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), 20 ms)
↳2 CpxTRS
    ↳3 CpxTrsMatchBoundsTAProof (⇔, 184 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)), 135 ms)
        ↳14 CdtProblem
        ↳15 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 34 ms)
        ↳16 CdtProblem
        ↳17 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 27 ms)
        ↳18 CdtProblem
        ↳19 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)
        ↳20 CdtProblem
        ↳21 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)
        ↳22 CdtProblem
        ↳23 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 1 ms)
        ↳24 CdtProblem
        ↳25 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 17 ms)
        ↳26 CdtProblem
        ↳27 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳28 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, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
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, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))

(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))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
top(ok(X)) → top(active(X))
proper(tt) → ok(tt)
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U12(mark(X)) → mark(U12(X))
isNat(ok(X)) → ok(isNat(X))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U12(ok(X)) → ok(U12(X))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
proper(0) → ok(0)
U31(mark(X1), X2) → mark(U31(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
top(mark(X)) → top(proper(X))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))

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, 9, 10, 11]
transitions:
ok0(0) → 0
mark0(0) → 0
active0(0) → 0
tt0() → 0
00() → 0
U110(0, 0) → 1
U420(0, 0, 0) → 2
top0(0) → 3
proper0(0) → 4
U310(0, 0) → 5
U120(0) → 6
isNat0(0) → 7
s0(0) → 8
plus0(0, 0) → 9
U410(0, 0, 0) → 10
U210(0) → 11
U111(0, 0) → 12
ok1(12) → 1
U421(0, 0, 0) → 13
mark1(13) → 2
active1(0) → 14
top1(14) → 3
tt1() → 15
ok1(15) → 4
U311(0, 0) → 16
ok1(16) → 5
U121(0) → 17
mark1(17) → 6
isNat1(0) → 18
ok1(18) → 7
U421(0, 0, 0) → 19
ok1(19) → 2
U121(0) → 20
ok1(20) → 6
s1(0) → 21
ok1(21) → 8
U111(0, 0) → 22
mark1(22) → 1
s1(0) → 23
mark1(23) → 8
plus1(0, 0) → 24
mark1(24) → 9
U411(0, 0, 0) → 25
mark1(25) → 10
01() → 26
ok1(26) → 4
U311(0, 0) → 27
mark1(27) → 5
plus1(0, 0) → 28
ok1(28) → 9
proper1(0) → 29
top1(29) → 3
U211(0) → 30
mark1(30) → 11
U211(0) → 31
ok1(31) → 11
U411(0, 0, 0) → 32
ok1(32) → 10
ok1(12) → 12
ok1(12) → 22
mark1(13) → 13
mark1(13) → 19
ok1(15) → 29
ok1(16) → 16
ok1(16) → 27
mark1(17) → 17
mark1(17) → 20
ok1(18) → 18
ok1(19) → 13
ok1(19) → 19
ok1(20) → 17
ok1(20) → 20
ok1(21) → 21
ok1(21) → 23
mark1(22) → 12
mark1(22) → 22
mark1(23) → 21
mark1(23) → 23
mark1(24) → 24
mark1(24) → 28
mark1(25) → 25
mark1(25) → 32
ok1(26) → 29
mark1(27) → 16
mark1(27) → 27
ok1(28) → 24
ok1(28) → 28
mark1(30) → 30
mark1(30) → 31
ok1(31) → 30
ok1(31) → 31
ok1(32) → 25
ok1(32) → 32
active2(15) → 33
top2(33) → 3
active2(26) → 33

(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))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(z0, z1, z2))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
proper(tt) → ok(tt)
proper(0) → ok(0)
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
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))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
TOP(ok(z0)) → c4(TOP(active(z0)))
TOP(mark(z0)) → c5(TOP(proper(z0)), PROPER(z0))
PROPER(tt) → c6
PROPER(0) → c7
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
TOP(ok(z0)) → c4(TOP(active(z0)))
TOP(mark(z0)) → c5(TOP(proper(z0)), PROPER(z0))
PROPER(tt) → c6
PROPER(0) → c7
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
K tuples:none
Defined Rule Symbols:

U11, U42, top, proper, U31, U12, isNat, s, plus, U41, U21

Defined Pair Symbols:

U11', U42', TOP, PROPER, U31', U12', ISNAT, S, PLUS, U41', U21'

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21

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

Removed 3 trailing nodes:

PROPER(0) → c7
PROPER(tt) → c6
TOP(ok(z0)) → c4(TOP(active(z0)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U11(mark(z0), z1) → mark(U11(z0, z1))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(z0, z1, z2))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
proper(tt) → ok(tt)
proper(0) → ok(0)
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
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))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
TOP(mark(z0)) → c5(TOP(proper(z0)), PROPER(z0))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
TOP(mark(z0)) → c5(TOP(proper(z0)), PROPER(z0))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
K tuples:none
Defined Rule Symbols:

U11, U42, top, proper, U31, U12, isNat, s, plus, U41, U21

Defined Pair Symbols:

U11', U42', TOP, U31', U12', ISNAT, S, PLUS, U41', U21'

Compound Symbols:

c, c1, c2, c3, c5, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21

(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))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(z0, z1, z2))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
proper(tt) → ok(tt)
proper(0) → ok(0)
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
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))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
TOP(mark(z0)) → c5(TOP(proper(z0)))
S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
TOP(mark(z0)) → c5(TOP(proper(z0)))
K tuples:none
Defined Rule Symbols:

U11, U42, top, proper, U31, U12, isNat, s, plus, U41, U21

Defined Pair Symbols:

U11', U42', U31', U12', ISNAT, S, PLUS, U41', U21', TOP

Compound Symbols:

c, c1, c2, c3, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c5

(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))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(z0, z1, z2))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
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))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))

(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))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
TOP(mark(z0)) → c5(TOP(proper(z0)))
S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
TOP(mark(z0)) → c5(TOP(proper(z0)))
K tuples:none
Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U42', U31', U12', ISNAT, S, PLUS, U41', U21', TOP

Compound Symbols:

c, c1, c2, c3, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c5

(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.

TOP(mark(z0)) → c5(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))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
TOP(mark(z0)) → c5(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(ISNAT(x1)) = 0   
POL(PLUS(x1, x2)) = 0   
POL(S(x1)) = 0   
POL(TOP(x1)) = x1   
POL(U11'(x1, x2)) = 0   
POL(U12'(x1)) = 0   
POL(U21'(x1)) = 0   
POL(U31'(x1, x2)) = 0   
POL(U41'(x1, x2, x3)) = 0   
POL(U42'(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(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c2(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c3(x1)) = x1   
POL(c5(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   

(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))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
TOP(mark(z0)) → c5(TOP(proper(z0)))
S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
K tuples:

TOP(mark(z0)) → c5(TOP(proper(z0)))
Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U42', U31', U12', ISNAT, S, PLUS, U41', U21', TOP

Compound Symbols:

c, c1, c2, c3, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c5

(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.

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(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))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
TOP(mark(z0)) → c5(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(ISNAT(x1)) = 0   
POL(PLUS(x1, x2)) = 0   
POL(S(x1)) = x1   
POL(TOP(x1)) = 0   
POL(U11'(x1, x2)) = x1 + x2   
POL(U12'(x1)) = 0   
POL(U21'(x1)) = 0   
POL(U31'(x1, x2)) = x1   
POL(U41'(x1, x2, x3)) = 0   
POL(U42'(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(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c2(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c3(x1)) = x1   
POL(c5(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   

(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))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
TOP(mark(z0)) → c5(TOP(proper(z0)))
S tuples:

U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
K tuples:

TOP(mark(z0)) → c5(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U42', U31', U12', ISNAT, S, PLUS, U41', U21', TOP

Compound Symbols:

c, c1, c2, c3, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c5

(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.

U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(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))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
TOP(mark(z0)) → c5(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(ISNAT(x1)) = 0   
POL(PLUS(x1, x2)) = x1   
POL(S(x1)) = 0   
POL(TOP(x1)) = 0   
POL(U11'(x1, x2)) = x1   
POL(U12'(x1)) = 0   
POL(U21'(x1)) = x1   
POL(U31'(x1, x2)) = 0   
POL(U41'(x1, x2, x3)) = x1   
POL(U42'(x1, x2, x3)) = x1 + 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(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c2(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c3(x1)) = x1   
POL(c5(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   

(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))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
TOP(mark(z0)) → c5(TOP(proper(z0)))
S tuples:

U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(ok(z0)) → c21(U21'(z0))
K tuples:

TOP(mark(z0)) → c5(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U42', U31', U12', ISNAT, S, PLUS, U41', U21', TOP

Compound Symbols:

c, c1, c2, c3, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c5

(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.

PLUS(z0, mark(z1)) → c17(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))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
TOP(mark(z0)) → c5(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [1]   
POL(ISNAT(x1)) = 0   
POL(PLUS(x1, x2)) = [2]x2   
POL(S(x1)) = 0   
POL(TOP(x1)) = 0   
POL(U11'(x1, x2)) = [2]x2   
POL(U12'(x1)) = 0   
POL(U21'(x1)) = 0   
POL(U31'(x1, x2)) = [2]x1   
POL(U41'(x1, x2, x3)) = 0   
POL(U42'(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(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c2(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c3(x1)) = x1   
POL(c5(x1)) = x1   
POL(c8(x1)) = x1   
POL(c9(x1)) = x1   
POL(mark(x1)) = [2] + x1   
POL(ok(x1)) = x1   
POL(proper(x1)) = [1] + x1   
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))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
TOP(mark(z0)) → c5(TOP(proper(z0)))
S tuples:

U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(ok(z0)) → c21(U21'(z0))
K tuples:

TOP(mark(z0)) → c5(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U42', U31', U12', ISNAT, S, PLUS, U41', U21', TOP

Compound Symbols:

c, c1, c2, c3, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c5

(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.

