Runtime Complexity (innermost) proof of /tmp/tmpWWjF4R/PALINDROME_nosorts

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

0 CpxTRS

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

↳2 CdtProblem

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

↳4 CdtProblem

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

↳6 CdtProblem

↳7 CdtUsableRulesProof (⇔, 28 ms)

↳8 CdtProblem

↳9 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 783 ms)

↳10 CdtProblem

↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳12 CdtProblem

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

↳14 CdtProblem

↳15 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 4 ms)

↳16 CdtProblem

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

↳18 CdtProblem

↳19 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳20 CdtProblem

↳21 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 2 ms)

↳22 CdtProblem

↳23 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳24 CdtProblem

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

↳26 CdtProblem

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

↳28 CdtProblem

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

↳30 CdtProblem

↳31 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳32 CdtProblem

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

↳34 CdtProblem

↳35 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳36 CdtProblem

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

↳38 CdtProblem

↳39 CdtUnreachableProof (⇔, 0 ms)

↳40 CdtProblem

↳41 CdtUsableRulesProof (⇔, 0 ms)

↳42 CdtProblem

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

↳44 CdtProblem

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

↳46 BOUNDS(O(1), O(1))

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isNePal(__(I, __(P, I)))) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
active(__(z0, z1)) → __(active(z0), z1)
active(__(z0, z1)) → __(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
active(isNePal(z0)) → isNePal(active(z0))
__(mark(z0), z1) → mark(__(z0, z1))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNePal(mark(z0)) → mark(isNePal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
proper(__(z0, z1)) → __(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(isNePal(z0)) → isNePal(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(__(__(z0, z1), z2)) → c(__'(z0, __(z1, z2)), __'(z1, z2))
ACTIVE(__(z0, nil)) → c1
ACTIVE(__(nil, z0)) → c2
ACTIVE(and(tt, z0)) → c3
ACTIVE(isNePal(__(z0, __(z1, z0)))) → c4
ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c7(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(isNePal(z0)) → c8(ISNEPAL(active(z0)), ACTIVE(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(__(z0, z1)) → c16(__'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(nil) → c17
PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(tt) → c19
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(__(__(z0, z1), z2)) → c(__'(z0, __(z1, z2)), __'(z1, z2))
ACTIVE(__(z0, nil)) → c1
ACTIVE(__(nil, z0)) → c2
ACTIVE(and(tt, z0)) → c3
ACTIVE(isNePal(__(z0, __(z1, z0)))) → c4
ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c7(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(isNePal(z0)) → c8(ISNEPAL(active(z0)), ACTIVE(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(__(z0, z1)) → c16(__'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(nil) → c17
PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(tt) → c19
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, __, and, isNePal, proper, top

Defined Pair Symbols:

ACTIVE, __', AND, ISNEPAL, PROPER, TOP

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

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

Removed 6 trailing nodes:

PROPER(nil) → c17
ACTIVE(__(z0, nil)) → c1
ACTIVE(__(nil, z0)) → c2
ACTIVE(and(tt, z0)) → c3
ACTIVE(isNePal(__(z0, __(z1, z0)))) → c4
PROPER(tt) → c19

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
active(__(z0, z1)) → __(active(z0), z1)
active(__(z0, z1)) → __(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
active(isNePal(z0)) → isNePal(active(z0))
__(mark(z0), z1) → mark(__(z0, z1))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNePal(mark(z0)) → mark(isNePal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
proper(__(z0, z1)) → __(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(isNePal(z0)) → isNePal(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(__(__(z0, z1), z2)) → c(__'(z0, __(z1, z2)), __'(z1, z2))
ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c7(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(isNePal(z0)) → c8(ISNEPAL(active(z0)), ACTIVE(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(__(z0, z1)) → c16(__'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(__(__(z0, z1), z2)) → c(__'(z0, __(z1, z2)), __'(z1, z2))
ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c7(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(isNePal(z0)) → c8(ISNEPAL(active(z0)), ACTIVE(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(__(z0, z1)) → c16(__'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, __, and, isNePal, proper, top

Defined Pair Symbols:

ACTIVE, __', AND, ISNEPAL, PROPER, TOP

Compound Symbols:

c, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c18, c20, c21, c22

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

Split RHS of tuples not part of any SCC

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
active(__(z0, z1)) → __(active(z0), z1)
active(__(z0, z1)) → __(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
active(isNePal(z0)) → isNePal(active(z0))
__(mark(z0), z1) → mark(__(z0, z1))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNePal(mark(z0)) → mark(isNePal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
proper(__(z0, z1)) → __(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(isNePal(z0)) → isNePal(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c7(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(isNePal(z0)) → c8(ISNEPAL(active(z0)), ACTIVE(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(__(z0, z1)) → c16(__'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))

S tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c7(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(isNePal(z0)) → c8(ISNEPAL(active(z0)), ACTIVE(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(__(z0, z1)) → c16(__'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))

K tuples:none
Defined Rule Symbols:

active, __, and, isNePal, proper, top

Defined Pair Symbols:

ACTIVE, __', AND, ISNEPAL, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c18, c20, c21, c22, c1

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
active(__(z0, z1)) → __(active(z0), z1)
active(__(z0, z1)) → __(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
active(isNePal(z0)) → isNePal(active(z0))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNePal(mark(z0)) → mark(isNePal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
proper(__(z0, z1)) → __(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(isNePal(z0)) → isNePal(proper(z0))

Tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c7(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(isNePal(z0)) → c8(ISNEPAL(active(z0)), ACTIVE(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(__(z0, z1)) → c16(__'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))

S tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c7(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(isNePal(z0)) → c8(ISNEPAL(active(z0)), ACTIVE(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(__(z0, z1)) → c16(__'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))

K tuples:none
Defined Rule Symbols:

active, __, and, isNePal, proper

Defined Pair Symbols:

ACTIVE, __', AND, ISNEPAL, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c18, c20, c21, c22, c1

(9) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

proper(isNePal(z0)) → isNePal(proper(z0))
active(and(z0, z1)) → and(active(z0), z1)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
__(mark(z0), z1) → mark(__(z0, z1))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
__(z0, mark(z1)) → mark(__(z0, z1))
active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(and(tt, z0)) → mark(z0)
proper(__(z0, z1)) → __(proper(z0), proper(z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
active(isNePal(z0)) → isNePal(active(z0))
proper(nil) → ok(nil)
isNePal(ok(z0)) → ok(isNePal(z0))
proper(tt) → ok(tt)
active(__(z0, z1)) → __(active(z0), z1)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
isNePal(mark(z0)) → mark(isNePal(z0))
active(__(z0, z1)) → __(z0, active(z1))

And the Tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c7(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(isNePal(z0)) → c8(ISNEPAL(active(z0)), ACTIVE(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(__(z0, z1)) → c16(__'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))

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

POL(ACTIVE(x₁)) = 0
POL(AND(x₁, x₂)) = 0
POL(ISNEPAL(x₁)) = 0
POL(PROPER(x₁)) = 0
POL(TOP(x₁)) = [2]x₁
POL(__(x₁, x₂)) = [2] + [4]x₁ + x₂
POL(__'(x₁, x₂)) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = [4]x₁ + x₂
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c18(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c20(x₁, x₂)) = x₁ + x₂
POL(c21(x₁, x₂)) = x₁ + x₂
POL(c22(x₁, x₂)) = x₁ + x₂
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c6(x₁, x₂)) = x₁ + x₂
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁)) = x₁
POL(isNePal(x₁)) = [4]x₁
POL(mark(x₁)) = [1] + x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(tt) = [4]

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
active(__(z0, z1)) → __(active(z0), z1)
active(__(z0, z1)) → __(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
active(isNePal(z0)) → isNePal(active(z0))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNePal(mark(z0)) → mark(isNePal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
proper(__(z0, z1)) → __(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(isNePal(z0)) → isNePal(proper(z0))

Tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c7(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(isNePal(z0)) → c8(ISNEPAL(active(z0)), ACTIVE(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(__(z0, z1)) → c16(__'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))

S tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c7(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(isNePal(z0)) → c8(ISNEPAL(active(z0)), ACTIVE(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(__(z0, z1)) → c16(__'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))

K tuples:

TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, __, and, isNePal, proper

Defined Pair Symbols:

ACTIVE, __', AND, ISNEPAL, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c18, c20, c21, c22, c1

(11) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(and(z0, z1)) → c7(AND(active(z0), z1), ACTIVE(z0)) by

ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
active(__(z0, z1)) → __(active(z0), z1)
active(__(z0, z1)) → __(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
active(isNePal(z0)) → isNePal(active(z0))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNePal(mark(z0)) → mark(isNePal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
proper(__(z0, z1)) → __(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(isNePal(z0)) → isNePal(proper(z0))

Tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
ACTIVE(isNePal(z0)) → c8(ISNEPAL(active(z0)), ACTIVE(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(__(z0, z1)) → c16(__'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))

S tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
ACTIVE(isNePal(z0)) → c8(ISNEPAL(active(z0)), ACTIVE(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(__(z0, z1)) → c16(__'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))

K tuples:

TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, __, and, isNePal, proper

Defined Pair Symbols:

ACTIVE, __', AND, ISNEPAL, PROPER, TOP

Compound Symbols:

c5, c6, c8, c9, c10, c11, c12, c13, c14, c15, c16, c18, c20, c21, c22, c1, c7

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

Removed 1 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
active(__(z0, z1)) → __(active(z0), z1)
active(__(z0, z1)) → __(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
active(isNePal(z0)) → isNePal(active(z0))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNePal(mark(z0)) → mark(isNePal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
proper(__(z0, z1)) → __(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(isNePal(z0)) → isNePal(proper(z0))

Tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
ACTIVE(isNePal(z0)) → c8(ISNEPAL(active(z0)), ACTIVE(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(__(z0, z1)) → c16(__'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))

S tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
ACTIVE(isNePal(z0)) → c8(ISNEPAL(active(z0)), ACTIVE(z0))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(__(z0, z1)) → c16(__'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))

K tuples:

TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, __, and, isNePal, proper

Defined Pair Symbols:

ACTIVE, __', AND, ISNEPAL, PROPER, TOP

Compound Symbols:

c5, c6, c8, c9, c10, c11, c12, c13, c14, c15, c16, c18, c20, c21, c22, c1, c7, c7

(15) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(isNePal(z0)) → c8(ISNEPAL(active(z0)), ACTIVE(z0)) by

ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
active(__(z0, z1)) → __(active(z0), z1)
active(__(z0, z1)) → __(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
active(isNePal(z0)) → isNePal(active(z0))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNePal(mark(z0)) → mark(isNePal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
proper(__(z0, z1)) → __(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(isNePal(z0)) → isNePal(proper(z0))

Tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(__(z0, z1)) → c16(__'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))

S tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(__(z0, z1)) → c16(__'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))

K tuples:

TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, __, and, isNePal, proper

Defined Pair Symbols:

ACTIVE, __', AND, ISNEPAL, PROPER, TOP

Compound Symbols:

c5, c6, c9, c10, c11, c12, c13, c14, c15, c16, c18, c20, c21, c22, c1, c7, c7, c8

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

Removed 1 trailing tuple parts

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
active(__(z0, z1)) → __(active(z0), z1)
active(__(z0, z1)) → __(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
active(isNePal(z0)) → isNePal(active(z0))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNePal(mark(z0)) → mark(isNePal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
proper(__(z0, z1)) → __(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(isNePal(z0)) → isNePal(proper(z0))

Tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(__(z0, z1)) → c16(__'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)))

S tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(__(z0, z1)) → c16(__'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)))

K tuples:

TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, __, and, isNePal, proper

Defined Pair Symbols:

ACTIVE, __', AND, ISNEPAL, PROPER, TOP

Compound Symbols:

c5, c6, c9, c10, c11, c12, c13, c14, c15, c16, c18, c20, c21, c22, c1, c7, c7, c8, c8

(19) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(__(z0, z1)) → c16(__'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(__(x0, __(z0, z1))) → c16(__'(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(__(x0, nil)) → c16(__'(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(__(x0, and(z0, z1))) → c16(__'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(__(x0, tt)) → c16(__'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(__(x0, isNePal(z0))) → c16(__'(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(__(__(z0, z1), x1)) → c16(__'(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(__(nil, x1)) → c16(__'(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(__(and(z0, z1), x1)) → c16(__'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(__(tt, x1)) → c16(__'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(__(isNePal(z0), x1)) → c16(__'(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
active(__(z0, z1)) → __(active(z0), z1)
active(__(z0, z1)) → __(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
active(isNePal(z0)) → isNePal(active(z0))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNePal(mark(z0)) → mark(isNePal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
proper(__(z0, z1)) → __(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(isNePal(z0)) → isNePal(proper(z0))

Tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)))
PROPER(__(x0, __(z0, z1))) → c16(__'(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(__(x0, nil)) → c16(__'(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(__(x0, and(z0, z1))) → c16(__'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(__(x0, tt)) → c16(__'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(__(x0, isNePal(z0))) → c16(__'(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(__(__(z0, z1), x1)) → c16(__'(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(__(nil, x1)) → c16(__'(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(__(and(z0, z1), x1)) → c16(__'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(__(tt, x1)) → c16(__'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(__(isNePal(z0), x1)) → c16(__'(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))

S tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)))
PROPER(__(x0, __(z0, z1))) → c16(__'(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(__(x0, nil)) → c16(__'(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(__(x0, and(z0, z1))) → c16(__'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(__(x0, tt)) → c16(__'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(__(x0, isNePal(z0))) → c16(__'(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(__(__(z0, z1), x1)) → c16(__'(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(__(nil, x1)) → c16(__'(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(__(and(z0, z1), x1)) → c16(__'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(__(tt, x1)) → c16(__'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(__(isNePal(z0), x1)) → c16(__'(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))

K tuples:

TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, __, and, isNePal, proper

Defined Pair Symbols:

ACTIVE, __', AND, ISNEPAL, PROPER, TOP

Compound Symbols:

c5, c6, c9, c10, c11, c12, c13, c14, c15, c18, c20, c21, c22, c1, c7, c7, c8, c8, c16

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

Removed 4 trailing tuple parts

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
active(__(z0, z1)) → __(active(z0), z1)
active(__(z0, z1)) → __(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
active(isNePal(z0)) → isNePal(active(z0))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNePal(mark(z0)) → mark(isNePal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
proper(__(z0, z1)) → __(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(isNePal(z0)) → isNePal(proper(z0))

Tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)))
PROPER(__(x0, __(z0, z1))) → c16(__'(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(__(x0, and(z0, z1))) → c16(__'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(__(x0, isNePal(z0))) → c16(__'(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(__(__(z0, z1), x1)) → c16(__'(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(__(and(z0, z1), x1)) → c16(__'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(__(isNePal(z0), x1)) → c16(__'(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(__(x0, nil)) → c16(__'(proper(x0), ok(nil)), PROPER(x0))
PROPER(__(x0, tt)) → c16(__'(proper(x0), ok(tt)), PROPER(x0))
PROPER(__(nil, x1)) → c16(__'(ok(nil), proper(x1)), PROPER(x1))
PROPER(__(tt, x1)) → c16(__'(ok(tt), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)))
PROPER(__(x0, __(z0, z1))) → c16(__'(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(__(x0, and(z0, z1))) → c16(__'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(__(x0, isNePal(z0))) → c16(__'(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(__(__(z0, z1), x1)) → c16(__'(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(__(and(z0, z1), x1)) → c16(__'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(__(isNePal(z0), x1)) → c16(__'(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(__(x0, nil)) → c16(__'(proper(x0), ok(nil)), PROPER(x0))
PROPER(__(x0, tt)) → c16(__'(proper(x0), ok(tt)), PROPER(x0))
PROPER(__(nil, x1)) → c16(__'(ok(nil), proper(x1)), PROPER(x1))
PROPER(__(tt, x1)) → c16(__'(ok(tt), proper(x1)), PROPER(x1))

