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 (⇔, 138 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)), 122 ms)
        ↳14 CdtProblem
        ↳15 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 10 ms)
        ↳16 CdtProblem
        ↳17 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 47 ms)
        ↳18 CdtProblem
        ↳19 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)
        ↳20 CdtProblem
        ↳21 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 4 ms)
        ↳22 CdtProblem
        ↳23 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)
        ↳24 CdtProblem
        ↳25 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
        ↳26 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(p(0)) → mark(0)
active(p(s(X))) → mark(X)
active(leq(0, Y)) → mark(true)
active(leq(s(X), 0)) → mark(false)
active(leq(s(X), s(Y))) → mark(leq(X, Y))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(diff(X, Y)) → mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) → p(active(X))
active(s(X)) → s(active(X))
active(leq(X1, X2)) → leq(active(X1), X2)
active(leq(X1, X2)) → leq(X1, active(X2))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(diff(X1, X2)) → diff(active(X1), X2)
active(diff(X1, X2)) → diff(X1, active(X2))
p(mark(X)) → mark(p(X))
s(mark(X)) → mark(s(X))
leq(mark(X1), X2) → mark(leq(X1, X2))
leq(X1, mark(X2)) → mark(leq(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
diff(mark(X1), X2) → mark(diff(X1, X2))
diff(X1, mark(X2)) → mark(diff(X1, X2))
proper(p(X)) → p(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(leq(X1, X2)) → leq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) → diff(proper(X1), proper(X2))
p(ok(X)) → ok(p(X))
s(ok(X)) → ok(s(X))
leq(ok(X1), ok(X2)) → ok(leq(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
diff(ok(X1), ok(X2)) → ok(diff(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(p(0)) → mark(0)
active(p(s(X))) → mark(X)
active(leq(0, Y)) → mark(true)
active(leq(s(X), 0)) → mark(false)
active(leq(s(X), s(Y))) → mark(leq(X, Y))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(diff(X, Y)) → mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) → p(active(X))
active(s(X)) → s(active(X))
active(leq(X1, X2)) → leq(active(X1), X2)
active(leq(X1, X2)) → leq(X1, active(X2))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(diff(X1, X2)) → diff(active(X1), X2)
active(diff(X1, X2)) → diff(X1, active(X2))
proper(p(X)) → p(proper(X))
proper(s(X)) → s(proper(X))
proper(leq(X1, X2)) → leq(proper(X1), proper(X2))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) → diff(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))
leq(ok(X1), ok(X2)) → ok(leq(X1, X2))
proper(true) → ok(true)
top(ok(X)) → top(active(X))
p(ok(X)) → ok(p(X))
leq(X1, mark(X2)) → mark(leq(X1, X2))
diff(mark(X1), X2) → mark(diff(X1, X2))
s(ok(X)) → ok(s(X))
leq(mark(X1), X2) → mark(leq(X1, X2))
s(mark(X)) → mark(s(X))
proper(false) → ok(false)
proper(0) → ok(0)
diff(ok(X1), ok(X2)) → ok(diff(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
diff(X1, mark(X2)) → mark(diff(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]
transitions:
mark0(0) → 0
ok0(0) → 0
true0() → 0
active0(0) → 0
false0() → 0
00() → 0
p0(0) → 1
leq0(0, 0) → 2
proper0(0) → 3
top0(0) → 4
diff0(0, 0) → 5
s0(0) → 6
if0(0, 0, 0) → 7
p1(0) → 8
mark1(8) → 1
leq1(0, 0) → 9
ok1(9) → 2
true1() → 10
ok1(10) → 3
active1(0) → 11
top1(11) → 4
p1(0) → 12
ok1(12) → 1
leq1(0, 0) → 13
mark1(13) → 2
diff1(0, 0) → 14
mark1(14) → 5
s1(0) → 15
ok1(15) → 6
s1(0) → 16
mark1(16) → 6
false1() → 17
ok1(17) → 3
01() → 18
ok1(18) → 3
diff1(0, 0) → 19
ok1(19) → 5
if1(0, 0, 0) → 20
mark1(20) → 7
if1(0, 0, 0) → 21
ok1(21) → 7
proper1(0) → 22
top1(22) → 4
mark1(8) → 8
mark1(8) → 12
ok1(9) → 9
ok1(9) → 13
ok1(10) → 22
ok1(12) → 8
ok1(12) → 12
mark1(13) → 9
mark1(13) → 13
mark1(14) → 14
mark1(14) → 19
ok1(15) → 15
ok1(15) → 16
mark1(16) → 15
mark1(16) → 16
ok1(17) → 22
ok1(18) → 22
ok1(19) → 14
ok1(19) → 19
mark1(20) → 20
mark1(20) → 21
ok1(21) → 20
ok1(21) → 21
active2(10) → 23
top2(23) → 4
active2(17) → 23
active2(18) → 23

