 0
   iftimes(false, x, y) -> plus(y, times(p(x), y))
   p(s(x)) -> x
   p(0) -> 0
   shorter(nil, y) -> true
   shorter(cons(x, l), 0) -> false
   shorter(cons(x, l), s(y)) -> shorter(l, y)
   prod(l) -> if(shorter(l, 0), shorter(l, s(0)), l)
   if(true, b, l) -> s(0)
   if(false, b, l) -> if2(b, l)
   if2(true, l) -> car(l)
   if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l)))

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(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(nil) -> nil
   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(cons_car(x_1)) -> car(encArg(x_1))
   encArg(cons_cddr(x_1)) -> cddr(encArg(x_1))
   encArg(cons_cadr(x_1)) -> cadr(encArg(x_1))
   encArg(cons_isZero(x_1)) -> isZero(encArg(x_1))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_ifplus(x_1, x_2, x_3)) -> ifplus(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_iftimes(x_1, x_2, x_3)) -> iftimes(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_shorter(x_1, x_2)) -> shorter(encArg(x_1), encArg(x_2))
   encArg(cons_prod(x_1)) -> prod(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_if2(x_1, x_2)) -> if2(encArg(x_1), encArg(x_2))
   encode_car(x_1) -> car(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_cddr(x_1) -> cddr(encArg(x_1))
   encode_nil -> nil
   encode_cadr(x_1) -> cadr(encArg(x_1))
   encode_isZero(x_1) -> isZero(encArg(x_1))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_ifplus(x_1, x_2, x_3) -> ifplus(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_p(x_1) -> p(encArg(x_1))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_iftimes(x_1, x_2, x_3) -> iftimes(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_shorter(x_1, x_2) -> shorter(encArg(x_1), encArg(x_2))
   encode_prod(x_1) -> prod(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if2(x_1, x_2) -> if2(encArg(x_1), encArg(x_2))


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

(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:

   car(cons(x, l)) -> x
   cddr(nil) -> nil
   cddr(cons(x, nil)) -> nil
   cddr(cons(x, cons(y, l))) -> l
   cadr(cons(x, cons(y, l))) -> y
   isZero(0) -> true
   isZero(s(x)) -> false
   plus(x, y) -> ifplus(isZero(x), x, y)
   ifplus(true, x, y) -> y
   ifplus(false, x, y) -> s(plus(p(x), y))
   times(x, y) -> iftimes(isZero(x), x, y)
   iftimes(true, x, y) -> 0
   iftimes(false, x, y) -> plus(y, times(p(x), y))
   p(s(x)) -> x
   p(0) -> 0
   shorter(nil, y) -> true
   shorter(cons(x, l), 0) -> false
   shorter(cons(x, l), s(y)) -> shorter(l, y)
   prod(l) -> if(shorter(l, 0), shorter(l, s(0)), l)
   if(true, b, l) -> s(0)
   if(false, b, l) -> if2(b, l)
   if2(true, l) -> car(l)
   if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l)))

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

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(nil) -> nil
   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(cons_car(x_1)) -> car(encArg(x_1))
   encArg(cons_cddr(x_1)) -> cddr(encArg(x_1))
   encArg(cons_cadr(x_1)) -> cadr(encArg(x_1))
   encArg(cons_isZero(x_1)) -> isZero(encArg(x_1))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_ifplus(x_1, x_2, x_3)) -> ifplus(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_iftimes(x_1, x_2, x_3)) -> iftimes(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_shorter(x_1, x_2)) -> shorter(encArg(x_1), encArg(x_2))
   encArg(cons_prod(x_1)) -> prod(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_if2(x_1, x_2)) -> if2(encArg(x_1), encArg(x_2))
   encode_car(x_1) -> car(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_cddr(x_1) -> cddr(encArg(x_1))
   encode_nil -> nil
   encode_cadr(x_1) -> cadr(encArg(x_1))
   encode_isZero(x_1) -> isZero(encArg(x_1))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_ifplus(x_1, x_2, x_3) -> ifplus(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_p(x_1) -> p(encArg(x_1))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_iftimes(x_1, x_2, x_3) -> iftimes(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_shorter(x_1, x_2) -> shorter(encArg(x_1), encArg(x_2))
   encode_prod(x_1) -> prod(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if2(x_1, x_2) -> if2(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^1, INF).


