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

(3) CdtPolyRedPairProof (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:

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))
isNePal(mark(z0)) → mark(isNePal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
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))

And the 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))

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)) = [1] + [4]x₁ + x₂   
POL(__'(x₁, x₂)) = 0   
POL(active(x₁)) = x₁   
POL(and(x₁, x₂)) = [2] + [2]x₁ + x₂   
POL(c(x₁, x₂)) = x₁ + 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(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)) = x₁   
POL(mark(x₁)) = [2] + x₁   
POL(nil) = [1]   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = x₁   
POL(tt) = 0   

(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(ok(z0)) → c22(TOP(active(z0)), ACTIVE(z0))

K tuples:

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

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) 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)))
ACTIVE(and(x0, x1)) → c7

(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), 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(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(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)))
ACTIVE(and(x0, x1)) → c7

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(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(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)))
ACTIVE(and(x0, x1)) → c7

K tuples:

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

Defined Rule Symbols:

active, __, and, isNePal, proper, top

Defined Pair Symbols:

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

Compound Symbols:

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

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

Removed 1 trailing nodes:

ACTIVE(and(x0, x1)) → c7

(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))
__(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(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(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), 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(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(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, top

Defined Pair Symbols:

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

Compound Symbols:

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

(9) 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)))
ACTIVE(isNePal(x0)) → c8

(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))
__(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))
__'(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(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)))
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)))
ACTIVE(isNePal(x0)) → c8

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))
__'(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(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)))
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)))
ACTIVE(isNePal(x0)) → c8

K tuples:

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

Defined Rule Symbols:

active, __, and, isNePal, proper, top

Defined Pair Symbols:

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

Compound Symbols:

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

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

Removed 1 trailing nodes:

ACTIVE(isNePal(x0)) → c8

(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))
__(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))
__'(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(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)))
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), 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))
__'(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(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)))
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, top

Defined Pair Symbols:

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

Compound Symbols:

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

(13) 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))
PROPER(__(x0, x1)) → c16

(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))
__(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))
__'(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(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)))
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)))
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))
PROPER(__(x0, x1)) → c16

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))
__'(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(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)))
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)))
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))
PROPER(__(x0, x1)) → c16

K tuples:

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

Defined Rule Symbols:

active, __, and, isNePal, proper, top

Defined Pair Symbols:

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

Compound Symbols:

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

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

Removed 1 trailing nodes:

PROPER(__(x0, x1)) → c16

(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))
__(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))
__'(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(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)))
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)))
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), 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))
__'(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(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)))
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)))
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, top

Defined Pair Symbols:

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

Compound Symbols:

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

(17) 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))
PROPER(and(x0, x1)) → c18

(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))
__(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))
__'(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(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)))
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)))
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))
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))
PROPER(and(x0, x1)) → c18

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))
__'(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(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)))
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)))
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))
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))
PROPER(and(x0, x1)) → c18

K tuples:

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

Defined Rule Symbols:

active, __, and, isNePal, proper, top

Defined Pair Symbols:

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

Compound Symbols:

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

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

Removed 1 trailing nodes:

PROPER(and(x0, x1)) → c18

(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))
__(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))
__'(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(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)))
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)))
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))
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), 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))
__'(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(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)))
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)))
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))
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, top

Defined Pair Symbols:

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

Compound Symbols:

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

(21) 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)))
PROPER(isNePal(x0)) → c20

(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))
__(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))
__'(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(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)))
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)))
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))
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))
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)))
PROPER(isNePal(x0)) → c20

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))
__'(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(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)))
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)))
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))
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))
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)))
PROPER(isNePal(x0)) → c20

K tuples:

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

Defined Rule Symbols:

active, __, and, isNePal, proper, top

Defined Pair Symbols:

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

Compound Symbols:

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

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

Removed 1 trailing nodes:

PROPER(isNePal(x0)) → c20

(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))
__(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))
__'(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(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)))
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)))
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))
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))
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), 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))
__'(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(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)))
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)))
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))
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))
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, top

Defined Pair Symbols:

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

Compound Symbols:

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

(25) 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)))
TOP(mark(x0)) → c21

(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))
__(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))
__'(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(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)))
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)))
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))
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))
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)))
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)))
TOP(mark(x0)) → c21

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))
__'(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(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)))
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)))
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))
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))
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, top

Defined Pair Symbols:

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

Compound Symbols:

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

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

Removed 1 trailing nodes:

TOP(mark(x0)) → c21

(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))
__(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))
__'(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(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)))
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)))
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))
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))
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)))
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), 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))
__'(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(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)))
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)))
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))
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))
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:none
Defined Rule Symbols:

active, __, and, isNePal, proper, top

Defined Pair Symbols:

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

Compound Symbols:

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

(29) 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)))
TOP(ok(x0)) → c22

(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))
__(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))
__'(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(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)))
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)))
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))
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))
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)))
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)))
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)))
TOP(ok(x0)) → c22

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))
__'(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(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)))
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)))
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))
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))
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)))
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)))
TOP(ok(x0)) → c22

K tuples:none
Defined Rule Symbols:

active, __, and, isNePal, proper, top

Defined Pair Symbols:

