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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsTAProof [FINISHED, 62 ms]
    (6) BOUNDS(1, n^1)
(7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTRS
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) typed CpxTrs
    (11) OrderProof [LOWER BOUND(ID), 0 ms]
    (12) typed CpxTrs
    (13) RewriteLemmaProof [LOWER BOUND(ID), 433 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) typed CpxTrs
            (19) RewriteLemmaProof [LOWER BOUND(ID), 94 ms]
            (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 124 ms]
            (22) typed CpxTrs


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

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


The TRS R consists of the following rules:

   active(minus(0, Y)) -> mark(0)
   active(minus(s(X), s(Y))) -> mark(minus(X, Y))
   active(geq(X, 0)) -> mark(true)
   active(geq(0, s(Y))) -> mark(false)
   active(geq(s(X), s(Y))) -> mark(geq(X, Y))
   active(div(0, s(Y))) -> mark(0)
   active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
   active(if(true, X, Y)) -> mark(X)
   active(if(false, X, Y)) -> mark(Y)
   active(s(X)) -> s(active(X))
   active(div(X1, X2)) -> div(active(X1), X2)
   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
   s(mark(X)) -> mark(s(X))
   div(mark(X1), X2) -> mark(div(X1, X2))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(s(X)) -> s(proper(X))
   proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   proper(div(X1, X2)) -> div(proper(X1), proper(X2))
   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
   minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
   s(ok(X)) -> ok(s(X))
   geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
   div(ok(X1), ok(X2)) -> ok(div(X1, X2))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of top: proper, active
The following defined symbols can occur below the 0th argument of proper: proper, active
The following defined symbols can occur below the 0th argument of active: proper, active

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(minus(0, Y)) -> mark(0)
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0)) -> mark(true)
active(geq(0, s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0, s(Y))) -> mark(0)
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))

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

(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   s(mark(X)) -> mark(s(X))
   div(mark(X1), X2) -> mark(div(X1, X2))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   proper(0) -> ok(0)
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
   s(ok(X)) -> ok(s(X))
   geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
   div(ok(X1), ok(X2)) -> ok(div(X1, X2))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(4)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   s(mark(X)) -> mark(s(X))
   div(mark(X1), X2) -> mark(div(X1, X2))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   proper(0) -> ok(0)
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
   s(ok(X)) -> ok(s(X))
   geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
   div(ok(X1), ok(X2)) -> ok(div(X1, X2))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

(5) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4, 5, 6, 7]
transitions: 
mark0(0) -> 0
00() -> 0
ok0(0) -> 0
true0() -> 0
false0() -> 0
active0(0) -> 0
s0(0) -> 1
div0(0, 0) -> 2
if0(0, 0, 0) -> 3
proper0(0) -> 4
minus0(0, 0) -> 5
geq0(0, 0) -> 6
top0(0) -> 7
s1(0) -> 8
mark1(8) -> 1
div1(0, 0) -> 9
mark1(9) -> 2
if1(0, 0, 0) -> 10
mark1(10) -> 3
01() -> 11
ok1(11) -> 4
true1() -> 12
ok1(12) -> 4
false1() -> 13
ok1(13) -> 4
minus1(0, 0) -> 14
ok1(14) -> 5
s1(0) -> 15
ok1(15) -> 1
geq1(0, 0) -> 16
ok1(16) -> 6
div1(0, 0) -> 17
ok1(17) -> 2
if1(0, 0, 0) -> 18
ok1(18) -> 3
proper1(0) -> 19
top1(19) -> 7
active1(0) -> 20
top1(20) -> 7
mark1(8) -> 8
mark1(8) -> 15
mark1(9) -> 9
mark1(9) -> 17
mark1(10) -> 10
mark1(10) -> 18
ok1(11) -> 19
ok1(12) -> 19
ok1(13) -> 19
ok1(14) -> 14
ok1(15) -> 8
ok1(15) -> 15
ok1(16) -> 16
ok1(17) -> 9
ok1(17) -> 17
ok1(18) -> 10
ok1(18) -> 18
active2(11) -> 21
top2(21) -> 7
active2(12) -> 21
active2(13) -> 21

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

(6)
BOUNDS(1, n^1)

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

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

(8)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   active(minus(0', Y)) -> mark(0')
   active(minus(s(X), s(Y))) -> mark(minus(X, Y))
   active(geq(X, 0')) -> mark(true)
   active(geq(0', s(Y))) -> mark(false)
   active(geq(s(X), s(Y))) -> mark(geq(X, Y))
   active(div(0', s(Y))) -> mark(0')
   active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0'))
   active(if(true, X, Y)) -> mark(X)
   active(if(false, X, Y)) -> mark(Y)
   active(s(X)) -> s(active(X))
   active(div(X1, X2)) -> div(active(X1), X2)
   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
   s(mark(X)) -> mark(s(X))
   div(mark(X1), X2) -> mark(div(X1, X2))
   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
   proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
   proper(0') -> ok(0')
   proper(s(X)) -> s(proper(X))
   proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
   proper(true) -> ok(true)
   proper(false) -> ok(false)
   proper(div(X1, X2)) -> div(proper(X1), proper(X2))
   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
   minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
   s(ok(X)) -> ok(s(X))
   geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
   div(ok(X1), ok(X2)) -> ok(div(X1, X2))
   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

