WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, 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), 301 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), 74 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 94 ms]
        (16) BOUNDS(1, INF)


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__minus(0, Y) -> 0
   a__minus(s(X), s(Y)) -> a__minus(X, Y)
   a__geq(X, 0) -> true
   a__geq(0, s(Y)) -> false
   a__geq(s(X), s(Y)) -> a__geq(X, Y)
   a__div(0, s(Y)) -> 0
   a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
   a__if(true, X, Y) -> mark(X)
   a__if(false, X, Y) -> mark(Y)
   mark(minus(X1, X2)) -> a__minus(X1, X2)
   mark(geq(X1, X2)) -> a__geq(X1, X2)
   mark(div(X1, X2)) -> a__div(mark(X1), X2)
   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
   mark(0) -> 0
   mark(s(X)) -> s(mark(X))
   mark(true) -> true
   mark(false) -> false
   a__minus(X1, X2) -> minus(X1, X2)
   a__geq(X1, X2) -> geq(X1, X2)
   a__div(X1, X2) -> div(X1, X2)
   a__if(X1, X2, X3) -> if(X1, X2, X3)

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^1, INF).


The TRS R consists of the following rules:

   a__minus(0', Y) -> 0'
   a__minus(s(X), s(Y)) -> a__minus(X, Y)
   a__geq(X, 0') -> true
   a__geq(0', s(Y)) -> false
   a__geq(s(X), s(Y)) -> a__geq(X, Y)
   a__div(0', s(Y)) -> 0'
   a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
   a__if(true, X, Y) -> mark(X)
   a__if(false, X, Y) -> mark(Y)
   mark(minus(X1, X2)) -> a__minus(X1, X2)
   mark(geq(X1, X2)) -> a__geq(X1, X2)
   mark(div(X1, X2)) -> a__div(mark(X1), X2)
   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
   mark(0') -> 0'
   mark(s(X)) -> s(mark(X))
   mark(true) -> true
   mark(false) -> false
   a__minus(X1, X2) -> minus(X1, X2)
   a__geq(X1, X2) -> geq(X1, X2)
   a__div(X1, X2) -> div(X1, X2)
   a__if(X1, X2, X3) -> if(X1, X2, X3)

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

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

(4)
Obligation:
Innermost TRS:
Rules:
a__minus(0', Y) -> 0'
a__minus(s(X), s(Y)) -> a__minus(X, Y)
a__geq(X, 0') -> true
a__geq(0', s(Y)) -> false
a__geq(s(X), s(Y)) -> a__geq(X, Y)
a__div(0', s(Y)) -> 0'
a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
a__if(true, X, Y) -> mark(X)
a__if(false, X, Y) -> mark(Y)
mark(minus(X1, X2)) -> a__minus(X1, X2)
mark(geq(X1, X2)) -> a__geq(X1, X2)
mark(div(X1, X2)) -> a__div(mark(X1), X2)
mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
mark(0') -> 0'
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false
a__minus(X1, X2) -> minus(X1, X2)
a__geq(X1, X2) -> geq(X1, X2)
a__div(X1, X2) -> div(X1, X2)
a__if(X1, X2, X3) -> if(X1, X2, X3)

Types:
a__minus :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
0' :: 0':s:true:false:minus:div:geq:if
s :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
a__geq :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
true :: 0':s:true:false:minus:div:geq:if
false :: 0':s:true:false:minus:div:geq:if
a__div :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
a__if :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
div :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
minus :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
mark :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
geq :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
if :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
hole_0':s:true:false:minus:div:geq:if1_0 :: 0':s:true:false:minus:div:geq:if
gen_0':s:true:false:minus:div:geq:if2_0 :: Nat -> 0':s:true:false:minus:div:geq:if

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
a__minus, a__geq, mark

They will be analysed ascendingly in the following order:
a__minus < mark
a__geq < mark

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

(6)
Obligation:
Innermost TRS:
Rules:
a__minus(0', Y) -> 0'
a__minus(s(X), s(Y)) -> a__minus(X, Y)
a__geq(X, 0') -> true
a__geq(0', s(Y)) -> false
a__geq(s(X), s(Y)) -> a__geq(X, Y)
a__div(0', s(Y)) -> 0'
a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
a__if(true, X, Y) -> mark(X)
a__if(false, X, Y) -> mark(Y)
mark(minus(X1, X2)) -> a__minus(X1, X2)
mark(geq(X1, X2)) -> a__geq(X1, X2)
mark(div(X1, X2)) -> a__div(mark(X1), X2)
mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
mark(0') -> 0'
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false
a__minus(X1, X2) -> minus(X1, X2)
a__geq(X1, X2) -> geq(X1, X2)
a__div(X1, X2) -> div(X1, X2)
a__if(X1, X2, X3) -> if(X1, X2, X3)

