Runtime Complexity (innermost) proof of /tmp/tmpXiXpMF/MYNAT_nosorts

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS

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

↳2 CdtProblem

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

↳4 CdtProblem

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

↳6 CdtProblem

↳7 CdtUsableRulesProof (⇔, 22 ms)

↳8 CdtProblem

↳9 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))), 4380 ms)

↳10 CdtProblem

↳11 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))), 3530 ms)

↳12 CdtProblem

↳13 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))), 2625 ms)

↳14 CdtProblem

↳15 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))), 1712 ms)

↳16 CdtProblem

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

↳18 CdtProblem

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

↳20 CdtProblem

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

↳22 CdtProblem

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

↳24 CdtProblem

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

↳26 CdtProblem

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

↳28 CdtProblem

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

↳30 CdtProblem

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

↳32 CdtProblem

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

↳34 CdtProblem

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

↳36 CdtProblem

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

↳38 CdtProblem

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

↳40 CdtProblem

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

↳42 CdtProblem

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

↳44 CdtProblem

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

↳46 CdtProblem

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

↳48 CdtProblem

↳49 CdtUnreachableProof (⇔, 0 ms)

↳50 CdtProblem

↳51 CdtUsableRulesProof (⇔, 0 ms)

↳52 CdtProblem

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

↳54 CdtProblem

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

↳56 CdtProblem

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

↳58 CdtProblem

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

↳60 BOUNDS(O(1), O(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(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))
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))
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
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))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
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))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(ok(X)) → ok(s(X))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(and(tt, z0)) → c
ACTIVE(plus(z0, 0)) → c1
ACTIVE(plus(z0, s(z1))) → c2(S(plus(z0, z1)), PLUS(z0, z1))
ACTIVE(x(z0, 0)) → c3
ACTIVE(x(z0, s(z1))) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(tt) → c22
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(0) → c24
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(and(tt, z0)) → c
ACTIVE(plus(z0, 0)) → c1
ACTIVE(plus(z0, s(z1))) → c2(S(plus(z0, z1)), PLUS(z0, z1))
ACTIVE(x(z0, 0)) → c3
ACTIVE(x(z0, s(z1))) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(tt) → c22
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(0) → c24
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, and, plus, s, x, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, PROPER, TOP

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28

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

Removed 5 trailing nodes:

ACTIVE(and(tt, z0)) → c
ACTIVE(plus(z0, 0)) → c1
PROPER(0) → c24
PROPER(tt) → c22
ACTIVE(x(z0, 0)) → c3

(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(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0))
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))
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(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))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
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(x(z0, s(z1))) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(plus(z0, s(z1))) → c2(S(plus(z0, z1)), PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, and, plus, s, x, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, PROPER, TOP

Compound Symbols:

c2, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c23, c25, c26, c27, c28

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

Removed 2 trailing tuple parts

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

Tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

S tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

K tuples:none
Defined Rule Symbols:

active, and, plus, s, x, proper, top

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c23, c25, c26, c27, c28, c2, c4

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

S tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

K tuples:none
Defined Rule Symbols:

active, and, plus, s, x, proper

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c23, c25, c26, c27, c28, c2, c4

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

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

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

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

POL(0) = [2]
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)) = [2]x₁
POL(X(x₁, x₂)) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = [2]x₁·x₂ + x₁²
POL(c10(x₁, x₂)) = x₁ + x₂
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c23(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c25(x₁, x₂)) = x₁ + x₂
POL(c26(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c27(x₁, x₂)) = x₁ + x₂
POL(c28(x₁, x₂)) = x₁ + x₂
POL(c4(x₁)) = x₁
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c6(x₁, x₂)) = x₁ + x₂
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂)) = x₁ + x₂
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁
POL(plus(x₁, x₂)) = x₁ + [2]x₂
POL(proper(x₁)) = x₁
POL(s(x₁)) = [2] + x₁
POL(tt) = [1]
POL(x(x₁, x₂)) = [2]x₁ + [2]x₂ + x₁·x₂

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

Tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

S tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, and, plus, s, x, proper

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c23, c25, c26, c27, c28, c2, c4

