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

0 CpxTRS
↳1 NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CpxTRS
    ↳3 CpxTrsMatchBoundsTAProof (⇔, 94 ms)
    ↳4 BOUNDS(1, n^1)
    ↳5 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳6 CdtProblem
        ↳7 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳8 CdtProblem
        ↳9 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳10 CdtProblem
        ↳11 CdtUsableRulesProof (⇔, 0 ms)
        ↳12 CdtProblem
        ↳13 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 94 ms)
        ↳14 CdtProblem
        ↳15 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 14 ms)
        ↳16 CdtProblem
        ↳17 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 27 ms)
        ↳18 CdtProblem
        ↳19 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 26 ms)
        ↳20 CdtProblem
        ↳21 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 25 ms)
        ↳22 CdtProblem
        ↳23 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)
        ↳24 CdtProblem
        ↳25 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)
        ↳26 CdtProblem
        ↳27 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 16 ms)
        ↳28 CdtProblem
        ↳29 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)
        ↳30 CdtProblem
        ↳31 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳32 BOUNDS(1, 1)

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(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(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(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(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(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))
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(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
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))
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))

Rewrite Strategy: INNERMOST

(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)

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(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(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(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(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(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))

(2) Obligation:

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


The TRS R consists of the following rules:

U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
top(ok(X)) → top(active(X))
proper(tt) → ok(tt)
and(mark(X1), X2) → mark(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
s(ok(X)) → ok(s(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
proper(0) → ok(0)
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
top(mark(X)) → top(proper(X))

Rewrite Strategy: INNERMOST

(3) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

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]
transitions:
ok0(0) → 0
active0(0) → 0
tt0() → 0
mark0(0) → 0
00() → 0
U110(0, 0) → 1
top0(0) → 2
proper0(0) → 3
and0(0, 0) → 4
isNat0(0) → 5
s0(0) → 6
plus0(0, 0) → 7
U210(0, 0, 0) → 8
U111(0, 0) → 9
ok1(9) → 1
active1(0) → 10
top1(10) → 2
tt1() → 11
ok1(11) → 3
and1(0, 0) → 12
mark1(12) → 4
isNat1(0) → 13
ok1(13) → 5
and1(0, 0) → 14
ok1(14) → 4
s1(0) → 15
ok1(15) → 6
U111(0, 0) → 16
mark1(16) → 1
s1(0) → 17
mark1(17) → 6
plus1(0, 0) → 18
mark1(18) → 7
U211(0, 0, 0) → 19
ok1(19) → 8
01() → 20
ok1(20) → 3
plus1(0, 0) → 21
ok1(21) → 7
U211(0, 0, 0) → 22
mark1(22) → 8
proper1(0) → 23
top1(23) → 2
ok1(9) → 9
ok1(9) → 16
ok1(11) → 23
mark1(12) → 12
mark1(12) → 14
ok1(13) → 13
ok1(14) → 12
ok1(14) → 14
ok1(15) → 15
ok1(15) → 17
mark1(16) → 9
mark1(16) → 16
mark1(17) → 15
mark1(17) → 17
mark1(18) → 18
mark1(18) → 21
ok1(19) → 19
ok1(19) → 22
ok1(20) → 23
ok1(21) → 18
ok1(21) → 21
mark1(22) → 19
mark1(22) → 22
active2(11) → 24
top2(24) → 2
active2(20) → 24

(4) BOUNDS(1, n^1)

(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted Cpx (relative) TRS to CDT

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U11(mark(z0), z1) → mark(U11(z0, z1))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
proper(tt) → ok(tt)
proper(0) → ok(0)
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
TOP(ok(z0)) → c2(TOP(active(z0)))
TOP(mark(z0)) → c3(TOP(proper(z0)), PROPER(z0))
PROPER(tt) → c4
PROPER(0) → c5
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
TOP(ok(z0)) → c2(TOP(active(z0)))
TOP(mark(z0)) → c3(TOP(proper(z0)), PROPER(z0))
PROPER(tt) → c4
PROPER(0) → c5
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
K tuples:none
Defined Rule Symbols:

U11, top, proper, and, isNat, s, plus, U21

Defined Pair Symbols:

U11', TOP, PROPER, AND, ISNAT, S, PLUS, U21'

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15

(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing nodes:

TOP(ok(z0)) → c2(TOP(active(z0)))
PROPER(tt) → c4
PROPER(0) → c5

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U11(mark(z0), z1) → mark(U11(z0, z1))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
proper(tt) → ok(tt)
proper(0) → ok(0)
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
TOP(mark(z0)) → c3(TOP(proper(z0)), PROPER(z0))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
TOP(mark(z0)) → c3(TOP(proper(z0)), PROPER(z0))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
K tuples:none
Defined Rule Symbols:

U11, top, proper, and, isNat, s, plus, U21

Defined Pair Symbols:

U11', TOP, AND, ISNAT, S, PLUS, U21'

Compound Symbols:

c, c1, c3, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U11(mark(z0), z1) → mark(U11(z0, z1))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
proper(tt) → ok(tt)
proper(0) → ok(0)
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
K tuples:none
Defined Rule Symbols:

U11, top, proper, and, isNat, s, plus, U21

Defined Pair Symbols:

U11', AND, ISNAT, S, PLUS, U21', TOP

Compound Symbols:

c, c1, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c3

(11) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U11(mark(z0), z1) → mark(U11(z0, z1))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)
Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
K tuples:none
Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', AND, ISNAT, S, PLUS, U21', TOP

Compound Symbols:

c, c1, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c3

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
We considered the (Usable) Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)
And the Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AND(x1, x2)) = 0   
POL(ISNAT(x1)) = 0   
POL(PLUS(x1, x2)) = 0   
POL(S(x1)) = 0   
POL(TOP(x1)) = [2]x1   
POL(U11'(x1, x2)) = 0   
POL(U21'(x1, x2, x3)) = [2]x3   
POL(c(x1)) = x1   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c3(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1)) = x1   
POL(mark(x1)) = [2] + x1   
POL(ok(x1)) = [1] + x1   
POL(proper(x1)) = [2]   
POL(tt) = 0   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)
Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
K tuples:

U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', AND, ISNAT, S, PLUS, U21', TOP

Compound Symbols:

c, c1, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c3

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

TOP(mark(z0)) → c3(TOP(proper(z0)))
We considered the (Usable) Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)
And the Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AND(x1, x2)) = 0   
POL(ISNAT(x1)) = 0   
POL(PLUS(x1, x2)) = 0   
POL(S(x1)) = 0   
POL(TOP(x1)) = x1   
POL(U11'(x1, x2)) = 0   
POL(U21'(x1, x2, x3)) = 0   
POL(c(x1)) = x1   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c3(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1)) = x1   
POL(mark(x1)) = [1] + x1   
POL(ok(x1)) = 0   
POL(proper(x1)) = 0   
POL(tt) = 0   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)
Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
K tuples:

U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', AND, ISNAT, S, PLUS, U21', TOP

Compound Symbols:

c, c1, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c3

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AND(x1, x2)) = x2   
POL(ISNAT(x1)) = 0   
POL(PLUS(x1, x2)) = x1 + x2   
POL(S(x1)) = 0   
POL(TOP(x1)) = 0   
POL(U11'(x1, x2)) = 0   
POL(U21'(x1, x2, x3)) = x3   
POL(c(x1)) = x1   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c3(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1)) = x1   
POL(mark(x1)) = [1] + x1   
POL(ok(x1)) = [1] + x1   
POL(proper(x1)) = 0   
POL(tt) = 0   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)
Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
K tuples:

U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', AND, ISNAT, S, PLUS, U21', TOP

Compound Symbols:

c, c1, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c3

(19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AND(x1, x2)) = 0   
POL(ISNAT(x1)) = 0   
POL(PLUS(x1, x2)) = 0   
POL(S(x1)) = 0   
POL(TOP(x1)) = 0   
POL(U11'(x1, x2)) = 0   
POL(U21'(x1, x2, x3)) = x1   
POL(c(x1)) = x1   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c3(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1)) = x1   
POL(mark(x1)) = [1] + x1   
POL(ok(x1)) = x1   
POL(proper(x1)) = [1]   
POL(tt) = [1]   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)
Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
K tuples:

U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', AND, ISNAT, S, PLUS, U21', TOP

