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


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WORST_CASE(Omega(n^2), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2, INF).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 302 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 310 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), 66 ms]
        (18) BEST
            (19) proven lower bound
                (20) LowerBoundPropagationProof [FINISHED, 0 ms]
                (21) BOUNDS(n^2, INF)
            (22) typed CpxTrs
                (23) RewriteLemmaProof [LOWER BOUND(ID), 84 ms]
                (24) typed CpxTrs
                (25) RewriteLemmaProof [LOWER BOUND(ID), 2218 ms]
                (26) BOUNDS(1, INF)


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

(0)
Obligation:
The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   a__p(0) -> 0
   a__p(s(X)) -> mark(X)
   a__leq(0, Y) -> true
   a__leq(s(X), 0) -> false
   a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y))
   a__if(true, X, Y) -> mark(X)
   a__if(false, X, Y) -> mark(Y)
   a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0, s(diff(p(X), Y)))
   mark(p(X)) -> a__p(mark(X))
   mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2))
   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
   mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2))
   mark(0) -> 0
   mark(s(X)) -> s(mark(X))
   mark(true) -> true
   mark(false) -> false
   a__p(X) -> p(X)
   a__leq(X1, X2) -> leq(X1, X2)
   a__if(X1, X2, X3) -> if(X1, X2, X3)
   a__diff(X1, X2) -> diff(X1, X2)

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2))
   encArg(p(x_1)) -> p(encArg(x_1))
   encArg(leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2))
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__p(x_1)) -> a__p(encArg(x_1))
   encArg(cons_a__leq(x_1, x_2)) -> a__leq(encArg(x_1), encArg(x_2))
   encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__diff(x_1, x_2)) -> a__diff(encArg(x_1), encArg(x_2))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__p(x_1) -> a__p(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_a__leq(x_1, x_2) -> a__leq(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a__diff(x_1, x_2) -> a__diff(encArg(x_1), encArg(x_2))
   encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(encArg(x_1))
   encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))


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

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   a__p(0) -> 0
   a__p(s(X)) -> mark(X)
   a__leq(0, Y) -> true
   a__leq(s(X), 0) -> false
   a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y))
   a__if(true, X, Y) -> mark(X)
   a__if(false, X, Y) -> mark(Y)
   a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0, s(diff(p(X), Y)))
   mark(p(X)) -> a__p(mark(X))
   mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2))
   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
   mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2))
   mark(0) -> 0
   mark(s(X)) -> s(mark(X))
   mark(true) -> true
   mark(false) -> false
   a__p(X) -> p(X)
   a__leq(X1, X2) -> leq(X1, X2)
   a__if(X1, X2, X3) -> if(X1, X2, X3)
   a__diff(X1, X2) -> diff(X1, X2)

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2))
   encArg(p(x_1)) -> p(encArg(x_1))
   encArg(leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2))
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__p(x_1)) -> a__p(encArg(x_1))
   encArg(cons_a__leq(x_1, x_2)) -> a__leq(encArg(x_1), encArg(x_2))
   encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__diff(x_1, x_2)) -> a__diff(encArg(x_1), encArg(x_2))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__p(x_1) -> a__p(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_a__leq(x_1, x_2) -> a__leq(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a__diff(x_1, x_2) -> a__diff(encArg(x_1), encArg(x_2))
   encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(encArg(x_1))
   encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

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

(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   a__p(0) -> 0
   a__p(s(X)) -> mark(X)
   a__leq(0, Y) -> true
   a__leq(s(X), 0) -> false
   a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y))
   a__if(true, X, Y) -> mark(X)
   a__if(false, X, Y) -> mark(Y)
   a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0, s(diff(p(X), Y)))
   mark(p(X)) -> a__p(mark(X))
   mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2))
   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
   mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2))
   mark(0) -> 0
   mark(s(X)) -> s(mark(X))
   mark(true) -> true
   mark(false) -> false
   a__p(X) -> p(X)
   a__leq(X1, X2) -> leq(X1, X2)
   a__if(X1, X2, X3) -> if(X1, X2, X3)
   a__diff(X1, X2) -> diff(X1, X2)