The TRS R consists of the following rules:

   car(cons(x, l)) -> x
   cddr(nil) -> nil
   cddr(cons(x, nil)) -> nil
   cddr(cons(x, cons(y, l))) -> l
   cadr(cons(x, cons(y, l))) -> y
   isZero(0) -> true
   isZero(s(x)) -> false
   plus(x, y) -> ifplus(isZero(x), x, y)
   ifplus(true, x, y) -> y
   ifplus(false, x, y) -> s(plus(p(x), y))
   times(x, y) -> iftimes(isZero(x), x, y)
   iftimes(true, x, y) -> 0
   iftimes(false, x, y) -> plus(y, times(p(x), y))
   p(s(x)) -> x
   p(0) -> 0
   shorter(nil, y) -> true
   shorter(cons(x, l), 0) -> false
   shorter(cons(x, l), s(y)) -> shorter(l, y)
   prod(l) -> if(shorter(l, 0), shorter(l, s(0)), l)
   if(true, b, l) -> s(0)
   if(false, b, l) -> if2(b, l)
   if2(true, l) -> car(l)
   if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l)))

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

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(nil) -> nil
   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(cons_car(x_1)) -> car(encArg(x_1))
   encArg(cons_cddr(x_1)) -> cddr(encArg(x_1))
   encArg(cons_cadr(x_1)) -> cadr(encArg(x_1))
   encArg(cons_isZero(x_1)) -> isZero(encArg(x_1))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_ifplus(x_1, x_2, x_3)) -> ifplus(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_iftimes(x_1, x_2, x_3)) -> iftimes(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_shorter(x_1, x_2)) -> shorter(encArg(x_1), encArg(x_2))
   encArg(cons_prod(x_1)) -> prod(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_if2(x_1, x_2)) -> if2(encArg(x_1), encArg(x_2))
   encode_car(x_1) -> car(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_cddr(x_1) -> cddr(encArg(x_1))
   encode_nil -> nil
   encode_cadr(x_1) -> cadr(encArg(x_1))
   encode_isZero(x_1) -> isZero(encArg(x_1))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_ifplus(x_1, x_2, x_3) -> ifplus(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_p(x_1) -> p(encArg(x_1))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_iftimes(x_1, x_2, x_3) -> iftimes(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_shorter(x_1, x_2) -> shorter(encArg(x_1), encArg(x_2))
   encode_prod(x_1) -> prod(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if2(x_1, x_2) -> if2(encArg(x_1), encArg(x_2))

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

(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(6)
Obligation:
Analyzing the following TRS for decreasing loops:

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:

   car(cons(x, l)) -> x
   cddr(nil) -> nil
   cddr(cons(x, nil)) -> nil
   cddr(cons(x, cons(y, l))) -> l
   cadr(cons(x, cons(y, l))) -> y
   isZero(0) -> true
   isZero(s(x)) -> false
   plus(x, y) -> ifplus(isZero(x), x, y)
   ifplus(true, x, y) -> y
   ifplus(false, x, y) -> s(plus(p(x), y))
   times(x, y) -> iftimes(isZero(x), x, y)
   iftimes(true, x, y) -> 0
   iftimes(false, x, y) -> plus(y, times(p(x), y))
   p(s(x)) -> x
   p(0) -> 0
   shorter(nil, y) -> true
   shorter(cons(x, l), 0) -> false
   shorter(cons(x, l), s(y)) -> shorter(l, y)
   prod(l) -> if(shorter(l, 0), shorter(l, s(0)), l)
   if(true, b, l) -> s(0)
   if(false, b, l) -> if2(b, l)
   if2(true, l) -> car(l)
   if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l)))

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

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(nil) -> nil
   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(cons_car(x_1)) -> car(encArg(x_1))
   encArg(cons_cddr(x_1)) -> cddr(encArg(x_1))
   encArg(cons_cadr(x_1)) -> cadr(encArg(x_1))
   encArg(cons_isZero(x_1)) -> isZero(encArg(x_1))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_ifplus(x_1, x_2, x_3)) -> ifplus(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_iftimes(x_1, x_2, x_3)) -> iftimes(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_shorter(x_1, x_2)) -> shorter(encArg(x_1), encArg(x_2))
   encArg(cons_prod(x_1)) -> prod(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_if2(x_1, x_2)) -> if2(encArg(x_1), encArg(x_2))
   encode_car(x_1) -> car(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_cddr(x_1) -> cddr(encArg(x_1))
   encode_nil -> nil
   encode_cadr(x_1) -> cadr(encArg(x_1))
   encode_isZero(x_1) -> isZero(encArg(x_1))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_ifplus(x_1, x_2, x_3) -> ifplus(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_p(x_1) -> p(encArg(x_1))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_iftimes(x_1, x_2, x_3) -> iftimes(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_shorter(x_1, x_2) -> shorter(encArg(x_1), encArg(x_2))
   encode_prod(x_1) -> prod(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if2(x_1, x_2) -> if2(encArg(x_1), encArg(x_2))

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

(7) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

shorter(cons(x, l), s(y)) ->^+ shorter(l, y)

gives rise to a decreasing loop by considering the right hand sides subterm at position [].

The pumping substitution is [l / cons(x, l), y / s(y)].

The result substitution is [ ].




