Runtime Complexity (innermost) proof of /tmp/tmpdNoHrC/Ex1_2_AEL03

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), 0 ms)

↳2 CpxTRS

    ↳3 CpxTrsMatchBoundsTAProof (⇔, 315 ms)

    ↳4 BOUNDS(1, n^1)

    ↳5 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳6 CdtProblem

        ↳7 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

        ↳8 CdtProblem

        ↳9 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

        ↳10 CdtProblem

        ↳11 CdtUsableRulesProof (⇔, 0 ms)

        ↳12 CdtProblem

        ↳13 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 168 ms)

        ↳14 CdtProblem

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

        ↳16 CdtProblem

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

        ↳18 CdtProblem

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

        ↳20 CdtProblem

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

        ↳22 CdtProblem

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

        ↳24 CdtProblem

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

        ↳26 CdtProblem

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

        ↳28 CdtProblem

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

        ↳30 CdtProblem

        ↳31 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 6 ms)

        ↳32 CdtProblem

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

        ↳34 CdtProblem

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

        ↳36 CdtProblem

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

        ↳38 CdtProblem

        ↳39 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 12 ms)

        ↳40 CdtProblem

        ↳41 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)

        ↳42 CdtProblem

        ↳43 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 6 ms)

        ↳44 CdtProblem

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

        ↳46 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(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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

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

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))

(2) Obligation:

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

The TRS R consists of the following rules:

negrecip(mark(X)) → mark(negrecip(X))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
top(ok(X)) → top(active(X))
posrecip(mark(X)) → mark(posrecip(X))
from(ok(X)) → ok(from(X))
from(mark(X)) → mark(from(X))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
square(mark(X)) → mark(square(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
posrecip(ok(X)) → ok(posrecip(X))
square(ok(X)) → ok(square(X))
proper(rnil) → ok(rnil)
times(X1, mark(X2)) → mark(times(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
pi(mark(X)) → mark(pi(X))
proper(nil) → ok(nil)
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
plus(mark(X1), X2) → mark(plus(X1, X2))
proper(0) → ok(0)
cons(mark(X1), X2) → mark(cons(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(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]
transitions:
mark0(0) → 0
ok0(0) → 0
active0(0) → 0
rnil0() → 0
nil0() → 0
00() → 0
negrecip0(0) → 1
times0(0, 0) → 2
top0(0) → 3
posrecip0(0) → 4
from0(0) → 5
rcons0(0, 0) → 6
cons0(0, 0) → 7
2ndsneg0(0, 0) → 8
square0(0) → 9
2ndspos0(0, 0) → 10
proper0(0) → 11
pi0(0) → 12
plus0(0, 0) → 13
s0(0) → 14
negrecip1(0) → 15
mark1(15) → 1
times1(0, 0) → 16
ok1(16) → 2
active1(0) → 17
top1(17) → 3
posrecip1(0) → 18
mark1(18) → 4
from1(0) → 19
ok1(19) → 5
from1(0) → 20
mark1(20) → 5
rcons1(0, 0) → 21
mark1(21) → 6
negrecip1(0) → 22
ok1(22) → 1
cons1(0, 0) → 23
ok1(23) → 7
rcons1(0, 0) → 24
ok1(24) → 6
2ndsneg1(0, 0) → 25
mark1(25) → 8
square1(0) → 26
mark1(26) → 9
2ndspos1(0, 0) → 27
ok1(27) → 10
posrecip1(0) → 28
ok1(28) → 4
square1(0) → 29
ok1(29) → 9
rnil1() → 30
ok1(30) → 11
times1(0, 0) → 31
mark1(31) → 2
pi1(0) → 32
ok1(32) → 12
plus1(0, 0) → 33
ok1(33) → 13
plus1(0, 0) → 34
mark1(34) → 13
pi1(0) → 35
mark1(35) → 12
nil1() → 36
ok1(36) → 11
2ndsneg1(0, 0) → 37
ok1(37) → 8
2ndspos1(0, 0) → 38
mark1(38) → 10
s1(0) → 39
ok1(39) → 14
s1(0) → 40
mark1(40) → 14
01() → 41
ok1(41) → 11
cons1(0, 0) → 42
mark1(42) → 7
proper1(0) → 43
top1(43) → 3
mark1(15) → 15
mark1(15) → 22
ok1(16) → 16
ok1(16) → 31
mark1(18) → 18
mark1(18) → 28
ok1(19) → 19
ok1(19) → 20
mark1(20) → 19
mark1(20) → 20
mark1(21) → 21
mark1(21) → 24
ok1(22) → 15
ok1(22) → 22
ok1(23) → 23
ok1(23) → 42
ok1(24) → 21
ok1(24) → 24
mark1(25) → 25
mark1(25) → 37
mark1(26) → 26
mark1(26) → 29
ok1(27) → 27
ok1(27) → 38
ok1(28) → 18
ok1(28) → 28
ok1(29) → 26
ok1(29) → 29
ok1(30) → 43
mark1(31) → 16
mark1(31) → 31
ok1(32) → 32
ok1(32) → 35
ok1(33) → 33
ok1(33) → 34
mark1(34) → 33
mark1(34) → 34
mark1(35) → 32
mark1(35) → 35
ok1(36) → 43
ok1(37) → 25
ok1(37) → 37
mark1(38) → 27
mark1(38) → 38
ok1(39) → 39
ok1(39) → 40
mark1(40) → 39
mark1(40) → 40
ok1(41) → 43
mark1(42) → 23
mark1(42) → 42
active2(30) → 44
top2(44) → 3
active2(36) → 44
active2(41) → 44

