Runtime Complexity (innermost) proof of /tmp/tmp5klGcb/LISTUTILITIES_nosorts

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), 10 ms)

↳2 CpxTRS

    ↳3 CpxTrsMatchBoundsTAProof (⇔, 468 ms)

    ↳4 BOUNDS(1, n^1)

    ↳5 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 14 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)), 19 ms)

        ↳16 CdtProblem

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

        ↳18 CdtProblem

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

        ↳20 CdtProblem

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

        ↳22 CdtProblem

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

        ↳24 CdtProblem

        ↳25 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 11 ms)

        ↳26 CdtProblem

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

        ↳28 CdtProblem

        ↳29 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 40 ms)

        ↳30 CdtProblem

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

        ↳32 CdtProblem

        ↳33 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 8 ms)

        ↳34 CdtProblem

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

        ↳36 CdtProblem

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

        ↳38 BOUNDS(1, 1)

(0) Obligation:

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

The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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

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

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(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))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
take(mark(X1), X2) → mark(take(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
snd(ok(X)) → ok(snd(X))
pair(X1, mark(X2)) → mark(pair(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
snd(mark(X)) → mark(snd(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
take(X1, mark(X2)) → mark(take(X1, X2))
head(ok(X)) → ok(head(X))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
head(mark(X)) → mark(head(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
natsFrom(mark(X)) → mark(natsFrom(X))
natsFrom(ok(X)) → ok(natsFrom(X))
fst(mark(X)) → mark(fst(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
tail(mark(X)) → mark(tail(X))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
fst(ok(X)) → ok(fst(X))
proper(0) → ok(0)
cons(mark(X1), X2) → mark(cons(X1, X2))
top(mark(X)) → top(proper(X))

Rewrite Strategy: INNERMOST

(3) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]
transitions:
ok0(0) → 0
active0(0) → 0
mark0(0) → 0
tt0() → 0
nil0() → 0
00() → 0
top0(0) → 1
afterNth0(0, 0) → 2
take0(0, 0) → 3
cons0(0, 0) → 4
tail0(0) → 5
snd0(0) → 6
pair0(0, 0) → 7
and0(0, 0) → 8
sel0(0, 0) → 9
splitAt0(0, 0) → 10
U110(0, 0, 0, 0) → 11
proper0(0) → 12
head0(0) → 13
U120(0, 0) → 14
natsFrom0(0) → 15
fst0(0) → 16
s0(0) → 17
active1(0) → 18
top1(18) → 1
afterNth1(0, 0) → 19
mark1(19) → 2
take1(0, 0) → 20
mark1(20) → 3
cons1(0, 0) → 21
ok1(21) → 4
tail1(0) → 22
ok1(22) → 5
snd1(0) → 23
ok1(23) → 6
pair1(0, 0) → 24
mark1(24) → 7
and1(0, 0) → 25
ok1(25) → 8
snd1(0) → 26
mark1(26) → 6
take1(0, 0) → 27
ok1(27) → 3
sel1(0, 0) → 28
ok1(28) → 9
splitAt1(0, 0) → 29
mark1(29) → 10
sel1(0, 0) → 30
mark1(30) → 9
U111(0, 0, 0, 0) → 31
ok1(31) → 11
tt1() → 32
ok1(32) → 12
nil1() → 33
ok1(33) → 12
U111(0, 0, 0, 0) → 34
mark1(34) → 11
head1(0) → 35
ok1(35) → 13
pair1(0, 0) → 36
ok1(36) → 7
and1(0, 0) → 37
mark1(37) → 8
head1(0) → 38
mark1(38) → 13
U121(0, 0) → 39
ok1(39) → 14
U121(0, 0) → 40
mark1(40) → 14
natsFrom1(0) → 41
mark1(41) → 15
natsFrom1(0) → 42
ok1(42) → 15
fst1(0) → 43
mark1(43) → 16
afterNth1(0, 0) → 44
ok1(44) → 2
tail1(0) → 45
mark1(45) → 5
s1(0) → 46
ok1(46) → 17
s1(0) → 47
mark1(47) → 17
splitAt1(0, 0) → 48
ok1(48) → 10
fst1(0) → 49
ok1(49) → 16
01() → 50
ok1(50) → 12
cons1(0, 0) → 51
mark1(51) → 4
proper1(0) → 52
top1(52) → 1
mark1(19) → 19
mark1(19) → 44
mark1(20) → 20
mark1(20) → 27
ok1(21) → 21
ok1(21) → 51
ok1(22) → 22
ok1(22) → 45
ok1(23) → 23
ok1(23) → 26
mark1(24) → 24
mark1(24) → 36
ok1(25) → 25
ok1(25) → 37
mark1(26) → 23
mark1(26) → 26
ok1(27) → 20
ok1(27) → 27
ok1(28) → 28
ok1(28) → 30
mark1(29) → 29
mark1(29) → 48
mark1(30) → 28
mark1(30) → 30
ok1(31) → 31
ok1(31) → 34
ok1(32) → 52
ok1(33) → 52
mark1(34) → 31
mark1(34) → 34
ok1(35) → 35
ok1(35) → 38
ok1(36) → 24
ok1(36) → 36
mark1(37) → 25
mark1(37) → 37
mark1(38) → 35
mark1(38) → 38
ok1(39) → 39
ok1(39) → 40
mark1(40) → 39
mark1(40) → 40
mark1(41) → 41
mark1(41) → 42
ok1(42) → 41
ok1(42) → 42
mark1(43) → 43
mark1(43) → 49
ok1(44) → 19
ok1(44) → 44
mark1(45) → 22
mark1(45) → 45
ok1(46) → 46
ok1(46) → 47
mark1(47) → 46
mark1(47) → 47
ok1(48) → 29
ok1(48) → 48
ok1(49) → 43
ok1(49) → 49
ok1(50) → 52
mark1(51) → 21
mark1(51) → 51
active2(32) → 53
top2(53) → 1
active2(33) → 53
active2(50) → 53

