186.70/62.51 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml

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The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^3)).

0 CpxTRS

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

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↳2 CdtProblem

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

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↳4 CdtProblem

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↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))

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↳6 CdtProblem

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

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↳8 CdtProblem

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

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↳10 CdtProblem

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

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↳12 CdtProblem

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

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↳14 CdtProblem

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

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↳16 CdtProblem

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

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↳18 CdtProblem

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

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↳20 CdtProblem

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

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↳22 CdtProblem

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

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↳24 CdtProblem

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

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↳26 CdtProblem

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

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↳28 CdtProblem

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

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↳30 CdtProblem

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

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↳32 CdtProblem

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

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↳34 CdtProblem

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

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↳36 CdtProblem

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↳37 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))

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↳38 CdtProblem

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↳39 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

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↳40 CdtProblem

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↳41 CdtUnreachableProof (⇔)

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↳42 CdtProblem

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

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↳44 CdtProblem

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↳45 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))

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↳46 CdtProblem

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↳47 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))))

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↳48 CdtProblem

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↳49 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳50 CdtProblem

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↳51 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳52 CdtProblem

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↳53 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳54 CdtProblem

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↳55 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))

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↳56 CdtProblem

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↳57 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))

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↳58 CdtProblem

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

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↳60 BOUNDS(O(1), O(1))

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(0) Obligation:

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

active(and(tt, X)) → mark(X) 186.70/62.51
active(plus(N, 0)) → mark(N) 186.70/62.51
active(plus(N, s(M))) → mark(s(plus(N, M))) 186.70/62.51
active(x(N, 0)) → mark(0) 186.70/62.51
active(x(N, s(M))) → mark(plus(x(N, M), N)) 186.70/62.51
active(and(X1, X2)) → and(active(X1), X2) 186.70/62.51
active(plus(X1, X2)) → plus(active(X1), X2) 186.70/62.51
active(plus(X1, X2)) → plus(X1, active(X2)) 186.70/62.51
active(s(X)) → s(active(X)) 186.70/62.51
active(x(X1, X2)) → x(active(X1), X2) 186.70/62.51
active(x(X1, X2)) → x(X1, active(X2)) 186.70/62.51
and(mark(X1), X2) → mark(and(X1, X2)) 186.70/62.51
plus(mark(X1), X2) → mark(plus(X1, X2)) 186.70/62.51
plus(X1, mark(X2)) → mark(plus(X1, X2)) 186.70/62.51
s(mark(X)) → mark(s(X)) 186.70/62.51
x(mark(X1), X2) → mark(x(X1, X2)) 186.70/62.51
x(X1, mark(X2)) → mark(x(X1, X2)) 186.70/62.51
proper(and(X1, X2)) → and(proper(X1), proper(X2)) 186.70/62.51
proper(tt) → ok(tt) 186.70/62.51
proper(plus(X1, X2)) → plus(proper(X1), proper(X2)) 186.70/62.51
proper(0) → ok(0) 186.70/62.51
proper(s(X)) → s(proper(X)) 186.70/62.51
proper(x(X1, X2)) → x(proper(X1), proper(X2)) 186.70/62.51
and(ok(X1), ok(X2)) → ok(and(X1, X2)) 186.70/62.51
plus(ok(X1), ok(X2)) → ok(plus(X1, X2)) 186.70/62.51
s(ok(X)) → ok(s(X)) 186.70/62.51
x(ok(X1), ok(X2)) → ok(x(X1, X2)) 186.70/62.51
top(mark(X)) → top(proper(X)) 186.70/62.51
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

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(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 186.70/62.51
active(plus(z0, 0)) → mark(z0) 186.70/62.51
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 186.70/62.51
active(x(z0, 0)) → mark(0) 186.70/62.51
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 186.70/62.51
active(and(z0, z1)) → and(active(z0), z1) 186.70/62.51
active(plus(z0, z1)) → plus(active(z0), z1) 186.70/62.51
active(plus(z0, z1)) → plus(z0, active(z1)) 186.70/62.51
active(s(z0)) → s(active(z0)) 186.70/62.51
active(x(z0, z1)) → x(active(z0), z1) 186.70/62.51
active(x(z0, z1)) → x(z0, active(z1)) 186.70/62.51
and(mark(z0), z1) → mark(and(z0, z1)) 186.70/62.51
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 186.70/62.51
plus(mark(z0), z1) → mark(plus(z0, z1)) 186.70/62.51
plus(z0, mark(z1)) → mark(plus(z0, z1)) 186.70/62.51
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 186.70/62.51
s(mark(z0)) → mark(s(z0)) 186.70/62.51
s(ok(z0)) → ok(s(z0)) 186.70/62.51
x(mark(z0), z1) → mark(x(z0, z1)) 186.70/62.51
x(z0, mark(z1)) → mark(x(z0, z1)) 186.70/62.51
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 186.70/62.51
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 186.70/62.51
proper(tt) → ok(tt) 186.70/62.51
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 186.70/62.51
proper(0) → ok(0) 186.70/62.51
proper(s(z0)) → s(proper(z0)) 186.70/62.51
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 186.70/62.51
top(mark(z0)) → top(proper(z0)) 186.70/62.51
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, s(z1))) → c2(S(plus(z0, z1)), PLUS(z0, z1)) 186.70/62.51
ACTIVE(x(z0, s(z1))) → c4(PLUS(x(z0, z1), z0), X(z0, z1)) 186.70/62.51
ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0)) 186.70/62.51
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 186.70/62.51
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.51
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 186.70/62.51
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.51
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.51
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.51
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.51
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.51
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.51
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.51
S(mark(z0)) → c16(S(z0)) 186.70/62.51
S(ok(z0)) → c17(S(z0)) 186.70/62.51
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.51
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.51
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.51
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.51
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.51
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.51
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.51
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.51
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(plus(z0, s(z1))) → c2(S(plus(z0, z1)), PLUS(z0, z1)) 186.70/62.51
ACTIVE(x(z0, s(z1))) → c4(PLUS(x(z0, z1), z0), X(z0, z1)) 186.70/62.51
ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0)) 186.70/62.51
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 186.70/62.52
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.52
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 186.70/62.52
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.52
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.52
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.52
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.52
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.52
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.52
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.52
S(mark(z0)) → c16(S(z0)) 186.70/62.52
S(ok(z0)) → c17(S(z0)) 186.70/62.52
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.52
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.52
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.52
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.52
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.52
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.52
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.52
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.52
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

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

Removed 2 trailing tuple parts

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(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 186.70/62.52
active(plus(z0, 0)) → mark(z0) 186.70/62.52
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 186.70/62.52
active(x(z0, 0)) → mark(0) 186.70/62.52
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 186.70/62.52
active(and(z0, z1)) → and(active(z0), z1) 186.70/62.52
active(plus(z0, z1)) → plus(active(z0), z1) 186.70/62.52
active(plus(z0, z1)) → plus(z0, active(z1)) 186.70/62.52
active(s(z0)) → s(active(z0)) 186.70/62.52
active(x(z0, z1)) → x(active(z0), z1) 186.70/62.52
active(x(z0, z1)) → x(z0, active(z1)) 186.70/62.52
and(mark(z0), z1) → mark(and(z0, z1)) 186.70/62.52
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 186.70/62.52
plus(mark(z0), z1) → mark(plus(z0, z1)) 186.70/62.52
plus(z0, mark(z1)) → mark(plus(z0, z1)) 186.70/62.52
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 186.70/62.52
s(mark(z0)) → mark(s(z0)) 186.70/62.52
s(ok(z0)) → ok(s(z0)) 186.70/62.52
x(mark(z0), z1) → mark(x(z0, z1)) 186.70/62.52
x(z0, mark(z1)) → mark(x(z0, z1)) 186.70/62.52
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 186.70/62.52
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 186.70/62.52
proper(tt) → ok(tt) 186.70/62.52
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 186.70/62.52
proper(0) → ok(0) 186.70/62.52
proper(s(z0)) → s(proper(z0)) 186.70/62.52
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 186.70/62.52
top(mark(z0)) → top(proper(z0)) 186.70/62.52
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0)) 186.70/62.52
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 186.70/62.52
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.52
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 186.70/62.52
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.52
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.52
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.52
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.52
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.52
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.52
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.52
S(mark(z0)) → c16(S(z0)) 186.70/62.52
S(ok(z0)) → c17(S(z0)) 186.70/62.52
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.52
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.52
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.52
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.52
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.52
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.52
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.52
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.52
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.52
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.52
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

S tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0)) 186.70/62.52
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 186.70/62.52
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.52
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 186.70/62.52
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.52
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.52
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.52
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.52
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.52
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.52
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.52
S(mark(z0)) → c16(S(z0)) 186.70/62.52
S(ok(z0)) → c17(S(z0)) 186.70/62.52
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.52
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.52
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.52
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.52
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.52
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.52
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.52
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.52
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.52
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.52
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

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

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

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

We considered the (Usable) Rules:

active(and(tt, z0)) → mark(z0) 186.70/62.52
active(plus(z0, 0)) → mark(z0) 186.70/62.52
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 186.70/62.52
active(x(z0, 0)) → mark(0) 186.70/62.52
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 186.70/62.52
active(and(z0, z1)) → and(active(z0), z1) 186.70/62.52
active(plus(z0, z1)) → plus(active(z0), z1) 186.70/62.52
active(plus(z0, z1)) → plus(z0, active(z1)) 186.70/62.52
active(s(z0)) → s(active(z0)) 186.70/62.52
active(x(z0, z1)) → x(active(z0), z1) 186.70/62.52
active(x(z0, z1)) → x(z0, active(z1)) 186.70/62.52
x(z0, mark(z1)) → mark(x(z0, z1)) 186.70/62.52
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 186.70/62.52
x(mark(z0), z1) → mark(x(z0, z1)) 186.70/62.52
s(mark(z0)) → mark(s(z0)) 186.70/62.52
s(ok(z0)) → ok(s(z0)) 186.70/62.52
plus(z0, mark(z1)) → mark(plus(z0, z1)) 186.70/62.52
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 186.70/62.52
plus(mark(z0), z1) → mark(plus(z0, z1)) 186.70/62.52
and(mark(z0), z1) → mark(and(z0, z1)) 186.70/62.52
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 186.70/62.52
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 186.70/62.52
proper(tt) → ok(tt) 186.70/62.52
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 186.70/62.52
proper(0) → ok(0) 186.70/62.52
proper(s(z0)) → s(proper(z0)) 186.70/62.52
proper(x(z0, z1)) → x(proper(z0), proper(z1))

And the Tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0)) 186.70/62.52
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 186.70/62.52
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.52
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 186.70/62.52
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.52
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.52
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.52
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.52
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.52
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.52
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.52
S(mark(z0)) → c16(S(z0)) 186.70/62.52
S(ok(z0)) → c17(S(z0)) 186.70/62.52
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.52
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.52
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.52
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.52
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.52
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.53
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.53
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.53
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.53
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.53
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 186.70/62.53

POL(0) = [1] 186.70/62.53
POL(ACTIVE(x₁)) = 0 186.70/62.53
POL(AND(x₁, x₂)) = 0 186.70/62.53
POL(PLUS(x₁, x₂)) = 0 186.70/62.53
POL(PROPER(x₁)) = 0 186.70/62.53
POL(S(x₁)) = 0 186.70/62.53
POL(TOP(x₁)) = [2]x₁ 186.70/62.53
POL(X(x₁, x₂)) = 0 186.70/62.53
POL(active(x₁)) = x₁ 186.70/62.53
POL(and(x₁, x₂)) = [1] + x₁ + [2]x₂ 186.70/62.53
POL(c10(x₁, x₂)) = x₁ + x₂ 186.70/62.53
POL(c11(x₁)) = x₁ 186.70/62.53
POL(c12(x₁)) = x₁ 186.70/62.53
POL(c13(x₁)) = x₁ 186.70/62.53
POL(c14(x₁)) = x₁ 186.70/62.53
POL(c15(x₁)) = x₁ 186.70/62.53
POL(c16(x₁)) = x₁ 186.70/62.53
POL(c17(x₁)) = x₁ 186.70/62.53
POL(c18(x₁)) = x₁ 186.70/62.53
POL(c19(x₁)) = x₁ 186.70/62.53
POL(c2(x₁)) = x₁ 186.70/62.53
POL(c20(x₁)) = x₁ 186.70/62.53
POL(c21(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 186.70/62.53
POL(c23(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 186.70/62.53
POL(c25(x₁, x₂)) = x₁ + x₂ 186.70/62.53
POL(c26(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 186.70/62.53
POL(c27(x₁, x₂)) = x₁ + x₂ 186.70/62.53
POL(c28(x₁, x₂)) = x₁ + x₂ 186.70/62.53
POL(c4(x₁)) = x₁ 186.70/62.53
POL(c5(x₁, x₂)) = x₁ + x₂ 186.70/62.53
POL(c6(x₁, x₂)) = x₁ + x₂ 186.70/62.53
POL(c7(x₁, x₂)) = x₁ + x₂ 186.70/62.53
POL(c8(x₁, x₂)) = x₁ + x₂ 186.70/62.53
POL(c9(x₁, x₂)) = x₁ + x₂ 186.70/62.53
POL(mark(x₁)) = [1] + x₁ 186.70/62.53
POL(ok(x₁)) = x₁ 186.70/62.53
POL(plus(x₁, x₂)) = x₁ + [2]x₂ 186.70/62.53
POL(proper(x₁)) = x₁ 186.70/62.53
POL(s(x₁)) = [2] + x₁ 186.70/62.53
POL(tt) = 0 186.70/62.53
POL(x(x₁, x₂)) = [2] + x₁ + x₂ + x₁·x₂

186.70/62.53

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 186.70/62.53
active(plus(z0, 0)) → mark(z0) 186.70/62.53
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 186.70/62.53
active(x(z0, 0)) → mark(0) 186.70/62.53
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 186.70/62.53
active(and(z0, z1)) → and(active(z0), z1) 186.70/62.53
active(plus(z0, z1)) → plus(active(z0), z1) 186.70/62.53
active(plus(z0, z1)) → plus(z0, active(z1)) 186.70/62.53
active(s(z0)) → s(active(z0)) 186.70/62.53
active(x(z0, z1)) → x(active(z0), z1) 186.70/62.53
active(x(z0, z1)) → x(z0, active(z1)) 186.70/62.53
and(mark(z0), z1) → mark(and(z0, z1)) 186.70/62.53
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 186.70/62.53
plus(mark(z0), z1) → mark(plus(z0, z1)) 186.70/62.53
plus(z0, mark(z1)) → mark(plus(z0, z1)) 186.70/62.53
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 186.70/62.53
s(mark(z0)) → mark(s(z0)) 186.70/62.53
s(ok(z0)) → ok(s(z0)) 186.70/62.53
x(mark(z0), z1) → mark(x(z0, z1)) 186.70/62.53
x(z0, mark(z1)) → mark(x(z0, z1)) 186.70/62.53
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 186.70/62.53
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 186.70/62.53
proper(tt) → ok(tt) 186.70/62.53
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 186.70/62.53
proper(0) → ok(0) 186.70/62.53
proper(s(z0)) → s(proper(z0)) 186.70/62.53
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 186.70/62.53
top(mark(z0)) → top(proper(z0)) 186.70/62.53
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0)) 186.70/62.53
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 186.70/62.53
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.53
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 186.70/62.53
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.53
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.53
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.53
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.53
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.53
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.53
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.53
S(mark(z0)) → c16(S(z0)) 186.70/62.53
S(ok(z0)) → c17(S(z0)) 186.70/62.53
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.53
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.53
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.53
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.53
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.53
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.53
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.53
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.53
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.53
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.53
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

S tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0)) 186.70/62.53
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 186.70/62.53
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.53
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 186.70/62.53
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.53
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.53
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.53
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.53
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.53
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.53
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.53
S(mark(z0)) → c16(S(z0)) 186.70/62.53
S(ok(z0)) → c17(S(z0)) 186.70/62.53
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.53
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.53
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.53
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.53
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.53
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.53
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.53
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.53
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.53
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, 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

186.70/62.53

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

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

PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.53
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.53
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

We considered the (Usable) Rules:

active(and(tt, z0)) → mark(z0) 186.70/62.53
active(plus(z0, 0)) → mark(z0) 186.70/62.53
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 186.70/62.53
active(x(z0, 0)) → mark(0) 186.70/62.53
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 186.70/62.53
active(and(z0, z1)) → and(active(z0), z1) 186.70/62.53
active(plus(z0, z1)) → plus(active(z0), z1) 186.70/62.53
active(plus(z0, z1)) → plus(z0, active(z1)) 186.70/62.53
active(s(z0)) → s(active(z0)) 186.70/62.53
active(x(z0, z1)) → x(active(z0), z1) 186.70/62.53
active(x(z0, z1)) → x(z0, active(z1)) 186.70/62.53
x(z0, mark(z1)) → mark(x(z0, z1)) 186.70/62.53
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 186.70/62.53
x(mark(z0), z1) → mark(x(z0, z1)) 186.70/62.53
s(mark(z0)) → mark(s(z0)) 186.70/62.53
s(ok(z0)) → ok(s(z0)) 186.70/62.53
plus(z0, mark(z1)) → mark(plus(z0, z1)) 186.70/62.53
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 186.70/62.53
plus(mark(z0), z1) → mark(plus(z0, z1)) 186.70/62.53
and(mark(z0), z1) → mark(and(z0, z1)) 186.70/62.53
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 186.70/62.53
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 186.70/62.53
proper(tt) → ok(tt) 186.70/62.53
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 186.70/62.53
proper(0) → ok(0) 186.70/62.53
proper(s(z0)) → s(proper(z0)) 186.70/62.53
proper(x(z0, z1)) → x(proper(z0), proper(z1))

