/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^2), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 0 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 381 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs
        (13) RewriteLemmaProof [LOWER BOUND(ID), 64 ms]
        (14) BEST
            (15) proven lower bound
                (16) LowerBoundPropagationProof [FINISHED, 0 ms]
                (17) BOUNDS(n^2, INF)
            (18) typed CpxTrs
                (19) RewriteLemmaProof [LOWER BOUND(ID), 57 ms]
                (20) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS 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) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS 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
----------------------------------------

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

(4)
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)

Types:
a__p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
0' :: 0':s:true:false:p:diff:leq:if
s :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
mark :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
a__leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
true :: 0':s:true:false:p:diff:leq:if
false :: 0':s:true:false:p:diff:leq:if
a__if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
a__diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
hole_0':s:true:false:p:diff:leq:if1_0 :: 0':s:true:false:p:diff:leq:if
gen_0':s:true:false:p:diff:leq:if2_0 :: Nat -> 0':s:true:false:p:diff:leq:if

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

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

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

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

(6)
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)

Types:
a__p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
0' :: 0':s:true:false:p:diff:leq:if
s :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
mark :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
a__leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
true :: 0':s:true:false:p:diff:leq:if
false :: 0':s:true:false:p:diff:leq:if
a__if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
a__diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
hole_0':s:true:false:p:diff:leq:if1_0 :: 0':s:true:false:p:diff:leq:if
gen_0':s:true:false:p:diff:leq:if2_0 :: Nat -> 0':s:true:false:p:diff:leq:if


Generator Equations:
gen_0':s:true:false:p:diff:leq:if2_0(0) <=> 0'
gen_0':s:true:false:p:diff:leq:if2_0(+(x, 1)) <=> s(gen_0':s:true:false:p:diff:leq:if2_0(x))


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

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

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_0':s:true:false:p:diff:leq:if2_0(n4_0)) -> gen_0':s:true:false:p:diff:leq:if2_0(n4_0), rt in Omega(1 + n4_0)

Induction Base:
mark(gen_0':s:true:false:p:diff:leq:if2_0(0)) ->_R^Omega(1)
0'

Induction Step:
mark(gen_0':s:true:false:p:diff:leq:if2_0(+(n4_0, 1))) ->_R^Omega(1)
s(mark(gen_0':s:true:false:p:diff:leq:if2_0(n4_0))) ->_IH
s(gen_0':s:true:false:p:diff:leq:if2_0(c5_0))

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

(8)
Complex Obligation (BEST)

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

(9)
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)

Types:
a__p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
0' :: 0':s:true:false:p:diff:leq:if
s :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
mark :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
a__leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
true :: 0':s:true:false:p:diff:leq:if
false :: 0':s:true:false:p:diff:leq:if
a__if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
a__diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
hole_0':s:true:false:p:diff:leq:if1_0 :: 0':s:true:false:p:diff:leq:if
gen_0':s:true:false:p:diff:leq:if2_0 :: Nat -> 0':s:true:false:p:diff:leq:if


Generator Equations:
gen_0':s:true:false:p:diff:leq:if2_0(0) <=> 0'
gen_0':s:true:false:p:diff:leq:if2_0(+(x, 1)) <=> s(gen_0':s:true:false:p:diff:leq:if2_0(x))


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

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

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
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)

Types:
a__p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
0' :: 0':s:true:false:p:diff:leq:if
s :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
mark :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
a__leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
true :: 0':s:true:false:p:diff:leq:if
false :: 0':s:true:false:p:diff:leq:if
a__if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
a__diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
hole_0':s:true:false:p:diff:leq:if1_0 :: 0':s:true:false:p:diff:leq:if
gen_0':s:true:false:p:diff:leq:if2_0 :: Nat -> 0':s:true:false:p:diff:leq:if


Lemmas:
mark(gen_0':s:true:false:p:diff:leq:if2_0(n4_0)) -> gen_0':s:true:false:p:diff:leq:if2_0(n4_0), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s:true:false:p:diff:leq:if2_0(0) <=> 0'
gen_0':s:true:false:p:diff:leq:if2_0(+(x, 1)) <=> s(gen_0':s:true:false:p:diff:leq:if2_0(x))


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

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

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
a__leq(gen_0':s:true:false:p:diff:leq:if2_0(n1537_0), gen_0':s:true:false:p:diff:leq:if2_0(n1537_0)) -> true, rt in Omega(1 + n1537_0 + n1537_0^2)

Induction Base:
a__leq(gen_0':s:true:false:p:diff:leq:if2_0(0), gen_0':s:true:false:p:diff:leq:if2_0(0)) ->_R^Omega(1)
true

