Runtime Complexity (innermost) proof of /tmp/tmp16yhE_/ExIntrod_Zan97

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), 22 ms)

↳2 CpxTRS

    ↳3 CpxTrsMatchBoundsTAProof (⇔, 183 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^2)), 159 ms)

        ↳14 CdtProblem

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

        ↳16 CdtProblem

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

        ↳18 CdtProblem

        ↳19 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 16 ms)

        ↳20 CdtProblem

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

        ↳22 CdtProblem

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

        ↳24 CdtProblem

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

        ↳26 CdtProblem

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

        ↳28 BOUNDS(1, 1)

(0) Obligation:

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

The TRS R consists of the following rules:

active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
fact(mark(X)) → mark(fact(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
zero(mark(X)) → mark(zero(X))
s(mark(X)) → mark(s(X))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(X1, mark(X2)) → mark(prod(X1, X2))
p(mark(X)) → mark(p(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
fact(ok(X)) → ok(fact(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
zero(ok(X)) → ok(zero(X))
s(ok(X)) → ok(s(X))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
p(ok(X)) → ok(p(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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

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

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X)))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(prod(0, X)) → mark(0)
active(prod(s(X), Y)) → mark(add(Y, prod(X, Y)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(zero(0)) → mark(true)
active(zero(s(X))) → mark(false)
active(p(s(X))) → mark(X)
active(fact(X)) → fact(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(zero(X)) → zero(active(X))
active(s(X)) → s(active(X))
active(prod(X1, X2)) → prod(active(X1), X2)
active(prod(X1, X2)) → prod(X1, active(X2))
active(p(X)) → p(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
proper(fact(X)) → fact(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(zero(X)) → zero(proper(X))
proper(s(X)) → s(proper(X))
proper(prod(X1, X2)) → prod(proper(X1), proper(X2))
proper(p(X)) → p(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))

(2) Obligation:

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

The TRS R consists of the following rules:

p(mark(X)) → mark(p(X))
proper(true) → ok(true)
add(ok(X1), ok(X2)) → ok(add(X1, X2))
top(ok(X)) → top(active(X))
p(ok(X)) → ok(p(X))
zero(ok(X)) → ok(zero(X))
zero(mark(X)) → mark(zero(X))
prod(X1, mark(X2)) → mark(prod(X1, X2))
prod(mark(X1), X2) → mark(prod(X1, X2))
prod(ok(X1), ok(X2)) → ok(prod(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
s(ok(X)) → ok(s(X))
fact(ok(X)) → ok(fact(X))
s(mark(X)) → mark(s(X))
proper(false) → ok(false)
proper(0) → ok(0)
fact(mark(X)) → mark(fact(X))
add(mark(X1), X2) → mark(add(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
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]
transitions:
mark0(0) → 0
true0() → 0
ok0(0) → 0
active0(0) → 0
false0() → 0
00() → 0
p0(0) → 1
proper0(0) → 2
add0(0, 0) → 3
top0(0) → 4
zero0(0) → 5
prod0(0, 0) → 6
s0(0) → 7
fact0(0) → 8
if0(0, 0, 0) → 9
p1(0) → 10
mark1(10) → 1
true1() → 11
ok1(11) → 2
add1(0, 0) → 12
ok1(12) → 3
active1(0) → 13
top1(13) → 4
p1(0) → 14
ok1(14) → 1
zero1(0) → 15
ok1(15) → 5
zero1(0) → 16
mark1(16) → 5
prod1(0, 0) → 17
mark1(17) → 6
prod1(0, 0) → 18
ok1(18) → 6
add1(0, 0) → 19
mark1(19) → 3
s1(0) → 20
ok1(20) → 7
fact1(0) → 21
ok1(21) → 8
s1(0) → 22
mark1(22) → 7
false1() → 23
ok1(23) → 2
01() → 24
ok1(24) → 2
fact1(0) → 25
mark1(25) → 8
if1(0, 0, 0) → 26
mark1(26) → 9
if1(0, 0, 0) → 27
ok1(27) → 9
proper1(0) → 28
top1(28) → 4
mark1(10) → 10
mark1(10) → 14
ok1(11) → 28
ok1(12) → 12
ok1(12) → 19
ok1(14) → 10
ok1(14) → 14
ok1(15) → 15
ok1(15) → 16
mark1(16) → 15
mark1(16) → 16
mark1(17) → 17
mark1(17) → 18
ok1(18) → 17
ok1(18) → 18
mark1(19) → 12
mark1(19) → 19
ok1(20) → 20
ok1(20) → 22
ok1(21) → 21
ok1(21) → 25
mark1(22) → 20
mark1(22) → 22
ok1(23) → 28
ok1(24) → 28
mark1(25) → 21
mark1(25) → 25
mark1(26) → 26
mark1(26) → 27
ok1(27) → 26
ok1(27) → 27
active2(11) → 29
top2(29) → 4
active2(23) → 29
active2(24) → 29