K tuples:

TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, __, and, isNePal, proper

Defined Pair Symbols:

ACTIVE, __', AND, ISNEPAL, PROPER, TOP

Compound Symbols:

c5, c6, c9, c10, c11, c12, c13, c14, c15, c18, c20, c21, c22, c1, c7, c7, c8, c8, c16, c16

(23) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(and(z0, z1)) → c18(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(and(x0, __(z0, z1))) → c18(AND(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(and(x0, nil)) → c18(AND(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(and(x0, and(z0, z1))) → c18(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, tt)) → c18(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, isNePal(z0))) → c18(AND(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(and(__(z0, z1), x1)) → c18(AND(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(and(nil, x1)) → c18(AND(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c18(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(tt, x1)) → c18(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(isNePal(z0), x1)) → c18(AND(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
active(__(z0, z1)) → __(active(z0), z1)
active(__(z0, z1)) → __(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
active(isNePal(z0)) → isNePal(active(z0))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNePal(mark(z0)) → mark(isNePal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
proper(__(z0, z1)) → __(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(isNePal(z0)) → isNePal(proper(z0))

Tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)))
PROPER(__(x0, __(z0, z1))) → c16(__'(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(__(x0, and(z0, z1))) → c16(__'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(__(x0, isNePal(z0))) → c16(__'(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(__(__(z0, z1), x1)) → c16(__'(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(__(and(z0, z1), x1)) → c16(__'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(__(isNePal(z0), x1)) → c16(__'(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(__(x0, nil)) → c16(__'(proper(x0), ok(nil)), PROPER(x0))
PROPER(__(x0, tt)) → c16(__'(proper(x0), ok(tt)), PROPER(x0))
PROPER(__(nil, x1)) → c16(__'(ok(nil), proper(x1)), PROPER(x1))
PROPER(__(tt, x1)) → c16(__'(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(x0, __(z0, z1))) → c18(AND(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(and(x0, nil)) → c18(AND(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(and(x0, and(z0, z1))) → c18(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, tt)) → c18(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, isNePal(z0))) → c18(AND(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(and(__(z0, z1), x1)) → c18(AND(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(and(nil, x1)) → c18(AND(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c18(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(tt, x1)) → c18(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(isNePal(z0), x1)) → c18(AND(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))

S tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)))
PROPER(__(x0, __(z0, z1))) → c16(__'(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(__(x0, and(z0, z1))) → c16(__'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(__(x0, isNePal(z0))) → c16(__'(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(__(__(z0, z1), x1)) → c16(__'(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(__(and(z0, z1), x1)) → c16(__'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(__(isNePal(z0), x1)) → c16(__'(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(__(x0, nil)) → c16(__'(proper(x0), ok(nil)), PROPER(x0))
PROPER(__(x0, tt)) → c16(__'(proper(x0), ok(tt)), PROPER(x0))
PROPER(__(nil, x1)) → c16(__'(ok(nil), proper(x1)), PROPER(x1))
PROPER(__(tt, x1)) → c16(__'(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(x0, __(z0, z1))) → c18(AND(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(and(x0, nil)) → c18(AND(proper(x0), ok(nil)), PROPER(x0), PROPER(nil))
PROPER(and(x0, and(z0, z1))) → c18(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, tt)) → c18(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, isNePal(z0))) → c18(AND(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(and(__(z0, z1), x1)) → c18(AND(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(and(nil, x1)) → c18(AND(ok(nil), proper(x1)), PROPER(nil), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c18(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(tt, x1)) → c18(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(isNePal(z0), x1)) → c18(AND(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))

K tuples:

TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, __, and, isNePal, proper

Defined Pair Symbols:

ACTIVE, __', AND, ISNEPAL, PROPER, TOP

Compound Symbols:

c5, c6, c9, c10, c11, c12, c13, c14, c15, c20, c21, c22, c1, c7, c7, c8, c8, c16, c16, c18

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

Removed 4 trailing tuple parts

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
active(__(z0, z1)) → __(active(z0), z1)
active(__(z0, z1)) → __(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
active(isNePal(z0)) → isNePal(active(z0))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNePal(mark(z0)) → mark(isNePal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
proper(__(z0, z1)) → __(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(isNePal(z0)) → isNePal(proper(z0))

Tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)))
PROPER(__(x0, __(z0, z1))) → c16(__'(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(__(x0, and(z0, z1))) → c16(__'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(__(x0, isNePal(z0))) → c16(__'(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(__(__(z0, z1), x1)) → c16(__'(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(__(and(z0, z1), x1)) → c16(__'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(__(isNePal(z0), x1)) → c16(__'(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(__(x0, nil)) → c16(__'(proper(x0), ok(nil)), PROPER(x0))
PROPER(__(x0, tt)) → c16(__'(proper(x0), ok(tt)), PROPER(x0))
PROPER(__(nil, x1)) → c16(__'(ok(nil), proper(x1)), PROPER(x1))
PROPER(__(tt, x1)) → c16(__'(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(x0, __(z0, z1))) → c18(AND(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c18(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNePal(z0))) → c18(AND(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(and(__(z0, z1), x1)) → c18(AND(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c18(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNePal(z0), x1)) → c18(AND(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(and(x0, nil)) → c18(AND(proper(x0), ok(nil)), PROPER(x0))
PROPER(and(x0, tt)) → c18(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(nil, x1)) → c18(AND(ok(nil), proper(x1)), PROPER(x1))
PROPER(and(tt, x1)) → c18(AND(ok(tt), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)))
PROPER(__(x0, __(z0, z1))) → c16(__'(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(__(x0, and(z0, z1))) → c16(__'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(__(x0, isNePal(z0))) → c16(__'(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(__(__(z0, z1), x1)) → c16(__'(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(__(and(z0, z1), x1)) → c16(__'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(__(isNePal(z0), x1)) → c16(__'(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(__(x0, nil)) → c16(__'(proper(x0), ok(nil)), PROPER(x0))
PROPER(__(x0, tt)) → c16(__'(proper(x0), ok(tt)), PROPER(x0))
PROPER(__(nil, x1)) → c16(__'(ok(nil), proper(x1)), PROPER(x1))
PROPER(__(tt, x1)) → c16(__'(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(x0, __(z0, z1))) → c18(AND(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c18(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNePal(z0))) → c18(AND(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(and(__(z0, z1), x1)) → c18(AND(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c18(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNePal(z0), x1)) → c18(AND(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(and(x0, nil)) → c18(AND(proper(x0), ok(nil)), PROPER(x0))
PROPER(and(x0, tt)) → c18(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(nil, x1)) → c18(AND(ok(nil), proper(x1)), PROPER(x1))
PROPER(and(tt, x1)) → c18(AND(ok(tt), proper(x1)), PROPER(x1))

K tuples:

TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, __, and, isNePal, proper

Defined Pair Symbols:

ACTIVE, __', AND, ISNEPAL, PROPER, TOP

Compound Symbols:

c5, c6, c9, c10, c11, c12, c13, c14, c15, c20, c21, c22, c1, c7, c7, c8, c8, c16, c16, c18, c18

(27) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(isNePal(z0)) → c20(ISNEPAL(proper(z0)), PROPER(z0)) by

PROPER(isNePal(__(z0, z1))) → c20(ISNEPAL(__(proper(z0), proper(z1))), PROPER(__(z0, z1)))
PROPER(isNePal(nil)) → c20(ISNEPAL(ok(nil)), PROPER(nil))
PROPER(isNePal(and(z0, z1))) → c20(ISNEPAL(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNePal(tt)) → c20(ISNEPAL(ok(tt)), PROPER(tt))
PROPER(isNePal(isNePal(z0))) → c20(ISNEPAL(isNePal(proper(z0))), PROPER(isNePal(z0)))

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
active(__(z0, z1)) → __(active(z0), z1)
active(__(z0, z1)) → __(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
active(isNePal(z0)) → isNePal(active(z0))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNePal(mark(z0)) → mark(isNePal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
proper(__(z0, z1)) → __(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(isNePal(z0)) → isNePal(proper(z0))

Tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)))
PROPER(__(x0, __(z0, z1))) → c16(__'(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(__(x0, and(z0, z1))) → c16(__'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(__(x0, isNePal(z0))) → c16(__'(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(__(__(z0, z1), x1)) → c16(__'(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(__(and(z0, z1), x1)) → c16(__'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(__(isNePal(z0), x1)) → c16(__'(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(__(x0, nil)) → c16(__'(proper(x0), ok(nil)), PROPER(x0))
PROPER(__(x0, tt)) → c16(__'(proper(x0), ok(tt)), PROPER(x0))
PROPER(__(nil, x1)) → c16(__'(ok(nil), proper(x1)), PROPER(x1))
PROPER(__(tt, x1)) → c16(__'(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(x0, __(z0, z1))) → c18(AND(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c18(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNePal(z0))) → c18(AND(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(and(__(z0, z1), x1)) → c18(AND(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c18(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNePal(z0), x1)) → c18(AND(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(and(x0, nil)) → c18(AND(proper(x0), ok(nil)), PROPER(x0))
PROPER(and(x0, tt)) → c18(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(nil, x1)) → c18(AND(ok(nil), proper(x1)), PROPER(x1))
PROPER(and(tt, x1)) → c18(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(isNePal(__(z0, z1))) → c20(ISNEPAL(__(proper(z0), proper(z1))), PROPER(__(z0, z1)))
PROPER(isNePal(nil)) → c20(ISNEPAL(ok(nil)), PROPER(nil))
PROPER(isNePal(and(z0, z1))) → c20(ISNEPAL(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNePal(tt)) → c20(ISNEPAL(ok(tt)), PROPER(tt))
PROPER(isNePal(isNePal(z0))) → c20(ISNEPAL(isNePal(proper(z0))), PROPER(isNePal(z0)))

S tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)))
PROPER(__(x0, __(z0, z1))) → c16(__'(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(__(x0, and(z0, z1))) → c16(__'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(__(x0, isNePal(z0))) → c16(__'(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(__(__(z0, z1), x1)) → c16(__'(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(__(and(z0, z1), x1)) → c16(__'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(__(isNePal(z0), x1)) → c16(__'(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(__(x0, nil)) → c16(__'(proper(x0), ok(nil)), PROPER(x0))
PROPER(__(x0, tt)) → c16(__'(proper(x0), ok(tt)), PROPER(x0))
PROPER(__(nil, x1)) → c16(__'(ok(nil), proper(x1)), PROPER(x1))
PROPER(__(tt, x1)) → c16(__'(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(x0, __(z0, z1))) → c18(AND(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c18(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNePal(z0))) → c18(AND(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(and(__(z0, z1), x1)) → c18(AND(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c18(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNePal(z0), x1)) → c18(AND(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(and(x0, nil)) → c18(AND(proper(x0), ok(nil)), PROPER(x0))
PROPER(and(x0, tt)) → c18(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(nil, x1)) → c18(AND(ok(nil), proper(x1)), PROPER(x1))
PROPER(and(tt, x1)) → c18(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(isNePal(__(z0, z1))) → c20(ISNEPAL(__(proper(z0), proper(z1))), PROPER(__(z0, z1)))
PROPER(isNePal(nil)) → c20(ISNEPAL(ok(nil)), PROPER(nil))
PROPER(isNePal(and(z0, z1))) → c20(ISNEPAL(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNePal(tt)) → c20(ISNEPAL(ok(tt)), PROPER(tt))
PROPER(isNePal(isNePal(z0))) → c20(ISNEPAL(isNePal(proper(z0))), PROPER(isNePal(z0)))

K tuples:

TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, __, and, isNePal, proper

Defined Pair Symbols:

ACTIVE, __', AND, ISNEPAL, TOP, PROPER

Compound Symbols:

c5, c6, c9, c10, c11, c12, c13, c14, c15, c21, c22, c1, c7, c7, c8, c8, c16, c16, c18, c18, c20

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

Removed 2 trailing tuple parts

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
active(__(z0, z1)) → __(active(z0), z1)
active(__(z0, z1)) → __(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
active(isNePal(z0)) → isNePal(active(z0))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNePal(mark(z0)) → mark(isNePal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
proper(__(z0, z1)) → __(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(isNePal(z0)) → isNePal(proper(z0))

Tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)))
PROPER(__(x0, __(z0, z1))) → c16(__'(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(__(x0, and(z0, z1))) → c16(__'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(__(x0, isNePal(z0))) → c16(__'(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(__(__(z0, z1), x1)) → c16(__'(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(__(and(z0, z1), x1)) → c16(__'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(__(isNePal(z0), x1)) → c16(__'(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(__(x0, nil)) → c16(__'(proper(x0), ok(nil)), PROPER(x0))
PROPER(__(x0, tt)) → c16(__'(proper(x0), ok(tt)), PROPER(x0))
PROPER(__(nil, x1)) → c16(__'(ok(nil), proper(x1)), PROPER(x1))
PROPER(__(tt, x1)) → c16(__'(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(x0, __(z0, z1))) → c18(AND(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c18(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNePal(z0))) → c18(AND(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(and(__(z0, z1), x1)) → c18(AND(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c18(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNePal(z0), x1)) → c18(AND(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(and(x0, nil)) → c18(AND(proper(x0), ok(nil)), PROPER(x0))
PROPER(and(x0, tt)) → c18(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(nil, x1)) → c18(AND(ok(nil), proper(x1)), PROPER(x1))
PROPER(and(tt, x1)) → c18(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(isNePal(__(z0, z1))) → c20(ISNEPAL(__(proper(z0), proper(z1))), PROPER(__(z0, z1)))
PROPER(isNePal(and(z0, z1))) → c20(ISNEPAL(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNePal(isNePal(z0))) → c20(ISNEPAL(isNePal(proper(z0))), PROPER(isNePal(z0)))
PROPER(isNePal(nil)) → c20(ISNEPAL(ok(nil)))
PROPER(isNePal(tt)) → c20(ISNEPAL(ok(tt)))

S tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)))
PROPER(__(x0, __(z0, z1))) → c16(__'(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(__(x0, and(z0, z1))) → c16(__'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(__(x0, isNePal(z0))) → c16(__'(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(__(__(z0, z1), x1)) → c16(__'(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(__(and(z0, z1), x1)) → c16(__'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(__(isNePal(z0), x1)) → c16(__'(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(__(x0, nil)) → c16(__'(proper(x0), ok(nil)), PROPER(x0))
PROPER(__(x0, tt)) → c16(__'(proper(x0), ok(tt)), PROPER(x0))
PROPER(__(nil, x1)) → c16(__'(ok(nil), proper(x1)), PROPER(x1))
PROPER(__(tt, x1)) → c16(__'(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(x0, __(z0, z1))) → c18(AND(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c18(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNePal(z0))) → c18(AND(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(and(__(z0, z1), x1)) → c18(AND(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c18(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNePal(z0), x1)) → c18(AND(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(and(x0, nil)) → c18(AND(proper(x0), ok(nil)), PROPER(x0))
PROPER(and(x0, tt)) → c18(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(nil, x1)) → c18(AND(ok(nil), proper(x1)), PROPER(x1))
PROPER(and(tt, x1)) → c18(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(isNePal(__(z0, z1))) → c20(ISNEPAL(__(proper(z0), proper(z1))), PROPER(__(z0, z1)))
PROPER(isNePal(and(z0, z1))) → c20(ISNEPAL(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNePal(isNePal(z0))) → c20(ISNEPAL(isNePal(proper(z0))), PROPER(isNePal(z0)))
PROPER(isNePal(nil)) → c20(ISNEPAL(ok(nil)))
PROPER(isNePal(tt)) → c20(ISNEPAL(ok(tt)))

K tuples:

TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, __, and, isNePal, proper

Defined Pair Symbols:

ACTIVE, __', AND, ISNEPAL, TOP, PROPER

Compound Symbols:

c5, c6, c9, c10, c11, c12, c13, c14, c15, c21, c22, c1, c7, c7, c8, c8, c16, c16, c18, c18, c20, c20