(11) CdtRuleRemovalProof (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)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(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)) = x₁²
POL(X(x₁, x₂)) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = [2]x₁ + [2]x₂ + x₁²
POL(c10(x₁, x₂)) = x₁ + x₂
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c23(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c25(x₁, x₂)) = x₁ + x₂
POL(c26(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c27(x₁, x₂)) = x₁ + x₂
POL(c28(x₁, x₂)) = x₁ + x₂
POL(c4(x₁)) = x₁
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c6(x₁, x₂)) = x₁ + x₂
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂)) = x₁ + x₂
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁
POL(plus(x₁, x₂)) = x₁ + [2]x₂
POL(proper(x₁)) = x₁
POL(s(x₁)) = [2] + x₁
POL(tt) = [2]
POL(x(x₁, x₂)) = [2] + [2]x₁ + x₂ + x₁·x₂

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

Tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

S tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, x, proper

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c23, c25, c26, c27, c28, c2, c4

(13) CdtRuleRemovalProof (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(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

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

POL(0) = [2]
POL(ACTIVE(x₁)) = 0
POL(AND(x₁, x₂)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(PROPER(x₁)) = [3]x₁
POL(S(x₁)) = 0
POL(TOP(x₁)) = x₁ + [2]x₁²
POL(X(x₁, x₂)) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(c10(x₁, x₂)) = x₁ + x₂
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c23(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c25(x₁, x₂)) = x₁ + x₂
POL(c26(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c27(x₁, x₂)) = x₁ + x₂
POL(c28(x₁, x₂)) = x₁ + x₂
POL(c4(x₁)) = x₁
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c6(x₁, x₂)) = x₁ + x₂
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂)) = x₁ + x₂
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁
POL(plus(x₁, x₂)) = [3] + x₁ + [2]x₂
POL(proper(x₁)) = x₁
POL(s(x₁)) = [3] + x₁
POL(tt) = [1]
POL(x(x₁, x₂)) = [2]x₁ + x₂ + x₂² + [2]x₁·x₂

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

Tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

S tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, x, proper

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c23, c25, c26, c27, c28, c2, c4

(15) CdtRuleRemovalProof (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)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

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

POL(0) = [2]
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)) = x₁²
POL(X(x₁, x₂)) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = [1] + x₁ + x₂
POL(c10(x₁, x₂)) = x₁ + x₂
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c23(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c25(x₁, x₂)) = x₁ + x₂
POL(c26(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c27(x₁, x₂)) = x₁ + x₂
POL(c28(x₁, x₂)) = x₁ + x₂
POL(c4(x₁)) = x₁
POL(c5(x₁, x₂)) = x₁ + x₂
POL(c6(x₁, x₂)) = x₁ + x₂
POL(c7(x₁, x₂)) = x₁ + x₂
POL(c8(x₁, x₂)) = x₁ + x₂
POL(c9(x₁, x₂)) = x₁ + x₂
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁
POL(plus(x₁, x₂)) = x₁ + [2]x₂
POL(proper(x₁)) = x₁
POL(s(x₁)) = [1] + x₁
POL(tt) = 0
POL(x(x₁, x₂)) = x₁ + [2]x₂ + [2]x₁·x₂

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

Tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

S tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, x, proper

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c23, c25, c26, c27, c28, c2, c4

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

Use narrowing to replace ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0)) by

ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))

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

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))

S tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, x, proper

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, PROPER, TOP

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c23, c25, c26, c27, c28, c2, c4, c5

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

Removed 1 trailing tuple parts

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

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))

S tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, x, proper

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, PROPER, TOP

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c23, c25, c26, c27, c28, c2, c4, c5, c5

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

Use narrowing to replace ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) by

ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))

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

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))

S tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, x, proper

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, PROPER, TOP

Compound Symbols:

c6, c7, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c23, c25, c26, c27, c28, c2, c4, c5, c5, c8

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

Removed 1 trailing tuple parts

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

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

S tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, x, proper

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, PROPER, TOP

Compound Symbols:

c6, c7, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c23, c25, c26, c27, c28, c2, c4, c5, c5, c8, c8

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

Use narrowing to replace PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))

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

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))

S tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, x, proper

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, PROPER, TOP

Compound Symbols:

c6, c7, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c23, c25, c26, c27, c28, c2, c4, c5, c5, c8, c8, c21

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

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, x, proper

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, PROPER, TOP

Compound Symbols:

c6, c7, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c23, c25, c26, c27, c28, c2, c4, c5, c5, c8, c8, c21, c21

