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


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 444 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:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   minus(minus(x, y), z) -> minus(x, plus(y, z))
   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
   plus(zero, y) -> y
   plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
   id(x) -> x
   if(true, x, y) -> x
   if(false, x, y) -> y
   not(x) -> if(x, false, true)
   gt(s(x), zero) -> true
   gt(zero, y) -> false
   gt(s(x), s(y)) -> gt(x, y)

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(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(zero) -> zero
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encArg(cons_id(x_1)) -> id(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_sum(x_1) -> sum(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_id(x_1) -> id(encArg(x_1))
   encode_zero -> zero
   encode_true -> true
   encode_false -> false


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

(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:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   minus(minus(x, y), z) -> minus(x, plus(y, z))
   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
   plus(zero, y) -> y
   plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
   id(x) -> x
   if(true, x, y) -> x
   if(false, x, y) -> y
   not(x) -> if(x, false, true)
   gt(s(x), zero) -> true
   gt(zero, y) -> false
   gt(s(x), s(y)) -> gt(x, y)

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(zero) -> zero
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encArg(cons_id(x_1)) -> id(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_sum(x_1) -> sum(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_id(x_1) -> id(encArg(x_1))
   encode_zero -> zero
   encode_true -> true
   encode_false -> false

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:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   minus(minus(x, y), z) -> minus(x, plus(y, z))
   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
   plus(zero, y) -> y
   plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
   id(x) -> x
   if(true, x, y) -> x
   if(false, x, y) -> y
   not(x) -> if(x, false, true)
   gt(s(x), zero) -> true
   gt(zero, y) -> false
   gt(s(x), s(y)) -> gt(x, y)

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(zero) -> zero
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encArg(cons_id(x_1)) -> id(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_sum(x_1) -> sum(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_id(x_1) -> id(encArg(x_1))
   encode_zero -> zero
   encode_true -> true
   encode_false -> false

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:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   minus(minus(x, y), z) -> minus(x, plus(y, z))
   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
   plus(zero, y) -> y
   plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
   id(x) -> x
   if(true, x, y) -> x
   if(false, x, y) -> y
   not(x) -> if(x, false, true)
   gt(s(x), zero) -> true
   gt(zero, y) -> false
   gt(s(x), s(y)) -> gt(x, y)

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(zero) -> zero
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encArg(cons_id(x_1)) -> id(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_sum(x_1) -> sum(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_id(x_1) -> id(encArg(x_1))
   encode_zero -> zero
   encode_true -> true
   encode_false -> false

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

app(cons(x, l), k) ->^+ cons(x, app(l, k))

gives rise to a decreasing loop by considering the right hand sides subterm at position [1].

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

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:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   minus(minus(x, y), z) -> minus(x, plus(y, z))
   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
   plus(zero, y) -> y
   plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
   id(x) -> x
   if(true, x, y) -> x
   if(false, x, y) -> y
   not(x) -> if(x, false, true)
   gt(s(x), zero) -> true
   gt(zero, y) -> false
   gt(s(x), s(y)) -> gt(x, y)

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(zero) -> zero
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encArg(cons_id(x_1)) -> id(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_sum(x_1) -> sum(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_id(x_1) -> id(encArg(x_1))
   encode_zero -> zero
   encode_true -> true
   encode_false -> false

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

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

(11)
BOUNDS(n^1, INF)

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

(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:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   minus(minus(x, y), z) -> minus(x, plus(y, z))
   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
   plus(zero, y) -> y
   plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
   id(x) -> x
   if(true, x, y) -> x
   if(false, x, y) -> y
   not(x) -> if(x, false, true)
   gt(s(x), zero) -> true
   gt(zero, y) -> false
   gt(s(x), s(y)) -> gt(x, y)

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(nil) -> nil
   encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2))
   encArg(zero) -> zero
   encArg(true) -> true
   encArg(false) -> false
   encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2))
   encArg(cons_quot(x_1, x_2)) -> quot(encArg(x_1), encArg(x_2))
   encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2))
   encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2))
   encArg(cons_sum(x_1)) -> sum(encArg(x_1))
   encArg(cons_id(x_1)) -> id(encArg(x_1))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_not(x_1)) -> not(encArg(x_1))
   encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2))
   encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_quot(x_1, x_2) -> quot(encArg(x_1), encArg(x_2))
   encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2))
   encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2))
   encode_nil -> nil
   encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2))
   encode_sum(x_1) -> sum(encArg(x_1))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2))
   encode_not(x_1) -> not(encArg(x_1))
   encode_id(x_1) -> id(encArg(x_1))
   encode_zero -> zero
   encode_true -> true
   encode_false -> false

Rewrite Strategy: INNERMOST