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

Compound Symbols:

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

(31) CdtPolyRedPairProof (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(ok(x0)) → c22

We considered the (Usable) 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))
isNePal(mark(z0)) → mark(isNePal(z0))
isNePal(ok(z0)) → ok(isNePal(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
__(z0, mark(z1)) → mark(__(z0, z1))
__(ok(z0), ok(z1)) → ok(__(z0, z1))
__(mark(z0), z1) → mark(__(z0, z1))
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))

And the 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))
__'(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(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)))
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)))
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))
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))
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)))
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)))
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)))
TOP(ok(x0)) → c22

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)) = [4]   
POL(__(x₁, x₂)) = [4]   
POL(__'(x₁, x₂)) = 0   
POL(active(x₁)) = [4]x₁   
POL(and(x₁, x₂)) = [2] + x₁ + [3]x₂   
POL(c(x₁, x₂)) = x₁ + 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(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) = 0   
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)) = 0   
POL(mark(x₁)) = 0   
POL(nil) = [3]   
POL(ok(x₁)) = 0   
POL(proper(x₁)) = [1] + [4]x₁   
POL(tt) = [1]   

(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))
__(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))
__'(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(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)))
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)))
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))
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))
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)))
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)))
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)))
TOP(ok(x0)) → c22

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))
__'(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(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)))
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)))
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))
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))
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)))
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(ok(x0)) → c22

Defined Rule Symbols:

active, __, and, isNePal, proper, top

Defined Pair Symbols:

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

Compound Symbols:

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

(33) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

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

The compatible tree automaton used to show the Match(-raise)-Boundedness (for constructor-based start-terms) is represented by:
final states : [6, 2, 3, 4, 5, 1, 17, 18, 19, 22, 23, 26, 33, 34, 35, 36, 37]
transitions:
mark0(0) → 0
ok0(0) → 0
top0(0) → 6
__1(0, 0) → 7
mark1(7) → 2
__1(0, 0) → 2
and1(0, 0) → 8
mark1(8) → 3
and1(0, 0) → 3
isNePal1(0) → 9
mark1(9) → 4
isNePal1(0) → 4
ok1(10) → 5
ok1(11) → 5
__1(0, 0) → 12
ok1(12) → 2
and1(0, 0) → 13
ok1(13) → 3
isNePal1(0) → 14
ok1(14) → 4
proper1(0) → 15
top1(15) → 6
proper1(0) → 5
active1(0) → 16
top1(16) → 6
active1(0) → 1
__1(0, 0) → 17
and1(0, 0) → 18
isNePal1(0) → 19
proper1(0) → 22
active1(0) → 23
mark0(20) → 0
mark0(21) → 0
ok0(20) → 0
ok0(21) → 0
top0(20) → 6
top0(21) → 6
mark1(17) → 2
mark1(18) → 3
mark1(19) → 4
__1(20, 0) → 17
__1(0, 20) → 17
__1(21, 0) → 17
__1(0, 21) → 17
__1(20, 20) → 17
__1(20, 21) → 17
__1(21, 20) → 17
__1(21, 21) → 17
and1(20, 0) → 18
and1(0, 20) → 18
and1(21, 0) → 18
and1(0, 21) → 18
and1(20, 20) → 18
and1(20, 21) → 18
and1(21, 20) → 18
and1(21, 21) → 18
isNePal1(20) → 19
isNePal1(21) → 19
ok1(17) → 2
ok1(18) → 3
ok1(19) → 4
ok1(20) → 5
ok1(21) → 5
top1(22) → 6
top1(23) → 6
proper1(20) → 22
proper1(21) → 22
active1(20) → 23
active1(21) → 23
ok1(20) → 0
ok1(21) → 0
mark1(7) → 17
__1(20, 0) → 7
__1(21, 0) → 7
__1(0, 20) → 7
__1(20, 20) → 7
__1(21, 20) → 7
__1(0, 21) → 7
__1(20, 21) → 7
__1(21, 21) → 7
mark1(8) → 18
and1(20, 0) → 8
and1(21, 0) → 8
and1(0, 20) → 8
and1(20, 20) → 8
and1(21, 20) → 8
and1(0, 21) → 8
and1(20, 21) → 8
and1(21, 21) → 8
mark1(9) → 19
isNePal1(20) → 9
isNePal1(21) → 9
nil2() → 24
ok2(24) → 22
nil2() → 10
nil2() → 0
nil2() → 20
tt2() → 25
ok2(25) → 22
tt2() → 11
tt2() → 0
tt2() → 21
ok1(12) → 17
__1(0, 20) → 12
__1(0, 21) → 12
__1(20, 0) → 12
__1(20, 20) → 12
__1(20, 21) → 12
__1(21, 0) → 12
__1(21, 20) → 12
__1(21, 21) → 12
ok1(13) → 18
and1(0, 20) → 13
and1(0, 21) → 13
and1(20, 0) → 13
and1(20, 20) → 13
and1(20, 21) → 13
and1(21, 0) → 13
and1(21, 20) → 13
and1(21, 21) → 13
ok1(14) → 19
isNePal1(20) → 14
isNePal1(21) → 14
ok1(20) → 26
ok1(21) → 26
nil2() → 27
tt2() → 28
mark0(26) → 0
mark0(27) → 0
mark0(28) → 0
ok0(26) → 0
ok0(27) → 0
ok0(28) → 0
top0(26) → 6
top0(27) → 6
top0(28) → 6
mark1(17) → 17
mark1(18) → 18
mark1(19) → 19
__1(26, 0) → 17
__1(26, 20) → 17
__1(26, 21) → 17
__1(0, 26) → 17
__1(20, 26) → 17
__1(21, 26) → 17
__1(27, 0) → 17
__1(27, 20) → 17
__1(27, 21) → 17
__1(0, 27) → 17
__1(20, 27) → 17
__1(21, 27) → 17
__1(28, 0) → 17
__1(28, 20) → 17
__1(28, 21) → 17
__1(0, 28) → 17
__1(20, 28) → 17
__1(21, 28) → 17
__1(26, 26) → 17
__1(26, 27) → 17
__1(26, 28) → 17
__1(27, 26) → 17
__1(28, 26) → 17
and1(26, 0) → 18
and1(26, 20) → 18
and1(26, 21) → 18
and1(0, 26) → 18
and1(20, 26) → 18
and1(21, 26) → 18
and1(27, 0) → 18
and1(27, 20) → 18
and1(27, 21) → 18
and1(0, 27) → 18
and1(20, 27) → 18
and1(21, 27) → 18
and1(28, 0) → 18
and1(28, 20) → 18
and1(28, 21) → 18
and1(0, 28) → 18
and1(20, 28) → 18
and1(21, 28) → 18
and1(26, 26) → 18
and1(26, 27) → 18
and1(26, 28) → 18
and1(27, 26) → 18
and1(28, 26) → 18
isNePal1(26) → 19
ok1(17) → 17
ok1(18) → 18
ok1(19) → 19
ok1(27) → 26
ok1(28) → 26
proper1(26) → 22
proper1(27) → 22
proper1(28) → 22
active1(26) → 23
ok2(27) → 22
ok2(28) → 22
ok1(27) → 0
ok2(27) → 0
ok1(28) → 0
ok2(28) → 0
ok2(27) → 26
ok2(28) → 26
__2(27, 27) → 29
ok2(29) → 17
__2(27, 27) → 17
__2(27, 28) → 29
__2(27, 28) → 17
__2(28, 27) → 29
__2(28, 27) → 17
__2(28, 28) → 29
__2(28, 28) → 17
and2(27, 27) → 30
ok2(30) → 18
and2(27, 27) → 18
and2(27, 28) → 30
and2(27, 28) → 18
and2(28, 27) → 30
and2(28, 27) → 18
and2(28, 28) → 30
and2(28, 28) → 18
isNePal2(27) → 31
ok2(31) → 19
isNePal2(27) → 19
isNePal2(28) → 31
isNePal2(28) → 19
active2(27) → 32
top2(32) → 6
active2(27) → 23
active2(28) → 32
active2(28) → 23
ok2(27) → 33
ok2(28) → 33
__2(27, 27) → 34
__2(27, 28) → 34
__2(28, 27) → 34
__2(28, 28) → 34
and2(27, 27) → 35
and2(27, 28) → 35
and2(28, 27) → 35
and2(28, 28) → 35
isNePal2(27) → 36
isNePal2(28) → 36
active2(27) → 37
active2(28) → 37
mark0(33) → 0
ok0(33) → 0
top0(33) → 6
mark1(34) → 17
mark1(35) → 18
mark1(36) → 19
__1(27, 27) → 17
__1(27, 28) → 17
__1(28, 27) → 17
__1(28, 28) → 17
__1(33, 0) → 17
__1(33, 20) → 17
__1(33, 21) → 17
__1(33, 26) → 17
__1(33, 27) → 17
__1(33, 28) → 17
__1(0, 33) → 17
__1(20, 33) → 17
__1(21, 33) → 17
__1(26, 33) → 17
__1(27, 33) → 17
__1(28, 33) → 17
__1(33, 33) → 17
and1(27, 27) → 18
and1(27, 28) → 18
and1(28, 27) → 18
and1(28, 28) → 18
and1(33, 0) → 18
and1(33, 20) → 18
and1(33, 21) → 18
and1(33, 26) → 18
and1(33, 27) → 18
and1(33, 28) → 18
and1(0, 33) → 18
and1(20, 33) → 18
and1(21, 33) → 18
and1(26, 33) → 18
and1(27, 33) → 18
and1(28, 33) → 18
and1(33, 33) → 18
isNePal1(27) → 19
isNePal1(28) → 19
isNePal1(33) → 19
ok1(34) → 17
ok1(35) → 18
ok1(36) → 19
top1(33) → 6
top1(37) → 6
proper1(33) → 22
active1(27) → 23
active1(28) → 23
active1(33) → 23
ok2(34) → 17
ok2(35) → 18
ok2(36) → 19
top2(37) → 6

(34) BOUNDS(O(1), O(n^1))