Runtime Complexity (innermost) proof of /tmp/tmpY4COSz/MYNAT_nokinds_noand

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), 25 ms)

↳2 CpxTRS

    ↳3 CpxTrsMatchBoundsTAProof (⇔, 485 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)), 197 ms)

        ↳14 CdtProblem

        ↳15 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 44 ms)

        ↳16 CdtProblem

        ↳17 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 52 ms)

        ↳18 CdtProblem

        ↳19 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 41 ms)

        ↳20 CdtProblem

        ↳21 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 47 ms)

        ↳22 CdtProblem

        ↳23 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 36 ms)

        ↳24 CdtProblem

        ↳25 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 45 ms)

        ↳26 CdtProblem

        ↳27 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 51 ms)

        ↳28 CdtProblem

        ↳29 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 24 ms)

        ↳30 CdtProblem

        ↳31 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 32 ms)

        ↳32 CdtProblem

        ↳33 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 14 ms)

        ↳34 CdtProblem

        ↳35 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 27 ms)

        ↳36 CdtProblem

        ↳37 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 43 ms)

        ↳38 CdtProblem

        ↳39 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 21 ms)

        ↳40 CdtProblem

        ↳41 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 17 ms)

        ↳42 CdtProblem

        ↳43 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 7 ms)

        ↳44 CdtProblem

        ↳45 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 13 ms)

        ↳46 CdtProblem

        ↳47 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 22 ms)

        ↳48 CdtProblem

        ↳49 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 30 ms)

        ↳50 CdtProblem

        ↳51 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)

        ↳52 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, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(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(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(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))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(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(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
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))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(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, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(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(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(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(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(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))
top(ok(X)) → top(active(X))
U12(mark(X)) → mark(U12(X))
isNat(ok(X)) → ok(isNat(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U12(ok(X)) → ok(U12(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U61(ok(X)) → ok(U61(X))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
U41(mark(X1), X2) → mark(U41(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
proper(tt) → ok(tt)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
x(mark(X1), X2) → mark(x(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
x(ok(X1), ok(X2)) → ok(x(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))
x(X1, mark(X2)) → mark(x(X1, X2))
U32(ok(X)) → ok(U32(X))
proper(0) → ok(0)
U31(mark(X1), X2) → mark(U31(X1, X2))
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, 9, 10, 11, 12, 13, 14, 15, 16, 17]
transitions:
ok0(0) → 0
active0(0) → 0
mark0(0) → 0
tt0() → 0
00() → 0
U110(0, 0) → 1
top0(0) → 2
U120(0) → 3
isNat0(0) → 4
U710(0, 0, 0) → 5
U610(0) → 6
U520(0, 0, 0) → 7
U720(0, 0, 0) → 8
U510(0, 0, 0) → 9
U410(0, 0) → 10
plus0(0, 0) → 11
U210(0) → 12
proper0(0) → 13
U320(0) → 14
U310(0, 0) → 15
x0(0, 0) → 16
s0(0) → 17
U111(0, 0) → 18
ok1(18) → 1
active1(0) → 19
top1(19) → 2
U121(0) → 20
mark1(20) → 3
isNat1(0) → 21
ok1(21) → 4
U711(0, 0, 0) → 22
ok1(22) → 5
U121(0) → 23
ok1(23) → 3
U611(0) → 24
mark1(24) → 6
U711(0, 0, 0) → 25
mark1(25) → 5
U611(0) → 26
ok1(26) → 6
U521(0, 0, 0) → 27
mark1(27) → 7
U721(0, 0, 0) → 28
mark1(28) → 8
U511(0, 0, 0) → 29
mark1(29) → 9
U521(0, 0, 0) → 30
ok1(30) → 7
U411(0, 0) → 31
mark1(31) → 10
plus1(0, 0) → 32
ok1(32) → 11
plus1(0, 0) → 33
mark1(33) → 11
U211(0) → 34
mark1(34) → 12
U211(0) → 35
ok1(35) → 12
U721(0, 0, 0) → 36
ok1(36) → 8
tt1() → 37
ok1(37) → 13
U511(0, 0, 0) → 38
ok1(38) → 9
U321(0) → 39
mark1(39) → 14
U311(0, 0) → 40
ok1(40) → 15
x1(0, 0) → 41
mark1(41) → 16
U411(0, 0) → 42
ok1(42) → 10
x1(0, 0) → 43
ok1(43) → 16
s1(0) → 44
ok1(44) → 17
U111(0, 0) → 45
mark1(45) → 1
s1(0) → 46
mark1(46) → 17
U321(0) → 47
ok1(47) → 14
01() → 48
ok1(48) → 13
U311(0, 0) → 49
mark1(49) → 15
proper1(0) → 50
top1(50) → 2
ok1(18) → 18
ok1(18) → 45
mark1(20) → 20
mark1(20) → 23
ok1(21) → 21
ok1(22) → 22
ok1(22) → 25
ok1(23) → 20
ok1(23) → 23
mark1(24) → 24
mark1(24) → 26
mark1(25) → 22
mark1(25) → 25
ok1(26) → 24
ok1(26) → 26
mark1(27) → 27
mark1(27) → 30
mark1(28) → 28
mark1(28) → 36
mark1(29) → 29
mark1(29) → 38
ok1(30) → 27
ok1(30) → 30
mark1(31) → 31
mark1(31) → 42
ok1(32) → 32
ok1(32) → 33
mark1(33) → 32
mark1(33) → 33
mark1(34) → 34
mark1(34) → 35
ok1(35) → 34
ok1(35) → 35
ok1(36) → 28
ok1(36) → 36
ok1(37) → 50
ok1(38) → 29
ok1(38) → 38
mark1(39) → 39
mark1(39) → 47
ok1(40) → 40
ok1(40) → 49
mark1(41) → 41
mark1(41) → 43
ok1(42) → 31
ok1(42) → 42
ok1(43) → 41
ok1(43) → 43
ok1(44) → 44
ok1(44) → 46
mark1(45) → 18
mark1(45) → 45
mark1(46) → 44
mark1(46) → 46
ok1(47) → 39
ok1(47) → 47
ok1(48) → 50
mark1(49) → 40
mark1(49) → 49
active2(37) → 51
top2(51) → 2
active2(48) → 51