And the Tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0)) 186.70/62.53
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 186.70/62.54
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.54
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 186.70/62.54
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.54
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.54
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.54
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.54
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.54
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.54
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.54
S(mark(z0)) → c16(S(z0)) 186.70/62.54
S(ok(z0)) → c17(S(z0)) 186.70/62.54
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.54
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.54
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.54
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.54
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.54
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.54
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.54
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.54
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.54
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.54
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 186.70/62.54

POL(0) = [1] 186.70/62.54
POL(ACTIVE(x₁)) = 0 186.70/62.54
POL(AND(x₁, x₂)) = 0 186.70/62.54
POL(PLUS(x₁, x₂)) = 0 186.70/62.54
POL(PROPER(x₁)) = x₁ 186.70/62.54
POL(S(x₁)) = 0 186.70/62.54
POL(TOP(x₁)) = x₁² 186.70/62.54
POL(X(x₁, x₂)) = 0 186.70/62.54
POL(active(x₁)) = x₁ 186.70/62.54
POL(and(x₁, x₂)) = [1] + [2]x₁ + [2]x₂ + [2]x₁·x₂ 186.70/62.54
POL(c10(x₁, x₂)) = x₁ + x₂ 186.70/62.54
POL(c11(x₁)) = x₁ 186.70/62.54
POL(c12(x₁)) = x₁ 186.70/62.54
POL(c13(x₁)) = x₁ 186.70/62.54
POL(c14(x₁)) = x₁ 186.70/62.54
POL(c15(x₁)) = x₁ 186.70/62.54
POL(c16(x₁)) = x₁ 186.70/62.54
POL(c17(x₁)) = x₁ 186.70/62.54
POL(c18(x₁)) = x₁ 186.70/62.54
POL(c19(x₁)) = x₁ 186.70/62.54
POL(c2(x₁)) = x₁ 186.70/62.54
POL(c20(x₁)) = x₁ 186.70/62.54
POL(c21(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 186.70/62.54
POL(c23(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 186.70/62.54
POL(c25(x₁, x₂)) = x₁ + x₂ 186.70/62.54
POL(c26(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 186.70/62.54
POL(c27(x₁, x₂)) = x₁ + x₂ 186.70/62.54
POL(c28(x₁, x₂)) = x₁ + x₂ 186.70/62.54
POL(c4(x₁)) = x₁ 186.70/62.54
POL(c5(x₁, x₂)) = x₁ + x₂ 186.70/62.54
POL(c6(x₁, x₂)) = x₁ + x₂ 186.70/62.54
POL(c7(x₁, x₂)) = x₁ + x₂ 186.70/62.54
POL(c8(x₁, x₂)) = x₁ + x₂ 186.70/62.54
POL(c9(x₁, x₂)) = x₁ + x₂ 186.70/62.54
POL(mark(x₁)) = [2] + x₁ 186.70/62.54
POL(ok(x₁)) = x₁ 186.70/62.54
POL(plus(x₁, x₂)) = x₁ + [2]x₂ 186.70/62.54
POL(proper(x₁)) = x₁ 186.70/62.54
POL(s(x₁)) = [2] + x₁ 186.70/62.54
POL(tt) = [3] 186.70/62.54
POL(x(x₁, x₂)) = [1] + x₁ + [2]x₂ + [2]x₁·x₂

186.70/62.54

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 186.70/62.54
active(plus(z0, 0)) → mark(z0) 186.70/62.54
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 186.70/62.54
active(x(z0, 0)) → mark(0) 186.70/62.54
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 186.70/62.54
active(and(z0, z1)) → and(active(z0), z1) 186.70/62.54
active(plus(z0, z1)) → plus(active(z0), z1) 186.70/62.54
active(plus(z0, z1)) → plus(z0, active(z1)) 186.70/62.54
active(s(z0)) → s(active(z0)) 186.70/62.54
active(x(z0, z1)) → x(active(z0), z1) 186.70/62.54
active(x(z0, z1)) → x(z0, active(z1)) 186.70/62.54
and(mark(z0), z1) → mark(and(z0, z1)) 186.70/62.54
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 186.70/62.54
plus(mark(z0), z1) → mark(plus(z0, z1)) 186.70/62.54
plus(z0, mark(z1)) → mark(plus(z0, z1)) 186.70/62.54
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 186.70/62.54
s(mark(z0)) → mark(s(z0)) 186.70/62.54
s(ok(z0)) → ok(s(z0)) 186.70/62.54
x(mark(z0), z1) → mark(x(z0, z1)) 186.70/62.54
x(z0, mark(z1)) → mark(x(z0, z1)) 186.70/62.54
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 186.70/62.54
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 186.70/62.54
proper(tt) → ok(tt) 186.70/62.54
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 186.70/62.54
proper(0) → ok(0) 186.70/62.54
proper(s(z0)) → s(proper(z0)) 186.70/62.54
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 186.70/62.54
top(mark(z0)) → top(proper(z0)) 186.70/62.54
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0)) 186.70/62.54
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 186.70/62.54
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.54
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 186.70/62.54
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.54
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.54
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.54
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.54
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.54
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.54
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.54
S(mark(z0)) → c16(S(z0)) 186.70/62.55
S(ok(z0)) → c17(S(z0)) 186.70/62.55
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.55
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.55
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.55
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.55
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.55
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.55
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.55
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.55
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.55
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.55
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

S tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0)) 186.70/62.55
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 186.70/62.55
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.55
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 186.70/62.55
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.55
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.55
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.55
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.55
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.55
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.55
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.55
S(mark(z0)) → c16(S(z0)) 186.70/62.55
S(ok(z0)) → c17(S(z0)) 186.70/62.55
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.55
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.55
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.55
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.55
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.55
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.55
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.55
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.55
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.55
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

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

186.70/62.55

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

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

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

We considered the (Usable) Rules:

active(and(tt, z0)) → mark(z0) 186.70/62.55
active(plus(z0, 0)) → mark(z0) 186.70/62.55
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 186.70/62.55
active(x(z0, 0)) → mark(0) 186.70/62.55
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 186.70/62.55
active(and(z0, z1)) → and(active(z0), z1) 186.70/62.55
active(plus(z0, z1)) → plus(active(z0), z1) 186.70/62.55
active(plus(z0, z1)) → plus(z0, active(z1)) 186.70/62.55
active(s(z0)) → s(active(z0)) 186.70/62.55
active(x(z0, z1)) → x(active(z0), z1) 186.70/62.55
active(x(z0, z1)) → x(z0, active(z1)) 186.70/62.55
x(z0, mark(z1)) → mark(x(z0, z1)) 186.70/62.55
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 186.70/62.55
x(mark(z0), z1) → mark(x(z0, z1)) 186.70/62.55
s(mark(z0)) → mark(s(z0)) 186.70/62.55
s(ok(z0)) → ok(s(z0)) 186.70/62.55
plus(z0, mark(z1)) → mark(plus(z0, z1)) 186.70/62.55
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 186.70/62.55
plus(mark(z0), z1) → mark(plus(z0, z1)) 186.70/62.55
and(mark(z0), z1) → mark(and(z0, z1)) 186.70/62.55
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 186.70/62.55
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 186.70/62.55
proper(tt) → ok(tt) 186.70/62.55
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 186.70/62.55
proper(0) → ok(0) 186.70/62.55
proper(s(z0)) → s(proper(z0)) 186.70/62.55
proper(x(z0, z1)) → x(proper(z0), proper(z1))

And the Tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0)) 186.70/62.55
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 186.70/62.55
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.55
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 186.70/62.55
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.55
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.55
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.55
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.55
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.55
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.55
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.55
S(mark(z0)) → c16(S(z0)) 186.70/62.55
S(ok(z0)) → c17(S(z0)) 186.70/62.55
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.55
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.55
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.55
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.55
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.55
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.55
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.55
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.55
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.55
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.55
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 186.70/62.55

POL(0) = 0 186.70/62.55
POL(ACTIVE(x₁)) = 0 186.70/62.55
POL(AND(x₁, x₂)) = 0 186.70/62.55
POL(PLUS(x₁, x₂)) = 0 186.70/62.55
POL(PROPER(x₁)) = [1] + [2]x₁ 186.70/62.55
POL(S(x₁)) = 0 186.70/62.55
POL(TOP(x₁)) = x₁² 186.70/62.55
POL(X(x₁, x₂)) = 0 186.70/62.55
POL(active(x₁)) = x₁ 186.70/62.55
POL(and(x₁, x₂)) = [1] + x₁ + x₂ 186.70/62.55
POL(c10(x₁, x₂)) = x₁ + x₂ 186.70/62.55
POL(c11(x₁)) = x₁ 186.70/62.55
POL(c12(x₁)) = x₁ 186.70/62.55
POL(c13(x₁)) = x₁ 186.70/62.55
POL(c14(x₁)) = x₁ 186.70/62.55
POL(c15(x₁)) = x₁ 186.70/62.55
POL(c16(x₁)) = x₁ 186.70/62.55
POL(c17(x₁)) = x₁ 186.70/62.55
POL(c18(x₁)) = x₁ 186.70/62.55
POL(c19(x₁)) = x₁ 186.70/62.55
POL(c2(x₁)) = x₁ 186.70/62.55
POL(c20(x₁)) = x₁ 186.70/62.55
POL(c21(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 186.70/62.55
POL(c23(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 186.70/62.55
POL(c25(x₁, x₂)) = x₁ + x₂ 186.70/62.55
POL(c26(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 186.70/62.55
POL(c27(x₁, x₂)) = x₁ + x₂ 186.70/62.55
POL(c28(x₁, x₂)) = x₁ + x₂ 186.70/62.55
POL(c4(x₁)) = x₁ 186.70/62.55
POL(c5(x₁, x₂)) = x₁ + x₂ 186.70/62.55
POL(c6(x₁, x₂)) = x₁ + x₂ 186.70/62.55
POL(c7(x₁, x₂)) = x₁ + x₂ 186.70/62.55
POL(c8(x₁, x₂)) = x₁ + x₂ 186.70/62.55
POL(c9(x₁, x₂)) = x₁ + x₂ 186.70/62.55
POL(mark(x₁)) = [1] + x₁ 186.70/62.55
POL(ok(x₁)) = x₁ 186.70/62.55
POL(plus(x₁, x₂)) = [1] + x₁ + [2]x₂ 186.70/62.55
POL(proper(x₁)) = x₁ 186.70/62.55
POL(s(x₁)) = [1] + x₁ 186.70/62.55
POL(tt) = 0 186.70/62.55
POL(x(x₁, x₂)) = [1] + x₁ + [2]x₂ + [3]x₁·x₂

186.70/62.55

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 186.70/62.55
active(plus(z0, 0)) → mark(z0) 186.70/62.55
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 186.70/62.55
active(x(z0, 0)) → mark(0) 186.70/62.55
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 186.70/62.55
active(and(z0, z1)) → and(active(z0), z1) 186.70/62.55
active(plus(z0, z1)) → plus(active(z0), z1) 186.70/62.55
active(plus(z0, z1)) → plus(z0, active(z1)) 186.70/62.55
active(s(z0)) → s(active(z0)) 186.70/62.55
active(x(z0, z1)) → x(active(z0), z1) 186.70/62.55
active(x(z0, z1)) → x(z0, active(z1)) 186.70/62.55
and(mark(z0), z1) → mark(and(z0, z1)) 186.70/62.55
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 186.70/62.55
plus(mark(z0), z1) → mark(plus(z0, z1)) 186.70/62.55
plus(z0, mark(z1)) → mark(plus(z0, z1)) 186.70/62.55
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 186.70/62.55
s(mark(z0)) → mark(s(z0)) 186.70/62.55
s(ok(z0)) → ok(s(z0)) 186.70/62.55
x(mark(z0), z1) → mark(x(z0, z1)) 186.70/62.55
x(z0, mark(z1)) → mark(x(z0, z1)) 186.70/62.55
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 186.70/62.55
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 186.70/62.55
proper(tt) → ok(tt) 186.70/62.55
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 186.70/62.55
proper(0) → ok(0) 186.70/62.55
proper(s(z0)) → s(proper(z0)) 186.70/62.55
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 186.70/62.55
top(mark(z0)) → top(proper(z0)) 186.70/62.55
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0)) 186.70/62.55
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 186.70/62.55
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.55
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 186.70/62.55
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.55
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.55
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.55
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.55
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.55
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.55
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.55
S(mark(z0)) → c16(S(z0)) 186.70/62.55
S(ok(z0)) → c17(S(z0)) 186.70/62.55
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.55
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.55
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.55
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.55
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.55
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.55
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.55
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.55
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.55
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.55
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

S tuples:

ACTIVE(and(z0, z1)) → c5(AND(active(z0), z1), ACTIVE(z0)) 186.70/62.55
ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 186.70/62.55
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.55
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 186.70/62.55
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.55
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.55
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.55
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.55
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.55
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.55
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.55
S(mark(z0)) → c16(S(z0)) 186.70/62.55
S(ok(z0)) → c17(S(z0)) 186.70/62.55
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.55
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.55
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.55
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.55
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.55
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.55
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.55
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.55
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.55
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

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

186.70/62.55

(11) 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))) 186.70/62.55
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 186.70/62.55
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 186.70/62.55
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 186.70/62.55
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 186.70/62.55
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 186.70/62.55
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 186.70/62.55
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 186.70/62.55
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 186.70/62.55
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 186.70/62.55
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))

186.70/62.55

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 186.70/62.55
active(plus(z0, 0)) → mark(z0) 186.70/62.55
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 186.70/62.55
active(x(z0, 0)) → mark(0) 186.70/62.55
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 186.70/62.55
active(and(z0, z1)) → and(active(z0), z1) 186.70/62.55
active(plus(z0, z1)) → plus(active(z0), z1) 186.70/62.55
active(plus(z0, z1)) → plus(z0, active(z1)) 186.70/62.55
active(s(z0)) → s(active(z0)) 186.70/62.55
active(x(z0, z1)) → x(active(z0), z1) 186.70/62.55
active(x(z0, z1)) → x(z0, active(z1)) 186.70/62.55
and(mark(z0), z1) → mark(and(z0, z1)) 186.70/62.55
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 186.70/62.55
plus(mark(z0), z1) → mark(plus(z0, z1)) 186.70/62.55
plus(z0, mark(z1)) → mark(plus(z0, z1)) 186.70/62.55
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 186.70/62.55
s(mark(z0)) → mark(s(z0)) 186.70/62.55
s(ok(z0)) → ok(s(z0)) 186.70/62.55
x(mark(z0), z1) → mark(x(z0, z1)) 186.70/62.55
x(z0, mark(z1)) → mark(x(z0, z1)) 186.70/62.55
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 186.70/62.55
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 186.70/62.55
proper(tt) → ok(tt) 186.70/62.55
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 186.70/62.55
proper(0) → ok(0) 186.70/62.55
proper(s(z0)) → s(proper(z0)) 186.70/62.55
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 186.70/62.55
top(mark(z0)) → top(proper(z0)) 186.70/62.55
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 186.70/62.55
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.55
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 186.70/62.55
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.55
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.55
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.55
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.55
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.55
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.55
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.55
S(mark(z0)) → c16(S(z0)) 186.70/62.55
S(ok(z0)) → c17(S(z0)) 186.70/62.55
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.55
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.55
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.55
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.55
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.55
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.55
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.55
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.55
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.55
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.55
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 186.70/62.55
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1), ACTIVE(and(tt, z0))) 186.70/62.55
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 186.70/62.55
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 186.70/62.55
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 186.70/62.55
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 186.70/62.55
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 186.70/62.55
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 186.70/62.55
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 186.70/62.55
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 186.70/62.55
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 186.70/62.55
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)) 186.70/62.55
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.55
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 186.70/62.55
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.55
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.55
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.55
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.55
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.55
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.55
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.55
S(mark(z0)) → c16(S(z0)) 186.70/62.55
S(ok(z0)) → c17(S(z0)) 186.70/62.55
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.55
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.55
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.55
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.55
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.55
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 186.70/62.55
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1), ACTIVE(and(tt, z0))) 186.70/62.55
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 186.70/62.55
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 186.70/62.55
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 186.70/62.55
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 186.70/62.55
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 186.70/62.55
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 186.70/62.55
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 186.70/62.55
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 186.70/62.55
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 186.70/62.55
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)) 186.70/62.55
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.55
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.55
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.55
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

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

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

186.70/62.55

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

Removed 1 trailing tuple parts

186.70/62.55

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 186.70/62.55
active(plus(z0, 0)) → mark(z0) 186.70/62.55
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 186.70/62.55
active(x(z0, 0)) → mark(0) 186.70/62.55
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 186.70/62.55
active(and(z0, z1)) → and(active(z0), z1) 186.70/62.55
active(plus(z0, z1)) → plus(active(z0), z1) 186.70/62.55
active(plus(z0, z1)) → plus(z0, active(z1)) 186.70/62.55
active(s(z0)) → s(active(z0)) 186.70/62.55
active(x(z0, z1)) → x(active(z0), z1) 186.70/62.55
active(x(z0, z1)) → x(z0, active(z1)) 186.70/62.55
and(mark(z0), z1) → mark(and(z0, z1)) 186.70/62.55
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 186.70/62.55
plus(mark(z0), z1) → mark(plus(z0, z1)) 186.70/62.55
plus(z0, mark(z1)) → mark(plus(z0, z1)) 186.70/62.55
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 186.70/62.55
s(mark(z0)) → mark(s(z0)) 186.70/62.55
s(ok(z0)) → ok(s(z0)) 186.70/62.55
x(mark(z0), z1) → mark(x(z0, z1)) 186.70/62.55
x(z0, mark(z1)) → mark(x(z0, z1)) 186.70/62.55
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 186.70/62.55
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 186.70/62.55
proper(tt) → ok(tt) 186.70/62.55
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 186.70/62.55
proper(0) → ok(0) 186.70/62.55
proper(s(z0)) → s(proper(z0)) 186.70/62.55
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 186.70/62.55
top(mark(z0)) → top(proper(z0)) 186.70/62.55
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 186.70/62.55
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.55
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 186.70/62.55
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.55
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.55
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.55
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.55
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.55
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.55
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.55
S(mark(z0)) → c16(S(z0)) 186.70/62.55
S(ok(z0)) → c17(S(z0)) 186.70/62.55
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.55
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.55
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.55
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.55
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.55
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.55
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.55
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.55
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.55
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.55
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 186.70/62.55
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 186.70/62.55
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 186.70/62.55
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 186.70/62.55
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 186.70/62.55
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 186.70/62.55
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 186.70/62.55
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 186.70/62.55
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 186.70/62.55
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 186.70/62.55
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 186.70/62.55
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))

S tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 186.70/62.55
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.55
ACTIVE(s(z0)) → c8(S(active(z0)), ACTIVE(z0)) 186.70/62.55
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.55
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.55
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.55
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.55
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.55
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.55
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.55
S(mark(z0)) → c16(S(z0)) 186.70/62.55
S(ok(z0)) → c17(S(z0)) 186.70/62.55
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.55
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.55
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.55
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.55
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.55
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 186.70/62.55
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 186.70/62.55
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 186.70/62.55
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 186.70/62.55
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 186.70/62.55
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 186.70/62.55
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 186.70/62.55
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 186.70/62.55
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 186.70/62.55
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 186.70/62.55
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 186.70/62.55
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.55
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.57
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.57
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.57
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

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

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

186.70/62.57

(15) 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))) 186.70/62.57
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 186.70/62.57
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 186.70/62.57
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 186.70/62.57
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 186.70/62.57
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 186.70/62.57
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 186.70/62.57
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 186.70/62.57
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 186.70/62.57
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 186.70/62.57
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))

186.70/62.57

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 186.70/62.57
active(plus(z0, 0)) → mark(z0) 186.70/62.57
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 186.70/62.57
active(x(z0, 0)) → mark(0) 186.70/62.57
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 186.70/62.57
active(and(z0, z1)) → and(active(z0), z1) 186.70/62.57
active(plus(z0, z1)) → plus(active(z0), z1) 186.70/62.57
active(plus(z0, z1)) → plus(z0, active(z1)) 186.70/62.57
active(s(z0)) → s(active(z0)) 186.70/62.57
active(x(z0, z1)) → x(active(z0), z1) 186.70/62.57
active(x(z0, z1)) → x(z0, active(z1)) 186.70/62.57
and(mark(z0), z1) → mark(and(z0, z1)) 186.70/62.57
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 186.70/62.57
plus(mark(z0), z1) → mark(plus(z0, z1)) 186.70/62.57
plus(z0, mark(z1)) → mark(plus(z0, z1)) 186.70/62.57
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 186.70/62.57
s(mark(z0)) → mark(s(z0)) 186.70/62.57
s(ok(z0)) → ok(s(z0)) 186.70/62.57
x(mark(z0), z1) → mark(x(z0, z1)) 186.70/62.57
x(z0, mark(z1)) → mark(x(z0, z1)) 186.70/62.57
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 186.70/62.57
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 186.70/62.57
proper(tt) → ok(tt) 186.70/62.57
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 186.70/62.57
proper(0) → ok(0) 186.70/62.57
proper(s(z0)) → s(proper(z0)) 186.70/62.57
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 186.70/62.57
top(mark(z0)) → top(proper(z0)) 186.70/62.57
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 186.70/62.57
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.57
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.57
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.57
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.57
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.57
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.57
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.57
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.57
S(mark(z0)) → c16(S(z0)) 186.70/62.57
S(ok(z0)) → c17(S(z0)) 186.70/62.57
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.57
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.57
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.57
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.57
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.57
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.57
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.57
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.57
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.57
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.57
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 186.70/62.57
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 186.70/62.57
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 186.70/62.57
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 186.70/62.57
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 186.70/62.57
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 186.70/62.57
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 186.70/62.57
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 186.70/62.57
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 186.70/62.57
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 186.70/62.57
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 186.70/62.57
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 186.70/62.57
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)), ACTIVE(and(tt, z0))) 186.70/62.57
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 186.70/62.57
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 186.70/62.57
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 186.70/62.57
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 186.70/62.57
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 186.70/62.57
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 186.70/62.57
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 186.70/62.57
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 186.70/62.57
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 186.70/62.57
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)) 186.70/62.57
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.57
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.57
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.57
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.57
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.57
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.57
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.57
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.57
S(mark(z0)) → c16(S(z0)) 186.70/62.57
S(ok(z0)) → c17(S(z0)) 186.70/62.57
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.57
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.57
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.57
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.57
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.57
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 186.70/62.57
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 186.70/62.57
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 186.70/62.57
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 186.70/62.57
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 186.70/62.57
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 186.70/62.57
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 186.70/62.57
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 186.70/62.57
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 186.70/62.57
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 186.70/62.57
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 186.70/62.57
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 186.70/62.57
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)), ACTIVE(and(tt, z0))) 186.70/62.57
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 186.70/62.57
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 186.70/62.57
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 186.70/62.57
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 186.70/62.57
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 186.70/62.57
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 186.70/62.57
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 186.70/62.57
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 186.70/62.57
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 186.70/62.57
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)) 186.70/62.57
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.57
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.57
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.57
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

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

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

186.70/62.57

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

Removed 1 trailing tuple parts

186.70/62.57

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 186.70/62.57
active(plus(z0, 0)) → mark(z0) 186.70/62.57
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 186.70/62.57
active(x(z0, 0)) → mark(0) 186.70/62.57
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 186.70/62.57
active(and(z0, z1)) → and(active(z0), z1) 186.70/62.57
active(plus(z0, z1)) → plus(active(z0), z1) 186.70/62.57
active(plus(z0, z1)) → plus(z0, active(z1)) 186.70/62.57
active(s(z0)) → s(active(z0)) 186.70/62.57
active(x(z0, z1)) → x(active(z0), z1) 186.70/62.57
active(x(z0, z1)) → x(z0, active(z1)) 186.70/62.57
and(mark(z0), z1) → mark(and(z0, z1)) 186.70/62.57
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 186.70/62.57
plus(mark(z0), z1) → mark(plus(z0, z1)) 186.70/62.57
plus(z0, mark(z1)) → mark(plus(z0, z1)) 186.70/62.57
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 186.70/62.57
s(mark(z0)) → mark(s(z0)) 186.70/62.57
s(ok(z0)) → ok(s(z0)) 186.70/62.57
x(mark(z0), z1) → mark(x(z0, z1)) 186.70/62.57
x(z0, mark(z1)) → mark(x(z0, z1)) 186.70/62.57
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 186.70/62.57
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 186.70/62.57
proper(tt) → ok(tt) 186.70/62.57
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 186.70/62.57
proper(0) → ok(0) 186.70/62.57
proper(s(z0)) → s(proper(z0)) 186.70/62.57
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 186.70/62.57
top(mark(z0)) → top(proper(z0)) 186.70/62.57
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 186.70/62.57
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.57
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.57
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.57
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.57
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.57
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.57
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.57
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.57
S(mark(z0)) → c16(S(z0)) 186.70/62.57
S(ok(z0)) → c17(S(z0)) 186.70/62.57
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.57
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.57
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.57
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.57
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.57
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.57
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.57
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.57
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.57
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.57
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 186.70/62.57
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 186.70/62.57
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 186.70/62.57
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 186.70/62.59
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 186.70/62.59
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 186.70/62.59
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 186.70/62.59
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 186.70/62.59
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 186.70/62.59
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 186.70/62.59
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 186.70/62.59
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 186.70/62.59
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 186.70/62.59
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 186.70/62.59
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 186.70/62.59
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 186.70/62.59
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 186.70/62.59
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 186.70/62.59
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 186.70/62.59
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 186.70/62.59
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 186.70/62.59
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 186.70/62.59
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

S tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 186.70/62.59
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.59
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.59
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.59
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.59
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.59
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.59
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.59
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.59
S(mark(z0)) → c16(S(z0)) 186.70/62.59
S(ok(z0)) → c17(S(z0)) 186.70/62.59
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.59
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.59
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.59
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.59
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.59
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 186.70/62.59
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 186.70/62.59
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 186.70/62.59
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 186.70/62.59
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 186.70/62.59
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 186.70/62.59
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 186.70/62.59
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 186.70/62.59
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 186.70/62.59
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 186.70/62.59
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 186.70/62.59
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 186.70/62.59
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 186.70/62.59
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 186.70/62.59
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 186.70/62.59
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 186.70/62.59
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 186.70/62.59
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 186.70/62.59
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 186.70/62.59
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 186.70/62.59
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 186.70/62.59
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 186.70/62.59
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.59
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.59
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.59
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.59
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

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

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

186.70/62.59

(19) 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))) 186.70/62.59
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt)) 186.70/62.59
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 186.70/62.59
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 186.70/62.59
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 186.70/62.59
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 186.70/62.59
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 186.70/62.59
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1)) 186.70/62.59
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 186.70/62.59
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 186.70/62.59
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 186.70/62.59
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))

186.70/62.59

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 186.70/62.59
active(plus(z0, 0)) → mark(z0) 186.70/62.59
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 186.70/62.59
active(x(z0, 0)) → mark(0) 186.70/62.59
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 186.70/62.59
active(and(z0, z1)) → and(active(z0), z1) 186.70/62.59
active(plus(z0, z1)) → plus(active(z0), z1) 186.70/62.59
active(plus(z0, z1)) → plus(z0, active(z1)) 186.70/62.59
active(s(z0)) → s(active(z0)) 186.70/62.59
active(x(z0, z1)) → x(active(z0), z1) 186.70/62.59
active(x(z0, z1)) → x(z0, active(z1)) 186.70/62.59
and(mark(z0), z1) → mark(and(z0, z1)) 186.70/62.59
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 186.70/62.59
plus(mark(z0), z1) → mark(plus(z0, z1)) 186.70/62.59
plus(z0, mark(z1)) → mark(plus(z0, z1)) 186.70/62.59
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 186.70/62.59
s(mark(z0)) → mark(s(z0)) 186.70/62.59
s(ok(z0)) → ok(s(z0)) 186.70/62.59
x(mark(z0), z1) → mark(x(z0, z1)) 186.70/62.59
x(z0, mark(z1)) → mark(x(z0, z1)) 186.70/62.59
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 186.70/62.59
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 186.70/62.59
proper(tt) → ok(tt) 186.70/62.59
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 186.70/62.59
proper(0) → ok(0) 186.70/62.59
proper(s(z0)) → s(proper(z0)) 186.70/62.59
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 186.70/62.59
top(mark(z0)) → top(proper(z0)) 186.70/62.59
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 186.70/62.59
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.59
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.59
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.59
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.59
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.59
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.59
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.59
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.59
S(mark(z0)) → c16(S(z0)) 186.70/62.59
S(ok(z0)) → c17(S(z0)) 186.70/62.59
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.59
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.59
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.59
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.60
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.60
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.60
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.60
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.60
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.60
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 186.70/62.60
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 186.70/62.60
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 186.70/62.60
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 186.70/62.60
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 186.70/62.60
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 186.70/62.60
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 186.70/62.60
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 186.70/62.60
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 186.70/62.60
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 186.70/62.60
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 186.70/62.60
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 186.70/62.60
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 186.70/62.60
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 186.70/62.60
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0))) 186.70/62.60
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 186.70/62.60
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt)) 186.70/62.60
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 186.70/62.60
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 186.70/62.60
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 186.70/62.60
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 186.70/62.60
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 186.70/62.60
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1)) 186.70/62.60
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 186.70/62.60
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 186.70/62.60
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 186.70/62.60
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)) 186.70/62.60
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.60
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.60
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.60
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.60
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.60
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.60
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.60
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.60
S(mark(z0)) → c16(S(z0)) 186.70/62.60
S(ok(z0)) → c17(S(z0)) 186.70/62.60
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.60
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.60
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.60
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.60
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.60
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 186.70/62.60
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 186.70/62.60
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 186.70/62.60
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 186.70/62.60
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 186.70/62.60
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 186.70/62.60
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 186.70/62.60
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 186.70/62.60
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 186.70/62.60
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 186.70/62.60
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 186.70/62.60
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 186.70/62.60
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 186.70/62.60
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 186.70/62.60
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.60
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.60
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.60
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.60
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

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

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

186.70/62.60

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

Removed 4 trailing tuple parts

186.70/62.60

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 186.70/62.60
active(plus(z0, 0)) → mark(z0) 186.70/62.60
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 186.70/62.60
active(x(z0, 0)) → mark(0) 186.70/62.60
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 186.70/62.60
active(and(z0, z1)) → and(active(z0), z1) 186.70/62.60
active(plus(z0, z1)) → plus(active(z0), z1) 186.70/62.60
active(plus(z0, z1)) → plus(z0, active(z1)) 186.70/62.60
active(s(z0)) → s(active(z0)) 186.70/62.60
active(x(z0, z1)) → x(active(z0), z1) 186.70/62.60
active(x(z0, z1)) → x(z0, active(z1)) 186.70/62.60
and(mark(z0), z1) → mark(and(z0, z1)) 186.70/62.60
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 186.70/62.60
plus(mark(z0), z1) → mark(plus(z0, z1)) 186.70/62.60
plus(z0, mark(z1)) → mark(plus(z0, z1)) 186.70/62.60
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 186.70/62.60
s(mark(z0)) → mark(s(z0)) 186.70/62.60
s(ok(z0)) → ok(s(z0)) 186.70/62.60
x(mark(z0), z1) → mark(x(z0, z1)) 186.70/62.60
x(z0, mark(z1)) → mark(x(z0, z1)) 186.70/62.60
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 186.70/62.60
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 186.70/62.60
proper(tt) → ok(tt) 186.70/62.60
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 186.70/62.60
proper(0) → ok(0) 186.70/62.60
proper(s(z0)) → s(proper(z0)) 186.70/62.60
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 186.70/62.60
top(mark(z0)) → top(proper(z0)) 186.70/62.60
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 186.70/62.60
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.60
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.60
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.60
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.60
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.60
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.60
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.60
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.60
S(mark(z0)) → c16(S(z0)) 186.70/62.60
S(ok(z0)) → c17(S(z0)) 186.70/62.60
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.60
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.60
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.60
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.60
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.60
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.60
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.60
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.60
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.60
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 186.70/62.60
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 186.70/62.60
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 186.70/62.60
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 186.70/62.60
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 186.70/62.60
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 186.70/62.60
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 186.70/62.60
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 186.70/62.60
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 186.70/62.60
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 186.70/62.60
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 186.70/62.60
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 186.70/62.60
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 186.70/62.60
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 186.70/62.60
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0))) 186.70/62.60
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 186.70/62.60
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 186.70/62.60
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 186.70/62.60
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 186.70/62.60
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 186.70/62.60
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 186.70/62.60
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 186.70/62.60
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 186.70/62.60
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0)) 186.70/62.60
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0)) 186.70/62.60
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(x1)) 186.70/62.60
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)) 186.70/62.60
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.60
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.60
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.60
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.60
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.60
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.60
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.60
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.60
S(mark(z0)) → c16(S(z0)) 186.70/62.60
S(ok(z0)) → c17(S(z0)) 186.70/62.60
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.60
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.60
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.60
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.60
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.60
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 186.70/62.60
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 186.70/62.60
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 186.70/62.60
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 186.70/62.60
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 186.70/62.60
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 186.70/62.60
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 186.70/62.60
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 186.70/62.60
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 186.70/62.60
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 186.70/62.60
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 186.70/62.60
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 186.70/62.60
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 186.70/62.60
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 186.70/62.60
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.60
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.60
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.60
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.60
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

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

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

186.70/62.60

(23) 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))) 186.70/62.60
PROPER(plus(x0, tt)) → c23(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt)) 186.70/62.60
PROPER(plus(x0, plus(z0, z1))) → c23(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 186.70/62.60
PROPER(plus(x0, 0)) → c23(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 186.70/62.60
PROPER(plus(x0, s(z0))) → c23(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 186.70/62.60
PROPER(plus(x0, x(z0, z1))) → c23(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 186.70/62.60
PROPER(plus(and(z0, z1), x1)) → c23(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 186.70/62.60
PROPER(plus(tt, x1)) → c23(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1)) 186.70/62.60
PROPER(plus(plus(z0, z1), x1)) → c23(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 186.70/62.60
PROPER(plus(0, x1)) → c23(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 186.70/62.60
PROPER(plus(s(z0), x1)) → c23(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 186.70/62.60
PROPER(plus(x(z0, z1), x1)) → c23(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))