(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))
leq(ok(z0), ok(z1)) → ok(leq(z0, z1))
leq(z0, mark(z1)) → mark(leq(z0, z1))
leq(mark(z0), z1) → mark(leq(z0, z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(0) → ok(0)
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
diff(mark(z0), z1) → mark(diff(z0, z1))
diff(ok(z0), ok(z1)) → ok(diff(z0, z1))
diff(z0, mark(z1)) → mark(diff(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(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))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
PROPER(true) → c5
PROPER(false) → c6
PROPER(0) → c7
TOP(ok(z0)) → c8(TOP(active(z0)))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c16(IF(z0, z1, z2))
S tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
PROPER(true) → c5
PROPER(false) → c6
PROPER(0) → c7
TOP(ok(z0)) → c8(TOP(active(z0)))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c16(IF(z0, z1, z2))
K tuples:none
Defined Rule Symbols:

p, leq, proper, top, diff, s, if

Defined Pair Symbols:

P, LEQ, PROPER, TOP, DIFF, S, IF

Compound Symbols:

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

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

Removed 4 trailing nodes:

TOP(ok(z0)) → c8(TOP(active(z0)))
PROPER(0) → c7
PROPER(true) → c5
PROPER(false) → c6

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(mark(z0)) → mark(p(z0))
p(ok(z0)) → ok(p(z0))
leq(ok(z0), ok(z1)) → ok(leq(z0, z1))
leq(z0, mark(z1)) → mark(leq(z0, z1))
leq(mark(z0), z1) → mark(leq(z0, z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(0) → ok(0)
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
diff(mark(z0), z1) → mark(diff(z0, z1))
diff(ok(z0), ok(z1)) → ok(diff(z0, z1))
diff(z0, mark(z1)) → mark(diff(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(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))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c16(IF(z0, z1, z2))
S tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
TOP(mark(z0)) → c9(TOP(proper(z0)), PROPER(z0))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c16(IF(z0, z1, z2))
K tuples:none
Defined Rule Symbols:

p, leq, proper, top, diff, s, if

Defined Pair Symbols:

P, LEQ, TOP, DIFF, S, IF

Compound Symbols:

c, c1, c2, c3, c4, c9, c10, c11, c12, c13, c14, c15, c16

(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))
leq(ok(z0), ok(z1)) → ok(leq(z0, z1))
leq(z0, mark(z1)) → mark(leq(z0, z1))
leq(mark(z0), z1) → mark(leq(z0, z1))
proper(true) → ok(true)
proper(false) → ok(false)
proper(0) → ok(0)
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
diff(mark(z0), z1) → mark(diff(z0, z1))
diff(ok(z0), ok(z1)) → ok(diff(z0, z1))
diff(z0, mark(z1)) → mark(diff(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(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))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c16(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))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c16(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))
K tuples:none
Defined Rule Symbols:

p, leq, proper, top, diff, s, if

Defined Pair Symbols:

P, LEQ, DIFF, S, IF, TOP

Compound Symbols:

c, c1, c2, c3, c4, c10, c11, c12, c13, c14, c15, c16, 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))
leq(ok(z0), ok(z1)) → ok(leq(z0, z1))
leq(z0, mark(z1)) → mark(leq(z0, z1))
leq(mark(z0), z1) → mark(leq(z0, z1))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
diff(mark(z0), z1) → mark(diff(z0, z1))
diff(ok(z0), ok(z1)) → ok(diff(z0, z1))
diff(z0, mark(z1)) → mark(diff(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(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))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c16(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))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c16(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))
K tuples:none
Defined Rule Symbols:

proper

Defined Pair Symbols:

P, LEQ, DIFF, S, IF, TOP

Compound Symbols:

c, c1, c2, c3, c4, c10, c11, c12, c13, c14, c15, c16, c9

(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.

IF(ok(z0), ok(z1), ok(z2)) → c16(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))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c16(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(DIFF(x1, x2)) = 0   
POL(IF(x1, x2, x3)) = x2 + x3   
POL(LEQ(x1, x2)) = 0   
POL(P(x1)) = 0   
POL(S(x1)) = 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(c2(x1)) = x1   
POL(c3(x1)) = x1   
POL(c4(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   

(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))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c16(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))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c16(IF(z0, z1, z2))
Defined Rule Symbols:

proper

Defined Pair Symbols:

P, LEQ, DIFF, S, IF, TOP

Compound Symbols:

c, c1, c2, c3, c4, c10, c11, c12, c13, c14, c15, c16, 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.

