193.16/71.72 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 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))

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↳6 CdtProblem

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

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↳8 CdtProblem

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

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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 CdtNarrowingProof (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 CdtNarrowingProof (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 CdtNarrowingProof (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 CdtNarrowingProof (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 CdtNarrowingProof (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 CdtUnreachableProof (⇔)

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↳34 CdtProblem

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

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↳36 CdtProblem

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

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↳38 CdtProblem

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

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↳40 CdtProblem

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

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↳42 CdtProblem

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

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↳44 CdtProblem

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

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↳46 CdtProblem

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

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↳48 CdtProblem

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

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↳50 BOUNDS(O(1), O(1))

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(0) Obligation:

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

active(2nd(cons1(X, cons(Y, Z)))) → mark(Y) 193.16/71.72
active(2nd(cons(X, X1))) → mark(2nd(cons1(X, X1))) 193.16/71.72
active(from(X)) → mark(cons(X, from(s(X)))) 193.16/71.72
active(2nd(X)) → 2nd(active(X)) 193.16/71.72
active(cons(X1, X2)) → cons(active(X1), X2) 193.16/71.72
active(from(X)) → from(active(X)) 193.16/71.72
active(s(X)) → s(active(X)) 193.16/71.72
active(cons1(X1, X2)) → cons1(active(X1), X2) 193.16/71.72
active(cons1(X1, X2)) → cons1(X1, active(X2)) 193.16/71.72
2nd(mark(X)) → mark(2nd(X)) 193.16/71.72
cons(mark(X1), X2) → mark(cons(X1, X2)) 193.16/71.72
from(mark(X)) → mark(from(X)) 193.16/71.72
s(mark(X)) → mark(s(X)) 193.16/71.72
cons1(mark(X1), X2) → mark(cons1(X1, X2)) 193.16/71.72
cons1(X1, mark(X2)) → mark(cons1(X1, X2)) 193.16/71.72
proper(2nd(X)) → 2nd(proper(X)) 193.16/71.72
proper(cons(X1, X2)) → cons(proper(X1), proper(X2)) 193.16/71.72
proper(from(X)) → from(proper(X)) 193.16/71.72
proper(s(X)) → s(proper(X)) 193.16/71.72
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2)) 193.16/71.72
2nd(ok(X)) → ok(2nd(X)) 193.16/71.72
cons(ok(X1), ok(X2)) → ok(cons(X1, X2)) 193.16/71.72
from(ok(X)) → ok(from(X)) 193.16/71.72
s(ok(X)) → ok(s(X)) 193.16/71.72
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2)) 193.16/71.72
top(mark(X)) → top(proper(X)) 193.16/71.72
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(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.16/71.72
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.16/71.72
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.16/71.72
active(2nd(z0)) → 2nd(active(z0)) 193.16/71.72
active(cons(z0, z1)) → cons(active(z0), z1) 193.16/71.72
active(from(z0)) → from(active(z0)) 193.16/71.72
active(s(z0)) → s(active(z0)) 193.16/71.72
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.16/71.72
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.16/71.72
2nd(mark(z0)) → mark(2nd(z0)) 193.16/71.72
2nd(ok(z0)) → ok(2nd(z0)) 193.16/71.72
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.16/71.72
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.16/71.72
from(mark(z0)) → mark(from(z0)) 193.16/71.72
from(ok(z0)) → ok(from(z0)) 193.16/71.72
s(mark(z0)) → mark(s(z0)) 193.16/71.72
s(ok(z0)) → ok(s(z0)) 193.16/71.72
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.16/71.72
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.16/71.72
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.16/71.72
proper(2nd(z0)) → 2nd(proper(z0)) 193.16/71.72
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.16/71.72
proper(from(z0)) → from(proper(z0)) 193.16/71.72
proper(s(z0)) → s(proper(z0)) 193.16/71.72
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.16/71.72
top(mark(z0)) → top(proper(z0)) 193.16/71.72
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(2nd(cons(z0, z1))) → c1(2ND(cons1(z0, z1)), CONS1(z0, z1)) 193.16/71.72
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0)) 193.16/71.72
ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.16/71.72
2ND(mark(z0)) → c9(2ND(z0)) 193.16/71.72
2ND(ok(z0)) → c10(2ND(z0)) 193.16/71.72
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.16/71.72
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.16/71.72
FROM(mark(z0)) → c13(FROM(z0)) 193.16/71.72
FROM(ok(z0)) → c14(FROM(z0)) 193.16/71.72
S(mark(z0)) → c15(S(z0)) 193.16/71.72
S(ok(z0)) → c16(S(z0)) 193.16/71.72
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.16/71.72
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.16/71.72
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.16/71.72
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.16/71.72
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.16/71.72
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.16/71.72
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(2nd(cons(z0, z1))) → c1(2ND(cons1(z0, z1)), CONS1(z0, z1)) 193.16/71.72
ACTIVE(from(z0)) → c2(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0)) 193.16/71.72
ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.16/71.72
2ND(mark(z0)) → c9(2ND(z0)) 193.16/71.72
2ND(ok(z0)) → c10(2ND(z0)) 193.16/71.72
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.16/71.72
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.16/71.72
FROM(mark(z0)) → c13(FROM(z0)) 193.16/71.72
FROM(ok(z0)) → c14(FROM(z0)) 193.16/71.72
S(mark(z0)) → c15(S(z0)) 193.16/71.72
S(ok(z0)) → c16(S(z0)) 193.16/71.72
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.16/71.72
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.16/71.72
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.16/71.72
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.16/71.72
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.16/71.72
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.16/71.72
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

ACTIVE, 2ND, CONS, FROM, S, CONS1, PROPER, TOP

Compound Symbols:

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

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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(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.16/71.72
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.16/71.72
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.16/71.72
active(2nd(z0)) → 2nd(active(z0)) 193.16/71.72
active(cons(z0, z1)) → cons(active(z0), z1) 193.16/71.72
active(from(z0)) → from(active(z0)) 193.16/71.72
active(s(z0)) → s(active(z0)) 193.16/71.72
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.16/71.72
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.16/71.72
2nd(mark(z0)) → mark(2nd(z0)) 193.16/71.72
2nd(ok(z0)) → ok(2nd(z0)) 193.16/71.72
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.16/71.72
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.16/71.72
from(mark(z0)) → mark(from(z0)) 193.16/71.72
from(ok(z0)) → ok(from(z0)) 193.16/71.72
s(mark(z0)) → mark(s(z0)) 193.16/71.72
s(ok(z0)) → ok(s(z0)) 193.16/71.72
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.16/71.72
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.16/71.72
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.16/71.72
proper(2nd(z0)) → 2nd(proper(z0)) 193.16/71.72
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.16/71.72
proper(from(z0)) → from(proper(z0)) 193.16/71.72
proper(s(z0)) → s(proper(z0)) 193.16/71.72
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.16/71.72
top(mark(z0)) → top(proper(z0)) 193.16/71.72
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(2nd(cons(z0, z1))) → c1(2ND(cons1(z0, z1)), CONS1(z0, z1)) 193.16/71.72
ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.16/71.72
2ND(mark(z0)) → c9(2ND(z0)) 193.16/71.72
2ND(ok(z0)) → c10(2ND(z0)) 193.16/71.72
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.16/71.72
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.16/71.72
FROM(mark(z0)) → c13(FROM(z0)) 193.16/71.72
FROM(ok(z0)) → c14(FROM(z0)) 193.16/71.72
S(mark(z0)) → c15(S(z0)) 193.16/71.72
S(ok(z0)) → c16(S(z0)) 193.16/71.72
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.16/71.72
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.16/71.72
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.16/71.72
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.16/71.72
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.16/71.72
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.16/71.72
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(from(z0)) → c2(S(z0))

S tuples:

ACTIVE(2nd(cons(z0, z1))) → c1(2ND(cons1(z0, z1)), CONS1(z0, z1)) 193.16/71.72
ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.16/71.72
2ND(mark(z0)) → c9(2ND(z0)) 193.16/71.72
2ND(ok(z0)) → c10(2ND(z0)) 193.16/71.72
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.16/71.72
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.16/71.72
FROM(mark(z0)) → c13(FROM(z0)) 193.16/71.72
FROM(ok(z0)) → c14(FROM(z0)) 193.16/71.72
S(mark(z0)) → c15(S(z0)) 193.16/71.72
S(ok(z0)) → c16(S(z0)) 193.16/71.72
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.16/71.72
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.16/71.72
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.16/71.72
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.16/71.72
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.16/71.72
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.16/71.72
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(from(z0)) → c2(S(z0))

K tuples:none
Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

ACTIVE, 2ND, CONS, FROM, S, CONS1, PROPER, TOP

Compound Symbols:

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

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

Split RHS of tuples not part of any SCC

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(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.16/71.72
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.16/71.72
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.16/71.72
active(2nd(z0)) → 2nd(active(z0)) 193.16/71.72
active(cons(z0, z1)) → cons(active(z0), z1) 193.16/71.72
active(from(z0)) → from(active(z0)) 193.16/71.72
active(s(z0)) → s(active(z0)) 193.16/71.72
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.16/71.72
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.16/71.72
2nd(mark(z0)) → mark(2nd(z0)) 193.16/71.72
2nd(ok(z0)) → ok(2nd(z0)) 193.16/71.72
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.16/71.72
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.16/71.72
from(mark(z0)) → mark(from(z0)) 193.16/71.72
from(ok(z0)) → ok(from(z0)) 193.16/71.72
s(mark(z0)) → mark(s(z0)) 193.16/71.72
s(ok(z0)) → ok(s(z0)) 193.16/71.72
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.16/71.72
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.16/71.72
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.16/71.72
proper(2nd(z0)) → 2nd(proper(z0)) 193.16/71.72
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.16/71.72
proper(from(z0)) → from(proper(z0)) 193.16/71.72
proper(s(z0)) → s(proper(z0)) 193.16/71.72
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.16/71.72
top(mark(z0)) → top(proper(z0)) 193.16/71.72
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.16/71.72
2ND(mark(z0)) → c9(2ND(z0)) 193.16/71.72
2ND(ok(z0)) → c10(2ND(z0)) 193.16/71.72
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.16/71.72
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.16/71.72
FROM(mark(z0)) → c13(FROM(z0)) 193.16/71.72
FROM(ok(z0)) → c14(FROM(z0)) 193.16/71.72
S(mark(z0)) → c15(S(z0)) 193.16/71.72
S(ok(z0)) → c16(S(z0)) 193.16/71.72
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.16/71.72
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.16/71.72
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.16/71.72
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.16/71.72
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.16/71.72
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.16/71.72
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(from(z0)) → c2(S(z0)) 193.16/71.72
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.16/71.72
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1))

S tuples:

ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.16/71.72
2ND(mark(z0)) → c9(2ND(z0)) 193.16/71.72
2ND(ok(z0)) → c10(2ND(z0)) 193.16/71.72
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.16/71.72
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.16/71.72
FROM(mark(z0)) → c13(FROM(z0)) 193.16/71.72
FROM(ok(z0)) → c14(FROM(z0)) 193.16/71.72
S(mark(z0)) → c15(S(z0)) 193.16/71.72
S(ok(z0)) → c16(S(z0)) 193.16/71.72
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.16/71.72
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.16/71.72
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.16/71.72
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.16/71.72
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.16/71.72
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.16/71.72
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(from(z0)) → c2(S(z0)) 193.16/71.72
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.16/71.72
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1))

K tuples:none
Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

ACTIVE, 2ND, CONS, FROM, S, CONS1, PROPER, TOP

Compound Symbols:

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

193.16/71.72

(7) 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.

ACTIVE(from(z0)) → c2(S(z0)) 193.16/71.72
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.16/71.72
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1))

We considered the (Usable) Rules:

cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.16/71.72
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.16/71.72
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.16/71.72
active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.16/71.72
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.16/71.72
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.16/71.72
active(2nd(z0)) → 2nd(active(z0)) 193.16/71.72
active(cons(z0, z1)) → cons(active(z0), z1) 193.16/71.72
active(from(z0)) → from(active(z0)) 193.16/71.72
active(s(z0)) → s(active(z0)) 193.16/71.72
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.16/71.72
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.16/71.72
s(mark(z0)) → mark(s(z0)) 193.16/71.72
s(ok(z0)) → ok(s(z0)) 193.16/71.72
from(mark(z0)) → mark(from(z0)) 193.16/71.72
from(ok(z0)) → ok(from(z0)) 193.16/71.72
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.16/71.72
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.16/71.72
2nd(mark(z0)) → mark(2nd(z0)) 193.16/71.72
2nd(ok(z0)) → ok(2nd(z0)) 193.16/71.72
proper(2nd(z0)) → 2nd(proper(z0)) 193.16/71.72
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.16/71.72
proper(from(z0)) → from(proper(z0)) 193.16/71.72
proper(s(z0)) → s(proper(z0)) 193.16/71.72
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))

And the Tuples:

ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.16/71.72
2ND(mark(z0)) → c9(2ND(z0)) 193.16/71.72
2ND(ok(z0)) → c10(2ND(z0)) 193.16/71.72
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.16/71.72
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.16/71.72
FROM(mark(z0)) → c13(FROM(z0)) 193.16/71.72
FROM(ok(z0)) → c14(FROM(z0)) 193.16/71.72
S(mark(z0)) → c15(S(z0)) 193.16/71.72
S(ok(z0)) → c16(S(z0)) 193.16/71.72
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.16/71.72
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.16/71.72
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.16/71.72
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.16/71.72
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.16/71.72
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.16/71.72
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(from(z0)) → c2(S(z0)) 193.16/71.72
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.16/71.72
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 193.16/71.72

POL(2ND(x₁)) = 0 193.16/71.72
POL(2nd(x₁)) = [2]x₁ 193.16/71.72
POL(ACTIVE(x₁)) = [1] 193.16/71.72
POL(CONS(x₁, x₂)) = 0 193.16/71.72
POL(CONS1(x₁, x₂)) = 0 193.16/71.72
POL(FROM(x₁)) = 0 193.16/71.72
POL(PROPER(x₁)) = 0 193.16/71.72
POL(S(x₁)) = 0 193.16/71.72
POL(TOP(x₁)) = x₁ 193.16/71.72
POL(active(x₁)) = x₁ 193.16/71.72
POL(c(x₁)) = x₁ 193.16/71.72
POL(c10(x₁)) = x₁ 193.16/71.72
POL(c11(x₁)) = x₁ 193.16/71.72
POL(c12(x₁)) = x₁ 193.16/71.72
POL(c13(x₁)) = x₁ 193.16/71.72
POL(c14(x₁)) = x₁ 193.16/71.72
POL(c15(x₁)) = x₁ 193.16/71.72
POL(c16(x₁)) = x₁ 193.16/71.72
POL(c17(x₁)) = x₁ 193.16/71.72
POL(c18(x₁)) = x₁ 193.16/71.72
POL(c19(x₁)) = x₁ 193.16/71.72
POL(c2(x₁)) = x₁ 193.16/71.72
POL(c20(x₁, x₂)) = x₁ + x₂ 193.16/71.72
POL(c21(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 193.16/71.72
POL(c22(x₁, x₂)) = x₁ + x₂ 193.16/71.72
POL(c23(x₁, x₂)) = x₁ + x₂ 193.16/71.72
POL(c24(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 193.16/71.72
POL(c25(x₁, x₂)) = x₁ + x₂ 193.16/71.72
POL(c26(x₁, x₂)) = x₁ + x₂ 193.16/71.72
POL(c3(x₁, x₂)) = x₁ + x₂ 193.16/71.72
POL(c4(x₁, x₂)) = x₁ + x₂ 193.16/71.72
POL(c5(x₁, x₂)) = x₁ + x₂ 193.16/71.72
POL(c6(x₁, x₂)) = x₁ + x₂ 193.16/71.72
POL(c7(x₁, x₂)) = x₁ + x₂ 193.16/71.72
POL(c8(x₁, x₂)) = x₁ + x₂ 193.16/71.72
POL(c9(x₁)) = x₁ 193.16/71.72
POL(cons(x₁, x₂)) = [2]x₂ 193.16/71.72
POL(cons1(x₁, x₂)) = x₁ 193.16/71.72
POL(from(x₁)) = x₁ 193.16/71.72
POL(mark(x₁)) = 0 193.16/71.72
POL(ok(x₁)) = [1] + x₁ 193.16/71.72
POL(proper(x₁)) = 0 193.16/71.72
POL(s(x₁)) = [4]x₁

193.16/71.72

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.16/71.72
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.16/71.72
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.16/71.72
active(2nd(z0)) → 2nd(active(z0)) 193.16/71.72
active(cons(z0, z1)) → cons(active(z0), z1) 193.16/71.72
active(from(z0)) → from(active(z0)) 193.16/71.72
active(s(z0)) → s(active(z0)) 193.16/71.72
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.16/71.72
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.16/71.72
2nd(mark(z0)) → mark(2nd(z0)) 193.16/71.72
2nd(ok(z0)) → ok(2nd(z0)) 193.16/71.72
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.16/71.72
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.16/71.72
from(mark(z0)) → mark(from(z0)) 193.16/71.72
from(ok(z0)) → ok(from(z0)) 193.16/71.72
s(mark(z0)) → mark(s(z0)) 193.16/71.72
s(ok(z0)) → ok(s(z0)) 193.16/71.72
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.16/71.72
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.16/71.72
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.16/71.72
proper(2nd(z0)) → 2nd(proper(z0)) 193.16/71.72
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.16/71.72
proper(from(z0)) → from(proper(z0)) 193.16/71.72
proper(s(z0)) → s(proper(z0)) 193.16/71.72
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.16/71.72
top(mark(z0)) → top(proper(z0)) 193.16/71.72
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.16/71.72
2ND(mark(z0)) → c9(2ND(z0)) 193.16/71.72
2ND(ok(z0)) → c10(2ND(z0)) 193.16/71.72
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.16/71.72
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.16/71.72
FROM(mark(z0)) → c13(FROM(z0)) 193.16/71.72
FROM(ok(z0)) → c14(FROM(z0)) 193.16/71.72
S(mark(z0)) → c15(S(z0)) 193.16/71.72
S(ok(z0)) → c16(S(z0)) 193.16/71.72
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.16/71.72
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.16/71.72
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.16/71.72
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.16/71.72
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.16/71.72
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.16/71.72
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(from(z0)) → c2(S(z0)) 193.16/71.72
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.16/71.72
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1))

