/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 290 ms]
(4) CpxRelTRS
(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(6) TRS for Loop Detection
(7) DecreasingLoopProof [LOWER BOUND(ID), 8 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:

   active(p(0)) -> mark(0)
   active(p(s(X))) -> mark(X)
   active(leq(0, Y)) -> mark(true)
   active(leq(s(X), 0)) -> mark(false)
   active(leq(s(X), s(Y))) -> mark(leq(X, Y))
   active(if(true, X, Y)) -> mark(X)
   active(if(false, X, Y)) -> mark(Y)
   active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
   active(p(X)) -> p(active(X))
   active(s(X)) -> s(active(X))
   active(leq(X1, X2)) -> leq(active(X1), X2)
   active(leq(X1, X2)) -> leq(X1, active(X2))
   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
   active(diff(X1, X2)) -> diff(active(X1), X2)
   active(diff(X1, X2)) -> diff(X1, active(X2))
   p(mark(X)) -> mark(p(X))
   s(mark(X)) -> mark(s(X))
   leq(mark(X1), X2) -> mark(leq(X1, X2))
   leq(X1, mark(X2)) -> mark(leq(X1, X2))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   diff(mark(X1), X2) -> mark(diff(X1, X2))
   diff(X1, mark(X2)) -> mark(diff(X1, X2))
   proper(p(X)) -> p(proper(X))
   proper(0) -> ok(0)
   proper(s(X)) -> s(proper(X))
   proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
   proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
   p(ok(X)) -> ok(p(X))
   s(ok(X)) -> ok(s(X))
   leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(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(mark(x_1)) -> mark(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_0 -> 0
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2))
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))


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

(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:

   active(p(0)) -> mark(0)
   active(p(s(X))) -> mark(X)
   active(leq(0, Y)) -> mark(true)
   active(leq(s(X), 0)) -> mark(false)
   active(leq(s(X), s(Y))) -> mark(leq(X, Y))
   active(if(true, X, Y)) -> mark(X)
   active(if(false, X, Y)) -> mark(Y)
   active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
   active(p(X)) -> p(active(X))
   active(s(X)) -> s(active(X))
   active(leq(X1, X2)) -> leq(active(X1), X2)
   active(leq(X1, X2)) -> leq(X1, active(X2))
   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
   active(diff(X1, X2)) -> diff(active(X1), X2)
   active(diff(X1, X2)) -> diff(X1, active(X2))
   p(mark(X)) -> mark(p(X))
   s(mark(X)) -> mark(s(X))
   leq(mark(X1), X2) -> mark(leq(X1, X2))
   leq(X1, mark(X2)) -> mark(leq(X1, X2))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   diff(mark(X1), X2) -> mark(diff(X1, X2))
   diff(X1, mark(X2)) -> mark(diff(X1, X2))
   proper(p(X)) -> p(proper(X))
   proper(0) -> ok(0)
   proper(s(X)) -> s(proper(X))
   proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
   proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
   p(ok(X)) -> ok(p(X))
   s(ok(X)) -> ok(s(X))
   leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

   encArg(0) -> 0
   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_0 -> 0
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2))
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))

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:

   active(p(0)) -> mark(0)
   active(p(s(X))) -> mark(X)
   active(leq(0, Y)) -> mark(true)
   active(leq(s(X), 0)) -> mark(false)
   active(leq(s(X), s(Y))) -> mark(leq(X, Y))
   active(if(true, X, Y)) -> mark(X)
   active(if(false, X, Y)) -> mark(Y)
   active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
   active(p(X)) -> p(active(X))
   active(s(X)) -> s(active(X))
   active(leq(X1, X2)) -> leq(active(X1), X2)
   active(leq(X1, X2)) -> leq(X1, active(X2))
   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
   active(diff(X1, X2)) -> diff(active(X1), X2)
   active(diff(X1, X2)) -> diff(X1, active(X2))
   p(mark(X)) -> mark(p(X))
   s(mark(X)) -> mark(s(X))
   leq(mark(X1), X2) -> mark(leq(X1, X2))
   leq(X1, mark(X2)) -> mark(leq(X1, X2))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   diff(mark(X1), X2) -> mark(diff(X1, X2))
   diff(X1, mark(X2)) -> mark(diff(X1, X2))
   proper(p(X)) -> p(proper(X))
   proper(0) -> ok(0)
   proper(s(X)) -> s(proper(X))
   proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
   proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
   p(ok(X)) -> ok(p(X))
   s(ok(X)) -> ok(s(X))
   leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

   encArg(0) -> 0
   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_0 -> 0
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2))
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))

