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), 29 ms)
↳2 CpxTRS
    ↳3 CpxTrsMatchBoundsTAProof (⇔, 154 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)), 152 ms)
        ↳14 CdtProblem
        ↳15 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 44 ms)
        ↳16 CdtProblem
        ↳17 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 32 ms)
        ↳18 CdtProblem
        ↳19 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)
        ↳20 CdtProblem
        ↳21 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 36 ms)
        ↳22 CdtProblem
        ↳23 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 46 ms)
        ↳24 CdtProblem
        ↳25 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 14 ms)
        ↳26 CdtProblem
        ↳27 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳28 BOUNDS(1, 1)

(0) Obligation:

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


The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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

The following defined symbols can occur below the 0th argument of top: proper, active
The following defined symbols can occur below the 0th argument of proper: proper, active
The following defined symbols can occur below the 0th argument of active: proper, active

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
proper(dbl(X)) → dbl(proper(X))
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(dbl1(X)) → dbl1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))

(2) Obligation:

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


The TRS R consists of the following rules:

top(ok(X)) → top(active(X))
from(ok(X)) → ok(from(X))
dbl(mark(X)) → mark(dbl(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
dbl1(mark(X)) → mark(dbl1(X))
proper(01) → ok(01)
dbl1(ok(X)) → ok(dbl1(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
proper(nil) → ok(nil)
indx(mark(X1), X2) → mark(indx(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
quote(ok(X)) → ok(quote(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
s(ok(X)) → ok(s(X))
dbls(mark(X)) → mark(dbls(X))
proper(0) → ok(0)
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
dbl(ok(X)) → ok(dbl(X))
dbls(ok(X)) → ok(dbls(X))
quote(mark(X)) → mark(quote(X))
top(mark(X)) → top(proper(X))
s1(mark(X)) → mark(s1(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]
transitions:
ok0(0) → 0
active0(0) → 0
mark0(0) → 0
010() → 0
nil0() → 0
00() → 0
top0(0) → 1
from0(0) → 2
dbl0(0) → 3
sel10(0, 0) → 4
cons0(0, 0) → 5
dbl10(0) → 6
proper0(0) → 7
sel0(0, 0) → 8
indx0(0, 0) → 9
quote0(0) → 10
s10(0) → 11
s0(0) → 12
dbls0(0) → 13
active1(0) → 14
top1(14) → 1
from1(0) → 15
ok1(15) → 2
dbl1(0) → 16
mark1(16) → 3
sel11(0, 0) → 17
mark1(17) → 4
cons1(0, 0) → 18
ok1(18) → 5
dbl11(0) → 19
mark1(19) → 6
011() → 20
ok1(20) → 7
dbl11(0) → 21
ok1(21) → 6
sel1(0, 0) → 22
ok1(22) → 8
sel1(0, 0) → 23
mark1(23) → 8
nil1() → 24
ok1(24) → 7
indx1(0, 0) → 25
mark1(25) → 9
quote1(0) → 26
ok1(26) → 10
s11(0) → 27
ok1(27) → 11
sel11(0, 0) → 28
ok1(28) → 4
s1(0) → 29
ok1(29) → 12
dbls1(0) → 30
mark1(30) → 13
01() → 31
ok1(31) → 7
indx1(0, 0) → 32
ok1(32) → 9
dbl1(0) → 33
ok1(33) → 3
dbls1(0) → 34
ok1(34) → 13
quote1(0) → 35
mark1(35) → 10
proper1(0) → 36
top1(36) → 1
s11(0) → 37
mark1(37) → 11
ok1(15) → 15
mark1(16) → 16
mark1(16) → 33
mark1(17) → 17
mark1(17) → 28
ok1(18) → 18
mark1(19) → 19
mark1(19) → 21
ok1(20) → 36
ok1(21) → 19
ok1(21) → 21
ok1(22) → 22
ok1(22) → 23
mark1(23) → 22
mark1(23) → 23
ok1(24) → 36
mark1(25) → 25
mark1(25) → 32
ok1(26) → 26
ok1(26) → 35
ok1(27) → 27
ok1(27) → 37
ok1(28) → 17
ok1(28) → 28
ok1(29) → 29
mark1(30) → 30
mark1(30) → 34
ok1(31) → 36
ok1(32) → 25
ok1(32) → 32
ok1(33) → 16
ok1(33) → 33
ok1(34) → 30
ok1(34) → 34
mark1(35) → 26
mark1(35) → 35
mark1(37) → 27
mark1(37) → 37
active2(20) → 38
top2(38) → 1
active2(24) → 38
active2(31) → 38

