Runtime Complexity (innermost) proof of /tmp/tmpBKfyXE/Ex6_15_AEL02

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 (⇔, 386 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)), 225 ms)

        ↳14 CdtProblem

        ↳15 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 54 ms)

        ↳16 CdtProblem

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

        ↳18 CdtProblem

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

        ↳20 CdtProblem

        ↳21 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 34 ms)

        ↳22 CdtProblem

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

        ↳24 CdtProblem

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

        ↳26 CdtProblem

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

        ↳28 CdtProblem

        ↳29 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 26 ms)

        ↳30 CdtProblem

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

        ↳32 CdtProblem

        ↳33 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 9 ms)

        ↳34 CdtProblem

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

        ↳36 CdtProblem

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

        ↳38 CdtProblem

        ↳39 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 28 ms)

        ↳40 CdtProblem

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

        ↳42 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(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(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(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(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:

top(ok(X)) → top(active(X))
from(ok(X)) → ok(from(X))
from(mark(X)) → mark(from(X))
unquote1(mark(X)) → mark(unquote1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
unquote1(ok(X)) → ok(unquote1(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
unquote(ok(X)) → ok(unquote(X))
unquote(mark(X)) → mark(unquote(X))
first1(X1, mark(X2)) → mark(first1(X1, X2))
proper(01) → ok(01)
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
proper(nil) → ok(nil)
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
quote1(ok(X)) → ok(quote1(X))
first(mark(X1), X2) → mark(first(X1, X2))
proper(nil1) → ok(nil1)
cons1(mark(X1), X2) → mark(cons1(X1, X2))
quote(ok(X)) → ok(quote(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
s1(ok(X)) → ok(s1(X))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
proper(0) → ok(0)
first1(mark(X1), X2) → mark(first1(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
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, 14, 15, 16]
transitions:
ok0(0) → 0
active0(0) → 0
mark0(0) → 0
010() → 0
nil0() → 0
nil10() → 0
00() → 0
top0(0) → 1
from0(0) → 2
unquote10(0) → 3
sel10(0, 0) → 4
cons0(0, 0) → 5
first0(0, 0) → 6
unquote0(0) → 7
first10(0, 0) → 8
proper0(0) → 9
fcons0(0, 0) → 10
sel0(0, 0) → 11
cons10(0, 0) → 12
quote10(0) → 13
quote0(0) → 14
s10(0) → 15
s0(0) → 16
active1(0) → 17
top1(17) → 1
from1(0) → 18
ok1(18) → 2
from1(0) → 19
mark1(19) → 2
unquote11(0) → 20
mark1(20) → 3
sel11(0, 0) → 21
mark1(21) → 4
cons1(0, 0) → 22
ok1(22) → 5
unquote11(0) → 23
ok1(23) → 3
first1(0, 0) → 24
ok1(24) → 6
unquote1(0) → 25
ok1(25) → 7
unquote1(0) → 26
mark1(26) → 7
first11(0, 0) → 27
mark1(27) → 8
011() → 28
ok1(28) → 9
fcons1(0, 0) → 29
mark1(29) → 10
first1(0, 0) → 30
mark1(30) → 6
sel1(0, 0) → 31
ok1(31) → 11
sel1(0, 0) → 32
mark1(32) → 11
cons11(0, 0) → 33
mark1(33) → 12
fcons1(0, 0) → 34
ok1(34) → 10
first11(0, 0) → 35
ok1(35) → 8
nil1() → 36
ok1(36) → 9
cons11(0, 0) → 37
ok1(37) → 12
quote11(0) → 38
ok1(38) → 13
nil11() → 39
ok1(39) → 9
quote1(0) → 40
ok1(40) → 14
sel11(0, 0) → 41
ok1(41) → 4
s11(0) → 42
ok1(42) → 15
s1(0) → 43
ok1(43) → 16
s1(0) → 44
mark1(44) → 16
01() → 45
ok1(45) → 9
cons1(0, 0) → 46
mark1(46) → 5
proper1(0) → 47
top1(47) → 1
s11(0) → 48
mark1(48) → 15
ok1(18) → 18
ok1(18) → 19
mark1(19) → 18
mark1(19) → 19
mark1(20) → 20
mark1(20) → 23
mark1(21) → 21
mark1(21) → 41
ok1(22) → 22
ok1(22) → 46
ok1(23) → 20
ok1(23) → 23
ok1(24) → 24
ok1(24) → 30
ok1(25) → 25
ok1(25) → 26
mark1(26) → 25
mark1(26) → 26
mark1(27) → 27
mark1(27) → 35
ok1(28) → 47
mark1(29) → 29
mark1(29) → 34
mark1(30) → 24
mark1(30) → 30
ok1(31) → 31
ok1(31) → 32
mark1(32) → 31
mark1(32) → 32
mark1(33) → 33
mark1(33) → 37
ok1(34) → 29
ok1(34) → 34
ok1(35) → 27
ok1(35) → 35
ok1(36) → 47
ok1(37) → 33
ok1(37) → 37
ok1(38) → 38
ok1(39) → 47
ok1(40) → 40
ok1(41) → 21
ok1(41) → 41
ok1(42) → 42
ok1(42) → 48
ok1(43) → 43
ok1(43) → 44
mark1(44) → 43
mark1(44) → 44
ok1(45) → 47
mark1(46) → 22
mark1(46) → 46
mark1(48) → 42
mark1(48) → 48
active2(28) → 49
top2(49) → 1
active2(36) → 49
active2(39) → 49
active2(45) → 49