S tuples:

ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.16/71.72
2ND(mark(z0)) → c9(2ND(z0)) 193.16/71.72
2ND(ok(z0)) → c10(2ND(z0)) 193.16/71.72
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.16/71.72
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.16/71.72
FROM(mark(z0)) → c13(FROM(z0)) 193.16/71.72
FROM(ok(z0)) → c14(FROM(z0)) 193.16/71.72
S(mark(z0)) → c15(S(z0)) 193.16/71.72
S(ok(z0)) → c16(S(z0)) 193.16/71.72
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.16/71.72
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.16/71.72
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.16/71.72
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.16/71.72
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.16/71.72
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.16/71.72
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))

K tuples:

ACTIVE(from(z0)) → c2(S(z0)) 193.16/71.72
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.16/71.72
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

ACTIVE, 2ND, CONS, FROM, S, CONS1, PROPER, TOP

Compound Symbols:

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

193.16/71.72

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

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

TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.16/71.72
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))

We considered the (Usable) Rules:

cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.16/71.72
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.16/71.72
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.16/71.72
active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.16/71.72
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.16/71.72
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.16/71.72
active(2nd(z0)) → 2nd(active(z0)) 193.16/71.72
active(cons(z0, z1)) → cons(active(z0), z1) 193.16/71.72
active(from(z0)) → from(active(z0)) 193.16/71.72
active(s(z0)) → s(active(z0)) 193.16/71.72
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.16/71.72
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.16/71.72
s(mark(z0)) → mark(s(z0)) 193.16/71.72
s(ok(z0)) → ok(s(z0)) 193.16/71.72
from(mark(z0)) → mark(from(z0)) 193.16/71.72
from(ok(z0)) → ok(from(z0)) 193.16/71.72
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.16/71.72
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.16/71.72
2nd(mark(z0)) → mark(2nd(z0)) 193.16/71.72
2nd(ok(z0)) → ok(2nd(z0)) 193.16/71.72
proper(2nd(z0)) → 2nd(proper(z0)) 193.16/71.72
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.16/71.72
proper(from(z0)) → from(proper(z0)) 193.16/71.72
proper(s(z0)) → s(proper(z0)) 193.16/71.72
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1))

And the Tuples:

ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.16/71.72
2ND(mark(z0)) → c9(2ND(z0)) 193.16/71.72
2ND(ok(z0)) → c10(2ND(z0)) 193.16/71.72
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.16/71.72
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.16/71.72
FROM(mark(z0)) → c13(FROM(z0)) 193.16/71.72
FROM(ok(z0)) → c14(FROM(z0)) 193.16/71.72
S(mark(z0)) → c15(S(z0)) 193.16/71.72
S(ok(z0)) → c16(S(z0)) 193.16/71.72
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.16/71.72
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.16/71.72
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.16/71.72
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.16/71.72
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.16/71.72
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.16/71.72
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(from(z0)) → c2(S(z0)) 193.16/71.72
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.16/71.72
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 193.16/71.72

POL(2ND(x₁)) = 0 193.16/71.72
POL(2nd(x₁)) = x₁ 193.16/71.72
POL(ACTIVE(x₁)) = [1] 193.16/71.72
POL(CONS(x₁, x₂)) = 0 193.16/71.72
POL(CONS1(x₁, x₂)) = 0 193.16/71.72
POL(FROM(x₁)) = 0 193.16/71.72
POL(PROPER(x₁)) = 0 193.16/71.72
POL(S(x₁)) = 0 193.16/71.72
POL(TOP(x₁)) = [4]x₁ 193.16/71.72
POL(active(x₁)) = [4] + x₁ 193.16/71.72
POL(c(x₁)) = x₁ 193.16/71.72
POL(c10(x₁)) = x₁ 193.16/71.72
POL(c11(x₁)) = x₁ 193.16/71.72
POL(c12(x₁)) = x₁ 193.16/71.72
POL(c13(x₁)) = x₁ 193.16/71.72
POL(c14(x₁)) = x₁ 193.16/71.72
POL(c15(x₁)) = x₁ 193.16/71.72
POL(c16(x₁)) = x₁ 193.16/71.72
POL(c17(x₁)) = x₁ 193.16/71.72
POL(c18(x₁)) = x₁ 193.16/71.72
POL(c19(x₁)) = x₁ 193.16/71.72
POL(c2(x₁)) = x₁ 193.16/71.72
POL(c20(x₁, x₂)) = x₁ + x₂ 193.16/71.72
POL(c21(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 193.16/71.72
POL(c22(x₁, x₂)) = x₁ + x₂ 193.16/71.72
POL(c23(x₁, x₂)) = x₁ + x₂ 193.16/71.72
POL(c24(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 193.16/71.72
POL(c25(x₁, x₂)) = x₁ + x₂ 193.16/71.72
POL(c26(x₁, x₂)) = x₁ + x₂ 193.16/71.72
POL(c3(x₁, x₂)) = x₁ + x₂ 193.16/71.72
POL(c4(x₁, x₂)) = x₁ + x₂ 193.16/71.72
POL(c5(x₁, x₂)) = x₁ + x₂ 193.16/71.72
POL(c6(x₁, x₂)) = x₁ + x₂ 193.16/71.72
POL(c7(x₁, x₂)) = x₁ + x₂ 193.16/71.72
POL(c8(x₁, x₂)) = x₁ + x₂ 193.16/71.72
POL(c9(x₁)) = x₁ 193.16/71.72
POL(cons(x₁, x₂)) = x₁ 193.16/71.72
POL(cons1(x₁, x₂)) = x₁ + x₂ 193.16/71.72
POL(from(x₁)) = x₁ 193.16/71.72
POL(mark(x₁)) = [1] 193.16/71.72
POL(ok(x₁)) = [5] + x₁ 193.16/71.72
POL(proper(x₁)) = 0 193.16/71.72
POL(s(x₁)) = x₁

193.16/71.72

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.16/71.72
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.16/71.72
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.16/71.72
active(2nd(z0)) → 2nd(active(z0)) 193.16/71.72
active(cons(z0, z1)) → cons(active(z0), z1) 193.16/71.72
active(from(z0)) → from(active(z0)) 193.16/71.72
active(s(z0)) → s(active(z0)) 193.16/71.72
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.16/71.72
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.16/71.72
2nd(mark(z0)) → mark(2nd(z0)) 193.16/71.72
2nd(ok(z0)) → ok(2nd(z0)) 193.16/71.72
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.16/71.72
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.16/71.72
from(mark(z0)) → mark(from(z0)) 193.16/71.72
from(ok(z0)) → ok(from(z0)) 193.16/71.72
s(mark(z0)) → mark(s(z0)) 193.16/71.72
s(ok(z0)) → ok(s(z0)) 193.16/71.72
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.16/71.72
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.16/71.72
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.16/71.72
proper(2nd(z0)) → 2nd(proper(z0)) 193.16/71.72
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.16/71.72
proper(from(z0)) → from(proper(z0)) 193.16/71.72
proper(s(z0)) → s(proper(z0)) 193.16/71.72
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.16/71.72
top(mark(z0)) → top(proper(z0)) 193.16/71.72
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.16/71.72
2ND(mark(z0)) → c9(2ND(z0)) 193.16/71.72
2ND(ok(z0)) → c10(2ND(z0)) 193.16/71.72
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.16/71.72
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.16/71.72
FROM(mark(z0)) → c13(FROM(z0)) 193.16/71.72
FROM(ok(z0)) → c14(FROM(z0)) 193.16/71.72
S(mark(z0)) → c15(S(z0)) 193.16/71.72
S(ok(z0)) → c16(S(z0)) 193.16/71.72
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.16/71.72
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.16/71.72
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.16/71.72
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.16/71.72
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.16/71.72
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.16/71.72
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(from(z0)) → c2(S(z0)) 193.16/71.72
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.16/71.72
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1))

S tuples:

ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.16/71.72
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.16/71.72
2ND(mark(z0)) → c9(2ND(z0)) 193.16/71.72
2ND(ok(z0)) → c10(2ND(z0)) 193.16/71.72
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.16/71.72
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.16/71.72
FROM(mark(z0)) → c13(FROM(z0)) 193.16/71.72
FROM(ok(z0)) → c14(FROM(z0)) 193.16/71.72
S(mark(z0)) → c15(S(z0)) 193.16/71.72
S(ok(z0)) → c16(S(z0)) 193.16/71.72
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.16/71.72
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.16/71.72
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.16/71.72
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.16/71.72
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.16/71.72
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.56/71.82
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))

K tuples:

ACTIVE(from(z0)) → c2(S(z0)) 193.56/71.82
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.56/71.82
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.56/71.82
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.56/71.82
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

ACTIVE, 2ND, CONS, FROM, S, CONS1, PROPER, TOP

Compound Symbols:

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

193.56/71.82

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

Use narrowing to replace ACTIVE(2nd(z0)) → c3(2ND(active(z0)), ACTIVE(z0)) by

ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.82
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.82
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.82
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.82
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.56/71.82
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.82
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1)))

193.56/71.82

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.56/71.82
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.56/71.82
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.56/71.82
active(2nd(z0)) → 2nd(active(z0)) 193.56/71.82
active(cons(z0, z1)) → cons(active(z0), z1) 193.56/71.82
active(from(z0)) → from(active(z0)) 193.56/71.82
active(s(z0)) → s(active(z0)) 193.56/71.82
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.56/71.82
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.56/71.82
2nd(mark(z0)) → mark(2nd(z0)) 193.56/71.82
2nd(ok(z0)) → ok(2nd(z0)) 193.56/71.82
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.56/71.82
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.56/71.82
from(mark(z0)) → mark(from(z0)) 193.56/71.82
from(ok(z0)) → ok(from(z0)) 193.56/71.82
s(mark(z0)) → mark(s(z0)) 193.56/71.82
s(ok(z0)) → ok(s(z0)) 193.56/71.82
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.56/71.82
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.56/71.82
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.56/71.82
proper(2nd(z0)) → 2nd(proper(z0)) 193.56/71.82
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.56/71.82
proper(from(z0)) → from(proper(z0)) 193.56/71.82
proper(s(z0)) → s(proper(z0)) 193.56/71.82
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.56/71.82
top(mark(z0)) → top(proper(z0)) 193.56/71.82
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 193.56/71.82
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) 193.56/71.82
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0)) 193.56/71.82
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.56/71.82
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.56/71.82
2ND(mark(z0)) → c9(2ND(z0)) 193.56/71.82
2ND(ok(z0)) → c10(2ND(z0)) 193.56/71.82
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.56/71.82
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.56/71.82
FROM(mark(z0)) → c13(FROM(z0)) 193.56/71.82
FROM(ok(z0)) → c14(FROM(z0)) 193.56/71.82
S(mark(z0)) → c15(S(z0)) 193.56/71.82
S(ok(z0)) → c16(S(z0)) 193.56/71.82
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.56/71.82
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.56/71.82
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.56/71.82
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0)) 193.56/71.82
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.82
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.56/71.82
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.56/71.82
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.82
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.56/71.82
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0)) 193.56/71.82
ACTIVE(from(z0)) → c2(S(z0)) 193.56/71.82
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.56/71.82
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.56/71.82
ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.82
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.82
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.82
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.82
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.56/71.82
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.82
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1)))

S tuples:

ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) 193.56/71.82
ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) 193.56/71.82
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0)) 193.56/71.82
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.56/71.82
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.56/71.82
2ND(mark(z0)) → c9(2ND(z0)) 193.56/71.82
2ND(ok(z0)) → c10(2ND(z0)) 193.56/71.82
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.56/71.82
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.56/71.82
FROM(mark(z0)) → c13(FROM(z0)) 193.56/71.82
FROM(ok(z0)) → c14(FROM(z0)) 193.56/71.82
S(mark(z0)) → c15(S(z0)) 193.56/71.82
S(ok(z0)) → c16(S(z0)) 193.56/71.82
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.56/71.82
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.56/71.82
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.56/71.82
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0)) 193.56/71.82
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.82
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.56/71.82
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.56/71.82
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.82
ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.82
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.82
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.82
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.82
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.56/71.82
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.82
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1)))

K tuples:

ACTIVE(from(z0)) → c2(S(z0)) 193.56/71.82
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.56/71.82
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.56/71.82
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.56/71.82
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

ACTIVE, 2ND, CONS, FROM, S, CONS1, PROPER, TOP

Compound Symbols:

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

193.56/71.82

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

Use narrowing to replace ACTIVE(cons(z0, z1)) → c4(CONS(active(z0), z1), ACTIVE(z0)) by

ACTIVE(cons(2nd(cons1(z0, cons(z1, z2))), x1)) → c4(CONS(mark(z1), x1), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.82
ACTIVE(cons(2nd(cons(z0, z1)), x1)) → c4(CONS(mark(2nd(cons1(z0, z1))), x1), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.82
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(cons(2nd(z0), x1)) → c4(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 193.56/71.82
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.56/71.82
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.56/71.82
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(active(z0), z1), x1), ACTIVE(cons1(z0, z1))) 193.56/71.82
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(z0, active(z1)), x1), ACTIVE(cons1(z0, z1)))

193.56/71.82

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.56/71.82
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.56/71.82
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.56/71.82
active(2nd(z0)) → 2nd(active(z0)) 193.56/71.82
active(cons(z0, z1)) → cons(active(z0), z1) 193.56/71.82
active(from(z0)) → from(active(z0)) 193.56/71.82
active(s(z0)) → s(active(z0)) 193.56/71.82
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.56/71.82
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.56/71.82
2nd(mark(z0)) → mark(2nd(z0)) 193.56/71.82
2nd(ok(z0)) → ok(2nd(z0)) 193.56/71.82
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.56/71.82
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.56/71.82
from(mark(z0)) → mark(from(z0)) 193.56/71.82
from(ok(z0)) → ok(from(z0)) 193.56/71.82
s(mark(z0)) → mark(s(z0)) 193.56/71.82
s(ok(z0)) → ok(s(z0)) 193.56/71.82
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.56/71.82
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.56/71.82
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.56/71.82
proper(2nd(z0)) → 2nd(proper(z0)) 193.56/71.82
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.56/71.82
proper(from(z0)) → from(proper(z0)) 193.56/71.82
proper(s(z0)) → s(proper(z0)) 193.56/71.82
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.56/71.82
top(mark(z0)) → top(proper(z0)) 193.56/71.82
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) 193.56/71.82
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0)) 193.56/71.82
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.56/71.82
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.56/71.82
2ND(mark(z0)) → c9(2ND(z0)) 193.56/71.82
2ND(ok(z0)) → c10(2ND(z0)) 193.56/71.82
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.56/71.82
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.56/71.82
FROM(mark(z0)) → c13(FROM(z0)) 193.56/71.82
FROM(ok(z0)) → c14(FROM(z0)) 193.56/71.82
S(mark(z0)) → c15(S(z0)) 193.56/71.82
S(ok(z0)) → c16(S(z0)) 193.56/71.82
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.56/71.82
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.56/71.82
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.56/71.82
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0)) 193.56/71.82
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.82
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.56/71.82
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.56/71.82
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.82
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.56/71.82
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0)) 193.56/71.82
ACTIVE(from(z0)) → c2(S(z0)) 193.56/71.82
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.56/71.82
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.56/71.82
ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.82
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.82
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.82
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.82
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.56/71.82
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.82
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.82
ACTIVE(cons(2nd(cons1(z0, cons(z1, z2))), x1)) → c4(CONS(mark(z1), x1), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.82
ACTIVE(cons(2nd(cons(z0, z1)), x1)) → c4(CONS(mark(2nd(cons1(z0, z1))), x1), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.82
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(cons(2nd(z0), x1)) → c4(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 193.56/71.82
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.56/71.82
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.56/71.82
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(active(z0), z1), x1), ACTIVE(cons1(z0, z1))) 193.56/71.82
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(z0, active(z1)), x1), ACTIVE(cons1(z0, z1)))

S tuples:

ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) 193.56/71.82
ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0)) 193.56/71.82
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.56/71.82
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.56/71.82
2ND(mark(z0)) → c9(2ND(z0)) 193.56/71.82
2ND(ok(z0)) → c10(2ND(z0)) 193.56/71.82
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.56/71.82
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.56/71.82
FROM(mark(z0)) → c13(FROM(z0)) 193.56/71.82
FROM(ok(z0)) → c14(FROM(z0)) 193.56/71.82
S(mark(z0)) → c15(S(z0)) 193.56/71.82
S(ok(z0)) → c16(S(z0)) 193.56/71.82
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.56/71.82
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.56/71.82
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.56/71.82
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0)) 193.56/71.82
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.82
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.56/71.82
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.56/71.82
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.82
ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.82
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.82
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.82
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.82
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.56/71.82
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.82
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.82
ACTIVE(cons(2nd(cons1(z0, cons(z1, z2))), x1)) → c4(CONS(mark(z1), x1), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.82
ACTIVE(cons(2nd(cons(z0, z1)), x1)) → c4(CONS(mark(2nd(cons1(z0, z1))), x1), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.82
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(cons(2nd(z0), x1)) → c4(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 193.56/71.82
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.56/71.82
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.56/71.82
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(active(z0), z1), x1), ACTIVE(cons1(z0, z1))) 193.56/71.82
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(z0, active(z1)), x1), ACTIVE(cons1(z0, z1)))

K tuples:

ACTIVE(from(z0)) → c2(S(z0)) 193.56/71.82
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.56/71.82
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.56/71.82
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.56/71.82
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

ACTIVE, 2ND, CONS, FROM, S, CONS1, PROPER, TOP

Compound Symbols:

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

193.56/71.82

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

Use narrowing to replace ACTIVE(from(z0)) → c5(FROM(active(z0)), ACTIVE(z0)) by

ACTIVE(from(2nd(cons1(z0, cons(z1, z2))))) → c5(FROM(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.82
ACTIVE(from(2nd(cons(z0, z1)))) → c5(FROM(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.82
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(from(2nd(z0))) → c5(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.82
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.82
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.56/71.82
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.82
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1)))