(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U71(ok(z0), ok(z1), ok(z2)) → ok(U71(z0, z1, z2))
U71(mark(z0), z1, z2) → mark(U71(z0, z1, z2))
U61(mark(z0)) → mark(U61(z0))
U61(ok(z0)) → ok(U61(z0))
U52(mark(z0), z1, z2) → mark(U52(z0, z1, z2))
U52(ok(z0), ok(z1), ok(z2)) → ok(U52(z0, z1, z2))
U72(mark(z0), z1, z2) → mark(U72(z0, z1, z2))
U72(ok(z0), ok(z1), ok(z2)) → ok(U72(z0, z1, z2))
U51(mark(z0), z1, z2) → mark(U51(z0, z1, z2))
U51(ok(z0), ok(z1), ok(z2)) → ok(U51(z0, z1, z2))
U41(mark(z0), z1) → mark(U41(z0, z1))
U41(ok(z0), ok(z1)) → ok(U41(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
proper(tt) → ok(tt)
proper(0) → ok(0)
U32(mark(z0)) → mark(U32(z0))
U32(ok(z0)) → ok(U32(z0))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))

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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
PROPER(tt) → c24
PROPER(0) → c25
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))

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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
PROPER(tt) → c24
PROPER(0) → c25
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))

K tuples:none
Defined Rule Symbols:

U11, top, U12, isNat, U71, U61, U52, U72, U51, U41, plus, U21, proper, U32, U31, x, s

Defined Pair Symbols:

U11', TOP, U12', ISNAT, U71', U61', U52', U72', U51', U41', PLUS, U21', PROPER, U32', U31', X, S

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34

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

Removed 3 trailing nodes:

TOP(ok(z0)) → c2(TOP(active(z0)))
PROPER(tt) → c24
PROPER(0) → c25

(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U71(ok(z0), ok(z1), ok(z2)) → ok(U71(z0, z1, z2))
U71(mark(z0), z1, z2) → mark(U71(z0, z1, z2))
U61(mark(z0)) → mark(U61(z0))
U61(ok(z0)) → ok(U61(z0))
U52(mark(z0), z1, z2) → mark(U52(z0, z1, z2))
U52(ok(z0), ok(z1), ok(z2)) → ok(U52(z0, z1, z2))
U72(mark(z0), z1, z2) → mark(U72(z0, z1, z2))
U72(ok(z0), ok(z1), ok(z2)) → ok(U72(z0, z1, z2))
U51(mark(z0), z1, z2) → mark(U51(z0, z1, z2))
U51(ok(z0), ok(z1), ok(z2)) → ok(U51(z0, z1, z2))
U41(mark(z0), z1) → mark(U41(z0, z1))
U41(ok(z0), ok(z1)) → ok(U41(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
proper(tt) → ok(tt)
proper(0) → ok(0)
U32(mark(z0)) → mark(U32(z0))
U32(ok(z0)) → ok(U32(z0))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))