(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:

negrecip(mark(z0)) → mark(negrecip(z0))
negrecip(ok(z0)) → ok(negrecip(z0))
times(ok(z0), ok(z1)) → ok(times(z0, z1))
times(z0, mark(z1)) → mark(times(z0, z1))
times(mark(z0), z1) → mark(times(z0, z1))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
posrecip(mark(z0)) → mark(posrecip(z0))
posrecip(ok(z0)) → ok(posrecip(z0))
from(ok(z0)) → ok(from(z0))
from(mark(z0)) → mark(from(z0))
rcons(mark(z0), z1) → mark(rcons(z0, z1))
rcons(ok(z0), ok(z1)) → ok(rcons(z0, z1))
rcons(z0, mark(z1)) → mark(rcons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
2ndsneg(z0, mark(z1)) → mark(2ndsneg(z0, z1))
2ndsneg(ok(z0), ok(z1)) → ok(2ndsneg(z0, z1))
2ndsneg(mark(z0), z1) → mark(2ndsneg(z0, z1))
square(mark(z0)) → mark(square(z0))
square(ok(z0)) → ok(square(z0))
2ndspos(ok(z0), ok(z1)) → ok(2ndspos(z0, z1))
2ndspos(mark(z0), z1) → mark(2ndspos(z0, z1))
2ndspos(z0, mark(z1)) → mark(2ndspos(z0, z1))
proper(rnil) → ok(rnil)
proper(nil) → ok(nil)
proper(0) → ok(0)
pi(ok(z0)) → ok(pi(z0))
pi(mark(z0)) → mark(pi(z0))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))

Tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
TOP(ok(z0)) → c5(TOP(active(z0)))
TOP(mark(z0)) → c6(TOP(proper(z0)), PROPER(z0))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PROPER(rnil) → c24
PROPER(nil) → c25
PROPER(0) → c26
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))

S tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
TOP(ok(z0)) → c5(TOP(active(z0)))
TOP(mark(z0)) → c6(TOP(proper(z0)), PROPER(z0))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PROPER(rnil) → c24
PROPER(nil) → c25
PROPER(0) → c26
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))

K tuples:none
Defined Rule Symbols:

negrecip, times, top, posrecip, from, rcons, cons, 2ndsneg, square, 2ndspos, proper, pi, plus, s

Defined Pair Symbols:

NEGRECIP, TIMES, TOP, POSRECIP, FROM, RCONS, CONS, 2NDSNEG, SQUARE, 2NDSPOS, PROPER, PI, PLUS, 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

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

Removed 4 trailing nodes:

TOP(ok(z0)) → c5(TOP(active(z0)))
PROPER(nil) → c25
PROPER(rnil) → c24
PROPER(0) → c26

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

negrecip(mark(z0)) → mark(negrecip(z0))
negrecip(ok(z0)) → ok(negrecip(z0))
times(ok(z0), ok(z1)) → ok(times(z0, z1))
times(z0, mark(z1)) → mark(times(z0, z1))
times(mark(z0), z1) → mark(times(z0, z1))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
posrecip(mark(z0)) → mark(posrecip(z0))
posrecip(ok(z0)) → ok(posrecip(z0))
from(ok(z0)) → ok(from(z0))
from(mark(z0)) → mark(from(z0))
rcons(mark(z0), z1) → mark(rcons(z0, z1))
rcons(ok(z0), ok(z1)) → ok(rcons(z0, z1))
rcons(z0, mark(z1)) → mark(rcons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
2ndsneg(z0, mark(z1)) → mark(2ndsneg(z0, z1))
2ndsneg(ok(z0), ok(z1)) → ok(2ndsneg(z0, z1))
2ndsneg(mark(z0), z1) → mark(2ndsneg(z0, z1))
square(mark(z0)) → mark(square(z0))
square(ok(z0)) → ok(square(z0))
2ndspos(ok(z0), ok(z1)) → ok(2ndspos(z0, z1))
2ndspos(mark(z0), z1) → mark(2ndspos(z0, z1))
2ndspos(z0, mark(z1)) → mark(2ndspos(z0, z1))
proper(rnil) → ok(rnil)
proper(nil) → ok(nil)
proper(0) → ok(0)
pi(ok(z0)) → ok(pi(z0))
pi(mark(z0)) → mark(pi(z0))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))

Tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
TOP(mark(z0)) → c6(TOP(proper(z0)), PROPER(z0))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))

S tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
TOP(mark(z0)) → c6(TOP(proper(z0)), PROPER(z0))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))

K tuples:none
Defined Rule Symbols:

negrecip, times, top, posrecip, from, rcons, cons, 2ndsneg, square, 2ndspos, proper, pi, plus, s

Defined Pair Symbols:

NEGRECIP, TIMES, TOP, POSRECIP, FROM, RCONS, CONS, 2NDSNEG, SQUARE, 2NDSPOS, PI, PLUS, S

Compound Symbols:

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

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

Removed 1 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

negrecip(mark(z0)) → mark(negrecip(z0))
negrecip(ok(z0)) → ok(negrecip(z0))
times(ok(z0), ok(z1)) → ok(times(z0, z1))
times(z0, mark(z1)) → mark(times(z0, z1))
times(mark(z0), z1) → mark(times(z0, z1))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
posrecip(mark(z0)) → mark(posrecip(z0))
posrecip(ok(z0)) → ok(posrecip(z0))
from(ok(z0)) → ok(from(z0))
from(mark(z0)) → mark(from(z0))
rcons(mark(z0), z1) → mark(rcons(z0, z1))
rcons(ok(z0), ok(z1)) → ok(rcons(z0, z1))
rcons(z0, mark(z1)) → mark(rcons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
2ndsneg(z0, mark(z1)) → mark(2ndsneg(z0, z1))
2ndsneg(ok(z0), ok(z1)) → ok(2ndsneg(z0, z1))
2ndsneg(mark(z0), z1) → mark(2ndsneg(z0, z1))
square(mark(z0)) → mark(square(z0))
square(ok(z0)) → ok(square(z0))
2ndspos(ok(z0), ok(z1)) → ok(2ndspos(z0, z1))
2ndspos(mark(z0), z1) → mark(2ndspos(z0, z1))
2ndspos(z0, mark(z1)) → mark(2ndspos(z0, z1))
proper(rnil) → ok(rnil)
proper(nil) → ok(nil)
proper(0) → ok(0)
pi(ok(z0)) → ok(pi(z0))
pi(mark(z0)) → mark(pi(z0))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))

Tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

S tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

negrecip, times, top, posrecip, from, rcons, cons, 2ndsneg, square, 2ndspos, proper, pi, plus, s

Defined Pair Symbols:

NEGRECIP, TIMES, POSRECIP, FROM, RCONS, CONS, 2NDSNEG, SQUARE, 2NDSPOS, PI, PLUS, S, TOP

Compound Symbols:

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

(11) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

negrecip(mark(z0)) → mark(negrecip(z0))
negrecip(ok(z0)) → ok(negrecip(z0))
times(ok(z0), ok(z1)) → ok(times(z0, z1))
times(z0, mark(z1)) → mark(times(z0, z1))
times(mark(z0), z1) → mark(times(z0, z1))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
posrecip(mark(z0)) → mark(posrecip(z0))
posrecip(ok(z0)) → ok(posrecip(z0))
from(ok(z0)) → ok(from(z0))
from(mark(z0)) → mark(from(z0))
rcons(mark(z0), z1) → mark(rcons(z0, z1))
rcons(ok(z0), ok(z1)) → ok(rcons(z0, z1))
rcons(z0, mark(z1)) → mark(rcons(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
2ndsneg(z0, mark(z1)) → mark(2ndsneg(z0, z1))
2ndsneg(ok(z0), ok(z1)) → ok(2ndsneg(z0, z1))
2ndsneg(mark(z0), z1) → mark(2ndsneg(z0, z1))
square(mark(z0)) → mark(square(z0))
square(ok(z0)) → ok(square(z0))
2ndspos(ok(z0), ok(z1)) → ok(2ndspos(z0, z1))
2ndspos(mark(z0), z1) → mark(2ndspos(z0, z1))
2ndspos(z0, mark(z1)) → mark(2ndspos(z0, z1))
pi(ok(z0)) → ok(pi(z0))
pi(mark(z0)) → mark(pi(z0))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(rnil) → ok(rnil)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

S tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

proper

Defined Pair Symbols:

NEGRECIP, TIMES, POSRECIP, FROM, RCONS, CONS, 2NDSNEG, SQUARE, 2NDSPOS, PI, PLUS, S, TOP

Compound Symbols:

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

(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)) → c14(CONS(z0, z1))

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

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

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

POL(0) = [2]
POL(2NDSNEG(x₁, x₂)) = 0
POL(2NDSPOS(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = [2]x₂
POL(FROM(x₁)) = 0
POL(NEGRECIP(x₁)) = 0
POL(PI(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(POSRECIP(x₁)) = 0
POL(RCONS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(SQUARE(x₁)) = 0
POL(TIMES(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c4(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(rnil) = [2]

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(rnil) → ok(rnil)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

S tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

K tuples:

CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

NEGRECIP, TIMES, POSRECIP, FROM, RCONS, CONS, 2NDSNEG, SQUARE, 2NDSPOS, PI, PLUS, S, TOP

Compound Symbols:

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

(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.

CONS(mark(z0), z1) → c15(CONS(z0, z1))

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

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

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

POL(0) = 0
POL(2NDSNEG(x₁, x₂)) = 0
POL(2NDSPOS(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = x₁
POL(FROM(x₁)) = 0
POL(NEGRECIP(x₁)) = 0
POL(PI(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(POSRECIP(x₁)) = 0
POL(RCONS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(SQUARE(x₁)) = 0
POL(TIMES(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c4(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(rnil) = 0

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(rnil) → ok(rnil)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

S tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

K tuples:

CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

NEGRECIP, TIMES, POSRECIP, FROM, RCONS, CONS, 2NDSNEG, SQUARE, 2NDSPOS, PI, PLUS, S, TOP

Compound Symbols:

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

(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.

POSRECIP(mark(z0)) → c7(POSRECIP(z0))

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

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

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

POL(0) = 0
POL(2NDSNEG(x₁, x₂)) = 0
POL(2NDSPOS(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(NEGRECIP(x₁)) = 0
POL(PI(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(POSRECIP(x₁)) = x₁
POL(RCONS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(SQUARE(x₁)) = 0
POL(TIMES(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c4(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(rnil) = 0

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(rnil) → ok(rnil)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

S tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

K tuples:

CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

NEGRECIP, TIMES, POSRECIP, FROM, RCONS, CONS, 2NDSNEG, SQUARE, 2NDSPOS, PI, PLUS, S, TOP

Compound Symbols:

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

(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.