(31) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(__(z0, z1))) → c21(TOP(__(proper(z0), proper(z1))), PROPER(__(z0, z1)))
TOP(mark(nil)) → c21(TOP(ok(nil)), PROPER(nil))
TOP(mark(and(z0, z1))) → c21(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(tt)) → c21(TOP(ok(tt)), PROPER(tt))
TOP(mark(isNePal(z0))) → c21(TOP(isNePal(proper(z0))), PROPER(isNePal(z0)))

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
active(__(z0, z1)) → __(active(z0), z1)
active(__(z0, z1)) → __(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
active(isNePal(z0)) → isNePal(active(z0))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNePal(mark(z0)) → mark(isNePal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
proper(__(z0, z1)) → __(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(isNePal(z0)) → isNePal(proper(z0))

Tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)))
PROPER(__(x0, __(z0, z1))) → c16(__'(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(__(x0, and(z0, z1))) → c16(__'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(__(x0, isNePal(z0))) → c16(__'(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(__(__(z0, z1), x1)) → c16(__'(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(__(and(z0, z1), x1)) → c16(__'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(__(isNePal(z0), x1)) → c16(__'(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(__(x0, nil)) → c16(__'(proper(x0), ok(nil)), PROPER(x0))
PROPER(__(x0, tt)) → c16(__'(proper(x0), ok(tt)), PROPER(x0))
PROPER(__(nil, x1)) → c16(__'(ok(nil), proper(x1)), PROPER(x1))
PROPER(__(tt, x1)) → c16(__'(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(x0, __(z0, z1))) → c18(AND(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c18(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNePal(z0))) → c18(AND(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(and(__(z0, z1), x1)) → c18(AND(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c18(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNePal(z0), x1)) → c18(AND(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(and(x0, nil)) → c18(AND(proper(x0), ok(nil)), PROPER(x0))
PROPER(and(x0, tt)) → c18(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(nil, x1)) → c18(AND(ok(nil), proper(x1)), PROPER(x1))
PROPER(and(tt, x1)) → c18(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(isNePal(__(z0, z1))) → c20(ISNEPAL(__(proper(z0), proper(z1))), PROPER(__(z0, z1)))
PROPER(isNePal(and(z0, z1))) → c20(ISNEPAL(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNePal(isNePal(z0))) → c20(ISNEPAL(isNePal(proper(z0))), PROPER(isNePal(z0)))
PROPER(isNePal(nil)) → c20(ISNEPAL(ok(nil)))
PROPER(isNePal(tt)) → c20(ISNEPAL(ok(tt)))
TOP(mark(__(z0, z1))) → c21(TOP(__(proper(z0), proper(z1))), PROPER(__(z0, z1)))
TOP(mark(nil)) → c21(TOP(ok(nil)), PROPER(nil))
TOP(mark(and(z0, z1))) → c21(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(tt)) → c21(TOP(ok(tt)), PROPER(tt))
TOP(mark(isNePal(z0))) → c21(TOP(isNePal(proper(z0))), PROPER(isNePal(z0)))

S tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)))
PROPER(__(x0, __(z0, z1))) → c16(__'(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(__(x0, and(z0, z1))) → c16(__'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(__(x0, isNePal(z0))) → c16(__'(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(__(__(z0, z1), x1)) → c16(__'(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(__(and(z0, z1), x1)) → c16(__'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(__(isNePal(z0), x1)) → c16(__'(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(__(x0, nil)) → c16(__'(proper(x0), ok(nil)), PROPER(x0))
PROPER(__(x0, tt)) → c16(__'(proper(x0), ok(tt)), PROPER(x0))
PROPER(__(nil, x1)) → c16(__'(ok(nil), proper(x1)), PROPER(x1))
PROPER(__(tt, x1)) → c16(__'(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(x0, __(z0, z1))) → c18(AND(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c18(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNePal(z0))) → c18(AND(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(and(__(z0, z1), x1)) → c18(AND(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c18(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNePal(z0), x1)) → c18(AND(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(and(x0, nil)) → c18(AND(proper(x0), ok(nil)), PROPER(x0))
PROPER(and(x0, tt)) → c18(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(nil, x1)) → c18(AND(ok(nil), proper(x1)), PROPER(x1))
PROPER(and(tt, x1)) → c18(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(isNePal(__(z0, z1))) → c20(ISNEPAL(__(proper(z0), proper(z1))), PROPER(__(z0, z1)))
PROPER(isNePal(and(z0, z1))) → c20(ISNEPAL(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNePal(isNePal(z0))) → c20(ISNEPAL(isNePal(proper(z0))), PROPER(isNePal(z0)))
PROPER(isNePal(nil)) → c20(ISNEPAL(ok(nil)))
PROPER(isNePal(tt)) → c20(ISNEPAL(ok(tt)))

K tuples:

TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, __, and, isNePal, proper

Defined Pair Symbols:

ACTIVE, __', AND, ISNEPAL, TOP, PROPER

Compound Symbols:

c5, c6, c9, c10, c11, c12, c13, c14, c15, c22, c1, c7, c7, c8, c8, c16, c16, c18, c18, c20, c20, c21

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

Removed 2 trailing tuple parts

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
active(__(z0, z1)) → __(active(z0), z1)
active(__(z0, z1)) → __(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
active(isNePal(z0)) → isNePal(active(z0))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNePal(mark(z0)) → mark(isNePal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
proper(__(z0, z1)) → __(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(isNePal(z0)) → isNePal(proper(z0))

Tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)))
PROPER(__(x0, __(z0, z1))) → c16(__'(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(__(x0, and(z0, z1))) → c16(__'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(__(x0, isNePal(z0))) → c16(__'(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(__(__(z0, z1), x1)) → c16(__'(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(__(and(z0, z1), x1)) → c16(__'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(__(isNePal(z0), x1)) → c16(__'(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(__(x0, nil)) → c16(__'(proper(x0), ok(nil)), PROPER(x0))
PROPER(__(x0, tt)) → c16(__'(proper(x0), ok(tt)), PROPER(x0))
PROPER(__(nil, x1)) → c16(__'(ok(nil), proper(x1)), PROPER(x1))
PROPER(__(tt, x1)) → c16(__'(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(x0, __(z0, z1))) → c18(AND(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c18(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNePal(z0))) → c18(AND(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(and(__(z0, z1), x1)) → c18(AND(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c18(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNePal(z0), x1)) → c18(AND(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(and(x0, nil)) → c18(AND(proper(x0), ok(nil)), PROPER(x0))
PROPER(and(x0, tt)) → c18(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(nil, x1)) → c18(AND(ok(nil), proper(x1)), PROPER(x1))
PROPER(and(tt, x1)) → c18(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(isNePal(__(z0, z1))) → c20(ISNEPAL(__(proper(z0), proper(z1))), PROPER(__(z0, z1)))
PROPER(isNePal(and(z0, z1))) → c20(ISNEPAL(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNePal(isNePal(z0))) → c20(ISNEPAL(isNePal(proper(z0))), PROPER(isNePal(z0)))
PROPER(isNePal(nil)) → c20(ISNEPAL(ok(nil)))
PROPER(isNePal(tt)) → c20(ISNEPAL(ok(tt)))
TOP(mark(__(z0, z1))) → c21(TOP(__(proper(z0), proper(z1))), PROPER(__(z0, z1)))
TOP(mark(and(z0, z1))) → c21(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNePal(z0))) → c21(TOP(isNePal(proper(z0))), PROPER(isNePal(z0)))
TOP(mark(nil)) → c21(TOP(ok(nil)))
TOP(mark(tt)) → c21(TOP(ok(tt)))

S tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)))
PROPER(__(x0, __(z0, z1))) → c16(__'(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(__(x0, and(z0, z1))) → c16(__'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(__(x0, isNePal(z0))) → c16(__'(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(__(__(z0, z1), x1)) → c16(__'(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(__(and(z0, z1), x1)) → c16(__'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(__(isNePal(z0), x1)) → c16(__'(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(__(x0, nil)) → c16(__'(proper(x0), ok(nil)), PROPER(x0))
PROPER(__(x0, tt)) → c16(__'(proper(x0), ok(tt)), PROPER(x0))
PROPER(__(nil, x1)) → c16(__'(ok(nil), proper(x1)), PROPER(x1))
PROPER(__(tt, x1)) → c16(__'(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(x0, __(z0, z1))) → c18(AND(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c18(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNePal(z0))) → c18(AND(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(and(__(z0, z1), x1)) → c18(AND(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c18(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNePal(z0), x1)) → c18(AND(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(and(x0, nil)) → c18(AND(proper(x0), ok(nil)), PROPER(x0))
PROPER(and(x0, tt)) → c18(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(nil, x1)) → c18(AND(ok(nil), proper(x1)), PROPER(x1))
PROPER(and(tt, x1)) → c18(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(isNePal(__(z0, z1))) → c20(ISNEPAL(__(proper(z0), proper(z1))), PROPER(__(z0, z1)))
PROPER(isNePal(and(z0, z1))) → c20(ISNEPAL(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNePal(isNePal(z0))) → c20(ISNEPAL(isNePal(proper(z0))), PROPER(isNePal(z0)))
PROPER(isNePal(nil)) → c20(ISNEPAL(ok(nil)))
PROPER(isNePal(tt)) → c20(ISNEPAL(ok(tt)))

K tuples:

TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, __, and, isNePal, proper

Defined Pair Symbols:

ACTIVE, __', AND, ISNEPAL, TOP, PROPER

Compound Symbols:

c5, c6, c9, c10, c11, c12, c13, c14, c15, c22, c1, c7, c7, c8, c8, c16, c16, c18, c18, c20, c20, c21, c21

(35) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace TOP(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(__(__(z0, z1), z2))) → c22(TOP(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
TOP(ok(__(z0, nil))) → c22(TOP(mark(z0)), ACTIVE(__(z0, nil)))
TOP(ok(__(nil, z0))) → c22(TOP(mark(z0)), ACTIVE(__(nil, z0)))
TOP(ok(and(tt, z0))) → c22(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNePal(__(z0, __(z1, z0))))) → c22(TOP(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
TOP(ok(__(z0, z1))) → c22(TOP(__(active(z0), z1)), ACTIVE(__(z0, z1)))
TOP(ok(__(z0, z1))) → c22(TOP(__(z0, active(z1))), ACTIVE(__(z0, z1)))
TOP(ok(and(z0, z1))) → c22(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
TOP(ok(isNePal(z0))) → c22(TOP(isNePal(active(z0))), ACTIVE(isNePal(z0)))

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
active(__(z0, z1)) → __(active(z0), z1)
active(__(z0, z1)) → __(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
active(isNePal(z0)) → isNePal(active(z0))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNePal(mark(z0)) → mark(isNePal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
proper(__(z0, z1)) → __(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(isNePal(z0)) → isNePal(proper(z0))

Tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)))
PROPER(__(x0, __(z0, z1))) → c16(__'(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(__(x0, and(z0, z1))) → c16(__'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(__(x0, isNePal(z0))) → c16(__'(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(__(__(z0, z1), x1)) → c16(__'(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(__(and(z0, z1), x1)) → c16(__'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(__(isNePal(z0), x1)) → c16(__'(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(__(x0, nil)) → c16(__'(proper(x0), ok(nil)), PROPER(x0))
PROPER(__(x0, tt)) → c16(__'(proper(x0), ok(tt)), PROPER(x0))
PROPER(__(nil, x1)) → c16(__'(ok(nil), proper(x1)), PROPER(x1))
PROPER(__(tt, x1)) → c16(__'(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(x0, __(z0, z1))) → c18(AND(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c18(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNePal(z0))) → c18(AND(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(and(__(z0, z1), x1)) → c18(AND(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c18(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNePal(z0), x1)) → c18(AND(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(and(x0, nil)) → c18(AND(proper(x0), ok(nil)), PROPER(x0))
PROPER(and(x0, tt)) → c18(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(nil, x1)) → c18(AND(ok(nil), proper(x1)), PROPER(x1))
PROPER(and(tt, x1)) → c18(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(isNePal(__(z0, z1))) → c20(ISNEPAL(__(proper(z0), proper(z1))), PROPER(__(z0, z1)))
PROPER(isNePal(and(z0, z1))) → c20(ISNEPAL(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNePal(isNePal(z0))) → c20(ISNEPAL(isNePal(proper(z0))), PROPER(isNePal(z0)))
PROPER(isNePal(nil)) → c20(ISNEPAL(ok(nil)))
PROPER(isNePal(tt)) → c20(ISNEPAL(ok(tt)))
TOP(mark(__(z0, z1))) → c21(TOP(__(proper(z0), proper(z1))), PROPER(__(z0, z1)))
TOP(mark(and(z0, z1))) → c21(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNePal(z0))) → c21(TOP(isNePal(proper(z0))), PROPER(isNePal(z0)))
TOP(mark(nil)) → c21(TOP(ok(nil)))
TOP(mark(tt)) → c21(TOP(ok(tt)))
TOP(ok(__(__(z0, z1), z2))) → c22(TOP(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
TOP(ok(__(z0, nil))) → c22(TOP(mark(z0)), ACTIVE(__(z0, nil)))
TOP(ok(__(nil, z0))) → c22(TOP(mark(z0)), ACTIVE(__(nil, z0)))
TOP(ok(and(tt, z0))) → c22(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNePal(__(z0, __(z1, z0))))) → c22(TOP(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
TOP(ok(__(z0, z1))) → c22(TOP(__(active(z0), z1)), ACTIVE(__(z0, z1)))
TOP(ok(__(z0, z1))) → c22(TOP(__(z0, active(z1))), ACTIVE(__(z0, z1)))
TOP(ok(and(z0, z1))) → c22(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
TOP(ok(isNePal(z0))) → c22(TOP(isNePal(active(z0))), ACTIVE(isNePal(z0)))

S tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)))
PROPER(__(x0, __(z0, z1))) → c16(__'(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(__(x0, and(z0, z1))) → c16(__'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(__(x0, isNePal(z0))) → c16(__'(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(__(__(z0, z1), x1)) → c16(__'(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(__(and(z0, z1), x1)) → c16(__'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(__(isNePal(z0), x1)) → c16(__'(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(__(x0, nil)) → c16(__'(proper(x0), ok(nil)), PROPER(x0))
PROPER(__(x0, tt)) → c16(__'(proper(x0), ok(tt)), PROPER(x0))
PROPER(__(nil, x1)) → c16(__'(ok(nil), proper(x1)), PROPER(x1))
PROPER(__(tt, x1)) → c16(__'(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(x0, __(z0, z1))) → c18(AND(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c18(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNePal(z0))) → c18(AND(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(and(__(z0, z1), x1)) → c18(AND(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c18(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNePal(z0), x1)) → c18(AND(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(and(x0, nil)) → c18(AND(proper(x0), ok(nil)), PROPER(x0))
PROPER(and(x0, tt)) → c18(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(nil, x1)) → c18(AND(ok(nil), proper(x1)), PROPER(x1))
PROPER(and(tt, x1)) → c18(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(isNePal(__(z0, z1))) → c20(ISNEPAL(__(proper(z0), proper(z1))), PROPER(__(z0, z1)))
PROPER(isNePal(and(z0, z1))) → c20(ISNEPAL(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNePal(isNePal(z0))) → c20(ISNEPAL(isNePal(proper(z0))), PROPER(isNePal(z0)))
PROPER(isNePal(nil)) → c20(ISNEPAL(ok(nil)))
PROPER(isNePal(tt)) → c20(ISNEPAL(ok(tt)))
TOP(ok(__(__(z0, z1), z2))) → c22(TOP(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
TOP(ok(__(z0, nil))) → c22(TOP(mark(z0)), ACTIVE(__(z0, nil)))
TOP(ok(__(nil, z0))) → c22(TOP(mark(z0)), ACTIVE(__(nil, z0)))
TOP(ok(and(tt, z0))) → c22(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNePal(__(z0, __(z1, z0))))) → c22(TOP(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
TOP(ok(__(z0, z1))) → c22(TOP(__(active(z0), z1)), ACTIVE(__(z0, z1)))
TOP(ok(__(z0, z1))) → c22(TOP(__(z0, active(z1))), ACTIVE(__(z0, z1)))
TOP(ok(and(z0, z1))) → c22(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
TOP(ok(isNePal(z0))) → c22(TOP(isNePal(active(z0))), ACTIVE(isNePal(z0)))