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

   encArg(0) -> 0
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2))
   encArg(p(x_1)) -> p(encArg(x_1))
   encArg(leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2))
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__p(x_1)) -> a__p(encArg(x_1))
   encArg(cons_a__leq(x_1, x_2)) -> a__leq(encArg(x_1), encArg(x_2))
   encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__diff(x_1, x_2)) -> a__diff(encArg(x_1), encArg(x_2))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__p(x_1) -> a__p(encArg(x_1))
   encode_0 -> 0
   encode_s(x_1) -> s(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_a__leq(x_1, x_2) -> a__leq(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a__diff(x_1, x_2) -> a__diff(encArg(x_1), encArg(x_2))
   encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(encArg(x_1))
   encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

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

(5) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   a__p(0') -> 0'
   a__p(s(X)) -> mark(X)
   a__leq(0', Y) -> true
   a__leq(s(X), 0') -> false
   a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y))
   a__if(true, X, Y) -> mark(X)
   a__if(false, X, Y) -> mark(Y)
   a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0', s(diff(p(X), Y)))
   mark(p(X)) -> a__p(mark(X))
   mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2))
   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
   mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2))
   mark(0') -> 0'
   mark(s(X)) -> s(mark(X))
   mark(true) -> true
   mark(false) -> false
   a__p(X) -> p(X)
   a__leq(X1, X2) -> leq(X1, X2)
   a__if(X1, X2, X3) -> if(X1, X2, X3)
   a__diff(X1, X2) -> diff(X1, X2)

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

   encArg(0') -> 0'
   encArg(s(x_1)) -> s(encArg(x_1))
   encArg(true) -> true
   encArg(false) -> false
   encArg(diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2))
   encArg(p(x_1)) -> p(encArg(x_1))
   encArg(leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2))
   encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__p(x_1)) -> a__p(encArg(x_1))
   encArg(cons_a__leq(x_1, x_2)) -> a__leq(encArg(x_1), encArg(x_2))
   encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encArg(cons_a__diff(x_1, x_2)) -> a__diff(encArg(x_1), encArg(x_2))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__p(x_1) -> a__p(encArg(x_1))
   encode_0 -> 0'
   encode_s(x_1) -> s(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_a__leq(x_1, x_2) -> a__leq(encArg(x_1), encArg(x_2))
   encode_true -> true
   encode_false -> false
   encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
   encode_a__diff(x_1, x_2) -> a__diff(encArg(x_1), encArg(x_2))
   encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2))
   encode_p(x_1) -> p(encArg(x_1))
   encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2))
   encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
a__p(0') -> 0'
a__p(s(X)) -> mark(X)
a__leq(0', Y) -> true
a__leq(s(X), 0') -> false
a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y))
a__if(true, X, Y) -> mark(X)
a__if(false, X, Y) -> mark(Y)
a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0', s(diff(p(X), Y)))
mark(p(X)) -> a__p(mark(X))
mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2))
mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2))
mark(0') -> 0'
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false
a__p(X) -> p(X)
a__leq(X1, X2) -> leq(X1, X2)
a__if(X1, X2, X3) -> if(X1, X2, X3)
a__diff(X1, X2) -> diff(X1, X2)
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(true) -> true
encArg(false) -> false
encArg(diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2))
encArg(p(x_1)) -> p(encArg(x_1))
encArg(leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2))
encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__p(x_1)) -> a__p(encArg(x_1))
encArg(cons_a__leq(x_1, x_2)) -> a__leq(encArg(x_1), encArg(x_2))
encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__diff(x_1, x_2)) -> a__diff(encArg(x_1), encArg(x_2))
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encode_a__p(x_1) -> a__p(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_mark(x_1) -> mark(encArg(x_1))
encode_a__leq(x_1, x_2) -> a__leq(encArg(x_1), encArg(x_2))
encode_true -> true
encode_false -> false
encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_a__diff(x_1, x_2) -> a__diff(encArg(x_1), encArg(x_2))
encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2))
encode_p(x_1) -> p(encArg(x_1))
encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
a__p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
0' :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
s :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
mark :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
a__leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
true :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
false :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
a__if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
a__diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encArg :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_mark :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_0 :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_s :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_mark :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_true :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_false :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
hole_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark1_4 :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4 :: Nat -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
a__p, mark, a__leq, a__if, a__diff, encArg

They will be analysed ascendingly in the following order:
a__p = mark
a__p = a__leq
a__p = a__if
a__p = a__diff
a__p < encArg
mark = a__leq
mark = a__if
mark = a__diff
mark < encArg
a__leq = a__if
a__leq = a__diff
a__leq < encArg
a__if = a__diff
a__if < encArg
a__diff < encArg

