173.53/63.94 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^1)).

0 CpxTRS

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

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↳2 CdtProblem

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

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↳4 CdtProblem

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

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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 CdtNarrowingProof (BOTH BOUNDS(ID, ID))

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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 CdtUnreachableProof (⇔)

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↳30 CdtProblem

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

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↳32 CdtProblem

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

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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^1))))

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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 SIsEmptyProof (BOTH BOUNDS(ID, ID))

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↳42 BOUNDS(O(1), O(1))

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(0) Obligation:

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

active(2nd(cons(X, cons(Y, Z)))) → mark(Y) 173.53/63.94
active(from(X)) → mark(cons(X, from(s(X)))) 173.53/63.94
active(2nd(X)) → 2nd(active(X)) 173.53/63.94
active(cons(X1, X2)) → cons(active(X1), X2) 173.53/63.94
active(from(X)) → from(active(X)) 173.53/63.94
active(s(X)) → s(active(X)) 173.53/63.94
2nd(mark(X)) → mark(2nd(X)) 173.53/63.94
cons(mark(X1), X2) → mark(cons(X1, X2)) 173.53/63.94
from(mark(X)) → mark(from(X)) 173.53/63.94
s(mark(X)) → mark(s(X)) 173.53/63.94
proper(2nd(X)) → 2nd(proper(X)) 173.53/63.94
proper(cons(X1, X2)) → cons(proper(X1), proper(X2)) 173.53/63.94
proper(from(X)) → from(proper(X)) 173.53/63.94
proper(s(X)) → s(proper(X)) 173.53/63.94
2nd(ok(X)) → ok(2nd(X)) 173.53/63.94
cons(ok(X1), ok(X2)) → ok(cons(X1, X2)) 173.53/63.94
from(ok(X)) → ok(from(X)) 173.53/63.94
s(ok(X)) → ok(s(X)) 173.53/63.94
top(mark(X)) → top(proper(X)) 173.53/63.94
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(cons(z0, cons(z1, z2)))) → mark(z1) 173.53/63.94
active(from(z0)) → mark(cons(z0, from(s(z0)))) 173.53/63.94
active(2nd(z0)) → 2nd(active(z0)) 173.53/63.94
active(cons(z0, z1)) → cons(active(z0), z1) 173.53/63.94
active(from(z0)) → from(active(z0)) 173.53/63.94
active(s(z0)) → s(active(z0)) 173.53/63.94
2nd(mark(z0)) → mark(2nd(z0)) 173.53/63.94
2nd(ok(z0)) → ok(2nd(z0)) 173.53/63.94
cons(mark(z0), z1) → mark(cons(z0, z1)) 173.53/63.94
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 173.53/63.94
from(mark(z0)) → mark(from(z0)) 173.53/63.94
from(ok(z0)) → ok(from(z0)) 173.53/63.94
s(mark(z0)) → mark(s(z0)) 173.53/63.94
s(ok(z0)) → ok(s(z0)) 173.53/63.94
proper(2nd(z0)) → 2nd(proper(z0)) 173.53/63.94
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 173.53/63.94
proper(from(z0)) → from(proper(z0)) 173.53/63.94
proper(s(z0)) → s(proper(z0)) 173.53/63.94
top(mark(z0)) → top(proper(z0)) 173.53/63.94
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(from(z0)) → c1(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0)) 173.53/63.94
ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 173.53/63.94
2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.94
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.94
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.94
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.94
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.94
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.94
S(mark(z0)) → c12(S(z0)) 173.53/63.94
S(ok(z0)) → c13(S(z0)) 173.53/63.94
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 173.53/63.94
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(from(z0)) → c1(CONS(z0, from(s(z0))), FROM(s(z0)), S(z0)) 173.53/63.94
ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 173.53/63.94
2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.94
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.94
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.94
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.94
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.94
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.94
S(mark(z0)) → c12(S(z0)) 173.53/63.94
S(ok(z0)) → c13(S(z0)) 173.53/63.94
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 173.53/63.94
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19

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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(cons(z0, cons(z1, z2)))) → mark(z1) 173.53/63.94
active(from(z0)) → mark(cons(z0, from(s(z0)))) 173.53/63.94
active(2nd(z0)) → 2nd(active(z0)) 173.53/63.94
active(cons(z0, z1)) → cons(active(z0), z1) 173.53/63.94
active(from(z0)) → from(active(z0)) 173.53/63.94
active(s(z0)) → s(active(z0)) 173.53/63.94
2nd(mark(z0)) → mark(2nd(z0)) 173.53/63.94
2nd(ok(z0)) → ok(2nd(z0)) 173.53/63.94
cons(mark(z0), z1) → mark(cons(z0, z1)) 173.53/63.94
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 173.53/63.94
from(mark(z0)) → mark(from(z0)) 173.53/63.94
from(ok(z0)) → ok(from(z0)) 173.53/63.94
s(mark(z0)) → mark(s(z0)) 173.53/63.94
s(ok(z0)) → ok(s(z0)) 173.53/63.94
proper(2nd(z0)) → 2nd(proper(z0)) 173.53/63.94
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 173.53/63.94
proper(from(z0)) → from(proper(z0)) 173.53/63.94
proper(s(z0)) → s(proper(z0)) 173.53/63.94
top(mark(z0)) → top(proper(z0)) 173.53/63.94
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 173.53/63.94
2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.94
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.94
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.94
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.94
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.94
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.94
S(mark(z0)) → c12(S(z0)) 173.53/63.94
S(ok(z0)) → c13(S(z0)) 173.53/63.94
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 173.53/63.94
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c1(S(z0))

S tuples:

ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 173.53/63.94
2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.94
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.94
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.94
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.94
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.94
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.94
S(mark(z0)) → c12(S(z0)) 173.53/63.94
S(ok(z0)) → c13(S(z0)) 173.53/63.94
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 173.53/63.94
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c1(S(z0))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c1

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

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

TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c1(S(z0))

We considered the (Usable) Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(z1) 173.53/63.94
active(from(z0)) → mark(cons(z0, from(s(z0)))) 173.53/63.94
active(2nd(z0)) → 2nd(active(z0)) 173.53/63.94
active(cons(z0, z1)) → cons(active(z0), z1) 173.53/63.94
active(from(z0)) → from(active(z0)) 173.53/63.94
active(s(z0)) → s(active(z0)) 173.53/63.94
s(mark(z0)) → mark(s(z0)) 173.53/63.94
s(ok(z0)) → ok(s(z0)) 173.53/63.94
from(mark(z0)) → mark(from(z0)) 173.53/63.94
from(ok(z0)) → ok(from(z0)) 173.53/63.94
cons(mark(z0), z1) → mark(cons(z0, z1)) 173.53/63.94
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 173.53/63.94
2nd(mark(z0)) → mark(2nd(z0)) 173.53/63.94
2nd(ok(z0)) → ok(2nd(z0)) 173.53/63.94
proper(2nd(z0)) → 2nd(proper(z0)) 173.53/63.94
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 173.53/63.94
proper(from(z0)) → from(proper(z0)) 173.53/63.94
proper(s(z0)) → s(proper(z0))

And the Tuples:

ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 173.53/63.94
2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.94
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.94
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.94
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.94
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.94
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.94
S(mark(z0)) → c12(S(z0)) 173.53/63.94
S(ok(z0)) → c13(S(z0)) 173.53/63.94
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 173.53/63.94
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c1(S(z0))

The order we found is given by the following interpretation:
Polynomial interpretation : 173.53/63.94

POL(2ND(x₁)) = 0 173.53/63.94
POL(2nd(x₁)) = [4]x₁ 173.53/63.94
POL(ACTIVE(x₁)) = [2] 173.53/63.94
POL(CONS(x₁, x₂)) = 0 173.53/63.94
POL(FROM(x₁)) = 0 173.53/63.94
POL(PROPER(x₁)) = 0 173.53/63.94
POL(S(x₁)) = 0 173.53/63.94
POL(TOP(x₁)) = x₁ 173.53/63.94
POL(active(x₁)) = 0 173.53/63.94
POL(c1(x₁)) = x₁ 173.53/63.94
POL(c10(x₁)) = x₁ 173.53/63.94
POL(c11(x₁)) = x₁ 173.53/63.94
POL(c12(x₁)) = x₁ 173.53/63.94
POL(c13(x₁)) = x₁ 173.53/63.94
POL(c14(x₁, x₂)) = x₁ + x₂ 173.53/63.94
POL(c15(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 173.53/63.94
POL(c16(x₁, x₂)) = x₁ + x₂ 173.53/63.94
POL(c17(x₁, x₂)) = x₁ + x₂ 173.53/63.94
POL(c18(x₁, x₂)) = x₁ + x₂ 173.53/63.94
POL(c19(x₁, x₂)) = x₁ + x₂ 173.53/63.94
POL(c2(x₁, x₂)) = x₁ + x₂ 173.53/63.94
POL(c3(x₁, x₂)) = x₁ + x₂ 173.53/63.94
POL(c4(x₁, x₂)) = x₁ + x₂ 173.53/63.94
POL(c5(x₁, x₂)) = x₁ + x₂ 173.53/63.94
POL(c6(x₁)) = x₁ 173.53/63.94
POL(c7(x₁)) = x₁ 173.53/63.94
POL(c8(x₁)) = x₁ 173.53/63.94
POL(c9(x₁)) = x₁ 173.53/63.94
POL(cons(x₁, x₂)) = x₁ 173.53/63.94
POL(from(x₁)) = x₁ 173.53/63.94
POL(mark(x₁)) = 0 173.53/63.94
POL(ok(x₁)) = [4] 173.53/63.94
POL(proper(x₁)) = 0 173.53/63.94
POL(s(x₁)) = [4]x₁