186.70/62.60

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 186.70/62.60
active(plus(z0, 0)) → mark(z0) 186.70/62.60
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 186.70/62.60
active(x(z0, 0)) → mark(0) 186.70/62.60
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 186.70/62.60
active(and(z0, z1)) → and(active(z0), z1) 186.70/62.60
active(plus(z0, z1)) → plus(active(z0), z1) 186.70/62.60
active(plus(z0, z1)) → plus(z0, active(z1)) 186.70/62.60
active(s(z0)) → s(active(z0)) 186.70/62.60
active(x(z0, z1)) → x(active(z0), z1) 186.70/62.60
active(x(z0, z1)) → x(z0, active(z1)) 186.70/62.60
and(mark(z0), z1) → mark(and(z0, z1)) 186.70/62.60
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 186.70/62.60
plus(mark(z0), z1) → mark(plus(z0, z1)) 186.70/62.60
plus(z0, mark(z1)) → mark(plus(z0, z1)) 186.70/62.60
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 186.70/62.60
s(mark(z0)) → mark(s(z0)) 186.70/62.60
s(ok(z0)) → ok(s(z0)) 186.70/62.60
x(mark(z0), z1) → mark(x(z0, z1)) 186.70/62.60
x(z0, mark(z1)) → mark(x(z0, z1)) 186.70/62.60
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 186.70/62.60
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 186.70/62.60
proper(tt) → ok(tt) 186.70/62.60
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 186.70/62.60
proper(0) → ok(0) 186.70/62.60
proper(s(z0)) → s(proper(z0)) 186.70/62.60
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 186.70/62.60
top(mark(z0)) → top(proper(z0)) 186.70/62.60
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 186.70/62.60
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.60
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.60
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.60
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.60
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.60
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.60
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.60
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.60
S(mark(z0)) → c16(S(z0)) 186.70/62.60
S(ok(z0)) → c17(S(z0)) 186.70/62.60
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.60
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.60
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.60
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.60
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.60
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.60
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.60
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.60
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 186.70/62.60
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 186.70/62.60
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 186.70/62.60
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 186.70/62.60
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 186.70/62.60
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 186.70/62.60
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 186.70/62.60
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 186.70/62.60
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 186.70/62.60
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 186.70/62.60
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 186.70/62.60
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 186.70/62.60
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 186.70/62.60
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 186.70/62.60
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0))) 186.70/62.60
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 186.70/62.60
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 186.70/62.60
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 186.70/62.60
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 186.70/62.60
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 186.70/62.60
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 186.70/62.60
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 186.70/62.60
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 186.70/62.60
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0)) 186.70/62.60
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0)) 186.70/62.60
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(x1)) 186.70/62.60
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(x1)) 186.70/62.60
PROPER(plus(x0, and(z0, z1))) → c23(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 186.70/62.60
PROPER(plus(x0, tt)) → c23(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt)) 186.70/62.60
PROPER(plus(x0, plus(z0, z1))) → c23(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 186.70/62.60
PROPER(plus(x0, 0)) → c23(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 186.70/62.60
PROPER(plus(x0, s(z0))) → c23(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 186.70/62.60
PROPER(plus(x0, x(z0, z1))) → c23(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 186.70/62.60
PROPER(plus(and(z0, z1), x1)) → c23(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 186.70/62.60
PROPER(plus(tt, x1)) → c23(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1)) 186.70/62.60
PROPER(plus(plus(z0, z1), x1)) → c23(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 186.70/62.60
PROPER(plus(0, x1)) → c23(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 186.70/62.60
PROPER(plus(s(z0), x1)) → c23(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 186.70/62.60
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)) 186.70/62.60
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.60
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.60
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.60
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.60
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.60
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.60
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.60
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.60
S(mark(z0)) → c16(S(z0)) 186.70/62.60
S(ok(z0)) → c17(S(z0)) 186.70/62.60
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.60
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.60
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.60
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.60
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.60
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 186.70/62.60
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 186.70/62.60
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 186.70/62.60
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 186.70/62.60
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 186.70/62.60
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 186.70/62.60
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 186.70/62.60
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 186.70/62.60
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 186.70/62.60
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 186.70/62.60
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 186.70/62.60
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 186.70/62.60
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 186.70/62.60
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 186.70/62.60
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.60
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.60
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.60
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.60
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

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

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

186.70/62.60

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

Removed 4 trailing tuple parts

186.70/62.60

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 186.70/62.60
active(plus(z0, 0)) → mark(z0) 186.70/62.60
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 186.70/62.60
active(x(z0, 0)) → mark(0) 186.70/62.60
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 186.70/62.60
active(and(z0, z1)) → and(active(z0), z1) 186.70/62.60
active(plus(z0, z1)) → plus(active(z0), z1) 186.70/62.60
active(plus(z0, z1)) → plus(z0, active(z1)) 186.70/62.60
active(s(z0)) → s(active(z0)) 186.70/62.60
active(x(z0, z1)) → x(active(z0), z1) 186.70/62.60
active(x(z0, z1)) → x(z0, active(z1)) 186.70/62.60
and(mark(z0), z1) → mark(and(z0, z1)) 186.70/62.60
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 186.70/62.60
plus(mark(z0), z1) → mark(plus(z0, z1)) 186.70/62.60
plus(z0, mark(z1)) → mark(plus(z0, z1)) 186.70/62.60
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 186.70/62.60
s(mark(z0)) → mark(s(z0)) 186.70/62.60
s(ok(z0)) → ok(s(z0)) 186.70/62.60
x(mark(z0), z1) → mark(x(z0, z1)) 186.70/62.60
x(z0, mark(z1)) → mark(x(z0, z1)) 186.70/62.60
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 186.70/62.60
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 186.70/62.60
proper(tt) → ok(tt) 186.70/62.60
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 186.70/62.60
proper(0) → ok(0) 186.70/62.60
proper(s(z0)) → s(proper(z0)) 186.70/62.60
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 186.70/62.60
top(mark(z0)) → top(proper(z0)) 186.70/62.60
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 186.70/62.60
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.60
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.60
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.60
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.60
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.60
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.60
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.60
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.60
S(mark(z0)) → c16(S(z0)) 186.70/62.60
S(ok(z0)) → c17(S(z0)) 186.70/62.60
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.60
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.60
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.60
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 186.70/62.60
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 186.70/62.60
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 186.70/62.60
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.60
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.60
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 186.70/62.60
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 186.70/62.60
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 186.70/62.60
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 186.70/62.60
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 186.70/62.60
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 186.70/62.60
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 186.70/62.60
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 186.70/62.60
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 186.70/62.60
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 186.70/62.60
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 186.70/62.60
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 186.70/62.60
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 186.70/62.60
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 186.70/62.60
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0))) 186.70/62.60
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 186.70/62.60
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 186.70/62.60
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 186.70/62.60
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 186.70/62.60
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 186.70/62.60
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 186.70/62.60
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 186.70/62.60
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 186.70/62.60
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0)) 186.70/62.60
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0)) 186.70/62.60
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(x1)) 186.70/62.60
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(x1)) 186.70/62.60
PROPER(plus(x0, and(z0, z1))) → c23(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 186.70/62.60
PROPER(plus(x0, plus(z0, z1))) → c23(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 186.70/62.60
PROPER(plus(x0, s(z0))) → c23(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 186.70/62.60
PROPER(plus(x0, x(z0, z1))) → c23(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 186.70/62.60
PROPER(plus(and(z0, z1), x1)) → c23(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 186.70/62.60
PROPER(plus(plus(z0, z1), x1)) → c23(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 186.70/62.60
PROPER(plus(s(z0), x1)) → c23(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 186.70/62.60
PROPER(plus(x(z0, z1), x1)) → c23(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 186.70/62.60
PROPER(plus(x0, tt)) → c23(PLUS(proper(x0), ok(tt)), PROPER(x0)) 186.70/62.60
PROPER(plus(x0, 0)) → c23(PLUS(proper(x0), ok(0)), PROPER(x0)) 186.70/62.60
PROPER(plus(tt, x1)) → c23(PLUS(ok(tt), proper(x1)), PROPER(x1)) 186.70/62.60
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)) 186.70/62.60
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 186.70/62.60
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 186.70/62.60
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 186.70/62.60
AND(mark(z0), z1) → c11(AND(z0, z1)) 186.70/62.60
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 186.70/62.60
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 186.70/62.60
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 186.70/62.60
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 186.70/62.60
S(mark(z0)) → c16(S(z0)) 186.70/62.60
S(ok(z0)) → c17(S(z0)) 186.70/62.60
X(mark(z0), z1) → c18(X(z0, z1)) 186.70/62.60
X(z0, mark(z1)) → c19(X(z0, z1)) 186.70/62.60
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 186.70/62.60
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 186.70/62.60
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 186.70/62.60
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 186.70/62.60
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 186.70/62.60
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 186.70/62.60
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 186.70/62.60
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 186.70/62.60
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 186.70/62.60
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 186.70/62.60
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 186.70/62.60
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 186.70/62.60
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 186.70/62.60
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 186.70/62.60
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 187.06/62.62
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 187.06/62.62
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 187.06/62.62
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 187.06/62.62
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 187.06/62.62
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 187.06/62.62
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 187.06/62.62
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 187.06/62.62
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 187.06/62.62
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 187.06/62.62
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 187.06/62.62
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 187.06/62.62
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 187.06/62.62
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

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

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

187.06/62.62

(27) 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))) 187.06/62.62
PROPER(s(tt)) → c25(S(ok(tt)), PROPER(tt)) 187.06/62.62
PROPER(s(plus(z0, z1))) → c25(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 187.06/62.62
PROPER(s(0)) → c25(S(ok(0)), PROPER(0)) 187.06/62.62
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0))) 187.06/62.62
PROPER(s(x(z0, z1))) → c25(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))

187.06/62.62

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 187.06/62.62
active(plus(z0, 0)) → mark(z0) 187.06/62.62
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 187.06/62.62
active(x(z0, 0)) → mark(0) 187.06/62.62
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 187.06/62.62
active(and(z0, z1)) → and(active(z0), z1) 187.06/62.62
active(plus(z0, z1)) → plus(active(z0), z1) 187.06/62.62
active(plus(z0, z1)) → plus(z0, active(z1)) 187.06/62.62
active(s(z0)) → s(active(z0)) 187.06/62.62
active(x(z0, z1)) → x(active(z0), z1) 187.06/62.62
active(x(z0, z1)) → x(z0, active(z1)) 187.06/62.62
and(mark(z0), z1) → mark(and(z0, z1)) 187.06/62.62
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 187.06/62.62
plus(mark(z0), z1) → mark(plus(z0, z1)) 187.06/62.62
plus(z0, mark(z1)) → mark(plus(z0, z1)) 187.06/62.62
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 187.06/62.62
s(mark(z0)) → mark(s(z0)) 187.06/62.62
s(ok(z0)) → ok(s(z0)) 187.06/62.62
x(mark(z0), z1) → mark(x(z0, z1)) 187.06/62.62
x(z0, mark(z1)) → mark(x(z0, z1)) 187.06/62.62
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 187.06/62.62
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 187.06/62.62
proper(tt) → ok(tt) 187.06/62.62
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 187.06/62.62
proper(0) → ok(0) 187.06/62.62
proper(s(z0)) → s(proper(z0)) 187.06/62.62
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 187.06/62.62
top(mark(z0)) → top(proper(z0)) 187.06/62.62
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 187.06/62.62
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 187.06/62.62
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 187.06/62.62
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 187.06/62.62
AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.62
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.62
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.62
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.62
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.62
S(mark(z0)) → c16(S(z0)) 187.06/62.62
S(ok(z0)) → c17(S(z0)) 187.06/62.62
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.62
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.62
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 187.06/62.62
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 187.06/62.62
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 187.06/62.62
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 187.06/62.62
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 187.06/62.62
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 187.06/62.62
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 187.06/62.62
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 187.06/62.62
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 187.06/62.62
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 187.06/62.62
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 187.06/62.62
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 187.06/62.62
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 187.06/62.62
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 187.06/62.62
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 187.06/62.62
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 187.06/62.62
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 187.06/62.62
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 187.06/62.62
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 187.06/62.62
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 187.06/62.62
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 187.06/62.62
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 187.06/62.62
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 187.06/62.62
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 187.06/62.62
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 187.06/62.62
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 187.06/62.62
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 187.06/62.62
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0))) 187.06/62.62
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 187.06/62.62
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 187.06/62.62
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 187.06/62.62
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 187.06/62.62
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 187.06/62.62
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 187.06/62.62
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 187.06/62.62
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 187.06/62.62
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0)) 187.06/62.62
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0)) 187.06/62.62
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(x1)) 187.06/62.62
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(x1)) 187.06/62.62
PROPER(plus(x0, and(z0, z1))) → c23(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 187.06/62.62
PROPER(plus(x0, plus(z0, z1))) → c23(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 187.06/62.62
PROPER(plus(x0, s(z0))) → c23(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 187.06/62.62
PROPER(plus(x0, x(z0, z1))) → c23(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 187.06/62.62
PROPER(plus(and(z0, z1), x1)) → c23(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 187.06/62.62
PROPER(plus(plus(z0, z1), x1)) → c23(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 187.06/62.62
PROPER(plus(s(z0), x1)) → c23(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 187.06/62.62
PROPER(plus(x(z0, z1), x1)) → c23(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 187.06/62.62
PROPER(plus(x0, tt)) → c23(PLUS(proper(x0), ok(tt)), PROPER(x0)) 187.06/62.62
PROPER(plus(x0, 0)) → c23(PLUS(proper(x0), ok(0)), PROPER(x0)) 187.06/62.62
PROPER(plus(tt, x1)) → c23(PLUS(ok(tt), proper(x1)), PROPER(x1)) 187.06/62.62
PROPER(plus(0, x1)) → c23(PLUS(ok(0), proper(x1)), PROPER(x1)) 187.06/62.62
PROPER(s(and(z0, z1))) → c25(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 187.06/62.62
PROPER(s(tt)) → c25(S(ok(tt)), PROPER(tt)) 187.06/62.62
PROPER(s(plus(z0, z1))) → c25(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 187.06/62.62
PROPER(s(0)) → c25(S(ok(0)), PROPER(0)) 187.06/62.62
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0))) 187.06/62.62
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)) 187.06/62.62
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 187.06/62.62
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 187.06/62.62
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 187.06/62.62
AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.62
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.62
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.62
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.62
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.62
S(mark(z0)) → c16(S(z0)) 187.06/62.62
S(ok(z0)) → c17(S(z0)) 187.06/62.62
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.62
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.62
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 187.06/62.62
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 187.06/62.62
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 187.06/62.62
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 187.06/62.62
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 187.06/62.62
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 187.06/62.62
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 187.06/62.62
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 187.06/62.62
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 187.06/62.62
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 187.06/62.62
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 187.06/62.62
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 187.06/62.62
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 187.06/62.62
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 187.06/62.62
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 187.06/62.62
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 187.06/62.62
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 187.06/62.62
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 187.06/62.62
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 187.06/62.62
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 187.06/62.62
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 187.06/62.62
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 187.06/62.62
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 187.06/62.62
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 187.06/62.62
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 187.06/62.62
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 187.06/62.62
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 187.06/62.62
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 187.06/62.62
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 187.06/62.62
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

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

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

187.06/62.62

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

Removed 2 trailing tuple parts

187.06/62.62

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 187.06/62.62
active(plus(z0, 0)) → mark(z0) 187.06/62.62
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 187.06/62.62
active(x(z0, 0)) → mark(0) 187.06/62.62
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 187.06/62.62
active(and(z0, z1)) → and(active(z0), z1) 187.06/62.62
active(plus(z0, z1)) → plus(active(z0), z1) 187.06/62.62
active(plus(z0, z1)) → plus(z0, active(z1)) 187.06/62.62
active(s(z0)) → s(active(z0)) 187.06/62.62
active(x(z0, z1)) → x(active(z0), z1) 187.06/62.62
active(x(z0, z1)) → x(z0, active(z1)) 187.06/62.62
and(mark(z0), z1) → mark(and(z0, z1)) 187.06/62.62
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 187.06/62.62
plus(mark(z0), z1) → mark(plus(z0, z1)) 187.06/62.62
plus(z0, mark(z1)) → mark(plus(z0, z1)) 187.06/62.62
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 187.06/62.62
s(mark(z0)) → mark(s(z0)) 187.06/62.62
s(ok(z0)) → ok(s(z0)) 187.06/62.62
x(mark(z0), z1) → mark(x(z0, z1)) 187.06/62.62
x(z0, mark(z1)) → mark(x(z0, z1)) 187.06/62.62
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 187.06/62.62
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 187.06/62.62
proper(tt) → ok(tt) 187.06/62.62
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 187.06/62.62
proper(0) → ok(0) 187.06/62.62
proper(s(z0)) → s(proper(z0)) 187.06/62.62
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 187.06/62.62
top(mark(z0)) → top(proper(z0)) 187.06/62.62
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 187.06/62.62
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 187.06/62.62
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 187.06/62.62
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 187.06/62.62
AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.62
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.62
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.62
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.62
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.62
S(mark(z0)) → c16(S(z0)) 187.06/62.62
S(ok(z0)) → c17(S(z0)) 187.06/62.62
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.62
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.62
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 187.06/62.62
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 187.06/62.62
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 187.06/62.62
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 187.06/62.62
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 187.06/62.62
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 187.06/62.62
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 187.06/62.62
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 187.06/62.62
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 187.06/62.62
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 187.06/62.62
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 187.06/62.62
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 187.06/62.62
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 187.06/62.62
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 187.06/62.62
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 187.06/62.62
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 187.06/62.62
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 187.06/62.62
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 187.06/62.62
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 187.06/62.62
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 187.06/62.62
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 187.06/62.62
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 187.06/62.62
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 187.06/62.62
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 187.06/62.62
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 187.06/62.62
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 187.06/62.62
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 187.06/62.62
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0))) 187.06/62.62
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 187.06/62.62
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 187.06/62.62
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 187.06/62.62
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 187.06/62.62
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 187.06/62.62
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 187.06/62.62
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 187.06/62.62
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 187.06/62.62
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0)) 187.06/62.62
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0)) 187.06/62.62
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(x1)) 187.06/62.62
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(x1)) 187.06/62.62
PROPER(plus(x0, and(z0, z1))) → c23(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 187.06/62.62
PROPER(plus(x0, plus(z0, z1))) → c23(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 187.06/62.62
PROPER(plus(x0, s(z0))) → c23(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 187.06/62.62
PROPER(plus(x0, x(z0, z1))) → c23(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 187.06/62.62
PROPER(plus(and(z0, z1), x1)) → c23(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 187.06/62.62
PROPER(plus(plus(z0, z1), x1)) → c23(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 187.06/62.62
PROPER(plus(s(z0), x1)) → c23(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 187.06/62.62
PROPER(plus(x(z0, z1), x1)) → c23(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 187.06/62.62
PROPER(plus(x0, tt)) → c23(PLUS(proper(x0), ok(tt)), PROPER(x0)) 187.06/62.62
PROPER(plus(x0, 0)) → c23(PLUS(proper(x0), ok(0)), PROPER(x0)) 187.06/62.62
PROPER(plus(tt, x1)) → c23(PLUS(ok(tt), proper(x1)), PROPER(x1)) 187.06/62.62
PROPER(plus(0, x1)) → c23(PLUS(ok(0), proper(x1)), PROPER(x1)) 187.06/62.62
PROPER(s(and(z0, z1))) → c25(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 187.06/62.62
PROPER(s(plus(z0, z1))) → c25(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 187.06/62.62
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0))) 187.06/62.62
PROPER(s(x(z0, z1))) → c25(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1))) 187.06/62.63
PROPER(s(tt)) → c25(S(ok(tt))) 187.06/62.63
PROPER(s(0)) → c25(S(ok(0)))

S tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 187.06/62.63
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 187.06/62.63
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.63
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.63
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.63
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.63
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.63
S(mark(z0)) → c16(S(z0)) 187.06/62.63
S(ok(z0)) → c17(S(z0)) 187.06/62.63
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.63
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.63
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 187.06/62.63
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 187.06/62.63
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 187.06/62.63
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 187.06/62.63
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 187.06/62.63
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 187.06/62.63
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 187.06/62.63
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 187.06/62.63
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 187.06/62.63
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

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

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

187.06/62.63

(31) 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))) 187.06/62.63
PROPER(x(x0, tt)) → c26(X(proper(x0), ok(tt)), PROPER(x0), PROPER(tt)) 187.06/62.63
PROPER(x(x0, plus(z0, z1))) → c26(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(x(x0, 0)) → c26(X(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 187.06/62.63
PROPER(x(x0, s(z0))) → c26(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 187.06/62.63
PROPER(x(x0, x(z0, z1))) → c26(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 187.06/62.63
PROPER(x(and(z0, z1), x1)) → c26(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(x(tt, x1)) → c26(X(ok(tt), proper(x1)), PROPER(tt), PROPER(x1)) 187.06/62.63
PROPER(x(plus(z0, z1), x1)) → c26(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(x(0, x1)) → c26(X(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 187.06/62.63
PROPER(x(s(z0), x1)) → c26(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 187.06/62.63
PROPER(x(x(z0, z1), x1)) → c26(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))

187.06/62.63

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 187.06/62.63
active(plus(z0, 0)) → mark(z0) 187.06/62.63
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 187.06/62.63
active(x(z0, 0)) → mark(0) 187.06/62.63
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 187.06/62.63
active(and(z0, z1)) → and(active(z0), z1) 187.06/62.63
active(plus(z0, z1)) → plus(active(z0), z1) 187.06/62.63
active(plus(z0, z1)) → plus(z0, active(z1)) 187.06/62.63
active(s(z0)) → s(active(z0)) 187.06/62.63
active(x(z0, z1)) → x(active(z0), z1) 187.06/62.63
active(x(z0, z1)) → x(z0, active(z1)) 187.06/62.63
and(mark(z0), z1) → mark(and(z0, z1)) 187.06/62.63
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 187.06/62.63
plus(mark(z0), z1) → mark(plus(z0, z1)) 187.06/62.63
plus(z0, mark(z1)) → mark(plus(z0, z1)) 187.06/62.63
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 187.06/62.63
s(mark(z0)) → mark(s(z0)) 187.06/62.63
s(ok(z0)) → ok(s(z0)) 187.06/62.63
x(mark(z0), z1) → mark(x(z0, z1)) 187.06/62.63
x(z0, mark(z1)) → mark(x(z0, z1)) 187.06/62.63
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 187.06/62.63
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 187.06/62.63
proper(tt) → ok(tt) 187.06/62.63
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 187.06/62.63
proper(0) → ok(0) 187.06/62.63
proper(s(z0)) → s(proper(z0)) 187.06/62.63
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 187.06/62.63
top(mark(z0)) → top(proper(z0)) 187.06/62.63
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 187.06/62.63
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 187.06/62.63
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.63
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.63
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.63
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.63
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.63
S(mark(z0)) → c16(S(z0)) 187.06/62.63
S(ok(z0)) → c17(S(z0)) 187.06/62.63
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.63
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.63
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 187.06/62.63
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 187.06/62.63
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 187.06/62.63
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 187.06/62.63
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 187.06/62.63
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 187.06/62.63
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0))) 187.06/62.63
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 187.06/62.63
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 187.06/62.63
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 187.06/62.63
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 187.06/62.63
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0)) 187.06/62.63
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0)) 187.06/62.63
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(plus(x0, and(z0, z1))) → c23(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 187.06/62.63
PROPER(plus(x0, plus(z0, z1))) → c23(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(plus(x0, s(z0))) → c23(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 187.06/62.63
PROPER(plus(x0, x(z0, z1))) → c23(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 187.06/62.63
PROPER(plus(and(z0, z1), x1)) → c23(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(plus(plus(z0, z1), x1)) → c23(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(plus(s(z0), x1)) → c23(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 187.06/62.63
PROPER(plus(x(z0, z1), x1)) → c23(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(plus(x0, tt)) → c23(PLUS(proper(x0), ok(tt)), PROPER(x0)) 187.06/62.63
PROPER(plus(x0, 0)) → c23(PLUS(proper(x0), ok(0)), PROPER(x0)) 187.06/62.63
PROPER(plus(tt, x1)) → c23(PLUS(ok(tt), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(plus(0, x1)) → c23(PLUS(ok(0), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(s(and(z0, z1))) → c25(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 187.06/62.63
PROPER(s(plus(z0, z1))) → c25(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0))) 187.06/62.63
PROPER(s(x(z0, z1))) → c25(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1))) 187.06/62.63
PROPER(s(tt)) → c25(S(ok(tt))) 187.06/62.63
PROPER(s(0)) → c25(S(ok(0))) 187.06/62.63
PROPER(x(x0, and(z0, z1))) → c26(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 187.06/62.63
PROPER(x(x0, tt)) → c26(X(proper(x0), ok(tt)), PROPER(x0), PROPER(tt)) 187.06/62.63
PROPER(x(x0, plus(z0, z1))) → c26(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(x(x0, 0)) → c26(X(proper(x0), ok(0)), PROPER(x0), PROPER(0)) 187.06/62.63
PROPER(x(x0, s(z0))) → c26(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 187.06/62.63
PROPER(x(x0, x(z0, z1))) → c26(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 187.06/62.63
PROPER(x(and(z0, z1), x1)) → c26(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(x(tt, x1)) → c26(X(ok(tt), proper(x1)), PROPER(tt), PROPER(x1)) 187.06/62.63
PROPER(x(plus(z0, z1), x1)) → c26(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(x(0, x1)) → c26(X(ok(0), proper(x1)), PROPER(0), PROPER(x1)) 187.06/62.63
PROPER(x(s(z0), x1)) → c26(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 187.06/62.63
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)) 187.06/62.63
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 187.06/62.63
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.63
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.63
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.63
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.63
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.63
S(mark(z0)) → c16(S(z0)) 187.06/62.63
S(ok(z0)) → c17(S(z0)) 187.06/62.63
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.63
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.63
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 187.06/62.63
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 187.06/62.63
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 187.06/62.63
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 187.06/62.63
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 187.06/62.63
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 187.06/62.63
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 187.06/62.63
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 187.06/62.63
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 187.06/62.63
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

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

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

187.06/62.63

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

Removed 4 trailing tuple parts

187.06/62.63

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 187.06/62.63
active(plus(z0, 0)) → mark(z0) 187.06/62.63
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 187.06/62.63
active(x(z0, 0)) → mark(0) 187.06/62.63
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 187.06/62.63
active(and(z0, z1)) → and(active(z0), z1) 187.06/62.63
active(plus(z0, z1)) → plus(active(z0), z1) 187.06/62.63
active(plus(z0, z1)) → plus(z0, active(z1)) 187.06/62.63
active(s(z0)) → s(active(z0)) 187.06/62.63
active(x(z0, z1)) → x(active(z0), z1) 187.06/62.63
active(x(z0, z1)) → x(z0, active(z1)) 187.06/62.63
and(mark(z0), z1) → mark(and(z0, z1)) 187.06/62.63
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 187.06/62.63
plus(mark(z0), z1) → mark(plus(z0, z1)) 187.06/62.63
plus(z0, mark(z1)) → mark(plus(z0, z1)) 187.06/62.63
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 187.06/62.63
s(mark(z0)) → mark(s(z0)) 187.06/62.63
s(ok(z0)) → ok(s(z0)) 187.06/62.63
x(mark(z0), z1) → mark(x(z0, z1)) 187.06/62.63
x(z0, mark(z1)) → mark(x(z0, z1)) 187.06/62.63
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 187.06/62.63
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 187.06/62.63
proper(tt) → ok(tt) 187.06/62.63
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 187.06/62.63
proper(0) → ok(0) 187.06/62.63
proper(s(z0)) → s(proper(z0)) 187.06/62.63
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 187.06/62.63
top(mark(z0)) → top(proper(z0)) 187.06/62.63
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 187.06/62.63
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 187.06/62.63
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.63
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.63
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.63
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.63
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.63
S(mark(z0)) → c16(S(z0)) 187.06/62.63
S(ok(z0)) → c17(S(z0)) 187.06/62.63
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.63
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.63
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 187.06/62.63
TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 187.06/62.63
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 187.06/62.63
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 187.06/62.63
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 187.06/62.63
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 187.06/62.63
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0))) 187.06/62.63
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 187.06/62.63
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 187.06/62.63
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 187.06/62.63
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 187.06/62.63
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0)) 187.06/62.63
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0)) 187.06/62.63
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(plus(x0, and(z0, z1))) → c23(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 187.06/62.63
PROPER(plus(x0, plus(z0, z1))) → c23(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(plus(x0, s(z0))) → c23(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 187.06/62.63
PROPER(plus(x0, x(z0, z1))) → c23(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 187.06/62.63
PROPER(plus(and(z0, z1), x1)) → c23(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(plus(plus(z0, z1), x1)) → c23(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(plus(s(z0), x1)) → c23(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 187.06/62.63
PROPER(plus(x(z0, z1), x1)) → c23(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(plus(x0, tt)) → c23(PLUS(proper(x0), ok(tt)), PROPER(x0)) 187.06/62.63
PROPER(plus(x0, 0)) → c23(PLUS(proper(x0), ok(0)), PROPER(x0)) 187.06/62.63
PROPER(plus(tt, x1)) → c23(PLUS(ok(tt), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(plus(0, x1)) → c23(PLUS(ok(0), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(s(and(z0, z1))) → c25(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 187.06/62.63
PROPER(s(plus(z0, z1))) → c25(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0))) 187.06/62.63
PROPER(s(x(z0, z1))) → c25(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1))) 187.06/62.63
PROPER(s(tt)) → c25(S(ok(tt))) 187.06/62.63
PROPER(s(0)) → c25(S(ok(0))) 187.06/62.63
PROPER(x(x0, and(z0, z1))) → c26(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 187.06/62.63
PROPER(x(x0, plus(z0, z1))) → c26(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(x(x0, s(z0))) → c26(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 187.06/62.63
PROPER(x(x0, x(z0, z1))) → c26(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 187.06/62.63
PROPER(x(and(z0, z1), x1)) → c26(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(x(plus(z0, z1), x1)) → c26(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(x(s(z0), x1)) → c26(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 187.06/62.63
PROPER(x(x(z0, z1), x1)) → c26(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(x(x0, tt)) → c26(X(proper(x0), ok(tt)), PROPER(x0)) 187.06/62.63
PROPER(x(x0, 0)) → c26(X(proper(x0), ok(0)), PROPER(x0)) 187.06/62.63
PROPER(x(tt, x1)) → c26(X(ok(tt), proper(x1)), PROPER(x1)) 187.06/62.63
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)) 187.06/62.63
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 187.06/62.63
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.63
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.63
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.63
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.63
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.63
S(mark(z0)) → c16(S(z0)) 187.06/62.63
S(ok(z0)) → c17(S(z0)) 187.06/62.63
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.63
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.63
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 187.06/62.63
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 187.06/62.63
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 187.06/62.63
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 187.06/62.63
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 187.06/62.63
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 187.06/62.63
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 187.06/62.63
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 187.06/62.63
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 187.06/62.63
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

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

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

187.06/62.63

(35) 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))) 187.06/62.63
TOP(mark(tt)) → c27(TOP(ok(tt)), PROPER(tt)) 187.06/62.63
TOP(mark(plus(z0, z1))) → c27(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 187.06/62.63
TOP(mark(0)) → c27(TOP(ok(0)), PROPER(0)) 187.06/62.63
TOP(mark(s(z0))) → c27(TOP(s(proper(z0))), PROPER(s(z0))) 187.06/62.63
TOP(mark(x(z0, z1))) → c27(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))

187.06/62.63

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 187.06/62.63
active(plus(z0, 0)) → mark(z0) 187.06/62.63
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 187.06/62.63
active(x(z0, 0)) → mark(0) 187.06/62.63
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 187.06/62.63
active(and(z0, z1)) → and(active(z0), z1) 187.06/62.63
active(plus(z0, z1)) → plus(active(z0), z1) 187.06/62.63
active(plus(z0, z1)) → plus(z0, active(z1)) 187.06/62.63
active(s(z0)) → s(active(z0)) 187.06/62.63
active(x(z0, z1)) → x(active(z0), z1) 187.06/62.63
active(x(z0, z1)) → x(z0, active(z1)) 187.06/62.63
and(mark(z0), z1) → mark(and(z0, z1)) 187.06/62.63
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 187.06/62.63
plus(mark(z0), z1) → mark(plus(z0, z1)) 187.06/62.63
plus(z0, mark(z1)) → mark(plus(z0, z1)) 187.06/62.63
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 187.06/62.63
s(mark(z0)) → mark(s(z0)) 187.06/62.63
s(ok(z0)) → ok(s(z0)) 187.06/62.63
x(mark(z0), z1) → mark(x(z0, z1)) 187.06/62.63
x(z0, mark(z1)) → mark(x(z0, z1)) 187.06/62.63
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 187.06/62.63
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 187.06/62.63
proper(tt) → ok(tt) 187.06/62.63
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 187.06/62.63
proper(0) → ok(0) 187.06/62.63
proper(s(z0)) → s(proper(z0)) 187.06/62.63
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 187.06/62.63
top(mark(z0)) → top(proper(z0)) 187.06/62.63
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 187.06/62.63
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 187.06/62.63
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.63
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.63
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.63
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.63
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.63
S(mark(z0)) → c16(S(z0)) 187.06/62.63
S(ok(z0)) → c17(S(z0)) 187.06/62.63
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.63
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.63
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 187.06/62.63
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 187.06/62.63
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 187.06/62.63
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 187.06/62.63
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 187.06/62.63
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0))) 187.06/62.63
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 187.06/62.63
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 187.06/62.63
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 187.06/62.63
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 187.06/62.63
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0)) 187.06/62.63
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0)) 187.06/62.63
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(plus(x0, and(z0, z1))) → c23(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 187.06/62.63
PROPER(plus(x0, plus(z0, z1))) → c23(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(plus(x0, s(z0))) → c23(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 187.06/62.63
PROPER(plus(x0, x(z0, z1))) → c23(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 187.06/62.63
PROPER(plus(and(z0, z1), x1)) → c23(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(plus(plus(z0, z1), x1)) → c23(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(plus(s(z0), x1)) → c23(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 187.06/62.63
PROPER(plus(x(z0, z1), x1)) → c23(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(plus(x0, tt)) → c23(PLUS(proper(x0), ok(tt)), PROPER(x0)) 187.06/62.63
PROPER(plus(x0, 0)) → c23(PLUS(proper(x0), ok(0)), PROPER(x0)) 187.06/62.63
PROPER(plus(tt, x1)) → c23(PLUS(ok(tt), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(plus(0, x1)) → c23(PLUS(ok(0), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(s(and(z0, z1))) → c25(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 187.06/62.63
PROPER(s(plus(z0, z1))) → c25(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0))) 187.06/62.63
PROPER(s(x(z0, z1))) → c25(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1))) 187.06/62.63
PROPER(s(tt)) → c25(S(ok(tt))) 187.06/62.63
PROPER(s(0)) → c25(S(ok(0))) 187.06/62.63
PROPER(x(x0, and(z0, z1))) → c26(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 187.06/62.63
PROPER(x(x0, plus(z0, z1))) → c26(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(x(x0, s(z0))) → c26(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 187.06/62.63
PROPER(x(x0, x(z0, z1))) → c26(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 187.06/62.63
PROPER(x(and(z0, z1), x1)) → c26(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(x(plus(z0, z1), x1)) → c26(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(x(s(z0), x1)) → c26(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 187.06/62.63
PROPER(x(x(z0, z1), x1)) → c26(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(x(x0, tt)) → c26(X(proper(x0), ok(tt)), PROPER(x0)) 187.06/62.63
PROPER(x(x0, 0)) → c26(X(proper(x0), ok(0)), PROPER(x0)) 187.06/62.63
PROPER(x(tt, x1)) → c26(X(ok(tt), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(x(0, x1)) → c26(X(ok(0), proper(x1)), PROPER(x1)) 187.06/62.63
TOP(mark(and(z0, z1))) → c27(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 187.06/62.63
TOP(mark(tt)) → c27(TOP(ok(tt)), PROPER(tt)) 187.06/62.63
TOP(mark(plus(z0, z1))) → c27(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 187.06/62.63
TOP(mark(0)) → c27(TOP(ok(0)), PROPER(0)) 187.06/62.63
TOP(mark(s(z0))) → c27(TOP(s(proper(z0))), PROPER(s(z0))) 187.06/62.63
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)) 187.06/62.63
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 187.06/62.63
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.63
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.63
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.63
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.63
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.63
S(mark(z0)) → c16(S(z0)) 187.06/62.63
S(ok(z0)) → c17(S(z0)) 187.06/62.63
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.63
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.63
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 187.06/62.63
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 187.06/62.63
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 187.06/62.63
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 187.06/62.63
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 187.06/62.63
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 187.06/62.63
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 187.06/62.63
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 187.06/62.63
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 187.06/62.63
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