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

(10)
Obligation:
Innermost TRS:
Rules:
a__p(0') -> 0'
a__p(s(X)) -> mark(X)
a__leq(0', Y) -> true
a__leq(s(X), 0') -> false
a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y))
a__if(true, X, Y) -> mark(X)
a__if(false, X, Y) -> mark(Y)
a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0', s(diff(p(X), Y)))
mark(p(X)) -> a__p(mark(X))
mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2))
mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2))
mark(0') -> 0'
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false
a__p(X) -> p(X)
a__leq(X1, X2) -> leq(X1, X2)
a__if(X1, X2, X3) -> if(X1, X2, X3)
a__diff(X1, X2) -> diff(X1, X2)
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(true) -> true
encArg(false) -> false
encArg(diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2))
encArg(p(x_1)) -> p(encArg(x_1))
encArg(leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2))
encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__p(x_1)) -> a__p(encArg(x_1))
encArg(cons_a__leq(x_1, x_2)) -> a__leq(encArg(x_1), encArg(x_2))
encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__diff(x_1, x_2)) -> a__diff(encArg(x_1), encArg(x_2))
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encode_a__p(x_1) -> a__p(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_mark(x_1) -> mark(encArg(x_1))
encode_a__leq(x_1, x_2) -> a__leq(encArg(x_1), encArg(x_2))
encode_true -> true
encode_false -> false
encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_a__diff(x_1, x_2) -> a__diff(encArg(x_1), encArg(x_2))
encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2))
encode_p(x_1) -> p(encArg(x_1))
encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
a__p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
0' :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
s :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
mark :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
a__leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
true :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
false :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
a__if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
a__diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encArg :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_mark :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_0 :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_s :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_mark :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_true :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_false :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
hole_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark1_4 :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4 :: Nat -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark


Generator Equations:
gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(0) <=> 0'
gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(+(x, 1)) <=> s(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(x))


The following defined symbols remain to be analysed:
mark, a__p, a__leq, a__if, a__diff, encArg

They will be analysed ascendingly in the following order:
a__p = mark
a__p = a__leq
a__p = a__if
a__p = a__diff
a__p < encArg
mark = a__leq
mark = a__if
mark = a__diff
mark < encArg
a__leq = a__if
a__leq = a__diff
a__leq < encArg
a__if = a__diff
a__if < encArg
a__diff < encArg

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n4_4)) -> gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n4_4), rt in Omega(1 + n4_4)

Induction Base:
mark(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(0)) ->_R^Omega(1)
0'

Induction Step:
mark(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(+(n4_4, 1))) ->_R^Omega(1)
s(mark(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n4_4))) ->_IH
s(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(c5_4))

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

(12)
Complex Obligation (BEST)

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

(13)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
a__p(0') -> 0'
a__p(s(X)) -> mark(X)
a__leq(0', Y) -> true
a__leq(s(X), 0') -> false
a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y))
a__if(true, X, Y) -> mark(X)
a__if(false, X, Y) -> mark(Y)
a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0', s(diff(p(X), Y)))
mark(p(X)) -> a__p(mark(X))
mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2))
mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2))
mark(0') -> 0'
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false
a__p(X) -> p(X)
a__leq(X1, X2) -> leq(X1, X2)
a__if(X1, X2, X3) -> if(X1, X2, X3)
a__diff(X1, X2) -> diff(X1, X2)
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(true) -> true
encArg(false) -> false
encArg(diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2))
encArg(p(x_1)) -> p(encArg(x_1))
encArg(leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2))
encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__p(x_1)) -> a__p(encArg(x_1))
encArg(cons_a__leq(x_1, x_2)) -> a__leq(encArg(x_1), encArg(x_2))
encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__diff(x_1, x_2)) -> a__diff(encArg(x_1), encArg(x_2))
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encode_a__p(x_1) -> a__p(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_mark(x_1) -> mark(encArg(x_1))
encode_a__leq(x_1, x_2) -> a__leq(encArg(x_1), encArg(x_2))
encode_true -> true
encode_false -> false
encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_a__diff(x_1, x_2) -> a__diff(encArg(x_1), encArg(x_2))
encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2))
encode_p(x_1) -> p(encArg(x_1))
encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
a__p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
0' :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
s :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
mark :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
a__leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
true :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
false :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
a__if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
a__diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encArg :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_mark :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_0 :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_s :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_mark :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_true :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_false :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
hole_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark1_4 :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4 :: Nat -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark