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

(10)
Obligation:
TRS:
Rules:
active(minus(0', Y)) -> mark(0')
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0')) -> mark(true)
active(geq(0', s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0', s(Y))) -> mark(0')
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0'))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
s(mark(X)) -> mark(s(X))
div(mark(X1), X2) -> mark(div(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
s(ok(X)) -> ok(s(X))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:true:false:ok -> 0':mark:true:false:ok
minus :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok -> 0':mark:true:false:ok
s :: 0':mark:true:false:ok -> 0':mark:true:false:ok
geq :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
div :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
if :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
proper :: 0':mark:true:false:ok -> 0':mark:true:false:ok
ok :: 0':mark:true:false:ok -> 0':mark:true:false:ok
top :: 0':mark:true:false:ok -> top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat -> 0':mark:true:false:ok

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

(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
active, minus, geq, if, s, div, proper, top

They will be analysed ascendingly in the following order:
minus < active
geq < active
if < active
s < active
div < active
active < top
minus < proper
geq < proper
if < proper
s < proper
div < proper
proper < top

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

(12)
Obligation:
TRS:
Rules:
active(minus(0', Y)) -> mark(0')
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0')) -> mark(true)
active(geq(0', s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0', s(Y))) -> mark(0')
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0'))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
s(mark(X)) -> mark(s(X))
div(mark(X1), X2) -> mark(div(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
s(ok(X)) -> ok(s(X))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:true:false:ok -> 0':mark:true:false:ok
minus :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok -> 0':mark:true:false:ok
s :: 0':mark:true:false:ok -> 0':mark:true:false:ok
geq :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
div :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
if :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
proper :: 0':mark:true:false:ok -> 0':mark:true:false:ok
ok :: 0':mark:true:false:ok -> 0':mark:true:false:ok
top :: 0':mark:true:false:ok -> top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat -> 0':mark:true:false:ok


Generator Equations:
gen_0':mark:true:false:ok3_0(0) <=> 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) <=> mark(gen_0':mark:true:false:ok3_0(x))


The following defined symbols remain to be analysed:
minus, active, geq, if, s, div, proper, top

They will be analysed ascendingly in the following order:
minus < active
geq < active
if < active
s < active
div < active
active < top
minus < proper
geq < proper
if < proper
s < proper
div < proper
proper < top

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
if(gen_0':mark:true:false:ok3_0(+(1, n15_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) -> *4_0, rt in Omega(n15_0)

Induction Base:
if(gen_0':mark:true:false:ok3_0(+(1, 0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c))

Induction Step:
if(gen_0':mark:true:false:ok3_0(+(1, +(n15_0, 1))), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) ->_R^Omega(1)
mark(if(gen_0':mark:true:false:ok3_0(+(1, n15_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c))) ->_IH
mark(*4_0)

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

(14)
Complex Obligation (BEST)

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

(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
active(minus(0', Y)) -> mark(0')
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0')) -> mark(true)
active(geq(0', s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0', s(Y))) -> mark(0')
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0'))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
s(mark(X)) -> mark(s(X))
div(mark(X1), X2) -> mark(div(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
s(ok(X)) -> ok(s(X))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:true:false:ok -> 0':mark:true:false:ok
minus :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok -> 0':mark:true:false:ok
s :: 0':mark:true:false:ok -> 0':mark:true:false:ok
geq :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
div :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
if :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
proper :: 0':mark:true:false:ok -> 0':mark:true:false:ok
ok :: 0':mark:true:false:ok -> 0':mark:true:false:ok
top :: 0':mark:true:false:ok -> top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat -> 0':mark:true:false:ok


Generator Equations:
gen_0':mark:true:false:ok3_0(0) <=> 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) <=> mark(gen_0':mark:true:false:ok3_0(x))


The following defined symbols remain to be analysed:
if, active, s, div, proper, top

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

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

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

(17)
BOUNDS(n^1, INF)

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

(18)
Obligation:
TRS:
Rules:
active(minus(0', Y)) -> mark(0')
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0')) -> mark(true)
active(geq(0', s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0', s(Y))) -> mark(0')
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0'))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
s(mark(X)) -> mark(s(X))
div(mark(X1), X2) -> mark(div(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
s(ok(X)) -> ok(s(X))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:true:false:ok -> 0':mark:true:false:ok
minus :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok -> 0':mark:true:false:ok
s :: 0':mark:true:false:ok -> 0':mark:true:false:ok
geq :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
div :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
if :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
proper :: 0':mark:true:false:ok -> 0':mark:true:false:ok
ok :: 0':mark:true:false:ok -> 0':mark:true:false:ok
top :: 0':mark:true:false:ok -> top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat -> 0':mark:true:false:ok