173.53/63.94

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(z1) 173.53/63.94
active(from(z0)) → mark(cons(z0, from(s(z0)))) 173.53/63.94
active(2nd(z0)) → 2nd(active(z0)) 173.53/63.94
active(cons(z0, z1)) → cons(active(z0), z1) 173.53/63.94
active(from(z0)) → from(active(z0)) 173.53/63.94
active(s(z0)) → s(active(z0)) 173.53/63.94
2nd(mark(z0)) → mark(2nd(z0)) 173.53/63.94
2nd(ok(z0)) → ok(2nd(z0)) 173.53/63.94
cons(mark(z0), z1) → mark(cons(z0, z1)) 173.53/63.94
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 173.53/63.94
from(mark(z0)) → mark(from(z0)) 173.53/63.94
from(ok(z0)) → ok(from(z0)) 173.53/63.94
s(mark(z0)) → mark(s(z0)) 173.53/63.94
s(ok(z0)) → ok(s(z0)) 173.53/63.94
proper(2nd(z0)) → 2nd(proper(z0)) 173.53/63.94
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 173.53/63.94
proper(from(z0)) → from(proper(z0)) 173.53/63.94
proper(s(z0)) → s(proper(z0)) 173.53/63.94
top(mark(z0)) → top(proper(z0)) 173.53/63.94
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 173.53/63.94
2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.94
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.94
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.94
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.94
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.94
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.94
S(mark(z0)) → c12(S(z0)) 173.53/63.94
S(ok(z0)) → c13(S(z0)) 173.53/63.94
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 173.53/63.94
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c1(S(z0))

S tuples:

ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 173.53/63.94
2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.94
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.94
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.94
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.94
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.94
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.94
S(mark(z0)) → c12(S(z0)) 173.53/63.94
S(ok(z0)) → c13(S(z0)) 173.53/63.94
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 173.53/63.94
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))

K tuples:

TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c1(S(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c1

173.53/63.94

(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.

TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(z1) 173.53/63.94
active(from(z0)) → mark(cons(z0, from(s(z0)))) 173.53/63.94
active(2nd(z0)) → 2nd(active(z0)) 173.53/63.94
active(cons(z0, z1)) → cons(active(z0), z1) 173.53/63.94
active(from(z0)) → from(active(z0)) 173.53/63.94
active(s(z0)) → s(active(z0)) 173.53/63.94
s(mark(z0)) → mark(s(z0)) 173.53/63.94
s(ok(z0)) → ok(s(z0)) 173.53/63.94
from(mark(z0)) → mark(from(z0)) 173.53/63.94
from(ok(z0)) → ok(from(z0)) 173.53/63.94
cons(mark(z0), z1) → mark(cons(z0, z1)) 173.53/63.94
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 173.53/63.94
2nd(mark(z0)) → mark(2nd(z0)) 173.53/63.94
2nd(ok(z0)) → ok(2nd(z0)) 173.53/63.94
proper(2nd(z0)) → 2nd(proper(z0)) 173.53/63.94
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 173.53/63.94
proper(from(z0)) → from(proper(z0)) 173.53/63.94
proper(s(z0)) → s(proper(z0))

And the Tuples:

ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 173.53/63.94
2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.94
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.94
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.94
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.94
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.94
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.94
S(mark(z0)) → c12(S(z0)) 173.53/63.94
S(ok(z0)) → c13(S(z0)) 173.53/63.94
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 173.53/63.94
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c1(S(z0))

The order we found is given by the following interpretation:
Polynomial interpretation : 173.53/63.94

POL(2ND(x₁)) = 0 173.53/63.94
POL(2nd(x₁)) = x₁ 173.53/63.94
POL(ACTIVE(x₁)) = 0 173.53/63.94
POL(CONS(x₁, x₂)) = 0 173.53/63.94
POL(FROM(x₁)) = 0 173.53/63.94
POL(PROPER(x₁)) = 0 173.53/63.94
POL(S(x₁)) = 0 173.53/63.94
POL(TOP(x₁)) = x₁ 173.53/63.94
POL(active(x₁)) = [4] 173.53/63.94
POL(c1(x₁)) = x₁ 173.53/63.94
POL(c10(x₁)) = x₁ 173.53/63.94
POL(c11(x₁)) = x₁ 173.53/63.94
POL(c12(x₁)) = x₁ 173.53/63.94
POL(c13(x₁)) = x₁ 173.53/63.94
POL(c14(x₁, x₂)) = x₁ + x₂ 173.53/63.94
POL(c15(x₁, x₂, x₃)) = x₁ + x₂ + x₃ 173.53/63.94
POL(c16(x₁, x₂)) = x₁ + x₂ 173.53/63.94
POL(c17(x₁, x₂)) = x₁ + x₂ 173.53/63.94
POL(c18(x₁, x₂)) = x₁ + x₂ 173.53/63.94
POL(c19(x₁, x₂)) = x₁ + x₂ 173.53/63.94
POL(c2(x₁, x₂)) = x₁ + x₂ 173.53/63.94
POL(c3(x₁, x₂)) = x₁ + x₂ 173.53/63.94
POL(c4(x₁, x₂)) = x₁ + x₂ 173.53/63.94
POL(c5(x₁, x₂)) = x₁ + x₂ 173.53/63.94
POL(c6(x₁)) = x₁ 173.53/63.94
POL(c7(x₁)) = x₁ 173.53/63.94
POL(c8(x₁)) = x₁ 173.53/63.94
POL(c9(x₁)) = x₁ 173.53/63.94
POL(cons(x₁, x₂)) = x₁ 173.53/63.94
POL(from(x₁)) = x₁ 173.53/63.94
POL(mark(x₁)) = [2] 173.53/63.94
POL(ok(x₁)) = [4] 173.53/63.94
POL(proper(x₁)) = 0 173.53/63.94
POL(s(x₁)) = x₁

173.53/63.94

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(z1) 173.53/63.94
active(from(z0)) → mark(cons(z0, from(s(z0)))) 173.53/63.94
active(2nd(z0)) → 2nd(active(z0)) 173.53/63.94
active(cons(z0, z1)) → cons(active(z0), z1) 173.53/63.94
active(from(z0)) → from(active(z0)) 173.53/63.94
active(s(z0)) → s(active(z0)) 173.53/63.94
2nd(mark(z0)) → mark(2nd(z0)) 173.53/63.94
2nd(ok(z0)) → ok(2nd(z0)) 173.53/63.94
cons(mark(z0), z1) → mark(cons(z0, z1)) 173.53/63.94
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 173.53/63.94
from(mark(z0)) → mark(from(z0)) 173.53/63.94
from(ok(z0)) → ok(from(z0)) 173.53/63.94
s(mark(z0)) → mark(s(z0)) 173.53/63.94
s(ok(z0)) → ok(s(z0)) 173.53/63.94
proper(2nd(z0)) → 2nd(proper(z0)) 173.53/63.94
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 173.53/63.94
proper(from(z0)) → from(proper(z0)) 173.53/63.94
proper(s(z0)) → s(proper(z0)) 173.53/63.94
top(mark(z0)) → top(proper(z0)) 173.53/63.94
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 173.53/63.94
2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.94
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.94
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.94
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.94
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.94
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.94
S(mark(z0)) → c12(S(z0)) 173.53/63.94
S(ok(z0)) → c13(S(z0)) 173.53/63.94
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 173.53/63.94
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c1(S(z0))

S tuples:

ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 173.53/63.94
2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.94
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.94
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.94
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.94
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.94
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.94
S(mark(z0)) → c12(S(z0)) 173.53/63.94
S(ok(z0)) → c13(S(z0)) 173.53/63.94
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 173.53/63.94
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0))

K tuples:

TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c1(S(z0)) 173.53/63.94
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c1

173.53/63.94

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

Use narrowing to replace ACTIVE(2nd(z0)) → c2(2ND(active(z0)), ACTIVE(z0)) by

ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))

