/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 11 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsTAProof [FINISHED, 80 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), 423 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), 63 ms]
            (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 99 ms]
            (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 120 ms]
            (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 96 ms]
            (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 111 ms]
            (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 85 ms]
            (30) 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(U11(tt, M, N)) -> mark(U12(tt, M, N))
   active(U12(tt, M, N)) -> mark(s(plus(N, M)))
   active(U21(tt, M, N)) -> mark(U22(tt, M, N))
   active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
   active(plus(N, 0)) -> mark(N)
   active(plus(N, s(M))) -> mark(U11(tt, M, N))
   active(x(N, 0)) -> mark(0)
   active(x(N, s(M))) -> mark(U21(tt, M, N))
   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
   active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
   active(s(X)) -> s(active(X))
   active(plus(X1, X2)) -> plus(active(X1), X2)
   active(plus(X1, X2)) -> plus(X1, active(X2))
   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
   active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
   active(x(X1, X2)) -> x(active(X1), X2)
   active(x(X1, X2)) -> x(X1, active(X2))
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
   s(mark(X)) -> mark(s(X))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
   x(mark(X1), X2) -> mark(x(X1, X2))
   x(X1, mark(X2)) -> mark(x(X1, X2))
   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
   proper(tt) -> ok(tt)
   proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
   proper(0) -> ok(0)
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
   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(U11(tt, M, N)) -> mark(U12(tt, M, N))
active(U12(tt, M, N)) -> mark(s(plus(N, M)))
active(U21(tt, M, N)) -> mark(U22(tt, M, N))
active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
active(plus(N, 0)) -> mark(N)
active(plus(N, s(M))) -> mark(U11(tt, M, N))
active(x(N, 0)) -> mark(0)
active(x(N, s(M))) -> mark(U21(tt, M, N))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(s(X)) -> s(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))

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

(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:

   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
   s(mark(X)) -> mark(s(X))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
   x(mark(X1), X2) -> mark(x(X1, X2))
   x(X1, mark(X2)) -> mark(x(X1, X2))
   proper(tt) -> ok(tt)
   proper(0) -> ok(0)
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
   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:

   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
   s(mark(X)) -> mark(s(X))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
   x(mark(X1), X2) -> mark(x(X1, X2))
   x(X1, mark(X2)) -> mark(x(X1, X2))
   proper(tt) -> ok(tt)
   proper(0) -> ok(0)
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
   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, 8, 9]
transitions: 
mark0(0) -> 0
tt0() -> 0
ok0(0) -> 0
00() -> 0
active0(0) -> 0
U110(0, 0, 0) -> 1
U120(0, 0, 0) -> 2
s0(0) -> 3
plus0(0, 0) -> 4
U210(0, 0, 0) -> 5
U220(0, 0, 0) -> 6
x0(0, 0) -> 7
proper0(0) -> 8
top0(0) -> 9
U111(0, 0, 0) -> 10
mark1(10) -> 1
U121(0, 0, 0) -> 11
mark1(11) -> 2
s1(0) -> 12
mark1(12) -> 3
plus1(0, 0) -> 13
mark1(13) -> 4
U211(0, 0, 0) -> 14
mark1(14) -> 5
U221(0, 0, 0) -> 15
mark1(15) -> 6
x1(0, 0) -> 16
mark1(16) -> 7
tt1() -> 17
ok1(17) -> 8
01() -> 18
ok1(18) -> 8
U111(0, 0, 0) -> 19
ok1(19) -> 1
U121(0, 0, 0) -> 20
ok1(20) -> 2
s1(0) -> 21
ok1(21) -> 3
plus1(0, 0) -> 22
ok1(22) -> 4
U211(0, 0, 0) -> 23
ok1(23) -> 5
U221(0, 0, 0) -> 24
ok1(24) -> 6
x1(0, 0) -> 25
ok1(25) -> 7
proper1(0) -> 26
top1(26) -> 9
active1(0) -> 27
top1(27) -> 9
mark1(10) -> 10
mark1(10) -> 19
mark1(11) -> 11
mark1(11) -> 20
mark1(12) -> 12
mark1(12) -> 21
mark1(13) -> 13
mark1(13) -> 22
mark1(14) -> 14
mark1(14) -> 23
mark1(15) -> 15
mark1(15) -> 24
mark1(16) -> 16
mark1(16) -> 25
ok1(17) -> 26
ok1(18) -> 26
ok1(19) -> 10
ok1(19) -> 19
ok1(20) -> 11
ok1(20) -> 20
ok1(21) -> 12
ok1(21) -> 21
ok1(22) -> 13
ok1(22) -> 22
ok1(23) -> 14
ok1(23) -> 23
ok1(24) -> 15
ok1(24) -> 24
ok1(25) -> 16
ok1(25) -> 25
active2(17) -> 28
top2(28) -> 9
active2(18) -> 28

