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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 387 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 5 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 451 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), 163 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 93 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 124 ms]
        (22) typed CpxTrs
        (23) RewriteLemmaProof [LOWER BOUND(ID), 108 ms]
        (24) typed CpxTrs
        (25) RewriteLemmaProof [LOWER BOUND(ID), 2314 ms]
        (26) BOUNDS(1, INF)


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

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

(8)
Obligation:
Innermost TRS:
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))
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))

Types:
active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
0' :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
mark :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
true :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
false :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
ok :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encArg :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_0 :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_mark :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_true :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_false :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_ok :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
hole_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top1_4 :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4 :: Nat -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
active, leq, if, s, diff, p, proper, top, encArg

They will be analysed ascendingly in the following order:
leq < active
if < active
s < active
diff < active
p < active
active < top
active < encArg
leq < proper
leq < encArg
if < proper
if < encArg
s < proper
s < encArg
diff < proper
diff < encArg
p < proper
p < encArg
proper < top
proper < encArg
top < encArg

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

(10)
Obligation:
Innermost TRS:
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))
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))

Types:
active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
0' :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
mark :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
true :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
false :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
ok :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encArg :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_0 :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_mark :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_true :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_false :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_ok :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
hole_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top1_4 :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4 :: Nat -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top


Generator Equations:
gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(0) <=> 0'
gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(x, 1)) <=> mark(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(x))


The following defined symbols remain to be analysed:
leq, active, if, s, diff, p, proper, top, encArg

They will be analysed ascendingly in the following order:
leq < active
if < active
s < active
diff < active
p < active
active < top
active < encArg
leq < proper
leq < encArg
if < proper
if < encArg
s < proper
s < encArg
diff < proper
diff < encArg
p < proper
p < encArg
proper < top
proper < encArg
top < encArg

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
leq(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n4_4)), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(b)) -> *3_4, rt in Omega(n4_4)

Induction Base:
leq(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, 0)), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(b))

Induction Step:
leq(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, +(n4_4, 1))), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(b)) ->_R^Omega(1)
mark(leq(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n4_4)), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(b))) ->_IH
mark(*3_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:
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))
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))

Types:
active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
0' :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
mark :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
true :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
false :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
ok :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encArg :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_0 :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_mark :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_true :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_false :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_ok :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
hole_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top1_4 :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4 :: Nat -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top


Generator Equations:
gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(0) <=> 0'
gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(x, 1)) <=> mark(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(x))


The following defined symbols remain to be analysed:
leq, active, if, s, diff, p, proper, top, encArg

They will be analysed ascendingly in the following order:
leq < active
if < active
s < active
diff < active
p < active
active < top
active < encArg
leq < proper
leq < encArg
if < proper
if < encArg
s < proper
s < encArg
diff < proper
diff < encArg
p < proper
p < encArg
proper < top
proper < encArg
top < encArg

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
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))
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))

Types:
active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
0' :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
mark :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
true :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
false :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
ok :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encArg :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_0 :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_mark :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_true :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_false :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_ok :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
hole_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top1_4 :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4 :: Nat -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top


Lemmas:
leq(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n4_4)), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(b)) -> *3_4, rt in Omega(n4_4)


Generator Equations:
gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(0) <=> 0'
gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(x, 1)) <=> mark(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(x))


The following defined symbols remain to be analysed:
if, active, s, diff, p, proper, top, encArg

They will be analysed ascendingly in the following order:
if < active
s < active
diff < active
p < active
active < top
active < encArg
if < proper
if < encArg
s < proper
s < encArg
diff < proper
diff < encArg
p < proper
p < encArg
proper < top
proper < encArg
top < encArg

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
if(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n1772_4)), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(b), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(c)) -> *3_4, rt in Omega(n1772_4)

Induction Base:
if(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, 0)), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(b), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(c))

Induction Step:
if(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, +(n1772_4, 1))), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(b), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(c)) ->_R^Omega(1)
mark(if(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n1772_4)), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(b), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(c))) ->_IH
mark(*3_4)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(18)
Obligation:
Innermost TRS:
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))
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))

Types:
active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
0' :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
mark :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
true :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
false :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
ok :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encArg :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_0 :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_mark :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_true :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_false :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_ok :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
hole_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top1_4 :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4 :: Nat -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top


Lemmas:
leq(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n4_4)), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(b)) -> *3_4, rt in Omega(n4_4)
if(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n1772_4)), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(b), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(c)) -> *3_4, rt in Omega(n1772_4)


Generator Equations:
gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(0) <=> 0'
gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(x, 1)) <=> mark(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(x))


The following defined symbols remain to be analysed:
s, active, diff, p, proper, top, encArg