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

Use narrowing to replace PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(plus(x0, and(z0, z1))) → c23(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, tt)) → c23(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(plus(x0, plus(z0, z1))) → c23(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, 0)) → c23(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(plus(x0, s(z0))) → c23(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, x(z0, z1))) → c23(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(and(z0, z1), x1)) → c23(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(tt, x1)) → c23(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c23(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(0, x1)) → c23(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(plus(s(z0), x1)) → c23(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c23(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))

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

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c23(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, tt)) → c23(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(plus(x0, plus(z0, z1))) → c23(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, 0)) → c23(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(plus(x0, s(z0))) → c23(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, x(z0, z1))) → c23(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(and(z0, z1), x1)) → c23(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(tt, x1)) → c23(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c23(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(0, x1)) → c23(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(plus(s(z0), x1)) → c23(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c23(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))

S tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, x, proper

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, PROPER, TOP

Compound Symbols:

c6, c7, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c25, c26, c27, c28, c2, c4, c5, c5, c8, c8, c21, c21, c23

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

Removed 4 trailing tuple parts

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

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c23(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, plus(z0, z1))) → c23(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, s(z0))) → c23(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, x(z0, z1))) → c23(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(and(z0, z1), x1)) → c23(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c23(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c23(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c23(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c23(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c23(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c23(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c23(PLUS(ok(0), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, x, proper

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, PROPER, TOP

Compound Symbols:

c6, c7, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c25, c26, c27, c28, c2, c4, c5, c5, c8, c8, c21, c21, c23, c23

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

Use narrowing to replace PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) by

PROPER(s(and(z0, z1))) → c25(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(tt)) → c25(S(ok(tt)), PROPER(tt))
PROPER(s(plus(z0, z1))) → c25(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(0)) → c25(S(ok(0)), PROPER(0))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(x(z0, z1))) → c25(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))

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

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c23(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, plus(z0, z1))) → c23(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, s(z0))) → c23(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, x(z0, z1))) → c23(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(and(z0, z1), x1)) → c23(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c23(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c23(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c23(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c23(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c23(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c23(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c23(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(and(z0, z1))) → c25(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(tt)) → c25(S(ok(tt)), PROPER(tt))
PROPER(s(plus(z0, z1))) → c25(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(0)) → c25(S(ok(0)), PROPER(0))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(x(z0, z1))) → c25(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))

S tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, x, proper

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, PROPER, TOP

Compound Symbols:

c6, c7, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c26, c27, c28, c2, c4, c5, c5, c8, c8, c21, c21, c23, c23, c25

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

Removed 2 trailing tuple parts

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

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c23(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, plus(z0, z1))) → c23(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, s(z0))) → c23(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, x(z0, z1))) → c23(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(and(z0, z1), x1)) → c23(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c23(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c23(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c23(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c23(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c23(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c23(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c23(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(and(z0, z1))) → c25(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(plus(z0, z1))) → c25(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(x(z0, z1))) → c25(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c25(S(ok(tt)))
PROPER(s(0)) → c25(S(ok(0)))

S tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, x, proper

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, PROPER, TOP

Compound Symbols:

c6, c7, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c26, c27, c28, c2, c4, c5, c5, c8, c8, c21, c21, c23, c23, c25, c25

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

Use narrowing to replace PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(x(x0, and(z0, z1))) → c26(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, tt)) → c26(X(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(x(x0, plus(z0, z1))) → c26(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, 0)) → c26(X(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(x(x0, s(z0))) → c26(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, x(z0, z1))) → c26(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(and(z0, z1), x1)) → c26(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(tt, x1)) → c26(X(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c26(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(0, x1)) → c26(X(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(x(s(z0), x1)) → c26(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c26(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))

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

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c23(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, plus(z0, z1))) → c23(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, s(z0))) → c23(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, x(z0, z1))) → c23(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(and(z0, z1), x1)) → c23(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c23(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c23(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c23(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c23(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c23(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c23(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c23(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(and(z0, z1))) → c25(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(plus(z0, z1))) → c25(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(x(z0, z1))) → c25(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c25(S(ok(tt)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(x(x0, and(z0, z1))) → c26(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, tt)) → c26(X(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(x(x0, plus(z0, z1))) → c26(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, 0)) → c26(X(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(x(x0, s(z0))) → c26(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, x(z0, z1))) → c26(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(and(z0, z1), x1)) → c26(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(tt, x1)) → c26(X(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c26(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(0, x1)) → c26(X(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(x(s(z0), x1)) → c26(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c26(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))

S tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, x, proper

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, TOP, PROPER

Compound Symbols:

c6, c7, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c27, c28, c2, c4, c5, c5, c8, c8, c21, c21, c23, c23, c25, c25, c26

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

Removed 4 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(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0))
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))
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
x(mark(z0), z1) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
x(z0, mark(z1)) → mark(x(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))
proper(x(z0, z1)) → x(proper(z0), proper(z1))

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c23(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, plus(z0, z1))) → c23(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, s(z0))) → c23(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, x(z0, z1))) → c23(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(and(z0, z1), x1)) → c23(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c23(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c23(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c23(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c23(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c23(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c23(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c23(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(and(z0, z1))) → c25(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(plus(z0, z1))) → c25(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(x(z0, z1))) → c25(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c25(S(ok(tt)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(x(x0, and(z0, z1))) → c26(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, plus(z0, z1))) → c26(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, s(z0))) → c26(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, x(z0, z1))) → c26(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(and(z0, z1), x1)) → c26(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c26(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(s(z0), x1)) → c26(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c26(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(x0, tt)) → c26(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c26(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c26(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c26(X(ok(0), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, x, proper

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, TOP, PROPER

Compound Symbols:

c6, c7, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c27, c28, c2, c4, c5, c5, c8, c8, c21, c21, c23, c23, c25, c25, c26, c26

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

Use narrowing to replace TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(and(z0, z1))) → c27(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(tt)) → c27(TOP(ok(tt)), PROPER(tt))
TOP(mark(plus(z0, z1))) → c27(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(0)) → c27(TOP(ok(0)), PROPER(0))
TOP(mark(s(z0))) → c27(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(x(z0, z1))) → c27(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))

(42) 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(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0))
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))
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
x(mark(z0), z1) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
x(z0, mark(z1)) → mark(x(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))
proper(x(z0, z1)) → x(proper(z0), proper(z1))

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c23(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, plus(z0, z1))) → c23(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, s(z0))) → c23(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, x(z0, z1))) → c23(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(and(z0, z1), x1)) → c23(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c23(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c23(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c23(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c23(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c23(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c23(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c23(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(and(z0, z1))) → c25(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(plus(z0, z1))) → c25(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(x(z0, z1))) → c25(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c25(S(ok(tt)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(x(x0, and(z0, z1))) → c26(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, plus(z0, z1))) → c26(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, s(z0))) → c26(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, x(z0, z1))) → c26(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(and(z0, z1), x1)) → c26(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c26(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(s(z0), x1)) → c26(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c26(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(x0, tt)) → c26(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c26(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c26(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c26(X(ok(0), proper(x1)), PROPER(x1))
TOP(mark(and(z0, z1))) → c27(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(tt)) → c27(TOP(ok(tt)), PROPER(tt))
TOP(mark(plus(z0, z1))) → c27(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(0)) → c27(TOP(ok(0)), PROPER(0))
TOP(mark(s(z0))) → c27(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(x(z0, z1))) → c27(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))

S tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, x, proper

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, TOP, PROPER

Compound Symbols:

c6, c7, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c28, c2, c4, c5, c5, c8, c8, c21, c21, c23, c23, c25, c25, c26, c26, c27

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

Removed 2 trailing tuple parts

(44) 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(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0))
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))
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
x(mark(z0), z1) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
x(z0, mark(z1)) → mark(x(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))
proper(x(z0, z1)) → x(proper(z0), proper(z1))

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c23(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, plus(z0, z1))) → c23(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, s(z0))) → c23(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, x(z0, z1))) → c23(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(and(z0, z1), x1)) → c23(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c23(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c23(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c23(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c23(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c23(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c23(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c23(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(and(z0, z1))) → c25(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(plus(z0, z1))) → c25(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(x(z0, z1))) → c25(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c25(S(ok(tt)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(x(x0, and(z0, z1))) → c26(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, plus(z0, z1))) → c26(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, s(z0))) → c26(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, x(z0, z1))) → c26(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(and(z0, z1), x1)) → c26(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c26(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(s(z0), x1)) → c26(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c26(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(x0, tt)) → c26(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c26(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c26(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c26(X(ok(0), proper(x1)), PROPER(x1))
TOP(mark(and(z0, z1))) → c27(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(plus(z0, z1))) → c27(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(s(z0))) → c27(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(x(z0, z1))) → c27(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(mark(tt)) → c27(TOP(ok(tt)))
TOP(mark(0)) → c27(TOP(ok(0)))

S tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, x, proper

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, TOP, PROPER

Compound Symbols:

c6, c7, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c28, c2, c4, c5, c5, c8, c8, c21, c21, c23, c23, c25, c25, c26, c26, c27, c27

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

Use narrowing to replace TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(and(tt, z0))) → c28(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(plus(z0, 0))) → c28(TOP(mark(z0)), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c28(TOP(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
TOP(ok(x(z0, 0))) → c28(TOP(mark(0)), ACTIVE(x(z0, 0)))
TOP(ok(x(z0, s(z1)))) → c28(TOP(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
TOP(ok(and(z0, z1))) → c28(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
TOP(ok(plus(z0, z1))) → c28(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c28(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(s(z0))) → c28(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(x(z0, z1))) → c28(TOP(x(active(z0), z1)), ACTIVE(x(z0, z1)))
TOP(ok(x(z0, z1))) → c28(TOP(x(z0, active(z1))), ACTIVE(x(z0, z1)))

(46) 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(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0))
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))
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
x(mark(z0), z1) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
x(z0, mark(z1)) → mark(x(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))
proper(x(z0, z1)) → x(proper(z0), proper(z1))

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c23(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, plus(z0, z1))) → c23(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, s(z0))) → c23(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, x(z0, z1))) → c23(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(and(z0, z1), x1)) → c23(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c23(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c23(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c23(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c23(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c23(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c23(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c23(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(and(z0, z1))) → c25(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(plus(z0, z1))) → c25(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(x(z0, z1))) → c25(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c25(S(ok(tt)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(x(x0, and(z0, z1))) → c26(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, plus(z0, z1))) → c26(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, s(z0))) → c26(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, x(z0, z1))) → c26(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(and(z0, z1), x1)) → c26(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c26(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(s(z0), x1)) → c26(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c26(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(x0, tt)) → c26(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c26(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c26(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c26(X(ok(0), proper(x1)), PROPER(x1))
TOP(mark(and(z0, z1))) → c27(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(plus(z0, z1))) → c27(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(s(z0))) → c27(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(x(z0, z1))) → c27(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(mark(tt)) → c27(TOP(ok(tt)))
TOP(mark(0)) → c27(TOP(ok(0)))
TOP(ok(and(tt, z0))) → c28(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(plus(z0, 0))) → c28(TOP(mark(z0)), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c28(TOP(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
TOP(ok(x(z0, 0))) → c28(TOP(mark(0)), ACTIVE(x(z0, 0)))
TOP(ok(x(z0, s(z1)))) → c28(TOP(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
TOP(ok(and(z0, z1))) → c28(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
TOP(ok(plus(z0, z1))) → c28(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c28(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(s(z0))) → c28(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(x(z0, z1))) → c28(TOP(x(active(z0), z1)), ACTIVE(x(z0, z1)))
TOP(ok(x(z0, z1))) → c28(TOP(x(z0, active(z1))), ACTIVE(x(z0, z1)))

S tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))
TOP(ok(and(tt, z0))) → c28(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(plus(z0, 0))) → c28(TOP(mark(z0)), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c28(TOP(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
TOP(ok(x(z0, 0))) → c28(TOP(mark(0)), ACTIVE(x(z0, 0)))
TOP(ok(x(z0, s(z1)))) → c28(TOP(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
TOP(ok(and(z0, z1))) → c28(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
TOP(ok(plus(z0, z1))) → c28(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c28(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(s(z0))) → c28(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(x(z0, z1))) → c28(TOP(x(active(z0), z1)), ACTIVE(x(z0, z1)))
TOP(ok(x(z0, z1))) → c28(TOP(x(z0, active(z1))), ACTIVE(x(z0, z1)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0))
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0))
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

active, and, plus, s, x, proper

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, PROPER, TOP

Compound Symbols:

c6, c7, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c2, c4, c5, c5, c8, c8, c21, c21, c23, c23, c25, c25, c26, c26, c27, c27, c28

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

Removed 2 trailing nodes:

TOP(mark(tt)) → c27(TOP(ok(tt)))
TOP(mark(0)) → c27(TOP(ok(0)))

(48) 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(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0))
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))
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
x(mark(z0), z1) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
x(z0, mark(z1)) → mark(x(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))
proper(x(z0, z1)) → x(proper(z0), proper(z1))