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:

   active(p(0)) -> mark(0)
   active(p(s(X))) -> mark(X)
   active(leq(0, Y)) -> mark(true)
   active(leq(s(X), 0)) -> mark(false)
   active(leq(s(X), s(Y))) -> mark(leq(X, Y))
   active(if(true, X, Y)) -> mark(X)
   active(if(false, X, Y)) -> mark(Y)
   active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
   active(p(X)) -> p(active(X))
   active(s(X)) -> s(active(X))
   active(leq(X1, X2)) -> leq(active(X1), X2)
   active(leq(X1, X2)) -> leq(X1, active(X2))
   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
   active(diff(X1, X2)) -> diff(active(X1), X2)
   active(diff(X1, X2)) -> diff(X1, active(X2))
   p(mark(X)) -> mark(p(X))
   s(mark(X)) -> mark(s(X))
   leq(mark(X1), X2) -> mark(leq(X1, X2))
   leq(X1, mark(X2)) -> mark(leq(X1, X2))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   diff(mark(X1), X2) -> mark(diff(X1, X2))
   diff(X1, mark(X2)) -> mark(diff(X1, X2))
   proper(p(X)) -> p(proper(X))
   proper(0) -> ok(0)
   proper(s(X)) -> s(proper(X))
   proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
   proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
   p(ok(X)) -> ok(p(X))
   s(ok(X)) -> ok(s(X))
   leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

   encArg(0) -> 0
   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_0 -> 0
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2))
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))

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

p(mark(X)) ->^+ mark(p(X))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [X / mark(X)].

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:

   active(p(0)) -> mark(0)
   active(p(s(X))) -> mark(X)
   active(leq(0, Y)) -> mark(true)
   active(leq(s(X), 0)) -> mark(false)
   active(leq(s(X), s(Y))) -> mark(leq(X, Y))
   active(if(true, X, Y)) -> mark(X)
   active(if(false, X, Y)) -> mark(Y)
   active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
   active(p(X)) -> p(active(X))
   active(s(X)) -> s(active(X))
   active(leq(X1, X2)) -> leq(active(X1), X2)
   active(leq(X1, X2)) -> leq(X1, active(X2))
   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
   active(diff(X1, X2)) -> diff(active(X1), X2)
   active(diff(X1, X2)) -> diff(X1, active(X2))
   p(mark(X)) -> mark(p(X))
   s(mark(X)) -> mark(s(X))
   leq(mark(X1), X2) -> mark(leq(X1, X2))
   leq(X1, mark(X2)) -> mark(leq(X1, X2))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   diff(mark(X1), X2) -> mark(diff(X1, X2))
   diff(X1, mark(X2)) -> mark(diff(X1, X2))
   proper(p(X)) -> p(proper(X))
   proper(0) -> ok(0)
   proper(s(X)) -> s(proper(X))
   proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
   proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
   p(ok(X)) -> ok(p(X))
   s(ok(X)) -> ok(s(X))
   leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

   encArg(0) -> 0
   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_0 -> 0
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2))
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))

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:

   active(p(0)) -> mark(0)
   active(p(s(X))) -> mark(X)
   active(leq(0, Y)) -> mark(true)
   active(leq(s(X), 0)) -> mark(false)
   active(leq(s(X), s(Y))) -> mark(leq(X, Y))
   active(if(true, X, Y)) -> mark(X)
   active(if(false, X, Y)) -> mark(Y)
   active(diff(X, Y)) -> mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
   active(p(X)) -> p(active(X))
   active(s(X)) -> s(active(X))
   active(leq(X1, X2)) -> leq(active(X1), X2)
   active(leq(X1, X2)) -> leq(X1, active(X2))
   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
   active(diff(X1, X2)) -> diff(active(X1), X2)
   active(diff(X1, X2)) -> diff(X1, active(X2))
   p(mark(X)) -> mark(p(X))
   s(mark(X)) -> mark(s(X))
   leq(mark(X1), X2) -> mark(leq(X1, X2))
   leq(X1, mark(X2)) -> mark(leq(X1, X2))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   diff(mark(X1), X2) -> mark(diff(X1, X2))
   diff(X1, mark(X2)) -> mark(diff(X1, X2))
   proper(p(X)) -> p(proper(X))
   proper(0) -> ok(0)
   proper(s(X)) -> s(proper(X))
   proper(leq(X1, X2)) -> leq(proper(X1), proper(X2))
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
   proper(diff(X1, X2)) -> diff(proper(X1), proper(X2))
   p(ok(X)) -> ok(p(X))
   s(ok(X)) -> ok(s(X))
   leq(ok(X1), ok(X2)) -> ok(leq(X1, X2))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   diff(ok(X1), ok(X2)) -> ok(diff(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

   encArg(0) -> 0
   encArg(mark(x_1)) -> mark(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(ok(x_1)) -> ok(encArg(x_1))
   encArg(cons_active(x_1)) -> active(encArg(x_1))
   encArg(cons_p(x_1)) -> p(encArg(x_1))
   encArg(cons_s(x_1)) -> s(encArg(x_1))
   encArg(cons_leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2))
   encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2))
   encArg(cons_proper(x_1)) -> proper(encArg(x_1))
   encArg(cons_top(x_1)) -> top(encArg(x_1))
   encode_active(x_1) -> active(encArg(x_1))
   encode_p(x_1) -> p(encArg(x_1))
   encode_0 -> 0
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_s(x_1) -> s(encArg(x_1))
   encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2))
   encode_proper(x_1) -> proper(encArg(x_1))
   encode_ok(x_1) -> ok(encArg(x_1))
   encode_top(x_1) -> top(encArg(x_1))

Rewrite Strategy: INNERMOST