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


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 124 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), 417 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), 82 ms]
            (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 108 ms]
            (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 109 ms]
            (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 111 ms]
            (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 133 ms]
            (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 50 ms]
            (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 136 ms]
            (32) 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, N)) -> mark(N)
   active(U21(tt, M, N)) -> mark(s(plus(N, M)))
   active(U31(tt)) -> mark(0)
   active(U41(tt, M, N)) -> mark(plus(x(N, M), N))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNat(x(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
   active(plus(N, 0)) -> mark(U11(isNat(N), N))
   active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
   active(x(N, 0)) -> mark(U31(isNat(N)))
   active(x(N, s(M))) -> mark(U41(and(isNat(M), isNat(N)), M, N))
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(U21(X1, X2, X3)) -> U21(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(U31(X)) -> U31(active(X))
   active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
   active(x(X1, X2)) -> x(active(X1), X2)
   active(x(X1, X2)) -> x(X1, active(X2))
   active(and(X1, X2)) -> and(active(X1), X2)
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(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))
   U31(mark(X)) -> mark(U31(X))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   x(mark(X1), X2) -> mark(x(X1, X2))
   x(X1, mark(X2)) -> mark(x(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(U31(X)) -> U31(proper(X))
   proper(0) -> ok(0)
   proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   U31(ok(X)) -> ok(U31(X))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   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, N)) -> mark(N)
active(U21(tt, M, N)) -> mark(s(plus(N, M)))
active(U31(tt)) -> mark(0)
active(U41(tt, M, N)) -> mark(plus(x(N, M), N))
active(and(tt, X)) -> mark(X)
active(isNat(0)) -> mark(tt)
active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(isNat(s(V1))) -> mark(isNat(V1))
active(isNat(x(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(plus(N, 0)) -> mark(U11(isNat(N), N))
active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
active(x(N, 0)) -> mark(U31(isNat(N)))
active(x(N, s(M))) -> mark(U41(and(isNat(M), isNat(N)), M, N))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(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(U31(X)) -> U31(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U31(X)) -> U31(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))

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

(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) -> mark(U11(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(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))
   U31(mark(X)) -> mark(U31(X))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   x(mark(X1), X2) -> mark(x(X1, X2))
   x(X1, mark(X2)) -> mark(x(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(tt) -> ok(tt)
   proper(0) -> ok(0)
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   U31(ok(X)) -> ok(U31(X))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   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) -> mark(U11(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(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))
   U31(mark(X)) -> mark(U31(X))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   x(mark(X1), X2) -> mark(x(X1, X2))
   x(X1, mark(X2)) -> mark(x(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(tt) -> ok(tt)
   proper(0) -> ok(0)
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   U31(ok(X)) -> ok(U31(X))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   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, 10, 11]
transitions: 
mark0(0) -> 0
tt0() -> 0
ok0(0) -> 0
00() -> 0
active0(0) -> 0
U110(0, 0) -> 1
U210(0, 0, 0) -> 2
s0(0) -> 3
plus0(0, 0) -> 4
U310(0) -> 5
U410(0, 0, 0) -> 6
x0(0, 0) -> 7
and0(0, 0) -> 8
proper0(0) -> 9
isNat0(0) -> 10
top0(0) -> 11
U111(0, 0) -> 12
mark1(12) -> 1
U211(0, 0, 0) -> 13
mark1(13) -> 2
s1(0) -> 14
mark1(14) -> 3
plus1(0, 0) -> 15
mark1(15) -> 4
U311(0) -> 16
mark1(16) -> 5
U411(0, 0, 0) -> 17
mark1(17) -> 6
x1(0, 0) -> 18
mark1(18) -> 7
and1(0, 0) -> 19
mark1(19) -> 8
tt1() -> 20
ok1(20) -> 9
01() -> 21
ok1(21) -> 9
U111(0, 0) -> 22
ok1(22) -> 1
U211(0, 0, 0) -> 23
ok1(23) -> 2
s1(0) -> 24
ok1(24) -> 3
plus1(0, 0) -> 25
ok1(25) -> 4
U311(0) -> 26
ok1(26) -> 5
U411(0, 0, 0) -> 27
ok1(27) -> 6
x1(0, 0) -> 28
ok1(28) -> 7
and1(0, 0) -> 29
ok1(29) -> 8
isNat1(0) -> 30
ok1(30) -> 10
proper1(0) -> 31
top1(31) -> 11
active1(0) -> 32
top1(32) -> 11
mark1(12) -> 12
mark1(12) -> 22
mark1(13) -> 13
mark1(13) -> 23
mark1(14) -> 14
mark1(14) -> 24
mark1(15) -> 15
mark1(15) -> 25
mark1(16) -> 16
mark1(16) -> 26
mark1(17) -> 17
mark1(17) -> 27
mark1(18) -> 18
mark1(18) -> 28
mark1(19) -> 19
mark1(19) -> 29
ok1(20) -> 31
ok1(21) -> 31
ok1(22) -> 12
ok1(22) -> 22
ok1(23) -> 13
ok1(23) -> 23
ok1(24) -> 14
ok1(24) -> 24
ok1(25) -> 15
ok1(25) -> 25
ok1(26) -> 16
ok1(26) -> 26
ok1(27) -> 17
ok1(27) -> 27
ok1(28) -> 18
ok1(28) -> 28
ok1(29) -> 19
ok1(29) -> 29
ok1(30) -> 30
active2(20) -> 33
top2(33) -> 11
active2(21) -> 33