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

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

187.06/62.63

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

Removed 2 trailing tuple parts

187.06/62.63

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 187.06/62.63
active(plus(z0, 0)) → mark(z0) 187.06/62.63
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 187.06/62.63
active(x(z0, 0)) → mark(0) 187.06/62.63
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 187.06/62.63
active(and(z0, z1)) → and(active(z0), z1) 187.06/62.63
active(plus(z0, z1)) → plus(active(z0), z1) 187.06/62.63
active(plus(z0, z1)) → plus(z0, active(z1)) 187.06/62.63
active(s(z0)) → s(active(z0)) 187.06/62.63
active(x(z0, z1)) → x(active(z0), z1) 187.06/62.63
active(x(z0, z1)) → x(z0, active(z1)) 187.06/62.63
and(mark(z0), z1) → mark(and(z0, z1)) 187.06/62.63
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 187.06/62.63
plus(mark(z0), z1) → mark(plus(z0, z1)) 187.06/62.63
plus(z0, mark(z1)) → mark(plus(z0, z1)) 187.06/62.63
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 187.06/62.63
s(mark(z0)) → mark(s(z0)) 187.06/62.63
s(ok(z0)) → ok(s(z0)) 187.06/62.63
x(mark(z0), z1) → mark(x(z0, z1)) 187.06/62.63
x(z0, mark(z1)) → mark(x(z0, z1)) 187.06/62.63
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 187.06/62.63
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 187.06/62.63
proper(tt) → ok(tt) 187.06/62.63
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 187.06/62.63
proper(0) → ok(0) 187.06/62.63
proper(s(z0)) → s(proper(z0)) 187.06/62.63
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 187.06/62.63
top(mark(z0)) → top(proper(z0)) 187.06/62.63
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 187.06/62.63
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 187.06/62.63
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.63
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.63
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.63
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.63
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.63
S(mark(z0)) → c16(S(z0)) 187.06/62.63
S(ok(z0)) → c17(S(z0)) 187.06/62.63
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.63
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.63
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 187.06/62.63
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 187.06/62.63
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 187.06/62.63
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 187.06/62.63
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 187.06/62.63
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0))) 187.06/62.63
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 187.06/62.63
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 187.06/62.63
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 187.06/62.63
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 187.06/62.63
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0)) 187.06/62.63
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0)) 187.06/62.63
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(plus(x0, and(z0, z1))) → c23(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 187.06/62.63
PROPER(plus(x0, plus(z0, z1))) → c23(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(plus(x0, s(z0))) → c23(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 187.06/62.63
PROPER(plus(x0, x(z0, z1))) → c23(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 187.06/62.63
PROPER(plus(and(z0, z1), x1)) → c23(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(plus(plus(z0, z1), x1)) → c23(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(plus(s(z0), x1)) → c23(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 187.06/62.63
PROPER(plus(x(z0, z1), x1)) → c23(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(plus(x0, tt)) → c23(PLUS(proper(x0), ok(tt)), PROPER(x0)) 187.06/62.63
PROPER(plus(x0, 0)) → c23(PLUS(proper(x0), ok(0)), PROPER(x0)) 187.06/62.63
PROPER(plus(tt, x1)) → c23(PLUS(ok(tt), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(plus(0, x1)) → c23(PLUS(ok(0), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(s(and(z0, z1))) → c25(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 187.06/62.63
PROPER(s(plus(z0, z1))) → c25(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0))) 187.06/62.63
PROPER(s(x(z0, z1))) → c25(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1))) 187.06/62.63
PROPER(s(tt)) → c25(S(ok(tt))) 187.06/62.63
PROPER(s(0)) → c25(S(ok(0))) 187.06/62.63
PROPER(x(x0, and(z0, z1))) → c26(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 187.06/62.63
PROPER(x(x0, plus(z0, z1))) → c26(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(x(x0, s(z0))) → c26(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 187.06/62.63
PROPER(x(x0, x(z0, z1))) → c26(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 187.06/62.63
PROPER(x(and(z0, z1), x1)) → c26(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(x(plus(z0, z1), x1)) → c26(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(x(s(z0), x1)) → c26(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 187.06/62.63
PROPER(x(x(z0, z1), x1)) → c26(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(x(x0, tt)) → c26(X(proper(x0), ok(tt)), PROPER(x0)) 187.06/62.63
PROPER(x(x0, 0)) → c26(X(proper(x0), ok(0)), PROPER(x0)) 187.06/62.63
PROPER(x(tt, x1)) → c26(X(ok(tt), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(x(0, x1)) → c26(X(ok(0), proper(x1)), PROPER(x1)) 187.06/62.63
TOP(mark(and(z0, z1))) → c27(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 187.06/62.63
TOP(mark(plus(z0, z1))) → c27(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 187.06/62.63
TOP(mark(s(z0))) → c27(TOP(s(proper(z0))), PROPER(s(z0))) 187.06/62.63
TOP(mark(x(z0, z1))) → c27(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1))) 187.06/62.63
TOP(mark(tt)) → c27(TOP(ok(tt))) 187.06/62.63
TOP(mark(0)) → c27(TOP(ok(0)))

S tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 187.06/62.63
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 187.06/62.63
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.63
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.63
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.63
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.63
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.63
S(mark(z0)) → c16(S(z0)) 187.06/62.63
S(ok(z0)) → c17(S(z0)) 187.06/62.63
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.63
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.63
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 187.06/62.63
TOP(ok(z0)) → c28(TOP(active(z0)), ACTIVE(z0)) 187.06/62.63
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 187.06/62.63
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 187.06/62.63
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 187.06/62.63
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0)))

K tuples:

TOP(mark(z0)) → c27(TOP(proper(z0)), PROPER(z0)) 187.06/62.63
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 187.06/62.63
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 187.06/62.63
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 187.06/62.63
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

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

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

187.06/62.63

(39) 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))) 187.06/62.63
TOP(ok(plus(z0, 0))) → c28(TOP(mark(z0)), ACTIVE(plus(z0, 0))) 187.06/62.63
TOP(ok(plus(z0, s(z1)))) → c28(TOP(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
TOP(ok(x(z0, 0))) → c28(TOP(mark(0)), ACTIVE(x(z0, 0))) 187.06/62.63
TOP(ok(x(z0, s(z1)))) → c28(TOP(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 187.06/62.63
TOP(ok(and(z0, z1))) → c28(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1))) 187.06/62.63
TOP(ok(plus(z0, z1))) → c28(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 187.06/62.63
TOP(ok(plus(z0, z1))) → c28(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 187.06/62.63
TOP(ok(s(z0))) → c28(TOP(s(active(z0))), ACTIVE(s(z0))) 187.06/62.63
TOP(ok(x(z0, z1))) → c28(TOP(x(active(z0), z1)), ACTIVE(x(z0, z1))) 187.06/62.63
TOP(ok(x(z0, z1))) → c28(TOP(x(z0, active(z1))), ACTIVE(x(z0, z1)))

187.06/62.63

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 187.06/62.63
active(plus(z0, 0)) → mark(z0) 187.06/62.63
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 187.06/62.63
active(x(z0, 0)) → mark(0) 187.06/62.63
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 187.06/62.63
active(and(z0, z1)) → and(active(z0), z1) 187.06/62.63
active(plus(z0, z1)) → plus(active(z0), z1) 187.06/62.63
active(plus(z0, z1)) → plus(z0, active(z1)) 187.06/62.63
active(s(z0)) → s(active(z0)) 187.06/62.63
active(x(z0, z1)) → x(active(z0), z1) 187.06/62.63
active(x(z0, z1)) → x(z0, active(z1)) 187.06/62.63
and(mark(z0), z1) → mark(and(z0, z1)) 187.06/62.63
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 187.06/62.63
plus(mark(z0), z1) → mark(plus(z0, z1)) 187.06/62.63
plus(z0, mark(z1)) → mark(plus(z0, z1)) 187.06/62.63
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 187.06/62.63
s(mark(z0)) → mark(s(z0)) 187.06/62.63
s(ok(z0)) → ok(s(z0)) 187.06/62.63
x(mark(z0), z1) → mark(x(z0, z1)) 187.06/62.63
x(z0, mark(z1)) → mark(x(z0, z1)) 187.06/62.63
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 187.06/62.63
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 187.06/62.63
proper(tt) → ok(tt) 187.06/62.63
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 187.06/62.63
proper(0) → ok(0) 187.06/62.63
proper(s(z0)) → s(proper(z0)) 187.06/62.63
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 187.06/62.63
top(mark(z0)) → top(proper(z0)) 187.06/62.63
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(plus(z0, z1)) → c6(PLUS(active(z0), z1), ACTIVE(z0)) 187.06/62.63
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 187.06/62.63
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.63
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.63
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.63
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.63
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.63
S(mark(z0)) → c16(S(z0)) 187.06/62.63
S(ok(z0)) → c17(S(z0)) 187.06/62.63
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.63
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.63
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 187.06/62.63
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 187.06/62.63
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 187.06/62.63
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 187.06/62.63
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0))) 187.06/62.63
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 187.06/62.63
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 187.06/62.63
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 187.06/62.63
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 187.06/62.63
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0)) 187.06/62.63
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0)) 187.06/62.63
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(plus(x0, and(z0, z1))) → c23(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 187.06/62.63
PROPER(plus(x0, plus(z0, z1))) → c23(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(plus(x0, s(z0))) → c23(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 187.06/62.63
PROPER(plus(x0, x(z0, z1))) → c23(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 187.06/62.63
PROPER(plus(and(z0, z1), x1)) → c23(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(plus(plus(z0, z1), x1)) → c23(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(plus(s(z0), x1)) → c23(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 187.06/62.63
PROPER(plus(x(z0, z1), x1)) → c23(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(plus(x0, tt)) → c23(PLUS(proper(x0), ok(tt)), PROPER(x0)) 187.06/62.63
PROPER(plus(x0, 0)) → c23(PLUS(proper(x0), ok(0)), PROPER(x0)) 187.06/62.63
PROPER(plus(tt, x1)) → c23(PLUS(ok(tt), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(plus(0, x1)) → c23(PLUS(ok(0), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(s(and(z0, z1))) → c25(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 187.06/62.63
PROPER(s(plus(z0, z1))) → c25(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0))) 187.06/62.63
PROPER(s(x(z0, z1))) → c25(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1))) 187.06/62.63
PROPER(s(tt)) → c25(S(ok(tt))) 187.06/62.63
PROPER(s(0)) → c25(S(ok(0))) 187.06/62.63
PROPER(x(x0, and(z0, z1))) → c26(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 187.06/62.63
PROPER(x(x0, plus(z0, z1))) → c26(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(x(x0, s(z0))) → c26(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 187.06/62.63
PROPER(x(x0, x(z0, z1))) → c26(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 187.06/62.63
PROPER(x(and(z0, z1), x1)) → c26(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(x(plus(z0, z1), x1)) → c26(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(x(s(z0), x1)) → c26(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 187.06/62.63
PROPER(x(x(z0, z1), x1)) → c26(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(x(x0, tt)) → c26(X(proper(x0), ok(tt)), PROPER(x0)) 187.06/62.63
PROPER(x(x0, 0)) → c26(X(proper(x0), ok(0)), PROPER(x0)) 187.06/62.63
PROPER(x(tt, x1)) → c26(X(ok(tt), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(x(0, x1)) → c26(X(ok(0), proper(x1)), PROPER(x1)) 187.06/62.63
TOP(mark(and(z0, z1))) → c27(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 187.06/62.63
TOP(mark(plus(z0, z1))) → c27(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 187.06/62.63
TOP(mark(s(z0))) → c27(TOP(s(proper(z0))), PROPER(s(z0))) 187.06/62.63
TOP(mark(x(z0, z1))) → c27(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1))) 187.06/62.63
TOP(mark(tt)) → c27(TOP(ok(tt))) 187.06/62.63
TOP(mark(0)) → c27(TOP(ok(0))) 187.06/62.63
TOP(ok(and(tt, z0))) → c28(TOP(mark(z0)), ACTIVE(and(tt, z0))) 187.06/62.63
TOP(ok(plus(z0, 0))) → c28(TOP(mark(z0)), ACTIVE(plus(z0, 0))) 187.06/62.63
TOP(ok(plus(z0, s(z1)))) → c28(TOP(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
TOP(ok(x(z0, 0))) → c28(TOP(mark(0)), ACTIVE(x(z0, 0))) 187.06/62.63
TOP(ok(x(z0, s(z1)))) → c28(TOP(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 187.06/62.63
TOP(ok(and(z0, z1))) → c28(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1))) 187.06/62.63
TOP(ok(plus(z0, z1))) → c28(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 187.06/62.63
TOP(ok(plus(z0, z1))) → c28(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 187.06/62.63
TOP(ok(s(z0))) → c28(TOP(s(active(z0))), ACTIVE(s(z0))) 187.06/62.63
TOP(ok(x(z0, z1))) → c28(TOP(x(active(z0), z1)), ACTIVE(x(z0, z1))) 187.06/62.63
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)) 187.06/62.63
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 187.06/62.63
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.63
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.63
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.63
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.63
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.63
S(mark(z0)) → c16(S(z0)) 187.06/62.63
S(ok(z0)) → c17(S(z0)) 187.06/62.63
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.63
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.63
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 187.06/62.63
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 187.06/62.63
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 187.06/62.63
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 187.06/62.63
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0))) 187.06/62.63
TOP(ok(and(tt, z0))) → c28(TOP(mark(z0)), ACTIVE(and(tt, z0))) 187.06/62.63
TOP(ok(plus(z0, 0))) → c28(TOP(mark(z0)), ACTIVE(plus(z0, 0))) 187.06/62.63
TOP(ok(plus(z0, s(z1)))) → c28(TOP(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
TOP(ok(x(z0, 0))) → c28(TOP(mark(0)), ACTIVE(x(z0, 0))) 187.06/62.63
TOP(ok(x(z0, s(z1)))) → c28(TOP(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 187.06/62.63
TOP(ok(and(z0, z1))) → c28(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1))) 187.06/62.63
TOP(ok(plus(z0, z1))) → c28(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 187.06/62.63
TOP(ok(plus(z0, z1))) → c28(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 187.06/62.63
TOP(ok(s(z0))) → c28(TOP(s(active(z0))), ACTIVE(s(z0))) 187.06/62.63
TOP(ok(x(z0, z1))) → c28(TOP(x(active(z0), z1)), ACTIVE(x(z0, z1))) 187.06/62.63
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)) 187.06/62.63
PROPER(and(z0, z1)) → c21(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 187.06/62.63
PROPER(s(z0)) → c25(S(proper(z0)), PROPER(z0)) 187.06/62.63
PROPER(x(z0, z1)) → c26(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 187.06/62.63
PROPER(plus(z0, z1)) → c23(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

Defined Rule Symbols:

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

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

187.06/62.63

(41) 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)) 187.06/62.63
ACTIVE(plus(z0, z1)) → c7(PLUS(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
ACTIVE(x(z0, z1)) → c9(X(active(z0), z1), ACTIVE(z0)) 187.06/62.63
ACTIVE(x(z0, z1)) → c10(X(z0, active(z1)), ACTIVE(z1)) 187.06/62.63
ACTIVE(plus(z0, s(z1))) → c2(PLUS(z0, z1)) 187.06/62.63
ACTIVE(x(z0, s(z1))) → c4(X(z0, z1)) 187.06/62.63
ACTIVE(and(plus(z0, 0), x1)) → c5(AND(mark(z0), x1), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(and(plus(z0, s(z1)), x1)) → c5(AND(mark(s(plus(z0, z1))), x1), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(and(x(z0, 0), x1)) → c5(AND(mark(0), x1), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(and(x(z0, s(z1)), x1)) → c5(AND(mark(plus(x(z0, z1), z0)), x1), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(and(and(z0, z1), x1)) → c5(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(plus(z0, z1), x1)) → c5(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(and(s(z0), x1)) → c5(AND(s(active(z0)), x1), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(x(z0, z1), x1)) → c5(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(and(and(tt, z0), x1)) → c5(AND(mark(z0), x1)) 187.06/62.63
ACTIVE(s(plus(z0, 0))) → c8(S(mark(z0)), ACTIVE(plus(z0, 0))) 187.06/62.63
ACTIVE(s(plus(z0, s(z1)))) → c8(S(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
ACTIVE(s(x(z0, 0))) → c8(S(mark(0)), ACTIVE(x(z0, 0))) 187.06/62.63
ACTIVE(s(x(z0, s(z1)))) → c8(S(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 187.06/62.63
ACTIVE(s(and(z0, z1))) → c8(S(and(active(z0), z1)), ACTIVE(and(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(plus(z0, z1))) → c8(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 187.06/62.63
ACTIVE(s(s(z0))) → c8(S(s(active(z0))), ACTIVE(s(z0))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(active(z0), z1)), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(x(z0, z1))) → c8(S(x(z0, active(z1))), ACTIVE(x(z0, z1))) 187.06/62.63
ACTIVE(s(and(tt, z0))) → c8(S(mark(z0))) 187.06/62.63
PROPER(and(x0, and(z0, z1))) → c21(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 187.06/62.63
PROPER(and(x0, plus(z0, z1))) → c21(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(and(x0, s(z0))) → c21(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 187.06/62.63
PROPER(and(x0, x(z0, z1))) → c21(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 187.06/62.63
PROPER(and(and(z0, z1), x1)) → c21(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(and(plus(z0, z1), x1)) → c21(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(and(s(z0), x1)) → c21(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 187.06/62.63
PROPER(and(x(z0, z1), x1)) → c21(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(and(x0, tt)) → c21(AND(proper(x0), ok(tt)), PROPER(x0)) 187.06/62.63
PROPER(and(x0, 0)) → c21(AND(proper(x0), ok(0)), PROPER(x0)) 187.06/62.63
PROPER(and(tt, x1)) → c21(AND(ok(tt), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(and(0, x1)) → c21(AND(ok(0), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(plus(x0, and(z0, z1))) → c23(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 187.06/62.63
PROPER(plus(x0, plus(z0, z1))) → c23(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(plus(x0, s(z0))) → c23(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 187.06/62.63
PROPER(plus(x0, x(z0, z1))) → c23(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 187.06/62.63
PROPER(plus(and(z0, z1), x1)) → c23(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(plus(plus(z0, z1), x1)) → c23(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(plus(s(z0), x1)) → c23(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 187.06/62.63
PROPER(plus(x(z0, z1), x1)) → c23(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(plus(x0, tt)) → c23(PLUS(proper(x0), ok(tt)), PROPER(x0)) 187.06/62.63
PROPER(plus(x0, 0)) → c23(PLUS(proper(x0), ok(0)), PROPER(x0)) 187.06/62.63
PROPER(plus(tt, x1)) → c23(PLUS(ok(tt), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(plus(0, x1)) → c23(PLUS(ok(0), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(s(and(z0, z1))) → c25(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 187.06/62.63
PROPER(s(plus(z0, z1))) → c25(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(s(s(z0))) → c25(S(s(proper(z0))), PROPER(s(z0))) 187.06/62.63
PROPER(s(x(z0, z1))) → c25(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1))) 187.06/62.63
PROPER(s(tt)) → c25(S(ok(tt))) 187.06/62.63
PROPER(s(0)) → c25(S(ok(0))) 187.06/62.63
PROPER(x(x0, and(z0, z1))) → c26(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1))) 187.06/62.63
PROPER(x(x0, plus(z0, z1))) → c26(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1))) 187.06/62.63
PROPER(x(x0, s(z0))) → c26(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 187.06/62.63
PROPER(x(x0, x(z0, z1))) → c26(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1))) 187.06/62.63
PROPER(x(and(z0, z1), x1)) → c26(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(x(plus(z0, z1), x1)) → c26(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(x(s(z0), x1)) → c26(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 187.06/62.63
PROPER(x(x(z0, z1), x1)) → c26(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1)) 187.06/62.63
PROPER(x(x0, tt)) → c26(X(proper(x0), ok(tt)), PROPER(x0)) 187.06/62.63
PROPER(x(x0, 0)) → c26(X(proper(x0), ok(0)), PROPER(x0)) 187.06/62.63
PROPER(x(tt, x1)) → c26(X(ok(tt), proper(x1)), PROPER(x1)) 187.06/62.63
PROPER(x(0, x1)) → c26(X(ok(0), proper(x1)), PROPER(x1)) 187.06/62.63
TOP(mark(and(z0, z1))) → c27(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1))) 187.06/62.63
TOP(mark(plus(z0, z1))) → c27(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1))) 187.06/62.63
TOP(mark(s(z0))) → c27(TOP(s(proper(z0))), PROPER(s(z0))) 187.06/62.63
TOP(mark(x(z0, z1))) → c27(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1))) 187.06/62.63
TOP(ok(and(tt, z0))) → c28(TOP(mark(z0)), ACTIVE(and(tt, z0))) 187.06/62.63
TOP(ok(plus(z0, 0))) → c28(TOP(mark(z0)), ACTIVE(plus(z0, 0))) 187.06/62.63
TOP(ok(plus(z0, s(z1)))) → c28(TOP(mark(s(plus(z0, z1)))), ACTIVE(plus(z0, s(z1)))) 187.06/62.63
TOP(ok(x(z0, 0))) → c28(TOP(mark(0)), ACTIVE(x(z0, 0))) 187.06/62.63
TOP(ok(x(z0, s(z1)))) → c28(TOP(mark(plus(x(z0, z1), z0))), ACTIVE(x(z0, s(z1)))) 187.06/62.63
TOP(ok(and(z0, z1))) → c28(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1))) 187.06/62.63
TOP(ok(plus(z0, z1))) → c28(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1))) 187.06/62.63
TOP(ok(plus(z0, z1))) → c28(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1))) 187.06/62.63
TOP(ok(s(z0))) → c28(TOP(s(active(z0))), ACTIVE(s(z0))) 187.06/62.63
TOP(ok(x(z0, z1))) → c28(TOP(x(active(z0), z1)), ACTIVE(x(z0, z1))) 187.06/62.63
TOP(ok(x(z0, z1))) → c28(TOP(x(z0, active(z1))), ACTIVE(x(z0, z1)))

187.06/62.63

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 187.06/62.63
active(plus(z0, 0)) → mark(z0) 187.06/62.63
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 187.06/62.63
active(x(z0, 0)) → mark(0) 187.06/62.63
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 187.06/62.63
active(and(z0, z1)) → and(active(z0), z1) 187.06/62.63
active(plus(z0, z1)) → plus(active(z0), z1) 187.06/62.63
active(plus(z0, z1)) → plus(z0, active(z1)) 187.06/62.63
active(s(z0)) → s(active(z0)) 187.06/62.63
active(x(z0, z1)) → x(active(z0), z1) 187.06/62.63
active(x(z0, z1)) → x(z0, active(z1)) 187.06/62.63
and(mark(z0), z1) → mark(and(z0, z1)) 187.06/62.63
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 187.06/62.63
plus(mark(z0), z1) → mark(plus(z0, z1)) 187.06/62.63
plus(z0, mark(z1)) → mark(plus(z0, z1)) 187.06/62.63
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 187.06/62.63
s(mark(z0)) → mark(s(z0)) 187.06/62.63
s(ok(z0)) → ok(s(z0)) 187.06/62.63
x(mark(z0), z1) → mark(x(z0, z1)) 187.06/62.63
x(z0, mark(z1)) → mark(x(z0, z1)) 187.06/62.63
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 187.06/62.63
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 187.06/62.63
proper(tt) → ok(tt) 187.06/62.63
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 187.06/62.63
proper(0) → ok(0) 187.06/62.63
proper(s(z0)) → s(proper(z0)) 187.06/62.63
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 187.06/62.63
top(mark(z0)) → top(proper(z0)) 187.06/62.63
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.63
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.63
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.63
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.63
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.63
S(mark(z0)) → c16(S(z0)) 187.06/62.63
S(ok(z0)) → c17(S(z0)) 187.06/62.63
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.63
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.63
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 187.06/62.63
TOP(mark(tt)) → c27(TOP(ok(tt))) 187.06/62.63
TOP(mark(0)) → c27(TOP(ok(0)))

S tuples:

AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.63
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.63
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.63
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.63
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.63
S(mark(z0)) → c16(S(z0)) 187.06/62.63
S(ok(z0)) → c17(S(z0)) 187.06/62.63
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.63
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.63
X(ok(z0), ok(z1)) → c20(X(z0, z1))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

AND, PLUS, S, X, TOP

Compound Symbols:

c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c27

187.06/62.63

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

Removed 2 trailing tuple parts

187.06/62.63

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 187.06/62.63
active(plus(z0, 0)) → mark(z0) 187.06/62.63
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 187.06/62.63
active(x(z0, 0)) → mark(0) 187.06/62.63
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 187.06/62.63
active(and(z0, z1)) → and(active(z0), z1) 187.06/62.63
active(plus(z0, z1)) → plus(active(z0), z1) 187.06/62.63
active(plus(z0, z1)) → plus(z0, active(z1)) 187.06/62.63
active(s(z0)) → s(active(z0)) 187.06/62.63
active(x(z0, z1)) → x(active(z0), z1) 187.06/62.63
active(x(z0, z1)) → x(z0, active(z1)) 187.06/62.63
and(mark(z0), z1) → mark(and(z0, z1)) 187.06/62.63
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 187.06/62.63
plus(mark(z0), z1) → mark(plus(z0, z1)) 187.06/62.63
plus(z0, mark(z1)) → mark(plus(z0, z1)) 187.06/62.63
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 187.06/62.63
s(mark(z0)) → mark(s(z0)) 187.06/62.63
s(ok(z0)) → ok(s(z0)) 187.06/62.63
x(mark(z0), z1) → mark(x(z0, z1)) 187.06/62.63
x(z0, mark(z1)) → mark(x(z0, z1)) 187.06/62.63
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 187.06/62.63
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 187.06/62.63
proper(tt) → ok(tt) 187.06/62.63
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 187.06/62.63
proper(0) → ok(0) 187.06/62.63
proper(s(z0)) → s(proper(z0)) 187.06/62.63
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 187.06/62.63
top(mark(z0)) → top(proper(z0)) 187.06/62.63
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.63
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.63
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.63
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.63
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.63
S(mark(z0)) → c16(S(z0)) 187.06/62.63
S(ok(z0)) → c17(S(z0)) 187.06/62.63
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.63
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.63
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 187.06/62.63
TOP(mark(tt)) → c27 187.06/62.63
TOP(mark(0)) → c27

S tuples:

AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.63
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.63
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.63
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.63
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.63
S(mark(z0)) → c16(S(z0)) 187.06/62.63
S(ok(z0)) → c17(S(z0)) 187.06/62.63
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.63
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.63
X(ok(z0), ok(z1)) → c20(X(z0, z1))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

AND, PLUS, S, X, TOP

Compound Symbols:

c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c27

187.06/62.63

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

Removed 2 trailing nodes:

TOP(mark(0)) → c27 187.06/62.63
TOP(mark(tt)) → c27

187.06/62.63

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 187.06/62.63
active(plus(z0, 0)) → mark(z0) 187.06/62.63
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 187.06/62.63
active(x(z0, 0)) → mark(0) 187.06/62.63
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 187.06/62.63
active(and(z0, z1)) → and(active(z0), z1) 187.06/62.63
active(plus(z0, z1)) → plus(active(z0), z1) 187.06/62.63
active(plus(z0, z1)) → plus(z0, active(z1)) 187.06/62.63
active(s(z0)) → s(active(z0)) 187.06/62.63
active(x(z0, z1)) → x(active(z0), z1) 187.06/62.63
active(x(z0, z1)) → x(z0, active(z1)) 187.06/62.63
and(mark(z0), z1) → mark(and(z0, z1)) 187.06/62.63
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 187.06/62.63
plus(mark(z0), z1) → mark(plus(z0, z1)) 187.06/62.63
plus(z0, mark(z1)) → mark(plus(z0, z1)) 187.06/62.63
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 187.06/62.63
s(mark(z0)) → mark(s(z0)) 187.06/62.63
s(ok(z0)) → ok(s(z0)) 187.06/62.63
x(mark(z0), z1) → mark(x(z0, z1)) 187.06/62.63
x(z0, mark(z1)) → mark(x(z0, z1)) 187.06/62.63
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 187.06/62.63
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 187.06/62.63
proper(tt) → ok(tt) 187.06/62.63
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 187.06/62.63
proper(0) → ok(0) 187.06/62.63
proper(s(z0)) → s(proper(z0)) 187.06/62.63
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 187.06/62.63
top(mark(z0)) → top(proper(z0)) 187.06/62.63
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.63
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.63
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.63
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.63
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.63
S(mark(z0)) → c16(S(z0)) 187.06/62.63
S(ok(z0)) → c17(S(z0)) 187.06/62.63
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.63
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.63
X(ok(z0), ok(z1)) → c20(X(z0, z1))

S tuples:

AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.63
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.63
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.63
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.63
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.63
S(mark(z0)) → c16(S(z0)) 187.06/62.63
S(ok(z0)) → c17(S(z0)) 187.06/62.63
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.63
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.63
X(ok(z0), ok(z1)) → c20(X(z0, z1))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

AND, PLUS, S, X

Compound Symbols:

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

187.06/62.63

(47) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^3))) transformation)

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

S(mark(z0)) → c16(S(z0)) 187.06/62.63
S(ok(z0)) → c17(S(z0))

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

AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.63
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.63
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.63
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.63
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.63
S(mark(z0)) → c16(S(z0)) 187.06/62.63
S(ok(z0)) → c17(S(z0)) 187.06/62.63
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.63
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.63
X(ok(z0), ok(z1)) → c20(X(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 187.06/62.63

POL(AND(x₁, x₂)) = 0 187.06/62.63
POL(PLUS(x₁, x₂)) = 0 187.06/62.63
POL(S(x₁)) = x₁ + x₁² + x₁³ 187.06/62.63
POL(X(x₁, x₂)) = 0 187.06/62.63
POL(c11(x₁)) = x₁ 187.06/62.63
POL(c12(x₁)) = x₁ 187.06/62.63
POL(c13(x₁)) = x₁ 187.06/62.63
POL(c14(x₁)) = x₁ 187.06/62.63
POL(c15(x₁)) = x₁ 187.06/62.63
POL(c16(x₁)) = x₁ 187.06/62.63
POL(c17(x₁)) = x₁ 187.06/62.63
POL(c18(x₁)) = x₁ 187.06/62.63
POL(c19(x₁)) = x₁ 187.06/62.63
POL(c20(x₁)) = x₁ 187.06/62.63
POL(mark(x₁)) = [1] + x₁ 187.06/62.63
POL(ok(x₁)) = [1] + x₁

187.06/62.63

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 187.06/62.63
active(plus(z0, 0)) → mark(z0) 187.06/62.63
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 187.06/62.63
active(x(z0, 0)) → mark(0) 187.06/62.63
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 187.06/62.63
active(and(z0, z1)) → and(active(z0), z1) 187.06/62.63
active(plus(z0, z1)) → plus(active(z0), z1) 187.06/62.63
active(plus(z0, z1)) → plus(z0, active(z1)) 187.06/62.63
active(s(z0)) → s(active(z0)) 187.06/62.63
active(x(z0, z1)) → x(active(z0), z1) 187.06/62.63
active(x(z0, z1)) → x(z0, active(z1)) 187.06/62.63
and(mark(z0), z1) → mark(and(z0, z1)) 187.06/62.63
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 187.06/62.63
plus(mark(z0), z1) → mark(plus(z0, z1)) 187.06/62.63
plus(z0, mark(z1)) → mark(plus(z0, z1)) 187.06/62.63
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 187.06/62.63
s(mark(z0)) → mark(s(z0)) 187.06/62.63
s(ok(z0)) → ok(s(z0)) 187.06/62.63
x(mark(z0), z1) → mark(x(z0, z1)) 187.06/62.63
x(z0, mark(z1)) → mark(x(z0, z1)) 187.06/62.63
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 187.06/62.63
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 187.06/62.63
proper(tt) → ok(tt) 187.06/62.63
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 187.06/62.63
proper(0) → ok(0) 187.06/62.63
proper(s(z0)) → s(proper(z0)) 187.06/62.63
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 187.06/62.63
top(mark(z0)) → top(proper(z0)) 187.06/62.63
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.63
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.63
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.63
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.63
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.63
S(mark(z0)) → c16(S(z0)) 187.06/62.63
S(ok(z0)) → c17(S(z0)) 187.06/62.63
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.63
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.63
X(ok(z0), ok(z1)) → c20(X(z0, z1))

S tuples:

AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.63
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.63
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.63
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.63
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.63
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.63
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.63
X(ok(z0), ok(z1)) → c20(X(z0, z1))

K tuples:

S(mark(z0)) → c16(S(z0)) 187.06/62.63
S(ok(z0)) → c17(S(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

AND, PLUS, S, X

Compound Symbols:

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

187.06/62.63

(49) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.63
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.63
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.63
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)) 187.06/62.63
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.63
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.63
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.63
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.63
S(mark(z0)) → c16(S(z0)) 187.06/62.63
S(ok(z0)) → c17(S(z0)) 187.06/62.63
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.63
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.63
X(ok(z0), ok(z1)) → c20(X(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 187.06/62.63

POL(AND(x₁, x₂)) = [4]x₁ 187.06/62.63
POL(PLUS(x₁, x₂)) = 0 187.06/62.63
POL(S(x₁)) = [5]x₁ 187.06/62.63
POL(X(x₁, x₂)) = x₂ 187.06/62.63
POL(c11(x₁)) = x₁ 187.06/62.63
POL(c12(x₁)) = x₁ 187.06/62.63
POL(c13(x₁)) = x₁ 187.06/62.63
POL(c14(x₁)) = x₁ 187.06/62.63
POL(c15(x₁)) = x₁ 187.06/62.63
POL(c16(x₁)) = x₁ 187.06/62.63
POL(c17(x₁)) = x₁ 187.06/62.63
POL(c18(x₁)) = x₁ 187.06/62.63
POL(c19(x₁)) = x₁ 187.06/62.63
POL(c20(x₁)) = x₁ 187.06/62.63
POL(mark(x₁)) = [1] + x₁ 187.06/62.63
POL(ok(x₁)) = [1] + x₁