193.56/71.82

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.56/71.82
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.56/71.82
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.56/71.82
active(2nd(z0)) → 2nd(active(z0)) 193.56/71.82
active(cons(z0, z1)) → cons(active(z0), z1) 193.56/71.82
active(from(z0)) → from(active(z0)) 193.56/71.82
active(s(z0)) → s(active(z0)) 193.56/71.82
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.56/71.82
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.56/71.82
2nd(mark(z0)) → mark(2nd(z0)) 193.56/71.82
2nd(ok(z0)) → ok(2nd(z0)) 193.56/71.82
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.56/71.82
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.56/71.82
from(mark(z0)) → mark(from(z0)) 193.56/71.82
from(ok(z0)) → ok(from(z0)) 193.56/71.82
s(mark(z0)) → mark(s(z0)) 193.56/71.82
s(ok(z0)) → ok(s(z0)) 193.56/71.82
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.56/71.82
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.56/71.82
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.56/71.82
proper(2nd(z0)) → 2nd(proper(z0)) 193.56/71.82
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.56/71.82
proper(from(z0)) → from(proper(z0)) 193.56/71.82
proper(s(z0)) → s(proper(z0)) 193.56/71.82
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.56/71.82
top(mark(z0)) → top(proper(z0)) 193.56/71.82
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0)) 193.56/71.82
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.56/71.82
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.56/71.82
2ND(mark(z0)) → c9(2ND(z0)) 193.56/71.82
2ND(ok(z0)) → c10(2ND(z0)) 193.56/71.82
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.56/71.82
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.56/71.82
FROM(mark(z0)) → c13(FROM(z0)) 193.56/71.82
FROM(ok(z0)) → c14(FROM(z0)) 193.56/71.82
S(mark(z0)) → c15(S(z0)) 193.56/71.82
S(ok(z0)) → c16(S(z0)) 193.56/71.82
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.56/71.82
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.56/71.82
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.56/71.82
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0)) 193.56/71.82
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.82
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.56/71.82
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.56/71.82
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.82
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.56/71.82
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0)) 193.56/71.82
ACTIVE(from(z0)) → c2(S(z0)) 193.56/71.82
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.56/71.82
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.56/71.82
ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.82
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.82
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.82
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.82
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.56/71.82
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.82
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.82
ACTIVE(cons(2nd(cons1(z0, cons(z1, z2))), x1)) → c4(CONS(mark(z1), x1), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.82
ACTIVE(cons(2nd(cons(z0, z1)), x1)) → c4(CONS(mark(2nd(cons1(z0, z1))), x1), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.82
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(cons(2nd(z0), x1)) → c4(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 193.56/71.82
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.56/71.82
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.56/71.82
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(active(z0), z1), x1), ACTIVE(cons1(z0, z1))) 193.56/71.82
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(z0, active(z1)), x1), ACTIVE(cons1(z0, z1))) 193.56/71.82
ACTIVE(from(2nd(cons1(z0, cons(z1, z2))))) → c5(FROM(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.82
ACTIVE(from(2nd(cons(z0, z1)))) → c5(FROM(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.82
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(from(2nd(z0))) → c5(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.82
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.82
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.56/71.82
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.56/71.82
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.82
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1)))

S tuples:

ACTIVE(s(z0)) → c6(S(active(z0)), ACTIVE(z0)) 193.56/71.82
ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.56/71.82
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.56/71.82
2ND(mark(z0)) → c9(2ND(z0)) 193.56/71.82
2ND(ok(z0)) → c10(2ND(z0)) 193.56/71.82
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.56/71.83
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.56/71.83
FROM(mark(z0)) → c13(FROM(z0)) 193.56/71.83
FROM(ok(z0)) → c14(FROM(z0)) 193.56/71.83
S(mark(z0)) → c15(S(z0)) 193.56/71.83
S(ok(z0)) → c16(S(z0)) 193.56/71.83
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.56/71.83
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.56/71.83
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.56/71.83
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0)) 193.56/71.83
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.83
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.56/71.83
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.56/71.83
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.83
ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.83
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.83
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.83
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.83
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.83
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.56/71.83
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.56/71.83
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.83
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.83
ACTIVE(cons(2nd(cons1(z0, cons(z1, z2))), x1)) → c4(CONS(mark(z1), x1), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.83
ACTIVE(cons(2nd(cons(z0, z1)), x1)) → c4(CONS(mark(2nd(cons1(z0, z1))), x1), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.83
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.56/71.83
ACTIVE(cons(2nd(z0), x1)) → c4(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 193.56/71.83
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.56/71.83
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.56/71.83
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.56/71.83
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(active(z0), z1), x1), ACTIVE(cons1(z0, z1))) 193.56/71.83
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(z0, active(z1)), x1), ACTIVE(cons1(z0, z1))) 193.56/71.83
ACTIVE(from(2nd(cons1(z0, cons(z1, z2))))) → c5(FROM(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.83
ACTIVE(from(2nd(cons(z0, z1)))) → c5(FROM(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.83
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.83
ACTIVE(from(2nd(z0))) → c5(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.83
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.83
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.56/71.83
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.56/71.83
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.83
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1)))

K tuples:

ACTIVE(from(z0)) → c2(S(z0)) 193.56/71.85
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.56/71.85
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.56/71.85
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.56/71.85
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

ACTIVE, 2ND, CONS, FROM, S, CONS1, PROPER, TOP

Compound Symbols:

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

193.56/71.85

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

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

ACTIVE(s(2nd(cons1(z0, cons(z1, z2))))) → c6(S(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.85
ACTIVE(s(2nd(cons(z0, z1)))) → c6(S(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.85
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.85
ACTIVE(s(2nd(z0))) → c6(S(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.85
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.85
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0))) 193.56/71.85
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0))) 193.56/71.85
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.85
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1)))

193.56/71.85

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.56/71.85
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.56/71.85
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.56/71.85
active(2nd(z0)) → 2nd(active(z0)) 193.56/71.85
active(cons(z0, z1)) → cons(active(z0), z1) 193.56/71.85
active(from(z0)) → from(active(z0)) 193.56/71.85
active(s(z0)) → s(active(z0)) 193.56/71.85
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.56/71.85
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.56/71.85
2nd(mark(z0)) → mark(2nd(z0)) 193.56/71.85
2nd(ok(z0)) → ok(2nd(z0)) 193.56/71.85
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.56/71.85
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.56/71.85
from(mark(z0)) → mark(from(z0)) 193.56/71.85
from(ok(z0)) → ok(from(z0)) 193.56/71.85
s(mark(z0)) → mark(s(z0)) 193.56/71.85
s(ok(z0)) → ok(s(z0)) 193.56/71.85
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.56/71.85
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.56/71.85
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.56/71.85
proper(2nd(z0)) → 2nd(proper(z0)) 193.56/71.85
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.56/71.85
proper(from(z0)) → from(proper(z0)) 193.56/71.85
proper(s(z0)) → s(proper(z0)) 193.56/71.85
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.56/71.85
top(mark(z0)) → top(proper(z0)) 193.56/71.85
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.56/71.85
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.56/71.85
2ND(mark(z0)) → c9(2ND(z0)) 193.56/71.85
2ND(ok(z0)) → c10(2ND(z0)) 193.56/71.85
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.56/71.85
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.56/71.85
FROM(mark(z0)) → c13(FROM(z0)) 193.56/71.85
FROM(ok(z0)) → c14(FROM(z0)) 193.56/71.85
S(mark(z0)) → c15(S(z0)) 193.56/71.85
S(ok(z0)) → c16(S(z0)) 193.56/71.85
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.56/71.85
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.56/71.85
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.56/71.85
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0)) 193.56/71.85
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.85
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.56/71.85
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.56/71.85
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.85
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.56/71.85
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0)) 193.56/71.85
ACTIVE(from(z0)) → c2(S(z0)) 193.56/71.85
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.56/71.85
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.56/71.85
ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.85
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.85
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.85
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.85
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.85
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.56/71.85
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.56/71.85
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.85
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.85
ACTIVE(cons(2nd(cons1(z0, cons(z1, z2))), x1)) → c4(CONS(mark(z1), x1), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.85
ACTIVE(cons(2nd(cons(z0, z1)), x1)) → c4(CONS(mark(2nd(cons1(z0, z1))), x1), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.85
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.56/71.85
ACTIVE(cons(2nd(z0), x1)) → c4(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 193.56/71.85
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.56/71.85
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.56/71.85
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.56/71.85
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(active(z0), z1), x1), ACTIVE(cons1(z0, z1))) 193.56/71.85
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(z0, active(z1)), x1), ACTIVE(cons1(z0, z1))) 193.56/71.85
ACTIVE(from(2nd(cons1(z0, cons(z1, z2))))) → c5(FROM(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.85
ACTIVE(from(2nd(cons(z0, z1)))) → c5(FROM(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.85
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.85
ACTIVE(from(2nd(z0))) → c5(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.85
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.85
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.56/71.85
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.56/71.85
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.85
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.85
ACTIVE(s(2nd(cons1(z0, cons(z1, z2))))) → c6(S(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.85
ACTIVE(s(2nd(cons(z0, z1)))) → c6(S(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.85
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.85
ACTIVE(s(2nd(z0))) → c6(S(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.85
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.85
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0))) 193.56/71.85
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0))) 193.56/71.85
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.85
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1)))

S tuples:

ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.56/71.85
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.56/71.85
2ND(mark(z0)) → c9(2ND(z0)) 193.56/71.85
2ND(ok(z0)) → c10(2ND(z0)) 193.56/71.85
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.56/71.85
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.56/71.85
FROM(mark(z0)) → c13(FROM(z0)) 193.56/71.85
FROM(ok(z0)) → c14(FROM(z0)) 193.56/71.85
S(mark(z0)) → c15(S(z0)) 193.56/71.85
S(ok(z0)) → c16(S(z0)) 193.56/71.85
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.56/71.85
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.56/71.85
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.56/71.85
PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0)) 193.56/71.85
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.85
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.56/71.85
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.56/71.85
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.85
ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.85
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.85
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.85
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.85
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.85
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.56/71.85
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.56/71.85
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.85
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.85
ACTIVE(cons(2nd(cons1(z0, cons(z1, z2))), x1)) → c4(CONS(mark(z1), x1), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.85
ACTIVE(cons(2nd(cons(z0, z1)), x1)) → c4(CONS(mark(2nd(cons1(z0, z1))), x1), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.85
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.56/71.85
ACTIVE(cons(2nd(z0), x1)) → c4(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 193.56/71.85
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.56/71.85
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.56/71.85
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.56/71.85
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(active(z0), z1), x1), ACTIVE(cons1(z0, z1))) 193.56/71.85
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(z0, active(z1)), x1), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(from(2nd(cons1(z0, cons(z1, z2))))) → c5(FROM(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(from(2nd(cons(z0, z1)))) → c5(FROM(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(from(2nd(z0))) → c5(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(s(2nd(cons1(z0, cons(z1, z2))))) → c6(S(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(s(2nd(cons(z0, z1)))) → c6(S(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(s(2nd(z0))) → c6(S(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1)))

K tuples:

ACTIVE(from(z0)) → c2(S(z0)) 193.56/71.87
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.56/71.87
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.56/71.87
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.56/71.87
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

ACTIVE, 2ND, CONS, FROM, S, CONS1, PROPER, TOP

Compound Symbols:

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

193.56/71.87

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

Use narrowing to replace PROPER(2nd(z0)) → c20(2ND(proper(z0)), PROPER(z0)) by

PROPER(2nd(2nd(z0))) → c20(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.87
PROPER(2nd(cons(z0, z1))) → c20(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.87
PROPER(2nd(from(z0))) → c20(2ND(from(proper(z0))), PROPER(from(z0))) 193.56/71.87
PROPER(2nd(s(z0))) → c20(2ND(s(proper(z0))), PROPER(s(z0))) 193.56/71.87
PROPER(2nd(cons1(z0, z1))) → c20(2ND(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))

193.56/71.87

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.56/71.87
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.56/71.87
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.56/71.87
active(2nd(z0)) → 2nd(active(z0)) 193.56/71.87
active(cons(z0, z1)) → cons(active(z0), z1) 193.56/71.87
active(from(z0)) → from(active(z0)) 193.56/71.87
active(s(z0)) → s(active(z0)) 193.56/71.87
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.56/71.87
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.56/71.87
2nd(mark(z0)) → mark(2nd(z0)) 193.56/71.87
2nd(ok(z0)) → ok(2nd(z0)) 193.56/71.87
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.56/71.87
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.56/71.87
from(mark(z0)) → mark(from(z0)) 193.56/71.87
from(ok(z0)) → ok(from(z0)) 193.56/71.87
s(mark(z0)) → mark(s(z0)) 193.56/71.87
s(ok(z0)) → ok(s(z0)) 193.56/71.87
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.56/71.87
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.56/71.87
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.56/71.87
proper(2nd(z0)) → 2nd(proper(z0)) 193.56/71.87
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.56/71.87
proper(from(z0)) → from(proper(z0)) 193.56/71.87
proper(s(z0)) → s(proper(z0)) 193.56/71.87
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.56/71.87
top(mark(z0)) → top(proper(z0)) 193.56/71.87
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.56/71.87
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.56/71.87
2ND(mark(z0)) → c9(2ND(z0)) 193.56/71.87
2ND(ok(z0)) → c10(2ND(z0)) 193.56/71.87
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.56/71.87
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.56/71.87
FROM(mark(z0)) → c13(FROM(z0)) 193.56/71.87
FROM(ok(z0)) → c14(FROM(z0)) 193.56/71.87
S(mark(z0)) → c15(S(z0)) 193.56/71.87
S(ok(z0)) → c16(S(z0)) 193.56/71.87
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.56/71.87
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.56/71.87
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.56/71.87
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.87
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.56/71.87
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.56/71.87
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.87
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.56/71.87
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0)) 193.56/71.87
ACTIVE(from(z0)) → c2(S(z0)) 193.56/71.87
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.56/71.87
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.56/71.87
ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(cons(2nd(cons1(z0, cons(z1, z2))), x1)) → c4(CONS(mark(z1), x1), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(cons(2nd(cons(z0, z1)), x1)) → c4(CONS(mark(2nd(cons1(z0, z1))), x1), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(cons(2nd(z0), x1)) → c4(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(active(z0), z1), x1), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(z0, active(z1)), x1), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(from(2nd(cons1(z0, cons(z1, z2))))) → c5(FROM(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(from(2nd(cons(z0, z1)))) → c5(FROM(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(from(2nd(z0))) → c5(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(s(2nd(cons1(z0, cons(z1, z2))))) → c6(S(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(s(2nd(cons(z0, z1)))) → c6(S(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(s(2nd(z0))) → c6(S(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.87
PROPER(2nd(2nd(z0))) → c20(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.87
PROPER(2nd(cons(z0, z1))) → c20(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.87
PROPER(2nd(from(z0))) → c20(2ND(from(proper(z0))), PROPER(from(z0))) 193.56/71.87
PROPER(2nd(s(z0))) → c20(2ND(s(proper(z0))), PROPER(s(z0))) 193.56/71.87
PROPER(2nd(cons1(z0, z1))) → c20(2ND(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))

S tuples:

ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.56/71.87
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.56/71.87
2ND(mark(z0)) → c9(2ND(z0)) 193.56/71.87
2ND(ok(z0)) → c10(2ND(z0)) 193.56/71.87
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.56/71.87
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.56/71.87
FROM(mark(z0)) → c13(FROM(z0)) 193.56/71.87
FROM(ok(z0)) → c14(FROM(z0)) 193.56/71.87
S(mark(z0)) → c15(S(z0)) 193.56/71.87
S(ok(z0)) → c16(S(z0)) 193.56/71.87
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.56/71.87
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.56/71.87
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.56/71.87
PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.87
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.56/71.87
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.56/71.87
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.87
ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(cons(2nd(cons1(z0, cons(z1, z2))), x1)) → c4(CONS(mark(z1), x1), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(cons(2nd(cons(z0, z1)), x1)) → c4(CONS(mark(2nd(cons1(z0, z1))), x1), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(cons(2nd(z0), x1)) → c4(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(active(z0), z1), x1), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(z0, active(z1)), x1), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(from(2nd(cons1(z0, cons(z1, z2))))) → c5(FROM(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(from(2nd(cons(z0, z1)))) → c5(FROM(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(from(2nd(z0))) → c5(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(s(2nd(cons1(z0, cons(z1, z2))))) → c6(S(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(s(2nd(cons(z0, z1)))) → c6(S(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(s(2nd(z0))) → c6(S(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.87
PROPER(2nd(2nd(z0))) → c20(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.87
PROPER(2nd(cons(z0, z1))) → c20(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.87
PROPER(2nd(from(z0))) → c20(2ND(from(proper(z0))), PROPER(from(z0))) 193.56/71.87
PROPER(2nd(s(z0))) → c20(2ND(s(proper(z0))), PROPER(s(z0))) 193.56/71.87
PROPER(2nd(cons1(z0, z1))) → c20(2ND(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))

K tuples:

ACTIVE(from(z0)) → c2(S(z0)) 193.56/71.87
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.56/71.87
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.56/71.87
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.56/71.87
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

ACTIVE, 2ND, CONS, FROM, S, CONS1, PROPER, TOP

Compound Symbols:

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

193.56/71.87

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

Use narrowing to replace PROPER(cons(z0, z1)) → c21(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(cons(x0, 2nd(z0))) → c21(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 193.56/71.87
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.56/71.87
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.56/71.87
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.56/71.87
PROPER(cons(x0, cons1(z0, z1))) → c21(CONS(proper(x0), cons1(proper(z0), proper(z1))), PROPER(x0), PROPER(cons1(z0, z1))) 193.56/71.87
PROPER(cons(2nd(z0), x1)) → c21(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 193.56/71.87
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.56/71.87
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.56/71.87
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.56/71.87
PROPER(cons(cons1(z0, z1), x1)) → c21(CONS(cons1(proper(z0), proper(z1)), proper(x1)), PROPER(cons1(z0, z1)), PROPER(x1))