(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))
afterNth(z0, mark(z1)) → mark(afterNth(z0, z1))
afterNth(mark(z0), z1) → mark(afterNth(z0, z1))
afterNth(ok(z0), ok(z1)) → ok(afterNth(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
tail(ok(z0)) → ok(tail(z0))
tail(mark(z0)) → mark(tail(z0))
snd(ok(z0)) → ok(snd(z0))
snd(mark(z0)) → mark(snd(z0))
pair(z0, mark(z1)) → mark(pair(z0, z1))
pair(mark(z0), z1) → mark(pair(z0, z1))
pair(ok(z0), ok(z1)) → ok(pair(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
and(mark(z0), z1) → mark(and(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))
splitAt(mark(z0), z1) → mark(splitAt(z0, z1))
splitAt(z0, mark(z1)) → mark(splitAt(z0, z1))
splitAt(ok(z0), ok(z1)) → ok(splitAt(z0, z1))
U11(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U11(z0, z1, z2, z3))
U11(mark(z0), z1, z2, z3) → mark(U11(z0, z1, z2, z3))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)
head(ok(z0)) → ok(head(z0))
head(mark(z0)) → mark(head(z0))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
U12(mark(z0), z1) → mark(U12(z0, z1))
natsFrom(mark(z0)) → mark(natsFrom(z0))
natsFrom(ok(z0)) → ok(natsFrom(z0))
fst(mark(z0)) → mark(fst(z0))
fst(ok(z0)) → ok(fst(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))
AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
PROPER(tt) → c27
PROPER(nil) → c28
PROPER(0) → c29
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))

S tuples:

TOP(ok(z0)) → c(TOP(active(z0)))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
PROPER(tt) → c27
PROPER(nil) → c28
PROPER(0) → c29
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))

K tuples:none
Defined Rule Symbols:

top, afterNth, take, cons, tail, snd, pair, and, sel, splitAt, U11, proper, head, U12, natsFrom, fst, s

Defined Pair Symbols:

TOP, AFTERNTH, TAKE, CONS, TAIL, SND, PAIR, AND, SEL, SPLITAT, U11', PROPER, HEAD, U12', NATSFROM, FST, 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, c38, c39

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

Removed 4 trailing nodes:

PROPER(0) → c29
TOP(ok(z0)) → c(TOP(active(z0)))
PROPER(nil) → c28
PROPER(tt) → c27