187.06/62.63

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 187.06/62.63
active(plus(z0, 0)) → mark(z0) 187.06/62.63
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 187.06/62.63
active(x(z0, 0)) → mark(0) 187.06/62.63
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 187.06/62.63
active(and(z0, z1)) → and(active(z0), z1) 187.06/62.63
active(plus(z0, z1)) → plus(active(z0), z1) 187.06/62.63
active(plus(z0, z1)) → plus(z0, active(z1)) 187.06/62.63
active(s(z0)) → s(active(z0)) 187.06/62.63
active(x(z0, z1)) → x(active(z0), z1) 187.06/62.63
active(x(z0, z1)) → x(z0, active(z1)) 187.06/62.63
and(mark(z0), z1) → mark(and(z0, z1)) 187.06/62.63
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 187.06/62.63
plus(mark(z0), z1) → mark(plus(z0, z1)) 187.06/62.63
plus(z0, mark(z1)) → mark(plus(z0, z1)) 187.06/62.63
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 187.06/62.63
s(mark(z0)) → mark(s(z0)) 187.06/62.63
s(ok(z0)) → ok(s(z0)) 187.06/62.63
x(mark(z0), z1) → mark(x(z0, z1)) 187.06/62.63
x(z0, mark(z1)) → mark(x(z0, z1)) 187.06/62.63
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 187.06/62.63
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 187.06/62.63
proper(tt) → ok(tt) 187.06/62.63
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 187.06/62.64
proper(0) → ok(0) 187.06/62.64
proper(s(z0)) → s(proper(z0)) 187.06/62.64
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 187.06/62.64
top(mark(z0)) → top(proper(z0)) 187.06/62.64
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.64
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.64
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.64
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.64
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.64
S(mark(z0)) → c16(S(z0)) 187.06/62.64
S(ok(z0)) → c17(S(z0)) 187.06/62.64
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.64
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.64
X(ok(z0), ok(z1)) → c20(X(z0, z1))

S tuples:

PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.64
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.64
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.64
X(mark(z0), z1) → c18(X(z0, z1))

K tuples:

S(mark(z0)) → c16(S(z0)) 187.06/62.64
S(ok(z0)) → c17(S(z0)) 187.06/62.64
AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.64
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.64
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.64
X(ok(z0), ok(z1)) → c20(X(z0, z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

AND, PLUS, S, X

Compound Symbols:

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

187.06/62.64

(51) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

PLUS(mark(z0), z1) → c13(PLUS(z0, z1))

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

AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.64
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.64
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.64
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.64
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.64
S(mark(z0)) → c16(S(z0)) 187.06/62.64
S(ok(z0)) → c17(S(z0)) 187.06/62.64
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.64
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.64
X(ok(z0), ok(z1)) → c20(X(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 187.06/62.64

POL(AND(x₁, x₂)) = [5]x₁ + [3]x₂ 187.06/62.64
POL(PLUS(x₁, x₂)) = [2]x₁ 187.06/62.64
POL(S(x₁)) = [3]x₁ 187.06/62.64
POL(X(x₁, x₂)) = [3]x₂ 187.06/62.64
POL(c11(x₁)) = x₁ 187.06/62.64
POL(c12(x₁)) = x₁ 187.06/62.64
POL(c13(x₁)) = x₁ 187.06/62.64
POL(c14(x₁)) = x₁ 187.06/62.64
POL(c15(x₁)) = x₁ 187.06/62.64
POL(c16(x₁)) = x₁ 187.06/62.64
POL(c17(x₁)) = x₁ 187.06/62.64
POL(c18(x₁)) = x₁ 187.06/62.64
POL(c19(x₁)) = x₁ 187.06/62.64
POL(c20(x₁)) = x₁ 187.06/62.64
POL(mark(x₁)) = [1] + x₁ 187.06/62.64
POL(ok(x₁)) = x₁

187.06/62.64

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 187.06/62.64
active(plus(z0, 0)) → mark(z0) 187.06/62.64
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 187.06/62.64
active(x(z0, 0)) → mark(0) 187.06/62.64
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 187.06/62.64
active(and(z0, z1)) → and(active(z0), z1) 187.06/62.64
active(plus(z0, z1)) → plus(active(z0), z1) 187.06/62.64
active(plus(z0, z1)) → plus(z0, active(z1)) 187.06/62.64
active(s(z0)) → s(active(z0)) 187.06/62.64
active(x(z0, z1)) → x(active(z0), z1) 187.06/62.64
active(x(z0, z1)) → x(z0, active(z1)) 187.06/62.64
and(mark(z0), z1) → mark(and(z0, z1)) 187.06/62.64
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 187.06/62.64
plus(mark(z0), z1) → mark(plus(z0, z1)) 187.06/62.64
plus(z0, mark(z1)) → mark(plus(z0, z1)) 187.06/62.64
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 187.06/62.64
s(mark(z0)) → mark(s(z0)) 187.06/62.64
s(ok(z0)) → ok(s(z0)) 187.06/62.64
x(mark(z0), z1) → mark(x(z0, z1)) 187.06/62.64
x(z0, mark(z1)) → mark(x(z0, z1)) 187.06/62.64
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 187.06/62.64
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 187.06/62.64
proper(tt) → ok(tt) 187.06/62.64
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 187.06/62.64
proper(0) → ok(0) 187.06/62.64
proper(s(z0)) → s(proper(z0)) 187.06/62.64
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 187.06/62.64
top(mark(z0)) → top(proper(z0)) 187.06/62.64
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.64
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.64
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.64
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.64
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.64
S(mark(z0)) → c16(S(z0)) 187.06/62.64
S(ok(z0)) → c17(S(z0)) 187.06/62.64
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.64
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.64
X(ok(z0), ok(z1)) → c20(X(z0, z1))

S tuples:

PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.64
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.64
X(mark(z0), z1) → c18(X(z0, z1))

K tuples:

S(mark(z0)) → c16(S(z0)) 187.06/62.64
S(ok(z0)) → c17(S(z0)) 187.06/62.64
AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.64
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.64
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.64
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 187.06/62.64
PLUS(mark(z0), z1) → c13(PLUS(z0, z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

AND, PLUS, S, X

Compound Symbols:

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

187.06/62.64

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

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

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)) 187.06/62.64
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.64
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.64
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.64
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.64
S(mark(z0)) → c16(S(z0)) 187.06/62.64
S(ok(z0)) → c17(S(z0)) 187.06/62.64
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.64
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.64
X(ok(z0), ok(z1)) → c20(X(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 187.06/62.64

POL(AND(x₁, x₂)) = [5]x₁ + [5]x₂ 187.06/62.64
POL(PLUS(x₁, x₂)) = [4]x₂ 187.06/62.64
POL(S(x₁)) = [5]x₁ 187.06/62.64
POL(X(x₁, x₂)) = [3]x₂ 187.06/62.64
POL(c11(x₁)) = x₁ 187.06/62.64
POL(c12(x₁)) = x₁ 187.06/62.64
POL(c13(x₁)) = x₁ 187.06/62.64
POL(c14(x₁)) = x₁ 187.06/62.64
POL(c15(x₁)) = x₁ 187.06/62.64
POL(c16(x₁)) = x₁ 187.06/62.64
POL(c17(x₁)) = x₁ 187.06/62.64
POL(c18(x₁)) = x₁ 187.06/62.64
POL(c19(x₁)) = x₁ 187.06/62.64
POL(c20(x₁)) = x₁ 187.06/62.64
POL(mark(x₁)) = [1] + x₁ 187.06/62.64
POL(ok(x₁)) = x₁

187.06/62.64

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 187.06/62.64
active(plus(z0, 0)) → mark(z0) 187.06/62.64
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 187.06/62.64
active(x(z0, 0)) → mark(0) 187.06/62.64
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 187.06/62.64
active(and(z0, z1)) → and(active(z0), z1) 187.06/62.64
active(plus(z0, z1)) → plus(active(z0), z1) 187.06/62.64
active(plus(z0, z1)) → plus(z0, active(z1)) 187.06/62.64
active(s(z0)) → s(active(z0)) 187.06/62.64
active(x(z0, z1)) → x(active(z0), z1) 187.06/62.64
active(x(z0, z1)) → x(z0, active(z1)) 187.06/62.64
and(mark(z0), z1) → mark(and(z0, z1)) 187.06/62.64
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 187.06/62.64
plus(mark(z0), z1) → mark(plus(z0, z1)) 187.06/62.64
plus(z0, mark(z1)) → mark(plus(z0, z1)) 187.06/62.64
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 187.06/62.64
s(mark(z0)) → mark(s(z0)) 187.06/62.64
s(ok(z0)) → ok(s(z0)) 187.06/62.64
x(mark(z0), z1) → mark(x(z0, z1)) 187.06/62.64
x(z0, mark(z1)) → mark(x(z0, z1)) 187.06/62.64
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 187.06/62.64
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 187.06/62.64
proper(tt) → ok(tt) 187.06/62.64
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 187.06/62.64
proper(0) → ok(0) 187.06/62.64
proper(s(z0)) → s(proper(z0)) 187.06/62.64
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 187.06/62.64
top(mark(z0)) → top(proper(z0)) 187.06/62.64
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.64
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.64
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.64
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.64
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.64
S(mark(z0)) → c16(S(z0)) 187.06/62.64
S(ok(z0)) → c17(S(z0)) 187.06/62.64
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.64
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.64
X(ok(z0), ok(z1)) → c20(X(z0, z1))

S tuples:

PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.64
X(mark(z0), z1) → c18(X(z0, z1))

K tuples:

S(mark(z0)) → c16(S(z0)) 187.06/62.64
S(ok(z0)) → c17(S(z0)) 187.06/62.64
AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.64
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.64
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.64
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 187.06/62.64
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.64
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

AND, PLUS, S, X

Compound Symbols:

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

187.06/62.64

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

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

PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))

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

AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.64
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.64
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.64
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.64
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.64
S(mark(z0)) → c16(S(z0)) 187.06/62.64
S(ok(z0)) → c17(S(z0)) 187.06/62.64
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.64
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.64
X(ok(z0), ok(z1)) → c20(X(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 187.06/62.64

POL(AND(x₁, x₂)) = [5]x₁ + [5]x₂ 187.06/62.64
POL(PLUS(x₁, x₂)) = x₂ 187.06/62.64
POL(S(x₁)) = [3]x₁ 187.06/62.64
POL(X(x₁, x₂)) = [5]x₂ 187.06/62.64
POL(c11(x₁)) = x₁ 187.06/62.64
POL(c12(x₁)) = x₁ 187.06/62.64
POL(c13(x₁)) = x₁ 187.06/62.64
POL(c14(x₁)) = x₁ 187.06/62.64
POL(c15(x₁)) = x₁ 187.06/62.64
POL(c16(x₁)) = x₁ 187.06/62.64
POL(c17(x₁)) = x₁ 187.06/62.64
POL(c18(x₁)) = x₁ 187.06/62.64
POL(c19(x₁)) = x₁ 187.06/62.64
POL(c20(x₁)) = x₁ 187.06/62.64
POL(mark(x₁)) = [1] + x₁ 187.06/62.64
POL(ok(x₁)) = [1] + x₁

187.06/62.64

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 187.06/62.64
active(plus(z0, 0)) → mark(z0) 187.06/62.64
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 187.06/62.64
active(x(z0, 0)) → mark(0) 187.06/62.64
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 187.06/62.64
active(and(z0, z1)) → and(active(z0), z1) 187.06/62.64
active(plus(z0, z1)) → plus(active(z0), z1) 187.06/62.64
active(plus(z0, z1)) → plus(z0, active(z1)) 187.06/62.64
active(s(z0)) → s(active(z0)) 187.06/62.64
active(x(z0, z1)) → x(active(z0), z1) 187.06/62.64
active(x(z0, z1)) → x(z0, active(z1)) 187.06/62.64
and(mark(z0), z1) → mark(and(z0, z1)) 187.06/62.64
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 187.06/62.64
plus(mark(z0), z1) → mark(plus(z0, z1)) 187.06/62.64
plus(z0, mark(z1)) → mark(plus(z0, z1)) 187.06/62.64
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 187.06/62.64
s(mark(z0)) → mark(s(z0)) 187.06/62.64
s(ok(z0)) → ok(s(z0)) 187.06/62.64
x(mark(z0), z1) → mark(x(z0, z1)) 187.06/62.64
x(z0, mark(z1)) → mark(x(z0, z1)) 187.06/62.64
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 187.06/62.64
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 187.06/62.64
proper(tt) → ok(tt) 187.06/62.64
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 187.06/62.64
proper(0) → ok(0) 187.06/62.64
proper(s(z0)) → s(proper(z0)) 187.06/62.64
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 187.06/62.64
top(mark(z0)) → top(proper(z0)) 187.06/62.64
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.64
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.64
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.64
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.64
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.64
S(mark(z0)) → c16(S(z0)) 187.06/62.64
S(ok(z0)) → c17(S(z0)) 187.06/62.64
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.64
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.64
X(ok(z0), ok(z1)) → c20(X(z0, z1))

S tuples:

X(mark(z0), z1) → c18(X(z0, z1))

K tuples:

S(mark(z0)) → c16(S(z0)) 187.06/62.64
S(ok(z0)) → c17(S(z0)) 187.06/62.64
AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.64
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.64
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.64
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 187.06/62.64
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.64
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.64
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

AND, PLUS, S, X

Compound Symbols:

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

187.06/62.64

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

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

X(mark(z0), z1) → c18(X(z0, z1))

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

AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.64
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.64
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.64
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.64
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.64
S(mark(z0)) → c16(S(z0)) 187.06/62.64
S(ok(z0)) → c17(S(z0)) 187.06/62.64
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.64
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.64
X(ok(z0), ok(z1)) → c20(X(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 187.06/62.64

POL(AND(x₁, x₂)) = x₁ + x₂ + [3]x₂² + [3]x₁·x₂ + [3]x₁² 187.06/62.64
POL(PLUS(x₁, x₂)) = x₁ + x₂ + [3]x₂² + [3]x₁·x₂ + [3]x₁² 187.06/62.64
POL(S(x₁)) = [3]x₁ + [3]x₁² 187.06/62.64
POL(X(x₁, x₂)) = [2]x₁ + x₂ + [3]x₂² + [3]x₁·x₂ 187.06/62.64
POL(c11(x₁)) = x₁ 187.06/62.64
POL(c12(x₁)) = x₁ 187.06/62.64
POL(c13(x₁)) = x₁ 187.06/62.64
POL(c14(x₁)) = x₁ 187.06/62.64
POL(c15(x₁)) = x₁ 187.06/62.64
POL(c16(x₁)) = x₁ 187.06/62.64
POL(c17(x₁)) = x₁ 187.06/62.64
POL(c18(x₁)) = x₁ 187.06/62.64
POL(c19(x₁)) = x₁ 187.06/62.64
POL(c20(x₁)) = x₁ 187.06/62.64
POL(mark(x₁)) = [2] + x₁ 187.06/62.64
POL(ok(x₁)) = x₁

187.06/62.64

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(and(tt, z0)) → mark(z0) 187.06/62.64
active(plus(z0, 0)) → mark(z0) 187.06/62.64
active(plus(z0, s(z1))) → mark(s(plus(z0, z1))) 187.06/62.64
active(x(z0, 0)) → mark(0) 187.06/62.64
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0)) 187.06/62.64
active(and(z0, z1)) → and(active(z0), z1) 187.06/62.64
active(plus(z0, z1)) → plus(active(z0), z1) 187.06/62.64
active(plus(z0, z1)) → plus(z0, active(z1)) 187.06/62.64
active(s(z0)) → s(active(z0)) 187.06/62.64
active(x(z0, z1)) → x(active(z0), z1) 187.06/62.64
active(x(z0, z1)) → x(z0, active(z1)) 187.06/62.64
and(mark(z0), z1) → mark(and(z0, z1)) 187.06/62.64
and(ok(z0), ok(z1)) → ok(and(z0, z1)) 187.06/62.64
plus(mark(z0), z1) → mark(plus(z0, z1)) 187.06/62.64
plus(z0, mark(z1)) → mark(plus(z0, z1)) 187.06/62.64
plus(ok(z0), ok(z1)) → ok(plus(z0, z1)) 187.06/62.64
s(mark(z0)) → mark(s(z0)) 187.06/62.64
s(ok(z0)) → ok(s(z0)) 187.06/62.64
x(mark(z0), z1) → mark(x(z0, z1)) 187.06/62.64
x(z0, mark(z1)) → mark(x(z0, z1)) 187.06/62.64
x(ok(z0), ok(z1)) → ok(x(z0, z1)) 187.06/62.64
proper(and(z0, z1)) → and(proper(z0), proper(z1)) 187.06/62.64
proper(tt) → ok(tt) 187.06/62.64
proper(plus(z0, z1)) → plus(proper(z0), proper(z1)) 187.06/62.64
proper(0) → ok(0) 187.06/62.64
proper(s(z0)) → s(proper(z0)) 187.06/62.64
proper(x(z0, z1)) → x(proper(z0), proper(z1)) 187.06/62.64
top(mark(z0)) → top(proper(z0)) 187.06/62.64
top(ok(z0)) → top(active(z0))

Tuples:

AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.64
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.64
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.64
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.64
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.64
S(mark(z0)) → c16(S(z0)) 187.06/62.64
S(ok(z0)) → c17(S(z0)) 187.06/62.64
X(mark(z0), z1) → c18(X(z0, z1)) 187.06/62.64
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.64
X(ok(z0), ok(z1)) → c20(X(z0, z1))

S tuples:none
K tuples:

S(mark(z0)) → c16(S(z0)) 187.06/62.64
S(ok(z0)) → c17(S(z0)) 187.06/62.64
AND(mark(z0), z1) → c11(AND(z0, z1)) 187.06/62.64
AND(ok(z0), ok(z1)) → c12(AND(z0, z1)) 187.06/62.64
X(z0, mark(z1)) → c19(X(z0, z1)) 187.06/62.64
X(ok(z0), ok(z1)) → c20(X(z0, z1)) 187.06/62.64
PLUS(mark(z0), z1) → c13(PLUS(z0, z1)) 187.06/62.64
PLUS(z0, mark(z1)) → c14(PLUS(z0, z1)) 187.06/62.64
PLUS(ok(z0), ok(z1)) → c15(PLUS(z0, z1)) 187.06/62.64
X(mark(z0), z1) → c18(X(z0, z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

AND, PLUS, S, X

Compound Symbols:

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

187.06/62.64

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

The set S is empty

187.06/62.64

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

187.06/62.64