(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:

p(mark(z0)) → mark(p(z0))
p(ok(z0)) → ok(p(z0))
proper(true) → ok(true)
proper(false) → ok(false)
proper(0) → ok(0)
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
zero(ok(z0)) → ok(zero(z0))
zero(mark(z0)) → mark(zero(z0))
prod(z0, mark(z1)) → mark(prod(z0, z1))
prod(mark(z0), z1) → mark(prod(z0, z1))
prod(ok(z0), ok(z1)) → ok(prod(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
fact(ok(z0)) → ok(fact(z0))
fact(mark(z0)) → mark(fact(z0))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))

Tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
PROPER(true) → c2
PROPER(false) → c3
PROPER(0) → c4
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
TOP(ok(z0)) → c8(TOP(active(z0)))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))

S tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
PROPER(true) → c2
PROPER(false) → c3
PROPER(0) → c4
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
TOP(ok(z0)) → c8(TOP(active(z0)))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))

K tuples:none
Defined Rule Symbols:

p, proper, add, top, zero, prod, s, fact, if

Defined Pair Symbols:

P, PROPER, ADD, TOP, ZERO, PROD, S, FACT, IF

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20

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

Removed 4 trailing nodes:

PROPER(false) → c3
PROPER(true) → c2
TOP(ok(z0)) → c8(TOP(active(z0)))
PROPER(0) → c4

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(mark(z0)) → mark(p(z0))
p(ok(z0)) → ok(p(z0))
proper(true) → ok(true)
proper(false) → ok(false)
proper(0) → ok(0)
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
zero(ok(z0)) → ok(zero(z0))
zero(mark(z0)) → mark(zero(z0))
prod(z0, mark(z1)) → mark(prod(z0, z1))
prod(mark(z0), z1) → mark(prod(z0, z1))
prod(ok(z0), ok(z1)) → ok(prod(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
fact(ok(z0)) → ok(fact(z0))
fact(mark(z0)) → mark(fact(z0))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))

Tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))

S tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))

K tuples:none
Defined Rule Symbols:

p, proper, add, top, zero, prod, s, fact, if

Defined Pair Symbols:

P, ADD, TOP, ZERO, PROD, S, FACT, IF

Compound Symbols:

c, c1, c5, c6, c7, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20

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

Removed 1 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(mark(z0)) → mark(p(z0))
p(ok(z0)) → ok(p(z0))
proper(true) → ok(true)
proper(false) → ok(false)
proper(0) → ok(0)
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
zero(ok(z0)) → ok(zero(z0))
zero(mark(z0)) → mark(zero(z0))
prod(z0, mark(z1)) → mark(prod(z0, z1))
prod(mark(z0), z1) → mark(prod(z0, z1))
prod(ok(z0), ok(z1)) → ok(prod(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
fact(ok(z0)) → ok(fact(z0))
fact(mark(z0)) → mark(fact(z0))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))

Tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))

S tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

p, proper, add, top, zero, prod, s, fact, if

Defined Pair Symbols:

P, ADD, ZERO, PROD, S, FACT, IF, TOP

Compound Symbols:

c, c1, c5, c6, c7, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c9

(11) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

p(mark(z0)) → mark(p(z0))
p(ok(z0)) → ok(p(z0))
add(ok(z0), ok(z1)) → ok(add(z0, z1))
add(z0, mark(z1)) → mark(add(z0, z1))
add(mark(z0), z1) → mark(add(z0, z1))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
zero(ok(z0)) → ok(zero(z0))
zero(mark(z0)) → mark(zero(z0))
prod(z0, mark(z1)) → mark(prod(z0, z1))
prod(mark(z0), z1) → mark(prod(z0, z1))
prod(ok(z0), ok(z1)) → ok(prod(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
fact(ok(z0)) → ok(fact(z0))
fact(mark(z0)) → mark(fact(z0))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(true) → ok(true)
proper(false) → ok(false)
proper(0) → ok(0)

Tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))

S tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

proper

Defined Pair Symbols:

P, ADD, ZERO, PROD, S, FACT, IF, TOP

Compound Symbols:

c, c1, c5, c6, c7, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c9

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

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

TOP(mark(z0)) → c9(TOP(proper(z0)))

We considered the (Usable) Rules:

proper(true) → ok(true)
proper(false) → ok(false)
proper(0) → ok(0)

And the Tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))

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

POL(0) = [1]
POL(ADD(x₁, x₂)) = 0
POL(FACT(x₁)) = 0
POL(IF(x₁, x₂, x₃)) = 0
POL(P(x₁)) = 0
POL(PROD(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = [2]x₁²
POL(ZERO(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(c20(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(false) = [1]
POL(mark(x₁)) = [2]
POL(ok(x₁)) = 0
POL(proper(x₁)) = 0
POL(true) = 0

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(true) → ok(true)
proper(false) → ok(false)
proper(0) → ok(0)

Tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))

S tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c9(TOP(proper(z0)))

Defined Rule Symbols:

proper

Defined Pair Symbols:

P, ADD, ZERO, PROD, S, FACT, IF, TOP

Compound Symbols:

c, c1, c5, c6, c7, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c9

(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.

IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))

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

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))

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

POL(0) = 0
POL(ADD(x₁, x₂)) = 0
POL(FACT(x₁)) = 0
POL(IF(x₁, x₂, x₃)) = x₂ + x₃
POL(P(x₁)) = 0
POL(PROD(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(ZERO(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(c20(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(false) = 0
POL(mark(x₁)) = 0
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = 0
POL(true) = 0

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(true) → ok(true)
proper(false) → ok(false)
proper(0) → ok(0)

Tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))

S tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c9(TOP(proper(z0)))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

P, ADD, ZERO, PROD, S, FACT, IF, TOP

Compound Symbols:

c, c1, c5, c6, c7, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c9

(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.

ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))

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

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))

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

POL(0) = 0
POL(ADD(x₁, x₂)) = x₁ + x₂
POL(FACT(x₁)) = 0
POL(IF(x₁, x₂, x₃)) = 0
POL(P(x₁)) = 0
POL(PROD(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(ZERO(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(c20(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(false) = 0
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(true) = 0

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(true) → ok(true)
proper(false) → ok(false)
proper(0) → ok(0)

Tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))

S tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c9(TOP(proper(z0)))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

P, ADD, ZERO, PROD, S, FACT, IF, TOP

Compound Symbols:

c, c1, c5, c6, c7, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c9

(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.

PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))

We considered the (Usable) Rules:

proper(true) → ok(true)
proper(false) → ok(false)
proper(0) → ok(0)

And the Tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))

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

POL(0) = 0
POL(ADD(x₁, x₂)) = 0
POL(FACT(x₁)) = [2]x₁
POL(IF(x₁, x₂, x₃)) = x₁ + [2]x₂
POL(P(x₁)) = 0
POL(PROD(x₁, x₂)) = x₂
POL(S(x₁)) = 0
POL(TOP(x₁)) = [3]x₁
POL(ZERO(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(c20(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(false) = 0
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = [1]
POL(true) = 0

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(true) → ok(true)
proper(false) → ok(false)
proper(0) → ok(0)

Tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))

S tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))

K tuples:

TOP(mark(z0)) → c9(TOP(proper(z0)))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

P, ADD, ZERO, PROD, S, FACT, IF, TOP

Compound Symbols:

c, c1, c5, c6, c7, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c9

(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.

ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))

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

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))

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

POL(0) = 0
POL(ADD(x₁, x₂)) = x₂
POL(FACT(x₁)) = 0
POL(IF(x₁, x₂, x₃)) = 0
POL(P(x₁)) = 0
POL(PROD(x₁, x₂)) = x₁
POL(S(x₁)) = x₁
POL(TOP(x₁)) = 0
POL(ZERO(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(c20(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(false) = 0
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = 0
POL(true) = 0

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(true) → ok(true)
proper(false) → ok(false)
proper(0) → ok(0)

Tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))

S tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))

K tuples:

TOP(mark(z0)) → c9(TOP(proper(z0)))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

P, ADD, ZERO, PROD, S, FACT, IF, TOP

Compound Symbols:

c, c1, c5, c6, c7, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c9

(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.

P(mark(z0)) → c(P(z0))

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

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))

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

POL(0) = 0
POL(ADD(x₁, x₂)) = 0
POL(FACT(x₁)) = 0
POL(IF(x₁, x₂, x₃)) = 0
POL(P(x₁)) = [2]x₁
POL(PROD(x₁, x₂)) = [2]x₂
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(ZERO(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(c20(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(false) = 0
POL(mark(x₁)) = [2] + x₁
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(true) = 0

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(true) → ok(true)
proper(false) → ok(false)
proper(0) → ok(0)

Tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))

S tuples:

P(ok(z0)) → c1(P(z0))

K tuples:

TOP(mark(z0)) → c9(TOP(proper(z0)))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
P(mark(z0)) → c(P(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

P, ADD, ZERO, PROD, S, FACT, IF, TOP

Compound Symbols:

c, c1, c5, c6, c7, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c9

(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.

P(ok(z0)) → c1(P(z0))

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

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))

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

POL(0) = 0
POL(ADD(x₁, x₂)) = 0
POL(FACT(x₁)) = 0
POL(IF(x₁, x₂, x₃)) = 0
POL(P(x₁)) = x₁
POL(PROD(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(ZERO(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(c20(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(false) = 0
POL(mark(x₁)) = x₁
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = 0
POL(true) = 0

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(true) → ok(true)
proper(false) → ok(false)
proper(0) → ok(0)

Tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))

S tuples:none
K tuples:

TOP(mark(z0)) → c9(TOP(proper(z0)))
IF(ok(z0), ok(z1), ok(z2)) → c20(IF(z0, z1, z2))
ADD(z0, mark(z1)) → c6(ADD(z0, z1))
ADD(mark(z0), z1) → c7(ADD(z0, z1))
PROD(z0, mark(z1)) → c12(PROD(z0, z1))
PROD(ok(z0), ok(z1)) → c14(PROD(z0, z1))
FACT(ok(z0)) → c17(FACT(z0))
FACT(mark(z0)) → c18(FACT(z0))
IF(mark(z0), z1, z2) → c19(IF(z0, z1, z2))
ADD(ok(z0), ok(z1)) → c5(ADD(z0, z1))
ZERO(ok(z0)) → c10(ZERO(z0))
ZERO(mark(z0)) → c11(ZERO(z0))
PROD(mark(z0), z1) → c13(PROD(z0, z1))
S(ok(z0)) → c15(S(z0))
S(mark(z0)) → c16(S(z0))
P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

P, ADD, ZERO, PROD, S, FACT, IF, TOP

Compound Symbols:

c, c1, c5, c6, c7, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c9

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

(28) BOUNDS(1, 1)