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
afterNth(z0, mark(z1)) → mark(afterNth(z0, z1))
afterNth(mark(z0), z1) → mark(afterNth(z0, z1))
afterNth(ok(z0), ok(z1)) → ok(afterNth(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
tail(ok(z0)) → ok(tail(z0))
tail(mark(z0)) → mark(tail(z0))
snd(ok(z0)) → ok(snd(z0))
snd(mark(z0)) → mark(snd(z0))
pair(z0, mark(z1)) → mark(pair(z0, z1))
pair(mark(z0), z1) → mark(pair(z0, z1))
pair(ok(z0), ok(z1)) → ok(pair(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
and(mark(z0), z1) → mark(and(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))
splitAt(mark(z0), z1) → mark(splitAt(z0, z1))
splitAt(z0, mark(z1)) → mark(splitAt(z0, z1))
splitAt(ok(z0), ok(z1)) → ok(splitAt(z0, z1))
U11(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U11(z0, z1, z2, z3))
U11(mark(z0), z1, z2, z3) → mark(U11(z0, z1, z2, z3))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)
head(ok(z0)) → ok(head(z0))
head(mark(z0)) → mark(head(z0))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
U12(mark(z0), z1) → mark(U12(z0, z1))
natsFrom(mark(z0)) → mark(natsFrom(z0))
natsFrom(ok(z0)) → ok(natsFrom(z0))
fst(mark(z0)) → mark(fst(z0))
fst(ok(z0)) → ok(fst(z0))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))

Tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))

S tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))

K tuples:none
Defined Rule Symbols:

top, afterNth, take, cons, tail, snd, pair, and, sel, splitAt, U11, proper, head, U12, natsFrom, fst, s

Defined Pair Symbols:

TOP, AFTERNTH, TAKE, CONS, TAIL, SND, PAIR, AND, SEL, SPLITAT, U11', HEAD, U12', NATSFROM, FST, S

Compound Symbols:

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, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39

(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))
afterNth(z0, mark(z1)) → mark(afterNth(z0, z1))
afterNth(mark(z0), z1) → mark(afterNth(z0, z1))
afterNth(ok(z0), ok(z1)) → ok(afterNth(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
tail(ok(z0)) → ok(tail(z0))
tail(mark(z0)) → mark(tail(z0))
snd(ok(z0)) → ok(snd(z0))
snd(mark(z0)) → mark(snd(z0))
pair(z0, mark(z1)) → mark(pair(z0, z1))
pair(mark(z0), z1) → mark(pair(z0, z1))
pair(ok(z0), ok(z1)) → ok(pair(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
and(mark(z0), z1) → mark(and(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))
splitAt(mark(z0), z1) → mark(splitAt(z0, z1))
splitAt(z0, mark(z1)) → mark(splitAt(z0, z1))
splitAt(ok(z0), ok(z1)) → ok(splitAt(z0, z1))
U11(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U11(z0, z1, z2, z3))
U11(mark(z0), z1, z2, z3) → mark(U11(z0, z1, z2, z3))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)
head(ok(z0)) → ok(head(z0))
head(mark(z0)) → mark(head(z0))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
U12(mark(z0), z1) → mark(U12(z0, z1))
natsFrom(mark(z0)) → mark(natsFrom(z0))
natsFrom(ok(z0)) → ok(natsFrom(z0))
fst(mark(z0)) → mark(fst(z0))
fst(ok(z0)) → ok(fst(z0))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))

Tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

top, afterNth, take, cons, tail, snd, pair, and, sel, splitAt, U11, proper, head, U12, natsFrom, fst, s

Defined Pair Symbols:

AFTERNTH, TAKE, CONS, TAIL, SND, PAIR, AND, SEL, SPLITAT, U11', HEAD, U12', NATSFROM, FST, S, TOP

Compound Symbols:

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, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, 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))
afterNth(z0, mark(z1)) → mark(afterNth(z0, z1))
afterNth(mark(z0), z1) → mark(afterNth(z0, z1))
afterNth(ok(z0), ok(z1)) → ok(afterNth(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
tail(ok(z0)) → ok(tail(z0))
tail(mark(z0)) → mark(tail(z0))
snd(ok(z0)) → ok(snd(z0))
snd(mark(z0)) → mark(snd(z0))
pair(z0, mark(z1)) → mark(pair(z0, z1))
pair(mark(z0), z1) → mark(pair(z0, z1))
pair(ok(z0), ok(z1)) → ok(pair(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
and(mark(z0), z1) → mark(and(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))
splitAt(mark(z0), z1) → mark(splitAt(z0, z1))
splitAt(z0, mark(z1)) → mark(splitAt(z0, z1))
splitAt(ok(z0), ok(z1)) → ok(splitAt(z0, z1))
U11(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U11(z0, z1, z2, z3))
U11(mark(z0), z1, z2, z3) → mark(U11(z0, z1, z2, z3))
head(ok(z0)) → ok(head(z0))
head(mark(z0)) → mark(head(z0))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
U12(mark(z0), z1) → mark(U12(z0, z1))
natsFrom(mark(z0)) → mark(natsFrom(z0))
natsFrom(ok(z0)) → ok(natsFrom(z0))
fst(mark(z0)) → mark(fst(z0))
fst(ok(z0)) → ok(fst(z0))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

proper

Defined Pair Symbols:

AFTERNTH, TAKE, CONS, TAIL, SND, PAIR, AND, SEL, SPLITAT, U11', HEAD, U12', NATSFROM, FST, S, TOP

Compound Symbols:

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, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, 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.

CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))

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

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

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

POL(0) = [2]
POL(AFTERNTH(x₁, x₂)) = 0
POL(AND(x₁, x₂)) = [3]x₂
POL(CONS(x₁, x₂)) = x₂
POL(FST(x₁)) = 0
POL(HEAD(x₁)) = 0
POL(NATSFROM(x₁)) = 0
POL(PAIR(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(SEL(x₁, x₂)) = 0
POL(SND(x₁)) = 0
POL(SPLITAT(x₁, x₂)) = 0
POL(TAIL(x₁)) = 0
POL(TAKE(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂, x₃, x₄)) = [2]x₂
POL(U12'(x₁, x₂)) = [2]x₂
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(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(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = 0
POL(nil) = [2]
POL(ok(x₁)) = [3] + x₁
POL(proper(x₁)) = [2] + [3]x₁
POL(tt) = [2]

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

K tuples:

CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

AFTERNTH, TAKE, CONS, TAIL, SND, PAIR, AND, SEL, SPLITAT, U11', HEAD, U12', NATSFROM, FST, S, TOP

Compound Symbols:

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, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, 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.

TOP(mark(z0)) → c1(TOP(proper(z0)))

We considered the (Usable) Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

And the Tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(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(AFTERNTH(x₁, x₂)) = 0
POL(AND(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = 0
POL(FST(x₁)) = 0
POL(HEAD(x₁)) = 0
POL(NATSFROM(x₁)) = 0
POL(PAIR(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(SEL(x₁, x₂)) = 0
POL(SND(x₁)) = 0
POL(SPLITAT(x₁, x₂)) = 0
POL(TAIL(x₁)) = 0
POL(TAKE(x₁, x₂)) = 0
POL(TOP(x₁)) = x₁
POL(U11'(x₁, x₂, x₃, x₄)) = 0
POL(U12'(x₁, x₂)) = 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(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(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(c38(x₁)) = x₁
POL(c39(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]
POL(nil) = 0
POL(ok(x₁)) = 0
POL(proper(x₁)) = 0
POL(tt) = 0

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))

K tuples:

CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))

Defined Rule Symbols:

proper

Defined Pair Symbols:

AFTERNTH, TAKE, CONS, TAIL, SND, PAIR, AND, SEL, SPLITAT, U11', HEAD, U12', NATSFROM, FST, S, TOP

Compound Symbols:

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, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, 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.

TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAIL(mark(z0)) → c11(TAIL(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
S(mark(z0)) → c39(S(z0))

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

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(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(AFTERNTH(x₁, x₂)) = 0
POL(AND(x₁, x₂)) = x₁
POL(CONS(x₁, x₂)) = x₂
POL(FST(x₁)) = 0
POL(HEAD(x₁)) = x₁
POL(NATSFROM(x₁)) = x₁
POL(PAIR(x₁, x₂)) = x₁ + x₂
POL(S(x₁)) = x₁
POL(SEL(x₁, x₂)) = 0
POL(SND(x₁)) = 0
POL(SPLITAT(x₁, x₂)) = x₂
POL(TAIL(x₁)) = x₁
POL(TAKE(x₁, x₂)) = x₂
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂, x₃, x₄)) = x₁ + x₂
POL(U12'(x₁, x₂)) = x₁ + x₂
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(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(c38(x₁)) = x₁
POL(c39(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(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(tt) = 0

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
HEAD(ok(z0)) → c30(HEAD(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))

K tuples:

CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAIL(mark(z0)) → c11(TAIL(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
S(mark(z0)) → c39(S(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

AFTERNTH, TAKE, CONS, TAIL, SND, PAIR, AND, SEL, SPLITAT, U11', HEAD, U12', NATSFROM, FST, S, TOP

Compound Symbols:

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, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, 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.

PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))

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

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(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(AFTERNTH(x₁, x₂)) = 0
POL(AND(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = x₂
POL(FST(x₁)) = x₁
POL(HEAD(x₁)) = 0
POL(NATSFROM(x₁)) = x₁
POL(PAIR(x₁, x₂)) = x₁ + x₂
POL(S(x₁)) = 0
POL(SEL(x₁, x₂)) = 0
POL(SND(x₁)) = 0
POL(SPLITAT(x₁, x₂)) = x₁
POL(TAIL(x₁)) = 0
POL(TAKE(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂, x₃, x₄)) = 0
POL(U12'(x₁, x₂)) = 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(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(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(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(nil) = 0
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = 0
POL(tt) = 0

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
HEAD(ok(z0)) → c30(HEAD(z0))
S(ok(z0)) → c38(S(z0))

K tuples:

CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAIL(mark(z0)) → c11(TAIL(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
S(mark(z0)) → c39(S(z0))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

AFTERNTH, TAKE, CONS, TAIL, SND, PAIR, AND, SEL, SPLITAT, U11', HEAD, U12', NATSFROM, FST, S, TOP

Compound Symbols:

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, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, 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.

TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))

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

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(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(AFTERNTH(x₁, x₂)) = 0
POL(AND(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = x₂
POL(FST(x₁)) = 0
POL(HEAD(x₁)) = 0
POL(NATSFROM(x₁)) = 0
POL(PAIR(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(SEL(x₁, x₂)) = x₁
POL(SND(x₁)) = 0
POL(SPLITAT(x₁, x₂)) = 0
POL(TAIL(x₁)) = 0
POL(TAKE(x₁, x₂)) = x₁
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂, x₃, x₄)) = 0
POL(U12'(x₁, x₂)) = x₂
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(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(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(nil) = 0
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = 0
POL(tt) = 0

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
HEAD(ok(z0)) → c30(HEAD(z0))
S(ok(z0)) → c38(S(z0))

K tuples:

CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAIL(mark(z0)) → c11(TAIL(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
S(mark(z0)) → c39(S(z0))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

AFTERNTH, TAKE, CONS, TAIL, SND, PAIR, AND, SEL, SPLITAT, U11', HEAD, U12', NATSFROM, FST, S, TOP

Compound Symbols:

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, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c1

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

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

SEL(z0, mark(z1)) → c20(SEL(z0, z1))

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

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(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(AFTERNTH(x₁, x₂)) = 0
POL(AND(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = 0
POL(FST(x₁)) = x₁
POL(HEAD(x₁)) = 0
POL(NATSFROM(x₁)) = 0
POL(PAIR(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(SEL(x₁, x₂)) = x₂
POL(SND(x₁)) = 0
POL(SPLITAT(x₁, x₂)) = 0
POL(TAIL(x₁)) = 0
POL(TAKE(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂, x₃, x₄)) = x₂
POL(U12'(x₁, x₂)) = 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(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(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(c38(x₁)) = x₁
POL(c39(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(ok(x₁)) = x₁
POL(proper(x₁)) = [1] + x₁
POL(tt) = 0

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
HEAD(ok(z0)) → c30(HEAD(z0))
S(ok(z0)) → c38(S(z0))

K tuples:

CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAIL(mark(z0)) → c11(TAIL(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
S(mark(z0)) → c39(S(z0))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

AFTERNTH, TAKE, CONS, TAIL, SND, PAIR, AND, SEL, SPLITAT, U11', HEAD, U12', NATSFROM, FST, S, TOP

Compound Symbols:

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, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, 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.