(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))
from(mark(z0)) → mark(from(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(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))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
unquote(ok(z0)) → ok(unquote(z0))
unquote(mark(z0)) → mark(unquote(z0))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
first1(mark(z0), z1) → mark(first1(z0, z1))
proper(01) → ok(01)
proper(nil) → ok(nil)
proper(nil1) → ok(nil1)
proper(0) → ok(0)
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
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))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
quote1(ok(z0)) → ok(quote1(z0))
quote(ok(z0)) → ok(quote(z0))
s1(ok(z0)) → ok(s1(z0))
s1(mark(z0)) → mark(s1(z0))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))

Tuples:

TOP(ok(z0)) → c(TOP(active(z0)))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
PROPER(01) → c19
PROPER(nil) → c20
PROPER(nil1) → c21
PROPER(0) → c22
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(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))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
PROPER(01) → c19
PROPER(nil) → c20
PROPER(nil1) → c21
PROPER(0) → c22
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))

K tuples:none
Defined Rule Symbols:

top, from, unquote1, sel1, cons, first, unquote, first1, proper, fcons, sel, cons1, quote1, quote, s1, s

Defined Pair Symbols:

TOP, FROM, UNQUOTE1, SEL1, CONS, FIRST, UNQUOTE, FIRST1, PROPER, FCONS, SEL, CONS1, QUOTE1, QUOTE, S1, 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, c35, c36, c37

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

Removed 5 trailing nodes:

PROPER(0) → c22
TOP(ok(z0)) → c(TOP(active(z0)))
PROPER(nil) → c20
PROPER(nil1) → c21
PROPER(01) → c19

(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))
from(mark(z0)) → mark(from(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(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))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
unquote(ok(z0)) → ok(unquote(z0))
unquote(mark(z0)) → mark(unquote(z0))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
first1(mark(z0), z1) → mark(first1(z0, z1))
proper(01) → ok(01)
proper(nil) → ok(nil)
proper(nil1) → ok(nil1)
proper(0) → ok(0)
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
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))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
quote1(ok(z0)) → ok(quote1(z0))
quote(ok(z0)) → ok(quote(z0))
s1(ok(z0)) → ok(s1(z0))
s1(mark(z0)) → mark(s1(z0))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))

Tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))

S tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))

K tuples:none
Defined Rule Symbols:

top, from, unquote1, sel1, cons, first, unquote, first1, proper, fcons, sel, cons1, quote1, quote, s1, s

Defined Pair Symbols:

TOP, FROM, UNQUOTE1, SEL1, CONS, FIRST, UNQUOTE, FIRST1, FCONS, SEL, CONS1, QUOTE1, QUOTE, S1, S

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37

(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))
from(mark(z0)) → mark(from(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(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))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
unquote(ok(z0)) → ok(unquote(z0))
unquote(mark(z0)) → mark(unquote(z0))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
first1(mark(z0), z1) → mark(first1(z0, z1))
proper(01) → ok(01)
proper(nil) → ok(nil)
proper(nil1) → ok(nil1)
proper(0) → ok(0)
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
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))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
quote1(ok(z0)) → ok(quote1(z0))
quote(ok(z0)) → ok(quote(z0))
s1(ok(z0)) → ok(s1(z0))
s1(mark(z0)) → mark(s1(z0))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))

Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

top, from, unquote1, sel1, cons, first, unquote, first1, proper, fcons, sel, cons1, quote1, quote, s1, s

Defined Pair Symbols:

FROM, UNQUOTE1, SEL1, CONS, FIRST, UNQUOTE, FIRST1, FCONS, SEL, CONS1, QUOTE1, QUOTE, S1, S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, 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))
from(mark(z0)) → mark(from(z0))
unquote1(mark(z0)) → mark(unquote1(z0))
unquote1(ok(z0)) → ok(unquote1(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))
cons(mark(z0), z1) → mark(cons(z0, z1))
first(ok(z0), ok(z1)) → ok(first(z0, z1))
first(z0, mark(z1)) → mark(first(z0, z1))
first(mark(z0), z1) → mark(first(z0, z1))
unquote(ok(z0)) → ok(unquote(z0))
unquote(mark(z0)) → mark(unquote(z0))
first1(z0, mark(z1)) → mark(first1(z0, z1))
first1(ok(z0), ok(z1)) → ok(first1(z0, z1))
first1(mark(z0), z1) → mark(first1(z0, z1))
fcons(z0, mark(z1)) → mark(fcons(z0, z1))
fcons(ok(z0), ok(z1)) → ok(fcons(z0, z1))
fcons(mark(z0), z1) → mark(fcons(z0, z1))
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))
cons1(z0, mark(z1)) → mark(cons1(z0, z1))
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1))
cons1(mark(z0), z1) → mark(cons1(z0, z1))
quote1(ok(z0)) → ok(quote1(z0))
quote(ok(z0)) → ok(quote(z0))
s1(ok(z0)) → ok(s1(z0))
s1(mark(z0)) → mark(s1(z0))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(nil1) → ok(nil1)
proper(0) → ok(0)

Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

proper

Defined Pair Symbols:

FROM, UNQUOTE1, SEL1, CONS, FIRST, UNQUOTE, FIRST1, FCONS, SEL, CONS1, QUOTE1, QUOTE, S1, S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, 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.

FIRST(mark(z0), z1) → c13(FIRST(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(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) = 0
POL(CONS(x₁, x₂)) = 0
POL(CONS1(x₁, x₂)) = 0
POL(FCONS(x₁, x₂)) = 0
POL(FIRST(x₁, x₂)) = x₁
POL(FIRST1(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(QUOTE(x₁)) = 0
POL(QUOTE1(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(UNQUOTE(x₁)) = 0
POL(UNQUOTE1(x₁)) = 0
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(c2(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(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(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(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(nil1) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = [1] + x₁

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(nil1) → ok(nil1)
proper(0) → ok(0)

Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

K tuples:

FIRST(mark(z0), z1) → c13(FIRST(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

FROM, UNQUOTE1, SEL1, CONS, FIRST, UNQUOTE, FIRST1, FCONS, SEL, CONS1, QUOTE1, QUOTE, S1, S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, 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.

QUOTE(ok(z0)) → c33(QUOTE(z0))

We considered the (Usable) Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(nil1) → ok(nil1)

And the Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(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(CONS1(x₁, x₂)) = 0
POL(FCONS(x₁, x₂)) = 0
POL(FIRST(x₁, x₂)) = 0
POL(FIRST1(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(QUOTE(x₁)) = x₁
POL(QUOTE1(x₁)) = 0
POL(S(x₁)) = 0
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = 0
POL(SEL1(x₁, x₂)) = 0
POL(TOP(x₁)) = [2]x₁
POL(UNQUOTE(x₁)) = 0
POL(UNQUOTE1(x₁)) = 0
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(c2(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(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(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(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(nil) = [1]
POL(nil1) = 0
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = [2]

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(nil1) → ok(nil1)
proper(0) → ok(0)

Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

K tuples:

FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
QUOTE(ok(z0)) → c33(QUOTE(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

FROM, UNQUOTE1, SEL1, CONS, FIRST, UNQUOTE, FIRST1, FCONS, SEL, CONS1, QUOTE1, QUOTE, S1, S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, 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.

CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))

We considered the (Usable) Rules:none
And the Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(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)) = x₂
POL(CONS1(x₁, x₂)) = 0
POL(FCONS(x₁, x₂)) = 0
POL(FIRST(x₁, x₂)) = x₁
POL(FIRST1(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(QUOTE(x₁)) = 0
POL(QUOTE1(x₁)) = x₁
POL(S(x₁)) = 0
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = x₁
POL(SEL1(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(UNQUOTE(x₁)) = 0
POL(UNQUOTE1(x₁)) = 0
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(c2(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(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(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(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(nil1) = 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(nil1) → ok(nil1)
proper(0) → ok(0)

Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

K tuples:

FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
QUOTE(ok(z0)) → c33(QUOTE(z0))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

FROM, UNQUOTE1, SEL1, CONS, FIRST, UNQUOTE, FIRST1, FCONS, SEL, CONS1, QUOTE1, QUOTE, S1, S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, 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.

TOP(mark(z0)) → c1(TOP(proper(z0)))

We considered the (Usable) Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(nil1) → ok(nil1)

And the Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(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(CONS1(x₁, x₂)) = 0
POL(FCONS(x₁, x₂)) = 0
POL(FIRST(x₁, x₂)) = 0
POL(FIRST1(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(QUOTE(x₁)) = [3]x₁
POL(QUOTE1(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(UNQUOTE(x₁)) = 0
POL(UNQUOTE1(x₁)) = 0
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(c2(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(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(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(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)) = [3]
POL(nil) = 0
POL(nil1) = [1]
POL(ok(x₁)) = x₁
POL(proper(x₁)) = [1]

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(nil1) → ok(nil1)
proper(0) → ok(0)

Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))

K tuples:

FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
QUOTE(ok(z0)) → c33(QUOTE(z0))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

Defined Rule Symbols:

proper

Defined Pair Symbols:

FROM, UNQUOTE1, SEL1, CONS, FIRST, UNQUOTE, FIRST1, FCONS, SEL, CONS1, QUOTE1, QUOTE, S1, S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, 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.

FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
S(mark(z0)) → c37(S(z0))

We considered the (Usable) Rules:none
And the Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(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(CONS1(x₁, x₂)) = [2]x₁
POL(FCONS(x₁, x₂)) = [2]x₁ + x₂
POL(FIRST(x₁, x₂)) = 0
POL(FIRST1(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(QUOTE(x₁)) = [2]x₁
POL(QUOTE1(x₁)) = 0
POL(S(x₁)) = x₁
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = [2]x₂
POL(SEL1(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(UNQUOTE(x₁)) = 0
POL(UNQUOTE1(x₁)) = 0
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(c2(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(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(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(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(nil1) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(nil1) → ok(nil1)
proper(0) → ok(0)

Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))

K tuples:

FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
QUOTE(ok(z0)) → c33(QUOTE(z0))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
S(mark(z0)) → c37(S(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

FROM, UNQUOTE1, SEL1, CONS, FIRST, UNQUOTE, FIRST1, FCONS, SEL, CONS1, QUOTE1, QUOTE, S1, S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, 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.

UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
S1(mark(z0)) → c35(S1(z0))

We considered the (Usable) Rules:none
And the Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(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(CONS1(x₁, x₂)) = 0
POL(FCONS(x₁, x₂)) = 0
POL(FIRST(x₁, x₂)) = x₁
POL(FIRST1(x₁, x₂)) = x₁
POL(FROM(x₁)) = 0
POL(QUOTE(x₁)) = 0
POL(QUOTE1(x₁)) = 0
POL(S(x₁)) = 0
POL(S1(x₁)) = x₁
POL(SEL(x₁, x₂)) = 0
POL(SEL1(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(UNQUOTE(x₁)) = x₁
POL(UNQUOTE1(x₁)) = 0
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(c2(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(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(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(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) = [1]
POL(nil1) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = [1]