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

(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(U11(tt, M, N)) -> mark(U12(tt, M, N))
   active(U12(tt, M, N)) -> mark(s(plus(N, M)))
   active(U21(tt, M, N)) -> mark(U22(tt, M, N))
   active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
   active(plus(N, 0')) -> mark(N)
   active(plus(N, s(M))) -> mark(U11(tt, M, N))
   active(x(N, 0')) -> mark(0')
   active(x(N, s(M))) -> mark(U21(tt, M, N))
   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
   active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
   active(s(X)) -> s(active(X))
   active(plus(X1, X2)) -> plus(active(X1), X2)
   active(plus(X1, X2)) -> plus(X1, active(X2))
   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
   active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
   active(x(X1, X2)) -> x(active(X1), X2)
   active(x(X1, X2)) -> x(X1, active(X2))
   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
   s(mark(X)) -> mark(s(X))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
   U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
   x(mark(X1), X2) -> mark(x(X1, X2))
   x(X1, mark(X2)) -> mark(x(X1, X2))
   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
   proper(tt) -> ok(tt)
   proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
   proper(0') -> ok(0')
   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
   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(U11(tt, M, N)) -> mark(U12(tt, M, N))
active(U12(tt, M, N)) -> mark(s(plus(N, M)))
active(U21(tt, M, N)) -> mark(U22(tt, M, N))
active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
active(plus(N, 0')) -> mark(N)
active(plus(N, s(M))) -> mark(U11(tt, M, N))
active(x(N, 0')) -> mark(0')
active(x(N, s(M))) -> mark(U21(tt, M, N))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(s(X)) -> s(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
s(mark(X)) -> mark(s(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
x(mark(X1), X2) -> mark(x(X1, X2))
x(X1, mark(X2)) -> mark(x(X1, X2))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))
proper(0') -> ok(0')
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
s(ok(X)) -> ok(s(X))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
x(ok(X1), ok(X2)) -> ok(x(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: tt:mark:0':ok -> tt:mark:0':ok
U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok -> tt:mark:0':ok
U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
s :: tt:mark:0':ok -> tt:mark:0':ok
plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
0' :: tt:mark:0':ok
proper :: tt:mark:0':ok -> tt:mark:0':ok
ok :: tt:mark:0':ok -> tt:mark:0':ok
top :: tt:mark:0':ok -> top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok

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

(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
active, U12, s, plus, U22, x, U11, U21, proper, top

They will be analysed ascendingly in the following order:
U12 < active
s < active
plus < active
U22 < active
x < active
U11 < active
U21 < active
active < top
U12 < proper
s < proper
plus < proper
U22 < proper
x < proper
U11 < proper
U21 < proper
proper < top

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

(12)
Obligation:
TRS:
Rules:
active(U11(tt, M, N)) -> mark(U12(tt, M, N))
active(U12(tt, M, N)) -> mark(s(plus(N, M)))
active(U21(tt, M, N)) -> mark(U22(tt, M, N))
active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
active(plus(N, 0')) -> mark(N)
active(plus(N, s(M))) -> mark(U11(tt, M, N))
active(x(N, 0')) -> mark(0')
active(x(N, s(M))) -> mark(U21(tt, M, N))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(s(X)) -> s(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
s(mark(X)) -> mark(s(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
x(mark(X1), X2) -> mark(x(X1, X2))
x(X1, mark(X2)) -> mark(x(X1, X2))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))
proper(0') -> ok(0')
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
s(ok(X)) -> ok(s(X))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
x(ok(X1), ok(X2)) -> ok(x(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: tt:mark:0':ok -> tt:mark:0':ok
U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok -> tt:mark:0':ok
U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
s :: tt:mark:0':ok -> tt:mark:0':ok
plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
0' :: tt:mark:0':ok
proper :: tt:mark:0':ok -> tt:mark:0':ok
ok :: tt:mark:0':ok -> tt:mark:0':ok
top :: tt:mark:0':ok -> top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))