CONS(mark(z0), z1) → c9(CONS(z0, z1))

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

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(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(AFTERNTH(x₁, x₂)) = 0
POL(AND(x₁, x₂)) = x₁
POL(CONS(x₁, x₂)) = x₁
POL(FST(x₁)) = 0
POL(HEAD(x₁)) = 0
POL(NATSFROM(x₁)) = 0
POL(PAIR(x₁, x₂)) = x₂
POL(S(x₁)) = 0
POL(SEL(x₁, x₂)) = 0
POL(SND(x₁)) = 0
POL(SPLITAT(x₁, x₂)) = 0
POL(TAIL(x₁)) = 0
POL(TAKE(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂, x₃, x₄)) = 0
POL(U12'(x₁, x₂)) = 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(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(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(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(tt) = 0

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
HEAD(ok(z0)) → c30(HEAD(z0))
S(ok(z0)) → c38(S(z0))

K tuples:

CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAIL(mark(z0)) → c11(TAIL(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
S(mark(z0)) → c39(S(z0))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

AFTERNTH, TAKE, CONS, TAIL, SND, PAIR, AND, SEL, SPLITAT, U11', HEAD, U12', NATSFROM, FST, S, TOP

Compound Symbols:

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, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, 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.

AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
SND(mark(z0)) → c13(SND(z0))

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

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(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(AFTERNTH(x₁, x₂)) = x₁
POL(AND(x₁, x₂)) = x₂
POL(CONS(x₁, x₂)) = x₁
POL(FST(x₁)) = 0
POL(HEAD(x₁)) = 0
POL(NATSFROM(x₁)) = 0
POL(PAIR(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(SEL(x₁, x₂)) = 0
POL(SND(x₁)) = x₁
POL(SPLITAT(x₁, x₂)) = 0
POL(TAIL(x₁)) = 0
POL(TAKE(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂, x₃, x₄)) = 0
POL(U12'(x₁, x₂)) = 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(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(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(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(tt) = 0

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
HEAD(ok(z0)) → c30(HEAD(z0))
S(ok(z0)) → c38(S(z0))

K tuples:

CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAIL(mark(z0)) → c11(TAIL(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
S(mark(z0)) → c39(S(z0))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
SND(mark(z0)) → c13(SND(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

AFTERNTH, TAKE, CONS, TAIL, SND, PAIR, AND, SEL, SPLITAT, U11', HEAD, U12', NATSFROM, FST, S, TOP

Compound Symbols:

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, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, 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.

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))

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

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(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(AFTERNTH(x₁, x₂)) = x₂
POL(AND(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = 0
POL(FST(x₁)) = 0
POL(HEAD(x₁)) = 0
POL(NATSFROM(x₁)) = 0
POL(PAIR(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(SEL(x₁, x₂)) = 0
POL(SND(x₁)) = 0
POL(SPLITAT(x₁, x₂)) = 0
POL(TAIL(x₁)) = 0
POL(TAKE(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂, x₃, x₄)) = 0
POL(U12'(x₁, x₂)) = 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(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(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(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(tt) = 0

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
HEAD(ok(z0)) → c30(HEAD(z0))
S(ok(z0)) → c38(S(z0))

K tuples:

CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAIL(mark(z0)) → c11(TAIL(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
S(mark(z0)) → c39(S(z0))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
SND(mark(z0)) → c13(SND(z0))
AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

AFTERNTH, TAKE, CONS, TAIL, SND, PAIR, AND, SEL, SPLITAT, U11', HEAD, U12', NATSFROM, FST, S, TOP

Compound Symbols:

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, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, 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.

SND(ok(z0)) → c12(SND(z0))

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

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(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(AFTERNTH(x₁, x₂)) = 0
POL(AND(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = 0
POL(FST(x₁)) = 0
POL(HEAD(x₁)) = 0
POL(NATSFROM(x₁)) = 0
POL(PAIR(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(SEL(x₁, x₂)) = 0
POL(SND(x₁)) = x₁
POL(SPLITAT(x₁, x₂)) = x₂
POL(TAIL(x₁)) = 0
POL(TAKE(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂, x₃, x₄)) = x₂
POL(U12'(x₁, x₂)) = 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(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(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(c38(x₁)) = x₁
POL(c39(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(ok(x₁)) = [2] + x₁
POL(proper(x₁)) = 0
POL(tt) = 0

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
HEAD(ok(z0)) → c30(HEAD(z0))
S(ok(z0)) → c38(S(z0))

K tuples:

CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAIL(mark(z0)) → c11(TAIL(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
S(mark(z0)) → c39(S(z0))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
SND(mark(z0)) → c13(SND(z0))
AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
SND(ok(z0)) → c12(SND(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

AFTERNTH, TAKE, CONS, TAIL, SND, PAIR, AND, SEL, SPLITAT, U11', HEAD, U12', NATSFROM, FST, S, TOP

Compound Symbols:

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, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, 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.