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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))

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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))

K tuples:none
Defined Rule Symbols:

U11, top, U12, isNat, U71, U61, U52, U72, U51, U41, plus, U21, proper, U32, U31, x, s

Defined Pair Symbols:

U11', TOP, U12', ISNAT, U71', U61', U52', U72', U51', U41', PLUS, U21', U32', U31', X, S

Compound Symbols:

c, c1, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34

(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U71(ok(z0), ok(z1), ok(z2)) → ok(U71(z0, z1, z2))
U71(mark(z0), z1, z2) → mark(U71(z0, z1, z2))
U61(mark(z0)) → mark(U61(z0))
U61(ok(z0)) → ok(U61(z0))
U52(mark(z0), z1, z2) → mark(U52(z0, z1, z2))
U52(ok(z0), ok(z1), ok(z2)) → ok(U52(z0, z1, z2))
U72(mark(z0), z1, z2) → mark(U72(z0, z1, z2))
U72(ok(z0), ok(z1), ok(z2)) → ok(U72(z0, z1, z2))
U51(mark(z0), z1, z2) → mark(U51(z0, z1, z2))
U51(ok(z0), ok(z1), ok(z2)) → ok(U51(z0, z1, z2))
U41(mark(z0), z1) → mark(U41(z0, z1))
U41(ok(z0), ok(z1)) → ok(U41(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
proper(tt) → ok(tt)
proper(0) → ok(0)
U32(mark(z0)) → mark(U32(z0))
U32(ok(z0)) → ok(U32(z0))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))

Tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

U11, top, U12, isNat, U71, U61, U52, U72, U51, U41, plus, U21, proper, U32, U31, x, s

Defined Pair Symbols:

U11', U12', ISNAT, U71', U61', U52', U72', U51', U41', PLUS, U21', U32', U31', X, S, TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, 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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U71(ok(z0), ok(z1), ok(z2)) → ok(U71(z0, z1, z2))
U71(mark(z0), z1, z2) → mark(U71(z0, z1, z2))
U61(mark(z0)) → mark(U61(z0))
U61(ok(z0)) → ok(U61(z0))
U52(mark(z0), z1, z2) → mark(U52(z0, z1, z2))
U52(ok(z0), ok(z1), ok(z2)) → ok(U52(z0, z1, z2))
U72(mark(z0), z1, z2) → mark(U72(z0, z1, z2))
U72(ok(z0), ok(z1), ok(z2)) → ok(U72(z0, z1, z2))
U51(mark(z0), z1, z2) → mark(U51(z0, z1, z2))
U51(ok(z0), ok(z1), ok(z2)) → ok(U51(z0, z1, z2))
U41(mark(z0), z1) → mark(U41(z0, z1))
U41(ok(z0), ok(z1)) → ok(U41(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
U32(mark(z0)) → mark(U32(z0))
U32(ok(z0)) → ok(U32(z0))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', ISNAT, U71', U61', U52', U72', U51', U41', PLUS, U21', U32', U31', X, S, TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, 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.

ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(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)) = 0
POL(U21'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂, x₃)) = x₂ + x₃
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = x₃
POL(U72'(x₁, x₂, x₃)) = x₂
POL(X(x₁, x₂)) = 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(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(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

(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

K tuples:

ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', ISNAT, U71', U61', U52', U72', U51', U41', PLUS, U21', U32', U31', X, S, TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, 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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(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(U32'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = 0
POL(U72'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = 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(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))

K tuples:

ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
TOP(mark(z0)) → c3(TOP(proper(z0)))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', ISNAT, U71', U61', U52', U72', U51', U41', PLUS, U21', U32', U31', X, S, TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, 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.

U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(ISNAT(x₁)) = [2]x₁
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = [2]x₁
POL(U52'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = x₁ + [2]x₃
POL(U72'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = 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(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(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

(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))

K tuples:

ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', ISNAT, U71', U61', U52', U72', U51', U41', PLUS, U21', U32', U31', X, S, TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, 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.

U12'(mark(z0)) → c4(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(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)) = 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(U32'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = 0
POL(U72'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = 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(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(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) = 0

(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
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))
U12'(ok(z0)) → c5(U12'(z0))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))

K tuples:

ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
U12'(mark(z0)) → c4(U12'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', ISNAT, U71', U61', U52', U72', U51', U41', PLUS, U21', U32', U31', X, S, TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, 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.

PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U31'(mark(z0), z1) → c29(U31'(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(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)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁)) = 0
POL(U31'(x₁, x₂)) = x₁
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂, x₃)) = x₃
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = 0
POL(U72'(x₁, x₂, x₃)) = x₂
POL(X(x₁, x₂)) = 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(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(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

(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
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))
U12'(ok(z0)) → c5(U12'(z0))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))

K tuples:

ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
U12'(mark(z0)) → c4(U12'(z0))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', ISNAT, U71', U61', U52', U72', U51', U41', PLUS, U21', U32', U31', X, S, TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, 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.

U72'(mark(z0), z1, z2) → c13(U72'(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(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)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = 0
POL(U72'(x₁, x₂, x₃)) = x₁
POL(X(x₁, x₂)) = 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(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(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) = [1]

(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
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))
U12'(ok(z0)) → c5(U12'(z0))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))

K tuples:

ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
U12'(mark(z0)) → c4(U12'(z0))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', ISNAT, U71', U61', U52', U72', U51', U41', PLUS, U21', U32', U31', X, S, TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, 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.

U32'(mark(z0)) → c26(U32'(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(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)) = 0
POL(U21'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = x₁
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = 0
POL(U72'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = 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(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(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

(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
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))
U12'(ok(z0)) → c5(U12'(z0))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))

K tuples:

ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
U12'(mark(z0)) → c4(U12'(z0))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U32'(mark(z0)) → c26(U32'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', ISNAT, U71', U61', U52', U72', U51', U41', PLUS, U21', U32', U31', X, S, TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, 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.

U52'(mark(z0), z1, z2) → c11(U52'(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(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)) = 0
POL(U21'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = x₃
POL(U52'(x₁, x₂, x₃)) = x₁ + x₂
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = x₂
POL(U72'(x₁, x₂, x₃)) = x₂ + x₃
POL(X(x₁, x₂)) = 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(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(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

(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
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))
U12'(ok(z0)) → c5(U12'(z0))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))

K tuples:

ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
U12'(mark(z0)) → c4(U12'(z0))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U32'(mark(z0)) → c26(U32'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', ISNAT, U71', U61', U52', U72', U51', U41', PLUS, U21', U32', U31', X, S, TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, 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.

U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(ISNAT(x₁)) = [2]x₁
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁)) = x₁
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = [3]x₃
POL(U52'(x₁, x₂, x₃)) = [2]x₃
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = [3]x₃
POL(U72'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = 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(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(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

(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
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))
U12'(ok(z0)) → c5(U12'(z0))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))

K tuples:

ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
U12'(mark(z0)) → c4(U12'(z0))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U32'(mark(z0)) → c26(U32'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', ISNAT, U71', U61', U52', U72', U51', U41', PLUS, U21', U32', U31', X, S, TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c3

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

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

U31'(ok(z0), ok(z1)) → c28(U31'(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(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)) = 0
POL(U21'(x₁)) = 0
POL(U31'(x₁, x₂)) = x₁
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = x₂
POL(U52'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = 0
POL(U72'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = 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(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(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

(32) 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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
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))
U12'(ok(z0)) → c5(U12'(z0))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
U32'(ok(z0)) → c27(U32'(z0))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))

K tuples:

ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
U12'(mark(z0)) → c4(U12'(z0))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U32'(mark(z0)) → c26(U32'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', ISNAT, U71', U61', U52', U72', U51', U41', PLUS, U21', U32', U31', X, S, TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c3

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

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

U61'(mark(z0)) → c9(U61'(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(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)) = 0
POL(U21'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = x₃
POL(U52'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = x₁
POL(U71'(x₁, x₂, x₃)) = x₁
POL(U72'(x₁, x₂, x₃)) = x₃
POL(X(x₁, x₂)) = 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(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(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

(34) 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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
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))
U12'(ok(z0)) → c5(U12'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
U32'(ok(z0)) → c27(U32'(z0))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))

K tuples:

ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
U12'(mark(z0)) → c4(U12'(z0))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U32'(mark(z0)) → c26(U32'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U61'(mark(z0)) → c9(U61'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', ISNAT, U71', U61', U52', U72', U51', U41', PLUS, U21', U32', U31', X, S, TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c3

(35) 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))
S(mark(z0)) → c34(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(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
POL(U12'(x₁)) = 0
POL(U21'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = x₂
POL(U52'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = 0
POL(U72'(x₁, x₂, x₃)) = x₂
POL(X(x₁, x₂)) = 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(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(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) = 0