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c23(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, plus(z0, z1))) → c23(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, s(z0))) → c23(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, x(z0, z1))) → c23(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(and(z0, z1), x1)) → c23(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c23(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c23(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c23(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c23(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c23(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c23(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c23(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(and(z0, z1))) → c25(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(plus(z0, z1))) → c25(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(x(z0, z1))) → c25(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c25(S(ok(tt)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(x(x0, and(z0, z1))) → c26(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, plus(z0, z1))) → c26(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, s(z0))) → c26(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, x(z0, z1))) → c26(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(and(z0, z1), x1)) → c26(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c26(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(s(z0), x1)) → c26(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c26(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(x0, tt)) → c26(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c26(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c26(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c26(X(ok(0), proper(x1)), PROPER(x1))
TOP(mark(and(z0, z1))) → c27(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(plus(z0, z1))) → c27(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(s(z0))) → c27(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(x(z0, z1))) → c27(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(ok(and(tt, z0))) → c28(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(plus(z0, 0))) → c28(TOP(mark(z0)), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c28(TOP(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
TOP(ok(x(z0, 0))) → c28(TOP(mark(0)), ACTIVE(x(z0, 0)))
TOP(ok(x(z0, s(z1)))) → c28(TOP(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
TOP(ok(and(z0, z1))) → c28(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
TOP(ok(plus(z0, z1))) → c28(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c28(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(s(z0))) → c28(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(x(z0, z1))) → c28(TOP(x(active(z0), z1)), ACTIVE(x(z0, z1)))
TOP(ok(x(z0, z1))) → c28(TOP(x(z0, active(z1))), ACTIVE(x(z0, z1)))

S tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))
TOP(ok(and(tt, z0))) → c28(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(plus(z0, 0))) → c28(TOP(mark(z0)), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c28(TOP(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
TOP(ok(x(z0, 0))) → c28(TOP(mark(0)), ACTIVE(x(z0, 0)))
TOP(ok(x(z0, s(z1)))) → c28(TOP(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
TOP(ok(and(z0, z1))) → c28(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
TOP(ok(plus(z0, z1))) → c28(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c28(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(s(z0))) → c28(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(x(z0, z1))) → c28(TOP(x(active(z0), z1)), ACTIVE(x(z0, z1)))
TOP(ok(x(z0, z1))) → c28(TOP(x(z0, active(z1))), ACTIVE(x(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, and, plus, s, x, proper

Defined Pair Symbols:

ACTIVE, AND, PLUS, S, X, PROPER, TOP

Compound Symbols:

c6, c7, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c2, c4, c5, c5, c8, c8, c21, c21, c23, c23, c25, c25, c26, c26, c27, c28

(49) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1))
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(plus(x0, and(z0, z1))) → c23(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, plus(z0, z1))) → c23(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, s(z0))) → c23(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, x(z0, z1))) → c23(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(and(z0, z1), x1)) → c23(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c23(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c23(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c23(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c23(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c23(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c23(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c23(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(s(and(z0, z1))) → c25(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(plus(z0, z1))) → c25(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(x(z0, z1))) → c25(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(tt)) → c25(S(ok(tt)))
PROPER(s(0)) → c25(S(ok(0)))
PROPER(x(x0, and(z0, z1))) → c26(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, plus(z0, z1))) → c26(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, s(z0))) → c26(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, x(z0, z1))) → c26(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(and(z0, z1), x1)) → c26(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c26(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(s(z0), x1)) → c26(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c26(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(x0, tt)) → c26(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c26(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c26(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c26(X(ok(0), proper(x1)), PROPER(x1))
TOP(mark(and(z0, z1))) → c27(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(plus(z0, z1))) → c27(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(s(z0))) → c27(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(x(z0, z1))) → c27(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(ok(and(tt, z0))) → c28(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(plus(z0, 0))) → c28(TOP(mark(z0)), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c28(TOP(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1))))
TOP(ok(x(z0, 0))) → c28(TOP(mark(0)), ACTIVE(x(z0, 0)))
TOP(ok(x(z0, s(z1)))) → c28(TOP(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1))))
TOP(ok(and(z0, z1))) → c28(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
TOP(ok(plus(z0, z1))) → c28(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c28(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(s(z0))) → c28(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(x(z0, z1))) → c28(TOP(x(active(z0), z1)), ACTIVE(x(z0, z1)))
TOP(ok(x(z0, z1))) → c28(TOP(x(z0, active(z1))), ACTIVE(x(z0, z1)))