193.56/71.87

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.56/71.87
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.56/71.87
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.56/71.87
active(2nd(z0)) → 2nd(active(z0)) 193.56/71.87
active(cons(z0, z1)) → cons(active(z0), z1) 193.56/71.87
active(from(z0)) → from(active(z0)) 193.56/71.87
active(s(z0)) → s(active(z0)) 193.56/71.87
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.56/71.87
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.56/71.87
2nd(mark(z0)) → mark(2nd(z0)) 193.56/71.87
2nd(ok(z0)) → ok(2nd(z0)) 193.56/71.87
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.56/71.87
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.56/71.87
from(mark(z0)) → mark(from(z0)) 193.56/71.87
from(ok(z0)) → ok(from(z0)) 193.56/71.87
s(mark(z0)) → mark(s(z0)) 193.56/71.87
s(ok(z0)) → ok(s(z0)) 193.56/71.87
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.56/71.87
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.56/71.87
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.56/71.87
proper(2nd(z0)) → 2nd(proper(z0)) 193.56/71.87
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.56/71.87
proper(from(z0)) → from(proper(z0)) 193.56/71.87
proper(s(z0)) → s(proper(z0)) 193.56/71.87
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.56/71.87
top(mark(z0)) → top(proper(z0)) 193.56/71.87
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.56/71.87
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.56/71.87
2ND(mark(z0)) → c9(2ND(z0)) 193.56/71.87
2ND(ok(z0)) → c10(2ND(z0)) 193.56/71.87
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.56/71.87
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.56/71.87
FROM(mark(z0)) → c13(FROM(z0)) 193.56/71.87
FROM(ok(z0)) → c14(FROM(z0)) 193.56/71.87
S(mark(z0)) → c15(S(z0)) 193.56/71.87
S(ok(z0)) → c16(S(z0)) 193.56/71.87
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.56/71.87
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.56/71.87
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.56/71.87
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.56/71.87
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.56/71.87
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.87
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.56/71.87
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0)) 193.56/71.87
ACTIVE(from(z0)) → c2(S(z0)) 193.56/71.87
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.56/71.87
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.56/71.87
ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(cons(2nd(cons1(z0, cons(z1, z2))), x1)) → c4(CONS(mark(z1), x1), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(cons(2nd(cons(z0, z1)), x1)) → c4(CONS(mark(2nd(cons1(z0, z1))), x1), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(cons(2nd(z0), x1)) → c4(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(active(z0), z1), x1), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(z0, active(z1)), x1), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(from(2nd(cons1(z0, cons(z1, z2))))) → c5(FROM(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(from(2nd(cons(z0, z1)))) → c5(FROM(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(from(2nd(z0))) → c5(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(s(2nd(cons1(z0, cons(z1, z2))))) → c6(S(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(s(2nd(cons(z0, z1)))) → c6(S(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(s(2nd(z0))) → c6(S(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.87
PROPER(2nd(2nd(z0))) → c20(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.87
PROPER(2nd(cons(z0, z1))) → c20(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.87
PROPER(2nd(from(z0))) → c20(2ND(from(proper(z0))), PROPER(from(z0))) 193.56/71.87
PROPER(2nd(s(z0))) → c20(2ND(s(proper(z0))), PROPER(s(z0))) 193.56/71.87
PROPER(2nd(cons1(z0, z1))) → c20(2ND(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.56/71.87
PROPER(cons(x0, 2nd(z0))) → c21(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 193.56/71.87
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.56/71.87
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.56/71.87
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.56/71.87
PROPER(cons(x0, cons1(z0, z1))) → c21(CONS(proper(x0), cons1(proper(z0), proper(z1))), PROPER(x0), PROPER(cons1(z0, z1))) 193.56/71.87
PROPER(cons(2nd(z0), x1)) → c21(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 193.56/71.87
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.56/71.87
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.56/71.87
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.56/71.87
PROPER(cons(cons1(z0, z1), x1)) → c21(CONS(cons1(proper(z0), proper(z1)), proper(x1)), PROPER(cons1(z0, z1)), PROPER(x1))

S tuples:

ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.56/71.87
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.56/71.87
2ND(mark(z0)) → c9(2ND(z0)) 193.56/71.87
2ND(ok(z0)) → c10(2ND(z0)) 193.56/71.87
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.56/71.87
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.56/71.87
FROM(mark(z0)) → c13(FROM(z0)) 193.56/71.87
FROM(ok(z0)) → c14(FROM(z0)) 193.56/71.87
S(mark(z0)) → c15(S(z0)) 193.56/71.87
S(ok(z0)) → c16(S(z0)) 193.56/71.87
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.56/71.87
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.56/71.87
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.56/71.87
PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) 193.56/71.87
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.56/71.87
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.87
ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(cons(2nd(cons1(z0, cons(z1, z2))), x1)) → c4(CONS(mark(z1), x1), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(cons(2nd(cons(z0, z1)), x1)) → c4(CONS(mark(2nd(cons1(z0, z1))), x1), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(cons(2nd(z0), x1)) → c4(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(active(z0), z1), x1), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(z0, active(z1)), x1), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(from(2nd(cons1(z0, cons(z1, z2))))) → c5(FROM(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(from(2nd(cons(z0, z1)))) → c5(FROM(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(from(2nd(z0))) → c5(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(s(2nd(cons1(z0, cons(z1, z2))))) → c6(S(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(s(2nd(cons(z0, z1)))) → c6(S(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(s(2nd(z0))) → c6(S(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.87
PROPER(2nd(2nd(z0))) → c20(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.87
PROPER(2nd(cons(z0, z1))) → c20(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.87
PROPER(2nd(from(z0))) → c20(2ND(from(proper(z0))), PROPER(from(z0))) 193.56/71.87
PROPER(2nd(s(z0))) → c20(2ND(s(proper(z0))), PROPER(s(z0))) 193.56/71.87
PROPER(2nd(cons1(z0, z1))) → c20(2ND(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.56/71.87
PROPER(cons(x0, 2nd(z0))) → c21(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 193.56/71.87
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.56/71.87
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.56/71.87
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.56/71.87
PROPER(cons(x0, cons1(z0, z1))) → c21(CONS(proper(x0), cons1(proper(z0), proper(z1))), PROPER(x0), PROPER(cons1(z0, z1))) 193.56/71.87
PROPER(cons(2nd(z0), x1)) → c21(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 193.56/71.87
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.56/71.87
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.56/71.87
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.56/71.87
PROPER(cons(cons1(z0, z1), x1)) → c21(CONS(cons1(proper(z0), proper(z1)), proper(x1)), PROPER(cons1(z0, z1)), PROPER(x1))

K tuples:

ACTIVE(from(z0)) → c2(S(z0)) 193.56/71.87
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.56/71.87
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.56/71.87
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.56/71.87
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

ACTIVE, 2ND, CONS, FROM, S, CONS1, PROPER, TOP

Compound Symbols:

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

193.56/71.87

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

Use narrowing to replace PROPER(from(z0)) → c22(FROM(proper(z0)), PROPER(z0)) by

PROPER(from(2nd(z0))) → c22(FROM(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.87
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.87
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0))) 193.56/71.87
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 193.56/71.87
PROPER(from(cons1(z0, z1))) → c22(FROM(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))

193.56/71.87

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.56/71.87
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.56/71.87
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.56/71.87
active(2nd(z0)) → 2nd(active(z0)) 193.56/71.87
active(cons(z0, z1)) → cons(active(z0), z1) 193.56/71.87
active(from(z0)) → from(active(z0)) 193.56/71.87
active(s(z0)) → s(active(z0)) 193.56/71.87
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.56/71.87
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.56/71.87
2nd(mark(z0)) → mark(2nd(z0)) 193.56/71.87
2nd(ok(z0)) → ok(2nd(z0)) 193.56/71.87
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.56/71.87
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.56/71.87
from(mark(z0)) → mark(from(z0)) 193.56/71.87
from(ok(z0)) → ok(from(z0)) 193.56/71.87
s(mark(z0)) → mark(s(z0)) 193.56/71.87
s(ok(z0)) → ok(s(z0)) 193.56/71.87
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.56/71.87
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.56/71.87
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.56/71.87
proper(2nd(z0)) → 2nd(proper(z0)) 193.56/71.87
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.56/71.87
proper(from(z0)) → from(proper(z0)) 193.56/71.87
proper(s(z0)) → s(proper(z0)) 193.56/71.87
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.56/71.87
top(mark(z0)) → top(proper(z0)) 193.56/71.87
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.56/71.87
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.56/71.87
2ND(mark(z0)) → c9(2ND(z0)) 193.56/71.87
2ND(ok(z0)) → c10(2ND(z0)) 193.56/71.87
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.56/71.87
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.56/71.87
FROM(mark(z0)) → c13(FROM(z0)) 193.56/71.87
FROM(ok(z0)) → c14(FROM(z0)) 193.56/71.87
S(mark(z0)) → c15(S(z0)) 193.56/71.87
S(ok(z0)) → c16(S(z0)) 193.56/71.87
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.56/71.87
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.56/71.87
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.56/71.87
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.56/71.87
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.87
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.56/71.87
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0)) 193.56/71.87
ACTIVE(from(z0)) → c2(S(z0)) 193.56/71.87
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.56/71.87
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.56/71.87
ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(cons(2nd(cons1(z0, cons(z1, z2))), x1)) → c4(CONS(mark(z1), x1), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(cons(2nd(cons(z0, z1)), x1)) → c4(CONS(mark(2nd(cons1(z0, z1))), x1), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(cons(2nd(z0), x1)) → c4(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(active(z0), z1), x1), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(z0, active(z1)), x1), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(from(2nd(cons1(z0, cons(z1, z2))))) → c5(FROM(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(from(2nd(cons(z0, z1)))) → c5(FROM(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(from(2nd(z0))) → c5(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(s(2nd(cons1(z0, cons(z1, z2))))) → c6(S(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(s(2nd(cons(z0, z1)))) → c6(S(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(s(2nd(z0))) → c6(S(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.87
PROPER(2nd(2nd(z0))) → c20(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.87
PROPER(2nd(cons(z0, z1))) → c20(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.87
PROPER(2nd(from(z0))) → c20(2ND(from(proper(z0))), PROPER(from(z0))) 193.56/71.87
PROPER(2nd(s(z0))) → c20(2ND(s(proper(z0))), PROPER(s(z0))) 193.56/71.87
PROPER(2nd(cons1(z0, z1))) → c20(2ND(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.56/71.87
PROPER(cons(x0, 2nd(z0))) → c21(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 193.56/71.87
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.56/71.87
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.56/71.87
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.56/71.87
PROPER(cons(x0, cons1(z0, z1))) → c21(CONS(proper(x0), cons1(proper(z0), proper(z1))), PROPER(x0), PROPER(cons1(z0, z1))) 193.56/71.87
PROPER(cons(2nd(z0), x1)) → c21(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 193.56/71.87
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.56/71.87
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.56/71.87
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.56/71.87
PROPER(cons(cons1(z0, z1), x1)) → c21(CONS(cons1(proper(z0), proper(z1)), proper(x1)), PROPER(cons1(z0, z1)), PROPER(x1)) 193.56/71.87
PROPER(from(2nd(z0))) → c22(FROM(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.87
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.87
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0))) 193.56/71.87
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 193.56/71.87
PROPER(from(cons1(z0, z1))) → c22(FROM(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))

S tuples:

ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.56/71.87
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.56/71.87
2ND(mark(z0)) → c9(2ND(z0)) 193.56/71.87
2ND(ok(z0)) → c10(2ND(z0)) 193.56/71.87
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.56/71.87
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.56/71.87
FROM(mark(z0)) → c13(FROM(z0)) 193.56/71.87
FROM(ok(z0)) → c14(FROM(z0)) 193.56/71.87
S(mark(z0)) → c15(S(z0)) 193.56/71.87
S(ok(z0)) → c16(S(z0)) 193.56/71.87
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.56/71.87
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.56/71.87
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.56/71.87
PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) 193.56/71.87
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.87
ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(cons(2nd(cons1(z0, cons(z1, z2))), x1)) → c4(CONS(mark(z1), x1), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(cons(2nd(cons(z0, z1)), x1)) → c4(CONS(mark(2nd(cons1(z0, z1))), x1), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(cons(2nd(z0), x1)) → c4(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(active(z0), z1), x1), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(z0, active(z1)), x1), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(from(2nd(cons1(z0, cons(z1, z2))))) → c5(FROM(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(from(2nd(cons(z0, z1)))) → c5(FROM(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(from(2nd(z0))) → c5(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(s(2nd(cons1(z0, cons(z1, z2))))) → c6(S(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.87
ACTIVE(s(2nd(cons(z0, z1)))) → c6(S(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.87
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(s(2nd(z0))) → c6(S(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.87
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.87
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0))) 193.56/71.87
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0))) 193.56/71.87
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.87
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.87
PROPER(2nd(2nd(z0))) → c20(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.87
PROPER(2nd(cons(z0, z1))) → c20(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.87
PROPER(2nd(from(z0))) → c20(2ND(from(proper(z0))), PROPER(from(z0))) 193.56/71.87
PROPER(2nd(s(z0))) → c20(2ND(s(proper(z0))), PROPER(s(z0))) 193.56/71.87
PROPER(2nd(cons1(z0, z1))) → c20(2ND(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.56/71.88
PROPER(cons(x0, 2nd(z0))) → c21(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 193.56/71.88
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.56/71.88
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.56/71.88
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.56/71.88
PROPER(cons(x0, cons1(z0, z1))) → c21(CONS(proper(x0), cons1(proper(z0), proper(z1))), PROPER(x0), PROPER(cons1(z0, z1))) 193.56/71.88
PROPER(cons(2nd(z0), x1)) → c21(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 193.56/71.88
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.56/71.88
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.56/71.88
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.56/71.88
PROPER(cons(cons1(z0, z1), x1)) → c21(CONS(cons1(proper(z0), proper(z1)), proper(x1)), PROPER(cons1(z0, z1)), PROPER(x1)) 193.56/71.88
PROPER(from(2nd(z0))) → c22(FROM(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.88
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.88
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0))) 193.56/71.88
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 193.56/71.88
PROPER(from(cons1(z0, z1))) → c22(FROM(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))

K tuples:

ACTIVE(from(z0)) → c2(S(z0)) 193.56/71.88
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.56/71.88
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.56/71.88
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.56/71.88
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

ACTIVE, 2ND, CONS, FROM, S, CONS1, PROPER, TOP

Compound Symbols:

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

193.56/71.88

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

Use narrowing to replace PROPER(s(z0)) → c23(S(proper(z0)), PROPER(z0)) by

PROPER(s(2nd(z0))) → c23(S(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.88
PROPER(s(cons(z0, z1))) → c23(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.88
PROPER(s(from(z0))) → c23(S(from(proper(z0))), PROPER(from(z0))) 193.56/71.88
PROPER(s(s(z0))) → c23(S(s(proper(z0))), PROPER(s(z0))) 193.56/71.88
PROPER(s(cons1(z0, z1))) → c23(S(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))

193.56/71.88

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.56/71.88
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.56/71.88
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.56/71.88
active(2nd(z0)) → 2nd(active(z0)) 193.56/71.88
active(cons(z0, z1)) → cons(active(z0), z1) 193.56/71.88
active(from(z0)) → from(active(z0)) 193.56/71.88
active(s(z0)) → s(active(z0)) 193.56/71.88
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.56/71.88
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.56/71.88
2nd(mark(z0)) → mark(2nd(z0)) 193.56/71.88
2nd(ok(z0)) → ok(2nd(z0)) 193.56/71.88
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.56/71.88
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.56/71.88
from(mark(z0)) → mark(from(z0)) 193.56/71.88
from(ok(z0)) → ok(from(z0)) 193.56/71.88
s(mark(z0)) → mark(s(z0)) 193.56/71.88
s(ok(z0)) → ok(s(z0)) 193.56/71.88
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.56/71.88
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.56/71.88
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.56/71.88
proper(2nd(z0)) → 2nd(proper(z0)) 193.56/71.88
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.56/71.88
proper(from(z0)) → from(proper(z0)) 193.56/71.88
proper(s(z0)) → s(proper(z0)) 193.56/71.88
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.56/71.88
top(mark(z0)) → top(proper(z0)) 193.56/71.88
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.56/71.88
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.56/71.88
2ND(mark(z0)) → c9(2ND(z0)) 193.56/71.88
2ND(ok(z0)) → c10(2ND(z0)) 193.56/71.88
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.56/71.88
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.56/71.88
FROM(mark(z0)) → c13(FROM(z0)) 193.56/71.88
FROM(ok(z0)) → c14(FROM(z0)) 193.56/71.88
S(mark(z0)) → c15(S(z0)) 193.56/71.88
S(ok(z0)) → c16(S(z0)) 193.56/71.88
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.56/71.88
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.56/71.88
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.56/71.88
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.88
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.56/71.88
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0)) 193.56/71.88
ACTIVE(from(z0)) → c2(S(z0)) 193.56/71.88
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.56/71.88
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.56/71.88
ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.88
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.88
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.88
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.88
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.88
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.56/71.88
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.56/71.88
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.88
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.88
ACTIVE(cons(2nd(cons1(z0, cons(z1, z2))), x1)) → c4(CONS(mark(z1), x1), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.88
ACTIVE(cons(2nd(cons(z0, z1)), x1)) → c4(CONS(mark(2nd(cons1(z0, z1))), x1), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.88
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.56/71.88
ACTIVE(cons(2nd(z0), x1)) → c4(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 193.56/71.88
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.56/71.88
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.56/71.88
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.56/71.88
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(active(z0), z1), x1), ACTIVE(cons1(z0, z1))) 193.56/71.88
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(z0, active(z1)), x1), ACTIVE(cons1(z0, z1))) 193.56/71.88
ACTIVE(from(2nd(cons1(z0, cons(z1, z2))))) → c5(FROM(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.88
ACTIVE(from(2nd(cons(z0, z1)))) → c5(FROM(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.88
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.88
ACTIVE(from(2nd(z0))) → c5(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.88
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.88
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.56/71.88
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.56/71.88
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.88
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.88
ACTIVE(s(2nd(cons1(z0, cons(z1, z2))))) → c6(S(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.88
ACTIVE(s(2nd(cons(z0, z1)))) → c6(S(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.88
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.88
ACTIVE(s(2nd(z0))) → c6(S(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.88
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.88
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0))) 193.56/71.88
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0))) 193.56/71.88
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.88
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.88
PROPER(2nd(2nd(z0))) → c20(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.88
PROPER(2nd(cons(z0, z1))) → c20(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.88
PROPER(2nd(from(z0))) → c20(2ND(from(proper(z0))), PROPER(from(z0))) 193.56/71.88
PROPER(2nd(s(z0))) → c20(2ND(s(proper(z0))), PROPER(s(z0))) 193.56/71.88
PROPER(2nd(cons1(z0, z1))) → c20(2ND(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.56/71.88
PROPER(cons(x0, 2nd(z0))) → c21(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 193.56/71.88
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.56/71.88
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.56/71.88
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.56/71.88
PROPER(cons(x0, cons1(z0, z1))) → c21(CONS(proper(x0), cons1(proper(z0), proper(z1))), PROPER(x0), PROPER(cons1(z0, z1))) 193.56/71.88
PROPER(cons(2nd(z0), x1)) → c21(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 193.56/71.88
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.56/71.88
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.56/71.88
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.56/71.88
PROPER(cons(cons1(z0, z1), x1)) → c21(CONS(cons1(proper(z0), proper(z1)), proper(x1)), PROPER(cons1(z0, z1)), PROPER(x1)) 193.56/71.88
PROPER(from(2nd(z0))) → c22(FROM(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.88
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.88
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0))) 193.56/71.88
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 193.56/71.88
PROPER(from(cons1(z0, z1))) → c22(FROM(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.56/71.88
PROPER(s(2nd(z0))) → c23(S(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.88
PROPER(s(cons(z0, z1))) → c23(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.88
PROPER(s(from(z0))) → c23(S(from(proper(z0))), PROPER(from(z0))) 193.56/71.88
PROPER(s(s(z0))) → c23(S(s(proper(z0))), PROPER(s(z0))) 193.56/71.88
PROPER(s(cons1(z0, z1))) → c23(S(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))