K tuples:

TOP(mark(z0)) → c21(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, __, and, isNePal, proper

Defined Pair Symbols:

ACTIVE, __', AND, ISNEPAL, PROPER, TOP

Compound Symbols:

c5, c6, c9, c10, c11, c12, c13, c14, c15, c1, c7, c7, c8, c8, c16, c16, c18, c18, c20, c20, c21, c21, c22

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

Removed 2 trailing nodes:

TOP(mark(tt)) → c21(TOP(ok(tt)))
TOP(mark(nil)) → c21(TOP(ok(nil)))

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
active(__(z0, z1)) → __(active(z0), z1)
active(__(z0, z1)) → __(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
active(isNePal(z0)) → isNePal(active(z0))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNePal(mark(z0)) → mark(isNePal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
proper(__(z0, z1)) → __(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(isNePal(z0)) → isNePal(proper(z0))

Tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)))
PROPER(__(x0, __(z0, z1))) → c16(__'(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(__(x0, and(z0, z1))) → c16(__'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(__(x0, isNePal(z0))) → c16(__'(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(__(__(z0, z1), x1)) → c16(__'(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(__(and(z0, z1), x1)) → c16(__'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(__(isNePal(z0), x1)) → c16(__'(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(__(x0, nil)) → c16(__'(proper(x0), ok(nil)), PROPER(x0))
PROPER(__(x0, tt)) → c16(__'(proper(x0), ok(tt)), PROPER(x0))
PROPER(__(nil, x1)) → c16(__'(ok(nil), proper(x1)), PROPER(x1))
PROPER(__(tt, x1)) → c16(__'(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(x0, __(z0, z1))) → c18(AND(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c18(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNePal(z0))) → c18(AND(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(and(__(z0, z1), x1)) → c18(AND(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c18(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNePal(z0), x1)) → c18(AND(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(and(x0, nil)) → c18(AND(proper(x0), ok(nil)), PROPER(x0))
PROPER(and(x0, tt)) → c18(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(nil, x1)) → c18(AND(ok(nil), proper(x1)), PROPER(x1))
PROPER(and(tt, x1)) → c18(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(isNePal(__(z0, z1))) → c20(ISNEPAL(__(proper(z0), proper(z1))), PROPER(__(z0, z1)))
PROPER(isNePal(and(z0, z1))) → c20(ISNEPAL(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNePal(isNePal(z0))) → c20(ISNEPAL(isNePal(proper(z0))), PROPER(isNePal(z0)))
PROPER(isNePal(nil)) → c20(ISNEPAL(ok(nil)))
PROPER(isNePal(tt)) → c20(ISNEPAL(ok(tt)))
TOP(mark(__(z0, z1))) → c21(TOP(__(proper(z0), proper(z1))), PROPER(__(z0, z1)))
TOP(mark(and(z0, z1))) → c21(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNePal(z0))) → c21(TOP(isNePal(proper(z0))), PROPER(isNePal(z0)))
TOP(ok(__(__(z0, z1), z2))) → c22(TOP(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
TOP(ok(__(z0, nil))) → c22(TOP(mark(z0)), ACTIVE(__(z0, nil)))
TOP(ok(__(nil, z0))) → c22(TOP(mark(z0)), ACTIVE(__(nil, z0)))
TOP(ok(and(tt, z0))) → c22(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNePal(__(z0, __(z1, z0))))) → c22(TOP(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
TOP(ok(__(z0, z1))) → c22(TOP(__(active(z0), z1)), ACTIVE(__(z0, z1)))
TOP(ok(__(z0, z1))) → c22(TOP(__(z0, active(z1))), ACTIVE(__(z0, z1)))
TOP(ok(and(z0, z1))) → c22(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
TOP(ok(isNePal(z0))) → c22(TOP(isNePal(active(z0))), ACTIVE(isNePal(z0)))

S tuples:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)))
PROPER(__(x0, __(z0, z1))) → c16(__'(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(__(x0, and(z0, z1))) → c16(__'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(__(x0, isNePal(z0))) → c16(__'(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(__(__(z0, z1), x1)) → c16(__'(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(__(and(z0, z1), x1)) → c16(__'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(__(isNePal(z0), x1)) → c16(__'(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(__(x0, nil)) → c16(__'(proper(x0), ok(nil)), PROPER(x0))
PROPER(__(x0, tt)) → c16(__'(proper(x0), ok(tt)), PROPER(x0))
PROPER(__(nil, x1)) → c16(__'(ok(nil), proper(x1)), PROPER(x1))
PROPER(__(tt, x1)) → c16(__'(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(x0, __(z0, z1))) → c18(AND(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c18(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNePal(z0))) → c18(AND(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(and(__(z0, z1), x1)) → c18(AND(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c18(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNePal(z0), x1)) → c18(AND(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(and(x0, nil)) → c18(AND(proper(x0), ok(nil)), PROPER(x0))
PROPER(and(x0, tt)) → c18(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(nil, x1)) → c18(AND(ok(nil), proper(x1)), PROPER(x1))
PROPER(and(tt, x1)) → c18(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(isNePal(__(z0, z1))) → c20(ISNEPAL(__(proper(z0), proper(z1))), PROPER(__(z0, z1)))
PROPER(isNePal(and(z0, z1))) → c20(ISNEPAL(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNePal(isNePal(z0))) → c20(ISNEPAL(isNePal(proper(z0))), PROPER(isNePal(z0)))
PROPER(isNePal(nil)) → c20(ISNEPAL(ok(nil)))
PROPER(isNePal(tt)) → c20(ISNEPAL(ok(tt)))
TOP(ok(__(__(z0, z1), z2))) → c22(TOP(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
TOP(ok(__(z0, nil))) → c22(TOP(mark(z0)), ACTIVE(__(z0, nil)))
TOP(ok(__(nil, z0))) → c22(TOP(mark(z0)), ACTIVE(__(nil, z0)))
TOP(ok(and(tt, z0))) → c22(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNePal(__(z0, __(z1, z0))))) → c22(TOP(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
TOP(ok(__(z0, z1))) → c22(TOP(__(active(z0), z1)), ACTIVE(__(z0, z1)))
TOP(ok(__(z0, z1))) → c22(TOP(__(z0, active(z1))), ACTIVE(__(z0, z1)))
TOP(ok(and(z0, z1))) → c22(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
TOP(ok(isNePal(z0))) → c22(TOP(isNePal(active(z0))), ACTIVE(isNePal(z0)))

K tuples:none
Defined Rule Symbols:

active, __, and, isNePal, proper

Defined Pair Symbols:

ACTIVE, __', AND, ISNEPAL, PROPER, TOP

Compound Symbols:

c5, c6, c9, c10, c11, c12, c13, c14, c15, c1, c7, c7, c8, c8, c16, c16, c18, c18, c20, c20, c21, c22