Types:
a__minus :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
0' :: 0':s:true:false:minus:div:geq:if
s :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
a__geq :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
true :: 0':s:true:false:minus:div:geq:if
false :: 0':s:true:false:minus:div:geq:if
a__div :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
a__if :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
div :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
minus :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
mark :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
geq :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
if :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
hole_0':s:true:false:minus:div:geq:if1_0 :: 0':s:true:false:minus:div:geq:if
gen_0':s:true:false:minus:div:geq:if2_0 :: Nat -> 0':s:true:false:minus:div:geq:if


Generator Equations:
gen_0':s:true:false:minus:div:geq:if2_0(0) <=> 0'
gen_0':s:true:false:minus:div:geq:if2_0(+(x, 1)) <=> s(gen_0':s:true:false:minus:div:geq:if2_0(x))


The following defined symbols remain to be analysed:
a__minus, a__geq, mark

They will be analysed ascendingly in the following order:
a__minus < mark
a__geq < mark

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
a__minus(gen_0':s:true:false:minus:div:geq:if2_0(n4_0), gen_0':s:true:false:minus:div:geq:if2_0(n4_0)) -> gen_0':s:true:false:minus:div:geq:if2_0(0), rt in Omega(1 + n4_0)

Induction Base:
a__minus(gen_0':s:true:false:minus:div:geq:if2_0(0), gen_0':s:true:false:minus:div:geq:if2_0(0)) ->_R^Omega(1)
0'

Induction Step:
a__minus(gen_0':s:true:false:minus:div:geq:if2_0(+(n4_0, 1)), gen_0':s:true:false:minus:div:geq:if2_0(+(n4_0, 1))) ->_R^Omega(1)
a__minus(gen_0':s:true:false:minus:div:geq:if2_0(n4_0), gen_0':s:true:false:minus:div:geq:if2_0(n4_0)) ->_IH
gen_0':s:true:false:minus:div:geq:if2_0(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__minus(0', Y) -> 0'
a__minus(s(X), s(Y)) -> a__minus(X, Y)
a__geq(X, 0') -> true
a__geq(0', s(Y)) -> false
a__geq(s(X), s(Y)) -> a__geq(X, Y)
a__div(0', s(Y)) -> 0'
a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
a__if(true, X, Y) -> mark(X)
a__if(false, X, Y) -> mark(Y)
mark(minus(X1, X2)) -> a__minus(X1, X2)
mark(geq(X1, X2)) -> a__geq(X1, X2)
mark(div(X1, X2)) -> a__div(mark(X1), X2)
mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
mark(0') -> 0'
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false
a__minus(X1, X2) -> minus(X1, X2)
a__geq(X1, X2) -> geq(X1, X2)
a__div(X1, X2) -> div(X1, X2)
a__if(X1, X2, X3) -> if(X1, X2, X3)

Types:
a__minus :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
0' :: 0':s:true:false:minus:div:geq:if
s :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
a__geq :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
true :: 0':s:true:false:minus:div:geq:if
false :: 0':s:true:false:minus:div:geq:if
a__div :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
a__if :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
div :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
minus :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
mark :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
geq :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
if :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
hole_0':s:true:false:minus:div:geq:if1_0 :: 0':s:true:false:minus:div:geq:if
gen_0':s:true:false:minus:div:geq:if2_0 :: Nat -> 0':s:true:false:minus:div:geq:if


Generator Equations:
gen_0':s:true:false:minus:div:geq:if2_0(0) <=> 0'
gen_0':s:true:false:minus:div:geq:if2_0(+(x, 1)) <=> s(gen_0':s:true:false:minus:div:geq:if2_0(x))


The following defined symbols remain to be analysed:
a__minus, a__geq, mark

They will be analysed ascendingly in the following order:
a__minus < mark
a__geq < mark

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
Innermost TRS:
Rules:
a__minus(0', Y) -> 0'
a__minus(s(X), s(Y)) -> a__minus(X, Y)
a__geq(X, 0') -> true
a__geq(0', s(Y)) -> false
a__geq(s(X), s(Y)) -> a__geq(X, Y)
a__div(0', s(Y)) -> 0'
a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
a__if(true, X, Y) -> mark(X)
a__if(false, X, Y) -> mark(Y)
mark(minus(X1, X2)) -> a__minus(X1, X2)
mark(geq(X1, X2)) -> a__geq(X1, X2)
mark(div(X1, X2)) -> a__div(mark(X1), X2)
mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
mark(0') -> 0'
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false
a__minus(X1, X2) -> minus(X1, X2)
a__geq(X1, X2) -> geq(X1, X2)
a__div(X1, X2) -> div(X1, X2)
a__if(X1, X2, X3) -> if(X1, X2, X3)