S tuples:

ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.56/71.88
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.56/71.88
2ND(mark(z0)) → c9(2ND(z0)) 193.56/71.88
2ND(ok(z0)) → c10(2ND(z0)) 193.56/71.88
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.56/71.88
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.56/71.88
FROM(mark(z0)) → c13(FROM(z0)) 193.56/71.88
FROM(ok(z0)) → c14(FROM(z0)) 193.56/71.88
S(mark(z0)) → c15(S(z0)) 193.56/71.88
S(ok(z0)) → c16(S(z0)) 193.56/71.88
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.56/71.88
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.56/71.88
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.56/71.88
PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 193.56/71.88
ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.89
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.89
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.89
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.89
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.56/71.89
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(cons(2nd(cons1(z0, cons(z1, z2))), x1)) → c4(CONS(mark(z1), x1), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.89
ACTIVE(cons(2nd(cons(z0, z1)), x1)) → c4(CONS(mark(2nd(cons1(z0, z1))), x1), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.89
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(cons(2nd(z0), x1)) → c4(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 193.56/71.89
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.56/71.89
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.56/71.89
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(active(z0), z1), x1), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(z0, active(z1)), x1), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(from(2nd(cons1(z0, cons(z1, z2))))) → c5(FROM(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.89
ACTIVE(from(2nd(cons(z0, z1)))) → c5(FROM(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.89
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(from(2nd(z0))) → c5(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.89
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.89
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.56/71.89
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(s(2nd(cons1(z0, cons(z1, z2))))) → c6(S(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.89
ACTIVE(s(2nd(cons(z0, z1)))) → c6(S(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.89
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(s(2nd(z0))) → c6(S(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.89
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.89
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0))) 193.56/71.89
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.89
PROPER(2nd(2nd(z0))) → c20(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.89
PROPER(2nd(cons(z0, z1))) → c20(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.89
PROPER(2nd(from(z0))) → c20(2ND(from(proper(z0))), PROPER(from(z0))) 193.56/71.89
PROPER(2nd(s(z0))) → c20(2ND(s(proper(z0))), PROPER(s(z0))) 193.56/71.89
PROPER(2nd(cons1(z0, z1))) → c20(2ND(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.56/71.89
PROPER(cons(x0, 2nd(z0))) → c21(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 193.56/71.89
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.56/71.89
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.56/71.89
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.56/71.89
PROPER(cons(x0, cons1(z0, z1))) → c21(CONS(proper(x0), cons1(proper(z0), proper(z1))), PROPER(x0), PROPER(cons1(z0, z1))) 193.56/71.89
PROPER(cons(2nd(z0), x1)) → c21(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.56/71.89
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons(cons1(z0, z1), x1)) → c21(CONS(cons1(proper(z0), proper(z1)), proper(x1)), PROPER(cons1(z0, z1)), PROPER(x1)) 193.56/71.89
PROPER(from(2nd(z0))) → c22(FROM(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.89
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.89
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0))) 193.56/71.89
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 193.56/71.89
PROPER(from(cons1(z0, z1))) → c22(FROM(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.56/71.89
PROPER(s(2nd(z0))) → c23(S(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.89
PROPER(s(cons(z0, z1))) → c23(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.89
PROPER(s(from(z0))) → c23(S(from(proper(z0))), PROPER(from(z0))) 193.56/71.89
PROPER(s(s(z0))) → c23(S(s(proper(z0))), PROPER(s(z0))) 193.56/71.89
PROPER(s(cons1(z0, z1))) → c23(S(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))

K tuples:

ACTIVE(from(z0)) → c2(S(z0)) 193.56/71.89
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.56/71.89
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.56/71.89
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.56/71.89
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

ACTIVE, 2ND, CONS, FROM, S, CONS1, PROPER, TOP

Compound Symbols:

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

193.56/71.89

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

Use narrowing to replace PROPER(cons1(z0, z1)) → c24(CONS1(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(cons1(x0, 2nd(z0))) → c24(CONS1(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 193.56/71.89
PROPER(cons1(x0, cons(z0, z1))) → c24(CONS1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.56/71.89
PROPER(cons1(x0, from(z0))) → c24(CONS1(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.56/71.89
PROPER(cons1(x0, s(z0))) → c24(CONS1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.56/71.89
PROPER(cons1(x0, cons1(z0, z1))) → c24(CONS1(proper(x0), cons1(proper(z0), proper(z1))), PROPER(x0), PROPER(cons1(z0, z1))) 193.56/71.89
PROPER(cons1(2nd(z0), x1)) → c24(CONS1(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons1(cons(z0, z1), x1)) → c24(CONS1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.56/71.89
PROPER(cons1(from(z0), x1)) → c24(CONS1(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons1(s(z0), x1)) → c24(CONS1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons1(cons1(z0, z1), x1)) → c24(CONS1(cons1(proper(z0), proper(z1)), proper(x1)), PROPER(cons1(z0, z1)), PROPER(x1))

193.56/71.89

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.56/71.89
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.56/71.89
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.56/71.89
active(2nd(z0)) → 2nd(active(z0)) 193.56/71.89
active(cons(z0, z1)) → cons(active(z0), z1) 193.56/71.89
active(from(z0)) → from(active(z0)) 193.56/71.89
active(s(z0)) → s(active(z0)) 193.56/71.89
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.56/71.89
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.56/71.89
2nd(mark(z0)) → mark(2nd(z0)) 193.56/71.89
2nd(ok(z0)) → ok(2nd(z0)) 193.56/71.89
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.56/71.89
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.56/71.89
from(mark(z0)) → mark(from(z0)) 193.56/71.89
from(ok(z0)) → ok(from(z0)) 193.56/71.89
s(mark(z0)) → mark(s(z0)) 193.56/71.89
s(ok(z0)) → ok(s(z0)) 193.56/71.89
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.56/71.89
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.56/71.89
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.56/71.89
proper(2nd(z0)) → 2nd(proper(z0)) 193.56/71.89
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.56/71.89
proper(from(z0)) → from(proper(z0)) 193.56/71.89
proper(s(z0)) → s(proper(z0)) 193.56/71.89
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.56/71.89
top(mark(z0)) → top(proper(z0)) 193.56/71.89
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.56/71.89
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.56/71.89
2ND(mark(z0)) → c9(2ND(z0)) 193.56/71.89
2ND(ok(z0)) → c10(2ND(z0)) 193.56/71.89
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.56/71.89
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.56/71.89
FROM(mark(z0)) → c13(FROM(z0)) 193.56/71.89
FROM(ok(z0)) → c14(FROM(z0)) 193.56/71.89
S(mark(z0)) → c15(S(z0)) 193.56/71.89
S(ok(z0)) → c16(S(z0)) 193.56/71.89
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.56/71.89
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.56/71.89
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.56/71.89
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.56/71.89
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0)) 193.56/71.89
ACTIVE(from(z0)) → c2(S(z0)) 193.56/71.89
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.56/71.89
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.56/71.89
ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.89
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.89
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.89
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.89
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.56/71.89
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(cons(2nd(cons1(z0, cons(z1, z2))), x1)) → c4(CONS(mark(z1), x1), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.89
ACTIVE(cons(2nd(cons(z0, z1)), x1)) → c4(CONS(mark(2nd(cons1(z0, z1))), x1), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.89
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(cons(2nd(z0), x1)) → c4(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 193.56/71.89
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.56/71.89
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.56/71.89
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(active(z0), z1), x1), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(z0, active(z1)), x1), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(from(2nd(cons1(z0, cons(z1, z2))))) → c5(FROM(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.89
ACTIVE(from(2nd(cons(z0, z1)))) → c5(FROM(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.89
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(from(2nd(z0))) → c5(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.89
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.89
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.56/71.89
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(s(2nd(cons1(z0, cons(z1, z2))))) → c6(S(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.89
ACTIVE(s(2nd(cons(z0, z1)))) → c6(S(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.89
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(s(2nd(z0))) → c6(S(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.89
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.89
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0))) 193.56/71.89
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.89
PROPER(2nd(2nd(z0))) → c20(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.89
PROPER(2nd(cons(z0, z1))) → c20(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.89
PROPER(2nd(from(z0))) → c20(2ND(from(proper(z0))), PROPER(from(z0))) 193.56/71.89
PROPER(2nd(s(z0))) → c20(2ND(s(proper(z0))), PROPER(s(z0))) 193.56/71.89
PROPER(2nd(cons1(z0, z1))) → c20(2ND(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.56/71.89
PROPER(cons(x0, 2nd(z0))) → c21(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 193.56/71.89
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.56/71.89
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.56/71.89
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.56/71.89
PROPER(cons(x0, cons1(z0, z1))) → c21(CONS(proper(x0), cons1(proper(z0), proper(z1))), PROPER(x0), PROPER(cons1(z0, z1))) 193.56/71.89
PROPER(cons(2nd(z0), x1)) → c21(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.56/71.89
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons(cons1(z0, z1), x1)) → c21(CONS(cons1(proper(z0), proper(z1)), proper(x1)), PROPER(cons1(z0, z1)), PROPER(x1)) 193.56/71.89
PROPER(from(2nd(z0))) → c22(FROM(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.89
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.89
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0))) 193.56/71.89
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 193.56/71.89
PROPER(from(cons1(z0, z1))) → c22(FROM(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.56/71.89
PROPER(s(2nd(z0))) → c23(S(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.89
PROPER(s(cons(z0, z1))) → c23(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.89
PROPER(s(from(z0))) → c23(S(from(proper(z0))), PROPER(from(z0))) 193.56/71.89
PROPER(s(s(z0))) → c23(S(s(proper(z0))), PROPER(s(z0))) 193.56/71.89
PROPER(s(cons1(z0, z1))) → c23(S(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.56/71.89
PROPER(cons1(x0, 2nd(z0))) → c24(CONS1(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 193.56/71.89
PROPER(cons1(x0, cons(z0, z1))) → c24(CONS1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.56/71.89
PROPER(cons1(x0, from(z0))) → c24(CONS1(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.56/71.89
PROPER(cons1(x0, s(z0))) → c24(CONS1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.56/71.89
PROPER(cons1(x0, cons1(z0, z1))) → c24(CONS1(proper(x0), cons1(proper(z0), proper(z1))), PROPER(x0), PROPER(cons1(z0, z1))) 193.56/71.89
PROPER(cons1(2nd(z0), x1)) → c24(CONS1(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons1(cons(z0, z1), x1)) → c24(CONS1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.56/71.89
PROPER(cons1(from(z0), x1)) → c24(CONS1(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons1(s(z0), x1)) → c24(CONS1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons1(cons1(z0, z1), x1)) → c24(CONS1(cons1(proper(z0), proper(z1)), proper(x1)), PROPER(cons1(z0, z1)), PROPER(x1))

S tuples:

ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.56/71.89
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.56/71.89
2ND(mark(z0)) → c9(2ND(z0)) 193.56/71.89
2ND(ok(z0)) → c10(2ND(z0)) 193.56/71.89
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.56/71.89
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.56/71.89
FROM(mark(z0)) → c13(FROM(z0)) 193.56/71.89
FROM(ok(z0)) → c14(FROM(z0)) 193.56/71.89
S(mark(z0)) → c15(S(z0)) 193.56/71.89
S(ok(z0)) → c16(S(z0)) 193.56/71.89
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.56/71.89
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.56/71.89
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.56/71.89
ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.89
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.89
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.89
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.89
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.56/71.89
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(cons(2nd(cons1(z0, cons(z1, z2))), x1)) → c4(CONS(mark(z1), x1), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.89
ACTIVE(cons(2nd(cons(z0, z1)), x1)) → c4(CONS(mark(2nd(cons1(z0, z1))), x1), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.89
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(cons(2nd(z0), x1)) → c4(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 193.56/71.89
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.56/71.89
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.56/71.89
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(active(z0), z1), x1), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(z0, active(z1)), x1), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(from(2nd(cons1(z0, cons(z1, z2))))) → c5(FROM(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.89
ACTIVE(from(2nd(cons(z0, z1)))) → c5(FROM(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.89
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(from(2nd(z0))) → c5(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.89
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.89
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.56/71.89
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(s(2nd(cons1(z0, cons(z1, z2))))) → c6(S(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.89
ACTIVE(s(2nd(cons(z0, z1)))) → c6(S(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.89
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(s(2nd(z0))) → c6(S(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.89
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.89
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0))) 193.56/71.89
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.89
PROPER(2nd(2nd(z0))) → c20(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.89
PROPER(2nd(cons(z0, z1))) → c20(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.89
PROPER(2nd(from(z0))) → c20(2ND(from(proper(z0))), PROPER(from(z0))) 193.56/71.89
PROPER(2nd(s(z0))) → c20(2ND(s(proper(z0))), PROPER(s(z0))) 193.56/71.89
PROPER(2nd(cons1(z0, z1))) → c20(2ND(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.56/71.89
PROPER(cons(x0, 2nd(z0))) → c21(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 193.56/71.89
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.56/71.89
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.56/71.89
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.56/71.89
PROPER(cons(x0, cons1(z0, z1))) → c21(CONS(proper(x0), cons1(proper(z0), proper(z1))), PROPER(x0), PROPER(cons1(z0, z1))) 193.56/71.89
PROPER(cons(2nd(z0), x1)) → c21(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.56/71.89
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons(cons1(z0, z1), x1)) → c21(CONS(cons1(proper(z0), proper(z1)), proper(x1)), PROPER(cons1(z0, z1)), PROPER(x1)) 193.56/71.89
PROPER(from(2nd(z0))) → c22(FROM(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.89
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.89
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0))) 193.56/71.89
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 193.56/71.89
PROPER(from(cons1(z0, z1))) → c22(FROM(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.56/71.89
PROPER(s(2nd(z0))) → c23(S(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.89
PROPER(s(cons(z0, z1))) → c23(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.89
PROPER(s(from(z0))) → c23(S(from(proper(z0))), PROPER(from(z0))) 193.56/71.89
PROPER(s(s(z0))) → c23(S(s(proper(z0))), PROPER(s(z0))) 193.56/71.89
PROPER(s(cons1(z0, z1))) → c23(S(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.56/71.89
PROPER(cons1(x0, 2nd(z0))) → c24(CONS1(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 193.56/71.89
PROPER(cons1(x0, cons(z0, z1))) → c24(CONS1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.56/71.89
PROPER(cons1(x0, from(z0))) → c24(CONS1(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.56/71.89
PROPER(cons1(x0, s(z0))) → c24(CONS1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.56/71.89
PROPER(cons1(x0, cons1(z0, z1))) → c24(CONS1(proper(x0), cons1(proper(z0), proper(z1))), PROPER(x0), PROPER(cons1(z0, z1))) 193.56/71.89
PROPER(cons1(2nd(z0), x1)) → c24(CONS1(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons1(cons(z0, z1), x1)) → c24(CONS1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.56/71.89
PROPER(cons1(from(z0), x1)) → c24(CONS1(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons1(s(z0), x1)) → c24(CONS1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons1(cons1(z0, z1), x1)) → c24(CONS1(cons1(proper(z0), proper(z1)), proper(x1)), PROPER(cons1(z0, z1)), PROPER(x1))

K tuples:

ACTIVE(from(z0)) → c2(S(z0)) 193.56/71.89
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.56/71.89
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.56/71.89
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.56/71.89
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

ACTIVE, 2ND, CONS, FROM, S, CONS1, TOP, PROPER

Compound Symbols:

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

193.56/71.89

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

Use narrowing to replace TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(2nd(z0))) → c25(TOP(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.89
TOP(mark(cons(z0, z1))) → c25(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.89
TOP(mark(from(z0))) → c25(TOP(from(proper(z0))), PROPER(from(z0))) 193.56/71.89
TOP(mark(s(z0))) → c25(TOP(s(proper(z0))), PROPER(s(z0))) 193.56/71.89
TOP(mark(cons1(z0, z1))) → c25(TOP(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))