(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:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
from(ok(z0)) → ok(from(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
proper(01) → ok(01)
proper(nil) → ok(nil)
proper(0) → ok(0)
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
quote(ok(z0)) → ok(quote(z0))
quote(mark(z0)) → mark(quote(z0))
s1(ok(z0)) → ok(s1(z0))
s1(mark(z0)) → mark(s1(z0))
s(ok(z0)) → ok(s(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
Tuples:

TOP(ok(z0)) → c(TOP(active(z0)))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
FROM(ok(z0)) → c2(FROM(z0))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
PROPER(01) → c11
PROPER(nil) → c12
PROPER(0) → c13
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
S tuples:

TOP(ok(z0)) → c(TOP(active(z0)))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
FROM(ok(z0)) → c2(FROM(z0))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
PROPER(01) → c11
PROPER(nil) → c12
PROPER(0) → c13
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
K tuples:none
Defined Rule Symbols:

top, from, dbl, sel1, cons, dbl1, proper, sel, indx, quote, s1, s, dbls

Defined Pair Symbols:

TOP, FROM, DBL, SEL1, CONS, DBL1, PROPER, SEL, INDX, QUOTE, S1, S, DBLS

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

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

Removed 4 trailing nodes:

PROPER(nil) → c12
PROPER(01) → c11
PROPER(0) → c13
TOP(ok(z0)) → c(TOP(active(z0)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
from(ok(z0)) → ok(from(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
proper(01) → ok(01)
proper(nil) → ok(nil)
proper(0) → ok(0)
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
quote(ok(z0)) → ok(quote(z0))
quote(mark(z0)) → mark(quote(z0))
s1(ok(z0)) → ok(s1(z0))
s1(mark(z0)) → mark(s1(z0))
s(ok(z0)) → ok(s(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
Tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
FROM(ok(z0)) → c2(FROM(z0))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
S tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
FROM(ok(z0)) → c2(FROM(z0))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
K tuples:none
Defined Rule Symbols:

top, from, dbl, sel1, cons, dbl1, proper, sel, indx, quote, s1, s, dbls

Defined Pair Symbols:

TOP, FROM, DBL, SEL1, CONS, DBL1, SEL, INDX, QUOTE, S1, S, DBLS

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25

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

Removed 1 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
from(ok(z0)) → ok(from(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
proper(01) → ok(01)
proper(nil) → ok(nil)
proper(0) → ok(0)
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
quote(ok(z0)) → ok(quote(z0))
quote(mark(z0)) → mark(quote(z0))
s1(ok(z0)) → ok(s1(z0))
s1(mark(z0)) → mark(s1(z0))
s(ok(z0)) → ok(s(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))
Tuples:

FROM(ok(z0)) → c2(FROM(z0))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
S tuples:

FROM(ok(z0)) → c2(FROM(z0))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
K tuples:none
Defined Rule Symbols:

top, from, dbl, sel1, cons, dbl1, proper, sel, indx, quote, s1, s, dbls

Defined Pair Symbols:

FROM, DBL, SEL1, CONS, DBL1, SEL, INDX, QUOTE, S1, S, DBLS, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c1