The following defined symbols remain to be analysed:
U12, active, s, plus, U22, x, U11, U21, proper, top

They will be analysed ascendingly in the following order:
U12 < active
s < active
plus < active
U22 < active
x < active
U11 < active
U21 < active
active < top
U12 < proper
s < proper
plus < proper
U22 < proper
x < proper
U11 < proper
U21 < proper
proper < top

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)

Induction Base:
U12(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))

Induction Step:
U12(gen_tt:mark:0':ok3_0(+(1, +(n5_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
mark(U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':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(U11(tt, M, N)) -> mark(U12(tt, M, N))
active(U12(tt, M, N)) -> mark(s(plus(N, M)))
active(U21(tt, M, N)) -> mark(U22(tt, M, N))
active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
active(plus(N, 0')) -> mark(N)
active(plus(N, s(M))) -> mark(U11(tt, M, N))
active(x(N, 0')) -> mark(0')
active(x(N, s(M))) -> mark(U21(tt, M, N))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(s(X)) -> s(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
s(mark(X)) -> mark(s(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
x(mark(X1), X2) -> mark(x(X1, X2))
x(X1, mark(X2)) -> mark(x(X1, X2))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))
proper(0') -> ok(0')
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
s(ok(X)) -> ok(s(X))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
x(ok(X1), ok(X2)) -> ok(x(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: tt:mark:0':ok -> tt:mark:0':ok
U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok -> tt:mark:0':ok
U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
s :: tt:mark:0':ok -> tt:mark:0':ok
plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
0' :: tt:mark:0':ok
proper :: tt:mark:0':ok -> tt:mark:0':ok
ok :: tt:mark:0':ok -> tt:mark:0':ok
top :: tt:mark:0':ok -> top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))


The following defined symbols remain to be analysed:
U12, active, s, plus, U22, x, U11, U21, proper, top

They will be analysed ascendingly in the following order:
U12 < active
s < active
plus < active
U22 < active
x < active
U11 < active
U21 < active
active < top
U12 < proper
s < proper
plus < proper
U22 < proper
x < proper
U11 < proper
U21 < proper
proper < top

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

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

(17)
BOUNDS(n^1, INF)

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

(18)
Obligation:
TRS:
Rules:
active(U11(tt, M, N)) -> mark(U12(tt, M, N))
active(U12(tt, M, N)) -> mark(s(plus(N, M)))
active(U21(tt, M, N)) -> mark(U22(tt, M, N))
active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
active(plus(N, 0')) -> mark(N)
active(plus(N, s(M))) -> mark(U11(tt, M, N))
active(x(N, 0')) -> mark(0')
active(x(N, s(M))) -> mark(U21(tt, M, N))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(s(X)) -> s(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
s(mark(X)) -> mark(s(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
x(mark(X1), X2) -> mark(x(X1, X2))
x(X1, mark(X2)) -> mark(x(X1, X2))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))
proper(0') -> ok(0')
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
s(ok(X)) -> ok(s(X))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
x(ok(X1), ok(X2)) -> ok(x(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: tt:mark:0':ok -> tt:mark:0':ok
U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok -> tt:mark:0':ok
U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
s :: tt:mark:0':ok -> tt:mark:0':ok
plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
0' :: tt:mark:0':ok
proper :: tt:mark:0':ok -> tt:mark:0':ok
ok :: tt:mark:0':ok -> tt:mark:0':ok
top :: tt:mark:0':ok -> top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))


The following defined symbols remain to be analysed:
s, active, plus, U22, x, U11, U21, proper, top

They will be analysed ascendingly in the following order:
s < active
plus < active
U22 < active
x < active
U11 < active
U21 < active
active < top
s < proper
plus < proper
U22 < proper
x < proper
U11 < proper
U21 < proper
proper < top

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
s(gen_tt:mark:0':ok3_0(+(1, n1708_0))) -> *4_0, rt in Omega(n1708_0)

Induction Base:
s(gen_tt:mark:0':ok3_0(+(1, 0)))