193.56/71.89

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.56/71.89
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.56/71.89
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.56/71.89
active(2nd(z0)) → 2nd(active(z0)) 193.56/71.89
active(cons(z0, z1)) → cons(active(z0), z1) 193.56/71.89
active(from(z0)) → from(active(z0)) 193.56/71.89
active(s(z0)) → s(active(z0)) 193.56/71.89
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.56/71.89
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.56/71.89
2nd(mark(z0)) → mark(2nd(z0)) 193.56/71.89
2nd(ok(z0)) → ok(2nd(z0)) 193.56/71.89
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.56/71.89
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.56/71.89
from(mark(z0)) → mark(from(z0)) 193.56/71.89
from(ok(z0)) → ok(from(z0)) 193.56/71.89
s(mark(z0)) → mark(s(z0)) 193.56/71.89
s(ok(z0)) → ok(s(z0)) 193.56/71.89
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.56/71.89
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.56/71.89
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.56/71.89
proper(2nd(z0)) → 2nd(proper(z0)) 193.56/71.89
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.56/71.89
proper(from(z0)) → from(proper(z0)) 193.56/71.89
proper(s(z0)) → s(proper(z0)) 193.56/71.89
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.56/71.89
top(mark(z0)) → top(proper(z0)) 193.56/71.89
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.56/71.89
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.56/71.89
2ND(mark(z0)) → c9(2ND(z0)) 193.56/71.89
2ND(ok(z0)) → c10(2ND(z0)) 193.56/71.89
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.56/71.89
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.56/71.89
FROM(mark(z0)) → c13(FROM(z0)) 193.56/71.89
FROM(ok(z0)) → c14(FROM(z0)) 193.56/71.89
S(mark(z0)) → c15(S(z0)) 193.56/71.89
S(ok(z0)) → c16(S(z0)) 193.56/71.89
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.56/71.89
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.56/71.89
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.56/71.89
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0)) 193.56/71.89
ACTIVE(from(z0)) → c2(S(z0)) 193.56/71.89
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.56/71.89
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.56/71.89
ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.89
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.89
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.89
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.89
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.56/71.89
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(cons(2nd(cons1(z0, cons(z1, z2))), x1)) → c4(CONS(mark(z1), x1), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.89
ACTIVE(cons(2nd(cons(z0, z1)), x1)) → c4(CONS(mark(2nd(cons1(z0, z1))), x1), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.89
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(cons(2nd(z0), x1)) → c4(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 193.56/71.89
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.56/71.89
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.56/71.89
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(active(z0), z1), x1), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(z0, active(z1)), x1), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(from(2nd(cons1(z0, cons(z1, z2))))) → c5(FROM(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.89
ACTIVE(from(2nd(cons(z0, z1)))) → c5(FROM(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.89
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(from(2nd(z0))) → c5(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.89
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.89
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.56/71.89
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(s(2nd(cons1(z0, cons(z1, z2))))) → c6(S(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.89
ACTIVE(s(2nd(cons(z0, z1)))) → c6(S(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.89
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(s(2nd(z0))) → c6(S(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.89
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.89
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0))) 193.56/71.89
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.56/71.89
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.56/71.89
PROPER(2nd(2nd(z0))) → c20(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.89
PROPER(2nd(cons(z0, z1))) → c20(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.89
PROPER(2nd(from(z0))) → c20(2ND(from(proper(z0))), PROPER(from(z0))) 193.56/71.89
PROPER(2nd(s(z0))) → c20(2ND(s(proper(z0))), PROPER(s(z0))) 193.56/71.89
PROPER(2nd(cons1(z0, z1))) → c20(2ND(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.56/71.89
PROPER(cons(x0, 2nd(z0))) → c21(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 193.56/71.89
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.56/71.89
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.56/71.89
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.56/71.89
PROPER(cons(x0, cons1(z0, z1))) → c21(CONS(proper(x0), cons1(proper(z0), proper(z1))), PROPER(x0), PROPER(cons1(z0, z1))) 193.56/71.89
PROPER(cons(2nd(z0), x1)) → c21(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.56/71.89
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons(cons1(z0, z1), x1)) → c21(CONS(cons1(proper(z0), proper(z1)), proper(x1)), PROPER(cons1(z0, z1)), PROPER(x1)) 193.56/71.89
PROPER(from(2nd(z0))) → c22(FROM(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.89
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.89
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0))) 193.56/71.89
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 193.56/71.89
PROPER(from(cons1(z0, z1))) → c22(FROM(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.56/71.89
PROPER(s(2nd(z0))) → c23(S(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.89
PROPER(s(cons(z0, z1))) → c23(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.89
PROPER(s(from(z0))) → c23(S(from(proper(z0))), PROPER(from(z0))) 193.56/71.89
PROPER(s(s(z0))) → c23(S(s(proper(z0))), PROPER(s(z0))) 193.56/71.89
PROPER(s(cons1(z0, z1))) → c23(S(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.56/71.89
PROPER(cons1(x0, 2nd(z0))) → c24(CONS1(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 193.56/71.89
PROPER(cons1(x0, cons(z0, z1))) → c24(CONS1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.56/71.89
PROPER(cons1(x0, from(z0))) → c24(CONS1(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.56/71.89
PROPER(cons1(x0, s(z0))) → c24(CONS1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.56/71.89
PROPER(cons1(x0, cons1(z0, z1))) → c24(CONS1(proper(x0), cons1(proper(z0), proper(z1))), PROPER(x0), PROPER(cons1(z0, z1))) 193.56/71.89
PROPER(cons1(2nd(z0), x1)) → c24(CONS1(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons1(cons(z0, z1), x1)) → c24(CONS1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.56/71.89
PROPER(cons1(from(z0), x1)) → c24(CONS1(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons1(s(z0), x1)) → c24(CONS1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.56/71.89
PROPER(cons1(cons1(z0, z1), x1)) → c24(CONS1(cons1(proper(z0), proper(z1)), proper(x1)), PROPER(cons1(z0, z1)), PROPER(x1)) 193.56/71.89
TOP(mark(2nd(z0))) → c25(TOP(2nd(proper(z0))), PROPER(2nd(z0))) 193.56/71.89
TOP(mark(cons(z0, z1))) → c25(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.56/71.89
TOP(mark(from(z0))) → c25(TOP(from(proper(z0))), PROPER(from(z0))) 193.56/71.89
TOP(mark(s(z0))) → c25(TOP(s(proper(z0))), PROPER(s(z0))) 193.56/71.89
TOP(mark(cons1(z0, z1))) → c25(TOP(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1)))

S tuples:

ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.56/71.89
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.56/71.89
2ND(mark(z0)) → c9(2ND(z0)) 193.56/71.89
2ND(ok(z0)) → c10(2ND(z0)) 193.56/71.89
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.56/71.89
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.56/71.89
FROM(mark(z0)) → c13(FROM(z0)) 193.56/71.89
FROM(ok(z0)) → c14(FROM(z0)) 193.56/71.89
S(mark(z0)) → c15(S(z0)) 193.56/71.89
S(ok(z0)) → c16(S(z0)) 193.56/71.89
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.56/71.89
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.56/71.89
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.56/71.89
ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.56/71.89
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.56/71.89
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.56/71.89
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.56/71.89
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.56/71.90
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.56/71.90
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.56/71.90
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(cons(2nd(cons1(z0, cons(z1, z2))), x1)) → c4(CONS(mark(z1), x1), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.95/71.91
ACTIVE(cons(2nd(cons(z0, z1)), x1)) → c4(CONS(mark(2nd(cons1(z0, z1))), x1), ACTIVE(2nd(cons(z0, z1)))) 193.95/71.91
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(cons(2nd(z0), x1)) → c4(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 193.95/71.91
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.95/71.91
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.95/71.91
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(active(z0), z1), x1), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(z0, active(z1)), x1), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(from(2nd(cons1(z0, cons(z1, z2))))) → c5(FROM(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.95/71.91
ACTIVE(from(2nd(cons(z0, z1)))) → c5(FROM(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.95/71.91
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(from(2nd(z0))) → c5(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 193.95/71.91
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.95/71.91
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.95/71.91
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(s(2nd(cons1(z0, cons(z1, z2))))) → c6(S(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.95/71.91
ACTIVE(s(2nd(cons(z0, z1)))) → c6(S(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.95/71.91
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(s(2nd(z0))) → c6(S(2nd(active(z0))), ACTIVE(2nd(z0))) 193.95/71.91
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.95/71.91
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0))) 193.95/71.91
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.95/71.91
PROPER(2nd(2nd(z0))) → c20(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 193.95/71.91
PROPER(2nd(cons(z0, z1))) → c20(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.95/71.91
PROPER(2nd(from(z0))) → c20(2ND(from(proper(z0))), PROPER(from(z0))) 193.95/71.91
PROPER(2nd(s(z0))) → c20(2ND(s(proper(z0))), PROPER(s(z0))) 193.95/71.91
PROPER(2nd(cons1(z0, z1))) → c20(2ND(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.95/71.91
PROPER(cons(x0, 2nd(z0))) → c21(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 193.95/71.91
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.95/71.91
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.95/71.91
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.95/71.91
PROPER(cons(x0, cons1(z0, z1))) → c21(CONS(proper(x0), cons1(proper(z0), proper(z1))), PROPER(x0), PROPER(cons1(z0, z1))) 193.95/71.91
PROPER(cons(2nd(z0), x1)) → c21(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.95/71.91
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons(cons1(z0, z1), x1)) → c21(CONS(cons1(proper(z0), proper(z1)), proper(x1)), PROPER(cons1(z0, z1)), PROPER(x1)) 193.95/71.91
PROPER(from(2nd(z0))) → c22(FROM(2nd(proper(z0))), PROPER(2nd(z0))) 193.95/71.91
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.95/71.91
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0))) 193.95/71.91
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 193.95/71.91
PROPER(from(cons1(z0, z1))) → c22(FROM(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.95/71.91
PROPER(s(2nd(z0))) → c23(S(2nd(proper(z0))), PROPER(2nd(z0))) 193.95/71.91
PROPER(s(cons(z0, z1))) → c23(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.95/71.91
PROPER(s(from(z0))) → c23(S(from(proper(z0))), PROPER(from(z0))) 193.95/71.91
PROPER(s(s(z0))) → c23(S(s(proper(z0))), PROPER(s(z0))) 193.95/71.91
PROPER(s(cons1(z0, z1))) → c23(S(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.95/71.91
PROPER(cons1(x0, 2nd(z0))) → c24(CONS1(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 193.95/71.91
PROPER(cons1(x0, cons(z0, z1))) → c24(CONS1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.95/71.91
PROPER(cons1(x0, from(z0))) → c24(CONS1(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.95/71.91
PROPER(cons1(x0, s(z0))) → c24(CONS1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.95/71.91
PROPER(cons1(x0, cons1(z0, z1))) → c24(CONS1(proper(x0), cons1(proper(z0), proper(z1))), PROPER(x0), PROPER(cons1(z0, z1))) 193.95/71.91
PROPER(cons1(2nd(z0), x1)) → c24(CONS1(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons1(cons(z0, z1), x1)) → c24(CONS1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.95/71.91
PROPER(cons1(from(z0), x1)) → c24(CONS1(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons1(s(z0), x1)) → c24(CONS1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons1(cons1(z0, z1), x1)) → c24(CONS1(cons1(proper(z0), proper(z1)), proper(x1)), PROPER(cons1(z0, z1)), PROPER(x1))

K tuples:

ACTIVE(from(z0)) → c2(S(z0)) 193.95/71.91
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.95/71.91
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.95/71.91
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.95/71.91
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

ACTIVE, 2ND, CONS, FROM, S, CONS1, TOP, PROPER

Compound Symbols:

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

193.95/71.91

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

Use narrowing to replace TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(2nd(cons1(z0, cons(z1, z2))))) → c26(TOP(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.95/71.91
TOP(ok(2nd(cons(z0, z1)))) → c26(TOP(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.95/71.91
TOP(ok(from(z0))) → c26(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.95/71.91
TOP(ok(2nd(z0))) → c26(TOP(2nd(active(z0))), ACTIVE(2nd(z0))) 193.95/71.91
TOP(ok(cons(z0, z1))) → c26(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.95/71.91
TOP(ok(from(z0))) → c26(TOP(from(active(z0))), ACTIVE(from(z0))) 193.95/71.91
TOP(ok(s(z0))) → c26(TOP(s(active(z0))), ACTIVE(s(z0))) 193.95/71.91
TOP(ok(cons1(z0, z1))) → c26(TOP(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.95/71.91
TOP(ok(cons1(z0, z1))) → c26(TOP(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1)))

193.95/71.91

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.95/71.91
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.95/71.91
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.95/71.91
active(2nd(z0)) → 2nd(active(z0)) 193.95/71.91
active(cons(z0, z1)) → cons(active(z0), z1) 193.95/71.91
active(from(z0)) → from(active(z0)) 193.95/71.91
active(s(z0)) → s(active(z0)) 193.95/71.91
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.95/71.91
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.95/71.91
2nd(mark(z0)) → mark(2nd(z0)) 193.95/71.91
2nd(ok(z0)) → ok(2nd(z0)) 193.95/71.91
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.95/71.91
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.95/71.91
from(mark(z0)) → mark(from(z0)) 193.95/71.91
from(ok(z0)) → ok(from(z0)) 193.95/71.91
s(mark(z0)) → mark(s(z0)) 193.95/71.91
s(ok(z0)) → ok(s(z0)) 193.95/71.91
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.95/71.91
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.95/71.91
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.95/71.91
proper(2nd(z0)) → 2nd(proper(z0)) 193.95/71.91
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.95/71.91
proper(from(z0)) → from(proper(z0)) 193.95/71.91
proper(s(z0)) → s(proper(z0)) 193.95/71.91
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.95/71.91
top(mark(z0)) → top(proper(z0)) 193.95/71.91
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.95/71.91
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.95/71.91
2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.95/71.91
ACTIVE(from(z0)) → c2(S(z0)) 193.95/71.91
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.95/71.91
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.95/71.91
ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.95/71.91
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.95/71.91
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.95/71.91
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.95/71.91
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.95/71.91
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(cons(2nd(cons1(z0, cons(z1, z2))), x1)) → c4(CONS(mark(z1), x1), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.95/71.91
ACTIVE(cons(2nd(cons(z0, z1)), x1)) → c4(CONS(mark(2nd(cons1(z0, z1))), x1), ACTIVE(2nd(cons(z0, z1)))) 193.95/71.91
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(cons(2nd(z0), x1)) → c4(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 193.95/71.91
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.95/71.91
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.95/71.91
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(active(z0), z1), x1), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(z0, active(z1)), x1), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(from(2nd(cons1(z0, cons(z1, z2))))) → c5(FROM(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.95/71.91
ACTIVE(from(2nd(cons(z0, z1)))) → c5(FROM(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.95/71.91
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(from(2nd(z0))) → c5(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 193.95/71.91
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.95/71.91
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.95/71.91
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(s(2nd(cons1(z0, cons(z1, z2))))) → c6(S(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.95/71.91
ACTIVE(s(2nd(cons(z0, z1)))) → c6(S(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.95/71.91
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(s(2nd(z0))) → c6(S(2nd(active(z0))), ACTIVE(2nd(z0))) 193.95/71.91
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.95/71.91
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0))) 193.95/71.91
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.95/71.91
PROPER(2nd(2nd(z0))) → c20(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 193.95/71.91
PROPER(2nd(cons(z0, z1))) → c20(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.95/71.91
PROPER(2nd(from(z0))) → c20(2ND(from(proper(z0))), PROPER(from(z0))) 193.95/71.91
PROPER(2nd(s(z0))) → c20(2ND(s(proper(z0))), PROPER(s(z0))) 193.95/71.91
PROPER(2nd(cons1(z0, z1))) → c20(2ND(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.95/71.91
PROPER(cons(x0, 2nd(z0))) → c21(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 193.95/71.91
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.95/71.91
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.95/71.91
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.95/71.91
PROPER(cons(x0, cons1(z0, z1))) → c21(CONS(proper(x0), cons1(proper(z0), proper(z1))), PROPER(x0), PROPER(cons1(z0, z1))) 193.95/71.91
PROPER(cons(2nd(z0), x1)) → c21(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.95/71.91
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons(cons1(z0, z1), x1)) → c21(CONS(cons1(proper(z0), proper(z1)), proper(x1)), PROPER(cons1(z0, z1)), PROPER(x1)) 193.95/71.91
PROPER(from(2nd(z0))) → c22(FROM(2nd(proper(z0))), PROPER(2nd(z0))) 193.95/71.91
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.95/71.91
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0))) 193.95/71.91
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 193.95/71.91
PROPER(from(cons1(z0, z1))) → c22(FROM(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.95/71.91
PROPER(s(2nd(z0))) → c23(S(2nd(proper(z0))), PROPER(2nd(z0))) 193.95/71.91
PROPER(s(cons(z0, z1))) → c23(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.95/71.91
PROPER(s(from(z0))) → c23(S(from(proper(z0))), PROPER(from(z0))) 193.95/71.91
PROPER(s(s(z0))) → c23(S(s(proper(z0))), PROPER(s(z0))) 193.95/71.91
PROPER(s(cons1(z0, z1))) → c23(S(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.95/71.91
PROPER(cons1(x0, 2nd(z0))) → c24(CONS1(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 193.95/71.91
PROPER(cons1(x0, cons(z0, z1))) → c24(CONS1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.95/71.91
PROPER(cons1(x0, from(z0))) → c24(CONS1(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.95/71.91
PROPER(cons1(x0, s(z0))) → c24(CONS1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.95/71.91
PROPER(cons1(x0, cons1(z0, z1))) → c24(CONS1(proper(x0), cons1(proper(z0), proper(z1))), PROPER(x0), PROPER(cons1(z0, z1))) 193.95/71.91
PROPER(cons1(2nd(z0), x1)) → c24(CONS1(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons1(cons(z0, z1), x1)) → c24(CONS1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.95/71.91
PROPER(cons1(from(z0), x1)) → c24(CONS1(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons1(s(z0), x1)) → c24(CONS1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons1(cons1(z0, z1), x1)) → c24(CONS1(cons1(proper(z0), proper(z1)), proper(x1)), PROPER(cons1(z0, z1)), PROPER(x1)) 193.95/71.91
TOP(mark(2nd(z0))) → c25(TOP(2nd(proper(z0))), PROPER(2nd(z0))) 193.95/71.91
TOP(mark(cons(z0, z1))) → c25(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.95/71.91
TOP(mark(from(z0))) → c25(TOP(from(proper(z0))), PROPER(from(z0))) 193.95/71.91
TOP(mark(s(z0))) → c25(TOP(s(proper(z0))), PROPER(s(z0))) 193.95/71.91
TOP(mark(cons1(z0, z1))) → c25(TOP(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.95/71.91
TOP(ok(2nd(cons1(z0, cons(z1, z2))))) → c26(TOP(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.95/71.91
TOP(ok(2nd(cons(z0, z1)))) → c26(TOP(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.95/71.91
TOP(ok(from(z0))) → c26(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.95/71.91
TOP(ok(2nd(z0))) → c26(TOP(2nd(active(z0))), ACTIVE(2nd(z0))) 193.95/71.91
TOP(ok(cons(z0, z1))) → c26(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.95/71.91
TOP(ok(from(z0))) → c26(TOP(from(active(z0))), ACTIVE(from(z0))) 193.95/71.91
TOP(ok(s(z0))) → c26(TOP(s(active(z0))), ACTIVE(s(z0))) 193.95/71.91
TOP(ok(cons1(z0, z1))) → c26(TOP(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.95/71.91
TOP(ok(cons1(z0, z1))) → c26(TOP(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1)))

