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(and(tt, X)) → mark(X)
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))
active(and(X1, X2)) → and(active(X1), X2)
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(s(X)) → s(active(X))
and(mark(X1), X2) → mark(and(X1, X2))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
s(mark(X)) → mark(s(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(ok(X)) → ok(s(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(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, s(z1))) → c2(S(plus(z0, z1)), PLUS(z0, z1))
ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(plus(z0, s(z1))) → c2(S(plus(z0, z1)), PLUS(z0, z1))
ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))

S tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))

K tuples:none
Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

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

(5) 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)) → c19(TOP(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))

And the Tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))

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

POL(0) = [1]   
POL(ACTIVE(x₁)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(PLUS(x₁, x₂)) = 0   
POL(PROPER(x₁)) = 0   
POL(S(x₁)) = 0   
POL(TOP(x₁)) = [4]x₁   
POL(active(x₁)) = x₁   
POL(and(x₁, x₂)) = x₁ + x₂   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c16(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c18(x₁, x₂)) = x₁ + x₂   
POL(c19(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c20(x₁, x₂)) = x₁ + x₂   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(ok(x₁)) = x₁   
POL(plus(x₁, x₂)) = [2]x₁ + [2]x₂   
POL(proper(x₁)) = x₁   
POL(s(x₁)) = [4] + x₁   
POL(tt) = [1]   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))

S tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

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

(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))

And the Tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))

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

POL(0) = [1]   
POL(ACTIVE(x₁)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(PLUS(x₁, x₂)) = 0   
POL(PROPER(x₁)) = x₁   
POL(S(x₁)) = 0   
POL(TOP(x₁)) = [2]x₁²   
POL(active(x₁)) = x₁   
POL(and(x₁, x₂)) = [2]x₁ + x₂ + [2]x₁·x₂ + [2]x₁²   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c16(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c18(x₁, x₂)) = x₁ + x₂   
POL(c19(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c20(x₁, x₂)) = x₁ + x₂   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [2] + x₁   
POL(ok(x₁)) = x₁   
POL(plus(x₁, x₂)) = x₁ + x₂ + x₂²   
POL(proper(x₁)) = x₁   
POL(s(x₁)) = [2] + x₁   
POL(tt) = [1]   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))

S tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

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

(9) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

We considered the (Usable) Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))

And the Tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))

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

POL(0) = 0   
POL(ACTIVE(x₁)) = 0   
POL(AND(x₁, x₂)) = 0   
POL(PLUS(x₁, x₂)) = 0   
POL(PROPER(x₁)) = [1] + [2]x₁   
POL(S(x₁)) = 0   
POL(TOP(x₁)) = x₁²   
POL(active(x₁)) = x₁   
POL(and(x₁, x₂)) = [2] + x₁ + x₂   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c16(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(c18(x₁, x₂)) = x₁ + x₂   
POL(c19(x₁, x₂)) = x₁ + x₂   
POL(c2(x₁)) = x₁   
POL(c20(x₁, x₂)) = x₁ + x₂   
POL(c3(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [2] + x₁   
POL(ok(x₁)) = x₁   
POL(plus(x₁, x₂)) = [2] + x₁ + x₂ + [2]x₂² + x₁·x₂   
POL(proper(x₁)) = x₁   
POL(s(x₁)) = [2] + [2]x₁   
POL(tt) = 0   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))

S tuples:

ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace ACTIVE(and(z0, z1)) → c3(AND(active(z0), z1), ACTIVE(z0)) by

ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x0, x1)) → c3

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x0, x1)) → c3

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x0, x1)) → c3

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing nodes:

ACTIVE(and(x0, x1)) → c3

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing tuple parts

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0)) by

ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x0)) → c6

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x0)) → c6

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x0)) → c6

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing nodes:

ACTIVE(s(x0)) → c6

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing tuple parts

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, x1)) → c14

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, x1)) → c14

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
PROPER(and(z0, z1)) → c14(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing nodes:

PROPER(and(x0, x1)) → c14

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

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

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

Removed 4 trailing tuple parts

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(plus(x0, and(z0, z1))) → c16(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, tt)) → c16(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(plus(x0, plus(z0, z1))) → c16(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, 0)) → c16(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(plus(x0, s(z0))) → c16(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(and(z0, z1), x1)) → c16(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(tt, x1)) → c16(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c16(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(0, x1)) → c16(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(plus(s(z0), x1)) → c16(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x0, x1)) → c16

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c16(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, tt)) → c16(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(plus(x0, plus(z0, z1))) → c16(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, 0)) → c16(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(plus(x0, s(z0))) → c16(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(and(z0, z1), x1)) → c16(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(tt, x1)) → c16(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c16(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(0, x1)) → c16(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(plus(s(z0), x1)) → c16(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x0, x1)) → c16

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c16(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