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

(8)
Complex Obligation (BEST)

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

(9)
Obligation:
Proved the lower bound n^1 for the following 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:

   car(cons(x, l)) -> x
   cddr(nil) -> nil
   cddr(cons(x, nil)) -> nil
   cddr(cons(x, cons(y, l))) -> l
   cadr(cons(x, cons(y, l))) -> y
   isZero(0) -> true
   isZero(s(x)) -> false
   plus(x, y) -> ifplus(isZero(x), x, y)
   ifplus(true, x, y) -> y
   ifplus(false, x, y) -> s(plus(p(x), y))
   times(x, y) -> iftimes(isZero(x), x, y)
   iftimes(true, x, y) -> 0
   iftimes(false, x, y) -> plus(y, times(p(x), y))
   p(s(x)) -> x
   p(0) -> 0
   shorter(nil, y) -> true
   shorter(cons(x, l), 0) -> false
   shorter(cons(x, l), s(y)) -> shorter(l, y)
   prod(l) -> if(shorter(l, 0), shorter(l, s(0)), l)
   if(true, b, l) -> s(0)
   if(false, b, l) -> if2(b, l)
   if2(true, l) -> car(l)
   if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l)))

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

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(nil) -> nil
   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(cons_car(x_1)) -> car(encArg(x_1))
   encArg(cons_cddr(x_1)) -> cddr(encArg(x_1))
   encArg(cons_cadr(x_1)) -> cadr(encArg(x_1))
   encArg(cons_isZero(x_1)) -> isZero(encArg(x_1))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_ifplus(x_1, x_2, x_3)) -> ifplus(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_iftimes(x_1, x_2, x_3)) -> iftimes(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_shorter(x_1, x_2)) -> shorter(encArg(x_1), encArg(x_2))
   encArg(cons_prod(x_1)) -> prod(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_if2(x_1, x_2)) -> if2(encArg(x_1), encArg(x_2))
   encode_car(x_1) -> car(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_cddr(x_1) -> cddr(encArg(x_1))
   encode_nil -> nil
   encode_cadr(x_1) -> cadr(encArg(x_1))
   encode_isZero(x_1) -> isZero(encArg(x_1))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_ifplus(x_1, x_2, x_3) -> ifplus(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_p(x_1) -> p(encArg(x_1))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_iftimes(x_1, x_2, x_3) -> iftimes(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_shorter(x_1, x_2) -> shorter(encArg(x_1), encArg(x_2))
   encode_prod(x_1) -> prod(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if2(x_1, x_2) -> if2(encArg(x_1), encArg(x_2))

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

(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(11)
BOUNDS(n^1, INF)

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(12)
Obligation:
Analyzing the following TRS for decreasing loops:

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:

   car(cons(x, l)) -> x
   cddr(nil) -> nil
   cddr(cons(x, nil)) -> nil
   cddr(cons(x, cons(y, l))) -> l
   cadr(cons(x, cons(y, l))) -> y
   isZero(0) -> true
   isZero(s(x)) -> false
   plus(x, y) -> ifplus(isZero(x), x, y)
   ifplus(true, x, y) -> y
   ifplus(false, x, y) -> s(plus(p(x), y))
   times(x, y) -> iftimes(isZero(x), x, y)
   iftimes(true, x, y) -> 0
   iftimes(false, x, y) -> plus(y, times(p(x), y))
   p(s(x)) -> x
   p(0) -> 0
   shorter(nil, y) -> true
   shorter(cons(x, l), 0) -> false
   shorter(cons(x, l), s(y)) -> shorter(l, y)
   prod(l) -> if(shorter(l, 0), shorter(l, s(0)), l)
   if(true, b, l) -> s(0)
   if(false, b, l) -> if2(b, l)
   if2(true, l) -> car(l)
   if2(false, l) -> prod(cons(times(car(l), cadr(l)), cddr(l)))

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

   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(nil) -> nil
   encArg(0) -> 0
   encArg(true) -> true
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(false) -> false
   encArg(cons_car(x_1)) -> car(encArg(x_1))
   encArg(cons_cddr(x_1)) -> cddr(encArg(x_1))
   encArg(cons_cadr(x_1)) -> cadr(encArg(x_1))
   encArg(cons_isZero(x_1)) -> isZero(encArg(x_1))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_ifplus(x_1, x_2, x_3)) -> ifplus(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2))
   encArg(cons_iftimes(x_1, x_2, x_3)) -> iftimes(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_shorter(x_1, x_2)) -> shorter(encArg(x_1), encArg(x_2))
   encArg(cons_prod(x_1)) -> prod(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_if2(x_1, x_2)) -> if2(encArg(x_1), encArg(x_2))
   encode_car(x_1) -> car(encArg(x_1))
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_cddr(x_1) -> cddr(encArg(x_1))
   encode_nil -> nil
   encode_cadr(x_1) -> cadr(encArg(x_1))
   encode_isZero(x_1) -> isZero(encArg(x_1))
   encode_0 -> 0
   encode_true -> true
   encode_s(x_1) -> s(encArg(x_1))
   encode_false -> false
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_ifplus(x_1, x_2, x_3) -> ifplus(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_p(x_1) -> p(encArg(x_1))
   encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2))
   encode_iftimes(x_1, x_2, x_3) -> iftimes(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_shorter(x_1, x_2) -> shorter(encArg(x_1), encArg(x_2))
   encode_prod(x_1) -> prod(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_if2(x_1, x_2) -> if2(encArg(x_1), encArg(x_2))

Rewrite Strategy: INNERMOST