2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))

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

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

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

POL(0) = [1]
POL(2NDSNEG(x₁, x₂)) = 0
POL(2NDSPOS(x₁, x₂)) = x₁
POL(CONS(x₁, x₂)) = x₂
POL(FROM(x₁)) = 0
POL(NEGRECIP(x₁)) = 0
POL(PI(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(POSRECIP(x₁)) = 0
POL(RCONS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(SQUARE(x₁)) = 0
POL(TIMES(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c4(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(rnil) = [1]

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(rnil) → ok(rnil)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

S tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

K tuples:

CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

NEGRECIP, TIMES, POSRECIP, FROM, RCONS, CONS, 2NDSNEG, SQUARE, 2NDSPOS, PI, PLUS, S, TOP

Compound Symbols:

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

(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.

TOP(mark(z0)) → c6(TOP(proper(z0)))

We considered the (Usable) Rules:

proper(nil) → ok(nil)
proper(0) → ok(0)
proper(rnil) → ok(rnil)

And the Tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

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

POL(0) = 0
POL(2NDSNEG(x₁, x₂)) = 0
POL(2NDSPOS(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(NEGRECIP(x₁)) = 0
POL(PI(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(POSRECIP(x₁)) = 0
POL(RCONS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(SQUARE(x₁)) = 0
POL(TIMES(x₁, x₂)) = 0
POL(TOP(x₁)) = x₁
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c4(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)) = 0
POL(proper(x₁)) = x₁
POL(rnil) = [1]

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(rnil) → ok(rnil)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

S tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))

K tuples:

CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
TOP(mark(z0)) → c6(TOP(proper(z0)))

Defined Rule Symbols:

proper

Defined Pair Symbols:

NEGRECIP, TIMES, POSRECIP, FROM, RCONS, CONS, 2NDSNEG, SQUARE, 2NDSPOS, PI, PLUS, S, TOP

Compound Symbols:

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

(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.

2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(mark(z0)) → c28(PI(z0))

We considered the (Usable) Rules:

proper(nil) → ok(nil)
proper(0) → ok(0)
proper(rnil) → ok(rnil)

And the Tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

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

POL(0) = 0
POL(2NDSNEG(x₁, x₂)) = 0
POL(2NDSPOS(x₁, x₂)) = [2]x₂
POL(CONS(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(NEGRECIP(x₁)) = 0
POL(PI(x₁)) = [2]x₁
POL(PLUS(x₁, x₂)) = 0
POL(POSRECIP(x₁)) = 0
POL(RCONS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(SQUARE(x₁)) = 0
POL(TIMES(x₁, x₂)) = 0
POL(TOP(x₁)) = [2]x₁
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [2] + x₁
POL(nil) = [1]
POL(ok(x₁)) = x₁
POL(proper(x₁)) = [2]
POL(rnil) = [1]

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(rnil) → ok(rnil)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

S tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))

K tuples:

CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
TOP(mark(z0)) → c6(TOP(proper(z0)))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(mark(z0)) → c28(PI(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

NEGRECIP, TIMES, POSRECIP, FROM, RCONS, CONS, 2NDSNEG, SQUARE, 2NDSPOS, PI, PLUS, S, TOP

Compound Symbols:

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

(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.

POSRECIP(ok(z0)) → c8(POSRECIP(z0))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))

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

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

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

POL(0) = 0
POL(2NDSNEG(x₁, x₂)) = 0
POL(2NDSPOS(x₁, x₂)) = x₂
POL(CONS(x₁, x₂)) = [2]x₂
POL(FROM(x₁)) = 0
POL(NEGRECIP(x₁)) = 0
POL(PI(x₁)) = [2]x₁
POL(PLUS(x₁, x₂)) = x₁
POL(POSRECIP(x₁)) = x₁
POL(RCONS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(SQUARE(x₁)) = [2]x₁
POL(TIMES(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c4(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(rnil) = 0

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(rnil) → ok(rnil)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

S tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))

K tuples:

CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
TOP(mark(z0)) → c6(TOP(proper(z0)))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(mark(z0)) → c28(PI(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

NEGRECIP, TIMES, POSRECIP, FROM, RCONS, CONS, 2NDSNEG, SQUARE, 2NDSPOS, PI, PLUS, S, TOP

Compound Symbols:

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

(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.