Types:
a__minus :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
0' :: 0':s:true:false:minus:div:geq:if
s :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
a__geq :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
true :: 0':s:true:false:minus:div:geq:if
false :: 0':s:true:false:minus:div:geq:if
a__div :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
a__if :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
div :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
minus :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
mark :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
geq :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
if :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
hole_0':s:true:false:minus:div:geq:if1_0 :: 0':s:true:false:minus:div:geq:if
gen_0':s:true:false:minus:div:geq:if2_0 :: Nat -> 0':s:true:false:minus:div:geq:if


Lemmas:
a__minus(gen_0':s:true:false:minus:div:geq:if2_0(n4_0), gen_0':s:true:false:minus:div:geq:if2_0(n4_0)) -> gen_0':s:true:false:minus:div:geq:if2_0(0), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s:true:false:minus:div:geq:if2_0(0) <=> 0'
gen_0':s:true:false:minus:div:geq:if2_0(+(x, 1)) <=> s(gen_0':s:true:false:minus:div:geq:if2_0(x))


The following defined symbols remain to be analysed:
a__geq, mark

They will be analysed ascendingly in the following order:
a__geq < mark

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
a__geq(gen_0':s:true:false:minus:div:geq:if2_0(n752_0), gen_0':s:true:false:minus:div:geq:if2_0(n752_0)) -> true, rt in Omega(1 + n752_0)

Induction Base:
a__geq(gen_0':s:true:false:minus:div:geq:if2_0(0), gen_0':s:true:false:minus:div:geq:if2_0(0)) ->_R^Omega(1)
true

Induction Step:
a__geq(gen_0':s:true:false:minus:div:geq:if2_0(+(n752_0, 1)), gen_0':s:true:false:minus:div:geq:if2_0(+(n752_0, 1))) ->_R^Omega(1)
a__geq(gen_0':s:true:false:minus:div:geq:if2_0(n752_0), gen_0':s:true:false:minus:div:geq:if2_0(n752_0)) ->_IH
true

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

(14)
Obligation:
Innermost TRS:
Rules:
a__minus(0', Y) -> 0'
a__minus(s(X), s(Y)) -> a__minus(X, Y)
a__geq(X, 0') -> true
a__geq(0', s(Y)) -> false
a__geq(s(X), s(Y)) -> a__geq(X, Y)
a__div(0', s(Y)) -> 0'
a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
a__if(true, X, Y) -> mark(X)
a__if(false, X, Y) -> mark(Y)
mark(minus(X1, X2)) -> a__minus(X1, X2)
mark(geq(X1, X2)) -> a__geq(X1, X2)
mark(div(X1, X2)) -> a__div(mark(X1), X2)
mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
mark(0') -> 0'
mark(s(X)) -> s(mark(X))
mark(true) -> true
mark(false) -> false
a__minus(X1, X2) -> minus(X1, X2)
a__geq(X1, X2) -> geq(X1, X2)
a__div(X1, X2) -> div(X1, X2)
a__if(X1, X2, X3) -> if(X1, X2, X3)

Types:
a__minus :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
0' :: 0':s:true:false:minus:div:geq:if
s :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
a__geq :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
true :: 0':s:true:false:minus:div:geq:if
false :: 0':s:true:false:minus:div:geq:if
a__div :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
a__if :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
div :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
minus :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
mark :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
geq :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
if :: 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if -> 0':s:true:false:minus:div:geq:if
hole_0':s:true:false:minus:div:geq:if1_0 :: 0':s:true:false:minus:div:geq:if
gen_0':s:true:false:minus:div:geq:if2_0 :: Nat -> 0':s:true:false:minus:div:geq:if


Lemmas:
a__minus(gen_0':s:true:false:minus:div:geq:if2_0(n4_0), gen_0':s:true:false:minus:div:geq:if2_0(n4_0)) -> gen_0':s:true:false:minus:div:geq:if2_0(0), rt in Omega(1 + n4_0)
a__geq(gen_0':s:true:false:minus:div:geq:if2_0(n752_0), gen_0':s:true:false:minus:div:geq:if2_0(n752_0)) -> true, rt in Omega(1 + n752_0)


Generator Equations:
gen_0':s:true:false:minus:div:geq:if2_0(0) <=> 0'
gen_0':s:true:false:minus:div:geq:if2_0(+(x, 1)) <=> s(gen_0':s:true:false:minus:div:geq:if2_0(x))


The following defined symbols remain to be analysed:
mark
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(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_0':s:true:false:minus:div:geq:if2_0(n1592_0)) -> gen_0':s:true:false:minus:div:geq:if2_0(n1592_0), rt in Omega(1 + n1592_0)

Induction Base:
mark(gen_0':s:true:false:minus:div:geq:if2_0(0)) ->_R^Omega(1)
0'

Induction Step:
mark(gen_0':s:true:false:minus:div:geq:if2_0(+(n1592_0, 1))) ->_R^Omega(1)
s(mark(gen_0':s:true:false:minus:div:geq:if2_0(n1592_0))) ->_IH
s(gen_0':s:true:false:minus:div:geq:if2_0(c1593_0))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(16)
BOUNDS(1, INF)