S tuples:

ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.95/71.91
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.95/71.91
2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1)) 193.95/71.91
ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.95/71.91
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.95/71.91
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.95/71.91
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.95/71.91
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.95/71.91
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(cons(2nd(cons1(z0, cons(z1, z2))), x1)) → c4(CONS(mark(z1), x1), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.95/71.91
ACTIVE(cons(2nd(cons(z0, z1)), x1)) → c4(CONS(mark(2nd(cons1(z0, z1))), x1), ACTIVE(2nd(cons(z0, z1)))) 193.95/71.91
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(cons(2nd(z0), x1)) → c4(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 193.95/71.91
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.95/71.91
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.95/71.91
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(active(z0), z1), x1), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(z0, active(z1)), x1), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(from(2nd(cons1(z0, cons(z1, z2))))) → c5(FROM(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.95/71.91
ACTIVE(from(2nd(cons(z0, z1)))) → c5(FROM(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.95/71.91
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(from(2nd(z0))) → c5(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 193.95/71.91
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.95/71.91
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.95/71.91
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(s(2nd(cons1(z0, cons(z1, z2))))) → c6(S(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.95/71.91
ACTIVE(s(2nd(cons(z0, z1)))) → c6(S(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.95/71.91
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(s(2nd(z0))) → c6(S(2nd(active(z0))), ACTIVE(2nd(z0))) 193.95/71.91
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.95/71.91
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0))) 193.95/71.91
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.95/71.91
PROPER(2nd(2nd(z0))) → c20(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 193.95/71.91
PROPER(2nd(cons(z0, z1))) → c20(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.95/71.91
PROPER(2nd(from(z0))) → c20(2ND(from(proper(z0))), PROPER(from(z0))) 193.95/71.91
PROPER(2nd(s(z0))) → c20(2ND(s(proper(z0))), PROPER(s(z0))) 193.95/71.91
PROPER(2nd(cons1(z0, z1))) → c20(2ND(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.95/71.91
PROPER(cons(x0, 2nd(z0))) → c21(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 193.95/71.91
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.95/71.91
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.95/71.91
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.95/71.91
PROPER(cons(x0, cons1(z0, z1))) → c21(CONS(proper(x0), cons1(proper(z0), proper(z1))), PROPER(x0), PROPER(cons1(z0, z1))) 193.95/71.91
PROPER(cons(2nd(z0), x1)) → c21(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.95/71.91
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons(cons1(z0, z1), x1)) → c21(CONS(cons1(proper(z0), proper(z1)), proper(x1)), PROPER(cons1(z0, z1)), PROPER(x1)) 193.95/71.91
PROPER(from(2nd(z0))) → c22(FROM(2nd(proper(z0))), PROPER(2nd(z0))) 193.95/71.91
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.95/71.91
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0))) 193.95/71.91
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 193.95/71.91
PROPER(from(cons1(z0, z1))) → c22(FROM(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.95/71.91
PROPER(s(2nd(z0))) → c23(S(2nd(proper(z0))), PROPER(2nd(z0))) 193.95/71.91
PROPER(s(cons(z0, z1))) → c23(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.95/71.91
PROPER(s(from(z0))) → c23(S(from(proper(z0))), PROPER(from(z0))) 193.95/71.91
PROPER(s(s(z0))) → c23(S(s(proper(z0))), PROPER(s(z0))) 193.95/71.91
PROPER(s(cons1(z0, z1))) → c23(S(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.95/71.91
PROPER(cons1(x0, 2nd(z0))) → c24(CONS1(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 193.95/71.91
PROPER(cons1(x0, cons(z0, z1))) → c24(CONS1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.95/71.91
PROPER(cons1(x0, from(z0))) → c24(CONS1(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.95/71.91
PROPER(cons1(x0, s(z0))) → c24(CONS1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.95/71.91
PROPER(cons1(x0, cons1(z0, z1))) → c24(CONS1(proper(x0), cons1(proper(z0), proper(z1))), PROPER(x0), PROPER(cons1(z0, z1))) 193.95/71.91
PROPER(cons1(2nd(z0), x1)) → c24(CONS1(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons1(cons(z0, z1), x1)) → c24(CONS1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.95/71.91
PROPER(cons1(from(z0), x1)) → c24(CONS1(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons1(s(z0), x1)) → c24(CONS1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons1(cons1(z0, z1), x1)) → c24(CONS1(cons1(proper(z0), proper(z1)), proper(x1)), PROPER(cons1(z0, z1)), PROPER(x1))

K tuples:

ACTIVE(from(z0)) → c2(S(z0)) 193.95/71.91
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.95/71.91
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.95/71.91
TOP(mark(z0)) → c25(TOP(proper(z0)), PROPER(z0)) 193.95/71.91
TOP(ok(z0)) → c26(TOP(active(z0)), ACTIVE(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

ACTIVE, 2ND, CONS, FROM, S, CONS1, PROPER, TOP

Compound Symbols:

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

193.95/71.91

(33) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(cons1(z0, z1)) → c7(CONS1(active(z0), z1), ACTIVE(z0)) 193.95/71.91
ACTIVE(cons1(z0, z1)) → c8(CONS1(z0, active(z1)), ACTIVE(z1)) 193.95/71.91
ACTIVE(from(z0)) → c2(S(z0)) 193.95/71.91
ACTIVE(2nd(cons(z0, z1))) → c(2ND(cons1(z0, z1))) 193.95/71.91
ACTIVE(2nd(cons(z0, z1))) → c(CONS1(z0, z1)) 193.95/71.91
ACTIVE(2nd(2nd(cons1(z0, cons(z1, z2))))) → c3(2ND(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.95/71.91
ACTIVE(2nd(2nd(cons(z0, z1)))) → c3(2ND(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.95/71.91
ACTIVE(2nd(from(z0))) → c3(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(2nd(2nd(z0))) → c3(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 193.95/71.91
ACTIVE(2nd(cons(z0, z1))) → c3(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.95/71.91
ACTIVE(2nd(from(z0))) → c3(2ND(from(active(z0))), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(2nd(s(z0))) → c3(2ND(s(active(z0))), ACTIVE(s(z0))) 193.95/71.91
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(2nd(cons1(z0, z1))) → c3(2ND(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(cons(2nd(cons1(z0, cons(z1, z2))), x1)) → c4(CONS(mark(z1), x1), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.95/71.91
ACTIVE(cons(2nd(cons(z0, z1)), x1)) → c4(CONS(mark(2nd(cons1(z0, z1))), x1), ACTIVE(2nd(cons(z0, z1)))) 193.95/71.91
ACTIVE(cons(from(z0), x1)) → c4(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(cons(2nd(z0), x1)) → c4(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 193.95/71.91
ACTIVE(cons(cons(z0, z1), x1)) → c4(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 193.95/71.91
ACTIVE(cons(from(z0), x1)) → c4(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(cons(s(z0), x1)) → c4(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 193.95/71.91
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(active(z0), z1), x1), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(cons(cons1(z0, z1), x1)) → c4(CONS(cons1(z0, active(z1)), x1), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(from(2nd(cons1(z0, cons(z1, z2))))) → c5(FROM(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.95/71.91
ACTIVE(from(2nd(cons(z0, z1)))) → c5(FROM(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.95/71.91
ACTIVE(from(from(z0))) → c5(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(from(2nd(z0))) → c5(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 193.95/71.91
ACTIVE(from(cons(z0, z1))) → c5(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.95/71.91
ACTIVE(from(from(z0))) → c5(FROM(from(active(z0))), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(from(s(z0))) → c5(FROM(s(active(z0))), ACTIVE(s(z0))) 193.95/71.91
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(from(cons1(z0, z1))) → c5(FROM(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(s(2nd(cons1(z0, cons(z1, z2))))) → c6(S(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.95/71.91
ACTIVE(s(2nd(cons(z0, z1)))) → c6(S(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.95/71.91
ACTIVE(s(from(z0))) → c6(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(s(2nd(z0))) → c6(S(2nd(active(z0))), ACTIVE(2nd(z0))) 193.95/71.91
ACTIVE(s(cons(z0, z1))) → c6(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.95/71.91
ACTIVE(s(from(z0))) → c6(S(from(active(z0))), ACTIVE(from(z0))) 193.95/71.91
ACTIVE(s(s(z0))) → c6(S(s(active(z0))), ACTIVE(s(z0))) 193.95/71.91
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.95/71.91
ACTIVE(s(cons1(z0, z1))) → c6(S(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1))) 193.95/71.91
PROPER(2nd(2nd(z0))) → c20(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 193.95/71.91
PROPER(2nd(cons(z0, z1))) → c20(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.95/71.91
PROPER(2nd(from(z0))) → c20(2ND(from(proper(z0))), PROPER(from(z0))) 193.95/71.91
PROPER(2nd(s(z0))) → c20(2ND(s(proper(z0))), PROPER(s(z0))) 193.95/71.91
PROPER(2nd(cons1(z0, z1))) → c20(2ND(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.95/71.91
PROPER(cons(x0, 2nd(z0))) → c21(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 193.95/71.91
PROPER(cons(x0, cons(z0, z1))) → c21(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.95/71.91
PROPER(cons(x0, from(z0))) → c21(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.95/71.91
PROPER(cons(x0, s(z0))) → c21(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.95/71.91
PROPER(cons(x0, cons1(z0, z1))) → c21(CONS(proper(x0), cons1(proper(z0), proper(z1))), PROPER(x0), PROPER(cons1(z0, z1))) 193.95/71.91
PROPER(cons(2nd(z0), x1)) → c21(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons(cons(z0, z1), x1)) → c21(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.95/71.91
PROPER(cons(from(z0), x1)) → c21(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons(s(z0), x1)) → c21(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons(cons1(z0, z1), x1)) → c21(CONS(cons1(proper(z0), proper(z1)), proper(x1)), PROPER(cons1(z0, z1)), PROPER(x1)) 193.95/71.91
PROPER(from(2nd(z0))) → c22(FROM(2nd(proper(z0))), PROPER(2nd(z0))) 193.95/71.91
PROPER(from(cons(z0, z1))) → c22(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.95/71.91
PROPER(from(from(z0))) → c22(FROM(from(proper(z0))), PROPER(from(z0))) 193.95/71.91
PROPER(from(s(z0))) → c22(FROM(s(proper(z0))), PROPER(s(z0))) 193.95/71.91
PROPER(from(cons1(z0, z1))) → c22(FROM(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.95/71.91
PROPER(s(2nd(z0))) → c23(S(2nd(proper(z0))), PROPER(2nd(z0))) 193.95/71.91
PROPER(s(cons(z0, z1))) → c23(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.95/71.91
PROPER(s(from(z0))) → c23(S(from(proper(z0))), PROPER(from(z0))) 193.95/71.91
PROPER(s(s(z0))) → c23(S(s(proper(z0))), PROPER(s(z0))) 193.95/71.91
PROPER(s(cons1(z0, z1))) → c23(S(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.95/71.91
PROPER(cons1(x0, 2nd(z0))) → c24(CONS1(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 193.95/71.91
PROPER(cons1(x0, cons(z0, z1))) → c24(CONS1(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 193.95/71.91
PROPER(cons1(x0, from(z0))) → c24(CONS1(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 193.95/71.91
PROPER(cons1(x0, s(z0))) → c24(CONS1(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 193.95/71.91
PROPER(cons1(x0, cons1(z0, z1))) → c24(CONS1(proper(x0), cons1(proper(z0), proper(z1))), PROPER(x0), PROPER(cons1(z0, z1))) 193.95/71.91
PROPER(cons1(2nd(z0), x1)) → c24(CONS1(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons1(cons(z0, z1), x1)) → c24(CONS1(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 193.95/71.91
PROPER(cons1(from(z0), x1)) → c24(CONS1(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons1(s(z0), x1)) → c24(CONS1(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 193.95/71.91
PROPER(cons1(cons1(z0, z1), x1)) → c24(CONS1(cons1(proper(z0), proper(z1)), proper(x1)), PROPER(cons1(z0, z1)), PROPER(x1)) 193.95/71.91
TOP(mark(2nd(z0))) → c25(TOP(2nd(proper(z0))), PROPER(2nd(z0))) 193.95/71.91
TOP(mark(cons(z0, z1))) → c25(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 193.95/71.91
TOP(mark(from(z0))) → c25(TOP(from(proper(z0))), PROPER(from(z0))) 193.95/71.91
TOP(mark(s(z0))) → c25(TOP(s(proper(z0))), PROPER(s(z0))) 193.95/71.91
TOP(mark(cons1(z0, z1))) → c25(TOP(cons1(proper(z0), proper(z1))), PROPER(cons1(z0, z1))) 193.95/71.91
TOP(ok(2nd(cons1(z0, cons(z1, z2))))) → c26(TOP(mark(z1)), ACTIVE(2nd(cons1(z0, cons(z1, z2))))) 193.95/71.91
TOP(ok(2nd(cons(z0, z1)))) → c26(TOP(mark(2nd(cons1(z0, z1)))), ACTIVE(2nd(cons(z0, z1)))) 193.95/71.91
TOP(ok(from(z0))) → c26(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 193.95/71.91
TOP(ok(2nd(z0))) → c26(TOP(2nd(active(z0))), ACTIVE(2nd(z0))) 193.95/71.91
TOP(ok(cons(z0, z1))) → c26(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 193.95/71.91
TOP(ok(from(z0))) → c26(TOP(from(active(z0))), ACTIVE(from(z0))) 193.95/71.91
TOP(ok(s(z0))) → c26(TOP(s(active(z0))), ACTIVE(s(z0))) 193.95/71.91
TOP(ok(cons1(z0, z1))) → c26(TOP(cons1(active(z0), z1)), ACTIVE(cons1(z0, z1))) 193.95/71.91
TOP(ok(cons1(z0, z1))) → c26(TOP(cons1(z0, active(z1))), ACTIVE(cons1(z0, z1)))

193.95/71.91

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.95/71.91
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.95/71.91
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.95/71.91
active(2nd(z0)) → 2nd(active(z0)) 193.95/71.91
active(cons(z0, z1)) → cons(active(z0), z1) 193.95/71.91
active(from(z0)) → from(active(z0)) 193.95/71.91
active(s(z0)) → s(active(z0)) 193.95/71.91
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.95/71.91
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.95/71.91
2nd(mark(z0)) → mark(2nd(z0)) 193.95/71.91
2nd(ok(z0)) → ok(2nd(z0)) 193.95/71.91
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.95/71.91
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.95/71.91
from(mark(z0)) → mark(from(z0)) 193.95/71.91
from(ok(z0)) → ok(from(z0)) 193.95/71.91
s(mark(z0)) → mark(s(z0)) 193.95/71.91
s(ok(z0)) → ok(s(z0)) 193.95/71.91
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.95/71.91
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.95/71.91
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.95/71.91
proper(2nd(z0)) → 2nd(proper(z0)) 193.95/71.91
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.95/71.91
proper(from(z0)) → from(proper(z0)) 193.95/71.91
proper(s(z0)) → s(proper(z0)) 193.95/71.91
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.95/71.91
top(mark(z0)) → top(proper(z0)) 193.95/71.91
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

S tuples:

2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

K tuples:none
Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

2ND, CONS, FROM, S, CONS1

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

193.95/71.91

(35) 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.

FROM(mark(z0)) → c13(FROM(z0))

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

2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 193.95/71.91

POL(2ND(x₁)) = 0 193.95/71.91
POL(CONS(x₁, x₂)) = 0 193.95/71.91
POL(CONS1(x₁, x₂)) = 0 193.95/71.91
POL(FROM(x₁)) = [4]x₁ 193.95/71.91
POL(S(x₁)) = 0 193.95/71.91
POL(c10(x₁)) = x₁ 193.95/71.91
POL(c11(x₁)) = x₁ 193.95/71.91
POL(c12(x₁)) = x₁ 193.95/71.91
POL(c13(x₁)) = x₁ 193.95/71.91
POL(c14(x₁)) = x₁ 193.95/71.91
POL(c15(x₁)) = x₁ 193.95/71.91
POL(c16(x₁)) = x₁ 193.95/71.91
POL(c17(x₁)) = x₁ 193.95/71.91
POL(c18(x₁)) = x₁ 193.95/71.91
POL(c19(x₁)) = x₁ 193.95/71.91
POL(c9(x₁)) = x₁ 193.95/71.91
POL(mark(x₁)) = [4] + x₁ 193.95/71.91
POL(ok(x₁)) = x₁

193.95/71.91

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.95/71.91
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.95/71.91
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.95/71.91
active(2nd(z0)) → 2nd(active(z0)) 193.95/71.91
active(cons(z0, z1)) → cons(active(z0), z1) 193.95/71.91
active(from(z0)) → from(active(z0)) 193.95/71.91
active(s(z0)) → s(active(z0)) 193.95/71.91
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.95/71.91
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.95/71.91
2nd(mark(z0)) → mark(2nd(z0)) 193.95/71.91
2nd(ok(z0)) → ok(2nd(z0)) 193.95/71.91
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.95/71.91
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.95/71.91
from(mark(z0)) → mark(from(z0)) 193.95/71.91
from(ok(z0)) → ok(from(z0)) 193.95/71.91
s(mark(z0)) → mark(s(z0)) 193.95/71.91
s(ok(z0)) → ok(s(z0)) 193.95/71.91
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.95/71.91
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.95/71.91
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.95/71.91
proper(2nd(z0)) → 2nd(proper(z0)) 193.95/71.91
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.95/71.91
proper(from(z0)) → from(proper(z0)) 193.95/71.91
proper(s(z0)) → s(proper(z0)) 193.95/71.91
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.95/71.91
top(mark(z0)) → top(proper(z0)) 193.95/71.91
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

S tuples:

2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

K tuples:

FROM(mark(z0)) → c13(FROM(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

2ND, CONS, FROM, S, CONS1

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

193.95/71.91

(37) 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.