(50) 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(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0))
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))
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
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(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
x(mark(z0), z1) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
x(z0, mark(z1)) → mark(x(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))
proper(x(z0, z1)) → x(proper(z0), proper(z1))

Tuples:

AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))

S tuples:

AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))

K tuples:none
Defined Rule Symbols:

active, and, plus, s, x, proper

Defined Pair Symbols:

AND, PLUS, S, X

Compound Symbols:

c11, c12, c13, c14, c15, c16, c17, c18, c19, c20

(51) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))

S tuples:

AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

AND, PLUS, S, X

Compound Symbols:

c11, c12, c13, c14, c15, c16, c17, c18, c19, c20

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

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

AND(mark(z0), z1) → c11(AND(z0, z1))
S(mark(z0)) → c16(S(z0))
X(z0, mark(z1)) → c19(X(z0, z1))

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

AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))

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

POL(AND(x₁, x₂)) = [4]x₁
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = x₁
POL(X(x₁, x₂)) = x₂
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))

S tuples:

AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))

K tuples:

AND(mark(z0), z1) → c11(AND(z0, z1))
S(mark(z0)) → c16(S(z0))
X(z0, mark(z1)) → c19(X(z0, z1))

Defined Rule Symbols:none

Defined Pair Symbols:

AND, PLUS, S, X

Compound Symbols:

c11, c12, c13, c14, c15, c16, c17, c18, c19, c20

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

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

AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))

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

AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))

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

POL(AND(x₁, x₂)) = [2]x₁ + x₂
POL(PLUS(x₁, x₂)) = [3]x₁
POL(S(x₁)) = [5]x₁
POL(X(x₁, x₂)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(mark(x₁)) = [5] + x₁
POL(ok(x₁)) = [1] + x₁

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))

S tuples:

PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))

K tuples:

AND(mark(z0), z1) → c11(AND(z0, z1))
S(mark(z0)) → c16(S(z0))
X(z0, mark(z1)) → c19(X(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))

Defined Rule Symbols:none

Defined Pair Symbols:

AND, PLUS, S, X

Compound Symbols:

c11, c12, c13, c14, c15, c16, c17, c18, c19, c20

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

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

PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))

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

AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))

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

POL(AND(x₁, x₂)) = x₁ + [4]x₂
POL(PLUS(x₁, x₂)) = [2]x₁ + [2]x₂
POL(S(x₁)) = [5]x₁
POL(X(x₁, x₂)) = [4]x₁ + [4]x₂
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

AND(mark(z0), z1) → c11(AND(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(mark(z0)) → c16(S(z0))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(z0, mark(z1)) → c19(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))

S tuples:none
K tuples:

AND(mark(z0), z1) → c11(AND(z0, z1))
S(mark(z0)) → c16(S(z0))
X(z0, mark(z1)) → c19(X(z0, z1))
AND(ok(z0), ok(z1)) → c12(AND(z0, z1))
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))
S(ok(z0)) → c17(S(z0))
X(mark(z0), z1) → c18(X(z0, z1))
X(ok(z0), ok(z1)) → c20(X(z0, z1))
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))

Defined Rule Symbols:none

Defined Pair Symbols:

AND, PLUS, S, X

Compound Symbols:

c11, c12, c13, c14, c15, c16, c17, c18, c19, c20

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

The set S is empty

(0) Obligation:

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

(2) Obligation:

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

(4) Obligation:

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

(6) Obligation:

(7) CdtUsableRulesProof (EQUIVALENT transformation)

(8) Obligation:

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

(10) Obligation:

(11) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))) transformation)

(12) Obligation:

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

(14) Obligation:

(15) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))) transformation)

(16) Obligation:

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

(18) Obligation:

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

(20) Obligation:

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

(22) Obligation:

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

(24) Obligation:

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

(26) Obligation:

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

(28) Obligation:

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

(30) Obligation:

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

(32) Obligation:

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

(34) Obligation:

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

(36) Obligation:

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

(38) Obligation:

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

(40) Obligation:

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

(42) Obligation:

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

(44) Obligation:

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

(46) Obligation:

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

(48) Obligation:

(49) CdtUnreachableProof (EQUIVALENT transformation)

(50) Obligation:

(51) CdtUsableRulesProof (EQUIVALENT transformation)

(52) Obligation:

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

(54) Obligation:

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

(56) Obligation:

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

(58) Obligation:

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

(60) BOUNDS(O(1), O(1))