173.53/63.94

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(z1) 173.53/63.94
active(from(z0)) → mark(cons(z0, from(s(z0)))) 173.53/63.94
active(2nd(z0)) → 2nd(active(z0)) 173.53/63.94
active(cons(z0, z1)) → cons(active(z0), z1) 173.53/63.94
active(from(z0)) → from(active(z0)) 173.53/63.94
active(s(z0)) → s(active(z0)) 173.53/63.94
2nd(mark(z0)) → mark(2nd(z0)) 173.53/63.94
2nd(ok(z0)) → ok(2nd(z0)) 173.53/63.94
cons(mark(z0), z1) → mark(cons(z0, z1)) 173.53/63.94
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 173.53/63.94
from(mark(z0)) → mark(from(z0)) 173.53/63.94
from(ok(z0)) → ok(from(z0)) 173.53/63.94
s(mark(z0)) → mark(s(z0)) 173.53/63.94
s(ok(z0)) → ok(s(z0)) 173.53/63.94
proper(2nd(z0)) → 2nd(proper(z0)) 173.53/63.94
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 173.53/63.94
proper(from(z0)) → from(proper(z0)) 173.53/63.94
proper(s(z0)) → s(proper(z0)) 173.53/63.94
top(mark(z0)) → top(proper(z0)) 173.53/63.94
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 173.53/63.94
2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.94
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.94
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.94
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.94
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.94
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.94
S(mark(z0)) → c12(S(z0)) 173.53/63.94
S(ok(z0)) → c13(S(z0)) 173.53/63.94
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 173.53/63.94
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c1(S(z0)) 173.53/63.94
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))

S tuples:

ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 173.53/63.94
2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.94
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.94
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.94
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.94
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.94
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.94
S(mark(z0)) → c12(S(z0)) 173.53/63.94
S(ok(z0)) → c13(S(z0)) 173.53/63.94
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 173.53/63.94
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) 173.53/63.94
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0)))

K tuples:

TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c1(S(z0)) 173.53/63.94
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c1, c2

173.53/63.94

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

Use narrowing to replace ACTIVE(cons(z0, z1)) → c3(CONS(active(z0), z1), ACTIVE(z0)) by

ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0)))

173.53/63.94

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(z1) 173.53/63.94
active(from(z0)) → mark(cons(z0, from(s(z0)))) 173.53/63.94
active(2nd(z0)) → 2nd(active(z0)) 173.53/63.94
active(cons(z0, z1)) → cons(active(z0), z1) 173.53/63.94
active(from(z0)) → from(active(z0)) 173.53/63.94
active(s(z0)) → s(active(z0)) 173.53/63.94
2nd(mark(z0)) → mark(2nd(z0)) 173.53/63.94
2nd(ok(z0)) → ok(2nd(z0)) 173.53/63.94
cons(mark(z0), z1) → mark(cons(z0, z1)) 173.53/63.94
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 173.53/63.94
from(mark(z0)) → mark(from(z0)) 173.53/63.94
from(ok(z0)) → ok(from(z0)) 173.53/63.94
s(mark(z0)) → mark(s(z0)) 173.53/63.94
s(ok(z0)) → ok(s(z0)) 173.53/63.94
proper(2nd(z0)) → 2nd(proper(z0)) 173.53/63.94
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 173.53/63.94
proper(from(z0)) → from(proper(z0)) 173.53/63.94
proper(s(z0)) → s(proper(z0)) 173.53/63.94
top(mark(z0)) → top(proper(z0)) 173.53/63.94
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 173.53/63.94
2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.94
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.94
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.94
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.94
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.94
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.94
S(mark(z0)) → c12(S(z0)) 173.53/63.94
S(ok(z0)) → c13(S(z0)) 173.53/63.94
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 173.53/63.94
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c1(S(z0)) 173.53/63.94
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0)))

S tuples:

ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 173.53/63.94
2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.94
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.94
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.94
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.94
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.94
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.94
S(mark(z0)) → c12(S(z0)) 173.53/63.94
S(ok(z0)) → c13(S(z0)) 173.53/63.94
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 173.53/63.94
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) 173.53/63.94
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0)))

K tuples:

TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c1(S(z0)) 173.53/63.94
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c1, c2, c3

173.53/63.94

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

Use narrowing to replace ACTIVE(from(z0)) → c4(FROM(active(z0)), ACTIVE(z0)) by

ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))

173.53/63.94

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(z1) 173.53/63.94
active(from(z0)) → mark(cons(z0, from(s(z0)))) 173.53/63.94
active(2nd(z0)) → 2nd(active(z0)) 173.53/63.94
active(cons(z0, z1)) → cons(active(z0), z1) 173.53/63.94
active(from(z0)) → from(active(z0)) 173.53/63.94
active(s(z0)) → s(active(z0)) 173.53/63.94
2nd(mark(z0)) → mark(2nd(z0)) 173.53/63.94
2nd(ok(z0)) → ok(2nd(z0)) 173.53/63.94
cons(mark(z0), z1) → mark(cons(z0, z1)) 173.53/63.94
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 173.53/63.94
from(mark(z0)) → mark(from(z0)) 173.53/63.94
from(ok(z0)) → ok(from(z0)) 173.53/63.94
s(mark(z0)) → mark(s(z0)) 173.53/63.94
s(ok(z0)) → ok(s(z0)) 173.53/63.94
proper(2nd(z0)) → 2nd(proper(z0)) 173.53/63.94
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 173.53/63.94
proper(from(z0)) → from(proper(z0)) 173.53/63.94
proper(s(z0)) → s(proper(z0)) 173.53/63.94
top(mark(z0)) → top(proper(z0)) 173.53/63.94
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 173.53/63.94
2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.94
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.94
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.94
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.94
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.94
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.94
S(mark(z0)) → c12(S(z0)) 173.53/63.94
S(ok(z0)) → c13(S(z0)) 173.53/63.94
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 173.53/63.94
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c1(S(z0)) 173.53/63.94
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))

S tuples:

ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) 173.53/63.94
2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.94
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.94
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.94
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.94
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.94
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.94
S(mark(z0)) → c12(S(z0)) 173.53/63.94
S(ok(z0)) → c13(S(z0)) 173.53/63.94
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 173.53/63.94
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) 173.53/63.94
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0)))

K tuples:

TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c1(S(z0)) 173.53/63.94
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c1, c2, c3, c4

173.53/63.94

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

Use narrowing to replace ACTIVE(s(z0)) → c5(S(active(z0)), ACTIVE(z0)) by

ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))

173.53/63.94

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(z1) 173.53/63.94
active(from(z0)) → mark(cons(z0, from(s(z0)))) 173.53/63.94
active(2nd(z0)) → 2nd(active(z0)) 173.53/63.94
active(cons(z0, z1)) → cons(active(z0), z1) 173.53/63.94
active(from(z0)) → from(active(z0)) 173.53/63.94
active(s(z0)) → s(active(z0)) 173.53/63.94
2nd(mark(z0)) → mark(2nd(z0)) 173.53/63.94
2nd(ok(z0)) → ok(2nd(z0)) 173.53/63.94
cons(mark(z0), z1) → mark(cons(z0, z1)) 173.53/63.94
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 173.53/63.94
from(mark(z0)) → mark(from(z0)) 173.53/63.94
from(ok(z0)) → ok(from(z0)) 173.53/63.94
s(mark(z0)) → mark(s(z0)) 173.53/63.94
s(ok(z0)) → ok(s(z0)) 173.53/63.94
proper(2nd(z0)) → 2nd(proper(z0)) 173.53/63.94
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 173.53/63.94
proper(from(z0)) → from(proper(z0)) 173.53/63.94
proper(s(z0)) → s(proper(z0)) 173.53/63.94
top(mark(z0)) → top(proper(z0)) 173.53/63.94
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.94
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.94
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.94
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.94
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.94
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.94
S(mark(z0)) → c12(S(z0)) 173.53/63.94
S(ok(z0)) → c13(S(z0)) 173.53/63.94
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 173.53/63.94
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c1(S(z0)) 173.53/63.94
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))

S tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.94
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.94
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.94
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.94
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.94
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.94
S(mark(z0)) → c12(S(z0)) 173.53/63.94
S(ok(z0)) → c13(S(z0)) 173.53/63.94
PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 173.53/63.94
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) 173.53/63.94
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0)))

K tuples:

TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c1(S(z0)) 173.53/63.94
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c1, c2, c3, c4, c5

173.53/63.94

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

Use narrowing to replace PROPER(2nd(z0)) → c14(2ND(proper(z0)), PROPER(z0)) by

PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.94
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.94
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0))) 173.53/63.94
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0)))