Induction Step:
a__leq(gen_0':s:true:false:p:diff:leq:if2_0(+(n1537_0, 1)), gen_0':s:true:false:p:diff:leq:if2_0(+(n1537_0, 1))) ->_R^Omega(1)
a__leq(mark(gen_0':s:true:false:p:diff:leq:if2_0(n1537_0)), mark(gen_0':s:true:false:p:diff:leq:if2_0(n1537_0))) ->_L^Omega(1 + n1537_0)
a__leq(gen_0':s:true:false:p:diff:leq:if2_0(n1537_0), mark(gen_0':s:true:false:p:diff:leq:if2_0(n1537_0))) ->_L^Omega(1 + n1537_0)
a__leq(gen_0':s:true:false:p:diff:leq:if2_0(n1537_0), gen_0':s:true:false:p:diff:leq:if2_0(n1537_0)) ->_IH
true

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

(14)
Complex Obligation (BEST)

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

(15)
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)

Types:
a__p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
0' :: 0':s:true:false:p:diff:leq:if
s :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
mark :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
a__leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
true :: 0':s:true:false:p:diff:leq:if
false :: 0':s:true:false:p:diff:leq:if
a__if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
a__diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
hole_0':s:true:false:p:diff:leq:if1_0 :: 0':s:true:false:p:diff:leq:if
gen_0':s:true:false:p:diff:leq:if2_0 :: Nat -> 0':s:true:false:p:diff:leq:if


Lemmas:
mark(gen_0':s:true:false:p:diff:leq:if2_0(n4_0)) -> gen_0':s:true:false:p:diff:leq:if2_0(n4_0), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s:true:false:p:diff:leq:if2_0(0) <=> 0'
gen_0':s:true:false:p:diff:leq:if2_0(+(x, 1)) <=> s(gen_0':s:true:false:p:diff:leq:if2_0(x))


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

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

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

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

(17)
BOUNDS(n^2, INF)

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

(18)
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)

Types:
a__p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
0' :: 0':s:true:false:p:diff:leq:if
s :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
mark :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
a__leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
true :: 0':s:true:false:p:diff:leq:if
false :: 0':s:true:false:p:diff:leq:if
a__if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
a__diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
hole_0':s:true:false:p:diff:leq:if1_0 :: 0':s:true:false:p:diff:leq:if
gen_0':s:true:false:p:diff:leq:if2_0 :: Nat -> 0':s:true:false:p:diff:leq:if


Lemmas:
mark(gen_0':s:true:false:p:diff:leq:if2_0(n4_0)) -> gen_0':s:true:false:p:diff:leq:if2_0(n4_0), rt in Omega(1 + n4_0)
a__leq(gen_0':s:true:false:p:diff:leq:if2_0(n1537_0), gen_0':s:true:false:p:diff:leq:if2_0(n1537_0)) -> true, rt in Omega(1 + n1537_0 + n1537_0^2)


Generator Equations:
gen_0':s:true:false:p:diff:leq:if2_0(0) <=> 0'
gen_0':s:true:false:p:diff:leq:if2_0(+(x, 1)) <=> s(gen_0':s:true:false:p:diff:leq:if2_0(x))


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

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

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_0':s:true:false:p:diff:leq:if2_0(n2399_0)) -> gen_0':s:true:false:p:diff:leq:if2_0(n2399_0), rt in Omega(1 + n2399_0)

Induction Base:
mark(gen_0':s:true:false:p:diff:leq:if2_0(0)) ->_R^Omega(1)
0'

Induction Step:
mark(gen_0':s:true:false:p:diff:leq:if2_0(+(n2399_0, 1))) ->_R^Omega(1)
s(mark(gen_0':s:true:false:p:diff:leq:if2_0(n2399_0))) ->_IH
s(gen_0':s:true:false:p:diff:leq:if2_0(c2400_0))

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:
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)

Types:
a__p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
0' :: 0':s:true:false:p:diff:leq:if
s :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
mark :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
a__leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
true :: 0':s:true:false:p:diff:leq:if
false :: 0':s:true:false:p:diff:leq:if
a__if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
a__diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
diff :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
p :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
leq :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
if :: 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if -> 0':s:true:false:p:diff:leq:if
hole_0':s:true:false:p:diff:leq:if1_0 :: 0':s:true:false:p:diff:leq:if
gen_0':s:true:false:p:diff:leq:if2_0 :: Nat -> 0':s:true:false:p:diff:leq:if


Lemmas:
mark(gen_0':s:true:false:p:diff:leq:if2_0(n2399_0)) -> gen_0':s:true:false:p:diff:leq:if2_0(n2399_0), rt in Omega(1 + n2399_0)
a__leq(gen_0':s:true:false:p:diff:leq:if2_0(n1537_0), gen_0':s:true:false:p:diff:leq:if2_0(n1537_0)) -> true, rt in Omega(1 + n1537_0 + n1537_0^2)


Generator Equations:
gen_0':s:true:false:p:diff:leq:if2_0(0) <=> 0'
gen_0':s:true:false:p:diff:leq:if2_0(+(x, 1)) <=> s(gen_0':s:true:false:p:diff:leq:if2_0(x))


The following defined symbols remain to be analysed:
a__p

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