ISNAT(ok(z0)) → c12(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))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
TOP(mark(z0)) → c5(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(ISNAT(x1)) = x1   
POL(PLUS(x1, x2)) = 0   
POL(S(x1)) = 0   
POL(TOP(x1)) = 0   
POL(U11'(x1, x2)) = [3]x2   
POL(U12'(x1)) = 0   
POL(U21'(x1)) = 0   
POL(U31'(x1, x2)) = [2]x2   
POL(U41'(x1, x2, x3)) = 0   
POL(U42'(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(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c2(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c3(x1)) = x1   
POL(c5(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   

(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))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
TOP(mark(z0)) → c5(TOP(proper(z0)))
S tuples:

U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(ok(z0)) → c21(U21'(z0))
K tuples:

TOP(mark(z0)) → c5(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
ISNAT(ok(z0)) → c12(ISNAT(z0))
Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U42', U31', U12', ISNAT, S, PLUS, U41', U21', TOP

Compound Symbols:

c, c1, c2, c3, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c5

(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.

U12'(ok(z0)) → c11(U12'(z0))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(ok(z0)) → c21(U21'(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))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
TOP(mark(z0)) → c5(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(ISNAT(x1)) = x1   
POL(PLUS(x1, x2)) = x1   
POL(S(x1)) = 0   
POL(TOP(x1)) = 0   
POL(U11'(x1, x2)) = 0   
POL(U12'(x1)) = x1   
POL(U21'(x1)) = x1   
POL(U31'(x1, x2)) = 0   
POL(U41'(x1, x2, x3)) = x1 + x2   
POL(U42'(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(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c2(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c3(x1)) = x1   
POL(c5(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   

(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))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
TOP(mark(z0)) → c5(TOP(proper(z0)))
S tuples:

U12'(mark(z0)) → c10(U12'(z0))
K tuples:

TOP(mark(z0)) → c5(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
ISNAT(ok(z0)) → c12(ISNAT(z0))
U12'(ok(z0)) → c11(U12'(z0))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(ok(z0)) → c21(U21'(z0))
Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U42', U31', U12', ISNAT, S, PLUS, U41', U21', TOP

Compound Symbols:

c, c1, c2, c3, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c5

(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.

U12'(mark(z0)) → c10(U12'(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))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
TOP(mark(z0)) → c5(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(ISNAT(x1)) = 0   
POL(PLUS(x1, x2)) = 0   
POL(S(x1)) = 0   
POL(TOP(x1)) = 0   
POL(U11'(x1, x2)) = 0   
POL(U12'(x1)) = x1   
POL(U21'(x1)) = 0   
POL(U31'(x1, x2)) = 0   
POL(U41'(x1, x2, x3)) = 0   
POL(U42'(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(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c2(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c3(x1)) = x1   
POL(c5(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   

(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))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
U12'(mark(z0)) → c10(U12'(z0))
U12'(ok(z0)) → c11(U12'(z0))
ISNAT(ok(z0)) → c12(ISNAT(z0))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
U21'(ok(z0)) → c21(U21'(z0))
TOP(mark(z0)) → c5(TOP(proper(z0)))
S tuples:none
K tuples:

TOP(mark(z0)) → c5(TOP(proper(z0)))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U42'(ok(z0), ok(z1), ok(z2)) → c3(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c8(U31'(z0, z1))
U31'(mark(z0), z1) → c9(U31'(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
U42'(mark(z0), z1, z2) → c2(U42'(z0, z1, z2))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
U41'(mark(z0), z1, z2) → c18(U41'(z0, z1, z2))
U21'(mark(z0)) → c20(U21'(z0))
PLUS(z0, mark(z1)) → c17(PLUS(z0, z1))
ISNAT(ok(z0)) → c12(ISNAT(z0))
U12'(ok(z0)) → c11(U12'(z0))
PLUS(ok(z0), ok(z1)) → c16(PLUS(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c19(U41'(z0, z1, z2))
U21'(ok(z0)) → c21(U21'(z0))
U12'(mark(z0)) → c10(U12'(z0))
Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U42', U31', U12', ISNAT, S, PLUS, U41', U21', TOP

Compound Symbols:

c, c1, c2, c3, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c5

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

The set S is empty

(28) BOUNDS(1, 1)