(11) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
from(ok(z0)) → ok(from(z0))
dbl(mark(z0)) → mark(dbl(z0))
dbl(ok(z0)) → ok(dbl(z0))
sel1(mark(z0), z1) → mark(sel1(z0, z1))
sel1(z0, mark(z1)) → mark(sel1(z0, z1))
sel1(ok(z0), ok(z1)) → ok(sel1(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
dbl1(mark(z0)) → mark(dbl1(z0))
dbl1(ok(z0)) → ok(dbl1(z0))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
indx(mark(z0), z1) → mark(indx(z0, z1))
indx(ok(z0), ok(z1)) → ok(indx(z0, z1))
quote(ok(z0)) → ok(quote(z0))
quote(mark(z0)) → mark(quote(z0))
s1(ok(z0)) → ok(s1(z0))
s1(mark(z0)) → mark(s1(z0))
s(ok(z0)) → ok(s(z0))
dbls(mark(z0)) → mark(dbls(z0))
dbls(ok(z0)) → ok(dbls(z0))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(0) → ok(0)
Tuples:

FROM(ok(z0)) → c2(FROM(z0))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
S tuples:

FROM(ok(z0)) → c2(FROM(z0))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
K tuples:none
Defined Rule Symbols:

proper

Defined Pair Symbols:

FROM, DBL, SEL1, CONS, DBL1, SEL, INDX, QUOTE, S1, S, DBLS, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c1

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

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

TOP(mark(z0)) → c1(TOP(proper(z0)))
We considered the (Usable) Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(0) → ok(0)
And the Tuples:

FROM(ok(z0)) → c2(FROM(z0))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(01) = 0   
POL(CONS(x1, x2)) = 0   
POL(DBL(x1)) = 0   
POL(DBL1(x1)) = 0   
POL(DBLS(x1)) = 0   
POL(FROM(x1)) = 0   
POL(INDX(x1, x2)) = 0   
POL(QUOTE(x1)) = 0   
POL(S(x1)) = 0   
POL(S1(x1)) = 0   
POL(SEL(x1, x2)) = 0   
POL(SEL1(x1, x2)) = 0   
POL(TOP(x1)) = x1   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c2(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c22(x1)) = x1   
POL(c23(x1)) = x1   
POL(c24(x1)) = x1   
POL(c25(x1)) = x1   
POL(c3(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(nil) = 0   
POL(ok(x1)) = 0   
POL(proper(x1)) = 0   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(0) → ok(0)
Tuples:

FROM(ok(z0)) → c2(FROM(z0))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
S tuples:

FROM(ok(z0)) → c2(FROM(z0))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
Defined Rule Symbols:

proper

Defined Pair Symbols:

FROM, DBL, SEL1, CONS, DBL1, SEL, INDX, QUOTE, S1, S, DBLS, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c1

(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.

DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
We considered the (Usable) Rules:none
And the Tuples:

FROM(ok(z0)) → c2(FROM(z0))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(01) = 0   
POL(CONS(x1, x2)) = x1   
POL(DBL(x1)) = x1   
POL(DBL1(x1)) = x1   
POL(DBLS(x1)) = x1   
POL(FROM(x1)) = 0   
POL(INDX(x1, x2)) = 0   
POL(QUOTE(x1)) = x1   
POL(S(x1)) = x1   
POL(S1(x1)) = x1   
POL(SEL(x1, x2)) = x1   
POL(SEL1(x1, x2)) = x2   
POL(TOP(x1)) = 0   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c2(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c22(x1)) = x1   
POL(c23(x1)) = x1   
POL(c24(x1)) = x1   
POL(c25(x1)) = x1   
POL(c3(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(nil) = 0   
POL(ok(x1)) = [1] + x1   
POL(proper(x1)) = 0   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(0) → ok(0)
Tuples:

FROM(ok(z0)) → c2(FROM(z0))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
S tuples:

FROM(ok(z0)) → c2(FROM(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
Defined Rule Symbols:

proper

Defined Pair Symbols:

FROM, DBL, SEL1, CONS, DBL1, SEL, INDX, QUOTE, S1, S, DBLS, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c1

(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.