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

(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, N)) -> mark(N)
   active(U21(tt, M, N)) -> mark(s(plus(N, M)))
   active(U31(tt)) -> mark(0')
   active(U41(tt, M, N)) -> mark(plus(x(N, M), N))
   active(and(tt, X)) -> mark(X)
   active(isNat(0')) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNat(x(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
   active(plus(N, 0')) -> mark(U11(isNat(N), N))
   active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
   active(x(N, 0')) -> mark(U31(isNat(N)))
   active(x(N, s(M))) -> mark(U41(and(isNat(M), isNat(N)), M, N))
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(U21(X1, X2, X3)) -> U21(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(U31(X)) -> U31(active(X))
   active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
   active(x(X1, X2)) -> x(active(X1), X2)
   active(x(X1, X2)) -> x(X1, active(X2))
   active(and(X1, X2)) -> and(active(X1), X2)
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U21(mark(X1), X2, X3) -> mark(U21(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))
   U31(mark(X)) -> mark(U31(X))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   x(mark(X1), X2) -> mark(x(X1, X2))
   x(X1, mark(X2)) -> mark(x(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(U31(X)) -> U31(proper(X))
   proper(0') -> ok(0')
   proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
   s(ok(X)) -> ok(s(X))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   U31(ok(X)) -> ok(U31(X))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   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, N)) -> mark(N)
active(U21(tt, M, N)) -> mark(s(plus(N, M)))
active(U31(tt)) -> mark(0')
active(U41(tt, M, N)) -> mark(plus(x(N, M), N))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(isNat(s(V1))) -> mark(isNat(V1))
active(isNat(x(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(plus(N, 0')) -> mark(U11(isNat(N), N))
active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
active(x(N, 0')) -> mark(U31(isNat(N)))
active(x(N, s(M))) -> mark(U41(and(isNat(M), isNat(N)), M, N))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(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(U31(X)) -> U31(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
U11(mark(X1), X2) -> mark(U11(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(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))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
x(mark(X1), X2) -> mark(x(X1, X2))
x(X1, mark(X2)) -> mark(x(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U31(X)) -> U31(proper(X))
proper(0') -> ok(0')
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
s(ok(X)) -> ok(s(X))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
x(ok(X1), ok(X2)) -> ok(x(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
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 :: tt:mark:0':ok
mark :: 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
s :: tt:mark:0':ok -> tt:mark:0':ok
plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U31 :: tt:mark:0':ok -> tt:mark:0':ok
0' :: tt:mark:0':ok
U41 :: 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
and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
isNat :: tt:mark:0':ok -> 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, s, plus, x, and, isNat, U11, U21, U31, U41, proper, top

They will be analysed ascendingly in the following order:
s < active
plus < active
x < active
and < active
isNat < active
U11 < active
U21 < active
U31 < active
U41 < active
active < top
s < proper
plus < proper
x < proper
and < proper
isNat < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
proper < top