173.53/63.94

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(z1) 173.53/63.94
active(from(z0)) → mark(cons(z0, from(s(z0)))) 173.53/63.94
active(2nd(z0)) → 2nd(active(z0)) 173.53/63.94
active(cons(z0, z1)) → cons(active(z0), z1) 173.53/63.94
active(from(z0)) → from(active(z0)) 173.53/63.94
active(s(z0)) → s(active(z0)) 173.53/63.94
2nd(mark(z0)) → mark(2nd(z0)) 173.53/63.94
2nd(ok(z0)) → ok(2nd(z0)) 173.53/63.94
cons(mark(z0), z1) → mark(cons(z0, z1)) 173.53/63.94
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 173.53/63.94
from(mark(z0)) → mark(from(z0)) 173.53/63.94
from(ok(z0)) → ok(from(z0)) 173.53/63.94
s(mark(z0)) → mark(s(z0)) 173.53/63.94
s(ok(z0)) → ok(s(z0)) 173.53/63.94
proper(2nd(z0)) → 2nd(proper(z0)) 173.53/63.94
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 173.53/63.94
proper(from(z0)) → from(proper(z0)) 173.53/63.94
proper(s(z0)) → s(proper(z0)) 173.53/63.94
top(mark(z0)) → top(proper(z0)) 173.53/63.94
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.94
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.94
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.94
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.94
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.94
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.94
S(mark(z0)) → c12(S(z0)) 173.53/63.94
S(ok(z0)) → c13(S(z0)) 173.53/63.94
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 173.53/63.94
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c1(S(z0)) 173.53/63.94
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 173.53/63.94
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.94
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.94
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0))) 173.53/63.94
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0)))

S tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.94
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.94
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.94
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.94
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.94
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.94
S(mark(z0)) → c12(S(z0)) 173.53/63.94
S(ok(z0)) → c13(S(z0)) 173.53/63.94
PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) 173.53/63.94
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) 173.53/63.94
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 173.53/63.94
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.94
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.94
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0))) 173.53/63.94
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0)))

K tuples:

TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c1(S(z0)) 173.53/63.94
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13, c15, c16, c17, c18, c19, c1, c2, c3, c4, c5, c14

173.53/63.94

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

Use narrowing to replace PROPER(cons(z0, z1)) → c15(CONS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 173.53/63.94
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 173.53/63.94
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 173.53/63.94
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 173.53/63.94
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 173.53/63.94
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 173.53/63.94
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 173.53/63.94
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))

173.53/63.94

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(z1) 173.53/63.94
active(from(z0)) → mark(cons(z0, from(s(z0)))) 173.53/63.94
active(2nd(z0)) → 2nd(active(z0)) 173.53/63.94
active(cons(z0, z1)) → cons(active(z0), z1) 173.53/63.94
active(from(z0)) → from(active(z0)) 173.53/63.94
active(s(z0)) → s(active(z0)) 173.53/63.94
2nd(mark(z0)) → mark(2nd(z0)) 173.53/63.94
2nd(ok(z0)) → ok(2nd(z0)) 173.53/63.94
cons(mark(z0), z1) → mark(cons(z0, z1)) 173.53/63.94
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 173.53/63.94
from(mark(z0)) → mark(from(z0)) 173.53/63.94
from(ok(z0)) → ok(from(z0)) 173.53/63.94
s(mark(z0)) → mark(s(z0)) 173.53/63.94
s(ok(z0)) → ok(s(z0)) 173.53/63.94
proper(2nd(z0)) → 2nd(proper(z0)) 173.53/63.94
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 173.53/63.94
proper(from(z0)) → from(proper(z0)) 173.53/63.94
proper(s(z0)) → s(proper(z0)) 173.53/63.94
top(mark(z0)) → top(proper(z0)) 173.53/63.94
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.94
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.94
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.94
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.94
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.94
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.94
S(mark(z0)) → c12(S(z0)) 173.53/63.94
S(ok(z0)) → c13(S(z0)) 173.53/63.94
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c1(S(z0)) 173.53/63.94
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 173.53/63.94
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.94
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.94
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0))) 173.53/63.94
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0))) 173.53/63.94
PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 173.53/63.94
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 173.53/63.94
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 173.53/63.94
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 173.53/63.94
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 173.53/63.94
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 173.53/63.94
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 173.53/63.94
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))

S tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.94
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.94
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.94
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.94
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.94
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.94
S(mark(z0)) → c12(S(z0)) 173.53/63.94
S(ok(z0)) → c13(S(z0)) 173.53/63.94
PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) 173.53/63.94
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) 173.53/63.94
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 173.53/63.94
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.94
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.94
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0))) 173.53/63.94
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0))) 173.53/63.94
PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 173.53/63.94
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 173.53/63.94
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 173.53/63.94
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 173.53/63.94
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 173.53/63.94
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 173.53/63.94
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 173.53/63.94
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))

K tuples:

TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c1(S(z0)) 173.53/63.94
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13, c16, c17, c18, c19, c1, c2, c3, c4, c5, c14, c15

173.53/63.94

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

Use narrowing to replace PROPER(from(z0)) → c16(FROM(proper(z0)), PROPER(z0)) by

PROPER(from(2nd(z0))) → c16(FROM(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.94
PROPER(from(cons(z0, z1))) → c16(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.94
PROPER(from(from(z0))) → c16(FROM(from(proper(z0))), PROPER(from(z0))) 173.53/63.94
PROPER(from(s(z0))) → c16(FROM(s(proper(z0))), PROPER(s(z0)))

173.53/63.94

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(z1) 173.53/63.94
active(from(z0)) → mark(cons(z0, from(s(z0)))) 173.53/63.94
active(2nd(z0)) → 2nd(active(z0)) 173.53/63.94
active(cons(z0, z1)) → cons(active(z0), z1) 173.53/63.94
active(from(z0)) → from(active(z0)) 173.53/63.94
active(s(z0)) → s(active(z0)) 173.53/63.94
2nd(mark(z0)) → mark(2nd(z0)) 173.53/63.94
2nd(ok(z0)) → ok(2nd(z0)) 173.53/63.94
cons(mark(z0), z1) → mark(cons(z0, z1)) 173.53/63.94
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 173.53/63.94
from(mark(z0)) → mark(from(z0)) 173.53/63.94
from(ok(z0)) → ok(from(z0)) 173.53/63.94
s(mark(z0)) → mark(s(z0)) 173.53/63.94
s(ok(z0)) → ok(s(z0)) 173.53/63.94
proper(2nd(z0)) → 2nd(proper(z0)) 173.53/63.94
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 173.53/63.94
proper(from(z0)) → from(proper(z0)) 173.53/63.94
proper(s(z0)) → s(proper(z0)) 173.53/63.94
top(mark(z0)) → top(proper(z0)) 173.53/63.94
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.94
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.94
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.94
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.94
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.94
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.94
S(mark(z0)) → c12(S(z0)) 173.53/63.94
S(ok(z0)) → c13(S(z0)) 173.53/63.94
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 173.53/63.94
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.94
ACTIVE(from(z0)) → c1(S(z0)) 173.53/63.94
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0))) 173.53/63.94
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.94
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.94
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.94
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 173.53/63.94
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 173.53/63.94
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.94
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.94
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0))) 173.53/63.94
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0))) 173.53/63.94
PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 173.53/63.94
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 173.53/63.94
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 173.53/63.95
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 173.53/63.95
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 173.53/63.95
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 173.53/63.95
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 173.53/63.95
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 173.53/63.95
PROPER(from(2nd(z0))) → c16(FROM(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(from(cons(z0, z1))) → c16(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(from(from(z0))) → c16(FROM(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(from(s(z0))) → c16(FROM(s(proper(z0))), PROPER(s(z0)))

S tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.95
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.95
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.95
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.95
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.95
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.95
S(mark(z0)) → c12(S(z0)) 173.53/63.95
S(ok(z0)) → c13(S(z0)) 173.53/63.95
PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) 173.53/63.95
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0))) 173.53/63.95
PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 173.53/63.95
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 173.53/63.95
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 173.53/63.95
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 173.53/63.95
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 173.53/63.95
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 173.53/63.95
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 173.53/63.95
PROPER(from(2nd(z0))) → c16(FROM(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(from(cons(z0, z1))) → c16(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(from(from(z0))) → c16(FROM(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(from(s(z0))) → c16(FROM(s(proper(z0))), PROPER(s(z0)))

K tuples:

TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.95
ACTIVE(from(z0)) → c1(S(z0)) 173.53/63.95
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13, c17, c18, c19, c1, c2, c3, c4, c5, c14, c15, c16

173.53/63.95

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

Use narrowing to replace PROPER(s(z0)) → c17(S(proper(z0)), PROPER(z0)) by

PROPER(s(2nd(z0))) → c17(S(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(s(cons(z0, z1))) → c17(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(s(from(z0))) → c17(S(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(s(s(z0))) → c17(S(s(proper(z0))), PROPER(s(z0)))