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(nil1) → ok(nil1)
proper(0) → ok(0)

Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
S1(ok(z0)) → c34(S1(z0))
S(ok(z0)) → c36(S(z0))

K tuples:

FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
QUOTE(ok(z0)) → c33(QUOTE(z0))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
S(mark(z0)) → c37(S(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
S1(mark(z0)) → c35(S1(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

FROM, UNQUOTE1, SEL1, CONS, FIRST, UNQUOTE, FIRST1, FCONS, SEL, CONS1, QUOTE1, QUOTE, S1, S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, 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.

FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(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(CONS1(x₁, x₂)) = x₂
POL(FCONS(x₁, x₂)) = 0
POL(FIRST(x₁, x₂)) = x₂
POL(FIRST1(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(QUOTE(x₁)) = 0
POL(QUOTE1(x₁)) = 0
POL(S(x₁)) = 0
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = x₁
POL(SEL1(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(UNQUOTE(x₁)) = 0
POL(UNQUOTE1(x₁)) = 0
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(c2(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(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(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(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(nil1) = 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(nil1) → ok(nil1)
proper(0) → ok(0)

Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
S1(ok(z0)) → c34(S1(z0))
S(ok(z0)) → c36(S(z0))

K tuples:

FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
QUOTE(ok(z0)) → c33(QUOTE(z0))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
S(mark(z0)) → c37(S(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
S1(mark(z0)) → c35(S1(z0))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

FROM, UNQUOTE1, SEL1, CONS, FIRST, UNQUOTE, FIRST1, FCONS, SEL, CONS1, QUOTE1, QUOTE, S1, S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c1

(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.

SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
S1(ok(z0)) → c34(S1(z0))

We considered the (Usable) Rules:none
And the Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(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(CONS1(x₁, x₂)) = x₁ + x₂
POL(FCONS(x₁, x₂)) = x₁
POL(FIRST(x₁, x₂)) = x₁
POL(FIRST1(x₁, x₂)) = x₁
POL(FROM(x₁)) = 0
POL(QUOTE(x₁)) = 0
POL(QUOTE1(x₁)) = 0
POL(S(x₁)) = 0
POL(S1(x₁)) = x₁
POL(SEL(x₁, x₂)) = 0
POL(SEL1(x₁, x₂)) = x₁ + x₂
POL(TOP(x₁)) = 0
POL(UNQUOTE(x₁)) = x₁
POL(UNQUOTE1(x₁)) = 0
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(c2(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(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(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(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(nil) = 0
POL(nil1) = 0
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = 0

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(nil1) → ok(nil1)
proper(0) → ok(0)

Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
S(ok(z0)) → c36(S(z0))

K tuples:

FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
QUOTE(ok(z0)) → c33(QUOTE(z0))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
S(mark(z0)) → c37(S(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
S1(mark(z0)) → c35(S1(z0))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
S1(ok(z0)) → c34(S1(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

FROM, UNQUOTE1, SEL1, CONS, FIRST, UNQUOTE, FIRST1, FCONS, SEL, CONS1, QUOTE1, QUOTE, S1, S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c1

(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.

CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(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]x₁
POL(CONS1(x₁, x₂)) = 0
POL(FCONS(x₁, x₂)) = 0
POL(FIRST(x₁, x₂)) = 0
POL(FIRST1(x₁, x₂)) = [2]x₂
POL(FROM(x₁)) = 0
POL(QUOTE(x₁)) = 0
POL(QUOTE1(x₁)) = 0
POL(S(x₁)) = 0
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = [2]x₁
POL(SEL1(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(UNQUOTE(x₁)) = 0
POL(UNQUOTE1(x₁)) = 0
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(c2(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(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(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(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(nil) = [2]
POL(nil1) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = [2]x₁

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(nil1) → ok(nil1)
proper(0) → ok(0)

Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
S(ok(z0)) → c36(S(z0))

K tuples:

FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
QUOTE(ok(z0)) → c33(QUOTE(z0))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
S(mark(z0)) → c37(S(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
S1(mark(z0)) → c35(S1(z0))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
S1(ok(z0)) → c34(S1(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

FROM, UNQUOTE1, SEL1, CONS, FIRST, UNQUOTE, FIRST1, FCONS, SEL, CONS1, QUOTE1, QUOTE, S1, S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c1

(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.