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

(12)
Obligation:
TRS:
Rules:
active(U11(tt, N)) -> mark(N)
active(U21(tt, M, N)) -> mark(s(plus(N, M)))
active(U31(tt)) -> mark(0')
active(U41(tt, M, N)) -> mark(plus(x(N, M), N))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(isNat(s(V1))) -> mark(isNat(V1))
active(isNat(x(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(plus(N, 0')) -> mark(U11(isNat(N), N))
active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
active(x(N, 0')) -> mark(U31(isNat(N)))
active(x(N, s(M))) -> mark(U41(and(isNat(M), isNat(N)), M, N))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(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(U31(X)) -> U31(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
U11(mark(X1), X2) -> mark(U11(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(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))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
x(mark(X1), X2) -> mark(x(X1, X2))
x(X1, mark(X2)) -> mark(x(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U31(X)) -> U31(proper(X))
proper(0') -> ok(0')
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
s(ok(X)) -> ok(s(X))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
x(ok(X1), ok(X2)) -> ok(x(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
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 :: tt:mark:0':ok
mark :: 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
s :: tt:mark:0':ok -> tt:mark:0':ok
plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U31 :: tt:mark:0':ok -> tt:mark:0':ok
0' :: tt:mark:0':ok
U41 :: 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
and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
isNat :: tt:mark:0':ok -> 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:
s, active, plus, x, and, isNat, U11, U21, U31, U41, proper, top

They will be analysed ascendingly in the following order:
s < active
plus < active
x < active
and < active
isNat < active
U11 < active
U21 < active
U31 < active
U41 < active
active < top
s < proper
plus < proper
x < proper
and < proper
isNat < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
proper < top

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

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

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

Induction Step:
s(gen_tt:mark:0':ok3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1)
mark(s(gen_tt:mark:0':ok3_0(+(1, n5_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).
----------------------------------------

(14)
Complex Obligation (BEST)

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

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

TRS:
Rules:
active(U11(tt, N)) -> mark(N)
active(U21(tt, M, N)) -> mark(s(plus(N, M)))
active(U31(tt)) -> mark(0')
active(U41(tt, M, N)) -> mark(plus(x(N, M), N))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(isNat(s(V1))) -> mark(isNat(V1))
active(isNat(x(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(plus(N, 0')) -> mark(U11(isNat(N), N))
active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
active(x(N, 0')) -> mark(U31(isNat(N)))
active(x(N, s(M))) -> mark(U41(and(isNat(M), isNat(N)), M, N))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(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(U31(X)) -> U31(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
U11(mark(X1), X2) -> mark(U11(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(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))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
x(mark(X1), X2) -> mark(x(X1, X2))
x(X1, mark(X2)) -> mark(x(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U31(X)) -> U31(proper(X))
proper(0') -> ok(0')
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
s(ok(X)) -> ok(s(X))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
x(ok(X1), ok(X2)) -> ok(x(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
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 :: tt:mark:0':ok
mark :: 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
s :: tt:mark:0':ok -> tt:mark:0':ok
plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U31 :: tt:mark:0':ok -> tt:mark:0':ok
0' :: tt:mark:0':ok
U41 :: 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
and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
isNat :: tt:mark:0':ok -> 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:
s, active, plus, x, and, isNat, U11, U21, U31, U41, proper, top

They will be analysed ascendingly in the following order:
s < active
plus < active
x < active
and < active
isNat < active
U11 < active
U21 < active
U31 < active
U41 < active
active < top
s < proper
plus < proper
x < proper
and < proper
isNat < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
proper < top

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

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

(17)
BOUNDS(n^1, INF)