173.53/63.95

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(z1) 173.53/63.95
active(from(z0)) → mark(cons(z0, from(s(z0)))) 173.53/63.95
active(2nd(z0)) → 2nd(active(z0)) 173.53/63.95
active(cons(z0, z1)) → cons(active(z0), z1) 173.53/63.95
active(from(z0)) → from(active(z0)) 173.53/63.95
active(s(z0)) → s(active(z0)) 173.53/63.95
2nd(mark(z0)) → mark(2nd(z0)) 173.53/63.95
2nd(ok(z0)) → ok(2nd(z0)) 173.53/63.95
cons(mark(z0), z1) → mark(cons(z0, z1)) 173.53/63.95
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 173.53/63.95
from(mark(z0)) → mark(from(z0)) 173.53/63.95
from(ok(z0)) → ok(from(z0)) 173.53/63.95
s(mark(z0)) → mark(s(z0)) 173.53/63.95
s(ok(z0)) → ok(s(z0)) 173.53/63.95
proper(2nd(z0)) → 2nd(proper(z0)) 173.53/63.95
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 173.53/63.95
proper(from(z0)) → from(proper(z0)) 173.53/63.95
proper(s(z0)) → s(proper(z0)) 173.53/63.95
top(mark(z0)) → top(proper(z0)) 173.53/63.95
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.95
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.95
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.95
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.95
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.95
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.95
S(mark(z0)) → c12(S(z0)) 173.53/63.95
S(ok(z0)) → c13(S(z0)) 173.53/63.95
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) 173.53/63.95
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.95
ACTIVE(from(z0)) → c1(S(z0)) 173.53/63.95
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0))) 173.53/63.95
PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 173.53/63.95
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 173.53/63.95
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 173.53/63.95
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 173.53/63.95
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 173.53/63.95
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 173.53/63.95
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 173.53/63.95
PROPER(from(2nd(z0))) → c16(FROM(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(from(cons(z0, z1))) → c16(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(from(from(z0))) → c16(FROM(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(from(s(z0))) → c16(FROM(s(proper(z0))), PROPER(s(z0))) 173.53/63.95
PROPER(s(2nd(z0))) → c17(S(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(s(cons(z0, z1))) → c17(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(s(from(z0))) → c17(S(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(s(s(z0))) → c17(S(s(proper(z0))), PROPER(s(z0)))

S tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.95
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.95
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.95
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.95
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.95
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.95
S(mark(z0)) → c12(S(z0)) 173.53/63.95
S(ok(z0)) → c13(S(z0)) 173.53/63.95
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0))) 173.53/63.95
PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 173.53/63.95
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 173.53/63.95
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 173.53/63.95
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 173.53/63.95
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 173.53/63.95
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 173.53/63.95
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 173.53/63.95
PROPER(from(2nd(z0))) → c16(FROM(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(from(cons(z0, z1))) → c16(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(from(from(z0))) → c16(FROM(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(from(s(z0))) → c16(FROM(s(proper(z0))), PROPER(s(z0))) 173.53/63.95
PROPER(s(2nd(z0))) → c17(S(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(s(cons(z0, z1))) → c17(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(s(from(z0))) → c17(S(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(s(s(z0))) → c17(S(s(proper(z0))), PROPER(s(z0)))

K tuples:

TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.95
ACTIVE(from(z0)) → c1(S(z0)) 173.53/63.95
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13, c18, c19, c1, c2, c3, c4, c5, c14, c15, c16, c17

173.53/63.95

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

Use narrowing to replace TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(2nd(z0))) → c18(TOP(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
TOP(mark(cons(z0, z1))) → c18(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
TOP(mark(from(z0))) → c18(TOP(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
TOP(mark(s(z0))) → c18(TOP(s(proper(z0))), PROPER(s(z0)))

173.53/63.95

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(z1) 173.53/63.95
active(from(z0)) → mark(cons(z0, from(s(z0)))) 173.53/63.95
active(2nd(z0)) → 2nd(active(z0)) 173.53/63.95
active(cons(z0, z1)) → cons(active(z0), z1) 173.53/63.95
active(from(z0)) → from(active(z0)) 173.53/63.95
active(s(z0)) → s(active(z0)) 173.53/63.95
2nd(mark(z0)) → mark(2nd(z0)) 173.53/63.95
2nd(ok(z0)) → ok(2nd(z0)) 173.53/63.95
cons(mark(z0), z1) → mark(cons(z0, z1)) 173.53/63.95
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 173.53/63.95
from(mark(z0)) → mark(from(z0)) 173.53/63.95
from(ok(z0)) → ok(from(z0)) 173.53/63.95
s(mark(z0)) → mark(s(z0)) 173.53/63.95
s(ok(z0)) → ok(s(z0)) 173.53/63.95
proper(2nd(z0)) → 2nd(proper(z0)) 173.53/63.95
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 173.53/63.95
proper(from(z0)) → from(proper(z0)) 173.53/63.95
proper(s(z0)) → s(proper(z0)) 173.53/63.95
top(mark(z0)) → top(proper(z0)) 173.53/63.95
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.95
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.95
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.95
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.95
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.95
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.95
S(mark(z0)) → c12(S(z0)) 173.53/63.95
S(ok(z0)) → c13(S(z0)) 173.53/63.95
TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.95
ACTIVE(from(z0)) → c1(S(z0)) 173.53/63.95
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0))) 173.53/63.95
PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 173.53/63.95
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 173.53/63.95
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 173.53/63.95
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 173.53/63.95
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 173.53/63.95
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 173.53/63.95
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 173.53/63.95
PROPER(from(2nd(z0))) → c16(FROM(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(from(cons(z0, z1))) → c16(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(from(from(z0))) → c16(FROM(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(from(s(z0))) → c16(FROM(s(proper(z0))), PROPER(s(z0))) 173.53/63.95
PROPER(s(2nd(z0))) → c17(S(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(s(cons(z0, z1))) → c17(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(s(from(z0))) → c17(S(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(s(s(z0))) → c17(S(s(proper(z0))), PROPER(s(z0))) 173.53/63.95
TOP(mark(2nd(z0))) → c18(TOP(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
TOP(mark(cons(z0, z1))) → c18(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
TOP(mark(from(z0))) → c18(TOP(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
TOP(mark(s(z0))) → c18(TOP(s(proper(z0))), PROPER(s(z0)))

S tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.95
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.95
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.95
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.95
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.95
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.95
S(mark(z0)) → c12(S(z0)) 173.53/63.95
S(ok(z0)) → c13(S(z0)) 173.53/63.95
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0))) 173.53/63.95
PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 173.53/63.95
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 173.53/63.95
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 173.53/63.95
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 173.53/63.95
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 173.53/63.95
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 173.53/63.95
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 173.53/63.95
PROPER(from(2nd(z0))) → c16(FROM(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(from(cons(z0, z1))) → c16(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(from(from(z0))) → c16(FROM(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(from(s(z0))) → c16(FROM(s(proper(z0))), PROPER(s(z0))) 173.53/63.95
PROPER(s(2nd(z0))) → c17(S(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(s(cons(z0, z1))) → c17(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(s(from(z0))) → c17(S(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(s(s(z0))) → c17(S(s(proper(z0))), PROPER(s(z0)))

K tuples:

TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.95
ACTIVE(from(z0)) → c1(S(z0)) 173.53/63.95
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13, c19, c1, c2, c3, c4, c5, c14, c15, c16, c17, c18

173.53/63.95

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

Use narrowing to replace TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(2nd(cons(z0, cons(z1, z2))))) → c19(TOP(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
TOP(ok(from(z0))) → c19(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
TOP(ok(2nd(z0))) → c19(TOP(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
TOP(ok(cons(z0, z1))) → c19(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
TOP(ok(from(z0))) → c19(TOP(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
TOP(ok(s(z0))) → c19(TOP(s(active(z0))), ACTIVE(s(z0)))