HEAD(ok(z0)) → c30(HEAD(z0))

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

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(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(AFTERNTH(x₁, x₂)) = 0
POL(AND(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = 0
POL(FST(x₁)) = 0
POL(HEAD(x₁)) = x₁
POL(NATSFROM(x₁)) = 0
POL(PAIR(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(SEL(x₁, x₂)) = [2]x₂
POL(SND(x₁)) = 0
POL(SPLITAT(x₁, x₂)) = x₁
POL(TAIL(x₁)) = 0
POL(TAKE(x₁, x₂)) = [2]x₁
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂, x₃, x₄)) = [2]x₁ + x₃ + [2]x₄
POL(U12'(x₁, x₂)) = 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(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(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(c38(x₁)) = x₁
POL(c39(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(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = 0
POL(tt) = 0

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
S(ok(z0)) → c38(S(z0))

K tuples:

CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAIL(mark(z0)) → c11(TAIL(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
S(mark(z0)) → c39(S(z0))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
SND(mark(z0)) → c13(SND(z0))
AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
SND(ok(z0)) → c12(SND(z0))
HEAD(ok(z0)) → c30(HEAD(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

AFTERNTH, TAKE, CONS, TAIL, SND, PAIR, AND, SEL, SPLITAT, U11', HEAD, U12', NATSFROM, FST, S, TOP

Compound Symbols:

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, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, 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.

AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
S(ok(z0)) → c38(S(z0))

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

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(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(AFTERNTH(x₁, x₂)) = x₁ + x₂
POL(AND(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = 0
POL(FST(x₁)) = 0
POL(HEAD(x₁)) = 0
POL(NATSFROM(x₁)) = 0
POL(PAIR(x₁, x₂)) = 0
POL(S(x₁)) = x₁
POL(SEL(x₁, x₂)) = 0
POL(SND(x₁)) = 0
POL(SPLITAT(x₁, x₂)) = 0
POL(TAIL(x₁)) = x₁
POL(TAKE(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂, x₃, x₄)) = 0
POL(U12'(x₁, x₂)) = 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(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(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(c38(x₁)) = x₁
POL(c39(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(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = [1]
POL(tt) = 0

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(ok(z0)) → c30(HEAD(z0))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:none
K tuples:

CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
AND(ok(z0), ok(z1)) → c17(AND(z0, z1))
U11'(ok(z0), ok(z1), ok(z2), ok(z3)) → c25(U11'(z0, z1, z2, z3))
U12'(ok(z0), ok(z1)) → c32(U12'(z0, z1))
TOP(mark(z0)) → c1(TOP(proper(z0)))
TAKE(z0, mark(z1)) → c7(TAKE(z0, z1))
TAIL(mark(z0)) → c11(TAIL(z0))
PAIR(z0, mark(z1)) → c14(PAIR(z0, z1))
PAIR(mark(z0), z1) → c15(PAIR(z0, z1))
AND(mark(z0), z1) → c18(AND(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U11'(mark(z0), z1, z2, z3) → c26(U11'(z0, z1, z2, z3))
HEAD(mark(z0)) → c31(HEAD(z0))
U12'(mark(z0), z1) → c33(U12'(z0, z1))
NATSFROM(mark(z0)) → c34(NATSFROM(z0))
S(mark(z0)) → c39(S(z0))
PAIR(ok(z0), ok(z1)) → c16(PAIR(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
NATSFROM(ok(z0)) → c35(NATSFROM(z0))
FST(mark(z0)) → c36(FST(z0))
FST(ok(z0)) → c37(FST(z0))
TAKE(mark(z0), z1) → c5(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c6(TAKE(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
AFTERNTH(mark(z0), z1) → c3(AFTERNTH(z0, z1))
SND(mark(z0)) → c13(SND(z0))
AFTERNTH(z0, mark(z1)) → c2(AFTERNTH(z0, z1))
SND(ok(z0)) → c12(SND(z0))
HEAD(ok(z0)) → c30(HEAD(z0))
AFTERNTH(ok(z0), ok(z1)) → c4(AFTERNTH(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
S(ok(z0)) → c38(S(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

AFTERNTH, TAKE, CONS, TAIL, SND, PAIR, AND, SEL, SPLITAT, U11', HEAD, U12', NATSFROM, FST, S, TOP

Compound Symbols:

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, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c1

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

(38) BOUNDS(1, 1)