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

(18)
Obligation:
TRS:
Rules:
active(U11(tt, N)) -> mark(N)
active(U21(tt, M, N)) -> mark(s(plus(N, M)))
active(U31(tt)) -> mark(0')
active(U41(tt, M, N)) -> mark(plus(x(N, M), N))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(isNat(s(V1))) -> mark(isNat(V1))
active(isNat(x(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(plus(N, 0')) -> mark(U11(isNat(N), N))
active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
active(x(N, 0')) -> mark(U31(isNat(N)))
active(x(N, s(M))) -> mark(U41(and(isNat(M), isNat(N)), M, N))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(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(U31(X)) -> U31(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
U11(mark(X1), X2) -> mark(U11(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(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))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
x(mark(X1), X2) -> mark(x(X1, X2))
x(X1, mark(X2)) -> mark(x(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U31(X)) -> U31(proper(X))
proper(0') -> ok(0')
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
s(ok(X)) -> ok(s(X))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
x(ok(X1), ok(X2)) -> ok(x(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
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 :: tt:mark:0':ok
mark :: 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
s :: tt:mark:0':ok -> tt:mark:0':ok
plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U31 :: tt:mark:0':ok -> tt:mark:0':ok
0' :: tt:mark:0':ok
U41 :: 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
and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
isNat :: tt:mark:0':ok -> 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:
s(gen_tt:mark:0':ok3_0(+(1, n5_0))) -> *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:
plus, active, x, and, isNat, U11, U21, U31, U41, proper, top

They will be analysed ascendingly in the following order:
plus < active
x < active
and < active
isNat < active
U11 < active
U21 < active
U31 < active
U41 < active
active < top
plus < proper
x < proper
and < proper
isNat < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
proper < top

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_tt:mark:0':ok3_0(+(1, n437_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n437_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, +(n437_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
mark(plus(gen_tt:mark:0':ok3_0(+(1, n437_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).
----------------------------------------

(20)
Obligation:
TRS:
Rules:
active(U11(tt, N)) -> mark(N)
active(U21(tt, M, N)) -> mark(s(plus(N, M)))
active(U31(tt)) -> mark(0')
active(U41(tt, M, N)) -> mark(plus(x(N, M), N))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(isNat(s(V1))) -> mark(isNat(V1))
active(isNat(x(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(plus(N, 0')) -> mark(U11(isNat(N), N))
active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
active(x(N, 0')) -> mark(U31(isNat(N)))
active(x(N, s(M))) -> mark(U41(and(isNat(M), isNat(N)), M, N))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(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(U31(X)) -> U31(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
U11(mark(X1), X2) -> mark(U11(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(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))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
x(mark(X1), X2) -> mark(x(X1, X2))
x(X1, mark(X2)) -> mark(x(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U31(X)) -> U31(proper(X))
proper(0') -> ok(0')
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
s(ok(X)) -> ok(s(X))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
x(ok(X1), ok(X2)) -> ok(x(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
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 :: tt:mark:0':ok
mark :: 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
s :: tt:mark:0':ok -> tt:mark:0':ok
plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U31 :: tt:mark:0':ok -> tt:mark:0':ok
0' :: tt:mark:0':ok
U41 :: 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
and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
isNat :: tt:mark:0':ok -> 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:
s(gen_tt:mark:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
plus(gen_tt:mark:0':ok3_0(+(1, n437_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n437_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, and, isNat, U11, U21, U31, U41, proper, top

They will be analysed ascendingly in the following order:
x < active
and < active
isNat < active
U11 < active
U21 < active
U31 < active
U41 < active
active < top
x < proper
and < proper
isNat < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
proper < top

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
x(gen_tt:mark:0':ok3_0(+(1, n2049_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n2049_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, +(n2049_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
mark(x(gen_tt:mark:0':ok3_0(+(1, n2049_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, N)) -> mark(N)
active(U21(tt, M, N)) -> mark(s(plus(N, M)))
active(U31(tt)) -> mark(0')
active(U41(tt, M, N)) -> mark(plus(x(N, M), N))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(isNat(s(V1))) -> mark(isNat(V1))
active(isNat(x(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(plus(N, 0')) -> mark(U11(isNat(N), N))
active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
active(x(N, 0')) -> mark(U31(isNat(N)))
active(x(N, s(M))) -> mark(U41(and(isNat(M), isNat(N)), M, N))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(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(U31(X)) -> U31(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
U11(mark(X1), X2) -> mark(U11(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(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))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
x(mark(X1), X2) -> mark(x(X1, X2))
x(X1, mark(X2)) -> mark(x(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U31(X)) -> U31(proper(X))
proper(0') -> ok(0')
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
s(ok(X)) -> ok(s(X))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
x(ok(X1), ok(X2)) -> ok(x(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
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 :: tt:mark:0':ok
mark :: 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
s :: tt:mark:0':ok -> tt:mark:0':ok
plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U31 :: tt:mark:0':ok -> tt:mark:0':ok
0' :: tt:mark:0':ok
U41 :: 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
and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
isNat :: tt:mark:0':ok -> 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:
s(gen_tt:mark:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
plus(gen_tt:mark:0':ok3_0(+(1, n437_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n437_0)
x(gen_tt:mark:0':ok3_0(+(1, n2049_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n2049_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:
and, active, isNat, U11, U21, U31, U41, proper, top