173.53/63.95

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(z1) 173.53/63.95
active(from(z0)) → mark(cons(z0, from(s(z0)))) 173.53/63.95
active(2nd(z0)) → 2nd(active(z0)) 173.53/63.95
active(cons(z0, z1)) → cons(active(z0), z1) 173.53/63.95
active(from(z0)) → from(active(z0)) 173.53/63.95
active(s(z0)) → s(active(z0)) 173.53/63.95
2nd(mark(z0)) → mark(2nd(z0)) 173.53/63.95
2nd(ok(z0)) → ok(2nd(z0)) 173.53/63.95
cons(mark(z0), z1) → mark(cons(z0, z1)) 173.53/63.95
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 173.53/63.95
from(mark(z0)) → mark(from(z0)) 173.53/63.95
from(ok(z0)) → ok(from(z0)) 173.53/63.95
s(mark(z0)) → mark(s(z0)) 173.53/63.95
s(ok(z0)) → ok(s(z0)) 173.53/63.95
proper(2nd(z0)) → 2nd(proper(z0)) 173.53/63.95
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 173.53/63.95
proper(from(z0)) → from(proper(z0)) 173.53/63.95
proper(s(z0)) → s(proper(z0)) 173.53/63.95
top(mark(z0)) → top(proper(z0)) 173.53/63.95
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.95
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.95
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.95
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.95
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.95
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.95
S(mark(z0)) → c12(S(z0)) 173.53/63.95
S(ok(z0)) → c13(S(z0)) 173.53/63.95
ACTIVE(from(z0)) → c1(S(z0)) 173.53/63.95
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0))) 173.53/63.95
PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 173.53/63.95
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 173.53/63.95
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 173.53/63.95
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 173.53/63.95
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 173.53/63.95
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 173.53/63.95
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 173.53/63.95
PROPER(from(2nd(z0))) → c16(FROM(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(from(cons(z0, z1))) → c16(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(from(from(z0))) → c16(FROM(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(from(s(z0))) → c16(FROM(s(proper(z0))), PROPER(s(z0))) 173.53/63.95
PROPER(s(2nd(z0))) → c17(S(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(s(cons(z0, z1))) → c17(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(s(from(z0))) → c17(S(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(s(s(z0))) → c17(S(s(proper(z0))), PROPER(s(z0))) 173.53/63.95
TOP(mark(2nd(z0))) → c18(TOP(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
TOP(mark(cons(z0, z1))) → c18(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
TOP(mark(from(z0))) → c18(TOP(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
TOP(mark(s(z0))) → c18(TOP(s(proper(z0))), PROPER(s(z0))) 173.53/63.95
TOP(ok(2nd(cons(z0, cons(z1, z2))))) → c19(TOP(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
TOP(ok(from(z0))) → c19(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
TOP(ok(2nd(z0))) → c19(TOP(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
TOP(ok(cons(z0, z1))) → c19(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
TOP(ok(from(z0))) → c19(TOP(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
TOP(ok(s(z0))) → c19(TOP(s(active(z0))), ACTIVE(s(z0)))

S tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.95
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.95
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.95
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.95
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.95
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.95
S(mark(z0)) → c12(S(z0)) 173.53/63.95
S(ok(z0)) → c13(S(z0)) 173.53/63.95
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0))) 173.53/63.95
PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 173.53/63.95
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 173.53/63.95
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 173.53/63.95
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 173.53/63.95
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 173.53/63.95
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 173.53/63.95
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 173.53/63.95
PROPER(from(2nd(z0))) → c16(FROM(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(from(cons(z0, z1))) → c16(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(from(from(z0))) → c16(FROM(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(from(s(z0))) → c16(FROM(s(proper(z0))), PROPER(s(z0))) 173.53/63.95
PROPER(s(2nd(z0))) → c17(S(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(s(cons(z0, z1))) → c17(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(s(from(z0))) → c17(S(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(s(s(z0))) → c17(S(s(proper(z0))), PROPER(s(z0)))

K tuples:

TOP(ok(z0)) → c19(TOP(active(z0)), ACTIVE(z0)) 173.53/63.95
ACTIVE(from(z0)) → c1(S(z0)) 173.53/63.95
TOP(mark(z0)) → c18(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

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

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13, c1, c2, c3, c4, c5, c14, c15, c16, c17, c18, c19

173.53/63.95

(29) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(from(z0)) → c1(S(z0)) 173.53/63.95
ACTIVE(2nd(2nd(cons(z0, cons(z1, z2))))) → c2(2ND(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(2nd(from(z0))) → c2(2ND(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(2nd(2nd(z0))) → c2(2ND(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(2nd(cons(z0, z1))) → c2(2ND(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(2nd(from(z0))) → c2(2ND(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(2nd(s(z0))) → c2(2ND(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(cons(2nd(cons(z0, cons(z1, z2))), x1)) → c3(CONS(mark(z1), x1), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(cons(from(z0), x1)) → c3(CONS(mark(cons(z0, from(s(z0)))), x1), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(cons(2nd(z0), x1)) → c3(CONS(2nd(active(z0)), x1), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(cons(cons(z0, z1), x1)) → c3(CONS(cons(active(z0), z1), x1), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(cons(from(z0), x1)) → c3(CONS(from(active(z0)), x1), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(cons(s(z0), x1)) → c3(CONS(s(active(z0)), x1), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(from(2nd(cons(z0, cons(z1, z2))))) → c4(FROM(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(from(from(z0))) → c4(FROM(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(from(2nd(z0))) → c4(FROM(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(from(cons(z0, z1))) → c4(FROM(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(from(from(z0))) → c4(FROM(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(from(s(z0))) → c4(FROM(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
ACTIVE(s(2nd(cons(z0, cons(z1, z2))))) → c5(S(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
ACTIVE(s(from(z0))) → c5(S(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(s(2nd(z0))) → c5(S(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
ACTIVE(s(cons(z0, z1))) → c5(S(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
ACTIVE(s(from(z0))) → c5(S(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
ACTIVE(s(s(z0))) → c5(S(s(active(z0))), ACTIVE(s(z0))) 173.53/63.95
PROPER(2nd(2nd(z0))) → c14(2ND(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(2nd(cons(z0, z1))) → c14(2ND(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(2nd(from(z0))) → c14(2ND(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(2nd(s(z0))) → c14(2ND(s(proper(z0))), PROPER(s(z0))) 173.53/63.95
PROPER(cons(x0, 2nd(z0))) → c15(CONS(proper(x0), 2nd(proper(z0))), PROPER(x0), PROPER(2nd(z0))) 173.53/63.95
PROPER(cons(x0, cons(z0, z1))) → c15(CONS(proper(x0), cons(proper(z0), proper(z1))), PROPER(x0), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(cons(x0, from(z0))) → c15(CONS(proper(x0), from(proper(z0))), PROPER(x0), PROPER(from(z0))) 173.53/63.95
PROPER(cons(x0, s(z0))) → c15(CONS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0))) 173.53/63.95
PROPER(cons(2nd(z0), x1)) → c15(CONS(2nd(proper(z0)), proper(x1)), PROPER(2nd(z0)), PROPER(x1)) 173.53/63.95
PROPER(cons(cons(z0, z1), x1)) → c15(CONS(cons(proper(z0), proper(z1)), proper(x1)), PROPER(cons(z0, z1)), PROPER(x1)) 173.53/63.95
PROPER(cons(from(z0), x1)) → c15(CONS(from(proper(z0)), proper(x1)), PROPER(from(z0)), PROPER(x1)) 173.53/63.95
PROPER(cons(s(z0), x1)) → c15(CONS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1)) 173.53/63.95
PROPER(from(2nd(z0))) → c16(FROM(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(from(cons(z0, z1))) → c16(FROM(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(from(from(z0))) → c16(FROM(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(from(s(z0))) → c16(FROM(s(proper(z0))), PROPER(s(z0))) 173.53/63.95
PROPER(s(2nd(z0))) → c17(S(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
PROPER(s(cons(z0, z1))) → c17(S(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
PROPER(s(from(z0))) → c17(S(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
PROPER(s(s(z0))) → c17(S(s(proper(z0))), PROPER(s(z0))) 173.53/63.95
TOP(mark(2nd(z0))) → c18(TOP(2nd(proper(z0))), PROPER(2nd(z0))) 173.53/63.95
TOP(mark(cons(z0, z1))) → c18(TOP(cons(proper(z0), proper(z1))), PROPER(cons(z0, z1))) 173.53/63.95
TOP(mark(from(z0))) → c18(TOP(from(proper(z0))), PROPER(from(z0))) 173.53/63.95
TOP(mark(s(z0))) → c18(TOP(s(proper(z0))), PROPER(s(z0))) 173.53/63.95
TOP(ok(2nd(cons(z0, cons(z1, z2))))) → c19(TOP(mark(z1)), ACTIVE(2nd(cons(z0, cons(z1, z2))))) 173.53/63.95
TOP(ok(from(z0))) → c19(TOP(mark(cons(z0, from(s(z0))))), ACTIVE(from(z0))) 173.53/63.95
TOP(ok(2nd(z0))) → c19(TOP(2nd(active(z0))), ACTIVE(2nd(z0))) 173.53/63.95
TOP(ok(cons(z0, z1))) → c19(TOP(cons(active(z0), z1)), ACTIVE(cons(z0, z1))) 173.53/63.95
TOP(ok(from(z0))) → c19(TOP(from(active(z0))), ACTIVE(from(z0))) 173.53/63.95
TOP(ok(s(z0))) → c19(TOP(s(active(z0))), ACTIVE(s(z0)))