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))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c16(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(DIFF(x1, x2)) = 0   
POL(IF(x1, x2, x3)) = 0   
POL(LEQ(x1, x2)) = 0   
POL(P(x1)) = 0   
POL(S(x1)) = 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(c2(x1)) = x1   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   
POL(c9(x1)) = x1   
POL(false) = 0   
POL(mark(x1)) = [1]   
POL(ok(x1)) = 0   
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))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c16(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))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c16(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))
Defined Rule Symbols:

proper

Defined Pair Symbols:

P, LEQ, DIFF, S, IF, TOP

Compound Symbols:

c, c1, c2, c3, c4, c10, c11, c12, c13, c14, c15, c16, 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.

LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(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))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c16(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(DIFF(x1, x2)) = x2   
POL(IF(x1, x2, x3)) = x2 + [2]x3   
POL(LEQ(x1, x2)) = x2   
POL(P(x1)) = 0   
POL(S(x1)) = x1   
POL(TOP(x1)) = [3]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(c2(x1)) = x1   
POL(c3(x1)) = x1   
POL(c4(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   

(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))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c16(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))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c16(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
Defined Rule Symbols:

proper

Defined Pair Symbols:

P, LEQ, DIFF, S, IF, TOP

Compound Symbols:

c, c1, c2, c3, c4, c10, c11, c12, c13, c14, c15, c16, 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.

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
IF(mark(z0), z1, z2) → c15(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))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c16(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(DIFF(x1, x2)) = 0   
POL(IF(x1, x2, x3)) = x1   
POL(LEQ(x1, x2)) = x2   
POL(P(x1)) = x1   
POL(S(x1)) = 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(c2(x1)) = x1   
POL(c3(x1)) = x1   
POL(c4(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   

(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))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c16(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))
S tuples:

LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c16(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
Defined Rule Symbols:

proper

Defined Pair Symbols:

P, LEQ, DIFF, S, IF, TOP

Compound Symbols:

c, c1, c2, c3, c4, c10, c11, c12, c13, c14, c15, c16, 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.

DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c16(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(DIFF(x1, x2)) = x1 + x2   
POL(IF(x1, x2, x3)) = 0   
POL(LEQ(x1, x2)) = 0   
POL(P(x1)) = 0   
POL(S(x1)) = 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(c2(x1)) = x1   
POL(c3(x1)) = x1   
POL(c4(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   

(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))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c16(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))
S tuples:

LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c16(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
Defined Rule Symbols:

proper

Defined Pair Symbols:

P, LEQ, DIFF, S, IF, TOP

Compound Symbols:

c, c1, c2, c3, c4, c10, c11, c12, c13, c14, c15, c16, 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.

LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c16(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(DIFF(x1, x2)) = 0   
POL(IF(x1, x2, x3)) = 0   
POL(LEQ(x1, x2)) = x1   
POL(P(x1)) = 0   
POL(S(x1)) = 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(c2(x1)) = x1   
POL(c3(x1)) = x1   
POL(c4(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   

(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))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c16(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))
S tuples:none
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c16(IF(z0, z1, z2))
TOP(mark(z0)) → c9(TOP(proper(z0)))
LEQ(ok(z0), ok(z1)) → c2(LEQ(z0, z1))
LEQ(z0, mark(z1)) → c3(LEQ(z0, z1))
DIFF(ok(z0), ok(z1)) → c11(DIFF(z0, z1))
DIFF(z0, mark(z1)) → c12(DIFF(z0, z1))
S(ok(z0)) → c13(S(z0))
S(mark(z0)) → c14(S(z0))
P(mark(z0)) → c(P(z0))
P(ok(z0)) → c1(P(z0))
IF(mark(z0), z1, z2) → c15(IF(z0, z1, z2))
DIFF(mark(z0), z1) → c10(DIFF(z0, z1))
LEQ(mark(z0), z1) → c4(LEQ(z0, z1))
Defined Rule Symbols:

proper

Defined Pair Symbols:

P, LEQ, DIFF, S, IF, TOP

Compound Symbols:

c, c1, c2, c3, c4, c10, c11, c12, c13, c14, c15, c16, c9

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

The set S is empty

(26) BOUNDS(1, 1)