Induction Step:
s(gen_tt:mark:0':ok3_0(+(1, +(n1708_0, 1)))) ->_R^Omega(1)
mark(s(gen_tt:mark:0':ok3_0(+(1, n1708_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(U11(tt, M, N)) -> mark(U12(tt, M, N))
active(U12(tt, M, N)) -> mark(s(plus(N, M)))
active(U21(tt, M, N)) -> mark(U22(tt, M, N))
active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
active(plus(N, 0')) -> mark(N)
active(plus(N, s(M))) -> mark(U11(tt, M, N))
active(x(N, 0')) -> mark(0')
active(x(N, s(M))) -> mark(U21(tt, M, N))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(s(X)) -> s(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
s(mark(X)) -> mark(s(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
x(mark(X1), X2) -> mark(x(X1, X2))
x(X1, mark(X2)) -> mark(x(X1, X2))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))
proper(0') -> ok(0')
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
s(ok(X)) -> ok(s(X))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
x(ok(X1), ok(X2)) -> ok(x(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: tt:mark:0':ok -> tt:mark:0':ok
U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok -> tt:mark:0':ok
U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
s :: tt:mark:0':ok -> tt:mark:0':ok
plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
0' :: tt:mark:0':ok
proper :: tt:mark:0':ok -> tt:mark:0':ok
ok :: tt:mark:0':ok -> tt:mark:0':ok
top :: tt:mark:0':ok -> top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
s(gen_tt:mark:0':ok3_0(+(1, n1708_0))) -> *4_0, rt in Omega(n1708_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))


The following defined symbols remain to be analysed:
plus, active, U22, x, U11, U21, proper, top

They will be analysed ascendingly in the following order:
plus < active
U22 < active
x < active
U11 < active
U21 < active
active < top
plus < proper
U22 < proper
x < proper
U11 < proper
U21 < proper
proper < top

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_tt:mark:0':ok3_0(+(1, n2306_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n2306_0)

Induction Base:
plus(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))

Induction Step:
plus(gen_tt:mark:0':ok3_0(+(1, +(n2306_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
mark(plus(gen_tt:mark:0':ok3_0(+(1, n2306_0)), gen_tt:mark:0':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(U11(tt, M, N)) -> mark(U12(tt, M, N))
active(U12(tt, M, N)) -> mark(s(plus(N, M)))
active(U21(tt, M, N)) -> mark(U22(tt, M, N))
active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
active(plus(N, 0')) -> mark(N)
active(plus(N, s(M))) -> mark(U11(tt, M, N))
active(x(N, 0')) -> mark(0')
active(x(N, s(M))) -> mark(U21(tt, M, N))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(s(X)) -> s(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
s(mark(X)) -> mark(s(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
x(mark(X1), X2) -> mark(x(X1, X2))
x(X1, mark(X2)) -> mark(x(X1, X2))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))
proper(0') -> ok(0')
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
s(ok(X)) -> ok(s(X))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
x(ok(X1), ok(X2)) -> ok(x(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: tt:mark:0':ok -> tt:mark:0':ok
U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok -> tt:mark:0':ok
U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
s :: tt:mark:0':ok -> tt:mark:0':ok
plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
0' :: tt:mark:0':ok
proper :: tt:mark:0':ok -> tt:mark:0':ok
ok :: tt:mark:0':ok -> tt:mark:0':ok
top :: tt:mark:0':ok -> top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
s(gen_tt:mark:0':ok3_0(+(1, n1708_0))) -> *4_0, rt in Omega(n1708_0)
plus(gen_tt:mark:0':ok3_0(+(1, n2306_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n2306_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))


The following defined symbols remain to be analysed:
U22, active, x, U11, U21, proper, top

They will be analysed ascendingly in the following order:
U22 < active
x < active
U11 < active
U21 < active
active < top
U22 < proper
x < proper
U11 < proper
U21 < proper
proper < top

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U22(gen_tt:mark:0':ok3_0(+(1, n4168_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n4168_0)

Induction Base:
U22(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))

Induction Step:
U22(gen_tt:mark:0':ok3_0(+(1, +(n4168_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
mark(U22(gen_tt:mark:0':ok3_0(+(1, n4168_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':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).
----------------------------------------