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(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]x₁
POL(CONS1(x₁, x₂)) = 0
POL(FCONS(x₁, x₂)) = 0
POL(FIRST(x₁, x₂)) = [2]x₁
POL(FIRST1(x₁, x₂)) = 0
POL(FROM(x₁)) = [2]x₁
POL(QUOTE(x₁)) = 0
POL(QUOTE1(x₁)) = 0
POL(S(x₁)) = 0
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = 0
POL(SEL1(x₁, x₂)) = [2]x₁
POL(TOP(x₁)) = 0
POL(UNQUOTE(x₁)) = 0
POL(UNQUOTE1(x₁)) = 0
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(c2(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(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(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(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(nil) = 0
POL(nil1) = 0
POL(ok(x₁)) = [2] + x₁
POL(proper(x₁)) = 0

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(nil1) → ok(nil1)
proper(0) → ok(0)

Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
S(ok(z0)) → c36(S(z0))

K tuples:

FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
QUOTE(ok(z0)) → c33(QUOTE(z0))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
S(mark(z0)) → c37(S(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
S1(mark(z0)) → c35(S1(z0))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
S1(ok(z0)) → c34(S1(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

FROM, UNQUOTE1, SEL1, CONS, FIRST, UNQUOTE, FIRST1, FCONS, SEL, CONS1, QUOTE1, QUOTE, S1, S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c1

(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.

SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(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(CONS1(x₁, x₂)) = 0
POL(FCONS(x₁, x₂)) = 0
POL(FIRST(x₁, x₂)) = 0
POL(FIRST1(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(QUOTE(x₁)) = 0
POL(QUOTE1(x₁)) = 0
POL(S(x₁)) = 0
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = x₁
POL(SEL1(x₁, x₂)) = x₂
POL(TOP(x₁)) = 0
POL(UNQUOTE(x₁)) = 0
POL(UNQUOTE1(x₁)) = 0
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(c2(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(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(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(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(nil1) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(nil1) → ok(nil1)
proper(0) → ok(0)

Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
S(ok(z0)) → c36(S(z0))

K tuples:

FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
QUOTE(ok(z0)) → c33(QUOTE(z0))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
S(mark(z0)) → c37(S(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
S1(mark(z0)) → c35(S1(z0))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
S1(ok(z0)) → c34(S1(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

FROM, UNQUOTE1, SEL1, CONS, FIRST, UNQUOTE, FIRST1, FCONS, SEL, CONS1, QUOTE1, QUOTE, S1, S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c1

(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.

UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))

We considered the (Usable) Rules:none
And the Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(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(CONS1(x₁, x₂)) = 0
POL(FCONS(x₁, x₂)) = x₂
POL(FIRST(x₁, x₂)) = 0
POL(FIRST1(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(QUOTE(x₁)) = 0
POL(QUOTE1(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(UNQUOTE(x₁)) = 0
POL(UNQUOTE1(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(c2(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(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(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(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(nil1) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(nil1) → ok(nil1)
proper(0) → ok(0)

Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
S(ok(z0)) → c36(S(z0))

K tuples:

FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
QUOTE(ok(z0)) → c33(QUOTE(z0))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
S(mark(z0)) → c37(S(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
S1(mark(z0)) → c35(S1(z0))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
S1(ok(z0)) → c34(S1(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

FROM, UNQUOTE1, SEL1, CONS, FIRST, UNQUOTE, FIRST1, FCONS, SEL, CONS1, QUOTE1, QUOTE, S1, S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c1

(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)) → c36(S(z0))

We considered the (Usable) Rules:none
And the Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(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(CONS1(x₁, x₂)) = 0
POL(FCONS(x₁, x₂)) = 0
POL(FIRST(x₁, x₂)) = x₂
POL(FIRST1(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(QUOTE(x₁)) = 0
POL(QUOTE1(x₁)) = 0
POL(S(x₁)) = x₁
POL(S1(x₁)) = 0
POL(SEL(x₁, x₂)) = 0
POL(SEL1(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(UNQUOTE(x₁)) = 0
POL(UNQUOTE1(x₁)) = 0
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(c2(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(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(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(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(nil) = 0
POL(nil1) = 0
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = 0