Lemmas:
if(gen_0':mark:true:false:ok3_0(+(1, n15_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) -> *4_0, rt in Omega(n15_0)


Generator Equations:
gen_0':mark:true:false:ok3_0(0) <=> 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) <=> mark(gen_0':mark:true:false:ok3_0(x))


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

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

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
s(gen_0':mark:true:false:ok3_0(+(1, n1368_0))) -> *4_0, rt in Omega(n1368_0)

Induction Base:
s(gen_0':mark:true:false:ok3_0(+(1, 0)))

Induction Step:
s(gen_0':mark:true:false:ok3_0(+(1, +(n1368_0, 1)))) ->_R^Omega(1)
mark(s(gen_0':mark:true:false:ok3_0(+(1, n1368_0)))) ->_IH
mark(*4_0)

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

(20)
Obligation:
TRS:
Rules:
active(minus(0', Y)) -> mark(0')
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0')) -> mark(true)
active(geq(0', s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0', s(Y))) -> mark(0')
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0'))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
s(mark(X)) -> mark(s(X))
div(mark(X1), X2) -> mark(div(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
s(ok(X)) -> ok(s(X))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:true:false:ok -> 0':mark:true:false:ok
minus :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok -> 0':mark:true:false:ok
s :: 0':mark:true:false:ok -> 0':mark:true:false:ok
geq :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
div :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
if :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
proper :: 0':mark:true:false:ok -> 0':mark:true:false:ok
ok :: 0':mark:true:false:ok -> 0':mark:true:false:ok
top :: 0':mark:true:false:ok -> top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat -> 0':mark:true:false:ok


Lemmas:
if(gen_0':mark:true:false:ok3_0(+(1, n15_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) -> *4_0, rt in Omega(n15_0)
s(gen_0':mark:true:false:ok3_0(+(1, n1368_0))) -> *4_0, rt in Omega(n1368_0)


Generator Equations:
gen_0':mark:true:false:ok3_0(0) <=> 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) <=> mark(gen_0':mark:true:false:ok3_0(x))


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

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

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
div(gen_0':mark:true:false:ok3_0(+(1, n1924_0)), gen_0':mark:true:false:ok3_0(b)) -> *4_0, rt in Omega(n1924_0)

Induction Base:
div(gen_0':mark:true:false:ok3_0(+(1, 0)), gen_0':mark:true:false:ok3_0(b))

Induction Step:
div(gen_0':mark:true:false:ok3_0(+(1, +(n1924_0, 1))), gen_0':mark:true:false:ok3_0(b)) ->_R^Omega(1)
mark(div(gen_0':mark:true:false:ok3_0(+(1, n1924_0)), gen_0':mark:true:false:ok3_0(b))) ->_IH
mark(*4_0)

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

(22)
Obligation:
TRS:
Rules:
active(minus(0', Y)) -> mark(0')
active(minus(s(X), s(Y))) -> mark(minus(X, Y))
active(geq(X, 0')) -> mark(true)
active(geq(0', s(Y))) -> mark(false)
active(geq(s(X), s(Y))) -> mark(geq(X, Y))
active(div(0', s(Y))) -> mark(0')
active(div(s(X), s(Y))) -> mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0'))
active(if(true, X, Y)) -> mark(X)
active(if(false, X, Y)) -> mark(Y)
active(s(X)) -> s(active(X))
active(div(X1, X2)) -> div(active(X1), X2)
active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
s(mark(X)) -> mark(s(X))
div(mark(X1), X2) -> mark(div(X1, X2))
if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
proper(minus(X1, X2)) -> minus(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(geq(X1, X2)) -> geq(proper(X1), proper(X2))
proper(true) -> ok(true)
proper(false) -> ok(false)
proper(div(X1, X2)) -> div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) -> ok(minus(X1, X2))
s(ok(X)) -> ok(s(X))
geq(ok(X1), ok(X2)) -> ok(geq(X1, X2))
div(ok(X1), ok(X2)) -> ok(div(X1, X2))
if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:true:false:ok -> 0':mark:true:false:ok
minus :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok -> 0':mark:true:false:ok
s :: 0':mark:true:false:ok -> 0':mark:true:false:ok
geq :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
div :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
if :: 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok -> 0':mark:true:false:ok
proper :: 0':mark:true:false:ok -> 0':mark:true:false:ok
ok :: 0':mark:true:false:ok -> 0':mark:true:false:ok
top :: 0':mark:true:false:ok -> top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat -> 0':mark:true:false:ok


Lemmas:
if(gen_0':mark:true:false:ok3_0(+(1, n15_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) -> *4_0, rt in Omega(n15_0)
s(gen_0':mark:true:false:ok3_0(+(1, n1368_0))) -> *4_0, rt in Omega(n1368_0)
div(gen_0':mark:true:false:ok3_0(+(1, n1924_0)), gen_0':mark:true:false:ok3_0(b)) -> *4_0, rt in Omega(n1924_0)


Generator Equations:
gen_0':mark:true:false:ok3_0(0) <=> 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) <=> mark(gen_0':mark:true:false:ok3_0(x))


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

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