They will be analysed ascendingly in the following order:
and < active
isNat < active
U11 < active
U21 < active
U31 < active
U41 < active
active < top
and < proper
isNat < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
proper < top

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
and(gen_tt:mark:0':ok3_0(+(1, n3967_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3967_0)

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

Induction Step:
and(gen_tt:mark:0':ok3_0(+(1, +(n3967_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
mark(and(gen_tt:mark:0':ok3_0(+(1, n3967_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).
----------------------------------------

(24)
Obligation:
TRS:
Rules:
active(U11(tt, N)) -> mark(N)
active(U21(tt, M, N)) -> mark(s(plus(N, M)))
active(U31(tt)) -> mark(0')
active(U41(tt, M, N)) -> mark(plus(x(N, M), N))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(isNat(s(V1))) -> mark(isNat(V1))
active(isNat(x(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(plus(N, 0')) -> mark(U11(isNat(N), N))
active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
active(x(N, 0')) -> mark(U31(isNat(N)))
active(x(N, s(M))) -> mark(U41(and(isNat(M), isNat(N)), M, N))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(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(U31(X)) -> U31(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
U11(mark(X1), X2) -> mark(U11(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(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))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
x(mark(X1), X2) -> mark(x(X1, X2))
x(X1, mark(X2)) -> mark(x(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U31(X)) -> U31(proper(X))
proper(0') -> ok(0')
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
s(ok(X)) -> ok(s(X))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
x(ok(X1), ok(X2)) -> ok(x(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
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 :: tt:mark:0':ok
mark :: 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
s :: tt:mark:0':ok -> tt:mark:0':ok
plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U31 :: tt:mark:0':ok -> tt:mark:0':ok
0' :: tt:mark:0':ok
U41 :: 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
and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
isNat :: tt:mark:0':ok -> 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:
s(gen_tt:mark:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
plus(gen_tt:mark:0':ok3_0(+(1, n437_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n437_0)
x(gen_tt:mark:0':ok3_0(+(1, n2049_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n2049_0)
and(gen_tt:mark:0':ok3_0(+(1, n3967_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3967_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:
isNat, active, U11, U21, U31, U41, proper, top

They will be analysed ascendingly in the following order:
isNat < active
U11 < active
U21 < active
U31 < active
U41 < active
active < top
isNat < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
proper < top

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U11(gen_tt:mark:0':ok3_0(+(1, n6014_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n6014_0)

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

Induction Step:
U11(gen_tt:mark:0':ok3_0(+(1, +(n6014_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
mark(U11(gen_tt:mark:0':ok3_0(+(1, n6014_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, N)) -> mark(N)
active(U21(tt, M, N)) -> mark(s(plus(N, M)))
active(U31(tt)) -> mark(0')
active(U41(tt, M, N)) -> mark(plus(x(N, M), N))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(isNat(s(V1))) -> mark(isNat(V1))
active(isNat(x(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(plus(N, 0')) -> mark(U11(isNat(N), N))
active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
active(x(N, 0')) -> mark(U31(isNat(N)))
active(x(N, s(M))) -> mark(U41(and(isNat(M), isNat(N)), M, N))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(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(U31(X)) -> U31(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
U11(mark(X1), X2) -> mark(U11(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(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))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
x(mark(X1), X2) -> mark(x(X1, X2))
x(X1, mark(X2)) -> mark(x(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U31(X)) -> U31(proper(X))
proper(0') -> ok(0')
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
s(ok(X)) -> ok(s(X))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
x(ok(X1), ok(X2)) -> ok(x(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
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 :: tt:mark:0':ok
mark :: 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
s :: tt:mark:0':ok -> tt:mark:0':ok
plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U31 :: tt:mark:0':ok -> tt:mark:0':ok
0' :: tt:mark:0':ok
U41 :: 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
and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
isNat :: tt:mark:0':ok -> 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:
s(gen_tt:mark:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
plus(gen_tt:mark:0':ok3_0(+(1, n437_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n437_0)
x(gen_tt:mark:0':ok3_0(+(1, n2049_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n2049_0)
and(gen_tt:mark:0':ok3_0(+(1, n3967_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3967_0)
U11(gen_tt:mark:0':ok3_0(+(1, n6014_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n6014_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, U31, U41, proper, top