173.53/63.95

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(z1) 173.53/63.95
active(from(z0)) → mark(cons(z0, from(s(z0)))) 173.53/63.95
active(2nd(z0)) → 2nd(active(z0)) 173.53/63.95
active(cons(z0, z1)) → cons(active(z0), z1) 173.53/63.95
active(from(z0)) → from(active(z0)) 173.53/63.95
active(s(z0)) → s(active(z0)) 173.53/63.95
2nd(mark(z0)) → mark(2nd(z0)) 173.53/63.95
2nd(ok(z0)) → ok(2nd(z0)) 173.53/63.95
cons(mark(z0), z1) → mark(cons(z0, z1)) 173.53/63.95
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 173.53/63.95
from(mark(z0)) → mark(from(z0)) 173.53/63.95
from(ok(z0)) → ok(from(z0)) 173.53/63.95
s(mark(z0)) → mark(s(z0)) 173.53/63.95
s(ok(z0)) → ok(s(z0)) 173.53/63.95
proper(2nd(z0)) → 2nd(proper(z0)) 173.53/63.95
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 173.53/63.95
proper(from(z0)) → from(proper(z0)) 173.53/63.95
proper(s(z0)) → s(proper(z0)) 173.53/63.95
top(mark(z0)) → top(proper(z0)) 173.53/63.95
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.95
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.95
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.95
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.95
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.95
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.95
S(mark(z0)) → c12(S(z0)) 173.53/63.95
S(ok(z0)) → c13(S(z0))

S tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.95
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.95
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.95
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.95
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.95
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.95
S(mark(z0)) → c12(S(z0)) 173.53/63.95
S(ok(z0)) → c13(S(z0))

K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

2ND, CONS, FROM, S

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13

173.53/63.95

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

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

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.95
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.95
S(mark(z0)) → c12(S(z0)) 173.53/63.95
S(ok(z0)) → c13(S(z0))

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

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.95
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.95
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.95
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.95
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.95
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.95
S(mark(z0)) → c12(S(z0)) 173.53/63.95
S(ok(z0)) → c13(S(z0))

The order we found is given by the following interpretation:
Polynomial interpretation : 173.53/63.95

POL(2ND(x₁)) = [2]x₁ 173.53/63.95
POL(CONS(x₁, x₂)) = 0 173.53/63.95
POL(FROM(x₁)) = 0 173.53/63.95
POL(S(x₁)) = x₁ 173.53/63.95
POL(c10(x₁)) = x₁ 173.53/63.95
POL(c11(x₁)) = x₁ 173.53/63.95
POL(c12(x₁)) = x₁ 173.53/63.95
POL(c13(x₁)) = x₁ 173.53/63.95
POL(c6(x₁)) = x₁ 173.53/63.95
POL(c7(x₁)) = x₁ 173.53/63.95
POL(c8(x₁)) = x₁ 173.53/63.95
POL(c9(x₁)) = x₁ 173.53/63.96
POL(mark(x₁)) = [1] + x₁ 173.53/63.96
POL(ok(x₁)) = [1] + x₁

173.53/63.96

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(z1) 173.53/63.96
active(from(z0)) → mark(cons(z0, from(s(z0)))) 173.53/63.96
active(2nd(z0)) → 2nd(active(z0)) 173.53/63.96
active(cons(z0, z1)) → cons(active(z0), z1) 173.53/63.96
active(from(z0)) → from(active(z0)) 173.53/63.96
active(s(z0)) → s(active(z0)) 173.53/63.96
2nd(mark(z0)) → mark(2nd(z0)) 173.53/63.96
2nd(ok(z0)) → ok(2nd(z0)) 173.53/63.96
cons(mark(z0), z1) → mark(cons(z0, z1)) 173.53/63.96
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 173.53/63.96
from(mark(z0)) → mark(from(z0)) 173.53/63.96
from(ok(z0)) → ok(from(z0)) 173.53/63.96
s(mark(z0)) → mark(s(z0)) 173.53/63.96
s(ok(z0)) → ok(s(z0)) 173.53/63.96
proper(2nd(z0)) → 2nd(proper(z0)) 173.53/63.96
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 173.53/63.96
proper(from(z0)) → from(proper(z0)) 173.53/63.96
proper(s(z0)) → s(proper(z0)) 173.53/63.96
top(mark(z0)) → top(proper(z0)) 173.53/63.96
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.96
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.96
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.96
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.96
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.96
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.96
S(mark(z0)) → c12(S(z0)) 173.53/63.96
S(ok(z0)) → c13(S(z0))

S tuples:

CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.96
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.96
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.96
FROM(ok(z0)) → c11(FROM(z0))

K tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.96
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.96
S(mark(z0)) → c12(S(z0)) 173.53/63.96
S(ok(z0)) → c13(S(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

2ND, CONS, FROM, S

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13

173.53/63.96

(33) 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(mark(z0), z1) → c8(CONS(z0, z1))

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

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.96
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.96
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.96
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.96
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.96
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.96
S(mark(z0)) → c12(S(z0)) 173.53/63.96
S(ok(z0)) → c13(S(z0))

The order we found is given by the following interpretation:
Polynomial interpretation : 173.53/63.96

POL(2ND(x₁)) = [5]x₁ 173.53/63.96
POL(CONS(x₁, x₂)) = [2]x₁ 173.53/63.96
POL(FROM(x₁)) = 0 173.53/63.96
POL(S(x₁)) = [3]x₁ 173.53/63.96
POL(c10(x₁)) = x₁ 173.53/63.96
POL(c11(x₁)) = x₁ 173.53/63.96
POL(c12(x₁)) = x₁ 173.53/63.96
POL(c13(x₁)) = x₁ 173.53/63.96
POL(c6(x₁)) = x₁ 173.53/63.96
POL(c7(x₁)) = x₁ 173.53/63.96
POL(c8(x₁)) = x₁ 173.53/63.96
POL(c9(x₁)) = x₁ 173.53/63.96
POL(mark(x₁)) = [1] + x₁ 173.53/63.96
POL(ok(x₁)) = x₁

173.53/63.96

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(z1) 173.53/63.96
active(from(z0)) → mark(cons(z0, from(s(z0)))) 173.53/63.96
active(2nd(z0)) → 2nd(active(z0)) 173.53/63.96
active(cons(z0, z1)) → cons(active(z0), z1) 173.53/63.96
active(from(z0)) → from(active(z0)) 173.53/63.96
active(s(z0)) → s(active(z0)) 173.53/63.96
2nd(mark(z0)) → mark(2nd(z0)) 173.53/63.96
2nd(ok(z0)) → ok(2nd(z0)) 173.53/63.96
cons(mark(z0), z1) → mark(cons(z0, z1)) 173.53/63.96
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 173.53/63.96
from(mark(z0)) → mark(from(z0)) 173.53/63.96
from(ok(z0)) → ok(from(z0)) 173.53/63.96
s(mark(z0)) → mark(s(z0)) 173.53/63.96
s(ok(z0)) → ok(s(z0)) 173.53/63.96
proper(2nd(z0)) → 2nd(proper(z0)) 173.53/63.96
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 173.53/63.96
proper(from(z0)) → from(proper(z0)) 173.53/63.96
proper(s(z0)) → s(proper(z0)) 173.53/63.96
top(mark(z0)) → top(proper(z0)) 173.53/63.96
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.96
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.96
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.96
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.96
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.96
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.96
S(mark(z0)) → c12(S(z0)) 173.53/63.96
S(ok(z0)) → c13(S(z0))

S tuples:

CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.96
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.96
FROM(ok(z0)) → c11(FROM(z0))

K tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.96
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.96
S(mark(z0)) → c12(S(z0)) 173.53/63.96
S(ok(z0)) → c13(S(z0)) 173.53/63.96
CONS(mark(z0), z1) → c8(CONS(z0, z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

2ND, CONS, FROM, S

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13

173.53/63.96

(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.

CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))

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

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.96
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.96
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.96
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.96
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.96
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.96
S(mark(z0)) → c12(S(z0)) 173.53/63.96
S(ok(z0)) → c13(S(z0))

The order we found is given by the following interpretation:
Polynomial interpretation : 173.53/63.96

POL(2ND(x₁)) = 0 173.53/63.96
POL(CONS(x₁, x₂)) = x₂ 173.53/63.96
POL(FROM(x₁)) = 0 173.53/63.96
POL(S(x₁)) = 0 173.53/63.96
POL(c10(x₁)) = x₁ 173.53/63.96
POL(c11(x₁)) = x₁ 173.53/63.96
POL(c12(x₁)) = x₁ 173.53/63.96
POL(c13(x₁)) = x₁ 173.53/63.96
POL(c6(x₁)) = x₁ 173.53/63.96
POL(c7(x₁)) = x₁ 173.53/63.96
POL(c8(x₁)) = x₁ 173.53/63.96
POL(c9(x₁)) = x₁ 173.53/63.96
POL(mark(x₁)) = [1] 173.53/63.96
POL(ok(x₁)) = [1] + x₁

173.53/63.96

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(z1) 173.53/63.96
active(from(z0)) → mark(cons(z0, from(s(z0)))) 173.53/63.96
active(2nd(z0)) → 2nd(active(z0)) 173.53/63.96
active(cons(z0, z1)) → cons(active(z0), z1) 173.53/63.96
active(from(z0)) → from(active(z0)) 173.53/63.96
active(s(z0)) → s(active(z0)) 173.53/63.96
2nd(mark(z0)) → mark(2nd(z0)) 173.53/63.96
2nd(ok(z0)) → ok(2nd(z0)) 173.53/63.96
cons(mark(z0), z1) → mark(cons(z0, z1)) 173.53/63.96
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 173.53/63.96
from(mark(z0)) → mark(from(z0)) 173.53/63.96
from(ok(z0)) → ok(from(z0)) 173.53/63.96
s(mark(z0)) → mark(s(z0)) 173.53/63.96
s(ok(z0)) → ok(s(z0)) 173.53/63.96
proper(2nd(z0)) → 2nd(proper(z0)) 173.53/63.96
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 173.53/63.96
proper(from(z0)) → from(proper(z0)) 173.53/63.96
proper(s(z0)) → s(proper(z0)) 173.53/63.96
top(mark(z0)) → top(proper(z0)) 173.53/63.96
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.96
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.96
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.96
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.96
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.96
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.96
S(mark(z0)) → c12(S(z0)) 173.53/63.96
S(ok(z0)) → c13(S(z0))

S tuples:

FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.96
FROM(ok(z0)) → c11(FROM(z0))

K tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.96
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.96
S(mark(z0)) → c12(S(z0)) 173.53/63.96
S(ok(z0)) → c13(S(z0)) 173.53/63.96
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.96
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1))

Defined Rule Symbols:

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

Defined Pair Symbols:

2ND, CONS, FROM, S

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13

173.53/63.96

(37) 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)) → c10(FROM(z0))

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

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.96
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.96
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.96
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.96
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.96
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.96
S(mark(z0)) → c12(S(z0)) 173.53/63.96
S(ok(z0)) → c13(S(z0))

The order we found is given by the following interpretation:
Polynomial interpretation : 173.53/63.96

POL(2ND(x₁)) = [5]x₁ 173.53/63.96
POL(CONS(x₁, x₂)) = [3]x₁ + [3]x₂ 173.53/63.96
POL(FROM(x₁)) = [2]x₁ 173.53/63.96
POL(S(x₁)) = [5]x₁ 173.53/63.96
POL(c10(x₁)) = x₁ 173.53/63.96
POL(c11(x₁)) = x₁ 173.53/63.96
POL(c12(x₁)) = x₁ 173.53/63.96
POL(c13(x₁)) = x₁ 173.53/63.96
POL(c6(x₁)) = x₁ 173.53/63.96
POL(c7(x₁)) = x₁ 173.53/63.96
POL(c8(x₁)) = x₁ 173.53/63.96
POL(c9(x₁)) = x₁ 173.53/63.96
POL(mark(x₁)) = [1] + x₁ 173.53/63.96
POL(ok(x₁)) = x₁

173.53/63.96

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(z1) 173.53/63.96
active(from(z0)) → mark(cons(z0, from(s(z0)))) 173.53/63.96
active(2nd(z0)) → 2nd(active(z0)) 173.53/63.96
active(cons(z0, z1)) → cons(active(z0), z1) 173.53/63.96
active(from(z0)) → from(active(z0)) 173.53/63.96
active(s(z0)) → s(active(z0)) 173.53/63.96
2nd(mark(z0)) → mark(2nd(z0)) 173.53/63.96
2nd(ok(z0)) → ok(2nd(z0)) 173.53/63.96
cons(mark(z0), z1) → mark(cons(z0, z1)) 173.53/63.96
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 173.53/63.96
from(mark(z0)) → mark(from(z0)) 173.53/63.96
from(ok(z0)) → ok(from(z0)) 173.53/63.96
s(mark(z0)) → mark(s(z0)) 173.53/63.96
s(ok(z0)) → ok(s(z0)) 173.53/63.96
proper(2nd(z0)) → 2nd(proper(z0)) 173.53/63.96
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 173.53/63.96
proper(from(z0)) → from(proper(z0)) 173.53/63.96
proper(s(z0)) → s(proper(z0)) 173.53/63.96
top(mark(z0)) → top(proper(z0)) 173.53/63.96
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.96
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.96
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.96
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.96
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.96
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.96
S(mark(z0)) → c12(S(z0)) 173.53/63.96
S(ok(z0)) → c13(S(z0))

S tuples:

FROM(ok(z0)) → c11(FROM(z0))

K tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.96
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.96
S(mark(z0)) → c12(S(z0)) 173.53/63.96
S(ok(z0)) → c13(S(z0)) 173.53/63.96
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.96
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.96
FROM(mark(z0)) → c10(FROM(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

2ND, CONS, FROM, S

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13

173.53/63.96

(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.

FROM(ok(z0)) → c11(FROM(z0))

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

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.96
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.96
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.96
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.96
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.96
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.96
S(mark(z0)) → c12(S(z0)) 173.53/63.96
S(ok(z0)) → c13(S(z0))

The order we found is given by the following interpretation:
Polynomial interpretation : 173.53/63.96

POL(2ND(x₁)) = [3]x₁ 173.53/63.96
POL(CONS(x₁, x₂)) = [3]x₁ + [5]x₂ 173.53/63.96
POL(FROM(x₁)) = [3]x₁ 173.53/63.96
POL(S(x₁)) = [5]x₁ 173.53/63.96
POL(c10(x₁)) = x₁ 173.53/63.96
POL(c11(x₁)) = x₁ 173.53/63.96
POL(c12(x₁)) = x₁ 173.53/63.96
POL(c13(x₁)) = x₁ 173.53/63.96
POL(c6(x₁)) = x₁ 173.53/63.96
POL(c7(x₁)) = x₁ 173.53/63.96
POL(c8(x₁)) = x₁ 173.53/63.96
POL(c9(x₁)) = x₁ 173.53/63.96
POL(mark(x₁)) = x₁ 173.53/63.96
POL(ok(x₁)) = [1] + x₁

173.53/63.96

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(2nd(cons(z0, cons(z1, z2)))) → mark(z1) 173.53/63.96
active(from(z0)) → mark(cons(z0, from(s(z0)))) 173.53/63.96
active(2nd(z0)) → 2nd(active(z0)) 173.53/63.96
active(cons(z0, z1)) → cons(active(z0), z1) 173.53/63.96
active(from(z0)) → from(active(z0)) 173.53/63.96
active(s(z0)) → s(active(z0)) 173.53/63.96
2nd(mark(z0)) → mark(2nd(z0)) 173.53/63.96
2nd(ok(z0)) → ok(2nd(z0)) 173.53/63.96
cons(mark(z0), z1) → mark(cons(z0, z1)) 173.53/63.96
cons(ok(z0), ok(z1)) → ok(cons(z0, z1)) 173.53/63.96
from(mark(z0)) → mark(from(z0)) 173.53/63.96
from(ok(z0)) → ok(from(z0)) 173.53/63.96
s(mark(z0)) → mark(s(z0)) 173.53/63.96
s(ok(z0)) → ok(s(z0)) 173.53/63.96
proper(2nd(z0)) → 2nd(proper(z0)) 173.53/63.96
proper(cons(z0, z1)) → cons(proper(z0), proper(z1)) 173.53/63.96
proper(from(z0)) → from(proper(z0)) 173.53/63.96
proper(s(z0)) → s(proper(z0)) 173.53/63.96
top(mark(z0)) → top(proper(z0)) 173.53/63.96
top(ok(z0)) → top(active(z0))

Tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.96
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.96
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.96
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.96
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.96
FROM(ok(z0)) → c11(FROM(z0)) 173.53/63.96
S(mark(z0)) → c12(S(z0)) 173.53/63.96
S(ok(z0)) → c13(S(z0))

S tuples:none
K tuples:

2ND(mark(z0)) → c6(2ND(z0)) 173.53/63.96
2ND(ok(z0)) → c7(2ND(z0)) 173.53/63.96
S(mark(z0)) → c12(S(z0)) 173.53/63.96
S(ok(z0)) → c13(S(z0)) 173.53/63.96
CONS(mark(z0), z1) → c8(CONS(z0, z1)) 173.53/63.96
CONS(ok(z0), ok(z1)) → c9(CONS(z0, z1)) 173.53/63.96
FROM(mark(z0)) → c10(FROM(z0)) 173.53/63.96
FROM(ok(z0)) → c11(FROM(z0))

Defined Rule Symbols:

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

Defined Pair Symbols:

2ND, CONS, FROM, S

Compound Symbols:

c6, c7, c8, c9, c10, c11, c12, c13

173.53/63.96

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

The set S is empty

173.53/63.96

(42) BOUNDS(O(1), O(1))

173.53/63.96