They will be analysed ascendingly in the following order:
s < active
diff < active
p < active
active < top
active < encArg
s < proper
s < encArg
diff < proper
diff < encArg
p < proper
p < encArg
proper < top
proper < encArg
top < encArg

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
s(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n4981_4))) -> *3_4, rt in Omega(n4981_4)

Induction Base:
s(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, 0)))

Induction Step:
s(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, +(n4981_4, 1)))) ->_R^Omega(1)
mark(s(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n4981_4)))) ->_IH
mark(*3_4)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(20)
Obligation:
Innermost TRS:
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))
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))

Types:
active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
0' :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
mark :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
true :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
false :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
ok :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encArg :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_0 :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_mark :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_true :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_false :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_ok :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
hole_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top1_4 :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4 :: Nat -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top


Lemmas:
leq(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n4_4)), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(b)) -> *3_4, rt in Omega(n4_4)
if(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n1772_4)), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(b), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(c)) -> *3_4, rt in Omega(n1772_4)
s(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n4981_4))) -> *3_4, rt in Omega(n4981_4)


Generator Equations:
gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(0) <=> 0'
gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(x, 1)) <=> mark(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(x))


The following defined symbols remain to be analysed:
diff, active, p, proper, top, encArg

They will be analysed ascendingly in the following order:
diff < active
p < active
active < top
active < encArg
diff < proper
diff < encArg
p < proper
p < encArg
proper < top
proper < encArg
top < encArg

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
diff(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n5977_4)), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(b)) -> *3_4, rt in Omega(n5977_4)

Induction Base:
diff(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, 0)), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(b))

Induction Step:
diff(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, +(n5977_4, 1))), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(b)) ->_R^Omega(1)
mark(diff(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n5977_4)), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(b))) ->_IH
mark(*3_4)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(22)
Obligation:
Innermost TRS:
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))
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))

Types:
active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
0' :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
mark :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
true :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
false :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
ok :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encArg :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_0 :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_mark :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_true :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_false :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_ok :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
hole_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top1_4 :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4 :: Nat -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top


Lemmas:
leq(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n4_4)), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(b)) -> *3_4, rt in Omega(n4_4)
if(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n1772_4)), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(b), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(c)) -> *3_4, rt in Omega(n1772_4)
s(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n4981_4))) -> *3_4, rt in Omega(n4981_4)
diff(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n5977_4)), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(b)) -> *3_4, rt in Omega(n5977_4)


Generator Equations:
gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(0) <=> 0'
gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(x, 1)) <=> mark(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(x))


The following defined symbols remain to be analysed:
p, active, proper, top, encArg

They will be analysed ascendingly in the following order:
p < active
active < top
active < encArg
p < proper
p < encArg
proper < top
proper < encArg
top < encArg

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
p(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n8657_4))) -> *3_4, rt in Omega(n8657_4)

Induction Base:
p(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, 0)))

Induction Step:
p(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, +(n8657_4, 1)))) ->_R^Omega(1)
mark(p(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n8657_4)))) ->_IH
mark(*3_4)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(24)
Obligation:
Innermost TRS:
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))
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))

Types:
active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
0' :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
mark :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
true :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
false :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
ok :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encArg :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
cons_top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_active :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_p :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_0 :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_mark :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_s :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_leq :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_true :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_false :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_if :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_diff :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_proper :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_ok :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
encode_top :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
hole_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top1_4 :: 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top
gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4 :: Nat -> 0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top


Lemmas:
leq(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n4_4)), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(b)) -> *3_4, rt in Omega(n4_4)
if(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n1772_4)), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(b), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(c)) -> *3_4, rt in Omega(n1772_4)
s(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n4981_4))) -> *3_4, rt in Omega(n4981_4)
diff(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n5977_4)), gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(b)) -> *3_4, rt in Omega(n5977_4)
p(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(1, n8657_4))) -> *3_4, rt in Omega(n8657_4)


Generator Equations:
gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(0) <=> 0'
gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(x, 1)) <=> mark(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(x))


The following defined symbols remain to be analysed:
active, proper, top, encArg

They will be analysed ascendingly in the following order:
active < top
active < encArg
proper < top
proper < encArg
top < encArg

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(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(n86276_4)) -> gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(n86276_4), rt in Omega(0)

Induction Base:
encArg(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(+(n86276_4, 1))) ->_R^Omega(0)
mark(encArg(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(n86276_4))) ->_IH
mark(gen_0':mark:true:false:ok:cons_active:cons_p:cons_s:cons_leq:cons_if:cons_diff:cons_proper:cons_top2_4(c86277_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)