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(nil1) → ok(nil1)
proper(0) → ok(0)

Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))

K tuples:

FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
QUOTE(ok(z0)) → c33(QUOTE(z0))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
S(mark(z0)) → c37(S(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
S1(mark(z0)) → c35(S1(z0))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
S1(ok(z0)) → c34(S1(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
S(ok(z0)) → c36(S(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

FROM, UNQUOTE1, SEL1, CONS, FIRST, UNQUOTE, FIRST1, FCONS, SEL, CONS1, QUOTE1, QUOTE, S1, S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c1

(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.

UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))

We considered the (Usable) Rules:none
And the Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(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(CONS1(x₁, x₂)) = 0
POL(FCONS(x₁, x₂)) = 0
POL(FIRST(x₁, x₂)) = 0
POL(FIRST1(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(QUOTE(x₁)) = 0
POL(QUOTE1(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(UNQUOTE(x₁)) = 0
POL(UNQUOTE1(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(c2(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(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(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(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(nil) = 0
POL(nil1) = 0
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = 0

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(01) → ok(01)
proper(nil) → ok(nil)
proper(nil1) → ok(nil1)
proper(0) → ok(0)

Tuples:

FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
QUOTE(ok(z0)) → c33(QUOTE(z0))
S1(ok(z0)) → c34(S1(z0))
S1(mark(z0)) → c35(S1(z0))
S(ok(z0)) → c36(S(z0))
S(mark(z0)) → c37(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:none
K tuples:

FIRST(mark(z0), z1) → c13(FIRST(z0, z1))
QUOTE(ok(z0)) → c33(QUOTE(z0))
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))
FIRST(ok(z0), ok(z1)) → c11(FIRST(z0, z1))
SEL(ok(z0), ok(z1)) → c26(SEL(z0, z1))
SEL(mark(z0), z1) → c28(SEL(z0, z1))
QUOTE1(ok(z0)) → c32(QUOTE1(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))
FCONS(z0, mark(z1)) → c23(FCONS(z0, z1))
FCONS(mark(z0), z1) → c25(FCONS(z0, z1))
SEL(z0, mark(z1)) → c27(SEL(z0, z1))
CONS1(mark(z0), z1) → c31(CONS1(z0, z1))
S(mark(z0)) → c37(S(z0))
UNQUOTE(mark(z0)) → c15(UNQUOTE(z0))
FIRST1(mark(z0), z1) → c18(FIRST1(z0, z1))
S1(mark(z0)) → c35(S1(z0))
FIRST(z0, mark(z1)) → c12(FIRST(z0, z1))
CONS1(z0, mark(z1)) → c29(CONS1(z0, z1))
SEL1(ok(z0), ok(z1)) → c8(SEL1(z0, z1))
UNQUOTE(ok(z0)) → c14(UNQUOTE(z0))
FIRST1(ok(z0), ok(z1)) → c17(FIRST1(z0, z1))
FCONS(ok(z0), ok(z1)) → c24(FCONS(z0, z1))
CONS1(ok(z0), ok(z1)) → c30(CONS1(z0, z1))
S1(ok(z0)) → c34(S1(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
FIRST1(z0, mark(z1)) → c16(FIRST1(z0, z1))
FROM(ok(z0)) → c2(FROM(z0))
FROM(mark(z0)) → c3(FROM(z0))
SEL1(mark(z0), z1) → c6(SEL1(z0, z1))
SEL1(z0, mark(z1)) → c7(SEL1(z0, z1))
UNQUOTE1(mark(z0)) → c4(UNQUOTE1(z0))
S(ok(z0)) → c36(S(z0))
UNQUOTE1(ok(z0)) → c5(UNQUOTE1(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

FROM, UNQUOTE1, SEL1, CONS, FIRST, UNQUOTE, FIRST1, FCONS, SEL, CONS1, QUOTE1, QUOTE, S1, S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c1

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

(42) BOUNDS(1, 1)