FROM(ok(z0)) → c2(FROM(z0))
We considered the (Usable) Rules:none
And the Tuples:

FROM(ok(z0)) → c2(FROM(z0))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(01) = 0   
POL(CONS(x1, x2)) = [2]x2   
POL(DBL(x1)) = 0   
POL(DBL1(x1)) = 0   
POL(DBLS(x1)) = 0   
POL(FROM(x1)) = [2]x1   
POL(INDX(x1, x2)) = 0   
POL(QUOTE(x1)) = 0   
POL(S(x1)) = 0   
POL(S1(x1)) = 0   
POL(SEL(x1, x2)) = 0   
POL(SEL1(x1, x2)) = 0   
POL(TOP(x1)) = 0   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c2(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c22(x1)) = x1   
POL(c23(x1)) = x1   
POL(c24(x1)) = x1   
POL(c25(x1)) = x1   
POL(c3(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(nil) = 0   
POL(ok(x1)) = [1] + x1   
POL(proper(x1)) = 0   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(0) → ok(0)
Tuples:

FROM(ok(z0)) → c2(FROM(z0))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
S tuples:

SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
FROM(ok(z0)) → c2(FROM(z0))
Defined Rule Symbols:

proper

Defined Pair Symbols:

FROM, DBL, SEL1, CONS, DBL1, SEL, INDX, QUOTE, S1, S, DBLS, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c1

(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.

INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

FROM(ok(z0)) → c2(FROM(z0))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(01) = 0   
POL(CONS(x1, x2)) = x1 + x2   
POL(DBL(x1)) = 0   
POL(DBL1(x1)) = 0   
POL(DBLS(x1)) = 0   
POL(FROM(x1)) = 0   
POL(INDX(x1, x2)) = x2   
POL(QUOTE(x1)) = 0   
POL(S(x1)) = 0   
POL(S1(x1)) = 0   
POL(SEL(x1, x2)) = 0   
POL(SEL1(x1, x2)) = 0   
POL(TOP(x1)) = 0   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c2(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c22(x1)) = x1   
POL(c23(x1)) = x1   
POL(c24(x1)) = x1   
POL(c25(x1)) = x1   
POL(c3(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(nil) = 0   
POL(ok(x1)) = [1] + x1   
POL(proper(x1)) = 0   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(0) → ok(0)
Tuples:

FROM(ok(z0)) → c2(FROM(z0))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
S tuples:

SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
FROM(ok(z0)) → c2(FROM(z0))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
Defined Rule Symbols:

proper

Defined Pair Symbols:

FROM, DBL, SEL1, CONS, DBL1, SEL, INDX, QUOTE, S1, S, DBLS, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c1

(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.

SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

FROM(ok(z0)) → c2(FROM(z0))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(01) = [1]   
POL(CONS(x1, x2)) = 0   
POL(DBL(x1)) = 0   
POL(DBL1(x1)) = 0   
POL(DBLS(x1)) = 0   
POL(FROM(x1)) = 0   
POL(INDX(x1, x2)) = x2   
POL(QUOTE(x1)) = 0   
POL(S(x1)) = 0   
POL(S1(x1)) = 0   
POL(SEL(x1, x2)) = 0   
POL(SEL1(x1, x2)) = x1   
POL(TOP(x1)) = 0   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c2(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c22(x1)) = x1   
POL(c23(x1)) = x1   
POL(c24(x1)) = x1   
POL(c25(x1)) = x1   
POL(c3(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(nil) = 0   
POL(ok(x1)) = x1   
POL(proper(x1)) = x1   

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(0) → ok(0)
Tuples:

FROM(ok(z0)) → c2(FROM(z0))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
S tuples:

SEL(z0, mark(z1)) → c15(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
FROM(ok(z0)) → c2(FROM(z0))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
Defined Rule Symbols:

proper

Defined Pair Symbols:

FROM, DBL, SEL1, CONS, DBL1, SEL, INDX, QUOTE, S1, S, DBLS, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c1

(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.