S(ok(z0)) → c32(S(z0))

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

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

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

POL(0) = 0
POL(2NDSNEG(x₁, x₂)) = 0
POL(2NDSPOS(x₁, x₂)) = x₁
POL(CONS(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(NEGRECIP(x₁)) = 0
POL(PI(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(POSRECIP(x₁)) = 0
POL(RCONS(x₁, x₂)) = 0
POL(S(x₁)) = x₁
POL(SQUARE(x₁)) = 0
POL(TIMES(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c4(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(rnil) = 0

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(rnil) → ok(rnil)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

S tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
S(mark(z0)) → c33(S(z0))

K tuples:

CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
TOP(mark(z0)) → c6(TOP(proper(z0)))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(mark(z0)) → c28(PI(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

NEGRECIP, TIMES, POSRECIP, FROM, RCONS, CONS, 2NDSNEG, SQUARE, 2NDSPOS, PI, PLUS, S, TOP

Compound Symbols:

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

(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.

TIMES(mark(z0), z1) → c4(TIMES(z0, z1))

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

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

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

POL(0) = 0
POL(2NDSNEG(x₁, x₂)) = 0
POL(2NDSPOS(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = x₂
POL(FROM(x₁)) = 0
POL(NEGRECIP(x₁)) = 0
POL(PI(x₁)) = 0
POL(PLUS(x₁, x₂)) = x₁
POL(POSRECIP(x₁)) = x₁
POL(RCONS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(SQUARE(x₁)) = 0
POL(TIMES(x₁, x₂)) = x₁
POL(TOP(x₁)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c4(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(rnil) = 0

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(rnil) → ok(rnil)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

S tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
S(mark(z0)) → c33(S(z0))

K tuples:

CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
TOP(mark(z0)) → c6(TOP(proper(z0)))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(mark(z0)) → c28(PI(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

NEGRECIP, TIMES, POSRECIP, FROM, RCONS, CONS, 2NDSNEG, SQUARE, 2NDSPOS, PI, PLUS, S, TOP

Compound Symbols:

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

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

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

FROM(mark(z0)) → c10(FROM(z0))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
S(mark(z0)) → c33(S(z0))

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

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

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

POL(0) = 0
POL(2NDSNEG(x₁, x₂)) = 0
POL(2NDSPOS(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = 0
POL(FROM(x₁)) = x₁
POL(NEGRECIP(x₁)) = 0
POL(PI(x₁)) = 0
POL(PLUS(x₁, x₂)) = x₂
POL(POSRECIP(x₁)) = 0
POL(RCONS(x₁, x₂)) = 0
POL(S(x₁)) = x₁
POL(SQUARE(x₁)) = 0
POL(TIMES(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c4(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(rnil) = 0

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(rnil) → ok(rnil)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

S tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
FROM(ok(z0)) → c9(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))

K tuples:

CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
TOP(mark(z0)) → c6(TOP(proper(z0)))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(mark(z0)) → c28(PI(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
S(mark(z0)) → c33(S(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

NEGRECIP, TIMES, POSRECIP, FROM, RCONS, CONS, 2NDSNEG, SQUARE, 2NDSPOS, PI, PLUS, S, TOP

Compound Symbols:

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

(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.

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))

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

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

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

POL(0) = 0
POL(2NDSNEG(x₁, x₂)) = 0
POL(2NDSPOS(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = x₁
POL(FROM(x₁)) = 0
POL(NEGRECIP(x₁)) = x₁
POL(PI(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(POSRECIP(x₁)) = x₁
POL(RCONS(x₁, x₂)) = x₁
POL(S(x₁)) = 0
POL(SQUARE(x₁)) = 0
POL(TIMES(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c4(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(rnil) = 0

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(rnil) → ok(rnil)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

S tuples:

TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
FROM(ok(z0)) → c9(FROM(z0))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))

K tuples:

CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
TOP(mark(z0)) → c6(TOP(proper(z0)))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(mark(z0)) → c28(PI(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
S(mark(z0)) → c33(S(z0))
NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

NEGRECIP, TIMES, POSRECIP, FROM, RCONS, CONS, 2NDSNEG, SQUARE, 2NDSPOS, PI, PLUS, S, TOP

Compound Symbols:

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

(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.

TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))

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

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

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

POL(0) = [1]
POL(2NDSNEG(x₁, x₂)) = x₂
POL(2NDSPOS(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(NEGRECIP(x₁)) = 0
POL(PI(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(POSRECIP(x₁)) = 0
POL(RCONS(x₁, x₂)) = x₂
POL(S(x₁)) = 0
POL(SQUARE(x₁)) = 0
POL(TIMES(x₁, x₂)) = x₂
POL(TOP(x₁)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c4(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)) = [1] + x₁
POL(rnil) = [1]

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(rnil) → ok(rnil)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

S tuples:

TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
FROM(ok(z0)) → c9(FROM(z0))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))

K tuples:

CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
TOP(mark(z0)) → c6(TOP(proper(z0)))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(mark(z0)) → c28(PI(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
S(mark(z0)) → c33(S(z0))
NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

NEGRECIP, TIMES, POSRECIP, FROM, RCONS, CONS, 2NDSNEG, SQUARE, 2NDSPOS, PI, PLUS, S, TOP

Compound Symbols:

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

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

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

TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))

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

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

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

POL(0) = 0
POL(2NDSNEG(x₁, x₂)) = 0
POL(2NDSPOS(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = [2]x₂
POL(FROM(x₁)) = 0
POL(NEGRECIP(x₁)) = 0
POL(PI(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(POSRECIP(x₁)) = 0
POL(RCONS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(SQUARE(x₁)) = 0
POL(TIMES(x₁, x₂)) = x₂
POL(TOP(x₁)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c4(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(rnil) = 0

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(rnil) → ok(rnil)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

S tuples:

FROM(ok(z0)) → c9(FROM(z0))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))

K tuples:

CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
TOP(mark(z0)) → c6(TOP(proper(z0)))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(mark(z0)) → c28(PI(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
S(mark(z0)) → c33(S(z0))
NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

NEGRECIP, TIMES, POSRECIP, FROM, RCONS, CONS, 2NDSNEG, SQUARE, 2NDSPOS, PI, PLUS, S, TOP

Compound Symbols:

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

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

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

FROM(ok(z0)) → c9(FROM(z0))

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

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

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

POL(0) = 0
POL(2NDSNEG(x₁, x₂)) = 0
POL(2NDSPOS(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = 0
POL(FROM(x₁)) = x₁
POL(NEGRECIP(x₁)) = 0
POL(PI(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(POSRECIP(x₁)) = 0
POL(RCONS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(SQUARE(x₁)) = 0
POL(TIMES(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c4(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(rnil) = 0

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(rnil) → ok(rnil)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

S tuples:

2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))

K tuples:

CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
TOP(mark(z0)) → c6(TOP(proper(z0)))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(mark(z0)) → c28(PI(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
S(mark(z0)) → c33(S(z0))
NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
FROM(ok(z0)) → c9(FROM(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

NEGRECIP, TIMES, POSRECIP, FROM, RCONS, CONS, 2NDSNEG, SQUARE, 2NDSPOS, PI, PLUS, S, TOP

Compound Symbols:

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

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

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

2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))

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

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

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

POL(0) = 0
POL(2NDSNEG(x₁, x₂)) = x₁
POL(2NDSPOS(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(NEGRECIP(x₁)) = 0
POL(PI(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(POSRECIP(x₁)) = 0
POL(RCONS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(SQUARE(x₁)) = 0
POL(TIMES(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c4(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(rnil) = 0

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(rnil) → ok(rnil)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

S tuples:

2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))

K tuples:

CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
TOP(mark(z0)) → c6(TOP(proper(z0)))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(mark(z0)) → c28(PI(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
S(mark(z0)) → c33(S(z0))
NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
FROM(ok(z0)) → c9(FROM(z0))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