They will be analysed ascendingly in the following order:
U21 < active
U31 < active
U41 < active
active < top
U21 < proper
U31 < proper
U41 < proper
proper < top

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U21(gen_tt:mark:0':ok3_0(+(1, n8345_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n8345_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, +(n8345_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, n8345_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, N)) -> mark(N)
active(U21(tt, M, N)) -> mark(s(plus(N, M)))
active(U31(tt)) -> mark(0')
active(U41(tt, M, N)) -> mark(plus(x(N, M), N))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(isNat(s(V1))) -> mark(isNat(V1))
active(isNat(x(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(plus(N, 0')) -> mark(U11(isNat(N), N))
active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
active(x(N, 0')) -> mark(U31(isNat(N)))
active(x(N, s(M))) -> mark(U41(and(isNat(M), isNat(N)), M, N))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(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(U31(X)) -> U31(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
U11(mark(X1), X2) -> mark(U11(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(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))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
x(mark(X1), X2) -> mark(x(X1, X2))
x(X1, mark(X2)) -> mark(x(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U31(X)) -> U31(proper(X))
proper(0') -> ok(0')
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
s(ok(X)) -> ok(s(X))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
x(ok(X1), ok(X2)) -> ok(x(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
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 :: tt:mark:0':ok
mark :: 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
s :: tt:mark:0':ok -> tt:mark:0':ok
plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U31 :: tt:mark:0':ok -> tt:mark:0':ok
0' :: tt:mark:0':ok
U41 :: 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
and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
isNat :: tt:mark:0':ok -> 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:
s(gen_tt:mark:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
plus(gen_tt:mark:0':ok3_0(+(1, n437_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n437_0)
x(gen_tt:mark:0':ok3_0(+(1, n2049_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n2049_0)
and(gen_tt:mark:0':ok3_0(+(1, n3967_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3967_0)
U11(gen_tt:mark:0':ok3_0(+(1, n6014_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n6014_0)
U21(gen_tt:mark:0':ok3_0(+(1, n8345_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n8345_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:
U31, active, U41, proper, top

They will be analysed ascendingly in the following order:
U31 < active
U41 < active
active < top
U31 < proper
U41 < proper
proper < top

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U31(gen_tt:mark:0':ok3_0(+(1, n12486_0))) -> *4_0, rt in Omega(n12486_0)

Induction Base:
U31(gen_tt:mark:0':ok3_0(+(1, 0)))

Induction Step:
U31(gen_tt:mark:0':ok3_0(+(1, +(n12486_0, 1)))) ->_R^Omega(1)
mark(U31(gen_tt:mark:0':ok3_0(+(1, n12486_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).
----------------------------------------