(24)
Obligation:
TRS:
Rules:
active(U11(tt, M, N)) -> mark(U12(tt, M, N))
active(U12(tt, M, N)) -> mark(s(plus(N, M)))
active(U21(tt, M, N)) -> mark(U22(tt, M, N))
active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
active(plus(N, 0')) -> mark(N)
active(plus(N, s(M))) -> mark(U11(tt, M, N))
active(x(N, 0')) -> mark(0')
active(x(N, s(M))) -> mark(U21(tt, M, N))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(s(X)) -> s(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
s(mark(X)) -> mark(s(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
x(mark(X1), X2) -> mark(x(X1, X2))
x(X1, mark(X2)) -> mark(x(X1, X2))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))
proper(0') -> ok(0')
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
s(ok(X)) -> ok(s(X))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
x(ok(X1), ok(X2)) -> ok(x(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: tt:mark:0':ok -> tt:mark:0':ok
U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok -> tt:mark:0':ok
U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
s :: tt:mark:0':ok -> tt:mark:0':ok
plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
0' :: tt:mark:0':ok
proper :: tt:mark:0':ok -> tt:mark:0':ok
ok :: tt:mark:0':ok -> tt:mark:0':ok
top :: tt:mark:0':ok -> top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
s(gen_tt:mark:0':ok3_0(+(1, n1708_0))) -> *4_0, rt in Omega(n1708_0)
plus(gen_tt:mark:0':ok3_0(+(1, n2306_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n2306_0)
U22(gen_tt:mark:0':ok3_0(+(1, n4168_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n4168_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))


The following defined symbols remain to be analysed:
x, active, U11, U21, proper, top

They will be analysed ascendingly in the following order:
x < active
U11 < active
U21 < active
active < top
x < proper
U11 < proper
U21 < proper
proper < top

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
x(gen_tt:mark:0':ok3_0(+(1, n7257_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n7257_0)

Induction Base:
x(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))

Induction Step:
x(gen_tt:mark:0':ok3_0(+(1, +(n7257_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
mark(x(gen_tt:mark:0':ok3_0(+(1, n7257_0)), gen_tt:mark:0':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).
----------------------------------------

(26)
Obligation:
TRS:
Rules:
active(U11(tt, M, N)) -> mark(U12(tt, M, N))
active(U12(tt, M, N)) -> mark(s(plus(N, M)))
active(U21(tt, M, N)) -> mark(U22(tt, M, N))
active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
active(plus(N, 0')) -> mark(N)
active(plus(N, s(M))) -> mark(U11(tt, M, N))
active(x(N, 0')) -> mark(0')
active(x(N, s(M))) -> mark(U21(tt, M, N))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(s(X)) -> s(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
s(mark(X)) -> mark(s(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
x(mark(X1), X2) -> mark(x(X1, X2))
x(X1, mark(X2)) -> mark(x(X1, X2))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))
proper(0') -> ok(0')
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
s(ok(X)) -> ok(s(X))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
x(ok(X1), ok(X2)) -> ok(x(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: tt:mark:0':ok -> tt:mark:0':ok
U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok -> tt:mark:0':ok
U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
s :: tt:mark:0':ok -> tt:mark:0':ok
plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
0' :: tt:mark:0':ok
proper :: tt:mark:0':ok -> tt:mark:0':ok
ok :: tt:mark:0':ok -> tt:mark:0':ok
top :: tt:mark:0':ok -> top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
s(gen_tt:mark:0':ok3_0(+(1, n1708_0))) -> *4_0, rt in Omega(n1708_0)
plus(gen_tt:mark:0':ok3_0(+(1, n2306_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n2306_0)
U22(gen_tt:mark:0':ok3_0(+(1, n4168_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n4168_0)
x(gen_tt:mark:0':ok3_0(+(1, n7257_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n7257_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))


The following defined symbols remain to be analysed:
U11, active, U21, proper, top

They will be analysed ascendingly in the following order:
U11 < active
U21 < active
active < top
U11 < proper
U21 < proper
proper < top

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U11(gen_tt:mark:0':ok3_0(+(1, n9829_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n9829_0)

Induction Base:
U11(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))

Induction Step:
U11(gen_tt:mark:0':ok3_0(+(1, +(n9829_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
mark(U11(gen_tt:mark:0':ok3_0(+(1, n9829_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':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).
----------------------------------------