Generator Equations:
gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(0) <=> 0'
gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(+(x, 1)) <=> s(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(x))


The following defined symbols remain to be analysed:
mark, a__p, a__leq, a__if, a__diff, encArg

They will be analysed ascendingly in the following order:
a__p = mark
a__p = a__leq
a__p = a__if
a__p = a__diff
a__p < encArg
mark = a__leq
mark = a__if
mark = a__diff
mark < encArg
a__leq = a__if
a__leq = a__diff
a__leq < encArg
a__if = a__diff
a__if < encArg
a__diff < encArg

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
a__p(0') -> 0'
a__p(s(X)) -> mark(X)
a__leq(0', Y) -> true
a__leq(s(X), 0') -> false
a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y))
a__if(true, X, Y) -> mark(X)
a__if(false, X, Y) -> mark(Y)
a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0', s(diff(p(X), Y)))
mark(p(X)) -> a__p(mark(X))
mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2))
mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2))
mark(0') -> 0'
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false
a__p(X) -> p(X)
a__leq(X1, X2) -> leq(X1, X2)
a__if(X1, X2, X3) -> if(X1, X2, X3)
a__diff(X1, X2) -> diff(X1, X2)
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(true) -> true
encArg(false) -> false
encArg(diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2))
encArg(p(x_1)) -> p(encArg(x_1))
encArg(leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2))
encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__p(x_1)) -> a__p(encArg(x_1))
encArg(cons_a__leq(x_1, x_2)) -> a__leq(encArg(x_1), encArg(x_2))
encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__diff(x_1, x_2)) -> a__diff(encArg(x_1), encArg(x_2))
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encode_a__p(x_1) -> a__p(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_mark(x_1) -> mark(encArg(x_1))
encode_a__leq(x_1, x_2) -> a__leq(encArg(x_1), encArg(x_2))
encode_true -> true
encode_false -> false
encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_a__diff(x_1, x_2) -> a__diff(encArg(x_1), encArg(x_2))
encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2))
encode_p(x_1) -> p(encArg(x_1))
encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
a__p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
0' :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
s :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
mark :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
a__leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
true :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
false :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
a__if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
a__diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encArg :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_mark :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_0 :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_s :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_mark :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_true :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_false :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
hole_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark1_4 :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4 :: Nat -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark


Lemmas:
mark(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n4_4)) -> gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n4_4), rt in Omega(1 + n4_4)


Generator Equations:
gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(0) <=> 0'
gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(+(x, 1)) <=> s(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(x))


The following defined symbols remain to be analysed:
a__p, a__leq, a__if, a__diff, encArg

They will be analysed ascendingly in the following order:
a__p = mark
a__p = a__leq
a__p = a__if
a__p = a__diff
a__p < encArg
mark = a__leq
mark = a__if
mark = a__diff
mark < encArg
a__leq = a__if
a__leq = a__diff
a__leq < encArg
a__if = a__diff
a__if < encArg
a__diff < encArg

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
a__leq(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n1745_4), gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n1745_4)) -> true, rt in Omega(1 + n1745_4 + n1745_4^2)

Induction Base:
a__leq(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(0), gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(0)) ->_R^Omega(1)
true

Induction Step:
a__leq(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(+(n1745_4, 1)), gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(+(n1745_4, 1))) ->_R^Omega(1)
a__leq(mark(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n1745_4)), mark(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n1745_4))) ->_L^Omega(1 + n1745_4)
a__leq(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n1745_4), mark(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n1745_4))) ->_L^Omega(1 + n1745_4)
a__leq(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n1745_4), gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n1745_4)) ->_IH
true

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

(18)
Complex Obligation (BEST)