(30)
Obligation:
TRS:
Rules:
active(U11(tt, N)) -> mark(N)
active(U21(tt, M, N)) -> mark(s(plus(N, M)))
active(U31(tt)) -> mark(0')
active(U41(tt, M, N)) -> mark(plus(x(N, M), N))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(isNat(s(V1))) -> mark(isNat(V1))
active(isNat(x(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(plus(N, 0')) -> mark(U11(isNat(N), N))
active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
active(x(N, 0')) -> mark(U31(isNat(N)))
active(x(N, s(M))) -> mark(U41(and(isNat(M), isNat(N)), M, N))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(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(U31(X)) -> U31(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
U11(mark(X1), X2) -> mark(U11(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(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))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
x(mark(X1), X2) -> mark(x(X1, X2))
x(X1, mark(X2)) -> mark(x(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U31(X)) -> U31(proper(X))
proper(0') -> ok(0')
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
s(ok(X)) -> ok(s(X))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
x(ok(X1), ok(X2)) -> ok(x(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
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 :: tt:mark:0':ok
mark :: 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
s :: tt:mark:0':ok -> tt:mark:0':ok
plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U31 :: tt:mark:0':ok -> tt:mark:0':ok
0' :: tt:mark:0':ok
U41 :: 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
and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
isNat :: tt:mark:0':ok -> 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:
s(gen_tt:mark:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
plus(gen_tt:mark:0':ok3_0(+(1, n437_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n437_0)
x(gen_tt:mark:0':ok3_0(+(1, n2049_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n2049_0)
and(gen_tt:mark:0':ok3_0(+(1, n3967_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3967_0)
U11(gen_tt:mark:0':ok3_0(+(1, n6014_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n6014_0)
U21(gen_tt:mark:0':ok3_0(+(1, n8345_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n8345_0)
U31(gen_tt:mark:0':ok3_0(+(1, n12486_0))) -> *4_0, rt in Omega(n12486_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:
U41, active, proper, top

They will be analysed ascendingly in the following order:
U41 < active
active < top
U41 < proper
proper < top

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U41(gen_tt:mark:0':ok3_0(+(1, n13818_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n13818_0)

Induction Base:
U41(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))

Induction Step:
U41(gen_tt:mark:0':ok3_0(+(1, +(n13818_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
mark(U41(gen_tt:mark:0':ok3_0(+(1, n13818_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).
----------------------------------------

(32)
Obligation:
TRS:
Rules:
active(U11(tt, N)) -> mark(N)
active(U21(tt, M, N)) -> mark(s(plus(N, M)))
active(U31(tt)) -> mark(0')
active(U41(tt, M, N)) -> mark(plus(x(N, M), N))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(isNat(s(V1))) -> mark(isNat(V1))
active(isNat(x(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
active(plus(N, 0')) -> mark(U11(isNat(N), N))
active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
active(x(N, 0')) -> mark(U31(isNat(N)))
active(x(N, s(M))) -> mark(U41(and(isNat(M), isNat(N)), M, N))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U21(X1, X2, X3)) -> U21(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(U31(X)) -> U31(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(x(X1, X2)) -> x(active(X1), X2)
active(x(X1, X2)) -> x(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
U11(mark(X1), X2) -> mark(U11(X1, X2))
U21(mark(X1), X2, X3) -> mark(U21(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))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
x(mark(X1), X2) -> mark(x(X1, X2))
x(X1, mark(X2)) -> mark(x(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
proper(s(X)) -> s(proper(X))
proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
proper(U31(X)) -> U31(proper(X))
proper(0') -> ok(0')
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) -> x(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
s(ok(X)) -> ok(s(X))
plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
x(ok(X1), ok(X2)) -> ok(x(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
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 :: tt:mark:0':ok
mark :: 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
s :: tt:mark:0':ok -> tt:mark:0':ok
plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
U31 :: tt:mark:0':ok -> tt:mark:0':ok
0' :: tt:mark:0':ok
U41 :: 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
and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
isNat :: tt:mark:0':ok -> 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:
s(gen_tt:mark:0':ok3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
plus(gen_tt:mark:0':ok3_0(+(1, n437_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n437_0)
x(gen_tt:mark:0':ok3_0(+(1, n2049_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n2049_0)
and(gen_tt:mark:0':ok3_0(+(1, n3967_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3967_0)
U11(gen_tt:mark:0':ok3_0(+(1, n6014_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n6014_0)
U21(gen_tt:mark:0':ok3_0(+(1, n8345_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n8345_0)
U31(gen_tt:mark:0':ok3_0(+(1, n12486_0))) -> *4_0, rt in Omega(n12486_0)
U41(gen_tt:mark:0':ok3_0(+(1, n13818_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n13818_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