NEGRECIP, TIMES, POSRECIP, FROM, RCONS, CONS, 2NDSNEG, SQUARE, 2NDSPOS, PI, PLUS, S, TOP

Compound Symbols:

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

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

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

2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))

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

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

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

POL(0) = 0
POL(2NDSNEG(x₁, x₂)) = x₂
POL(2NDSPOS(x₁, x₂)) = 0
POL(CONS(x₁, x₂)) = 0
POL(FROM(x₁)) = 0
POL(NEGRECIP(x₁)) = 0
POL(PI(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(POSRECIP(x₁)) = 0
POL(RCONS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(SQUARE(x₁)) = 0
POL(TIMES(x₁, x₂)) = 0
POL(TOP(x₁)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c4(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(rnil) = 0

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(rnil) → ok(rnil)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
FROM(ok(z0)) → c9(FROM(z0))
FROM(mark(z0)) → c10(FROM(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PI(mark(z0)) → c28(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
S(mark(z0)) → c33(S(z0))
TOP(mark(z0)) → c6(TOP(proper(z0)))

S tuples:none
K tuples:

CONS(ok(z0), ok(z1)) → c14(CONS(z0, z1))
CONS(mark(z0), z1) → c15(CONS(z0, z1))
POSRECIP(mark(z0)) → c7(POSRECIP(z0))
2NDSPOS(mark(z0), z1) → c22(2NDSPOS(z0, z1))
TOP(mark(z0)) → c6(TOP(proper(z0)))
2NDSPOS(z0, mark(z1)) → c23(2NDSPOS(z0, z1))
PI(mark(z0)) → c28(PI(z0))
POSRECIP(ok(z0)) → c8(POSRECIP(z0))
SQUARE(mark(z0)) → c19(SQUARE(z0))
SQUARE(ok(z0)) → c20(SQUARE(z0))
2NDSPOS(ok(z0), ok(z1)) → c21(2NDSPOS(z0, z1))
PI(ok(z0)) → c27(PI(z0))
PLUS(ok(z0), ok(z1)) → c29(PLUS(z0, z1))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
S(ok(z0)) → c32(S(z0))
TIMES(mark(z0), z1) → c4(TIMES(z0, z1))
FROM(mark(z0)) → c10(FROM(z0))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
S(mark(z0)) → c33(S(z0))
NEGRECIP(mark(z0)) → c(NEGRECIP(z0))
NEGRECIP(ok(z0)) → c1(NEGRECIP(z0))
RCONS(mark(z0), z1) → c11(RCONS(z0, z1))
RCONS(ok(z0), ok(z1)) → c12(RCONS(z0, z1))
TIMES(z0, mark(z1)) → c3(TIMES(z0, z1))
RCONS(z0, mark(z1)) → c13(RCONS(z0, z1))
2NDSNEG(z0, mark(z1)) → c16(2NDSNEG(z0, z1))
TIMES(ok(z0), ok(z1)) → c2(TIMES(z0, z1))
FROM(ok(z0)) → c9(FROM(z0))
2NDSNEG(mark(z0), z1) → c18(2NDSNEG(z0, z1))
2NDSNEG(ok(z0), ok(z1)) → c17(2NDSNEG(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

NEGRECIP, TIMES, POSRECIP, FROM, RCONS, CONS, 2NDSNEG, SQUARE, 2NDSPOS, PI, PLUS, S, TOP

Compound Symbols:

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

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

The set S is empty

(0) Obligation:

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

(2) Obligation:

(3) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

(4) BOUNDS(1, n^1)

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

(6) Obligation:

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

(8) Obligation:

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

(10) Obligation:

(11) CdtUsableRulesProof (EQUIVALENT transformation)

(12) Obligation:

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

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

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

(20) Obligation:

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

(22) Obligation:

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

(24) Obligation:

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

(26) Obligation:

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

(28) Obligation:

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

(30) Obligation:

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

(32) Obligation:

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

(34) Obligation:

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

(36) Obligation:

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

(38) Obligation:

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

(40) Obligation:

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

(42) Obligation:

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

(44) Obligation:

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

(46) BOUNDS(1, 1)