(36) 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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U12'(ok(z0)) → c5(U12'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
U32'(ok(z0)) → c27(U32'(z0))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))

K tuples:

ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
U12'(mark(z0)) → c4(U12'(z0))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U32'(mark(z0)) → c26(U32'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U61'(mark(z0)) → c9(U61'(z0))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
S(mark(z0)) → c34(S(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', ISNAT, U71', U61', U52', U72', U51', U41', PLUS, U21', U32', U31', X, S, TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c3

(37) 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)) → c33(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(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)) = [2]x₁
POL(S(x₁)) = [2]x₁
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁)) = 0
POL(U31'(x₁, x₂)) = [2]x₁
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = [2]x₁
POL(U52'(x₁, x₂, x₃)) = [2]x₁ + [2]x₂
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = x₂ + [2]x₃
POL(U72'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = [2]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(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(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

(38) 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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U12'(ok(z0)) → c5(U12'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
U32'(ok(z0)) → c27(U32'(z0))
X(z0, mark(z1)) → c32(X(z0, z1))

K tuples:

ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
U12'(mark(z0)) → c4(U12'(z0))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U32'(mark(z0)) → c26(U32'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U61'(mark(z0)) → c9(U61'(z0))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
S(mark(z0)) → c34(S(z0))
S(ok(z0)) → c33(S(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', ISNAT, U71', U61', U52', U72', U51', U41', PLUS, U21', U32', U31', X, S, TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c3

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

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

U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(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)) = [2]x₁
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁)) = [2]x₁
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂)) = [2]x₁
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = 0
POL(U72'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = 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(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(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)) = [2] + x₁
POL(proper(x₁)) = 0
POL(tt) = 0

(40) 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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U12'(ok(z0)) → c5(U12'(z0))
U61'(ok(z0)) → c10(U61'(z0))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
U32'(ok(z0)) → c27(U32'(z0))
X(z0, mark(z1)) → c32(X(z0, z1))

K tuples:

ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
U12'(mark(z0)) → c4(U12'(z0))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U32'(mark(z0)) → c26(U32'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U61'(mark(z0)) → c9(U61'(z0))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
S(mark(z0)) → c34(S(z0))
S(ok(z0)) → c33(S(z0))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', ISNAT, U71', U61', U52', U72', U51', U41', PLUS, U21', U32', U31', X, S, TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c3

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

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

X(z0, mark(z1)) → c32(X(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(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)) = 0
POL(U21'(x₁)) = 0
POL(U31'(x₁, x₂)) = x₂
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = x₁
POL(U52'(x₁, x₂, x₃)) = x₃
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = 0
POL(U72'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = x₂
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(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(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

(42) 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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
U12'(ok(z0)) → c5(U12'(z0))
U61'(ok(z0)) → c10(U61'(z0))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
U32'(ok(z0)) → c27(U32'(z0))

K tuples:

ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
U12'(mark(z0)) → c4(U12'(z0))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U32'(mark(z0)) → c26(U32'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U61'(mark(z0)) → c9(U61'(z0))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
S(mark(z0)) → c34(S(z0))
S(ok(z0)) → c33(S(z0))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', ISNAT, U71', U61', U52', U72', U51', U41', PLUS, U21', U32', U31', X, S, TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c3

(43) 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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(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)) = x₂
POL(U12'(x₁)) = 0
POL(U21'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂)) = x₂
POL(U51'(x₁, x₂, x₃)) = x₂ + x₃
POL(U52'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = x₂
POL(U72'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = 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(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(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

(44) 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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U12'(ok(z0)) → c5(U12'(z0))
U61'(ok(z0)) → c10(U61'(z0))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
U32'(ok(z0)) → c27(U32'(z0))

K tuples:

ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
U12'(mark(z0)) → c4(U12'(z0))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U32'(mark(z0)) → c26(U32'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U61'(mark(z0)) → c9(U61'(z0))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
S(mark(z0)) → c34(S(z0))
S(ok(z0)) → c33(S(z0))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', ISNAT, U71', U61', U52', U72', U51', U41', PLUS, U21', U32', U31', X, S, TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c3

(45) 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)) → c20(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(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)) = x₂
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂)) = x₂
POL(U12'(x₁)) = 0
POL(U21'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = 0
POL(U41'(x₁, x₂)) = x₂
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = 0
POL(U72'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = 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(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(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

(46) 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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U12'(ok(z0)) → c5(U12'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U32'(ok(z0)) → c27(U32'(z0))