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

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

Innermost TRS:
Rules:
a__p(0') -> 0'
a__p(s(X)) -> mark(X)
a__leq(0', Y) -> true
a__leq(s(X), 0') -> false
a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y))
a__if(true, X, Y) -> mark(X)
a__if(false, X, Y) -> mark(Y)
a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0', s(diff(p(X), Y)))
mark(p(X)) -> a__p(mark(X))
mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2))
mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2))
mark(0') -> 0'
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false
a__p(X) -> p(X)
a__leq(X1, X2) -> leq(X1, X2)
a__if(X1, X2, X3) -> if(X1, X2, X3)
a__diff(X1, X2) -> diff(X1, X2)
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(true) -> true
encArg(false) -> false
encArg(diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2))
encArg(p(x_1)) -> p(encArg(x_1))
encArg(leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2))
encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__p(x_1)) -> a__p(encArg(x_1))
encArg(cons_a__leq(x_1, x_2)) -> a__leq(encArg(x_1), encArg(x_2))
encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__diff(x_1, x_2)) -> a__diff(encArg(x_1), encArg(x_2))
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encode_a__p(x_1) -> a__p(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_mark(x_1) -> mark(encArg(x_1))
encode_a__leq(x_1, x_2) -> a__leq(encArg(x_1), encArg(x_2))
encode_true -> true
encode_false -> false
encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_a__diff(x_1, x_2) -> a__diff(encArg(x_1), encArg(x_2))
encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2))
encode_p(x_1) -> p(encArg(x_1))
encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
a__p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
0' :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
s :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
mark :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
a__leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
true :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
false :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
a__if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
a__diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encArg :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_mark :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_0 :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_s :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_mark :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_true :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_false :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
hole_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark1_4 :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4 :: Nat -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark


Lemmas:
mark(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n4_4)) -> gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n4_4), rt in Omega(1 + n4_4)


Generator Equations:
gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(0) <=> 0'
gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(+(x, 1)) <=> s(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(x))


The following defined symbols remain to be analysed:
a__leq, a__if, a__diff, encArg

They will be analysed ascendingly in the following order:
a__p = mark
a__p = a__leq
a__p = a__if
a__p = a__diff
a__p < encArg
mark = a__leq
mark = a__if
mark = a__diff
mark < encArg
a__leq = a__if
a__leq = a__diff
a__leq < encArg
a__if = a__diff
a__if < encArg
a__diff < encArg

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

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

(21)
BOUNDS(n^2, INF)

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

(22)
Obligation:
Innermost TRS:
Rules:
a__p(0') -> 0'
a__p(s(X)) -> mark(X)
a__leq(0', Y) -> true
a__leq(s(X), 0') -> false
a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y))
a__if(true, X, Y) -> mark(X)
a__if(false, X, Y) -> mark(Y)
a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0', s(diff(p(X), Y)))
mark(p(X)) -> a__p(mark(X))
mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2))
mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2))
mark(0') -> 0'
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false
a__p(X) -> p(X)
a__leq(X1, X2) -> leq(X1, X2)
a__if(X1, X2, X3) -> if(X1, X2, X3)
a__diff(X1, X2) -> diff(X1, X2)
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(true) -> true
encArg(false) -> false
encArg(diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2))
encArg(p(x_1)) -> p(encArg(x_1))
encArg(leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2))
encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__p(x_1)) -> a__p(encArg(x_1))
encArg(cons_a__leq(x_1, x_2)) -> a__leq(encArg(x_1), encArg(x_2))
encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__diff(x_1, x_2)) -> a__diff(encArg(x_1), encArg(x_2))
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encode_a__p(x_1) -> a__p(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_mark(x_1) -> mark(encArg(x_1))
encode_a__leq(x_1, x_2) -> a__leq(encArg(x_1), encArg(x_2))
encode_true -> true
encode_false -> false
encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_a__diff(x_1, x_2) -> a__diff(encArg(x_1), encArg(x_2))
encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2))
encode_p(x_1) -> p(encArg(x_1))
encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
a__p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
0' :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
s :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
mark :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
a__leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
true :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
false :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
a__if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
a__diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encArg :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_mark :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_0 :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_s :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_mark :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_true :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_false :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
hole_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark1_4 :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4 :: Nat -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark


Lemmas:
mark(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n4_4)) -> gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n4_4), rt in Omega(1 + n4_4)
a__leq(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n1745_4), gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n1745_4)) -> true, rt in Omega(1 + n1745_4 + n1745_4^2)


Generator Equations:
gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(0) <=> 0'
gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(+(x, 1)) <=> s(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(x))


The following defined symbols remain to be analysed:
a__if, a__p, mark, a__diff, encArg

They will be analysed ascendingly in the following order:
a__p = mark
a__p = a__leq
a__p = a__if
a__p = a__diff
a__p < encArg
mark = a__leq
mark = a__if
mark = a__diff
mark < encArg
a__leq = a__if
a__leq = a__diff
a__leq < encArg
a__if = a__diff
a__if < encArg
a__diff < encArg