(28)
Obligation:
TRS:
Rules:
active(U11(tt, M, N)) -> mark(U12(tt, M, N))
active(U12(tt, M, N)) -> mark(s(plus(N, M)))
active(U21(tt, M, N)) -> mark(U22(tt, M, N))
active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
active(plus(N, 0')) -> mark(N)
active(plus(N, s(M))) -> mark(U11(tt, M, N))
active(x(N, 0')) -> mark(0')
active(x(N, s(M))) -> mark(U21(tt, M, N))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(s(X)) -> s(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
s(mark(X)) -> mark(s(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
x(mark(X1), X2) -> mark(x(X1, X2))
x(X1, mark(X2)) -> mark(x(X1, X2))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))
proper(0') -> ok(0')
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
s(ok(X)) -> ok(s(X))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
x(ok(X1), ok(X2)) -> ok(x(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: tt:mark:0':ok -> tt:mark:0':ok
U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok -> tt:mark:0':ok
U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
s :: tt:mark:0':ok -> tt:mark:0':ok
plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
0' :: tt:mark:0':ok
proper :: tt:mark:0':ok -> tt:mark:0':ok
ok :: tt:mark:0':ok -> tt:mark:0':ok
top :: tt:mark:0':ok -> top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
s(gen_tt:mark:0':ok3_0(+(1, n1708_0))) -> *4_0, rt in Omega(n1708_0)
plus(gen_tt:mark:0':ok3_0(+(1, n2306_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n2306_0)
U22(gen_tt:mark:0':ok3_0(+(1, n4168_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n4168_0)
x(gen_tt:mark:0':ok3_0(+(1, n7257_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n7257_0)
U11(gen_tt:mark:0':ok3_0(+(1, n9829_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n9829_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))


The following defined symbols remain to be analysed:
U21, active, proper, top

They will be analysed ascendingly in the following order:
U21 < active
active < top
U21 < proper
proper < top

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U21(gen_tt:mark:0':ok3_0(+(1, n13989_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n13989_0)

Induction Base:
U21(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))

Induction Step:
U21(gen_tt:mark:0':ok3_0(+(1, +(n13989_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
mark(U21(gen_tt:mark:0':ok3_0(+(1, n13989_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':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).
----------------------------------------

(30)
Obligation:
TRS:
Rules:
active(U11(tt, M, N)) -> mark(U12(tt, M, N))
active(U12(tt, M, N)) -> mark(s(plus(N, M)))
active(U21(tt, M, N)) -> mark(U22(tt, M, N))
active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
active(plus(N, 0')) -> mark(N)
active(plus(N, s(M))) -> mark(U11(tt, M, N))
active(x(N, 0')) -> mark(0')
active(x(N, s(M))) -> mark(U21(tt, M, N))
active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
active(s(X)) -> s(active(X))
active(plus(X1, X2)) -> plus(active(X1), X2)
active(plus(X1, X2)) -> plus(X1, active(X2))
active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
s(mark(X)) -> mark(s(X))
plus(mark(X1), X2) -> mark(plus(X1, X2))
plus(X1, mark(X2)) -> mark(plus(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
x(mark(X1), X2) -> mark(x(X1, X2))
x(X1, mark(X2)) -> mark(x(X1, X2))
proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))
proper(0') -> ok(0')
U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
s(ok(X)) -> ok(s(X))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
x(ok(X1), ok(X2)) -> ok(x(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: tt:mark:0':ok -> tt:mark:0':ok
U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok -> tt:mark:0':ok
U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
s :: tt:mark:0':ok -> tt:mark:0':ok
plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
0' :: tt:mark:0':ok
proper :: tt:mark:0':ok -> tt:mark:0':ok
ok :: tt:mark:0':ok -> tt:mark:0':ok
top :: tt:mark:0':ok -> top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
s(gen_tt:mark:0':ok3_0(+(1, n1708_0))) -> *4_0, rt in Omega(n1708_0)
plus(gen_tt:mark:0':ok3_0(+(1, n2306_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n2306_0)
U22(gen_tt:mark:0':ok3_0(+(1, n4168_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n4168_0)
x(gen_tt:mark:0':ok3_0(+(1, n7257_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n7257_0)
U11(gen_tt:mark:0':ok3_0(+(1, n9829_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n9829_0)
U21(gen_tt:mark:0':ok3_0(+(1, n13989_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n13989_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':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