2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))

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

2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 193.95/71.91

POL(2ND(x₁)) = x₁ + x₁² + x₁³ 193.95/71.91
POL(CONS(x₁, x₂)) = x₂² + x₁·x₂ + x₁² + x₁³ + x₁²·x₂ + x₁·x₂² + x₂³ 193.95/71.91
POL(CONS1(x₁, x₂)) = x₂² + x₁·x₂ + x₁² + x₁³ + x₁²·x₂ + x₁·x₂² + x₂³ 193.95/71.91
POL(FROM(x₁)) = x₁ + x₁² + x₁³ 193.95/71.91
POL(S(x₁)) = x₁ + x₁² + x₁³ 193.95/71.91
POL(c10(x₁)) = x₁ 193.95/71.91
POL(c11(x₁)) = x₁ 193.95/71.91
POL(c12(x₁)) = x₁ 193.95/71.91
POL(c13(x₁)) = x₁ 193.95/71.91
POL(c14(x₁)) = x₁ 193.95/71.91
POL(c15(x₁)) = x₁ 193.95/71.91
POL(c16(x₁)) = x₁ 193.95/71.91
POL(c17(x₁)) = x₁ 193.95/71.91
POL(c18(x₁)) = x₁ 193.95/71.91
POL(c19(x₁)) = x₁ 193.95/71.91
POL(c9(x₁)) = x₁ 193.95/71.91
POL(mark(x₁)) = [1] + x₁ 193.95/71.91
POL(ok(x₁)) = x₁

193.95/71.91

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.95/71.91
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.95/71.91
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.95/71.91
active(2nd(z0)) → 2nd(active(z0)) 193.95/71.91
active(cons(z0, z1)) → cons(active(z0), z1) 193.95/71.91
active(from(z0)) → from(active(z0)) 193.95/71.91
active(s(z0)) → s(active(z0)) 193.95/71.91
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.95/71.91
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.95/71.91
2nd(mark(z0)) → mark(2nd(z0)) 193.95/71.91
2nd(ok(z0)) → ok(2nd(z0)) 193.95/71.91
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.95/71.91
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.95/71.91
from(mark(z0)) → mark(from(z0)) 193.95/71.91
from(ok(z0)) → ok(from(z0)) 193.95/71.91
s(mark(z0)) → mark(s(z0)) 193.95/71.91
s(ok(z0)) → ok(s(z0)) 193.95/71.91
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.95/71.91
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.95/71.91
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.95/71.91
proper(2nd(z0)) → 2nd(proper(z0)) 193.95/71.91
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.95/71.91
proper(from(z0)) → from(proper(z0)) 193.95/71.91
proper(s(z0)) → s(proper(z0)) 193.95/71.91
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.95/71.91
top(mark(z0)) → top(proper(z0)) 193.95/71.91
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

S tuples:

2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

K tuples:

FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

2ND, CONS, FROM, S, CONS1

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

193.95/71.91

(39) 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.

2ND(ok(z0)) → c10(2ND(z0))

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

2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 193.95/71.91

POL(2ND(x₁)) = [2]x₁ 193.95/71.91
POL(CONS(x₁, x₂)) = 0 193.95/71.91
POL(CONS1(x₁, x₂)) = 0 193.95/71.91
POL(FROM(x₁)) = 0 193.95/71.91
POL(S(x₁)) = 0 193.95/71.91
POL(c10(x₁)) = x₁ 193.95/71.91
POL(c11(x₁)) = x₁ 193.95/71.91
POL(c12(x₁)) = x₁ 193.95/71.91
POL(c13(x₁)) = x₁ 193.95/71.91
POL(c14(x₁)) = x₁ 193.95/71.91
POL(c15(x₁)) = x₁ 193.95/71.91
POL(c16(x₁)) = x₁ 193.95/71.91
POL(c17(x₁)) = x₁ 193.95/71.91
POL(c18(x₁)) = x₁ 193.95/71.91
POL(c19(x₁)) = x₁ 193.95/71.91
POL(c9(x₁)) = x₁ 193.95/71.91
POL(mark(x₁)) = x₁ 193.95/71.91
POL(ok(x₁)) = [1] + x₁

193.95/71.91

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.95/71.91
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.95/71.91
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.95/71.91
active(2nd(z0)) → 2nd(active(z0)) 193.95/71.91
active(cons(z0, z1)) → cons(active(z0), z1) 193.95/71.91
active(from(z0)) → from(active(z0)) 193.95/71.91
active(s(z0)) → s(active(z0)) 193.95/71.91
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.95/71.91
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.95/71.91
2nd(mark(z0)) → mark(2nd(z0)) 193.95/71.91
2nd(ok(z0)) → ok(2nd(z0)) 193.95/71.91
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.95/71.91
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.95/71.91
from(mark(z0)) → mark(from(z0)) 193.95/71.91
from(ok(z0)) → ok(from(z0)) 193.95/71.91
s(mark(z0)) → mark(s(z0)) 193.95/71.91
s(ok(z0)) → ok(s(z0)) 193.95/71.91
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.95/71.91
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.95/71.91
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.95/71.91
proper(2nd(z0)) → 2nd(proper(z0)) 193.95/71.91
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.95/71.91
proper(from(z0)) → from(proper(z0)) 193.95/71.91
proper(s(z0)) → s(proper(z0)) 193.95/71.91
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.95/71.91
top(mark(z0)) → top(proper(z0)) 193.95/71.91
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

S tuples:

CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

K tuples:

FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

2ND, CONS, FROM, S, CONS1

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

193.95/71.91

(41) 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.

CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))

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

2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 193.95/71.91

POL(2ND(x₁)) = 0 193.95/71.91
POL(CONS(x₁, x₂)) = x₂ 193.95/71.91
POL(CONS1(x₁, x₂)) = 0 193.95/71.91
POL(FROM(x₁)) = 0 193.95/71.91
POL(S(x₁)) = 0 193.95/71.91
POL(c10(x₁)) = x₁ 193.95/71.91
POL(c11(x₁)) = x₁ 193.95/71.91
POL(c12(x₁)) = x₁ 193.95/71.91
POL(c13(x₁)) = x₁ 193.95/71.91
POL(c14(x₁)) = x₁ 193.95/71.91
POL(c15(x₁)) = x₁ 193.95/71.91
POL(c16(x₁)) = x₁ 193.95/71.91
POL(c17(x₁)) = x₁ 193.95/71.91
POL(c18(x₁)) = x₁ 193.95/71.91
POL(c19(x₁)) = x₁ 193.95/71.91
POL(c9(x₁)) = x₁ 193.95/71.91
POL(mark(x₁)) = 0 193.95/71.91
POL(ok(x₁)) = [1] + x₁

193.95/71.91

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.95/71.91
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.95/71.91
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.95/71.91
active(2nd(z0)) → 2nd(active(z0)) 193.95/71.91
active(cons(z0, z1)) → cons(active(z0), z1) 193.95/71.91
active(from(z0)) → from(active(z0)) 193.95/71.91
active(s(z0)) → s(active(z0)) 193.95/71.91
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.95/71.91
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.95/71.91
2nd(mark(z0)) → mark(2nd(z0)) 193.95/71.91
2nd(ok(z0)) → ok(2nd(z0)) 193.95/71.91
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.95/71.91
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.95/71.91
from(mark(z0)) → mark(from(z0)) 193.95/71.91
from(ok(z0)) → ok(from(z0)) 193.95/71.91
s(mark(z0)) → mark(s(z0)) 193.95/71.91
s(ok(z0)) → ok(s(z0)) 193.95/71.91
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.95/71.91
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.95/71.91
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.95/71.91
proper(2nd(z0)) → 2nd(proper(z0)) 193.95/71.91
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.95/71.91
proper(from(z0)) → from(proper(z0)) 193.95/71.91
proper(s(z0)) → s(proper(z0)) 193.95/71.91
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.95/71.91
top(mark(z0)) → top(proper(z0)) 193.95/71.91
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

S tuples:

FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

K tuples:

FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

2ND, CONS, FROM, S, CONS1

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

193.95/71.91

(43) 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.

FROM(ok(z0)) → c14(FROM(z0))

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

2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 193.95/71.91

POL(2ND(x₁)) = [3]x₁ 193.95/71.91
POL(CONS(x₁, x₂)) = [5]x₁ + [3]x₂ 193.95/71.91
POL(CONS1(x₁, x₂)) = 0 193.95/71.91
POL(FROM(x₁)) = [2]x₁ 193.95/71.91
POL(S(x₁)) = 0 193.95/71.91
POL(c10(x₁)) = x₁ 193.95/71.91
POL(c11(x₁)) = x₁ 193.95/71.91
POL(c12(x₁)) = x₁ 193.95/71.91
POL(c13(x₁)) = x₁ 193.95/71.91
POL(c14(x₁)) = x₁ 193.95/71.91
POL(c15(x₁)) = x₁ 193.95/71.91
POL(c16(x₁)) = x₁ 193.95/71.91
POL(c17(x₁)) = x₁ 193.95/71.91
POL(c18(x₁)) = x₁ 193.95/71.91
POL(c19(x₁)) = x₁ 193.95/71.91
POL(c9(x₁)) = x₁ 193.95/71.91
POL(mark(x₁)) = x₁ 193.95/71.91
POL(ok(x₁)) = [1] + x₁

193.95/71.91

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.95/71.91
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.95/71.91
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.95/71.91
active(2nd(z0)) → 2nd(active(z0)) 193.95/71.91
active(cons(z0, z1)) → cons(active(z0), z1) 193.95/71.91
active(from(z0)) → from(active(z0)) 193.95/71.91
active(s(z0)) → s(active(z0)) 193.95/71.91
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.95/71.91
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.95/71.91
2nd(mark(z0)) → mark(2nd(z0)) 193.95/71.91
2nd(ok(z0)) → ok(2nd(z0)) 193.95/71.91
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.95/71.91
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.95/71.91
from(mark(z0)) → mark(from(z0)) 193.95/71.91
from(ok(z0)) → ok(from(z0)) 193.95/71.91
s(mark(z0)) → mark(s(z0)) 193.95/71.91
s(ok(z0)) → ok(s(z0)) 193.95/71.91
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.95/71.91
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.95/71.91
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.95/71.91
proper(2nd(z0)) → 2nd(proper(z0)) 193.95/71.91
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.95/71.91
proper(from(z0)) → from(proper(z0)) 193.95/71.91
proper(s(z0)) → s(proper(z0)) 193.95/71.91
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.95/71.91
top(mark(z0)) → top(proper(z0)) 193.95/71.91
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

S tuples:

S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

K tuples:

FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

2ND, CONS, FROM, S, CONS1

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

193.95/71.91

(45) 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(ok(z0)) → c16(S(z0))

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

2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 193.95/71.91

POL(2ND(x₁)) = x₁ + x₁² + x₁³ 193.95/71.91
POL(CONS(x₁, x₂)) = x₁ + x₂ + x₂² + x₁·x₂ + x₁² + x₁³ + x₁²·x₂ + x₁·x₂² + x₂³ 193.95/71.91
POL(CONS1(x₁, x₂)) = 0 193.95/71.91
POL(FROM(x₁)) = x₁ + x₁² + x₁³ 193.95/71.91
POL(S(x₁)) = x₁ + x₁² + x₁³ 193.95/71.91
POL(c10(x₁)) = x₁ 193.95/71.91
POL(c11(x₁)) = x₁ 193.95/71.91
POL(c12(x₁)) = x₁ 193.95/71.91
POL(c13(x₁)) = x₁ 193.95/71.91
POL(c14(x₁)) = x₁ 193.95/71.91
POL(c15(x₁)) = x₁ 193.95/71.91
POL(c16(x₁)) = x₁ 193.95/71.91
POL(c17(x₁)) = x₁ 193.95/71.91
POL(c18(x₁)) = x₁ 193.95/71.91
POL(c19(x₁)) = x₁ 193.95/71.91
POL(c9(x₁)) = x₁ 193.95/71.91
POL(mark(x₁)) = [1] + x₁ 193.95/71.91
POL(ok(x₁)) = [1] + x₁

193.95/71.91

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.95/71.91
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.95/71.91
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.95/71.91
active(2nd(z0)) → 2nd(active(z0)) 193.95/71.91
active(cons(z0, z1)) → cons(active(z0), z1) 193.95/71.91
active(from(z0)) → from(active(z0)) 193.95/71.91
active(s(z0)) → s(active(z0)) 193.95/71.91
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.95/71.91
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.95/71.91
2nd(mark(z0)) → mark(2nd(z0)) 193.95/71.91
2nd(ok(z0)) → ok(2nd(z0)) 193.95/71.91
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.95/71.91
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.95/71.91
from(mark(z0)) → mark(from(z0)) 193.95/71.91
from(ok(z0)) → ok(from(z0)) 193.95/71.91
s(mark(z0)) → mark(s(z0)) 193.95/71.91
s(ok(z0)) → ok(s(z0)) 193.95/71.91
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.95/71.91
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.95/71.91
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.95/71.91
proper(2nd(z0)) → 2nd(proper(z0)) 193.95/71.91
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.95/71.91
proper(from(z0)) → from(proper(z0)) 193.95/71.91
proper(s(z0)) → s(proper(z0)) 193.95/71.91
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.95/71.91
top(mark(z0)) → top(proper(z0)) 193.95/71.91
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

S tuples:

CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

K tuples:

FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

2ND, CONS, FROM, S, CONS1

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

193.95/71.91

(47) 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.

CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

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

2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation : 193.95/71.91

POL(2ND(x₁)) = [3]x₁ 193.95/71.91
POL(CONS(x₁, x₂)) = [5]x₁ + [5]x₂ 193.95/71.91
POL(CONS1(x₁, x₂)) = [2]x₁ 193.95/71.91
POL(FROM(x₁)) = [3]x₁ 193.95/71.91
POL(S(x₁)) = [3]x₁ 193.95/71.91
POL(c10(x₁)) = x₁ 193.95/71.91
POL(c11(x₁)) = x₁ 193.95/71.91
POL(c12(x₁)) = x₁ 193.95/71.91
POL(c13(x₁)) = x₁ 193.95/71.91
POL(c14(x₁)) = x₁ 193.95/71.91
POL(c15(x₁)) = x₁ 193.95/71.91
POL(c16(x₁)) = x₁ 193.95/71.91
POL(c17(x₁)) = x₁ 193.95/71.91
POL(c18(x₁)) = x₁ 193.95/71.91
POL(c19(x₁)) = x₁ 193.95/71.91
POL(c9(x₁)) = x₁ 193.95/71.91
POL(mark(x₁)) = x₁ 193.95/71.91
POL(ok(x₁)) = [1] + x₁

193.95/71.91

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons1(z0, cons(z1, z2)))) → mark(z1) 193.95/71.91
active(2nd(cons(z0, z1))) → mark(2nd(cons1(z0, z1))) 193.95/71.91
active(from(z0)) → mark(cons(z0, from(s(z0)))) 193.95/71.91
active(2nd(z0)) → 2nd(active(z0)) 193.95/71.91
active(cons(z0, z1)) → cons(active(z0), z1) 193.95/71.91
active(from(z0)) → from(active(z0)) 193.95/71.91
active(s(z0)) → s(active(z0)) 193.95/71.91
active(cons1(z0, z1)) → cons1(active(z0), z1) 193.95/71.91
active(cons1(z0, z1)) → cons1(z0, active(z1)) 193.95/71.91
2nd(mark(z0)) → mark(2nd(z0)) 193.95/71.91
2nd(ok(z0)) → ok(2nd(z0)) 193.95/71.91
cons(mark(z0), z1) → mark(cons(z0, z1)) 193.95/71.91
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 193.95/71.91
from(mark(z0)) → mark(from(z0)) 193.95/71.91
from(ok(z0)) → ok(from(z0)) 193.95/71.91
s(mark(z0)) → mark(s(z0)) 193.95/71.91
s(ok(z0)) → ok(s(z0)) 193.95/71.91
cons1(mark(z0), z1) → mark(cons1(z0, z1)) 193.95/71.91
cons1(z0, mark(z1)) → mark(cons1(z0, z1)) 193.95/71.91
cons1(ok(z0), ok(z1)) → ok(cons1(z0, z1)) 193.95/71.91
proper(2nd(z0)) → 2nd(proper(z0)) 193.95/71.91
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 193.95/71.91
proper(from(z0)) → from(proper(z0)) 193.95/71.91
proper(s(z0)) → s(proper(z0)) 193.95/71.91
proper(cons1(z0, z1)) → cons1(proper(z0), proper(z1)) 193.95/71.91
top(mark(z0)) → top(proper(z0)) 193.95/71.91
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

S tuples:none
K tuples:

FROM(mark(z0)) → c13(FROM(z0)) 193.95/71.91
2ND(mark(z0)) → c9(2ND(z0)) 193.95/71.91
CONS(mark(z0), z1) → c11(CONS(z0, z1)) 193.95/71.91
S(mark(z0)) → c15(S(z0)) 193.95/71.91
CONS1(mark(z0), z1) → c17(CONS1(z0, z1)) 193.95/71.91
CONS1(z0, mark(z1)) → c18(CONS1(z0, z1)) 193.95/71.91
2ND(ok(z0)) → c10(2ND(z0)) 193.95/71.91
CONS(ok(z0), ok(z1)) → c12(CONS(z0, z1)) 193.95/71.91
FROM(ok(z0)) → c14(FROM(z0)) 193.95/71.91
S(ok(z0)) → c16(S(z0)) 193.95/71.91
CONS1(ok(z0), ok(z1)) → c19(CONS1(z0, z1))

Defined Rule Symbols:

active, 2nd, cons, from, s, cons1, proper, top

Defined Pair Symbols:

2ND, CONS, FROM, S, CONS1

Compound Symbols:

c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

193.95/71.91

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

The set S is empty

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(50) BOUNDS(O(1), O(1))

193.95/71.91