Compound Symbols:

c, c1, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c3

(21) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
We considered the (Usable) Rules:none
And the Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AND(x1, x2)) = 0   
POL(ISNAT(x1)) = 0   
POL(PLUS(x1, x2)) = [2]x2   
POL(S(x1)) = x1   
POL(TOP(x1)) = 0   
POL(U11'(x1, x2)) = 0   
POL(U21'(x1, x2, x3)) = [2]x2 + [2]x3   
POL(c(x1)) = x1   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c3(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1)) = x1   
POL(mark(x1)) = [2] + x1   
POL(ok(x1)) = [2] + x1   
POL(proper(x1)) = 0   
POL(tt) = 0   

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)
Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
K tuples:

U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', AND, ISNAT, S, PLUS, U21', TOP

Compound Symbols:

c, c1, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c3

(23) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

AND(mark(z0), z1) → c6(AND(z0, z1))
We considered the (Usable) Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)
And the Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AND(x1, x2)) = x1   
POL(ISNAT(x1)) = 0   
POL(PLUS(x1, x2)) = 0   
POL(S(x1)) = 0   
POL(TOP(x1)) = [2]x1   
POL(U11'(x1, x2)) = 0   
POL(U21'(x1, x2, x3)) = 0   
POL(c(x1)) = x1   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c3(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1)) = x1   
POL(mark(x1)) = [1] + x1   
POL(ok(x1)) = x1   
POL(proper(x1)) = 0   
POL(tt) = 0   

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)
Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
K tuples:

U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
AND(mark(z0), z1) → c6(AND(z0, z1))
Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', AND, ISNAT, S, PLUS, U21', TOP

Compound Symbols:

c, c1, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c3

(25) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U11'(mark(z0), z1) → c1(U11'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [1]   
POL(AND(x1, x2)) = 0   
POL(ISNAT(x1)) = 0   
POL(PLUS(x1, x2)) = 0   
POL(S(x1)) = 0   
POL(TOP(x1)) = 0   
POL(U11'(x1, x2)) = x1   
POL(U21'(x1, x2, x3)) = 0   
POL(c(x1)) = x1   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c3(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1)) = x1   
POL(mark(x1)) = [1] + x1   
POL(ok(x1)) = x1   
POL(proper(x1)) = [1] + x1   
POL(tt) = 0   

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)
Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
K tuples:

U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
AND(mark(z0), z1) → c6(AND(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', AND, ISNAT, S, PLUS, U21', TOP

Compound Symbols:

c, c1, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c3

(27) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

ISNAT(ok(z0)) → c8(ISNAT(z0))
We considered the (Usable) Rules:none
And the Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AND(x1, x2)) = 0   
POL(ISNAT(x1)) = x1   
POL(PLUS(x1, x2)) = 0   
POL(S(x1)) = 0   
POL(TOP(x1)) = 0   
POL(U11'(x1, x2)) = 0   
POL(U21'(x1, x2, x3)) = 0   
POL(c(x1)) = x1   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c3(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1)) = x1   
POL(mark(x1)) = 0   
POL(ok(x1)) = [1] + x1   
POL(proper(x1)) = 0   
POL(tt) = 0   

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)
Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
K tuples:

U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
AND(mark(z0), z1) → c6(AND(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', AND, ISNAT, S, PLUS, U21', TOP

Compound Symbols:

c, c1, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c3

(29) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AND(x1, x2)) = 0   
POL(ISNAT(x1)) = 0   
POL(PLUS(x1, x2)) = 0   
POL(S(x1)) = 0   
POL(TOP(x1)) = 0   
POL(U11'(x1, x2)) = x1   
POL(U21'(x1, x2, x3)) = 0   
POL(c(x1)) = x1   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c11(x1)) = x1   
POL(c12(x1)) = x1   
POL(c13(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c3(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1)) = x1   
POL(mark(x1)) = x1   
POL(ok(x1)) = [1] + x1   
POL(proper(x1)) = 0   
POL(tt) = 0   

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)
Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
AND(mark(z0), z1) → c6(AND(z0, z1))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
S tuples:none
K tuples:

U21'(ok(z0), ok(z1), ok(z2)) → c14(U21'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))
AND(ok(z0), ok(z1)) → c7(AND(z0, z1))
PLUS(mark(z0), z1) → c11(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c12(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c13(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c15(U21'(z0, z1, z2))
S(ok(z0)) → c9(S(z0))
S(mark(z0)) → c10(S(z0))
AND(mark(z0), z1) → c6(AND(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', AND, ISNAT, S, PLUS, U21', TOP

Compound Symbols:

c, c1, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c3

(31) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

(32) BOUNDS(1, 1)