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n3493_4)) -> gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n3493_4), rt in Omega(1 + n3493_4)

Induction Base:
mark(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(0)) ->_R^Omega(1)
0'

Induction Step:
mark(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(+(n3493_4, 1))) ->_R^Omega(1)
s(mark(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n3493_4))) ->_IH
s(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(c3494_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:
a__p(0') -> 0'
a__p(s(X)) -> mark(X)
a__leq(0', Y) -> true
a__leq(s(X), 0') -> false
a__leq(s(X), s(Y)) -> a__leq(mark(X), mark(Y))
a__if(true, X, Y) -> mark(X)
a__if(false, X, Y) -> mark(Y)
a__diff(X, Y) -> a__if(a__leq(mark(X), mark(Y)), 0', s(diff(p(X), Y)))
mark(p(X)) -> a__p(mark(X))
mark(leq(X1, X2)) -> a__leq(mark(X1), mark(X2))
mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
mark(diff(X1, X2)) -> a__diff(mark(X1), mark(X2))
mark(0') -> 0'
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false
a__p(X) -> p(X)
a__leq(X1, X2) -> leq(X1, X2)
a__if(X1, X2, X3) -> if(X1, X2, X3)
a__diff(X1, X2) -> diff(X1, X2)
encArg(0') -> 0'
encArg(s(x_1)) -> s(encArg(x_1))
encArg(true) -> true
encArg(false) -> false
encArg(diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2))
encArg(p(x_1)) -> p(encArg(x_1))
encArg(leq(x_1, x_2)) -> leq(encArg(x_1), encArg(x_2))
encArg(if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__p(x_1)) -> a__p(encArg(x_1))
encArg(cons_a__leq(x_1, x_2)) -> a__leq(encArg(x_1), encArg(x_2))
encArg(cons_a__if(x_1, x_2, x_3)) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
encArg(cons_a__diff(x_1, x_2)) -> a__diff(encArg(x_1), encArg(x_2))
encArg(cons_mark(x_1)) -> mark(encArg(x_1))
encode_a__p(x_1) -> a__p(encArg(x_1))
encode_0 -> 0'
encode_s(x_1) -> s(encArg(x_1))
encode_mark(x_1) -> mark(encArg(x_1))
encode_a__leq(x_1, x_2) -> a__leq(encArg(x_1), encArg(x_2))
encode_true -> true
encode_false -> false
encode_a__if(x_1, x_2, x_3) -> a__if(encArg(x_1), encArg(x_2), encArg(x_3))
encode_a__diff(x_1, x_2) -> a__diff(encArg(x_1), encArg(x_2))
encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2))
encode_p(x_1) -> p(encArg(x_1))
encode_leq(x_1, x_2) -> leq(encArg(x_1), encArg(x_2))
encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3))

Types:
a__p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
0' :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
s :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
mark :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
a__leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
true :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
false :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
a__if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
a__diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encArg :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_a__diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
cons_mark :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_0 :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_s :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_mark :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_true :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_false :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_a__diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_diff :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_p :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_leq :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
encode_if :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
hole_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark1_4 :: 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark
gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4 :: Nat -> 0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark


Lemmas:
mark(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n3493_4)) -> gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n3493_4), rt in Omega(1 + n3493_4)
a__leq(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n1745_4), gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n1745_4)) -> true, rt in Omega(1 + n1745_4 + n1745_4^2)


Generator Equations:
gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(0) <=> 0'
gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(+(x, 1)) <=> s(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(x))


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

They will be analysed ascendingly in the following order:
a__p = mark
a__p = a__leq
a__p = a__if
a__p = a__diff
a__p < encArg
mark = a__leq
mark = a__if
mark = a__diff
mark < encArg
a__leq = a__if
a__leq = a__diff
a__leq < encArg
a__if = a__diff
a__if < encArg
a__diff < encArg

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
encArg(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n5238_4)) -> gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n5238_4), rt in Omega(0)

Induction Base:
encArg(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(0)) ->_R^Omega(0)
0'

Induction Step:
encArg(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(+(n5238_4, 1))) ->_R^Omega(0)
s(encArg(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(n5238_4))) ->_IH
s(gen_0':s:true:false:p:diff:leq:if:cons_a__p:cons_a__leq:cons_a__if:cons_a__diff:cons_mark2_4(c5239_4))

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

(26)
BOUNDS(1, INF)