K tuples:

ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
U12'(mark(z0)) → c4(U12'(z0))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U32'(mark(z0)) → c26(U32'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U61'(mark(z0)) → c9(U61'(z0))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
S(mark(z0)) → c34(S(z0))
S(ok(z0)) → c33(S(z0))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', ISNAT, U71', U61', U52', U72', U51', U41', PLUS, U21', U32', U31', X, S, TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c3

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

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

U61'(ok(z0)) → c10(U61'(z0))
U32'(ok(z0)) → c27(U32'(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(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)) = 0
POL(U21'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U32'(x₁)) = x₁
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = x₁
POL(U71'(x₁, x₂, x₃)) = x₃
POL(U72'(x₁, x₂, x₃)) = x₂ + x₃
POL(X(x₁, x₂)) = 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(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(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

(48) 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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U12'(ok(z0)) → c5(U12'(z0))

K tuples:

ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
U12'(mark(z0)) → c4(U12'(z0))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U32'(mark(z0)) → c26(U32'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U61'(mark(z0)) → c9(U61'(z0))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
S(mark(z0)) → c34(S(z0))
S(ok(z0)) → c33(S(z0))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
U61'(ok(z0)) → c10(U61'(z0))
U32'(ok(z0)) → c27(U32'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', ISNAT, U71', U61', U52', U72', U51', U41', PLUS, U21', U32', U31', X, S, TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c3

(49) 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)) → c5(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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(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(U32'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U52'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = 0
POL(U72'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = 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(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(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

(50) 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))
U12'(mark(z0)) → c4(U12'(z0))
U12'(ok(z0)) → c5(U12'(z0))
ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U61'(mark(z0)) → c9(U61'(z0))
U61'(ok(z0)) → c10(U61'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U32'(mark(z0)) → c26(U32'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
S(ok(z0)) → c33(S(z0))
S(mark(z0)) → c34(S(z0))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:none
K tuples:

ISNAT(ok(z0)) → c6(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c7(U71'(z0, z1, z2))
U52'(ok(z0), ok(z1), ok(z2)) → c12(U52'(z0, z1, z2))
U72'(ok(z0), ok(z1), ok(z2)) → c14(U72'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
TOP(mark(z0)) → c3(TOP(proper(z0)))
U71'(mark(z0), z1, z2) → c8(U71'(z0, z1, z2))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
X(mark(z0), z1) → c30(X(z0, z1))
X(ok(z0), ok(z1)) → c31(X(z0, z1))
U12'(mark(z0)) → c4(U12'(z0))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U31'(mark(z0), z1) → c29(U31'(z0, z1))
U72'(mark(z0), z1, z2) → c13(U72'(z0, z1, z2))
U32'(mark(z0)) → c26(U32'(z0))
U52'(mark(z0), z1, z2) → c11(U52'(z0, z1, z2))
U21'(mark(z0)) → c22(U21'(z0))
U21'(ok(z0)) → c23(U21'(z0))
U31'(ok(z0), ok(z1)) → c28(U31'(z0, z1))
U61'(mark(z0)) → c9(U61'(z0))
U11'(mark(z0), z1) → c1(U11'(z0, z1))
S(mark(z0)) → c34(S(z0))
S(ok(z0)) → c33(S(z0))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
X(z0, mark(z1)) → c32(X(z0, z1))
U11'(ok(z0), ok(z1)) → c(U11'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
U61'(ok(z0)) → c10(U61'(z0))
U32'(ok(z0)) → c27(U32'(z0))
U12'(ok(z0)) → c5(U12'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U11', U12', ISNAT, U71', U61', U52', U72', U51', U41', PLUS, U21', U32', U31', X, S, TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c3

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

The set S is empty

(0) Obligation:

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

(2) Obligation:

(3) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

(4) BOUNDS(1, n^1)

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

(6) Obligation:

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

(8) Obligation:

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

(10) Obligation:

(11) CdtUsableRulesProof (EQUIVALENT transformation)

(12) Obligation:

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

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

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

(20) Obligation:

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

(22) Obligation:

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

(24) Obligation:

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

(26) Obligation:

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

(28) Obligation:

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

(30) Obligation:

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

(32) Obligation:

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

(34) Obligation:

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

(36) Obligation:

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

(38) Obligation:

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

(40) Obligation:

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

(42) Obligation:

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

(44) Obligation:

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

(46) Obligation:

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

(48) Obligation:

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

(50) Obligation:

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

(52) BOUNDS(1, 1)