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

FROM(ok(z0)) → c2(FROM(z0))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [1]   
POL(01) = [1]   
POL(CONS(x1, x2)) = 0   
POL(DBL(x1)) = 0   
POL(DBL1(x1)) = 0   
POL(DBLS(x1)) = 0   
POL(FROM(x1)) = 0   
POL(INDX(x1, x2)) = 0   
POL(QUOTE(x1)) = 0   
POL(S(x1)) = 0   
POL(S1(x1)) = 0   
POL(SEL(x1, x2)) = [2]x2   
POL(SEL1(x1, x2)) = 0   
POL(TOP(x1)) = 0   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c2(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c22(x1)) = x1   
POL(c23(x1)) = x1   
POL(c24(x1)) = x1   
POL(c25(x1)) = x1   
POL(c3(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(nil) = 0   
POL(ok(x1)) = [1] + x1   
POL(proper(x1)) = [2]   

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(0) → ok(0)
Tuples:

FROM(ok(z0)) → c2(FROM(z0))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
S tuples:

INDX(mark(z0), z1) → c17(INDX(z0, z1))
K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
FROM(ok(z0)) → c2(FROM(z0))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
Defined Rule Symbols:

proper

Defined Pair Symbols:

FROM, DBL, SEL1, CONS, DBL1, SEL, INDX, QUOTE, S1, S, DBLS, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c1

(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.

INDX(mark(z0), z1) → c17(INDX(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

FROM(ok(z0)) → c2(FROM(z0))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(01) = 0   
POL(CONS(x1, x2)) = 0   
POL(DBL(x1)) = 0   
POL(DBL1(x1)) = 0   
POL(DBLS(x1)) = 0   
POL(FROM(x1)) = 0   
POL(INDX(x1, x2)) = x1   
POL(QUOTE(x1)) = 0   
POL(S(x1)) = 0   
POL(S1(x1)) = 0   
POL(SEL(x1, x2)) = 0   
POL(SEL1(x1, x2)) = 0   
POL(TOP(x1)) = 0   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c2(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c22(x1)) = x1   
POL(c23(x1)) = x1   
POL(c24(x1)) = x1   
POL(c25(x1)) = x1   
POL(c3(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(nil) = 0   
POL(ok(x1)) = x1   
POL(proper(x1)) = 0   

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(0) → ok(0)
Tuples:

FROM(ok(z0)) → c2(FROM(z0))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
S tuples:none
K tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)))
DBL(mark(z0)) → c3(DBL(z0))
DBL(ok(z0)) → c4(DBL(z0))
SEL1(z0, mark(z1)) → c6(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c7(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
DBL1(mark(z0)) → c9(DBL1(z0))
DBL1(ok(z0)) → c10(DBL1(z0))
SEL(ok(z0), ok(z1)) → c14(SEL(z0, z1))
SEL(mark(z0), z1) → c16(SEL(z0, z1))
QUOTE(ok(z0)) → c19(QUOTE(z0))
QUOTE(mark(z0)) → c20(QUOTE(z0))
S1(ok(z0)) → c21(S1(z0))
S1(mark(z0)) → c22(S1(z0))
S(ok(z0)) → c23(S(z0))
DBLS(mark(z0)) → c24(DBLS(z0))
DBLS(ok(z0)) → c25(DBLS(z0))
FROM(ok(z0)) → c2(FROM(z0))
INDX(ok(z0), ok(z1)) → c18(INDX(z0, z1))
SEL1(mark(z0), z1) → c5(SEL1(z0, z1))
SEL(z0, mark(z1)) → c15(SEL(z0, z1))
INDX(mark(z0), z1) → c17(INDX(z0, z1))
Defined Rule Symbols:

proper

Defined Pair Symbols:

FROM, DBL, SEL1, CONS, DBL1, SEL, INDX, QUOTE, S1, S, DBLS, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c1

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

The set S is empty

(28) BOUNDS(1, 1)