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

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

Removed 1 trailing nodes:

PROPER(plus(x0, x1)) → c16

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c16(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, tt)) → c16(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(plus(x0, plus(z0, z1))) → c16(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, 0)) → c16(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(plus(x0, s(z0))) → c16(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(and(z0, z1), x1)) → c16(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(tt, x1)) → c16(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c16(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(0, x1)) → c16(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(plus(s(z0), x1)) → c16(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

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

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

Removed 4 trailing tuple parts

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c16(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, plus(z0, z1))) → c16(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, s(z0))) → c16(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(and(z0, z1), x1)) → c16(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c16(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c16(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c16(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c16(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c16(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c16(PLUS(ok(0), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, PROPER, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0)) by

PROPER(s(and(z0, z1))) → c18(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(tt)) → c18(S(ok(tt)), PROPER(tt))
PROPER(s(plus(z0, z1))) → c18(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(0)) → c18(S(ok(0)), PROPER(0))
PROPER(s(s(z0))) → c18(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(x0)) → c18

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c16(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, plus(z0, z1))) → c16(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, s(z0))) → c16(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(and(z0, z1), x1)) → c16(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c16(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c16(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c16(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c16(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c16(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c16(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(and(z0, z1))) → c18(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(tt)) → c18(S(ok(tt)), PROPER(tt))
PROPER(s(plus(z0, z1))) → c18(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(0)) → c18(S(ok(0)), PROPER(0))
PROPER(s(s(z0))) → c18(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(x0)) → c18

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c18(S(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, TOP, PROPER

Compound Symbols:

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

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

Removed 1 trailing nodes:

PROPER(s(x0)) → c18

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c16(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, plus(z0, z1))) → c16(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, s(z0))) → c16(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(and(z0, z1), x1)) → c16(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c16(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c16(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c16(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c16(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c16(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c16(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(and(z0, z1))) → c18(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(tt)) → c18(S(ok(tt)), PROPER(tt))
PROPER(s(plus(z0, z1))) → c18(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(0)) → c18(S(ok(0)), PROPER(0))
PROPER(s(s(z0))) → c18(S(s(proper(z0))), PROPER(s(z0)))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, TOP, PROPER

Compound Symbols:

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

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

Removed 2 trailing tuple parts

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(and(z0, z1)) → and(active(z0), z1)
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(s(z0)) → s(active(z0))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
proper(s(z0)) → s(proper(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c14(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c14(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c14(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(and(z0, z1), x1)) → c14(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c14(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c14(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c14(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c14(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c14(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c14(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c16(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, plus(z0, z1))) → c16(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, s(z0))) → c16(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(and(z0, z1), x1)) → c16(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c16(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c16(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c16(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c16(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c16(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c16(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(and(z0, z1))) → c18(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(plus(z0, z1))) → c18(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(s(z0))) → c18(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(tt)) → c18(S(ok(tt)))
PROPER(s(0)) → c18(S(ok(0)))

S tuples:

ACTIVE(plus(z0, z1)) → c4(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c5(PLUS(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c7(AND(z0, z1))
AND(ok(z0), ok(z1)) → c8(AND(z0, z1))
PLUS(mark(z0), z1) → c9(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c10(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c11(PLUS(z0, z1))
S(mark(z0)) → c12(S(z0))
S(ok(z0)) → c13(S(z0))
TOP(ok(z0)) → c20(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c3(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c3(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c3(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c3(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c3(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(and(tt, z0), x1)) → c3(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c6(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c6(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c6(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c6(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(and(tt, z0))) → c6(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c19(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, TOP, PROPER

Compound Symbols:

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

(41) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

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

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3, 4, 5, 6]
transitions:
tt0() → 0
mark0(0) → 0
00() → 0
ok0(0) → 0
active0(0) → 1
and0(0, 0) → 2
plus0(0, 0) → 3
s0(0) → 4
proper0(0) → 5
top0(0) → 6
and1(0, 0) → 7
mark1(7) → 2
plus1(0, 0) → 8
mark1(8) → 3
s1(0) → 9
mark1(9) → 4
tt1() → 10
ok1(10) → 5
01() → 11
ok1(11) → 5
and1(0, 0) → 12
ok1(12) → 2
plus1(0, 0) → 13
ok1(13) → 3
s1(0) → 14
ok1(14) → 4
proper1(0) → 15
top1(15) → 6
active1(0) → 16
top1(16) → 6
mark1(7) → 7
mark1(7) → 12
mark1(8) → 8
mark1(8) → 13
mark1(9) → 9
mark1(9) → 14
ok1(10) → 15
ok1(11) → 15
ok1(12) → 7
ok1(12) → 12
ok1(13) → 8
ok1(13) → 13
ok1(14) → 9
ok1(14) → 14
active2(10) → 17
top2(17) → 6
active2(11) → 17

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