(39) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

ACTIVE(__(z0, z1)) → c5(__'(active(z0), z1), ACTIVE(z0))
ACTIVE(__(z0, z1)) → c6(__'(z0, active(z1)), ACTIVE(z1))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z0, __(z1, z2)))
ACTIVE(__(__(z0, z1), z2)) → c1(__'(z1, z2))
ACTIVE(and(__(__(z0, z1), z2), x1)) → c7(AND(mark(__(z0, __(z1, z2))), x1), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(and(__(z0, nil), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(z0, nil)))
ACTIVE(and(__(nil, z0), x1)) → c7(AND(mark(z0), x1), ACTIVE(__(nil, z0)))
ACTIVE(and(isNePal(__(z0, __(z1, z0))), x1)) → c7(AND(mark(tt), x1), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(active(z0), z1), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(__(z0, z1), x1)) → c7(AND(__(z0, active(z1)), x1), ACTIVE(__(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c7(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(isNePal(z0), x1)) → c7(AND(isNePal(active(z0)), x1), ACTIVE(isNePal(z0)))
ACTIVE(and(and(tt, z0), x1)) → c7(AND(mark(z0), x1))
ACTIVE(isNePal(__(__(z0, z1), z2))) → c8(ISNEPAL(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
ACTIVE(isNePal(__(z0, nil))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(z0, nil)))
ACTIVE(isNePal(__(nil, z0))) → c8(ISNEPAL(mark(z0)), ACTIVE(__(nil, z0)))
ACTIVE(isNePal(isNePal(__(z0, __(z1, z0))))) → c8(ISNEPAL(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(active(z0), z1)), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(__(z0, z1))) → c8(ISNEPAL(__(z0, active(z1))), ACTIVE(__(z0, z1)))
ACTIVE(isNePal(and(z0, z1))) → c8(ISNEPAL(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(isNePal(isNePal(z0))) → c8(ISNEPAL(isNePal(active(z0))), ACTIVE(isNePal(z0)))
ACTIVE(isNePal(and(tt, z0))) → c8(ISNEPAL(mark(z0)))
PROPER(__(x0, __(z0, z1))) → c16(__'(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(__(x0, and(z0, z1))) → c16(__'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(__(x0, isNePal(z0))) → c16(__'(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(__(__(z0, z1), x1)) → c16(__'(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(__(and(z0, z1), x1)) → c16(__'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(__(isNePal(z0), x1)) → c16(__'(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(__(x0, nil)) → c16(__'(proper(x0), ok(nil)), PROPER(x0))
PROPER(__(x0, tt)) → c16(__'(proper(x0), ok(tt)), PROPER(x0))
PROPER(__(nil, x1)) → c16(__'(ok(nil), proper(x1)), PROPER(x1))
PROPER(__(tt, x1)) → c16(__'(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(x0, __(z0, z1))) → c18(AND(proper(x0), __(proper(z0), proper(z1))), PROPER(x0), PROPER(__(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c18(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNePal(z0))) → c18(AND(proper(x0), isNePal(proper(z0))), PROPER(x0), PROPER(isNePal(z0)))
PROPER(and(__(z0, z1), x1)) → c18(AND(__(proper(z0), proper(z1)), proper(x1)), PROPER(__(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c18(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNePal(z0), x1)) → c18(AND(isNePal(proper(z0)), proper(x1)), PROPER(isNePal(z0)), PROPER(x1))
PROPER(and(x0, nil)) → c18(AND(proper(x0), ok(nil)), PROPER(x0))
PROPER(and(x0, tt)) → c18(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(nil, x1)) → c18(AND(ok(nil), proper(x1)), PROPER(x1))
PROPER(and(tt, x1)) → c18(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(isNePal(__(z0, z1))) → c20(ISNEPAL(__(proper(z0), proper(z1))), PROPER(__(z0, z1)))
PROPER(isNePal(and(z0, z1))) → c20(ISNEPAL(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNePal(isNePal(z0))) → c20(ISNEPAL(isNePal(proper(z0))), PROPER(isNePal(z0)))
PROPER(isNePal(nil)) → c20(ISNEPAL(ok(nil)))
PROPER(isNePal(tt)) → c20(ISNEPAL(ok(tt)))
TOP(mark(__(z0, z1))) → c21(TOP(__(proper(z0), proper(z1))), PROPER(__(z0, z1)))
TOP(mark(and(z0, z1))) → c21(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNePal(z0))) → c21(TOP(isNePal(proper(z0))), PROPER(isNePal(z0)))
TOP(ok(__(__(z0, z1), z2))) → c22(TOP(mark(__(z0, __(z1, z2)))), ACTIVE(__(__(z0, z1), z2)))
TOP(ok(__(z0, nil))) → c22(TOP(mark(z0)), ACTIVE(__(z0, nil)))
TOP(ok(__(nil, z0))) → c22(TOP(mark(z0)), ACTIVE(__(nil, z0)))
TOP(ok(and(tt, z0))) → c22(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNePal(__(z0, __(z1, z0))))) → c22(TOP(mark(tt)), ACTIVE(isNePal(__(z0, __(z1, z0)))))
TOP(ok(__(z0, z1))) → c22(TOP(__(active(z0), z1)), ACTIVE(__(z0, z1)))
TOP(ok(__(z0, z1))) → c22(TOP(__(z0, active(z1))), ACTIVE(__(z0, z1)))
TOP(ok(and(z0, z1))) → c22(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
TOP(ok(isNePal(z0))) → c22(TOP(isNePal(active(z0))), ACTIVE(isNePal(z0)))

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
active(__(z0, z1)) → __(active(z0), z1)
active(__(z0, z1)) → __(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
active(isNePal(z0)) → isNePal(active(z0))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNePal(mark(z0)) → mark(isNePal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
proper(__(z0, z1)) → __(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(isNePal(z0)) → isNePal(proper(z0))

Tuples:

__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))

S tuples:

__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))

K tuples:none
Defined Rule Symbols:

active, __, and, isNePal, proper

Defined Pair Symbols:

__', AND, ISNEPAL

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15

(41) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

active(__(__(z0, z1), z2)) → mark(__(z0, __(z1, z2)))
active(__(z0, nil)) → mark(z0)
active(__(nil, z0)) → mark(z0)
active(and(tt, z0)) → mark(z0)
active(isNePal(__(z0, __(z1, z0)))) → mark(tt)
active(__(z0, z1)) → __(active(z0), z1)
active(__(z0, z1)) → __(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
active(isNePal(z0)) → isNePal(active(z0))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNePal(mark(z0)) → mark(isNePal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
proper(__(z0, z1)) → __(proper(z0), proper(z1))
proper(nil) → ok(nil)
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(isNePal(z0)) → isNePal(proper(z0))

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))

S tuples:

__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

__', AND, ISNEPAL

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15

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

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

__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))

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

__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))

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

POL(AND(x₁, x₂)) = [3]x₁
POL(ISNEPAL(x₁)) = [5]x₁
POL(__'(x₁, x₂)) = [4]x₁ + [4]x₂
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(c9(x₁)) = x₁
POL(mark(x₁)) = [5] + x₁
POL(ok(x₁)) = [5] + x₁

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))

S tuples:none
K tuples:

__'(mark(z0), z1) → c9(__'(z0, z1))
__'(z0, mark(z1)) → c10(__'(z0, z1))
__'(ok(z0), ok(z1)) → c11(__'(z0, z1))
AND(mark(z0), z1) → c12(AND(z0, z1))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
ISNEPAL(mark(z0)) → c14(ISNEPAL(z0))
ISNEPAL(ok(z0)) → c15(ISNEPAL(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

__', AND, ISNEPAL

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15

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

The set S is empty

(0) Obligation:

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

(2) Obligation:

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

(4) Obligation:

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

(6) Obligation:

(7) CdtUsableRulesProof (EQUIVALENT transformation)

(8) Obligation:

(9) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

(10) Obligation:

(11) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(12) Obligation:

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

(14) Obligation:

(15) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(16) Obligation:

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

(18) Obligation:

(19) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(20) Obligation:

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

(22) Obligation:

(23) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(24) Obligation:

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

(26) Obligation:

(27) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(28) Obligation:

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

(30) Obligation:

(31) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(32) Obligation:

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

(34) Obligation:

(35) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(36) Obligation:

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

(38) Obligation:

(39) CdtUnreachableProof (EQUIVALENT transformation)

(40) Obligation:

(41) CdtUsableRulesProof (EQUIVALENT transformation)

(42) Obligation:

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

(44) Obligation:

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

(46) BOUNDS(O(1), O(1))